text stringlengths 4 2.78M |
|---|
---
abstract: |
We provide a critical atomistic evidence of pseudoelastic behavior in complex solid-solution BCC Mo-W-Ta-Ti-Zr alloy. Prior to this work, only limited single-crystal BCC solids of pure metals and quaternary alloys have shown pseudoelastic behavior at low temperatures and high strain rates. The deformation mechanisms investigated using classical molecular simulations under tensile-compressive loading reveal temperature-dependent pseudoelastic behavior aided by twinning during the loading-unloading cycle. The pseudoelasticity is found to be independent of loading directions with identical cyclic deformation characteristics during uniaxial loading. Additionally, temperature variation from 77 to 1500 K enhances the elastic strain recovery in the alloy.\
**Keywords:** deformation, pseudoelasticity, high entropy alloy, twinning, molecular simulations
author:
- Aayush Sharma
- 'Valery I. Levitas'
- Prashant Singh
- Anup Basak
- Ganesh Balasubramanian
- 'Duane D. Johnson'
title: 'Twinning-induced pseudoelastic behavior in (MoW)$_{85}$(TaTi)$_{7.5}$Zr$_{7.5}$'
---
Martensitic transformations (MTs) are first-order diffusionless transformations that are observed in ferroelectric and ferromagnetic alloys and play the central role in exhibiting the pseudoelasticity or superelasticity in shape-memory alloys.[@Kaushik2003; @1_] Since the pioneering work of Kurdyumov and coworkers about the nature of the MTs,[@4; @5] a deep understanding exists that has made it possible to create a new class of materials showcasing pseudoelasticity.[@6] Under mechanical load and reduction in temperature, the austenite (parent phase) transforms into a martensite (product phase), which consists of complex microstructures such as twinned martensite, wedges, and twins within twins.[@10; @2_] When the load is removed and the temperature is increased, the martensite transforms back into the parent phase and the initial shape is recovered. Such special transformations are responsible for the pseudoelasticity in alloys.[@11] However, pinning of the austenite-martensite and martensite-martensite interfaces by dislocations or other crystal defects often gives rise to irreversible martensitic transformations that may inhibit complete pseudoelasticity.[@10; @11]
Recently, much attention has been on identifying pseudoelastic alloys, due to their diverse application, especially as actuation devices and bio-compatible stents.[@Zhang2005] Eliminating the toxicity issue in some of existing pseudoelastic materials, e.g., Ni-Ti, poses a challenge to the biomedical industry.[@Zhou2004-01; @Takahashi2002] All that being said, the complex microstructures of these pseudoelastic alloys make it difficult to experimentally characterize the reversible stress-induced deformation products. To discover an alternate non-toxic alloy, we perform extensive research on novel refractory-based Mo-W-Ta-Ti-Zr high entropy alloys (HEAs).[@Yeh2004] HEAs are of intense interest due to their remarkable mechanical behavior, structural strength, resistance to fatigue, oxidation, corrosion, and wear,[@Gao2016; @Miracle2017; @Miracle2017_1] with a potential view of employing these materials in systems ranging across defense equipment, naval architecture, and high-temperature applications.[@Gao2016; @Singh2018]
Our recent investigation on novel refractory Mo-W-Ta-Ti-Zr complex solid-solution alloy (CSA) reveals interesting electronic and mechanical characteristics, alongside global and local stability.[@Singh2018; @PRM] While sweeping through 5-dimensional composition space, we zeroed on one such composition, (MoW)$_{85}$(TaTi)$_{7.5}$Zr$_{7.5}$ (MWTTZ), which exhibits pseudoelastic deformation under applied strain. The understanding of the control mechanism in MWTTZ alloy, along with revealing the twinning process, makes an interesting case study.
A cuboidal simulation domain is constructed by random distribution of Mo, Ta, W, Ti and Zr atoms in a BCC lattice (Fig. \[struct\]) of (95.9014 $\times$ 95.9014 $\times$ 95.9014) Å$^{3}$, with a lattice constant of, a$_{0}$ = 3.19 Å.[@Singh2018] The mole fractions of the different elements constitutes a total of 54000 atoms. Periodic boundary conditions are imposed in all the directions. The intermolecular interactions are described using the assimilated Embedded Atomic Method (EAM),[@Zhou2004; @AS2016] and validated previously.[@Singh2018] We employ the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) simulation package,[@Plimton1995] for calculations. For visualization, the common neighbor analysis (CNA)[@Stukowski2012] is used in “OVITO" the Open Visualization Tool.[@Stukowski2010]
![Simulation cell of (a) Tungsten (W) and (b) quinary (MoW)$_{85}$(TaTi)$_{7.5}$Zr$_{7.5}$ (MWTTZ) alloy. Common neighbor analysis reveals a BCC coordination among the 54,000 atoms for both. Elements are visualized through different colors: Mo=green, W=red, Ta=yellow, Ti=black, Zr=blue.[]{data-label="struct"}](Fig1-structure.png)
With large-scale atomistic simulations, we reveal twinning and detwinning phenomena under loading and unloading of the MWTTZ alloy. Similar loading-unloading hysteresis originating from twinning process has been found in nanocrystalline tungsten (W).[@Wang2015] In these materials, twin boundaries act as effective barriers to dislocation slip, which in turn increases the yield strength and ductility. Previously such behavior were limited to a couple of refractory-based quaternary alloys.[@Hagihara2016; @Hagihara2017] However, we did not find any reports on pseudoelastic behavior in quinary (five-component) alloys. The deformation mechanisms for the quaternary (Ti-Nb-Ta-Zr) are found to be dependent on composition and vary between slip and twinning modes.[@Hagihara2016; @Hagihara2017] MWTTZ is the first quinary CSA that exhibits desired characteristics of pseudoelastic deformation, and holds great promise for biomedical applications.
The energy minimization of the structure carried out using the conjugate-gradient algorithm with energy tolerance of 10$^{-15}$ and force tolerance of 10$^{-15}$ (eV/Å) results in a geometrically optimized configuration for the HEA. The structure is initialized at 4000 K under an isothermal-isobaric (NPT) ensemble at a pressure of 0 MPa for 90 picoseconds (ps). The alloy is rapidly quenched from 4000 K to different temperatures (300 to 1500 K) in 10 ns, and the structure is equilibrated for an additional 90 ps. The quenched HEA is further simulated under the NPT and NVT (canonical) ensembles, successively for 10 ps and 20 ps, respectively. The pressure (0 MPa), and temperature constraints are imposed by the Nóse-Hoover barostat and thermostat with coupling times for both set at 1 ps. Finally, the structure is equilibrated for 10 ps under the microcanonical ensemble (NVE) to complete the quenching process. A time step of 0.001 ps is maintained throughout all our simulations. A quasistatic uniaxial loading-unloading in the $<$100$>$ direction is applied to analyze the deformation mechanisms. At each loading step, we expand the simulation box at a rate of 0.01 ps$^{-1}$, and the strain expressed on the sample is the true strain. Subsequently, the deformed alloy is equilibrated under NPT and NVE ensembles for 90 and 50 ps, respectively, after each loading step.
For any alloy, analyzing the phase stability is an important criteria. Based on valence electron composition ($4 < VEC < 6$), size-effect ($\le$6.6%), and mixing (formation) energy $\Delta E_f$ (-15 mRy $\leq$ $\Delta E_f$ $\leq$ 5 mRy) calculated from first-principles shows that the HEA is energetically stable, and will form a complex solid-solution.[@Singh2018] Throughout our calculations, we consider the solid-solution phase.
In Fig. \[struct\], the crystal lattice of (a) pure tungsten (W), and (b) quinary MWTTZ is shown. The common neighbor analysis (CNA) performed on the undeformed MWTTZ alloy reveals a body centered cubic (BCC) coordination.
\[\]
[]{}
Alongside phase stability we also investigate the alloy’s (MWTT) mechanical stability, based on standard criteria:[@Slaughter2002]
$$B > 0; \quad {C}' > 0; \quad C_{44} > 0; \\
\label{Eqn:stability-1}$$
where the bulk modulus ($B$) and shear modulus $C'$ are given by: $$B = \frac{C_{11}+2C_{12}}{3}; { \ \ \ \ } {C}'=\frac{C_{11}-C_{12}}{2} \\
\label{Eqn:stability-2}$$ C$_{11}$, C$_{12}$, and C$_{44}$ are three independent elastic constants in cubic crystals which relate all six components of the stress (’$\sigma$’) tensor with six components of the strain (’$\epsilon$’) tensor.[@Slaughter2002] These constants are usually derived from the total-energy calculations representing the single-crystal elastic properties. We find that the MWTTZ alloy satisfies the criteria for both (a) phase and (b) mechanical stability.
![Temperature-dependent pseudoelastic behavior observed in (MoW)$_{0.85}$(Zr(TaTi))$_{0.15}$ (MWTTZ) at (a) 77 K, (b) 300 K, (c) 700 K, and (d) 1100 K. The MWTTZ shows extreme thermal sensitivity. A large hysteresis found at 77 K slowly decreases with increase in temperature, and disappears at very high temperatures ($>$1100 K).[]{data-label="fig:thermal-stress-strain"}](Fig3_1.png)
To capture the correct values of the elastic constants and relevant parameters of MWTTZ, we consider the small-displacement method.[@Shinoda2004] We found a drastic drop in C$_{11}$ from 395 GPa at 77 K to 336 GPa at 1100 K, however, C$_{12}$ and C$_{44}$ show slight increase from 179$\rightarrow$191 GPa and 123 $\rightarrow$130 GPa, respectively, with increase in temperature. In a physical sense, C$_{11}$ shows reduction in longitudinal elastic behavior, whereas C$_{12}$ and C$_{44}$ show slow increase with temperature in off-diagonal and elastic shear characteristic of MWTTZ, respectively. A longitudinal strain produces a change in volume without a change in shape and is related to the pressure, which reflects a larger change in C$_{11}$. In contrast, a transverse strain or shearing causes a change in shape without a change in volume. So, C$_{12}$ and C$_{44}$ are less sensitive to pressure than C$_{11}$. The shear modulus (Eq. \[Eqn:stability-2\]) drops from 108 GPa at 77 K to 60.5 GPa at 1100 K, which clearly comes from the reduction in longitudinal elastic constant C$_{11}$.
We perform quasistatic tensile loading and unloading (compression) on the MWTTZ alloy to investigate the behavior of the alloy under external strain. Uniaxial loading is widely used to reveal the deformation characteristics of a material. We present one such deformation curve in Fig. \[fig:tension-compression\] at 300 K. MWTTZ follows elastic limit till 0.02 strain value with stress of 6.16 GPa. Beyond which, instability drives MWTTZ away from the elastic regime. As stress piles up, twinning is observed in the crystalline lattice (T10 in Fig.\[fig:tension-compression\]). The first major stress drop is observed at T10 showing deviation from the perfect BCC coordination. At T10, with a major stress drop, twinning relieves the stress. The findings are also observed through the CNA mapping. The CNA analysis in the present study helps us to track the nucleation of twins, its growth, as well as detwinning in a loading-unloading cycle. Along with a first set of twins, we also observe a set of cross-twins in the MWTTZ alloy (T19 inset Fig. \[fig:tension-compression\]). During unloading (compression) we find that the twins disappears (C7 inset Fig. \[fig:tension-compression\]). Detwinning is characterized by an abrupt rise in stress levels and beyond C7, (C8 to C11 inset Fig. \[fig:tension-compression\]), we find that the material is following its elastic loading curve.
The maximum shear prior to twinning (T9) was 0.17, while during twinning (T10), and for the second twin (cross-twin) it was 0.29, and 0.31 respectively. It is interesting to note from literature that BCC solids (1/2 atoms shuffle) show a twinning shear value of 0.35.[@Mahajan1995]
To analyze the effect of thermal fluctuation on the pseudoelasticity observed in MWTTZ complex solid-solution alloy, we perform temperature (77 K to 1500 K) dependent loading-unloading deformations. In Fig. \[fig:thermal-stress-strain\](a)&(b), we show that the first twin formation is observed at $\approx$0.065 strain, while the cross-twins appear at $\approx$0.12 strain levels. With further increase in temperature, Fig. \[fig:thermal-stress-strain\](c)&(d), MWTTZ allows only cross-twins at higher strain ($\approx$0.12), which softens the elastic modes compared to low-temperature cases.[@Mahajan1973] Clearly, the large thermal fluctuations at higher temperature require higher external strain for the twin nucleation. The soften elastic modes at higher temperature, as shown in Fig. \[fig:thermal-stress-strain\], leads to a reduction in the loading-unloading hysteresis. In MWTTZ, the temperature can be use as a control parameter to tune the pseudoelastic trend. Recent studies show that elemental ‘W’ in single crystal form is known to exhibit deformation twins at negative temperatures (T $<$ 0$^\circ$C). Studies also suggest that an increase in the purity would facilitate the twinning process.[@Savitskii1970] In our MD analysis, we observe twinning during tensile loading for both pure tungsten, and the quinary MWTTZ [(see supplementary)]{}.
![Variation of shear strain during (tensile) loading cycle along $\left[111\right]$. (a) T1 (b) T9 (c) T17 and (d) T19 represents the quasistatic loading steps at different strains. Thickness of the twin layer grows from 3 atomic layers (T9) to around 4 atomic layers (T17). Further strain hardening leads to the formation of cross-twinning. The magnitude of stress required for initial twin nucleation is higher than that of its propagation.[]{data-label="fig:shear-strain"}](Fig-4-new.png)
To investigate the evolution of shear strain for MWTTZ, we show twins along the (111) plane, in Fig.\[fig:shear-strain\]. With no load, as anticipated, atoms experience zero shear (Fig.\[fig:shear-strain\]a), with further increase in uniaxial loading ($<$100$>$ (x)) twins appear (Fig. \[fig:shear-strain\]b; T9). The twinning layers are about 3 atomic layers thick which marginally increases to 4 atomic layers at T17 (Fig. \[fig:shear-strain\]c). Any increase in quasistatic ($<$100$>$) load beyond this leads to the formation of additional cross-twins for the MWTTZ HEA. The thickness of both the original and the cross-twins is found to be approximately 3 atomic layers. The features of twins, cross-twins and reverse twinning or detwinning is reproduced for the uniaxial loading along any of the three directions: $<$100$>$ or $<$010$>$ or $<$001$>$. For clarity, we have specifically discussed the $<$100$>$ case. It is also worth mentioning that within the strain limits (small-strain regime) considered in the present study, biaxial loading ($<$110$>$ or $<$101$>$ or $<$011$>$) of the quinary MWTTZ alloy does not yield evidence of twinning and detwinning.
To conclude, we provide evidence for the pseudoelastic behavior in quinary high-entropy alloys using atomistic simulations for the very first time. Our calculations reveal the presence of pseduoelasticity in (MoW)$_{85}$(TaTi)$_{7.5}$Zr$_{7.5}$. We observe strong temperature dependent twinning and detwinning process during the loading-unloading (tension-compression) cycle. The twinning-detwinning feature is responsible for the pseduoelastic behavior in (MoW)$_{85}$(TaTi)$_{7.5}$Zr$_{7.5}$, which persists over a wide range of temperatures from 77 to 1500 K. In our findings, the elastic modulus softens with increasing temperature, which reduces the hysteresis and possibly can be used as a control parameter. Psuedoelastic materials have many possible applications including but not limited to actuators in devices and/or bio-medical implants.
AS and GB thank the Office of Naval Research (ONR) for support through the grants N00014-16-1-2548 and N00014-18-1-2484. The research was supported by a grant of computer time from the DoD High Performance Computing Modernization Program at Army Engineer Research and Development Center. The work at the Ames Laboratory was funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Ames Laboratory is operated for the U.S. DOE by Iowa State University under Contract No. DE-AC02-07CH11358.
[100]{} K. Bhattacharya, [*microstructure of martensite : Why it forms and how it gives rise to the shape-memory effect*]{}, 1st ed., Oxford university Press, Great Britain, p. 46-83 (2003). H.K.D.H. Bhadeshia, [*Martensite in Steels. Available*]{}. at https://www.phase-trans.msm.cam.ac.uk/2002/martensite.html, pp. 1-12 (2002). A.L. Roytburd, and G.V. Kurdyumov, Mat. Sci. Eng. [**39**]{}, 141-167 (1979). W.J. Buehler, J.W. Gilfrich, and R.C. Wiley, J Appl. Phys. [**34**]{}, 1475-1477 (1963). K. Otsuka, C.M. Wayman, [*Shape memory materials*]{}. University Press, Cambridge, (1998). J.D. Eisenwasser, and L.C. Brown, Metall. Trans. [**3**]{}, 1359-1363 (1972). G.S. Firstov, J. Van Humbeeck, and Yu.N. Koval, J. Intelligent Material Systems and Structures [**17**]{}, 1041-1047 (2006). J. Ma, I. Karaman, and R. D. Noebe, International Materials Reviews [**55**]{}, 257-315 (2010). L. C. Zhang, T. Zhou, S. P. Alpay, and M. Aindow, App. Phys. Lett. [**87**]{}, 241909 (1-3) (2005). T. Zhou, M. Aindow, S. P. Alpay, M. J. Blackburn, and M. H. Wu, Scr. Mater. [**50**]{}, 343-348 (2004). E. Takahashi, T. Sakurai, S. Watanabe, N. Masahashi, and S. Hanada, Mater. Trans., JIM [**43**]{}, 2978-2983 (2002). J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau, S.Y. Chang, Adv. Eng. Mater. [**6**]{}, 299-303 (2004). M. C. Gao, J-W. Yeh, P. K. Liaw, Y. Zhang (Ed.), [*High-Entropy Alloys: Fundamentals and Applications*]{}, 1st ed., Springer International Publishing, Switzerland, pp. 333-366 (2016). S. Gorsse, D. B. Miracle, O. N. Senkov, Acta Mater. [**135**]{}, 177-187 (2017). D. B. Miracle, O. N. Senkov, Acta Mater. [**122**]{}, 448-511(2017). P. Singh, A. Sharma, A.V. Smirnov, M.S. Diallo, P.K. Ray, G. Balasubramanian, and D.D. Johnson, npj Comp. Mater., [**4**]{}, 16 (1-8) (2018). P. Singh, A.V. Smirnov, and D.D. Johnson, Phys. Rev. Mater. [**2**]{}, 055004 (1-5) (2018). X.W. Zhou, R.A. Johnson, and H.N.G. Wadley, Phys. Rev. B, [**69**]{}, 144113 (1-10) (2004). A. Sharma, P. Singh, P.K. Liaw, D.D. Johnson and G. Balasubramanian, [*Sci. Rep.*]{} [**6**]{}, 31028 (1-11) (2016). S. Plimpton, J. Comp. Phys [**1**]{}, 1-19 (1995). A. Stukowski, Modelling Simul. Mater. Sci. Eng. [**20**]{}, 045021 (1-15) (2012). A. Stukowski, Modelling Simul. Mater. Sci. Eng. [**18**]{}, 015012 (1-7) (2010). J. Wang, Z. Zeng, C. R. Weinberger, Z. Zhang, T. Zhu, and S. X. Mao, Nat. Mater. [**14**]{}, 594-600 (2015). K. Hagihara, T. Nakano, H. Maki, Y. Umakoshi, and M. Niinomi, Sci. Rep. [**6**]{}, 29779 (1-10) (2016). K. Hagihara, T. Nakano, Int. J. Plast. [**98**]{}, 27-44 (2017). W. S. Slaughter, [*The Linearized Theory of Elasticity*]{}, 1st ed., Springer, New York, 2002. Shinoda, Shiga, and Mikami, Phys. Rev. B [**69**]{}, 134103 (1-8) (2004). J. W. Christian, and S. Mahajan, Prog. Mater. Sci. [**39**]{}, 1-157 (1995). S. Mahajan, and D. F. Williams, Int. Metal. Rev. [**18**]{}, 43-61 (1973). E. M. Savitskii [*Physical Metallurgy of Refractory Metals and Alloys*]{}, Ch. V (1970).
|
---
abstract: 'In this note, we propose a simple way of constructing HDG+ projections on polyhedral elements. The projections enable us to analyze the Lehrenfeld-Schöberl HDG (HDG+) methods in a very concise manner, and make many existing analysis techniques of standard HDG methods reusable for HDG+. The novelty here is an alternative way of constructing the projections without using $M$-decomposition as a middle step. This extends our previous results \[S. Du and F.-J. Sayas, SpringerBriefs in Mathematics (2019)\] (elliptic problems) and \[S. Du and F.-J. Sayas, arXiv:1903.11766\] (elasticity) to polyhedral meshes.'
author:
- |
Shukai Du[^1] and Francisco-Javier Sayas\
Department of Mathematical Sciences, University of Delaware
bibliography:
- 'maxref.bib'
title: A note on devising HDG+ projections on polyhedral elements
---
Introduction
============
The Lehrenfeld-Schöberl HDG (HDG+) methods [@LeSc:2010] have recently gained considerable interest since they superconverge on polyhedral meshes in addition to the easiness of implementation. In [@DuSa:2019] (elliptic problems) and [@du2019new] (elasticity), we proposed mathematical tools to incorporate the analysis of the HDG+ methods into the projection-based error analysis setting [@CoGoSa:2010]. In this way, we can reuse existing analysis techniques and avoid repeated or unnecessary arguments. In [@DuSa:2019] and [@du2019new], the projections were devised for simplicial elements. In this paper, we extend the results to polyhedral elements.
To motivate the discussion, let us review some existing works. For mixed finite element methods (or simply mixed methods), the core in their design and analysis is the local projection operators; see, for instance, [@RaTh:1977] for the Raviart-Thomas (RT) projection, [@BrDoMa:1985] for the Brezzi-Douglas-Marini (BDM) projection, and [@Ne:1980; @Ne:1986] for the Nédélec projection. These projections satisfy certain commutativity properties that can be used to analyze the numerical methods in a very concise way. Inspired by the mixed method projections, the first HDG projection was devised in [@CoGoLa:2009]. It enables us to analyze a wide class of HDG methods in an unified, and also simple and concise manner. Since for both the mixed methods and the HDG methods, the core in their error analysis is the specially devised projections that are tailored to the numerical schemes, this way of analysis is often referred to as the “projection-based error analysis” (PBEA).
PBEA has been widely used to analyze HDG methods. See, for instance, the error analysis of the HDG methods for heat/fractional diffusion [@ChCo:2012; @CoMu:2015], acoustic waves [@CoQu:2014; @CoFuHuJiSaSa:2018], Stokes equations [@CoGoNgPeSa:2011], Helmholtz equations [@GrMo:2011]. On the other hand, new HDG projections have been devised, incorporating more variants of HDG methods into the PBEA setting; see the work of $M$-decomposition [@CoFuSa:2017], an mathematical tool to systematically devise superconvergent HDG methods on polyhedral meshes. Since all $M$-decomposition HDG methods have associated HDG projections, all of their analysis can be incorporated into the PBEA setting.
Despite the wide and successful applications of HDG projections, the error analysis of some important HDG methods can [*not*]{} be incorporated into the PBEA setting until very recently. An important example is the HDG+ method, proposed first by Lehrenfeld and Schöberl [@LeSc:2010] and then analyzed by Oikawa [@Oi:2014] in the setting of elliptic diffusion. The method uses $P_k^d$-$P_{k+1}$-$P_k$ to approximate the flux-primal-trace triplet, and it achieves optimal convergence for all variables on general polyhedral meshes. Compared to the standard $P_k^d$-$P_k$-$P_k$ HDG method, the HDG+ method is as efficient as the standard method, since the two methods share the same size of the global systems. Moreover, the HDG+ method does not suffer from the problem of losing convergence order, which is observed for the standard HDG method on non-simplicial polyhedral meshes, or for elastic problems with strong symmetric stress formulation. Finally, the HDG+ method is extremely easy to implement, since it is a simple tweak of the standard HDG method.
As is mentioned before, most of the existing error analysis of the HDG+ methods (see, for instance, [@HuPrSa:2017; @Oi:2014; @QiShSh:2018; @QiSh:2016]) can not be incorporated into the PBEA setting. This makes their error analysis less concise compared to those HDG methods that can be analyzed by HDG projections. More importantly, this leads to a scattered style of error analysis and makes it hard for us to reuse the existing projection-based analysis techniques that were established in a decade. All the above indicates the necessity to develop mathematical tools to incorporate the error analysis of HDG+ methods into the PBEA setting. In this way, many existing works using HDG projections, such as the analysis of the HDG methods for various types of evolutionary equations and Helmholtz equations (see, for instance, [@ChCo:2012; @CoMu:2015; @CoQu:2014; @GrMo:2011; @CoFuHuJiSaSa:2018]), can be automatically reused for the design and analysis of the HDG+ methods.
Following this idea, we have devised the HDG+ projections in [@DuSa:2019] for ellptic problems and in [@du2019new] for elasticity with strong symmetric stress formulation. We have sucessfully used the projections, combined with some existing analysis techniques of the standard HDG methods, to derive the error estimates of the HDG+ methods for heat diffusion and acoustic waves in [@DuSa:2019], and for time-harmonic and transient elastic waves in [@du2019new]. For simplicity, we have limited the discussions on simplicial meshes in [@DuSa:2019; @du2019new]. In this paper, we extend the results to polyhedral meshes by using an alternative way of constructing the projections without using $M$-decomposition [@CoFuSa:2017] as a middle step.
We finally give an outline for the rest of the paper. In Section \[sec:hdgp\_pj\_ell\], we devise the HDG+ projection for elliptic problems. We also demonstrate how to use the projection to analyze the HDG+ method for a model problem. In Section \[sec:hdgp\_pj\_elas\], we devise the HDG+ projection for elasticity. We will not demonstrate its usage, since this has been done in [@du2019new]. The projection we devise here satisfies [@du2019new Theorem 2.1] and it will render all the analysis and estimates in [@du2019new Sections 5,6&7] valid for general polyhedral meshes.
The projection for elliptic problems {#sec:hdgp_pj_ell}
====================================
In this section, we devise the HDG+ projection and demonstrate how to use it to derive the error estimates for the HDG+ method. Note that the first analysis of the HDG+ method was obtain in [@Oi:2014]. However, our proof here is quite different from the proof in [@Oi:2014]. Instead, as we will demonstrate in Section \[sec:prf\_est\], the proof we obtained is very similar to those used in [@CoGoSa:2010], thanks to the introduction of the HDG+ projection. In this way, we are able to reuse the existing projection-based error analysis to analyze the HDG+ method in a very concise way. Consequently, we can unify the analysis of the standard HDG and HDG+ methods.
\
[**Notation.**]{} Let us first introduce some notation that will be used throughout the paper. Let $\Omega\subset{\mathbb}R^d$ ($d=2,3$) be a polyhedral domain with Lipschitz continuous boundary. We consider a triangulation of $\Omega$ denoted by ${\mathcal}T_h$. For each element $K\in{\mathcal}T_h$, we use the standard notation $h_K$ as the diameter of $K$, and introduce the parameter $\gamma_K$ that describes the shape-regularity of $K$, as is done in [@DiDr:2017 Appendix]. Let ${\mathcal}E_K$ and ${\mathcal}E_h$ denote the collections of all the faces of $K$ and ${\mathcal}T_h$, respectively. We assume that there is a fixed positive constant $\gamma_0$ such that $\gamma_0\ge\gamma_K$ for all $K\in{\mathcal}T_h$ (consequently the shape-regularity of ${\mathcal}T_h$ is controlled). We write $h:=\max_{K\in{\mathcal}T_h}h_K$ as the mesh-size and $h_{\mathrm}{min}:=\min_{K\in{\mathcal}T_h}h_K$ as the smallest diameter among all elements.
Let ${\mathcal}P_k(X)$ denote the polynomial space of degree $k$ on $X$ and let $\Pi_k:L^2(X)\rightarrow{\mathcal}P_k(X)$ be the corresponding $L^2$ projection. Here $X$ can be an element $K$ or a face of $K$. Let ${\mathcal}R_k({\partial}K):=\prod_{F\in{\mathcal}E_K}{\mathcal}P_k(F)$ and let ${\mathrm}P_M: \prod_{K\in{\mathcal}T_h}L^2({\partial}K)\rightarrow\prod_{K\in{\mathcal}T_h}{\mathcal}R_k({\partial}K)$ be the corresponding $L^2$ projection. We finally introduce the following notation for the discrete inner products on ${\mathcal}T_h$ and ${\partial}{\mathcal}T_h$: $$\begin{aligned}
(*_1,*_2)_{{\mathcal}T_h}=\sum_{K\in{\mathcal}T_h}(*_1,*_2)_K,\quad
{\langle *_1,*_2\rangle}_{{\partial}{\mathcal}T_h}=\sum_{K\in{\mathcal}T_h}{\langle *_1,*_2\rangle}_{{\partial}K},\end{aligned}$$ where $(\cdot,\cdot)_K$ and ${\langle \cdot,\cdot\rangle}_{{\partial}K}$ denote the $L_2$ inner products on $K$ and ${\partial}K$, respectively.
\
[**Model problem.**]{} In this section, we consider the following steady-state diffusion equations:
\[eq:ell\_pde\] $$\begin{aligned}
{5}
\kappa^{-1}{\mathbf}q+\nabla u &= 0 &\quad& {\mathrm}{in}\ \Omega,\\
{\nabla\cdot}{\mathbf}q &= f && {\mathrm}{in}\ \Omega,\\
u &= g && {\mathrm}{on}\ \Gamma:={\partial}\Omega,\end{aligned}$$
where the parameter $\kappa\in L^\infty(\Omega)$ is uniformly positive, the forcing term $f\in L^2(\Omega)$ and the Dirichlet data $g\in H^{1/2}(\Gamma)$. We also introduce the notion of elliptic regularity: $$\begin{aligned}
\label{eq:ell_reg}
\|\kappa\nabla u\|_{1,\Omega}+\|u\|_{2,\Omega}\le C_{\mathrm}{reg} \|{\nabla\cdot}(\kappa\nabla u)\|_\Omega\end{aligned}$$ holds for any $u\in H^1(\Omega)$ such that the right term of the above inequality is finite, where $C_{\mathrm}{reg}$ is a positive constant depending only on $\kappa$ and $\Omega$.
\
[**HDG+ method.**]{} Let us first define the approximation spaces: $$\begin{aligned}
{\mathbf}V_h:=\prod_{K\in{\mathcal}T_h}{\mathcal}P_k(K)^d,\quad
W_h:=\prod_{K\in{\mathcal}T_h}{\mathcal}P_{k+1}(K),\quad
M_h:=\prod_{F\in{\mathcal}E_h}{\mathcal}P_k(F).\end{aligned}$$ The HDG+ scheme is defined as follows: find $({\mathbf}q_h,u_h,\widehat{u}_h)\in {\mathbf}V_h\times W_h\times M_h$ such that
\[eq:HDG\_s\] $$\begin{aligned}
\label{eq:HDG_s_1}
(\kappa^{-1}{\mathbf}q_h,{\mathbf}r)_{{\mathcal}T_h} - (u_h,{\nabla\cdot}{\mathbf}r)_{{\mathcal}T_h} + {\langle \widehat{u}_h,{\mathbf}r\cdot{\mathbf}n\rangle}_{{\partial}{\mathcal}T_h} &= 0,\\
\label{eq:HDG_s_2}
({\nabla\cdot}{\mathbf}q_h,w)_{{\mathcal}T_h} + {\langle \tau{\mathrm}P_M(u_h-\widehat{u}_h),w\rangle}_{{\partial}{\mathcal}T_h}
&=(f,w)_{{\mathcal}T_h},\\
\label{eq:HDG_s_3}
-{\langle {\mathbf}q_h\cdot{\mathbf}n+\tau(u_h-\widehat{u}_h),\mu\rangle}_{{\partial}{\mathcal}T_h\backslash\Gamma} &= 0,\\
\label{eq:HDG_s_4}
{\langle \widehat{u}_h,\mu\rangle}_{\Gamma} &= {\langle g,\mu\rangle}_\Gamma,\end{aligned}$$
for all $({\mathbf}r,w,\mu)\in{\mathbf}V_h\times W_h\times M_h$. The stabilization function $\tau\in\prod_{K\in{\mathcal}T_h}{\mathcal}R_0({\partial}K)$ and it satisfies $c_1h_K^{-1}\le \tau\big|_{{\partial}K}\le c_2h_K^{-1}$ for all $K\in{\mathcal}T_h$, where $c_1$ and $c_2$ are two fixed positive constants.
Main results
------------
We now present the main results in this section – the HDG+ projection (Theorem \[thm:hdgp\_pj\_ell\]) and its application (Theorem \[thm:est\_ell\]). Their proofs can be found in Section \[sec:ell\_hdg+\_pfs\] and Section \[sec:prf\_est\].
\
[**HDG+ projection.**]{} The HDG+ projection is defined as follows:
\[eq:def\_hdg+\_ell\] $$\begin{aligned}
\label{eq:def_hdg+_ell_1}
{\boldsymbol}\Pi_K({\mathbf}q,u):=({\boldsymbol}\Pi_k^g{\mathbf}q,\Pi_{k+1}u)\in{\mathcal}P_k(K)^d\times{\mathcal}P_{k+1}(K),\end{aligned}$$ where ${\boldsymbol}\Pi_k^g:H^1(K)^d\rightarrow{\mathcal}P_k(K)^d$ is defined by solving $$\begin{aligned}
\label{eq:def_hdg+_ell_2}
({\boldsymbol}\Pi_k^g{\mathbf}q-{\mathbf}q,{\mathbf}r)_K &=0\qquad\forall{\mathbf}r\in\nabla{\mathcal}P_k(K)\oplus (\nabla{\mathcal}P_{k+1}(K))^{\perp_k},\\
\label{eq:def_hdg+_ell_3}
({\boldsymbol}\Pi_k^g{\mathbf}q-{\mathbf}q,\nabla w)_K&={\langle {\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)-{\mathbf}q\cdot{\mathbf}n,w\rangle}_{{\partial}K}
\qquad\forall w\in {\mathcal}P_k(K)^{\perp_{k+1}}.\end{aligned}$$ In the above equations, $(\cdot)^{\perp_m}$ represents the orthogonal complement in the background space ${\mathcal}P_k(K)^d$ (for ) or ${\mathcal}P_m(K)$ (for ).
We also define an operator: $$\begin{aligned}
\label{eq:def_hdg+_ell_4}
\delta_{\pm\tau}^{\Pi_K}({\mathbf}q,u):={\boldsymbol}\Pi_K{\mathbf}q\cdot{\mathbf}n-{\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)
\pm\tau({\mathrm}P_M\Pi_K u-{\mathrm}P_Mu)\in{\mathcal}R_k({\partial}K).\end{aligned}$$
We call $\delta_{\pm\tau}^{\Pi_K}$ the [*boundary remainder*]{} of ${\boldsymbol}\Pi_K$.
\[thm:hdgp\_pj\_ell\] The projection ${\boldsymbol}\Pi_K$ and the remainder $\delta_{\pm\tau}^{\Pi_K}({\mathbf}q,u)$ are well defined by and they satisfy
\[eq:hdgp\_pj\_ell\_eqn\] $$\begin{aligned}
{5}
\label{eq:hdgp_pj_ell_eqn_1}
(\Pi_Ku - u,v)_K &= 0 &\quad& \forall v\in{\mathcal}P_{k-1}(K),\\
\label{eq:hdgp_pj_ell_eqn_2}
{\langle {\boldsymbol}\Pi_K{\mathbf}q\cdot{\mathbf}n-{\mathbf}q\cdot{\mathbf}n\pm\tau(\Pi_K u-u),\mu\rangle}_{{\partial}K} &=
{\langle \delta_{\pm\tau}^{\Pi_K}({\mathbf}q,u),\mu\rangle}_{{\partial}K}
&&\forall \mu\in{\mathcal}R_k({\partial}K),\\
\label{eq:hdgp_pj_ell_eqn_3}
({\nabla\cdot}({\boldsymbol}\Pi_K{\mathbf}q-{\mathbf}q),w)_K\pm{\langle \tau{\mathrm}P_M(\Pi_Ku-u),w\rangle}_{{\partial}K}
&= {\langle \delta_{\pm\tau}^{\Pi_K}({\mathbf}q,u), w\rangle}_{{\partial}K}
&&\forall w\in{\mathcal}P_{k+1}(K).\end{aligned}$$
Furthermore, $$\begin{aligned}
\label{eq:hdgp_pj_ell_eqn_conv}
\|{\boldsymbol}\Pi_K{\mathbf}q-{\mathbf}q\|_K+h_K^{-1}\|\Pi_K u-u\|_K + h_K^{1/2}\|\delta_{\pm\tau}^{\Pi_K}({\mathbf}q,u)\|_{{\partial}K}
\le Ch_K^m(|{\mathbf}q|_{m,K}+|u|_{m+1,K}),\end{aligned}$$ where $m\in[1,k+1]$. Here, the constant $C$ depends only on $k$, $\gamma_K$, and $c_2$.
Note that in Theorem \[thm:hdgp\_pj\_ell\], equations do not define the HDG+ projection. However, they are exactly what we need for the error analysis. The boundary remainder operator $\delta_\tau^{\Pi_K}$ can be regarded as an indicator for how much the projection $\Pi_K$ resembles a HDG projection or a mixed method projection. For instance, if ${\boldsymbol}\Pi_K$ is the classical HDG projection [@CoGoSa:2010], then we have $\delta_\tau^{\Pi^{\mathrm}{HDG}}=0$. This can be easily obtained by using [@CoGoSa:2010 Eqn. (2.1c)]. Similarly, we have $\delta_{\tau=0}^{\Pi^{\mathrm}{RT}}=0$ and $\delta_{\tau=0}^{\Pi^{\mathrm}{BDM}}=0$, where ${\boldsymbol}\Pi^{\mathrm}{RT}$ and ${\boldsymbol}\Pi^{\mathrm}{BDM}$ represent the Raviart-Thomas and the BDM projection, respectively.
The key idea behind the HDG+ projection is to find weaker but still sufficient conditions to carry out a projection-based error analysis. For the classical HDG projection, the boundary remainder is zero, and the equations that define the projection are also the equations that we use for the error analysis. However, these two properties are not necessary, especially if we want to extend the projection-based error analysis to more variants of HDG methods. Taking the HDG+ method as an example, the guideline for devising the projection now becomes the following: among all the projections that satisfy the equations , find one such that its approximation property is optimal, and its boundary remainder is as small as possible. As we will see soon, there is no need to enforce the boundary remainder to be zero, which is the case the standard HDG projection. In fact, a small enough boundary remainder is sufficient for optimal convergence of the method. In this way, we can devise HDG projections more flexibly, and generalize the classical projection-based error analysis of HDG methods [@CoGoSa:2010].
\
[**Error estimates.**]{} By using , we define the element-wise projections and the boundary remainder of the exact solutions: $$\begin{aligned}
{5}
&{\boldsymbol}\Pi{\mathbf}q:=\prod_{K\in{\mathcal}T_h}{\boldsymbol}\Pi_K{\mathbf}q,&\quad&
\Pi u:=\prod_{K\in{\mathcal}T_h}\Pi_Ku,&\quad&
\delta_\tau({\mathbf}q,u):=\prod_{K\in{\mathcal}T_h}\delta_\tau^{\Pi_K}({\mathbf}q,u).\end{aligned}$$ We also define the norm $\|\cdot\|_h$ by $\|\mu\|_h^2:=\sum_{K\in{\mathcal}T_h}h_K\|\mu\|_{{\partial}K}^2$ for any $\mu\in L^2({\partial}{\mathcal}T_h):=\prod_{K\in{\mathcal}T_h}L^2({\partial}K)$.
\[thm:est\_ell\] There holds $$\begin{aligned}
\label{eq:est_qh}
\|{\boldsymbol}\Pi{\mathbf}q-{\mathbf}q_h\|_{{\mathcal}T_h}&\le C_1\left(
\|{\boldsymbol}\Pi{\mathbf}q-{\mathbf}q\|_{{\mathcal}T_h}+\|\tau^{-1/2}\delta_\tau({\mathbf}q,u)\|_{{\partial}{\mathcal}T_h}
\right).\end{aligned}$$ If $k\ge1$, and the regularity assumption holds, then we have $$\begin{aligned}
\label{eq:est_uh}
\|\Pi u-u_h\|_{{\mathcal}T_h}\le
C_2\,h\left(
\|{\boldsymbol}\Pi{\mathbf}q-{\mathbf}q_h\|_{{\mathcal}T_h}+\|\tau^{-1/2}\delta_\tau({\mathbf}q,u)\|_{{\partial}{\mathcal}T_h}
\right),\\
\label{eq:est_uhhat}
\|{\mathrm}P_Mu-u_h\|_h\le C_2\,h(1+\frac{h}{h_{\mathrm}{min}})\left(
\|{\boldsymbol}\Pi{\mathbf}q-{\mathbf}q_h\|_{{\mathcal}T_h}+\|\tau^{-1/2}\delta_\tau({\mathbf}q,u)\|_{{\partial}{\mathcal}T_h}
\right).\end{aligned}$$ Here, $C_1$ depends only on $\kappa$, and $C_2$ depends additionally on $k$, $\gamma_0$, and $C_{\mathrm}{reg}$.
Note that by and the fact that $\tau\big|_{{\partial}K}\approx h_K^{-1}$, we have $\|\tau^{-1/2}\delta_\tau({\mathbf}q,u)\|_{{\partial}{\mathcal}T_h}\lesssim h^{k+1}$ for smooth enough exact solutions. Therefore, Theorem \[thm:est\_ell\] shows that the HDG+ method is optimal for both ${\mathbf}q_h$ and $u_h$, and $\|{\mathrm}Pu-\widehat{u}_h\|_h$ achieves the superconvergence rate of ${\mathcal}O(h^{k+2})$ since $\widehat{u}_h\big|_F\in{\mathcal}P_k(F)$.
Theorem \[thm:est\_ell\] can be easily proved by adopting a very similar analysis used in [@CoGoSa:2010], combined with the HDG+ projection. We show how this is done in Section \[sec:prf\_est\].
Proof of Theorem \[thm:hdgp\_pj\_ell\] {#sec:ell_hdg+_pfs}
--------------------------------------
In this subsection, we prove Theorem \[thm:hdgp\_pj\_ell\]. We begin by presenting a lemma that gives a collection of lifting/inverse inequalities and convergence properties about $L^2$ projections. These inequalities will be used extensively in the paper.
\[lm:bs\_inq\] If $u\in{\mathcal}P_k(K)$, then
\[eq:bs\_inq\] $$\begin{aligned}
\label{eq:bs_inq_1}
\|u\|_{{\partial}K}\le C h_K^{-1/2} \|u\|_K,\quad
\|\nabla u\|_K\le C h_K^{-1}\|u\|_K.\end{aligned}$$ If $u\in H^1(K)$, then $$\begin{aligned}
\label{eq:bs_inq_2}
\|\Pi_ku-u\|_K\le C h_K^m|u|_{m,K},\quad
\|\Pi_ku-u\|_{{\partial}K}\le C h_K^{m-1/2}|u|_{m,K}.\end{aligned}$$
Here, $m\in[1,k+1]$, and the constant $C$ depends only on $k$ and $\gamma_K$.
See [@DiDr:2017 Appendix].
We next prove that the projection ${\boldsymbol}\Pi_k^g$ is well defined by and , and it converges optimally.
\[prop:g+\] The projection ${\boldsymbol}\Pi_k^g$ is well defined by and , and we have $$\begin{aligned}
\label{eq:g+_conv}
h_K^{1/2}\|{\boldsymbol}\Pi_k^g{\mathbf}q-{\mathbf}q\|_{{\partial}K}+\|{\boldsymbol}\Pi_k^g{\mathbf}q-{\mathbf}q\|_K\le
Ch_K^m|{\mathbf}q|_{m,K},\end{aligned}$$ where $m\in[1,k+1]$. Here, the constant $C$ depends only on $k$ and $\gamma_K$.
In this proof, we use the sign ‘$\lesssim$’ to hide a constant that depends only on $k$ and $\gamma_K$. First note that and define a square system. We next prove the convergence equation , from which the unique solvability of and follows automatically. Let ${\boldsymbol}\varepsilon_q:={\boldsymbol}\Pi_k^g{\mathbf}q-{\boldsymbol}\Pi_k{\mathbf}q\in{\mathcal}P_k(K)^d$. By and , we have
$$\begin{aligned}
\label{eq:cv_p_0}
({\boldsymbol}\varepsilon_q,{\mathbf}r)_K &=0\qquad\forall{\mathbf}r\in\nabla{\mathcal}P_k(K)\oplus (\nabla{\mathcal}P_{k+1}(K))^{\perp_k},\\
\label{eq:cv_p_3}
({\boldsymbol}\varepsilon_q,\nabla w)_K&={\langle {\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)-{\mathbf}q\cdot{\mathbf}n,w\rangle}_{{\partial}K}
\qquad\forall w\in {\mathcal}P_k(K)^{\perp_{k+1}}.\end{aligned}$$
The above equations imply that $$\begin{aligned}
\label{eq:cv_p_6}
({\boldsymbol}\varepsilon_q,\nabla w)_K&={\langle {\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)-{\mathbf}q\cdot{\mathbf}n,w\rangle}_{{\partial}K}
\qquad\forall w\in {\mathcal}P_{k+1}(K).\end{aligned}$$ We now decompose ${\boldsymbol}\varepsilon_q$ into the summation ${\boldsymbol}\varepsilon_q={\boldsymbol}\varepsilon_q^1+{\boldsymbol}\varepsilon_q^2$, where ${\boldsymbol}\varepsilon_q^1\in\nabla{\mathcal}P_{k+1}(K)$ and ${\boldsymbol}\varepsilon_q^2\in(\nabla{\mathcal}P_{k+1}(K))^{\perp_{k}}$. By we have $\|{\boldsymbol}\varepsilon_q\|_K^2=({\boldsymbol}\varepsilon_q,{\boldsymbol}\varepsilon_q^1)_K$. Since ${\boldsymbol}\varepsilon_q^1\in\nabla{\mathcal}P_{k+1}(K)$, we can write ${\boldsymbol}\varepsilon_q^1=\nabla(p+c)$ for some $p\in{\mathcal}P_{k+1}(K)$ and arbitrary constant $c$. This with gives $$\begin{aligned}
\|{\boldsymbol}\varepsilon_q\|_K^2&=({\boldsymbol}\varepsilon_q,\nabla(p+c))_K={\langle {\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)-{\mathbf}q\cdot{\mathbf}n,p+c\rangle}_{{\partial}K}\\
&\lesssim h_K^{-1/2}\|{\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)-{\mathbf}q\cdot{\mathbf}n\|_{{\partial}K}\|p+c\|_{K}.\end{aligned}$$ We now choose the constant $c=-\Pi_0 p$ and obtain $$\begin{aligned}
\|{\boldsymbol}\varepsilon_q\|_K^2\lesssim h_K^{1/2}\|{\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)-{\mathbf}q\cdot{\mathbf}n\|_{{\partial}K}\|\nabla p\|_K\le h_K^{1/2}\|{\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)-{\mathbf}q\cdot{\mathbf}n\|_{{\partial}K}\|{\boldsymbol}\varepsilon_q\|_K.\end{aligned}$$ This completes the proof.
We are now ready to prove Theorem \[thm:hdgp\_pj\_ell\]. By Proposition \[prop:g+\], we know ${\boldsymbol}\Pi_K$ and $\delta_{\pm\tau}^{\Pi_K}$ are well defined. We next prove that ${\boldsymbol}\Pi_K$ satisfies equations . Equation holds obviously since $\Pi_K u=\Pi_{k+1}u$. Equation holds by the definition . To prove , first note that $$\begin{aligned}
\nonumber
&({\nabla\cdot}({\boldsymbol}\Pi_k^g{\mathbf}q-{\mathbf}q),w)_K
\pm{\langle \tau{\mathrm}P_M(\Pi_{k+1}u-u),w\rangle}_{{\partial}K}\\
\label{eq:cv_p_9}
&\qquad={\langle ({\boldsymbol}\Pi_k^g{\mathbf}q-{\mathbf}q)\cdot{\mathbf}n\pm\tau{\mathrm}P_M(\Pi_{k+1}u-u),w\rangle}_{{\partial}K}
-({\boldsymbol}\Pi_k^g{\mathbf}q-{\mathbf}q,\nabla w)_K,\end{aligned}$$ for all $w\in{\mathcal}P_{k+1}(K)$. By and , we have $$\begin{aligned}
\label{eq:cv_p_12}
(\Pi_k^g{\mathbf}q-{\mathbf}q,\nabla w)_K={\langle {\mathrm}P_M({\mathbf}q\cdot{\mathbf}n)-{\mathbf}q\cdot{\mathbf}n,w\rangle}_{{\partial}K}
\qquad\forall w\in {\mathcal}P_{k+1}(K).\end{aligned}$$ Now follows by using and .
We next prove . By and , we know that ${\boldsymbol}\Pi_K{\mathbf}q={\boldsymbol}\Pi_k^g{\mathbf}q$ and $\Pi_K u=\Pi_{k+1}u$ converge optimally. It only remains to estimate the boundary remainder. By the definition and the fact that $\|\tau\|_{L^\infty({\partial}K)}\le c_2h_K^{-1}$, we have $$\begin{aligned}
\|\delta_{\pm\tau}^{\Pi_K}({\mathbf}q,u)\|_{{\partial}K}\le \|{\mathrm}P_M({\boldsymbol}\Pi_k^g{\mathbf}q\cdot{\mathbf}n-{\mathbf}q\cdot{\mathbf}n)\|_{{\partial}K}+c_2h_K^{-1}\|{\mathrm}P_M(\Pi_{k+1}u-u)\|_{{\partial}K}.\end{aligned}$$ By and again, we complete the proof.
Proof of Theorem \[thm:est\_ell\] {#sec:prf_est}
---------------------------------
In this subsection, we give a step-by-step proof for Theorem \[thm:est\_ell\]. The proof will be very similar to those used in [@CoGoSa:2010], thanks to the introduction of the HDG+ projection. In this way, we are able to reuse the existing projection-based error analysis for the analysis of the HDG+ method.
\
[**Step 1: Error equations.**]{} We first define the error terms: $$\begin{aligned}
{5}
{\boldsymbol}\varepsilon_h^q:={\boldsymbol}\Pi{\mathbf}q-{\mathbf}q_h\in{\mathbf}V_h,\quad \varepsilon_h^u:=\Pi u-u_h\in W_h,\quad \widehat{\varepsilon}_h^{\,u}:={\mathrm}P_Mu-\widehat{u}_h\in M_h. \end{aligned}$$ Now, by testing with $({\mathbf}r,w,\mu)\in{\mathbf}V_h\times W_h\times M_h$ and then using , we obtain the projection equations:
\[eq:prj\_eqns\] $$\begin{aligned}
\label{eq:prj_eqns_1}
(\kappa^{-1}{\boldsymbol}\Pi{\mathbf}q,{\mathbf}r)_{{\mathcal}T_h} - (\Pi u,{\nabla\cdot}{\mathbf}r)_{{\mathcal}T_h} + {\langle {\mathrm}P_Mu,{\mathbf}r\cdot{\mathbf}n\rangle}_{{\partial}{\mathcal}T_h}
&= (\kappa^{-1}({\boldsymbol}\Pi{\mathbf}q-{\mathbf}q),{\mathbf}r)_{{\mathcal}T_h},\\
\label{eq:prj_eqns_2}
({\nabla\cdot}{\boldsymbol}\Pi{\mathbf}q,w)_{{\mathcal}T_h} + {\langle \tau{\mathrm}P_M(\Pi u-{\mathrm}P_Mu),w\rangle}_{{\partial}{\mathcal}T_h}
&=(f,w)_{{\mathcal}T_h}+{\langle \delta_\tau({\mathbf}q,u), w\rangle}_{{\partial}{\mathcal}T_h},\\
\label{eq:prj_eqns_3}
-{\langle {\boldsymbol}\Pi{\mathbf}q\cdot{\mathbf}n+\tau(\Pi u-u),\mu\rangle}_{{\partial}{\mathcal}T_h\backslash\Gamma} &= -{\langle \delta_\tau({\mathbf}q,u),\mu\rangle}_{{\partial}{\mathcal}T_h\backslash\Gamma},\\
\label{eq:prj_eqns_4}
{\langle {\mathrm}P_Mu,\mu\rangle}_{\Gamma} &= {\langle g,\mu\rangle}_\Gamma,\end{aligned}$$ for all $({\mathbf}r,w,\mu)\in{\mathbf}V_h\times W_h\times M_h$.
In the above equations, , , and are obtained by using , , and , respectively. The equation holds obviously since ${\mathrm}P_M\big|_{{\partial}K}$ is the $L^2$ projection to ${\mathcal}R_k({\partial}K)$ for all $K\in{\mathcal}T_h$.
By taking the difference between and , we obtain the error equations:
\[eq:err\_eqns\] $$\begin{aligned}
\label{eq:err_eqns_1}
(\kappa^{-1}{\boldsymbol}\varepsilon_h^q,{\mathbf}r)_{{\mathcal}T_h} - (\varepsilon_h^u,{\nabla\cdot}{\mathbf}r)_{{\mathcal}T_h} + {\langle \widehat{\varepsilon}_h^{\,u},{\mathbf}r\cdot{\mathbf}n\rangle}_{{\partial}{\mathcal}T_h} &= (\kappa^{-1}({\boldsymbol}\Pi{\mathbf}q-{\mathbf}q),{\mathbf}r)_{{\mathcal}T_h},\\
\label{eq:err_eqns_2}
({\nabla\cdot}{\boldsymbol}\varepsilon_h^q,w)_{{\mathcal}T_h} + {\langle \tau{\mathrm}P_M(\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u}),w\rangle}_{{\partial}{\mathcal}T_h}
&={\langle \delta_\tau({\mathbf}q,u), w\rangle}_{{\partial}{\mathcal}T_h},\\
\label{eq:err_eqns_3}
-{\langle {\boldsymbol}\varepsilon_h^q\cdot{\mathbf}n+\tau(\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u}),\mu\rangle}_{{\partial}{\mathcal}T_h\backslash\Gamma} &= -{\langle \delta_\tau({\mathbf}q,u),\mu\rangle}_{{\partial}{\mathcal}T_h\backslash\Gamma},\\
\label{eq:err_eqns_4}
{\langle \widehat{\varepsilon}_h^{\,u},\mu\rangle}_{\Gamma} &= 0,\end{aligned}$$ for all $({\mathbf}r,w,\mu)\in{\mathbf}V_h\times W_h\times M_h$.
\
[**Step 2: Energy identity.**]{} By testing the error equations with ${\mathbf}r={\boldsymbol}\varepsilon_h^q$, $w=\varepsilon_h^u$, $\mu=\widehat{\varepsilon}_h^{\,u}$ in - and adding the equations, then using , which suggests that $\widehat{\varepsilon}_h^{\,u}\big|_\Gamma=0$, we obtain the following energy identity: $$\begin{aligned}
\label{eq:ene_id}
(\kappa^{-1}{\boldsymbol}\varepsilon_h^q,{\boldsymbol}\varepsilon_h^q)_{{\mathcal}T_h}+{\langle \tau{\mathrm}P_M(\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u}),\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u}\rangle}_{{\partial}{\mathcal}T_h}=(\kappa^{-1}({\boldsymbol}\Pi{\mathbf}q-{\mathbf}q),{\boldsymbol}\varepsilon_h^q)_{{\mathcal}T_h}+{\langle \delta_\tau({\mathbf}q,u),\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u}\rangle}_{{\partial}{\mathcal}T_h}.\end{aligned}$$ By using the energy identity , we easily obtain $$\begin{aligned}
\label{eq:est_qh_jump}
\|\kappa^{-1/2}{\boldsymbol}\varepsilon_h^q\|_{{\mathcal}T_h}^2+\|\tau^{1/2}{\mathrm}P_M(\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u})\|_{{\partial}{\mathcal}T_h}^2
\le \|\kappa^{-1/2}({\boldsymbol}\Pi{\mathbf}q-{\mathbf}q)\|_{{\mathcal}T_h}^2
+\|\tau^{-1/2}\delta_\tau({\mathbf}q,u)\|_{{\partial}{\mathcal}T_h}^2.\end{aligned}$$ This proves . We are next going to prove and .
\
[**Step 3: Duality identity.**]{} We first introduce the duality equations of :
\[eq:dual\_pde\] $$\begin{aligned}
{5}
\kappa^{-1}{\boldsymbol}\psi - \nabla \phi &= {\mathbf}0 &\qquad& {\mathrm}{in}\ \Omega,\\
-{\nabla\cdot}{\boldsymbol}\psi & = \theta&&{\mathrm}{in}\ \Omega,\\
\phi & =0 &&{\mathrm}{on}\ \Gamma,\end{aligned}$$
We next define the projections and the boundary remainder of the solutions of the duality equations : $$\begin{aligned}
{5}
{\boldsymbol}\Pi{\boldsymbol}\psi:=\prod_{K\in{\mathcal}T_h}{\boldsymbol}\Pi_K{\boldsymbol}\psi,\quad
\Pi\phi:=\prod_{K\in{\mathcal}T_h}\Pi_K\phi,\quad
\delta_{-\tau}({\boldsymbol}\psi,\phi):=\prod_{K\in{\mathcal}T_h}\delta_{-\tau}^{\Pi_K}({\boldsymbol}\psi,\phi).\end{aligned}$$ Note that we used $-\tau$ to define the boundary remainder. By testing with $({\mathbf}r,w,\mu)\in{\mathbf}V_h\times W_h\times M_h$ and then using , we obtain the following equations in a similar way we obtained :
\[eq:dual\_pj\_eqns\] $$\begin{aligned}
\label{eq:dual_pj_eqns_1}
(\kappa^{-1}{\boldsymbol}\Pi{\boldsymbol}\psi,{\mathbf}r)_{{\mathcal}T_h} + (\Pi\phi,{\nabla\cdot}{\mathbf}r)_{{\mathcal}T_h} - {\langle {\mathrm}P_M\phi,{\mathbf}r\cdot{\mathbf}n\rangle}_{{\partial}{\mathcal}T_h} &= (\kappa^{-1}({\boldsymbol}\Pi{\boldsymbol}\psi-{\boldsymbol}\psi),{\mathbf}r)_{{\mathcal}T_h},\\
\label{eq:dual_pj_eqns_2}
-({\nabla\cdot}{\boldsymbol}\Pi{\boldsymbol}\psi,w)_{{\mathcal}T_h} + {\langle \tau{\mathrm}P_M(\Pi\phi-{\mathrm}P_M\phi),w\rangle}_{{\partial}{\mathcal}T_h}
&=(\theta,w)_{{\mathcal}T_h}-{\langle \delta_{-\tau}({\boldsymbol}\psi,\phi), w\rangle}_{{\partial}{\mathcal}T_h},\\
\label{eq:dual_pj_eqns_3}
{\langle {\boldsymbol}\Pi{\boldsymbol}\psi\cdot{\mathbf}n-\tau(\Pi\phi-\phi),\mu\rangle}_{{\partial}{\mathcal}T_h\backslash\Gamma} &= {\langle \delta_{-\tau}({\boldsymbol}\psi,\phi),\mu\rangle}_{{\partial}{\mathcal}T_h\backslash\Gamma},\\
\label{eq:dual_pj_eqns_4}
{\langle {\mathrm}P_M\phi,\mu\rangle}_{\Gamma} &= 0,\end{aligned}$$
for all $({\mathbf}r,w,\mu)\in{\mathbf}V_h\times W_h\times M_h$. Now we test - with ${\mathbf}r={\boldsymbol}\Pi\psi$, $w=\Pi\phi$, $\mu={\mathrm}P_M\phi$, test - with ${\mathbf}r={\boldsymbol}\varepsilon_h^q$, $w=\varepsilon_h^u$, $\mu=\widehat{\varepsilon}_h^{\,u}$, and use and , which imply $\widehat{\varepsilon}_h^{\,u}\big|_\Gamma={\mathrm}P_M\phi\big|_\Gamma=0$. Comparing the two sets of equations, we obtain $$\begin{aligned}
&(\kappa^{-1}({\boldsymbol}\Pi{\mathbf}q-{\mathbf}q),{\boldsymbol}\Pi{\boldsymbol}\psi)_{{\mathcal}T_h}+{\langle \delta_\tau({\mathbf}q,u),\Pi\phi-{\mathrm}P_M\phi\rangle}_{{\partial}{\mathcal}T_h}\\
&\qquad =(\kappa^{-1}({\boldsymbol}\Pi{\boldsymbol}\psi-{\boldsymbol}\psi),{\boldsymbol}\varepsilon_h^q)_{{\mathcal}T_h}+(\theta,\varepsilon_h^u)_{{\mathcal}T_h}
-{\langle \delta_{-\tau}({\boldsymbol}\psi,\phi),\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u}\rangle}_{{\partial}{\mathcal}T_h}.\end{aligned}$$ Assuming $k\ge1$ and rearranging the terms of the above identity, we have the following duality identity: $$\begin{aligned}
\nonumber
(\theta,\varepsilon_h^u)_{{\mathcal}T_h}
&=(({\boldsymbol}\Pi{\mathbf}q-{\mathbf}q),\nabla\phi-{\boldsymbol}\Pi_0\nabla\phi)_{{\mathcal}T_h}
+(\kappa^{-1}({\boldsymbol}\Pi{\boldsymbol}\psi-{\boldsymbol}\psi),{\mathbf}q_h-{\mathbf}q)_{{\mathcal}T_h}\\
\label{eq:dual_id}
&\quad +{\langle \delta_\tau({\mathbf}q,u),{\mathrm}P_M\Pi \phi-{\mathrm}P_M\phi\rangle}_{{\partial}{\mathcal}T_h}
+{\langle \delta_{-\tau}({\boldsymbol}\psi,\phi),{\mathrm}P_M\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u}\rangle}_{{\partial}{\mathcal}T_h}.\end{aligned}$$ Note that in the above equation, we have used $({\boldsymbol}\Pi{\mathbf}q-{\mathbf}q,{\boldsymbol}\Pi_0\nabla\phi)=0$, which holds because of and the assumption $k\ge1$.
\
[**Step 4: Estimating $u_h$ and $\widehat{u}_h$.**]{} By with $m=1$ and then using , we have $$\begin{aligned}
&\|\nabla\phi-{\boldsymbol}\Pi_0\nabla\phi\|_{{\mathcal}T_h}+\|{\boldsymbol}\Pi{\boldsymbol}\psi-{\boldsymbol}\psi\|_{{\mathcal}T_h}
+\|\tau^{1/2}({\mathrm}P_M\Pi\phi-{\mathrm}P_M\phi)\|_{{\partial}{\mathcal}T_h}+\|\tau^{-1/2}\delta_{-\tau}({\boldsymbol}\psi,\phi)\|_{{\partial}{\mathcal}T_h}\\
&\qquad\lesssim h(|{\boldsymbol}\psi|_{1,\Omega}+|\phi|_{2,\Omega})\lesssim h\|\theta\|_{{\mathcal}T_h}.\end{aligned}$$ Taking $\theta=\varepsilon_h^u$ in , we have $$\begin{aligned}
\|\varepsilon_h^u\|_{{\mathcal}T_h}\lesssim h\left(\|{\boldsymbol}\Pi{\mathbf}q-{\mathbf}q\|_{{\mathcal}T_h}+\|{\mathbf}q_h-{\mathbf}q\|_{{\mathcal}T_h}+\|\tau^{-1/2}\delta_\tau({\mathbf}q,u)\|_{{\partial}{\mathcal}T_h}+\|\tau^{1/2}({\mathrm}P_M\varepsilon_h^u-\widehat{\varepsilon}_h^{\,u})\|_{{\partial}{\mathcal}T_h}\right).\end{aligned}$$ The above inequality with implies .
It now only remains to estimate the term $\|{\mathrm}Pu-\widehat{u}_h\|_h$. First note that $$\begin{aligned}
\label{eq:prf_15}
\|\widehat{\varepsilon}_h^{\,u}\|_h^2=\sum_{K\in{\mathcal}T_h}h_K\|\widehat{\varepsilon}_h^{\,u}\|_{{\partial}K}^2
\approx\sum_{K\in{\mathcal}T_h}h_K^2\|\tau^{1/2}\widehat{\varepsilon}_h^{\,u}\|_{{\partial}K}^2
\le h^2\|\tau^{1/2}\widehat{\varepsilon}_h^{\,u}\|_{{\partial}{\mathcal}T_h}^2.\end{aligned}$$ By , we have $$\begin{aligned}
\label{eq:prf_11}
\|\tau^{1/2}\widehat{\varepsilon}_h^{\,u}\|_{{\partial}{\mathcal}T_h}\lesssim \|\tau^{1/2}{\mathrm}P_M\varepsilon_h^u\|_{{\partial}{\mathcal}T_h}+\|{\boldsymbol}\Pi{\mathbf}q-{\mathbf}q\|_{{\mathcal}T_h}+\|\tau^{-1/2}\delta_\tau({\mathbf}q,u)\|_{{\partial}{\mathcal}T_h}.\end{aligned}$$ By using , we can estimate the term $\|\tau^{1/2}{\mathrm}P_M\varepsilon_h^u\|_{{\partial}{\mathcal}T_h}$ as follows: $$\begin{aligned}
\nonumber
\|\tau^{1/2}{\mathrm}P_M\varepsilon_h^u\|_{{\partial}{\mathcal}T_h}^2
&=\sum_{K\in{\mathcal}T_h}\|\tau^{1/2}{\mathrm}P_M(\Pi u-u_h)\|_{{\partial}K}^2
\lesssim \sum_{K\in{\mathcal}T_h}h_K^{-2}\|\Pi u-u_h\|_{K}^2\\
\label{eq:prf_13}
&\lesssim h_{\mathrm}{min}^{-2}h^2\left(
\|{\boldsymbol}\Pi{\mathbf}q-{\mathbf}q\|_{{\mathcal}T_h}
+\|\tau^{-1/2}\delta_\tau({\mathbf}q,u)\|_{{\partial}{\mathcal}T_h}
\right)^2.\end{aligned}$$ Combining , , and , we obtain . This completes the proof.
The projection for elasticity {#sec:hdgp_pj_elas}
=============================
Main results
------------
In [@du2019new], we devised the HDG+ projection for elasticity on simplicial elements. In this section, we extend the projection (see [@du2019new Theorem 2.1]) to polyhedral elements. This new projection will render all the analysis and estimates in [@du2019new Sections 5,6&7] valid for general polyhedral meshes. (The three sections in [@du2019new] cover the error analysis of the HDG+ methods for steady-state elasticity, time-harmonic elastodynamics, and transient elastic waves, respectively.)
We define the HDG+ projection for elasticity as follows:
\[eq:def\_hdg+\_elas\] $$\begin{aligned}
\label{eq:def_hdg+_elas_1}
{\boldsymbol}\Pi_K({\boldsymbol}\sigma,{\boldsymbol}u):= ({\boldsymbol}\Pi_k^{\mathrm}{sg}{\boldsymbol}\sigma,{\boldsymbol}\Pi_{k+1}{\boldsymbol}u)\in{\mathcal}P_k(K;{\mathbb}R_{\mathrm}{sym}^{d\times d})\times{\mathcal}P_{k+1}(K;{\mathbb}R^d),\end{aligned}$$ where ${\boldsymbol}\Pi_k^{\mathrm}{sg}: H^1(K;{\mathbb}R_{\mathrm}{sym}^{d\times d})\rightarrow {\mathcal}P_k(K;{\mathbb}R_{\mathrm}{sym}^{d\times d})$ is defined by solving $$\begin{aligned}
\label{eq:def_hdg+_elas_2}
({\boldsymbol}\Pi_k^{sg}{\boldsymbol}\sigma-{\boldsymbol}\sigma,{\boldsymbol}\theta)_K &=0\qquad\forall{\boldsymbol}\theta\in{\boldsymbol}\varepsilon({\mathcal}P_k(K;{\mathbb}R^d))\oplus {\boldsymbol}\varepsilon({\mathcal}P_{k+1}(K;{\mathbb}R^d))^{\perp_k},\\
\label{eq:def_hdg+_elas_3}
({\boldsymbol}\Pi_k^{sg}{\boldsymbol}\sigma-{\boldsymbol}\sigma,{\boldsymbol}\varepsilon({\boldsymbol}v))_K&={\langle {\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\sigma{\mathbf}n)-{\boldsymbol}\sigma{\mathbf}n,{\boldsymbol}v\rangle}_{{\partial}K}
\qquad\forall {\boldsymbol}v\in {\mathcal}P_k(K;{\mathbb}R^d)^{\perp_{k+1}}.\end{aligned}$$ In the above equations, $(\cdot)^{\perp_m}$ represents the orthogonal complement in the background space ${\mathcal}P_m(K;{\mathbb}R_{\mathrm}{sym}^{d\times d})$ (for ) or ${\mathcal}P_m(K;{\mathbb}R^d)$ (for ), the notation ${\boldsymbol}\varepsilon({\boldsymbol}v):=\nabla{\boldsymbol}v+(\nabla{\boldsymbol}v)^\perp$ represents the symmetric gradient, and ${\boldsymbol}{{\mathrm}P}_M:L^2({\partial}K;{\mathbb}R^d)\rightarrow{\mathcal}R_k({\partial}K;{\mathbb}R^d)$ is the $L^2$ projection to the range space.
We define the associated boundary remainder as follows: $$\begin{aligned}
\label{eq:def_hdg+_elas_4}
{\boldsymbol}\delta_{\pm\tau}^{\Pi_K}({\boldsymbol}\sigma,{\boldsymbol}u):= -({\boldsymbol}\Pi_K{\boldsymbol}\sigma\,{\mathbf}n-{\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\sigma{\mathbf}n))\pm{\boldsymbol}\tau({\boldsymbol}{{\mathrm}P}_M{\boldsymbol}\Pi_K{\boldsymbol}u-{\boldsymbol}{{\mathrm}P}_M{\boldsymbol}u)\in{\mathcal}R_k({\partial}K;{\mathbb}R^d),\end{aligned}$$
where ${\boldsymbol}\tau\in{\mathcal}R_0({\partial}K;{\mathbb}R_{\mathrm}{sym}^{d\times d})$ satisfying [@du2019new Eqn. (2.1)], namely, $c_1h_K^{-1}\|{\boldsymbol}\mu\|_{{\partial}K}^2\le{\langle {\boldsymbol}\tau{\boldsymbol}\mu,{\boldsymbol}\mu\rangle}_{{\partial}K}\le c_2h_K^{-1}\|{\boldsymbol}\mu\|_{{\partial}K}^2$ for all ${\boldsymbol}\mu\in L^2({\partial}K;{\mathbb}R^d)$ and two fixed positive constants $c_1$ and $c_2$.
The main result in this section is the following theorem. For notational convenience, we hide the dependence of ${\boldsymbol}\delta_{\pm\tau}^{\Pi_K}$ on $({\boldsymbol}\sigma,{\boldsymbol}u)$.
\[thm:pj\_elas\] The projection ${\boldsymbol}\Pi_K$ and the remainder ${\boldsymbol}\delta_{\pm\tau}^{\Pi_K}$ are well defined by and they satisfy
\[eq:hdgp\_pj\_el\] $$\begin{aligned}
{5}
\label{eq:hdgp_pj_el_1}
({\boldsymbol}\Pi_K{\boldsymbol}u -{\boldsymbol}u,{\boldsymbol}v)_K &= 0 &\quad& \forall{\boldsymbol}v\in{\mathcal}P_{k-1}(K;{\mathbb}R^d),\\
\label{eq:hdgp_pj_el_2}
{\langle -({\boldsymbol}\Pi_K{\boldsymbol}\sigma\,{\mathbf}n-{\boldsymbol}\sigma{\mathbf}n)\pm{\boldsymbol}\tau({\boldsymbol}\Pi_K{\boldsymbol}u-{\boldsymbol}u),{\boldsymbol}\mu\rangle}_{{\partial}K} &=
{\langle {\boldsymbol}\delta_{\pm\tau}^{\Pi_K},{\boldsymbol}\mu\rangle}_{{\partial}K}
&&\forall {\boldsymbol}\mu\in{\mathcal}R_k({\partial}K;{\mathbb}R^d),\\
\label{eq:hdgp_pj_el_3}
-({\nabla\cdot}({\boldsymbol}\Pi_K{\boldsymbol}\sigma-{\boldsymbol}\sigma),{\boldsymbol}w)_K\pm{\langle {\boldsymbol}\tau{\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\Pi_K{\boldsymbol}u-{\boldsymbol}u),{\boldsymbol}w\rangle}_{{\partial}K}
&= {\langle {\boldsymbol}\delta_{\pm\tau}^{\Pi_K},{\boldsymbol}w\rangle}_{{\partial}K}
&&\forall {\boldsymbol}w\in{\mathcal}P_{k+1}(K;{\mathbb}R^d).\end{aligned}$$
Furthermore, $$\begin{aligned}
\label{eq:hdgp_pj_el_conv}
\|{\boldsymbol}\Pi_K{\boldsymbol}\sigma-{\boldsymbol}\sigma\|_K+h_K^{-1}\|{\boldsymbol}\Pi_K{\boldsymbol}u-{\boldsymbol}u\|_K + h_K^{1/2}\|{\boldsymbol}\delta_{\pm\tau}^{\Pi_K}\|_{{\partial}K}
\le Ch_K^m(|{\boldsymbol}\sigma|_{m,K}+|{\boldsymbol}u|_{m+1,K}),\end{aligned}$$ where $m\in[1,k+1]$. Here, the constant $C$ depends only on $k$, $\gamma_K$, and $c_2$.
Note that the two boundary remainders ${\boldsymbol}\delta_{+\tau}^{\Pi_K}$ and ${\boldsymbol}\delta_{-\tau}^{\Pi_K}$ correspond to the HDG+ projection and the adjoint projection in [@du2019new Theorem 2.1], respectively. We also remark that we have used the HDG+ projection to define the initial velocity for the semi-discrete HDG+ scheme in [@du2019new]. Therefore, equations provide a way of calculating the initial conditions for the semi-discrete scheme for elastic waves.
Proof of Theorem \[thm:pj\_elas\]
---------------------------------
In this subsection, we prove Theorem \[thm:pj\_elas\]. The proof here will be similar to the proof of Theorem \[thm:hdgp\_pj\_ell\] in Section \[sec:ell\_hdg+\_pfs\].
\[prop:sg+\] The projection ${\boldsymbol}\Pi_k^{sg}$ is well defined by and , and we have $$\begin{aligned}
\label{eq:sg+_conv}
h_K^{1/2}\|{\boldsymbol}\Pi_k^{sg}{\boldsymbol}\sigma-{\boldsymbol}\sigma\|_{{\partial}K}+\|{\boldsymbol}\Pi_k^{sg}{\boldsymbol}\sigma-{\boldsymbol}\sigma\|_K\le
Ch_K^m|{\boldsymbol}\sigma|_{m,K},\end{aligned}$$ where $m\in[1,k+1]$. Here, the constant $C$ depends only on $k$ and the shape-regularity constant $\gamma_K$.
We can easily verify that and define a square system. We next prove the convergence equation , from which the unique solvability of and follows automatically. Let ${\boldsymbol}\varepsilon_\sigma:={\boldsymbol}\Pi_k^{\mathrm}{sg}{\boldsymbol}\sigma-{\boldsymbol}\Pi_k{\boldsymbol}\sigma$. By and , we have
\[eq:strpj\_pf\_1\] $$\begin{aligned}
\label{eq:strpj_pf_1a}
({\boldsymbol}\varepsilon_\sigma,{\boldsymbol}\theta)_K &=0\qquad\forall{\boldsymbol}\theta\in{\boldsymbol}\varepsilon({\mathcal}P_k(K;{\mathbb}R^d))\oplus {\boldsymbol}\varepsilon({\mathcal}P_{k+1}(K;{\mathbb}R^d))^{\perp_k},\\
\label{eq:strpj_pf_1b}
({\boldsymbol}\varepsilon_\sigma,{\boldsymbol}\varepsilon({\boldsymbol}v))_K&={\langle {\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\sigma{\mathbf}n)-{\boldsymbol}\sigma{\mathbf}n,{\boldsymbol}v\rangle}_{{\partial}K}
\qquad\forall {\boldsymbol}v\in {\mathcal}P_k(K;{\mathbb}R^d)^{\perp_{k+1}}.\end{aligned}$$
The above equations imply that $$\begin{aligned}
\label{eq:strpj_pf_2}
({\boldsymbol}\varepsilon_\sigma,{\boldsymbol}\varepsilon({\boldsymbol}v))_K={\langle {\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\sigma{\mathbf}n)-{\boldsymbol}\sigma{\mathbf}n,{\boldsymbol}v\rangle}_{{\partial}K}
\qquad\forall {\boldsymbol}v\in {\mathcal}P_{k+1}(K;{\mathbb}R^d).\end{aligned}$$ We now decompose ${\boldsymbol}\varepsilon_\sigma$ into the summation ${\boldsymbol}\varepsilon_\sigma={\boldsymbol}\varepsilon_\sigma^1+\varepsilon_\sigma^2$, where $\varepsilon_\sigma^1\in {\boldsymbol}\varepsilon({\mathcal}P_{k+1}(K;{\mathbb}R^d))$ and $\varepsilon_\sigma^2\in {\boldsymbol}\varepsilon({\mathcal}P_{k+1}(K;{\mathbb}R^d))^{\perp_k}$. Since ${\boldsymbol}\varepsilon_\sigma^1\in{\boldsymbol}\varepsilon({\mathcal}P_{k+1}(K;{\mathbb}R^d))$, we can write ${\boldsymbol}\varepsilon_\sigma^1={\boldsymbol}\varepsilon({\boldsymbol}p+{\boldsymbol}m)$ for some ${\boldsymbol}p\in{\mathcal}P_{k+1}(K;{\mathbb}R^d)$ and arbitrary rigid motion ${\boldsymbol}m\in{\mathcal}M$. By and we have $$\begin{aligned}
\|{\boldsymbol}\varepsilon_\sigma\|_{K}^2=({\boldsymbol}\varepsilon_\sigma,{\boldsymbol}\varepsilon_\sigma^1)_K
=({\boldsymbol}\varepsilon_\sigma,{\boldsymbol}\varepsilon({\boldsymbol}p+{\boldsymbol}m))_K
={\langle {\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\sigma{\mathbf}n)-{\boldsymbol}\sigma{\mathbf}n,{\boldsymbol}p+{\boldsymbol}m\rangle}_{{\partial}K}.\end{aligned}$$ We next apply [@QiShSh:2018 Lemma 4.1] to the term ${\boldsymbol}p+{\boldsymbol}m$ and then obtain $$\begin{aligned}
\|{\boldsymbol}\varepsilon_\sigma\|_{K}^2\lesssim h_K^{1/2}\|{\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\sigma{\mathbf}n)-{\boldsymbol}\sigma{\mathbf}n\|_{{\partial}K}\|{\boldsymbol}\varepsilon({\boldsymbol}p)\|_K\le h_K^{1/2}\|{\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\sigma{\mathbf}n)-{\boldsymbol}\sigma{\mathbf}n\|_{{\partial}K}\|{\boldsymbol}\varepsilon_\sigma\|_K.\end{aligned}$$ This completes the proof.
Let us now prove Theorem \[thm:pj\_elas\]. By Proposition \[prop:sg+\], we know ${\boldsymbol}\Pi_K$ and ${\boldsymbol}\delta_{\pm\tau}^{\Pi_K}$ are well defined by . We next prove equations . Equations and hold obviously by the definitions and .
To prove , first note that $$\begin{aligned}
\nonumber
&-({\nabla\cdot}({\boldsymbol}\Pi_k^{\mathrm}{sg}{\boldsymbol}\sigma-{\boldsymbol}\sigma),{\boldsymbol}w)_K
\pm{\langle {\boldsymbol}\tau{\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\Pi_{k+1}{\boldsymbol}u-{\boldsymbol}u),{\boldsymbol}w\rangle}_{{\partial}K}\\
\label{eq:pf_17}
&\qquad={\langle -({\boldsymbol}\Pi_k^{\mathrm}{sg}{\boldsymbol}\sigma\,{\mathbf}n-{\boldsymbol}\sigma{\mathbf}n)\pm{\boldsymbol}\tau{\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\Pi_{k+1}{\boldsymbol}u-{\boldsymbol}u),{\boldsymbol}w\rangle}_{{\partial}K}+({\boldsymbol}\Pi_k^{\mathrm}{sg}{\boldsymbol}\sigma-{\boldsymbol}\sigma,{\boldsymbol}\varepsilon({\boldsymbol}w))_K,\end{aligned}$$ for all ${\boldsymbol}w\in{\mathcal}P_{k+1}(K;{\mathbb}R^d)$. By and , we obtain $$\begin{aligned}
\label{eq:pf_19}
({\boldsymbol}\Pi_k^{\mathrm}{sg}{\boldsymbol}\sigma-{\boldsymbol}\sigma,{\boldsymbol}\varepsilon({\boldsymbol}v))_K={\langle {\boldsymbol}{{\mathrm}P}_M({\boldsymbol}\sigma{\mathbf}n)-{\boldsymbol}\sigma{\mathbf}n,{\boldsymbol}v\rangle}_{{\partial}K}
\qquad\forall {\boldsymbol}v\in {\mathcal}P_{k+1}(K;{\mathbb}R^d).\end{aligned}$$ Equations and imply .
The convergence property holds because of equations and , and the fact that ${\langle {\boldsymbol}\tau{\boldsymbol}\mu,{\boldsymbol}\mu\rangle}_{{\partial}K}\le c_2 h_K^{-1}\|{\boldsymbol}\mu\|_{{\partial}K}^2$ for all ${\boldsymbol}\mu\in L^2({\partial}K;{\mathbb}R^d)$. This completes the proof.
Conclusions {#conclusions .unnumbered}
===========
We have devised two new HDG+ projections on polyhedral elements, extending our previous results in [@DuSa:2019] for elliptic problems and the results in [@du2019new] for elasticity to polyhedral meshes. The projections here are constructed in a different way without using the $M$-decomposition as a middle step. Consequently, the construction is more straightforward. Future work of interest involves incorporating the HDG methods with Hybrid-High Order stabilization functions [@CoDiEr:2016] into the projection-based error analysis setting.
[**Acknowledgments.**]{} This work was partially supported by the NSF grant DMS-1818867. Shukai Du would like to thank B. Cockburn for the discussions that lead the paper to a better form, and for supporting his visit to the University of Minnesota, where the paper was written.
[^1]: Email: shukaidu@udel.edu
|
---
abstract: 'The duration distribution of 947 GRBs observed by [*Swift*]{}/BAT, as well as its subsample of 347 events with measured redshift, allowing to examine the durations in both the observer and rest frames, are examined. Using a maximum log-likelihood method, mixtures of two and three standard Gaussians are fitted to each sample, and the adequate model is chosen based on the value of the difference in the log-likelihoods, Akaike information criterion and Bayesian information criterion. It is found that a two-Gaussian is a better description than a three-Gaussian, and that the presumed intermediate-duration class is unlikely to be present in the [*Swift*]{} duration data.'
address: 'Astronomical Observatory, Jagiellonian University, Orla 171, Kraków, Poland'
author:
- Mariusz Tarnopolski
bibliography:
- 'mybibfile.bib'
title: 'Analysis of the observed and intrinsic durations of *Swift*/BAT gamma-ray bursts'
---
gamma-ray burst: general,methods: data analysis,methods: statistical
Introduction {#intro}
============
Gamma-ray bursts (GRBs) were detected by military satellites *Vela* in late 1960’s. @mazets first pointed out hints for a bimodal distribution of $T_b$ (taken to be the time interval within which fall $80\%-90\%$ of the measured GRB’s intensity) drawn for 143 events detected in the KONUS experiment. Burst and Transient Source Explorer (BATSE) onboard the Compton Gamma Ray Observatory (*CGRO*) provided data that were further investigated by @kouve, and led to establishing the common classification of GRBs into short ($T_{90}<2\,{\rm s}$) and long ($T_{90}>2\,{\rm s}$), where $T_{90}$ is the time during which 90% of the burst’s fluence is accumulated, referred to as the duration of a GRB. The progenitors of long GRBs are associated with supernovae related with collapse of massive stars [@woosley]. Progenitors of short GRBs are thought to be NS-NS or NS-BH mergers [@nakar], and no connection between short GRBs and supernovae has been proven [@zhang5]. It was observed that durations $T_{90}$ seem to exhibit log-normal distributions which were thereafter fitted to short and long GRBs [@mcbreen; @koshut; @kouve2; @horvath02].
The existence of an intermediate-duration GRB class, consisting of GRBs with $T_{90}$ in the range $2-10\,{\rm s}$, was put forward [@horvath98; @mukh] based on the analysis of BATSE 3B data. It was supported [@horvath02; @chatto] with the use of the complete BATSE dataset. Evidence for a third log-normal component was also found in [*Swift*]{}/BAT data [@horvath08; @zhang2; @huja; @horvath10]. Interestingly, @zitouni re-examined the BATSE current catalog as well as the [*Swift*]{} dataset, and found that a mixture of three Gaussians (3-G) fits the $\log T_{90}$ data from [*Swift*]{} better than a two-Gaussian (2-G), while in the case of BATSE statistical tests did not support the presence of a third component (hereinafter, the $\log T_{90}$ distributions are considered, and are shortly referred to as durations as well). Regarding [*Fermi*]{}/GBM [@gruber; @kienlin], a 3-G is a better fit than a 2-G,[^1] however the presence of a third group in the duration distribution was found to be unlikely [@Tarnopolski; @Tarnopolski2], which was based on the fact that the $\log T_{90}$ distribution is bimodal, i.e. it exhibits two local maxima [@Tarnopolski], and that a mixture of two skewed components follows the data better than a standard three-Gaussian [@Tarnopolski2].
The [*Swift*]{} data were re-examined by @bromberg, and they found that a limit of $0.8\,{\rm s}$ is more suitable for the GRBs observed by [*Swift*]{} than the conventional $2\,{\rm s}$ limit of @kouve. It should be stressed that @bromberg applied a different approach than @kouve and @Tarnopolski3: a functional form of the $T_{90}$ distribution different from the commonly used phenomenological log-normal distribution, coming from a physical model for the short duration collapsar distribution, and by means of exceeding a probability threshold that a GRB with a given $T_{90}$ is a non-collapsar. Interestingly, the limits for BATSE and [*Fermi*]{} data are consistent with the $2\,{\rm s}$ limit, and also with the results obtained by @Tarnopolski3, where based on the well-established conjecture that durations $T_{90}$ are log-normally distributed, the limit between short and long GRBs may be placed at the local minimum, which is detector-dependent. Finally, many works in which a 2-G was fitted to the $\log T_{90}$ distribution showed a significant overlap of components corresponding to short and long GRBs [@mcbreen; @koshut; @horvath02; @zhang2; @huja; @bromberg; @barnacka; @Tarnopolski3; @zitouni].
The aim of this paper is to analyze the current dataset of [*Swift*]{}/BAT GRBs, and to test whether a greater sample of 947 events leads to conclusions other than @zitouni arrived at for a set of 757 events. Moreover, a relevant increase of GRBs with measured redshift—347 compared to 248 GRBs examined by @zitouni—provides an opportunity for a re-evaluation of the GRB properties that are, after moving to the rest frame, not affected by cosmological factors. This paper is organized in the following manner. In Section \[meth\] the datasets, fitting method and statistical criteria used to infer the validity of the models applied are described. Section \[res\] presents the results of fitting a 2-G and 3-G to the whole sample of 947 GRBs, as well as a subsample of 347 events in both the observer and rest frames. Section \[disc\] is devoted to discussion, and gathers concluding remarks.
Methods {#meth}
=======
Dataset {#data}
-------
The [*Swift*]{} dataset contains 947 GRBs[^2] with measured duration $T_{90}$, of which 9% are short (87 events). 347 GRBs have their redshift known, and those constitute the second sample examined herein. It consists of 324 long GRBs and 23 short ones. A scatter plot of this subsample on a $\log T_{90}-z$ plane is drawn in Fig. \[fig1\]. The median redshift for short and long GRBs is equal to $\tilde{z}_{\rm short}=0.72$ and $\tilde{z}_{\rm long}=1.90$, respectively. The intrinsic durations are calculated according to $$T^{\rm int}_{90}=\frac{T^{\rm obs}_{90}}{1+z}.
\label{eq1}$$ Distributions of the $\log T_{90}$ for the observed and intrinsic durations are examined hereinafter, and are displayed together with the distribution of the whole sample in Fig. \[fig2\].
![A scatter plot of the redshift versus the observed duration of the [*Swift*]{} subsample. Vertical dotted line denotes the limitting value of $2\,{\rm s}$ between short and long GRBs, and the horizontal dashed lines mark the medians of the respective classes, with values written in the plot.[]{data-label="fig1"}](figA.pdf){width="50.00000%"}
![Distributions of the examined samples: the whole sample (solid black); observed (dashed red) and intrinsic (dotted blue) durations in the subsample of GRBs with known redshift. The distributions of observed durations for both samples (all GRBs and those with measured redshift) are similar to each other.[]{data-label="fig2"}](figB.pdf){width="50.00000%"}
Fitting method {#fit}
--------------
Two standard fitting techniques are commonly applied: $\chi^2$ fitting [@voinov] and maximum likelihood (ML, @kendall). For the first, data needs to be binned, and despite various binning rules are known (e.g. Freedman-Diaconis, Scott, Knuth etc.), they still leave place for ambiguity, as it might happen that the fit may be statistically significant on a given significance level for a number of binnings [@huja; @koen; @Tarnopolski]. The ML method is not affected by this issue and is therefore applied herein. However, for display purposes, the binning was chosen based on the Freedman-Diaconis rule.
Having a distribution with a probability density function (PDF) given by $f=f(x;\theta)$ (possibly a mixture), where $\theta=\left\{\theta_i\right\}_{i=1}^p$ is a set of parameters, the log-likelihood function is defined as $$\mathcal{L}_p(\theta)=\sum\limits_{i=1}^N\ln f(x_i;\theta),
\label{eq2}$$ where $\left\{x_i\right\}_{i=1}^N$ are the datapoints from the sample to which a distribution is fitted. The fitting is performed by searching a set of parameters $\hat{\theta}$ for which the log-likelihood is maximized. When nested models are considered, the maximal value of the log-likelihood function $\mathcal{L}_{\rm max}\equiv\mathcal{L}_p(\hat{\theta})$ increases when the number of parameters $p$ increases.
A mixture of $k$ standard normal (Gaussian) distributions: $$f_k(x) = \sum\limits_{i=1}^k \frac{A_i}{\sqrt{2\pi}\sigma_i}\exp\left(-\frac{(x-\mu_i)^2}{2\sigma_i^2}\right),
\label{eq9}$$ is considered. It is described by $p=3k-1$ free parameters: $k$ pairs $(\mu_i,\sigma_i)$ and $k-1$ weights $A_i$, satysfying $\sum_{i=1}^k A_i=1$ due to normalization of a PDF. Therefore, $p=5$ for a 2-G, and $p=8$ for a 3-G.
Statistical criteria {#crit}
--------------------
If one has two fits such that $\mathcal{L}_{p_2,{\rm max}} > \mathcal{L}_{p_1,{\rm max}}$, then twice their difference, $2\Delta\mathcal{L}_{\rm max}=2(\mathcal{L}_{p_2,{\rm max}}-\mathcal{L}_{p_1,{\rm max}})$, is distributed like $\chi^2(\Delta p)$, where $\Delta p=p_2-p_1>0$ is the difference in the number of parameters [@kendall; @horvath02]. If a $p$-value associated with the value of $\chi^2(\Delta p)$ does not exceed the significance level $\alpha$, one of the fits (with higher $\mathcal{L}_{\rm max}$) is statistically better than the other. For instance, for a 2-G and a 3-G, $\Delta p=3$, and despite that, according to Footnote \[fn1\], $\mathcal{L}_{\rm max,\,3-G} > \mathcal{L}_{\rm max,\,2-G}$ holds always, twice their difference provides a decisive $p$-value.
For nested as well as non-nested models, the Akaike information criterion ($AIC$) [@akaike; @burnham; @liddle] may be applied. The $AIC$ is defined as $$AIC=2p-2\mathcal{L}_{p,{\rm max}}.
\label{eq3}$$ A preferred model is the one that minimizes $AIC$. The formulation of $AIC$ penalizes the use of an excessive number of parameters, hence discourages overfitting. It prefers models with fewer parameters, as long as the others do not provide a substantially better fit. The expression for $AIC$ consists of two competing terms: the first measuring the model complexity (number of free parameters), and the second measuring the goodness of fit (or more precisely, the lack of thereof). Among candidate models with $AIC_i$, let $AIC_{\rm min}$ denote the smallest. Then, $$Pr_i=\exp\left(-\frac{\Delta_i}{2}\right),
\label{eq4}$$ where $\Delta_i=AIC_i-AIC_{\rm min}$, can be interpreted as the relative (compared to $AIC_{\rm min}$) probability that the $i$th model minimizes the $AIC$.[^3]
The $AIC$ is suitable when $N/p$ is large, i.e. when $N/p>40$ [@burnham see also references therein]. When this condition is not fulfilled, a second order bias correction is introduced, resulting in a small-sample version of the $AIC$, called $AIC_c$: $$AIC_c=2p-2\mathcal{L}_{p,{\rm max}}+\frac{2p(p+1)}{N-p-1}.
\label{eq5}$$ The relative probability is computed similarly to when $AIC$ is used, i.e. Eq. (\[eq4\]) is valid when one takes $\Delta_i=AIC_{c,i}-AIC_{c,{\rm min}}$. Thence, $$Pr_i=\exp\left(-\frac{AIC_{c,i}-AIC_{c,{\rm min}}}{2}\right).
\label{eq6}$$ Obviously, $AIC_c$ converges to $AIC$ when $N$ is large.
It is important to note that this method allows to choose a model that is best among a given set, but does not allow to state that this model is the best among all possible ones. Hence, the probabilities computed by means of Eq. (\[eq6\]) are the relative (with respect to a model with $AIC_{c,{\rm min}}$) probabilities that the data is better described by a model with $AIC_{c,i}$. What is essential in assesing the goodness of a fit in the $AIC$ method is the difference, $\Delta_i=AIC_{c,i}-AIC_{c,{\rm min}}$, not the absolute value of an $AIC_{c,i}$.[^4] If $\Delta_i<2$, then there is substantial support for the $i$th model, and the proposition that it is a proper description is highly probable. If $2<\Delta_i<4$, then there is strong support for the $i$th model. When $4<\Delta_i<7$, there is considerably less support, and models with $\Delta_i>10$ have essentially no support [@burnham].
The Bayesian information criterion ($BIC$) was introduced by @schwarz, and is defined as $$BIC=p\ln N-2\mathcal{L}_{p,{\rm max}}.
\label{eq7}$$ As was the case in the $AIC$ (or $AIC_c$), a preferred model is the one that minimizes $BIC$, which also penalizes the usage of an excessive number of free parameters. The most striking difference between the two is that the penalization term, $k\ln N$, is greater than the corresponding term from the $AIC$, i.e. $2p$, for $N\geq 8$. Hence, the penalization in case of the $BIC$ is much more stringent, especially for large samples.
The probability in favor of the $i$th model, relative to a model with $BIC_{\rm min}$, is defined in the same manner as it was for $AIC$: $$Pr_i=\exp\left(-\frac{\Delta_i}{2}\right),
\label{eq8}$$ where $\Delta_i=BIC_i-BIC_{\rm min}$ in this case, and the support for the $i$th model (or evidence against it) also depends on the differences: if $\Delta_i<2$, then there is substantial support for the $i$th model (or the evidence against it is worth only a bare mention). When $2<\Delta_i<6$, then there is positive evidence against the $i$th model. If $6<\Delta_i<10$, the evidence is strong, and models with $\Delta_i>10$ yield a very strong evidence against the $i$th model (essentially no support, @kass).
Despite apparent similarities between the $AIC$ and $BIC$, they answer different questions, as they are derived based on different assumptions. $AIC$ tries to select a model that most adequately describes reality (in the form of the data under examination). This means that in fact the model being a real description of the data is never considered. On the contrary, $BIC$ tries to find the true model among the set of candidates. Because $BIC$ is more stringent, it has a tendency to underfit, while $AIC$, as a more liberal method, is inclined towards overfitting. This leads sometimes to pointing different models by the two criteria, which happens rarely, but is due to the fact that they try to satisfy different conditions.
Results {#res}
=======
All *Swift* GRBs {#res1}
----------------
Using the ML method from Sect. \[fit\], the fitting of a 2-G and 3-G to the duration distribution of 947 GRBs is performed. The results, in graphical form, are shown in Fig. \[fig3\]. The two-component mixture is unimodal, but with a prominent tail on the left side. The three-component fit is bimodal, with a clear shoulder on the left side of the peak related to long GRBs. The local minimum is placed at $T_{90}=1.07\,{\rm s}$. It is interesting to note that this value is consistent with the short-long GRB limit from @bromberg. The overall shape of the curve is in agreement with previous results [@horvath08; @zhang2; @Tarnopolski2; @zitouni].
![Distributions fitted to $\log T^{\rm obs}_{90}$ of all [*Swift*]{} GRBs. Color dashed curves are the components of the (black solid) mixture distribution. The panels show mixtures of (a) two and (b) three standard Gaussians.[]{data-label="fig3"}](figE.pdf){width="50.00000%"}
The parameters of the fits are gathered in Table \[tbl1\]. Twice the difference in $\mathcal{L}_{p,{\rm max}}$ is equal to 22.336, what corresponds to a $p$-value of $6\times 10^{-5}$, indicating that a 3-G is a highly significant improvement over a 2-G. This is confirmed with the $AIC_c$ approach, as their difference is equal to 16.246, what gives a probability of $3\times 10^{-4}$ that the 2-G might in fact be a better description than a 3-G. However, the results of the $BIC$ give a much lower significance—the difference is only 1.776, corresponding to a probability of 0.41 in favor of the 2-G. Nevertheless, all criteria pointed at a 3-G as a better model among the two under consideration.
$i$ $\mu_i$ $\sigma_i$ $A_i$ $\mathcal{L}_{\rm max}$ $AIC_c$ $BIC$
----- ---------- ------------ ------- ------------------------- -------------- --------------
1 0.083 0.781 0.155
2 1.657 0.521 0.845
1 $-0.407$ 0.529 0.093
2 0.878 0.322 0.190 $-$**1024.183** **2064.519** **2103.192**
3 1.793 0.434 0.717
: Parameters of the fits for the observed durations of 947 [*Swift*]{} GRBs. The values in favor of a respective model are marked in bold.[]{data-label="tbl1"}
Subsample of *Swift* GRBs with measured redshift {#res2}
------------------------------------------------
### Observed durations {#res21}
The observed durations of the redshift-equipped GRB subsample are examined in the same way as in the previous Section \[res1\]. The fitted curves, displayed in Fig. \[fig4\], resemble the ones obtained for the complete GRB sample. The fitted parameters, gathered in Table \[tbl2\], are in good agreement, too. The difference in $\mathcal{L}_{p,{\rm max}}$ multiplied by two, being equal to 6.522, corresponds to a relatively high probability of 0.09. The difference in $AIC_c$, equal to 0.270, is negligible, and hence based on it one can not rule out any of the fits—the relative probability is 0.87. On the other hand, the lower $BIC$ was achieved by a 2-G, and the difference is a prominent 11.027, which gives a probability of $4\times 10^{-3}$. Hence, the overall evidence against a 3-G is very strong—it follows that based on $BIC$, a 2-G is a better model.
![The same as Fig. \[fig3\], but for a subsample of GRBs with measured redshift.[]{data-label="fig4"}](figC.pdf){width="50.00000%"}
$i$ $\mu_i$ $\sigma_i$ $A_i$ $\mathcal{L}_{\rm max}$ $AIC_c$ $BIC$
----- ---------- ------------ ------- ------------------------- ------------- ---------
1 0.745 0.894 0.221
2 1.739 0.530 0.779
1 $-0.019$ 0.664 0.089
2 0.803 0.236 0.114 $-$**369.020** **754.467** 784.835
3 1.793 0.434 0.797
: Parameters of the fits for the observed durations of 347 [*Swift*]{} GRBs with measured redshift.[]{data-label="tbl2"}
### Intrinsic durations {#res22}
In the case of the intrinsic durations, the fits displayed in Fig. \[fig5\] reveal a systematic shift, compared to the distribution of $T^{\rm obs}_{90}$, towards shorter durations. While a 2-G is again unimodal and skewed leftwards, the 3-G shows a peak at $T_{90}=1.42\,{\rm s}$, between the regions of short and long GRBs. The parameters of the fits are gathered in Table \[tbl3\].
![The same as Fig. \[fig4\], but in the rest frame.[]{data-label="fig5"}](figD.pdf){width="50.00000%"}
The doubled difference of $\mathcal{L}_{p,{\rm max}}$ is equal to 1.504, what implies a high relative probability of 0.68. The difference in $AIC_c$ is 4.747, what hints toward a 2-G with a relative probability of 0.09 that a 3-G is in fact a better model. In this case, the $BIC$ yield a difference of 16.045, which gives a strong support in favor of a 2-G. The relative probability that the 3-G might be the more appropriate description of the data, is only $3\times 10^{-4}$. Overall, the $AIC_c$ and $BIC$ are in agreement, and without taking into account the doubled difference of $\mathcal{L}_{p,{\rm max}}$, because it provides no evidence in favor of any model, it turns out that a 2-G is a better model than a 3-G.
$i$ $\mu_i$ $\sigma_i$ $A_i$ $\mathcal{L}_{\rm max}$ $AIC_c$ $BIC$
----- ---------- ------------ ------- ------------------------- --------- ---------
1 0.913 0.802 0.705
2 1.487 0.326 0.295
1 $-0.480$ 0.521 0.068
2 0.353 0.244 0.141 $-$**378.892** 774.210 804.579
3 1.347 0.538 0.791
: Parameters of the fits for the intrinsic durations of 347 [*Swift*]{} GRBs.[]{data-label="tbl3"}
Discussion and conclusions {#disc}
==========================
The duration distribution of 947 GRBs observed by [*Swift*]{}, and a subsample consisting of 347 events with measured redshift, were investigated. The redshifts allowed to examine the intrinsic durations, i.e. in the rest frame, as well. Mixtures of two and three standard Gaussians were fitted. For each sample, the best fit was chosen based on the value of the difference in the log-likelihoods doubled, Akaike information criterion and Bayesian information criterion. The main conclusions are as follows:
1. All three criteria point at a 3-G as an adequate description of the $\log T_{90}$ distribution of all 947 [*Swift*]{} GRBs. The fit is bimodal, what is in good agreement with the two well established populations (mergers for short, and collapsars for long GRBs). This might suggest that the commonly applied log-normal distribution is not a good model for the observed duration distribution.
2. For a subsample of 347 GRBs with measured redshift, the analyses of the observed and intrinsic durations yielded results being in quite good agreement. While only the $BIC$ hints at a unimodal 2-G in the case of $\log T^{\rm obs}_{90}$, the other two criteria did not yield support for any of the models strong enough to infer their plausibility, but with a low ratio of short to long GRBs ($<1:14$) in this [*Swift*]{} subsample, combined with the well known overlap of durations of short and long GRBs and the relative smallness of the subsample, this is not that surprising.
3. The intrinsic durations, $\log T^{\rm int}_{90}$, are best described by a unimodal 2-G, too. Both $AIC_c$ and $BIC$ yield strong support in favor of a two-component Gaussian, while the criterion based on log-likelihoods did not provide a conclusive outcome. Hence, in all three samples the presence of two populations was found to be likely, and the evidence against a three-component description is strong.
4. In case of $T^{\rm obs}_{90}$ in the redshift-equipped subsample of 347 GRBs, the $AIC_c$ and $BIC$ lead to different conclusions: the $AIC_c$ pointed at a 3-G (with very weak evidence against a 2-G though), and $BIC$ yielded a very strong support for the 2-G. While this seems to be contradictive, it needs to be interpreted in the context of each criterion: according to $AIC_c$, a is a slightly better choice in describing the data, and because the difference in the models is negligible, the $BIC$ strongly favors the simpler one, i.e. the one with fewer parameters. This illustrates that any statistical criterion can not be a stand-alone determinant in inferring the underlying model, but [*(i)*]{} needs to be interpreted in the light of its conditions, and [*(ii)*]{} it is useful to apply different tools and analyze their output in relation to each other.
References {#references .unnumbered}
==========
[^1]: Adding parameters to a nested model always results in a better fit (in the sense of a lower $\chi^2$ or a higher maximum log-likelihood) due to more freedom given to the model to follow the data, i.e. due to introducing more free parameters. The important question is whether this improvement is statistically significant, and whether the model is justified\[fn1\].
[^2]: <http://swift.gsfc.nasa.gov/archive/grb_table.html>, accessed on September 30, 2015.
[^3]: Relative probabilities normalized to unity are called the Akaike weights, $w_i=\frac{\exp\left(-\Delta_i/2\right)}{\sum\limits_i \exp\left(-\Delta_i/2\right)}$. In Bayesian language, Akaike weight corresponds to the posterior probability of a model (under assumption of different prior probabilities; see @biesiada).
[^4]: The $AIC$ value contains scaling constants coming from the log-likelihood $\mathcal{L}$, and so $\Delta_i$ are free of such constants [@burnham]. One might consider $\Delta_i=AIC_{c,i}-AIC_{c,{\rm min}}$ a rescaling transformation that forces the best model to have $\Delta_{\rm min}:=0$.
|
---
author:
- |
\
[^1]
bibliography:
- 'IEEEabrv.bib'
- 'Referencelist.bib'
title: 'Co-Primary Multi-Operator Resource Sharing for Small Cell Networks'
---
Introduction
============
cellular systems are experiencing a growth data rate demands from users and it is expected that this trend will continue to speed up in the near future. Even though the fourth generation (4G) is still in its infancy, yet growing rapidly, the interest has already moved toward fifth generation (5G) networks. The continuing growth in demand for better coverage and capacity enhancements is pushing the industry to look ahead at how networks can meet future extreme capacity and performance demands [@ericsson13; @ericsson14].
5G mobile communication systems are expected to revolutionize everything seen so far in wireless systems. The requirements for 5G vary by application but will include data rates ranging from very low sensor data to very high video content delivery, stringent low latency requirements, low energy consumption, and high reliability [@5Gvision]. All of these technological requirements are expected to be achieved while keeping similar or lower cost than today’s technologies. 5G is likely to integrate enhancements in legacy radio access technologies with new developments in the areas of multiple access, waveform design, interference management, access protocols, network architecture and virtualization, massive MIMO, full-duplex radio technology, low latency, device-to-device (D2D) and machine type communication (MTC), etc [@metis].
In addition to radio access technology advances, network capacity and connectivity can be improved by network densification (mainly via small cell deployment) and by harnessing broader spectrum allocations. In addition to small cell deployments, there are many other techniques and systems that can improve coverage and data rates, in densely populated indoor environments. These techniques include the deployment of radio remote heads (RRHs), distributed antenna systems (DAS), WiFi access points, etc. The use of LTE small cells offers several advantages over such systems. Compared to DAS, LTE small cells are both cheaper and less complex to deploy [@Zhang_femtocells], and compared to WiFi, LTE small cells offer better performance, more efficient use of resources, and are well designed to support a substantial number of users [@huawei1].
Future networks are expected to include innovative ways of sharing both content and spectrum. This can be seen from observing current trends [@nsnCoPSS]. Mobile network operators (MNOs), which we will refer to as operators (OPs) hereafter in current wireless communication networks have commonly acquired exclusive usage rights for certain frequency bands and have little incentives to share it with other operators, despite significant research and regulatory efforts. This might be due to the lack of joint technological and business consideration. However, due to high cost and spectrum scarcity it can be expected that efficient use of spectrum in 5G networks will rather rely on sharing than exclusive licenses.
A recently proposed novel spectrum sharing mechanism towards 5G systems is the so-called co-primary spectrum sharing (CoPSS), where any OP is allowed to utilize shared spectrum allocated for 5G cellular systems. In [@metis2], CoPSS is defined as a spectrum access model where primary license holders agree on the joint use of (or parts of) their licensed spectrum. This would be possible in the small cells domain only where base stations coverage is similar to today’s WiFi access points and the frequency band is dedicated to small cell use. Depending on the expected time-frame for 5G roll-out, there are different views on the 5G system concept. The next World Radiocommunication Conference 2015 (WRC-15) will be quite important in setting the directions towards the next standard because it is evident that the next generation standard must be open enough to allow new spectrum sharing methods and drastically new technologies not even known during the development phase [@wrc]. Lot of discussion is going on regarding spectrum sharing at 3.5 GHz band in small cells which is in agenda of WRC-15 [@nsnCoPSS].
In [@CoPPS_business], enabling/limiting factors for CoPSS are discussed. Therein, the current scarcity of spectrum and new business potential, especially in hotspots and small cells, are seen as enabling factors for CoPSS. The limited availability of suitable spectrum for sharing, a low level of technical/business knowledge among OPs regarding CoPSS, and a lack of rules to coordinate sharing between OPs with similar customer profiles, are seen as limiting factors. The findings suggest that substantial further research is required, not only from a technical perspective, but also from business perspective.
Multi-operator spectrum sharing has been considered in many research papers over the years [@ss0c; @ss1c; @ss2c; @ss2j]. In [@saphyre], various aspects of inter-operator resource sharing have been studied such as analyzing and developing new self-organizing physical layer resource sharing models, analyzing efficient co-ordination mechanisms and developing a framework for infrastructure sharing. In [@saphyre2], the potential gain of spectrum sharing between cellular operators in terms of network efficiency is investigated. In [@ss1j], inter-operator sharing of cellular resources including capacity, spectrum and base stations is investigated. Therein, realistic sharing processes and architecture are proposed compatible with LTE.
To the best of our knowledge, the potential of CoPSS in LTE indoor multi-operator small cell base station (SBS) network has not been investigated. In this paper, CoPSS at the physical resource block (PRB) level is studied. Spectrum sharing at PRB level is challenging because different OPs’ SBSs have to be synchronized. However this type of spectrum sharing guarantees more efficient utilization of the spectrum. Coarser granularity component carrier level resource sharing may be a more practical approach when multi-operator networks are not jointly synchronized.
The main focus of this work is on CoPSS between SBSs belonging to different OPs. A dense indoor network deployment, consisting of multiple SBSs per building operated by three independent OPs, is considered. Traffic in the network is heterogeneous, i.e. a mix of full buffer and continuous constant rate traffic. Four CoPSS algorithms are proposed, and their performance is evaluated. These algorithms enable CoPSS when SBSs are not using 100% of their bandwidth. A given SBS is not fully utilizing its bandwidth when it can provide the required data rate for all the users without using 100% of its bandwidth, i.e. minimum bandwidth usage by SBS is ensured by utilizing the maximum transmission power and the highest order modulation and coding scheme (MCS) possible for all transmissions. From an energy efficiency perspective it may be beneficial for a SBS to utilize its full bandwidth for all transmissions. However, from CoPSS point of view it is more beneficial to keep the bandwidth usage to a minimum. Typically SBSs are placed in densely populated environments without frequency planning, and have time-variant traffic profiles. The aim is to use spectrum more efficiently, in order to reduce future spectrum requirements and increase the capacity of small cell networks.
Each SBS OP has its own dedicated spectrum, and each OP can define a percentage of how much spectrum they are willing to share. The idea of CoPSS is that spectrum is shared orthogonally and equally between operators. This way interference can be avoided and spectrum utilization is maximized. In three proposed algorithms, unused resources are shared equally between overloaded OPs for a given time instant, short term fairness among overloaded SBSs can be guaranteed. However, long term fairness between OPs cannot be guaranteed, i.e. the equal amount of the average loaned/rented spectrum usage over a given time period. Therefore, there is a need for a spectrum sharing framework that optimizes the usage of spectrum over a long-time period, which we address in our future work. In this paper we focus on the possible gains in the achieved throughput when OPs have similar traffic patterns.
The core of the extensive LTE-A system level network simulator has been built according to the International Telecommunication Union’s system level simulation guidelines [@WINII] and calibrated and rigorously evaluated in selected macro and microcell environments [@Pennanen], [@Haataja]. The simulator is extended to incorporate indoor femtocells, calibrated and verified in [@befemto2] and previously utilized in [@Luoto1], [@Luoto2].
This paper is organized as follows. The system and link model is defined in Section \[sec:model\]. Section \[sec:sharing\] describes the channel quality information (CQI) model for CoPSS. In Section \[sec:algorithms\], the actual CoPSS algorithms are elaborated. Section \[sec:results\] provides numerical results of the proposed CQI model and CoPSS algorithms. Finally, Section \[sec:conclusion\] concludes the paper.
System and Link Model {#sec:model}
=====================
Consider the downlink of an Orthogonal Frequency-Division Multiple Access (OFDMA) SBS network where $V$ SBSs are deployed. Each SBS has $N_\text{t}$ transmit antennas (Tx), which serve $U$ users each with $N_\text{r}$ receive antennas (Rx). The frequency domain resource consists of $N_\text{c}$ subcarriers, where 12 subcarriers are forming a PRB. It is assumed that the spectral allocations of the SBSs are orthogonal to the macro network layer and thus only the small cell traffic is modeled. Total system bandwidth is 10 MHz at 2 GHz center frequency and it is equally divided among the OPs.
In the system model SBSs form graphs. It is assumed that the SBSs communicate with each other if the distance is less than or equal to 50 meters. Let $G_l = (V_l,E_l)$ denote the graph, where $l = [0,\dots,L]$ is the number of graphs, the number of vertices in the graph (in this case SBSs) is $V_l = \{v_0,\dots,v_n\}$ and the edges in the graph ($v_i$ is connected to SBS $v_j$) $E_{l} = \{(v_i,v_j)\}$. Let $\mathcal{K} = \{1,\dots,K\}$ denotes the set of OPs. We define function $\text{OP}(\cdot)$ which maps SBS $v_i$ to respective operator, i.e. $\forall v_i, \text{OP}(v_i) \in \mathcal{K}$.
If a graph has $n$ vertices we have an $n \times n$ matrix $\textbf{A}$ which is called an adjacency matrix. The matrix $\textbf{A}$ is defined by $$\textbf{A}_{ij} = \begin{cases}
\text{1} & \quad \text{if $\text{SBS}_i$ $\leftrightarrow$ $\text{SBS}_j$}\\
\text{0} & \quad \text{otherwise}\\
\end{cases}$$ where $i$ and $j$ are SBSs indices. When $\textbf{A}_{ij} =1$, $\text{SBS}_i$ and $\text{SBS}_j$ communicate successfully with one another. This matrix is formed by the central controller $v_0$ when each SBS reports its adjacent vector, which indicates the wireless connections of the SBS to other SBSs. OP/SBS is willing to share its bandwidth if they are not utilizing it fully, we define bandwidth utilization $\text{BWU}(v_i), \forall v_i \in (V_l-v_0)$, and each OP defines a sharing factor $S = [0,\dots,1]$ indicating how much they are willing to share if part of the bandwidth is free.
The link model between a SBS and a user is illustrated in Fig. \[link\_model\]. Because a link-to-system interface (L2S) is used in the simulations, coding/decoding and modulation/demodulation are omitted. Antenna gain, path loss and shadowing loss are calculated for all links. Each user is then paired to a SBS. A geometry-based stochastic channel model [@WINII; @3GPP3] is used to model fast fading. Channel parameters are determined stochastically, based on the statistical distributions extracted from channel measurements [@itur2135]. SBS related assumptions for links are adopted from the [@befemto2]: all links are assumed to be non-line-of-sight (NLOS) and users are always inside buildings.
The link model starts from the scheduler that is responsible for resource allocation between users. Throughout simulations proportional fair scheduling is used. The scheduler utilizes CQI information transmitted by user. Based on the CQI information resource allocation is performed. The CQI provides information to the SBS about the link adaptation parameters. In the simulator, CQI is estimated from the received signal and for each user signal to interference and noise ratio (SINR) is calculated for every PRB. In order to model a practical closed loop system, periodic and delayed CQI is assumed. After scheduling, MCS selection is performed for scheduled users. The CQI modeling is explained in details in Section \[sec:sharing\]. Finally, before the data is sent over the fading channel, transmitter side spatial and OFDM processing are performed. The cyclic prefix is assumed to be longer than the multipath delay spread, and thus inter-symbol-interference is not considered.
At the receiver, perfect frequency and time synchronization is assumed. Link-to-system mapping is performed using mutual information effective SINR mapping (MIESM) [@MIESM]. This significantly reduces the computational overhead compared with exact modeling of the radio links, while still providing sufficiently accurate results. In the link-to-system interface, SINR is calculated and it is mapped to corresponding average mutual information. Based on the MIESM value, the frame error probability (FEP) is approximated according to a predefined frame error rate (FER) curve of used MCS. Based on the FER, successful and erroneous frames can be detected, and hybrid automatic repeat request (HARQ) can take the control for retransmissions. Acknowledgement (ACK) or negative acknowledgement (NACK) message is sent back to the SBS to signal the success or failure of the transmission, respectively. When a predefined number of channel samples have been simulated the results are calculated.
CQI modeling for Co-Primary Spectrum Sharing {#sec:sharing}
============================================
As mentioned in Section \[sec:model\], for each user the CQI is estimated from the received signal with SINR calculated for every PRB. When the SBS supports CoPSS, users have to calculate the CQI over the other operator’s bandwidth. Here, user equipment (UE) is required to receive/request reference signals from the other operator’s SBSs. In this case a user can only receive wideband reference signals from other OPs SBSs, because we assume that OPs are not willing to share operator specific reference signals. This means that users can only estimate if there are other OPs’ SBSs nearby but they may not estimate the SINR accurately for each PRB when spectrum is shared. We propose a CQI model in which it is enough to know the BWU from other SBSs/OPs in order to make accurate CQI estimation. Fig. \[CQI\_modeling\] shows an example of CQI modeling for UE1 when CoPSS is either supported or not (utilization of the central controller is explained in Section \[sec:algorithms\]). Each operator has a bandwidth of 4 PRBs. UE1 is connected to SBS1/OP1, UE2 is connected to SBS2/OP2 and UE3 is connected to SBS3/OP3. Without any sharing UE1 is not aware of any interference in the network. Let us assume that 50% of the bandwidth is shared, now UE1 can access OP2’s and OP3’s resources and vice versa. Without any coordination, when UE1 calculates the SINR for the CQI reporting it assumes that 50% of its own OP’s bandwidth is interference free and 50% experiences interference from SBSs 2-3. UE1 also assumes that shared PRBs of other OPs are used when the SINR is calculated. The reason is that UE1 receives the wideband reference signals from other SBSs but it does not know whether the shared resources are used or not. This means that effectively a user makes a worst case estimate for the CQI, which gives the wort case performance that can be achieved if the resources are used as estimated. If users could make accurate estimation from other OPs bandwidth for each PRB this assumption is relaxed.
When there is coordination, each SBS receives the BWU of other SBSs. This information is included in the wideband reference signals that UEs are requesting from SBSs in the vicinity error free. In this example SBS2 and SBS3 transmit their BWU to UE1. Now UE1 can estimate the channel accurately and transmit an accurate CQI to SBS1. Without coordination UE1 would detect only two interference free PRBs, but with coordination seven interference free PRBs are detected.
In order for UE to predict which part of the bandwidth is not occupied, it has to know in which manner the SBS/OP allocates PRBs to its users. In order to minimize signaling overhead, in this work it is assumed that each SBS/OP starts allocating PRBs from the beginning or from the end of its bandwidth. Arbitrary allocations would require detailed resource allocation information exchange, significantly increasing the signaling overhead. When a UE knows how much bandwidth a SBS/OP is willing to share and what is the BWU, the UE can predict which part of the bandwidth is free and which part is occupied. This way the UE can estimate the CQI more accurately.
Co-Primary Spectrum Sharing algorithms {#sec:algorithms}
======================================
With accurate CQI estimations, we propose three centralized and one decentralized algorithms for CoPSS. The proposed algorithms use moderate amount of shared information among OPs/SBSs and they do not require long iterative information exchange processes. Thus, the proposed CoPSS algorithms are practical.
In Algorithm \[random\], the free shared PRBs are randomly assigned to the SBSs in the building. The idea is that SBSs are connected to the central controller (as shown in Fig. \[CQI\_modeling\]) and there is no connection between SBSs resulting a single graph $G_l$ per building. This algorithm is time sensitive as there is a possibility that a randomly selected SBS from the graph can not exploit extra resources. Given that $v_{j'}$ is the selected SBS, the available free shared PRBs from OP $k$ to any SBS $v_j$ are given by: $${w}_{jk}=\begin{cases}\left\lfloor\min(W_k,S)\right\rfloor\times Q, & \text{if} ~ v_j=v_{j'}\\ 0 & \text{otherwise}, \label{allocation}
\end{cases}$$ where $Q$ is the number of PRBs[^2], $S$ is the sharing factor and $W_k = 1-\max_{\substack{v_i \in \{v | \text{OP}(v)=k\}}}\big(\text{BWU}(v_i)\big)$ is the number of free PRBs at OP $k$. Thus, the total amount of free PRBs for SBS $v_j$ is $\sum_{k \in \mathcal{K}} w_{jk}$.
Each SBS $v_i$ reports its BWU$(v_i)$ and sharing factor $S$ to the central controller $v_0$. $v_0$ picks $v_j \in (V_l-v_0)$ with probability $\frac{1}{|V_l-v_0|}$. Allocate PRBs based on (\[allocation\]). $v_0$ does not allocate any resources.
\[random\]
In Algorithm \[equal\], the free PRBs are equally assigned to overloaded SBSs in the building. It is assumed that a SBS is overloaded if the whole bandwidth is utilized, i.e., BWU is one hundred percent. Sharing is performed in a centralized manner using the central controller. Therefore, we define new set $v^+ = \{v_i|\text{BWU}(v_i)=1\}$ which includes all the overloaded SBSs. Here, free shared PRBs from OP $k$ to SBS $v_j$ are $${w}_{jk}=\begin{cases}\left\lfloor\frac{1}{|v^+|}\min(W_k,S)\right\rfloor\times Q, & \text{if} ~ v_j=v_{j'}\\ 0 & \text{otherwise}, \label{allocation2}
\end{cases}$$ and the total amount of free PRBs for SBS $v_j$ is $\sum_{k \in \mathcal{K}} w_{jk}$.
Each SBS $v_i$ reports its BWU$(v_i)$ and sharing factor $S$ to the central controller $v_0$. Central controller $v_0$ creates set $v^+$. Allocate PRBs based on (\[allocation2\]). $v_0$ does not allocate any resources.
\[equal\]
Algorithm \[decentralized\] aims to share resources equally between SBSs/OPs. The difference is that now SBSs are not connected to the central controller, but only to the SBSs in the vicinity, i.e sharing is done in a decentralized manner. We let $\mathcal{N}(v_i)$ denotes the set of neighbor vertices of $v_i$ and from (\[neighbor\]) we define two different sets, $\mathcal{N_\text{ol}}(v_i)$ for overloaded neighbors, and $\mathcal{N_\text{nol}}(v_i)$ for not overloaded neighbors, $$\begin{aligned}
&\mathcal{N}(v_i)= \{v| v \in V_l, (v,v_i) \in E_l \} \label{neighbor}, \\
&\mathcal{N_\text{ol}}(v_i)= \{v| v \in \mathcal{N}(v_i), \text{BWU}(v)=1\} \label{ol},\end{aligned}$$ and $$\begin{aligned}
&\mathcal{N_\text{nol}}(v_i)= \{v| v \in \mathcal{N}(v_i), \text{BWU}(v)<1\}. \label{nol}\end{aligned}$$
From (\[ol\]), we define a set of OPs which are overloaded neighbors and rest of the OPs are not overloaded $$\hat{\mathcal{K}}_i= \{\text{OP}(v)| \text{OP}(v) \in \mathcal{K}, v \in \mathcal{N_\text{ol}}\},$$ and $$\check{\mathcal{K}}_i= \mathcal{K}\backslash (\hat{\mathcal{K}_i} \cup \{\text{OP}(v_i)\}),$$ respectively. Now we can define free shared PRBs $w_{j}$ from neighbors $\check{\mathcal{K}}_i$ to SBS $v_j$, $${w}_{j}=\sum_{\forall k \in \check{\mathcal{K}}_i} \left\lfloor \frac{\min\Bigg({1-\max_{\substack{v \in \mathcal{N}_\text{nol}(v_i)\\\text{OP}(v)=k}}\Big(\text{BWU}(v)\Big)},S\Bigg)\times Q}{|\hat{\mathcal{K}_{{i}}} \cup\{\text{OP}(v_i)\}|}\right\rfloor. \label{allocation3}$$
Each $v_i$ reports BWU$(v_i)$ and sharing factor $S$ to all $v_j$ s.t $v_j \in V_l$ $(v_i,v_j)\in E_l$.
Each $v_i$ analyzes received reports.
$v_i$ allocates $|\mathcal{K}\backslash\text{OP}(v_i)|\times Q$ PRBs.
Allocate PRBs based on (\[allocation3\]).
\[decentralized\]
Fig. \[algorithm3\] shows an example of how resources are shared. It is assumed that each OP has its own index and based on the index they know which part from the bandwidth resources can be taken from. In this example, 50% of OP3 bandwidth is free. OP1 knows that half of the shared resources can be utilized in this case PRB3. Similarly OP2 knows that PRB4 can be utilized. Based on the number of overloaded SBSs/OPs, the unused portion of the bandwidth is divided equally. Allocations are interference free for each SBS/OP if the graph is a fully connected. For example a connection between SBS2 and SBS3 is not present, OP1 may utilize PRB3, but OP2 would see that OP3 is absent and may utilize 50% of the resources, in this case PRBs 3 and 4. This means that PRB3 is utilized by OP1 and OP3 which results in interference.
In Algorithm \[graph\], aforementioned interference problem can be avoided as each SBS reports its connections to other SBS and BWU to the central controller. The central controller then forms an adjacent matrix. Utilizing the information from the adjacency matrix the central controller can generate interference free resource allocations as illustrated in the interference avoidance step in Algorithm \[graph\]. Now we can define free shared PRBs $w_{j}$ from neighbors to SBS $v_j$ as
$${w}_{j}=\sum_{\forall k \in \check{\mathcal{K}}_i} \left\lfloor \frac{\min\Bigg({1-\max_{\substack{v \in \mathcal{N}_\text{nol}(v_i)\\\text{OP}(v)=k}}\Big(\text{BWU}(v)\Big)},S\Bigg)\times Q}{|\mathcal{N_\text{ol}}(v_i)|+1}\right\rfloor. \label{allocation4}$$
Each SBS reports the bandwidth utilization (BWU), sharing percentage and adjacent vector to central controller.
Central controller forms adjacent matrix **A** and adjacent matrix of overloaded SBSs **Â**.
$\exists~ v_i$ s.t. BWU$(v_i)=1$.
$v_i$ allocates $|\mathcal{K}\backslash\text{OP}(v_i)|\times Q$ PRBs.
Allocate PRBs based on (\[allocation3\]).
[$\hat{n}$ overloaded SBSs, **Â** $\in \mathbb{R}^{\hat{n}\times\hat{n}}$]{} *PRB allocation step*
Allocate PRBs $w_i$ based on (\[allocation4\]). Remove allocated PRBs from the available free resources.
*Interference avoidance step*
$w_i \leftarrow w_i \backslash (w_i \cap w_j), j=1,\ldots,\hat{n}, j\neq i$.
\[graph\]
The round trip-delay in the coordination methods is 5 ms and it is assumed that each OP uses the same maximum allowed sharing percentage[^3]. Backhaul links between SBSs or connections to the central controller are assumed to be ideal.
System Level Performance Results {#sec:results}
================================
System level simulations are particularly useful for studying network related issues such as resource allocation, interference management and mobility management. In this work, a multi-operator LTE-A system level simulator is used to model a cellular network consisting of an indoor SBS with multiple OPs.
The simulator uses a hexagonal macro layout which includes 21 sectors and in each sector there is a building of the size 120 m x 120 m as shown in Fig. \[Layout\]. Small cell layouts are shown in Figs. \[Even\_layout\] (fixed layout) and \[Random\_layout2\] (random layout). The building has one open corridor across it and in total 20 rooms, size 24 m x 24 m. Internal wall attenuation is 5 dB per wall.
When the locations of the SBSs are fixed, they are placed in the center of the building and each OP has one SBS per building, users are evenly distributed and each of them is connected to the own OP’s SBS. When SBSs are randomly distributed, the number of SBSs in the building is based on deployment probability. In this layout users are located a maximum of 20 m from the SBS (inner circle). In both layouts, the number of users connected to each SBS varies between one and two. The main reason to use two different layouts is to illustrate the applicability of the CoPSS concept in both planned (fixed) and unplanned (random) SBS deployment scenarios.
In the simulations, two different traffic models are used, full buffer and continuous constant rate transmission. With continuous constant rate transmission, two different target bit-rates are used; 4 Mb/s target rate is referred to as multimedia stream (e.g., on-demand video service) and 1 Mb/s target rate is referred to as constant rate (e.g., users with a limited speed data connection).
Table \[params\] summarizes some simulation parameters and assumptions which are used through simulations. Traffic in the network is constant, and movement of users is not modeled. This means that delay does not have a big impact on the performance because SBS resource allocation stays quite consistent throughout simulations.
**Parameter** **Assumption**
--------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------
Duplex mode FDD
System bandwidth 10 MHz (divided equally between OPs)
Number of PRBs 16 per SBS/OP
Number of users 1-2 user per SBS
Antenna configurations 1 Tx, 2 Rx
Receivers MRC
HARQ Chase combining
SBS transmission power 20 dBm
Feedback CQI period 6 ms
Feedback CQI delay 2 ms
Traffic models **Full buffer** (10% full buffer traffic) and **Continuous constant rate transmission** (50% constant rate traffic and 40% multimedia stream traffic)
Internal wall attenuation 5 dB
: Simulator parameters and assumptions.[]{data-label="params"}
Algorithms \[random\]-\[decentralized\] are used for both network layouts and Algorithm \[graph\] is only used for the random layout. For each CoPSS algorithm, the target is to allocate only shared resources that are unused within the network. It should be noted that CoPSS is highly sensitive to the network load and to different traffic types. In these simulations the network load (1-2 users per SBS) is relatively low, however for a high network load (i.e., all resources utilized) Algorithms \[random\] - \[decentralized\] in the fixed layout and Algorithms \[random\] - \[equal\] in the random layout would not provide any gain because only unused PRBs are shared between OPs/SBSs. The reason for zero gains in the fixed layout is that when all resources are utilized, and all OPs are colocated (\[allocation\]), (\[allocation2\]) and (\[allocation3\]) are always equal to zero. Similarly, (\[allocation\]) and (\[allocation2\]) are equal to zero in the random layout. However, the graphs in the random layout can have OPs that are not colocated and thus, (\[allocation3\]) and (\[allocation4\]) provide non-zero gains.
CQI coordination for CoPSS
--------------------------
Fig. \[Equal\_CQI\_analysis\_SINR\] shows cumulative distribution functions (CDFs) of SINR as estimated for the CQI with and without coordination, and as experienced at the receiver, when 50% bandwidth is shared. The SINR in the CQI and in the receiver is the mean SINR over the allocated PRBs. It is assumed that UE can report the PRB based CQI information and SBS then averages out the SINR with allocated PRBs and then selects one MCS level that is used for the transmission. The CDF shows that when the users are able to receive information about the bandwidth utilization of other SBSs/OPs in the vicinity SINR increases 3 dB (between 0 dB and 15 dB).
In Fig. \[Equal\_CQI\_analysis\_SINR\], it can be seen that the SINR at the receiver saturates around 28 dB. The reason is that in the receiver side error vector magnitude (EVM) is used to model hardware imperfection, which is assumed to have a value of 4%. The EVM error to the received SINR can be written as: $$SINR_\text{out} = 1/\Big((1/SINR_\text{in})+(EVM_\text{\%}/100)^2\Big)$$ where $SINR_\text{in}$ is the received SINR in linear scale and $EVM_\text{\%}$ is the percentage EVM.
Fig. \[Equal\_CQI\_analysis\_SP\] shows the mean throughput when the sharing percentage increased from 0% to 100% with and without CQI coordination. The used CoPSS algorithm is equal sharing. The SBSs are fixed in the center of the building. Results show that when the sharing percentage is increased and there is no CQI coordination, achieved mean throughput starts to decrease particulary for the full buffer and multimedia stream users. When 100% of the bandwidth is shared full buffer and multimedia stream users achieve throughput of 1 Mb/s, i.e. 5 Mb/s loss for the full buffer users and 2.5 Mb/s loss for the multimedia users compared with the case when 0% of the bandwidth is shared. When 100% of the bandwidth is shared the equal CoPSS provides a 3.8 Mb/s increase in the mean throughput for full buffer users when compared to the case without the CoPSS. When the sharing percentage is increased and there is CQI coordination, the multimedia stream and the constant rate users do not achieve any gain in mean throughput because most of the users can achieve the target bit rate, i.e. achieved throughput gain averages out. The CDFs of throughput are analyzed in Section \[sec:results\].
Fig. \[Random\_CQI\_analysis\_SP\] shows the mean throughput results when SBSs are randomly distributed in the building. Results show that when the sharing percentage is increased and there is no CQI coordination, the achieved mean throughput for full buffer users starts to increase, but for the multimedia stream users there is reduction in throughput. The reason is that for full buffer users the huge increase in available PRBs outweighs the loss by underestimated CQI, but for the lower data rate users the underestimation of the CQI leads to throughput reduction when the sharing percentage is increased. When 100% of the bandwidth is shared with CQI coordination the equal CoPSS provides a 9.0 Mb/s increase in the mean throughput for full buffer users, when compared with the case without sharing.
These results show that coordination is needed between OPs if CoPSS is supported in the network. Without coordination, quality of the service can not be guaranteed and CoPSS can even result in a loss in performance. In the rest of the discussion it is assumed that there is a coordination between the OPs/SBSs in the vicinity of one another.
CoPSS in the fixed layout
-------------------------
First the different CoPSS algorithms are analyzed in the fixed network given in Fig. \[Even\_layout\] where all the SBS are collocated and interconnected, i.e there is a simple complete graph $G_l$ per building. In this network layout, it is crucial that simultaneous use of the shared PRBs is avoided. Because the SBSs are close to each other the serving signal and the interference signal would have approximately the same strength, leading to a high FER. Decentralized sharing and equal sharing in the fixed layout should provide similar performance (all the SBS are collocated and interconnected) if there is a common protocol between OPs defining how shared resources can be utilized.
Fig. \[Equal\_algorithm\_analysis\_SP\] shows the mean throughput of full buffer users for each CoPSS algorithms with sharing percentage from 0% to 100%. When 0% of the bandwidth is shared mean throughput is 6.0 Mb/s. It can be clearly seen that all the CoPSS algorithms result in throughput gains, increasing with the sharing percentage. As expected, Algorithm 1 provides the lowest gain, a 1.5 Mb/s increment to mean throughput when 100% of the bandwidth is shared. Algorithm 2 provides a 3.7 Mb/s gain. As mentioned in Section \[sec:algorithms\], Algorithm 2 and Algorithm 3 provide very similar performance because each SBS has the same knowledge as the central controller.
CoPSS provides substantial gains for full buffer users. Figs. \[Equal\_CQI\_analysis\_SP\] and \[Random\_CQI\_analysis\_SP\] imply that only full buffer users achieve some gain from CoPSS. Figs. \[Equal\_algorithm\_analysis\_throughput1\] and \[Equal\_algorithm\_analysis\_throughput2\] show the CDF of throughput for the constant rate and multimedia stream users when 50% of the bandwidth is shared. From these figures it can be seen that all the CoPSS methods provide gain over the case when the spectrum is not shared, for users with all traffic types. Fig. \[Equal\_algorithm\_analysis\_throughput2\] shows for example at the 20% point on the CDF there is a 0.7 Mb/s gain in throughput when Algorithm 3 is compared to no spectrum sharing.
Fig. \[Equal\_algorithm\_analysis\_throughput3\] shows the CDF of throughput for full buffer users. The theoretical maximum throughput of a user (when the SBS is only serving one user and the highest MCS is used) is around 10 Mb/s. The CDF shows that a user will achieve a throughput of 10 Mb/s with a probability of 18%. For example at the 90% point on the CDF achieved gains are; 2.0 Mb/s for Algorithm 1, 5.8 Mb/s for Algorithm 2, and 5.7 Mb/s for Algorithm 3. When compared to the theoretical maximum throughput of 10 Mb/s without sharing, the gain is significant. Table \[gains1\] summarizes the achievable gains of cell edge users (5% from CDFs) when the CoPSS is supported.
**Algorithm** **Constant rate** **Multimedia stream** **Full buffer**
--------------- ------------------- ----------------------- -----------------
1 0.080 0.180 0.096
2 0.154 0.159 0.327
3 0.110 0.265 0.261
: Cell edge user throughput gain \[Mb/s\] with the CoPSS in the fixed layout.[]{data-label="gains1"}
CoPSS in the random layout
--------------------------
In the random layout (Fig. \[Random\_layout2\]), a connection is formed when the coverage area of two SBSs overlap with one another, in this case a maximum distance 20 m + 5 m range is used. It is assumed that within this distance if same resources are used, users will experience high interference from neighboring SBSs. When users are within 20 m range, the SBS is working as a local hotspot, and allows for higher data rates and spectral effciency resulting in better user experience. When a SBS does not detect the presence of any SBS belong to a particular OP within its detection range, it assumes those OPs’ resources to be free and exploitable.
Fig. \[Random\_algorithm\_analysis\_SP\] shows the mean throughput of full buffer users for each CoPSS algorithm with sharing percentage from 0% to 100%. When the results are compared with the fixed layout results it can be clearly seen that the achieved rates are higher because users are now closer to SBS. When 0% of the bandwidth is shared mean throughput is 7.0 Mb/s. Algorithm 1 provides the lowest gain, a 5.7 Mb/s improvement to mean throughput when 100% of the bandwidth is shared, while Algorithm 2 provides a 8.8 Mb/s gain. The achieved gain from Algorithm 3 is 9.6 Mb/s, and the Algorithm 4 results in the highest gain 11.6 Mb/s. The reason of Algorithm 4 providing higher gains compared to Algorithm 3 is explained in Section \[sec:algorithms\] and Fig. \[algorithm3\].
In the fixed layout, although there are no significant gains for low-data rate users, all the CoPSS methods result higher gains over the scenario with no spectrum sharing. In the random layout the gains from CoPSS are higher than in the fixed layout. Fig. \[Random\_algorithm\_analysis\_throughput1\] and \[Random\_algorithm\_analysis\_throughput2\] show the CDF of throughput for the constant rate and multimedia stream users when 50% of the bandwidth is shared. In Fig. \[Random\_algorithm\_analysis\_throughput2\], there is 10% probability of achieving less than 2.6 Mb/s, which is reduced to 4% in the case of sharing (Algorithm 4).
Fig. \[Random\_algorithm\_analysis\_throughput3\] shows the CDF of throughput for the full buffer users. At the 50% point on the CDF the achieved rates are: 7.8 Mb/s for No sharing, 10.5 Mb/s for Algorithm 1, 11.6 Mb/s for Algorithm 2, 15.2 Mb/s for Algorithm 3 and Algorithm 4 18.8 Mb/s. The achieved gains are significant compared to the no spectrum sharing scenario with the theocratical maximum of 10 Mb/s. Table \[gains2\] summarizes the gains of cell edge users when the CoPSS is supported.
**Algorithm** **Constant rate** **Multimedia stream** **Full buffer**
--------------- ------------------- ----------------------- -----------------
1 0.117 1.022 1.171
2 0.137 1.594 1.811
3 0.144 2.008 1.869
4 0.141 2.403 3.129
: Cell edge user throughput gain \[Mb/s\] with the CoPSS in the random layout.[]{data-label="gains2"}
CoPSS behavior with higher network load
---------------------------------------
As discussed earlier, a higher network load limits the achievable throughput gains using CoPSS. Table \[network\_load\] shows the mean achieved throughput of full buffer users, for an increasing network load, when 100% of each OPs bandwidth is shared. The results clearly show that the gain in average throughput when utilizing CoPSS decreases with the network load. However, utilizing CoPSS does result in non-negligible increased throughput even in the case of a high network load.
**Users** **No sharing** **Algo. 1** **Algo. 2** **Algo. 3** **Algo. 4**
----------- ---------------- ------------- ------------- ------------- -------------
1-2 7.04 12.71 15.84 16.50 18.64
1-4 4.84 9.11 10.68 11.64 13.07
1-6 3.34 6.27 7.30 9.08 9.69
: Achieved mean throughput \[Mb/s\] with CoPSS in the random layout with different network loads.[]{data-label="network_load"}
Although Algorithm \[random\] is totally random, it exhibits significant throughput gains compared to the no sharing method. Thus, even a simple CoPSS can help to improve capacity in SBS network scenarios. This type of sharing does not guarantee that resources are shared equally between SBSs/OPs during one time instant, but each SBS has an equal chance to be chosen. Algorithms \[equal\] and \[decentralized\] in the fixed network layout provide similar performance because all the SBS are collocated and interconnected. Generally, if all SBSs are connected, our proposed algorithm provides substantial throughput gain without central controller.
In the random layout, Algorithms \[equal\] and \[decentralized\] exhibit different performance. In this case, the decentralized Algorithm \[decentralized\] outperforms the centralized Algorithm \[equal\]. This is due to the reason that the decentralized algorithm does not share resources equally within each building, but resources are shared between SBSs that are within communication range of one another. In this case, an isolated SBS achieves significant gains in throughput even for a low sharing percentage. When Algorithm \[decentralized\] and Algorithm \[graph\] are compared, the centralized algorithm provides better performance as explained in Section \[sec:algorithms\]. The decentralized Algorithm \[decentralized\] provides substantial gains as compared to the no sharing case, with an average gain of more than 120%. However, the centralized Algorithm \[graph\] only results in additional average gain of 12% over Algorithm \[decentralized\]. Given that the performance of the decentralized Algorithm \[decentralized\] is so close to that of the centralized Algorithm \[graph\], we come to the conclusion that Algorithm \[decentralized\] is the most suitable for all the aforementioned scenarios.
The proposed algorithms do not require complex computation, or extensive signaling between SBSs. Algorithms \[equal\] - \[graph\] reach stable point quickly, and the only delay is the coordination delay between SBSs/OPs. This is because there is no requirement for iterative information exchange between SBSs/OPs, due to the common rules between SBSs/OPs, which determine how spectrum is shared.
Conclusion {#sec:conclusion}
==========
We have proposed and evaluated four different approaches toward co-primary multi-operator spectrum sharing in small cell indoor environment with mixed traffic distribution. The framework has been established under the LTE-A compliant system simulation platform where the system throughput performance has been rigorously assessed. Provided numerical results confirm the high potential co-primary spectrum sharing can offer to increase system throughput in the multi-operator setting. The results reveal the utmost importance of channel quality signaling among OPs in order to take full advantage of shared resources. It was also shown that, the connection based centralized and decentralized algorithms outperform simpler random and equal sharing schemes. This paper is a foundation for further studies. In our future work, we will study CoPSS with time variant network traffic, develop algorithms that ensure long term fairness between OPs and we will consider the economic part of the spectrum sharing in more detail.
[^1]: This research was supported by the Finnish Funding Agency for Technology and Innovation (TEKES), Nokia Networks, Anite Telecoms, Huawei Technologies, Broadcom Communications Finland, Elektrobit Wireless Communications and Infotech Oulu Graduate School. Kari Horneman, Ling Yu and Eric Galloix from Nokia Networks earn special thanks for suggesting this research direction and for giving invaluable feedback.
[^2]: Notation $\lfloor \cdot \rfloor$ defines the operation of round towards negative infinity.
[^3]: Percentages could be different but results are easier to analyze when same sharing percentage is used because we do not have to look at the gains achieved for each OP individually.
|
---
abstract: 'The generation of the plasma current resulting from Bremsstrahlung absorption is considered. It is shown that the electric current is higher than the naive estimates assuming that electrons absorb only the photon momentum and using the Spitzer conductivity would suggest. The current enhancement is in part because electrons get the recoil momentum from the Coulomb field of ions during the absorption and in part because the electromagnetic power is absorbed asymmetrically within the electron velocity distribution space.'
author:
- 'Vadim R. Munirov'
- 'Nathaniel J. Fisch'
bibliography:
- 'bremsstrahlung.bib'
date: 22 August 2017
title: Inverse Bremsstrahlung current drive
---
Introduction
============
In the presence of external electromagnetic field colliding electrons and ions absorb the incoming radiation through the process known as inverse Bremsstrahlung. In Bremsstrahlung absorption, the electron receives additional recoil momentum from the ion besides the momentum of the photon. Therefore, plasma electrons absorb more than just the photon momentum from the incoming radiation. The generated current is then larger than one would get by assuming that electrons absorb just the photon momentum. It was shown in [@Pashinin1978] that this increase in current is equal to $8/5$.
However, the recoil is not the only mechanism that will increase the current. Plasma electrons absorb the radiation asymmetrically in velocity space; specifically, electrons co-moving with the incoming photons will absorb slightly more power than electrons going in the opposite direction. Even in the absence of net momentum absorption, this asymmetric absorption in power can lead to current drive. This is because the collision frequency in plasma is speed dependent. Thus, upon absorbing energy electrons going in the direction of the incoming radiation will experience less resistance from the plasma than electrons going in the opposite direction resulting in current. This is called the asymmetric resistivity current drive effect and is mostly known with respect to cyclotron absorption used to drive toroidal current in tokamaks [@FischBoozer1980; @Fisch1987]. Moreover, even without the asymmetric resistivity effect the fluid approximation is less precise in considering current generation as opposed to momentum input, because it assumes that all electrons get equal push in the same direction, which is not the case for Bremsstrahlung absorption. In fact, the ability of electrons to retain current is sensitive to both its location in velocity space and the direction in which it is being pushed.
In this paper we rederive the result for the momentum absorption rate and calculate the additional increase in current due to the current drive effect. To derive the current drive effect, it will be necessary to consider in detail how exactly the momentum is absorbed within the electron velocity space. To do this we use the formalism developed by Tsytovich [@Tsytovich1992; @Tsytovich1995; @Tsytovich1996].
Probability of Bremsstrahlung
=============================
Consider Bremsstrahlung absorption for particles $\alpha$ (electrons) due to the Coulomb collisions with much heavier particles $\beta$ (ions). To satisfy the conservation laws of momentum and energy, in each act of the Bremsstrahlung absorption some recoil momentum must be transferred from the electron to ions. We can write down the momentum balance during inverse Bremsstrahlung as follows:
$$\begin{aligned}
\mathbf{p}_{\alpha}^{\prime} & =\mathbf{p}_{\alpha}+\hbar\mathbf{k}-\hbar\mathbf{q},\\
\mathbf{p}_{\beta}^{\prime} & =\mathbf{p}_{\beta}+\hbar\mathbf{q},\end{aligned}$$
where the primed values correspond to the quantities after the absorption, $\mathbf{k}$ is the wave vector of the photon, and $\mathbf{q}$ is the recoil wave vector transferred from the electron to the ion. The conservation of energy is
$$\varepsilon_{\mathbf{p}_{\alpha}}^{\alpha}+\varepsilon_{\mathbf{p}_{\beta}}^{\beta}+\hbar\omega_{\mathbf{k}}=\varepsilon_{\mathbf{p}_{\alpha}+\hbar\mathbf{k}-\hbar\mathbf{q}}^{\alpha}+\varepsilon_{\mathbf{p}_{\beta}+\hbar\mathbf{q}}^{\beta}.$$
Here, we will use the diffusion approximation, when $\hbar\mathbf{k}$, $\hbar\mathbf{q}$ are small in comparison with the particle momentum $(\hbar\mathbf{k},\:\hbar\mathbf{q}\ll\mathbf{p}_{\alpha})$. In this approximation, the energy conservation is simplified to
$$\omega_{\mathbf{k}}=\left(\mathbf{k}-\mathbf{q}\right)\mathbf{v}_{\alpha}+\mathbf{q}\mathbf{v}_{\beta}.\label{eq:cond_1}$$
Now consider the direct process of spontaneous Bremsstrahlung emission. The momentum balance can be written as:
$$\begin{aligned}
\mathbf{p}_{\alpha}^{\prime} & =\mathbf{p}_{\alpha}-\hbar\mathbf{k}+\hbar\mathbf{q},\\
\mathbf{p}_{\beta}^{\prime} & =\mathbf{p}_{\beta}-\hbar\mathbf{q}.\end{aligned}$$
With such a definition of the recoil momentum **$\mathbf{q}$** (notice different signs in the definition of **q** for emission and absorption), the energy conservation yields the same relationship between velocities of the particles and parameters of the photon as for the inverse process (Eq. (\[eq:cond\_1\])).
A schematic diagram of the two processes is shown in Fig. \[fig01\]. Essentially inverse Bremsstrahlung can be considered as Compton scattering, by the incoming electron, of the incoming photon $\mathbf{k}$ into the virtual photon of the Coulomb field $\mathbf{q}$ (see Fig. \[fig01\_a\]), while the Bremsstrahlung emission can be considered as Compton scattering of the virtual photons of the Coulomb field on the incoming electron (see Fig. \[fig01\_b\]).
It is clear, that due to time reversal symmetry, the transition probability of the inverse and direct processes are related to each other:
$$w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{IBr}\left(\mathbf{k},\mathbf{q}\right)=w_{\mathbf{p}_{\alpha}+\hbar\mathbf{k}-\hbar\mathbf{q},\mathbf{p}_{\beta}+\hbar\mathbf{q}}^{Br}\left(\mathbf{k},\mathbf{q}\right).$$
Here $w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)$ and $w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{IBr}\left(\mathbf{k},\mathbf{q}\right)$ are the probabilities of spontaneous Bremsstrahlung emission and inverse Bremsstrahlung per unit time within $d\mathbf{k}d\mathbf{q}$. Note that these probabilities must contain condition (\[eq:cond\_1\]) as the argument of the delta function.
One must remember that, in the presence of external radiation, the true absorption due to inverse Bremsstrahlung is always accompanied by the process of stimulated emission. For example, for electromagnetic waves ($\omega=kc$) and infinitely massive ions ($\mathbf{v}_{\beta}=0$), condition (\[eq:cond\_1\]) implies that for inverse Bremsstrahlung the change in the parallel momentum of the electron is approximately $\hbar\omega/v$, while for stimulated Bremsstrahlung emission this change is approximately $-\hbar\omega/v$. However, these two processes do not completely compensate each other because their probabilities are slightly different.
More generally, the evolution of the distribution function $f_{\mathbf{p}_{\alpha}}^{\alpha}$ due to the processes of inverse Bremsstrahlung and stimulated Bremsstrahlung emission is described by [@Tsytovich1992]
$$\begin{gathered}
\frac{\partial f_{\mathbf{p}_{\alpha}}^{\alpha}}{\partial t}=-\int w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{IBr}\left(\mathbf{k},\mathbf{q}\right)f_{\mathbf{p}_{\alpha}}^{\alpha}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}\\
+\int w_{\mathbf{p}_{\alpha}-\hbar\mathbf{k}+\hbar\mathbf{q},\mathbf{p}_{\beta}-\hbar\mathbf{q}}^{IBr}\left(\mathbf{k},\mathbf{q}\right)f_{\mathbf{p}_{\alpha}-\hbar\mathbf{k}+\hbar\mathbf{q}}^{\alpha}f_{\mathbf{p}_{\beta}-\hbar\mathbf{q}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}\\
-\int w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)f_{\mathbf{p}_{\alpha}}^{\alpha}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}\\
+\int w_{\mathbf{p}_{\alpha}+\hbar\mathbf{k}-\hbar\mathbf{q},\mathbf{p}_{\beta}+\hbar\mathbf{q}}^{Br}\left(\mathbf{k},\mathbf{q}\right)f_{\mathbf{p}_{\alpha}+\hbar\mathbf{k}-\hbar\mathbf{q}}^{\alpha}f_{\mathbf{p}_{\beta}+\hbar\mathbf{q}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}.\end{gathered}$$
Following Tsytovich [@Tsytovich1992; @Tsytovich1995; @Tsytovich1996], after Taylor expansion for $\hbar\mathbf{k},\:\hbar\mathbf{q}\ll\mathbf{p}_{\alpha}$ we get the Fokker-Planck equation for the evolution of $f_{\mathbf{p}_{\alpha}}^{\alpha}$:
$$\frac{\partial f_{\mathbf{p}_{\alpha}}^{\alpha}}{\partial t}=\frac{\partial}{\partial\mathbf{p}_{\alpha}}\cdot\mathbf{S}_{\mathbf{p}_{\alpha}}=\frac{\partial}{\partial\mathbf{p}_{\alpha}}\cdot\left(\widehat{D}_{\alpha}\frac{\partial f_{\mathbf{p}_{\alpha}}^{\alpha}}{\partial\mathbf{p}_{\alpha}}+\mathbf{F}_{\alpha}f_{\mathbf{p}_{\alpha}}^{\alpha}\right),\label{eq:FP}$$
where
$$\widehat{D}_{\alpha}=\int\hbar^{2}\left(\mathbf{k}-\mathbf{q}\right)\left(\mathbf{k}-\mathbf{q}\right)w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}n_{\mathbf{k}}f_{\mathbf{p}_{\beta}}^{\beta}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta},$$
$$\mathbf{F}_{\alpha}=\int\hbar^{2}\left(\mathbf{k}-\mathbf{q}\right)\left(\mathbf{q}\cdot\frac{\partial f_{\mathbf{p}_{\beta}}^{\beta}}{\partial\mathbf{p}_{\beta}}\right)w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}.$$
The normalization is such that the density of particles is $n_{\alpha}=\int f_{\mathbf{p}_{\alpha}}^{\alpha}d\mathbf{p}_{\alpha}=\int f_{\mathbf{v}_{\alpha}}^{\alpha}d\mathbf{v}_{\alpha},$ and the total number of photons per volume is $N_{ph}=\int n_{\mathbf{k}}d\mathbf{k}$, and $n_{\mathbf{k}}$ is the number of photons within $d\mathbf{k}$.
The probability of spontaneous Bremsstrahlung emission for electromagnetic waves $(\omega=kc)$ keeping terms of the order of $\mathbf{k}\mathbf{v}/\omega\sim v/c$ is given by [@Tsytovich1996]
$$\begin{gathered}
w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)=\frac{2e_{\alpha}^{4}e_{\beta}^{2}\delta\left[\omega_{\mathbf{k}}-\left(\mathbf{k}-\mathbf{q}\right)\mathbf{v}_{\alpha}-\mathbf{q}\mathbf{v}_{\beta}\right]}{\hbar\pi^{2}m_{\alpha}^{2}q^{4}\left(\omega_{\mathbf{k}}-\mathbf{k}\mathbf{v}_{\alpha}\right)^{2}\left.\frac{\partial\left(\varepsilon\omega^{2}\right)}{\partial\omega}\right|_{\omega=\omega_{\mathbf{k}}}\varepsilon_{\mathbf{q},\mathbf{q}\mathbf{v}_{\beta}}^{2}}\\
\times\left|\left[\mathbf{e}_{k}\times\mathbf{q}\right]+\frac{\mathbf{k}\mathbf{q}}{\omega_{\mathbf{k}}-\mathbf{k}\mathbf{v}_{\alpha}}\left[\mathbf{e}_{k}\times\mathbf{v}\right]\right|^{2}.\label{eq:prob}\end{gathered}$$
This expression is only correct for Bremsstrahlung ignoring the polarization effects. By polarization effects we mean that the plasma environment in which the electron finds itself is influenced by the presence of the electron. This approximation is good for dilute plasma. In general, the probability of Bremsstrahlung is proportional to $\left|\left[\mathbf{e}_{k}\times\left(\mathbf{M}^{\alpha}+\mathbf{M}^{\beta}+\mathbf{M}^{\alpha\beta}\right)\right]\right|^{2}$, where $\mathbf{M}^{\alpha}$ is the emission due to oscillation of $\alpha$ particles in the screened field of $\beta$ charges, $\mathbf{M}^{\beta}$ is the emission due to oscillation of $\beta$ particles in the screened field of $\alpha$ charges, and $\mathbf{M}^{\alpha\beta}$ is the emission due to oscillation of the polarization clouds around particles $\alpha$ and $\beta$. While $\mathbf{M}^{\beta}$ is small due to the high ion mass, the term $\mathbf{M}^{\alpha\beta}$ can be comparable with $\mathbf{M}^{\alpha}$. Moreover, polarization effects may make electron-electron and ion-ion collisions important as well. The polarization effects are especially important for longitudinal waves, and must be almost always taken into account for them (we consider only transverse electromagnetic waves here) [@Tsytovich1992; @Tsytovich1995; @Tsytovich1996]. In Eq. (\[eq:prob\]) the polarization effects are ignored and only $\mathbf{M}^{\alpha}$ term is retained; this requires the plasma to be tenuous enough. Another approximation used in Eq. (\[eq:prob\]) is non-relativistic velocities. In all subsequent calculations, we also take unity dielectric function ($\varepsilon\approx1$), which is a good approximation for tenuous plasma. We will also ignore plasma dispersive effects and take $\omega_{\mathbf{k}}=\omega=kc$, and assume an infinite ion mass and set $\mathbf{v}_{\beta}=0$, $\mathbf{v}_{\alpha}=\mathbf{v}$.
Momentum change
===============
In this section let us calculate the rate of momentum change for electrons during Bremsstrahlung absorption.
From Eq. (\[eq:FP\]) we can calculate the rate of momentum absorption due to Bremsstrahlung as:
$$\frac{d\mathbf{p}_{V}^{\alpha}}{dt}=-\int\mathbf{S}_{\mathbf{p}_{\alpha}}d\mathbf{p}_{\alpha},\label{eq:S_p}$$
so $-\mathbf{S}_{\mathbf{p}_{\alpha}}$ has the meaning of the rate of momentum absorption per $d\mathbf{p}_{\alpha}$ by electrons with momentum between $\mathbf{p}_{\alpha}$ and $\mathbf{p}_{\alpha}+d\mathbf{p}_{\alpha}$.
For plasma with a spherically symmetric distribution function and infinitely massive ions ($\mathbf{v}_{\beta}=0$) we can take advantage of condition (\[eq:cond\_1\]) and write
$$\frac{d\mathbf{p}_{V}^{\alpha}}{dt}=\int\hbar\left(\mathbf{k}-\mathbf{q}\right)\frac{\hbar\omega_{\mathbf{k}}}{v_{\alpha}}\frac{\partial f_{\mathbf{p}_{\alpha}}^{\alpha}}{\partial p_{\alpha}}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}d\mathbf{p}_{\alpha}.\label{eq:dpdt_recoil}$$
This suggests that the probability of the total absorption (inverse Bremsstrahlung plus stimulated Bremsstrahlung emission) in plasma with a spherically symmetric distribution function is proportional to the probability of spontaneous Bremsstrahlung emission and is $\left(\hbar\omega_{\mathbf{k}}/v_{\alpha}\right)\left(\partial\ln f_{\mathbf{p}_{\alpha}}^{\alpha}/\partial p_{\alpha}\right)w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)$. For plasma near equilibrium with Maxwell distribution function, which for convenience we will consider, this probability becomes $\left(\hbar\omega_{\mathbf{k}}/T\right)w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)$ and is actually correct even for the finite ion mass.
Consider the incoming electromagnetic radiation that consists of photons with $\mathbf{k}=k\mathbf{e}_{z}$ and of the total intensity $I=c\int\hbar\omega n_{\mathbf{k}}d\mathbf{k}$. Because of the condition (\[eq:cond\_1\]) the recoil momentum can be divided into the parts parallel and perpendicular to the velocity component: $$\mathbf{q}=-\frac{\omega-\mathbf{k}\mathbf{v}}{v^{2}}\mathbf{v}+\mathbf{q}_{\perp}.$$
Then the rate of momentum absorption directed along the $z$-axis can be written as
$$\begin{gathered}
\frac{d\mathbf{p}_{V,z}^{\alpha}}{dt}=\int\hbar\left(k+\frac{\omega}{v}\frac{v_{z}}{v}-\frac{\mathbf{k}\mathbf{v}}{v}\frac{v_{z}}{v}-q_{\perp_{z}}\right)\\
\times\frac{\hbar\omega}{T}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}f_{\mathbf{p}_{\alpha}}^{\alpha}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}d\mathbf{p}_{\alpha}.\label{eq:dpdt_qdecomp}\end{gathered}$$
To calculate the probability of Bremsstrahlung (\[eq:prob\]) we express
$$\begin{gathered}
\left|\left[\mathbf{e}_{k}\times\mathbf{q}\right]+\frac{\mathbf{k}\mathbf{q}}{\omega-\mathbf{k}\mathbf{v}}\left[\mathbf{e}_{k}\times\mathbf{v}\right]\right|^{2}\\
=\left|\left[\mathbf{e}_{z}\times\mathbf{q}_{\perp}\right]+\left(-\frac{\omega}{v^{2}}+\frac{\mathbf{k}\mathbf{q}_{\perp}}{\omega-\mathbf{k}\mathbf{v}}\right)\left[\mathbf{e}_{z}\times\mathbf{v}\right]\right|^{2}\\
=q_{\perp}^{2}-q_{\perp_{z}}^{2}+\frac{\omega^{2}}{v^{2}}\frac{v_{\perp}^{2}}{v^{2}}+2\frac{\omega}{v^{2}}q_{\perp_{z}}\left(v_{z}-v_{\perp}\beta_{\perp}\right)-2q_{\perp_{z}}^{2}\beta_{z},\end{gathered}$$
where we introduced $\boldsymbol{\beta}=\mathbf{v}/c$, used the expression for the scalar quadruple product $\left[\mathbf{e}_{z}\times\mathbf{q}_{\perp}\right]\cdot\left[\mathbf{e}_{z}\times\mathbf{v}\right]=-q_{\perp_{z}}v_{z}$, and kept only the first order terms.
We can write the $z$-axis projection of the perpendicular to the velocity component of the recoil momentum as $q_{\perp_{z}}=q_{\perp}\sin\theta\sin\varphi_{q_{\perp}}$, where $\theta$ is the angle between velocity and the $z$-axis, i.e. $v_{z}=v\cos\theta$ and $v_{\perp}=v\sin\theta$, while $\varphi_{q_{\perp}}$ is the polar angle of $q_{\perp}$ in the plane perpendicular to $\mathbf{v}$. We then integrate over $\varphi_{q_{\perp}}$ from $0$ to $2\pi$ and over $dq_{\parallel}q_{\perp}dq_{\perp}$. When we integrate over $dq_{\perp}$ it is necessary to introduce a cutoff to get rid of a logarithmic divergence. For definiteness, we will use the quantum mechanical cutoff ($q_{max}=m_{\alpha}v/\hbar$), which is correct when the Born approximation can be applied ($v\gg e^{2}/\hbar$). In the opposite classical limit ($v\ll e^{2}/\hbar$) the proper cutoff is $q_{max}=m_{\alpha}v^{2}/e_{\alpha}e_{\beta}$ and the conclusions of the paper should remain true but all logarithmic factors should be replaced with $\ln\left(m_{\alpha}v^{3}/\omega e_{\alpha}e_{\beta}\right)$.
Keeping only the leading logarithmic terms, the probability of Bremsstrahlung integrated over $d\mathbf{q}$ is then
$$\begin{gathered}
\int w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)d\mathbf{q}\\
\approx\frac{e_{\alpha}^{4}e_{\beta}^{2}}{\pi\hbar m_{\alpha}^{2}\omega^{3}v}\left(1+\frac{v_{z}^{2}}{v^{2}}+4\beta_{z}\frac{v_{z}^{2}}{v^{2}}\right)\ln\left(\frac{m_{\alpha}v^{2}}{\hbar\omega}\right),\label{eq:prob_integrated}\end{gathered}$$
which determines the absorbed power, and
$$\begin{gathered}
\int q_{\perp_{z}}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)d\mathbf{q}\\
\approx\frac{\omega}{c}\frac{e_{\alpha}^{4}e_{\beta}^{2}}{\pi\hbar m_{\alpha}^{2}\omega^{3}v}2\frac{v_{\perp}^{2}}{v^{2}}\left(\frac{cv_{z}}{v^{2}}+2\frac{v_{z}^{2}}{v^{2}}-\frac{v_{\perp}^{2}}{v^{2}}\right)\ln\left(\frac{m_{\alpha}v^{2}}{\hbar\omega}\right),\label{eq:q_perb_prob_integrated}\end{gathered}$$
which determines the amount of momentum change in the direction perpendicular to the velocity. This is needed to calculate the current. Note that while it is not necessary to retain the first order terms in Eq. (\[eq:prob\_integrated\]) to calculate the absorbed power, one needs to keep them while calculating current. Note also in Eq. (\[eq:prob\_integrated\]) that electrons moving in the direction of the photon $(\beta_{z}>0)$ are more likely to absrob energy than electrons moving in the opposite direction $(\beta_{z}<0)$. This is consistent with the picture that an electron moving in the direction of the photon can absorb its energy through a smaller angle scatter than would an electron moving in the opposite direction.
From Eqs. (\[eq:prob\_integrated\]) and (\[eq:q\_perb\_prob\_integrated\]) we can write the rate of momentum absorption as:
$$\begin{gathered}
\frac{d\mathbf{p}_{V,z}^{\alpha}}{dt}=\int\frac{\hbar\omega}{c}\left(1+\frac{cv_{z}}{v^{2}}-\frac{v_{z}^{2}}{v^{2}}\right)\frac{\hbar\omega}{T}\\
\times\frac{n_{\beta}e_{\alpha}^{4}e_{\beta}^{2}}{\pi\hbar m_{\alpha}^{2}\omega^{3}v}\left(1+\frac{v_{z}^{2}}{v^{2}}+4\beta_{z}\frac{v_{z}^{2}}{v^{2}}\right)\ln\left(\frac{m_{\alpha}v^{2}}{\hbar\omega}\right)f_{\mathbf{p}}^{\alpha}d\mathbf{p}n_{\mathbf{k}}d\mathbf{k}\\
-\int\frac{\hbar\omega}{c}\frac{\hbar\omega}{T}\frac{n_{\beta}e_{\alpha}^{4}e_{\beta}^{2}}{\pi\hbar m_{\alpha}^{2}\omega^{3}v}\\
\times2\frac{v_{\perp}^{2}}{v^{2}}\left(\frac{cv_{z}}{v^{2}}+2\frac{v_{z}^{2}}{v^{2}}-\frac{v_{\perp}^{2}}{v^{2}}\right)\ln\left(\frac{m_{\alpha}v^{2}}{\hbar\omega}\right)f_{\mathbf{p}}^{\alpha}d\mathbf{p}n_{\mathbf{k}}d\mathbf{k}.\end{gathered}$$
Integrating over angle $\theta$ we get
$$\frac{d\mathbf{p}_{V,z}^{\alpha}}{dt}=\frac{32}{15}\int\frac{\hbar\omega}{T}\frac{n_{\beta}e_{\alpha}^{4}e_{\beta}^{2}}{\pi cm_{\alpha}^{2}\omega^{2}v}\ln\left(\frac{m_{\alpha}v^{2}}{\hbar\omega}\right)f_{v}^{\alpha}d\mathbf{v}n_{\mathbf{k}}d\mathbf{k}.$$
Therefore,
$$\frac{d\mathbf{p}_{V,z}^{\alpha}}{dt}=\frac{8}{5}\frac{\alpha I}{c}.\label{eq:dpdt}$$
Here $\alpha$ is the effective absorption coefficient:
$$\alpha\approx\frac{4}{3}\sqrt{\frac{2}{\pi}}\frac{n_{\alpha}n_{\beta}e_{\alpha}^{4}e_{\beta}^{2}}{\pi cm_{\alpha}^{3}\omega^{2}v_{th}^{3}}\ln\left(\frac{2T}{\hbar\omega}\right),$$
where $v_{th}^{2}=T/m_{\alpha}$. This absorption coefficient determines the total absorbed power density: $P_{V}^{abs}=\alpha I$.
If we ignored the recoil momentum and assumed that electrons absorb just the incoming photon momentum $\hbar\mathbf{k}$, then the rate of momentum change would be:
$$\frac{d\mathbf{p}_{V,z}^{\mathbf{k}}}{dt}=\int\hbar k\frac{\hbar\omega}{T}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{\alpha,\beta}f_{\mathbf{p}_{\alpha}}^{\alpha}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}d\mathbf{p}_{\alpha}=\frac{\alpha I}{c}.\label{eq:dpdt_naive}$$
Thus, due to the recoil, electrons get $8/5$ times more momentum than they would have got absorbing only the photon momentum, which is consistent with the result obtained in [@Pashinin1978]. This conclusion is true for any spherically symmetric distribution function, not just a Maxwellian. This additional momentum absorbed by electrons (as a whole) is in the direction of the incoming radiation. The ions (as a whole), on the other hand, absorb momentum in the opposite to the incoming radiation direction such that the total rate of momentum absorption for plasma is equal to the rate of photon momentum absorption:
$$\frac{d\mathbf{p}_{V,z}^{\alpha}}{dt}+\frac{d\mathbf{p}_{V,z}^{\beta}}{dt}=\frac{8}{5}\frac{\alpha I}{c}-\frac{3}{5}\frac{\alpha I}{c}=\frac{d\mathbf{p}_{V,z}^{\mathbf{k}}}{dt}=\hbar k\frac{dN_{ph}^{abs}}{dt}=\frac{\alpha I}{c}.$$
It is curious that after averaging for spherically symmetric distribution functions the last two terms in Eq. (\[eq:dpdt\_qdecomp\]) cancel each other and the rate of momentum absorption becomes just
$$\begin{gathered}
\frac{d\mathbf{p}_{V,z}^{\alpha}}{dt}=\int\hbar\left(k+\frac{\omega}{v}\frac{v_{z}}{v}\right)\\
\times\frac{\hbar\omega}{T}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}f_{\mathbf{p}_{\alpha}}^{\alpha}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}d\mathbf{p}_{\alpha},\label{eq:dpdt_sph}\end{gathered}$$
where integration of $w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)$ over $d\mathbf{q}$ can be done independently to get (\[eq:prob\_integrated\]). $\int w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)d\mathbf{q}$ has a zero order term, which is even in $v_{z}$, and a first order term $O\left(\beta_{z}\right)$, which is odd in $v_{z}$. In Eq. (\[eq:dpdt\_sph\]) the first term $k=\omega/c$ is the momentum of the absorbed photon and it is much smaller than the momentum coming from the recoil $\left(\omega/v\right)\left(v_{z}/v\right)$. However, the photon term $k=\omega/c$ is the same for all electrons and is multiplied by the zero order term in $\int w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)d\mathbf{q}$, while the recoil term, which depends on the velocity projection $v_{z}$, has contribution only from the first order term in $\int w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)d\mathbf{q}$, because the zero order term is the same for oppositely going electrons and so gives zero contribution after averaging over the distribution function. Thus, after multiplication by the probability both terms give contributions of equal order. The coefficient next to the first order term in $\int w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}\left(\mathbf{k},\mathbf{q}\right)d\mathbf{q}$ is positive, which comes from the fact that Bremsstrahlung emission is the most pronounced in the direction of the electron velocity [@Jackson1999]. Since also the recoil term is proportional to $v_{z}$, we can immediately conclude that the averaged momentum gained by electrons due to the recoil is in the positive $z$-axis direction.
Inverse Bremsstrahlung Current
==============================
The time evolution of the current density can be put as
$$\frac{d\mathbf{j}}{dt}=-\frac{e}{m_{e}}\frac{d\mathbf{p}_{V}^{e}}{dt}-\nu_{Sp}\mathbf{j}.\label{eq:current_evol_naive}$$
This is a fluid approach, since it takes into account only how much momentum is absorbed by electrons, not which electrons absorb the momentum.
The collision frequency $\nu_{Sp}$ in Eq. (\[eq:current\_evol\_naive\]) corresponds to the Spitzer conductivity and can be approximated by the following empirical formula [@Diver2001]:
$$\nu_{Sp}=\frac{Z}{3}\sqrt{\frac{2}{\pi}}\left(0.295+\frac{0.39}{0.85+Z}\right)\frac{\Gamma}{v_{th}^{3}},$$
where $\Gamma=\omega_{p}^{4}\ln\Lambda/4\pi n$ and $Z$ is the ion charge. From Eq. (\[eq:current\_evol\_naive\]) the stationary current density is
$$\mathbf{j}_{fluid}=-\frac{e}{m_{e}}\nu_{Sp}^{-1}\frac{d\mathbf{p}_{V}^{e}}{dt}.\label{eq:j}$$
Since the current density in the fluid approximation is proportional to the rate of momentum absorption, the current corrected for the recoil is $8/5$ times higher than the simple fluid estimate ignoring the recoil, and is equal to
$$j_{fluid}=-\frac{8}{5}\frac{e}{m_{e}}\frac{\alpha I}{c}\nu_{Sp}^{-1}=-\frac{20.4}{Z\left(1+\frac{1.32}{0.85+Z}\right)}\frac{ev_{th}^{3}}{m_{e}\Gamma}\frac{\alpha I}{c}.\label{eq:j_fluid}$$
![\[fig02\]The probability of Bremsstrahlung absorption in arbitrary units versus the angle between the electron velocity and the incoming photon direction $\cos\theta=v_{z}/v$ for $\beta=0.08$.](fig02){width="45.00000%"}
However, the Spitzer conductivity is strictly applicable only to the current produced by dc electric field, when all electrons get equal acceleration in the same direction. The current generation due to inverse Bremsstrahlung is not equivalent to the action of dc electric field because different electrons absorb different amount of power and are pushed in different directions.
One example of the kinetic effects is the additional current due to asymmetric absorption of radiation. Fig. \[fig02\] shows the integrated probability of absorption within $d\theta$ given by Eq. (\[eq:prob\_integrated\]) for electrons lying on the circle with radius $\beta=0.08$ in velocity space. We see that the electrons going in the direction of the incoming photons ($0\leq\theta<\pi/2$) absorb more radiation than electrons going in the opposite direction ($\pi/2<\theta\leq\pi$). This asymmetric absorption will create additional current because the collision frequency in plasma is speed dependent and thus electrons going in the direction of the incoming radiation will experience less resistance from the plasma than electrons going in the opposite direction resulting in more current.
Fig. \[fig03\] shows, averaged over all possible recoils, the rate of momentum absorption along the $z$-axis by an electron with $\beta=0.08$ versus $\cos\theta=v_{z}/v$. $-S_{p,z}$ is defined by Eq. (\[eq:dpdt\_recoil\]) and determines the rate of momentum absorption taking into account the recoil effect. $-S_{k,z}$ is defined by Eq. (\[eq:dpdt\_naive\]) and determines the rate of momentum absorption assuming that only the photon momentum is absorbed. We can see that the recoil effect not only changes the integrated (average) rate of momentum absorption but radically alters the distribution of the absorbed momentum in velocity space. For $-S_{k,z}$ the momentum absorption rate is always positive, i.e. along the $z$-axis, and does not strongly depend on $\cos\theta$, while for $-S_{p,z}$ the momentum absorption rate varies greatly with $\cos\theta$ both in magnitude and sign. In considering Bremsstrahlung absorption by a particular electron, the natural directions are along the electron velocity and perpendicular to the electron velocity. When $\left|\cos\left(\theta\right)\right|$ is close to 1, the velocity of the electron is either parallel or antiparallel to the direction of the incoming photon and so the change in momentum along the $z$-axis is determined mostly by the recoil parallel to the velocity, which is about $\left(\hbar\omega/v\right)\left(v_{z}/v\right)$ in each act of the Bremsstrahlung, as was shown previously. For smaller values of $\left|\cos\left(\theta\right)\right|$ the change in momentum along the $z$-axis is mostly determined by the recoil perpendicular to the electron velocity. This is why the absorption rate shown in Fig. \[fig03\] changes sign.
![\[fig03\]The momentum absorption rate per electron as a function of $\cos\theta=v_{z}/v$ for $\beta=0.08$: along the $z$-axis taking into account the recoil (solid blue), along the $z$-axis taking into account only the photon momentum (dashed red).](fig03){width="1\columnwidth"}
In general, the distribution function will evolve both under the influence of Bremsstrahlung absorption and under the influence of collisions:
$$\frac{\partial f_{\mathbf{p}}^{e}}{\partial t}=\left(\frac{\partial f_{\mathbf{p}}^{e}}{\partial t}\right)_{Br}+\left(\frac{\partial f_{\mathbf{p}}^{e}}{\partial t}\right)_{coll},$$
and the time-evolution of the current should be described more completely than Eq. (\[eq:current\_evol\_naive\]) does by $$\frac{d\mathbf{j}}{dt}=-e\int\mathbf{v}\frac{\partial f_{\mathbf{p}}^{e}}{\partial t}d\mathbf{p}.$$
Following [@Fisch1987] we can write the current density at time $t$ as the rate of pushing electrons times the ensemble-averaged current difference:
$$\begin{gathered}
j_{cd}\left(t\right)=\sum_{\mathbf{v},\triangle\mathbf{v}}\int_{0}^{t}d\tau\frac{P_{V}\left(\tau,\mathbf{v},\triangle\mathbf{v}\right)}{\triangle\varepsilon}\\
\times\left\langle q_{e}v_{z}\left(t-\tau,\mathbf{v}+\triangle\mathbf{v}\right)-q_{e}v_{z}\left(t-\tau,\mathbf{v}\right)\right\rangle \\
\underset{\triangle\mathbf{v}\rightarrow0}{=}\sum_{\mathbf{v},\triangle\mathbf{v}}\int_{0}^{t}d\tau\frac{P_{V}\left(\tau,\mathbf{v},\triangle\mathbf{v}\right)}{\triangle\varepsilon}\triangle\mathbf{v}\cdot\frac{\partial\left\langle qv_{z}\left(t-\tau,\mathbf{v}\right)\right\rangle }{\partial\mathbf{v}}\label{eq:j_t_cd}\end{gathered}$$
If the power is independent of time we can put integration inside the ensemble-averaged current and write for a steady-state current:
$$j_{cd}=\int\left[-\frac{e}{m_{e}}\frac{\hbar\left(\mathbf{k}-\mathbf{q}\right)\cdot\partial\chi/\partial\mathbf{v}}{\hbar\omega}\right]dP_{V}\left(\mathbf{v},\mathbf{k},\mathbf{q}\right),\label{eq:j_cd_int}$$
where we expressed infinitesimal changes in energy and velocity through $\omega,$ $\mathbf{k}$, $\mathbf{q}$, changed from summation to integration, and introduced a Green’s function: $\chi=\int_{0}^{\infty}\left\langle v_{z}\left(\tau,\mathbf{v}\right)\right\rangle d\tau$. In most cases it is possible to express the Green’s function as $\chi(\mathbf{v})=v_{z}\nu^{-1}\left(v\right)$, where $\nu^{-1}$ can be thought of as an effective collision frequency [@FidoneGranataJohner1988].
The expression in square brackets of Eq. (\[eq:j\_cd\_int\]) can be understood as incremental current drive efficiency. Thus, to find the generated current one needs to average the incremental current drive efficiency over the power density absorbed:
$$\begin{gathered}
j_{cd}=\int\left(\frac{\delta j_{z}}{\delta P_{V}}\right)dP_{V}\\
=\frac{e}{m_{e}}\int\frac{\left(\mathbf{k}-\mathbf{q}\right)\cdot\partial\chi/\partial\mathbf{v}}{\omega}\\
\times\frac{m_{e}v^{2}}{2}\frac{\partial}{\partial\mathbf{v}}\cdot\hbar\left(\mathbf{k}-\mathbf{q}\right)\frac{\hbar\omega}{m_{e}T}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}f_{\mathbf{p}_{\alpha}}^{\alpha}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}d\mathbf{p}_{\alpha}\\
=\frac{e}{m_{e}}\int\left[\frac{\nu^{-1}}{\omega}\left(k_{z}-q_{z}\right)+\frac{\partial\nu^{-1}}{\partial v}\frac{v_{z}}{v}\right]\\
\times\frac{m_{e}v^{2}}{2}\frac{\partial}{\partial\mathbf{v}}\cdot\hbar\left(\mathbf{k}-\mathbf{q}\right)\frac{\hbar\omega}{m_{e}T}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}f_{\mathbf{p}_{\alpha}}^{\alpha}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}d\mathbf{p}_{\alpha}.\label{eq:j_cd}\end{gathered}$$
The first term in square brackets of Eq. (\[eq:j\_cd\]), which is proportional to $k_{z}-q_{z}$, is the usual current due to momentum injection along the $z$-axis, while the second term, which is proportional to $\partial\nu^{-1}/\partial v$, is the current due to asymmetric absorption.
One might want to calculate the generated current by summing the incremental currents instead:
$$\begin{gathered}
j_{cd,res}=\int\delta j_{z}=-\frac{e}{m}\int\hbar\left(\mathbf{k}-\mathbf{q}\right)\cdot\frac{\partial\chi}{\partial\mathbf{v}}\\
\times\frac{\hbar\omega}{T}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}f_{\mathbf{p}_{\alpha}}^{\alpha}f_{\mathbf{p}_{\beta}}^{\beta}n_{\mathbf{k}}d\mathbf{k}d\mathbf{q}d\mathbf{p}_{\beta}d\mathbf{p}_{\alpha}\\
=e\int\mathbf{S}_{\mathbf{v}}\cdot\frac{\partial\chi}{\partial\mathbf{v}}d\mathbf{v},\label{eq:j_cd_naive}\end{gathered}$$
where we used the wave induced flux in velocity space $\mathbf{S_{\mathbf{v}}}=m_{e}^{2}\mathbf{S}_{\mathbf{p}}$. Eq. (\[eq:j\_cd\_naive\]) follows from Eq. (\[eq:j\_t\_cd\]) if the power absorbed is localized around certain velocity. Therefore, Eqs. (\[eq:j\_cd\]) and (\[eq:j\_cd\_naive\]) are identical when the absorption is localized in the velocity space, but they produce different results otherwise. In the present problem all electrons are pushed by the incoming electromagnetic field and Eq. (\[eq:j\_cd\_naive\]) miscalculates the generated current density.
After integration by parts, Eq. (\[eq:j\_cd\]) can be written as
$$\begin{gathered}
j_{cd}=-\frac{e}{2}\int\frac{\partial\left(v\frac{\partial\nu^{-1}}{\partial v}\right)}{\partial v}\frac{v_{z}}{v}\hbar\omega\frac{\hbar\omega}{m_{e}T}N_{ph}n_{\beta}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}f_{\mathbf{v}}^{e}d\mathbf{q}d\mathbf{v}\\
-e\int\frac{\partial\left(v\nu^{-1}\right)}{\partial v}\hbar\left(k_{z}-q_{z}\right)\frac{\hbar\omega}{m_{e}T}N_{ph}n_{\beta}w_{\mathbf{p}_{\alpha},\mathbf{p}_{\beta}}^{Br}f_{\mathbf{v}}^{e}d\mathbf{q}d\mathbf{v}.\end{gathered}$$
The Green’s function and the corresponding effective collision frequency $\nu$, generally speaking, can be found only numerically. However, the high-velocity approximation exists [@KarneyFisch1985; @Fisch1987]:
$$\nu^{-1}=\frac{v^{3}}{\Gamma\left(5+Z\right)}+\frac{9v_{th}^{2}v}{\Gamma\left(5+Z\right)\left(3+Z\right)}.\label{eq:freq_inv}$$
This expression has two shortcomings. First, it uses the high-velocity approximation both for electron-electron and electron-ion collisions. While for electron-ion collisions this approximation is always good, it is less so for electron-electron collisions. Since it is mostly thermal electrons that absorb through Bremsstrahlung, the high-velocity approximation will noticeably underestimate the current for low $Z$ plasma. Second, this expression violates the momentum conservation in electron-electron collisions. Thus, we expect that Eq. (\[eq:freq\_inv\]) is a good approximation for high $Z$ plasma, but for low $Z$ plasma the error in the current can be appreciable.
After straightforward calculations using $\nu$ defined by Eq. (\[eq:freq\_inv\]) we obtain from Eq. (\[eq:j\_cd\]):
$$j_{cd}=-\frac{34.2}{5+Z}\frac{ev_{th}^{3}}{m_{e}\Gamma}\frac{\alpha I}{c}-\frac{39.5}{\left(5+Z\right)\left(3+Z\right)}\frac{ev_{th}^{3}}{m_{e}\Gamma}\frac{\alpha I}{c},\label{eq:j_cd_fin}$$
while Eq. (\[eq:j\_cd\_naive\]) would only give factors 12.8 and 24.8 respectively in the above formula.
For comparison, in the fluid approximation the current density corrected for the recoil, which is given by Eq. (\[eq:j\_fluid\]), can be represented as
$$j_{fluid}=e\nu_{Sp}^{-1}\int S_{v,z}d\mathbf{v}.$$
We can clearly see that Eq. (\[eq:j\_cd\]) has an additional term that is responsible for the current due to asymmetric absorption.
Because of the use of the high-velocity and momentum conservation violating approximation for $\nu$, Eq. (\[eq:j\_cd\_fin\]) underestimates the current, especially for small $Z$. Reckoning that electron-electron collisions conserve current, to remedy this problem we propose an alternative hybrid expression, where the part of the current in Eq. (\[eq:j\_cd\]) proportional to $k_{z}-q_{z}$ is substituted by the fluid expression Eq. (\[eq:j\_fluid\]), while the part proportional to $\partial\nu^{-1}/\partial v$ is left unchanged:
$$\begin{gathered}
j_{hybrid}=j_{fluid}-\frac{e}{m_{e}}\int\frac{\partial\nu^{-1}}{\partial v}\frac{v_{z}}{v}dP_{V}\left(\mathbf{v},\mathbf{k},\mathbf{q}\right)\\
=j_{fluid}-\frac{19.2}{5+Z}\frac{ev_{th}^{3}}{m_{e}\Gamma}\frac{\alpha I}{c}-\frac{12.4}{\left(5+Z\right)\left(3+Z\right)}\frac{ev_{th}^{3}}{m_{e}\Gamma}\frac{\alpha I}{c}.\label{eq:j_hybrid}\end{gathered}$$
If all electrons were to absorb equal amount of power, then the part of the current in Eq. (\[eq:j\_cd\]) proportional to $k_{z}-q_{z}$ would be exactly given by the fluid expression Eq. (\[eq:j\_fluid\]). In case of Bremsstrahlung absorption it is mostly thermal electrons that absorb radiation and the fluid formula overestimates the corresponding part of the current. On the other hand, the second part of Eq. (\[eq:j\_hybrid\]) underestimates the current because of the high-velocity limit for $\nu$. So all in all, Eq. (\[eq:j\_hybrid\]) can be a decent approximation for the current for all values of $Z$.
![\[fig04\]The generated current density versus the ion charge $Z$: fluid approximation with the Spitzer conductivity given by Eq. (\[eq:j\_fluid\]) (solid blue), current drive approximation keeping only the first term in Eq. (\[eq:j\_cd\_fin\]) (dotted red), current drive approximation keeping both terms in Eq. (\[eq:j\_cd\_fin\]) (dashed orange), hybrid current given by Eq. (\[eq:j\_hybrid\]) (dash-dotted green).](fig04){width="1\columnwidth"}
Fig. \[fig04\] shows the generated current given by the fluid formula (\[eq:j\_fluid\]), by the current drive formula (\[eq:j\_cd\_fin\]) keeping one and two terms in Eq. (\[eq:j\_cd\_fin\]), and by the hybrid expression (\[eq:j\_hybrid\]) versus the ion charge $Z$. We see that for small $Z$ the current drive formula substantially underestimates current making it even lower than the fluid prediction. However, starting already with $Z=4$ the current drive estimate (\[eq:j\_cd\_fin\]) gives higher current. For higher $Z$, when electron-electron collisions become negligible, the ratio of the current drive prediction to the Spitzer becomes stable and for infinite $Z$ is around 1.7, so that for high $Z$ the generated current with the recoil and kinetic effects taken into account is at least 2.7 higher than the naive fluid estimate without recoil would suggest. The hybrid expression is 1.3 times larger than the fluid estimate even for $Z=1$ and for $Z$ going to infinity the increase is about 2. To get better and definite results for small $Z$ plasma it is necessary to use more accurate than Eq. (\[eq:freq\_inv\]) estimate of the effective collision frequency $\nu$ or perform computer simulations.
Summary
=======
We analytically considered the generation of the plasma current resulting from electron-ion Bremsstrahlung absorption using the following approximations: the polarization effects in Bremsstrahlung are negligible; velocities are non-relativistic; recoil and photon momenta are small in comparison with the electron momentum; ions have infinite mass; waves are electromagnetic with the dispersion relation $\omega=kc$; and the plasma dielectric function is close to one. The laser intensity is not too high, so that the quiver velocity $eE/m\omega$ is much smaller than the thermal velocity. We also note that the logarithmic dependence on velocity has been ignored throughout the paper and $\ln\left(m_{\alpha}v^{2}/\hbar\omega\right)$ has been substituted by $\ln\left(2T/\hbar\omega\right)$ in all the equations.
We investigated how the momentum and energy are absorbed by electrons within the velocity space and confirmed the result obtained in [@Pashinin1978], namely that the averaged momentum absorption by electrons with the recoil taken into account is $8/5$ times higher than the momentum absorption assuming that electrons absorb just the photon momentum. In addition, we demonstrated that for high $Z$ plasma the actual current with the kinetic effects taken into account is at least 2.7 times higher than the naive fluid estimates without recoil would suggest, both because electrons get the recoil momentum from the Coulomb field of ions during the absorption and because electrons absorb power asymmetrically. We also proposed a hybrid expression of fluid and kinetic descriptions for the current that can be a good approximation for all values of $Z$.
The calculation of the current generated from Bremsstrahlung absorption is a fundamental problem of the basic plasma physics. Thus, the results here ought to be of interest in the different areas where radiation driven currents and the generated magnetic fields are important. Areas in which these effects might be important include the radiation driven magnetic field in astrophysics [@Munirov2017a; @Widrow2002; @Durrive2017] and laboratory experiments that use lasers to drive current [@Kruer1988], in particular for applications to inertial confinement fusion.
This work was supported by NNSA Grant No. DENA0002948.
|
---
abstract: 'We report specific heat and thermal conductivity of gadolinium- and yttrium-doped amorphous silicon thin films measured using silicon-nitride membrane-based microcalorimeters. Addition of gadolinium or yttrium to the amorphous silicon network reduces the thermal conductivity over a wide temperature range while significantly increasing the specific heat. This result indicates that a large number of non-propagating states are added to the vibrational spectrum that are most likely caused either by localized vibration of the dopant atom in a Si cage, or softening of the material forming the cage structures. High-resolution cross-sectional electron micrographs reveal columnar features in Gd-doped material which do not appear in pure amorphous silicon. Scattering from both the nanoscaled columns and the filled-cage structures play a role in the reduced thermal conductivity in the rare-earth doped amorphous semiconductor. The overall result is an amorphous solid with a large bump in $C/T^{3}$ and no plateau in thermal conductivity.'
author:
- 'B. L. Zink'
- 'R. Islam'
- 'David J. Smith'
- 'F. Hellman'
title: 'Excess Modes and Enhanced Scattering in Rare-Earth Doped Amorphous Silicon Thin Films'
---
Introduction
============
A wide range of amorphous materials have similar features in specific heat, thermal conductivity and various spectroscopies that suggest a common physical origin. The most notable of these features are a linear term in the specific heat, $C$, usually observed below $2$ K, thermal conductivity $k\propto T^{1.8}$ in the same temperature range, a broad peak in $C/T^{3}$ between $10$ and $50$ K which is larger than in the corresponding crystalline phase, and a plateau in $k$ over the same temperature range.[@pohl] The peak in $C/T^{3}$ is often correlated with excess vibrational density of states seen in neutron or Raman scattering, referred to either as a “boson peak"[@MalinovskySSC86] or “excess modes."[@FabianPRL96; @FeldmanPRB99] Despite several decades of study, no microscopic theory exists which offers a complete explanation of these phenomena. The standard tunneling model is a widely accepted description of the behavior of $C$ and $k$ below $2$ K. In this model the linear term in $C$ is attributed to a constant density of Schottky-like two-level systems (TLS), with the behavior of $k$ explained by scattering from these states.[@TLS; @PhillipsJLTP1972] This model offers little explanation of the physical nature or origin of the very closely spaced energy levels. There have been several efforts to develop physical models which also explain the $C/T^{3}$ peak and $k$ plateau. Examples are the soft-potential model,[@BuchenauandRamos; @BuchenauPRB1992] the fracton model,[@AlexanderPRB83; @JagannathanPRB89] and a model relating dynamics at the glass transition to the low temperature phenomena.[@LubchenkoPNAS03; @LubchenkoPRL01]
Amorphous silicon ($a$-Si) forms a tetrahedrally bonded continuous random network and can be made only in thin-film form by evaporation, sputtering, and various chemical vapor deposition techniques. Under-coordinated Si atoms form dangling bond defects, which can be reduced by introducing hydrogen either during deposition or after film growth. The hydrogenated material, $a$-Si:H, shows improved electronic properties and has several important industrial applications for large area microelectronic devices. $a$-Si is often studied theoretically, in part because of its relatively simple structure, but also due to several predicted and observed deviations from typical amorphous solids. Both theory and experimental evidence suggest a strikingly low density of two-level systems, a relatively large change in the Debye temperature between crystalline and amorphous silicon, and a relatively small peak in $C/T^{3}$ in $a$-Si, deviating less from the Debye model than the crystalline phase.[@TLS; @LiuPRL97; @BergaSi; @FeldmanPRB99; @ZinkPRL06] Properties ranging from electrical conductivity to density of TLS also show a strong dependence on the $a$-Si film growth method.[@PohlRMP02] Another example is the thermal conductivity plateau, though due to the difficulty of thermal measurements on thin-film samples, measurements of $k$ in the expected temperature range of the plateau exist for only three samples: an $\approx50$ $\mu$m thick sputtered sample reported by Pompe et al.[@Pompe] that shows a fairly well defined plateau, and much thinner $130$ and $277$ nm thick films recently reported by our group grown by e-beam evaporation that show no evidence of the plateau.[@ZinkPRL06] Both results can be reasonably explained by the theory presented by Feldman[@FeldmanPRB99] with different treatment of low-$Q$ scattering.
Amorphous rare-earth-silicon alloys ($a$-RE$_{x}$Si$_{1-x}$, with RE=Gd, Y, Tb, etc.) prepared by e-beam co-deposition, have shown many phenomena related to the interaction between local RE magnetic moments and conduction electrons including a very large negative magnetoresistance at low temperatures.[@prevpaper; @pengpaper] A negative Hall coefficient[@TeizerPRB2003] and thermopower[@ThermoPunpub] indicate that introducing the large, heavy RE adds carriers to the electron band, with the Gd ions contributing localized $S=7/2$ magnetic moments, while Y provides a non-magnetic counterpart with nearly identical ionic radius and valency. Recent computational and experimental results shed some light on the structure of $a$-Gd$_{x}$Si$_{1-x}$ and its non-magnetic analog, $a$-Y$_{x}$Si$_{1-x}$. Local density functional theory simulations of $a$-Y$_{x}$Si$_{1-x}$ suggest that Y$^{3+}$ ions are surrounded by low-coordinated Si, leaving them in a “cage” of dangling bonds.[@parrinello] X-ray absorption fine structure (XAFS) experiments on $a$-Gd$_{x}$Si$_{1-x}$ using both the Si and Gd absorption edges give a similar picture, with Si atoms as nearest-neighbors to the Gd ions and no clustering.[@XAFS] The XAFS studies also indicate that neither the Si coordination number nor the Gd-Si distance changes with Gd composition, suggesting a Si cage but no dangling bonds. Electron spin resonance (ESR) measurements indicate that the addition of even small amounts of Gd to the Si matrix eliminates the dangling bonds.[@OseroffESR] Cages in $a$-Gd$_{x}$Si$_{1-x}$ are further supported by calculations using the full potential linearized augmented plane wave (FLAPW) method.[@WuFLAPW]
This filled-cage structure is reminiscent of the filled skutterudite antimonides, such as CeFe$_{4}$Sb$_{12}$[@MorelliJAP95] or RE$_{1-y}$Fe$_{4-x}$Co$_{x}$Sb$_{12}$ (RE=La, Ce)[@SalesPRB97], or Sr or Eu doped Ge clathrates,[@CohnPRL99] where the heavy La, Ce, Sr, or Eu dopant atoms fill cages in the host crystal structure. Slack originally proposed that “rattling" movement of the heavy dopant atom in the anharmonic potential of the cage would strongly scatter phonons but not charge carriers, resulting in a material with a high thermoelectric figure of merit.[@SlackCRCchap] Although there is some debate about the detailed mechanism of the phonon scattering,[@FeldmanPRB2000; @CaoPRB04] it is clear that the filled-cage structures in these materials dramatically alter the vibrational spectrum, reducing the thermal conductivity over a wide temperature range (at least $2-300$ K), and increasing the specific heat.[@KeppensNature98; @FeldmanPRB2000; @HermannPRL03]
In this paper we present specific heat and thermal conductivity measurements of gadolinium- and yttrium-doped amorphous silicon thin films from $3-100$ K, as well as high-resolution cross-section transmission-electron microscope (XTEM) observations of the films. We compare these measurements to data on $a$-Si films grown by the same technique and literature values for crystalline silicon to probe the nature of vibrational states and scattering in rare-earth doped amorphous silicon.
Experiment
==========
Thin film $a$-Y$_{x}$Si$_{1-x}$ and $a$-Gd$_{x}$Si$_{1-x}$ samples were e-beam co-evaporated from separate Y, Gd, and Si crucibles onto amorphous Si-N membrane-based microcalorimeters and amorphous Si-N coated Si substrates. The microcalorimeters and substrates were held near room temperature throughout the deposition, promoting the growth of amorphous films. Typical deposition pressures were $\leq 2 \times 10^{-8}$ Torr. The films on substrates were used to measure the film thickness via profilometry, the composition by Rutherford backscattering (RBS), the dangling bond density via ESR, the sound velocity,[@LeePRB05] and for XTEM imaging. Both XAFS[@XAFS] and RBS measurements indicate atomic densities consistent with pore-free films. A picosecond ultrasonic measurement[@LeePRB05] of the longitudinal sound velocity in a $a$-Gd$_{x}$Si$_{1-x}$ film gave $v_{L}=(5.39 \pm 0.44) \times 10^{5}$ cm/s, lower than the value measured in our $a$-Si, $v_{L}=(7.51\pm 0.30) \times 10^{5}$ cm/s. Detailed description of the microcalorimetry techniques for measuring $C$ and $k$, including the determination and subtraction of background contributions, appear elsewhere.[@microcal; @KappaPaper; @RevazTCA05]
Results and Discussion
======================
Figures \[TEMlow\] and \[TEMhigh\] compare XTEM images for $a$-Si and $a$-Gd$_{18}$Si$_{82}$ films. The low magnification images (Fig. \[TEMlow\]) show a featureless $a$-Si film, whereas the $a$-Gd$_{18}$Si$_{82}$ film grown under identical conditions shows a vertically streaked appearance indicative of a columnar structure. The higher magnification images in Fig. \[TEMhigh\] also show featureless amorphous silicon, and $3-4$ nm columnar features in amorphous gadolinium-silicon. Note that all images suggest dense, pore-free, clearly amorphous films at the atomic level. Columnar microstructure, such as that seen in $a$-Gd$_{18}$Si$_{82}$, is a common outcome of the vapor deposition process for evaporated films when atomic mobility at the growth surface is low.
Figure \[kcomp\] compares $k$ of $a$-Y$_{x}$Si$_{1-x}$ and $a$-Gd$_{x}$Si$_{1-x}$ films to Pompe et al.’s sputtered $50$ $\mu$m thick $a$-Si film[@Pompe] and e-beam evaporated thin-film $a$-Si. The top axis indicates the estimated wavelength of the vibrations which carry the most heat in the dominant phonon approximation,[@LambdaDomNoteReSi] calculated for $a$-Y$_{12}$Si$_{88}$. The e-beam $a$-Si shows a lower $k$ than the extremely thick sputtered film and no plateau. Addition of Y or Gd to the material further reduces $k$ over the entire temperature range measured, and the three alloy films all have the same $k$ within error bars at all $T$. This reduction occurs despite the addition of electrons to the material, which increases the electrical conductivity dramatically. However, this is not surprising, since a Wiedemann-Franz law estimation of the electronic contribution suggests that $k$ is totally dominated by vibrational excitations.[@BZthesis] In addition to the lack of composition dependence, $k$ of $a$-Gd$_{x}$Si$_{1-x}$ (which shows a huge negative magnetoresistance at low T) showed no measurable change in applied magnetic fields up to $8$ Tesla.
Figure \[CoverT3\] compares $C/T^{3}$ vs. $T$ (in J/g K$^{4}$) for $a$-Y$_{9}$Si$_{91}$ to thin-film $a$-Si[@ZinkPRL06] and bulk crystalline silicon. The dashed line is the Debye specific heat function, $C_{D}$, for $\theta _{D}=487$ K. $C/T^{3}$ for $a$-Y$_{9}$Si$_{91}$ shows a much larger bump than that seen in either $a$-Si or crystalline Si. $a$-Y$_{x}$Si$_{1-x}$ for a broad range of $x$ have very similar specific heat above $60$ $K$, and all show a large maxima in $C/T^{3}$ equal to or greater than that shown here. Samples with larger $x$ have electronic terms, $\gamma T$ (which appear as $\gamma/T^{2}$ on a $C/T^{3}$ plot), and an additional contribution to $C$ occurs in samples near the metal-insulator transition.[@BZthesis] We have previously reported the specific heat of $a$-Gd$_{x}$Si$_{1-x}$, which is similar to values for $a$-Y$_{x}$Si$_{1-x}$ above $60$ K, but is dominated at lower temperatures by large contributions from magnetic degrees of freedom.[@myPRL; @Ternery; @BZthesis]
Figure \[CompDep\] compares $k$ vs $T$ below $100$ K for $a$-Si and $a$-Y$_{x}$Si$_{1-x}$ to two porous glasses, a porous silica (Vycor) sample with 29% porosity[@WatsonPRB03] (dashed line), and a glassy borosilicate material formed by fusing an array of capillaries ($\triangledown$).[@ZaitlinPRB75] The data for both porous materials has been corrected for the missing volume of the pores. Neither of these materials have a $k$ plateau in the typical temperature range, though Vycor shows an apparent plateau at much lower temperature. The reduction of $k$ in these porous materials is due to enhanced damping or scattering of long-wavelength modes caused by the pores.[@GracePRB86; @ZaitlinPRB75] The addition of Gd- and Y- to $a$-Si reduces $k$ by a quantitatively similar factor as does the addition of pores to $a$-SiO$_{2}$, but must rely on a different mechanism for scattering of heat carriers.
The structural evidence from XTEM (shown in Figs. \[TEMlow\] and \[TEMhigh\]), XAFS measurements, and simulations suggests two likely mechanisms for enhanced scattering of propagating vibrational excitations in $a$-RE$_{x}$Si$_{1-x}$ compared to $a$-Si: effects of the columnar structural features, and rattling of the caged Y or Gd dopants. The measurements of $C$ and $k$ presented here indicate that both mechanisms play a role in the thermal properties of the rare-earth doped amorphous silicon.
Figure \[MFP\]a compares the estimated mean free path of vibrations for amorphous Si and $a$-Y$_{x}$Si$_{1-x}$ films. This is given by $3k/C_{D}v_{D}$, where $v_{D}$ is the estimated Debye velocity and $C_{D}$ is the corresponding specific heat contribution from propagating modes, estimated here using $\theta_{D}=487$ K measured for $a$-Si.[@ZinkPRL06] In $a$-Y$_{12}$Si$_{88}$ MFP is roughly $\propto T^{-1.65}$ throughout the measured temperature range, while in $a$-Si the exponent increases below $\sim 15$ K so that $MFP\propto T^{-2.08}$. $15$ K is the temperature where $\lambda _{dom}$ is roughly equal to the spacing of the columnar features seen in Figs. \[TEMlow\]b and \[TEMhigh\]b. Fig. \[MFP\]b makes a similar comparison between filled and unfilled skutterudites. The mean free path is again estimated using $3k/C_{D}v_{D}$, where $k$, $v_{D}$, and $C_{D}$ values reported by Sales et al.[@SalesPRB97] Filling the cage in the skutterudite crystal with the heavy La dopant reduces the estimated mean free path over the whole measured temperature range. One interpretation of our data is that the overall suppression of $k$ in $a$-Y$_{x}$Si$_{1-x}$ is due to the effect of the rattling of caged rare-earth dopants, while the reduced temperature dependence of the MFP below $15$ K is the effect of the columnar structural features. This structural scattering need not be directly analogous to grain-boundary scattering in a crystal, but rather the effect of a softer region of the matrix occurring due to density fluctuations with the $3-4$ nm spacing seen in Figs. \[TEMlow\]b and \[TEMhigh\]b.
Figure \[Crat\] compares the ratio of the measured $C$ to the Debye contribution, $C_{D}$, for crystalline silicon, $a$-Si, and $a$-Y$_{9}$Si$_{91}$. In typical amorphous solids such as $a$-SiO$_{2}$, $C$ is significantly larger than $C_{D}$, as a result of the excess modes.[@pohl] We previously reported that the situation is very different in $a$-Si, where $C$ more closely matches the Debye model than does crystalline Si.[@ZinkPRL06] As shown in Fig. \[Crat\], the additional non-propagating vibrational states introduced by adding Y to the matrix cause $C/C_{D}$ for $a$-Y$_{9}$Si$_{91}$ to slightly exceed the value for crystalline Si, approaching the expectation for a typical amorphous material. It is also interesting to note that for $a$-Y$_{9}$Si$_{91}$, $a$-Y$_{13}$Si$_{87}$, and $a$-Y$_{21}$Si$_{79}$, the height of the peak in $C/T^{3}$, $P_{c}$, scales with its position in temperature, $T_{max}$, as $P_{c}\propto T_{\mathrm{max}}^{-1.6}$, which agrees with the scaling observed by Liu and Lohneysen for a wide range of amorphous solids.[@LiuEPL96] Our recent $a$-Si measurement does not match this scaling behavior particularly well, indicating again that $a$-Si is a somewhat atypical amorphous material, while the additional non-propagating vibrational states cause $a$-Y$_{x}$Si$_{1-x}$ to show more typical behavior.
The exact nature of these states is unknown, though it seems likely that excess modes are added due to locally softened Si cages around the heavy dopants or to anharmonic rattling of the dopant in the cage or both. It is clear from the reduced $k$ that these excess modes do not carry heat. Several authors have reported a similar increase in the specific heat of filled skutterudites when compared to the “empty" host crystal. In the case of the skutterudites, the excess $C$ can be explained by the contribution of one or more Einstein modes, which each contribute a term $C_{\mathrm{E}}=(\theta_{\mathrm{E}}/T)^{2}e^{(\theta_{\mathrm{E}}/T)}/(e^{(\theta_{\mathrm{E}}/T)}-1)^{2}$, where $\theta_{\mathrm{E}}$ is the Einstein temperature. The observation of these Einstein contributions provides direct evidence of the localized “rattling" of the filler atom, and is corroborated by inelastic neutron scattering data, resonant ultrasound spectroscopy, and simulations.[@KeppensNature98; @FeldmanPRB2000; @HermannPRL03] Following the “rattling" atom analogy, we fit the measured specific heat of $a$-Y$_{9}$Si$_{91}$ to the equation $C=C($$a$-Si$)+A_{1} C_{\mathrm{E1}}+A_{2} C_{\mathrm{E2}} + A_{3} C_{\mathrm{E3}}$. A fit of this type assumes that the filling of the cages has little effect on the elastic properties of the host material. Though we believe this is true to good approximation for the $a$-Y$_{9}$Si$_{91}$ sample, available evidence suggests a possible reduction in $\theta_{\mathrm{D}}$ for more heavily Y-doped $a$-Si.[@BZthesis] This type of effect can have important implications for the physical interpretation of the resulting fit parameters.[@FeldmanPRB2000] The solid line in Fig. \[CoverT3\] represents the fit with $A_{1}=1.0$ J/mol K, $\theta_{\mathrm{E1}}=194$ K, $A_{2}=0.83$ J/mol K, $\theta_{\mathrm{E2}}=100$ K, and $A_{3}=0.086$ J/mol K, $\theta_{\mathrm{E3}}=52$ K. The data can be fit somewhat less well by two Einstein modes ($\sim108$ and $51$ K) , but is very poorly modeled with a single Einstein contribution.[@aSi02fitnote] As a comparison, Keppens, et al. used Einstein modes at $70$ and $200$ K to explain the contribution of the rattling La atom in La$_{0.9}$Fe$_{3}$Co$_{}$Sb$_{12}$,[@KeppensNature98] while Hermann, et al. needed only a single Einstein mode at $53$ K to describe thallium rattling in similar antimony skutterudites.[@HermannPRL03] The similarity of these results to the Y-doped amorphous Si data presented here suggests that similar physics drives the excess modes and reduced $k$ in these two classes of materials.
Regardless of the exact nature of the contributions to $C$ in $a$-Y$_{x}$Si$_{1-x}$, the combined effect of the addition of excess modes and the enhanced scattering is an amorphous material with a large bump in $C/T^{3}$ but no corresponding plateau in $k$. We conclude that these two phenomena, which are often both observed in a given amorphous material, are not necessarily the work of the same physical mechanism. This suggests an interesting future experiment. It is currently a matter of debate whether the $k$ plateau and $C/T^{3}$ bump and the related excess modes can be explained within a single theoretical framework with the TLS that dominate the low $T$ properties of most amorphous insulators. Our work shows that addition of heavy Y dopants to $a$-Si, a material which has few excess modes and very low contributions from TLS, adds excess modes to the material. Study of $a$-Y$_{x}$Si$_{1-x}$ below $2$ K could indicate whether TLS have returned with the addition of Y, which would provide evidence of correlation between TLS and excess modes. This also suggests that addition of heavy dopants could provide a potentially tunable method for adding excess modes to $a$-Si, allowing systematic study of the vibrational excitations in amorphous insulators.
Conclusions
===========
In summary, we measured specific heat and thermal conductivity of Gd- and Y-doped amorphous Si thin films. Addition of the heavy dopant atoms introduces scattering which reduces the thermal conductivity below values measured for pure $a$-Si. At the same time, a large number of excess vibrational modes are added, resulting in a bump in $C/T^{3}$ which is much larger than that in the pure amorphous material and also larger than in crystalline Si, in better agreement with expected behavior of $C$ in amorphous insulators. This bump can similar manner as the contribution of rattling modes to filled antimony skutterudites, suggesting that similar physics drives the thermal behavior of these rather different systems. Furthermore, the ability to add excess modes to $a$-Si suggests that continuing study of heavy-atom doped amorphous silicon could enable systematic probe of the correlation between tunneling systems and excess modes in amorphous materials.
Acknowledgments
===============
We thank B. Maranville, M. Liu and M. Wong for the RBS measurements, S. Oseroff and C. Rettori for the ESR measurement, D. Cahill for the sound velocity measurement and many helpful comments, B. Pohl, A. Migliori and R. Dynes for fruitful discussions, and the NSF and LANL for support. We also acknowledge use of facilities in the John M. Cowley Center for High Resolution Electron Microscopy at Arizona State University.
[47]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, in **, edited by (, , ).
, , , , , , ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , , , , , , ****, ().
, , , ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , ****, ().
, , , .
, ****, ().
, , , , , , ****, ().
, , , .
, .
, ****, ().
, , , , , ****, ().
, , , , , ****, ().
, in **, edited by (, , ), p. .
, , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , ****, ().
, , , , ****, ().
, , , ****, ().
, ****, (),
, Ph.D. thesis, ().
, eds., **, National Bureau of Standards Monograph 21 (, , ).
, , , , ****, ().
, , , , ****, ().
, ****, (), .
, ****, ().
, ****, ().
, ****, ().
|
---
abstract: 'The use of effective Darboux transformations for general classes Lax pairs is discussed. The general construction of “binary” Darboux transformations preserving certain properties of the operator, such as self-adjointness, is given. The classes of Darboux transformations found include the multicomponent BKP and CKP reductions of the KP hierarchy.'
author:
- |
J J C NIMMO\
Department of Mathematics, University of Glasgow,\
Glasgow G12 8QW, Scotland
title: |
DARBOUX TRANSFORMATIONS FROM REDUCTIONS\
OF THE KP HIERARCHY
---
Introduction
============
Darboux transformations define a mapping between the solutions of a linear differential equations and a similar equation containing different coefficients. Since integrable nonlinear evolution equations frequently arise as the compatibility condition for a pair for such equations, Darboux transformations may be used to construct families of exact solutions of the nonlinear equations. Typically these are multi-soliton solutions. A good introduction to this topic, including the following example, is given in the monograph by Matveev and Salle.
As an example which motivates the work to be presented, consider the Lax pair[^1] $$L=i\d_y+\d^2+u,\ \ M=\d_t+4\d^3+3\d u+3u\d+3i\d^{-1}(u_y).$$ for the variant of the Kadomtsev-Petviashvili equation known as KPI $$(u_u+6uu_x+u_{xxx})_x)-3u_{yy}=0,$$ in which $u$ is a real variable. This means that $[L,M]=0$ if and only if $u$ satisfies KPI.
For all non-zero $\theta$ such that $L(\theta)=M(\theta)=0$, a Darboux transformation is defined by the operator $G=\theta\d\theta^{-1}$ in the sense that $$L(\psi)=M(\psi)=0\implies\t L(G(\psi))=\t M(G(\psi))=0,$$ where $\t L$ and $\t M$ are the operators obtained from $L$ and $M$ by replacing $u$ by $\t
u=u+2(\log\theta)_{xx}$. This result is readily proved by observing that $$\t LG=GL\ \ \mbox{and}\ \ \t MG=GM,$$ i.e. $$\t L=GLG^{-1}\ \ \mbox{and}\ \ \t M=GMG^{-1}.$$ In this way the Darboux transformation manifests itself as a (differential) gauge transformation. It also follows that $$[L,M]=0\implies[\t L,\t M]=0,$$ i.e. $\t u$ satisfies KPI whenever $u$ does. Hence the Darboux transformation induces an auto-Bäcklund transformation.
There is a problem with this transformation however. For almost all $u$, since $\theta$ is the solution of a complex equation, $\t u$ is not real and so we do not obtain solutions of KPI. At the root of the problem is the fact that, while $L$ and $M$ are self-adjoint, $\t L$ and $\t M$ are not. In order to overcome this problem one may use a “binary” transformation (to be defined in the next section) which does preserve the self-adjointness of $L$ and $M$.
This paper is concerned with the use of binary transformations to preserve the structure of two classes of operators with matrix coefficients and arbitrary order.
The structure of the binary transformation
==========================================
For an (matrix) operator $L$, let $S=\{\theta,\
\text{non-singular}:L(\theta)=0\}$ (and define $\t S$, $S^\ad$ for operators $\t L$, $L^\ad$ etc.). A (formally invertible) gauge transformation $G_\theta$, for $\theta\in
S$, defines a mapping $$G_\theta\colon S\to \t S,\ \ \text{where}\ \ \t L=G_\theta
LG_\theta^{-1}.$$
Consider also the (formal) adjoint operator $G_\theta^\ad$. (Taking the formal adjoint is, as usual, the linear operation defined by $$(a\d^i)^\ad=(-1)^i\d^ia^\ad,$$ for a matrix $a$, where $a^\ad$ denotes the Hermitian conjugate of $a$.) Since $\t L^\ad=G_\theta^{\ad^{-1}}L^\ad G_\theta^\ad$, we have $$G_\theta^\ad\colon \t S^\ad\to S^\ad.$$
By determining the kernel of $G_\theta^\ad$ we obtain some nontrivial solution in $\t S^\ad$. Typically, we may identify this subset in terms of $\theta$ and denote a member as $i(\theta)$. For example, in the classical case when $G_\theta=\theta\d\theta^{-1}$ and $G_\theta^\ad=-\theta^{\ad^{-1}}\d\theta^\ad$, we find that $$G_\theta^\ad(\rho)=0\iff\rho=\theta^{\ad^{-1}}c,$$ where $c$ is independent of $x$.
We represent this situation in the diagram below. $$\begin{diagram}
S\rto^{G_\theta}\xdotted[1,1]+<-2mm,2mm>|<{\rotate\tip}|>\tip
\save\go+<0mm,-5mm>\drop{\scriptstyle\theta}\restore&\t S\\
S^\ad&\lto_{G_\theta^\ad}\t S^\ad\save\go+<0mm,-5mm>
\drop{\scriptstyle i(\theta)}\restore
\end{diagram}.$$
To describe the general form of the binary transformation we consider operators $L$, $\t L$ and $\h L$ and the corresponding sets of non-singular solutions matrices $S$, $\t S$ and $\h S$. Let $\theta\in
S$ and $\h\theta\in \h S$ be such that $G_\theta\colon S\to\t S$ and $G_{\hat\theta}\colon \h S\to\t S$. Then we get the mapping $$G_{\hat\theta}^{-1}G_\theta^{\phantom{\hat\theta}}\colon S\to\h S.$$
The difficulty with this definition of a transformation is that to define it we need one of the solutions we are trying to determine, namely $\h\theta$! To overcome this, we use the fact that there corresponds to $\h\theta\in\h S$ a solution $i(\h\theta)\in\t S^\ad$ and then use the mapping $G_\theta^{\ad^{-1}}\colon S^\ad\to\t S^\ad$ to obtain $\h\theta=i^{-1}(G_\theta^{\ad^{-1}}(\phi))$ for any $\phi\in S^\ad$. This is shown in the diagram below. $$\spreaddiagramrows{-1pc}
\begin{diagram}
S\rto^{G_\theta}&\t S&\lto_{G_{\hat\theta}}\h
S\xdotted[1,-1]!<5mm,0mm>|<{\rotate\tip}|>\tip\\
S^\ad&\lto_{G^\ad_\theta}\t S^\ad&\\
\save\go+<0mm,2mm>\Drop{\phi}\xto[0,1]+<-3mm,2mm>|<\stop\restore&
\save\go+<1mm,2mm>\Drop{i(\h\theta)}\restore&
\end{diagram}$$
In this way we obtain the definition of a general binary transformation.
Consider an operator $L$ and gauge operator $G_\theta$, where $\theta\in S$, such that $G_\theta^\ad(i(\theta))=0$. For each $\phi\in S^\ad$, define $$G_{\theta,\phi}=G_{\hat\theta}^{-1}G_\theta^{\phantom{\hat\theta}},$$ where $\h\theta=i^{-1}(G_\theta^{\ad^{-1}}(\phi))$. Then $$G_{\theta,\phi}\colon S\to \h S,$$ where $\h L=G_{\theta,\phi}^{\phantom{-1}}LG_{\theta,\phi}^{-1}$, is called a *binary transformation*.
In the next section we will consider two concrete examples of such a binary transformation.
Now suppose that the operator $L$ has a constraint of the form $$L^\ad R^\ad=RL,$$ where $R$ is in some formally invertible (matrix differential) operator[^2]. We wish to find binary transformations that preserve this constraint. That is—using the notation of the above definition—we want $\h L$ to satisfy the constraint whenever $L$ does.
Examples for the choice of $R$ include
- $R=I$. $L$ is self-adjoint. An example of the application of this is quoted in the introduction.
- $R=iI$. $L$ is skew-adjoint. This corresponds to the CKP reduction of the KP hierarchy[@Date; @et; @al; @1] and the reduction of the Kuperschmidt “$k=0$” non-standard hierarchy.
- $R=\d$. This corresponds to the BKP[@Date; @et; @al; @2] or the Kuperschmidt “$k=1$” reduction.
The binary transformation we will discuss in the next section will preserve generalizations of these three reductions.
Let the gauge transformation $G$, such that $\h L=GLG^{-1}$, preserve the constraint $L^\ad=RLR^{\ad^{-1}}$. Then $$\h L^\ad-R\h LR^{\ad^{-1}}=0$$ which means that $$G^{\ad^{-1}}L^\ad G^\ad-RGLG^{-1}R^{\ad^{-1}}=
G^{\ad^{-1}}RLR^{\ad^{-1}}G^\ad-RGLG^{-1}R^{\ad^{-1}}=0.$$ This leads to the single condition $RG=G^{\ad^{-1}}R$.
Note that the relation between $L$ and its adjoint imposes a relationship between the solution sets $S$ and $S^\ad$. In particular, for each $\theta\in S$, $R^\ad(\theta)\in S^\ad$. Hence, in the case of a binary transformation $G=G_{\theta,\phi}$, we may make the choice $\phi=R^\ad(\theta)$.
Darboux transformations for general operators
=============================================
In this section we describe two classes of Darboux transformation for general classes of matrix differential operators of arbitrary order. The first is originally due to Matveev[@Matveev] and has also been considered recently by Oevel[@Oevel]. We will present a very simple proof of this result. The second was found by Oevel & Rogers in the case of scalar operators in the context of Sato theory. We will derive a more general version here.
In both cases, the results are remarkably general. There is however a serious drawback. There is, in this general case, absolutely no guarantee that the transformed operator we have the same “form” as the original and so only in special cases does one get a transformation that induces an auto-Bäcklund transformation.
First, consider $$L=\d_t+\sum_{i=0}^nu_i\d^i\ \ \mbox{and}\ \ \t L=\d_t+\sum_{i=0}^n\t
u_i\d^i,$$ where $u_i$ and $\t u_i$ are $N\times N$ (not necessarily constant) matrices. Let the operator $G$ be such that $$\t L=GLG^{-1}=L+[G,L]G^{-1}.$$ Hence $G$ must satisfy $$[G,L]G^{-1}=\sum_{i=0}^n (\t u_i-u_i)\d^i.$$ Taking $G=\theta\d\theta^{-1}$, where $\theta$ is a non-singular $N\times N$ matrix, and hence $G^{-1}=\theta\d^{-1}\theta^{-1}$, we get $$[G,L]G^{-1}=[G,L]\theta\d^{-1}\theta^{-1}=
\sum_{i=0}^n a_i\d^{i-1}\theta^{-1},$$ for some matrices $a_i$. For $i=1,\ldots,n$, $a_i=\t u_{i-1}-u_{i-1}$ and in order that $G$ define a Darboux transformation we must have $$a_0=0.$$ This condition gives $[G,L](\theta)=0$ i.e. $G(L(\theta))=0$ since $G(\theta)=0$. Hence we only need require that $L(\theta)=\theta C$, for some $x$-independent matrix $C$. Note that if $L(\theta)=0$ then for $\theta'=\theta\exp(\d_t^{-1}(C))$, $L(\theta')=\theta'C$. Also, $G_{\theta'}=G_\theta$ and so we may suppose, without loss of generality, that $C=0$[^3]. Thus we find that $\theta\in
S$.
The second case we consider is $$L=\d_t+\sum_{i=1}^nu_i\d^i\ \ \text{and}\ \ \t L=\d_t+\sum_{i=1}^n\t
u_i\d^i,$$ where $u_i$ and $\t u_i$ are again $N\times N$ matrices. Note that the multiplicative term in $L$ and $\t L$ is omitted.
As in the first case, a gauge operator $G$ must satisfy $$[G,L]G^{-1}=\sum_{i=1}^n\t u_i-u_i.$$ There are now two simple choices. First, let $G=G_\theta^{(1)}=\theta^{-1}$, an (invertible) $N\times N$ matrix. Then $$[G,L]G^{-1}=\sum_{i=0}^n a_i\d^i,$$ and so $a_0=[G,L](\theta)=G(L(\theta))=0$, i.e. $\theta\in S$.
Second, let $G=G^{(2)}_\rho=\rho_x^{-1}\d$, where $\rho_x$ is an invertible $N\times N$ matrix. Now $$[G,L]G^{-1}=[G,L]\d^{-1}\rho_x=\sum_{i=1}^{n+1} a_i\d^{i-1}\rho_x,$$ and we must have $a_1=0$, i.e. $[G,L](\d^{-1}(\rho_x))=G(L(\rho))=0$. Thus $L(\rho)=C$, an $x$-independent matrix. Again, we may take $C=0$ without loss of generality and so $\rho\in S$.
As in the scalar case, it is the composition of the two gauge transformations which is of most interest, and we take $G_\theta=G^{(2)}_{G^{(1)}_\theta(1)}G^{(1)}_\theta=(\theta^{-1})_x^{-1}
\d\theta^{-1}$.
Binary transformations and reductions
=====================================
To determine the binary transformations $G_{\theta,\phi}$ corresponding to the two Darboux transformations found above we must determine two additional things: the mapping $i\colon \hat S\to\t S^\ad$ and then the element $\h\theta\in\h
S$ in terms of $\theta$ and $\phi$.
First consider $L=\d_t+\sum_{i=0}^nu_i\d^i$, $G_\theta=\theta\d\theta^{-1}$. Here the condition $G_\theta^\ad(i(\theta))=-\theta^{\ad^{-1}}\d(\theta^\ad i(\theta))=0$ is satisfied by the choice $i(\theta)=\theta^{\ad^{-1}}$. Further, $$\begin{aligned}
\h\theta&=&\left(G_\theta^{\ad^{-1}}(\phi)\right)^{\ad^{-1}}\\
&=&-\left(\theta^{\ad^{-1}}\d^{-1}(\theta^\ad\phi)\right)^{\ad^{-1}}\\
&=&-\theta\Omega^{-1},\end{aligned}$$ where $\Omega=\d^{-1}(\phi^\ad\theta)$. It may be shown that for all operators $L=\sum_{i=0}^nu_i\d^i$, $\Omega$ is exact in the sense that $d\Omega=\phi^\ad\theta
dx+A(u_1,\ldots,u_n,\theta,\phi)dt$[@Oevel].
In this case the binary transformation is $$\begin{aligned}
G_{\theta,\phi}=G_{\hat\theta}^{-1}G_\theta^{\phantom{-1}}
&=&\theta\Omega^{-1}\d^{-1}\Omega\d\theta^{-1}\\
&=&\theta\Omega^{-1}\d^{-1}(\d\Omega-\Omega_x)\theta^{-1}\\
&=&1-\theta\Omega^{-1}\d^{-1}\phi^\ad.\end{aligned}$$ For discussion of the reduction we will also need $$G^{\ad^{-1}}=1-\phi\Omega^{\ad^{-1}}\d^{-1}\theta^\ad.$$
Now suppose that $L$ satisfies the constraint $L^\ad R^\ad=RL$ where $R=A$, a (not necessarily constant) matrix. Then we may choose $\phi=R^\ad(\theta)=A^\ad\theta$. The condition $RG_{\theta,\phi}=G_{\theta,\phi}^{\ad^{-1}}R$ now gives $$\begin{aligned}
A-A\theta\Omega^{-1}\d^{-1}\phi^\ad=A-\phi\Omega^{\ad^{-1}}\d^{-1}\theta^\ad A
&\iff&A\theta\Omega^{-1}\d^{-1}\theta^\ad
A=A^\ad\theta\Omega^{\ad^{-1}}\d^{-1}\theta^\ad A\\
&\iff&A^\ad=\pm A.\end{aligned}$$
This establishes the following theorem.
\[0\] Let the matrix operator $L=\sum_{i=0}^n u_i\d^i$ satisfy the constraint $$L^\ad A=AL,$$ where $A$ is an Hermitian or skew-Hermitian matrix. Then the binary transformation $$G=1-\theta\Omega^{-1}\d^{-1}\theta^\ad A$$ where $\Omega=\d^{-1}(\theta^\ad A\theta)$, preserves the above constraint, i.e. $\h L=GLG^{-1}$ satisfies $\h L^\ad A=A\h L$.
For the second case, $L=\sum_{i=1}^n u_i\d^i$, $G_\theta=(\theta^{-1})_x^{-1}\d\theta^{-1}$ and hence $i(\theta)=(\theta^{\ad^{-1}})_x$.
To determine the binary transformation it is notationally convenient to write an element of $S^\ad$ as $\phi_x$ rather than $\phi$ as we did above. Also, it is necessary to introduce two integrals $$\Omega=\d^{-1}(\phi^\ad\theta_x)\ \ \mbox{and}\ \
\Omega'=\d^{-1}(\phi_x^\ad\theta),$$ where $$\Omega+\Omega'=\phi^\ad\theta.$$ Now $$\begin{aligned}
i(\h\theta)=(\h\theta^{\ad^{-1}})_x
&=&G_\theta^{\ad^{-1}}(\phi_x)\\
&=&-(\theta^{\ad^{-1}})_x\d^{-1}(\theta^\ad\phi_x)\end{aligned}$$ and so $$(\h\theta^{-1})_x=-\d^{-1}(\phi^\ad_x\theta)(\theta^{-1})_x=
-\Omega'(\theta^{-1})_x.$$ Integrating by parts and taking inverses, we get $$\begin{aligned}
\h\theta&=&\left(-\Omega'\theta^{-1}+\d^{-1}(\Omega'_x\theta^{-1})\right)^{-1}\\
&=&\left(\phi^\ad-\Omega'\theta^{-1}\right)^{-1}\\
&=&\theta\Omega^{-1}.\end{aligned}$$
We may now obtain $$\begin{aligned}
G_{\theta,\phi_x}
&=&\h\theta\d^{-1}(\h\theta^{-1})_x(\theta^{-1})_x^{-1}\d\theta^{-1}\\
&=&-\theta\Omega^{-1}\d^{-1}\Omega'\d\theta^{-1}\\
&=&1-\theta\Omega^{-1}\d^{-1}\phi^\ad\d,\end{aligned}$$ and in a similar way $$G_{\theta,\phi_x}^{\ad^{-1}}=1-\d\phi\Omega'{}^{\ad^{-1}}\d^{-1}\theta^\ad.$$
Suppose that $L$ satisfies the constraint $L^\ad R^\ad=RL$ where $R=A\d$, $A$ a matrix, and choose $\phi_x=R^\ad(\theta)=-(A^\ad\theta)_x$, i.e. $\phi=-A^\ad\theta$. The condition $RG_{\theta,\phi_x}=G_{\theta,\phi_x}^{\ad^{-1}}R$ is $$A\d-A\d\theta\Omega^{-1}\d^{-1}\theta^\ad A\d=A\d-
\d A^\ad\theta\Omega'{}^{\ad^{-1}}\d^{-1}\theta^\ad A\d
\iff A^\ad=\pm A\ \mbox{and}\ A_x=0.$$ With these conditions on $A$, $\Omega=\pm\Omega'{}^\ad$.
This establishes a second theorem.
\[1\] Let the matrix operator $L=\sum_{i=1}^n u_i\d^i$ satisfy the constraint $$L^\ad A\d+A\d L=0,$$ where $A$ is an $x$-independent Hermitian or skew-Hermitian matrix. Then the binary transformation $$G=1-\theta\Omega^{-1}\d^{-1}\theta^\ad A\d$$ where[^4] $\Omega=\d^{-1}(\theta^\ad
A\theta_x)$, preserves the above constraint, i.e. $\h L=GLG^{-1}$ satisfies $\h L^\ad A\d+A\d\h L=0$.
Examples
========
Davey-Stewartson I
------------------
This system has Lax pair $$L=\d_y+\alpha\d+Q,\ \ M=i\d_t+\alpha\d^2+\frac12(Q\d+\d Q+\alpha Q_y)+D,$$ where $Q=\left(\begin{array}{cc}0&u\\\epsilon\bar u&0\end{array}\right)$ and $D=\left(\begin{array}{cc}U&0\\0&V\end{array}\right)$ is real.
Let $A=\left(\begin{array}{cc}1&0\\0&-\epsilon\end{array}\right)$. Then $$L^\ad(iA)^\ad=(iA)L,\ \ M^\ad A^\ad=AL.$$ Hence we may use Theorem \[0\] (with $A=I$) to obtain a binary transformation. This transformation has been used to obtain a wide class of solutions including dromions[@Nimmo1].
Sawada-Kotera equation
----------------------
The equation is a reduction of the BKP equation and so has Lax pair admitting the BKP reduction: $$L=(\d^2+3u)\d,\ \ M=\d_t+(9\d^5-15\d
u\d+30(\d^2u+u\d^2)+15u^2)\d,$$ where $$L^\ad\d+\d L=M^\ad\d+\d M=0.$$ We may use Theorem \[1\] (with $A=I$) to obtain the binary transformation. Note that this is given by $G=\theta^{-1}\d^{-1}\theta^2\d\theta^{-1}$ and coincides the with the well-known “Darboux” transformation.
Modified Novikov-Veselov equation
---------------------------------
This system belongs to the two component BKP hierarchy and has a Lax pair $$L=\d_y+S\d,\ \ M=\d_t+(S\d^2+T\d+\d T+U)\d,$$ where $S=\left(\begin{array}{cc}\cos u&\sin u\\\sin u&-\cos
u\end{array}\right)$, and $T$ and $U$ are give skew-symmetric and symmetric real matrices respectively. Again $$L^\ad\d+\d L=M^\ad\d+\d M=0$$ and Theorem \[1\] gives the binary transformation.
Conclusions
===========
We have discussed the general construction of binary transformations from Darboux transformations. In particular we have carried this out for two classes of operators. More importantly, we have shown that the well studied reductions of these classes (the multi-component BKP and CKP reductions) are among those that the binary transformations preserve.
Iteration of the binary transformation is, of course, possible and leads to closed-form expressions for solutions—of the linear problems and for the integrable systems that are their compatibility conditions—in terms of (multi-component) Grammian determinants. In the case of the reduction described in Theorem \[1\], one may see how these Grammians are transformed into Pfaffians by the reduction process. These features will be discussed in more details elsewhere[@Nimmo4].
Acknowledgements
================
I wish to thank Ralph Willox and Walter Oevel for useful discussions relating to this work and the Royal Society of London for financial support enabling my attendance at this workshop.
[1]{}
[^1]: Here and below $\d=\d/\d x$ and $\d_y=\d/\d y$ and so on.
[^2]: It is tempting to look for a constraint of the form $L^\ad S=RL$ but this in fact corresponds to two constraints since on taking adjoints $L^\ad R^\ad=S^\ad L$.
[^3]: Note that if $L$ is an ordinary differential operator, then taking $C\ne0$ is a genuine generalization. For example, this is exploited in the classical “discrete eigenvalue adding” Darboux transformation for the time-independent Schrödinger operator.
[^4]: For a better notation we have replaced $\Omega$ with $-\Omega$ in the statement of the theorem
|
---
abstract: 'We show that a $\IP$-object and simple configurations of $\IP$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.'
author:
- Andreas Hochenegger
- Andreas Krug
title: 'Formality of $\IP$-objects'
---
[^1] [^2]
Introduction {#introduction .unnumbered}
============
In recent decades, triangulated categories have become very popular in representation theory and algebraic geometry. Given an object $E$ in a $\kk$-linear triangulated category $\cT$, one can consider the triangulated subcategory generated by $E$ inside $\cT$. Its complexity depends strongly on the graded endomorphism algebra $\End^*(E) = \bigoplus_{i\in\IZ} \Hom(E,E[i])$.
For example, let $E\in \cT$ be an exceptional object, that is, $\End^*(E)=\kk$. In this case, $E$ generates a category equivalent to the derived category of vector spaces, which can be regarded as the smallest and simplest $\kk$-linear triangulated category.
In general, due to a result by Keller [@Keller-derivingdg], the generated category $\genby E$ can always be identified with the derived category $\D(B)$ of some differential graded (dg) algebra $B$ whose graded cohomology algebra coincides with the graded endomorphism algebra: $\Ho^*(B)\cong \End^*(E)$. (Depending on the exact definition of the category generated by one object, we may have to replace the derived category $\D(B)$ by its subcategory of compact objects, but we will ignore this issue in this introduction and return to it in .)
Of course, the situation is most pleasant if we already have $$\begin{aligned}
\tag{$\ast$}
\label{eq:goodsituation}
\genby E = \D(\End^*(E))\end{aligned}$$ so that the generated category only depends on the graded endomorphism algebra but not on the ambient category $\cT$. In this paper, we provide two situations in which holds: For $E$ the direct sum of $\IP$-objects that form a tree (in particular, if $E$ is a single $\IP$-object) and for $E=\reg_X$ the structure sheaf of a smooth projective variety.
It follows from Keller’s result that a sufficient condition for to hold is that the graded algebra $A \coloneqq \End^*(E)$ is *intrinsically formal*. This means that every dg-algebra $B$ with $\Ho^*(B)=A$ is actually quasi-isomorphic to $A$. A very useful sufficient criterion for intrinsic formality in terms of vanishing of Hochschild cohomology was given by Kadeishvili [@Kadeishvili]. This was used by Seidel and Thomas [@Seidel-Thomas] to prove intrinsic formality of the endomorphism algebra of $A_n$-configurations of spherical objects. The endomorphism algebra of a single spherical object is of the form $\End^*(E)=\kk\oplus \kk[-d]$ for some $d\in \IZ$. So, spherical objects are arguably the second simplest type of objects in triangulated categories after exceptional objects. Besides this, the main reason for interest in spherical objects is the fact that they induce autoequivalences, so-called *spherical twists*, of triangulated categories. We also want to mention that Keller, Yang and Zhou studied the Hall algebra of a triangulated category generated by a single spherical object in [@Keller-Yang-Zhou].
The notion of spherical objects was generalised by Huybrechts and Thomas [@Huybrechts-Thomas] to that of $\IP$-objects. These objects again induce twist autoequivalences. Furthermore, they play an important role in the theory of hyperkähler manifolds, but appear also in symplectic geometry; see, for example, [@Mak-Wu]. The graded endomorphism algebra of a $\IP$-object is still rather simple: namely it is generated by one element. More precisely, for $n,k$ positive integers, an object $P\in \cT$ is called a *$\IP^n[k]$-like object* if $$\End^*(P)=\kk[t]/t^{n+1} \quad\text{with}\quad \deg(t)=k\,.$$ Such an object is called a *$\IP^n[k]$-object* (or just *$\IP$-object*) if it is additionally a Calabi–Yau object; see for details.
\[main:pn-single\] Let $P$ be a $\IP^n[k]$-like object with $k\ge 2$. Then $\End^*(P)$ is intrinsically formal so that $\genby{P} \cong \D(\End^*(P))$ is independent of the ambient triangulated category.
One application is that the associated $\IP$-twist can be written as the twist along a spherical functor $F\colon \D(\kk[t])\to \cT$; see . This is actually a result due to Segal [@Segal Prop. 4.2]; we provide that the formality assumption there is always given.
A *tree* of $\IP^n[k]$-objects in a triangulated category $\cT$ is given by a collection of $\IP^n[k]$-objects $P_i\in \cT$, one for every vertex of a connected graph without loops, such that $\dim_{\kk}\Hom^*(P_i, P_j)=1$ if $i$ and $j$ are adjacent in the graph and $\Hom^*(P_i, P_j)=0$ otherwise.
\[main:pn-many\] Let $\{P_1,\ldots,P_m\}$ be a tree of $\IP^n[k]$-objects with either $n$ even and $k \geq 2$ or $n=1$ and $k \geq 4$ (the spherical case). Then $\genby{P_1,\ldots,P_m} \cong \D(\End^*(\bigoplus_i P_i))$ is independent of the ambient triangulated category.
Our proof uses Kadeishvili’s criterion for intrinsic formality together with a description of minimal resolutions of graded algebras due to Butler and King [@Butler-King].
might be useful in order to prove a faithfulness result for actions induced by $A_m$-configurations of $\IP$-objects; see for some more explanation on this.
Let $X$ be a smooth projective variety over $\IC$. A distinguished object in its bounded derived category of coherent sheaves $\Db(\Coh(X))$ is given by the structure sheaf $\reg_X$. We use the formality of the Dolbeault complex to prove the following result.
\[main:structuresheaf\] The category generated by $\reg_X$ in $\Db(\Coh(X))$ only depends on the graded algebra $\End^*(\reg_X)\cong \Ho^*(\reg_X)$. More precisely, $$\genby{\reg_X}\cong \D(\Ho^*(\reg_X))\,.$$
This result may be of interest for the conjecture that the graded algebra $\Ho^*(\reg_X)$ is a derived invariant of smooth projective varieties.
This paper is organised as follows. In we fix the notation and collect well-known facts on triangulated categories and dg-algebras. In we recall the definition of $\IP$-objects and their associated twist. Then, in , we prove that $\genby P \cong \D(\End^*(P))$ for a $\IP$-object $P$. We review the description of the terms of minimal resolutions of graded algebras due to Butler and King [@Butler-King], in the following . We go through the main steps of its proof to make sure that the results hold in our graded setting. In , we prove using the description of the terms of the minimal resolutions in order to obtain the vanishing of the relevant Hochschild cohomology. Actually our results on configurations of $\IP^n[k]$-like objects are more general than stated above; see and \[prop:spherical-moregeneral\]. We prove in , actually for compact complex manifolds satisfying the $\partial\bar\partial$-Lemma. In the final we give a general construction which produces trees of $\IP$-objects out of trees of spherical objects. As a geometric application, we explicitly construct trees of $\IP$-objects on Hilbert schemes of points on surfaces; see .
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Daniel Huybrechts for asking about the formality of $\IP$-objects and for helpful comments. Moreover, we are grateful for comments and suggestions from Ben Anthes, Elena Martinengo, Sönke Rollenske, Theo Raedschelders, Paolo Stellari, Greg Stevenson, Olaf Schnürer and an anonymous referee. Finally, we mention that Gufang Zhao had already obtained a partial result on the formality of a single $\IP^n$-object with $n \leq 4$.
Triangulated categories, dg-algebras and Hochschild cohomology {#sec:preliminaries}
==============================================================
Conventions on algebras
-----------------------
The letter $\kk$ will denote an algebraically closed field. All our algebras $A$ will be $\kk$-algebras, and whenever we speak of a graded algebra we mean a graded $\kk$-algebra $A = \bigoplus_{i\in \IZ} A^i$.
For a (graded) $\kk$-algebra $A$, we denote by the (graded) tensor product $A^e \coloneqq A \otimes_\kk A^{op}$ its *enveloping algebra*. Whenever we speak in the following about ideals or modules over some (not necessarily commutative) algebra, we refer to finitely generated left ideals or modules. The formalism of enveloping algebras allows us to speak of $A\hh A$-bimodules as left $A^e$-modules.
Conventions on triangulated categories {#sec:conv-tricat}
--------------------------------------
All triangulated categories are assumed to be $\kk$-linear, and subcategories thereof to be triangulated and full. The shift functor will be denoted by $[1]$. All triangles are meant to be distinguished and denoted by $A\to B\to C$, hiding the morphism $C \to A[1]$. We write $\Hom^*(A,B) = \bigoplus_{i\in\IZ} \Hom(A,B[i])[-i]$ for the derived homomorphisms in a triangulated category: this is a complex equipped with the zero differential. In contrast, $\Hom^\bullet(A,B)$ will be the homomorphism complex if $A$ and $B$ are objects of a dg-category, usually an enhancement of a triangulated category. In the literature this is sometimes also denoted by $\RHom(A,B)$.
All functors between triangulated categories are meant to be exact. In particular, we will abuse notation and write $\otimes$ for the derived functor $\otimes^L$, using the same symbol as for the functor between abelian categories.
dg-algebras and Hochschild cohomology {#sec:hochschild}
-------------------------------------
An *dg-algebra* $A$ (over $\kk$) consists of a graded $\kk$-vector space $$A = \bigoplus_{i\in \IZ} A^i$$ and graded $\kk$-linear maps
- $d \colon A \to A$ of degree $1$ and
- $m \colon A \otimes A \to A$ of degree $0$
satisfying the following compatibilities:
- $d^2=0$, so $d$ is a *differential*;
- $m(m\otimes\id) = m(\id\otimes m)$, so $m$ is an associative multiplication;
- $d(m(a,b)) = m(da,b) + (-1)^{\deg a} m(a,db)$ for homogeneous elements (the Leibniz rule).
A graded algebra is a dg-algebra with $d=0$ and $m(a,b) = a\cdot b$ its multiplication. Let $A$ be a dg-algebra. Then $m$ induces a multiplication on the cohomology $\Ho(A):=\oplus_{i\in \IZ}A^i$ of $A$ with respect to $d$. So $\Ho(A)$ is a graded algebra.
A morphism $\phi\colon A \to B$ of dg-algebras is a $\kk$-linear map compatible with differential and multiplication, that is, $\phi(d_Aa) = d_B(\phi(a))$ and $\phi(m_A(a,b)) = m_B(\phi(a),\phi(b))$. Note that $\phi$ induces a map on cohomology $\Ho(\phi)\colon \Ho(A) \to \Ho(B)$.
\[def:quasi-isomorphism\] Let $\phi\colon A \to B$ be a morphism of dg-algebras. Then $\phi$ is called a *quasi-isomorphism* if $\Ho(\phi)\colon \Ho(A) \to \Ho(B)$ is an isomorphism of graded $\kk$-algebras.
We say that two dg-algebras are *quasi-isomorphic* if they can be connected by a finite zigzag of quasi-isomorphisms.
A graded algebra $A$ is called *intrinsically formal* if any two dg-algebras with cohomology $A$ are quasi-isomorphic; or equivalently, if any dg-algebra $B$ with $\Ho(B)=A$ is already quasi-isomorphic to $A$.
Recall that we denote by $A^e = A \otimes_\kk A\op$ the enveloping algebra of $A$. Note that $A$ has a natural $A^e$-module structure by multiplication from left and right. Given $q\in \IZ$ and two graded $A^e$-modules $N$ and $M$, we write $\Hom_{A^e}^q(N,M)$ for the $A^e$-module homomorphisms which are homogeneous of degree $q$.
\[def:bar\] The complex $$B^\bullet\colon \cdots \to A^{\otimes (q+2)} \xto{d^{-q}} A^{\otimes (q+1)} \to \cdots \to A^{\otimes 2} \to 0,$$ with $A^{\otimes(q+2)}$ in degree $q$, is called the *Bar resolution* of $A$ as an $A^e$-module, where $d = d^{-q}$ is given by $$d(a_1 \otimes \cdots \otimes a_{q+2}) = \sum_i \pm a_1 \otimes \cdots \otimes a_i \cdot a_{i+1} \otimes \cdots \otimes a_{q+2}.$$ Note that the differentials are of degree zero.
Let $A$ be a graded algebra and $M$ a graded $A^e$-module. The *Hochschild cohomology* of $A$ with values in $M$ is given by $$\HH^{p,q}(A,M) \coloneqq \Ho^p( \Hom_{A^e}^q(B^\bullet,M)) \quad\text{for $p,q\in \IZ$.}$$
Our main tool for proving intrinsic formality of certain graded algebras will be the following result due to Kadeishvili.
\[prop:intrinsically-formal\] Let $A$ be a graded algebra. If $\HH^{q,2-q}(A,A)$ vanishes for $q>2$, then $A$ is intrinsically formal.
For an $A^e$-module $M = \bigoplus M^q$, its *shift in degree* by $i$ is the $A^e$-module $M\ds{i}$ with $M\ds{i}^q = M^{q+i}$.
\[rem:HHres\] Note that $\Hom_{A^e}^q(A,A)=\Hom_{A^e}^0(A,A(q))$. Moreover, graded $A^e$-modules, with $\Hom_{A^e}^0$ as morphisms, form an abelian category $\textsf{gr}A^e$. There, the Bar resolution is a projective resolution of the $A^e$-module $A$. It follows that $\HH^{p,q}(A,M) = \Ext^p_{\textsf{gr}A^e}(A,M(q))$. Hence, for any other projective resolution $P^\bullet$ of $A$ as a graded $A^e$-module, we also have $$\HH^{p,q}(A,M)=\Ho^p\bigl(\Hom_{A^e}^q(P^\bullet, M)\bigr)= \Ho^p\bigl(\Hom_{A^e}^0(P^\bullet, M(q))\bigr).$$
Derived categories and dg-categories
------------------------------------
In this paper we will encounter two types of derived categories. First, for an abelian category $\cA$ there is the category $\D(\cA)$ which is the localisation of the homotopy category of complexes with values in $\cA$ at the class of quasi-isomorphisms. In our examples, the abelian category $\cA$ will be a category of (coherent or quasi-coherent) $\reg_X$-modules over a variety or manifold $X$. For details on $\D(\cA)$ see, for example, [@Huybrechts Ch. 2].
Let us very quickly recall some facts and fix notation concerning dg-categories and enhancements; see, for example, [@Kuz-Lunts §3] for details. A *dg-category* is a $\kk$-linear category $\cE$ whose Hom-spaces are dg-$\kk$-modules and the compositions are compatible with the dg-structure. The *homotopy category* $\Ho^0(\cE)$ is defined to have the same objects as $\cE$ and morphisms $\Hom_{\Ho^0(\cE)}(E, F):=\Ho^0(\Hom_{\cE}(E,F))$. The category $\Dgmod(\cE)$ of (right) dg-modules over $\cE$ is defined as the category of dg-functors from $\cE\op$ to the category of dg-modules over $\kk$. Its homotopy category $\Ho^0(\Dgmod(\cE))$ carries the structure of a triangulated category. The dg-category $\cE$ is called *pretriangulated* if the image of the Yoneda embedding $\Ho^0(\cE)\hookrightarrow \Ho^0(\Dgmod(\cE))$ is a triangulated subcategory.
Given a triangulated category $\cT$, a *dg-enhancement* of $\cT$ is a pretriangulated dg-category $\cE$ together with an exact equivalence $\Phi\colon \Ho^0(\EE) \isom \cT$.
We can consider every dg-algebra $A$ as a dg-category with one object. Then the homotopy category $\Ho^0(\Dgmod(\cE))$ of dg-modules over that category agrees with the usual notion of the category of dg-modules over the algebra $A$. The derived category of the a dg-algebra $A$ is defined as $\D(A):=\Ho^0(\Dgmod(A))[\mathsf{qis}^{-1}]$, the category $\Dgmod(A)$ of right dg-modules over $A$ localised at the class of quasi-isomorphisms; for details see, for example, [@Keller-tilting §8]. If the dg-algebra $A$ is concentrated in degree 0 (i.e. it is an ordinary algebra), then $\D(A)=\D(\Mod(A))$ where the latter is the derived category of the abelian category $\Mod(A)$ in the sense explained above.
Formality and triangulated categories {#sec:formality-triangulated}
-------------------------------------
The following results are well known to experts and can be found in essence or in part in, for example, the survey [@Keller-dg] by Keller, the lecture notes [@Toen] by Toën or the book [@Bernstein-Lunts §10] by Bernstein and Lunts.
We recall terminology. Let $\cT$ be a triangulated category. The category $\cT$ is called *cocomplete* if arbitrary direct summands exist. It is called *idempotent complete* if every idempotent endomorphism splits. An object $T\in \cT$ is called *compact* if for every set $\{Y_i\}$ of objects in $\cT$ the natural morphism $$\medoplus{i} \Hom(T, Y_i)\to \Hom(T, \medoplus{i} Y_i)$$ is an isomorphism. We write $\cT^c\subset \cT$ for the full subcategory of compact objects. It is a *thick* (i.e. closed under direct summands) triangulated subcategory of $\cT$.
For an object $T\in \cT$, we write $\Genby{T}$ for the smallest cocomplete triangulated subcategory of $\cT$ containing $T$. We write $\genby{T}$ for the smallest thick triangulated subcategory of $\cT$ containing $T$. If $T$ is compact, then $\genby{T}\subset \cT^c$. There is an inclusion $\genby{T} \subset \Genby{T}$, since every cocomplete triangulated category is thick; see [@Neeman-book Prop. 1.6.8].
The compact objects in the derived category $\D(\QCoh(X))$ of coherent sheaves on a separated scheme of finite type over $\kk$ coincide with the *perfect* objects, that is, objects locally quasi-isomorphic to bounded complexes of locally free sheaves of finite rank.
The following statement can be found in a similar form in [@Keller-derivingdg §4.2] or [@Lunts-Orlov Prop. 1.16 & 1.17].
\[thm:generatedcat\] Let $\cT$ be a triangulated category with a dg-enhancement given by a dg-category $\EE$ and an equivalence $\Phi\colon \Ho^0(\EE) \isom \cT$. Let $T\in \cT$ be a compact object, $E\in \EE$ some object with $\Phi(E)\cong T$, and consider the dg-algebra $B=\Hom^\bullet(E,E)$.
1. \[thm:i\] Let $\cT$ be idempotent complete. Then there is an exact equivalence $\genby{T}\cong \D(B)^c$.
2. Let $\cT$ be cocomplete. Then there are exact equivalences $$\Genby{T} \cong \D(B)\quad \text{and} \quad \genby{T} \cong \D(B)^c\,.$$
This follows from [@Lunts-Schn Prop. B.1] by plugging in $P=I=E$ and $z=\id_E$.
The first part of the theorem applies to any bounded derived category $\Db(\cA)$ of an abelian category $\cA$. On the one hand, $\Db(\cA)$ admits a (not necessarily unique) dg-enhancement. On the other hand, $\Db(\cA)$ is automatically idempotent complete, which holds even if $\cA$ is just an idempotent complete exact category by [@Balmer-Schlichting Thm. 2.8].
\[cor:intformal\] Let $\cT$ be a cocomplete dg-enhanced triangulated category and $T\in \cT^c$ a compact object. If the graded algebra $A = \End^*(T)$ is intrinsically formal, then $$\Genby{T}\cong \D(A) \quad\text{and}\quad \genby{T}\cong \D(A)^c\,.$$
Let $E \in \EE$ be some object with $\Phi(E)=T$ and $B = \Hom^\bullet(E,E)$. Then $\Genby{T} \cong \D(B)$ by . As by assumption $A$ is intrinsically formal, $\D(B) \cong \D(A) = \D(\Mod(A))$ by [@Bernstein-Lunts Thm. 10.12.5.1]. Restricting to compact objects, this yields that $\genby{T} = \D(B)^c = \D(\Mod(A))^c$.
Let $A$ be an intrinsically formal graded algebra. By the corollary above, the categories generated by objects $T\in \cT$ with $\End^*(D)=A$ are all equivalent. In particular, the category generated by such an object is independent of the ambient cocomplete dg-enhanced triangulated category $\cT$.
$\IP$-objects {#sec:P}
=============
Definition and basic examples
-----------------------------
\[def:P\] Let $P$ be an object in a $\kk$-linear triangulated category $\cT$.
- If $\End^*(P) \cong \kk[t]/t^{n+1}$ as graded $\kk$-algebras with $\deg(t)=k$, then we call $P$ a *$\IP^n[k]$-like object*.
- If a $\IP^n[k]$-like object $P$ is also $nk$-Calabi–Yau object, i.e. $\Hom^*(P,\blank) = \Hom^*(\blank,P[nk])^\vee$ functorially, then $P$ is called a *$\IP^n[k]$-object*.
In some cases we will omit the integers $n,k$ and just speak of $\IP$-like objects or $\IP$-objects.
$\IP^1[k]$-objects are well-known as *spherical objects*; see [@Seidel-Thomas]. Without the Calabi–Yau property they are called *spherelike objects* and studied in [@Hochenegger-Kalck-Ploog; @Hochenegger-Kalck-Ploog-RT] by Kalck, Ploog and the first author.
$\IP^n[2]$-objects are known as *$\IP^n$-objects* and studied in [@Huybrechts-Thomas] by Huybrechts and Thomas. The focus there is on hyperkähler manifolds, whose structure sheaves are $\IP^n$-objects. Another standard example of a $\IP^n$-object is the structure sheaf $\reg_Z\in \D(X)$ of the centre $\reg_{\IP^n}\cong Z\subset X$ of a Mukai flop of a variety of dimension $\dim X=2n$.
The terminology $\IP^n[k]$ was introduced by the second author in [@Krug], where examples of varieties are also given, whose structure sheaves are $\IP^n[k]$-objects.
A $\IP$-like object $P$ is already a $\IP$-object in $\genby{P}$. Therefore our main question about the independence of $\genby{P}$ of the ambient category does not rely on the Calabi–Yau property. As a (possibly misleading) consequence, in [@Keller-Yang-Zhou] the Calabi–Yau property of spherical objects is never mentioned.
Let $X$ be a variety of dimension $nk$ such that $\OO_X$ is a $\IP^n[k]$-like object in $\Db(X)$, that is, $\End^*(\OO_X) = \kk[t]/t^{n+1}$ and $\deg(t)=k$. Note that $\End^*(\OO_X) = H^*(\OO_X)$ as graded $\kk$-algebras, where the Yoneda product on the left becomes the cup product on the right. As the cup product is graded commutative, $k$ odd implies immediately $t^2=0$, so $n=1$ and $\OO_X$ is spherelike. Consequently, $n>1$ is only possible for even $k$.
However, the graded endomorphism algebra $\End^*(E)$ of an arbitrary object $E\in \cT$ does not need to be graded commutative. In fact, there are examples of $\IP^n[k]$-like objects with $n\ge 2$ and $k$ odd. For a trivial example, consider the dg-algebra $A = \kk[t]/t^{n+1}$ with trivial differential and $\deg(t)=k$, where $n \ge 0$ and $k$ are integers. Then $A$ is a $\IP^n[k]$-object inside $\D(A)$. For examples of $\IP^n[1]$-objects of geometric origin, see [@Addington Ex. 4.2 (5) & (6)].
Associated $\IP$-twists {#sec:p-twists}
-----------------------
In this subsection we assume that $\cT$ is a $\kk$-linear triangulated category that admits a dg-enhancement and that the $\IP^n[k]$-object $P\in \cT$ is *proper*, that is, $\Hom^*(P,F)$ is a finite-dimensional graded vector space for all $F \in \cT$. Under these assumptions there is an autoequivalence $\PPP_P\colon \cT\isom \cT$, the *$\IP$-twist along $P$*, whose construction, due to [@Huybrechts-Thomas], we sketch in the following. Whenever we speak about the $\IP$-twist associated to a $\IP$-like object in later sections, we will tacitly assume that these assumptions are met.
So, let $P$ be a $\IP^n[k]$-object and $t$ be a non-zero element of $\Ext^k(P,P)$. Using this generator, one can define the upper triangle for any $F \in \cT$: $$\begin{tikzcd}
\Hom^*(P,F)\otimes P[-k] \ar[r, "H", "t\otimes \id - \id \otimes t"'] \ar[dr, "\mathrm{ev} \circ H"'] &[3em] \Hom^*(P,F) \otimes P \ar[r] \ar[d, "\mathrm{ev}"] & \Cone(H) \ar[ld, dashrightarrow] \\
& F \ar[dl, dashrightarrow]\\
\PPP_P(F)
\end{tikzcd}$$ As the composition $\mathrm{ev}\circ H = 0$, the arrow $\Cone(H) \dashrightarrow F$ exists. Completing this arrow to a triangle gives the double cone which we denote by $\PPP_P(F)$.
The dg-enhancement of $\cT$ is necessary to actually define the $\IP$-twist $\PPP_P$ as a functor. In the case of spherical twists see [@Anno-Logvinenko] for a proper treatment and [@Hochenegger-Kalck-Ploog §3.1] for a rough idea. In the geometric setting, Fourier-Mukai kernels allow us to circumvent dg-categories; see [@Huybrechts-Thomas §2], where it is also shown that, in the geometric set-up, the above double cone construction leads to a *unique* autoequivalence $\PPP_P$. The uniqueness of $\PPP_P$ in the general case is proved in [@Anno-Logvinenko-Pn].
Let $P$ be a $\IP$-object. Then the associated $\IP$-twist $\PPP_P$ is an autoequivalence.
\[rem:twist\] In the case of a spherical object, the $\IP^1$-twist associated to it is the square of the spherical twist; see [@Huybrechts-Thomas Prop. 2.9].
Formality of single $\IP$-objects {#sec:pn-single}
---------------------------------
The following proposition is in the introduction.
\[prop:pn-single\] Let $P$ be a $\IP^n[k]$-like object in a cocomplete $\kk$-linear dg-enhanced triangulated category with $n,k$ positive integers. Then there are equivalences $$\Genby P \cong \D(\End^*(P)) \quad \text{and}\quad \genby P \cong \D(\End^*(P))^c\,.$$
By the definition of a $\IP$-like object, $\End^*(P)=\kk[t]/t^{n+1}$ with $\deg t=k$. Hence, the result follows by together with the following lemma.
\[lem:Pformal\] For $n, k$ positive integers, the graded algebra $\kk[t]/t^{n+1}$ with $\deg t=k$ is intrinsically formal.
In order to apply the criterion of for intrinsic formality, we have to show the vanishing of the Hochschild cohomology groups $\HH^{q, 2-q}(A,A)$ for $q>2$.
There is the well-known 2-periodic free resolution $$\cdots \to\uA^e \xto{v} \uA^e \xto{u} \uA^e \xto{v} \uA^e \xto{u} \uA^e \xto{m} \uA \to 0$$ of the underlying non-graded algebra $\uA=\kk[t]/t^{n+1}$ considered as the diagonal bimodule over itself. Here, $m$ is multiplication in $A$, $u$ is multiplication by $t \otimes 1 - 1 \otimes t$, and $v$ is multiplication by $t^n \otimes 1 + t^{n-1} \otimes t + \cdots + 1 \otimes t^n$; see [@Weibel Ex. 9.1.4]. Now one can check easily that this becomes a graded free resolution $$\cdots
\xto{v} A^e\bigl(-(n+2)k\bigr) \xto{u} A^e\bigl(-(n+1)k\bigr) \xto{v} A^e\bigl(-k\bigr) \xto{u} A^e \xto{m} A \to 0\,.$$ So we obtain a graded free resolution $F^\bullet$ of the $A^e$-module $A$ where $$F^q=\begin{cases}
A^e\bigl(-i(n+1)k\bigr) & \text{for $q=2i$ even,}\\
A^e\bigl(-(i(n+1)+1)k\bigr) & \text{for $q=2i+1$ odd.}
\end{cases}$$ By , $\HH^{q, 2-q}(A, A)$ is a subquotient of $\Hom_{A^e}^0(F^q, A(2-q))$ so it is sufficient to show that the latter vanishes for $q>2$. For $q=2i$ even, $$\Hom_{A^e}^0(F^q, A(2-q))=\Hom_{A^e}^0\bigl(A^e, A(2-2i+i(n+1)k)\bigr)=A^{2-2i+i(n+1)k}\,.$$ We have $2-2i+i(n+1)k=2+i(nk+k-2)>nk$ for $i\geq2$. But $A$ is concentrated in degrees between $0$ and $nk$, so we get $$\Hom_{A^e}^0(F^{2i}, A(2-2i))=A^{2-2i+i(n+1)k}=0\,.$$ The verification that $\Hom_{A^e}^0(F^q, A(2-q))=0$ for $q>2$ odd is similar.
\[cor:pntwist-spherical-functor\] Let $P$ be a $\IP^n$-object in $\Db(X)$ and $B = \kk[t]$ where $t$ has degree $2$. Then the functor $F \colon \Db(B) \to \Db(X), B \mapsto P$ is spherical and the spherical twist along $F$ is the $\IP$-twist along $P$.
This is proved in [@Segal Prop. 4.2] under the assumption that $\End^\bullet(P)$ is formal. By , this assumption is always satisfied.
To be precise, Segal’s assumption is that the dg-algebra $\End^\bullet(P)$ is formal as a dg-module over $B$, so we have to show that this is implied by its formality as a dg-algebra. The $B$-module structure is given by choosing an isomorphism $\End^*(P)\cong \kk[s]/s^{n+1}$ and an element $u \in \End^\bullet(P)$ whose cohomology class is mapped to $s$ under this isomorphism. Then $t^l$ acts on $\End^\bullet(P)$ by multiplication by $u^l$. Now, we know that $\End^\bullet(P)$ is formal as a dg-algebra. Hence there is a roof $$\begin{tikzcd}[column sep=small]
& W^\bullet \ar[dl, "f"'] \ar[dr, "g"] \\
\End^\bullet(P) && \kk[s]/s^{n+1}
\end{tikzcd}$$ where $f$ and $g$ are quasi-isomorphisms of dg-algebras and $f$ is surjective. Indeed one can take $f\colon W^\bullet\to \End^\bullet(P)$ to be a cofibrant replacement with respect to the structure of a model category on the category of augmented dg-algebras as described in [@Keller-functorcats §4.2].
Let $v\in W^\bullet$ be a preimage of $u$ under $f$. Then we can equip $W^\bullet$ with the structure of a $B$-algebra by letting $t$ act by $v$ so that $f$ becomes an quasi-isomorphism of $B$-modules. Furthermore, the cohomology class of $v$ is non-zero, hence $g(v)$ is a non-zero multiple of $s$. Therefore, $g$ is a quasi-isomorphism of $B$-modules, too.
Minimal resolutions of graded algebras {#sec:min}
======================================
In this section we describe a minimal resolution for certain graded algebras in terms of a tensor presentation, following Eilenberg [@Eilenberg] and Butler and King [@Butler-King]. We use this for the computation of the Hochschild cohomology which leads to a sufficient condition for these algebras to be intrinsically formal.
Separably augmented algebras and resolutions of diagonal bimodules
------------------------------------------------------------------
We recall that a $\kk$-algebra $R$ is separable if and only if there is an element $p = \sum x_i \otimes y_i \in R^e$ (called *separability idempotent*), such that $ap=pa$ for all $a \in R$ and $\sum x_iy_i=1$ in $R$. For general facts on separable algebras see the textbook by Weibel [@Weibel §9.2].
We denote by $\IN$ the semigroup of non-negative integers. A *separably augmented $\kk$-algebra* is an $\IN$-graded $\kk$-algebra such that $R=A^0$ is a separable $\kk$-algebra.
\[rem:Eilenbergalgebra\] Note that $A^+ = \bigoplus_{i>0} A^i$ is the *homogeneous radical* of $A$, that is, the intersection of all homogeneous maximal ideals of $A$. Indeed, every homogeneous maximal ideal of $A$ is of the form $\fm=\fm_0\oplus A^+$. Hence, every separably augmented algebra satisfies the assumptions of [@Eilenberg §2].
For a separable $\kk$-algebra $R$ and an $R\hh R$-bimodule $V$, we denote by $T(V) = R \oplus V \oplus (V \otimes_R V) \oplus \ldots$ its (free) tensor algebra over $R$. Moreover let $J \subset T(V)$ be the two-sided ideal generated by $V$. If $V$ carries an $\IN_+$-grading, where $\IN_+$ denotes the positive integers, the tensor algebra inherits a canonical $\IN$-grading given by $$\deg(v_1\otimes v_2\otimes \dots\otimes v_n)=\deg(v_1)+\deg(v_2)+\dots+\deg(v_n)\,.$$ Then $T(V)^0=R$, so that $T(V)$ is a separably augmented algebra with $T(V)^+=J$. Conversely, every separably augmented algebra with $A^0=R$ has a graded surjection $T(V)\onto A$ for some $R\hh R$-bimodule $V$, for example $V=A^+$. Given a separably augmented algebra $A$ and a graded surjection $T(V)\onto A$ with kernel $I$, we call the induced isomorphism $A \cong T(V)/I$ a *tensor presentation* of $A$.
Replacing $V$ by $V/(V\cap I)$ if necessary, we may always choose a presentation such that $I \subset J^2$. In the following, we will also assume that the inclusion $J^n \subset I$ holds for some $n\geq 2$. This is automatic if $A$ is of finite dimension over $\kk$, as it will be in the applications.
\[prop:minimal-resolution\] Let $A$ be a separably augmented algebra with $A^0=R$. A minimal resolution $P^\bullet$ of $A$ as a graded $A^e$-module has terms $P^m = A \otimes_R \Tor^A_m(R,R) \otimes_R A$.
Moreover, suppose that $A \cong T(V)/I$ is a tensor presentation with $J^n\subset I\subset J^2$ for some $n\ge 2$. Then there are isomorphisms of graded $R$-algebras $$\Tor^A_{2p}(R,R) = \frac{I^p \cap J I^{p-1} J}{J I^p + I^p J}
\quad \text{and} \quad
\Tor^A_{2p+1}(R,R) = \frac{J I^p \cap I^p J}{I^{p+1} + J I^p J}.$$ where the grading on the left-hand side is induced by the grading on $A$ and the one on the right-hand side is induced by the grading on $T(V)$.
This follows by setting $L=T(V)^e$ in [@Butler-King Prop. 2.4]. Unfortunately, Butler and King assume that the grading of $A$ is induced by the natural grading of $T(V)$ (i.e. the elements of $V$ have degree $1$), which will never be the case in our applications.
However, one can check that every step of the proof of [@Butler-King Prop. 2.4] works in our general graded set-up. Indeed, the proof of the equality $P^m = A \otimes_R \Tor^A_m(R,R) \otimes_R A$ mainly refers to Eilenberg [@Eilenberg], who works in the general graded setting of separably augmented algebras throughout; compare . The arguments in [@Butler-King] needed for this equality can all be turned into arguments that also work in our graded set-up using the fact that an object in the category of graded $A$-modules is projective if and only if the underlying non-graded $\underline A$-module is projective; see, for example, [@Eilenberg §1].
The computation of $\Tor^A_m(R,R)$ in terms of the ideals $I$ and $J$ is done using the projective resolution $$\cdots \to \frac{JI^n}{JI^{n+1}} \to \frac{I^n}{I^{n+1}} \to \frac{JI^{n-1}}{JI^n} \to \cdots \to \frac{JI}{I} \to A \to R \to 0\,;$$ see also [@Bongartz] for details. Its differentials are induced by the inclusions of the homogeneous ideals $I$ and $J$, hence are graded homomorphisms.
\[rem:minimal-resolution-simplification\] Let $P^\bullet \to A$ be the minimal resolution of $A$ as in . Note that there is a natural isomorphism $$\Hom_{A^e}(P^q, \blank ) = \Hom_{R^e}(\Tor^A_q(R,R),\blank)\,.$$
Degree criterion for intrinsic formality
----------------------------------------
For a graded module $M$ we define the *maximal degree* of $M$ as $$\maxdeg(M) \coloneqq \max\{\deg(m) \mid \text{non-zero homogeneous } m \in M \}$$ and analogously the *minimal degree* $\mindeg(M)$.
\[prop:max-min-degree\] Let $A$ be a separably augmented algebra and let $q\in \IN$. If $$\maxdeg(A)+q-2 < \mindeg(\Tor_q^A(R,R))$$ then the Hochschild cohomology $\HH^{q,2-q}(A,A)$ vanishes. In particular, if this inequality holds for all $q>2$, then $A$ is intrinsically formal.
By and \[rem:minimal-resolution-simplification\], the Hochschild cohomology $\HH^{q,2-q}(A,A)$ is a subquotient of $$\Hom_{R^e}( \Tor_q^A(R,R), A\ds{2-q})\,.$$ Hence, it is sufficient to show the vanishing of this $\Hom$-space. But there cannot be any non-zero homomorphism of degree zero, since the minimal degree of the source is smaller than the maximal degree of the target.
Recall that $\HH^{q,2-q}(A,A)=0$ for $q>2$ implies intrinsic formality of $A$ by .
We will use to prove intrinsic formality of a given separably augmented algebra using a suitable tensor representation, namely in the case of endomorphism algebras of configurations of $\IP$-objects.
\[rem:mindeg\] Let $A$ be an $\IN$-graded $\kk$-algebra $P$ a graded $A$-module, and $M,N \subset P$ graded submodules. Then the following rules hold:
- $\mindeg(M+N) = \min \{ \mindeg(M),\mindeg(N) \}$; hence $\mindeg(M) = \min\{ \deg(m_i) \}$ for $M = {}_A\genby{m_1,\ldots,m_l}$ with $m_i$ homogeneous;
- $\mindeg(M\cap N) \geq \max\{ \mindeg(M),\mindeg(N) \}$;
- $\mindeg(P/M) \geq \mindeg(P)$, with equality if $\mindeg(M) > \mindeg(P)$.
If $I,J \subset A$ are ideals then we have additionally:
- $\mindeg(I\cdot J) \ge \mindeg(I) + \mindeg(J)$.
Let $A$ be a separably augmented algebra with tensor representation $A\cong T(V)/I$ and let $J=A^+$ so that $\mindeg(J)=\mindeg(V)$. By and the above rules, we get $$\begin{split}
\mindeg\Tor^A_{2p}(R,R) & \geq \mindeg(I^p \cap J I^{p-1} J) \\ & \geq
\max\{p\mindeg(I), 2\mindeg(J)+(p-1)\mindeg(I)\}\,,\\
\mindeg\Tor^A_{2p+1}(R,R) & \geq \mindeg(J I^p \cap I^p J) \geq p\mindeg(I)+\mindeg(J)\,.
\end{split}$$
Configurations of $\IP$-objects {#sec:pobjects}
===============================
\[def:tree\] Let $\cT$ a triangulated category and let $Q$ be a graph. Our convention for a graph is that we allow at most one edge joining two given vertices $i\neq j$ and no edge from a vertex to itself. A *$Q$-configuration* of objects in $\cT$ is a collection of indecomposable objects $P_i$, one for every vertex $i$ of $Q$, such that, for all $i\neq j$, we have $\Hom^*(P_i,P_j)= \Hom^*(P_j,P_i)=0$ if $i$ and $j$ are not adjacent and $$\dim_{\kk} \Hom^*(P_i,P_j)= \dim_{\kk} \Hom^*(P_j,P_i)=1$$ if $i$ and $j$ are connected by an edge. A *tree* is a graph in the sense above without loops. Given a $Q$-configuration $\{P_i\}$ for a tree $Q$ we also say that the objects $P_i$ *form the tree $Q$*.
\[rem:tensorpres\] Let $n,k$ be positive integers and let $P_1,\ldots,P_m$ be a $Q$-configuration of $\IP^n[k]$-like objects such that $\Hom^{\leq 0}(P_i,P_j)$ is zero for $i\neq j$. Then $A = \bigoplus_j \End^j(\bigoplus_i P_i)$ is a separably augmented algebra, since $R = A^0 = {}_{\kk}\genby{e_1,\ldots,e_m}$ is spanned by the mutually orthogonal idempotents $e_i \coloneqq \id_{P_i}$.
For each $i$, denote $t_i$ a non-zero map in $\Ext^k(P_i,P_i)$, which is unique up to multiplication with a unit. By assumption, for any two $P_i$ and $P_j$ adjacent in $Q$, there is a unique positive degree $h_{ij}$ such that $\Ext^{h_{ij}}(P_i,P_j) = \kk \cdot a_{ij}$. Let $V \subset \End^*(\bigoplus_{i=1}^m P_i)$ be the graded $\kk$-subvector space spanned by all $t_i$ and $a_{ij}$. This gives a graded surjection $T(V) \onto A$, hence a tensor presentation $A = T(V)/I$ by some homogeneous ideal $I$.
Formality of configurations of $\IP$-objects
--------------------------------------------
\[prop:pn-moregeneral\] Let $Q$ be a graph and let $P_1,\ldots,P_m$ be a $Q$-configuration consisting of $\IP^n[k]$-like objects in a $\kk$-linear triangulated category with $n,k$ integers with $n,k\geq2$. Assume that there exists an integer $h$ with $\frac{nk}2\le h\le nk$ and $\gcd(k,h)>1$ such that $$\Hom^*(P_i,P_j) = \kk[-h] \quad \text{for all adjacent $P_i$ and $P_j$.}$$ Then $A = \End^*(\bigoplus_{i=1}^m P_i)$ is intrinsically formal.
We will use the tensor presentation of $A = T(V)/I$ as in together with the notation $e_i = \id_{P_i}$, $\End^k(P_i) = \kk \cdot t_i$ and $\Ext^h(P_i,P_j) = \kk \cdot a_{ij}$ for adjacent $P_i$ and $P_j$. Here $V$ is the graded vector space spanned by all $t_i$ and $a_{ij}$. Recall that $J = T(V)^+$.
Note that the homogeneous elements $1, t_i, \ldots, t_i^n, a_{ij}$ constitute a basis of $A$ as a $\kk$-vector space. Hence, $$\tag{$A$}
\label{eq:maxdegA}
\maxdeg(A) = nk \le 2h\,.$$
The only elements of $I$ not involving an $a_{ij}$ lie in the ideal generated by the elements $t_i^{n+1}$. Indeed $t_i^l\neq 0$ for $l<n+1$ by the definition of a $\IP^n[k]$-object. Furthermore, for $i\neq j$, the tensor product $t_i\otimes t_j$ already vanishes as an element of $V\otimes_R V\subset T(V)$ as $t_it_j = t_i\otimes t_j=(t_i e_i)\otimes t_j=t_i\otimes (e_i t_j)=0$. Hence, the minimal degrees of $I$ and $J$ are $$\begin{aligned}
\mindeg(I) \geq &\ \min\{ \deg(t_i^{n+1}), \deg(a_{ij}a_{jl}), \deg(a_{ij}t_j) \} = h+k\,, \\
\mindeg(J) = &\ \min\{ \deg(t_i), \deg(a_{ij}) \} = k\,,\end{aligned}$$ where the assumption that $n \geq 2$, hence $h\ge k$, is used. Hence, by , we get $$\begin{aligned}
\tag{e}
\label{eq:even}
\mindeg \Tor^A_{2p}(R,R) \geq &\ \max\{ p(h\!+\!k), 2k + (p\!-\!1)(h\!+\!k) \}=p(h+k),\\
\tag{o}
\label{eq:odd}
\mindeg \Tor^A_{2p+1}(R,R) \geq &\ p(h+k)+k\,.\end{aligned}$$
We can now confirm that the assumptions of are satisfied for $q\ge 4$. For $q=2p$ with $p\ge 2$, we have $$\maxdeg(A)+q-2 \overset{\eqref{eq:maxdegA}}{\le} 2h+2p-2\overset{(2\le p)}< ph+2p \overset{(2\le k)}\le ph+pk \overset{\eqref{eq:even}} \leq \mindeg\Tor_{2p}^A(R,R)\,.$$ Similarly, for $q=2p+1$ with $p\ge 2$, $$\maxdeg(A)+q-2 \overset{\eqref{eq:maxdegA}} = 2h+2p-1\overset{(2\le p)} <
ph+2p \underset{(2\le k)} < ph+pk + k \underset{\eqref{eq:odd}} \leq \mindeg\Tor_{2p+1}^A(R,R)\,.$$
Hence, $\HH^{q,2-q}(A,A)$ vanishes for $q\ge 4$. In order to apply Kadeishvili’s criterion for intrinsic formality (see ), all that is left is to show that $\HH^{3,-1}(A,A)=0$. This can be done using the Bar resolution $B^\bullet$; see . Indeed, $A$ is concentrated in degrees divisible by $\gcd(k,\frac{nk}2)>1$. Hence, the same holds for $B^3=A^{\otimes 5}$. Thus, there is no non-trivial degree-zero homomorphism $A^{\otimes 5}\to A(-1)$.
The previous proposition does not cover the interesting case of configurations of spherical objects, which we treat in the following proposition.
\[prop:spherical-moregeneral\] Let $Q$ be a graph and let $P_1,\ldots,P_m$ be a $Q$-configuration consisting of $k$-spherelike objects in a $\kk$-linear triangulated category with $k\geq4$. Moreover, assume that, for adjacent $P_i$ and $P_j$, $$\Hom^*(P_i,P_j) = \kk[-h_{ij}] \quad\text{with } \abrunden{\frac k2}\le h_{ij}\le k \,.$$ Then $A = \End^*(\bigoplus_{i=1}^m P_i)$ is intrinsically formal.
This can be shown along the same lines as the proof of , but some of the estimates change. The minimal degrees of the ideals $I$ and $J$ become $$\begin{aligned}
\mindeg(I) & \geq \min\{ \deg(t_i^{2}), \deg(a_{ij}a_{jl}), \deg(a_{ij}t_l) \} \ge 2 h \,,\\
\mindeg(J) & = \min\{ \deg(t_i), \deg(a_{ij}) \} \ge h\,,\end{aligned}$$ where we abbreviate $h \coloneqq \abrunden{\frac k2}$. Note that $h\geq2$ by the assumption on $k$. Hence, the minimal degrees in the minimal projective resolution are now $$\begin{aligned}
\tag{e}
\label{eq:even3}
\mindeg \Tor^A_{2p}(R,R) \geq &\ \max\{ 2ph, 2h + 2(p-1)h \}=2ph\,,\\
\tag{o}
\label{eq:odd3}
\mindeg \Tor^A_{2p+1}(R,R) \geq &\ 2ph+h\,.\end{aligned}$$ Furthermore, note that $$\tag{$A$}
\label{eq:maxdeg3}
\maxdeg(A) = k \leq 2h+1\,.$$
We will check that the assumptions of are satisfied for $q > 2$, which concludes the proof. Indeed, for $q=2p$ with $p\ge 2$, $$\begin{aligned}
\maxdeg(A)+q-2\overset{\eqref{eq:maxdeg3}} \leq 2h+2p-1 < 2h+2p \leq 2ph \overset{\eqref{eq:even3}} \leq \mindeg\Tor_{2p}^A(R,R)\,.\end{aligned}$$ To see this, we still need $h+p\leq ph$. This inequality is equivalent to $\frac{h}{h-1} \leq p$, as $h\geq 2$, and holds as $p\geq 2$. Similarly, for $q=2p+1$ with $p\ge 1$, we have $$\begin{aligned}
\maxdeg(A)+q-2\overset{\eqref{eq:maxdeg3}} \leq 2h+2p < 2ph+h \overset{\eqref{eq:even}} \leq \mindeg\Tor_{2p+1}^A(R,R)\,.\end{aligned}$$ Here, the middle $2h+2p < 2ph+h$ is equivalent to $\frac{h}{h-1} < 2p$, as $h\geq2$, hence the inequality holds due to $p \geq1$.
\[rem:CYproperty\] If we assume in that the $P_i$ are $\IP^n[k]$-objects (not just $\IP$-like), the assumption $\frac{nk}2\le h\le nk$ already implies $h=\frac{nk}2$; compare the proof of below. In this case, the assumption $\gcd(k,h)>1$ is automatically fulfilled if $n$ is even or $k$ is a multiple of $4$.
Similarly, if we assume the objects in to be spherical, the assumption $\abrunden{\frac k2}\le h_{ij}\le k$ already implies $h\in {\textstyle \left\{\abrunden{\frac k2}, \aufrunden{\frac k2} \right\}}$.
For $k=1$, the assertion of has to be false. To see a counterexample, consider an elliptic curve $E$. Then $\Db(\Coh(E))$ is generated by the $A_2$-sequence of $1$-spherical sheaves $\reg_E$ and $\reg_p$ for any $p\in E$. Indeed, out of these two sheaves one can construct all the line bundles $\reg(n\cdot p)$ by successive cones, and the line bundles $\reg(n\cdot p)$ contain an ample sequence. Now, one can check that the graded endomorphism algebra $\End^*(\reg_E\oplus \reg_p)$ is the same, regardless of the chosen elliptic curve $E$ and point $p\in E$. However, two non-isomorphic elliptic curves $E\not\cong E'$ always have non-equivalent bounded derived categories; see, for example, [@Huybrechts Cor. 5.46].
For $k=2$ and $3$, we still expect intrinsic formality, as in the case of algebras coming from $A_m$-configurations of such objects; see [@Seidel-Thomas].
The following corollary in combination with gives in the introduction.
\[cor:tree\] Let $\{P_i\}$ be a tree of $\IP^n[k]$-objects in a cocomplete dg-enhanced triangulated category $\cT$ with
- either $n,k\ge 2$, $nk$ even, and $\gcd(k,{\textstyle \frac{nk}2})>1$;
- or $n=1$ and $k\ge 4$.
Then the thick subcategory $\genby{\{P_i\}}$ is independent of the ambient category $\cT$.
Replacing the objects $P_i$ by appropriate shifts $P_i[n_i]$, we may assume that they satisfy the assumptions on $\Hom^*(P_i, P_j)$ of and , respectively. For the restrictions on $n$ and $k$ see . Denote by $Q$ the underlying tree. We may start with some edge $i\in Q$ and set $n_{i}=0$. By definition of a $Q$-configuration of objects, for adjacent $i$ and $j$, we have $$\Hom^*(P_i,P_j)=\kk[-a]\,,\ \Hom^*(P_j, P_i)=\kk[-b]$$ for some $a,b\in \IZ$. Note that we assume the objects to be $\IP$-objects (not just $\IP$-like). Hence, Serre duality gives $a+b=nk$. Hence, in the case where $nk$ is even, we may set $n_j=a-h$, so after replacing $P_j$ by $P_j[a-h]$ we get $$\Hom^*(P_i,P_j)=\kk[-h]= \Hom^*(P_j, P_i)\,.$$ Since, by assumption, $Q$ has no loops, there is no obstruction to extending this procedure to the whole of $Q$. The case where $n=1$ and $k\ge 5$ is odd works similarly; compare [@Seidel-Thomas §4c].
Now we can use and \[prop:spherical-moregeneral\] together with to conclude that $\genby{\{P_i\}}\cong \D(A)^c$ where $A=\End(\bigoplus_i P_i)$.
There are results, analogous to those of this subsection, on the formality of the endomorphism algebras of configurations of $\IP^n[k]$-objects for $k$ negative. To see this, one can use that a non-positively graded algebra is intrinsically formal as soon as $$\mindeg(A)+q-2 > \maxdeg(\Tor_q^A(R,R))\quad \text{for $q\ge 3$}\,,$$ which is analogous to .
We chose to concentrate on $\IP^n[k]$-objects with $k$ positive, since those with negative $k$ are rare in practice, for example, negative Calabi–Yau objects cannot appear in derived categories of smooth varieties; compare [@Hochenegger-Kalck-Ploog-RT Lem. 1.7].
Actions induced by $A_m$-configurations of $\IP$-objects {#subsec:faithful}
--------------------------------------------------------
Seidel and Thomas [@Seidel-Thomas] used the formality of $\End(\bigoplus_i P_i)$, where $P_1,\dots, P_m$ is an $A_m$-configuration of $k$-spherical objects in a cocomplete dg-enhanced triangulated category $\cT$, in order to prove that the induced action of the braid group $B_{m+1}$ on $\cT$ is faithful. This means that the subgroup $\genby{\TTT_{P_1},\dots,\TTT_{P_m}} \subset \Aut(\cT)$ generated by the spherical twists is isomorphic to $B_{m+1}$. Consider the $P_i$ as $\IP^1[k]$-objects and the associated $\IP$-twists are then the square of the spherical twists: $\PPP_{P_i}\cong \TTT_{P_i}^{\,2}$; compare . It follows from the description of the group spanned by the squares of the standard generators of the braid group [@Collins--squares] that the only relations between the $\IP$-twists are the commutativity relations $$\PT_i\PT_j=\PT_j\PT_i\quad \text{for $|i-j|>1$.}$$ Hence, it makes sense to conjecture the following more general faithfulness result.
Let $P_1,\dots, P_m$ be an $A_m$-configuration of $\IP^n[k]$-objects with $k\ge 2$. Then the only relations between the associated $\IP$-twists $\PT_i:=\PT_{E_i}\in \Aut(\cT)$ are the commutativity relations $$\PT_i\PT_j=\PT_j\PT_i\quad \text{for $|i-j|>1$.}$$
It is easy to see that, for two $\IP^n[k]$-objects with vanishing graded Hom-space between them, the associated $\IP$-twists commute; see [@Krug-autos Cor. 2.5]. Hence, the unknown and probably difficult part of the conjecture is that there are no further relations between the twists associated to an $A_m$-configuration of $\IP$-objects.
By , it would be sufficient to consider one particular example of an $A_m$-configuration of $\IP^n[k]$-objects in order to prove (or disprove) the conjecture for a fixed value of $m$, $n$ and $k$ with $nk$ even and $\gcd(k,\frac {nk}2)>1$.
The triangulated category generated by the structure sheaf {#sec:OX}
==========================================================
Let $X$ be a smooth projective variety over $\IC$. Note that the graded endomorphism algebra $\End^*(\reg_X)$ coincides with the cohomology algebra $\Ho^*(\reg_X)$ where the multiplication is given by the cup product. This algebra, sometimes called the *homological unit* of $X$ is conjectured to be a derived invariant of the variety $X$; see [@Abuaf-homologicalunit]. In this section we will show that the generated thick triangulated category $\langle \reg_X\rangle\subset \Db(\Coh(X))$ only depends on this graded algebra.
Actually, we show this statement for any compact complex manifold $X$ which *satisfies the $\partial\bar\partial$-lemma*, that is, $X$ has the following property.
*Let $\omega$ be a complex-valued differential form on $X$ which is $\partial$-closed and $\bar\partial$-closed. If $\omega$ is $\partial$-exact or $\bar\partial$-exact, then it is already $\partial\bar\partial$-exact, which means that there is a differential form $\chi$ with $\partial\bar\partial \chi=\omega$.*
Every compact Kähler manifold satisfies the $\partial\bar\partial$-Lemma; see, for example, [@Voisin-book1 §6.1]. However, there are compact complex manifolds satisfying the $\partial\bar\partial$-Lemma which are not Kähler. Still they share many properties of compact Kähler manifolds; to get an impression see [@Deligneetal], [@Angella], [@Anthesetal].
\[thm:unit\] Let $Y$ be a compact complex manifold satisfying the $\partial\bar\partial$-Lemma. Let $\Db_{\textrm{coh}}(Y)$ be the subcategory of complexes with bounded and coherent cohomology in $\D(\Mod(Y))$ and $\genby{\OO_Y}\subset \Db_{\textrm{coh}}(Y)$ the thick subcategory generated by $\reg_Y$. Then there is an equivalence $\genby{\OO_Y}\cong \D(\Ho^*(\reg_Y))^c$.
In this case, by the GAGA principle, we get an equivalence $\Db(\Coh(X))\cong \Db_{\textrm{coh}}(\Xan)$; see [@Caldararu-thesis Thm. 2.2.10]. Consequently, we also obtain $\genby{\OO_X}\cong \D(\Ho^*(\reg_X))^c$.
We denote by $A^{0,\bullet}$ the *Dolbeault complex* on the compact complex manifold $Y$. Its terms $A^{0,p}$ are the anti-holomorphic $p$-forms on $Y$ and its differential is given by $\bar \partial$.
\[lem:Dolbeault\] Let $Y$ be a connected complex manifold satisfying the $\partial\bar\partial$-Lemma. Then the Dolbeault complex $A^{0,\bullet}$ on $Y$ is formal.
Note that $\OO_Y$ is a compact object in $\Db_{\textrm{coh}}(Y)$. Thus, in view of and , it is enough to find a pretriangulated dg-category $\EE$ together with an exact equivalence $\alpha\colon \Ho^0(\EE)\to \Db_{\textrm{coh}}(Y)$ and an object $E\in \EE$ with $\alpha(E)\cong \OO_Y$ and $\Hom_{\EE}^\bullet (E,E)\cong A^{0,\bullet}$.
A dg-enhancement of $\Db_{\textrm{coh}}(Y)$ is given by the category $\EE=\cP_A$ of [@Block-dg]; see, in particular, [@Block-dg Thm. 4.3]. The objects of $\cP_A$ are given by pairs $(M, \nabla)$ consisting of a graded module $M$ over $\cA=A^{0,0}$ and a connection $\nabla\colon M\otimes_\cA A^{0,\bullet}$ satisfying some additional conditions; see [@Block-dg Def. 2.4] for details. We consider the object $E=(\cA, \bar\partial)\in \cP_A$. Indeed, $\alpha(E)=\cA_Y^{0,\bullet}$ where $\alpha\colon \Ho^0(\cP_A)\isom\Db_{\textrm{coh}}(Y)$ is the equivalence constructed in [@Block-dg Lem. 4.5] and $\cA_Y^{0,\bullet}$ denotes the Dolbeault complex of *sheaves* (not their global sections as in $A^{0,\bullet}$). The complex $\cA_Y^{0,\bullet}$ is a resolution of $\OO_Y$; see, for example, [@Voisin-book1 Prop. 4.19]. Hence, $\alpha(\EE)\cong \OO_Y$. The fact that $\Hom_{\EE}^\bullet(E,E)\cong A^{0,\bullet}$ follows directly from the definition of the Hom-complexes in the category $\cP_A$; see [@Block-dg Def. 2.4].
Examples of configurations of $\IP$-objects {#sec:example}
===========================================
In the following examples, we assume that the characteristic of the field $\kk$ does not divide the order $n!$ of the group $\sym_n$.
Trees of $\IP$-like objects on symmetric quotient stacks
--------------------------------------------------------
We recall a construction of $\IP^n[k]$-like objects from $k$-spherelike objects, which is essentially due to Ploog and Sosna in [@Ploog-Sosna]. Let $X$ be a smooth projective variety and $E$ be a $k$-spherelike object in $\Db(X)$. Consider the $n$-fold cartesian product $X^n$ with its projections $\pi_i \colon X^n \to X$. Then we define $$E^{\boxtimes n} = \pi_1^* E \otimes \cdots \otimes \pi_n^* E \in \Db(X^n).$$ There is a natural action on $X^n$ by the permutation group $\sym_n$. Actually, we can turn $E^{\boxtimes n}$ into an object $E\poslinn$ in the equivariant derived category $\Db_{\sym_n}(X^n)$ by equipping $E^{\boxtimes n}$ with the canonical linearisation given by permutation of the tensor factors. By $E\neglinn\in\Db_{\sym_n}(X^n)$, we denote the object $E^{\boxtimes n}$ equipped with the linearisation which differs from the canonical one by the non-trivial character (also known as *sign* or *alternating representation*) $\alt$ of $\sym_n$. Let $[X^n/\sym_n]$ be the quotient stack of $X^n$ by the permutation action of $\sym_n$. By the definition of sheaves on a quotient stack, there is an equivalence $\Db_{\sym_n}(X^n) \cong \Db([X^n/\sym_n])$.
The above construction also works for $E$ inside $\Db(A) = \Db(\mod(A))$ where $A$ is a $\kk$-algebra. Instead of the cartesian product $X^n$ we consider $A^{\otimes n}$ and instead of the equivariant derived category we consider the derived category $\Db(\sym_n \# A^{\otimes n})$ of the algebra $\sym_n \# A^{\otimes n}$ which is known as the *skew group algebra* in the literature.
More generally, there is the concept of the symmetric power $S^n\cT$ of a (dg-enhanced) triangulated category $\cT$ due to Ganter and Kapranov [@Ganter-Kapranov]. This covers both of the constructions above, since $S^n\Db(X)\cong\Db_{\sym_n}(X^n)$ and $S^n\Db(A)\cong\Db(\sym_n\# A)$ for a variety $X$ and an algebra $A$, respectively.
\[rem:geometric-interpretation\] If $X$ is a surface, then $\Db_{\sym_n}(X^n)$ has a very geometric interpretation. Namely, the derived McKay correspondence of Bridgeland, King and Reid [@Bridgeland-King-Reid] and Haiman [@Haiman] gives an equivalence $\Db_{\sym_n}(X^n)\cong \Db(X^{[n]})$ where $X^{[n]}$ denotes the Hilbert scheme of $n$ points on $X$; see also [@Sca §1.3]. In we will describe the corresponding $\IP$-objects in $\Db(X^{[n]})$.
\[Prop:gradedhom\] Let $n,k\in \IN$ with $n\ge 2$ and $k$ even. Let $X$ be a smooth projective variety and let $E$ be a $k$-spherelike object in $\Db(X)$. Then $E\poslinn, E\neglinn \in \Db_{\sym_n}(X^n)$ are $\IP^n[k]$-like objects.
Moreover, if $E$ is a $k$-spherical object (e.g. if $E$ is spherelike and $X$ is a Calabi–Yau variety of dimension $k$), then $E\poslinn$ and $E\neglinn$ are $\IP^n[k]$-objects.
Finally, let $E, F\in \Db(X)$ be objects with $\Hom^*(E,F)=\kk[-m]$ for some $m\in \IN$. Then $$\begin{aligned}
\Hom^*(E\poslinn, F\poslinn)= \Hom^*(E\neglinn, F\neglinn)=\begin{cases}\kk[-nm] \quad &\text{if $m$ is even,} \\ 0 \quad &\text{if $m$ is odd.}
\end{cases}
\\
\Hom^*(E\neglinn, F\poslinn)= \Hom^*(E\poslinn, F\neglinn)=\begin{cases}0\quad &\text{if $m$ is even,}\\ \kk[-mn] \quad &\text{if $m$ is odd.}
\end{cases}\end{aligned}$$
All of this follows from the equivariant Künneth formula which says that $$\begin{aligned}
\Hom^*(E\poslinn, F\poslinn)= \Hom^*(E\neglinn, F\neglinn)&= S^n\Hom^*(E,F)\,.
\\
\Hom^*(E\neglinn, F\poslinn)= \Hom^*(E\poslinn, F\neglinn)&= \medwedge{n}\Hom^*(E,F)\,.\end{aligned}$$ Note that both the symmetric and the exterior product are formed in the graded sense. For example, if $m\in \IN$ is odd, then $$S^n(\kk[-m])\cong \bigl(\medwedge{n} \kk\bigr)[-mn]=0\,.\qedhere$$
\[Cor:inducedtree\] Let $\{E_i\}$ be $k$-spherelike objects in $\Db(X)$ which form a tree $Q$. Then there is a choice of signs $\eps_i=\pm 1$ such that the $\IP^n[k]$-like objects $E_1^{\eps_1\linearised{n}},\ldots,E_m^{\eps_m\linearised{n}}$ form the same tree $Q$. Additionally, if the $E_i$ are $k$-spherical, then this is a configuration of $\IP^n[k]$-objects.
We start with some vertex $i_0$ of $Q$ and set $F_{i_0} \coloneqq E_{i_0}\poslinn$. Then, given an adjacent $j$ of $i_0$, the graded Hom-space $\Hom^*(E_{i_0}, E_j)$ is one-dimensional, hence concentrated in one degree, say $d$. We set $\eps_j \coloneqq (-1)^{d}$ and $F_j \coloneqq E_j^{\eps_{j}\linearised{n}}$. By , $F_{i_0}$ and $F_j$ are $\IP^n[k]$-like objects and $\Hom^*(F_{i_0}, F_j)=\kk[dn]$ is one-dimensional. In other words, $F_{i_0}$ and $F_j$ form an $A_1$-tree of $\IP^n[k]$-like objects. Since $Q$ is a tree, we can continue inductively and end up with a $Q$-tree $\{F_j=E_j^{\eps_j\linearised{n}}\}$ of $\IP^n[k]$-like objects.
\[rem:closing-condition\] The corollary yields also more general configurations of $\IP$-like objects, provided that for each cycle the signs can be attributed consistently. We do not spell out the details, but provide an example.
By a result of Kodaira, cycles of $(-2)$-curves $C_i$ appear as singular fibres in elliptic fibrations of surfaces; see the book [@Barth-etal §V.7] by Barth, Hulek, Peters and van den Ven. Hence, such a cycle forms a cycle of spherical objects $\OO_{C_i}$, as $\Hom^*(\OO_{C_i},\OO_{C_j})$ is non-zero if and only if the curves $C_i$ and $C_j$ intersect. Consequently, these objects induce a cycle of $\IP^n$-objects, provided the cycle is of even length.
A geometric example of a tree of $\IP$-objects {#app:geometric-pns}
----------------------------------------------
Let $Q$ be a tree and $X$ be a smooth quasi-projective surface together with a $Q$-configuration of $(-2)$-curves. This means that, for every vertex $i$ of the tree $Q$, there is a $(-2)$-curve $\IP^1\cong C_i\subset X$, $C_i$ and $C_j$ intersect in one point if there is an edge joining $i$ and $j$ and they do not intersect otherwise. Note that such a configuration might not exist for any given tree $Q$; see [@Hochenegger-Ploog §6] for sufficient criteria.
The objects $\OO_{C_i} \in \Db(X)$ form a $Q$-tree of $2$-spherical objects with $\Hom^*(\OO_{C_i}, \OO_{C_j})=\kk[-1]$ for adjacent $i$ and $j$; see [@Seidel-Thomas Ex. 3.5]. By , there is an induced $Q$-tree of $\IP^n$-objects of the form $\OO_{C_i}^{\eps_i\linearised n}$ in $\D^b_{\sym_n}(X^n)$. Here we have to choose opposite signs $\eps_i =-\eps_j$ for adjacent $i$ and $j$, as the graded Hom-space is concentrated in the odd degree 1.
We use the derived McKay correspondence as mentioned in (recall that we omit $R$ and $L$ in front of derived functors) $$\Phi \coloneqq p_*\circ q^*\colon \Db(X^{[n]})\isom \Db_{\sym_n}(X^n)$$ to interpret this as a tree of $\IP^n$-objects on the Hilbert scheme $X^{[n]}$. Here we denote by $q\colon \cZ\to X^{[n]}$ and $p\colon \cZ\to X^n$ the projections from the universal family of $\sym_n$-clusters $\cZ\subset X^{[n]}\times X^n$. Hence there is a commutative diagram $$\begin{tikzcd}
\cZ \ar[r, "p"'] \ar[d, "q"'] & X^n \ar[d, "\pi"] \\
X^{[n]} \ar[r, "\mu"] & X^{(n)}
\end{tikzcd}$$ where $X^{(n)} \coloneqq X^n/\sym_n$ is the symmetric product, $\pi$ is the $\sym_n$-quotient morphism, and $\mu$ is the Hilbert–Chow morphism. Furthermore, $\cZ\cong (X^{[n]}\times_{ X^{(n)}}X^n)_{\mathsf{red}}$ is the reduced fibre product of this diagram. Note that every closed subscheme $C$ of $X$ induces a canonical closed embedding $C^{[n]}\hookrightarrow X^{[n]}$.
\[prop:CMcKay\] For $C\subset X$ a smooth curve, we have $\Phi(\OO_{C^{[n]}})\cong \OO_C\poslinn$.
For a smooth curve $C$, the Hilbert–Chow morphism $C^{[n]}\to C^{(n)}$ is an isomorphism. So, $C^{[n]} \isom C^{(n)} \into X^{(n)}$ is a closed embedding with image $C^{(n)}$. Consequently, using the diagram above, $p\colon \cZ\to X^n$ maps $q^{-1}C^{[n]}$ isomorphically to $C^n$. Hence, $\Phi(\OO_{C^{[n]}})\cong \OO_{C^n}\cong\OO_C^{\linearised n}$.
Let $i$ and $j$ be two adjacent vertices of $Q$. Then $C_i\cap C_j$ is a reduced point, hence cannot contain a subscheme of length $n\ge 2$. Thus, $C_i^{[n]}$ and $C_j^{[n]}$ do not intersect inside $X^{[n]}$. So we see geometrically that $\Hom^*(\OO_{C_i}\poslinn, \OO_{C_j}\poslinn)=0$ for support reasons.
For $n=2$, we give a concrete description of the image of $\OO_{C}\neglinn$ under the McKay correspondence, where $C=C_i$ is one of the rational curves. Denote by $\delta\colon X\to X^{(2)}$ the diagonal embedding into the symmetric product. Then $E \coloneqq \mu^{-1}\delta(X)$ is the exceptional divisor of $X^{[2]}\to X^{(2)}$. There is a line bundle $\LL\in \Pic(X^{[2]})$ such that $\LL^2\cong \OO(E)$; see [@Lehn-chern Lem. 3.7].
We summarise this situation in the following diagram, consisting of a blow-up square on the right and its restriction to $C$, so both are cartesian: $$\tag{$\square$}
\label{diag:E}
\begin{gathered}
\begin{tikzcd}
\llap{$\mu^{-1}\delta(C) \eqqcolon\,$} \Sigma_C \ar[r, hook] \ar[d] & E \ar[r, hook, "\iota"'] \ar[d, "\rho"'] & X^{[2]} \rlap{$\,= \Bl_{\delta(X)}(X^{(2)})$} \ar[d, "\mu"] \\
C \ar[r, hook] & X \ar[r, hook, "\delta"] & X^{(2)}
\end{tikzcd}
\end{gathered}$$ Note that the restrictions $\rho\colon E\to X$ and $\Sigma_C \to C$ of $\mu$ are $\IP^1$-bundles (actually, $\Sigma_C$ is isomorphic to the Hirzebruch surface $\Sigma_4$).
\[prop:YC\] Denote by $Y_C$ the closed subscheme $\Sigma_C\cup C^{[2]}$ in $X^{[2]}$. Then $\Phi(\OO_{Y_C}\otimes \LL)\cong \OO_C\neglin{2}$.
Before we prove the proposition, we need to recall another feature of the Hilbert scheme of two points, namely that the functor $$\Theta \coloneqq \iota_*\rho^* \colon \Db(X)\to \Db(X^{[2]})$$ is spherical; see [@Krug-autos Rem. 4.3] or [@Krug-Ploog-Sosna Thm 4.26(ii)]. This means, in particular, that the associated twist functor $\TTT_{\Theta}$, defined by the triangle of functors $$\Theta\Theta^R \xrightarrow{\eps} \id \to \TTT_{\Theta}\,,$$ where $\eps$ is the counit of adjunction, is an autoequivalence of $\Db(X^{[n]})$. The right adjoint $\Theta^R$ of $\Theta$ is given by $$\tag{$\ast$}
\label{eq:right-adjoint}
\Theta^R\cong \rho_* \iota^!\cong \rho_*\bigl(\iota^*(\blank)\otimes \OO_E(E)\bigr)[-1]\,.$$
The subvarieties $C^{[2]}$ and $E$ of $X^{[2]}$ intersect transversally and $\rho$ maps the scheme-theoretic intersection $E \cap C^{[2]}$ isomorphically to $\delta(C)\subset X^{(2)}$.
The second assertion implies the first one since transversality means that the scheme-theoretic intersection $E\cap C^{[2]}$ is reduced and of the expected dimension $1$.
The composition $C^{[2]}\hookrightarrow X^{[2]}\xrightarrow \rho X^{(2)}$ is a closed embedding with image $C^{(2)}\subset X^{(2)}$. Using the right cartesian diagram in , it follows that $\rho$ maps $E\cap C^{[2]}$ isomorphically to the scheme-theoretic intersection $\delta(X)\cap C^{(2)}$.
Hence, we only have to prove that $\delta(X)\cap C^{(2)}=\delta(C)$. This question is local in the analytic topology so that we may assume that $X=\Spec \IC[x_1,x_2]$ and $C=\Spec \IC[x_1]$. We set $s_i=x_i+y_i$ and $t_i=x_i-y_i$ so that $X^2= \Spec\IC[s_1,s_2,t_1,t_2]$ with the natural action of $\sym_2=\langle \tau\rangle$ given by $\tau\cdot s_i=s_i$ and $\tau\cdot t_i=-t_i$. Therefore, $$\reg(X^{(2)})=\IC[s_1,s_2,t_1,t_2]^{\sym_2}=\IC[s_1,s_2,t_1^2, t_1t_2,t_2^2]\,.$$ The ideal of $\delta(X)\subset X^{(2)}$ is given by $I=(t_1,t_2)^{\sym_2}=(t_1^2, t_1t_2,t_2^2)$ and the ideal of $C^{(2)}\subset X^{(2)}$ is given by $J=(s_2,t_2)^{\sym_2}=(s_2, t_2^2)$. Hence, $$\delta(X)\cap C^{(2)}= \Spec\bigl( \IC[s_1,s_2,t_1^2, t_1t_2,t_2^2]/(I+J)\bigr) =\Spec \IC[s_1]=\delta(C)\,.\qedhere$$
\[lem:OCtwist\] For the spherical twist $\TTT_\Theta$, we have $\TTT_\Theta(\OO_{C^{[2]}}(-E))\cong \OO_{Y_C}$.
By $\reg_{C^{[2]}}(-E)$ we mean $\left.\reg_{X^{[2]}}(-E)\right|_{C^{[2]}}$. Using and the previous lemma, we compute $$\Theta^R(\OO_{C^{[2]}}(-E))\cong \OO_C[-1]\,.$$ Note that $\Theta(\OO_C)\cong \OO_{\Sigma_C}$, so the twist triangle applied to $\OO_{C^{[2]}}(-E)$ becomes, after shift, $$\OO_{C^{[2]}}(-E)\to \TTT_\Theta(\OO_{C^{[2]}}(-E))\to \OO_{\Sigma_C}\,.$$ The long exact cohomology sequence shows that $\cH^i\bigl(\TTT_\Theta(\OO_{C^{[2]}}(-E))\bigr)=0$ for $i\neq 0$. Hence, the triangle reduces to the short exact sequence $$\begin{aligned}
\label{eq:es1}
0\to \OO_{C^{[2]}}(-E\cap C^{[2]})\to \cH^0\bigl(\TTT_\Theta(\OO_{C^{[2]}}(-E))\bigr)\to \OO_{\Sigma_C}\to 0.\end{aligned}$$ As $\Sigma_C \cap C^{[2]} = E \cap C^{[2]}$, there is the canonical short exact sequence $$\begin{aligned}
\label{eq:es2}
0\to \OO_{C^{[2]}}(-E\cap C^{[2]})\to \reg_{Y_C}\to \OO_{\Sigma_C}\to 0.\end{aligned}$$ One can compute that $\Ext^1(\reg_{\Sigma_C},\OO_{C^{[2]}}(-\Sigma_C\cap C^{[2]}))=\IC$, using for example [@CKS-ext Thm. A.1]. It follows that and coincide, so $$\TTT_\Theta(\OO_{C^{[2]}}(-E))\cong \cH^0\bigl(\TTT_\Theta(\OO_{C^{[2]}}(-E))\bigr)\cong\OO_{Y_C}\,.\qedhere$$
Combining the formulae of [@Krug-Ploog-Sosna Thm. 4.26], we get an isomorphism of functors $$\Phi^{-1}(\Phi(\blank)\otimes \alt)\cong \LL\otimes \TTT_\Theta(\blank\otimes \LL^{-2})\,,$$ where $\LL^2 = \OO(E)$. Combining this with gives $$\Phi^{-1}(\OO_C\neglin{2})\cong \LL\otimes \TTT_\Theta(\OO_{C^{[2]}}\otimes \LL^{-2})\,.$$ Now the assertion follows by .
For $C_i, C_j\in X$ two $(-2)$-curves which intersect in one point, $Y_{C_i}$ and $C_j^{[2]}$ intersect transversally in one point of $X^{[2]}$. This confirms geometrically that $$\Hom^*_{\Db_{\sym_2}(X^2)}(\OO_{C_i}\neglin{2},\OO_{C_j}\poslin{2})\cong \Hom^*_{\Db(X^{[2]})}(\OO_{Y_{C_i}}\otimes\LL,\OO_{C_j^{[2]}})=\IC[-2]\,;$$ compare .
*Contact:* `andreas.hochenegger@unimi.it`\
`andkrug@mathematik.uni-marburg.de`
[^1]: MSC 2010: 18E30, 14F05, 16E40
[^2]: Keywords: $\IP$-object; formality of endomorphism algebras; triangulated category
|
---
author:
- |
Samuel Zbarsky\
Carnegie Mellon University\
sa\_zbarsky@yahoo.com\
bibliography:
- 'zbarskybib.bib'
date: 'Mathematics Subject Classifications: 05A15, 05D05'
title: '**The Maximum Number of Subset Divisors of a Given Size**'
---
Abstract
========
If $s$ is a positive integer and $A$ is a set of positive integers, we say that $B$ is an $s$-divisor of $A$ if $\sum_{b\in B} b\mid s\sum_{a\in A} a$. We study the maximal number of $k$-subsets of an $n$-element set that can be $s$-divisors. We provide a counterexample to a conjecture of Huynh that for $s=1$, the answer is $\binom{n-1}{k}$ with only finitely many exceptions, but prove that adding a necessary condition makes this true. Moreover, we show that under a similar condition, the answer is $\binom{n-1}{k}$ with only finitely many exceptions for each $s$.
Introduction
============
If $X$ is a set of positive integers, let $\sum X$ denote $\sum_{x\in X}x$. Let $A$ be a finite subset of the positive integers. The elements of $A$ are $a_1<a_2<\cdots<a_n$ and let $B$ be a subset of $A$. We say that $B$ is a *divisor* of $A$ if $\sum B\mid\sum A$. We define $d_k(A)$ to be the number of $k$-subset divisors of $A$ and let $d(k,n)$ be the maximum value of $d_k(A)$ over all sets $A$ of $n$ positive integers.
Similarly, for $s\ge 1$ a positive integer, we say that $B$ is an *s-divisor* of $A$ if $\sum B\mid s\sum A$. We define $d^s_k(A)$ to be the number of $k$-subset $s$-divisors of $A$ and let $d^s(k,n)$ be the maximum value of $d^s_k(A)$ over all sets $A$ of $n$ positive integers.
Note that the concepts of divisor and 1-divisor coincide. Also, if $B$ is a divisor of $A$, then $B$ is an $s$-divisor of $A$ for all $s$, so $d^s_k(A)\ge d_k(A)$ and $d^s(k,n)\ge d(k,n)$
Huynh [@huynh14] notes that for any values of $a_1,\ldots,a_{n-1}$, we can pick such an $a_n$ that any $k$-subset of $\{a_1,\ldots,a_{n-1}\}$ will be an $A$-divisor. Therefore $d(k,n)\ge \binom{n-1}{k}$ for all $1\le k\le n$. This motivates the definition that $A$ is a *k-anti-pencil* if the set of $k$-subset divisors of $A$ is $\binom{A\wo\{a_n\}}{k}$. We similarly define $A$ to be a $(k,s)$-*anti-pencil* if the set of $k$-subset $s$-divisors of $A$ is $\binom{A\wo\{a_n\}}{k}$.
Huynh [@huynh14] also formulates the following conjecture (Conjecture 22).
\[huynhconj\] For all but finitely many values of $k$ and $n$, $d(k,n)=\binom{n-1}{k}$.
In this paper, we provide infinite families of counterexamples, but prove that, with the exception of these families, the conjecture is true. This gives us the following modified form.
\[huynhconjmod\] For all but finitely many integer pairs $(k,n)$ with $1<k<n$, $d(k,n)=\binom{n-1}{k}$.
For convenience, we now rescale, dividing every element of $A$ by $\sum A$, so that now the elements of $A$ are positive rational numbers and $\sum A=1$. Under this rescaling, $B\subseteq A$ is a divisor of $A$ if and only if $\sum B=\frac{1}{m}$ for some positive integer $m$ and $B$ is an $s$-divisor of $A$ if and only if $\sum B=\frac{s}{m}$ for some positive integer $m$. Clearly, the values of $d(k,n)$ and $d^s(k,n)$ do not change.
The $k<n$ condition in Conjecture \[huynhconjmod\] is necessary since it is easy to see that $d(n,n)=1>\binom{n-1}{n}$. Also, if $$A=\left\{\half,\frac{1}{4},\ldots,\frac{1}{2^{n-2}},\frac{1}{3(2^{n-1})},\frac{1}{3(2^{n-2})}\right\}$$ then $\sum A=1$, so $d_1(A)=n$ and $d(1,n)\ge n>\binom{n-1}{1}$. Therefore the $1<k$ condition is necessary.
However, we prove that these families cover all but finitely many exceptions.
\[s1case\] For all but finitely many pairs $(k,n)$, if $1<k<n$, $|A|=n$, and $d_k(n)\ge\binom{n-1}{k}$, then $A$ is a $k$-anti-pencil.
Note that this immediately implies Conjecture \[huynhconjmod\].
If we are interested in $s$-divisors, we get another family of exceptions. If $s\ge 2$, $a_n=\frac{1}{s+1}$ and $a_{n-1}=\frac{2}{s+2}$, then $d^s_{n-1}(A)\ge 2$, so $d^s(n-1,n)\ge 2>\binom{n-1}{n-1}$. However, we prove that these cover all but finitely many exceptions.
\[generalcase\] Fix $s\ge 1$. For all but finitely many pairs $(k,n)$ (with the number of these pairs depending on $s$), if $1<k<n-1$, $|A|=n$, and $d^s_k(n)\ge\binom{n-1}{k}$, then $A$ is a $(k,s)$-anti-pencil.
Note that this immediately implies the following corollary.
\[generalcasecorr\] Fix $s\ge 1$. Then $d^s(k,n)=\binom{n-1}{k}$ for all but finitely many pairs $(k,n)$ with $1<k<n-1$ (with the number of these pairs depending on $s$).
We will prove Theorem \[generalcase\]. In the $s=1$ case, where $k=n-1$, if $i\le n-1$, then $\sum(A\wo\{a_i\})>\half$, so $A\wo\{a_i\}$ is not a divisor of $A$. This, together with the $s=1$ case of Theorem \[generalcase\], gives us Theorem \[s1case\].
Lemmas
======
Take a $d$-dimensional lattice cube with $n$ lattice points per edge. Define a poset on the lattice points by $(x_1,\ldots,x_d)\le(y_1,\ldots,y_d)$ if $x_i\le y_i$ for all $i$.
The largest antichain in this poset has at most $(n+d-2)^{d-1}\sqrt{\frac{2}{d}}$ elements.
First, we need some definitions.
The *width* of a poset is the size of its largest antichain. If $P$ is a finite poset, we say that $P$ is *ranked* if there exists a function $\rho:P\to \Z$ satisfying $\rho(y)=\rho(x)+1$ if $y$ covers $x$ in $P$ (i.e. $y>x$, and there is no $z\in P$ with $y>z>x$). If $\rho(x)=i$, then $x$ is said to have *rank i*. Let $P_i$ denote the set of elements of $P$ of rank $i$. We say $P$ is *rank-symmetric rank-unimodal* if there exists some $c\in\Z$ with $|P_i|\le|P_{i+1}|$ when $i<c$ and $|P_{2c-i}|=|P_i|$ for all $i\in \Z$. A ranked poset $P$ is called *strongly Sperner* if for any positive integer $s$, the largest subset of $P$ that has no $(s+1)$-chain is the union of the $s$ largest $P_i$.
Proctor, Saks, and Sturtevant [@ProctorSturtevant1980] prove that the class of rank-symmetric rank-unimodal strongly Sperner posets is closed under products.
Since a linear ordering of length $n$ is rank-symmetric rank-unimodal strongly Sperner, so is a product of $d$ of them (the lattice cube).
Center the cube on the origin by translation in $\R^d$. Let $U$ be the set of elements whose coordinates sum to 0. Since the poset is rank-symmetric rank-unimodal strongly Sperner, its width is at most the size of $P_c$, which is $|U|$.
For each $y=(y_1,\ldots,y_d)\in U$, let $S_y$ be the set of points $(x_1,\ldots,x_d)$ with $|x_i-y_i|<\half$ for $1\le i\le d-1$ (note that this does not include the last index) which lie on the hyperplane given by $x_1+\cdots+x_d=0$. If $y,z$ are distinct elements of $U$, then $S_y$ and $S_z$ are clearly disjoint. Also, the projection of $S_y$ onto the hyperplane given by $x_d=0$ is a unit $(d-1)$-dimensional hypercube, which has volume 1. Thus the volume of $S_y$ is $\sqrt{d}$ and the volume of $\bigcup_{y\in U}S_y$ is $|U|\sqrt{d}$.
On the other hand, if $(x_1,\ldots,x_d)\in S_y$, then $|x_i-y_i|<\half$ for $1\le i\le d-1$ and $|x_d-y_d|\le\sum_{i=1}^{d-1}|x_i-y_i|<\half(d-1)$. Thus $(x_1,\ldots,x_d)$ lies in the cube of edge length $(n-1)+(d-1)=n+d-2$ centered at the origin. Therefore $\bigcup_{y\in U}S_y$ lies in the intersection of a cube of edge length $n+d-2$ with a hyperplane through its center (the origin).
Ball [@Ball1986] shows that the volume of the intersection of a unit hypercube of arbitrary dimension with a hyperplane through its center is at most $\sqrt{2}$. Therefore the volume of $\bigcup_{y\in U}S_y$ is at most $(n+d-2)^{d-1}\sqrt{2}$, so $$|U|\le (n+d-2)^{d-1}\sqrt{\frac{2}{d}}.$$
Let $X=\{x_1<\cdots<x_n\}$ be any set of positive integers. If $B,C\in \binom{X}{d}$, then we say that $B\le C$ if we can write $B=\{b_1,\ldots,b_d\}$ and $C=\{c_1,\ldots,c_d\}$ with $b_i\le c_i$ for all $1\le i\le d$. Whenever we compare subsets of $A$, we will be using this partial order.
\[simplex\] Fix $d>1$. For $n$ sufficiently large, the width of the partial order defined above is less than $\frac{2}{\sqrt{d}}\frac{1}{n}\left|\binom{X}{d}\right|$.
Let $U$ be a maximum antichain of the partial order. Take the partial order of $X^d$, which coincides with the cube partial order. Let $U'=\{(y_1,\ldots,y_d)\in X^d\mid \{y_1,\ldots,y_d\}\in U\}$. Note that this means, in particular, that all elements of any $k$-tuple in $U'$ are distinct. If $(y_1,\ldots,y_d),(z_1,\ldots,z_d)\in U'$ with $(y_1,\ldots,y_d)<(z_1,\ldots,z_d)$, then we get that $\{y_i\}\le\{z_i\}$ and $\sum_{i=1}^d y_i< \sum_{i=1}^d z_i$, so $\{y_i\}\ne\{z_i\}$, so $\{y_i\}<\{z_i\}$, which is impossible. Thus $U'$ is an antichain of $X^d$ of size $d!|U|$ and $$|U|\le\frac{1}{d!}(n+d-2)^{d-1}\sqrt{\frac{2}{d}}.$$ Then $\left|\binom{X}{d}\right|=\binom{n}{d}$ gives us $$\frac{|U|}{|\binom{X}{d}|}\le \frac{(n+d-2)^{d-1}\sqrt{\frac{2}{d}}}{n(n-1)\dotsm(n-d+1)}.$$ For sufficiently large $n$, $\left(\frac{n+d-2}{n-d+1}\right)^{d-1}<\sqrt{2}$, so $\frac{|U|}{|\binom{X}{d}|}<\frac{2}{\sqrt{d}}\frac{1}{n}$.
Let $d(n)$ denote the number of divisors of $n$.
\[divisors\] For any positive integer $k$, $d(n)=O(n^\frac{1}{k})$.
There are finitely many primes $p<2^k$, so there must be some constant $C$ such that for any $p<2^k$ and any positive integer $m$, $d(p^m)=m+1\le C(p^m)^\frac{1}{k}$.
For $p>2^k$, $d(p^m)=m+1\le 2^m\le (p^m)^\frac{1}{k}$. Thus if $n=\prod_{i=1}^j p_i^{m_i}$ for distinct prime $p_i$, then $$d(n)=\prod_{i=1}^j d\left(p_i^{m_i}\right)\le C^{2^k}\prod_{i=1}^j \left(p_i^{m_i}\right)^\frac{1}{k}\le C^{2^k}n^\frac{1}{k}=O\left(n^\frac{1}{k}\right).$$
\[fracsum\] Fix positive integers $k, m, a, b$. Then for positive integers $n$, the number of pairs of positive integers $(x,y)$ such that $\frac{m}{n}=\frac{a}{x}+\frac{b}{y}$ and all three fractions are in lowest terms is at most $O(n^\frac{1}{k})$.
Assume $\frac{m}{n}=\frac{a}{x}+\frac{b}{y}$. Let $p=\gcd(n,x)$, with $n=tp$ and $x=wp$. Then $$\frac{b}{y}=\frac{m}{n}-\frac{a}{x}=\frac{mw-at}{twp}.$$ Letting $q=\gcd(mw-at,twp)$, we get $$\label{eq:getw}
mw-at=qb.$$ For any choice of $n,p,q$, gives at most one possible value of $w$, thus at most one value of $x$, and thus at most one value of $(x,y)$.
The definition of $q$ gives us $q\mid p$. Then for a given $n$, both $p$ and $q$ are divisors of $n$, so by Lemma \[divisors\] there are $O(n^{\frac{1}{2k}})$ possible values for $p$ and $O(n^{\frac{1}{2k}})$ values for $q$, so there are $O(n^{\frac{1}{k}})$ values for $(p,q)$ and $O(n^{\frac{1}{k}})$ pairs of numbers $(x,y)$.
Proof of Theorem \[generalcase\]
================================
Assume that $|A|=n$, $d^s_k(A)\ge\binom{n-1}{k}$, and that $A$ is not a $(k,s)$-anti-pencil. Note that then some $B\ni a_n$ has $\sum B\le\frac{s}{s+1}$, so since $1<k$, we have $a_n<\frac{s}{s+1}$. We will use this in all the cases below. Also, the number of $k$-subsets of $A$ that are not $s$-divisors is at most $\binom{n}{k}-\binom{n}{k-1}=\binom{n-1}{k-1}$.
\[chains\] If $B$ and $C$ are $k$-subsets of $A$ with $B<C$, then $\sum B<\sum C$. Note that if $B_0<B_1<\cdots<B_m$ are all divisors of $A$ and $\sum B_m<s/q$, then $\sum B_0<s/(q+m)$. Therefore if $a\in B_0$, then $a<s/(q+m)$. Since $k<n$, $\sum B_m<s/s$, so we automatically get that $a<s/(s+m)$
Each of the subsections below is a separate case.
k small
-------
Fix $2\le k$ and let $n>>k$.
For $1\le i_1,\ldots,i_k\le n$, call the ordered $k$-tuple $(i_1,\ldots,i_k)$ *repetitive* if not all entries are distinct. Call it *good* if all entries are distinct and $\{a_{i_j}\}$ is an $s$-divisor. Otherwise, call the ordered $k$-tuple *bad*.
We will first restrict our attention to $k$-tuples where $i_k\ge n-1$. Among these, $O(n^{k-2})$ are repetitive. Also, $O(n^{k-2})$ include both $n$ and $n-1$ among their components. Of the remainder, at most $(k-1)!\binom{n-1}{k-1}\le n^{k-1}$ are bad. Thus at least 1/3 of the $k$-tuples $(i_1,\ldots,i_k)$ satisfying $i_k\ge n-1$ are good.
By the Pigeonhole Principle, there are some values $j_2,\ldots,j_k$ with $j_k\ge n-1$ such that the chain $\{(1,j_2,\ldots,j_k),\ldots,(n,j_2,\ldots,j_k)\}\subset U$ has at least $n/3$ good $k$-tuples. This gives us a chain of $k$-subset $s$-divisors of length at least $n/3$. Thus $a_{n-1}\le\frac{3s}{n}$.
Let $B=\{a_i\mid i>\left(1-\frac{1}{9s^2}\right)n\}$. If $a_i\in B$, then $$1=\sum A=\sum_{i=1}^n a_i<na_i+\frac{n}{9s^2}a_{n-1}+a_n<na_i +\frac{1}{3s}+\frac{s}{s+1}$$ so $na_i>\frac{1}{6s}$ and $a_i>\frac{1}{6sn}$.
Thus any $s$-divisors that is a subset of $B$ must sum to some $\frac{s}{m}>\frac{1}{6sn}$, so there are at most $6s^2n$ distinct values that $m$ can take. Thus there are at most $6s^2n$ distinct values that an $s$-divisor that is a subset of $B$ can sum to.
If $D\in\binom{B}{k-2}$ and $r=\frac{s}{m}$ for some positive integer $m$, call $D$ an *r-stem* if there are at least $\frac{1}{10000s^6}n$ pairs $\{x,y\}\subset B\wo D$ with $\sum (D\cup\{x,y\})=r$. Call such pairs *tails* of $D$. If two tails of $D$ are $\{x,y\}$ and $\{x,z\}$, then the sum condition gives us $y=z$, so tails of $D$ are pairwise disjoint.
Now let $B_0=B$. Note that $|B_0|>\frac{1}{10s^2}n$. As long as $|B_{i-1}|\ge\frac{1}{20s^2}n$, $B_{i-1}$ has at most $\binom{n-1}{k-1}$ subsets which are not $s$-divisors of $A$, so it has at least $\half\binom{|B_{i-1}|}{k}$ $k$-subsets that are $s$-divisors. Since these take on at most $6s^2n$ values, there must be some positive integer $m_i$ such that at least $\frac{1}{12s^2n}\binom{|B_{i-1}|}{k}$ $k$-subsets of $B_{i-1}$ sum to $r_i=\frac{s}{m_i}$.
If we randomly choose $D_i\in\binom{B_{i-1}}{k-2}$, the expected value for the number of pairs $\{x,y\}\subset B_{i-1}\wo D_i$ with $\sum(D_i\cup\{x,y\})=r_i$ is at least $\frac{1}{12s^2n}\binom{|B_{i-1}|-(k-2)}{2}$. Thus we will choose a $D_i$ such that the number of these pairs is at least $\frac{1}{12s^2n}\binom{|B_{i-1}|-(k-2)}{2}$. Since $$\frac{1}{12s^2n}\binom{|B_{i-1}|-(k-2)}{2}\ge \frac{1}{25s^2n}\left(|B_{i-1}|\right)^2\ge \frac{1}{25s^2n(20s^2)^2}n^2\ge \frac{1}{10000s^6}n,$$ $D_i$ satisfies the definition of an $r_i$-stem.
Let $B_i=B_{i-1}\wo D_i$. Then for $i\le\frac{1}{20ks^2}n$, $D_i$ is an $r_i$-stem and all the $D_i$ are disjoint.
Since the number of $k$-subsets of $A$ which are not $s$-divisors is less than $\binom{\frac{1}{20ks^2}n}{k}$, we know that there must exist disjoint $D_{i_1},\ldots,D_{i_k}$ such that any set consisting of one element of each $D_{i_j}$ will be an $s$-divisor. Note that in the $k=2$ case, $D_{i_1}=D_{i_2}=\emptyset$. Partition $\bigcup_{j=1}^k D_{i_j}$ into $k-2$ such sets $C_1,\ldots,C_{k-2}$.
Let $p=\ceil{\frac{1}{10000s^6}n/(2k)}=\ceil{\frac{1}{20000s^6k}n}$. For $1\le j\le k$, we want to choose $T^j_1,\ldots,T^j_p$ to be tails of $D_{i_j}$. We will choose them for $j=1$, then for $j=2$, and so on. When we choose $\{T^j_\ell\}$, we will make each of these tails disjoint from each of the $k$ stems, as well as from the already chosen tails. This is possible since $$\left|\bigcup_{h=1}^k D_{i_h}\cup \bigcup_{h=1}^{j-1} \bigcup_{\ell=1}^p T^h_\ell\right|=\left|\bigcup_{h=1}^k D_{i_h}\right|+\sum_{h=1}^{j-1} \left|\bigcup_{\ell=1}^p T^h_\ell\right|=k(k-2)+2(j-1)p\le k(k-2)+2(k-1)p.$$ Since any element in a stem or in a previously chosen tail can be in at most one tail of $D_{i_j}$, at most $k(k-2)+2(k-1)p$ tails are eliminated, so there must be at least $p$ tails still available to choose from. We say that a choice of $k$ tails $\{T^j_{i_j}\}_{j=1}^k$ for each stem is *fortuitous* if $\{x^j_{i_j}\}_{j=1}^k$ and $\{y^j_{i_j}\}_{j=1}^k$ are both $s$-divisors. There are $p^k>n^k/(20000s^6k)^k$ choices of tails, and at most $\binom{n-1}{k-1}$ of them are not fortuitous. Thus at least $\half$ of possible choices are fortuitous.
By the Pigeonhole Principle, we can choose $i_1,\ldots,i_{k-1}$ so that there are at least $p/2$ choices for $i$ which make $\{T^1_{i_1},\ldots,T^{k-1}_{i_{k-1}},T^k_i\}$ fortuitous.
Note that different choices of $i$ give us different values of $x^k_i$ and therefore different values of $\sum_{j=1}^k x^j_{i_j}$, so $\sum_{j=1}^k x^j_{i_j}$ can take on at least $$p/2=\Omega(n)$$ different values.
On the other hand, if we are given a fortuitous choice of tails $\{T^j_{i_j}\}$, then \_[j=1]{}\^[k-2]{}C\_j +\_[j=1]{}\^k x\^j\_[i\_j]{} +\_[j=1]{}\^k y\^j\_[i\_j]{}&=\_[j=1]{}\^[k]{}(D\_[i\_j]{}{x\^j\_[i\_j]{},y\^j\_[i\_j]{}})\
\_[j=1]{}\^k x\^j\_[i\_j]{} +\_[j=1]{}\^k y\^j\_[i\_j]{}&=\_[j=1]{}\^[k]{}r\_[i\_j]{}-\_[j=1]{}\^[k-2]{}C\_j. The right hand side does not depend on our choice of tails. Also, since each $r_{i_j}$ and each $\sum C_j$ has denominator at most $6s^2n$, the right hand side has denominator at most $(6s^2n)^{2k}$. Since both $\sum_{j=1}^k x^j_{i_j}$ and $\sum_{j=1}^k y^j_{i_j}$ are $s$-divisors, there are at most $s^2$ possibilities for their numerators. For each such possibility, by Lemma \[fracsum\], $\sum_{j=1}^k x^j_{i_j}$ can take on at most $$O\left(\left(6s^2n)^{2k}\right)^\frac{1}{4k}\right)=O\left(sn^\half\right)$$ different values. Thus $\sum_{j=1}^k x^j_{i_j}$ can take on at most $$O\left(s^3n^\half\right)$$ different values, contradicting the upper bound above.
n>=3/2k, k sufficiently large
--------------------------------
Let $d=\ceil{\left(s(s+1)/0.03\right)^2}$. Assume that $k$ is sufficiently large relative $d$ and that $n\ge\frac{3}{2}k$.
Let $T_2$ be the set of $k$-subsets of $A$ that include both $a_{n-1}$ and $a_n$. Let $T_1$ be the set of $k$-subsets of $A$ that include one of $a_{n-1}$ or $a_n$, but not both. Define $U_1$ and $U_2$ similarly, but with $(k-d)$-subsets.
For $S\in U_t$, let $P_S=\{B\in T_t\mid S\subset B\}$ (the set of $k$-subsets obtainable by adding $d$ elements of $A$ less than $a_{n-1}$ to $S$). Note that an element of $T_t$ is contained in $P_S$ for exactly $\binom{k-t}{d}$ values of $S$. Thus if $\alpha|T_t|$ elements of $T_t$ are $s$-divisors, then there is some $S\in U_t$ so that at least $\alpha|P_S|$ elements of $P_S$ are $s$-divisors.
Now note that the disjoint union $T_1\cup T_2$ is the set of all $k$-subsets whose greatest element is at least $a_{n-1}$, so $$|T_1\cup T_2|=\binom{n}{k}-\binom{n-2}{k}$$ and the fraction of the elements of $T_1\cup T_2$ which are not $s$-divisors is at most &=\
&=\
&=\
&=\
&=\
&=\
&0.76 for sufficiently large $k$. Therefore, if we set $\alpha=0.24$, then for $t=1$ or $t=2$, the fraction of elements of $T_t$ that are $s$-divisors is at least $\alpha$, so for some $S$, the fraction of elements of $P_S$ which are $s$-divisors is at least $\alpha=0.24$.
Note that the partial order of $P_S$ is the same as the partial order of $\binom{A\wo S\wo\{a_{n-1},a_n\}}{d}$, so by Lemma \[simplex\], its width is at most $\frac{2}{\sqrt{d}}\frac{1}{n-k-2}|P_S|$. Then, by Mirsky’s theorem, there is a chain of $k$-subset $s$-divisors in $P_S$ of length at least $$\frac{\alpha|P_S|}{\frac{2}{\sqrt{d}}\frac{1}{n-k-2}|P_S|}=0.12\sqrt{d}(n-k-2)\ge (0.03\sqrt{d})n. % here I use that $n$ is big$$ But then the first element of the chain includes $a_{n-1}$ or $a_n$, so by Remark \[chains\], $a_{n-1}\le \frac{s}{(0.03\sqrt{d})n}$. Then $$\sum_{i=1}^{n-1} a_i\le \frac{s}{0.03\sqrt{d}}$$ and, since $a_n<\frac{s}{s+1}$, $$\sum_{i=1}^n a_i<1$$ yielding a contradiction.
2/3n<k<n-c, k sufficiently large
--------------------------------------
Let $d=(6s^2+3s)^2$. Assume that $k$ is sufficiently large and that $\frac{2}{3}n<k<n-d$.
Randomly arrange the elements of $A$ around a circle. Let $M$ be the set of $k$-subsets of $A$ consisting of $k$ consecutive elements around the circle, and let $N=\{B\in M\mid \sum B\le\frac{1}{2(s+1)}\}$. If $B, C\in N$ and they are shifted relative each other by at least $n-k-1$, then $|A\wo(B\cup C)|\le 1$, so $$\sum A\le\sum(A\wo(B\cup C))+\sum B+\sum C<\frac{s}{s+1}+\frac{1}{2(s+1)}+\frac{1}{2(s+1)}=1,$$ which is impossible.
Thus any two elements of $N$ are shifted by at most $n-k-2$ around the circle. This gives us $|N|\le n-k-1$. Since any $k$-subset of $A$ summing to at most $\frac{1}{2(s+1)}$ has equal probability of being in $N$, this tells us that the number of $k$-subsets with sum at most $\frac{1}{2(s+1)}$ is at most $\frac{n-k-1}{n}\binom{n}{k}$. Thus there are at least $$\binom{n-1}{k}-\frac{n-k-1}{n}\binom{n}{k}=\frac{n-k}{n}\binom{n}{k}-\frac{n-k-1}{n}\binom{n}{k}=\frac{1}{n}\binom{n}{k}$$ $k$-subsets which are $s$-divisors of $A$ and have a sum of elements greater than $\frac{1}{2(s+1)}$. The sum of elements of such a set is $\frac{s}{m}>\frac{1}{2(s+1)}$, so it can take on one of $2s(s+1)-s=2s^2+s$ values, so there must be some integer $m$ so that at least $\frac{1}{(2s^2+s)n}\binom{n}{k}$ of the $k$-subsets of $A$ sum to $\frac{s}{m}$. Thus at least $\frac{1}{(2s^2+s)n}\binom{n}{n-k}$ of the $(n-k)$-subsets of $A$ sum to $1-\frac{s}{m}$.
If $S\in\binom{A}{n-k-d}$, let $P_S$ be the set of $(n-k)$-subsets obtainable by adding $d$ elements of $A$ to $S$. Note that any $(n-k)$-subset of $A$ is contained in $P_S$ for exactly $\binom{n-k}{d}$ values of $S$, so there is some $S$ so that at least $$\frac{1}{(2s^2+s)n}|P_S|$$ elements of $P_S$ sum to $1-\frac{s}{m}$. They must then form an antichain.
However, the partial order of $P_S$ is the same as the partial order of $\binom{A\wo S}{d}$, so by Lemma \[simplex\], its largest antichain has size less than $$\frac{2}{\sqrt{d}}\frac{1}{k+d}|P_S|<\frac{3}{n\sqrt{d}}|P_S|\le\frac{1}{(2s^2+s)n}|P_S|,$$ yielding a contradiction.
n-C<=k<n-1, k sufficiently large
--------------------------------------
Assume that $n-\left(6s^2+3s\right)^2\le k<n-1$. Let $u=n-k$. Thus $1<u\le \left(6s^2+3s\right)^2$, so $u$ can take on only finitely many values. Assume that $k$ is sufficiently large relative those values. Let $$Y=\left\{B\in\binom{A}{u}\;\middle|\; A\wo B\text{ is an }s\text{-divisor of } A\right\}.$$ By assumption, $|Y|\ge\binom{n-1}{k}=\binom{n-1}{u-1}$.
Let $q$ be as small as possible so that $a_{n-q}<\frac{1}{u(s+1)}$. Note that $q<u(s+1)$. If $B\in\binom{A}{u}$ and $b\le a_{n-q}$ for all $b\in B$, then $\sum B<\frac{1}{s+1}$, so $\sum (A\wo B)>\frac{s}{s+1}$ and $B\notin Y$. Thus every $B\in Y$ contains at least one of the $q$ greatest elements of $A$.
The number of $u$-subsets of $A$ containing at least 2 of the $q$ greatest elements of $A$ is bounded by $$2^q\binom{n-q}{u-2}< 2^{u(s+1)}\binom{n}{u-2}<\half|Y|,$$so at least half of the elements of $Y$ contain exactly one of the $q$ greatest elements of $A$.
Thus there must be some $a_i$ which is one of the $q$ greatest elements of $A$ such that at least $\frac{1}{2u(s+1)}\binom{n-1}{u-1}$ elements of $Y$ include $a_i$ and no other of the $q$ largest elements.
If $B$ is such an element of $Y$, then $$\sum B<a_i+(u-1)\frac{1}{u(s+1)}<\frac{s}{s+1}+\frac{u-1}{u(s+1)}=1-\frac{1}{u(s+1)}.$$ Since $\sum B$ must be of the form $1-\frac{s}{m}$ for some positive integer $m$, we get fewer than $s(s+1)u$ possible values of $m$. Thus there must be some value of $m$ so that there are at least $$\frac{1}{2u^2s(s+1)^2}\binom{n-1}{u-1}$$ different $u$-subsets of $A$ which include $a_i$ and sum to $1-\frac{s}{m}$. However, if we have a collection of that many $u$-subsets of $A$ that contain $a_i$, then some 2 of them will share $u-1$ elements and thus have different sum. This gives us a contradiction.
Conclusion
==========
For $k$ sufficiently large, all $n$ are covered by one of the three last cases. For $k$ small, all but finitely values of $n$ are covered by the first case.
In the statement of Theorem \[s1case\] and Theorem \[generalcase\], “all but finitely many” cannot be omitted. For example, Huynh[@huynh14] notes that $n=4, k=2$, $A=\{\frac{1}{24},\frac{5}{24},\frac{7}{24},\frac{11}{24}\}$ gives $d_k(A)=4>\binom{n-1}{k}$. As $s$ increases, the number of such exceptions grows; in fact, it is easy to see that any $n, k, A$, will be an exception for sufficiently large $s$.
We could follow the proof and trace out the upper bounds on $n$ such that $(k,n)$ is an exception; however these will probably be far from optimal (for instance, for $s=1$, $(2,4)$ is likely the only exception). It would be interesting to get a good bound on the number of such exceptions, or on how large $n$ can be in terms of $s$ for $(k,n)$ to be an exception.
In this paper, we are counting $B\in\binom{A}{k}$ such that $\sum B=\frac{s}{m}$. If we instead counted $B$ such that $\sum B<\frac{k}{n}$, this problem becomes equivalent to the Manickam-Miklós-Singhi conjecture:
For positive integers $n, k$ with $n\ge 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is also nonnegative.
The equivalence is given by taking the complement of $B$ and applying a linear transformation.
The MMS conjecture has been proven for $k\mid n$ [@ManickamMiklos1988], $n\ge 10^{46}k$ [@Pokrovskiy2013], and $n\ge 8k^2$ [@ChowdhurySarkisShariari2013], however there are pairs $(n,k)$ such that it does not hold. This suggests a more general problem.
Fix $S\subseteq [0,1]$ and positive integers $n$ and $k$. If $A$ is a set of positive reals, let $d_k(S,A)$ be the number of subsets $B\in\binom{A}{k}$ such that $\sum B=S$. Let $d(S,k,n)$ be the maximal value of $d(S,k,n)$ over all $A$ with $|A|=n$ and $\sum A=1$. For what $S,k,n$ do we get $d(S,k,n)=\binom{n-1}{k}$? Furthermore, when does $d_k(S,A)\ge\binom{n-1}{k}$ imply that $A$ is an $k$-anti-pencil?
This paper addresses this problem for $S=\{\frac{s}{m}\mid m\in \Zp\}$, while the MMS conjecture deals with this problem for $S=(0,k/n)$. Another example of a set for which this problem might be interesting is a set of the form $S=(0,\alpha k/n)\cup \{\frac{s}{m}\mid m\in \Zp\}$, which combines the theorem of this paper with the MMS conjecture.
Acknowledgements
================
This research was conducted as part of the University of Minnesota Duluth REU program, supported by NSA grant H98230-13-1-0273 and NSF grant 1358659. I would like to thank Joe Gallian for his advice and support. I would also like to thank Adam Hesterberg and Timothy Chow for helpful discussions and suggestions. I would also like to thank Brian Scott for a useful answer on Math Stack Exchange (http://math.stackexchange.com/questions/299770/width-of-a-product-of-chains).
|
.5cm
\[section\] \[lemma\][Teorema]{} \[lemma\][Proposición]{}
\[lemma\][Afirmación]{} \[lemma\][Corolario]{}
\[lemma\][Definición]{}
\[lemma\][Observación]{}
.
.
[Particiones de Markov para difeomorfismos de Anosov - ]{} Eleonora Catsigeras
[ ÍNDICE]{}
0\. RESUMEN
1\. DEFINICIONES Y RESULTADOS PREVIOS
- Difeomorfismos de Anosov
- Expansividad
- Estabilidad topológica
- Conjunto estable e inestable
- Variedades invariantes
- Intersección de variedades invariantes
- Forma local del producto
2\. RECTÁNGULOS Y PARTICIONES DE MARKOV
- Rectángulo de Markov
- Borde de un rectángulo
- Propiedades de los rectángulos de Markov
- Definición de partición de Markov
- Borde de la partición
- Propiedades de las particiones de Markov
3\. SEMICONJUGACIÓN CON EL SHIFT
- Espacio de las sucesiones y función shift
- Conjugación y semiconjugación
- Semiconjugación entre el difeomorfismo de Anosov y el shift.
- Conjuntos estable e inestable en el espacio de las sucesiones.
- Construcción de un cubrimiento con rectángulos
- Propiedades del cubrimiento.
4\. MÉTODO CONSTRUCTIVO PARA LA PARTICIÓN
- Método constructivo para la partición.
- Segundo refinamiento.
- Densidad de un conjunto cubierto por el refinamiento.
- Obtención de la partición.
- Densidad de las variedades estable e inestable.
5\. TEOREMA DE SINAI
- Enunciado
- Lema
- Demostración
6\. DINÁMICA SIMBÓLICA
- Matriz de transición
- Lema
- Teorema de semiconjugación
- Coclusión
7\. REFERENCIAS BIBLIOGRÁFICAS
[Particiones de Markov para difeomorfismos de Anosov - ]{} Eleonora Catsigeras
[RESUMEN ]{}
En este libro se define Partición de Markov de una variedad compacta y riemanniana $M$ para un difeomorfismo $f$ de $M$ en $M$, cuando este difeomorfismo pertenece a cierta clase particular llamada “de Anosov”. El motivo es demostrar la existencia de particiones de Markov. Está dirigido a estudiantes y egresados de nivel de grado universitario en Matemática.
En la parte 1 se exponen definiciones y teoremas que se asumen conocidos, referentes a los difeomorfismos llamados de Anosov. Casi todos los resultados expuestos en la sección 1 se enuncian sin demostración porque no son el motivo de esta presentación. Pueden encontrarse en las referencias [@2], [@3] y [@4].
En la parte 2 se define partición de Markov de una variedad $M$ para un difeomorfismo $f$ en $M$. La definición está referida solamente a los difeomorfismos de Anosov, aunque es aplicable a una clase más general de difeomorfismos en la variedad. (Véase [@5]).
La demostración de la existencia de una partición de Markov (Teorema de Sinai [@6]) se concluye en la parte 5 de este trabajo y se basa en la construcción de un cubrimiento adecuado de la variedad que se luego refinado apropiadamente. Este método constructivo fue extraído del libro de R. Bowen ([@1]).
También se extrajo de R. Bowen ([@1]) la presentación de la dinámica simbólica que se expone en la última parte de esta monografía.
.
[Particiones de Markov para difeomorfismos de Anosov - ]{} Eleonora Catsigeras
Definiciones y Resultados Previos
=================================
Se asume que $M$ es una variedad de clase $C^1$ compacta y Riemanniana y que $f:M \mapsto M$ es un dofeomorfismo de clase $C^1$ en $M$.
[**1.1 Difeomorfismo de Anosov**]{}
*$f$ es un *difeomorfismo de Anosov *si existen constantes $K$ y $\lambda$; $K>0$, $0 < \lambda < 1$; y subespacios $S_x,
U_x$ de $T_x M$, tales que:***
- $S_x, U_x$ varían continuamente con $x$
- $S_x \oplus U_x = T_xM$
- $S_x, U_x$ son invariantes con $f$, es decir: $f'_xS_x = S_{f(x)}, f'_xU_x = U_{f(x)} \; \forall x \in M$
- $\|(f^n)'_x s_x\| \leq K \lambda ^n \|s_x\| \; \forall
s_x \in S_x, \; \forall n \geq 0, \; \forall x \in M$
- $\|(f^n)'_x u_x\| \leq K \lambda ^{-n} \|u_x\| \; \forall
u_x \in U_x, \; \forall n \leq 0, \; \forall x \in M$
*Las dimensiones de $S_x$ y $U_x$ son constantes en las componentes conexas de $M$ debido a la condición i. de la definición anterior. El fibrado tangente $TM$ es la suma directa de los dos subfibrados $S$ y $U$ invariantes con $f$, llamados subfibrado estable“ e inestable” respectivamente. Si $(x,s_x) \in S$, entonces su norma decrece (más que exponencialmente con tasa $+\log \lambda <0$) cuando $n \rightarrow + \infty$. Si $(x,u_x) \in U$, entonces su norma crece (más que exponencialmente con tasa $-\log \lambda >0$) cuando $n \rightarrow + \infty$, pues sustituyendo en v. $m = -n,
y = f^{-m}(x), u_y = (f^{-m})'_xu_x$ resulta: $$\|(f^m)'_y u_y\| \geq \frac{1}{K} \lambda ^{-m}\|u_y\|\; \; \forall u_y \in U_y, \; \forall
m \geq 0, \; \forall y \in M$$ Si $f$ es un difeomorfismo de Anosov, también lo es $f^{-1}$, y el subfibrado estable para $f^{-1}$ es el inestable para $f$ y viceversa, como se observa de la definición de difeomorfismo de Anosov.*
[**1.2 Expansividad**]{}
\[121\] *Un difeomorfismo $f: M \mapsto M$ es *expansivo *si existe una constante $\rho >0$, llamada *constante de expansividad, *tal que $${\mbox{$\,$dist$\,$}}(f^nx, f^ny)\leq \rho \; \forall n \in {\mbox{$Z\!\!\!Z$}}\;\; \mbox {si y solo si }
\;\; x= y$$ Una sucesión bi-infinita $\{y_n\}_{n \in {\mbox{$Z\!\!\!Z$}}}, \; y_n
\in M $, se dice que *$\epsilon-$acompaña *a otra $\{x_n\}_{n \in {\mbox{$Z\!\!\!Z$}}}, \; x_n \in M$, si ${\mbox{$\,$dist$\,$}}(x_n, y_n) \leq
\epsilon \; \forall n \in {\mbox{$Z\!\!\!Z$}}$. La expansividad de un difeomorfismo significa que para cierto $\rho >0 $ suficientemente pequeño, dos órbitas diferentes nunca se $\rho-$acompañan.*******
Un propiedad conocida es la siguiente:
Todo difeomorfismo de Anosov es expansivo.
[**1.3 Estabilidad topológica.**]{}
*Un difeomorfismo $f: M \mapsto M$ es *topológicamente estable *si dado $\epsilon > 0$ existe un $C^0$ entorno ${\cal
V}$ de $f$ tal que para todo $g \in {\cal V}$ existe una semiconjugación $h: M \mapsto M$ entre $g$ y $f $ (i.e. h es continua, sobreyectiva y cumple $h \circ g = f \circ h$) tal que ${\mbox{$\,$dist$\,$}}(h(x),x) < \epsilon \;\; \forall x \in M$.***
Si $f$ es topológicamente estable, si ${\cal V}$ es un $C^0$ entorno de $f$ como en la definición anterior y si $g \in {cal V}$ entonces $$h (g^n(x))= f^n(h(x)) \;\; \forall x \in M \;\; \forall n \in {\mbox{$Z\!\!\!Z$}}$$ La estabilidad topológica de $f$ significa que si $g$ está suficientemente próxima de $f$ (en la topología $C^0$) entonces las órbitas de $g$ están $\epsilon$-acompañadas por las de $f$ y $\epsilon$ acompañan a todas las órbitas de $f$ (ya que la transformación $h$ es sobreyectiva.)
\[133\] *[**(Pugh)** ]{} *Un difeomorfismo $f: M \mapsto M$ es topológicamente estable si y sólo si dado $\epsilon >0$ existe $\delta >0$ tal que toda sucesión bi-infinita de puntos $\{y_n\}{n
\in {\mbox{$Z\!\!\!Z$}}}, \; y_n \in M$ que cumple ${\mbox{$\,$dist$\,$}}(y_{n+1}, f(y_n))< \delta
\; \forall n \in {\mbox{$Z\!\!\!Z$}}$ está $\epsilon$-acompañada por una órbita de $f$**
*Sea $f: M \mapsto M$ invertible. Una sucesión bi-infinita $\{y_n\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ de puntos de $M$ se llama *$\delta$-pesudo-órbita de $f$ *si ${\mbox{$\,$dist$\,$}}(f(y_n), y_{n+1})<
\delta \; \forall n \in {\mbox{$Z\!\!\!Z$}}$.***
Se concluye que un difeomorfismo $f: M \mapsto M$ es topológicamente estable si y solo si dado $\epsilon >0$ existe $\delta >0$ tal que toda $\delta$-pseudo-órbita está $\epsilon$-acompañada.
\[135\] Los difeomorfismos de Anosov son topológicamente estables.
[**1.4 Conjuntos estable e inestable.**]{}
\[141\] *Sea $f: M \mapsto M$ un difeomorfismo. Se llama *conjunto estable *de $f$ por el punto $x \in M$ a $$W^s(x)= \{y \in M: {\mbox{$\,$dist$\,$}}(f^n(y), f^n(x)) _{n \rightarrow + \infty} \rightarrow 0\}$$ Se llama *conjunto inestable *de $f$ por el punto $x \in M$ a $$W^u(x)= \{y \in M: {\mbox{$\,$dist$\,$}}(f^n(y), f^n(x)) _{n \rightarrow - \infty} \rightarrow 0\}$$*****
Si $f$ es un difeomorfismo de Anosov entonces *$$W^s=\{ y \in
M: \limsup_{n \rightarrow + \infty} \frac {1}{n} \log {\mbox{$\,$dist$\,$}}(f^n(x), f^n(y)) \leq \log \lambda\}$$ $$W^u=\{y \in M:
\limsup_{n \rightarrow + \infty} \frac {1}{n} \log {\mbox{$\,$dist$\,$}}(f^{-n}(x), f^{-n}(y)) \leq \log \lambda\}$$ *donde $\lambda$ es la misma constante $0< \lambda < 1$ de la definición de Anosov.**
El teorema anterior significa que para los difeomorfismos de Anosov, dos órbitas que en el futuro (o en el pasado) se acercan de modo que su distancia tienda a cero, entonces lo hacen más que exponencialmente con tasa $\log \lambda$. En efecto: $\lim sup_{n
\rightarrow + \infty} \frac {1}{n} \log {\mbox{$\,$dist$\,$}}(f^n(x), f^n(y))
\leq \log \lambda$ implica $${\mbox{$\,$dist$\,$}}(f^n(x),f^n(y))< A e ^{-n
\gamma} \; \; \forall n \geq 0$$ donde $A$ es un número positivo y $\gamma$ es un número real positivo elegido de modo que $\gamma <
-\log \lambda$.
*Dos conjuntos estables distintos son disjuntos pues si $z \in
W^s(x)\cap W^s(x') \neq \emptyset$ entonces, a partir de la definición de conjunto estable y la propiedad triangular de la distancia se tiene $W^s(z) \subset W^s(x) \cap W^s(x')$ y $W^s(x) \cup W^s(x') \subset W^s(z)$. Luego $W^s(x) = W^s(x')$.*
Los mismo vale para los conjuntos inestables.
\[145\] Si $f$ es un difeomorfismo expansivo con constante de expansividad $\rho$, entonces: *$$W^s(x)= \{y \in M: {\mbox{$\,$dist$\,$}}(f^n(x), f^n(y)) \leq \rho \; \forall n \mbox{ suficientemente
grande }\}$$ $$W^u(x)= \{y \in M: {\mbox{$\,$dist$\,$}}(f^{-n}(x), f^{-n}(y)) \leq \rho \; \forall n
\mbox{ suficientemente
grande }\}$$*
[*Demostración:* ]{}De la definición de conjunto estable se obtiene que $$W^s(x) \subset \{y\in M: {\mbox{$\,$dist$\,$}}(f^n(x),f^n(y)) \leq
\rho \; \forall n \mbox{ suficientemente grande}\})$$ Para demostrar la otra inclusión supongamos por absurdo que existe $y
\in M$ tal que $${\mbox{$\,$dist$\,$}}(f^n(x), f^n(y)) \leq \rho \; \forall n \geq N , \; {\mbox{$\,$dist$\,$}}(f^n(x),f^n(y))_{
n \rightarrow + \infty} \not \rightarrow 0$$ Entonces existe una sucesión $n_k \rightarrow + \infty$ tal que ${\mbox{$\,$dist$\,$}}(f^{n_k}(x),
f^{n_k}(y))\geq \epsilon >0 \; \forall k$. Podemos elegir $n_k$ de modo que $f^{n_k}(x)$ y $ f^{n_k}(y)$ sean convergentes (por la compacidad de $M$). Luego si $p \in {\mbox{$Z\!\!\!Z$}}$ se tiene $${\mbox{$\,$dist$\,$}}(f^{n_k+p}(x),f^{n_k+p}(y))\leq \rho \;\; \forall n_k >N -p$$ Cuando $k \rightarrow + \infty$ se tiene ${\mbox{$\,$dist$\,$}}(f^p(x_0),
f^p(y_0))\leq \rho \;\forall p \in {\mbox{$Z\!\!\!Z$}}$ donde $x_0 = \lim
f^{n_k}(x), \; y_0 = \lim f^{n_k}(y)$. Por la expansividad $x_0 =
y_0$, contradiciendo la elección de $n_k$, pues $${\mbox{$\,$dist$\,$}}(f^{n_k}(x), f^{n_k}(y))\geq \epsilon >0 \; \;\;\; \Box$$
\[146\] *Se observa que la proposición anterior sigue siendo válida si se sustituye la constante de expansividad $\rho$ por cualquier otra constante $\epsilon
>0, \epsilon \leq \rho$. Entonces $$W^s(x) = \bigcup _{n \in {\mbox{$Z\!\!\!Z$}}} \{y \in M: {\mbox{$\,$dist$\,$}}(f^n(x), f^n(y))\leq \epsilon \;
\forall n \geq N\}=$$ $$= \bigcup _{n \geq k} \{y \in M: {\mbox{$\,$dist$\,$}}(f^n(x), f^n(y))\leq \epsilon \; \forall n \geq N\} \;\; \;\forall
k \in {\mbox{$Z\!\!\!Z$}}$$*
*Se llama *$\epsilon-$conjunto estable *de $f$ por el punto $x \in M$ al conjunto $$W^s_\epsilon (x)= \{y \in M: {\mbox{$\,$dist$\,$}}(f^n(x), f^n(y))\leq \epsilon \; \forall n \geq 0\}$$***
Se llama *$\epsilon-$conjunto inestable *de $f$ por el punto $x \in M$ al conjunto $$W^u_\epsilon (x)= \{y \in M: {\mbox{$\,$dist$\,$}}(f^n(x), f^n(y))\leq \epsilon \; \forall n \leq 0\}$$**
*\[148\] Si $f$ es un difeomorfismo expansivo es fácil ver, a partir de la definición anterior y de \[145\] y \[146\] que:*
- $y \in W^s_{\epsilon } (x) \; \Leftrightarrow \; x \in W^s_{\epsilon}
(y); \;\;\;\; y \in W^u_{\epsilon } (x) \; \Leftrightarrow \; x
\in W^u_{\epsilon} (y)$.
- Si $\epsilon \leq \rho$ entonces $ W^s_\epsilon (x) \cap
W^u_\epsilon (x) = \{x\}$.
- $f(W^s_\epsilon(x)) = \{z \in M: {\mbox{$\,$dist$\,$}}(f^{n}(x), f^{n-1}(z))\leq \epsilon \;
\forall n \geq 0\} = $ $$= \{z \in M: {\mbox{$\,$dist$\,$}}(f^{n-1}(f(x)),
f^{n-1}(z))\leq \epsilon \; \forall n \geq 1\}\cap \{z \in M:
{\mbox{$\,$dist$\,$}}(x, f^{-1}(z))\leq \epsilon \} =$$ Luego $$f(W^s_\epsilon(x)) = W^s_{\epsilon }(f(x)) \cap f(\overline
B_{\epsilon }(x))$$ Análogamente $$W^u_\epsilon(f(x)) =
f(W^u_{\epsilon }(x)) \cap \overline B_{\epsilon }(f(x))$$
- $f(W^s_{\epsilon}(x)) \subset W^s_{\epsilon }(f(x))$ y $f(W^u_{\epsilon}(x)) \supset
W^u_{\epsilon }(f(x))$
- Si $\epsilon \leq \rho$ (véase \[146\]) entonces: $$W^s (x) = \bigcup _{N \geq 0}
f^{-N}(W^s_{\epsilon} (f^N(x))) = \bigcup _{N \in {\mbox{$Z\!\!\!Z$}}}
f^{-N}(W^s_{\epsilon } (f^N(x)))$$ En particular $$W^s_{\epsilon }(x) \subset W^s(x),
\;\;\; W^u_{\epsilon }(x) \subset W^u(x)$$
[**1.5 Variedades invariantes.**]{}
El teorema que sigue justifica el nombre de *variedad invariante estable *(respectivamente *inestable*) que recibe el conjunto estable (respectivamente inestable) cuando $f$ es un difeomorfismo de Anosov. Se enuncia sin demostración, la cual puede encontrarse en la referencia [@3].****
\[151\] Sea $f: M \mapsto M$ un difeomorfismo de Anosov. Entonces los conjuntos $W^s(x)$ y $W^u (x)$ son $C^1$ variedades inmersas en $M$, que pasan por $x$, tangentes en $x$ a los subespacios $S_x$ y $U_x$ respectivamente.
Se observa de la definición \[141\] que la partición de $M $ en las variedades estables e inestables es invariante por $f$; más precisamente: $f(W^s(x)) = W^s(f(x)); \; f(W^u(x)) = W^u(f(x))$
*Cuando $f$ es un difeomorfismo de Anosov, entonces $W^s(x)$ y $W^u(x)$ son variedades inmersas en $M$ según afirma el teorema \[151\]. Sin embargo no son necesariamente subvariedades de $M$: la inclusión es continua pero no necesariamente un homeomorfismo sobre su imagen (la topología en $W^s(x)$ podría ser estrcitamente más fina que la inducida por $M$).*
Sea $f: M \mapsto M$ un difeomorfismo de Anosov. Si $\epsilon >0$ es suficientemente pequeño entonces:
- Para todo $N \in {\mbox{$Z\!\!\!Z$}}$ el conjunto $f^N(W^s_{\epsilon }(f^{-N}(x)))$ es un entorno de $x$ en la variedad $W^s(x)$ homeomorfo a una bola en el subespacio $S_x$
- La topología en $f^N(W^s_{\epsilon }(f^{-N}(x)))$ como entorno en la variedad $W^s(x)$ es la misma que la inducida en él por la topología de $M$
La demostración se encuentra en l referencia bibliográfica \[3\]. Es parte de la demostración del teorema \[151\] de existencia variedades invariantes.
En particular $W^s_{\epsilon } (x)$ es un entorno de $x$ en $W^s(x)$ y la topología en $W^s_{\epsilon }(x)$ como subconjunto de $W^s(x)$ es la misma que la inducida por la topología de $M$ como subconjunto de $M$.
Aplicando el teorema anterior a $f^{-1}$ en lugar de $f$ y observando que por definición las variedades estables de $f^{-1}$ son las inestables de $f$, se obtiene que $W^u_{\epsilon } (x)$ es un entorno de $x$ en $W^u(x)$ y su topología como subconjunto de $W^u(x)$ es la misma que la inducida por la topología de $M$.
En virtud de la primera parte del teorema anterior, los $epsilon-$ conjuntos estable e inestable se llaman $epsilon-$variedades estable e inestable, y son subvariedades de $M$ (variedades encajadas, con la topología inducida por la de $M$).
De la segunda parte del teorema anterior se desprende que:
- Si $U$ es un abierto de $W^s_{\epsilon }(x)$ (como variedad estable), entonces existe $B$ abierto en $M$ tal que $U = B \cap W^s_{\epsilon }(x)$
- Si $x_n, x \in W^s_{\epsilon }(x)$ y si ${\mbox{$\,$dist$\,$}}(x_n, x) \rightarrow
0$ en $M$, entonces $x_n \rightarrow x$ en $W^s_{\epsilon }(x)$ (y recíprocamente).
[**1.6 Intersección de variedades invariantes.**]{}
En la observación \[148\] obtuvimos que $W^s_{\epsilon } (x) \cap W^u{\epsilon } (x) = \{x\}$ si $0
<\epsilon < \rho$. Probaremos a continuación que cuando $x$ e $y$ están suficientemente próximos y $\epsilon >0 $ es pequeño, entonces $W^s_{\epsilon }\cap W^u{\epsilon } (y) $ consiste en un único punto que denotaremos como $[x,y]$. En lo que sigue $f: M
\mapsto M$ denota un difeomorfismo de Anosov.
\[161\] Dado $0<\epsilon \leq \rho /2$ existe $\delta >0$ tal que $W^s_{\epsilon } (x) \cap W^u_{\epsilon }(y)$ consiste en un único punto, para todos $x,y \in M$ tales que ${\mbox{$\,$dist$\,$}}(x,y) < \delta$.
[*Demostración:* ]{} Por los teoremas \[133\] y \[135\] existe $\delta _1
>0 $ tal que toda $\delta _1$ pseudo-órbita de $f$ está $\epsilon
/2$ acompañada por una órbita de $f$.
Tomemos $delta = \min (\delta _1, \epsilon /2)$ y dos puntos $x,y
\in M$ tales que ${\mbox{$\,$dist$\,$}}(x,y) < \delta$.
La sucesión bi-infinita $\{y_n\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ definida por $y_n =
f^n(x)$ si $n \geq 0 $, $ y_n = f^n (y_n)$ si $n < 0 $ es una $\delta _1$ pseudo-órbita. Entonces existe $z \in M$ que cumple ${\mbox{$\,$dist$\,$}}(f^n(z), f^n(x)) \leq \epsilon /2 \; \forall n \geq 0,
{\mbox{$\,$dist$\,$}}(f^n(z), f^n(y)) \leq \epsilon /2 \; \forall n < 0$. Además ${\mbox{$\,$dist$\,$}}(z,y) \leq {\mbox{$\,$dist$\,$}}(z,x) + {\mbox{$\,$dist$\,$}}(x,y) \leq \epsilon /2 +
\delta \leq \epsilon$.
Entonces $z \in W^s_{\epsilon }(x) \cap W^u_{\epsilon }(y)$.
Además $z$ es único en $W^s_{\epsilon }(x) \cap W^u_{\epsilon
}(y)$ debido a la expansividad de $f$. $\;\; \Box$
*Llamaremos función *corchete *a $$[\cdot, \cdot]: \{(x,y) \in M^2: {\mbox{$\,$dist$\,$}}(x,y < \delta)\} \mapsto M$$ definida por $$[x,y]= W^s_{\epsilon }(x) \cap W^u_{\epsilon }(y)$$***
Se observa que $${\mbox{$\,$dist$\,$}}(f^n([x,y]), f^n(x))\leq \epsilon \;
\forall n \geq 0, \;\;\; {\mbox{$\,$dist$\,$}}(f^n([x,y]), f^n(y))\leq \epsilon
\; \forall n \leq 0$$ debido a la definición de la función corchete y a la definición de las $\epsilon$-variedades estable e inestable.
La función corchete $[\cdot, \cdot]$ es continua.
[*Demostración:* ]{} Sea $(x_n, y_n ) \rightarrow (x,y)$ en $M^2$ tales que ${\mbox{$\,$dist$\,$}}(x,y) < \delta$. Como $M$ es compacta puede elegirse $(x_n, y_n)$ de modo que $[x_n,y_n]$ sea convergente en $M$. Sea $z_n= [x_n,y_n] \rightarrow z \in M$. Basta demostrar que $z = [x,y]$.
Como $z_n \in W^s_{\epsilon }(x_n)$ entonces ${\mbox{$\,$dist$\,$}}(f^p(z_n),
f^p(x_n))\leq \epsilon \; \forall p \geq 0$. Dejando fijo $p$ y haciendo $n \rightarrow \infty$, por la continuidad de $f$ se tiene que ${\mbox{$\,$dist$\,$}}(f^p(z), f^p(x))\leq \epsilon \; \forall p \geq
0$, de donde $z \in W^s_{\epsilon }(x)$.
Análogamente se obtiene $z \in W^u_{\epsilon }(y)$, de donde $z
\in W^s_{\epsilon } (x) \cap W^u_{\epsilon }(y) = [x,y] \;\; \Box$
[**1.7 Forma local del producto**]{}
\[171\] Sea $f: M \mapsto M$ un difeomorfismo de Anosov. Existe una constante $\overline \epsilon >0 $ tal que para todo $x \in M$ el producto $W^u_{\overline \epsilon }(x) \times W^s _{\overline
\epsilon }(x)$ es homeomorfo a un entorno de $x $ en $M$.
[*Demostración:* ]{} Elijamos $0<\epsilon < \rho /2$ (donde $\rho$ es la constante de expansividad de $f$). Sea $\delta >0$ elegido según el teorema \[161\] y sea $0 < \overline \epsilon < \min
(\delta /2, \epsilon )$. Resulta: $$W^u_{\overline \epsilon } (x) \times W^s_{\overline \epsilon} (x)
\subset \{(z,y) \in M^2: {\mbox{$\,$dist$\,$}}(z,y)< \delta \}$$ Puede aplicarse la función corchete a puntos en $W^u_{\overline \epsilon}
(x) \times W^s_{\overline \epsilon} (x)
\subset M^2$.
Sea $\varphi = \left . [\cdot, \cdot]\right |_{
W^u_{\overline \epsilon } (x) \times W^s_{\overline \epsilon}
(x)}$. La aplicación $\varphi$ es continua porque es la restricción de una función continua. Es inyectiva pues si $[z,z'] = [\overline z,
\overline z ']$ donde $z, \overline z \in W^u_{\overline \epsilon}(x), \;
z', \overline z' \in W^s_{\overline \epsilon}(x)$ entonces $${\mbox{$\,$dist$\,$}}(f^n(z), f^n(\overline z))\leq 2 \overline \epsilon < 2 \epsilon \;
\forall n \geq 0$$ $${\mbox{$\,$dist$\,$}}(f^n(z), f^n(\overline z))\leq 2 \overline \epsilon < 2 \epsilon \;
\forall n \leq 0$$ Luego por la expansividad de $f$ se tiene $z = \overline z$. Análogamente se obtiene $z' = \overline z'$.
La $\overline \epsilon$-variedad estable $W^s_{\overline \epsilon}(x)=
\{y \in M: {\mbox{$\,$dist$\,$}}(f^n(y), f^n(x))\leq \overline \epsilon \; \forall n \geq
0\}$ es cerrada en $M$ que es compacta, luego es compacta. Análogamente es compacta $W^u_{\overline \epsilon}(x)$. Así $\varphi$ es continua e inyectiva con dominio compacto $W^u_{\overline \epsilon}(x)\times W^s_{\overline
\epsilon}(x)$ a $M$. Como el dominio de $\varphi$ y su codominio son variedades de la misma dimensión finita, $\varphi$ es un homeomorfismo sobre su imagen.
En efecto $\varphi (x,x) = x,
\; \dim W^s_{\overline \epsilon}(x) = \dim S_x, \; \dim
W^u_{\overline \epsilon}(x) = \dim U_x, \; S_x \oplus U_x = T_x M$ de donde $\dim (W^u_{\overline \epsilon}(x)\times W^s_{\overline
\epsilon}(x))= \dim M$.
Siendo $W^u_{\overline \epsilon}(x)\times W^s_{\overline
\epsilon}(x)$ un entorno de $(x,x)$ en $M^2$, su imagen homeomorfa es un entorno de $x \in M. \;\; \Box$
[Particiones de Markov para difeomorfismos de Anosov - ]{} Eleonora Catsigeras
Rectángulos y particiones de Markov
===================================
Sea $f$ un difeomorfismo de Anosov en una variedad compacta y Riemanniana $M$. Se definirá *Partición de Markov *para $f$. Es un cubrimiento finito de $M$ por cierta clase de cerrados llamados *rectángulos, *con interiores dos a dos disjuntos, y que cumplen condiciones que los vinculan a la dinámica de $f$, es decir, al espacio de órbitas de $f$.****
Comenzaremos definiendo *rectángulo *y demostrando algunas propiedades que serán utilizadas más adelante.**
[**2.1 Definición de rectángulo**]{}
Sea $\epsilon >0$ tal que $0 < \epsilon \rho /4$ (donde $\rho $ denota la constante de expansividad de $f$, definida en \[121\]. y sea $\delta >0$ elegido como en \[161\].
*Un subconjunto $R$ no vacío de $M$ se llama *rectángulo *para $f $ si tiene diámetro menor que $\delta$ y además $[x,y]\in
R\; \forall x,y \in R$. Es decir: si $x,y \in R$ entonces existe un único $z \in W^s_{\epsilon }(x) \cap W^s_{\epsilon }(y) =
[x,y]$ y además $z \in R$.***
*Un rectángulo $R$ es propio si $R = \overline{ \mbox{int}R }$.*
[**Ejemplos:** ]{}
- A partir de la definición obsérvese que si $\overline \epsilon >0$ se elige suficientemente pequeño entonces es un rectángulo el entorno $V$ de $x$ en $M
$ que tiene forma local del producto según el teorema \[171\] (es decir $V$ es homeomorfo a $W^u_{\overline \epsilon }(x) \times W^s_{\overline \epsilon
}(x)$).
- Si $x$ e $y$ son dos puntos próximos en $M$ y si $U$ y $V$ son entornos de $x$ e $y$ respectivamente, como en el ejemplo anterior, entonces $U \cup V \cup [U,V] \cup [V,U]$ es un rectángulo.
- Si $U$ y $V$ son dos rectángulos no disjuntos, entonces $U \cap V$ es un rectángulo.
*Sea $R$ un rectángulo y $x \in R$. Se llama $\epsilon$-variedad estable de $x$ en $R$ a $$W^s(x,R)= W^s_{\epsilon }(x) \cap R$$ Análogamente: $$W^u(x,R)= W^u_{\epsilon }(x) \cap R$$*
\[215\] Sean $x,y \in R$, $R$ rectángulo. Entonces $y \in W^s(x,R)$ si y solo si $W^s(x,R) = W^s(y,R)$.
[**Demostración:** ]{} Como $y \in W^s(y,R)$ es inmediato que $W^s(x,R ) \subset W^s(y, R)$ implica $y \in W^s(x,R)$.
Para el recíproco alcanza probar que $y \in W^s(x,R)$ implica $W^s(x,R) \subset W^s(y,R)$ (pues por simetría $y \in W^s(x,R)$ si y solo si $x \in W^s(y,R)$).
Probemos entonces que $y,z \in W^s(x, R)$ implica $z \in
W^s(y,R)$:
Sea $w = [z,y]= W^s_{\epsilon }(z) \cap W^u_{\epsilon } (y)$. Como $z \in W^s_{\epsilon } (x)$. Entonces $w = W^s_{2 \epsilon }(x )
\cap W^u_{\epsilon} (y)$.
Además $y \in W^s_{epsilon } (x)$. Entonces $w = W^s_{3 \epsilon
}(y ) \cap W^u_{\epsilon} (y)$. Siendo $3 \epsilon > \rho$, donde $\rho$ es la constante de expansividad, se tiene que $w = y$ o sea: $$y = [z,x] = W^s_{\epsilon }(z) \cap W^u_{\epsilon } (y)$$ de donde $y \in W^s_{epsilon } (z)$, o lo que es lo mismo $z \in
W^s_{\epsilon } (y)$ como queríamos demostrar. $ \Box $
*Se llama *Borde Estable *de un rectángulo $R$ al conjunto $$\partial ^s R = \{x \in R: x \not \in
\mbox{int}W^u(x,R) \mbox { en } W^u_{\epsilon }(x) \}$$ Se llama *Borde Inestable *de un rectángulo $R$ al conjunto $$\partial ^u R = \{x \in R: x \not \in \mbox{int}W^s(x,R) \mbox {
en } W^s_{\epsilon }(x) \}$$*****
Demostraremos que para los rectángulos $R$ cerrados, el borde topológico de $R$ es $\partial ^s R \cup \partial ^u R$. Además para justificar el nombre de borde estable, demostraremos que $\partial ^s(R)$ está formado por la unión de $\epsilon$-variedades estables en $R$:
$y \in \partial ^s R$ implica $W^s(y,R)\subset \partial ^sR$.
$y \in \partial ^u R$ implica $W^u(y,R)\subset \partial ^uR$
[**Demostración:** ]{} Por absurdo sea $x \in W^s(y, R)\setminus
\partial ^sR$. Entonces $x \in \mbox {int} W^u (x,R)$ en $W^u
_{\epsilon } (x)$, o sea existe un entorno $V$ de $x$ en $W^u_{\epsilon }(x)$ contenido en $R$.
Sea $\varphi (z)= [z,x]= W^s_{\epsilon }(z) \cap W^u _{\epsilon}
(x)$ definido para los puntos $z \in W^u {\epsilon }(y)$ que están a distancia menor que $\delta$ de $x$.
$\varphi$ es continua pues es la restricción de $[\cdot, \cdot]$.
Además $\varphi (y)= [y,x]= x$ porque $x \in W^s_{\epsilon }(y)$.
Entonces $\varphi ^{-1}(V)$ es un abierto de $W^u_{\epsilon }(y)$ que contiene a $y$. Además si $z \in \varphi ^{-1} (V)$ entonces $\varphi (z) = u \in V \subset R$. Luego $[z,x]= u$, de donde $z
\in W^s_{\epsilon } (u)$. Como $z \in W^u _{\epsilon }(y)$ se obtiene que $z = [u,y]$ con $u,y \in R$. Entonces por definición de rectángulo $z \in R$. Se tiene así que $\varphi ^{-1} (V)
\subset R$.
Se ha hallado un entorno de $y$ en $W^u_{\epsilon }(y)$ contenido en $R$. Entonces $y \in \mbox {int} W^u(y,R)$ en $W^u_{\epsilon
}(y)$, o sea $y \not \in \partial ^sR$ contradiciendo la hipótesis. $\Box$
\[223\] Si $R$ es un rectángulo cerrado entonces $\partial R = \partial ^s
R \cup \partial ^u R$.
[**Demostración:** ]{} Veremos que $\mbox{int}R = R \setminus
(\partial ^sR \cup \partial ^u R)$ Según la definición de borde estable e inestable tenemos que $$R \setminus (\partial ^s R \cup \partial ^u R) =
\{ x \in R : x \in \mbox{int } W^u(x,R) \mbox{ en } W^u_{\epsilon
} (x); \;\; x \in \mbox{int } W^s(x,R) \mbox{ en } W^s_{\epsilon
}(x)\}$$ Sea $y \in \mbox{int }R$, sea $B$ un entorno de $y$ en $M$ contenido en $R$. Tenemos que $$y \in W^u_{\epsilon } (y) \cap B \supset W^u_{\epsilon }(y) \cap R = W^u(y,R)$$ Luego $W^u_{\epsilon }(y) \cap B$ es un entorno de y en $W^u_{\epsilon }(y)$ que está contenido en $W^u(y,R)$, y entonces $y \not \in \partial ^sR$.
De igual forma tenemos que $y \not \in \partial ^u R$ , con lo cual deducimos que $$\mbox{int }R \subset R \setminus (\partial ^sR \cup \partial ^uR)$$ Recíprocamente: Si $y \in \mbox{int }W^u(y,R)$ en $W^u_{\epsilon
}(y)$ entonces existe un entorno de $y$ en $W^u_{\epsilon }(y)$ que está contenido en $R$. Llamemos $V$ a su intersección con $W^u{\overline \epsilon }(y)$, siendo $\overline \epsilon $ >0 elegido como en el Teorema \[171\].
$V$ es un entorno de $y$ en $W^u_{\epsilon }(y)$ porque $W^u{\overline \epsilon }(y)$ lo es. Además $V \subset R \cap
W^u{\overline \epsilon }(y)$. De igual forma hallemos $U$ entorno de $y$en $W^s_{\epsilon }(y)$ contenido de $R \cap W^s{\overline
\epsilon }(y) $.
Por el Teorema \[171\] $[V,U]$ es un entorno de $y$ en $M$. Como $V,U \subset R$, por la definición de rectángulo se deduce que $[V,U] \subset R$. Entonces $y \in \mbox{int }R$. Luego $R \setminus (\partial ^s R \cup \partial ^u R) \subset \mbox{int
}R.
\; \;\;\ \Box$
[**2.3 Propiedades de los rectángulos** ]{}
Algunas propiedades que se demuestran a continuación serán utilizadas en los parágrafos siguientes:
Si $R$ es un rectángulo entonces también los son $\overline R$ y $\mbox{int }R$ cuando no es vacío.
Si $R_1$ y $R_2$ son rectángulos tales que $R_1 \cap R_2 \not =
\emptyset$ entonces $R_1 \cap R_2$ también es un rectángulo.
[*Demostración:* ]{} Sean $x,y \in \overline R, \; x_n = \rightarrow x, \;
y_n \rightarrow y, \; x_n, y_n \in R$. Probemos que $[x,y]\in \overline R for
all x,y \in \overline R$. Se tiene por la continuidad de la función corchete que $[x,y]= \lim [x_n,
y_n]$. Según la definición de rectángulo se sabe que $[x_n, y_n] \in
R$, ya que $x_n,y_n \in R$. Entonces $[x,y]\in \overline R$ como se quería.
Sean ahora $x,y \in \mbox{int }R$. Entonces $[x,y]\in R$. Por la expansividad de $f$ se tiene que $x = [[x,y],x]$ e $y =
[y,[x,y]]$.
Sean $V_x, V_y$ entornos de $x$ e $y$ respectivamente, ambos contenidos en $R$. Por la continuidad de la función corchete existe $V$ entorno de $[x,y]$ en $M$ tal que $[V,x] \subset V_x, \;\;
[y,V] \subset V_y$. Entonces para todo $z \in V$ se cumple $$[[z,x],[y,z]] \in R$$ Siendo $\epsilon < \rho /4$ tenemos que $z= [[z,x],[y,z]]$ y entonces $V \subset R$. Luego $[x,y]\in \mbox{int }R$ como se quería.
La intersección no vacía de rectángulos es un rectángulo como se ve inmediatamente a partir de la definición de rectángulo. $\Box$
[**2.4 Definición de Partición de Markov** ]{}
*Una *partición por cerrados *de una variedad $M$ es un cubrimiento finito de $M$ por cerrados $R_i$ con interiores dos a dos disjuntos. El diámetro de la partición es el máximo de los diámetros de los conjuntos cerrados que la componen.***
*\[242\] Una partición por cerrados de $M$ es una *partición de Markov *para el difeomorfismo de Anosov $f$ si está constituida por rectángulos propios $R_1, R_2, \ldots, R_m$ y si para todo $ x \in
\mbox{int }R_i \cap f^{-1} \mbox{int }R_j$ se cumple:***
- $fW^s(x, R_i) \subset W^s(fx, R_j)$
- $ fW^u(x, R_i) \supset W^u(fx, R_j)$
*\[244\] La partición de Markov está vinculada a $f$ a través de la definición de rectángulo y de las condiciones (i) y (ii). Si $\mbox{int }R_i \cap f^{-1} \mbox{int }R_j \neq \emptyset$ entonces $f(R_i)$ se obtiene de $R_i$ al aplicarle $f$ comprimiendo las variedades $\epsilon$- estables y dilatando las inestables.*
Sea una partición ${\cal R}$ en $m$ subconjuntos de $M$(${\cal R
}$ no es necesariamente de Markov), con diámetro $\beta < \rho$ ( $\rho $ es la constante de expansividad de $f$). Se cumple:
- $\cap _{n \in {\mbox{$Z\!\!\!Z$}}} f^{-n} R_{j_n} $ consta a lo sumo de un punto (donde $j_n$ es una sucesión bi-infinita de números en $\{1,2, \ldots, m\}$). Esto es porque$$x, y \in \bigcap _{n \in
{\mbox{$Z\!\!\!Z$}}} f^{-n} R_{j_n} \;\; \Rightarrow {\mbox{$\,$dist$\,$}}(f^n x, f^n y) <
\rho\;\; \forall n \in {\mbox{$Z\!\!\!Z$}}\;\; \Rightarrow x= y$$
- Sea $\Sigma$ el conjunto de las sucesiones $\{j_n\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ tales que $\cap _{n \in {\mbox{$Z\!\!\!Z$}}} R_{j_n } \neq \emptyset$. Por lo observado antes existe una función $\Pi : \sigma \mapsto M$ definida por $$\Pi (\{ j_n \}_{n \in {\mbox{$Z\!\!\!Z$}}})= \bigcap _{n \in {\mbox{$Z\!\!\!Z$}}} R_{j_n}$$ $\Pi $ es sobreyectiva pues toda órbita $\{f^n x\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ está cubierta por conjuntos de la partición. Así dado $x \in M $ existe alguna sucesión $j_n$ tal que $f^n(x) \in R_{j_n } \; \;
\forall n \in {\mbox{$Z\!\!\!Z$}}$. Luego $x \in \cap _{n \in {\mbox{$Z\!\!\!Z$}}} f^{-n} R_{j_n}$.
- Si $x,y \in \cap _{n \geq 0 } f^{-n} R_{j_n}$ entonces ${\mbox{$\,$dist$\,$}}(f^n x, f^n y) \leq \beta \; \forall n \geq 0$. Luego $y
\in W^s _{\beta } (x) \cap R_{j_0 }$. Hemos probado que para cualquier partición ${\cal R }$ se cumple: $$x \in \cap _{n \geq 0 } f^{-n} R_{j_n } \; \Rightarrow
\cap _{n \geq 0 } f^{-n} R_{j_n } \subset W^x _{\beta }(x) \cap
R_{j_0 }$$
*Si $x = \Pi \{j_n\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ entonces $f^n (x) \in R_{j_n
} \forall n \in {\mbox{$Z\!\!\!Z$}}$, o sea $f^n (fx) \in R_{n_{j+1}} \; \forall n
\in {\mbox{$Z\!\!\!Z$}}$, de donde $fx = \Pi \{j_{n+1}\}_{n \in {\mbox{$Z\!\!\!Z$}}}$.*
Llamemos *shift *a la transformación $\sigma : \Sigma
\mapsto \Sigma $ tal que a la sucesión $\{j_n\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ hace corresponder la sucesión $\{j_{n+1}\}_{n \in {\mbox{$Z\!\!\!Z$}}}$.**
Hemos obtenido que $\Pi \circ \sigma = f \circ \Pi$, es decir, conmuta el siguiente diagrama $$\begin{array}{ccccc}
& & \sigma & & \\
& \Sigma & \rightarrow & \Sigma & \\
\Pi & \downarrow & & \downarrow & \Pi \\
& M & \rightarrow & M & \\
& & f & & \\
\end{array}$$ Además como $\Sigma$ y $f$ son invertibles se cumple que $\Pi \circ \Sigma ^n =
f^n \circ \Pi \; \forall n \in {\mbox{$Z\!\!\!Z$}}$. Luego:
*La función sobreyectiva $\Pi$ lleva órbitas del shift en órbitas de $f$. ***
Sea ${\cal R } = \{R_1, \ldots R_m \}$ una partición de Markov y sea $x \in M$ un punto cuya órbita por $f$ no corta a los bordes de los cerrados $R_j$ de la partición, o sea $ x \in \cap _{n \in
{\mbox{$Z\!\!\!Z$}}} f^{-n } (M \setminus \partial \cup _{j = 1 } ^{m} \partial
R_j)$. Sea $\{j_n\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ una sucesión tal que $f^n x \in
R_{j_n } \; \forall n \in {\mbox{$Z\!\!\!Z$}}$ (o sea $\Pi (\{j_n\}_{n \in {\mbox{$Z\!\!\!Z$}}})=
x$). Ahora, por construcción tenemos que $f^n x \in \mbox{int
}R_{j_n }$ y por definición de partición cerrada $\{j_n\}_{n \in
{\mbox{$Z\!\!\!Z$}}}$ es única. La función $\Pi $ es inyectiva sobre el conjunto $\cap _{n \in {\mbox{$Z\!\!\!Z$}}} f^{-n } (M \setminus \cup _{j = 1 } ^{m}
\partial R_j)$. A continuación veremos que ese conjunto es denso en $M$ e invariante bajo $f$.
[**2.5 Borde de la partición de Markov** ]{}
*Sea ${\cal R} = \{R_1, \ldots, R_m\}$ una partición de $M$ por cerrados. Se llama *Borde de ${\cal R}$ *al conjunto $$\partial {\cal R} = \bigcup _{j= 1}^m \partial R_j$$ Si ${\cal R } $ es una partición de Markov se llama *Borde estable de ${\cal
R}$ *a $$\partial ^s {\cal R} = \cup _{j = 1 }^m \partial ^s
R_j$$ y se llama *Borde inestable de ${\cal R}$ *a $$\partial ^u {\cal R} = \cup _{j = 1 }^m \partial ^u R_j$$ Se observa de la proposición \[223\] lo siguiente: $$\partial {\cal R } = \partial ^s {\cal R} \cup \partial ^u {\cal R}$$*******
\[252\] *Sea ${\cal R } = \{R_1, \ldots, R_m \}$ una partición por cerrados de $M$. Entonces:*
- $\partial {\cal R}$ tiene interior vacío y es cerrado.
- $\cup _{j = 1} ^m \mbox {int } R_j$ es abierto y denso en $M$.
- $A= M \setminus \cup _{n \in {\mbox{$Z\!\!\!Z$}}} f^{-n } (\partial {\cal
R})$ es denso en $M$ e invariante por $f$.
[*Demostración:* ]{} (1) $\partial R_j$ es cerrado con interior vacío. La unión finita de conjuntos en una variedad que son cerrados con interior vacío es cerrada con interior vacío.
\(2) Tomando el complemento $(\partial {\cal R}) ^c$ es abierto y denso en $M$. Pero si $y \in (\partial {\cal R}) ^c$ entonces $y
\in \cap _{j = 1}^m (\partial {R}_j) ^c$. Como ${\cal R} $ cubre a $M$ existe $j $ tal que $y \mbox{int } R_j$. Entonces $(\partial
{\cal R}) ^c \subset \cup _{j = 1}^m \mbox {int } R_j$. Deducimos que $\cup _{j = 1}^m \mbox {int } R_j$ es abierto y denso en $M$.
(3)$A = \cap _ {n \in {\mbox{$Z\!\!\!Z$}}} (f^{-n} \partial {R}_j) ^c = \cap _{n
\in {\mbox{$Z\!\!\!Z$}}} f^{-n} ((\partial {R}_j) ^c)$ es denso en $M$ porque es la intersección numerable de abiertos densos. Además $A$ es invariante por $f$ porque $$f^{-1}(A ) = f^{-1} (\bigcap _{n \in
{\mbox{$Z\!\!\!Z$}}} f^{-n} ((\partial {R}_j) ^c)) = \bigcap _{n \in {\mbox{$Z\!\!\!Z$}}} f^{-n-1}
((\partial {R}_j) ^c) = A \;\; \Box$$
[**2.6 Propiedades de las particiones de Markov** ]{}
La siguiente proposición permite aplicar las condiciones (i) y (ii) de la definición \[242\] de partición de Markov a otros puntos $x \in M$ que no están necesariamente en $\mbox{int } R_i
\cap f^{-1}(\mbox {int }R_j)$.
\[261\] Sea ${\cal R }= \{R_1, \ldots, R_m \}$ una partición de Markov de $M$ para $f$. Si $ \mbox{int } R_i \cap f^{-1}(\mbox {int }R_j)
\neq \emptyset$ entonces para todo $y \in R_i \cap f^{-1}R_j $ se cumple:
- $f(W^s(y, R_i)) \subset W^s(fy, R_j)$
- $f(W^u(y, R_i)) \supset W^u(fy, R_j)$
[*Demostración:* ]{} Sean $x \in \mbox{int } R_i \cap f^{-1}(\mbox
{int }R_j), \; \; y \in R_i \cap f^{-1}(R_j)$. Es inmediato, a partir de la definición de rectángulo y sabiendo que $x,y \in R_i$ lo siguiente: $$W^s(y, R_i) = \{[y,z] : z \in W^s (x, R_i)\} = \{[y,z] : fz \in f W^s (x, R_i)\}$$ Como $x \in \mbox{int } R_i \cap f^{-1}(\mbox {int }R_j)$ tenemos por la condición (i) de la definición \[242\] que se cumple: $$f(W^s(x, R_i)) \subset W^s(fx, R_j) \subset R_j$$ Entonces $y,z \in R_i, \;\; fy, fz \in R_j$. Así ${\mbox{$\,$dist$\,$}}(y,z) \leq
\mbox{ diam } R_i \leq \delta < \epsilon $. Análogamente ${\mbox{$\,$dist$\,$}}(fy, fz) < \epsilon$, de donde $f[y,z] = [fy, fz ]$. Luego $$fW^s (y, R_i) = \{ f[y,z]: z \in W^s (x, R_i)\}=
\{[fy,fz]: z \in W^s (x, R_i)\}=$$ $$= \{[fy,w]: w \in f W^s(x,
R_i)\} \subset \{ [fy, w]: w \in W^s (fx, R_j)\} = W^s(fy, R_j)$$ donde $fx, fy \in R_j$. Deducimos que $fW^s (y, R_i) \subset W^s
(fy, R_j)$. Análogamente $fW^u (y, R_i) \supset W^u (fy, R_j)\;\;
\Box$
Si $\{R_{j_n}\}_{n \geq 0 }$ es una sucesión de rectángulos de una partición de Markov ${\cal R }$ con diámtero $\beta >0 $ suficientemente pequeño y tales que $\mbox { int } R_{j_n} \cap
f^{-1} \mbox { int }R_{j_{n+1}} \neq \emptyset \; \; \forall n
\geq 0$ entonces $$x \in \bigcap _{n \geq 0} f^{-n} R_{j_n} \Rightarrow \bigcap _{n \geq 0 } f^{-n} R_{j_n}
= W^s _{\epsilon } (x, R_{j_0})$$
[*Demostración:* ]{} Por lo observado en \[244\] c) si se elige el diámetro de la partición de Markov menor que $\epsilon >0$ obtenemos $\bigcap _{n \geq 0 } f^{-n} R_{j_n} \subset W^s_{\epsilon } (x,
R_{j_0})$. Sea $y \in W^s_{\epsilon } (x, R_{j_0})$. Por \[261\] se tiene $$fy \in W^s _{\epsilon }(fx, R_{j_1}), \;\;\;
f^n y \in W^s_{\epsilon } (f^n y, R_{j_n}) \; \forall n \geq 0$$ Luego $y \in \bigcap _{n \geq 0 } f^{-n} R_{j_n}\; \Box$
Si ${\cal R}$ es una partición de Markov para el difeomorfismo de Anosov $f$ entonces: $$f(\partial ^s{\cal R}) \subset \partial ^s {\cal R}$$ $$f(\partial ^u{\cal R}) \supset \partial ^u {\cal R}$$
[*Demostración:* ]{} Sea $x \in \partial ^s {\cal R} = \bigcup
_{i= 1} ^m \partial ^s R_i$. Sea $i$ tal que $x \in \partial ^s
R_i$. En la proposición \[252\] se probó que $\bigcup _{j=1}^m
\mbox { int } R_j$ es denso en $M$. Entonces también lo es su preimagen por el difeomorfismo $f$. Luego $$(f^{-1} \bigcup
_{j=1}^m \mbox{ int } R_j ) \bigcap \mbox { int }R_i$$ es denso en $R_i$. Sea entonces $x_n \in (f^{-1} \bigcup
_{j=1}^m \mbox{ int } R_j ) \bigcap \mbox { int }R_i \; \forall n
$ tal que $x_n \rightarrow x$. Tenemos que $f(x_n) \in \bigcup
_{j=1}^m \mbox { int }R_j \; \forall n \geq 0 $, pero $j$ solo puede tomar una cantidad finita de valores. Luego, existe una subsucesión, que por comodidad seguimos llamando $x_n$, y un índice $j$ tal que $f(x_n) \in \mbox{ int }R_j \; \forall n \geq
0$.
El rectángulo $R_j$ es cerrado. Entonces $f(x) = \lim f (x_n) \in
R_j$. Tenemos entonces $$x_n \in (f^{-1} \mbox{ int } R_j )
\bigcap \mbox { int }R_i$$ Luego por \[261\] se cumple $$f (W^u(x, R_i)) \supset W^u (fx, R_j)$$ Supongamos por absurdo que $fx \not \in \partial ^s R_j$. Existe un entorno $V$ de $fx$ en $W^u_{\epsilon } (fx)$ contenido en $R_j
\bigcap W^u _{\epsilon } (fx)$. Entonces $$f^{-1}(W^u_{\epsilon } (fx))\subset W^u_{\epsilon } (x)$$ Así $x \in \mbox{int } W^u (x, R_i)$ en $W^u_{\epsilon }(x)$, o sea, $x \not \in \partial R_i$ contra lo supuesto.
De igual forma se prueba que $f(\partial ^u {\cal R}) \supset
\partial ^u{\cal R}\; \; \Box$
.
[Particiones de Markov para difeomorfismos de Anosov - ]{} Eleonora Catsigeras
Semiconjugación con el shift
============================
[**3.1 Espacio de funciones bi-infinitas y función shift** ]{}
Sea ${ P} = \{p_1, \ldots , p_m\}$ un conjunto finito de puntos en $M$. Se denota con $P^{{\mbox{$Z\!\!\!Z$}}}$ al espacio de las sucesiones bi-infinitas de puntos en $P$. Tomando en $P$ la topología discreta y en $P^{{\mbox{$Z\!\!\!Z$}}}$ la topología producto asociada a ella, por el teorema de Tychonov, el espacio $P^{{\mbox{$Z\!\!\!Z$}}}$ es compacto y metrizable.
Sea $q = \{q_j\}_{j \in {\mbox{$Z\!\!\!Z$}}}, \; q_j \in P, \; q \in P^{{\mbox{$Z\!\!\!Z$}}}$. Una base local de abiertos en $q$ está formada por los abiertos $$I_N (q) = \{q' \in P^{{\mbox{$Z\!\!\!Z$}}}: q_j = q'_j \, \; \forall |j| \leq N \}$$ Una métrica en $P^{{\mbox{$Z\!\!\!Z$}}}$ está dada por $${\mbox{$\,$dist$\,$}}(q,q') = \sum _{n \in {\mbox{$Z\!\!\!Z$}}} \frac {{\mbox{$\,$dist$\,$}}(q_n, q'_n)}{2^{|n|}}$$
*La función o transformación *shift, *denotada como $\sigma : P^{{\mbox{$Z\!\!\!Z$}}} \mapsto P^{{\mbox{$Z\!\!\!Z$}}}$, es la trasnformación definida por $\sigma (q) = q' $ donde $q' _n = q_{n+1}, \; \forall n \in
{\mbox{$Z\!\!\!Z$}}$.***
Se observa que la función shift $\sigma$ aplicada a $q$ consiste en un corrimiento a la izquierda de los términos de $q$: el mismo término $q_0$ que antes ocupaba el lugar 0, después de aplicarle $\sigma$ ocupará el lugar $-1$ (es decir es $q'_{-1}$), el término $q_1$ que antes ocupaba el lugar 1 pasará a ocupar el lugar 0 (es decir será $q' _0$) y así $q_j = q'_{j-1}$ para todo $j \in {\mbox{$Z\!\!\!Z$}}$. Es fácil demostrar que $\sigma : P^{{\mbox{$Z\!\!\!Z$}}} \mapsto
P^{{\mbox{$Z\!\!\!Z$}}}$ es un homeomorfismo.
[**3.2 Semiconjugación** ]{}
Sean $M, M'$ dos espacios topológicos, y sean $f, f'$ dos homeomorfismos en $M$ y $M'$ respectivamente.
*Una función $\theta: M' \mapsto M$ se llama *semiconjugación *de $f$ con $f'$ si cumple:***
- $\theta$ es continua y sobreyectiva
- $f \circ \theta = \theta \circ f'$
Se observa que $$f \circ \theta = \theta \circ f'\; \; \; \Rightarrow \; \; \;
f ^n \circ \theta = \theta \circ f'^n \; \; \forall n \in {\mbox{$Z\!\!\!Z$}}$$ Luego, toda órbita en $M'$ según $f'$ es llevada por $\theta $ a alguna única órbita por $f$ en $M$ y toda órbita en $M$ por $f$ es corresponde a alguna (no necesariamente única) órbita por $f'$ en $M'$.
*Una semiconjugación se llama *conjugación *entre $f$ y $f'$ si es un homeomorfismo.***
[**3.3 Semiconjugación de los difeomorfismos de Anosov con el shift**]{}
Sea $\beta >0 $ arbitrario dado. Sea $f: M \mapsto M $ un difeomorfismo de Anosov. Por el teorema \[135\] el difeomorfismo $f$ es topológicamente estable. Elijamos $\alpha >0 $ tal que toda pseudo-órbita de $f$ está $\beta $ acompañada por una órbita de $f$.
Sea $0< \gamma < \min (\beta, \alpha /2)$ tal que $${\mbox{$\,$dist$\,$}}(x,y )<
\gamma \; \Rightarrow \; {\mbox{$\,$dist$\,$}}(fx, fy) < \alpha /2$$ Tal número $\gamma $ existe porque $f$ es continua en $M$ compacta.
Siendo $M$ compacta existe un conjunto finito $P = \{p_1, \ldots,
p_m\}$ de puntos de $M$, centros de bolas de radio $\gamma$ que cubren $M$. Dado $x \in M$ existe $p_j \in P$ tal que ${\mbox{$\,$dist$\,$}}(x,
p_j) < \gamma$. Es decir $P$ es un conjunto $\gamma$-denso en $M$.
Sea $\Sigma (P) = \{q \in P^{{\mbox{$Z\!\!\!Z$}}}: {\mbox{$\,$dist$\,$}}(fq_j, f q _{j+1})< \alpha
\; \forall j \in {\mbox{$Z\!\!\!Z$}}\}$
$\Sigma (P)$ es el conjunto de las $\alpha $-pseudo-órbitas de $f$ que están formadas con puntos de $P$.
Si además elegimos $\beta < \rho /2$, donde $\rho$ es la constante de expansividad de $f$, se cumple, en virtud de la estabilidad topológica de $f$ dada por el teorema \[135\], lo siguiente:
Para todo $q \in \Sigma (P)$ existe un único $\theta (q) \in M$ tal que $$dist (f^n(\theta (q)), q_n) \leq \beta \; \; \forall n
\in Z$$
La aplicación $\theta : \Sigma (P) \mapsto M$ es sobreyectiva.
[*Demostración:* ]{} Sea $x \in M$. Demostremos que existe algún $q \in \Sigma (P)$ tal que $x = \theta (q)$.
La órbita $\{f^n(x)\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ se puede aproximar por $q =
\{q_n\}_ {n \in {\mbox{$Z\!\!\!Z$}}} \in P^{{\mbox{$Z\!\!\!Z$}}}$ de modo que $${\mbox{$\,$dist$\,$}}(f^n(x), q_n)
\leq \gamma \; \; \forall n \in {\mbox{$Z\!\!\!Z$}}$$ porque $P$ es $\gamma$-denso en $M$.
Entonces ${\mbox{$\,$dist$\,$}}(fq_n, q_{n+1}) \leq {\mbox{$\,$dist$\,$}}(fq_n, f^{n+1} x) +
\gamma$. De acuerdo a la elección de $\gamma$, siendo ${\mbox{$\,$dist$\,$}}(f^n
x, q_n) < \gamma$, se cumple ${\mbox{$\,$dist$\,$}}(f^{n+1} x, f q_n)< \alpha
/2$.
Así ${\mbox{$\,$dist$\,$}}(f q_n, q_{n+1}) \leq \alpha /2 + \gamma$.
Siendo $\gamma < \alpha /2$ se cumple ${\mbox{$\,$dist$\,$}}(f q_n, q_{n+1}) <
\alpha$, o sea $\{q_n\}$ es una $\alpha$-pseudo-órbita de $f$. Luego $q \in \Sigma (P)$.
Como $\{f^nx\}_{n \in {\mbox{$Z\!\!\!Z$}}}$ es una órbita que $\gamma $-acompaña a $q$ por construcción, y siendo $\alpha < \beta$ resulta ${\mbox{$\,$dist$\,$}}(f^n x, q_n) < \beta \; \forall n \in Z$. Entonces $x = \theta
(q)$ como se quería demostrar. $\Box$
La aplicación $\theta : \Sigma (P) \mapsto M$ es continua
[*Demostración:* ]{} Sea $q^n \rightarrow q \in \Sigma (P), \; \;
\theta (q^n) = x_n \in M$. La sucesión $x_n$ puede suponerse convergente $x_n \rightarrow x_0 $debido a la compacidad de la variedad $M$.
Se tiene que ${\mbox{$\,$dist$\,$}}(q^n_j, f^j (x_n)) \leq \beta \; \; \forall j
\in {\mbox{$Z\!\!\!Z$}}$ por la construcción de la función $\theta$.
Sea $j \in {\mbox{$Z\!\!\!Z$}}$ fijo. Como $q^n \rightarrow q \in \Sigma (P)$, existe $N(j)$ tal que para todo $n > N(j)$ se cumple $q^n \in I_j
(q)$, es decir $q^n _i = q_i \; \forall |i| \leq j$.
De lo anterior se deduce que para todo $n > N(j)$: $${\mbox{$\,$dist$\,$}}(q_j, f^j (x_n)) \leq \beta$$ Tomando $n \rightarrow \infty$, en virtud de la continuidad de $f$ se deduce que $${\mbox{$\,$dist$\,$}}(q_j, f^j (x_0)) \leq \beta \; \; \forall j \in {\mbox{$Z\!\!\!Z$}}$$ Entonces por construcción de la función $\theta$ se cumple que $x_0 = \theta (q)$ y luego $\theta (q^n ) \rightarrow \theta
(q).\; \; \Box$
\[334\] La función $\theta : \Sigma (P) \mapsto M$ es una semiconjugación del difeomorfismo de Anosov $f: M \mapsto M$ con el shift $\sigma
$ restringido a $\Sigma (P)$
[*Demostración:* ]{} La función shift $\sigma $, cuando restringida a $\Sigma (P)$, tiene codominio en $\Sigma (P)$ pues $\Sigma (P)$ es invariante bajo $\sigma$. En otras palabras $\left
. \sigma \right |
_{\Sigma (P)}: \Sigma (P) \mapsto \Sigma (P)$.
Para demostrar que $\theta$ es una semiconjugación entre $f$ y $\left
. \sigma \right |
_{\Sigma (P)}$ alcanza demostrar que el diagrama siguiente conmuta, pues ya se sabe que $\theta$ es continua y sobreyectiva:
$$\begin{array}{ccccc}
& & \sigma & & \\
& \Sigma (P) & \mapsto & \Sigma (P) & \\
\theta & \downarrow & & \downarrow & \theta \\
& M & \mapsto & M & \\
& & f & & \\
\end{array}$$ Sea $q \in \Sigma (P)$. Sean $x = \theta (q), \; q' = \sigma (q)$. Alcanza probar que $f(x) = \theta (q')$. Sea $x' = \theta (q')$. Entonces $${\mbox{$\,$dist$\,$}}(f^n(x'), q'_n) \leq \beta \; \; \forall n \in
{\mbox{$Z\!\!\!Z$}}$$ Pero $q'_n = q_{n+1}$ pues $q' = \sigma (q)$. Entonces $${\mbox{$\,$dist$\,$}}(f^n(x'), q_{n+1}) \leq \beta \; \; \forall n \in
{\mbox{$Z\!\!\!Z$}}$$ $${\mbox{$\,$dist$\,$}}(f^{n+1}(f^{-1} (x')), q_{n+1}) \leq \beta \; \; \forall n \in
{\mbox{$Z\!\!\!Z$}}$$ Luego $f^{-1}(x') = \theta (q) = x$, de donde $ x' = f(x)$ como queríamos probar. $\; \Box$
[**3.4 Conjuntos estable e inestable en el espacio de sucesiones**]{}
Sea $q \in \Sigma (P)$ donde $\Sigma (P)$ es el subconjunto de $P^{{\mbox{$Z\!\!\!Z$}}}$ (sucesiones bi-infinitas) definido en la sección 3.3
*Se llama *conjunto estable *por $q$ en $\Sigma (P)$ a: $$\widehat W^s(q) = \{q' \in \Sigma (P): {\mbox{$\,$dist$\,$}}(\sigma q, \sigma q') \rightarrow
_{n \rightarrow + \infty} 0\}$$***
*Se llama *conjunto inestable *por $q$ en $\Sigma (P)$ a: $$\widehat W^u(q) = \{q' \in \Sigma (P): {\mbox{$\,$dist$\,$}}(\sigma q, \sigma q') \rightarrow
_{n \rightarrow - \infty} 0\}$$***
*Se sabe que $dist (q, q') = \sum _{j \in {\mbox{$Z\!\!\!Z$}}} {\mbox{$\,$dist$\,$}}(q_j, q'_j)
/2^{|j|}$. Luego: $$dist (\sigma ^n q, \sigma ^n q') = \sum _{j
\in {\mbox{$Z\!\!\!Z$}}} {\mbox{$\,$dist$\,$}}(q_{n+j}, q'_{n+j}) /2^{|j|}$$ Si ${\mbox{$\,$dist$\,$}}(\sigma ^n
q, \sigma ^n q') \rightarrow 0$ entonces existe $N$ tal que $\forall n >N$ se cumple $$\sum _{j \in {\mbox{$Z\!\!\!Z$}}} \frac {{\mbox{$\,$dist$\,$}}(q_{n+j}, q' _{n+j})}{2^{|j|}} < \min \{{\mbox{$\,$dist$\,$}}(p_i,
p_j): i \neq j, p_i, p_j \in P\}$$ Lo anterior se cumple si y solo si $q_n = q' _n$ para todo $n$ suficientemente grande. Luego: $$\widehat W^s (q) = \{q' \in \Sigma (P): q'_n= q_n\; \forall n \mbox{ suficientemente grande}
\}=$$ $$\widehat W^s (q) = \bigcup_{N \geq 0} \{q' \in \Sigma
(P): q'_n= q_n\; \forall n \geq N \}$$ Análogamente: $$\widehat W^u (q) = \bigcup_{N \leq 0} \{q' \in \Sigma (P):
q'_n= q_n\; \forall n \leq N \}$$*
*Se llama *$0-$conjunto estable (e inestable) *por $q$ a $$\widehat W^s_0(q)= \{q' \in \Sigma (P): q'_n = q_n \; \forall n \geq 0\}$$ (respectivamente a: $$\widehat W^u_0(q)= \{q' \in \Sigma (P): q'_n = q_n \; \forall n \leq 0\} )$$***
**
- $\widehat W^s_0(q) \subset \widehat W^s(q), \; \; \;
\widehat W^u_0(q) \subset \widehat W^u(q)$
- $\sigma \widehat W^s_0(q) \subset \widehat W_0 ^s(\sigma q), \; \; \;
\sigma \widehat W^u_0(q) \supset \widehat W^u_0 (\sigma q)$
- Si $q, q'\in \Sigma (P)$ y si $q_0 = q'_0$, entonces puede construirse una única $q"$ tal que $q"_j = q_j \; \forall j
\geq 0, \; \; \; q"_j = q'_j \; \forall j \leq 0$. Se cumple $$q" = \widehat W^s_0 (q) \bigcap \widehat W^u _0 (q')$$
La función corchete en el espacio de sucesiones es: $$[,]: \{(q,q') \in (\Sigma (P))^2: q_0 = q' _0\} \mapsto \Sigma (P)$$ definida por $$[q,q'] = \widehat W _0 ^s (q) \bigcap \widehat W
^u_0 (q')$$ o sea $q" = [q,q'] $ si y solo si $q"_j = q_j \;
\forall j \geq 0, \; \; \; q"_j = q'_j \; \forall j \leq 0$.
\[346\] *Sea $\theta $ la semiconjugación definida en la sección 3.3. Si $\beta $ es suficientemente pequeño entonces $$\theta [q,q'] = [\theta q, \theta q']\; \forall q, q' \in \Sigma (P) \mbox { tales que }
q_0 = q' _0$$*
[*Demostración:* ]{} Sean $q, q' \in \Sigma (P)$ tales que $q_0 =
q' _0$. Llamemos $x = \theta (q), \; \; y = \theta (q'), \; \; z =
\theta [q,q']$. Hay que demostrar que $z = [x,y]$ o sea que $z =
W^s_{\epsilon }(x) \cap W^u_{\epsilon }(y)$.
Por construcción y por definición de la función $\theta $ se cumple: $${\mbox{$\,$dist$\,$}}(f^n x, q_n) \leq \beta \; \; \; {\mbox{$\,$dist$\,$}}(f^n y, q'_n) \leq \beta \; \; \; \forall n \in {\mbox{$Z\!\!\!Z$}}$$ $${\mbox{$\,$dist$\,$}}(f^n z, q_n) \leq \beta \; \forall n \geq 0, \; \; {\mbox{$\,$dist$\,$}}(f^n z, q' _n ) \leq
\beta \; \forall n \leq 0$$ porque $z = \theta [q,q']$.
Entonces $${\mbox{$\,$dist$\,$}}(f^n x, f^n z) \leq 2 \beta \; \forall n \geq 0,
\; \; \; {\mbox{$\,$dist$\,$}}(f^n y, f^n z) \leq 2 \beta \; \forall n \leq 0$$ Eligiendo $2 \beta < \min (\epsilon, \delta /2)$ se tiene que $z
\in W^s_{\epsilon } (x) \cap W^u_ {\epsilon } (y)$. Además ${\mbox{$\,$dist$\,$}}(x,y) \leq {\mbox{$\,$dist$\,$}}(x,z) + {\mbox{$\,$dist$\,$}}(y,z) \leq 4 \beta < \delta$ Entonces por la expansividad de $f$, el punto $z \in M$ es el único en $W^s_{\epsilon } (x) \cap W^u_ {\epsilon } (y). \; \; \;
\; \Box$
La semiconjugación definida en la sección 3.3 conmuta con la función corchete.
[**3.5 Construcción de un cubrimiento con rectángulos**]{}
\[351\] Sea $\beta >0 $ dado suficientemente pequeño, como en la Proposición \[346\]. Si $p_s \in P$ (según la sección 3.3) entonces $T_s = \theta \{q \in \Sigma (P): q_0 = p_s\} $ es un rectángulo cerrado de $M$ con diámetro a lo sumo $2 \beta $.
[*Demostración:* ]{} Si $q, q'$ cumplen $q_0 = q' _0 = p _s$ entonces también se cumple por construcción de $[q,q']$ que $[q,q']_0 = p _s$.
Para demostrar que $T_s$ es un rectángulo en $M$ alcanza tomar $x,y \in T_s$ y demostrar que $[x,y] \in T_s$. Pero $x,y \in T_s
\Rightarrow \theta (q) = x, \theta (q') = y $ con $q_0 = q' _0 =
p_s$. Luego según \[346\] se tiene $[x,y] = \theta [q,q']$. Entonces $[x,y]\in T_s$ (por construcción de $T_s$).
Además ${\mbox{$\,$dist$\,$}}(x,y) < {\mbox{$\,$dist$\,$}}(x, q_0) + {\mbox{$\,$dist$\,$}}(q'0, y) < 2 \beta$. Entonces $\mbox{diam} T_s \leq 2 \beta$.
$T_s$ es cerrado porque es la imagen continua de $\{q \in \Sigma (P): q_0 = p_s\}$ que es compacto en $P^{{\mbox{$Z\!\!\!Z$}}}$ (ya que es cerrado en el espacio compacto $P^{{\mbox{$Z\!\!\!Z$}}}$). $\Box$
La familia de rectángulos $\tau = \{T_i\}_{i = 1, \ldots, m}$ (construidos según la proposición \[351\]) es un cubrimiento finito de $M$ por rectángulos cerrados de diámetro a lo sumo $2 \beta$.
[*Demostración:* ]{} Alcanza ver que $\cup _{i=1} ^m T_i = M$. Como $\theta$ es sobreyectiva, todo punto $x \in M$ es $x = \theta
(q)$ con $q \in \Sigma (P)$. Sea $q_0 \in P = \{p_1, \ldots,
p_m\}$. Entonces existe un subíndice $s = 1, 2, \ldots, m$ tal que $q_0 = p_s$, o sea $x \in T_s$. Luego $M = \cup _{i=1} ^m T_i $ como se quería probar. $\Box$
[**3.6 Propiedades del cubrimiento por rectángulos**]{}
\[361\] Si $x = \theta (q)$ con $q_0 = p_s$ entonces $\theta (\widehat W_0
^s q) = W^x (x, T_s)$
[*Demostración:* ]{} Si $y \in \theta (\widehat W_0 ^s q) $, entonces $y = \theta (q')$ con $q'_j = q_j , \forall j \geq 0$. Así $[\theta q, \theta q'] = [x,y] = \theta [q,q']$ (por la Proposición \[346\]).
Como $q_j' = q_j \; \forall j \geq 0$ tenemos que $[q,q']= q'$.
Entonces $[x,y] = \theta q' = y$, de donde $y \in W^s(x, T_s)$.
Hemos probado que $\theta (\widehat W_0 ^s q) \subset W^x (x,
T_s)$. Recíprocamente, si $y \in W^s(x, T_s) \subset T_s$ entonces existe $q'$ con $q'_0 = p_s$ tal que $y = \theta (q')$ (por construcción del rectángulo $T_s$).
Además si $y \in W^s(x, T_s)$ entonces $y = [x,y]$.
Aplicando la proposición \[346\] se obtiene:
$$y = [x,y] = [\theta q, \theta q'] = \theta [q,q'] \in \theta (\widehat W^s_0 q)$$ Hemos probado entonces que $W^s (x, T_s) \subset \theta (\widehat
W^s_0 q). \; \; \Box$
*La proposición anterior caracteriza las $\epsilon$-variedades estables en los rectángulos $T_s$ de $M$: la semiconjugación $\theta$ lleva $0-$ variedades estables en el espacio $\Sigma (P)$ en $\epsilon$- variedades estables en rectángulos $T_s$ de $M$.*
La siguiente proposición será utilizada en la sección 5 de este trabajo para demostrar el teorema de Sinai (existencia de una partición de Markov para $f$, difeomorfismo de Anosov).
Sea $q \in \Sigma (P)$ con $q_0 = p_s, q_1 = p_t$ (según la definición al principio de la sección 3.3). Sea $x = \theta (q)$. Entonces:
- $f W^s(x, T_s) \subset W^s(fx, T_t)$
- $f W^u(x, T_s) \supset W^u(fx, T_t)$
[*Demostración:* ]{} Se tiene $fx = \theta (\sigma q), \; (\sigma
q)_0 = q_1 = p _t , \; \; x = \theta (q), \; q_0 = p_s$.
Por la Proposición \[361\] tenemos que $W^s(x, T_s) = \theta
(\widehat W^s_0 q), \; \; W^s(fx, T_t)= \theta (\widehat W^s_0
(\sigma q)$.
Por el Teorema \[334\] $f W^s(x, T_s) = f \circ \theta (\widehat
W ^s _0 q) = \theta \sigma (\widehat W^s_0 q)$.
De la definición de $\widehat W ^s _0 (q)$ y de la definición de la función shift $\sigma$, es inmediato que $\sigma (\widehat
W^s_0 q )\subset \widehat W^s _0 (\sigma q))$. Entonces: $$f W^s(x, T_s) \subset \theta (\widehat W^s_0 \sigma q) = W^s(fx, T_t))$$ En forma similar, utilizando conjuntos inestables en vez de estables, se prueba (ii). $\; \Box$
*Este procedimiento ha permitido construir un cubrimiento $\tau
= \{T_1, \ldots, T_m\}$ de $M$ por rectángulos cerrados de diámetro menor que un número positivo dado y que cumplen las condiciones (i) y (ii) de la proposición anterior. Estas condiciones son similares a las exigidas en la definición de partición de Markov en el parágrafo \[242\].*
El cubrimiento $\tau$ no es necesariamente una partición de Markov porque los interiores de los rectángulos de $\tau$ no son en general disjuntos dos y a dos y los rectángulos no son necesariamente propios. A partir del cubrimiento $\tau$, que cumple (i) y (ii), refinándolo apropiadamente, se construirá una partición de Markov.
[Particiones de Markov para difeomorfismos de Anosov - ]{} Eleonora Catsigeras
Método constructivo de la partición
===================================
En la sección 3.5 se construyó un cubrimiento finito $\tau $ de la variedad $M$, con rectángulos cerrados $\{T_1, T_2, \ldots, T_m\}$ para el difeomorfismo de Anosov $f$, que cumplen las condiciones (i) y (ii) de la Definición \[242\] de Partición de Markov.
En esta sección se refinará el curbrimiento $\tau$ para obtener ahora una partición ${\cal R}$ por cerrados de $M$ (con interiores dos a dos disjuntos) que sean además conjuntos propios (cada cerrado es la adherencia de su interior).
Finalmente en la sección 5 se demostrará que esa partición ${\cal
R}$ es una partición de Markov para $f$.
[**4.1 Primer refinamiento del cubrimiento**]{}
A partir del cubrimiento $\{T_i\}_{i= 1, 2, \ldots, m} = \tau$ definamos otro cubrimiento más fino, de la siguiente forma:
*Sean $T_j, T_k \in \tau$. Se definen:*
- $T_{jk}^1 = \{x \in T_j: W^u(x, T_j) \cap T_k \neq \emptyset, \;
W^s(x, T_j) \cap T_k \neq \emptyset\}$
- $T_{jk}^2 = \{x \in T_j: W^u(x, T_j) \cap T_k \neq \emptyset, \;
W^s(x, T_j) \cap T_k = \emptyset\}$
- $T_{jk}^3 = \{x \in T_j: W^u(x, T_j) \cap T_k = \emptyset, \;
W^s(x, T_j) \cap T_k \neq \emptyset\}$
- $T_{jk}^4 = \{x \in T_j: W^u(x, T_j) \cap T_k = \emptyset, \;
W^s(x, T_j) \cap T_k = \emptyset\}$
**
- $T_j $ es la unión disjunta $\cup _{n=1}^4 T_{jk}^n$.
- Si $n_1 \neq n_2$ entonces $\mbox{int}T_{jk}^{n_1} \cap T_{jk}^{n_2} =
\emptyset = T_{jk}^{n_1} \cap T_{jk}^{n_2}$.
- $T_{jk}^1 = T_j \cap T_k$. En efecto: $T_j \cap T_k \subset
T^1_{jk}$. Además si $x \in T_{jk}^1$ entonces existen $y,z \in
T_j \cap T_k$ tales que $y \in W^u_{\epsilon} (x), \; z \in
W^s_{\epsilon }(x)$. Luego $x = [ x,y]$. Pero por definición de rectángulo, como $y,z \in T_j \cap T_k$, entonces $x = [z,y] \in
T_j \cap T_k$.
*Si $T_{jk}^n \neq \emptyset$ entonces $T_{jk}^n$ es un rectángulo.*
[*Demostración:* ]{} Sean $x,y \in T_{jk}^n \subset T_j$. Entonces $z \in [x,y] \in T_j$ porque $T_j$ es un rectángulo.
Por la proposición \[215\], como $z \in W^s(x, T_j)$, tenemos que $W^s (z, T_j) = W^s (x, T_j)$. Análogamente $W^u(z, T_j) = W^u
(y, T_j)$. Entonces $W^s (z, T_j)$ y $W^u (z, T_j)$ cortan a $T_k
$ si y solo si lo hacen $W^s(x, T_j)$ o respectivamente $W^u(y,
T_j)$. Luego $z \in T^n_{jk}$ como queríamos. $\; \Box$
[**4.2 Segundo refinamiento de la partición**]{}
Los rectángulos $T_{jk}^n$ construidos al principio de la sección 4.1 cubren a $M$ pero no tienen necesariamente interiores disjuntos. Tampoco son todos propios porque no son todos cerrados. Construiremos un refinamiento ${\cal R}$ de $\{T_{jk }^n\}$.
En primer lugar hay que observar que un punto $x \in M$ puede pertenecer a varios rectángulos de la familia $\{T_{jk }^n\}$.
*Dado $x \in M$ sea $$H(x) = \{(j,k,n): x \in T_{j,k}^n\}$$ $$R(x) = \bigcap _{(j,k,n) \in H(x) } \mbox{ int } \overline T_{j,k}^n$$ donde $R(x)$ podría ser vacío. $$Z^* = \{x \in M: x \in \mbox { int } T_{j,k}^n \; \; \forall (j,k,n) \in H(x))\}$$ $${\cal R} = \{ \overline {R(x)}\, \}_{x \in Z^*}$$*
**
- ${\cal R}$ es una familia finita, porque $H(x)$ es un subconjunto del conjunto finito de todos los posibles índices $\{(j,k,n): 1 \leq j \leq m, 1 \ leq k \leq m, \ leq n \leq 4\}$
- $ x \in Z^*$ si y solo si toda vez que $x \in
T_{j,k}^n$ se cumple $x \in \mbox{ int } T_{j,k}^n$.
- Si $x \in Z^*$ entonces $x \in R (x)$; luego en ese caso $R(x) \neq \emptyset$.
- Si $x \in Z^*$ entonces $R(x)$ es un rectángulo abierto, porque no es vacío y es intersección finita de rectángulos abiertos.
- Si $x \in Z^* $ entonces $\overline {R(x)}$ es un rectángulo propio pues $$\overline{R(x)} \supset \overline{\mbox{ int } \overline{R(x)}}
\supset \overline{\mbox{ int } {R(x)}} = \overline{ {R(x)}}$$ pues $R(x)$ es abierto. Entonces $\overline
{R(x)} = \overline{\mbox{ int } \overline{R(x)}}$.
De las observaciones anteriores se deduce que ${\cal R}$ es una familia finita de rectángulos propios que cubren $Z^*$. Probaremos que ${\cal R}$ es una partición de $M$ (más aún será una partición de Markov), para lo cual demostraremos que:
- La familia ${\cal R}$ cubre a $M$ lo que se demostrará en la sección 4.4.
- Dos rectángulos de la familiar ${\cal R}$ que sean diferentes tienen interiores disjuntos (esto se demostrará en la sección 4.4).
Las afirmaciones (I) y (II) se ´probarán a partir de las definiciones de los rectángulos $T_{j,k}^n$ y del conjunto $Z^*$. En especial la densidad de $Z^*$ en $M$ que se demuestra a continuación juega un papel importante en la prueba.
[**4.3 Densidad del conjunto $Z^*$ cubierto por la partición.**]{}
El conjunto $Z^*$ definido en la sección *4.2 *es abierto y denso en $M$. Además *$$Z^* = \bigcap _{j,k = 1}^m \left
( \bigcup _{n= 1}^4 \mbox{ int }T_{j, k}^n \cup T_j^c \right )$$***
[*Demostración:* ]{}
Por definición $$Z^* = \{x \in M: x \in \mbox{ int } T_{jk}^n \; \forall (j,k,n) \in H(x)\}
= \{ x \in M: x \in T_{jk}^n \Rightarrow \mbox{ int } T_{jk}^n\}$$ Se probó que $M = \cap _{j=1}^m T_j$. Además por construcción $T_j
$ es la unión disjunta $ \cup _{n=1}^4 T_{j,k}^n \; \forall
(j,k)$.
Fijados $j,k (1 \leq j,k \leq m)$ y dado un punto $x \in M$ se cumple $x \in T_j^c \cup \cup_{n=1}^4 T_{jk}^n$. Si el punto $x \in
Z^*$ entonces $ x \in T_j^c \cup \cup_{n=1}^4 T_{jk}^n \; \forall j, k$
Y recíprocamente, si $x \in T_j^c \cup \cup_{n=1}^4 T_{jk}^n \;
\forall j, k$ entonces $$x \in \{x \in M: x \in T_{jk}^n \Rightarrow x \in \mbox{ int } T_{jk}^n\} = Z^*$$
Luego $$Z^* = \bigcap _{j,k = 1}^m \left ( T_j^c \bigcup
\bigcup_{n=1}^4 T_{jk}^n \right)$$
Entonces $Z^*$ es abierto, por se intersección finita de abiertos.
Para demostrar que $Z^* $ es denso en $M$ alcanza probar que cada uno de los siguientes abiertos $$Z_{jk}^* = T_j^c \bigcup \bigcup _{n=1}^4 T_{jk}^n$$ es denso en $M$.
Para eso es suficiente tomar un abierto cualquiera no vacío contenido en $T_j$ y probar que corta a $\bigcup _{n=1}^4 \mbox{
int } T_{jk}^n$.
El lema que sigue permite caracterizar $\bigcup_{n=1}^4\mbox{ int
} T_{jk}^n$ como intersección de dos abiertos densos en $\mbox {
int } T_j$. Luego por el teorema de Baire, $\bigcup_{n=1}^4\mbox{
int } T_{jk}^n$ será denso en $\mbox { int } T_j$ como queremos.
Dados $j,k$ sean *$$A _{j,k} = \mbox{ int } \{x \in T_j: W^s(x,T_j) \cap T_k = \emptyset\} \; \cup
\; \mbox{ int } \{x \in T_j : W^s (x, T_j) \cap T_k \neq \emptyset\}$$ $$B_{j,k} = \mbox{ int } \{x \in T_j: W^u(x,T_j) \cap T_k = \emptyset\} \;
\cup \;
\mbox{ int } \{x \in T_j : W^u (x, T_j) \cap T_k \neq \emptyset\}$$ *Entonces**
- $A_{j,k} \cap B_{j,k} = \cup_{n=1}^4 \mbox { int } T_{j,k}^n$
- $A_{j,k}$ y $B_{j,k}$ *son abiertos y densos en *$\mbox{ int }T_{j}$**
[*Demostración:* ]{} 1) Teniendo en cuenta las definiciones en la sección 4.1 se tiene que $$A_{j,k} \cap B_{j,k} \cap T{j,k}^n = \mbox {int } T_{j,k}^n \; \forall j,k,n$$ Entonces $$A_{j,k} \cap B_{j,k} \cap \cup _{n=1}^4 T_{j,k}^n = \cup _{n=1}^4 \mbox{ int } T_{j,k} ^n
\; \; \forall j,k$$ Sabemos que $\cup_{n=1}^4 T_{j,k}^n = T_j \ni
A_{j,k} \cap B_{j,k} $ y entonces $A_{j,k} \cap B_{j,k} = \cup
_{n=1}^4 \mbox { int } T_{j,k}^n \; \forall j,k$
2\) $A_{j,k}$ y $B_{j,k}$ son abiertos por construcción. Mostremos que $A_{j,k}$ es denso en $\mbox{ int } T_j$. Sea $V$ un abierto no vacío contenido en $T_j$. Basta probar que $V$ contiene algún punto de $A_{j,k}$. $$V = \{ y \in V: W^s(y, T_j) \cap T_k = \emptyset\}\; \cup \; \{
y \in V: W^s(y, T_j) \cap T_k \neq \emptyset\}$$ 1er. caso) Si el primero de esos subconjuntos es vacío, entonces el otro es el abierto $V$ y tiene entonces todos sus puntos interiores: $V
\subset A_{j,k}$.
2do caso) Si existe $y \in V$ tal que $W^s (y, T_j) \cap T_k =
\emptyset$ probemos que $y \in \mbox { int } \{ x \in V: W^s (y,
T_j) \cap T_k = \emptyset \} \subset A_{j,k}$:
En efecto, si así no fuera existirían $Y_n \rightarrow y$ en $T_j$ tales que $Z_n \in W^s(y_n, T_j) \cap T_k \neq \emptyset$. Tomando subsucesiones convergentes, tendríamos $z_n \rightarrow z$ en $T_j
\cap T_k$ (porque $T_j, T_k$ son cerrados). Luego $$z \in W^s_{\epsilon }(y_n) \Rightarrow {\mbox{$\,$dist$\,$}}(f^p z_n, f^p y_n) \leq \epsilon
\; \; \forall p \geq 0$$ Por continuidad ${\mbox{$\,$dist$\,$}}(f^pz, f^p y) \leq
\epsilon \; \forall p \geq 0$ y entonces $z \in W^s_{\epsilon }
(y)\cap T_j \cap T_k$. Luego $W^s(y,T_j) \cap T_k \neq \emptyset $ contra lo supuesto. $\; \; \Box$
[**4.4 Demostración de que ${\cal R}$ es una partición de $M$**]{}
En la sección 4.2 se construyó una familia ${\cal R} =
\{\overline{R(x)} \;\}_{x \in Z^*}$ finita de rectángulos propios. Para demostrar que ${\cal R} $ es una partición por cerrados de $M$ falta probar que
- ${\cal R}$ cubre $M$.
- Los interiores de dos rectángulos distintos de ${\cal
R}$ son disjuntos.
${\cal R } = \{ \overline {R(x)}\; \}_{x \in Z^*}$ definido según la sección 4.1 es un cubrimiento de $M$
[*Demostración:* ]{} Como la familia ${\cal R }$ es finita se tiene: $$\bigcup _{x \in Z^*} \overline {R(x)} \; = \overline{ \bigcup _{x \in Z^*} R(x)} \;$$ Por la sección 4.2 si $x \in Z^*$ entonces $x \in R(x)$. Entonces $\cup _{x \in Z^*} R (x) \supset Z^*$ de donde $$\overline {\bigcup _{x \in Z^*} R(x)}\; \; \supset \; \overline{Z^*} \; \; = \; M$$ debido a la densidad de $Z^*$ en $M$. $\; \; \Box$
Sea $R(x)$ definido en la sección 4.2. Se cumple
- $\mbox { int } \overline{R(x)} \; \; = R(x)$
- *Si $x,y $ son tales que $R(x) \cap R(y) \neq \emptyset$ entonces $R(x)= R(y)$.*
[*Demostración :* ]{} 1) Por la definición en la sección 4.2 se tiene: $$R(x) = \bigcap _{(j,k,n) \in H(x) } \mbox { int } \overline{T_{j,k} ^n}\;$$ Por otro lado $R(x)$ es abierto contenido en $\overline{R(x)}\;$ y entonces $R(x) \subset \mbox { int } \overline {R(x)}\;$.
Además $R(x) \subset \cap_{(j,k,n) \in H(x)} \overline {T_{j,k}^n}
\; $ que es un cerrado. Luego $\overline{R(x)}\; \subset \overline {T_{j,k}^n}
\; \; \; \forall (j,k,n) \in H(x) $.
Se deduce que $$\mbox{ int } \overline{R(x)} \; \subset \mbox { int } \overline {T_{j,k}^n}
\; \; \; \forall (j,k,n) \in H(x)$$ de donde
$$\mbox{ int } \overline{R(x)} \; \subset \bigcap _{(j,k,n) \in H(x)}
\mbox { int } \overline {T_{j,k}^n}
\; \; = R(x)$$
2\) $R(x) cap R(y)$ es abierto no vacío y como $Z^*$ es denso en $M
$ contiene algún punto $z \in Z^*$. Alcanza probar que $R(x) =
R(z) = R(y)$. Para eso es suficiente demostrar que si $z \in Z^*$ con $z \in R(x)$ entonces $H(x) = H(z)$ (por la definición en la sección 4.2).
Demostremos primero que $H(x) \subset H(z)$:
$(j,k,n) \in H(x) \Rightarrow x \in T_{j,k}^n \Rightarrow z \in
R(x) \subset \overline {T_{j,k}^n} \; \; \subset T_j =
\cap_{n=1}^4 T_{j,k}^n $. Luego existe $n_1$ tal que $z \in
T_{j,k}^{n_1}$. Pero $z \in R(x) = \overline {T_{j,k}^n} \; $, luego $$\mbox { int } T_{j,k}^{n_1} \cap \overline { T_{j,k} ^n} \; \neq \emptyset$$ de donde $n_1 = n$. Luego $z \in T_{j,k}^n$ y $(j,k,n) \in H(z)$ como se quería probar.
Ahora probemos que $H(z) \subset H(x)$:
Sea $(i,h,m) \in H(z)$, lo que implica $z \in T_{i,h}^m \subset
T_i$, de donde $z \in T_i$. Sea $j $ tal que $x \in T_j = \cup
_{n=1}^4 T_{j,i}^n$. Existe $n$ tal que $x \in T_{j,i} ^n$. Como $H(x) \subset H(z)$ entonces $z \in T_{j,i}^n \subset T_j$. Pero entonces $z \in T_j \cap T_i = T_{j,i}^1$, o sea $n=1$. Luego $x
\in T_{j,i}^1 = T_j \cap T_i \subset T_i = \cup_{m=1}^4
T_{i,h}^m$.
Luego existe $m_1$ tal que $x \in T_{i,h} ^{m_1}$. Como $H(x)
\subset H(z) $ entonces $z \in T_{i,h}^{m_1}$. Pero por hipótesis $z \in T_{i,h}^m$ y entonces $m = m_1$ y $x \in T_{i,h}^m$ de donde $(i,h,m) \in H(x)$ como se quería probar. $\; \; \Box$
[**4.5 Densidad de $Z^*$ en las variedades estable e inestables**]{}
En la sección 4.3 se probó que $Z^*$ es denso en $M$ y además que $$Z^* = \bigcap _{j,k = 1}^m T_j ^c \cup (A_{j,k} \cap B_{j,k})$$ Se probará que $Z^*$ es además denso en las $\epsilon$- variedades estables e inestables que pasan por algún punto de $Z^*$. Este resultado se utilizará luego en la demostración del teorema de Sinai.
Sean $A_{j,k}$ y $B_{j,k}$ definidos en la sección 4.3.
Si $x \in A_{j,k}$ entonces $\mbox{ int } T_j \cap W^s_{\epsilon}
(x) \subset A_{j,k}$.
Si $x \in B_{j,k}$ entonces $\mbox{ int } T_j \cap W^u_{\epsilon}
(x) \subset B_{j,k}$.
[*Demostración:* ]{} Si $x \in A_{j,k}$ existe un entorno $V$ de $x$ en $T_j$ tal que para todo $y \in V$ $W^s(y, T_j)$ corta a $T_k$ si y solo si lo hace $W^s(x, T_j)$ (por definición del abierto $A_{j,k}$ en la sección 4.3).
Sea $w \in \mbox{ int } T_j \cap W^s_{\epsilon } (x)$. Tenemos que $x = [w,x]$. Existe un entorno $W$ de $w$ en $M$ tal que $[W, x]
\subset V$ por la continuidad de la función corchete. Podemos elegir $W \subset T_j$ porque $w \in \mbox{ int } T_j$. Si $z \in
W $ entonces $[z,x] = y \in V$. Luego $W^s(z, T_j) = W^s(y, T_j)$. Entonces para todo $z \in W$: $W^s(z, T_j)$ corta a $T_k$ si y solo si lo hace $W^s(x, T_j)$. De donde $w \in W \subset A_{j,k}$.
Análogamente se tiene que $\mbox{ int } T_j \cap W^u_{\epsilon} (x) \subset B_{j,k}$ cuando $x \in B_{j,k}$. $\;\; \Box$
Sean $x \in Z^*$ y $R(x)$ definidos en la sección 4.2.
$Z^*$ es denso en $W^s(x, R(x))$ y en $W^u(x, R(x))$.
[*Demostración:* ]{} Dado un abierto $V$ en $W^s(x, R(x))$ demostremos que contiene algún punto de $Z^*$. Si $w \in V$, entonces $w \in W^s_{\epsilon } (x) \cap R(x), \; w = [x,w]$. Por continuidad de la función corchete existe un entorno $U$ de $w$ en $M$ tal que $[x,U] \subset V$.
Como $w \in R(x)$ y $R(x)$ es abierto podemos suponer que $U
\subset R(x)$.
Sabemos que $Z^*$ es denso en $M$ por lo demostrado en la sección 4.3. Existe $y \in U \cap Z^*$. Basta demostrar que el punto $z$ definido como $z = [x,y]$ pertenece a $Z^*$.
En efecto por lo demostrado en la sección 4.3: $Z^* = \cap
_{j,k=1} ^m T_j ^c \cup (A_{j,k} \cap B_{j,k})$. Basta demostrar que si $z \in T_j$ entonces $z \in A_{j,k} \cap B_{j,k} \; \; \forall
k$.
Por construcción $z = [x,y]\; \; , \; x,y \in Z^* \cap R(x)$. Sea $j $ tal que $z \in T_j$. Sea $i$ tal que $x \in T_i$. Se tiene que $y,z \in R(x) \subset \mbox{ int } \overline{ T_i} \; = \mbox { int }
T_i$ pues $T_i $ es cerrado. Así $x,y \in Z^* \cap T_i$. Por la caracterización de $Z^*$: $x,y \in A_{i,j} \cap B_{i,j}$.
Además $z \in W^s_{\epsilon }(x)\cap \mbox{ int } T_i, \; \; \;
z \in W^u _{\epsilon } (y) \cap \mbox{ int } T_i$. Aplicando el lema de la sección 4.3 se tiene que $z \in A_{i,j}\cap B_ {i,j}
= \cup _{n=1}^4 \mbox { int } T_{i,j} ^n$.
Como $z \in T_j \cap T_i = T_{i,j}^1$ entonces $z \mbox{ int }
T_{i,j}^1$. Como $x \in T_i$ entonces $x \in T_ {i,j}^n$ para algún $n$. Luego $y,z \in R(x) \subset overline{T_{i,j}^n} \;$. Luego $z \in \overline{T_{i.j}^n} \; \cap \mbox{ int } T_{i,j}
^1\neq \emptyset $ de donde $n = 1$.
Hemos probado que para todo $j$ tal que $z \in T_j$ se cumple $x,y \in
T_j$. Entonces $z \in T_j \Rightarrow x \in T_j \Rightarrow z \in R(x) \subset \mbox{ int }
T_j$. Luego $z \in W^s_{\epsilon }(x) \cap \mbox { int } T_j \; \; \; , z \in W^u_{\epsilon }
(y) \cap \mbox { int } T_j$. Como $x, y \in Z^* \cap T_j$ entonces $x,y \in A_{j,k} \cap B_{j,k} \; \; \forall k$. Aplicando de nuevo el lema de la sección 4.3 se deduce $z \in A_{j,k} \cap B_{j,k} \;
\; \; \forall k$. $ \; \; \Box$
Si $x \in f^{-1} (Z^*)$ entonces $f^{-1} (Z^*)$ es denso en $W^s(x, R(x))$.
[*Demostración:* ]{} Sea $V$ un abierto no vacío de $W^s(x,
R(x))$. Encontraremos un punto $y \in f^{-1} (Z^*) \cap V$.
$V= U \cap W^s_{\epsilon } (x)$ donde $U$ es un abierto de $M$.
$f(V) = f(U) \cap f(W^s_{\epsilon }(x))$ porque $f$ es invertible.
$f(V) = f(U) \cap W^s_{\epsilon } (fx) \cap f(\overline {B}_{\epsilon
}(x))$. Luego:
$f(V)$ es un entorno no vacío en $W^s_{\epsilon }(fx)$, de donde también lo es $f(V) \cap R(fx)$ porque $R(fx)$ es un abierto de $M$.
Por la proposición anterior existe $z \in Z^*$ en $f(V) \cap
R(fx)$. Sea $y = f^{-1}(z)$. Por construcción $y \in f^{-1} (Z^*) \cap
V$. $\; \; \Box$
[Particiones de Markov para difeomorfismos de Anosov - ]{} Eleonora Catsigeras
Teorema de Sinai
================
En la parte 4 se construyó una partición ${\cal R}$ por rectángulos propios. Porbaremos que ${\cal R}$ es una partición de Markov de $M$ para el difeomorfismo $f$ de Anosovo con lo cual quedará probada la existencia de particiones de Markov.
[**5.1 Enunciado del teorema de Sinai**]{}
\[511\] Si $f$ es un difeomorfismo de Anosov en $M$ entonces, dado $\beta
>0$ existe una partición de Markov de $M$ para $f$ con diámetro menor que $\beta$.
[**5.2 Lemma**]{}
Sea $\tau = \{T_1, T_2, \ldots, T_m\}$ el cubrimiento definido en la sección 3.5 con diámetro suficientemente pequeño y sean $Z^*,
{\cal R}$ definidos en la sección 4.2 a partir de $\tau$.
Si $x,y \in Z^*$ y si $y \in f^{-1}(Z^*) \cap R(x) \cap
W^s_{\epsilon }(x)$ entonces $fy \in R(fx)$.
[*Demostración:* ]{} $R(f(x)) = \cap _{(j,k,n)} \in H(fx) \mbox {
int } \overline {T_{j,k}^n} \; $ por la definición en la sección 4.2.
Basta probar que $fx \in T_{j,k}^n \Rightarrow fy \in \mbox { int } \overline {T_{j,k}^n}\;
$. Probemos primero que $fx \in T_j \Rightarrow fy \in T_j$:
$fx \in T_j \Rightarrow fx = \theta (\sigma q)$ con $q_1= p_j, \; q_o =
p_h$. Luego $x = \theta (q) \Rightarrow x \in T_h$.
$y \in W^s_{\epsilon }(x) \cap R(x) \subset W^s_{\epsilon }(x) \cap T_h = W^s(x,
T_h)$.
Aplicando la proposición de la sección 3.6 e obtiene $fy \in W^s(fx, T_j) \subset
T_j$.
Ahora probemos que $fx \in t_{j,k}^n \Rightarrow fy \in
T_{j,k}^n$:
$fx \in T_{j,k}^n \subset T_j \Rightarrow fy \in T_j = \cup _{n=1}^4
T_{j,k}^{n}$. Por absurdo supongamos que $fy \in T_{j,k}^{n_1}$ con $n_1 \neq n$. Por hipótesis $y \in W_{\epsilon }^s (x)$. Aplicando \[148\] se tiene $fy \in W^s_{\epsilon } (fx)$. Luego de la proposición \[215\] se obtiene $W^s (fx, T_j) = W^s (fy,
T_j)$. La hipótesis de absurdo y la definición de los rectángulos $T_{j,k}^n $ en la sección 4.4 implican que estrictamente uno de los conjuntos $W^u(fx, T_j), \; W^u (fy, T_j)$ corta a $T_k$.
Supongamos para fijar ideas que $W^u (fx, T_j) \cap T_k \neq \emptyset,
\; \; W^s (fy, T_j) \cap T_k = \emptyset$.
Como $fx \in T_j$ por \[351\] existe $q \in \Sigma (P)$ tal que $(\sigma q )_0 = p_j, \; fx = \theta (\sigma q= f (\theta q))$. Llamando $P_i = q_0$ se tiene $x = \theta (q)$. Luego $x \in T_i$.
Por hipótesis $ y \in R(x)\cap W^S_{\epsilon }(x)$. De lo anterior $R(x) \subset T_i$. Luego $y \in W^s(x, T_i)$.
Por lo supuesto, existe $fz \in W^u (fx, T_j) \cap T_k$. Aplicando la proposición de la sección 3.6 se tiene que $z \in W^u (x,
T_i)$.
Como además $fz \in T_k$, por \[351\] existe $\overline q \in
\Sigma (P)$ tal que $\sigma (overline {q})_0 = p_k, \; fz = \theta
(\sigma \overline q) = f \theta \overline q$. Llamando $p_h =
\overline q_0$ se tiene $z \in \theta (\overline q)$. Luego $z \in
T_h$.
Por otro lado como $x,y \in T_i = \cup_{n=1}^4 T_{i,h}^n$, existen $n_1, n_2$ tales que $x \in T_{i,h}^{n_1}, \; y \in
T_{i,h}^{n_2}$. Luego $y \in R(x) \subset
\overline{T_{i,h}}^{n_1}\;$. Como $y \in Z^*$ entonces $y \in
\mbox{ int } T_{i,h}^{n_2}$. Por lo observado en la sección 4.1 $n_1 = n_2$.
Sabiendo que $z \in W^u (x, T_i) \cap T_h \neq \emptyset$ y que $x,y \in T_{i,h}^{n_1}$ se obtiene que existe $w' \in W^u (y, T_i)
\cap T_h \neq \emptyset$.
Sea $w = [z, w'] \in T_i \cap T_h$ porque $z, w' \in T_i \cap
T_h$. Como $w' \in W^u_{\epsilon } (y)$ se tiene $w= [z, w'] = [z,y] \in T_i \cap
T_h$.
Si el diámetro de los rectángulos $T_i$ se elige suficientemente pequeño, por la continuidad uniforme de $f$ en $M$ compacta se cumple que $w \in W^u_{\epsilon }(y), \; \; w,y \in T_i \Rightarrow {\mbox{$\,$dist$\,$}}(f^pW, f^py)
\leq \epsilon \; \; \forall p \geq 0, \; \; {\mbox{$\,$dist$\,$}}(fw, fy) < \epsilon \Rightarrow
fw \in W^u_{\epsilon}(fy)$.
Como $w \in W^s (z, T_h)$ se tiene $f w \in W^s(fz, T_k)$.
Luego $f w = [fz, fy] \in T_j$ porque $fy, fz \in T_j$, de donde $fw \in W^u (fy, T_j) \cap T_k \neq \emptyset$ contra lo supuesto.
Se observa que se han utilizado hasta aquí todas las hipótesis excepto que $y \in f^{-1}(Z^*)$ y que la misma demostración puede realizarse permutando $x $ e $y$, pues todas las hípótesis utilizadas son simétricas en $x$ e $y$. Entonces $y \in R(x)\cap Z^* \Rightarrow
y \in R(y)\cap R(x) \neq \emptyset \Rightarrow R(y) = R(x)$ y como $x \in Z^*$ entonces $x \in R(x) = R(y)$. Además $y \in W^s_{\epsilon }(x)
\Rightarrow x \in W^s_{\epsilon } (y)$ por definición. De allí que la suposición del principio no era restrictiva.
Finalmente demostremos que $fx \in T_{j,k}^n \Rightarrow fy \in \mbox{ int } \overline{ T_{j,k}}
^n$:
Tenemos que $f x \in T_{j,k}^n \Rightarrow fy \in T_{j,k}^n $. Además por hipótesis $fy \in Z^*$ o sea $fy \in T_{j,k} ^n \Rightarrow
fy \in \mbox{ int } T_{j,k} ^n \subset \mbox{ int } \overline {T_{j,k}}
^n\; \; \; \Box$.
[**5.3 Demostración del teorema de Sinai**]{}
La partición ${\cal R}$ de $M$ construida en la sección 4.4 está formada por rectángulos propios y es un refinamiento del cubrimiento $\tau$ construido en la sección 3.5. Luego: $$\mbox{ diam } {\cal R} \leq \max_{T_i \in \tau } \mbox{ diam } T_i$$ Por la proposición \[351\] $\mbox{ diam } T_i \leq 2 \beta$ donde $\beta$ es un número positivo arbitrario.
Para terminar de demostrar el teorema de Sinai solo hace falta verificar que ${\cal R}$ cumple las condiciones (i) y (ii) de la definición de partición de Markov (\[242\]), lo cual se demuestra a continuación:
*Si $x \in \mbox{ int } R_i \cap f^{-1} \mbox{ int } R_j$ con $R_i,
R_j \in {\cal R}$ entonces*
\(i) $fW^s(x, R_i) \subset W^s(fx, R_j)$
\(ii) $fW^u(x, R_i) \supset W^u(fx, R_j)$
[*Demostración:* ]{} Por lo visto en \[148\] $fW^s_{\epsilon
}(x) \subset W^s_{\epsilon } (fx)$. Para demostrar (i) alcanza probar que $f W^s(x, R_i) \subset R_j$. Probémoslo primero en el caso particular que $$x \in Z^* \cap f^{-1} Z^* \cap \mbox{ int } R_i \cap f^{-1} \mbox{ int } R_j$$
Por lo visto en la sección 4.4 $R(x ) = \mbox{ int } R(x), \; \;
R(fx) = \mbox{ int } R(fx)$. Por lo visto en la sección 4.5 $Z^*
\cap f^{-1} Z^*$ es denso en $W^s (x, R_i)$. Dado $y \in W^s (x,
R_i)$ existe $y_n \rightarrow y $ con $y_n \in Z^* \cap f^{-1} Z^*
\cap W^s (x, R(x))$. Por el lema de la sección 5.2 se tiene $f Y_n
\in R(f(x)) \; \forall n$. Luego $f y = \lim f y_n \in \overline
{ R(fx)} = R_j$ o sea $f W^s (x, R_i) \subset R_j$ como queríamos probar.
Ahora probémoslo en general:
Si $x \in \mbox{ int } R_i \cap f^{-1} \mbox{ int } R_j$ sea $$\overline x \in Z^* \cap f^{-1} Z^* \cap \mbox{ int } R_i \cap f^{-1} \mbox { int } R_j$$ Existe tal $x$ porque $Z^* \cap f^{-1} Z^*$ es denso en $M$ al ser intersección de abiertos densos.
$W^s(x, R_i) = \{ [x, \overline y]: \overline y \in W^s (\overline
x, R_i)\}\; \; \; f W^s(x, R_i) = \{ f[x, \overline y]: \overline
y \in W^s (\overline x, R_i)\} $. Como $f[x, \overline y] = [fx, f
\overline y]$ se obtiene: $$fW^s(x, R_i)= \{ [fx, f \overline y]: \overline y \in W^s (\overline x, R_i)\}=
\{[fx, w]: w \in fW^s(\overline x, R_i)\}$$ $$\subset \{[fx, w]: w \in W^s(f \overline x, R_j)\}= W^s(fx, R_j)$$
Hemos probado que $fW^s(x, R_i) \subset W^s (fx, R_j)$.
La afirmación (ii) se prueba de (i) aplicándola al difeomorfismo $f^{-1}$ recordando que las variedades estables de $f^{-1}$ son las inestables de $f$. $ \; \; \; \Box$
[Particiones de Markov para difeomorfismos de Anosov - ]{} Eleonora Catsigeras
Dinámica Simbólica
==================
[**6.1 Matriz de transición**]{}
Sea ${\cal R} = \{R_1, R_2, \ldots, R_m\}$ una partición de la variedad $M$.
*La matriz de transición $A$ de la partición es una matriz $m
\times m$ tal que $A_{i,j} = 1 $ sin $\mbox{ int } R_i \cap f^{-1}
\mbox{ int } R_j \neq \emptyset$ y $A_{i, j} = 0$ en caso contrario.*
*Si $A$ es la matriz de transición de la partición ${\cal R}$ denotaremos con $\Sigma _A$ al conjunto de sucesiones bi-infinitas $\{R_{a_i}\}_{i \in {\mbox{$Z\!\!\!Z$}}}$ de rectángulos de ${\cal R}$ tales que dos rectángulos $R_{a_i}$ y $R_{a_{i+1}}$ consecutivos cumplen: $$\mbox{ int } R_{a_i} \cap f^{-1} \mbox{ int } R_{\_{i+1}} \neq \emptyset$$ Luego $$\Sigma _A = \{ a \in \{1,2,\ldots,m\}^{{\mbox{$Z\!\!\!Z$}}}: A_{a_i a_{i+1}}=1\}$$*
**
- Si ${\cal R}$ es una partición de Markov, aplicando \[261\] se obtiene, para todo $x \in R_i \cap f^{-1} R_j$ cuando $A_{ij} =1$:
- $fW^s(x, R_i) \subset W^s(fx, R_j)$
- $fW^u(x, R_i) \supset W^u (fx, R_j)$
- $\Sigma _A$ es invariante por el shift $\sigma$ pues $$a \in \Sigma _A \Rightarrow A_{a_i a_{i+1}} =1 \; \forall i, \; \sigma (a)_i = a_{i+1}, \;
\sigma (a) _{i+1} = a_{i+2}, \;$$ $$A_{a_{i+1}a_{i+2}} =1 \;
\forall i \Rightarrow \sigma (a) \in \Sigma _A$$
- En \[244\] se observó que si existe, es único el punto $x \in \cap _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j} R_{n_j}$ con $\{n_j\}_{j \in
{\mbox{$Z\!\!\!Z$}}}$ sucesión cualquiera bi-infinita.
Cuando ${\cal R} $ es una partición de Markov demostraremos que
- $a \in \Sigma _A \Rightarrow \; \exists x = \cap {j \in {\mbox{$Z\!\!\!Z$}}} f^{-j} R_{a_j}$
- La función $\pi: \Sigma _A \mapsto M \; \;
\pi (x) = \cap _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j} R_{a_j}$ es una semiconjugación de $f$ con el shift.
$\pi$ llevará entonces continuamente y sobreyectivamente las órbitas del shift en $\Sigma _A$ (llamada “dinámica simbólica”) en las órbitas de $f$ en $M$.
Además en la sección 2.4 se observó que si $x \not \in \cup _{j
\in Z} f^j \partial {\cal R}$ entonces existe una única sucesión bi-infinita $\{n_j\}_{j \in {\mbox{$Z\!\!\!Z$}}}$ tal que $f^j (x ) \in R_{n_j} \;
\forall j \in {\mbox{$Z\!\!\!Z$}}$ (es decir $x = \cap _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j}
R_{n_j}$). Eso significa que $\pi$ es además inyectiva en los puntos de $M \setminus \cup_{j \in {\mbox{$Z\!\!\!Z$}}} f^j \partial {\cal R}$, que como se vio en \[252\] es denso en $M$.
[**6.2 Lema**]{}
Sea ${\cal R} $ una partición de Markov y $A$ su matriz de transición. Sea $a \in \Sigma _A$. Entonces: $$K_N(a) = \bigcap_{j = -N}^{j = N} f^{-j} R_{a_j}$$ es un rectángulo cerrado.
[*Demostración:* ]{} Supongamos en primer lugar que $K_N (a) \neq
\emptyset$ y demostremos la tesis en este caso: $K_N (a)$ es cerrado por ser intersección finita de cerrados $R_i$. Si $x,y \in
K_N(a) = \cap _{-N}^N f^{-j} R_{a_j}$ entonces $f^jx, f^j y \in
R_{a_j} \; \forall |j| \leq N$. Sea $w = [x,y] \subset R_{a_0}$. Se cumple $w \in W^s(x, R_{a_0}), \; w \in W^u (y, R_{a_0})$. Aplicando la proposición \[261\] se obtiene: $$fw \in W^s (fx, R_{a_1})\; \;, \; \; f^{-1} w \in W^u (f^{-1} y, R_{a_{-1}})$$ $$f^jw \in W^s (f^jx, R_{a_j})\; \;, \; \; f^{-j} w \in W^u (f^{-j} y, R_{a_{-j}})
\; \; \forall 0 \leq j \leq N$$ Se observa que para poder aplicar la proposición \[261\] se usan las hipótesis $a \in \Sigma _A$ y la partición ${\cal R}$ es de Markov.
Luego $ f^j w \in R_{a_j} \; \forall |j| \leq N$ o sea $w \in K_N
(a)$.
Demostremos ahora que $K_N(a) \neq \emptyset \; \; \forall a \in
\Sigma _A$: $$K_N (a) = f^N R_{a_{-N}} \cap f^{N-1} R_{a_{-N +1}} \cap \ldots \cap f^{-N+1} R_{a_{N-1}}
\cap f^{-N} R_{a_N} =$$ $$= f^N (R_{a_{-N}} \cap f^{-1} R_{a_{-N+1}} \cap
\ldots \cap f^{-2N} R_{a_N})$$ Es suficiente demostrar que para todo $b \in \Sigma _A $ y para todo $n \geq 0$ $$R_{b_0} \cap f^{-1} R_{b_1} \cap \ldots \cap f^{-n} R_{b_n} \neq \emptyset$$ Por inducción completa sobre $n$: Cuando $n= 0$ el conjunto $R_{b_0} \neq \emptyset$ por ser un rectángulo.
Sabiendo por hipótesis de inducción que existe $y \in R_{b_1} \cap
\ldots \cap f^{-n+1} R_{b_n}$ encontremos $z \in R_{b_0} \cap
f^{-1} (R_{b_1} \cap \ldots \cap f^{-n+1} R_{b_n})$:
$b \in \Sigma _A \Rightarrow \mbox{int} R_{b_0} \cap f^{-1}
\mbox{int } R_{b_1} \neq \emptyset \; \Rightarrow \; \exists
f^{-1} x \in R_{b_0} \cap f^{-1} R_{b_1}$.
Sea $z = [y,x]$. Se tiene $z \in R_{b_1}$ porque $x,y \in
R_{b_1}$. Además $z \in W^s (y, R_{b_1})$. Por \[261\]: $$fz \in W^s (fy, R_{b_2}), \ldots, f^{n-1}z \in W^s (f^{n-1}y, R_{b_n})$$ Así $z \in R_{b_1} \cap \ldots \cap f^{-n+1} R_{b_n}$. Además $z
\in W^u (x: R_{b_1})$. Por \[261\] $f^{-1}z \in W^u (f^{-1}x,
R_{b_0}) \subset R_{b_0}$. Así se tiene $$f^{-1}z \in R_{b_0} \cap f^{-1} (R_{b_1} \cap \ldots \cap f^{-n+1} R_{b_n})
\;\;\; \Box$$
[**6.3 Teorema de semiconjugación**]{}
Sea ${\cal R} = \{R_i\}_{i= 1, \ldots, m }$ una partición de Markov de la variedad $M$ para el difeomorfismo $f$.
Sea $A$ la matriz de transición y $\Sigma _A$ el subespacio de las sucesiones bi-infinitas definido en la sección 6.1.
Entonces
- Para todo $a \in \Sigma _A$ existe único $x = \cap _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j} (R_{a_j})$
- La función $\pi: \Sigma _A \mapsto M $ definida por $\pi (a) =
\cap _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j} (R_{a_j})$ es una semiconjugación de $f$ con el shift $\sigma$ en $\Sigma _A$.
- $\pi |_{\pi^{-1} (\cap _{j \in Z} f^{-j} (\partial {\cal
R}) ^c)}$ es inyectiva, es decir $\pi$ es inyectiva en los puntos de $M$ cuyas órbitas no cortan al borde $\partial {\cal R}$ de la partición.
[*Demostración:* ]{} Sea $K_N(a) = \bigcap _{|j| \leq N } f^{-j}
R_{a_j}$.
$K_N(a)$ es cerrado no vacío por el lema de la sección 6.2
$K_{N}(a) \supset K_{N+1}(a)$ por construcción.
Siendo $M$ compacta por la propiedad de las intersecciones finitas se sabe que el conjunto $$K(a) = \bigcap _{\-infty} ^{+\infty}
f^{-j} R_{a_j} = \bigcap _{N=1} ^{+\infty} K_N(a) \neq \emptyset$$ lo cual prueba que existe $x \in K(a)$.
$x \in K(a)$ es único porque si $x,y \in K(a)$ entonces $f^j (x),
f^j(y) \in R_{a_j} \; \forall j \in {\mbox{$Z\!\!\!Z$}}$. Pero $R_{a_j}$ tiene diámetro a lo sumo igual al de la partición de Markov que puede elegirse menor que la constante de expansividad de $f$ , resultando $x = y$.
Se ha probado la parte 1) de la tesis.
Ahora probemos la parte 2): $$\pi (\sigma (a)) = \bigcap _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j} R_{a_{j+1}} = f (\bigcap _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j-1} R_
{a_{j+1}}) = f(\pi (a))$$ Entonces $$\pi \circ \sigma (a) = f
\circ \pi (a) \; \; \forall \; a \in \Sigma _A$$
Por lo tanto es conmutativo el siguiente diagrama $$\begin{array}{ccccc}
& & \sigma & & \\
& \Sigma _A & \mapsto & \Sigma _A & \\
\pi & \downarrow & & \downarrow & \pi \\
& M & \mapsto & M & \\
& & f & & \\
\end{array}$$ Para demostrar que $\pi$ es una semiconjugación hay que probar que $\pi: \Sigma _A \mapsto M$ es continua y sobreyectiva.
$\pi$ es continua pues si $a^n \rightarrow a \in \Sigma _A$, llamando $\pi (a^n) = x_n \in M $ y eligiendo una subsucesión convergente tenemos $x_n \rightarrow x \in M$ y además: $$x_n = \bigcap _{-\infty}^{+\infty} f^{-j} R_{a_j ^n}$$ Como $a^n \rightarrow a$, dado $p >0$ existe $N >0$ tal que $a_j^n
= a_j \; \forall n >N, \; \; \forall \; |j|\leq p$. Luego para todo $n >N$ el punto $x_n \in \bigcap _{-p}^p f^{-j} R_{a_j}$ que es un cerrado. Entonces $x = \lim x_n \in \bigcap _{-p}^p f^{-j}
R_{a_j}\; \; \forall \; p > 0$. Así $x \in \bigcap _{-\infty }^{+
\infty} f^{-j} R_{a_j}= \pi (a)$.
$\pi $ es sobreyectiva pues si $x \in M \setminus \bigcup _{n \in
{\mbox{$Z\!\!\!Z$}}} f^j \partial {\cal R}$ la órbita por $x$ no corta al borde $\partial {\cal R}$ de la partición. Entonces sea $a_j (x)$ el único subíndice tal que $f^j(x) \in \mbox{ int } R_{a_j(x)} $. Como $f^j (x)\in \mbox{ int } R_{a_j(x)} \bigcap f^{-1} \mbox {
int } R_{a_{j+1} (x)}$ tenemos que $A_{a_j a_{j+1}} =1$ de donde $a(x) \in \Sigma _A$.
Entonces por construcción: $$x \in \bigcap _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j}
R_{a_j(x)} = \pi (a)\; \; \forall a \in \Sigma _A$$.
Hemos probado que $\pi (\Sigma _A) \supset M \setminus \bigcup_{n
\in Z} f^j \partial {\cal R}$. Por \[252\] el conjunto $ M
\setminus \bigcup_{n \in Z} f^j \partial {\cal R}$ es denso en $M$.
Además $\Sigma _A$ es compacto porque es cerrado contenido en el espacio compacto $\{1,2,\ldots, m\} ^{{\mbox{$Z\!\!\!Z$}}}$. Luego $\pi (\Sigma
_A)$ es cerrado en $M$. Como contiene a un conjunto denso en $M$ y es cerrado en $M$ es $M$, lo cual prueba que $\pi$ es sobreyectiva.
Probemos ahora la parte 3): Si $a,a' \in \Sigma _A$ tales que $\pi
(a) = \pi (a') = x \in M \setminus \bigcup_{n \in Z} f^j \partial
{\cal R} $ entonces $f^j x \in \mbox{ int }R_{a_j} \cap \mbox{ int
} R_{a'_j} \; \; \forall \; j \in {\mbox{$Z\!\!\!Z$}}$.
Por la definición de la sección 2.4 los rectángulos distintos tienen interiores disjuntos. Entonces $a_j = a'_j$. Luego $a= a'$ y la transformación $\pi$ restringida a la preimagen por $\pi $ de $M \setminus \bigcup_{n \in Z} f^j \partial {\cal R}$, es inyectiva. $\; \; \Box$
[**6.4 Conclusión**]{}
El teorema anterior permite construir una semiconjugación $\pi$ del difeomorfismo de Ansosov $f$ con el shift $\sigma$ en el subespacio de la sucesiones bi-infinitas $\Sigma _A $. Además $\pi$ es inyectiva en un conjunto denso en $M$.
Ya se observó en la sección 2.4 que si ${\cal R }$ es un partición por cerrados cualquiera de $M$, aunque no sea de Markov, existe una función $\pi$ sobreyectiva que puede demostrarse que es continua usando la misma prueba de la parte 2) del teorema anterior, tal que conmuta el siguiente diagrama: $$\begin{array}{cccccc}
& & \sigma & & & \\
& \pi ^{-1} (M) & \mapsto & \pi ^{-1} (M) & \subset & \{1,2,\ldots,m\}^{{\mbox{$Z\!\!\!Z$}}} \\
& \pi \; \downarrow & & \downarrow \; \pi & & \\
& M & \mapsto & M & & \\
& & f & & & \\
\end{array}$$ Además $\pi$ es inyectiva en $M \setminus \cup _{j \in {\mbox{$Z\!\!\!Z$}}} f^{-j}
\partial {\cal R}$ que es denso en $M$.
En el caso que ${\cal R}$ sea además una partición de Markov se agrega que la semiconjugación tiene dominio en $\Sigma _A$. El subshift $\sigma |_{\Sigma _A} $ está definido en el subconjunto compacto de las sucesiones bi-infinitas que cumplen $A_{a_i
a_{i+1}} = 1$.
Se llama dinámica simbólica a la dinámica del shift en $\Sigma
_A$. La existencia de una partición de Markov en $M$ para $f$ asegura la existencia de la dinámica simbólica con la cual $f$ es semiconjugada.
Finalmente se observa que en la definición de rectángulo, en la de partición de Markov y en la demostración del teorema de Sinai, no se utiliza la diferenciabilidad de $f$ sino solo sus propiedades topológicas. Es por lo tanto aplicable a una clase más general que los difeomorfismos de Anosov: los homeomorfismos expansivos topológicamente estables.
[2]{}
R. Bowen: [*Equilibrium states and the ergodic theory of Anosov diffeomorphisms.*]{} [Lecture Notes in Math. [**470.**]{} Springer-Verlag ]{} [1975]{}
J. Lewowicz: [*Lyapunov functions and Topological Stability.*]{} [Journ. of Diff. Eq. [ **38**]{}]{} [1980]{}
J. Lewowicz: [*Invariant manifolds for regular points.*]{} [Pacific Journ. of Math. [ **96**]{}]{} [1981]{}
Hirsch-Pugh: [*Stable manifolds and hyperbolic sets.*]{} [Proc. Symp. in Pure Math. [ **14**]{}]{} [1970]{}
R. Bowen: [*Markov partitions for Axiom A diffeomorphisms.*]{} [Amer. Journ. of Math.. [ **92**]{}]{} [1970]{}
Y. Sinai: [*Construction of Markov partitions.*]{} [Funct. Anal. and its appl. [ **2**]{}]{} [1968]{}
|
---
abstract: 'Quantum illumination (QI) is an entanglement-enhanced sensing system whose performance advantage over a comparable classical system survives its usage in an entanglement-breaking scenario plagued by loss and noise. In particular, QI’s error-probability exponent for discriminating between equally-likely hypotheses of target absence or presence is 6dB higher than that of the optimum classical system using the same transmitted power. This performance advantage, however, presumes that the target return, when present, has known amplitude and phase, a situation that seldom occurs in lidar applications. At lidar wavelengths, most target surfaces are sufficiently rough that their returns are speckled, i.e., they have Rayleigh-distributed amplitudes and uniformly-distributed phases. QI’s optical parametric amplifier receiver—which affords a 3dB better-than-classical error-probability exponent for a return with known amplitude and phase—fails to offer any performance gain for Rayleigh-fading targets. We show that the sum-frequency generation receiver \[Phys. Rev. Lett. [**118**]{}, 040801 (2017)\]—whose error-probability exponent for a nonfading target achieves QI’s full 6dB advantage over optimum classical operation—outperforms the classical system for Rayleigh-fading targets. In this case, QI’s advantage is subexponential: its error probability is lower than the classical system’s by a factor of $1/\ln(M\bar{\kappa}N_S/N_B)$, when $M\bar{\kappa}N_S/N_B \gg 1$, with $M\gg 1$ being the QI transmitter’s time-bandwidth product, $N_S \ll 1$ its brightness, $\bar{\kappa}$ the target return’s average intensity, and $N_B$ the background light’s brightness.'
author:
- 'Quntao Zhuang$^{1,2}$'
- Zheshen Zhang$^1$
- 'Jeffrey H. Shapiro$^1$'
title: 'Quantum illumination for enhanced detection of Rayleigh-fading targets'
---
Quantum illumination (QI) [@Sacchi_2005_1; @Sacchi_2005_2; @Lloyd2008; @Tan2008; @Lopaeva_2013; @Guha2009; @Ragy2014; @Zheshen_15; @Barzanjeh_2015] uses entanglement to outperform the optimum classical-illumination (CI) system for detecting the presence of a weakly-reflecting target that is embedded in a very noisy background, despite that environment’s destroying the initial entanglement [@footnote0]. With optimum quantum reception, QI’s error-probability exponent—set by the quantum Chernoff bound (QCB) [@Audenaert2007]—is 6dB higher [@Tan2008] than that of the optimum CI system, i.e., a coherent-state transmitter and a homodyne receiver. Until recently, the sole structured receiver for QI that outperformed CI—Guha and Erkmen’s optical parametric amplifier (OPA) receiver [@Guha2009]—offered only a 3dB increase in error-probability exponent. In Ref. [@Zhuang_2017], we showed that the sum-frequency generation (SFG) receiver’s error-probability exponent reached QI’s QCB. Moreover, augmenting that receiver with feed-forward (FF) operations yielded the FF-SFG receiver [@Zhuang_2017], whose performance, for a low-brightness transmitter, matched QI’s Helstrom limit for both the target-detection error probability and the Neyman-Pearson criterion’s receiver operating characteristic (ROC) [@zhuang2017entanglement].
Prior QI performance analyses [@Tan2008; @Guha2009; @Zhuang_2017; @zhuang2017entanglement] have all assumed that the target return has known amplitude and phase, something that seldom occurs in lidar applications. At lidar wavelengths, most target surfaces are sufficiently rough that their returns are speckled, i.e., they have Rayleigh-distributed amplitudes and uniformly-distributed phases [@Goodman1965; @Goodman1976; @Shapiro1981; @Shapiro1982]. It is crucial, therefore, to show that QI maintains a target-detection performance advantage over CI for a target return with random amplitude and phase.
In this paper, we compare QI and CI target detection for Rayleigh-fading targets in the flat-fading limit, when the complex-field envelope of the target return from a single transmitted pulse suffers multiplication by a time-independent Rayleigh-distributed random amplitude and a time-independent uniformly-distributed random phase shift. We show that QI with OPA reception fails to offer any performance advantage over CI in this case. QI with SFG reception does provide an advantage over CI: when $M\bar{\kappa}N_S/N_B \gg 1$, its error probability is a factor of $1/\ln(M\bar{\kappa}N_S/N_B)$ lower than that of optimum CI, which transmits a coherent state and uses heterodyne reception. Here, $M\gg 1$ is the QI transmitter’s time-bandwidth product, $N_S$ is its brightness, $\bar{\kappa}$ is the target return’s average intensity, and $N_B$ is the background light’s brightness.
[*QI target detection*]{}—. In QI, the transmitter illuminates the region of interest with a single-spatial-mode, $T$-s-long pulse of signal light produced by pulse carving the continuous-wave output of a spontaneous parametric downconverter (SPDC). The SPDC source is taken to have a $W$-Hz-bandwidth, flat-spectrum phase-matching function with $W \gg 1/T$. The resulting signal pulse is maximally entangled with a corresponding single-spatial-mode, $T$-s-long pulse of idler light that the transmitter retains for subsequent joint measurement with the light returned from the region of interest. The $M = TW \gg 1$ signal-idler mode pairs that comprise the transmitted signal and retained idler pulses are thus in independent, identically-distributed (iid), two-mode squeezed-vacuum states with average photon number $N_S \ll 1$ in each signal and idler mode. Let $\{\hat{a}_{S_m},\hat{a}_{I_m}\}$ be the photon-annihilation operators for the transmitter’s $M$ signal and idler modes, and $\{\hat{a}_{R_m}\}$ the photon-annihilation operators of the $M$ modes returned from the region of interest. The target-detection hypothesis test is to determine whether $h=0$ (target absent) or $h=1$ (target present) is true when: $\hat{a}_{R_m} = \hat{a}_{B_m}$, for $h =0$, and $\hat{a}_{R_m} = \sqrt{\kappa}\,e^{i\phi}\hat{a}_{S_m} + \sqrt{1-\kappa}\,\hat{a}_{B_m}$, for $h=1$. Here: the $\{\hat{a}_{B_m}\}$ are photon-annihilation operators for iid background-noise modes that are in the thermal state with average photon number $N_B \gg 1$ when $h=0$ and in the thermal state with average photon number $N_B/(1-\kappa)$ when $h=1$ [@footnote1]; $\kappa > 0$ is the target-return’s reflectivity; and $\phi$ is the target-return’s phase.
Previous theoretical work on QI target detection [@Tan2008; @Guha2009; @Barzanjeh_2015; @Zhuang_2017] has assumed known $\kappa$, $\phi = 0$ [@footnote2], and lossless idler storage. For equally-likely target absence or presence, QI with optimum quantum reception—realizable with FF-SFG [@Zhuang_2017]—has error probability $\Pr(e)_{\rm opt} \simeq e^{-M\kappa N_S/N_B}/2$, QI with OPA reception has error probability $\Pr(e)_{\rm OPA} \simeq e^{-M\kappa N_S/2N_B}/2$, and optimum CI has error probability $\Pr(e)_{\rm CI} \simeq e^{-M\kappa N_S/4N_B}/2$.
Lidar targets are almost always speckle targets, viz., $\sqrt{\kappa}$ and $\phi$ are statistically independent random variables whose respective probability density functions (pdfs) are $f_{\sqrt{\kappa}}(x) = 2xe^{-x^2/\bar{\kappa}}/\bar{\kappa}$, for $x >0$, and $f_\phi(y) = 1/2\pi$, for $0\le y\le 2\pi$, where $\bar{\kappa}$ is the target return’s average intensity. These statistics invalidate *all* of the error-probability expressions from the preceding paragraph. Worse, as will soon be seen, they preclude *any* QI receiver from obtaining a single-pulse error probability that decreases exponentially with increasing $M\bar{\kappa}N_S/N_B$. For that demonstration we will employ the QCB, an exponentially-tight upper bound on the error probability of optimum quantum reception for multiple-copy quantum state discrimination [@Audenaert2007].
[*The QCB applied to QI with Rayleigh fading*]{}—. Conditioned on knowledge of $h$, $\sqrt{\kappa}$, and $\phi$, the $\{\hat{a}_{R_m},\hat{a}_{I_m}\}$ mode pairs at the QI receiver are in the state $\hat{{\boldsymbol \rho}}_h(\sqrt{\kappa},\phi) = \otimes_{m=1}^M\hat{\rho}^{(m)}_h(\sqrt{\kappa},\phi)$, with $\hat{\rho}_h^{(m)}(\sqrt{\kappa},\phi)$ being the two-mode, zero-mean, Gaussian state whose Wigner covariance matrix is & \_h = , \[hk\] & where $N_B \gg 1\gg N_S$ has been used. In this covariance matrix: ${\bf I}$ is the $2\times 2$ identity matrix, and ${\mathbf R}_h={\rm Re}\!\left[e^{i\phi} \left({\mathbf Z}-i{\mathbf X}\right)\right]\delta_{h1}$, where $\delta_{hk}$ is the Kronecker delta function, and ${\mathbf Z}$ and ${\mathbf X}$ are $2\times 2$ Pauli matrices. It follows that the signature of target presence is the nonzero phase-sensitive cross correlation, $C_p=\sqrt{\kappa N_S\left(N_S+1\right)}$, between the returned signal and the retained idler modes.
Erroneous target-detection decisions can be either false-alarm errors, when target presence is declared but no target is present, or miss errors, when target absence is declared but a target is present. For a given target-detection system, the conditional probabilities for these errors to occur are the false-alarm probability $P_F$, and the miss probability $P_M = 1-P_D$, where $P_D$ is the detection probability, i.e., the probability that target presence is declared when a target is present. Almost all QI target detection analyses [@Tan2008; @Guha2009; @Barzanjeh_2015; @Zhuang_2017] have been Bayesian: assign prior probabilities, $\{\pi_h\}$, to $h=0$ and $h=1$, and minimize the error probability, $\Pr(e) = \pi_0P_F + \pi_1P_M$, typically for equiprobable hypotheses, $\pi_0 = \pi_1 = 1/2$. Owing to the difficulty of accurately assigning priors to target absence and presence, a better approach to optimizing target-detection performance is to apply the Neyman-Pearson performance criterion: maximize $P_D$ subject to a constraint on $P_F$. Only recently has this criterion been applied to QI target detection [@zhuang2017entanglement], and that work assumed knowledge of the target return’s amplitude and phase. In this paper, we will consider both performance criteria—minimizing $\Pr(e)$ and maximizing $P_D$ for a given $P_F$—for our Rayleigh-fading QI scenario.
In the Bayesian approach, the minimum error probability for QI target detection is set by the Helstrom limit [@Helstrom1969] for discriminating between the *unconditional* $h=0$ and $h=1$ states, $$\hat{\bar{{\boldsymbol \rho}}}_h = \int\!{\rm d}x\int\!{\rm d}y\,f_{\sqrt{\kappa}}(x)f_\phi(y)\hat{{\boldsymbol \rho}}_h(x,y).$$ This limit’s calculation requires diagonalizing $\pi_1\hat{\bar{{\boldsymbol \rho}}}_1-\pi_0\hat{\bar{{\boldsymbol \rho}}}_0$, so it is intractable for QI with Rayleigh fading, because $\hat{\bar{{\boldsymbol \rho}}}_1$ is not an $M$-fold product state. Nevertheless, applying the QCB will yield an informative result.
Let $D_{\pi_0}(\hat{{\boldsymbol \rho}}_0(x,y),\hat{{\boldsymbol \rho}}_1(x,y))$ denote the Helstrom limit for discriminating between $\hat{{\boldsymbol \rho}}_0(x,y)$ and $\hat{{\boldsymbol \rho}}_1(x,y)$ that occur with priors $\pi_0$ and $\pi_1$, and let $\xi_{\rm QCB}(\hat{{\boldsymbol \rho}}_0(x,y),\hat{{\boldsymbol \rho}}_1(x,y)) \equiv -\lim_{M\rightarrow \infty} {\ln[D_{\pi_0}(\hat{{\boldsymbol \rho}}_0(x,y),\hat{{\boldsymbol \rho}}_1(x,y))]}/M$ be the QCB on its error-probability exponent. Then, using the Helstrom limit’s being concave in quantum states (see Lemma 1 in the Appendix), we can show (see Lemma 2 in the Appendix) that the Helstrom limit’s error-probability exponent for QI target detection, $\xi_{\rm QI}\equiv-\lim_{M\to\infty}{\ln[D_{\pi_0}(\hat{\bar{{\boldsymbol \rho}}}_0,\hat{\bar{{\boldsymbol \rho}}}_1)]}/M$, vanishes, i.e., $\xi_{\rm QI} = 0$, for all $\pi_0\pi_1 \neq 0$. Having $\xi_{\rm QI} = 0$ implies that optimum quantum reception for QI target detection with Rayleigh fading has an error probability that decreases subexponentially with the number of signal-idler mode pairs that are employed. This subexponential error-probability behavior applies to *all* QI receivers, including the FF-SFG, SFG, and OPA receivers. Because OPA receivers are relatively easy to build [@Zheshen_15]—as opposed to the far more complicated SFG and FF-SFG receivers [@Zhuang_2017]—one might hope that QI with OPA reception would offer a performance advantage over optimum CI for the Rayleigh-fading scenario. We next show that such is not the case.
[*OPA reception for QI with Rayleigh fading*]{}—. It is difficult to get an analytic error-probability approximation for QI with OPA reception in the Rayleigh-fading scenario, so we will content ourselves with finding its SNR and comparing that result to the SNR for the optimum Rayleigh-fading CI system. The OPA receiver’s essence is converting QI’s phase-sensitive cross-correlation signature of target presence to an average photon-number signature that can be sensed with direct detection. In particular, the OPA receiver measures $\hat{N} \equiv\sum_{m=1}^M \hat{a}_m^\dagger\hat{a}_m$, where $\hat{a}_m = \sqrt{G}\,\hat{a}_{I_m} + \sqrt{G-1}\,\hat{a}_{R_m}^\dagger$ is the idler-port output of a low-gain ($\max(N_S/N_B,N_S/\kappa N_B^2) \ll G-1 \sim \sqrt{N_S}/N_B \ll 1$) OPA. Hence, we define its SNR to be ${\rm SNR}_{\rm OPA} \equiv [(\sum_{j=0}^1(-1)^j\langle\hat{N}\rangle_j) /(\sum_{j=0}^1\sqrt{{\rm Var}_j(\hat{N})})]^2$, where $\langle \hat{N}\rangle_j$ and ${\rm Var}_j(\hat{N})$ for $j=0,1$ are the conditional means and conditional variances of the $\hat{N}$ measurement given $h=j$.
For known $\kappa$ and $\phi =0$, we get $\langle\hat{N}\rangle_1 - \langle\hat{N}\rangle_0 \approx 2M\sqrt{G(G-1)\kappa N_S(N_S+1)}$. Combining this result with ${\rm Var}_j(\hat{N}) \approx \langle\hat{N}\rangle_j$ for the $\hat{N}$ measurement’s conditional variances, gives ${\rm SNR}_{\rm OPA} \approx M\kappa N_S/N_B$ when $N_S \ll 1$, $\kappa\ll 1$ is known, $\phi=0$, and $N_B \gg 1$. In the Rayleigh-fading case, the uniformly-distributed random phase destroys the phase-sensitive cross-correlation signature in $\langle \hat{N}\rangle_1$, leading to $\langle\hat{N}\rangle_1 - \langle\hat{N}\rangle_0 = M(G-1)\bar{\kappa}N_S$, and it adds $2M^2(G-1)\bar{\kappa}N_S$ to ${\rm Var}_1(\hat{N})$, hence giving us $${\rm SNR}_{\rm OPA} \approx \frac{M(G-1)(\bar{\kappa}N_S)^2/N_B}{(1+\sqrt{1+ 2M\bar{\kappa}N_S/N_B})^2},$$ which is much smaller than $M\bar{\kappa}N_S/N_B$, the ${\rm SNR}_{\rm OPA}$ for a known $\kappa = \bar{\kappa}$ and $\phi=0$ [@footnote3].
Optimum CI for Rayleigh fading does matched filtering of its heterodyne detector’s output followed by square-law envelope detection that yields an output, $R$, which is exponentially distributed under both $h=0$ and $h=1$ [@VanTrees1]. The SNR for this system, ${\rm SNR}_{\rm CI} \equiv [(\sum_{j=0}^1(-1)^j\langle R\rangle_j)/(\sum_{j=0}^1\sqrt{{\rm Var}_j(R)})]^2$, satisfies $${\rm SN}_{\rm CI} = (M\bar{\kappa} N_S/2N_B)/\left(1+M\bar{\kappa}N_S/2N_B\right)^2,$$ which is orders of magnitude greater than ${\rm SNR}_{\rm OPA}$ for Rayleigh fading in the interesting $M\bar{\kappa}N_S/N_B \gg 1$ operating regime.
[*SFG Reception for QI with Rayleigh Fading*]{}—. The SFG receiver [@Zhuang_2017] uses a succession of $K$ SFG stages. At the input to each such stage a beam splitter taps off a small fraction of the light returned from the region of interest to undergo SFG with the retained idler light. The returned-light output from that SFG process is then recombined with the portion remaining from that stage’s input beam-splitter and applied, along with the retained-idler output, to the next stage. Photon-counting measurements are performed on the SFG’s sum-frequency output and on the auxiliary output from the return-light beam splitter at the output of each SFG stage. These measurements are used to decide on target absence or presence. Figure \[SFGrcvr\] shows a schematic representation of the SFG receiver’s $k$th stage, for more details see Ref. [@Zhuang_2017].
![Schematic representation of the sum-frequency generation (SFG) receiver’s $k$th stage, showing only the $m$th mode pair, although all $M$ mode pairs are processed simultaneously. The $m$th mode pair of the returned light ($\hat{a}_{R_m}^{(k)}$) and the retained idler ($\hat{a}_{I_m}^{(k)}$) at the input to the $k$th stage is transformed into the corresponding mode pair at that stage’s output by means of SFG. Photon-counting measurements are made on the single-mode sum-frequency output ($\hat{b}^{(k)}$) and the auxiliary output modes ($\{\hat{a}_{E_m}^{(k)} : 1 \le m \le M\}$). The SFG receiver’s decision as to target absence or presence is based on the total of all the photon-counting measurements, i.e., $N_T \equiv \sum_{k=1}^K(N_b^{(k)} + N_E^{(k)}),$ where $N_b^{(k)}$ is the outcome of the $\hat{b}^{(k)\dagger}\hat{b}^{(k)}$ measurement, and $N_E^{(k)}$ is the outcome of the $\sum_{m=1}^M\hat{a}_{E_m}^{(k)\dagger}\hat{a}_{E_m}^{(k)}$ measurement.[]{data-label="SFGrcvr"}](Zhuang_QI_with_fading_Fig1.pdf){width="40.00000%"}
For known $\kappa$ and $\phi = 0$, SFG reception’s error probability achieves the QCB. The FF-SFG receiver [@Zhuang_2017] augments the SFG receiver with pre-SFG and post-SFG squeezers, whose parameters are chosen in accordance with a Bayesian update rule that is controlled by feed-forward information from the prior stages. FF-SFG reception reaches the Helstrom limit for QI target detection—in both the Bayesian and Neyman-Pearson settings—for known $\kappa$ and $\phi = 0$ [@Zhuang_2017; @zhuang2017entanglement]. Because its feed-forward operations exploit $\phi=0$, FF-SFG reception ceases to function effectively when $\phi$ is uniformly distributed. SFG reception, which eschews the use of feed-forward, *does* cope with random amplitude and phase, as we now show.
When $h=0$, the SFG receiver’s total photon count—i.e., $N_T\equiv \sum_{k=1}^K(N_b^{(k)} + N_E^{(k)})$ from Fig. \[SFGrcvr\]—is the sum of $M$ iid Bose-Einstein random variables, and has mean value $N_0 \simeq -N_S \ln(\epsilon)/2$ for $N_S \ll 1$. When $h=1$, and conditioned on the values of $\kappa$ and $\phi$, the statistics of the SFG receiver’s total photon count equal those for direct detection of the coherent state $|\sqrt{(1-\epsilon)M\kappa N_S/N_B}\,e^{i\phi}\rangle$ embedded in a weak thermal-noise background of average photon number $N_0 \ll 1$. In these expressions, $\epsilon \ll 1$ is chosen to obtain good performance, see [@Zhuang_2017] for details. When $M\kappa N_S/N_B \gg N_0$, the thermal contribution to the $h=1$ statistics can be neglected. Then, averaging the $h=1$ conditional state over the $\sqrt{\kappa}$ and $\phi$ statistics results in a thermal state with average photon number $N_1 = (1-\epsilon)M\bar{\kappa}N_S/N_B$, implying that the SFG receiver has reduced Rayleigh-fading QI target detection to discriminating between two thermal states, $\hat{\sigma}_{0} = \sum_{n=0}^\infty [N_0^n/(N_0+1)^{(n+1)}]\ket{n}\bra{n}$ and $\hat{\sigma}_{1} = \sum_{n=0}^\infty [N_1^n/(N_1+1)^{(n+1)}]\ket{n}\bra{n}$, using photon-counting measurements. SFG reception’s minimum error-probability decision, $\tilde{h} = 0$ or 1, is therefore $\tilde{h}={\mathop{\mathrm{argmax}}}_h \pi_h \!\left[N_h^n/(N_h+1)^{(n+1)}\right]$, where $n$ is the observed photon count.
The preceding rule can be implemented as a threshold test: $\tilde{h} = 1$ if and only if $n > n_t$, where the threshold $n_t$ satisfies $\pi_0 N_0^{n_t}/(N_0+1)^{(n_t+1)}\ge \pi_1 N_1^{n_t}/(N_1+1)^{(n_t+1)}$ and $\pi_0 N_0^{n_t+1}/(N_0+1)^{(n_t+2)}< \pi_1 N_1^{n_t+1}/(N_1+1)^{(n_t+2)}$. SFG reception’s ROC—its $P_D$ versus $P_F$ behavior—can now be obtained analytically. For integer $n_t$, we have $P_F^{\rm SFG}= [N_0/(N_0+1)]^{n_t+1}$ and $P_D^{\rm SFG}=[N_1/(N_1+1)]^{n_t+1}$. ROC points intermediate between those generated with integer thresholds are then obtained from randomized tests [@VanTrees2].
The Bayesian approach’s error probability is easily found once its decision rule’s threshold $n_t$ is determined. Evaluating the false-alarm and detection probabilities for that threshold value, SFG reception’s error probability then follows from $
\Pr(e)_{\rm SFG}=\pi_0P_F^{\rm SFG}+\pi_1 (1-P_D^{\rm SFG}).
$ For $N_S\to 0$ with $\epsilon\ll1$, we find that $n_t=0$ and hence (e)\_[SFG]{}(e)\_[SFG]{}\^[N\_S0]{}\_1/(1+M|N\_S/N\_B). \[NS0\] This result’s algebraic scaling with $M$ is consistent with our earlier finding that optimum quantum reception for Rayleigh-fading QI target detection has an error probability that decreases subexponentially with increasing $M$.
[*QI versus CI for Rayleigh Fading*]{}—. We are now prepared to demonstrate that QI target detection with SFG reception enjoys a significant performance advantage over CI target detection in the Rayleigh-fading scenario. We start with the Neyman-Pearson criterion, for which we already have the ROC for QI with SFG reception. The ROC for CI target detection with a coherent-state transmitter and heterodyne detection is [@VanTrees1] $P_D^{\rm CI}=\left(P_F^{\rm CI}\right)^{1/{\left(1+M\bar{\kappa}N_S/N_B\right)}}.$ Figure \[Fig\_comparisons\_ROC\] compares two QI and CI ROCs. Similar to what was assumed in Refs. [@Tan2008; @Zhuang_2017], we took $\bar{\kappa} = 0.01$, $N_B=20$, and $\epsilon=0.01$ for both comparisons. In one case we assumed $N_S = 10^{-4}$ and $M = 10^{8.5}$, while in the other we chose $N_S = 10^{-2}$ and $M = 10^{6.5}$. Figure \[Fig\_comparisons\_ROC\] shows that QI target detection with SFG reception has a much higher detection probability than optimum CI target detection at low false-alarm probabilities.
![QI and CI ROCs for Rayleigh-fading target detection with $\bar{\kappa}=0.01$, $N_B = 20$, and $\epsilon=0.01$. (a) $N_S=10^{-4}$ and $M=10^{8.5}$. (b) $N_S=10^{-2}$ and $M=10^{6.5}$. \[Fig\_comparisons\_ROC\] ](Zhuang_QI_with_fading_Fig2a.pdf "fig:"){width="22.50000%"} ![QI and CI ROCs for Rayleigh-fading target detection with $\bar{\kappa}=0.01$, $N_B = 20$, and $\epsilon=0.01$. (a) $N_S=10^{-4}$ and $M=10^{8.5}$. (b) $N_S=10^{-2}$ and $M=10^{6.5}$. \[Fig\_comparisons\_ROC\] ](Zhuang_QI_with_fading_Fig2b.pdf "fig:"){width="22.50000%"}
Turning now to the Bayesian approach, we again have the QI result in hand, and we find optimum CI’s error probability from $\Pr(e)_{\rm CI}=\min_{P_F^{\rm CI}}[\pi_0 P_F^{\rm CI}+\pi_1(1-P_D^{\rm CI})]$. Figure \[Fig\_comparisons\] plots $\Pr(e)_{\rm SFG}$ and $\Pr(e)_{\rm CI}$ versus $\log_{10}(M)$ for equally-likely target absence or presence assuming $\bar{\kappa} = 0.01$, $N_B = 20$, and $\epsilon = 0.01$ for $N_S = 10^{-4}$ and $N_S = 10^{-2}$. Here we see that QI target detection with SFG reception offers a significantly lower error probability than optimum CI target detection. Indeed, for $MN_S \gg 1$ we obtain the asymptotic result (e)\_[CI]{}+O(), which is a factor of $\ln(M\bar{\kappa}N_S/N_B)$ higher than the corresponding result for $\Pr(e)_{\rm SFG}^{N_S\to0}$ when $M\bar{\kappa}N_S/N_B \gg 1$. Moreover, Fig. \[Fig\_comparisons\]a shows that $N_S = 10^{-4}$ is small enough to ensure $\Pr(e)_{\rm SFG} \approx \Pr(e)_{\rm SFG}^{N_S\to0}$ for the parameter values employed therein. At high enough $M$ values, however, the effect of background noise in the SFG process becomes significant and $\Pr(e)_{\rm SFG}$ begins to deviate from the ideal $N_S\to0$ result. The onset of this deviation occurs at lower $M$ values when $N_S=10^{-2}$, as seen in Fig. \[Fig\_comparisons\]b, because the background-noise effect on the SFG process is proportional to $N_S$ [@Zhuang_2017]. Nevertheless, QI’s advantage over CI persists. We also see that QI target detection’s robustness to noise is worse for Rayleigh fading than what our previous results [@Zhuang_2017] showed for known $\kappa$. This reduced robustness arises from noise having greater impact on Rayleigh-fading error probability—because $\kappa \ll \bar{\kappa}$ can occur—as opposed to its effect in a nonfading environment with $\kappa = \bar{\kappa}$.
![QI and CI error probabilities for Rayleigh-fading target detection with $\pi_0=\pi_1=1/2$, $\bar{\kappa} = 0.01$, $N_B=20$, and $\epsilon=0.01$. (a) $N_S=10^{-4}$. (b) $N_S=10^{-2}$. The slope discontinuity in $\Pr(e)_{\rm SFG}$ for $N_S = 10^{-2}$ is due to the its receiver’s photon-number threshold increasing from $n_t = 0$ to $n_t = 1$ at that point. \[Fig\_comparisons\] ](Zhuang_QI_with_fading_Fig3a.pdf "fig:"){width="22.50000%"} ![QI and CI error probabilities for Rayleigh-fading target detection with $\pi_0=\pi_1=1/2$, $\bar{\kappa} = 0.01$, $N_B=20$, and $\epsilon=0.01$. (a) $N_S=10^{-4}$. (b) $N_S=10^{-2}$. The slope discontinuity in $\Pr(e)_{\rm SFG}$ for $N_S = 10^{-2}$ is due to the its receiver’s photon-number threshold increasing from $n_t = 0$ to $n_t = 1$ at that point. \[Fig\_comparisons\] ](Zhuang_QI_with_fading_Fig3b.pdf "fig:"){width="22.50000%"}
[*Conclusions*]{}—. QI target detection is remarkable because it uses entanglement to outperform CI despite environmental loss and noise’s destroying that entanglement. Previously, both theory and experiment have demonstrated QI’s having an advantage over CI, but *only* for a target return with known amplitude and known phase. Yet lidar targets are generally speckle targets, so their target returns have Rayleigh-distributed amplitudes and uniformly-distributed phases. We have shown that SFG reception affords a target-detection performance advantage over optimum CI for this scenario, but its magnitude is much smaller than what QI provides for the nonfading situation. Nevertheless, our result brings QI target detection closer to practical application, although two major problems remain to be solved: implementing near-lossless idler-storage and near-unity efficiency SFG for low-brightness, broadband light.
Two final points now deserve mention. First, although we have limited our treatment to the Rayleigh-fading scenario, the SFG receiver’s immunity to a uniformly-distributed random phase means that it will also be effective against other fading distributions, e.g., the Rician fading that models a target return with both specular and diffuse components [@VanTrees1; @swerling1997radar]. Finally, because $N_B\gg 1$ most naturally occurs at microwave, rather than optical, wavelengths [@Barzanjeh_2015], SFG reception’s applicability to a variety of flat-fading scenarios makes it relevant for microwave as well as optical QI.
Q. Z. acknowledges support from the Claude E. Shannon Research Assistantship. Z. Z. and J. H. S. acknowledge support from Air Force Office of Scientific Research Grant No. FA9550-14-1-0052.
[*Appendix*]{}—. Here we prove the two lemmas that were used earlier.
[**Lemma 1**]{} *(Concavity of the Helstrom limit) Consider the problem of discriminating between states $\hat{\sigma}_0=\int\!{\rm d} {\bm x}\, f_{\bm X}({\bm x}) \hat{\rho}_0({\bm x})$ and $\hat{\sigma}_1=\int\! {\rm d} {\bm x}\, f_{\bm X}({\bm x}) \hat{\rho}_1({\bm x})$, where ${\bm X}$ is a random vector, that occur with prior probabilities $\pi_0$ and $\pi_1$. The Helstrom limit for this binary state-discrimination task satisfies $
D_{\pi_0}(\hat{\sigma}_0,\hat{\sigma}_1)
\ge \int\!{\rm d} {\bm x}\, f_{\bm X}({\bm x}) D_{\pi_0}(\hat{\rho}_0({\bm x}),\hat{\rho}_1({\bm x})).$*
[**Proof.**]{} Let $\hat{M}_0$ and $\hat{M}_1 = \hat{I}-\hat{M}_0$ be the Helstrom-limit positive operator-valued measurement for discriminating between $\hat{\sigma}_0$ and $\hat{\sigma}_1$ when those states’ prior probabilities are $\pi_0$ and $\pi_1$. Then we have that $$\begin{aligned}
\lefteqn{D_{\pi_0}(\hat{\sigma}_0,\hat{\sigma}_1) = \pi_0 {\rm tr}(\hat{M}_1\hat{\sigma}_0)+\pi_1{\rm tr}(\hat{M}_0\hat{\sigma}_1)} \nonumber \\
&=& \int\!{\rm d} {\bm x}\, f_{\bm X}({\bm x})
\{
\pi_0 {\rm tr}[\hat{M}_1\hat{\rho}_0({\bm x})] + \pi_1{\rm tr}[\hat{M}_0\hat{\rho}_1({\bm x})]\} \nonumber
\\
&\ge& \int\!{\rm d} {\bm x}\, f_{\bm X}({\bm x}) D_{\pi_0}(\hat{\rho}_0({\bm x}),\hat{\rho}_1({\bm x})), \nonumber\end{aligned}$$ and the proof is complete.
[**Lemma 2**]{} *(Error-probability exponent for QI with Rayleigh fading) For $h=0,1$, let $\hat{{\boldsymbol \rho}}_h(\sqrt{\kappa},\phi) = \otimes_{m=1}^M\hat{\rho}_h^{(m)}(\sqrt{\kappa},\phi)$, where $\hat{\rho}_h^{(m)}(\sqrt{\kappa},\phi)$ is the two-mode, zero-mean, Gaussian state whose Wigner covariance matrix is given by Eq. (\[hk\]), and let $\hat{\bar{{\boldsymbol \rho}}}_h$ be the unconditional density operators obtained by averaging $\hat{{\boldsymbol \rho}}_h(\sqrt{\kappa},\phi)$ over Rayleigh and uniform probability density functions for $\sqrt{\kappa}$ and $\phi$, respectively. Then, for all $\pi_0\pi_1 \neq 0$ we have $\xi_{\rm QI}\equiv-\lim_{M\to\infty}{\ln[D_{\pi_0}(\hat{\bar{{\boldsymbol \rho}}}_0,\hat{\bar{{\boldsymbol \rho}}}_1)]}/M = 0$.*
[**Proof.**]{} Because $\kappa \le 1$ is required for a passive target, i.e., one that only reflects, the Rayleigh pdf is really an approximation to $f_{\sqrt{\kappa}}(x) = 2xe^{-x^2/\bar{\kappa}}/\bar{\kappa}(1-e^{-1/\bar{\kappa}})$ for $0\le x\le 1$ that is very accurate in QI target detection’s $\bar{\kappa} \ll 1$ scenario. For proving Lemma 2, however, we need to employ the truncated pdf, so that Lemma 1 and the QCB’s exponential tightness for $M$-copy state discrimination gives us $$\begin{aligned}
\lefteqn{D_{\pi_0}(\hat{\bar{{\boldsymbol \rho}}}_0,\hat{\bar{{\boldsymbol \rho}}}_1)} \nonumber \\
&\ge& \int_0^1\!{\rm d}x\!\int_0^{2\pi}\!{\rm d}y\, \frac{2xe^{-x^2/\bar{\kappa}}}{2\pi \bar{\kappa}(1-e^{-1/\bar{\kappa}})}D_{\pi_0}(\hat{{\boldsymbol \rho}}_0(x,y),\hat{{\boldsymbol \rho}}_1(x,y)) \nonumber\\
&\ge& \int_0^1\!{\rm d}x\!\int_0^{2\pi}\!{\rm d}y\, \frac{2xe^{-x^2/\bar{\kappa}}}{2\pi \bar{\kappa}(1-e^{-1/\bar{\kappa}})} \nonumber \\
&&\hspace{.2in}\times\,\,C_{x,y}(M) e^{-M\xi_{\rm QCB}(\hat{\rho}_0(x,y),\hat{\rho}_1(x,y))}, \nonumber\end{aligned}$$ where the subunity prefactor, $C_{x,y}(M)$, is an algebraic function of $M$. Specifically, for all $0\le x\le 1$ and $0\le y\le 2\pi$, we have $\lim_{M\to\infty}\ln[C_{x,y}(M)]/M=0$. It follows that for every $\epsilon>0$ there is a finite $M_\epsilon(x,y)$ such that $C_{x,y}(M)\ge e^{-\epsilon M_\epsilon(x,y)}$ for all $M > M_\epsilon(x,y)$.
Because $\Omega \equiv \{0\le x\le 1, 0\le y\le 2\pi\}$ is a compact region, there is a *finite* $M^\star_\epsilon = \max_{(x,y)\in \Omega} M_\epsilon(x,y)$. So, for all $M > M^\star_\epsilon$ we have $$\begin{aligned}
D_{\pi_0}(\hat{\bar{{\boldsymbol \rho}}}_0,\hat{\bar{{\boldsymbol \rho}}}_1) &\ge& e^{-\epsilon M}\int_0^1\!{\rm d}x\!\int_0^{2\pi}\!{\rm d}y\, \frac{2xe^{-x^2/\bar{\kappa}}}{2\pi \bar{\kappa}(1-e^{-1/\bar{\kappa}})} \nonumber \\
&& \hspace{.2in}\times\,\,e^{-M\xi_{\rm QCB}(\hat{\rho}_0(x,y),\hat{\rho}_1(x,y))} \nonumber\end{aligned}$$ But $\min_{(x,y)\in \Omega}\xi_{\rm QCB}(\hat{\rho}_0(x,y),\hat{\rho}_1(x,y))$ occurs at $x=0$, where $\xi_{\rm QCB}(\hat{\rho}_0(0,y),\hat{\rho}_1(0,y)) = 0$, because $\hat{\bar{{\boldsymbol \rho}}}_0 = \hat{\bar{{\boldsymbol \rho}}}_1$ when the target return’s intensity vanishes. Thus, for any $0< \epsilon'<1$ we can define $\Omega_{\epsilon'} = \{(\sqrt{\kappa},\phi) : \xi_{\rm QCB}(\hat{\rho}_0(x,y),\hat{\rho}_1(x,y)) \le \epsilon'\}$, and then weaken our previous lower bound on the Helstrom limit to $$D_{\pi_0}(\hat{\bar{{\boldsymbol \rho}}}_0,\hat{\bar{{\boldsymbol \rho}}}_1) \ge e^{-(\epsilon+\epsilon') M}\Pr[(\sqrt{\kappa},\phi)\in \Omega_{\epsilon'}] > 0,\nonumber$$ where the last inequality follows from $\pi_0\pi_1 \neq 0$.
Applying this bound to the error-probability exponent then leads to $$\xi_{\rm QI}(\hat{\sigma}_0,\hat{\sigma}_1)
\equiv-\lim_{M\to\infty}{\ln[D_{\pi_0}(\hat{\bar{{\boldsymbol \rho}}}_0,\hat{\bar{{\boldsymbol \rho}}}_1)]}/M\nonumber \\
\le \epsilon + \epsilon'
\nonumber$$ Because this upper bound holds for all $\epsilon, \epsilon' > 0$, by continuity our proof is now complete.
[99]{}
M. F. Sacchi, [Phys. Rev. A]{} **[71]{}**, [062340]{} ([2005]{}).
M. F. Sacchi, [Phys. Rev. A]{} **[72]{}**, [014305]{} ([2005]{}).
S. Lloyd, [Science]{} **[321]{}**, [1463]{} ([2008]{}).
S.-H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola and J. H. Shapiro, [Phys. Rev. Lett.]{} **[101]{}**, [253601]{} ([2008]{}).
E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, [Phys. Rev. Lett.]{} **[110]{}**, [153603]{} ([2013]{}).
S. Guha and B. I. Erkmen, [Phys. Rev. A]{} **[80]{}**, [052310]{} ([2009]{}).
S. Ragy, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, M. G. A. Paris, G. Adesso, and M. Genovese, J. Opt. Soc. Am. B [**31**]{}, 2045 (2014).
Z. Zhang, S. Mouradian, F. N. C Wong, and J. H. Shapiro, [Phys. Rev. Lett.]{} **[114]{}**, [110506]{} ([2015]{}).
S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, [Phys. Rev. Lett.]{} **[114]{}**, [080503]{} ([2015]{}).
For brief discussions of QI in the broader context of quantum sensing see [@Genovese2016] or [@Braun2017].
M. Genovese, J. Opt. [**18**]{}, 073002 (2016).
D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell, and S. Pirnandola, arXiv:1701.05152 \[quant-ph\].
K. M. R. Audenaert, J. Calsamiglia, R. Muñoz-Tapia, E. Bagan, L. Masanes, A. Acín, and F. Verstraete, [Phys. Rev. Lett.]{} **[98]{}**, [160501]{} ([2007]{}).
Q. Zhuang, Z. Zhang, and J. H. Shapiro, [Phys. Rev. Lett.]{} **[118]{}**, [040801]{} ([2017]{}).
Q. Zhuang, Z. Zhang, and J. H. Shapiro, J. Opt. Soc. Am. B [**34**]{}, 1567 (2017).
J. W. Goodman, Proc. IEEE [**53**]{}, 1688 (1965).
J. W. Goodman, J. Opt. Soc. Am. [**66**]{}, 1145 (1976).
J. H. Shapiro, B. A. Capron, and R. C. Harney, Appl. Opt. [**20**]{}, 3292 (1981).
J. H. Shapiro, Appl. Opt. [**21**]{}, 3398 (1982).
Following [@Tan2008], we assume different average background photon numbers for $h=0$ and $h=1$ to preclude there being any passive signature of target presence. Because $\kappa \ll 1$ will prevail, there is little loss of generality in making this assumption.
These prior analyses are easily adapted to known $\kappa$ and known $\phi$, but in all that follows we will let $\phi = 0$ serve as a proxy for an arbitrary known phase value.
C. W. Helstrom, [J. Stat. Phys.]{} **[1]{}**, [231]{} ([1969]{}).
An alternative SNR calculation for Rayleigh-fading QI with OPA reception that exploits the ${\rm Var}_1(\hat{N})$ signature of target presence yields an even lower value, ${\rm SNR}_{\rm OPA} \simeq (G-1)(\bar{\kappa} N_S)^2/N_B$, when $M\bar{\kappa}N _S/N_B \gg 1$ and $(G-1)N_B \ll 1$.
H. L. Van Trees, *Detection, Estimation, and Modulation Theory, Part I* (Wiley, New York, 1968), Sec. 4.4.2.
H. L. Van Trees, *Detection, Estimation, and Modulation Theory, Part I* (Wiley, New York, 1968), Sec. 2.2.2.
P. Swerling, [IEEE Trans. Aerosp. Electron. Syst.]{} **[33]{}**, [698]{} ([1997]{}).
|
---
abstract: |
We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is ${\textsc{QMA}}$-hard, under randomized reductions, to distinguish between the cases when the players have a strategy using entanglement that succeeds with probability $1$ in the game, or when no such strategy succeeds with probability larger than $\frac{1}{2}$. This proves the “games quantum PCP conjecture” of Fitzsimons and the second author (ITCS’15), albeit under randomized reductions.
The core component in our reduction is a construction of a family of two-player games for testing $n$-qubit maximally entangled states. For any integer $n\geq
2$, we give such a game in which questions from the verifier are $O(\log n)$ bits long, and answers are $\operatorname{poly}(\log\log n)$ bits long. We show that for any constant ${\varepsilon}\geq 0$, any strategy that succeeds with probability at least $1-{\varepsilon}$ in the test must use a state that is within distance $\delta({\varepsilon}) =
O({\varepsilon}^c)$ from a state that is locally equivalent to a maximally entangled state on $n$ qubits, for some universal constant $c>0$. The construction is based on the classical plane-vs-point test for multivariate low-degree polynomials of Raz and Safra (STOC’97). We extend the classical test to the quantum regime by executing independent copies of the test in the generalized Pauli $X$ and $Z$ bases over ${\ensuremath{\mathbb{F}}}_q$, where $q$ is a sufficiently large prime power, and combine the two through a test for the Pauli twisted commutation relations.
Our main complexity-theoretic result is obtained by combining this family of games with techniques from the classical PCP literature. More specifically, we use constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC’05), and crucially rely on a linear property of such PCPs. Another consequence of our results is a deterministic reduction from the games quantum PCP conjecture to a suitable formulation of the constraint satisfaction quantum PCP conjecture.
author:
- 'Anand Natarajan[^1] Thomas Vidick[^2]'
bibliography:
- 'quantum\_pcp.bib'
title: |
Low-degree testing for quantum states,\
and a quantum entangled games PCP for QMA
---
Introduction
============
The PCP theorem [@AroLunMotSudSze98JACM; @AroSaf98JACM] makes a remarkable statement: any language that admits efficiently verifiable proofs of membership, i.e. any problem in ${\textsc{NP}}$, also admits proofs that can be verified by reading only a *constant* number of bits of the proof. Do similar encodings exist for problems that admit *quantum* proofs? Consider the local Hamiltonian problem. Is there a way to encode a witness for the minimal energy of a Hamiltonian in a way that the energy can be estimated to within inverse polynomial accuracy while accessing only a constant number of bits, or qubits, from the witness? The pursuit of this question, which, broadly speaking, asks for quantum extensions of the PCP theorem, has been one of the most fruitful and challenging problems animating quantum complexity theory in the past decade: it ties in to the theory of quantum error-correcting codes, has applications to quantum cryptography, and promises insights into the study of entanglement in ground states of local Hamiltonians [@AharonovAV13qpcp].
The question can be formalized in multiple ways. A first formulation, the “constraint satisfaction” variant of the quantum PCP (QPCP) conjecture [@AharonovILV09detectability], asks for the complexity of constant-factor approximations to the minimal energy of a local Hamiltonian $H$, normalized so that $\|H\|=1$. Despite considerable attention progress on the conjecture has been difficult [@AharonovE15commuting; @BrandaoH13product; @eldar2015local].
More recently a second formulation has been put forward. The “multiplayer games” variant of the QPCP conjecture, introduced in [@FV14], asks for the complexity of estimating, to within constant accuracy, the maximum success probability of provers (we use the terminology “provers” and “players” interchangeably) sharing entanglement in a multiplayer game, a quantity referred to as the *entangled value* of the game. The conjecture is a natural analogue of the “oracularized” formulation of the PCP theorem, which states that the maximum success probability of *classical* provers in a multiplayer game is ${\textsc{NP}}$-hard to approximate to within constant factors. (This can be thought of as a “scaled down” formulation of the equality ${\textsc{MIP}}= {\textsc{NEXP}}$ [@BabForLun91CC].)
In [@Vidick13xor], building on [@IV12] it was shown that the approximation problem for the entangled value of a multiplayer game remains ${\textsc{NP}}$-hard, provided there are at least three provers. This was extended to games with two provers only in [@NatarajanV17twoprover] (this result will be used as a building block in the present paper). In [@FV14; @ji2015classical] it was shown that inverse-polynomial approximations are ${\textsc{QMA}}$-hard (provided there are at least five provers), a result that is akin to a “quantum Cook-Levin theorem for entangled games.” These results motivate the following conjecture, first made in [@FV14]:
\[conj:qpcp\] Suppose given as input an explicit description of a classical multiplayer game. Then it is ${\textsc{QMA}}$-hard to determine whether provers sharing quantum entanglement (of arbitrary dimension) have optimal success probability at least $\frac{2}{3}$ or at most $\frac{1}{3}$ in the game.
We show that the conjecture holds, under randomized reductions.
\[conj:weak\_qpcp\] Suppose given as input an explicit description of a classical multiplayer game. Then it is ${\textsc{QMA}}$-hard, under randomized reductions, to determine whether provers sharing quantum entanglement (of arbitrary dimension) have optimal success probability at least $1$ or at most $\frac{1}{2}$ in the game.
Theorem \[conj:weak\_qpcp\] is stated and proved as Corollary \[cor:randomized\] in the body of the paper. The choice of constant $\frac{1}{2}$ in Theorem \[conj:weak\_qpcp\] is arbitrary, as for the kind of games we consider soundness amplification can be performed efficiently in parallel [@bavarian2017hardness].
We explain the need for a randomized reduction. Informally, the reason is that we do not know of a strong enough ${\textsc{QMA}}$-hardness result for the local Hamiltonian problem to initiate our reduction. In fact, we give two alternate formulations of Theorem \[conj:weak\_qpcp\] that would also establish the same ${\textsc{QMA}}$-hardness result, under deterministic reductions, provided that either:
1. it is ${\textsc{QMA}}$-hard to approximate the minimum energy of a local Hamiltonian in $Y$-free form (Definition \[def:gen-h\]) to within constant accuracy (this is a variant of the quantum PCP conjecture for local Hamiltonians), or
2. it is ${\textsc{QMA}}$ hard to approximate the ground energy of (not necessarily local) frustration-free Hamiltonian whose every term is a tensor product of generalized Pauli ${\tau}_X$ or ${\tau}_Z$ observables.
Note that point (i) amounts to a deterministic reduction from Conjecture \[conj:weak\_qpcp\] to the constraint satisfaction quantum PCP conjecture, and establishes the first proven relation between the two conjectures (see [@grilo2016pointer] for an incomparable result that relates stronger variants of both conjectures). Point (ii) is arguably a weaker assumption, as the gap is not required to be a constant and the terms of the Hamiltonian are not required to be local. However, due to the restriction that the Hamiltonian is frustration-free, it is currently not known whether the problem is ${\textsc{QMA}}$-hard (or even ${\textsc{QMA}}_1$-hard — though the frustration-free assumption can be relaxed to having exponentially small ground state energy).
Our results build on two main tools: a framework for protocols to test ground states, introduced in [@FV14] and further developed in [@ji2015classical; @NV17], and a new proof of soundness of the classical low-degree test of Raz and Safra against two entangled provers [@NatarajanV17twoprover]. The main result that underlies the complexity-theoretic applications is a two-prover test for $n$-qudit maximally entangled states, where each qudit has dimension $q=p^t = \operatorname{poly}\log(n)$ for a prime $p$ and integer $t$, that has inverse robustness independent of $n$ (for all ${\varepsilon}$ that are at least inverse polylogarithmic in $n$) and in which the verifier sends only $O(\log(n))$ bits to the provers, who reply with $O(\log\log n)$ bits each (Theorem \[thm:qld\]). This is an exponential improvement over all previous results, and provides the first robust entanglement test with sub-linear communication. While the ability to “test” structured objects with sub-linear efficiency has become customary in classical computer science, we find it remarkable that the framework for such tests may be extended to test such a complex object as quantum entanglement.
We first describe this test in more detail, before expanding on the complexity-theoretic consequences.
#### Efficient, robust entanglement tests.
The driving question behind our work is the following: “Is it possible to verify a quantum state using an amount of resources that scales sub-linearly in the number of qubits of the state?” We start with the “simplest” such state—the maximally entangled state. Results in self-testing have yielded increasingly efficient and robust tests for this state and other, more general families of highly entangled states. Here we loosely refer to the “efficiency” of a test as a measure of the total number of bits of communication involved in an execution of the test. The “robustness” of the test indicates how tightly success in the test characterizes the desired state: a test is $\delta({\varepsilon})$-robust if for all ${\varepsilon}\geq 0$, any strategy for the provers that succeeds with probability at least $1-{\varepsilon}$ in the test must use an entangled state that is within distance $\delta({\varepsilon})$ from the tested state (see Definition \[def:self-test\]). Using these measures, the best prior self-tests for a maximally entangled state of $n$ qubits are a test with communication $O(\log n)$ and robustness $O(n^{5/2}\sqrt{{\varepsilon}})$ [@chao2017test] and a test with communication $O(n)$ and robustness $O(\sqrt{{\varepsilon}})$ [@NV17]. Other recent results in this direction include [@OV16; @CN16; @Coladangelo16; @ColadangeloS17MS].
Our test is the first to combine robustness $\delta({\varepsilon}) = \operatorname{poly}({\varepsilon})$ that is independent of $n$, and logarithmic communication. Achieving both simultaneously is crucial to applications: constant (in $n$) robustness allows us to achieve gap-preserving reductions; logarithmic communication allows us to achieve efficient reductions.
As in previous results, the test is designed to constrain successful provers to use observables satisfying suitable relations; a statement about the entangled state follows by using that the state is stabilized by (a subset of) these observables. In the case of the maximally entangled state, the observables are all $n$-fold tensor products of Pauli observables. For reasons to be discussed below we test for qudits of dimension $q=p^t$ a prime power of order $q=\operatorname{poly}\log(n)$. This leads us to consider tensor products of single-qudit Pauli observables defined over the prime power field ${\ensuremath{\mathbb{F}_q}}$, which we denote using the symbol ${\tau}$: $$\label{eq:gen-pauli}
{\tau}_X(a) = \sum_{j \in {\ensuremath{\mathbb{F}_q}}} {|j+a\rangle}{\langlej|}\qquad\text{and}\qquad {\tau}_Z(b)= \sum_{j \in {\ensuremath{\mathbb{F}_q}}}
\omega^{{\mbox{\rm tr}}(b j)} {|j\rangle}{\langlej|}\;,$$ where $a,b\in{\ensuremath{\mathbb{F}_q}}$, $\omega = e^\frac{2i\pi}{p}$, addition and multiplication are over ${\ensuremath{\mathbb{F}_q}}$, and ${\mbox{\rm tr}}(\cdot)$ denotes the trace of ${\ensuremath{\mathbb{F}_q}}$ over ${ \ensuremath{{\mathbb{F}}}_p}$. The main difficulty we face is that there are $2\cdot q^n$ such observables, ${\tau}_X(a)={\tau}_X(a_1)\otimes \cdots\otimes {\tau}_X(a_n)$ and ${\tau}_Z(b)={\tau}_Z(b_1)\otimes \cdots \otimes {\tau}_Z(b_n)$ for $a,b\in {\ensuremath{\mathbb{F}_q}}^n$, an exponentially larger number than any test with polylogarithmic communication gives us direct access to. It is then natural to consider a test that certifies observables ${\tau}_X(a)$ and ${\tau}_Z(b)$ for $a,b\in T \subseteq {\ensuremath{\mathbb{F}_q}}^n $, where $|T|=\operatorname{poly}(n)$, and attempt to construct observables for all $a,b\in
{\ensuremath{\mathbb{F}_q}}^n $ in an inductive fashion, as is done in e.g. [@chao2017test], where $T$ is the set of all strings of Hamming weight at most $2$. Unfortunately, any naïve procedure will induce an error accumulation at each step of the induction, eventually resulting in a robustness parameter that depends polynomially on $n$ (as is the case in [@chao2017test]).
It is thus crucial to choose the set $T$ carefully — informally, it seems natural to require that this set behave in a “pseudorandom” way. We take direct inspiration from the classical proof of the PCP theorem, and use a set $T$ specified as the set of all codewords of a suitably chosen Reed-Muller code; this is the reason for using a sufficiently large qudit dimension $q$. Our proof eventually reduces the analysis to the soundness of the entangled-prover classical low-degree test [@NatarajanV17twoprover]. We explain the test, and its analysis, in more detail in Section \[sec:techniques\] below.
#### Testing ground states and a “gap preserving” reduction.
We sketch how our test for entanglement is applied to obtain results on the complexity of multiplayer entangled games. In the classical case, the proof that the value of a multiplayer game is at least as hard to approximate as the maximum fraction of constraints simultaneously satisfiable in a local constraint satisfaction problem proceeds via the technique of oracularization: the verifier selects a constraint at random and asks one prover for an assignment to all variables in the constraint and the other for an assignment to a single one of the variables. Given the provers’ answers, the verifier checks the natural satisfaction and consistency constraints. In the quantum case the analogous idea would require each prover to hold a copy of the ground state of a ${\textsc{QMA}}$-complete local Hamiltonian, and return qubits as requested by the verifier. This reduction does not work: it is not possible in general to check for “consistency” between the same qubit taken from two copies of an entangled state. In [@FV14] the idea was introduced of encoding the ground state using an error-correcting code and distributing a share to each prover. Subsequent work [@ji2015classical] showed that this idea can be used to show ${\textsc{QMA}}$-hardness of inverse-polynomial approximations to the entangled value of a multiplayer game. Unfortunately the reduction in [@ji2015classical] is not “gap-preserving”: a large promised energy gap in the starting instance of the local Hamiltonian problem does not lead to a large completeness-soundness gap in the resulting game. As a result, even assuming the “constraint satisfaction” QPCP does not lead to hardness for approximation factors larger than a fixed inverse polynomial. In [@NV17] we leveraged an entanglement test with constant robustness to achieve a gap-preserving reduction; unfortunately communication in the test is linear, resulting in a game with exponential size, so that no new complexity-theoretic consequence is obtained.
Armed with an exponentially more efficient entanglement test we are able to provide a much more effective reduction, yielding games of polynomial size from instances of the local Hamiltonian problem. The reduction follows similar lines as previous work, but with a new difficulty. Our entanglement test only certifies a specific family of observables: tensor products of generalized Pauli observables over ${\ensuremath{\mathbb{F}}}_q$, for $q$ a sufficiently large prime power. This requires us to initiate any direct reduction with a specific class of Hamiltonians, in so-called $Y$-free form (see Definition \[def:gen-h\]); informally, these are local Hamiltonians such that each local term is a tensor product of generalized ${\tau}_X$ and ${\tau}_Z$ observables. In the absence of general gap-preserving reductions between different variants of the local Hamiltonian problem (perturbation techniques [@CM13] do not generally preserve the promise gap) we obtain a reduction to the hardness of constant-factor approximations to the ground energy of local Hamiltonian of this form only. Nevertheless, even though the entanglement test requires a qudit dimension that scales (poly-logarithmically) with $n$, we show that any qubit Hamiltonian in $Y$-free form can be embedded in a Hamiltonian in $Y$-free form over qudits of dimension $2^t$ for any $t\geq 1$. As a result, we immediately obtain point (i) discussed earlier: that Conjecture \[conj:weak\_qpcp\] would follow from ${\textsc{QMA}}$-hardness of constant-factor approximations to local Hamiltonian whose every local term is a tensor product of ${\tau}_X$ and ${\tau}_Z$ Pauli observables (signed weights of up to poly-logarithmic size are allowed).
#### Composition and PCP.
To obtain strong results we develop more elaborate reductions, with the aim of removing the assumption on *locality* of the Hamiltonian whose ground state energy is being tested. As our entanglement test has direct access only to local Pauli observables, it cannot be used to evaluate the expectation value of non-local observables (acting on more than a constant number of qudits). We get around this as follows. Say the verifier would like to estimate the expectation value of a nonlocal tensor product observable such as ${\tau}_{X}(b)$, for some $b\in{\ensuremath{\mathbb{F}_q}}^n$. The verifier asks each prover to measure all its qudits in the $X$ basis, obtaining an outcome $a \in {\ensuremath{\mathbb{F}_q}}^n$, and report the value of the inner product $c = b \cdot a$. This provides the verifier with an estimate of the energy of ${\tau}_{X}(b)$. However, it remains to ensure that the outcome reported by the prover was obtained honestly, i.e. by measuring all qudits on which the observable acts, without having the ability to “read” all the single-qubit outcomes obtained. This sounds very similar to the kind of NP statements that PCPs are designed to allow efficient verification of, and indeed we employ classical PCP techniques, more specifically the notion of *PCP of proximity* (PCPP).
In order to verify that a prover honestly computed the inner product $c = b \cdot a$, the verifier asks it to provide PCPP of this fact. A PCPP for a language is a proof that a given input is in the language, which can be verified by making only a few queries to both the proof and the input. In our setting, the verifier asks each prover to compute a PCPP $\Pi$ for the claim that the measurement outcome string $a$ is in the language $L = \{x : b \cdot x = c\}$. This proof can be verified by making constantly-many queries to $\Pi$, together with constantly many queries to $a$. Both of these correspond to *local* measurements, either of the shared quantum state, or the proof string $\Pi$ generated from the measurement outcomes, and can thus be certified using our entanglement test.
There are two subtleties that arise. First, a PCPP (viewed as a nonlocal game) that is classically sound need not be sound against entangled provers. To address this, we perform a further layer of composition, encoding the PCPP proof $\Pi$ in a low-degree polynomial and querying this polynomial. Secondly, in our setting *completeness* does not automatically hold either. This is because each prover $j$ only has access to one share of the shared state, which is a qudit-by-qudit encoding of the actual QMA witness. The prover can thus only supply bits from a proof $\Pi_j$ computed from its share. As a result the usual method of transforming a PCP into a game, namely by querying multiple provers for locations in the proof and checking consistency between them, fails since even honest provers do not know each other’s measurement outcomes and thus cannot answer consistently. To surmount this obstacle, we exploit the linearity of the error correcting code, together with a linear PCPP construction from [@BGHSV05], for which the proof $\Pi$ is a linear function of the input $a$; the linearity holds as long as the language $L$ is itself specified by a set of linear equations, i.e. $L = \{x: Ax = b\}$. The linearity of the PCPP allows the verifier to check consistency between one prover’s answers and the appropriate linear combination of answers returned by the other provers.[^3].
With this PCPP-based protocol for measuring nonlocal Pauli observables in place, the proof of Theorem \[conj:weak\_qpcp\] follows: starting with a ${\textsc{QMA}}$-hard instance of the local Hamiltonian problem with inverse-polynomial promise gap, we amplify the gap by taking a large tensor product, and then randomly sample a polynomial subset of the exponentially many terms in the tensor product. By the matrix Chernoff bound [@AW02], with high probability this sampling preserves the promise gap, and the resulting nonlocal Hamiltonian can be tested using our protocol. (This random sampling is the source of the “randomized reductions” in Theorem \[conj:weak\_qpcp\].)
Finally, our PCPP-based protocol enables us to check not just one nonlocal term but also many terms at once, provided that they are all tensor products of Paulis in the same basis. This allows us to obtain a protocol that accommodates an inverse-polynomial promise gap for the ground energy, provided the Hamilton is frustration free (all of its terms are simultaneously satisfied in the ground state), and each of it terms can be expressed as a tensor product of generalized ${\tau}_X$ or ${\tau}_Z$ observables, acting on an arbitrary number of qudits (see Definition \[def:linear-xz\]). This shows point (ii) discussed earlier.
Techniques {#sec:techniques}
----------
Our main result, a robust entanglement test with logarithmic communication, can be stated informally as follows. For a formal statement, we refer to Theorem \[thm:qld\] in Section \[sec:qld\].
Let $n$ be an integer and $q = p^t$ a prime power such that $q = \Theta(\frac{\log^2 n}{\log \log n})$. Then there exists a two-prover test ${\textsc{q-lowdeg}}$ in which the verifier sends questions of length $\operatorname{poly}(\log n,\log q)$ and receives answers of length $O(\operatorname{poly}\log\log(n) \cdot \log q)$ such that the following hold:
1. *(Completeness:)* There exists a strategy for the provers based on sharing an $n$-qudit maximally entangled state, with qudits of local dimension $q$, and making measurements in the eigenbasis of tensor products of generalized ${\tau}_X$ or ${\tau}_Z$ observables over ${\ensuremath{\mathbb{F}_q}}$;
2. *(Soundness:)* For any ${\varepsilon}\geq 0$, any strategy that is accepted with probability at least $1-{\varepsilon}$ in the test must use an entangled state that is (up to local isometries) within distance $\delta =
\operatorname{poly}(\operatorname{poly}(p)\cdot\operatorname{poly}({\varepsilon}) )$ from an $n$-qudit maximally entangled state.[^4]
A typical setting of parameters for the theorem is to choose $p$ a constant, e.g. $p=2$, $t= \Theta(\log\log n)$, and ${\varepsilon}$ a small constant, which leads to constant soundness $\delta$.
The test mentioned in the theorem has three components: (a) a low-degree test in the $X$ basis; (b) a low-degree test in the $Z$ basis; (c) an anti-commutation test relating the two bases. Both (a) and (b) are direct adaptations of the “plane-vs-point” low-degree test from [@raz1997sub]. The basis label, $X$ or $Z$, asks the prover to measure its $n$ qudits in the simultaneous eigenbasis of the observables ${\tau}_X(a)$ or ${\tau}_Z(b)$ defined in respectively. The prover is then asked to encode the resulting outcome $a\in {\ensuremath{\mathbb{F}_q}}^n$ as a low-degree polynomial $g_a:{\ensuremath{\mathbb{F}_q}}^m \to {\ensuremath{\mathbb{F}_q}}$, where $m = O(\log n / \log\log n)$, and return either the evaluation of the polynomial at a randomly chosen point $x\in {\ensuremath{\mathbb{F}_q}}^m$, or its restriction to a randomly chosen two-dimensional subspace $s$ of ${\ensuremath{\mathbb{F}_q}}^m$. Part (c) is designed to enforce the “twisted commutation” relations ${\tau}_X(a){\tau}_Z(b) = \omega^{-{\mbox{\rm tr}}(ab)} {\tau}_Z(b){\tau}_X(a)$ satisfied by these observables. Before explaining the test and its analysis in greater detail, we first review the main steps that go into showing soundness of the classical low-degree test.
#### Classical low-degree tests.
The effectiveness of the classical low-degree test is based on the use of the following Reed-Muller encoding of an $n$-variable assignment $a=(a_1,\ldots,a_n)\in\{0,1\}^n$. First, integer values $h$ and $m$ are chosen so that $h^m \geq n$, and an injection ${\pi}:\{1,\ldots,n\} \to \{0,\ldots,h-1\}^m$ is fixed. Second, a finite field ${\ensuremath{\mathbb{F}_q}}$ is chosen such that $q\geq h$. Third, a function $g_a : {\ensuremath{\mathbb{F}_q}}^m\to {\ensuremath{\mathbb{F}_q}}$ is defined such that $g_a({\pi}(i)) = a_i$ for all $i\in\{1,\ldots,n\}$, and $g_a$ has degree at most $h$ in each of its $m$ variables; $g_a$ can be obtained by straightforward polynomial interpolation. Finally, the encoding of $a$ is defined as the concatenation of the evaluation table of $g_a$ at every point $x\in {\ensuremath{\mathbb{F}_q}}^m$ with a table describing the restriction of $g_a$ to every two-dimensional subspace $s\subseteq {\ensuremath{\mathbb{F}_q}}^m$. The encoding has roughly $q^{3m}$ entries, and each entry has size $O(d^2 \log q)$, where $d=mh$ is the total degree of $g_a$. Choosing $h \approx \log n$ and $m \approx \log n/\log\log n$ yields an encoding of quasi-polynomial size, $n^{O(\log n)}$, as long as $q$ is also polynomial in $n$.
When used for constructions of PCPs, the low-degree test provides an encoding that can be tested and evaluated while making only a small number of queries. This is achieved based on the following observations. First, the encoding can be checked by making only a constant number of queries: the test selects a pair $(x,s)$ such that $s$ is a uniformly random subspace and $x$ a uniformly random point in $s$, and checks consistency between the corresponding entries of the encoding. Second, the evaluation of $g_a$ at any point $z\in {\ensuremath{\mathbb{F}_q}}^m$ can be recovered by making $O(d)$ queries to the encoding in a way that each query is uniformly distributed: select a uniformly random line going through $z$, query $d+1$ points at random on the line, and interpolate to recover the value at $z$.
The analysis of the low-degree test described in the previous paragraph is not simple. The goal is to show that any table which passes the test with probability $1-{\varepsilon}$ must be close to the encoding of a polynomial of the form $g_a$, for some $a\in {\ensuremath{\mathbb{F}_q}}^n$. The proof is constructive: it recovers a low-degree polynomial $g_a$ through $m$ successive steps of interpolation. The case $m=2$ is immediate, since by definition the encoding contains the restriction of $g_a$ to any two-dimensional subspace. For general $m$, one selects $(d+1)$ parallel $(m-1)$-dimensional subspaces, applies the induction hypothesis to each, and interpolates to recover a $m$-variate polynomial defined over the whole space. The key difficulty in the analysis is to control the error: naïvely, it would, at best, double at each step, resulting in an unmanageable blow-up. The key innovation of the test, and its analysis, is a method to limit this blow-up by a procedure of “self-improvement”.
#### Entanglement tests.
Before moving on to our quantum low-degree test, it is useful to first recall the intuition behind our prior work [@NV17], which establishes a similar quantum analogue for the Hadamard encoding, which is based on the linearity test of Blum et al. [@BLR93].
In the linearity test, the assignment $a\in\{0,1\}^n$ is encoded as the evaluation table of the function $f_a : {\ensuremath{\mathbb{F}}}_2^n \to {\ensuremath{\mathbb{F}}}_2$, $f_a(x)=x\cdot a$. Each entry of the encoding is a single bit, but there are $2^n$ entries, thus the table has exponential size. The linearity test makes three queries, $x,y$ and $x+y$ for $x,y$ uniformly distributed in ${\ensuremath{\mathbb{F}}}_2^n$, and verifies that $f_a(x)+f_a(y)=f_a(x+y)$. The soundness analysis of the test is based on Fourier analysis; no induction is needed.
To turn the linearity test into a test for entanglement we first re-interpret it using the language of representation theory. The additive structure of ${\ensuremath{\mathbb{F}}}_2^n$ makes it into an abelian group, whose irreducible representations are the $2^n$ characters $\chi_a(x)=(-1)^{a\cdot x}$. An arbitrary table $f: {\ensuremath{\mathbb{F}}}_2^n \to {\ensuremath{\mathbb{F}}}_2$ can also be seen as a mapping $g=(-1)^f$ from the additive group of ${\ensuremath{\mathbb{F}}}_2^n $ to the $1$-dimensional unitary group, $U( {\ensuremath{\mathbb{C}}})$. A table $f$ which is accepted in the linearity test with probability $1-{\varepsilon}$ is an approximate representation of the group, in the sense that ${\ensuremath{\mathop{\textsc{E}}_{x,y}}} |g(x)g(y)-g(x+y)|^2 = O({\varepsilon})$, where the expectation is uniform. Thus the analysis of the linearity test exactly amounts to showing that approximate representations of abelian groups are close to exact representations (i.e. the characters, which precisely correspond to the linear functions).
We can try to apply the same reasoning to entangled-prover strategies. Using matrix-valued Fourier analysis it is possible to show that a quantum strategy which succeeds with probability $1-{\varepsilon}$ in an $X$-basis linearity test (resp. an $Z$-basis linearity test) implies the existence of observables for the provers which satisfy approximate linearity conditions $X(a)X(b)\approx X(a+b)$ (resp. $Z(a)Z(b)\approx Z(a+b)$), where the approximation holds on average over uniform $a,b\in{\ensuremath{\mathbb{F}}}_2^n$ and is measured using the state-dependent norm that is standard in testing. These relations by themselves do not imply anything “quantum”; in particular they are satisfied by one-dimensional observables $X(a)=Z(a)=(-1)^{f(a)}$. To obtain a truly quantum test we are missing a constraint relating the two bases: the Pauli (anti)-commutation relation $X(a)Z(b)=(-1)^{a\cdot b}Z(b)X(a)$. Enforcing this relation would allow us to frame the family of unitaries $\{\pm
X(a)Z(b),\,a,b\in{\ensuremath{\mathbb{F}}}_2^n\}$ as a representation of the Pauli group modulo complex phase (also known as the Weyl-Heisenberg group) and combine results on the stability of approximate representations [@gowers2015inverse] with information on the structure of irreducible representations of that group to conclude. This is what justifies the inclusion of part (c), an anti-commutation test, which can be based on e.g. the Mermin-Peres Magic Square game [@Arvind:02] to test for the desired anti-commutation relations.
#### A quantum low-degree test.
The previous outline of an entanglement test based on the BLR linearity test is implemented in [@NV17]. The use of the linearity test has two main advantages: (i) when executed in a single basis, its analysis with two entangled provers follows a direct argument using Fourier analysis; (ii) combining the linearity test in the $X$ and $Z$ bases naturally gives access to two families of observables $X(a)$ and $Z(b)$ for the provers, that can be used to specify an approximate representation of the $n$-qubit Weyl-Heisenberg group as described above, with the (anti)-commutation test certifying all required pairwise group relations.
To reduce the communication required in the test, it is natural to turn to low-degree tests: as described above, the latter only require poly-logarithmic, instead of linear, communication. Due to the fact that the test has only a quasi-polynomial number of questions, however, a strategy for the provers only involves a quasi-polynomial number of observables: how can one show that all exponentially many (anti)-commutation relations hold, in principle, between observables defined on the prover’s space, if the test itself only requires the existence of a tiny subset of these observables in order to be played?
This difficulty can be overcome as follows. From the classical analysis of the low-degree test, or rather its entanglement-resistant analogue [@NatarajanV17twoprover], it is possible to show that a strategy that succeeds in the $X$-basis (resp. $Z$-basis) low-degree test implies the existence of a family of observables $X(a)$ (resp. $Z(b)$), for $a\in {\ensuremath{\mathbb{F}_q}}^n$, that satisfy the commutation relations $X(a)X(b)=X(a+b)$ (resp. $Z(a)Z(b)=Z(a+b)$). Moreover, the use of an appropriate generalization of the Magic Square game over ${\ensuremath{\mathbb{Z}}}_2$, introduced in [@ColadangeloS17MS], to ${\ensuremath{\mathbb{Z}}}_s$, for any integer $s$, that allows us to test for the appropriate twisted commutation relation between any two observables that are actually queried in the test. The difficulty is to establish the right relations between observables $X(a)$ and $Z(b)$ that are not queried from the test, but whose existence follows from the independent application of the entangled-prover analysis of the low-degree test to the $X$- and $Z$- basis executions of the test.
Our solution proceeds in three steps. The first step consists in combining $X$ and $Z$ observables together into a single family of commuting observables. We do this by adjoining two ancilla systems for each prover, each initialized in a maximally entangled state local to the prover, and setting $\hat{X}(x) = X(x)_{{{\textsf{A}}}} \otimes {\tau}_X(x)_{{{\textsf{A'}}}} \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}_{{{\textsf{A''}}}}$, where ${\tau}_X(x)_{{{\textsf{A'}}}}$ denotes the $n$-qudit Pauli that the prover’s $X(x)$ is supposed to implement, for $x$ among the possible queries in the test. Defining $\hat{Z}(z)$ similarly, provided $\hat{X}(x)$ and $\hat{Z}(z)$ satisfy the (conjugate of) the twisted commutation relation satisfied by ${\tau}_X(x)$ and ${\tau}_Z(z)$ we have obtained a family of (approximately) commuting observables.
In the second step we use these commuting observables to define a strategy for the classical low-degree test, not over $m$-variate polynomials as the initial test requires, but over $2m$ variables, half of which are “$X$” variables, and half of which are “$Z$” variables. To construct such a strategy we have to define “points” and “subspace” measurements from the $\hat{X}(x)$ and $\hat{Z}(z)$, using the information that the initial observables $X(x)$ and $Z(z)$ came from a strategy for the provers that independently succeeded, with good probability, in the classical low-degree test. Once this has been completed we apply the analysis of the classical low-degree test against two entangled provers to recover a single family of measurements $\{\hat{S}^g\}$ with outcomes in the set of low-degree polynomials $g$ over ${\ensuremath{\mathbb{F}_q}}^{2m}$.
The last step consists in “pulling apart” the measurements obtained in the previous step to recover observables $\tilde{X}(x)$ and $\tilde{Z}(z)$, now defined for all $x,z\in{\ensuremath{\mathbb{F}_q}}^n$ (and not only the special subset used as queries in the test). Given the definition of $\hat{X}(x)$ from $X(x)$, it is natural to define $\tilde{X} =
({\ensuremath{\mathop{\rm Id}\nolimits}}_{{{\textsf{A}}}} \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}_{{{\textsf{A'}}}} \otimes {\tau}_X(x)_{{{\textsf{A''}}}})
\cdot \hat{X}(x)$, which has the effect of “undoing” the initial tensoring of $X(x)$ by a Pauli on ${{\textsf{A'}}}$ (this uses that the ancillas ${{\textsf{A'A''}}}$ are initialized in a maximally entangled state). It remains to argue that the exponentially many operators thus constructed approximately satisfy the Pauli twisted commutation relations. Once this has been established the result follows as in our previous work [@NV17], as it can be shown directly that such operators must be close to operators exactly satisfying all Pauli relations, whose only joint eigenvalue-$1$ eigenstate is the maximally entangled state.
#### Pauli observables over a prime power field.
To conclude this overview we briefly discuss some difficulties encountered while working with generalized Pauli observables over a prime power field. Had we restricted attention to prime fields the proof (and certainly the notation!) would have been somewhat simpler. The motivation for considering prime powers comes from the desire to allow embedding qubit Hamiltonians, which we can achieve if $q=2^t$, but did not see how to implement for odd values of $q$. Over prime power fields, we are faced with two possible definitions of generalized Pauli observables: the “clock” and “shift” operators mod $q$, with eigenvalues that are $q$-th roots of unity, and the definition , with eigenvalues that are $p$-th roots of unity. The former are more common in the literature and offer the convenience of allowing to encode a projective measurement with outcomes in ${\ensuremath{\mathbb{F}_q}}$ into a single generalized observable. However, they are not well-suited for describing strategies in the low-degree test, since they are defined in terms of addition and multiplication over ${\ensuremath{\mathbb{Z}}}_q$, whereas in the low degree test, all operations are performed over ${\ensuremath{\mathbb{F}_q}}$. Hence, we opted for the second definition, using families of $t$ such observables to encode a single measurement with outcomes in ${\ensuremath{\mathbb{F}_q}}\simeq { \ensuremath{{\mathbb{F}}}_p}^t$.
Further work
------------
There are several open problems raised by our work. Firstly, it would be interesting to expand the range of Hamiltonians for which we are able to give constant-gap interactive proofs, with the goal of eventually reaching a ${\textsc{QMA}}$-complete family, and thus a proof of Conjecture \[conj:qpcp\] based on a deterministic reduction. Secondly, a different route towards the proof of the conjecture would consist in establishing ${\textsc{QMA}}$-hardness results for either of the two classes of Hamiltonians described in Definition \[def:gen-h\] and Definition \[def:linear-xz\], for which we do already have a deterministic reduction to a game. As further motivation, we note that, if such a ${\textsc{QMA}}$-hardness result were achieved by constructing a “history Hamiltonian” from a polynomial quantum circuit—as in all such hardness results known—then by an observation of Fitzsimons and Hajdu[š]{}ek [@FitzsimonsH15], our results could be used to give an efficient delegation scheme for BQP in the “post-hoc” model. More broadly, the classical PCP theorem and MIP proof systems have become important tools in the design of delegated computation schemes (e.g. [@KRR14; @RRR16]), and we hope that similar applications may arise from the games variant of QPCP. Beyond the quantum games PCP conjecture, essentially resolved in this work, the complexity of the class ${\textsc{MIP}}^*$ of languages that have multi-prover interactive proof systems with entangled provers remains wildly open. Recent work of [@ji16nexp] introduces a “compression” technique, that allows him to obtain ${\textsc{MIP}}^*$ protocols for language in NEEXP (non-deterministic doubly-exponential time), albeit at the cost of an exponentially small completeness-soundness gap. Could our techniques be used to obtain the same result, for a constant gap? Such a result would provide an unconditional separation between ${\textsc{MIP}}$ and ${\textsc{MIP}}^*$.
In a different direction, it could be useful to extend our entanglement test to sub-constant error, in the same spirit as [@arnon2017noise; @arnon2017device]. Currently, all self-testing results we are aware of only provide guarantees in a regime where the success probability is close to $1$, which is arguably more challenging to demonstrate in experiments.
#### Organization.
We start with important notation and general preliminaries in Section \[sec:notation\]. The quantum low-degree test is stated in Section \[sec:qld\], and its soundness analysis is given in Section \[sec:soundness\]. In Section \[sec:code\] we extend the test to allow testing arbitrary states encoded using a suitable error-correcting code. Finally, Section \[sec:testing\] applies the test to prove Theorem \[conj:weak\_qpcp\], together with the two variants discussed as items (i) and (ii) in the introduction.
#### Acknowledgments.
AN was supported by NSF CAREER Grant CCF-1452616. TV was supported by NSF CAREER Grant CCF-1553477, AFOSR YIP award number FA9550-16-1-0495, a CIFAR Azrieli Global Scholar award, and the IQIM, an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028). Parts of this work was completed while both authors were hosted at the Institut Henri Poincaré in Paris, as part of the special semester on Analysis in Quantum Information Theory (Fall 2017), supported by NSF Grant DMS-1700168. The hospitality of the IHP is gratefully acknowledged.
Preliminaries {#sec:notation}
=============
Notation
--------
We use ${\mathcal{H}}$ to denote a finite-dimensional Hilbert space, ${{\mathrm{L}}}({\mathcal{H}})$ for the linear operators on ${\mathcal{H}}$, and ${{\mathrm{U}}}({\mathcal{H}})$ the set of unitary operators. Subscripts ${\mathcal{H}}_{{\textsf{A}}}$, ${\mathcal{H}}_{{\textsf{B}}}$ indicate distinct spaces. We use the notation $\operatorname{poly}(f(n))$ to denote $O(f^c(n))$ for some universal constant $c>0$ (which may vary each time the notation is used). Similarly, we write $\operatorname{poly}^{-1}(f(n))$ to denote $\Omega(f^{-c}(n))$. All parameters used in the paper will generally be a function of a single parameter $n$, and asymptotic notation $O(\cdot)$, $\Omega(\cdot)$, etc., should be understood as $n\to \infty$.
Finite fields and polynomials
-----------------------------
Throughout we use $p$ to denote a prime and $q = p^t$ a prime power. We let ${\ensuremath{\mathbb{F}_q}}$ denote the finite field with $q$ elements, and ${\ensuremath{\mathbb{Z}_p}}$ denote the cyclic group mod $p$. The additive group of ${ \ensuremath{{\mathbb{F}}}_p}$ coincides with ${\ensuremath{\mathbb{Z}}}_p$, but this is no longer the case for ${\ensuremath{\mathbb{F}_q}}$. The finite field trace is denoted by ${\mbox{\rm tr}}(a)$; it is a map from ${\ensuremath{\mathbb{F}_q}}$ to the prime subfield ${ \ensuremath{{\mathbb{F}}}_p}$, defined by ${\mbox{\rm tr}}(a) = \sum_{\ell = 0}^{t-1}
a^{p^{\ell}} $. The trace respects linear combinations with coefficients drawn from the prime subfield: ${\mbox{\rm tr}}(\alpha a + \beta b) = \alpha {\mbox{\rm tr}}(a) + \beta
{\mbox{\rm tr}}(b)$ for $\alpha, \beta \in { \ensuremath{{\mathbb{F}}}_p}$. A useful alternative view of ${\ensuremath{\mathbb{F}_q}}$ is as a $t$-dimensional vector space over ${ \ensuremath{{\mathbb{F}}}_p}$. Each element $e \in {\ensuremath{\mathbb{F}_q}}$ can be written as $e_1 b_1 + e_2 b_2 + \dots + e_{t} b_t$, where $(b_1, \dots,
b_t)$ is a basis for ${\ensuremath{\mathbb{F}_q}}$ over ${ \ensuremath{{\mathbb{F}}}_p}$ and the coefficients $e_\ell$ lie in the field of scalars ${ \ensuremath{{\mathbb{F}}}_p}$. This representation of ${\ensuremath{\mathbb{F}_q}}$ is convenient for addition, since one can add the individual components $e_\ell$ separately, but in general, it is hard to do multiplication. However, if $q$ is even or $q = p^t$ with both $p$ and $t$ odd there always exists a basis satisfying the property of *self-duality*, i.e. $$\label{eq:self-dual}
{\mbox{\rm tr}}(b_i b_j) \,=\, \delta_{ij}$$ for all $i,j\in\{1,\ldots,t\}$ (see e.g. [@menezes2013applications Theorem 1.9]). This property allows to express ${\mbox{\rm tr}}(ef)$, for $e, f \in {\ensuremath{\mathbb{F}_q}}$, as the inner product, over ${ \ensuremath{{\mathbb{F}}}_p}$, of their respective vector of components along the basis. As shown below, this property will make it convenient to express $q$-dimensional qudits as tensor products of $p$-dimensional qudits. For the remainder of the paper we only consider choices of $q$ such that ${\ensuremath{\mathbb{F}_q}}$ admits a self-dual basis over ${ \ensuremath{{\mathbb{F}}}_p}$.
For integer $d,m$ and a subspace $s\subset {\ensuremath{\mathbb{F}_q}}^m$ we let $\deg_d(s)$ denote the set of polynomials on $s$ of total degree at most $d$ (specified with respect to some fixed, implicit basis for $s$). We write $\omega = e^{\frac{2i\pi}{p}}$ for a fixed primitive $p$-th root of unity. Let $$\label{eq:def-epr-p}
{|{\textsc{EPR}}_q\rangle} = \frac{1}{\sqrt{q}}\sum_{i \in {\ensuremath{\mathbb{F}_q}}} {|i\rangle}\otimes {|i\rangle}\,\in\,{\ensuremath{\mathbb{C}}}^q\otimes{\ensuremath{\mathbb{C}}}^q\;.$$
#### Coordinates and polynomials.
Let $n\geq 1$ be an integer, and $h,m$ two integers such that $h^m \geq n$ and $h\leq q$. Throughout we fix an arbitrary injection ${\pi}:\{1,\ldots,n\}\to\{0,1,\ldots,h-1\}^m \subseteq {\ensuremath{\mathbb{F}_q}}^m$, where $n,h,m$ are integers such that $h^m \geq n$ that will be clear from context. For $x\in {\ensuremath{\mathbb{F}_q}}^m$ and $i\in\{1,\ldots,n\}$ define $$x_{{\pi}(i)}\,=\,\prod_{j=1}^m \frac{\prod_{\substack{k=0\\ k\neq {\pi}(i)_j}}^{h-1} (k-x_j)}{\prod_{\substack{k=0\\ k\neq {\pi}(i)_j}}^{h-1} (k-{\pi}(i)_j)}\,\in\,{\ensuremath{\mathbb{F}_q}}\;,$$ and let $x_{\pi}=(x_{{\pi}(1)},\ldots,x_{{\pi}(n)})\in{\ensuremath{\mathbb{F}_q}}^n$. Note that for $x\in \{0,1,\ldots,h-1\}^m$, $x_{{\pi}(i)}=1$ if $x={\pi}(i)$ and $x_{{\pi}(i)}=0$ otherwise. By ranging over all possible values for $x$ we obtain a subset of ${\ensuremath{\mathbb{F}_q}}^n$ of size $q^m$; we think of $x\mapsto x_{\pi}$ as a pseudo-random “coordinate expansion” map.
Let $g:{\ensuremath{\mathbb{F}_q}}^m\to{\ensuremath{\mathbb{F}_q}}$ be an $m$-variate polynomial of degree at most $h$ in each coordinate. Then by interpolation we can write $$\label{eq:interpolation}
g(x) = \sum_{i=1}^n x_{{\pi}(i)} g({\pi}(i)) = g\cdot x_{\pi}\;,$$ where we abuse notation and write $g$ for the vector $(g({\pi}(1)),\ldots,g({\pi}(n)))\in{\ensuremath{\mathbb{F}_q}}^n$. Conversely, for any $a\in{\ensuremath{\mathbb{F}_q}}^n$ we let $g_a$ be the $m$-variate polynomial of individual degree at most $h$ over ${\ensuremath{\mathbb{F}_q}}$ defined by $$\label{eq:def-ga}
g_a:\;x\in{\ensuremath{\mathbb{F}_q}}^m\,\mapsto\, \sum_i a_i x_{{\pi}(i)} \,=\, a\cdot x_{\pi}\;.$$ The map from ${\ensuremath{\mathbb{F}_q}}^n$ to ${\ensuremath{\mathbb{F}_q}}^{q^m}$ that maps $a$ to the evaluation table of $g_a$ is the $m$-variate Reed-Muller code of individual degree $h$. Note that $(g_a({\pi}(1)), \dots, g_a({\pi}(n))) = a$.
We recall the Schwartz-Zippel lemma [@Zippel79; @schwartz1980fast], which we will use repeatedly.
\[lem:sz\] Let $d,m\geq 1$ be integers and $r$ a non-zero polynomial in $m$ variables of total degree at most $d$ defined over the finite field ${\ensuremath{\mathbb{F}_q}}$. Then $r$ has at most $d|{\ensuremath{\mathbb{F}_q}}|^{m-1}$ zeros.
Pauli measurements and observables for qudits {#sec:qauli}
---------------------------------------------
To any projective measurement $\{M^a\}$ with outcomes $a \in {\ensuremath{\mathbb{Z}_p}}$ we can associate a generalized observable with eigenvalues that are $p$-th roots of unity: the unitary matrix $M = \sum_a \omega^a M^a$, where $\omega = e^{\frac{2i\pi}{ p}}$. The generalized Pauli operators over ${ \ensuremath{{\mathbb{F}}}_p}$ are a set of generalized observables indexed by a basis setting $X$ or $Z$ and an element $a$ or $b$ of ${ \ensuremath{{\mathbb{F}}}_p}$, with eigenvalues that are $p$-th roots of unity. They are given by $$\label{eq:pauli-fp}
\sigma_X(a) = \sum_{j \in { \ensuremath{{\mathbb{F}}}_p}} {|j + a\rangle} {\langlej|}\qquad \text{and}\qquad\sigma_Z(b) =\sum_{j \in { \ensuremath{{\mathbb{F}}}_p}} \omega^{bj} {|j\rangle} {\langlej|}\;,$$ where addition and multiplication are over ${ \ensuremath{{\mathbb{F}}}_p}$. These observables obey the “twisted commutation” relations $$\label{eq:twisted-fp}
\forall a,b\in{ \ensuremath{{\mathbb{F}}}_p},\qquad \sigma_X(a) \sigma_Z(b) \,=\, \omega^{-ab} \,\sigma_Z(b)\sigma_X(a)\;.$$ Similarly, over a field ${\ensuremath{\mathbb{F}_q}}$ we can define a set of generalized Pauli operators, indexed by a basis setting $X$ or $Z$ and an element of ${\ensuremath{\mathbb{F}_q}}$. There are different possible definitions for these operators. We choose them to have eigenvalues that are $p$-th roots of unity. For $a,b\in{\ensuremath{\mathbb{F}_q}}$ they are given by $${\tau}_X(a) = \sum_{j \in {\ensuremath{\mathbb{F}_q}}} {|j+a\rangle}{\langlej|}\qquad\text{and}\qquad {\tau}_Z(b)= \sum_{j \in {\ensuremath{\mathbb{F}_q}}}
\omega^{{\mbox{\rm tr}}(b j)} {|j\rangle}{\langlej|}\;,$$ where addition and multiplication are over ${\ensuremath{\mathbb{F}_q}}$. Powers of these observables obey the relation $$\forall W \in \{X,Z\},\, \forall a \in {\ensuremath{\mathbb{F}_q}},\,\forall b \in { \ensuremath{{\mathbb{F}}}_p},\qquad({\tau}_W(a))^b = {\tau}_W(a b) \;.$$ In particular, since $pa=0$ for any $a\in {\ensuremath{\mathbb{F}_q}}$ we get that that $({\tau}_W(a))^p = {\ensuremath{\mathop{\rm Id}\nolimits}}$ for any $a\in{\ensuremath{\mathbb{F}_q}}$. The observables obey analogous “twisted commutation” relations to , $$\label{eq:twisted-fq}
\forall a,b\in{\ensuremath{\mathbb{F}_q}},\qquad {\tau}_X(a) {\tau}_Z(b) = \omega^{-{\mbox{\rm tr}}(a b)} {\tau}_Z(b) {\tau}_X(a)\;.$$ It is clear from the definition that all of the ${\tau}_X$ operators commute with each other, and similarly all the ${\tau}_Z$ operators with each other. Thus, it is meaningful to speak of a common eigenbasis for all ${\tau}_X$ operators, and a common eigenbasis for all ${\tau}_Z$ operators. The common eigenbasis for the ${\tau}_Z$ operators is the computational basis. To map this basis to the common eigenbasis of the ${\tau}_X$ operators, one can apply the Fourier transform $$\label{eq:fourier-f}
F \,=\, \frac{1}{\sqrt{q}} \sum_{j ,k\in {\ensuremath{\mathbb{F}_q}}} \omega^{-{\mbox{\rm tr}}(jk)} {|j\rangle}{\langlek|}\;.$$ Explicitly, the eigenbases consist of the vectors ${|e_W\rangle}$ labeled by an element $e \in {\ensuremath{\mathbb{F}_q}}$ and $W \in \{X, Z\}$, given by $${|e_X\rangle} = \frac{1}{\sqrt{q}} \sum_j \omega^{- {\mbox{\rm tr}}(e j)} {|j\rangle}\;, \qquad
{|e_Z\rangle} = {|e\rangle}\;.$$ We denote the POVM whose elements are projectors onto basis vectors of the eigenbasis associated with the observables ${\tau}_W$ by $\{{\tau}_W^e\}_e$. Then the observables ${\tau}_W(a)$ can be written as $$\forall W \in\{X,Z\},\,\forall a \in {\ensuremath{\mathbb{F}_q}},\qquad{\tau}_W(a) = \sum_{e \in {\ensuremath{\mathbb{F}_q}}} \omega^{{\mbox{\rm tr}}(a e)} {\tau}_W^e\; .$$
For choices of $q$ such that ${\ensuremath{\mathbb{F}_q}}$ admits a self-dual basis $(b_1,\ldots,b_t)$, we can decompose a $q$-dimensional qudit (a “quqit”) as a tensor product of $t$ $p$-dimensional qudits (“qupits”). Based on this decomposition, for $W\in\{X,Z\}$ and $\ell\in\{1,\ldots,t\}$ we define the $W$-basis Pauli operator acting on the $\ell$-th qupit by $$\label{eq:pauli-l}
\forall a \in { \ensuremath{{\mathbb{F}}}_p},\qquad \sigma_{W,\ell}(a) \,=\, \sum_{e_1, \dots, e_t\in{ \ensuremath{{\mathbb{F}}}_p}} \omega^{ae_\ell} \,{\tau}^{(e_1 b_1 +
\dots + e_t b_t)}_W\,=\, {\tau}_W(a b_\ell) \;.$$ It can be verified by direct computation that for every $\ell\in\{1,\ldots,t\}$, $\sigma_{X, \ell}$ and $\sigma_{Z, \ell}$ obey the Pauli twisted commutation relations , and that when $\ell \neq \ell' \in \{1,\ldots,t\}$, $\sigma_{X, \ell}$ and $\sigma_{Z, \ell'}$ commute. Both of these facts also follow from noting that the transformation $F$ that maps ${|e_Z\rangle}$ to ${|e_X\rangle}$ decomposes as a tensor product over the qupits: $$\begin{aligned}
F &= \frac{1}{\sqrt{q}} \sum_{jk} \omega^{-{\mbox{\rm tr}}(jk)} {|j\rangle}{\langlek|} \\
&= \frac{1}{\sqrt{q}} \sum_{j_1, \dots j_t, k_1, \dots k_t} \omega^{- {\mbox{\rm tr}}(\sum_{\ell, \ell'}
j_\ell k_{\ell'} b_\ell b_{\ell'})} {|j_1\rangle} {\langlek_1|} {\otimes}\cdots {\otimes}{|j_t\rangle} {\langlek_t|} \\
&= \frac{1}{\sqrt{q}} \sum_{j_1, \dots j_t, k_1, \dots k_t} \omega^{ -\sum_{\ell, \ell'}
j_\ell k_{\ell'} {\mbox{\rm tr}}(b_\ell b_{\ell'})} {|j_1\rangle} {\langlek_1|} {\otimes}\cdots {\otimes}{|j_t\rangle} {\langlek_t|} \\
&= \frac{1}{\sqrt{q}}\sum_{j_1, \dots j_t, k_1, \dots k_t} \omega^{- \sum_{\ell}
j_\ell k_{\ell}} {|j_1\rangle} {\langlek_1|} {\otimes}\cdots {\otimes}{|j_t\rangle} {\langlek_t|} \\
&= \bigotimes_{\ell = 1}^{t} \Big( \frac{1}{\sqrt{p}}\sum_{j_\ell, k_\ell} \omega^{-j_\ell
k_\ell} {|j_\ell\rangle} {\langlek_\ell|} \Big)\;,\end{aligned}$$ where in going from the second to the third line we used the linearity of the trace and the fact that $j_\ell, k_{\ell'}$ are elements of the prime subfield ${ \ensuremath{{\mathbb{F}}}_p}$. We will sometimes consider the case where $p = 2$, in which case the $\sigma_{W,\ell}$ behave as the standard Pauli spin matrices acting on $t$ qubits, with the index $\ell$ labeling the qubit acted on. Also, it will be sometimes useful to allow the index $a$ to range over all of ${\ensuremath{\mathbb{F}_q}}$ instead of just ${ \ensuremath{{\mathbb{F}}}_p}$; extending we define $\sigma_{W,\ell}(a)$ to be ${\tau}_W(ab_\ell)$ for any $a \in {\ensuremath{\mathbb{F}_q}}$. For systems with many qudits, we will consider tensor products of the operators ${\tau}_W$. Slightly abusing notation, for $W \in \{X, Z\}$ and $a \in {\ensuremath{\mathbb{F}_q}}^n$ we denote by ${\tau}_W(a)$ the tensor product ${\tau}_W(a_1) {\otimes}\dots {\otimes}{\tau}_W(a_n)$. These obey the twisted commutation relations $$\forall a,b\in {\ensuremath{\mathbb{F}_q}}^n,\qquad {\tau}_X(a) {\tau}_Z(b) \,=\, \omega^{-{\mbox{\rm tr}}(a \cdot b)} {\tau}_Z(b) {\tau}_X(a)\;,$$ where $a \cdot b = \sum_{i=1}^{n} a_i b_i \in {\ensuremath{\mathbb{F}_q}}$. For $W\in\{X,Z\}$ and $e \in {\ensuremath{\mathbb{F}_q}}^n$ define the eigenstates $${|e_W\rangle} = {|(e_1)_W\rangle} {\otimes}\dots {\otimes}{|(e_n)_W\rangle}\;,$$ and associated rank-$1$ projectors ${\tau}_W^e$.
#### State-dependent distance.
For operators $A,B\in{{\mathrm{L}}}({\mathcal{H}})$, where ${\mathcal{H}}$ is a finite-dimensional Hilbert space, and a vector ${|\psi\rangle} \in {\mathcal{H}}\otimes {\mathcal{H}}'$, where ${\mathcal{H}}'$ is another finite-dimensional Hilbert space, we write $A\approx_\delta B$ for $\| (A-B)\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle} \|^2 = O(\delta)$. Note the state ${|\psi\rangle}$ and the space ${\mathcal{H}}'$ are usually kept implicit. We sometimes write the same with some free variables, e.g. $A_x^a \approx_\delta B_x^a$. By this we mean $${\ensuremath{\mathop{\textsc{E}}_{x}}}\sum_a \| (A_x^a-B_x^a)\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle}\|^2 \,=\, O(\delta)\;.$$ Variables appearing as subscript will most often be considered “inputs”, and should be averaged; superscripts are considered “answers” and should be summed over. Which is which will always be clear from context, including the distribution on inputs.
For a family of POVM $\{A_x^a\}$ acting on ${\mathcal{H}}_A$, we will say that $\{A_x^a\}$ is $\delta$-self-consistent if there exists a family of POVM $\{{{\color{MidnightBlue} \bm{A}}}_x^a\}$ acting on ${\mathcal{H}}_B$ such that $A_x^a \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}_B
\approx_\delta {\ensuremath{\mathop{\rm Id}\nolimits}}_A \otimes {{\color{MidnightBlue} \bm{A}}}_x^a$.[^5] Note that this definition relies on an implicit understanding of the space ${\mathcal{H}}_B$ and the operators $\{{{\color{MidnightBlue} \bm{A}}}_x^a\}$, and we will only use the terminology when the space and operators are clear from context. The following lemma relates two measures of consistency, defined via observables or the underlying projective measurement.
\[claim:obs-meas-cons\] Let $s$ be any integer, and let $\{A^a\}, \{B^b\}$ be projective measurements with outcomes $a,b\in{\ensuremath{\mathbb{Z}}}_s$. Let $A = \sum_a \omega_s^a A^a$ and $B= \sum_b \omega_s^b B^b$ where $\omega_s = \exp(2\pi i / s)$. Then for any state ${|\psi\rangle}$, $$\begin{aligned}
\frac{1}{2}\Big(1-\Re\big({\langle\psi|} A\otimes B^\dag {|\psi\rangle}\big)\Big)\,\leq\,\sum_{a\neq b} \big\| A^a\otimes B^b {|\psi\rangle}\big\|^2 \,\leq\, \frac{s^2}{2\pi^2}\Big(1-\Re\big({\langle\psi|} A\otimes B^\dag {|\psi\rangle}\big)\Big)\;.
$$
Expand $$\begin{aligned}
\big\|(A\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}- {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes B) {|\psi\rangle} \big\|^2
&= \sum_{a\neq b} \big|\omega_s^a-\omega_s^b\big|^2 \big\|A^a \otimes B^b {|\psi\rangle}\big\|^2\;,\end{aligned}$$ and use $\frac{2\pi}{s} \leq |\omega_s^a - \omega_s^b|\leq 2$ for all $a\neq b$.
Let $s$ be an integer, $\delta\geq 0$, ${|\psi\rangle}\in{\mathcal{H}}$ a state and $B\in{{\mathrm{L}}}({\mathcal{H}})$ a normal operator such that $\| (B^s - {\ensuremath{\mathop{\rm Id}\nolimits}})
{|\psi\rangle} \|^2 \leq \delta$. Then there exists a unitary $U$ with the same eigenvectors as $B$ such that $U^s={\ensuremath{\mathop{\rm Id}\nolimits}}$ and $$\big\| (B - U) {|\psi\rangle}\big\|^2
\,\leq \,4\,\delta\;.$$ \[claim:round\_to\_roots\]
Let $\lambda = re^{2i\pi\theta}$ be any complex number, where $r\in {\ensuremath{\mathbb{R}}}_+$ and $\theta\in[-1/2,1/2)$. Then $$\begin{aligned}
\big|re^{2i\pi\theta}-1\big| &\geq \max\big(|r-1|,\, \big|e^{2i\pi\theta}-1\big|\big)\notag\\
&\geq \max\big(|r-1|,\, 4|\theta|\big)\;.\label{eq:angle-bound}
\end{aligned}$$ Thus, by the triangle inequality, $$\begin{aligned}
\Big| \lambda - e^{\frac{2i\pi}{s} \lfloor s \theta \rfloor}\Big|
&\leq |r-1| +\Big|e^{2i\pi \theta }- e^{\frac{2i\pi}{s} \lfloor s \theta \rfloor}\Big|\notag\\
&\leq |r^s-1| + 8\Big| \theta - \Big[ \frac{1}{s} \lfloor s \theta \rfloor\Big]_1 \Big|\notag\\
&\leq \big| \lambda^s -1 \big| + \frac{2}{s} \big| \lambda^s -1 \big|\;,\label{eq:angle-bound-1}\end{aligned}$$ where in the second line we wrote $[x]_1$ for the representative of $x\bmod 1$ in $[-1/2,1/2)$, and the last line follows from .
Write the eigendecomposition of $B = \sum_i \lambda_i \Pi_i$, where $\Pi_i$ is a Hermitian projector and $\lambda_i$ a complex number. We include zero eigenvalues, so that $\sum_i \Pi_i = {\ensuremath{\mathop{\rm Id}\nolimits}}$. The assumption made in the claim can be written as $$\label{eq:angle-bound-0}
\| (B^s - {\ensuremath{\mathop{\rm Id}\nolimits}}) {|\psi\rangle} \|^2 = \Big\| \sum_i (\lambda_i^s - 1) \Pi_i
{|\psi\rangle} \Big\|^2 = \sum_i | \lambda_i^s - 1|^2 \|\Pi_i {|\psi\rangle}\|^2 \leq
\delta\;.$$ Let $\omega = e^{\frac{2\pi i}{s}}$. For each $i$, let $\omega^{a_i}$ be the closest $s$-th root of unity to $\lambda_i$. Define $U = \sum_i
\omega^{a_i} \Pi_i$. Then $$\begin{aligned}
\| (B - U) {|\psi\rangle} \|^2 &= \sum_i \big|\lambda_i - \omega^{a_i}\big|^2 \|\Pi_i
{|\psi\rangle} \|^2 \\
&\leq \sum_i \,4\,\big| \lambda_i^s - 1\big|^2 \| \Pi_i {|\psi\rangle} \|^2 \\
&\leq 4\,\delta\;,
\end{aligned}$$ where the second line uses , and the last is by .
Self-testing
------------
We use the language of multi-player self-tests (we will often call the players “provers” as well).
Let $k\geq 1$ be an integer. A *$k$-partite strategy* $S = ({|\psi\rangle},\mathcal{X},
\mathcal{A}, \mathcal{M})$ consists of finite question and answer sets $\mathcal{X}=X_1\times\cdots\times X_k$ and $\mathcal{A}=A_1\times\cdots\times A_k$ respectively, a $k$-partite quantum state ${|\psi\rangle} \in \mathcal{H}_1\otimes \cdots \otimes{\mathcal{H}}_k$, and for each $i\in\{1,\ldots,k\}$ a collection of measurement operators $\{M^a_x\}_{a \in A_i}$ on $\mathcal{H}_i$ and indexed by $x \in X_i$.[^6] We say that the strategy is *partial* if it only specifies measurement operators for a subset of the possible questions, or if it does not specify a state ${|\psi\rangle}$.
We reproduce a standard definition in self-testing.
\[def:self-test\] A *$k$-player self-test* with completeness $c$ and robustness $\delta({\varepsilon})$ for a (partial) strategy $S=({|\Psi\rangle},\mathcal{X},
\mathcal{A}, \mathcal{M})$ is a distribution $\pi$ on ${\mathcal{X}}$ and a family of coefficients $V(a_1,\ldots,a_k|x_1,\ldots,x_k)\in [0,1]$, for $(x_1,\ldots,x_k)\in{\mathcal{X}}$ and $(a_1,\ldots,a_k)\in{\mathcal{A}}$, such that the following hold:
- There exists a strategy $\hat{S}$ that extends the (partial) strategy $S$ and succeeds in the test with probability at least $c$; formally, $$\sum_{(x_1,\ldots,x_k)} \pi(x_1,\ldots,x_k) \sum_{a_1,\ldots,a_k} V(a_1,\ldots,a_k|x_1,\ldots,x_k) \,{\langle\psi|} \hat{M}_{x_1}^{a_1}\otimes \cdots \otimes \hat{M}_{x_k}^{a_k} {|\psi\rangle} \,\geq\, c\;.$$
- Any strategy with success at least $c-{\varepsilon}$ in the test must be $\delta({\varepsilon})$-close to the optimal strategy. Formally, for any strategy $\hat{S}=({|\hat{\psi}\rangle},\mathcal{X},
\mathcal{A}, \mathcal{\hat{M}})$ such that $$\sum_{(x_1,\ldots,x_k)} \pi(x_1,\ldots,x_k) \sum_{a_1,\ldots,a_k} V(a_1,\ldots,a_k|x_1,\ldots,x_k) \,{\langle\hat{\psi}|} \hat{M}_{x_1}^{a_1}\otimes \cdots \otimes \hat{M}_{x_k}^{a_k} {|\hat{\psi}\rangle} \,\geq\, c-{\varepsilon}\;,$$ there exists a local isometry $\Phi=\Phi_1\otimes\cdots\Phi_k$ and a state ${|{\textsc{aux}}\rangle}$ such that $$\big\| \Phi({|\hat{\psi}\rangle}) - {|{\textsc{aux}}\rangle}{|\psi\rangle}\big\| \,\leq\,\delta({\varepsilon})\;,$$ and $$\sum_{x_1,\ldots,x_k} \pi(x_1,\ldots,x_k) \sum_{a_1,\ldots,a_k}\, \big\| \Phi\big( \hat{M}_{x_1}^{a_1} \otimes \cdots\otimes \hat{M}_{x_k}^{a_k} {|\hat{\psi}\rangle}\big) - {|{\textsc{aux}}\rangle} M_{x_1}^{a_1}\otimes \cdots \otimes M_{x_k}^{a_k} {|\psi\rangle} \big\| \,\leq\, \delta({\varepsilon})\;.$$
In case $S$ only specifies a partial strategy, then the above expression is restricted to questions for which $S$ is defined.
The commutation test
--------------------
In designing self-tests, it is useful to have the ability to test commutation relations between pairs of observables applied by the provers. The following well-known test can be employed to certify that two observables commute:
\[thm:com\_test\] Let $s$ be an integer and ${\varepsilon}> 0$. There exists a two-player self-test ${\textsc{COM}}(M,N)$ with completeness $1$ and robustness $\delta({\varepsilon}) = O(s\sqrt{{\varepsilon}})$, for the (partial) strategy $S$ that uses commuting generalized observables $M$ and $N$ (with outcomes in ${\ensuremath{\mathbb{Z}}}_s$) for two special questions labelled $1$ and $2$, respectively. The test has $3$ questions per player and answers either in ${\ensuremath{\mathbb{Z}}}_s$ (for questions $1$ and $2$) or ${\ensuremath{\mathbb{Z}}}_s^2$ (for question $3$). Moreover, for any two commuting observables $A$ and $B$, there exists a strategy in which the first player uses the observables $M$ and $N$ for questions $1$ and $2$, using a shared state ${|\psi\rangle}$ that is a maximally entangled state of appropriate dimension.
The guarantees of the theorem are achieved by the following test, which is a simple instance of the idea of “oracularization” in multiprover interactive proofs. In the test, the verifier performs either of the following with equal probability $\frac{1}{2}$:
1. Send the first player a question $q$ chosen uniformly from $\{1, 2\}$, and send the second player the question $3$. Receive an answer $a
\in {\ensuremath{\mathbb{Z}}}_s$ from the first player and $(b_1, b_2) \in {\ensuremath{\mathbb{Z}}}_s^2$ from the second player. Accept if $a = b_q$, and reject otherwise.
2. Perform the same as in item 1., but with the players interchanged.
The analysis of this test is standard; see, e.g. [@coladangelo2017verifier Lemma 28].
The generalized Magic Square
----------------------------
In [@ColadangeloS17MS] a generalized version of the Magic Square game [@Arvind:02] is introduced and shown to robustly self-test generalized observables satisfying twisted commutation relations over ${\ensuremath{\mathbb{Z}}}_s$, for any integer $s$.
\[thm:ms-rigid\] Let $s$ be an integer and ${\varepsilon}>0$. There exists a two-player self-test ${\textsc{MS}}(X,Z)$, with completeness $1$ and robustness $\delta({\varepsilon}) = O(s^3\sqrt{{\varepsilon}})$, for the (partial) strategy $S$ that uses observables $\sigma_{X}$ and $\sigma_{Z}$ on two special questions labeled $X$ and $Z$ respectively. The test has $O(1)$ questions per player (including two questions labeled $X$ and $Z$) and answers in ${\ensuremath{\mathbb{Z}}}_s^2$. Furthermore, there is a strategy that succeeds with probability $1$ using only $\sigma_X$, $\sigma_Y$ and $\sigma_Z$ observables on two $s$-dimensional qudits per player initialized in ${|\psi\rangle} = {|{\textsc{EPR}}_s\rangle}\otimes {|{\textsc{EPR}}_s\rangle}$.
The classical low-degree test
-----------------------------
A stepping stone in our analysis is an extension of the “classical low-degree test” from [@Vidick13xor] to the case of only two provers.
\[thm:ml\] Let ${\varepsilon}> 0$, $m,d$ integers, and $q$ a prime power such that $q \geq
(dm/{\varepsilon})^{c}$ for a universal constant $c\geq 1$. There is a two-prover test, called the *classical low-degree test* ${\textsc{c-lowdeg}}(m,d,q)$, in which queries to the provers are chosen among affine subspaces $s\subseteq {\ensuremath{\mathbb{F}_q}}^m$, and answers are polynomials $r$ on $s$ of total degree at most $d$, such that the following holds. For any strategy for the provers using entangled state ${|\psi\rangle}$ and projective measurements $\{M_s^r\}$ that succeeds with probability at least $1-{\varepsilon}$ in the test there exists a POVM $\{S^g\}$, where $g$ ranges over the polynomials on ${\ensuremath{\mathbb{F}_q}}^m$ of total degree at most $d$, and a $\delta = \operatorname{poly}({\varepsilon})$ such that the following hold:
1. Approximate consistency with $M$: $${\ensuremath{\mathop{\textsc{E}}_{s}}}\, \sum_{g} \sum_{r \neq g|_s} {\langle\psi|} M_s^r
{\otimes}S^g {|\psi\rangle} \,\leq\, \delta,$$
2. Self-consistency: $$\sum_g {\langle\psi|} S^g {\otimes}({\ensuremath{\mathop{\rm Id}\nolimits}}- S^g) {|\psi\rangle} \,\leq\, \delta.$$
We let ${\pi_{\rm ld}}$ denote the distribution on questions used by the verifier in the low-degree test from Theorem \[thm:ml\]. This distribution is symmetric, and we slightly abuse notation by also writing ${\pi_{\rm ld}}$ for either marginal. We will use that the test from Theorem \[thm:ml\] that it satisfies the following properties:
- ${\pi_{\rm ld}}$ is a uniform mixture of the uniform distribution on pairs $(s,w)$ such that $s$ is an affine subspace of dimension $2$ in ${\ensuremath{\mathbb{F}_q}}^m$ and $w\in s$ is a uniformly random point in $s$, and its permutation $(w,s)$.
- Whenever provers in the test are queried for a pair of subspaces $(s,w)$, they are required to return a polynomial $r$ defined on $s$ and a value $a$ in ${\ensuremath{\mathbb{F}_q}}$ such that $r(w)=a$.
Theorem \[thm:ml\] assumes that the strategy employed by the provers in the test is invariant under permutation of the two provers. It will be convenient to allow non-symmetric strategies as well.
\[cor:ml\] Let $m,d,q$ and ${\varepsilon}$ be as in Theorem \[thm:ml\]. Let ${|\psi\rangle} \in {\mathcal{H}}_A
\otimes {\mathcal{H}}_B$ be a bipartite state, and $\{M_s^r\}$ and $\{{{\color{MidnightBlue} \bm{M}}}_s^r\}$ be POVMs on ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ respectively, such that the associated strategy for the provers succeeds in the test ${\textsc{c-lowdeg}}(m,d,q)$ from Theorem \[thm:ml\] with probability at least $1-{\varepsilon}$. Then there exist POVMs $\{S^g\}$ and $\{{{\color{MidnightBlue} \bm{S}}}^g\}$, where $g$ ranges over the polynomials on ${\ensuremath{\mathbb{F}_q}}^m$ of total degree at most $d$, defined on ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ respectively, and a $\delta = \operatorname{poly}({\varepsilon})$ such that the following relations hold, on average over $s\sim{\pi_{\rm ld}}$: $$\sum_{g}\,M_s^{ g_{|s}} {\otimes}{{\color{MidnightBlue} \bm{S}}}^g \,\approx_\delta\,{\ensuremath{\mathop{\rm Id}\nolimits}}\;,\qquad \sum_{g}\,S^g {\otimes}{{\color{MidnightBlue} \bm{M}}}_s^{ g_{|s}}\,\approx_\delta\,{\ensuremath{\mathop{\rm Id}\nolimits}}\;,$$ and $$S^g {\otimes}{\ensuremath{\mathop{\rm Id}\nolimits}}\,\approx_\delta \, {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{S}}}^g\;.$$
Extending ${{\textsf{A}}}$ or ${{\textsf{B}}}$ as needed, assume without loss of generality that ${\mathcal{H}}_{{{\textsf{A}}}}$ and ${\mathcal{H}}_{{{\textsf{B}}}}$ have the same dimension, and fix a canonical isomorphism between the two. Adjoin ancilla spaces ${\mathcal{H}}_{{{\textsf{A'}}}}$ and ${\mathcal{H}}_{{{\textsf{B'}}}}$, each isomorphic to ${\ensuremath{\mathbb{C}}}^2$. From an arbitrary strategy we can construct a symmetric one by letting $$\hat{M}_s^r \,=\, M_s^r \otimes {{|0\rangle}\!{\langle0|}}_{{{\textsf{A'}}}} + {{\color{MidnightBlue} \bm{M}}}_s^r \otimes {{|1\rangle}\!{\langle1|}}_{{{\textsf{A'}}}}\;,$$ and $${|\hat{\psi}\rangle} \,=\, \frac{1}{\sqrt{2}} \big( {|\psi\rangle}_{{{\textsf{AB}}}} \otimes {|0\rangle}_{{{\textsf{A'}}}} \otimes {|1\rangle}_{{{\textsf{B'}}}} +{|\psi'\rangle}_{{{\textsf{AB}}}} \otimes {|1\rangle}_{{{\textsf{A'}}}} \otimes {|0\rangle}_{{{\textsf{B'}}}} \big)\;,$$ where ${|\psi'\rangle}_{{{\textsf{AB}}}}$ is obtained by swapping registers ${{\textsf{A}}}$ and ${{\textsf{B}}}$ in ${|\psi\rangle}_{{{\textsf{AB}}}}$. Using that the test from Theorem \[thm:ml\] is symmetric, the success probability of this strategy is the same as that of the non-symmetric one. Applying Theorem \[thm:ml\] gives POVM $\{\hat{S}^g\}$ defined on ${{\textsf{AA'}}}$ that are consistent with the $\{\hat{M}_s^r\}$, on the state ${|\hat{\psi}\rangle}$. It then suffices to define $$S^g \,=\, \big({\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {\langle0|}_{{{\textsf{A'}}}}\big)\,S^g\,\big({\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {|0\rangle}_{{{\textsf{A'}}}}\big)\;,\qquad {{\color{MidnightBlue} \bm{S}}}^g \,=\, \big({\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {\langle1|}_{{{\textsf{A'}}}}\big)\,S^g\,\big({\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {|1\rangle}_{{{\textsf{A'}}}}\big)\;.$$
The length of questions in the low-degree test ${\textsc{c-lowdeg}}(m,d,q)$ from Theorem \[thm:ml\] is $O(m\log q)$, which for a choice of $q=\operatorname{poly}\log(n)$ is logarithmic in $n$. However, answers have length $O(d^2 \log q)$, which is super-logarithmic. To achieve reduced answer length it is standard to compose the test with itself: any answer $r$ from a prover is interpreted as an $n'=O(d^2 \log q)$-long string of bits, that can be encoded as a multilinear polynomial over ${\ensuremath{\mathbb{F}_q}}^{m'}$, for $m'$ such that $2^{m'} \geq n'$. Questions in the composed test are a subspace $s\subseteq{\ensuremath{\mathbb{F}_q}}^m$, together with a subspace $s'\subseteq {\ensuremath{\mathbb{F}_q}}^{m'}$, and answers are the restriction to $s'$ of the low-degree encoding of the polynomial $r$ that the prover would answer to the question $s$. The analysis of the composition is standard, and we state the result as the following theorem.
\[thm:2ml\] Let ${\varepsilon}> 0$, $m,d$ be integers, and $q$ a prime power such that $q \geq
(dm/{\varepsilon})^{c}$ for a universal constant $c\geq 1$. Let $n' = O(d^2)$ be the answer length (in number of ${\ensuremath{\mathbb{F}_q}}$-symbols) in ${\textsc{c-lowdeg}}(m,d,q)$, and $m'=\log(n')=O(\log\log n)$. There is a two-prover test, called the *composed classical low-degree test* ${\textsc{c-lowdeg}}^{(2)}(m,d,q)$, in which queries to the provers are chosen among, either pairs of affine subspaces $(s,s')\subseteq {\ensuremath{\mathbb{F}_q}}^m\times {\ensuremath{\mathbb{F}_q}}^{m'}$, or points in ${\ensuremath{\mathbb{F}_q}}^m$, and answers are, either multilinear polynomials $r'$ on $s'$, or values $a\in{\ensuremath{\mathbb{F}_q}}$, such that the following holds. For any strategy specified by a shared state ${|\psi\rangle} \in {\mathcal{H}}_A
\otimes {\mathcal{H}}_B$ and measurement operators $\{M_{s,s'}^{r'}\}$ and $\{{{\color{MidnightBlue} \bm{M}}}_{s,s'}^{r'}\}$ on ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ respectively, such that the associated strategy succeeds in the test ${\textsc{c-lowdeg}}^{(2)}(m,d,q)$ with probability at least $1-{\varepsilon}$, there exist POVM $\{S^g\}$ and $\{{{\color{MidnightBlue} \bm{S}}}^g\}$, where $g$ ranges over the polynomials on ${\ensuremath{\mathbb{F}_q}}^m$ of total degree at most $d$, defined on ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ respectively, and a $\delta = \operatorname{poly}({\varepsilon})$ such that the following relations hold, on average over $s\sim{\pi_{\rm ld}}$: $$\sum_{g}{\ensuremath{\mathop{\textsc{E}}_{s'}}} \,M_{s,s'}^{ g_{|s,s'}} {\otimes}{{\color{MidnightBlue} \bm{S}}}^g \,\approx_\delta\,{\ensuremath{\mathop{\rm Id}\nolimits}}\;,\qquad \sum_{g}{\ensuremath{\mathop{\textsc{E}}_{s'}}}\,S^g {\otimes}{{\color{MidnightBlue} \bm{M}}}_{s,s'}^{ g_{|s,s'}}\,\approx_\delta\,{\ensuremath{\mathop{\rm Id}\nolimits}}\;,$$ where the expectation is over an $s'$ as sampled in the test (conditioned on $s$), and $g_{|s,s'}$ denotes the polynomial on $s'$ obtained by restricting to $s'$ the low-degree extension of the description of the restriction $g_{|s}$ of $g$ to $s$. Furthermore, $$S^g {\otimes}{\ensuremath{\mathop{\rm Id}\nolimits}}\,\approx_\delta \, {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{S}}}^g\;.$$
The quantum low-degree test {#sec:qld}
===========================
Description of the test {#sec:lowdeg-protocol}
-----------------------
------------------------------------------------------------------------
\
Test ${\textsc{q-lowdeg}}^{(l)}(m,d,q)$. $m,d$ are integer, and $q=p^t$ is a prime power such that ${\ensuremath{\mathbb{F}_q}}$ admits a self-dual basis $(b_1,\ldots,b_t)$ over ${ \ensuremath{{\mathbb{F}}}_p}$. $l\in\{1,2\}$ is a parameter that indicates the level of the test.\
The verifier performs the following with equal probability:
1. Select $W\in\{X,Z\}$ uniformly at random and send $W$ to both provers. If $l=2$ execute the test ${\textsc{c-lowdeg}}^{(2)}(m,d,q)$ from Theorem \[thm:2ml\] with the provers. If $l=1$ execute the test ${\textsc{c-lowdeg}}(m,d,q)$ from Theorem \[thm:ml\]. Let $r$ be the polynomial returned by the first prover, and $r'$ by the second. If $W=X$, set $A=r$ and ${{\color{MidnightBlue} \bm{A'}}}=-r'$. If $W=Z$, set $A=r$ and ${{\color{MidnightBlue} \bm{A'}}}=r'$. Accept if and only if the pair of answers $(A,{{\color{MidnightBlue} \bm{A'}}})$ would have been accepted in the classical test.
2. Select $x,z\in{\ensuremath{\mathbb{F}_q}}^m$ and $u, u' \in {\ensuremath{\mathbb{F}_q}}$ uniformly at random, and let $a = {\mbox{\rm tr}}((u x_{\pi}) \cdot
(u' z_{\pi})) \in { \ensuremath{{\mathbb{F}}}_p}$.
- If $a = 0$, execute the self-test ${\textsc{COM}}$ (see Theorem \[thm:com\_test\]), replacing queries 1, 2, and 3 in the test by $(X, x)$, $(Z,z)$, and $(x,z,uu')$ respectively, and in the case of queries 1 and 2, replacing the prover’s answer $b \in {\ensuremath{\mathbb{F}_q}}$ by ${\mbox{\rm tr}}(u b)$ or ${\mbox{\rm tr}}(u' b) \in { \ensuremath{{\mathbb{F}}}_p}$, respectively, before making the same decision as the verifier in the test.
- If $a \neq
0$, execute the self-test ${\textsc{MS}}$ (see Theorem \[thm:ms-rigid\]) with the following modification: the question labeled $X$ is replaced by the query $(X, x)$ as in part (a), and the prover’s answer $b\in {\ensuremath{\mathbb{F}_q}}$ is replaced by ${\mbox{\rm tr}}(u
b)\in { \ensuremath{{\mathbb{F}}}_p}$; the question labeled $Z$ is replaced by the query $(Z, z)$ as in part (a), and the prover’s answer $b\in {\ensuremath{\mathbb{F}_q}}$ is replaced by $a^{-1}{\mbox{\rm tr}}(u' b)\in { \ensuremath{{\mathbb{F}}}_p}$.
------------------------------------------------------------------------
We denote our quantum low-degree test by ${\textsc{q-lowdeg}}^{(l)}$, for $l\in\{1,2\}$. Here $l$ denotes the “level” of the test, before ($l=1$) or after ($l=2$) composition. In general we also write ${\textsc{q-lowdeg}}$ for the “composed quantum low-degree test” ${\textsc{q-lowdeg}}^{(2)}$, which is the variant of the test with reduced answer size, and is the variant that will be used in our applications. The test is described in [Figure \[fig:protocol\]]{}. We show that the test is a self-test for the following class of Pauli strategies. To define the strategy, recall the definition of the POVM $\{{\tau}_W^a\}$ in Section \[sec:qauli\], defined for each $W\in \{X,Z\}$. For $s \subset {\ensuremath{\mathbb{F}_q}}^m$ either a point or a $2$-dimensional subspace, and $r$ a polynomial defined on $s$, define $$\label{eq:honest-m-def}
{\tau}_{W,s}^r = \sum_{a\in {\ensuremath{\mathbb{F}_q}}^n:\, (g_a)_{|s}=r}{\tau}_W^a\;,$$ where $g_{a}$ is defined in . Finally, for reasons that will become clear later, it is convenient to introduce $$\label{eq:honest-m-def-2}
{{\color{MidnightBlue} \bm{{\tau}}}}_{X,s}^r = {\tau}_{X,s}^{-r}\qquad\text{and}\qquad{{\color{MidnightBlue} \bm{{\tau}}}}_{Z,s}^{r} =
{\tau}_{Z,s}^r\;.$$
\[def:pauli-strategy\] Let $p$ be a prime, $t\geq 1$ an integer, and $q=p^t$. The low-degree Pauli strategy $S_{\textrm{P}}$ on $n$ qudits of local dimension $q$ is the strategy $({|\psi\rangle}, {\mathcal{X}}, {\mathcal{A}}, {\mathcal{M}})$ where ${|\psi\rangle} =
{|{\textsc{EPR}}_q\rangle}^{\otimes n}$, ${\mathcal{X}}= \{X, Z\} \times ({\mathcal{X}}_1 \cup {\mathcal{X}}_2)$, where ${\mathcal{X}}_1 = {\ensuremath{\mathbb{F}_q}}^m$ and ${\mathcal{X}}_2$ is the set of all two-dimensional subspaces of ${\ensuremath{\mathbb{F}_q}}^m$, ${\mathcal{A}}= {\mathcal{A}}_1 \cup {\mathcal{A}}_2$, where ${\mathcal{A}}_1 = {\ensuremath{\mathbb{F}_q}}$ and ${\mathcal{A}}_2 = \deg_d({\ensuremath{\mathbb{F}_q}}^2)$, and ${\mathcal{M}}= {\mathcal{M}}_1 \cup {\mathcal{M}}_2$, where ${\mathcal{M}}_1 = \{{\tau}_{W,w}^a\}\times\{{{\color{MidnightBlue} \bm{{\tau}}}}_{W,w}^a\}$ and ${\mathcal{M}}_2 =\{{\tau}_{W,s}^r\}\times\{{{\color{MidnightBlue} \bm{{\tau}}}}_{W,s}^r\}$, with ${\tau}_{W,w}^a$, ${\tau}_{W,s}^r$, and ${{\color{MidnightBlue} \bm{{\tau}}}}_{W,w}^a$, ${{\color{MidnightBlue} \bm{{\tau}}}}_{W,s}^r$ defined as in and respectively.
\[thm:qld\] Let $n\geq 1$ be an integer. Let $h,m$ be integer such that $h^m \geq n$, and let $d=hm$. Let $q = p^t$ be a prime power such that ${\ensuremath{\mathbb{F}_q}}$ admits a self-dual basis over ${ \ensuremath{{\mathbb{F}}}_p}$. Then for any ${\varepsilon}\geq 0$ the test ${\textsc{q-lowdeg}}^{(2)}(m, d, q)$ is a $2$-prover self-test for the low-degree Pauli strategy $S_{\textrm{P}}$ on $n$ qudits of local dimension $q$ with completeness $1$ and robustness $\delta = \operatorname{poly}(\operatorname{poly}(p)\cdot \operatorname{poly}({\varepsilon}) +
\operatorname{poly}(d/q))$. Moreover, the test has questions of length $O(m \log q)$ and answers of length $O(\log^2(d) \log (q))$.
Completeness of the test is shown in [Lemma \[lem:completeness\]]{} in Section \[sec:lowdeg-completeness\]. Soundness is shown in [Lemma \[lem:soundness\]]{} in Section \[sec:soundness\].
In a typical application of the test ${\textsc{q-lowdeg}}^{(2)}$, the parameters are chosen such that $m = \Theta(\frac{\log n}{\log \log n})$ and $h =
\Theta(\log n)$, resulting in $d = \Theta(\frac{\log^2(n)}{\log \log
n})$. Further, we chose $p$ to be constant and $q = \Theta(\frac{\log^2(n)}{\log \log
n})$ such that $d/q$ is a small constant. This results in a question length that is $O(\log n)$ and an answer length that is $\operatorname{poly}(\log\log n)$.
Completeness {#sec:lowdeg-completeness}
------------
The proof of the following lemma specifies the “honest” strategy that is expected of the provers in the quantum low-degree test.
\[lem:completeness\] For $m,d,q$ as in Theorem \[thm:qld\] the strategy $S_{\textrm{P}}$ introduced in Definition \[def:pauli-strategy\] can be extended to a strategy that succeeds with probability $1$ in the test ${\textsc{q-lowdeg}}(m,d,q)$.
Let $${|\psi_{\textsc{EPR}}\rangle} \,=\, \bigotimes_{j=1}^{n+1} {|{\textsc{EPR}}_q\rangle}\;,$$ where ${|{\textsc{EPR}}_q\rangle}$ is defined in . We first describe a strategy for the players assuming questions in part (a) of the test come from ${\textsc{c-lowdeg}}$, instead of the composed test ${\textsc{c-lowdeg}}^{(2)}$. Once a strategy for the former has been defined it is straightforward to adapt it to a strategy for the latter; this only requires classical post-processing.
To define the strategy we use the generalized Pauli operators and projections defined in Section \[sec:qauli\]. When queried for a subspace $s\subseteq
{ \ensuremath{{\mathbb{F}}}_p}^m$ in a basis $W\in\{X,Z\}$, the prover measures the first $n$ qudits using the projective measurement $\{{\tau}_W^a\}$ and returns the polynomial $(g_a)_{|s}$; this corresponds to the POVM described in .
To see that these measurements define a strategy which succeeds with probability $1$ in part (a) of the test, note that the state ${|{\textsc{EPR}}_q\rangle}$ is stabilized by ${\tau}_X(a) {\otimes}{\tau}_X(a)$ and ${\tau}_Z(b) {\otimes}{\tau}_Z(-b)$ for any $a, b \in {\ensuremath{\mathbb{F}_q}}$. Hence, if both provers measure the state ${|{\textsc{EPR}}_q\rangle}$ in the $X$ eigenbasis, and the first prover obtains an outcome $a\in{\ensuremath{\mathbb{F}_q}}$, the second prover will obtain the outcome $-a$; if they measure in the $Z$ eigenbasis, they will both always obtain the same outcome. As a consequence, the following consistency relations hold for any $s$: $$\begin{aligned}
\sum_{r\in \deg_d(s)} \,{\tau}_{X,s}^r {\otimes}{\tau}_{X,s}^{-r} \,{|\psi_{\textsc{EPR}}\rangle} &= {|\psi_{\textsc{EPR}}\rangle}\;,
\\
\sum_{r\in\deg_d(s)} \,{\tau}_{Z,s}^r {\otimes}{\tau}_{Z,s}^{r} \,{|\psi_{\textsc{EPR}}\rangle} &= {|\psi_{\textsc{EPR}}\rangle}\;.
\end{aligned}
\label{eq:epr_stab-0}$$ Thus whenever $W=X$ is selected in part (a) of the low-degree test the first prover’s answers are consistent with the negation of the second prover’s, as the verifier expects; in case $W=Z$ both provers’ answers are consistent.
Using the notation introduced in , the consistency relations become $$\begin{aligned}
\sum_q {\tau}_{W,s}^q {\otimes}{{\color{MidnightBlue} \bm{{\tau}}}}_{W,s}^q {|\psi_{\textsc{EPR}}\rangle} &=
{|\psi_{\textsc{EPR}}\rangle}\;, \\
\end{aligned}
\label{eq:epr_stab}$$ for any $W\in\{X,Z\}$.
To show completeness in part (b) of the test we introduce a family of generalized observables associated with the measurement performed by a prover in part (a) of the test when it is queried for a value at a single point $w\in{\ensuremath{\mathbb{F}_q}}^m$. The prover’s answer in this case is a value in ${\ensuremath{\mathbb{F}_q}}$. To the provers’ strategy for determining his answer we introduce a family of $q$ observables over ${ \ensuremath{{\mathbb{F}}}_p}$, indexed by $u\in {\ensuremath{\mathbb{F}_q}}$, each of which is associated with the value ${\mbox{\rm tr}}(ub)\in { \ensuremath{{\mathbb{F}}}_p}$, where $b\in{\ensuremath{\mathbb{F}_q}}$ is the answer obtained by the prover. We denote the corresponding for query $(W,w)$ by $W_u(w_{\pi})$: $$\begin{aligned}
W_u(w_{\pi}) &= \sum_{s \in {\ensuremath{\mathbb{F}_q}}^n}\, \omega^{ {\mbox{\rm tr}}(g_s(w) u)} \,{\tau}_W^a \label{eq:def-local-obs}\\
&= \sum_{s \in {\ensuremath{\mathbb{F}_q}}^n} \,\omega^{{\mbox{\rm tr}}(s\cdot (u w_{\pi}))} \,{\tau}_W^a \notag\\
&= {\tau}_W(u w_{\pi})\;.\notag\end{aligned}$$ From this expression it is clear that for any $x,z\in{\ensuremath{\mathbb{F}_q}}^m$, $u,u'\in{\ensuremath{\mathbb{F}_q}}$, and $a={\mbox{\rm tr}}((u x_{\pi}) \cdot
(u' z_{\pi}))$, $$X_u(x_{\pi}) Z_{u'}(a^{-1} z_{\pi}) =
\omega^{-{\mbox{\rm tr}}(a^{-1} (u x_{\pi})(u' z_{\pi}))} Z_{u'}(a^{-1} z_{\pi}) X_u(x_{\pi}) =
\omega^{-1} Z_{u'}(a^{-1} z_{\pi}) X_u(x_{\pi})\;.$$
Hence, the measurement operators corresponding to the questions labeled $X$ and $Z$ in part (b) of the test satisfy the required twisted commutation relation. It is then straightforward that each prover can implement a strategy that succeeds in part (b) of the test. In case the test ${\textsc{COM}}$ is executed this is immediate; in case it is ${\textsc{MS}}$ the provers may use the $(n+1)$-th qudit and a second pair of observables $(X',Z')$ satisfying the same twisted commutation relation, so that $(X,Z)$ and $(X',Z')$ together form a strategy which succeeds with probability $1$ in the test ${\textsc{MS}}$.
Soundness analysis {#sec:soundness}
==================
\[lem:soundness\] Let $n\geq 1$ be an integer and $m,h,d,$ and $q=p^t$ as in Theorem \[thm:qld\]. Let ${\varepsilon}\geq 0$. Suppose a strategy using state ${|\psi\rangle}_{{{\textsf{AB}}}} \in {\mathcal{H}}_{{{\textsf{A}}}} \otimes {\mathcal{H}}_{{{\textsf{B}}}}$ and projective measurements $\{M_{W,s,s'}^r\}$ and $\{M_{W,w}^a\}$ succeeds in test ${\textsc{q-lowdeg}}(m,d)$ with probability at least $1-{\varepsilon}$. Then there is a $\delta = \operatorname{poly}(\operatorname{poly}(p)\cdot \operatorname{poly}({\varepsilon}) +
\operatorname{poly}(d/q))$, isometries $V_D:
{\mathcal{H}}_{{{\textsf{D}}}} \to ({\ensuremath{\mathbb{C}}}^q)^{\otimes n}_{{{\textsf{D'}}}}\otimes {\mathcal{H}}_{{{\textsf{D''}}}}$ for $D\in \{A,B\}$, and a state ${|{\textsc{aux}}\rangle}\in{\mathcal{H}}_{{{\textsf{A''}}}} \otimes {\mathcal{H}}_{{{\textsf{B''}}}}$ such that $$\big\| V_A \otimes V_B{|\psi\rangle} - {|{\textsc{EPR}}_q\rangle}^{\otimes n} {|{\textsc{aux}}\rangle} \big\|^2 \,\leq\, \delta\;,$$ and for all $W\in \{X,Z\}$, $${\ensuremath{\mathop{\textsc{E}}_{w\in{\ensuremath{\mathbb{F}_q}}^m}}} \,\sum_{a\in {\ensuremath{\mathbb{F}_q}}}\, \big\|(V_A {\otimes}V_B)(M_{W,w}^a {\otimes}{\ensuremath{\mathop{\rm Id}\nolimits}}) {|\psi\rangle} -
({\tau}_{W,w}^a {\otimes}{\ensuremath{\mathop{\rm Id}\nolimits}}){|{\textsc{EPR}}_q\rangle}^{\otimes n} {|{\textsc{aux}}\rangle} \big\|^2 \, \leq \,
\delta\;.$$ Moreover, an analogous relation holds for the second prover’s operators.
The outline for the proof of Lemma \[lem:soundness\] is as follows:
1. In Section \[sec:strat\], we describe the conditions satisfied by any strategy that succeeds with high probability in the test.
2. In Section \[sec:hatx\], we adjoin an ancilla to each of the provers’ private registers, and define a set of approximately commuting “points” observables $\hat{X}_u(x_{\pi})$ and $\hat{Z}_u(z_{\pi})$, for each $u\in{\ensuremath{\mathbb{F}_q}}$, that act on the original shared state tensored with a maximally entangled state on the ancilla.
3. In Section \[sec:hatq\], we construct a family of joint measurements $\{\hat{Q}_{x,z}^{c}\}$ indexed by a pair of values $x, z \in {\ensuremath{\mathbb{F}_q}}^m$ and with outcomes $c \in {\ensuremath{\mathbb{F}_q}}$, obtained as a common refinement of the $2q$ approximately commuting observables $\hat{X}_{u}(x_{\pi})$ and $\hat{Z}_v(z_{\pi})$, for all $u,v\in{\ensuremath{\mathbb{F}_q}}$, defined in the previous step. Joint measurability is proved by showing that the observables satisfy approximate linearity relations, in the sense of implying a successful strategy in the two-prover linearity test from [@NV17].
4. From these joint measurements we define a strategy for the test ${\textsc{c-lowdeg}}(2m+2,d+1,q)$ over ${\ensuremath{\mathbb{F}_q}}^m\times{\ensuremath{\mathbb{F}_q}}^m \times {\ensuremath{\mathbb{F}_q}}^2$, denoted by $\{\hat{Q}_{s}^r\}$. Applying Theorem \[thm:2ml\], we deduce a single low-degree measurement $\{\hat{S}^g\}$.
5. We argue that $g$ must take the form $g(x,z,\alpha,\beta)=\alpha g_1(x) + \beta g_2(z)$ for low-degree polynomials $g_1,g_2:{\ensuremath{\mathbb{F}_q}}^m\to {\ensuremath{\mathbb{F}_q}}$. This allows us to recover commuting low-degree measurements $\{\hat{S}_X^{g_1}\}$ and $\{\hat{S}_Z^{g_2}\}$.
6. In Section \[sec:tildex\] we use the low-degree measurements obtained in the previous step to recover observables $\tilde{X}_\ell(x)$ and $\tilde{Z}_{\ell'}(z)$ defined for all points $x,z\in{ \ensuremath{{\mathbb{F}}}_p}^n$ and indices $\ell \in \{1,\ldots,t\}$. By construction these operators exactly satisfy the same twisted commutation relations as the “honest” generalized observables ${\tau}_X(b_\ell x)$ and ${\tau}_Z(b_{\ell'} z)$ (recall that $\{b_1,\ldots,b_t\}$ is a self-dual basis of ${\ensuremath{\mathbb{F}_q}}$ over ${ \ensuremath{{\mathbb{F}}}_p}$); moreover, we show that they are consistent with the provers’ original observables $X_{b_\ell} ,Z_{b_{\ell'}}$ at points of the form $x_{\pi},z_{\pi}$.
7. Finally, in Section \[sec:finish\_soundness\], we use that $\tilde{X}_\ell(x)\otimes \tilde{X}_\ell(x)$ and $\tilde{Z}_{\ell'}(z) \otimes \tilde{Z}_{\ell'}(z)$ approximately stabilize ${|\psi\rangle}$ to conclude that the provers’ shared state is close to the target state ${|{\textsc{EPR}}_q\rangle}^{\otimes n}$ (under the action of the appropriate isometry).
Arbitrary strategies in the test ${\textsc{q-lowdeg}}$ {#sec:strat}
------------------------------------------------------
We start with the following preliminary claim, which establishes basic properties of successful strategies in the test ${\textsc{q-lowdeg}}(m,d,q)$.
\[claim:strategies\] Let $m,d,q=p^t$, ${\varepsilon}$, ${|\psi\rangle}$ and $\{M_{W,s,s'}^r\}$ be as in Lemma \[lem:soundness\]. There exists $\delta_M = \operatorname{poly}(p)\cdot\operatorname{poly}({\varepsilon})$ such that the following hold. For $W\in\{X,Z\}$ and $s\subseteq {\ensuremath{\mathbb{F}_q}}^m$ there exist projective measurements $\{{M}_{W,s}^{r}\}_{r\in\deg_d(s)}$ and $\{{{\color{MidnightBlue} \bm{M}}}_{W,s}^{r}\}_{r\in\deg_d(s)}$ such that, on average over $s\sim{\pi_{\rm ld}}$, $$\label{eq:subspace-consistent}
M_{W,s}^r \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}\,\approx_{\delta_M} \, {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{M}}}_{W,s}^{r} \;,$$ and moreover the $\{{M}_{W,s}^{r}\}_{r\in\deg_d(s)}$ and $\{{{\color{MidnightBlue} \bm{M}}}_{W,s}^{r}\}_{r\in\deg_d(s)}$, together with the state ${|\psi\rangle}$, specify a strategy with success $1-{\varepsilon}'$ in the test ${\textsc{q-lowdeg}}^{(1)}(m,d,q)$, for some $\delta = \operatorname{poly}(p)\cdot\operatorname{poly}({\varepsilon})$.
For $W\in\{X,Z\}$, $u\in{\ensuremath{\mathbb{F}_q}}$, and $w\in {\ensuremath{\mathbb{F}_q}}^m$ define[^7] $$\label{eq:def-xz-obs}
W_u(w_{\pi}) \,=\, \sum_a \omega^{{\mbox{\rm tr}}(a u)} M_{W,w}^a\;, \qquad {{\color{MidnightBlue} \bm{W}}}_u(w_{\pi}) \,=\, \sum_a
\omega^{{\mbox{\rm tr}}(a u)} {{\color{MidnightBlue} \bm{M}}}_{W,w}^a\;.$$ Then for fixed $W$ and $w$, the $q$ observables $\{W_u(w_{\pi}),\,u\in{\ensuremath{\mathbb{F}_q}}\}$ pairwise commute. For any $a\in{\ensuremath{\mathbb{F}_q}}$, we can write the POVM elements $M_{W,w}^a$ and ${{\color{MidnightBlue} \bm{M}}}_{W,w}^a$ in terms of the observables $W_u(w_{\pi})$ and ${{\color{MidnightBlue} \bm{W}}}_u(w_{\pi})$ as follows:[^8] $$\label{eq:m-from-w}
M_{W,w}^a \,=\, {\ensuremath{\mathop{\textsc{E}}_{u\in{\ensuremath{\mathbb{F}_q}}}}} \,\omega^{-{\mbox{\rm tr}}(au)}\,W_u(w_{\pi})\;,\qquad {{\color{MidnightBlue} \bm{M}}}_{W,w}^a \,=\, {\ensuremath{\mathop{\textsc{E}}_{u\in{\ensuremath{\mathbb{F}_q}}}}} \,\omega^{-{\mbox{\rm tr}}(au)}\,{{\color{MidnightBlue} \bm{W}}}_u(w_{\pi})\;.$$ Moreover, on average over $w\in{\ensuremath{\mathbb{F}_q}}^m$ and for every $u\in{\ensuremath{\mathbb{F}_q}}$, $$\label{eq:xz-cons}
W_u(w_{\pi}) \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}\,\approx_{\delta_M} \,{\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{W}}}_u(w_{\pi})\;.$$ Finally, $$\label{eq:xz-ac}
X_u(x_{\pi}) Z_{u'}(z_{\pi}) \,\approx_{\delta_M}\, \omega^{{\mbox{\rm tr}}((u
x_{\pi}) \cdot (u' z_{\pi}))} Z_{u'}(z_{\pi}) X_u(x_{\pi})\;,$$ on average over uniformly random $x,z\in{\ensuremath{\mathbb{F}_q}}^m$ and $u,u' \in { \ensuremath{{\mathbb{F}}}_p}^t$.
For $s\subseteq {\ensuremath{\mathbb{F}_q}}^m$, $s'\subseteq{\ensuremath{\mathbb{F}_q}}^{m'}$ and $r\in\deg_d(s)$ let $${{\color{MidnightBlue} \bm{M}}}_{X,s,s'}^{r} \,=\, M_{X,s,s'}^{-r}\qquad\text{and}\qquad {{\color{MidnightBlue} \bm{M}}}_{Z,s,s'}^{r} \,=\, M_{Z,s}^{r}\;.$$ With this definition, follows from the assumption that the provers’ strategy succeeds with probability $1-O({\varepsilon})$ in part (a) of the test ${\textsc{q-lowdeg}}^{(2)}$. Moreover, for any fixed choice of $W$ and first part $s$ of the question $(s,s')$ in ${\textsc{c-lowdeg}}^{(2)}(m,d,q)$, the induced strategy is a successful strategy in ${\textsc{c-lowdeg}}^{(1)}(m',d',q)$, to which Theorem \[thm:ml\] can be applied. This defines the required POVM elements $\{{M}_{W,s}^{r}\}_{r\in\deg_d(s)}$ and $\{{{\color{MidnightBlue} \bm{M}}}_{W,s}^{r}\}_{r\in\deg_d(s)}$.
Commutation of the $\{W_u(w_{\pi}),\,u\in{\ensuremath{\mathbb{F}_q}}\}$ follows since $\{M_{W,w}^a\}$ are projective measurements. Eq. follows by expanding $W_u(w_{\pi})$ using the definition.
Using that the distribution ${\pi_{\rm ld}}$ from the low-degree test places constant probability on subspaces of dimension $0$, by Claim \[claim:obs-meas-cons\] the consistency conditions imply .
Finally, using Theorems \[thm:com\_test\] and \[thm:ms-rigid\] success with probability at least $1-{\varepsilon}$ in part (b) of the test implies that the observables defined in satisfy for $\delta_M = O(p^3\sqrt{{\varepsilon}})$.
Expanding the Hilbert space and defining commuting observables {#sec:hatx}
--------------------------------------------------------------
From the initial strategy of the provers, satisfying the properties expressed in Claim \[claim:strategies\], we define new observables on an extended Hilbert space that will be the main operators used in the proof.
\[lem:hats\] Let $m,d,q=p^t$, ${\varepsilon}$, ${|\psi\rangle}$ and $\{M_{W,s}^r\}$ be as in Lemma \[lem:soundness\], and $W_u(w_{\pi})$ as in Claim \[claim:strategies\]. There exists a state $${|\hat{\psi}\rangle}_{{{\textsf{AA'A''}}}{{\textsf{BB'B''}}}} \in {\mathcal{H}}_{{\textsf{A}}} \otimes ({\ensuremath{\mathbb{C}}}^q_{{{\textsf{A'}}}}\otimes {\ensuremath{\mathbb{C}}}^q_{{{\textsf{A''}}}})^{\otimes n} \otimes {\mathcal{H}}_{{\textsf{B}}} \otimes ({\ensuremath{\mathbb{C}}}^q_{{{\textsf{B'}}}}\otimes {\ensuremath{\mathbb{C}}}^q_{{{\textsf{B''}}}})^{\otimes n}\;,$$ and for $W\in\{X,Z\}$, $s\subseteq {\ensuremath{\mathbb{F}_q}}^m$, $u \in {\ensuremath{\mathbb{F}_q}}$, and $w\in{\ensuremath{\mathbb{F}_q}}^m$ there are POVM $\{\hat{M}_{W,s}^r\}_{r \in\deg_d(s)}$, $\{{{\color{MidnightBlue} \bm{\hat{M}}}}_{W,s}^r\}_{r\in\deg_d(s)}$ and observables $\hat{W}_u(w_{\pi})$, ${{\color{MidnightBlue} \bm{\hat{W}}}}_u(w_{\pi})$ on ${\mathcal{H}}_{{\textsf{A}}}\otimes
({\ensuremath{\mathbb{C}}}^q_{{{\textsf{A'}}}})^{\otimes n} $ and ${\mathcal{H}}_{{\textsf{B}}}\otimes
({\ensuremath{\mathbb{C}}}^q_{{{\textsf{B'}}}})^{\otimes n} $ respectively, $$\label{eq:def-hat-xz}
\hat{W}_u(w_{\pi}) \,=\, W_u(w_{\pi}) \otimes {{\color{MidnightBlue} \bm{{\tau}}}}_W(u
w_{\pi})\;,\qquad {{\color{MidnightBlue} \bm{\hat{W}}}}_u(w_{\pi}) \,= \,
{{\color{MidnightBlue} \bm{W}}}_u(w_{\pi}) {\otimes}{\tau}_W(u w_{\pi})\;,$$ such that the following hold for some $\delta_{\hat{M}}= \operatorname{poly}(\delta_M)$. On average over $(s,w)\sim {\pi_{\rm ld}}$ and for any $u \in {\ensuremath{\mathbb{F}_q}}$ and $W\in\{X,Z\}$, $$\label{eq:m-cons}
\begin{aligned}
\big(\hat{M}_{W,s}^r \big)_{{{\textsf{AA'}}}}\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}\,\approx_{\delta_{\hat{M}}}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes \big({{\color{MidnightBlue} \bm{\hat{M}}}}_{W,s}^r\big)_{{{\textsf{BA''}}}} \;,\\
\hat{W}_u(w_{\pi})_{{{\textsf{AA'}}}} \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}\,\approx_{\delta_{\hat{M}}}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{\hat{W}}}}_u(w_{\pi})_{{{\textsf{BA''}}}}\;,\\
\sum_{r\in\deg_d(s)} \big(\hat{M}_{W,s}^r\big)_{{{\textsf{AA'}}}}
\otimes \big({{\color{MidnightBlue} \bm{\hat{W}}}}_u^{{\mbox{\rm tr}}(r(w) u)}(w_{\pi})\big)_{{{\textsf{BA''}}}} \,\approx_{\delta_{\hat{M}}}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\;,\\
\sum_{r\in\deg_d(s)} \big(\hat{M}_{W,s}^r\big)_{{{\textsf{AA'}}}}
\otimes \big({{\color{MidnightBlue} \bm{\hat{M}}}}_{W,w}^{r(w)}\big)_{{{\textsf{BA''}}}} \,\approx_{\delta_{\hat{M}}}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\;,
\end{aligned}$$ where $$\label{eq:def-hatm-w}
\hat{M}_{W,w}^a \,=\, {\ensuremath{\mathop{\textsc{E}}_{u\in{\ensuremath{\mathbb{F}_q}}}}} \,\omega^{-{\mbox{\rm tr}}(au)} \,\hat{W}_u(w_{\pi})\;,\qquad {{\color{MidnightBlue} \bm{\hat{M}}}}_{W,w}^a \,=\, {\ensuremath{\mathop{\textsc{E}}_{u\in{\ensuremath{\mathbb{F}_q}}}}} \,\omega^{-{\mbox{\rm tr}}(au)}\,{{\color{MidnightBlue} \bm{\hat{W}}}}_u(w_{\pi})\;.$$ Finally, on average over uniformly random $x,z\in{\ensuremath{\mathbb{F}_q}}^m$ and $u, u' \in {\ensuremath{\mathbb{F}_q}}$, $$\label{eq:xz-com}
\hat{X}_u(x_{\pi})\hat{Z}_{u'}(z_{\pi}) \,\approx_{\delta_{\hat{M}}} \,\hat{Z}_{u'}(z_{\pi})\hat{X}_{u}(x_{\pi})\;.$$
We first define the state ${|\hat{\psi}\rangle}$. For this we enlarge the Hilbert space ${\mathcal{H}}_{{{\textsf{A}}}}\otimes {\mathcal{H}}_{{{\textsf{B}}}}$ in two ways. First we assume that each prover has access to a sufficiently large number $N$ of qubits initialized in the state ${|0\rangle}$. This allows us to apply Naimark’s dilation theorem to simulate a POVM measurement applied by the provers by a projective measurement, whenever it is convenient (we will always specify when we do so). Second, for each prover $D\in\{A,B\}$ we adjoin two ancilla registers ${{\textsf{D'}}},{{\textsf{D''}}}$ initialized in state $${|\psi_{\textsc{EPR}}\rangle}_{{{\textsf{D'D''}}}}\,=\,{|{\textsc{EPR}}_q\rangle}^{\otimes
n}_{{{\textsf{D'D''}}}}\;,$$ where ${|{\textsc{EPR}}_q\rangle}$ is defined in . The state of the enlarged system is $$\label{eq:ancilla-def}
{|\hat{\psi}\rangle}_{{{\textsf{AA'A''}}}{{\textsf{BB'B''}}}} \,=\, \big({|\psi\rangle} {\otimes}{|0\rangle}^{{\otimes}2N}\big)_{{{\textsf{AB}}}} {|\psi_{\textsc{EPR}}\rangle}_{{{\textsf{A'A''}}}}
{|\psi_{{\textsc{EPR}}}\rangle}_{{{\textsf{B'B''}}}}\;.$$ Next, for $W\in\{X,Z\}$, $s\subset {\ensuremath{\mathbb{F}_q}}^m$ and $r\in \deg_d(s')$ let $$\label{eq:hatm-def}
\hat{M}_{W,s}^r \,=\, \sum_{\substack{r',r''\in\deg_d(s):\\ r' + r''=r}}
M_{W,s}^{r'} \otimes {{\color{MidnightBlue} \bm{{\tau}}}}_{W,s}^{r''}\;,$$ where $\{{\tau}_{W,s}^{r''}\}$ is the “honest” subspace measurement defined in . Define complementary measurements $${{\color{MidnightBlue} \bm{\hat{M}}}}_{W,s}^r =
\sum_{\substack{r', r'' \in \deg_d(s):\\ r'+r''=r}} {{\color{MidnightBlue} \bm{M}}}_{W,s}^{r'} \otimes
{\tau}_{W,s}^{r''}\;.$$ The following claim establishes .
\[claim:hat-m-cons\] For $W\in\{X,Z\}$ the subspace measurements are self-consistent, and consistent with the point measurements: on average over $(s,w)\sim{\pi_{\rm ld}}$ and for any $u \in{\ensuremath{\mathbb{F}_q}}$, $$\begin{aligned}
\sum_{r\in\deg_d(s)} \big(\hat{M}_{W,s}^r \big)_{{{\textsf{AA'}}}}\otimes \big({{\color{MidnightBlue} \bm{\hat{M}}}}_{W,s}^r\big)_{{{\textsf{BA''}}}} &\approx_{\operatorname{poly}(\delta_M)} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,\\
\sum_{r\in\deg_d(s)} \big(\hat{M}_{W,s}^r\big)_{{{\textsf{AA'}}}} \otimes
\big({{\color{MidnightBlue} \bm{\hat{W}}}}_u^{{\mbox{\rm tr}}(r(w) u)}(w_{\pi})\big)_{{{\textsf{BA''}}}} &\approx_{\operatorname{poly}(\delta_M)} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,\\
\sum_{r\in\deg_d(s)} \big(\hat{M}_{W,s}^r\big)_{{{\textsf{AA'}}}}
\otimes \big({{\color{MidnightBlue} \bm{\hat{M}}}}_{W,w}^{r(w)}\big)_{{{\textsf{BA''}}}} \,\approx_{\delta_{\hat{M}}}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\;,\end{aligned}$$ and $$\hat{W}_u(w_{\pi})_{{{\textsf{AA'}}}} \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}\,\approx_{\operatorname{poly}(\delta_M)}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{\hat{W}}}}_u(w_{\pi})_{{{\textsf{BA''}}}}\;.$$
Note that $\{\hat{M}_{W,w}^a\}$ as defined in is a well-defined projective measurement, given the definition of the $\{\hat{W}_u(w_\pi)\}$ in .
For the first identity, decompose $\hat{M}_{W,s}^r$ using the definition and use self-consistency of $M_{W,s}^{r'}$, which is expressed in , and of ${\tau}_{W,s}^{r''}$, which follows since the ancilla introduced in is maximally entangled on ${{\textsf{A'A''}}}$.
For the second identity, decompose $\hat{M}_{W,s}^r$ and $\hat{W}_u^{{\mbox{\rm tr}}(u r(w))}(w_{\pi})$ using the definition to get $$\begin{aligned}
\sum_{r\in\deg_d(s)}& \hat{M}_{W,s}^r \otimes {{\color{MidnightBlue} \bm{\hat{W}}}}_u^{{\mbox{\rm tr}}(u
r(w))}(w_{\pi}) \\
&=\sum_{\substack{r',r'',a',a'':\\{\mbox{\rm tr}}( (r' + r'')(w) \cdot u) =(a' +
a'')}}\big(M_{W,s}^{r'}\big)_{{{\textsf{A}}}}
\otimes \big({{\color{MidnightBlue} \bm{{\tau}}}}_{W,s}^{r''}\big)_{{{\textsf{A'}}}}
\otimes \big({{\color{MidnightBlue} \bm{W}}}_u^{a'}(w_{\pi})\big)_{{{\textsf{B}}}}
\otimes \big({\tau}_W^{a''}(u w_{\pi})\big)_{{{\textsf{A''}}}} \\
&=\sum_{\substack{r',a':\\{\mbox{\rm tr}}(r'(w)\cdot u)=a'}}\big(M_{W,s}^{r'}\big)_{{{\textsf{A}}}} \otimes
\big({{\color{MidnightBlue} \bm{W}}}_u^{a'}(w_{\pi})\big)_{{{\textsf{B}}}} \\
&\approx_{\operatorname{poly}({\varepsilon})} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,\end{aligned}$$ where the second line uses consistency of ${{\color{MidnightBlue} \bm{{\tau}}}}_{W,s}^{r''}$ with ${\tau}_W ^{{\mbox{\rm tr}}(u r''(w))} (u w_{\pi})$ (which follows from the analysis of the honest strategy given in the proof of Lemma \[lem:completeness\]), and the third follows from success of the provers’ strategy in the low-degree test and the definition of ${{\color{MidnightBlue} \bm{W}}}_u(w_{\pi})$ in .
The third identity follows from the second and the definition .
Finally, the last relation follows from the first, specialized to $s=w$. Alternatively, combine consistency between $W_{{\textsf{A}}}$ and ${{\color{MidnightBlue} \bm{W}}}_{{\textsf{B}}}$ (shown in ) and of $({\tau}_W)_{{\textsf{A'}}}$ and $({{\color{MidnightBlue} \bm{{\tau}}}}_W)_{{\textsf{A''}}}$, which follows since the ancilla state is ${|\psi_{\textsc{EPR}}\rangle}_{{{\textsf{A'A''}}}}$.
The next claim establishes .
\[claim:hat-xz-commute\] On average over uniformly random $x, z\in{\ensuremath{\mathbb{F}_q}}^m$ and $u, u' \in {\ensuremath{\mathbb{F}_q}}$, $$\hat{X}_u(x_{\pi})\hat{Z}_{u'}(z_{\pi})\,\approx_{\operatorname{poly}(\delta_M)}\, \hat{Z}_{u'}(z_{\pi})\hat{X}_{u}(x_{\pi})\;.$$
Write $$\begin{aligned}
\hat{X}_u(x_{\pi})\hat{Z}_{u'} (z_{\pi}) &= X_{u} (x_{\pi})Z_{u'} (z_{\pi}) \otimes
{{\color{MidnightBlue} \bm{{\tau}}}}_X(u x_{\pi})
{{\color{MidnightBlue} \bm{{\tau}}}}_Z(u' z_{\pi})
\\
&= X_u(x_{\pi}) Z_{u'}(z_{\pi}) \otimes {\tau}_X(-u x_{\pi})
{\tau}_Z(u' z_{\pi}) \\
&\approx_{\delta_M} \big( \omega^{-{\mbox{\rm tr}}((u x_{\pi})\cdot (u' z_{\pi})}
Z_{u'}(z_{\pi})X_{u}(x_{\pi}) \big)\otimes \big(
\omega^{{\mbox{\rm tr}}((u x_{\pi})\cdot (u' z_{\pi}))}{\tau}_Z(u' z_{\pi})
{\tau}_X(-u x_{\pi})\big)\\
&= Z_{u'} (z_{\pi}) X_{u} (x_{\pi}) \otimes {{\color{MidnightBlue} \bm{{\tau}}}}_Z(u' z_{\pi})
{{\color{MidnightBlue} \bm{{\tau}}}}_X({u} x_{\pi}) \\
&= \hat{Z}_{u'}(z_{\pi})\hat{X}_{u}(x_{\pi})\;,\end{aligned}$$ where the approximation follows from in Claim \[claim:strategies\].
Combining $X$ and $Z$ measurements {#sec:hatq}
----------------------------------
In this section we combine the approximately commuting observables $\hat{X}_u(x_{\pi})$ and $\hat{Z}_u(z_{\pi})$ constructed in the proof of Lemma \[lem:hats\] into a single POVM. We then show that the POVM leads to a strategy for the classical low-degree test. Applying Theorem \[thm:ml\], we obtain a single POVM $\{\hat{S}^{g_1,g_2}\}_{g_1,g_2\in\deg_{d}(s)}$ which is simultaneously consistent with both families of observables, as shown in the following lemma.
\[lem:hat-s\] Let $m,d,q=p^t$ be as in Lemma \[lem:soundness\], and ${|\hat{\psi}\rangle}$, $\hat{W}_u(w_{\pi})$, ${{\color{MidnightBlue} \bm{\hat{W}}}}_u(w_{\pi})$ and $\delta_{\hat{M}}$ as in Lemma \[lem:hats\]. There exists projective measurements $\{\hat{S}_{{{\textsf{AA'}}}}^{g_1,g_2}\}$ and $\{{{\color{MidnightBlue} \bm{\hat{S}}}}_{{{\textsf{BA''}}}}^{g_1,g_2}\}$ with outcomes in the set of pairs $(g_1,g_2)$ of polynomials on ${\ensuremath{\mathbb{F}_q}}^m$ of total degree at most $d$ each such that, on average over uniformly random $x,z\in{\ensuremath{\mathbb{F}_q}}^m$, and for all $u \in {\ensuremath{\mathbb{F}_q}}$, $$\label{eq:hat-s-xz-cons}
\sum_{g_1, g_2} \hat{S}^{g_1,g_2} \otimes {{\color{MidnightBlue} \bm{\hat{X}}}}_{u}^{{\mbox{\rm tr}}(g_1(x) u)}(x_{\pi})\,\approx_{\delta_S}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\;,\qquad \sum_{g_1, g_2} \hat{S}^{g_1,g_2} \otimes {{\color{MidnightBlue} \bm{\hat{Z}}}}_{u}^{{\mbox{\rm tr}}(g_2(z) u)}(z_{\pi})\, \approx_{\delta_S}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\;,$$ for some $\delta_S = \operatorname{poly}(p)\cdot\operatorname{poly}(\delta_{\hat{M}})+\operatorname{poly}(d/q)$. Similar relations hold with ${{\color{MidnightBlue} \bm{\hat{S}}}}$ and $\hat{X}$, $\hat{Z}$ instead of $S$ and ${{\color{MidnightBlue} \bm{\hat{X}}}}$, ${{\color{MidnightBlue} \bm{\hat{Z}}}}$ respectively.
The proof of Lemma \[lem:hat-s\] proceeds in two steps. In the first step, for each pair $(u,v)\in {\ensuremath{\mathbb{F}_q}}^2$ we combine the approximately commuting observables $\hat{X}_u(x_{\pi})$ and $\hat{Z}_{v}(z_{\pi})$ into a single POVM $\{\hat{Q}_{xu,zv}^{a,b}\}$ that essentially measures in their joint eigenbasis. The following lemma shows how this can be done in general. (See [@glebsky2010almost] for a similar claim that applies to arbitrary unitaries but is restricted to the Frobenius norm.)
\[lem:approx-commute-u\] Let $\eta>0$, ${|\psi\rangle} \in {\mathcal{H}}\otimes {\mathcal{H}}'$, for finite-dimensional Hilbert spaces ${\mathcal{H}},{\mathcal{H}}'$, and $W_j\in{{\mathrm{U}}}({\mathcal{H}})$, ${{\color{MidnightBlue} \bm{W}}}_j
\in{{\mathrm{U}}}({\mathcal{H}}')$, for $j=1,\ldots,k$, be such that for all $j,\ell\in\{1,\ldots,k\}$, the powers $(W_j)^p = ({{\color{MidnightBlue} \bm{W}}}_j)^p = {\ensuremath{\mathop{\rm Id}\nolimits}}$, $\|(W_j\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}- {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{W}}}_j){|\psi\rangle}\|^2 \leq \eta$, and $\|(W_jW_\ell-W_\ell W_j)\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle}\|^2 \leq \eta$. Then there exists an $\eta' = \operatorname{poly}(p,k)\cdot\operatorname{poly}(\eta)$ and a POVM $\{Q^{a}\}_{a\in { \ensuremath{{\mathbb{F}}}_p}^k}$ such that $$Q^{a} \approx_{\eta'} \prod_j W_j^{a_j}\qquad\text{and}\qquad
\forall j,\quad W_j \,\approx_{\eta'}\, \sum_{a}\, \omega^{a_j}
Q^{a}\;,$$ where $W_j^{a_j}$ is the projector onto the eigenspace of $W_j$ associated with the eigenvalue $\omega^{a_j}$. Moreover, there exists a projective measurement $\{{{\color{MidnightBlue} \bm{Q}}}^{a}\}_{a \in
{ \ensuremath{{\mathbb{F}}}_p}^k}$ satisfying analogous relations with respect to ${{\color{MidnightBlue} \bm{W}}}_j$ and which is consistent with $\{Q^{a}\}$: $$Q^{a} \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}\,\approx_{\eta'}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{Q}}}^{a}\;.$$
Write the eigendecompositions $$W_j = \sum_{a_j\in{ \ensuremath{{\mathbb{F}}}_p}} \omega^{a_j} W^{a_j}\;,\qquad {{\color{MidnightBlue} \bm{W}}}_j = \sum_{a_j\in{ \ensuremath{{\mathbb{F}}}_p}} \omega^{a_j} {{\color{MidnightBlue} \bm{W}}}^{a_j}\;,$$ where $\omega = e^{2i\pi/p}$. Using Claim \[claim:obs-meas-cons\] the consistency assumption on $W_j$ implies that $\{W_j^{a_j}\}$ and $\{{{\color{MidnightBlue} \bm{W}}}_j^{a_j}\}$ are also consistent projective measurements, with error $O(p^2\eta)$. We use the following general claim.
\[claim:com-obs\] Let $\{P^a\}$ and $\{Q^b\}$, and $\{{{\color{MidnightBlue} \bm{P}}}^a\}$ and $\{{{\color{MidnightBlue} \bm{Q}}}^b\}$, be projective measurements with outcomes $a,b\in { \ensuremath{{\mathbb{F}}}_p}$ for a prime $p$ and such that $P^a \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}\approx_\eta {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{P}}}^a$ and $Q^b \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}\approx_\eta {\ensuremath{\mathop{\rm Id}\nolimits}}\otimes {{\color{MidnightBlue} \bm{Q}}}^b$ for some $\eta \geq 0$. Let $P = \sum_a \omega^a P^a$ and $Q = \sum_b \omega^b Q^b$. Then $$\sum_{a,b} \big\|(P^aQ^b-Q^bP^a) \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle}\big\|^2 \,\approx_{\operatorname{poly}(\eta)} O\Big(p^2\, \|(PQ-QP)\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle}\|^2\Big)\;.$$
Expand $$\begin{aligned}
{\langle\psi|} (PQ-QP)(PQ-QP)^\dagger \otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle} &= 2 - \sum_{a,b,c,d} \omega^{a+b-(c+d)}{\langle\psi|}\big( P^a Q^b P^c Q^d + Q^a P^b Q^c P^d \big)\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle} \notag\\
&\approx_{\operatorname{poly}(\eta)} 2- {\langle\psi|}\Big(\sum_{a}\omega^a P^a \Big)\otimes \Big(\sum_{b,c,d}\omega^{b-(c+d)}{{\color{MidnightBlue} \bm{Q}}}^d {{\color{MidnightBlue} \bm{P}}}^c {{\color{MidnightBlue} \bm{Q}}}^b\Big){|\psi\rangle}\notag\\
&\hskip2cm - {\langle\psi|}\Big(\sum_{a}\omega^{-a} P^a \Big)\otimes \Big(\sum_{b,c,d}\omega^{-b+(c+d)}{{\color{MidnightBlue} \bm{Q}}}^b {{\color{MidnightBlue} \bm{P}}}^c {{\color{MidnightBlue} \bm{Q}}}^d\Big){|\psi\rangle} \;.\label{eq:com-obs-1}\end{aligned}$$ Let $\delta=\frac{1}{2} \|(PQ-QP)\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle}\|^2$. From we get the consistency relation $$\begin{aligned}
\frac{1}{2}\sum_{a,b,c,d}\, {\langle\psi|}\big( \omega^{(a-c)+(b-d)} Q^d P^c Q^b + \omega^{(c-a)+(d-b)}Q^b P^c Q^d\big) \otimes {{\color{MidnightBlue} \bm{P}}}^{a} {|\psi\rangle}\, \approx_{\operatorname{poly}(\eta)} \,1-O(\delta)\;.\end{aligned}$$ We now repeat this argument, but replacing the observable $Q$ with its power $(Q)^{\alpha}$ for $\alpha \in { \ensuremath{{\mathbb{F}}}_p}$ (not to be confused with the POVM element $Q^a$). Starting from $\|(P(Q)^\alpha-(Q)^\alpha P)\otimes {\ensuremath{\mathop{\rm Id}\nolimits}}{|\psi\rangle}\|^2 = O(p\delta)$ for any $\alpha\in{ \ensuremath{{\mathbb{F}}}_p}$ gives $$\begin{aligned}
\frac{1}{2}\sum_{a,b,c,d}\, {\langle\psi|}\big( \omega^{(a-c)+\alpha (b-d)} Q^d P^c Q^b + \omega^{(c-a)+\alpha (d-b)}Q^b P^c Q^d\big) \otimes {{\color{MidnightBlue} \bm{P}}}^{a} {|\psi\rangle}\, \approx_{\operatorname{poly}(\eta)} \,1-O(p\delta)\;.\end{aligned}$$ Averaging the relations over all $\alpha\in{ \ensuremath{{\mathbb{F}}}_p}$ and using that $(b-d) { \ensuremath{{\mathbb{F}}}_p}= { \ensuremath{{\mathbb{F}}}_p}$ if $b-d\neq 0$ and $\{0\}$ otherwise yields $$\begin{aligned}
\frac{1}{2}\sum_{a,b,c}\, {\langle\psi|} \big(\omega^{a-c} + \omega^{c-a}\big) Q^b
P^a Q^b \otimes {{\color{MidnightBlue} \bm{P}}}^{a} {|\psi\rangle} \approx_{\operatorname{poly}(\eta)} 1-O(p^2\delta)\;.\end{aligned}$$ Using $|\omega^{a-c}+\omega^{c-a}|\leq 2-\Omega(1/p)$ for $a\neq c$ proves the claim.
We define the POVM elements $\{Q^a\}$ as $$Q^a \,=\, W_k^{a_k} \cdots W_1^{a_1} \cdots W_k^{a_k}\;,$$ and define $\{{{\color{MidnightBlue} \bm{Q}}}^a\}$ analogously. The two relations in the lemma then follow from the definition, the fact that $\{W_j^{a_j}\}$ are projections, and Claim \[claim:com-obs\]; similarly, consistency of $\{Q^a\}$ follows.
Lemma \[lem:approx-commute-u\] allows us to combine any pair of approximately commuting observables $\hat{X}_u(x_{\pi})$ and $\hat{Z}_v(z_{\pi})$ into a single POVM $\{\hat{Q}_{xu,zv}^{a,b}\}$, with outcomes $(a,b)\in { \ensuremath{{\mathbb{F}}}_p}^2$, that simultaneously refines both observables. In the following lemma we show that all $\{\hat{Q}_{xu,zv}^{a,b}\}$, for $u,v\in {\ensuremath{\mathbb{F}_q}}$, can be pieced together into a single POVM $\{\hat{Q}_{xz}^{a,b}\}$ with outcomes $(a, b) \in {\ensuremath{\mathbb{F}_q}}^2$ which simultaneously refines all $2 q$ observables $\hat{X}_u(x_{\pi})$ and $\hat{Z}_v(z_{\pi})$. Note that this is not trivial because we wish to avoid any dependence on $q$ when combining the $2q$ approximately commuting observables. To achieve this, we rely on the linearity test from [@NV17].
\[lem:hatq-meas-two-outcome\] For every $x,z\in{\ensuremath{\mathbb{F}_q}}^m$ there are projective measurements $\{\hat{Q}_{xz}^{a,b}\}_{a,b\in {\ensuremath{\mathbb{F}_q}}^2}$ and $\{{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}^{a,b}\}_{a,b\in {\ensuremath{\mathbb{F}_q}}^2}$ defined on ${{\textsf{AA'}}}$ and ${{\textsf{BA''}}}$ respectively such that the following hold, for some $\delta_Q = \operatorname{poly}(p)\cdot\operatorname{poly}(\delta_{\hat{M}})$:
1. The $\hat{Q}_{xz}$ are consistent with the ${{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}$: $$\sum_{a,b} \big(\hat{Q}_{xz}^{a,b}\big)_{{{\textsf{AA'}}}} \otimes \big({{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}^{a,b}\big)_{{{\textsf{BA''}}}}\, \approx_{\delta_Q} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,$$ on average over uniformly random $x,z\in{\ensuremath{\mathbb{F}_q}}^m$.
2. The $\hat{Q}_{xz\alpha\beta}$ are consistent with ${{\color{MidnightBlue} \bm{\hat{M}}}}_{X, x}$ and ${{\color{MidnightBlue} \bm{\hat{M}}}}_{Z, z}$: $$\sum_{a,b} \big(\hat{Q}_{xz}^{a,b}\big)_{{{\textsf{AA'}}}} \otimes \Big(
{{\color{MidnightBlue} \bm{\hat{M}}}}_{X, x}^a
{{\color{MidnightBlue} \bm{\hat{M}}}}_{Z, z}^{b}\Big)_{{{\textsf{BA''}}}} \approx_{\delta_Q} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,$$ on average over uniformly random $x,z\in{\ensuremath{\mathbb{F}_q}}^m$.
For every $x,z\in{\ensuremath{\mathbb{F}_q}}^m$ and $u,v\in {\ensuremath{\mathbb{F}_q}}$ we let $\{ \hat{Q}_{xu,zv}^{a,b} \}_{a,b\in{ \ensuremath{{\mathbb{F}}}_p}}$ and $\{{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xu,zv}^{a,b}\}_{a,b\in { \ensuremath{{\mathbb{F}}}_p}}$ be the projective measurements guaranteed by Lemma \[lem:approx-commute-u\] when the observables $P$ and $Q$ are chosen as $\hat{X}_u(x_{\pi})$ and $\hat{Z}_v(z_{\pi})$ respectively; the assumptions of the lemma are satisfied by and . We also define, for $c\in { \ensuremath{{\mathbb{F}}}_p}$, $$\label{eq:def-hatq-c}
\hat{Q}_{xu,zv}^{c} = \sum_{a,b:\, a + b = c} \hat{Q}_{xu,zv}^{a,b}\;,$$ and similarly define associated measurements $\{{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xu,zv}^{c}\}_{c\in{ \ensuremath{{\mathbb{F}}}_p}}$. In the following claim we identify ${\ensuremath{\mathbb{F}_q}}$ with the vector space ${ \ensuremath{{\mathbb{F}}}_p}^t$ to interpret this family of measurements as a strategy in the linearity test over ${ \ensuremath{{\mathbb{F}}}_p}^{2t}$, with the queries specified by $(u,v)\in{ \ensuremath{{\mathbb{F}}}_p}^{2t}$ and the answers $c\in{ \ensuremath{{\mathbb{F}}}_p}$.
\[claim:q-lin\] On average over $x,z\in{\ensuremath{\mathbb{F}_q}}^m$ the family of projective measurements $\{ \hat{Q}_{xu,zv}^{c} \}_{c\in{ \ensuremath{{\mathbb{F}}}_p}}$, indexed by $(u,v)\in { \ensuremath{{\mathbb{F}}}_p}^{2t}$, together with $\{{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xu,zv}^{c} \}_{c\in{ \ensuremath{{\mathbb{F}}}_p}}$, induces a strategy with success probability at least $1-\operatorname{poly}(\delta_{\hat{M}})$ in the two-prover linearity test from [@NV17 Section 3].
Consistency between $\hat{Q}_{xu,zv}^{c}$ and $ {{\color{MidnightBlue} \bm{\hat{Q}}}}_{xu,zv}^{c}$ is clear by the definition and Lemma \[lem:approx-commute-u\]. We need to verify (approximate) linearity. First note that from the definition of the observables $\hat{W}_u$ in , using that the underlying measurements are projective it follows that for $W\in\{X,Z\}$ the family of observables $\{\hat{W}_u(w_{\pi})\}_{u\in{ \ensuremath{{\mathbb{F}}}_p}^t}$ is linear, in the sense that for any $u,u'\in{ \ensuremath{{\mathbb{F}}}_p}^t$, $\hat{W}_u(w_{\pi})\hat{W}_{u'}(w_{\pi})=\hat{W}_{u+u'}(w_{\pi})$, where $u+u'$ is addition as elements of the vector space ${ \ensuremath{{\mathbb{F}}}_p}^t$. Using Lemma \[lem:approx-commute-u\] it follows that $\{ \hat{Q}_{xu,zv}^{c} \}$ is approximately linear in both $u$ and $v$, i.e. $${\ensuremath{\mathop{\textsc{E}}_{u,u',v,v'}}} \sum_{c,c'} \, \hat{Q}_{xu,zv}^{c} \hat{Q}_{xu',zv'}^{c'} \otimes {{\color{MidnightBlue} \bm{\hat{Q}}}}_{x(u+u'),z(v+v')}^{c+c'} \,\approx_{\operatorname{poly}(p,\delta_{\hat{M}})}\, {\ensuremath{\mathop{\rm Id}\nolimits}}\;.$$ Linearity for $\{ \hat{Q}_{xu,zv}^{c} \}$ then follows directly from the definition .
Applying [@NV17 Theorem 10] for every $x,z$ we find a single POVM $\{ \hat{Q}_{x,z}^{a,b} \}_{a,b\in{\ensuremath{\mathbb{F}_q}}^{2}}$ such that, on expectation over $u,v\in { \ensuremath{{\mathbb{F}}}_p}^t$, $\hat{Q}_{xu,zv}^{e} \approx \sum_{a,b:\, {\mbox{\rm tr}}(a \cdot u +b \cdot v) = e} \hat{Q}_{x,z}^{a,b}$.[^9] Similarly, there exists a $\{ {{\color{MidnightBlue} \bm{\hat{Q}}}}_{x,z}^{a,b} \}_{a,b\in{\ensuremath{\mathbb{F}_q}}^{2}}$ that is consistent with $\{ \hat{Q}_{x,z}^{a,b} \}_{a,b\in{\ensuremath{\mathbb{F}_q}}^{2}}$. The first item in the lemma now follows immediately from the consistency guarantees of the linearity test.
For the second item, note that from Lemma \[lem:approx-commute-u\] and the guarantees of the linearity test, it follows that $\hat{Q}_{x,z}^{(a,b)}$ is approximately consistent with a randomly chosen product of ${{\color{MidnightBlue} \bm{\hat{X}_u}}}$ and ${{\color{MidnightBlue} \bm{\hat{Z}_v}}}$ POVM elements. $$\begin{aligned}
{\ensuremath{\mathop{\rm Id}\nolimits}}&\approx_{\delta_{\hat{M}}'} {\ensuremath{\mathop{\textsc{E}}_{u,v}}} \sum_{(a,b)} (\hat{Q}_{x,z}^{a,b})_{{{\textsf{AA'}}}} {\otimes}\Big( \sum_{c,d : {\mbox{\rm tr}}(au + bv) = c +d} {{\color{MidnightBlue} \bm{\hat{X}}}}_u^{c} (x_\pi) {{\color{MidnightBlue} \bm{\hat{Z}}}}_v^{d} (z_\pi) \Big)_{{{\textsf{BA''}}}} \\
&= {\ensuremath{\mathop{\textsc{E}}_{u,v}}} \sum_{(a,b)} (\hat{Q}_{x,z}^{a,b})_{{{\textsf{AA'}}}} {\otimes}\Big( \sum_{a', b': {\mbox{\rm tr}}(au + bv) = {\mbox{\rm tr}}(a'u + b'v)} {{\color{MidnightBlue} \bm{\hat{M}}}}^{a'}_{X,x} {{\color{MidnightBlue} \bm{\hat{M}}}}^{b'}_{Z,z} \Big)_{{{\textsf{BA''}}}} \\
&= \sum_{(a,b)} (\hat{Q}_{x,z}^{a,b})_{{{\textsf{AA'}}}} {\otimes}\Big( {\ensuremath{\mathop{\textsc{E}}_{u,v}}} \sum_{a', b'} 1_{{\mbox{\rm tr}}(au + bv) = {\mbox{\rm tr}}(a'u + b' v)} {{\color{MidnightBlue} \bm{\hat{M}}}}^{a'}_{X,x} {{\color{MidnightBlue} \bm{\hat{M}}}}^{b'}_{Z,z} \Big)_{{{\textsf{BA''}}}} \\
&= \sum_{(a,b)} (\hat{Q}_{x,z}^{a,b})_{{{\textsf{AA'}}}} {\otimes}\Big( {{\color{MidnightBlue} \bm{\hat{M}}}}^{a}_{X,x} {{\color{MidnightBlue} \bm{\hat{M}}}}^{b}_{Z,z} + \frac{1}{p} ({\ensuremath{\mathop{\rm Id}\nolimits}}- {{\color{MidnightBlue} \bm{\hat{M}}}}^{a}_{X,x} {{\color{MidnightBlue} \bm{\hat{M}}}}^{b}_{Z,z}) \Big)_{{{\textsf{BA''}}}} \\
&= \frac{1}{p} {\ensuremath{\mathop{\rm Id}\nolimits}}+ \Big(1 - \frac{1}{p}\Big) \sum_{(a,b)} (\hat{Q}_{x,z}^{a,b})_{{{\textsf{AA'}}}} {\otimes}\Big({{\color{MidnightBlue} \bm{\hat{M}}}}^{a}_{X,x} {{\color{MidnightBlue} \bm{\hat{M}}}}^{b}_{Z,z} \Big)_{{{\textsf{BA''}}}}\;, \end{aligned}$$ where in the second to last equation we used the fact that any two distinct linear functions from ${ \ensuremath{{\mathbb{F}}}_p}^{2t}$ to ${ \ensuremath{{\mathbb{F}}}_p}$ agree with probability $1/p$. Moving the term proportional to ${\ensuremath{\mathop{\rm Id}\nolimits}}$ to the left hand side, we obtain $${\ensuremath{\mathop{\rm Id}\nolimits}}\approx_{O(\delta'_{\hat{M}})} \sum_{a,b} (\hat{Q}_{x,z}^{a,b})_{{{\textsf{AA'}}}} {\otimes}\Big({{\color{MidnightBlue} \bm{\hat{M}}}}^{a}_{X,x} {{\color{MidnightBlue} \bm{\hat{M}}}}^{b}_{Z,z} \Big)_{{{\textsf{BA''}}}}\;,$$ which is the desired consistency expression.
For a subsequent application of the low-degree test it is more convenient to have a POVM which takes values $c\in{\ensuremath{\mathbb{F}_q}}$, rather than $(a,b)\in{\ensuremath{\mathbb{F}_q}}^2$. In order not to lose information, the following lemma re-arranges $\hat{Q}_{xz}^{a,b}$ into a family of POVMs $\{\hat{Q}_{xz\alpha\beta}^c = \sum_{a,b:\,\alpha a + \beta b =c} \hat{Q}_{xz}^{a,b}\}$, and shows that this family obeys similar consistency properties.
\[lem:hatq-meas\] For every $\alpha,\beta\in{\ensuremath{\mathbb{F}_q}}$ and $x,z\in{\ensuremath{\mathbb{F}_q}}^m$ there are projective measurements $\{\hat{Q}_{xz\alpha\beta}^{c}\}_{c\in {\ensuremath{\mathbb{F}_q}}}$ and $\{{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz\alpha\beta}^{c}\}_{c\in {\ensuremath{\mathbb{F}_q}}}$ defined on ${{\textsf{AA'}}}$ and ${{\textsf{BA''}}}$ respectively such that the following hold, for $\delta_Q$ as in Lemma \[lem:hatq-meas-two-outcome\]:
1. The $\hat{Q}_{xz\alpha\beta}$ are consistent with the ${{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz\alpha\beta}$: $$\sum_c \big(\hat{Q}_{xz\alpha\beta}^c\big)_{{{\textsf{AA'}}}} \otimes \big({{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz\alpha\beta}^c\big)_{{{\textsf{BA''}}}}\, \approx_{\delta_Q} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,$$ on average over uniformly random $\alpha,\beta\in{\ensuremath{\mathbb{F}_q}}$ and $x,z\in{\ensuremath{\mathbb{F}_q}}^m$.
2. The $\hat{Q}_{xz\alpha\beta}$ are consistent with ${{\color{MidnightBlue} \bm{\hat{M}}}}_{X, x}$ and ${{\color{MidnightBlue} \bm{\hat{M}}}}_{Z, z}$: $$\sum_c \big(\hat{Q}_{xz\alpha\beta}^c\big)_{{{\textsf{AA'}}}} \otimes \Big(
\sum_{\substack{a, a':\\ \alpha a + \beta a'=c}} {{\color{MidnightBlue} \bm{\hat{M}}}}_{X, x}^a
{{\color{MidnightBlue} \bm{\hat{M}}}}_{Z, z}^{a'}\Big)_{{{\textsf{BA''}}}} \approx_{\delta_Q} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,$$ on average over uniformly random $\alpha,\beta\in{\ensuremath{\mathbb{F}_q}}$ and $x,z\in{\ensuremath{\mathbb{F}_q}}^m$.
Let $\hat{Q}_{x,z}^{a,b}$ and ${{\color{MidnightBlue} \bm{\hat{Q}}}}_{x,z}^{a,b}$ be the families of POVMs guaranteed by Lemma \[lem:hatq-meas-two-outcome\]. We “collapse” the $\hat{Q}$ (and analogously, the ${{\color{MidnightBlue} \bm{\hat{Q}}}}$) into a family of measurements with outcomes in ${\ensuremath{\mathbb{F}_q}}$ by defining, for every $\alpha,\beta\in{\ensuremath{\mathbb{F}_q}}$, $x,z\in{\ensuremath{\mathbb{F}_q}}^m$ and $c\in{\ensuremath{\mathbb{F}_q}}$, $$\label{eq:hat-q-def}
\hat{Q}_{xz\alpha\beta}^{c} \,=\,\sum_{a,b:\,\alpha a + \beta b = c} \hat{Q}_{x,z}^{a,b}\;.$$ With this definition, both items in the claim follows from the definition and corresponding items of Lemma \[lem:hatq-meas-two-outcome\].
The next step in the proof of Lemma \[lem:hat-s\] is to use the family of measurements $\{\hat{Q}_{xz\alpha\beta}^c\}$ to devise a strategy for the provers in the classical $(2m+2)$-variable degree-$(d+1)$ test. Towards this the following claim establishes the existence of appropriate subspace measurements.
\[claim:lines\] For every dimension-$2$ subspace $s \subseteq {\ensuremath{\mathbb{F}_q}}^{2m+2}$ there exists a POVM $\{\hat{Q}_{s}^r\}_r$, with outcomes $r\in \deg_{d+1}(s)$, such that, on average over a uniformly random $s$ and $(x,z,\alpha,\beta)\in s$, $$\sum_{r,c:\,r(x,z,\alpha,\beta)\neq c}\, {\langle\psi|} \hat{Q}_s^r \otimes
{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz\alpha\beta}^c {|\psi\rangle} \,=\, \operatorname{poly}(\delta_Q) +
O\big(d/q\big)\;.$$ Likewise, there exists $\{{{\color{MidnightBlue} \bm{\hat{Q}}}}_{s}^r\}$ that are consistent with $\{\hat{Q}_{xz\alpha\beta}^c\}$, on average.
The argument is entirely symmetric between $\hat{Q}$ and ${{\color{MidnightBlue} \bm{\hat{Q}}}}$. We give the construction for $\hat{Q}$.
Let $s\subset {\ensuremath{\mathbb{F}_q}}^{2m+2}$ be a two-dimensional linear subspace spanned by two vectors $y_1 = (x_1,z_1,\alpha_1,\beta_1),y_2=(x_2,z_2,\alpha_2,\beta_2)\in{\ensuremath{\mathbb{F}_q}}^{2m+2}$. Let $s',s'' \subset {\ensuremath{\mathbb{F}_q}}^m$ be the (at most) two-dimensional subspaces spanned by $\{x_1,x_2\}$ and $\{z_1,z_2\}$ respectively. Any point $(\lambda, \mu)$ in the subspace $s$ corresponds to a point in the full space $$\#(\lambda, \mu) = (\lambda x_1 + \mu x_2, \lambda z_1 + \mu z_2, \lambda \alpha_1 + \mu \alpha_2, \lambda \beta_1 + \mu \beta_2).$$ Let $g(x,y, \alpha, \beta) = \alpha p(x) + \beta q(y)$ be a $2m + 2$-variate polynomial. Its restriction $r$ to the subspace $s$ takes the values $$r(\lambda, \mu) = g(\#(\lambda, \mu)) = (\lambda \alpha_1 + \mu \alpha_2) p(\lambda x_1 + \mu x_2) + (\lambda \beta_1 + \mu \beta_2) q(\lambda z_1 + \mu z_2).$$ From this expression, we see that $r$ can be evaluated if we have access to the restrictions of $p$ to $s'$ and $q$ to $s''$, respectively. We will now construct a measurement that, given $s',
s''$ as input, jointly measures $p$ and $q$ in a manner that is consistent with the joint points measurement $\{\hat{Q}_{xz}^{a,b}\}$ guaranteed by Lemma \[lem:hatq-meas-two-outcome\]. Define $$T_{s',s''}^{p,q} = \hat{M}_{X,s'}^{p} \hat{M}_{Z,s''}^{q}
\hat{M}_{X,s'}^{p}. \label{eq:joint_subspace}$$ The collection $\{T_{s', s''}^{p,q}\}_{p,q}$ forms a valid POVM. Its inconsistency with $\hat{Q}_{xz}^{a,b}$ is: $$\begin{aligned}
&{\ensuremath{\mathop{\textsc{E}}_{s',s''}}} {\ensuremath{\mathop{\textsc{E}}_{x \in s', z \in s''}}} \sum_{\substack{a,b,p,q: \\ (a,b) \neq (f(x), q(z))}}
{\langle\psi|} T_{s', s''}^{p,q} {\otimes}{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}^{a,b} {|\psi\rangle}
\nonumber \\
&\qquad={\ensuremath{\mathop{\textsc{E}}_{s', s''}}} {\ensuremath{\mathop{\textsc{E}}_{x \in s', z \in s''}}} \sum_{\substack{a,b,p,q: \\ (a,b) \neq
(p(x), q(z))}}{\langle\psi|} \hat{M}_{X,s'}^{p} \hat{M}_{Z,s''}^{q} \hat{M}_{X,s'}^{p} {\otimes}{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}^{a,b} {|\psi\rangle} \nonumber \\
&\qquad= 1 - {\ensuremath{\mathop{\textsc{E}}_{s', s''}}} \sum_{p, q} {\ensuremath{\mathop{\textsc{E}}_{x \in s', z \in s''}}} {\langle\psi|} \hat{M}_{X,s'}^{p} \hat{M}_{Z,s''}^{q} \hat{M}_{X,s'}^{p} {\otimes}{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}^{p(x),q(z)} {|\psi\rangle} \nonumber\\
&\qquad \leq 1 - {\ensuremath{\mathop{\textsc{E}}_{s', s''}}} \sum_{p, q, p'} {\ensuremath{\mathop{\textsc{E}}_{x \in s', z \in s''}}} {\langle\psi|} \hat{M}_{X,s'}^{p} \hat{M}_{Z,s''}^{q} \hat{M}_{X,s'}^{p'} {\otimes}{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}^{p(x),q(z)} {|\psi\rangle} + \operatorname{poly}(\delta_Q) + O(d/q)
\nonumber \\
&\qquad = 1 - {\ensuremath{\mathop{\textsc{E}}_{s', s''}}} \sum_{p, q} {\ensuremath{\mathop{\textsc{E}}_{x \in s', z \in s''}}} {\langle\psi|} \hat{M}_{X,s'}^{p} \hat{M}_{Z,s''}^{q} {\otimes}{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}^{p(x),q(z)} {|\psi\rangle} + \operatorname{poly}(\delta_Q) + O(d/q) \nonumber\\
&\qquad = {\ensuremath{\mathop{\textsc{E}}_{s', s''}}} \sum_{p, q} \sum_{(a,b) \neq (p(x), q(z))} {\ensuremath{\mathop{\textsc{E}}_{x \in s', z \in s''}}} {\langle\psi|} \hat{M}_{X,s'}^{p} \hat{M}_{Z,s''}^{q} {\otimes}{{\color{MidnightBlue} \bm{\hat{Q}}}}_{xz}^{a,b} {|\psi\rangle} + \operatorname{poly}(\delta_Q) + O(d/q) \nonumber \\
&\qquad = \operatorname{poly}(\delta_Q) + O(d/q)\;. \label{eq:joint_subspace_points}\end{aligned}$$ Here, in the first equation we used . In the third equation we used item 2. of Lemma \[lem:hatq-meas\] to bound the error terms where $p'(x) \neq p(x)$ by $\operatorname{poly}(\delta_Q)$, and used Lemma \[lem:sz\] to bound the probability that $p(x) = p'(x)$ on a randomly chosen $x$ for distinct polynomials $p, p'$ by $O(d/q)$. Finally, in the sixth equation, we again used item 2. of Lemma \[lem:hatq-meas\].
Finally, we will use this to construct a joint subspace measurement $\{\hat{Q}_{s}^r\}_r$: $$\hat{Q}_{s}^r = \sum_{p, q: r = (\lambda \alpha_1 + \mu \alpha_2) p(\lambda
x_1 + \mu x_2) + (\lambda \beta_1 + \mu \beta_2) q(\lambda z_1 + \mu z_2)}
T_{s', s''}^{p,q}\;. \label{eq:hatq_subspace_def}$$ The consistency of $\hat{Q}_{s}^r$ with the re-arranged points measurement $\hat{Q}_{xz\alpha \beta}^{c}$ now follows directly from together with the consistency of $T_{s', s''}^{p,q}$ with $\hat{Q}_{xz}^{a,b}$ established in .
Claim \[claim:lines\] shows that the families of measurements $\{\hat{Q}_{s}^r\}_r$ and $\{\hat{Q}_{xz\alpha\beta}^c\}$ induce a strategy with success probability $1-{\varepsilon}'$ in the classical low-degree test, for some ${\varepsilon}'
= \operatorname{poly}(p) \cdot \operatorname{poly}(\delta_{\hat{M}})+O(d/q)$ (to obtain this error bound, recall that $\delta_Q = \operatorname{poly}(p) \cdot \operatorname{poly}(\delta_{\hat{M}})$). This allows us to apply Theorem \[thm:ml\] and conclude the proof of Lemma \[lem:hat-s\].
The proof is symmetric under exchanging $\hat{S}, \hat{X}, \hat{Z}$ with ${{\color{MidnightBlue} \bm{\hat{S}}}}, {{\color{MidnightBlue} \bm{\hat{X}}}}, {{\color{MidnightBlue} \bm{\hat{Z}}}}$, so we only present one direction. Claim \[claim:lines\] establishes the existence of a strategy $\{\hat{Q}_{s}^r,{{\color{MidnightBlue} \bm{\hat{Q}}}}_s^r\}$ which succeeds with probability $1-{\varepsilon}'$, for some ${\varepsilon}' = \operatorname{poly}(t)\cdot\operatorname{poly}(\delta_{\hat{M}})+O(d/q)$, in the classical low-degree test for $m'=2m+2$ and $d'=d+1$. Increasing the error artificially by replacing $O(d/q)$ with $md/q^{1/c}$ if needed, the assumption $q \geq (dm/{\varepsilon}')^c $ from Theorem \[thm:ml\] is satisfied. The theorem yields a POVM $\{\hat{S}^g\}$ with outcomes in the set of polynomials $g:{\ensuremath{\mathbb{F}_q}}^{2m+2}\to{\ensuremath{\mathbb{F}_q}}$ of degree at most $d+1$ that is self-consistent and consistent with the $\{\hat{Q}_{s}^r\}$ and the family of POVM $\{\hat{Q}_{xz\alpha\beta}^c\}$ defined in Lemma \[lem:hatq-meas\]. Moreover, applying Naimark’s dilation theorem (taking advantage of the assumption that the state ${|\hat{\psi}\rangle}$ defined in , by definition, contains sufficiently many ancilla ${|0\rangle}$ qubits), we can assume without loss of generality that $\{\hat{S}^g\}$ is a projective measurement.
In general there is no a priori guarantee that $g$ takes the form $g=\alpha g_1 + \beta g_2$ for $g_1$ (resp. $g_2$) a degree-$d$ polynomial in $x$ (resp. $z$) only. Let $\mathcal{G}$ denote the latter set of polynomials. We first show that the probability of obtaining an outcome $g$ that does not fall within the set $\mathcal{G}'$ of $(2m+2)$-variate polynomials that are linear in $\alpha$ and $\beta$ is small. Using consistency of $\{\hat{S}^g\}$ with the $\{\hat{Q}_{xz\alpha\beta}^c\}$, which follows from item 1. in Theorem \[thm:ml\], $$\begin{aligned}
\sum_{g \notin \mathcal{G}'} \hat{S}^g
&\approx_{\operatorname{poly}({\varepsilon}')} \sum_{g \notin \mathcal{G}'} {\ensuremath{\mathop{\textsc{E}}_{(x,z,\alpha,\beta)\in{\ensuremath{\mathbb{F}_q}}^{2m+2}}}} \hat{S}^g
{\otimes}{{\color{MidnightBlue} \bm{\hat{Q}}}}^{g(x,z,\alpha,\beta)}_{xz\alpha\beta} \notag\\
&\approx_{\operatorname{poly}(\delta_Q)} \sum_{g \notin \mathcal{G}'} \sum_{a,b} {\ensuremath{\mathop{\textsc{E}}_{x,z}}} \Big( {\ensuremath{\mathop{\textsc{E}}_{{\alpha},{\beta}}}} 1_{\alpha a + \beta b = g(x,z,\alpha,\beta)} \Big) \hat{S}^g
{\otimes}{{\color{MidnightBlue} \bm{\hat{M}}}}_{X, x}^{a} {{\color{MidnightBlue} \bm{\hat{M}}}}_{Z,z}^b\;,\label{eq:hats-1}
\end{aligned}$$ where the second approximation follows from item 2. in Lemma \[lem:hatq-meas\]. If $g$ contains a term of degree higher than $1$ in $\alpha$ or $\beta$ the expectation inside the brackets in is at most $O(d/q)$. This bounds the contribution of outcomes $g\notin \mathcal{G}'$. Next we show that outcomes in $\mathcal{G}'\backslash \mathcal{G}$ are unlikely, i.e. that $g_1$ should depend solely on $x$ and $g_2$ solely on $z$. (Note that either polynomial has degree at most $d$, since $g$ itself has degree at most $d+1$.) Towards this, starting from , write $$\begin{aligned}
\sum_{g = \alpha g_1 + \beta g_2\in \mathcal{G}'} \hat{S}^g
&\approx_{\operatorname{poly}({\varepsilon}',\delta_Q)} \sum_{g = \alpha g_1 + \beta g_2} \hat{S}^g
{\otimes}{\ensuremath{\mathop{\textsc{E}}_{z}}}\,\sum_b \,\Big( \sum_a {\ensuremath{\mathop{\textsc{E}}_{x}}}
1_{a=g_1(x,z)} 1_{b = g_2(x,z)} {{\color{MidnightBlue} \bm{\hat{M}}}}_{X, x}^{a}
\Big){{\color{MidnightBlue} \bm{\hat{M}}}}_{Z, z}^b\\
&\approx_{\operatorname{poly}(\delta_Q)} \sum_{g = \alpha g_1 + \beta g_2} \hat{S}^g {\otimes}{\ensuremath{\mathop{\textsc{E}}_{ z}}} \, \sum_b \, \sum_a \, {\ensuremath{\mathop{\textsc{E}}_{x}}} 1_{a=g_1(x,z)}
1_{b=g_2(x,z)} {{\color{MidnightBlue} \bm{\hat{Q}}}}^{a,b}_{x,z}\\
&\leq \sum_{g = \alpha g_1 + \beta g_2} \hat{S}^g {\otimes}{\ensuremath{\mathop{\textsc{E}}_{z}}} \, \sum_b \, {\ensuremath{\mathop{\textsc{E}}_{x}}}
1_{b=g_2(x,z)} \Big( \sum_a {{\color{MidnightBlue} \bm{\hat{Q}}}}^{a,b}_{x,z} \Big)\\
&\approx_{\operatorname{poly}(\delta_Q)} \sum_{g = \alpha g_1 + \beta g_2} \hat{S}^g
{\otimes}{\ensuremath{\mathop{\textsc{E}}_{z}}}\,\sum_b \,\Big( {\ensuremath{\mathop{\textsc{E}}_{x}}} 1_{b = g_2(x,z)}
\Big){{\color{MidnightBlue} \bm{\hat{M}}}}_{Z, z}^b\;,
\end{aligned}$$ where the inequality removes the first indicator by using positivity. If $g_2(x,z)$ depended on $x$, the indicator $1_{b=g_2(x,z)}$ appearing within the expression inside the brackets would have probability at most $O(d/q)$ to be satisfied, for a random choice of $b$ and $z$, on average over $x$. Thus outcomes $g=\alpha g_1 + \beta g_2$ for which $g_2$ depends on $x$ have small probability; similarly for $z$. The relation then follows directly from the above, Claim \[claim:hat-m-cons\], and the second item in Lemma \[lem:hatq-meas\].
Generalized $X$ and $Z$ observables {#sec:tildex}
-----------------------------------
In this section we complete the last main step of the proof. For $W\in\{X,Z\}$, $w\in{\ensuremath{\mathbb{F}_q}}^m$ and $u\in {\ensuremath{\mathbb{F}_q}}$ let ${W}_u(w_{\pi})$ be the observable defined in Claim \[claim:strategies\], and $\hat{W}_u(w_{\pi})$ defined in Lemma \[lem:hats\]. For convenience we consider a “basis” for these sets of observables, as $u$ ranges over ${ \ensuremath{{\mathbb{F}}}_p}^t$, by defining $$\label{eq:def-wl}
\forall \ell\in\{1,\ldots,t\}\;,\qquad {W}_\ell(w_{\pi})\,=\, {W}_{b_\ell}(w_{\pi})\;\qquad\text{and}\qquad \hat{W}_\ell(w_{\pi})\,=\, \hat{W}_{b_\ell}(w_{\pi})\;,$$ where $\{b_1,\ldots,b_t\}$ is a self-dual basis for ${\ensuremath{\mathbb{F}_q}}$ over ${ \ensuremath{{\mathbb{F}}}_p}$. The corresponding “true” Paulis $\sigma_{W, \ell}(w_{\pi})$ were defined in . Similarly define ${{\color{MidnightBlue} \bm{W}}}_\ell(w_{\pi})$ and ${{\color{MidnightBlue} \bm{\hat{W}}}}_\ell(w_{\pi})$. In this section we use these observables to construct a family of observables $\tilde{W}_\ell(a)$, for $a\in{\ensuremath{\mathbb{F}_q}}^n$, which satisfy appropriate self-consistency and twisted commutation relations, as expressed in the following lemma.
\[lem:xz-lowdeg\] Let $m,d,q=p^t$ be as in Lemma \[lem:soundness\], ${|\hat{\psi}\rangle}$ as in Lemma \[lem:hats\], $W_\ell(w_{\pi})$ and ${{\color{MidnightBlue} \bm{W}}}_\ell(w_{\pi})$, $\hat{W}_\ell(w_{\pi})$ and ${{\color{MidnightBlue} \bm{\hat{W}}}}_\ell(w_{\pi})$ as in , and $\{\hat{S}^{g_1,g_2}\}$, $\{{{\color{MidnightBlue} \bm{\hat{S}}}}^{g_1,g_2}\}$, and $\delta_S$ as in Lemma \[lem:hat-s\].
For every $a\in{\ensuremath{\mathbb{F}_q}}^n$ and $\ell \in \{1, \dots, t\}$ there are observables $\tilde{X}_\ell(a)$ acting on ${{\textsf{AA'A''}}}$ and ${{\color{MidnightBlue} \bm{\tilde{X}}}}_\ell(a)$ acting on ${{\textsf{BB'B''}}}$, and for every $b\in {\ensuremath{\mathbb{F}_q}}^n$ and $\ell' \in \{1, \dots , t\}$ observables $\tilde{Z}_{\ell'}(b)$ acting on ${{\textsf{AA'A''}}}$ and ${{\color{MidnightBlue} \bm{\tilde{Z}}}}_\ell(b)$ acting on ${{\textsf{BB'B''}}}$, such that the following properties hold for some $\delta_W = \operatorname{poly}(\delta_S)+\operatorname{poly}(d/q)$:
1. The families of observables $\{\tilde{X}_\ell(a),\tilde{Z}_{\ell'}(b),\,
a,b\in{\ensuremath{\mathbb{F}_q}}^n,\,\ell, \ell' \in \{1, \dots, t\} \}$ and $\{{{\color{MidnightBlue} \bm{\tilde{X}}}}_\ell(a), {{\color{MidnightBlue} \bm{\tilde{Z}}}}_{\ell'} (b), \,
a,b, \in {\ensuremath{\mathbb{F}_q}}^n,\, \ell, \ell' \in \{1, \dots, t\} \}$ exactly satisfy the same algebraic relations as the Pauli observables $\sigma_{W, \ell}$ over ${\ensuremath{\mathbb{F}_q}}$ defined in ;
2. On average over $x,z\in{\ensuremath{\mathbb{F}_q}}^m$, and for all $\ell, \ell' \in \{1, \dots, t\}$, $$\tilde{X}_{\ell}^e(x_{\pi}) \approx_{\delta_W} X_{\ell}^e(x_{\pi})\qquad
\text{and}\qquad \tilde{Z}_{\ell'}^e(z_{\pi}) \approx_{\delta_W} Z_{\ell'}^e(z_{\pi})\;,$$ and the analogous relations between ${{\color{MidnightBlue} \bm{\tilde{X}}}}_\ell,
{{\color{MidnightBlue} \bm{\tilde{Z}}}}_{\ell'}$ and ${{\color{MidnightBlue} \bm{X}}}_\ell, {{\color{MidnightBlue} \bm{Z}}}_{\ell'}$ hold;
3. The $\tilde{X}_{\ell}(a)$, $\tilde{Z}_{\ell'}(b)$ are approximately consistent with the ${{\color{MidnightBlue} \bm{\tilde{X}}}}_{\ell}(a), {{\color{MidnightBlue} \bm{\tilde{Z}}}}_{\ell'}(b)$: in expectation over $a,b\in{\ensuremath{\mathbb{F}_q}}^n$, $$\sum_{e \in { \ensuremath{{\mathbb{F}}}_p}} \,\tilde{X}_\ell^e(a) \otimes {{\color{MidnightBlue} \bm{\tilde{X}}}}_\ell^e(a)
\approx_{\delta_W} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,\qquad \sum_{e \in { \ensuremath{{\mathbb{F}}}_p}} \,\tilde{Z}_{\ell'}^e(b) \otimes {{\color{MidnightBlue} \bm{\tilde{Z}}}}_{\ell'}^e(b) \approx_{\delta_W} {\ensuremath{\mathop{\rm Id}\nolimits}}\;.$$
For any $a,b\in{\ensuremath{\mathbb{F}_q}}^n$ and $\ell \in \{1, \dots, t\}$ define $$\label{eq:def-tilde-xz}
\begin{aligned}
\tilde{X}_\ell(a)_{{\textsf{AA'A''}}} &= \Big(\sum_{g_1,g_2} \,\omega^{{\mbox{\rm tr}}( b_\ell(g_1\cdot a))}\,
\hat{S}^{g_1,g_2}_{{\textsf{AA'}}} \Big) \otimes \sigma_{X, \ell}(a)_{{\textsf{A''}}}^\dag\;,\quad
\tilde{Z}_\ell(b)_{{\textsf{AA'A''}}} = \Big(\sum_{g_1,g_2} \,\omega^{{\mbox{\rm tr}}(b_\ell(g_2\cdot
b))}
\,\hat{S}^{g_1,g_2}_{{\textsf{AA'}}}\Big) \otimes \sigma_{Z, \ell}(b)_{{\textsf{A''}}}^\dag\;, \\
{{\color{MidnightBlue} \bm{\tilde{X}}}}_{\ell}(a)_{{\textsf{BB'B''}}} &= \Big(\sum_{g_1,g_2} \,\omega^{{\mbox{\rm tr}}(b_\ell(g_1\cdot a))}\,
{{\color{MidnightBlue} \bm{\hat{S}}}}^{g_1,g_2}_{{{\textsf{BB'}}}} \Big) \otimes {{\color{MidnightBlue} \bm{\sigma}}}_{X,\ell}(a)_{{\textsf{B''}}}^\dag\;,\quad
{{\color{MidnightBlue} \bm{\tilde{Z}}}}_{\ell}(b)_{{{\textsf{BB'B''}}}} = \Big(\sum_{g_1,g_2} \,\omega^{{\mbox{\rm tr}}(b_\ell(g_2\cdot b))}
\,{{\color{MidnightBlue} \bm{\hat{S}}}}^{g_1,g_2}_{{\textsf{BB'}}} \Big) \otimes {{\color{MidnightBlue} \bm{\sigma}}}_{Z, \ell}(b)_{{\textsf{B''}}}^\dag\;,
\end{aligned}$$ where ${{\color{MidnightBlue} \bm{\hat{S}}}}^{g_1,g_2}_{{{\textsf{BB'}}}}$ is defined as ${{\color{MidnightBlue} \bm{\hat{S}}}}^{g_1,g_2}_{{{\textsf{BA'}}}}$, with the role of the register ${{\textsf{A'}}}$ replaced by ${{\textsf{B'}}}$ (note the two are isomorphic). From Lemma \[lem:hat-s\] we know that $\{\hat{S}^{g_1,g_2}\}$ is a projective measurement. In particular the first component of each of the observables defined in commute; hence the first item in the lemma follows from the fact that $\sigma_{X, \ell}(a)$ and $\sigma_{Z,\ell}(b)$ themselves satisfy the required Pauli relations.
Next we verify the second item. We give the proof for the $X$ observables; the arguments for $Z, {{\color{MidnightBlue} \bm{X}}}, {{\color{MidnightBlue} \bm{Z}}}$ are similar. Using the definition , on average over $x\in{\ensuremath{\mathbb{F}_q}}^m$, $$\begin{aligned}
\tilde{X}_\ell^a(x_{\pi}) &=\sum_{b,c \in { \ensuremath{{\mathbb{F}}}_p}:\,b - c=a}\Big(\sum_{g_1,g_2:\,
{\mbox{\rm tr}}(b_\ell g_1(x))=b}
\hat{S}_{{\textsf{AA'}}}^{g_1,g_2} \Big) \otimes
\sigma_{X, \ell}^c(x_{\pi})_{{\textsf{A''}}}\\
&\approx_{\operatorname{poly}(\delta_S)} \sum_{b,c \in { \ensuremath{{\mathbb{F}}}_p}:\,b-c=a}
\hat{X}_\ell^b(x_{\pi})_{{\textsf{AA'}}} \otimes \sigma_{X, \ell}^c(x_{\pi})_{{\textsf{A''}}} \\
&= \sum_{b',b'',c \in { \ensuremath{{\mathbb{F}}}_p}:\,b'+b'' - c=a} X_\ell^{b'}(x_{\pi})_{{\textsf{A}}} \otimes
{{\color{MidnightBlue} \bm{\sigma}}}_{X, \ell}^{b''}(x_{\pi})_{{\textsf{A'}}}\otimes \sigma_{X, \ell}^c(x_{\pi})_{{\textsf{A''}}} \\
& = X_{\ell}^a(x_{\pi}),\end{aligned}$$ where the second line follows from in Lemma \[lem:hat-s\] and the definition , the third uses the definition of $\hat{X}$ from Claim \[claim:strategies\], and the last uses the fact that ${|\hat{\psi}\rangle}$ defined in Lemma \[lem:hats\] is an EPR pair on ${{\textsf{A'A''}}}$ together with to set $b'' = c$ (so the equality holds in the state-dependent distance).
Finally we show the third item in the lemma. We show consistency for $\tilde{X}$; consistency for $\tilde{Z}$ is similar For ease of notation we write $g$ for $g_1$ and omit the outcome $g_2$, which in this argument is always summed over. Using the definition , $$\begin{aligned}
\sum_e\, \tilde{X}_\ell^e(a)_{{\textsf{AA'A''}}} \otimes {{\color{MidnightBlue} \bm{\tilde{X}}}}_\ell^e(a)_{{\textsf{BB'B''}}}
&= \sum_{\substack{g,c,g',c':\\ {\mbox{\rm tr}}(b_\ell(g \cdot a)) - c= {\mbox{\rm tr}}(b_\ell(g' \cdot a)) -c'}} {\hat{S}}^g_{{\textsf{AA'}}} \otimes \sigma_{X,\ell}^c(a)_{{\textsf{A''}}} \otimes {{\color{MidnightBlue} \bm{{\hat{S}}}}}^{g'}_{{\textsf{BB'}}} \otimes {{\color{MidnightBlue} \bm{\sigma}}}_{X,\ell}^{c'}(a)^\dag_{{\textsf{B''}}}\notag\\
&\approx_{\operatorname{poly}(\delta_S)} {\ensuremath{\mathop{\textsc{E}}_{x\in{\ensuremath{\mathbb{F}_q}}^m}}}\, \sum_{\substack{g,c,g',c':\\{\mbox{\rm tr}}(b_\ell( (g-g')\cdot a)) = c-c'}} {\hat{S}}^g_{{\textsf{AA'}}} \otimes \sigma_{X,\ell}^c(a)_{{\textsf{A''}}} \otimes {{\color{MidnightBlue} \bm{{\hat{S}}}}}^{g'}_{{\textsf{BB'}}} \otimes {{\color{MidnightBlue} \bm{\sigma}}}_{X,\ell}^{c'}(a)_{{\textsf{B''}}}\notag\\
&\hskip1cm \cdot \Big(\sum_{\substack{ r + s=g(x)\\ r' + s'=g'(x)}}
({M}_{X, x}^{r})_{{\textsf{A}}} \otimes
({{\color{MidnightBlue} \bm{{\tau}}}}_{X, x}^{s})_{{\textsf{A'}}} \otimes
({{\color{MidnightBlue} \bm{M}}}_{X, x}^{r'})_{{\textsf{B}}} \otimes ({\tau}_{X, x}^{s'})_{{\textsf{B'}}}\Big)\;,\label{eq:gen-xz-1}
\end{aligned}$$ where the approximation uses , , and and . Using consistency of $M$ with ${{\color{MidnightBlue} \bm{M}}}$, as shown in Claim \[claim:strategies\], additionally imposes the constraint that $r=r'$, i.e. $(g-g')(x)=s - s'$. Now, recall that $\sigma^c_{X,\ell}(a)_{{\textsf{A''}}}$ is implemented by measuring all $n$ qudits of register ${{\textsf{A''}}}$ in the $X$ basis, obtaining an outcome $h\in{\ensuremath{\mathbb{F}_q}}^n$, and reporting $c={\mbox{\rm tr}}(b_\ell(h\cdot a))$ as the outcome. Similarly, $({\tau}^s_{X,x})_{{{\textsf{A''}}}}$ is implemented by measuring all $n$ qudits of register ${{\textsf{A''}}}$ in the $X$ basis, obtaining an outcome $h\in{\ensuremath{\mathbb{F}_q}}^n$, and reporting $s=h(x) = h \cdot x_{\pi}$ as the outcome. Moreover, as the registers ${{\textsf{A'A''}}}$ and ${{\textsf{B'B''}}}$ are in EPR states, we can apply the exact consistency relations between Pauli measurements. This lets us rewrite as $$\begin{aligned}
&\approx_{\operatorname{poly}(\delta_M)} {\ensuremath{\mathop{\textsc{E}}_{x\in{\ensuremath{\mathbb{F}_q}}^m}}}\, \sum_{\substack{g,h,g',h':\\
{\mbox{\rm tr}}(b_\ell (g-g')\cdot a) = {\mbox{\rm tr}}(b_\ell (h-h')\cdot a) \\ (g-g')(x)=(h-h')(x)}} \sum_{r = (g-h)(x)} {\hat{S}}^g_{{\textsf{AA'}}} ({M}^{r}_{X,x})_{{\textsf{A}}} \otimes {{\color{MidnightBlue} \bm{{\hat{S}}}}}^{g'}_{{\textsf{BB'}}} ({{\color{MidnightBlue} \bm{M}}}^{r}_{X,x})_{{\textsf{B}}}
\otimes ({\tau}_X^{h})_{{\textsf{A''}}} \otimes ({{\color{MidnightBlue} \bm{{\tau}}}}_X^{h'})_{{\textsf{B''}}}\notag\\
&= {\ensuremath{\mathop{\textsc{E}}_{x\in{\ensuremath{\mathbb{F}_q}}^m}}}\, \sum_{\substack{g,h,g',h':\\ {\mbox{\rm tr}}(b_\ell (g-g')\cdot a)
= {\mbox{\rm tr}}(b_\ell (h-h')\cdot a)\\ (g-g')(x)=(h-h')(x)}} \sum_{\substack{ r =
g(x)\\ r'=g'(x)}} {\hat{S}}^g_{{\textsf{AA'}}} (\hat{M}^{r}_{X,x})_{{\textsf{AA'}}} \otimes {{\color{MidnightBlue} \bm{{\hat{S}}}}}^{g'}_{{\textsf{BB'}}} ({{\color{MidnightBlue} \bm{\hat{M}}}}^{r'}_{X,x})_{{\textsf{BB'}}}
\otimes ({\tau}_X^{h})_{{\textsf{A''}}} \otimes ({{\color{MidnightBlue} \bm{{\tau}}}}_X^{h'})_{{\textsf{B''}}}\notag\\
&\approx_{\operatorname{poly}(\delta_S)} {\ensuremath{\mathop{\textsc{E}}_{x\in{\ensuremath{\mathbb{F}_q}}^m}}}\, \sum_{\substack{g,h,g',h':\\
{\mbox{\rm tr}}(b_\ell (g-g')\cdot a) = {\mbox{\rm tr}}(b_\ell (h-h')\cdot a) \\ (g-g')(x)=(h-h')(x)}} {\hat{S}}^g_{{\textsf{AA'}}} \otimes {{\color{MidnightBlue} \bm{{\hat{S}}}}}^{g'}_{{\textsf{BB'}}}
\otimes ({\tau}_X^{h})_{{\textsf{A''}}} \otimes ({{\color{MidnightBlue} \bm{{\tau}}}}_X^{h'})_{{\textsf{B''}}}\;,\label{eq:gen-xz-2}
\end{aligned}$$ where the second line uses the definition of $\hat{M}$, and for the last approximation we removed the constraint on $r$ using . If $g - g'\neq h - h'$ then by Lemma \[lem:sz\] the condition $(g-g')(x)=(h-h')(x)$ holds at a random point $x\in{\ensuremath{\mathbb{F}_q}}^m$ with probability at most $O(d/q)$. If $g - g'=h - h'$ the constraint ${\mbox{\rm tr}}(b_\ell (g-g')\cdot
a) = {\mbox{\rm tr}}(b_\ell (h-h')\cdot a)$ is superfluous. Hence, under the expectation, we can replace the constraints in the summation in by the constraint $(g - g')(x) = (h - h')(x)$, incurring an error of $O(d/q)$: $$\begin{aligned}
&\approx_{\operatorname{poly}(d/q)} {\ensuremath{\mathop{\textsc{E}}_{x\in{\ensuremath{\mathbb{F}_q}}^m}}}\, \sum_{\substack{g,g',s, s':\\ (g-g')(x)=(s-s')}} {\hat{S}}^g_{{\textsf{AA'}}} \otimes {{\color{MidnightBlue} \bm{{\hat{S}}}}}^{g'}_{{\textsf{BB'}}}
\otimes ({\tau}^{s}_{X,x})_{{\textsf{A''}}} \otimes( {{\color{MidnightBlue} \bm{{\tau}}}}^{s'}_{X,x})_{{\textsf{B''}}}\\
&\approx_{\delta_S} {\ensuremath{\mathop{\textsc{E}}_{x\in{\ensuremath{\mathbb{F}_q}}^m}}}\, \sum_{\substack{e,e',s, s':\\
e - e'=s-s'}} (\hat{M}^{e}_{X,x})_{{\textsf{AA'}}} \otimes ({{\color{MidnightBlue} \bm{\hat{M}}}}^{e'}_{X,x})_{{\textsf{BB'}}}
\otimes ({\tau}^{s}_{X,x})_{{\textsf{A''}}} \otimes(
{{\color{MidnightBlue} \bm{{\tau}}}}^{s'}_{X,x})_{{\textsf{B''}}}\\
&= {\ensuremath{\mathop{\textsc{E}}_{x\in{\ensuremath{\mathbb{F}_q}}^m}}}\, \sum_{\substack{e,e',f, f', s, s':\\ e + f -(e' + f')=s-s'}}
(M^{e}_{X,x})_{{\textsf{A}}} \otimes ({{\color{MidnightBlue} \bm{M}}}^{e'}_{X,x})_{{\textsf{B}}}
\otimes ({{\color{MidnightBlue} \bm{{\tau}}}}^{f}_{X,x})_{{\textsf{A'}}} \otimes
({\tau}^{s}_{X,x})_{{\textsf{A''}}} \otimes ({{\color{MidnightBlue} \bm{{\tau}}}}^{f'}_{X,x})_{{\textsf{B'}}} \otimes(
{{\color{MidnightBlue} \bm{{\tau}}}}^{s'}_{X,x})_{{\textsf{B''}}}\\
&= {\ensuremath{\mathop{\textsc{E}}_{x \in {\ensuremath{\mathbb{F}_q}}^m}}} \, \sum_{e} (M_{X,x}^e)_A {\otimes}(M_{X,x}^{e})_B \\
&\approx_{\delta_M} {\ensuremath{\mathop{\rm Id}\nolimits}}\;,\end{aligned}$$ where the second line is by , the third by the definition of $\hat{M}$ (Lemma \[lem:hats\]), the fourth by consistency of ${\tau}$ on the EPR state, and the last is by self-consistency of $M$ (Claim \[claim:strategies\]).
Proof of Lemma \[lem:soundness\] {#sec:finish_soundness}
--------------------------------
We conclude the proof of Lemma \[lem:soundness\]. Lemma \[lem:xz-lowdeg\] shows that whenever there exists a strategy for the provers that succeeds in the test ${\textsc{q-lowdeg}}(m,d,q)$ with probability at least $1-{\varepsilon}$, for some ${\varepsilon}$ small enough with respect to $d/q$ so that the quantity $\delta_W$ in the lemma is a small constant, there exist operators $\tilde{X}_\ell(a)$ and $\tilde{Z}_\ell(b)$, for $a,b \in {\ensuremath{\mathbb{F}_q}}^n$, acting on the extended local spaces ${{\textsf{AA'A''}}}$ and ${{\textsf{BB'B''}}}$ respectively, that exactly satisfy the group relations of the generalized Pauli group (also known as the finite Heisenberg group).
Using these relations it is fairly straightforward to construct isometries $V_A$, $V_B$ acting on each prover’s space such that $V_A$ maps $\tilde{W}_\ell(a)_{{\textsf{AA'A''}}}$ to ${\ensuremath{\mathop{\rm Id}\nolimits}}_{{\textsf{AA'}}} {\otimes}\sigma_{W, \ell}(a)_{{\textsf{A''}}}$ for $W \in \{X, Z\}$, $\ell \in \{1, \dots, t\}$, and $a\in{\ensuremath{\mathbb{F}_q}}^n$, and likewise for $V_B$. The definition of $V_A$ and $V_B$ is analogous, so we drop the subscript. To explicitly define $V$, let $U$ be a unitary such that $U
\sigma_{W,\ell}(a)^\dag U^\dag = \sigma_{W,\ell}(a)$ for all $a\in{\ensuremath{\mathbb{F}_q}}^n$, $\ell \in \{1,\dots,t\}$ and $W\in\{X,Z\}$; such a $U$ can be explicitly defined through its expansion in the Pauli basis. Define $$V = \sum_{g_1 g_2} \hat{S}^{g_1, g_2}_{{\textsf{AA'}}} {\otimes}U {\tau}_X(g_2) {\tau}_Z(g_1)\;,$$ where in the notation ${\tau}_W(g)$ we interpret $g$ as the corresponding vector of coefficients in ${\ensuremath{\mathbb{F}_q}}^n$. To see that this accomplishes the desired map, using that $\{\hat{S}^{g_1g_2}\}$ is projective, evaluate $$\begin{aligned}
V \tilde{X}_{\ell}(a)_{{\textsf{AA'A''}}} V^\dag
&= \sum_{g_1, g_2} \omega^{{\mbox{\rm tr}}(b_\ell (g_1
\cdot a))} \hat{S}_{{\textsf{AA'}}}^{g_1, g_2} {\otimes}U{\tau}_X(g_2)
{\tau}_Z(g_1) \sigma_{X,\ell}(a)^\dag {\tau}_Z(g_1)^\dag
{\tau}_X(g_2)^\dag U^\dag \\
&= \sum_{g_1, g_2} \omega^{{\mbox{\rm tr}}(b_\ell (g_1
\cdot a))} \hat{S}_{{\textsf{AA'}}}^{g_1, g_2} {\otimes}U{\tau}_X(g_2)
{\tau}_Z(g_1) {\tau}_{X}(b_\ell a)^\dag {\tau}_Z(g_1)^\dag
{\tau}_X(g_2)^\dag U^\dag \\
&= \sum_{g_1, g_2} \hat{S}_{{\textsf{AA'}}}^{g_1, g_2} {\otimes}U
{\tau}_X(a b_\ell)^\dag U^\dag
\\
&= \sum_{g_1, g_2} \hat{S}_{{\textsf{AA'}}}^{g_1, g_2} {\otimes}U
\sigma_{X,\ell}(a)^\dag U^\dag \\
&= {\ensuremath{\mathop{\rm Id}\nolimits}}{\otimes}\sigma_{X,\ell}(a)\;.
\end{aligned}$$ A similar calculation can be done for $Z$.
The combination of all isometries considered in the analysis might have increased the local dimension by a large amount. However, using the outcome consistency between $\tilde{X}_\ell(a)$ and $X_\ell(a)$ (item 3. in Lemma \[lem:xz-lowdeg\]), we know that the state ${|\hat{\psi}\rangle}$ satisfies $${\ensuremath{\mathop{\textsc{E}}_{\ell \in \{1, \dots, t\}}}} {\ensuremath{\mathop{\textsc{E}}_{a \in {\ensuremath{\mathbb{F}_q}}^n }}} \,\sum_e \,{\langle\hat{\psi}|} \tilde{X}_{\ell}^e(a) {\otimes}{{\color{MidnightBlue} \bm{\tilde{X}}}}_{\ell}^e(a){|\hat{\psi}\rangle}\, \geq\, 1 - O(\delta_{W})\;,
\label{eq:tilde-x-con}$$ and an analogous property also holds for the $\tilde{Z}_\ell(b)$. Let $H_W = {\ensuremath{\mathop{\textsc{E}}_{\ell}}} {\ensuremath{\mathop{\textsc{E}}_{a}}}
\sum_e \sigma_{W,\ell}(a)^e {\otimes}{{\color{MidnightBlue} \bm{\sigma}}}_{W,\ell}(a)^e$ for $W \in \{X,
Z\}$, and ${|\psi'\rangle} = V_A\otimes V_B {|\hat{\psi}\rangle}$. In this notation, we can rewrite as $$\label{eq:hw-1}
{\langle\psi'|} H_W {|\psi'\rangle} \,\geq\, 1 - O(\delta_W)\;,$$ for $W \in \{X, Z\}$. By construction, the operators $H_W$ are Hermitian with $0
\leq H_W \leq {\ensuremath{\mathop{\rm Id}\nolimits}}$, and $H_X H_Z = H_ZH_X$. Hence, $H = H_X
H_Z$ is Hermitian and $0 \leq H \leq {\ensuremath{\mathop{\rm Id}\nolimits}}$. An application of the Cauchy-Schwarz inequality to yields $$\label{eq:hw-2}
{\langle\psi'|} H {|\psi'\rangle} \geq 1 -
O(\sqrt{\delta_W})\;.$$ Moreover, a direct calculation reveals that in fact $H = {|\psi_{\textsc{EPR}}\rangle}{\langle\psi_{\textsc{EPR}}|}$. First, we evaluate $H_W$: $$\begin{aligned}
H_W &= {\ensuremath{\mathop{\textsc{E}}_{a}}} {\ensuremath{\mathop{\textsc{E}}_{\ell}}} \sum_e \sigma_{W, \ell}^e(a) {\otimes}{{\color{MidnightBlue} \bm{\sigma}}}_{W,\ell}^e(a) \\
&= {\ensuremath{\mathop{\textsc{E}}_{a}}} {\ensuremath{\mathop{\textsc{E}}_{\ell}}} \sum_{h, h': {\mbox{\rm tr}}(b_\ell h \cdot a) ={\mbox{\rm tr}}(b_\ell h' \cdot a) } {\tau}_W^h{\otimes}{{\color{MidnightBlue} \bm{{\tau}}}}_W^{h'} \\
&= {\ensuremath{\mathop{\textsc{E}}_{a}}}{\ensuremath{\mathop{\textsc{E}}_{\ell}}} \sum_{h, h'} \omega^{{\mbox{\rm tr}}(b_\ell a \cdot (h - h'))} {\tau}_W^h {\otimes}{{\color{MidnightBlue} \bm{{\tau}}}}_W^{h'} \\
&= {\ensuremath{\mathop{\textsc{E}}_{a}}}{\ensuremath{\mathop{\textsc{E}}_{\ell}}} \sigma_{W, \ell}(a) {\otimes}{{\color{MidnightBlue} \bm{\sigma}}}_{W,\ell}(a)^\dag \\
&= {\ensuremath{\mathop{\textsc{E}}_{\ell}}} \Big( {\ensuremath{\mathop{\textsc{E}}_{a \in {\ensuremath{\mathbb{F}_q}}}}} (\sigma_{W,\ell}(a) {\otimes}{{\color{MidnightBlue} \bm{\sigma}}}_{W,\ell}^\dag(a))
\Big)^{{\otimes}n}\;, \\
&= {\ensuremath{\mathop{\textsc{E}}_{\ell}}} \Big( {\ensuremath{\mathop{\textsc{E}}_{a \in {\ensuremath{\mathbb{F}_q}}}}} ({\tau}_{W}(a b_\ell) {\otimes}{{\color{MidnightBlue} \bm{{\tau}}}}_W^\dag(a b_\ell))
\Big)^{{\otimes}n}\;, \\
&= \Big({\ensuremath{\mathop{\textsc{E}}_{a \in {\ensuremath{\mathbb{F}_q}}}}} ({\tau}_{W}(a) {\otimes}{{\color{MidnightBlue} \bm{{\tau}}}}_W^\dag(a))\Big)^{{\otimes}n},
\end{aligned}$$ where in going to the last line, we did a change of variables on the variable $a$, to absorb the factor of $b_\ell$. Now, we evaluate $H$: $$\begin{aligned}
H &= H_X H_Z \\
&= \Big({\ensuremath{\mathop{\textsc{E}}_{a,b \in {\ensuremath{\mathbb{F}_q}}}}} {\tau}_X(a) {\tau}_Z(b ) {\otimes}{{\color{MidnightBlue} \bm{{\tau}}}}_X^\dag (a) {{\color{MidnightBlue} \bm{{\tau}}}}_Z^\dag(b) \Big)^{{\otimes}n} \\
&= \Big({\ensuremath{\mathop{\textsc{E}}_{a,b \in {\ensuremath{\mathbb{F}_q}}}}} {\tau}_X(a) {\tau}_Z(b ) {\otimes}{\tau}_X(a) {\tau}_Z(-b) \Big)^{{\otimes}n} \\
&= \Big({\ensuremath{\mathop{\textsc{E}}_{a,b \in {\ensuremath{\mathbb{F}_q}}}}} \sum_{i, j} \omega^{{\mbox{\rm tr}}(i b )}
{|i + a\rangle}{\langlei|} {\otimes}\omega^{-{\mbox{\rm tr}}(j b)} {|j+a\rangle}
{\langlej|} \Big)^{{\otimes}n} \\
&= \Big({\ensuremath{\mathop{\textsc{E}}_{a \in {\ensuremath{\mathbb{F}_q}}}}} \sum_{i} {|i+a\rangle}
{\langlei|} {\otimes}{|i+a\rangle}{\langlei|} \Big)^{{\otimes}n} \\
&= {|\psi_{\textsc{EPR}}\rangle}{\langle\psi_{\textsc{EPR}}|}\;,
\end{aligned}$$ where in going from the fourth to the fifth line, we have used the fact that the summation over $b$ vanishes unless $j = i$, and for the last we use that ${\ensuremath{\mathop{\textsc{E}}_{a\in{\ensuremath{\mathbb{F}_q}}}}} {|i+a\rangle}{|i+a\rangle} =
q^{-1/2}{|\psi_{\textsc{EPR}}\rangle}$ for any $i\in{\ensuremath{\mathbb{F}_q}}$. Hence, from we conclude that $$\big\| {\langle \psi_{\textsc{EPR}}| \psi' \rangle} \big\|^2 \,\geq\, 1 - O(\sqrt{\delta_W})\;.$$ This completes the proof of Lemma \[lem:soundness\].
A test for codewords {#sec:code}
====================
In this section we show that the low-degree test ${\textsc{q-lowdeg}}$ can be combined with any weakly self-dual quantum CSS code ${\mathcal{C}}$ defined over ${\ensuremath{\mathbb{F}_q}}$ to obtain a self-test for states in the $n$-fold tensor product of the codespace, as well as certain tensor products of generalized Pauli observables on the codespace (including all single-qudit Pauli observables). The test uses as many provers as the dimension of the code.
CSS codes {#sec:codes}
---------
We consider weakly self-dual *Calderbank-Shor-Steane (CSS) codes* [@CalderbankShor96; @Steane96]. Let $C$ be a classical $[k,k']$ linear error-correcting code over ${\ensuremath{\mathbb{F}_q}}$, for a prime power $q$: $C$ is specified by a generator matrix $G \in {\ensuremath{\mathbb{F}_q}}^{k\times k'}$ and a parity check matrix $K\in {\ensuremath{\mathbb{F}_q}}^{(k-k')\times k}$ such that $C = \operatorname{Im}(G)=\ker(K)$. We say that $C$ is weakly self-dual if the dual code $C^\perp$, with generator matrix $K^T$, is such that $C\subseteq C^\perp$; equivalently, $G^T G=0$. To any such code $C$ we associate a subspace $\mathcal{C}$ of $({\ensuremath{\mathbb{C}}}^q)^{\otimes k}$ that is the simultaneous $+1$ eigenspace of a set of stabilizers $\{S_{W,j}\}_{W\in\{X,Z\},j\in\{1,\ldots,k'\}}$ such that $S_{W,j}$ is a tensor product of Pauli $W$ observables over ${\ensuremath{\mathbb{F}_q}}$ in the locations indicated by the $j$-th column of the generator matrix $G$, i.e. $$S_{W,j} = {\tau}_W(G_{1j}) {\otimes}{\tau}_W(G_{2j}) {\otimes}\dots
{\otimes}{\tau}_W(G_{kj}),$$ where $G_{ij}$ is the $(i,j)$-th entry of $G$. The condition that $G^TG=0$ implies that all the $S_{W,j}$ commute, so that $\mathcal{C}$ is well-defined. We refer to [@gottesman1999fault; @ketkar2006nonbinary] for more background on the theory of stabilizer codes over qudits.
A simple example of a weakly self-dual $2$-qudit code with dimension $1$ is the “EPR code” (our terminology) with generator matrix $G
= \begin{pmatrix} 1 \\ 1\end{pmatrix}$ in case $q=2$, and $G
= \begin{pmatrix} 1 \\ \sqrt{-1} \bmod q \end{pmatrix}$ for $q\equiv
1 \bmod 4$. The associated code subspace ${\mathcal{C}}$ has dimension $1$, and it is spanned by a maximally entangled state of two qudits.
\[ex:quad\_res\_code\] The $7$-qudit code is a weakly self-dual CSS code that has $k=7$, $k'=3$, and encodes one qudit over ${\ensuremath{\mathbb{F}_q}}$ for any prime power $q=p^e$ such that $p$ is a quadratic residue modulo $7$. For example $p=2$ is a quadratic residue modulo $7$. See Theorem 40 in [@ketkar2006nonbinary] for a more general construction.
The ${\textsc{code-check}}$ test {#sec:code-protocol}
--------------------------------
Let $n$ be an integer and $C$ a weakly self-dual $[k,k']$ linear code. Let ${\mathcal{C}}$ be the associated CSS code, as described in Section \[sec:codes\].[^10] The test ${\textsc{code-check}}(C,n)$ is summarized in Figure \[fig:code-test\]. The test builds on the (composed) low-degree test described in Figure \[fig:protocol\]. Recall the following properties of the honest strategy for the provers in the test (see Section \[sec:lowdeg-completeness\]):
- In the first part of the test, each prover is sent a query of the form $(W,s,s')$, where $W\in\{X,Z\}$ designates a choice of basis and $s$, $s'$ are the specification of a pair of subspaces. The honest prover measures each of his $n$ qudits in the basis $W$, obtaining a string $a\in{\ensuremath{\mathbb{F}_q}}^n$. From $a$, the prover computes the degree-$d$ polynomial $g_a$ specified in , and returns the restriction of the (suitably encoded) bivariate polynomial $(g_a)_{|s}$ to the subspace $s'$.
- In the second part of the test, the prover is sent a query of the form $(W_1,W_2)$, where $W_1,W_2\in\{X,Z,Y\}^n$ are commuting $n$-qudit observables. The honest prover jointly measures $W_1$ and $W_2$ and returns the pair of outcomes obtained.
We now describe the test ${\textsc{code-check}}(C,n)$. In the test, the verifier splits the $k$ provers into two groups. One prover, indexed by $t\in\{1,\ldots, k\}$, is chosen at random and called the “special prover”. The remaining $(k-1)$ provers are jointly called “composite prover”. In general a prover is not told whether it is the special prover, or a composite prover. In the test the verifier simulates queries from the two-prover low-degree test for the special and composite provers using the following scheme.
\[def:queries\] Let $G \in {\ensuremath{\mathbb{F}_q}}^{k\times k'}$ be the generator matrix for a $[k,k']$ weakly self-dual code $C$. Let $Q$ be a query in the test ${\textsc{q-lowdeg}}$.
1. The composite query associated with $Q$, denoted ${{\color{MidnightBlue} \bm{Q}}}$, is obtained by sending each prover forming the composite prover the query $Q$.
2. Given answers $(A_j)_{j \neq t}$ from the $(k-1)$ provers forming the composite prover, the composite answer ${{\color{MidnightBlue} \bm{A}}}$ is obtained by selecting a uniformly random vector $v$ in the column span of $G$ such that $v_t=1$, and computing the sum ${{\color{MidnightBlue} \bm{A}}} =
- \sum_{j \neq t} v_j A_j$. [^11]
This definition is consistent with the notation ${{\color{MidnightBlue} \bm{M}}}$ used in ${\textsc{q-lowdeg}}$; in both cases, the answers obtained from the composite prover (in the case of the two-player test, the second prover) are multiplied by the appropriate entry of the generator matrix of a code. The test ${\textsc{q-lowdeg}}$ differs only insofar as the EPR state is not a CSS code state, so the $X$ and $Z$ stabilizers are not identical. Moreover, in both cases, for honest strategies, the special and composite prover obtain the same outcome when given the same query. This fact is formalized in the following lemma.
\[lem:fq-linear\] Let $\ell \geq 1$ be an integer and $f: {\ensuremath{\mathbb{F}_q}}^n \to {\ensuremath{\mathbb{F}_q}}^\ell$ a linear function. Suppose that $k$ provers share a $(nk)$-qudit state ${|\Psi\rangle}$ that is a valid qudit-by-qudit encoding of an $n$-qudit state ${|\psi\rangle}$ according to a $k$-qudit self-dual CSS code $\mathcal{C}$. Let $W\in\{X,Z\}^n$ and suppose that for each $j\in\{1,\ldots,k\}$, the $j$-th prover measures the $i$-qudit of its share of the state in the basis $W_i$, for each $i\in\{1,\ldots,n\}$, to obtain an assignment $a_j \in {\ensuremath{\mathbb{F}_q}}^n$, and returns the value $y_j = f(a_j) \in {\ensuremath{\mathbb{F}_q}}^\ell$. Then for any index $t\in\{1,\ldots,k\}$ for the special prover, and vector $v\in{\ensuremath{\mathbb{F}_q}}^k$ chosen as in item 2. in Definition \[def:queries\], the special prover’s answer $y_t$ and the composite prover’s answer ${{\color{MidnightBlue} \bm{y}}} = -\sum_{j \neq t} v_j y_j$ are equal with certainty. \[lem:composite-linear\]
Before giving the proof, we note that the functions computed in the low-degree test, i.e. polynomials $g_a$ as in , evaluated on points or restricted to subspaces, are linear functions of $a$ of the form considered in the lemma.
It follows from the definition of $v$ and the stabilizer property of the code that $$\sum_{j} v_j a_j = 0\;.$$ Write the linear function $f$ as $f(a) = K a $, for $K\in{\ensuremath{\mathbb{F}_q}}^{\ell\times n}$. Then, a simple calculation shows that $${{\color{MidnightBlue} \bm{y}}} = -\sum_{j \neq t} v_j \,f(a_j) = - \sum_{j \neq t} v_j \,( K a_j) = K\Big(- \sum_{j \neq t} v_j \,a_j\Big) = K a = y\;.$$
------------------------------------------------------------------------
\
Test ${\textsc{code-check}}(C,n)$: Given is the generator matrix $G$ for a $[k,k']$ weakly self-dual linear code $C$ over ${\ensuremath{\mathbb{F}_q}}$, as described in Section \[sec:codes\], and $n$ an integer. Let $d = \lceil \log n \rceil \cdot \lceil
\frac{\log n}{\log \log n} \rceil$ and $m =\lceil \frac{\log(n)}{\log\log(n)} \rceil$.
- The verifier selects one of the $k$ provers at random and assigns it the label of “special prover”. All remaining provers are given the label of “composite prover”. (The provers are not told how they are labeled.) Let $t\in\{1,\ldots,k\}$ be the index of the special prover.
- The verifier executes the verifier for the test ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$ to generate a pair of queries $(Q,Q')$ for the two provers in that test. The verifier sends the query $Q$ to the special prover, and distributes the query ${{\color{MidnightBlue} \bm{Q'}}}$ to the composite prover. He receives answers $A$ and ${{\color{MidnightBlue} \bm{A'}}}$ respectively.
- The verifier accepts if and only if $(A,{{\color{MidnightBlue} \bm{A'}}})$ is a pair of valid answers to queries $(Q,Q')$ in the low-degree test.
------------------------------------------------------------------------
Analysis of the ${\textsc{code-check}}$ self-test
-------------------------------------------------
We first show completeness of the test ${\textsc{code-check}}$.
\[lem:code-completeness\] Let $C$ be a $[k,k']$ weakly self-dual linear code over ${\ensuremath{\mathbb{F}_q}}$, and $n$ an integer. Let ${\mathcal{C}}$ be the associated CSS code, as described in Section \[sec:codes\]. Then for any $(nk)$-qubit state ${|\Psi\rangle}\in {\mathcal{C}}^{\otimes n}$ there exists a strategy for the $k$ provers based on sharing ${|\Psi\rangle}$ and measuring tensor products of Pauli observables, such that the strategy is accepted with probability $1$ in the test ${\textsc{code-check}}(C,n)$.
Fix ${|\Psi\rangle}\in {\mathcal{C}}^{\otimes n}$. The strategy for the provers is simple: each prover directly applies the honest strategy in the test ${\textsc{q-lowdeg}}^{(2)}$, as described in Section \[sec:lowdeg-completeness\].
It remains to verify that the answers $(A,{{\color{MidnightBlue} \bm{A'}}})$ computed by the verifier in step (c) of the test ${\textsc{code-check}}$ are valid answers to $(Q,Q')$ in ${\textsc{q-lowdeg}}^{(2)}$. Fix a choice of codeword $v$ as made by the verifier in the computation of the composite query ${{\color{MidnightBlue} \bm{Q'}}}$ at step (b) of ${\textsc{code-check}}$ (see Definition \[def:queries\]). We make the following observations on the joint measurement performed by the provers that constitute the composite prover. Consider first queries of the form $Q'=(W,s,s')$. Upon receipt of such a query, the $i$-th prover that constitutes the composite prover measures each of its $n$ qudits using the projective measurement ${\tau}_{W}$ to obtain outcomes $a'_i=(a'_{i,1},\ldots,a'_{i,n})$, from which it computes a low-degree polynomial $g_{a'_i}$ as in . Since $a'\mapsto
g_{a'}$ is a linear function, the answer ${{\color{MidnightBlue} \bm{A'}}}$ computed by the verifier is the restriction to $s'$ of the (suitably encoded) bivariate polynomial $(g_{a'})_{|s}$, where $a' = \sum_{i \neq t} v_i a'_i$ is the outcome of an imaginary joint measurement performed by the composite prover using the measurement ${{\color{MidnightBlue} \bm{{\tau}}}}_{W}^{a'} = \sum_{a'_i : \sum_{i \neq t} v_i
a'_i = a'} \otimes_{i\neq t} {\tau}_{W}^{a'_i}$. The situation in case the query $Q'$ is taken from the second part of the low-degree test is similar. To conclude we show that for any choice of codeword $v$ made by the verifier, the provers’ strategy, conditioned on $v$, is isometric to the honest strategy for the low-degree test, as defined in the proof of Lemma \[lem:completeness\].
To see this, observe that by definition, for a fixed $v$, the operators $X \otimes {{\color{MidnightBlue} \bm{X}}}$ and $Z \otimes {{\color{MidnightBlue} \bm{Z}}}$ stabilize each group of $k$ qudits of ${|\Psi\rangle}$, where $X={\tau}_X(1)$, ${{\color{MidnightBlue} \bm{X}}} = \otimes_{i\neq t} {\tau}_X(v_i)$, and $Z={\tau}_Z(2)$, ${{\color{MidnightBlue} \bm{Z}}} = \otimes_{i\neq t} {{\tau}_Z}(v_i)$; indeed this is because $v$ defines both an $X$ and a $Z$ stabilizer for ${\mathcal{C}}$. Moreover, ${{\color{MidnightBlue} \bm{X}}}$ and ${{\color{MidnightBlue} \bm{Z}}}$ satisfy the same twisted commutation relation as ${{\color{MidnightBlue} \bm{{\tau}}}}_X$ and ${{\color{MidnightBlue} \bm{{\tau}}}}_Z$; this is because $v_t=1$ and $v\cdot v=0$ by weak self-duality. Thus there exists a local isometry acting jointly on all provers forming the composite prover, which maps ${{\color{MidnightBlue} \bm{X}}}\mapsto {{\color{MidnightBlue} \bm{{\tau}}}}_X$ and ${{\color{MidnightBlue} \bm{Z}}}\mapsto {{\color{MidnightBlue} \bm{{\tau}}}}_Z$. The image of ${|\Psi\rangle}$ under this isometry is stabilized by ${\tau}_X\otimes {{\color{MidnightBlue} \bm{{\tau}}}}_X$ and ${\tau}_Z \otimes {{\color{MidnightBlue} \bm{{\tau}}}}_Z$, hence must be the state ${|{\textsc{EPR}}_q\rangle}$. Lemma \[lem:completeness\] then lets us conclude that the above-defined strategy succeeds with probability $1$ in the test.
The next theorem shows soundness of the test ${\textsc{code-check}}$.
\[thm:codeword\_test\] Let $n,k,k'$ be integer. Let $q=p^t$ be a prime power such that ${\ensuremath{\mathbb{F}_q}}$ admits a self-dual basis over ${ \ensuremath{{\mathbb{F}}}_p}$. Let $C$ be a $[k,k']$ weakly self-dual linear code over ${\ensuremath{\mathbb{F}_q}}$, and ${\mathcal{C}}$ the associated CSS code. Let $m,d$ be as in Figure \[fig:code-test\]. Suppose a strategy using state ${|\Psi\rangle} \in \otimes_{i=1}^k {\mathcal{H}}_{i}$ and projective measurements $\{M_{W,w}^a\}$ for the special prover succeeds in test ${\textsc{code-check}}(C,n)$ with probability at least $1-{\varepsilon}$, for some ${\varepsilon}\geq 0$. Then there is a $\delta_C = \max(\operatorname{poly}(p)\cdot\operatorname{poly}({\varepsilon}),\operatorname{poly}(q^{-1}))$ and isometries $V_i:
{\mathcal{H}}_{i} \to ({\ensuremath{\mathbb{C}}}^q)^{\otimes n}\otimes {\mathcal{H}}'_{i}$ for $i\in \{1,\ldots,t\}$, and states ${|\psi\rangle}\in{\mathcal{C}}$ and ${|{\textsc{aux}}\rangle}\in \otimes_i {\mathcal{H}}'_{i} $ such that $$\big\| (\otimes_i V_i){|\Psi\rangle} - {|\psi\rangle} {|{\textsc{aux}}\rangle} \big\|^2 \,\leq\, \delta_C\;,$$ and for all $W\in \{X,Z\}$, $${\ensuremath{\mathop{\textsc{E}}_{w\in{\ensuremath{\mathbb{F}_q}}^m}}} \,\sum_{a\in {\ensuremath{\mathbb{F}_q}}}\, \big\|(\otimes_i V_i )(M_{W,w}^a {\otimes}{\ensuremath{\mathop{\rm Id}\nolimits}}) {|\Psi\rangle} -
({\tau}_{W,w}^a {\otimes}{\ensuremath{\mathop{\rm Id}\nolimits}}){|\psi\rangle} {|{\textsc{aux}}\rangle} \big\|^2 \, \leq \,
\delta_C\;.$$
Fix a strategy for the $k$ provers in the test that is accepted with probability at least $1-{\varepsilon}$. Fix any $t\in\{1,\ldots,k\}$. By combining the $(k-1)$ strategies employed by provers $\{1,\ldots,k\}\backslash\{t\}$, when they play the role of the composite prover, into a single strategy, we obtain a two-prover strategy for the test ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$ that has success probability at least $1-{\varepsilon}$. Applying Theorem \[thm:qld\] shows the self-testing claim for the observables applied by prover $t$, when designated as the special prover. The same applies for all $t\in\{1,\ldots,k\}$, proving the theorem.
\[rm:code\_check\_bits\] We record here the bit complexity of the protocol ${\textsc{code-check}}$. The test invokes the composed quantum low-degree test ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$ with $m = \lceil \frac{\log
n}{\log \log n} \rceil$ and $d = \Theta(
\frac{\log^2 n}{ \log \log n} )$. Hence, the number of bits in the verifier’s questions scales as $O(\frac{\log n}{\log \log n} \log q)$, and the number of bits in the provers’ responses scales as $O((\log \log
n)^2 \log q)$, so the overall bit complexity is $O(\frac{\log n}{\log
\log n} \log q)$.
Energy tests {#sec:testing}
============
In the previous section we gave a test that enforces that the provers’ shared state is close to a valid code state of an error-correcting code. In this section, we show how to further test any property of the encoded state that can be expressed in terms of a local Hamiltonian of the appropriate form. We achieve this by using interactive protocols to “command” the provers to measure a subset of the terms of a given Hamiltonian, perform a computation on the measurement outcomes, and return the result. We introduce the required tools from the classical PCP literature in Section \[sec:delcomp\], and adapt them to our setting in Section \[sec:sum-test\] and Section \[sec:multi-basis\]. In Section \[sec:constant\_gap\] we give a protocol to estimate the ground energy of a (not necessarily local) Hamiltonian up to constant precision, provided that the terms of the Hamiltonian have a certain form. As a consequence, we show that is ${\textsc{QMA}}$-hard to approximate the maximum success probability of a nonlocal game (i.e. one-round ${\textsc{MIP}}^*$ protocol) with logarithmic communication, either conditionally on the constraint satisfaction quantum PCP conjecture (Corollary \[cor:qma-generalizedXZ\]), or unconditionally, but under randomized reductions (Corollary \[cor:randomized\]). In Section \[sec:ff\] we use similar tools to give a protocol to estimate the ground energy of a class of (not necessarily local) frustration-free Hamiltonians up to inverse polynomial accuracy.
As a note on terminology, in previous sections, we introduced “tests” for states with certain properties, such as of being a valid codestate. In this section we provide tests for states that encode the answer to a computational problem (in particular, variants of the local Hamiltonian problem). To formulate these tests we use the language of interactive proofs, and often refer to a test for a property as an ${\textsc{MIP}}^*$ protocol for the corresponding computational problem. The notions of a test, a nonlocal game, and a one-round ${\textsc{MIP}}^*$ protocol are roughly synonymous, and their meaning should always be clear from context.
Classical PCPs for linear functions {#sec:delcomp}
-----------------------------------
The codeword test introduced in Section \[sec:code\] gives the verifier the ability to query a prover for a location in the low-degree encoding of the string of outcomes obtained by the prover when measuring all its qudits in the Pauli $X$ or $Z$ basis. For our applications, we would like to have the ability to command the provers to compute more general functions of their measurement outcomes. For example, upon obtaining an outcome $a \in {\ensuremath{\mathbb{F}_q}}^n$, we may want to ask the prover to compute the inner product $a \cdot b$ with a given string $b \in {\ensuremath{\mathbb{F}_q}}^n$ (that may not necessarily correspond to an entry in the low-degree encoding of $a$), agreed on in advance between the prover and the verifier. One approach to doing this is to use the sum-check protocol of [@lund1992algebraic], but this requires a logarithmic number of rounds of interaction, resulting in a polylogarithmic number of bits of communication. To achieve a protocol with logarithmic communication we rely on the following classical PCP construction, that can be extracted from [@BGHSV05].
\[thm:composedpcp\] Let $p = 2$, $n,t\geq 1$ integer such that $t=\Theta(\log\log n)$, and $q = 2^t$. For any $a, b \in {\ensuremath{\mathbb{F}_q}}^n$, there exists a proof $\Pi_{a,b} \in {\ensuremath{\mathbb{F}}}_q^{n'}$, with $n' = O(\operatorname{poly}(n))$, each of whose bits is an ${\ensuremath{\mathbb{F}}}_q$-linear function of $a$, such that the following holds. There exists an efficient test ${\textsc{lin}}(b)$ that uses $O(\log n)$ random bits and reads a total of $O(1)$ bits from $\Pi_{a,b}$ and from the evaluation table of the low-degree extension $g_a$ of $a$, as well as a value $c\in{\ensuremath{\mathbb{F}_q}}$, with the following properties:
1. Completeness: If $b \cdot a = c$ the test accepts with certainty.
2. Soundness: If $b \cdot a \neq c$, for any claimed proof $\Pi$, the test rejects with constant probability.
We use the language of “PCPs of proximity” from [@BGHSV05]. A PCP of proximity (PCPP) consists of an algorithm $V$ to verify that a given input $a \in
\{0,1\}^n$ (called the *assignment*) satisfies a *property* specified by a poly-sized Boolean circuit. The verifier $V$ is given access to $a$ and to an auxiliary proof string $\Phi$ of polynomial length, but is only allowed to query a small (e.g. constant) number of bits of $a$ and $\Phi$. The completeness and soundness requirements on the verifier are that whenever $a$ satisfies the property (the YES case), there exists a proof $\Phi_a$ that convinces the verifier $V$ to accept with certainty, and when $a$ is $(\delta n)$ far in Hamming distance from any string $a'$ satisfying the property (the NO case), then for all proofs $\Phi$, the verifier $V$ rejects with at least constant probability. The parameter $\delta$ is called the *proximity parameter* of the PCPP.
From [@BGHSV05 Theorem 3.3], with the parameter $t$ appearing in that theorem[^12] chosen to be a constant greater than $3$, and for a proximity parameter $\delta = \Theta(1/t)$, there exists a PCPP for properties encoded by circuits of size $O(n)$, consisting of a proof $\Pi$ and verifier $V$ that uses $O(\log^{2/t} n)$ random bits, reads $O(t) = O(1)$ bits of the proof and assignment, and in the NO case rejects with probability $\Omega(1/t)$. Moreover, from the discussion in Section 8.4 of [@BGHSV05] it follows that if the property can be expressed as an AND of linear constraints (i.e. of the form $b \cdot a
= c$ over ${\ensuremath{\mathbb{F}}}_2$), then the bits of the proof string $\Phi$ are ${\ensuremath{\mathbb{F}}}_2$-linear functions of the assignment, and the checks performed by the verifier $V$ are ${\ensuremath{\mathbb{F}}}_2$-linear constraints on $\Phi$.
Ideally, we would like to directly apply this PCPP to check that the string $a \in {\ensuremath{\mathbb{F}_q}}^{n}$, interpreted as a bit string in $\{0,1\}^{nt}$, satisfies the condition $ b \cdot a = c$, which is a linear condition over ${\ensuremath{\mathbb{F}_q}}$. To do this, we need to address two issues. First, the result of [@BGHSV05] applies to ${\ensuremath{\mathbb{F}}}_2$-linear conditions, and the proof string $\tilde{\Pi}$ is an ${\ensuremath{\mathbb{F}}}_2$-linear function of the input, whereas the present theorem requires linearity over ${\ensuremath{\mathbb{F}_q}}$. We resolve this by noting that the ${\ensuremath{\mathbb{F}_q}}$-linear condition $b \cdot a = c$ can be expressed as a conjunction of ${\ensuremath{\mathbb{F}}}_2$-linear conditions ${\mbox{\rm tr}}[(b \cdot a) \chi_i] = {\mbox{\rm tr}}[c \chi_i]$ for every element $\chi_i$ of a self-dual basis for ${\ensuremath{\mathbb{F}}}_q$ over ${\ensuremath{\mathbb{F}}}_2$. Moreover, any ${\ensuremath{\mathbb{F}}}_2$-linear function $f: {\ensuremath{\mathbb{F}}}_2^{tn} \to {\ensuremath{\mathbb{F}}}_2$ can be expressed a function $f: {\ensuremath{\mathbb{F}}}_q^{n} \to {\ensuremath{\mathbb{F}}}_2$ of the form ${\mbox{\rm tr}}[u \cdot a]$ for some $u \in {\ensuremath{\mathbb{F}}}_q^{n}$, and hence can be extended to an ${\ensuremath{\mathbb{F}}}_q$-linear function ${\underline{f}} : {\ensuremath{\mathbb{F}}}_q^n \to {\ensuremath{\mathbb{F}}}_q$ given by ${\underline{f}} = u \cdot a$. Applying this extension to each bit of $\Phi$, we can construct a proof $\Pi_{a,b} \in {\ensuremath{\mathbb{F}_q}}^{\operatorname{poly}(n)}$, such that each entry of $\Pi_{a,b}$ is an ${\ensuremath{\mathbb{F}_q}}$-linear function of $a$, and from which the PCPP verifier can recover the original proof $\Phi$ by taking the trace of each entry. Since $\tilde{\Pi}$ could be verified by querying a constant number of bits, $\Pi$ can be verified by querying a constant number of ${\ensuremath{\mathbb{F}_q}}$-valued entries.
The second issue concerns the soundness of the proof system. The statement of the present theorem requires soundness to hold against all non-satisfying $a$, not just those that satisfy the promise of the PCPP. Thus, instead of applying the PCPP directly to $a$, we apply it to the evaluation table of the low-degree encoding $g_a$ of the assignment, which has length $O(n \log n)$. The condition $b \cdot
a = c$ can be expressed as a linear condition on the evaluation table of $g_a$. Moreover, if $b \cdot a \neq c$, then the encoding $g_a$ differs from the encoding $g_{a'}$ of any $a'$ such that $b \cdot a' = c$ on at least a constant fraction of positions. (This follows from the Schwartz-Zippel lemma: if $a \neq a'$, then $g_a - g_{a'}$ is a nonzero polynomial and hence by Lemma \[lem:sz\], it cannot be $0$ on more than $d/|{\ensuremath{\mathbb{F}_q}}|$ fraction of the points.) Hence, the soundness promise holds on the encoded input $g_a$. Finally, to check that the part of the proof that corresponds to $g_a$ is a valid low-degree encoding, the verifier executes a standard low-degree test (such as the PCP version of the test ${\textsc{c-lowdeg}}$); this can be done using a constant number of additional queries to the evaluation table of $g_a$ (provided restrictions to lines and planes are included).
A test for non-local Pauli observables {#sec:sum-test}
--------------------------------------
Theorem \[thm:composedpcp\] specifies a test certifying that $a \cdot b =
c$, given $O(1)$ queries to a proof $\Pi_{a,b}$ whose bits are ${\ensuremath{\mathbb{F}}}_q$-linear functions of $a$. Based on this test, in Figure \[fig:sumgame\] we give a multiprover protocol ${\textsc{sum}}(C,W,b)$ in which the verifier commands one of $k$ provers (supposedly) sharing an encoding of a state ${|\psi\rangle}$ according to $\mathcal{C}$ to measure their share of the state in a specified basis ($X$ or $Z$) to obtain an assignment $a$, and report the value $a \cdot b$. The verifier checks that this value was computed correctly by using the test ${\textsc{lin}}(b)$ from Theorem \[thm:composedpcp\], together with the guarantees of the low-degree test. The test ${\textsc{lin}}(b)$ requires a constant number of queries to both $g_a$ and $\Pi_{a,b}$. In order to aggregate these queries, the verifier first asks the provers to encode $\Pi_{a,b}$ as a low-degree polynomial $h$; to query a constant number of entries of $\Pi_{a,b}$ the verifier then asks for the restriction $h|_s$ of $h$ to a curve $s$ that goes through all points to be queried. However, the number of bits required to specify the restriction $h|_s$ is, for our choice of parameters, polylogarithmic in $n$. To get around this we apply composition, in a similar way to the composed low degree test (Theorem \[thm:2ml\]). For concreteness, we explicitly state how to do this, following the variable substitution technique in [@Vidick13xor Section 3.1.2], which appeared earlier in [@DFKRS11 Section 4.4].
\[def:variable-substitution\] Define the substitution map $$\label{eq:substitution}
\#_d : {\ensuremath{\mathbb{F}_q}}\to {\ensuremath{\mathbb{F}_q}}^{\mu(d)}\;, \quad \#(x) = (x, x^2, x^4, \dots, x^d)\;,$$ where $\mu(d) = 2 \lceil \log (d+1) \rceil$. \[def:variable-substition\]
Under this map, any univariate degree-$d$ polynomial $f(x)$ can be viewed as a degree $\delta(d) = O(\log d)$, $\mu(d)$-variate polynomial $f(\#x)$, by formally identifying powers of $x$ with products of the substituted variables. Thus, instead of querying for the restriction $h|_s$, we view this restriction itself as a multivariate polynomial over ${\ensuremath{\mathbb{F}_q}}^{\mu(d)}$, and query the restriction of that polynomial to a curve over ${\ensuremath{\mathbb{F}_q}}^{\mu(d)}$. This can be described in logarithmically fewer bits. The precise form of the queries we make to the prover is specified in Figure \[fig:sumgame\].
------------------------------------------------------------------------
\
Test ${\textsc{sum}}(C,W, b = \{b_1,\ldots,b_k\})$:\
The verifier sends the basis label $W$ to all provers. In the test, the verifier sends pairs of questions, generally formatted as in the test ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$ and ${\textsc{c-lowdeg}}^{(2)}(m',d,q)$. We write the first question as $s$, and the second as $s'$. Note that, as in ${\textsc{q-lowdeg}}(m,d,q)$, $s$ (resp. $s'$) itself can consist of a point in ${\ensuremath{\mathbb{F}_q}}^m$ (resp. ${\ensuremath{\mathbb{F}_q}}^{m'}$), or a pair $(s_1,s_2)$ (resp. $s'_1,s'_2$) of subspaces.
1. Send the special prover a question $s=(s_1,s_2)$ distributed as in ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$ (conditioned on the basis choice $W$ having been made). Receive a polynomial $r\in\deg_d(s_2)$. Send the composite prover the composite query ${{\color{MidnightBlue} \bm{(b_{t'}, (s, s'))}}}$ consisting of the vector $b_{t'}$ as well as a question $(s,s')$, where $s'$ is distributed as in ${\textsc{c-lowdeg}}^{(2)}(m',d,q)$. Receive the composite answer ${{\color{MidnightBlue} \bm{(r',r'')}}}$. Reject if $r \neq {{\color{MidnightBlue} \bm{r'}}}$.
2. Send $b_{t}$, where $t$ is the index of the special prover, to both provers. Execute the tests ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$ and ${\textsc{c-lowdeg}}^{(2)}(m',d,q)$ in parallel. Accept if and only if the provers’ answers pass both tests.
3. Send $b_{t'}$ to both provers. Simulate the test ${\textsc{lin}}(b_{t'})$ from Theorem \[thm:composedpcp\] to obtain a tuple $(x_1,\ldots,x_\ell,i_1,\ldots,i_{\ell'})$ of queries, where $x_i \in {\ensuremath{\mathbb{F}_q}}^m$ are queries to $g_a$ and $i_j$ are indices of bits to be queried in the PCPP proof $\Pi_{a,b}$. Let $s_1$ be a constant-degree curve in ${\ensuremath{\mathbb{F}_q}}^m$ that goes through all the $x_i$, i.e. a constant-degree polynomial $s_1: {\ensuremath{\mathbb{F}_q}}\to {\ensuremath{\mathbb{F}_q}}^m$ whose image contains each point $x_i$, and likewise $s'_1$ a constant-degree curve in ${\ensuremath{\mathbb{F}_q}}^{m'}$ that goes through all ${\pi}(i_{j})$. Moreover, let $s_2$ be a constant-degree curve in ${\ensuremath{\mathbb{F}_q}}^{\mu(d)}$ that goes through all of the points $\#(s_1^{-1}(x_i))$, and let $s'_2$ be a constant-degree curve in ${\ensuremath{\mathbb{F}_q}}^{\mu(d')}$ that goes through all of the points $\#((s'_1)^{-1}({\pi}(i_{ij})))$, where $\#$ and $\mu(\cdot)$ are as in Definition \[def:variable-substitution\]. Perform one of the following tests with probability $\frac{1}{2}$ each.
1. Choose uniformly random points $y$ on $s_1$ and $y'$ on $s'_1$. Send $(y, y')$ to the special prover and ${{\color{MidnightBlue} \bm{((s, \#y), (s', \#y'))}}}$ to the composite prover. Receive answers $(\alpha, \beta) \in {\ensuremath{\mathbb{F}_q}}^2$ and ${{\color{MidnightBlue} \bm{(\gamma, \delta)}}}
\in {\ensuremath{\mathbb{F}_q}}^2$, respectively. If $\alpha = {{\color{MidnightBlue} \bm{\gamma}}}$ and $\beta =
{{\color{MidnightBlue} \bm{\delta}}}$, then accept, else reject.
2. Send $((s_1,s_2), (s'_1, s'_2))$ to the special prover, and two points ${{\color{MidnightBlue} \bm{(z,z')}}}$ to the composite prover, where $z$ is uniformly random in $s_2$ and $z'$ is uniformly random in $s'_2$. Receive from the special prover a pair of polynomials $(r, r')$, where $r$ is $\mu(d)$-variate and $r'$ is $\mu(d')$-variate, and from the composite prover a pair of values ${{\color{MidnightBlue} \bm{(\alpha, \beta)}}} \in {\ensuremath{\mathbb{F}_q}}^2$. If the answers are consistent on points $z, z'$ (i.e. $r(z) = {{\color{MidnightBlue} \bm{\alpha}}}$ and $r'(z') = {{\color{MidnightBlue} \bm{\beta}}}$) and if the entries of $\Pi$ and $g$ decoded from the answers $r$ and $r'$ on $s$ and $s'$ would be accepted in the test ${\textsc{lin}}(b)$, then accept, else reject.\
4. For $j\in\{1,\ldots,k\}$, send the $j$-th prover the vector $b_j$ and a query $((s_1,
s_2), (s'_1, s'_2))$ chosen as in part (c). Receive from each prover a value $c_j\in{\ensuremath{\mathbb{F}_q}}$, as well as a pair of polynomials $(r_j, r_j')$. If for each prover $j$, the entries of $\Pi$ and $g$ decoded from $r_j$ and $r_j'$ would be accepted in the test ${\textsc{lin}}(b_j)$, then return $\omega^{{\mbox{\rm tr}}[\sum_{j=1}^{k} c_j]}$.
------------------------------------------------------------------------
Before stating the completeness and soundness properties of the test ${\textsc{sum}}(C,W,b)$, we state the strategy followed by the honest prover.
\[def:sumgame-honest\] In the game ${\textsc{sum}}(C,W, b=\{b_1,\ldots,b_k\})$, the honest strategy is defined as follows:
- State: the provers share a state ${|\Psi\rangle}$ which is a qubit-by-qubit encoding of a state ${|\psi\rangle} \in ({\ensuremath{\mathbb{C}}}^q)^{\otimes
n}$ using the CSS code $\mathcal{C}$ derived from the self-dual classical code $C$.
- Measurements: each prover performs a measurement of each of its qudits in the basis $W$, to obtain as outcome a string $a\in{\ensuremath{\mathbb{F}_q}}^n$. Using $a$, the prover determines a polynomial $g_a : {\ensuremath{\mathbb{F}_q}}^m \to {\ensuremath{\mathbb{F}_q}}$. In addition, the prover may be sent a vector $b_j\in{\ensuremath{\mathbb{F}_q}}^n$. If this is the case, the prover replies with $c=a\cdot b_j\in{\ensuremath{\mathbb{F}_q}}$. In addition, it computes a polynomial $h_a : {\ensuremath{\mathbb{F}_q}}^{m'}\to {\ensuremath{\mathbb{F}_q}}$ that is a low-degree extension $h_a = g_{\Pi_{a,b_j}}$ for the PCPP proof $\Pi_{a,b_j}$ that verifies $c=a\cdot b_j$, as described in Theorem \[thm:composedpcp\]. Finally, the prover is sent a pair $(s,s')$. Here $s$ (resp. $s'$) may be: a question from the tests ${\textsc{q-lowdeg}}^{(2)}$ (resp. ${\textsc{c-lowdeg}}^{(2)}$), or, in part (c) of the test, a single point in ${\ensuremath{\mathbb{F}_q}}^{m}$ (resp. ${\ensuremath{\mathbb{F}_q}}^{m'})$, or a specification of a curve of constant degree in ${\ensuremath{\mathbb{F}_q}}^m$ (resp. ${\ensuremath{\mathbb{F}_q}}^{m'}$) together with a point or a curve in the space ${\ensuremath{\mathbb{F}_q}}^{\mu(d)}$ (resp. ${\ensuremath{\mathbb{F}_q}}^{\mu(d')}$). The honest prover responds with the appropriate restriction of $g_a$ (resp. $h_a$), composed with the variable substitution map whenever appropriate.
\[thm:sum-game\] Let $C$ be a $[k,k']$ weakly self-dual linear code. Let $n$ be an integer, and $d,m,q$ integer such that $q = 2^t$ and $d,m,\log q$ are polynomially bounded in $n$. Let ${\varepsilon}\geq d/q$. Let $b_1, \dots, b_k \in{\ensuremath{\mathbb{F}_q}}^n$. The procedure ${\textsc{sum}}(C,W, \{b_1,
\dots, b_k\})$ is a 1-round, $k$-prover interactive protocol with the following completeness and soundness properties.
1. *Completeness:* If the provers follow the strategy introduced in Definition \[def:sumgame-honest\], they pass the test with certainty.
2. *Soundness:* Suppose that a strategy for the provers is accepted with probability at least $1-{\varepsilon}$ in ${\textsc{sum}}(C,W, \{b_1, \dots, b_k\})$. Suppose further that the restriction of the strategy to questions formatted as in ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$ succeeds in that test with probability at least $1-{\varepsilon}$. Then there is a $\delta = \operatorname{poly}({\varepsilon}, \delta_{C})$, where $\delta_{C}$ is as specified in Theorem \[thm:codeword\_test\], such that the following holds. There exists an encoded state ${|\Psi\rangle} \in
({\ensuremath{\mathbb{C}}}^q)^{\otimes nk}$ such that the value returned by the verifier in step (d) of the protocol has expectation that is within $\delta$ of ${\langle\Psi|}\otimes_{i=1}^{k} {\tau}_{W}( b_i){|\Psi\rangle}$.
\[rk:sum-complexity\] The number of bits communicated to a prover in ${\textsc{sum}}$ is at most the number of bits needed in the (composed) low-degree test, plus the number of bits needed to specify a constant-degree curve over ${\ensuremath{\mathbb{F}_q}}^n$. By Theorem \[thm:qld\], both are at most $O(m \log q) = O(\frac{\log n}{\log \log n} \log q)$. The number of bits in the provers’ answers is at most the maximum of the number of bits needed in the (composed) low-degree test, and the the number of bits needed to specify a degree-$\delta(d) = O(\log d)$ polynomial restricted to a constant-degree curve. By Theorem \[thm:qld\], the former is at most $O(\log^2(d) \log(q))$, while the latter is at most $O(\log(d)\log(q))$.
Completeness follows from the definition of the honest strategy, Lemma \[lem:composite-linear\] and the completeness of ${\textsc{lin}}(b)$ as described in Theorem \[thm:composedpcp\].
We show soundness. A strategy for a prover in the test ${\textsc{sum}}(C,W,b)$ consists of a family of measurements $\{M_{b,s,s'}^{r,r'}\}$ used in response to questions of the form $(s,s')$. As the subscripts indicate, these measurements depend on the vector $b$ received by the prover. In addition, part (a) of the test involves cross-checking these measurements with a strategy for the quantum low-degree test ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$, in which the players are *not* told $b$. The strategy used for this test is described by measurements $\{N_{s}^{r}\}$, which do not depend on $b$.
We show the soundness of the test in two steps. First, we note that success in parts (a) and (b) of ${\textsc{sum}}(C,W, \{b_1, \dots, b_k\})$ implies that the measurements used by the players are close to product form: $$M_{b,s,s'}^{rr'} \approx_{\operatorname{poly}({\varepsilon})} A_{b,s}^r B_{b,s'}^{r'},$$ where $A_{b,s}^r = {\ensuremath{\mathop{\textsc{E}}_{s'}}}\sum_{r'} M_{b,ss'}^{rr'}$ and $B_{b,s'}^{r'} = {\ensuremath{\mathop{\textsc{E}}_{s}}}
\sum_{r} M_{b,ss'}^{rr'}$ are the measurements obtained by marginalizing the joint measurements $M_{b,ss'}^{r''}$. This follows from a standard “oracularization” analysis, similar to the one in the proof of Theorem \[thm:com\_test\].
From success in part (a) of the test it follows that the measurements $A_{b,s}^r$ must be $\operatorname{poly}({\varepsilon})$-close to the measurements $N_{s}^r$ used in the test ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$. By applying Theorem \[thm:codeword\_test\] to the strategy $N_{s}^r$, which is independent of $b$, this implies that, after applying a suitable isometry, $$A_{b,s}^r \approx_{\operatorname{poly}({\varepsilon}, \delta_C)} {\tau}_{W, s}^{r} \,=\,\sum_{a:\, (g_a)_{|s}=r} {\tau}_W^a \;,$$ where ${\tau}_{W,s}^{r}$ is the measurement used in the honest strategy for ${\textsc{q-lowdeg}}^{(2)}(m,d,q)$ as defined in Definition \[def:pauli-strategy\]. Moreover, this implies that the shared state is $\operatorname{poly}({\varepsilon}, \delta_C)$-close, under the isometry, to some encoded state ${|\Psi\rangle}$.
Moreover, success in part (b) of the test implies that the measurements $\{B_{b,s'}^{r'}\}$ must constitute a good strategy for the classical low-degree test ${\textsc{c-lowdeg}}^{(2)}(m',
d,q)$ (in which the players *are* informed of $b$ and $j$). This implies that there exists a measurement $\{B_{b}^{h}\}$ with outcomes $h$ that are $m'$-variate degree-$d$ polynomials over ${\ensuremath{\mathbb{F}_q}}$, such that $$B_{b, s'}^{r'} \approx \sum_{h : h|_{s'} = r'} B_{b}^{h}\;.$$
Finally, from part (c), we conclude that the set of outcomes $(g,h)$ that are such that the string $\Pi \in {\ensuremath{\mathbb{F}_q}}^{n'}$ for which $h$ is the low-degree extension, together with $g$, are not accepted by the PCPP verifier, must have low probability of being obtained when performing the corresponding measurement. Hence, by the soundness of the PCPP from Theorem \[thm:composedpcp\], it follows that, for each prover $j$, the polynomial $g$ obtained by this prover encodes an assignment $a_j$ which satisfies $b_j \cdot a_j = c_j$ with high probability. This implies that the expectation value of the output $\omega^{{\mbox{\rm tr}}[\sum_j c_j]}$ computed by the verifier in part (d) is close to the expectation value of the Pauli observable $ \otimes_{i=1}^{k} {\tau}_{W}(b_i)$, as desired.
Evaluating multiple-basis operators {#sec:multi-basis}
-----------------------------------
Building on the test ${\textsc{sum}}$, we design a test ${\textsc{eval}}$ that can measure operators which are tensor products of both $X$- and $Z$-basis Paulis. In anticipation of our application to testing Hamiltonians, we describe ${\textsc{eval}}$ as taking as input a distribution over logical operators to be measured. The process of translating these logical operators into physical operators to be measured by each prover is bundled into the test.
------------------------------------------------------------------------
\
Test ${\textsc{eval}}_\xi(C,\pi,{\overline{x}},{\overline{z}})$: Given is a distribution $\pi$ over $\{S\subseteq\{1,\ldots,n\}\}\times {\ensuremath{\mathbb{F}_q}}^n\times\{\pm 1\}$, a $[k,k']$ weakly self-dual linear code $C$, ${\overline{x}},{\overline{z}}\in{\ensuremath{\mathbb{F}_q}}^k$ such that ${\overline{X}}={\tau}_X({\overline{x}})$ and ${\overline{Z}}={\tau}_Z({\overline{z}})$ are ${\tau}_X$ and ${\tau}_Z$ logical operators for ${\mathcal{C}}$ respectively, and a parameter $0\leq \xi \leq 1$.\
The verifier samples $(S,u,\epsilon)\sim \pi$. The verifier performs one of the following two tests, the first with probability $(1-\xi)$, and the second with probability $\xi$:
1. (Test) Let $u_S$ and $u_{{\overline{S}}}$ be the substrings of $u$ indexed by $S$ and ${\overline{S}}$ respectively.
1. Do either of the following, with probability $1/2$ each:
1. Send either $S$, ${\overline{S}}$, $\emptyset$ or $\{1,\ldots,n\}$ to all provers, with probability $1/4$ each. Execute the test ${\textsc{code-check}}(C,n)$.
2. Do as in 1., except the sets sent to the special and composite provers are complemented ($S,{\overline{S}}$ or $\emptyset,{\overline{\emptyset}}$), and so is the choice of basis $W$ in ${\textsc{code-check}}(C,n)$.
2. Send $S$ to the special prover, and either $\emptyset$ or ${\overline{\emptyset}}$ to the composite prover. Choose a random vector $v \in {\ensuremath{\mathbb{F}_q}}^k$ in the column span of $G$, as in Definition \[def:queries\]. Execute the test ${\textsc{sum}}(C,X,\{v_1 s, v_2s, \dots, v_k s\})$ on query string $s=u_{{\overline{S}}}$ (case $\emptyset$) or $s=u_{{S}}$ (case ${\overline{\emptyset}}$). Reject if the test ${\textsc{sum}}$ rejects, or if the returned value is not $1$.
2. (Eval) Let $u_S$ and $u_{{\overline{S}}}$ be the substrings of $u$ indexed by $S$ and ${\overline{S}}$ respectively. For $i\in \{1,\ldots,k\}$ let $u_i$ be the string with substrings $u_{S,i}= {\overline{x}}_i u_S$ indexed on $S$ and $u_{{\overline{S}},i} = {\overline{z}}_i u_{{\overline{S}}}$ indexed on ${\overline{S}}$. Send all provers the set $S$. Execute part (d) of ${\textsc{sum}}(C,X,\{u_{1},
\dots u_{k}\})$ with all the provers. Let $E$ be the returned value. Return $\epsilon \cdot E$.
------------------------------------------------------------------------
The procedure ${\textsc{eval}}_\xi(C,\pi,{\overline{x}},{\overline{z}})$ is described in Figure \[fig:eval\]. It takes as input a $[k,k']$ weakly self-dual linear code $C$, a distribution[^13] $\pi$ over $\{S \subseteq\{1,\ldots,n\}\}\times {\ensuremath{\mathbb{F}_q}}^n\times\{\pm 1\}$, and strings $ {\overline{x}},{\overline{z}}\in{\ensuremath{\mathbb{F}_q}}^k$ such that the operators ${\overline{X}}={\tau}_X({\overline{x}})$ and ${\overline{Z}}={\tau}_Z({\overline{z}})$ are respectively logical ${\tau}_X$ and ${\tau}_Z$ operators for the CSS code ${\mathcal{C}}$ associated with $C$. To any triple $(S,u,{\varepsilon})$ in the support of $\pi$ we associate a qudit Pauli operator $$\label{eq:def-hs}
h_{S}(u) \,=\, \otimes_{i\in S} {\tau}_X(u_i) \otimes_{i\in {\overline{S}}} {\tau}_Z(u_i)\;.$$
The procedure is divided into a “test” and an “eval” part. The relative weight given to each part is governed by the parameter $0\leq
\xi \leq 1$. The goal of the testing part is to ensure that the provers’ answers in the evaluation part are distributed according to a distribution that can be obtained by performing Pauli measurements on the encoding of a fixed $n$-qudit state ${|\psi\rangle}$ using the CSS code ${\mathcal{C}}$ associated with $C$, where the Pauli ${\tau}_X$ and ${\tau}_Z$ are encoded using ${\overline{X}}$ and ${\overline{Z}}$ respectively. The test is formulated using the notion of “special” and “composite” prover introduced in Section \[sec:code-protocol\]; recall the scheme for distributing queries to the composite prover specified in Definition \[def:queries\].
\[lem:eval\] Let $C$ be a $[k,k']$ weakly self-dual linear code and ${\overline{X}}={\tau}_X({\overline{x}})$, ${\overline{Z}}={\tau}_Z({\overline{z}})$ logical ${\tau}_X$ and ${\tau}_Z$ operators for the associated CSS code ${\mathcal{C}}$ respectively. Let $n$ be an integer and $\pi$ a distribution over $\{S\subseteq\{1,\ldots,n\}\} \times {\ensuremath{\mathbb{F}_q}}^n \times\{\pm 1\}$, and let $\xi$ be a real number between $0$ and $1$. Then the procedure ${\textsc{eval}}_\xi(C, \pi , {\overline{x}},
{\overline{z}})$ has the following properties:
- *(Completeness)* For any state ${|\psi\rangle} \in ({\ensuremath{\mathbb{C}}}^q)^{\otimes n}$ there is a strategy for the provers that is accepted with probability $1$ in part (a) of the test, and such that the value returned by the verifier in part (b), conditioned on the choice of $(S,u,\epsilon)$, has expectation $ \epsilon\cdot \Re({\langle\psi|}h_S(u){|\psi\rangle})$, where $h_S(u)$ is defined in .
- *(Soundness)* Suppose a strategy for the provers is accepted with probability at least $1-{\varepsilon}$ in each of the “test” rounds of the procedure ${\textsc{eval}}_\xi(C,\pi,{\overline{x}},{\overline{z}})$. Then there exists a state ${|\psi\rangle} \in ({\ensuremath{\mathbb{C}}}^q)^{\otimes n}$ such that, on expectation over $(S,u,\epsilon)\sim \pi$, the value returned by the verifier in step (b) of the protocol, conditioned on the choice of $(S,u,\epsilon)$, has expectation that is within $\operatorname{poly}({\varepsilon},\delta_{C})$ of $ \epsilon\cdot \Re({\langle\psi|}h_S(u){|\psi\rangle})$, where $\delta_{C}$ is as specified in Theorem \[thm:codeword\_test\].
\[rm:eval\_bits\] The amount of bits communicated to any prover in the procedure ${\textsc{eval}}$ is at most the number of bits necessary to communicate an element sampled from $\pi$ (which scales as the logarithm of the support size of $\pi$) plus the maximum of the number of bits communicated in either the ${\textsc{sum}}$ or ${\textsc{code-check}}$ tests. It follows from Remark \[rk:sum-complexity\] that the former requires $O(m \log q) = O(\frac{\log n}{\log \log n} \log q )$ bits, and from Remark \[rm:code\_check\_bits\] that the latter similarly requires $O(\frac{\log n}{\log \log n} \log q)$ bits.
We first show the completeness property. Let ${|\psi\rangle} \in ({\ensuremath{\mathbb{C}}}^q)^{\otimes n}$ and ${|\Psi\rangle} \in ({\ensuremath{\mathbb{C}}}^q)^{\otimes nk}$ a qudit-by-qudit encoding of ${|\psi\rangle}$ according to ${\mathcal{C}}$. The strategy for the provers uses ${|\Psi\rangle}$ as a shared state. When a prover is sent a set $S$, it immediately applies an $F$ gate to all qudits in $S$. If sent a query from the test ${\textsc{code-check}}$, it applies the honest strategy for the test, as described in Lemma \[lem:code-completeness\]. If asked to execute the protocol ${\textsc{sum}}(C,X,\{s_1, \dots, s_k\})$, on a query string $s_j \in{\ensuremath{\mathbb{F}_q}}^n$, it measures all its qubits in the $X$ basis to obtain a string $a\in {\ensuremath{\mathbb{F}_q}}^n$ and then executes the protocol honestly, following the strategy specified in Definition \[def:sumgame-honest\].
We verify that this strategy succeeds in each of the sub-tests of part (a) with probability $1$. For (i) this is a direct consequence of success in ${\textsc{code-check}}$ and the fact that the code is self-dual; application of $F$ merely exchanges the role of the $X$ and $Z$ bases for all provers. For (ii) this follows from the completeness property in Theorem \[thm:sum-game\].
Regarding soundness, assume that a strategy for the provers succeeds with probability at least $1-{\varepsilon}$ in each of the sub-tests executed in part (a). Using (i)1., by [Theorem \[thm:codeword\_test\]]{}, for each $S$ the associated strategy is isometric to a $\delta_S$-extension of a Pauli ${\mathcal{C}}$-codeword strategy, where ${\ensuremath{\mathop{\textsc{E}}_{S\sim\pi}}}\delta_S=\delta_{C}({\varepsilon},q)$, as stated in the theorem. Note that in general the implied isometry depends on the choice of $S$. For the remainder of the proof, assume that a prover applies observable ${\tau}_{S,W}(w)$ when sent a query of the form $(W,w)$, after having been told the set $S$. Using the symmetry in the tests we may also assume that ${\tau}_{S,X}(w)={\tau}_{{\overline{S}},Z}(w)$ for all $w$ and $S$.
Next consider part (ii). Since the composite prover is not sent the set $S$, the value ${{\color{MidnightBlue} \bm{c}}}$ it claims also does not depend on $S$. Since the only way for the test ${\textsc{sum}}$ to return $1$ in part (d) of the test is for the values ${{\color{MidnightBlue} \bm{c}}}$ and $c$ to have identical trace, from the previous analysis it follows that we may assume that the value $c=a\cdot s$ returned by the special prover is obtained from the outcome obtained by a measurement of the composite prover in the eigenbasis of ${\tau}_{\emptyset,X}$ (case $s=u_S$) or ${\tau}_{\emptyset,Z}$ (case $s=u_{{\overline{S}}}$).
As a result, the distribution of claimed values obtained in part (b) of the test is close to what would be obtained if all provers were to perform a measurement in the eigenbasis of ${\tau}_{\emptyset,X}$ for the qudits in $S$, and ${\tau}_{\emptyset,Z}$ for the qudits in ${\overline{S}}$. By definition of the strings $u_{S,i}$ and $u_{{\overline{S}},i}$ that are actually sent to prover $i$, the resulting physical observable implements the logical $n$-qudit observable $h_S(u)$, as desired.
Efficient energy test for local Hamiltonians {#sec:constant_gap}
--------------------------------------------
We show how to use the ${\textsc{eval}}$ test to estimate the energy of a Hamiltonian up to constant accuracy, provided that the terms of the Hamiltonian are (not necessarily local) Pauli operators of a particular form, which we call $Y$-free. From this, we deduce two results in the direction of the quantum games PCP conjecture: we show ${\textsc{QMA}}$-hardness of approximating the maximum success probability of a nonlocal game with logarithmic communication, either conditionally, assuming the Local Hamiltonian problem is QMA-complete for a constant-error approximation (Corollary \[cor:qma-generalizedXZ\]), or unconditionally, under randomized reductions (Corollary \[cor:randomized\]).
\[def:gen-h\] Let $n$ be an integer and $q$ a prime power. We say that a Hamiltonian $H$ on $({\ensuremath{\mathbb{C}}}^q)^{\otimes n}$ is a Hamiltonian in $Y$-free form if $H$ can be expressed as $$\label{eq:gen-h}
H = {\ensuremath{\mathop{\textsc{E}}_{S \subseteq \{1, \dots, n\}, u \in {\ensuremath{\mathbb{F}_q}}^n}}} \,\frac{ \alpha_{S,u}
}{2}\,\big(h_S(u) + h_S(u)^\dagger\big)\;,$$ where each term $h_S(u)$ is a Pauli operator of the form described in , the weights $\alpha_{S,u} \in {\ensuremath{\mathbb{R}}}$ are such that $|\alpha_{S,u}|\leq 1$ for all $(S,u)$, and the expectation is taken according to a distribution $\pi$ with polynomial-size support.
The term $Y$-free refers to the fact that there are no Pauli $Y$ operators (i.e. products of $X$ and $Z$ acting on the same qudit) in any of the terms. As motivation for considering this class of Hamiltonians, we remark that in the case of $q = 2$, i.e. for qubits, our definition of $Y$-free Hamiltonians includes the generalized XZ model of [@CM13].[^14] In that reference, it was shown that the local Hamiltonian problem for the XZ model is ${\textsc{QMA}}$-complete, for an inverse-polynomial promise gap. The class of $Y$-free Hamiltonians is considerably more general as it imposes no limits on the locality of the terms in the Hamiltonian, and accommodates qudits of dimension up to $\operatorname{poly}(\log n)$.
The following lemma shows that it is possible to embed qubit Hamiltonians of the XZ model into qudit $Y$-free Hamiltonians with local dimension $q = 2^t$ for any $t$. This will be useful since the low-degree test requires fields of large enough size.
Given any Hamiltonian $H$ in the XZ model over qubits, and $q = 2^t$, there exists a Hamiltonian $H'$ in $Y$-free form over qudits of dimension $q$ with the same spectrum (up to multiplicity) as $H$. \[lem:embed\_qubit\_hamiltonian\]
Recall from Section \[sec:qauli\] that when $q = 2^t$ for any $t$, the field ${\ensuremath{\mathbb{F}_q}}$ admits a self-dual basis over ${\ensuremath{\mathbb{F}}}_2$, and a qudit of dimension $q$ decomposes as a tensor product of $t$ qubits. Moreover, qubit Pauli operators $\{\sigma_{W, \ell}\}_{\ell \in \{1, \dots, t\}}$ acting on a single “sub-qubit” can be recovered from the qudit Paulis by the formula $$\sigma_{W, \ell} = {\tau}_W(b_\ell),
\label{eq:embed_qubit_pauli}$$ where $\{b_1, \dots, b_t\}$ is a self-dual basis for ${\ensuremath{\mathbb{F}_q}}$ over ${\ensuremath{\mathbb{F}}}_2$. Extending this to multiple qudits, we can view a system of $n$ qudits of dimension $q=2^t$ each as a collection of $tn$ qubits of dimension $2$ each. Let us index these qubits by pairs $(i, \ell)$, where $i
\in \{1, \dots, n\}$ labels a qudit, and $\ell \in \{1, \dots, t\}$ labels a sub-qubit of the $i$th qudit. Then, given a qubit Hamiltonian $H$ over $n$ qubits, we construct the desired $H'$ by, for each qubit $X$ or $Z$ Pauli term in $H$ acting on qubits $i, j$, including the corresponding Pauli term acting on qubits $(i, 1)$ and $(j, 1)$. By this can be implemented by a generalized Pauli ${\tau}_X$ or ${\tau}_Z$ acting on qudits $i$ and $j$, and hence $H'$ is in $Y$-free form. Moreover, $H'$ decomposes as a tensor product $H
{\otimes}{\ensuremath{\mathop{\rm Id}\nolimits}}$ of $H$ acting on qubits $(1,1), (2,1), \dots, (n,1)$, and ${\ensuremath{\mathop{\rm Id}\nolimits}}$ acting on the remaining qubits. Hence $H'$ has the same spectrum (up to multiplicity) as $H$.
Given a Hamiltonian $H$ in $Y$-free form provided as input, we describe a test whose maximum success probability is linearly related to the minimum energy of the Hamiltonian. The test requires the honest provers to share an encoding of a minimum-energy eigenstate of $H$ according to a quantum code ${\mathcal{C}}$, and relies on the procedure ${\textsc{eval}}$ described in Figure \[fig:eval\] to estimate the energy of the encoded state under $H$. The energy test is described in Figure \[fig:energy\], and its guarantees are stated in Theorem \[thm:energy\] below.
------------------------------------------------------------------------
\
Test ${\textsc{energy}}_\xi(C,H)$: Given as input is a Hamiltonian in $Y$-free form, specified by real coefficients $\{\alpha_{S,u}\}$ as in , a $[k,k']$ weakly self-dual linear code $C$ such that the associated CSS code ${\mathcal{C}}$ encodes at least one logical qubit, and a parameter $0\leq \xi \leq 1$.
1. Let ${\overline{x}},{\overline{z}}\in{\ensuremath{\mathbb{F}_q}}^n$ be such that ${\tau}_X(a{\overline{x}})$ and ${\tau}_Z(b{\overline{z}})$ are logical ${\tau}_X(a)$ and ${\tau}_Z(b)$ operators for the code ${\mathcal{C}}$, respectively.
2. Let $\pi$ be the distribution over $\{S \subseteq \{1, \dots, n\}\} \times
{\ensuremath{\mathbb{F}_q}}^n \times \{\pm 1\}$ that is obtained by sampling $(S, u)$ uniformly, and returning $(S,u, \text{sign}(\alpha_{S,u}))$ with probability $|\alpha_{S,u}|$, and a symbol “$\perp$” with probability $1-|\alpha_{S,u}|$.
3. Execute ${\textsc{eval}}_\xi(C, \pi, {\overline{x}},{\overline{z}})$. If the element sampled from $\pi$ is $\perp$, then automatically accept in case part (a) of the test is executed, and reject in case part (b) is executed. Otherwise, if the test returns ACCEPT or REJECT, then accept or reject accordingly. Finally, if the test returns a value $e$ such that $\Re(e)\in [-1,1]$, then accept with probability $\frac{1}{2}(1-\Re(e))$.
------------------------------------------------------------------------
\[thm:energy\] Let $H$ be a Hamiltonian in $Y$-free form, $C$ a weakly self-dual linear code, and $0\leq \eta\leq 2$.
- *(Completeness)* If $\lambda_{\min}(H) \leq -1+\eta$, there is a strategy for the provers such that for any $0\leq \xi\leq 1$ the test ${\textsc{energy}}_\xi(C,H)$ accepts with probability at least $1-\frac{1}{2}\eta\xi$.
- *(Soundness)* If there exists a strategy with probability of success in the test ${\textsc{energy}}_\xi(C,H)$ at least $ 1- {\varepsilon}$, then $\lambda_{\min}(H) \leq -1+\eta'$ for $\eta' = \frac{2{\varepsilon}}{\xi} + \operatorname{poly}(\frac{{\varepsilon}}{1 - \xi},
\delta_C(\frac{{\varepsilon}}{1 - \xi}))$, where $\delta_C$ is as specified in Theorem \[thm:codeword\_test\].
\[thm:energy\_test\]
We show the completeness and soundness properties claimed in Theorem \[thm:energy\] in two separate lemma.
\[lem:energy-completeness\] Let $H$ be a Hamiltonian in $Y$-free form such that $H$ has an eigenstate ${|\psi\rangle}$ with associated eigenvalue $\lambda \in [-1,1]$, and $0\leq\xi\leq 1$. Then for any weakly self-dual linear code $C$ there is a strategy for the provers, based on sharing an encoding of the state ${|\psi\rangle}$ according to ${\mathcal{C}}$, whose success probability in the test ${\textsc{energy}}_\xi(C,H)$ is $ (1-\xi) + \frac{\xi}{2}(1-\lambda)$.
Let ${|\psi\rangle} \in ({\ensuremath{\mathbb{C}}}^q)^{\otimes n}$ be as in the lemma, and ${|\Psi\rangle}\in ({\ensuremath{\mathbb{C}}}^q)^{\otimes kn}$ a qudit-by-qudit encoding of ${|\psi\rangle}$ under the code $C$, where each individual qudit is encoded according to the logical operators ${\overline{x}}$, ${\overline{z}}$ used by the verifier in the test ${\textsc{energy}}$. When sent a query by the verifier, each prover applies the honest strategy in the test ${\textsc{code-check}}$, as specified in Lemma \[lem:code-completeness\]. By definition this strategy succeeds with probability $1$ in part (a) of ${\textsc{eval}}$.
Regarding part (b), it follows from the definition of the distribution $\pi$ and the completeness property of the procedure ${\textsc{eval}}$ stated in Lemma \[lem:eval\] that the probability of accepting in the third step of the procedure ${\textsc{energy}}$, conditioned on part (b) of ${\textsc{eval}}$ being executed by the verifier, is precisely $\frac{1}{2}(1-{\langle\psi|} H {|\psi\rangle})$.
\[lem:energy-soundness\] Let $H$ be a Hamiltonian in $Y$-free form, $0\leq\xi\leq 1$ and $C$ a weakly self-dual linear code. Suppose there exists a strategy for the provers whose success probability in the test ${\textsc{energy}}_\xi(C,H)$ is $1-{\varepsilon}$, for some ${\varepsilon}\leq\xi$. Then $H$ has an eigenvector with energy at most $-1 + \frac{2{\varepsilon}}{\xi} +
\operatorname{poly}(\frac{{\varepsilon}}{1 - \xi},\delta_{C})$.
By definition of the test ${\textsc{energy}}_\xi$, the provers’ strategy must succeed with probability at least $1-\frac{{\varepsilon}}{1-\xi} = 1-{\varepsilon}'$ in part (a) of ${\textsc{eval}}_\xi$. Applying Lemma \[lem:eval\], it follows that the value returned by the verifier in part (b) of the test is a random variable whose expectation is within $\operatorname{poly}({\varepsilon}',\delta_{C})$ of $\frac{1}{2}(1-{\langle\psi|}H{|\psi\rangle})$, for some state ${|\psi\rangle}$. Thus letting $\lambda = {\langle\psi|}H{|\psi\rangle}$ and $p_a$, $p_b$ the provers’ success probability in parts (a) and (b) of the test respectively we have $$\begin{aligned}
p_{\text{success}} &= \frac{(1 - \xi)}{2} p_a + \xi p_b \;,
\end{aligned}$$ thus $ p_{\text{success}} \geq 1 - {\varepsilon}$ implies that $\xi p_b \geq \xi - {\varepsilon}$. Using that $p_b \leq \frac{1}{2}(1-\lambda + \operatorname{poly}({\varepsilon}',\delta_C))$ yields $$\lambda \,\leq\, -1 + \frac{2{\varepsilon}}{\xi} + \operatorname{poly}({\varepsilon}', \delta_{C})\;,$$ giving the conclusion of the lemma.
\[cor:qma-generalizedXZ\] Assume the Local Hamiltonian problem for qubit Hamiltonians in the $XZ$ model with promise gap $b - a = \Omega(1)$ is ${\textsc{QMA}}$-complete. Then, there is a one-round, $7$-prover ${\textsc{MIP}}^*$ protocol for the class ${\textsc{QMA}}$ with $O(\log(n))$-bits of communication and constant completeness-soundness gap.
First, note that by the hardness assumption made in the corollary and Lemma \[lem:embed\_qubit\_hamiltonian\], it follows that estimating the ground energy of qudit Hamiltonians with local dimension $2^t$ in $Y$-free form with $\Omega(1)$ promise gap is ${\textsc{QMA}}$-hard for any choice of $t$. Thus, to establish the conclusion, it suffices to show that there exists an ${\textsc{MIP}}^*$ protocol with the desired parameters for this variant of the Local Hamiltonian Problem.
Let a Hamiltonian in $Y$-free form be given, scaled such that the energy threshold in the YES case is $-1 + \eta$, and in the NO case is $ -1 + \eta'$ with $\eta' - \eta = \Omega(1)$. Furthermore, let $q = 2^t$ where $t = \Theta(\log\log(n))$, and let $C$ be the quadratic residue code from Example \[ex:quad\_res\_code\], which has $k=7$. Using Theorem \[thm:energy\], the test ${\textsc{energy}}_\xi(C,H)$ succeeds with probability $p_{\text{YES}} \geq 1 - \frac{\eta \xi}{2}$ in the YES case, and in the NO case with probability $p_{\text{NO}} \leq 1- {\varepsilon}$ for ${\varepsilon}$ such that $$\frac{2{\varepsilon}}{\xi} + \operatorname{poly}\Big(\frac{{\varepsilon}}{1 - \xi}, \delta_C\Big) = \eta'\;, \label{eq:energy_soundness}$$ where $\delta_C = \max(\operatorname{poly}(\frac{{\varepsilon}}{1 - \xi}), \operatorname{poly}(q^{-1}))$. Denote the difference between these two probabilities by $\Delta$; it is given by $$\begin{aligned}
\Delta&= p_{\text{YES}} - p_{\text{NO}} \notag\\
&\geq {\varepsilon}- \frac{\eta \xi}{2} \notag
\\
&= \frac{\xi}{2}(\eta' - \eta) - \frac{\xi}{2} \operatorname{poly}(\frac{{\varepsilon}}{1 - \xi},
\delta_C)\;. \label{eq:energy_gap}
\end{aligned}$$ For $\Delta$ to be positive it suffices to ensure that the quantity $\operatorname{poly}(\frac{{\varepsilon}}{1 - \xi}, \delta_C)$ is less than, say, $\frac{1}{2}(\eta' - \eta)$, which is a constant. Given the definition of $\delta_C$ and our choice of $q \sim n^{\log n}$, this can be ensured for some constant ${\varepsilon}, \xi$ depending only on $\frac{1}{2}(\eta' -
\eta)$. This results in a constant $\Delta$. Therefore, the energy test ${\textsc{energy}}_\xi(C,H)$ constitutes a ${\textsc{MIP}}^*$ protocol for ${\textsc{QMA}}$ with a constant completeness-soundness gap. Regarding the communication cost, by plugging $q = O(\log \log
n)$ into the bounds for the test ${\textsc{eval}}$ given in Remark \[rm:eval\_bits\], we find that the test ${\textsc{energy}}_\xi(C,H)$ requires $O(\log n)$ bits communication.
Without making any assumptions, we can show a ${\textsc{QMA}}$-hardness result under randomized reductions.
\[cor:randomized\] It is ${\textsc{QMA}}$-hard under poly-time randomized Karp reductions to determine whether the maximum acceptance probability of a one-round ${\textsc{MIP}}^*$ protocol with logarithmic communication is at least $1$ or at most $ \frac{1}{2}$.
The idea for the proof of Corollary \[cor:randomized\] is to start with a ${\textsc{QMA}}$-hard instance of the Local Hamiltonian problem, with inverse-polynomial promise gap, and amplify this gap by taking a tensor power of the Hamiltonian. Expanding the tensor powers results in a Hamiltonian that is an average of exponentially many terms. We then apply the Ahlswede-Winter matrix Chernoff bound to randomly sub-sample a set of terms from the amplified Hamiltonian, yielding a Hamiltonian with polynomially many terms whose ground state energy can be tested using the ${\textsc{energy}}$ test.
Let $H$ be an $n$-qudit Hamiltonian with minimum energy $\lambda_{\min}(H)\geq 0$ and such that $\|H\|\leq 1$. Let $p(n), q(n)$ be polynomials such that $p(n) >
q(n)$ for all $n$. Let $$H' = {\ensuremath{\mathop{\rm Id}\nolimits}}^{\otimes a} - ({\ensuremath{\mathop{\rm Id}\nolimits}}- ( H - a^{-1} {\ensuremath{\mathop{\rm Id}\nolimits}}))^{\otimes a},\qquad \text{where}\qquad a = \Big(\frac{1}{q}-\frac{1}{p}\Big)^{-1}.$$ Then $H'$ is a (non-local) Hamiltonian over $an = O(np(n))$ qudits with norm $\|H'\|=O(1)$ and with each term having norm $O(1)$, such that if $\lambda_{\min}(H) \leq 1/p$, then $\lambda_{min}(H') \leq 1/2$, whereas if $\lambda_{\min}(H) \geq 1/q$, then $\lambda_{min}(H') \geq 1$. \[lem:amplify\]
We start by recalling that the Local Hamiltonian problem is ${\textsc{QMA}}$-complete for qubit Hamiltonians in the $XZ$ model, up to inverse-polynomial promise gap [@CM13]. Let $$H = {\ensuremath{\mathop{\textsc{E}}_{j \in \{1, \dots, \ell\}}}} H_j$$ be a given Hamiltonian on $n$ qubits from the $XZ$ model (also allowing terms that are multiples of the identity), with $\ell =
\operatorname{poly}(n)$ local terms $H_j$, normalized such that $0 \leq H \leq {\ensuremath{\mathop{\rm Id}\nolimits}}$. As can be seen from Definition \[def:gen-h\], this Hamiltonian can be equivalently viewed as a Hamiltonian in $Y$-free form acting on qubits. We aim to give a protocol that distinguishes between the cases $\lambda_{\min}(H) \leq 1/p$ (YES) or $\lambda_{\min}(H) \geq
1/q$ (NO), where $0 \leq 1/p < 1/q \leq 1$ and $p$ and $q$ are polynomial functions of $n$. By applying Lemma \[lem:amplify\] to $H$ and scaling down the resulting Hamiltonian, we obtain a new Hamiltonian $$H' = c\left({\ensuremath{\mathop{\rm Id}\nolimits}}^{\otimes a} - ({\ensuremath{\mathop{\rm Id}\nolimits}}- ( H - a^{-1}{\ensuremath{\mathop{\rm Id}\nolimits}}))^{\otimes
a}\right)$$ acting on $a n = \operatorname{poly}(n)$ qudits with norm $\|H'\| = 1$ and all of whose terms have norms bounded by $1$, such that $\lambda_{\min}(H') \leq c/2$ in the YES case and $\lambda_{\min}(H') \geq c$ in the NO case, for some constant $0 < c < 1$. For our purposes, it will be useful to express $H'$ as an average $$H' = {\ensuremath{\mathop{\textsc{E}}_{J \in \{1, \dots \ell'\}}}} H'_{J}\;,$$ where each term $H'_J$ is of the form $c {\ensuremath{\mathop{\rm Id}\nolimits}}^{{\otimes}a} - \alpha_J h_{S_J}(u_J)$ for some Pauli operator $h_{S_J}(u_J)$ and weight $\alpha_J \in [-1, 1]$. The number of terms in this decomposition is $\ell' = (\ell+1)^{a}$, which is exponential in $n$. This means that executing the test ${\textsc{energy}}(C,H')$ would require $\operatorname{poly}(n)$ bits of communication with the verifier, just to specify a single term in the Hamiltonian. To avoid this problem, we use randomness to sample a subset of the terms. First, rescale $H'$ so that all of the terms are positive and have norm at most $1$: $$H'' = {\ensuremath{\mathop{\textsc{E}}_{J \in \{1, \dots, \ell'\}}}} H''_{J}\;, \qquad H''_J = \frac{1}{2}
\big(H'_J + {\ensuremath{\mathop{\rm Id}\nolimits}}- c\big)\;.$$ This rescaled Hamiltonian satisfies $\lambda_{\min}(H'') = \frac{1}{2}(1
-c + \lambda_{\min}(H')) \geq \frac{1}{2}(1 - c) $. Now, let $H'''$ be a Hamiltonian obtained by uniformly sampling $m$ terms at random from $H''$, where $m$ is a parameter to be chosen. By the matrix Chernoff Bound [@AW02 Theorem 19], for any ${\varepsilon}\in [0, 1/2]$, $$\Pr[\lambda_{\min}(H''') \notin [(1-{\varepsilon}) \lambda_{\min}(H''),
(1+{\varepsilon}) \lambda_{\min}(H'')]] \leq 2 \cdot \exp\left(an \ln2 -m \frac{{\varepsilon}^2
\lambda_{\min}(H'')}{2\ln 2}\right).$$ In particular, taking ${\varepsilon}\leq c/(4 - 2c)$ and $m = \operatorname{poly}(n)$, we obtain that, with probability exponentially close to $1$, in the YES case $\lambda_{\min}(H''') \leq c_1$ and in the NO case $\lambda_{\min}(H''') \geq c_2$ where $c_2 - c_1 \geq c/8$. Moreover, $H'''$ is a $Y$-free Hamiltonian with polynomially many terms. Hence, by the same arguments as in the proof of Corollary \[cor:qma-generalizedXZ\], there exists a $7$-prover ${\textsc{MIP}}^*$ protocol with $O(\log(n))$-bit messages and constant completeness-soundness gap to estimate the ground energy of $H'''$ up to precision $c/16$, and hence to solve the Local Hamiltonian problem for $H$.
Energy test for frustration-free Hamiltonians with small gap {#sec:ff}
------------------------------------------------------------
In this section we show how the procedure ${\textsc{sum}}$ can be used in a different scenario than the one considered in the previous section: the case of an $n$-qubit Hamiltonian that is either frustration-free, or has ground state energy that is at most an inverse polynomial in $n$. The tests described in this section are more restrictive than those considered in the previous section, but they have the advantage of not relying on a randomized reduction. They apply to the following form of “linear XZ Hamiltonian”.
\[def:linear-xz\] A $n$-qudit Hamiltonian $H$, where each qudit has dimension a prime power $q$, is in *linear XZ form* if it can be written as $$H = {\ensuremath{\mathop{\textsc{E}}_{W\in\{X,Z\}, j \in \{1, \dots, \ell\}}}} \Pi_{W,j}\;,$$ where the expectation is taken under the uniform distribution, and for each $W \in \{X, Z\}$ and $j\in\{1,\ldots,\ell\}$ the term $\Pi_{W,j}$ is a projector that is diagonal in the basis $W$ (for each qubit), and such that the nullspace of $\Pi_{W,j}$ can be described by a collection of $t_{W,j}$ linear equations $\{s_{W,j,i} \cdot a = b_{W,j,i} ,\, i\in \{1,\ldots,t_{W,j}\}\}$ over ${\ensuremath{\mathbb{F}_q}}$, where here $a=(a_1,\ldots,a_n)\in{\ensuremath{\mathbb{F}_q}}^n$ specifies a basis state ${|a\rangle}_W$ in basis $W$ for the $n$ qudits, and $s_{W,j,1},\ldots,s_{W,j,t_j} \in {\ensuremath{\mathbb{F}_q}}^n$ and $b_{W,j,i}\in{\ensuremath{\mathbb{F}_q}}$ are coefficients of linear equations.
Note that a special case of the definition is one in which some of the $\Pi_{W,j}$ have rank $1$, since any fixed element $a\in{\ensuremath{\mathbb{F}_q}}^n$ can be uniquely specified by a system of $n$ linear equations.
A Hamiltonian $H$ in linear XZ form is specified by the collection of equations $\{(s_{W,j,i},b_{W,j,i}),\, W\in\{X,Z\}, j\in\{1,\ldots,\ell\},i\in\{1,\ldots,t_j\}$. We will be interested in the problem of deciding whether $H$ has ground state energy $0$, or at least some inverse polynomial in $n$, $\gamma(n)$. Replacing each $\Pi_{W,j}$ in $H$ by an average of $t_{W,j}$ terms, each associated with a single equation $(s_{W,j,i},b_{W,j,i})$, preserves the distinction between these two cases, up to a polynomial multiplicative scaling in $\gamma$. Therefore, for the remainder of this section we assume that $t_{W,j}=1$ for all $W,j$, and write $(s_{W,j},b_{W,j}) $ for $(s_{W,j,1},b_{W,j,1})$.
The main result of this section is an interactive protocol for deciding between the cases where a Hamiltonian $H$ in linear XZ form is frustration free, or has energy at least some inverse polynomial in $n$. The main ingredients for the protocol are the low-degree test from Theorem \[thm:qld\] and the test ${\textsc{sum}}$. As for the case of the $Y$-free Hamiltonians considered in Section \[sec:constant\_gap\], it would be straightforward to extend the results of this section to Hamiltonians as in Definition \[def:linear-xz\], but allowing a polynomial number of possible basis choices for the $n$ qudits, chosen among $\{X,Z\}^n$, instead of only $X^n$ and $Z^n$. For simplicity, we focus on the case of two bases only.
------------------------------------------------------------------------
\
Test ${\textsc{XZ}}_N(H)$: Given as input is a $n$-qudit Hamiltonian in linear $XZ$ form, where each qudit is of dimension a prime power $q$, and an integer $N$. Let $C$ be a $[k,k']$ weakly self-dual linear code over ${\ensuremath{\mathbb{F}_q}}$, known to all parties, such that $C$ encodes at least one qudit. The verifier performs one of the following, with probability $1/2$ each:
1. Select a basis $W\in\{X,Z\}$ uniformly at random. Select an equation $(s,b)$ as described in the proof of Theorem \[thm:xz-form\] (this depends on the parameter $N$). Choose ${\overline{w}} \in {\ensuremath{\mathbb{F}_q}}^k$ to be a random vector such that ${\tau}_W({\overline{w}})$ is a logical operator for ${\mathcal{C}}$, and execute the test ${\textsc{sum}}(C, W, \{ {\overline{w}}_1 s, {\overline{w}}_2 s, \dots, {\overline{w}}_k s\})$ with the provers. Reject if the protocol rejects, or if the linear combination $\sum_{i=1}^{k}
c_k$ of the claimed values is not equal to $E$. Else, accept.
2. Execute test ${\textsc{code-check}}(C,n')$ with the provers, where $n'= Nn$ is as in the proof of Theorem \[thm:xz-form\].
------------------------------------------------------------------------
We state the main result of this section.
\[thm:xz-form\] Let $n$ be an integer, $q = p^t$ a prime power such that $q=\Theta(\operatorname{poly}\log n)$, and $\gamma(n) = \Omega(\operatorname{poly}^{-1}(n))$. There exists a universal constant ${\varepsilon}_0 >0$ and $N = O(\operatorname{poly}(n))$ such that the following holds. For any $n$-qudit Hamiltonian $H$ in linear XZ form,
- If $H$ has ground state energy $0$, then there is a strategy for the provers that is accepted in the test ${\textsc{XZ}}_N(H)$ with probability $1$.
- If $H$ has ground state energy at least $\gamma(n)$, then no strategy for the provers is accepted with probability more than $1-{\varepsilon}_0$ in the test ${\textsc{XZ}}_N(H)$.
By basing the test on the code from Example \[ex:quad\_res\_code\], the test can be executed with $7$ provers and a total amount of communication between the verifier and the provers that is $O(\log n)$.
We start by amplifying the promise gap by taking a tensor product of $N$ copies of $H$, for $N = \lceil\delta\gamma^{-1}(n)\rceil$, for some $0<\delta\leq 1$ to be determined. For any $W\in\{X,Z\}$ and $j\in\{1,\ldots,\ell\}$ let $\tilde{\Pi}_{W,j} = {\ensuremath{\mathop{\rm Id}\nolimits}}- \Pi_{W,j}$. Define $$\begin{aligned}
H' &= {\ensuremath{\mathop{\rm Id}\nolimits}}- ({\ensuremath{\mathop{\rm Id}\nolimits}}- H)^{{\otimes}N} \notag\\
&= {\ensuremath{\mathop{\rm Id}\nolimits}}- ({\ensuremath{\mathop{\textsc{E}}_{(W,j)}}}({\ensuremath{\mathop{\rm Id}\nolimits}}- \Pi_{W,j}))^{{\otimes}N} \notag\\
&= {\ensuremath{\mathop{\rm Id}\nolimits}}- {\ensuremath{\mathop{\textsc{E}}_{W_1, j_1, \dots, W_N, j_N}}} \tilde{\Pi}_{W_1, j_1} {\otimes}\dots {\otimes}\tilde{\Pi}_{W_N, j_N}\;,\label{eq:hprime}
\end{aligned}$$ where all expectation are uniform over the appropriate sets. Then $H'\geq 0$. If $H$ is frustration-free then $H'$ is frustration-free as well. If $H$ has ground energy at least $\gamma$, then $H'$ has ground energy at least $1-e^{-\delta} \geq \delta/2$. Note that $H'$ is again a Hamiltonian in linear XZ form such that $H'$ acts on $n' = N n$ qudits.
For any $\vec{W}=(W_1,\ldots,W_N)$ and $\vec{j}=(j_1,\ldots,j_N)$ let $\tilde{\Pi}^X_{\vec{W},\vec{j}} = \otimes_{i=1}^N \tilde{\Pi}^X_{W_i,j_i}$, where $\tilde{\Pi}^X_{W,j} = \tilde{\Pi}_{W,j}$ if $W=X$ and $\tilde{\Pi}^X_{W,j} = {\ensuremath{\mathop{\rm Id}\nolimits}}$ otherwise. Define $\tilde{\Pi}^Z_{\vec{W},\vec{j}}$ similarly. From , we get $$\label{eq:hprime-2}
H'\,=\, {\ensuremath{\mathop{\rm Id}\nolimits}}- {\ensuremath{\mathop{\textsc{E}}_{\vec{W},\vec{j}}}} \tilde{\Pi}^X_{\vec{W},\vec{j}}\tilde{\Pi}^Z_{\vec{W},\vec{j}}\;.$$ We use the following claim:
\[claim:commute-bound\] Let $ A, B$ be two positive semidefinite operators such that $A, B \leq {\ensuremath{\mathop{\rm Id}\nolimits}}$ and $AB =
BA$. Then $${\ensuremath{\mathop{\rm Id}\nolimits}}-AB \,\leq\, \big({\ensuremath{\mathop{\rm Id}\nolimits}}-A\big) + \big({\ensuremath{\mathop{\rm Id}\nolimits}}-B\big)\;.$$ \[claim:commuting-pigeonhole\]
Note that since $B$ commutes with $A$, it must also commute with the positive square root of $A$. Hence $AB = A^{1/2} B A^{1/2} \geq 0$. Likewise, $({\ensuremath{\mathop{\rm Id}\nolimits}}- A)$ commutes with $({\ensuremath{\mathop{\rm Id}\nolimits}}- B)$, so $({\ensuremath{\mathop{\rm Id}\nolimits}}- A)({\ensuremath{\mathop{\rm Id}\nolimits}}- B) \geq 0$.
Starting from and applying Claim \[claim:commute-bound\], $$\begin{aligned}
H' &\leq 2{\ensuremath{\mathop{\textsc{E}}_{W\in\{X,Z\}}}}\Big( {\ensuremath{\mathop{\rm Id}\nolimits}}- {\ensuremath{\mathop{\textsc{E}}_{\vec{W},\vec{j}}}} \tilde{\Pi}^W_{\vec{W},\vec{j}}\Big)\;.\label{eq:hprime-3}\end{aligned}$$ For $W\in \{X,Z\}$ let $H'_W = {\ensuremath{\mathop{\rm Id}\nolimits}}- {\ensuremath{\mathop{\textsc{E}}_{\vec{W},\vec{j}}}} \tilde{\Pi}^W_{\vec{W},\vec{j}}$. If $H'$ has ground energy zero, then both $H'_Z$ and $H'_X$ have ground energy zero as well. If $H'$ has ground energy at least $\delta/2$, then for any vector ${|\psi\rangle}$, either ${\langle\psi|}H'_X{|\psi\rangle} \geq \delta/4$ or ${\langle\psi|}H'_Z{|\psi\rangle} \geq \delta/4$.
The goal of the test ${\textsc{XZ}}_N(H)$ described in Figure \[fig:xz-test\] is to distinguish between these two cases. To complete the description of the test we specify how the linear equation $(s,b)$ considered in item 1. of the test is obtained. First form the set $S=\{(s_\ell,b_\ell),\,1\leq\ell\leq t\}$ that is the union of all equations which specify the $+1$ eigenspace of each individual $\tilde{\Pi}_{W_i,j_i}$ such that $W_i=W$. Using the notation from Definition \[def:linear-xz\] we have $|S|=t = \sum_{i=1}^N
1_{W_i=W}t_{W,j_i}$, which is polynomial in $n$. Then $(s,b)$ is obtained by sampling $(\delta/8)$-biased random variables $(y_1,\ldots,y_t)\in{\ensuremath{\mathbb{F}_q}}^t$ and setting $s = \sum_\ell y_\ell s_\ell$ and $b=\sum_\ell y_\ell b_\ell$. We refer to e.g. [@azar1998approximating] for a construction of such random variables using $\operatorname{poly}\log( t, q,\delta^{-1})$ random bits; note that this use of $\delta$-biased random variables is analogous to their use in the classical exponential PCP for QUADEQ. Thus the amount of communication required to specify $s$ to a prover is $\operatorname{poly}\log(t,q,\delta^{-1})=\operatorname{poly}\log \log(n)$.
We show that the test ${\textsc{XZ}}_N(H)$ satisfies the requirements of the theorem. We first argue completeness, and then soundness.
\[claim:xz-completeness\] Suppose that $H$ has ground energy $0$. Then there is a strategy for the provers that is accepted with probability $1$ in the test ${\textsc{XZ}}_N(H)$.
Let ${|\psi\rangle}$ be a ground state of $H$. Let $C$ be the linear code used by the verifier. Consider the strategy for test ${\textsc{code-check}}(C,n')$ described in Lemma \[lem:code-completeness\], where the encoded state is the $n'$-qudit state ${|\psi\rangle}^{\otimes N}$. From the lemma it follows that the strategy succeeds with probability $1$ in item 2. of the test ${\textsc{XZ}}_N(H)$.
It remains to describe the behavior of the provers when elected to perform the test ${\textsc{sum}}$ in item 1. of ${\textsc{XZ}}_N(H)$. The prover $j$ first measures its share of each qudit in the basis $W$, obtaining outcomes $a_j\in{\ensuremath{\mathbb{F}_q}}^{n'}$. The prover then executes the honest behavior in test ${\textsc{sum}}(C, W, \{{\overline{w}}_1 s, \dots, {\overline{w}}_k s\} )$, with the claimed value being $c_j = {\overline{w}}_j s\cdot
a_j$. Moreover, since ${|\psi\rangle}$ is a ground state of $H$, the condition $\sum_{j} c_j = s \cdot \sum_{j} {\overline{w}}_j a_j = b$ is satisfied with certainty. Hence, the honest strategy is accepted with probability $1$.
The next claim shows soundness of the protocol.
\[claim:xz-soundness\] Suppose that $H$ has ground energy at least $\gamma$. Then any strategy for the provers in the test ${\textsc{XZ}}_N(H)$ is accepted with probability at most $1-\delta_s$, for some $\delta_s = \max(\operatorname{poly}(\delta),\operatorname{poly}(q^{-1}))$.
Fix a strategy for the provers that has success probability at least $1-{\varepsilon}$ in the test, for some ${\varepsilon}>0$. This consists of a state ${|\Psi\rangle}$ and measurement operators $\{M_s^q\}$ for the special prover. The strategy must succeed with probability at least $1-2{\varepsilon}$ in item 2. of test ${\textsc{XZ}}_N(H)$. Applying Theorem \[thm:codeword\_test\], it follows that there exists a state ${|\psi\rangle}\in({\ensuremath{\mathbb{C}}}^p)^{\otimes n'}$ such that, up to local isometries, ${|\Psi\rangle}$ is within distance $\delta_{C}$ of a valid $n'k$-qudit encoding of some state ${|\psi\rangle}$ according to the code $C$; under the same isometry, each prover’s measurement upon query $(W,w)$ is $\delta_{C}$-close to an application of the observable ${\tau}_W(w_{\pi})$ on the prover’s share of the encoding. Up to an increase of $\delta_{C}$ in the error we assume for the remainder of the proof that all provers apply the honest strategy when given queries distributed as in the test ${\textsc{code-check}}$.
We now analyze item 1. of test ${\textsc{XZ}}_N(H)$. By assumption, the honest strategy, based on state ${|\psi\rangle}$, succeeds with probability at least $1-O(\delta_{C})$ in this part of the test (we can assume $\delta_{C} \geq {\varepsilon}$ without loss of generality). Since $H$ has ground energy at least $\gamma$, by either ${\langle\psi|} H'_X{|\psi\rangle} \geq \delta/4$, or ${\langle\psi|}H'_Z{|\psi\rangle}\geq \delta/4$. Assume the former. This means that whenever each of the $n'$ qudits of ${|\psi\rangle}$ is measured in the $X$ basis, the probability that the outcome string $a$ is such that $a$ satisfies all linear equations $s_\ell\cdot a = b_\ell$, $\ell\in\{1,\ldots,t\}$, considered by the verifier, when the basis is chosen to be $W=X$, is at most $1-\delta/4$. By definition of the equation $(s,b)$, the probability (which now includes the verifier’s coin tosses in selecting $(s,b)$) that $s\cdot a =b$ is at most $1-\delta/8$. It follows from the soundness part of Theorem \[thm:sum-game\] that the provers must be rejected with probability $\operatorname{poly}(\delta) - O(\delta_{C})$. This gives a contradiction for any ${\varepsilon}$ small enough such that $O(\delta_{C}) \ll \operatorname{poly}(\delta)$.
To conclude the proof of the theorem, we choose the constant $\delta$ to be sufficiently small so that $\delta_s$ in Claim \[claim:xz-soundness\] is a positive constant.
[^1]: Center for Theoretical Physics, MIT, Cambridge, USA. email:`anandn@mit.edu`.
[^2]: Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, USA. email: `vidick@cms.caltech.edu`.
[^3]: We note that, just as in [@BGHSV05], we require linearity of the PCP in order for it to interface with a linear error correcting code.
[^4]: Here and throughout we use the notation $f(X)=\operatorname{poly}(h(X)))$ as an abbreviation for “there exists a universal constant $c>0$ such that $f(X) = O(h(X)^c)$ as $X\to 0$ (if $X={\varepsilon}$) or as $X\to\infty$ (if $X=n$); in the theorem $p,t$ and $q$ are all allowed to be implicitly functions of $n$, but not ${\varepsilon}$.
[^5]: We combine the use of bold font together with the “[ MidnightBlue]{}” color to (hopefully) make it easier to distinguish the operators, as well as make (online) reading of our paper a more colorful experience.
[^6]: Although this is left implicit in the notation, the measurement operators associated with different spaces need not be equal.
[^7]: The map $w\mapsto w_{{\pi}}$ from ${\ensuremath{\mathbb{F}_q}}^m$ to ${\ensuremath{\mathbb{F}_q}}^n$ is defined in Section \[sec:notation\].
[^8]: Note that here, and elsewhere, the superscript denotes the outcome of the measurement, not exponentiation.
[^9]: The results in [@NV17 Theorem 10] apply to the analysis of the linearity test over ${\ensuremath{\mathbb{F}}}_2^{2t}$, i.e. the case $p=2$ here. The same proof extends with, minor modifications, to the case of arbitrary prime $p$. We omit the details.
[^10]: All results in this and the next section can be obtained by restricting attention to the $7$-qubit code described in Example \[ex:quad\_res\_code\].
[^11]: The specific way in which this summation is performed depends on the form of the query $Q$. In general each $A_i$ is expected to be either a low-degree polynomial, or of a pair of values in ${\ensuremath{\mathbb{F}_q}}$. In both cases, there is a natural way to add up the answers in order to obtain an answer ${{\color{MidnightBlue} \bm{A}}}$ that is formatted as the prover’s answer in the low-degree test.
[^12]: Not to be confused with our parameter $t$, which controls the field size!
[^13]: There should be no confusion between $\pi$ and the coordinate expansion map ${\pi}$ used in previous sections.
[^14]: Called the $XY$ model in their convention; to convert to ours it suffices to relabel the Pauli $Y$ and $Z$ operators.
|
---
abstract: 'Aspect based sentiment analysis (ABSA) can provide more detailed information than general sentiment analysis, because it aims to predict the sentiment polarities of the given aspects or entities in text. We summarize previous approaches into two subtasks: aspect-category sentiment analysis (ACSA) and aspect-term sentiment analysis (ATSA). Most previous approaches employ long short-term memory and attention mechanisms to predict the sentiment polarity of the concerned targets, which are often complicated and need more training time. We propose a model based on convolutional neural networks and gating mechanisms, which is more accurate and efficient. First, the novel Gated Tanh-ReLU Units can selectively output the sentiment features according to the given aspect or entity. The architecture is much simpler than attention layer used in the existing models. Second, the computations of our model could be easily parallelized during training, because convolutional layers do not have time dependency as in LSTM layers, and gating units also work independently. The experiments on SemEval datasets demonstrate the efficiency and effectiveness of our models. [^1]'
author:
- |
Wei Xue [and]{.nodecor} Tao Li\
School of Computing and Information Sciences\
Florida International University, Miami, FL, USA\
bibliography:
- 'acl2018.bib'
title: Aspect Based Sentiment Analysis with Gated Convolutional Networks
---
Introduction
============
Opinion mining and sentiment analysis [@Pang:2008wj] on user-generated reviews can provide valuable information for providers and consumers. Instead of predicting the overall sentiment polarity, fine-grained aspect based sentiment analysis (ABSA) [@Liu:2012ke] is proposed to better understand reviews than traditional sentiment analysis. Specifically, we are interested in the sentiment polarity of aspect categories or target entities in the text. Sometimes, it is coupled with aspect term extractions [@xue2017mtna]. A number of models have been developed for ABSA, but there are two different subtasks, namely aspect-category sentiment analysis (ACSA) and aspect-term sentiment analysis (ATSA). The goal of ACSA is to predict the sentiment polarity with regard to the given aspect, which is one of a few predefined categories. On the other hand, the goal of ATSA is to identify the sentiment polarity concerning the target entities that appear in the text instead, which could be a multi-word phrase or a single word. The number of distinct words contributing to aspect terms could be more than a thousand. For example, in the sentence “*Average to good Thai food, but terrible delivery.*”, ATSA would ask the sentiment polarity towards the entity *Thai food*; while ACSA would ask the sentiment polarity toward the aspect *service*, even though the word *service* does not appear in the sentence.
Many existing models use LSTM layers [@Hochreiter:1997fq] to distill sentiment information from embedding vectors, and apply attention mechanisms [@Bahdanau:2014vz] to enforce models to focus on the text spans related to the given aspect/entity. Such models include Attention-based LSTM with Aspect Embedding (ATAE-LSTM) [@Wang:2016tf] for ACSA; Target-Dependent Sentiment Classification (TD-LSTM) [@Tang:2016th], Gated Neural Networks [@Zhang:2016th] and Recurrent Attention Memory Network (RAM) [@Chen:2017wv] for ATSA. Attention mechanisms has been successfully used in many NLP tasks. It first computes the alignment scores between context vectors and target vector; then carry out a weighted sum with the scores and the context vectors. However, the context vectors have to encode both the aspect and sentiment information, and the alignment scores are applied across all feature dimensions regardless of the differences between these two types of information. Both LSTM and attention layer are very time-consuming during training. LSTM processes one token in a step. Attention layer involves exponential operation and normalization of all alignment scores of all the words in the sentence [@Wang:2016tf]. Moreover, some models needs the positional information between words and targets to produce weighted LSTM [@Chen:2017wv], which can be unreliable in noisy review text. Certainly, it is possible to achieve higher accuracy by building more and more complicated LSTM cells and sophisticated attention mechanisms; but one has to hold more parameters in memory, get more hyper-parameters to tune and spend more time in training. In this paper, we propose a fast and effective neural network for ACSA and ATSA based on convolutions and gating mechanisms, which has much less training time than LSTM based networks, but with better accuracy.
For ACSA task, our model has two separate convolutional layers on the top of the embedding layer, whose outputs are combined by novel gating units. Convolutional layers with multiple filters can efficiently extract n-gram features at many granularities on each receptive field. The proposed gating units have two nonlinear gates, each of which is connected to one convolutional layer. With the given aspect information, they can selectively extract aspect-specific sentiment information for sentiment prediction. For example, in the sentence “*Average to good Thai food, but terrible delivery.*”, when the aspect *food* is provided, the gating units automatically ignore the negative sentiment of aspect *delivery* from the second clause, and only output the positive sentiment from the first clause. Since each component of the proposed model could be easily parallelized, it has much less training time than the models based on LSTM and attention mechanisms. For ATSA task, where the aspect terms consist of multiple words, we extend our model to include another convolutional layer for the target expressions. We evaluate our models on the SemEval datasets, which contains restaurants and laptops reviews with labels on aspect level. To the best of our knowledge, no CNN-based model has been proposed for aspect based sentiment analysis so far.
Related Work
============
We present the relevant studies into following two categories.
Neural Networks
---------------
Recently, neural networks have gained much popularity on sentiment analysis or sentence classification task. Tree-based recursive neural networks such as Recursive Neural Tensor Network [@Socher:2013ug] and Tree-LSTM [@Tai:2015wp], make use of syntactic interpretation of the sentence structure, but these methods suffer from time inefficiency and parsing errors on review text. Recurrent Neural Networks (RNNs) such as LSTM [@Hochreiter:1997fq] and GRU [@Chung:2014wf] have been used for sentiment analysis on data instances having variable length [@Tang:2015ts; @Xu:2016vb; @SiweiLai:2014to]. There are also many models that use convolutional neural networks (CNNs) [@Collobert:2011tk; @Kalchbrenner:2014wl; @Kim:2014vt; @Conneau:2016to] in NLP, which also prove that convolution operations can capture compositional structure of texts with rich semantic information without laborious feature engineering.
Aspect based Sentiment Analysis
-------------------------------
There is abundant research work on aspect based sentiment analysis. Actually, the name ABSA is used to describe two different subtasks in the literature. We classify the existing work into two main categories based on the descriptions of sentiment analysis tasks in SemEval 2014 Task 4 [@Pontiki:2014ex]: Aspect-Term Sentiment Analysis and Aspect-Category Sentiment Analysis.
**Aspect-Term Sentiment Analysis**. In the first category, sentiment analysis is performed toward the aspect terms that are labeled in the given sentence. A large body of literature tries to utilize the relation or position between the target words and the surrounding context words either by using the tree structure of dependency or by simply counting the number of words between them as a relevance information [@Chen:2017wv].
Recursive neural networks [@Lakkaraju:2014vy; @Dong:2014vd; @Wang:2016vm] rely on external syntactic parsers, which can be very inaccurate and slow on noisy texts like tweets and reviews, which may result in inferior performance. Recurrent neural networks are commonly used in many NLP tasks as well as in ABSA problem. TD-LSTM [@Tang:2016th] and gated neural networks [@Zhang:2016th] use two or three LSTM networks to model the left and right contexts of the given target individually. A fully-connected layer with gating units predicts the sentiment polarity with the outputs of LSTM layers. Memory network [@Weston:2014va] coupled with multiple-hop attention attempts to explicitly focus only on the most informative context area to infer the sentiment polarity towards the target word [@Tang:2016uz; @Chen:2017wv]. Nonetheless, memory network simply bases its knowledge bank on the embedding vectors of individual words [@Tang:2016uz], which makes itself hard to learn the opinion word enclosed in more complicated contexts. The performance is improved by using LSTM, attention layer and feature engineering with word distance between surrounding words and target words to produce target-specific memory [@Chen:2017wv].
**Aspect-Category Sentiment Analysis**. In this category, the model is asked to predict the sentiment polarity toward a predefined aspect category. Attention-based LSTM with Aspect Embedding [@Wang:2016tf] uses the embedding vectors of aspect words to selectively attend the regions of the representations generated by LSTMs.
Gated Convolutional Network with Aspect Embedding
=================================================
In this section, we present a new model for ACSA and ATSA, namely Gated Convolutional network with Aspect Embedding (GCAE), which is more efficient and simpler than recurrent network based models [@Wang:2016tf; @Tang:2016th; @Ma:2017jo; @Chen:2017wv]. Recurrent neural networks sequentially compose hidden vectors $\mathbf{h}_{i} = f (\mathbf{h}_{i-1})$, which does not enable parallelization over inputs. In the attention layer, softmax normalization also has to wait for all the alignment scores computed by a similarity function. Hence, they cannot take advantage of highly-parallelized modern hardware and libraries. Our model is built on convolutional layers and gating units. Each convolutional filter computes n-gram features at different granularities from the embedding vectors at each position individually. The gating units on top of the convolutional layers at each position are also independent from each other. Therefore, our model is more suitable to parallel computing. Moreover, our model is equipped with two kinds of effective filtering mechanisms: the gating units on top of the convolutional layers and the max pooling layer, both of which can accurately generate and select aspect-related sentiment features.
We first briefly review the vanilla CNN for text classification [@Kim:2014vt]. The model achieves state-of-the-art performance on many standard sentiment classification datasets [@Le:2017vg].
The CNN model consists of an embedding layer, a one-dimension convolutional layer and a max-pooling layer. The embedding layer takes the indices $w_i \in \{1, 2, \ldots, V\}$ of the input words and outputs the corresponding embedding vectors $\boldsymbol{v}_i \in \mathbb{R}^D$. $D$ denotes the dimension size of the embedding vectors. $V$ is the size of the word vocabulary. The embedding layer is usually initialized with pre-trained embeddings such as GloVe [@Pennington:2014uw], then they are fine-tuned during the training stage. The one-dimension convolutional layer convolves the inputs with multiple convolutional kernels of different widths. Each kernel corresponds a linguistic feature detector which extracts a specific pattern of n-gram at various granularities [@Kalchbrenner:2014wl]. Specifically, the input sentence is represented by a matrix through the embedding layer, $\mathbf{X} = [\boldsymbol{v}_1, \boldsymbol{v}_2, \ldots, \boldsymbol{v}_L]$, where $L$ is the length of the sentence with padding. A convolutional filter $\mathbf{W}_c \in \mathbb{R}^{D \times k}$ maps $k$ words in the receptive field to a single feature $c$. As we slide the filter across the whole sentence, we obtain a sequence of new features $\mathbf{c} = [c_1, c_2, \ldots, c_L]$. $$\label{c4eq:conv}
c_i = f(\mathbf{X}_{i:i+K} * \mathbf{W}_c + b_c) \quad,$$ where $b_c \in \mathbb{R}$ is the bias, $f$ is a non-linear activation function such as tanh function, $*$ denotes convolution operation. If there are $n_k$ filters of the same width $k$, the output features form a matrix $\mathbf{C} \in \mathbb{R}^{n_k \times L_k}$. For each convolutional filter, the max-over-time pooling layer takes the maximal value among the generated convolutional features, resulting in a fixed-size vector whose size is equal to the number of filters $n_k$. Finally, a softmax layer uses the vector to predict the sentiment polarity of the input sentence.
![Illustration of our model GCAE for ACSA task. A pair of convolutional neuron computes features for a pair of gates: tanh gate and ReLU gate. The ReLU gate receives the given aspect information to control the propagation of sentiment features. The outputs of two gates are element-wisely multiplied for the max pooling layer.[]{data-label="fig:model"}](gcae.pdf)
Figure \[fig:model\] illustrates our model architecture. The Gated Tanh-ReLU Units (GTRU) with aspect embedding are connected to two convolutional neurons at each position $t$. Specifically, we compute the features $c_i$ as $$\begin{aligned}
\label{eq:gcae_aspect}
a_i &= \text{relu}(\mathbf{X}_{i:i+k} * \mathbf{W}_a + \mathbf{V}_{a} \boldsymbol{v}_a + b_a) \\
\label{eq:gcae_sentiment}
s_i &= \text{tanh}(\mathbf{X}_{i:i+k} * \mathbf{W}_s + b_s) \\
\label{eq:gcae_mutiply}
c_i &= s_i \times a_i \quad ,\end{aligned}$$ where $\boldsymbol{v}_a$ is the embedding vector of the given aspect category in ACSA or computed by another CNN over aspect terms in ATSA. The two convolutions in Equation \[eq:gcae\_aspect\] and \[eq:gcae\_sentiment\] are the same as the convolution in the vanilla CNN, but the convolutional features $a_i$ receives additional aspect information $\boldsymbol{v}_a$ with ReLU activation function. In other words, $s_i$ and $a_i$ are responsible for generating sentiment features and aspect features respectively. The above max-over-time pooling layer generates a fixed-size vector $\boldsymbol{e} \in \mathbb{R}^{d_k}$, which keeps the most salient sentiment features of the whole sentence. The final fully-connected layer with softmax function uses the vector $\boldsymbol{e}$ to predict the sentiment polarity $\hat{y}$. The model is trained by minimizing the cross-entropy loss between the ground-truth $y$ and the predicted value $\hat{y}$ for all data samples. $$\mathcal{L} = - \sum_i \sum_j y_i^j \log \hat{y}_i^j \quad ,$$ where $i$ is the index of a data sample, $j$ is the index of a sentiment class.
Gating Mechanisms
=================
The proposed Gated Tanh-ReLU Units control the path through which the sentiment information flows towards the pooling layer. The gating mechanisms have proven to be effective in LSTM. In aspect based sentiment analysis, it is very common that different aspects with different sentiments appear in one sentence. The ReLU gate in Equation \[eq:gcae\_aspect\] does not have upper bound on positive inputs but strictly zero on negative inputs. Therefore, it can output a similarity score according to the relevance between the given aspect information $\boldsymbol{v}_a$ and the aspect feature $a_i$ at position $t$. If this score is zero, the sentiment features $s_i$ would be blocked at the gate; otherwise, its magnitude would be amplified accordingly. The max-over-time pooling further removes the sentiment features which are not significant over the whole sentence.
In language modeling [@Dauphin:2016uja; @Kalchbrenner:2016vf; @vandenOord:2016tk; @Gehring:2017tv], Gated Tanh Units (GTU) and Gated Linear Units (GLU) have shown effectiveness of gating mechanisms. GTU is represented by $\tanh(\mathbf{X}*\mathbf{W} + b) \times \sigma (\mathbf{X}*\mathbf{V} + c)$, in which the sigmoid gates control features for predicting the next word in a stacked convolutional block. To overcome the gradient vanishing problem of GTU, GLU uses $(\mathbf{X}*\mathbf{W} + b) \times \sigma (\mathbf{X}*\mathbf{V} + c)$ instead, so that the gradients would not be downscaled to propagate through many stacked convolutional layers. However, a neural network that has only one convolutional layer would not suffer from gradient vanish problem during training. We show that on text classification problem, our GTRU is more effective than these two gating units.
GCAE on ATSA
============
ATSA task is defined to predict the sentiment polarity of the aspect terms in the given sentence. We simply extend GCAE by adding a small convolutional layer on aspect terms, as shown in Figure \[fig:gcae\_atsa\]. In ACSA, the aspect information controlling the flow of sentiment features in GTRU is from one aspect word; while in ATSA, such information is provided by a small CNN on aspect terms $[w_i, w_{i+1}, \ldots, w_{i+k}]$. The additional CNN extracts the important features from multiple words while retains the ability of parallel computing.
Experiments
===========
Datasets and Experiment Preparation
-----------------------------------
We conduct experiments on public datasets from SemEval workshops [@Pontiki:2014ex], which consist of customer reviews about restaurants and laptops. Some existing work [@Wang:2016tf; @Ma:2017jo; @Chen:2017wv] removed “conflict” labels from four sentiment labels, which makes their results incomparable to those from the workshop report [@Kiritchenko:2014jw]. We reimplemented the compared methods, and used hyper-parameter settings described in these references.
The sentences which have different sentiment labels for different aspects or targets in the sentence are more common in review data than in standard sentiment classification benchmark. The sentence in Table \[tbl:testM\] shows the reviewer’s different attitude towards two aspects: food and delivery. Therefore, to access how the models perform on review sentences more accurately, we create small but difficult datasets, which are made up of the sentences having opposite or different sentiments on different aspects/targets. In Table \[tbl:testM\], the two identical sentences but with different sentiment labels are both included in the dataset. If a sentence has 4 aspect targets, this sentence would have 4 copies in the data set, each of which is associated with different target and sentiment label.
Sentence aspect category/term sentiment label
--------------------------------------------------- ---------------------- -----------------
Average to good Thai food, but terrible delivery. food positive
Average to good Thai food, but terrible delivery. delivery negative
For ACSA task, we conduct experiments on restaurant review data of SemEval 2014 Task 4. There are 5 aspects: food, price, service, ambience, and misc; 4 sentiment polarities: positive, negative, neutral, and conflict. By merging restaurant reviews of three years 2014 - 2016, we obtain a larger dataset called “Restaurant-Large”. Incompatibilities of data are fixed during merging. We replace conflict labels with neutral labels in the 2014 dataset. In the 2015 and 2016 datasets, there could be multiple pairs of “aspect terms” and “aspect category” in one sentence. For each sentence, let $p$ denote the number of positive labels minus the number of negative labels. We assign a sentence a positive label if $p > 0$, a negative label if $p<0$, or a neutral label if $p=0$. After removing duplicates, the statistics are show in Table \[tbl:r14r16\]. The resulting dataset has 8 aspects: [restaurant]{}, [food]{}, [drinks]{}, [ambience]{}, [service]{}, [price]{}, [misc]{} and [location]{}.
For ATSA task, we use restaurant reviews and laptop reviews from SemEval 2014 Task 4. On each dataset, we duplicate each sentence $n_a$ times, which is equal to the number of associated aspect categories (ACSA) or aspect terms (ATSA) [@Ruder:2016ve; @Ruder:2016ug]. The statistics of the datasets are shown in Table \[tbl:r14r16\].
The sizes of hard data sets are also shown in Table \[tbl:r14r16\]. The test set is designed to measure whether a model can detect multiple different sentiment polarities in one sentence toward different entities. Without such sentences, a classifier for overall sentiment classification might be good enough for the sentences associated with only one sentiment label.
----------------------- ------- ------ ------- ------ ------- ------ ------- ------
Train Test Train Test Train Test Train Test
Restaurant-Large 2710 1505 1198 680 757 241 - -
Restaurant-Large-Hard 182 92 178 81 107 61 - -
Restaurant-2014 2179 657 839 222 500 94 195 52
Restaurant-2014-Hard 139 32 136 26 50 12 40 19
----------------------- ------- ------ ------- ------ ------- ------ ------- ------
----------------- ------- ------ ------- ------ ------- ------ ------- ------
Train Test Train Test Train Test Train Test
Restaurant 2164 728 805 196 633 196 91 14
Restaurant-Hard 379 92 323 62 293 83 43 8
Laptop 987 341 866 128 460 169 45 16
Laptop-Hard 159 31 147 25 173 49 17 3
----------------- ------- ------ ------- ------ ------- ------ ------- ------
In our experiments, word embedding vectors are initialized with 300-dimension GloVe vectors which are pre-trained on unlabeled data of 840 billion tokens [@Pennington:2014uw]. Words out of the vocabulary of GloVe are randomly initialized with a uniform distribution $U(-0.25, 0.25)$. We use Adagrad [@Duchi:2011wu] with a batch size of 32 instances, default learning rate of $1e-2$, and maximal epochs of 30. We only fine tune early stopping with 5-fold cross validation on training datasets. All neural models are implemented in PyTorch.
Compared Methods
----------------
To comprehensively evaluate the performance of GCAE, we compare our model against the following models.
**NRC-Canada** [@Kiritchenko:2014jw] is the top method in SemEval 2014 Task 4 for ACSA and ATSA task. SVM is trained with extensive feature engineering: various types of n-grams, POS tags, and lexicon features. The sentiment lexicons improve the performance significantly, but it requires large scale labeled data: 183 thousand Yelp reviews, 124 thousand Amazon laptop reviews, 56 million tweets, and 3 sentiment lexicons labeled manually.
**CNN** [@Kim:2014vt] is widely used on text classification task. It cannot directly capture aspect-specific sentiment information on ACSA task, but it provides a very strong baseline for sentiment classification. We set the widths of filters to 3, 4, 5 with 100 features each.
**TD-LSTM** [@Tang:2016th] uses two LSTM networks to model the preceding and following contexts of the target to generate target-dependent representation for sentiment prediction.
**ATAE-LSTM** [@Wang:2016tf] is an attention-based LSTM for ACSA task. It appends the given aspect embedding with each word embedding as the input of LSTM, and has an attention layer above the LSTM layer.
**IAN** [@Ma:2017jo] stands for interactive attention network for ATSA task, which is also based on LSTM and attention mechanisms.
**RAM** [@Chen:2017wv] is a recurrent attention network for ATSA task, which uses LSTM and multiple attention mechanisms.
**GCN** stands for gated convolutional neural network, in which GTRU does not have the aspect embedding as an additional input.
Results and Analysis
--------------------
### ACSA
------------------ -------------------- -------------------- -------------------- --------------------
SVM\* - - 75.32 -
SVM + lexicons\* - - **82.93** -
ATAE-LSTM 83.91$\pm$0.49 66.32$\pm$2.28 78.29$\pm$0.68 45.62$\pm$0.90
CNN 84.28$\pm$0.15 50.43$\pm$0.38 79.47$\pm$0.32 44.94$\pm$0.01
GCN 84.48$\pm$0.06 50.08$\pm$0.31 **79.67$\pm$0.35** 44.49$\pm$1.52
GCAE **85.92$\pm$0.27** **70.75$\pm$1.19** 79.35$\pm$0.34 **50.55$\pm$1.83**
------------------ -------------------- -------------------- -------------------- --------------------
Following the SemEval workshop, we report the overall accuracy of all competing models over the test datasets of restaurant reviews as well as the hard test datasets. Every experiment is repeated five times. The mean and the standard deviation are reported in Table \[tbl:rest\_lap\].
LSTM based model ATAE-LSTM has the worst performance of all neural networks. Aspect-based sentiment analysis is to extract the sentiment information closely related to the given aspect. It is important to separate aspect information and sentiment information from the extracted information of sentences. The context vectors generated by LSTM have to convey the two kinds of information at the same time. Moreover, the attention scores generated by the similarity scoring function are for the entire context vector.
GCAE improves the performance by 1.1% to 2.5% compared with ATAE-LSTM. First, our model incorporates GTRU to control the sentiment information flow according to the given aspect information at each dimension of the context vectors. The element-wise gating mechanism works at fine granularity instead of exerting an alignment score to all the dimensions of the context vectors in the attention layer of other models. Second, GCAE does not generate a single context vector, but two vectors for aspect and sentiment features respectively, so that aspect and sentiment information is unraveled. By comparing the performance on the hard test datasets against CNN, it is easy to see the convolutional layer of GCAE is able to differentiate the sentiments of multiple entities.
Convolutional neural networks CNN and GCN are not designed for aspect based sentiment analysis, but their performance exceeds that of ATAE-LSTM.
The performance of SVM [@Kiritchenko:2014jw] depends on the availability of the features it can use. Without the large amount of sentiment lexicons, SVM perform worse than neural methods. With multiple sentiment lexicons, the performance is increased by 7.6%. This inspires us to work on leveraging sentiment lexicons in neural networks in the future.
The hard test datasets consist of replicated sentences with different sentiments towards different aspects. The models which cannot utilize the given aspect information such as CNN and GCN perform poorly as expected, but GCAE has higher accuracy than other neural network models. GCAE achieves 4% higher accuracy than ATAE-LSTM on Restaurant-Large and 5% higher on SemEval-2014 on ACSA task. However, GCN, which does not have aspect modeling part, has higher score than GCAE on the original restaurant dataset. It suggests that GCN performs better than GCAE when there is only one sentiment label in the given sentence, but not on the hard test dataset.
### ATSA
------------------ -------------------- -------------------- -------------------- --------------------
SVM\* 77.13 - 63.61 -
SVM + lexicons\* **80.16** - **70.49** -
TD-LSTM 73.44$\pm$1.17 56.48$\pm$2.46 62.23$\pm$0.92 46.11$\pm$1.89
ATAE-LSTM 73.74$\pm$3.01 50.98$\pm$2.27 64.38$\pm$4.52 40.39$\pm$1.30
IAN 76.34$\pm$0.27 55.16$\pm$1.97 68.49$\pm$0.57 44.51$\pm$0.48
RAM 76.97$\pm$0.64 55.85$\pm$1.60 68.48$\pm$0.85 45.37$\pm$2.03
GCAE **77.28$\pm$0.32** **56.73$\pm$0.56** **69.14$\pm$0.32** **47.06$\pm$2.45**
------------------ -------------------- -------------------- -------------------- --------------------
We apply the extended version of GCAE on ATSA task. On this task, the aspect terms are marked in the sentences and usually consist of multiple words. We compare IAN [@Ma:2017jo], RAM [@Chen:2017wv], TD-LSTM [@Tang:2016th], ATAE-LSTM [@Wang:2016tf], and our GCAE model in Table \[tbl:atsa\]. The models other than GCAE is based on LSTM and attention mechanisms. IAN has better performance than TD-LSTM and ATAE-LSTM, because two attention layers guides the representation learning of the context and the entity interactively. RAM also achieves good accuracy by combining multiple attentions with a recurrent neural network, but it needs more training time as shown in the following section. On the hard test dataset, GCAE has 1% higher accuracy than RAM on restaurant data and 1.7% higher on laptop data. GCAE uses the outputs of the small CNN over aspect terms to guide the composition of the sentiment features through the ReLU gate. Because of the gating mechanisms and the convolutional layer over aspect terms, GCAE outperforms other neural models and basic SVM. Again, large scale sentiment lexicons bring significant improvement to SVM.
Training Time
-------------
Model ATSA
--------- -------
ATAE 25.28
IAN 82.87
RAM 64.16
TD-LSTM 19.39
GCAE 3.33
: The time to converge in seconds on ATSA task.[]{data-label="tbl:acsa_time"}
We record the training time of all models until convergence on a validation set on a desktop machine with a 1080 Ti GPU, as shown in Table \[tbl:acsa\_time\]. LSTM based models take more training time than convolutional models. On ATSA task, because of multiple attention layers in IAN and RAM, they need even more time to finish the training. GCAE is much faster than other neural models, because neither convolutional operation nor GTRU has time dependency compared with LSTM and attention layer. Therefore, it is easier for hardware and libraries to parallel the computing process. Since the performance of SVM is retrieved from the original paper, we are not able to compare the training time of SVM.
Gating Mechanisms
-----------------
------ ----------- ----------- ----------- -----------
GTU 84.62 60.25 79.31 **51.93**
GLU 84.74 59.82 79.12 50.80
GTRU **85.92** **70.75** **79.35** 50.55
------ ----------- ----------- ----------- -----------
: The accuracy of different gating units on restaurant reviews on ACSA task.[]{data-label="tbl:gates"}
In this section, we compare GLU $(\mathbf{X}*\mathbf{W} + b) \times \sigma (\mathbf{X}*\mathbf{W}_a + \mathbf{V}\boldsymbol{v}_a + b_a)$ [@Dauphin:2016uja], GTU $\tanh(\mathbf{X}*\mathbf{W} + b) \times \sigma (\mathbf{X}*\mathbf{W}_a + \mathbf{V}\boldsymbol{v}_a + b_a)$ [@vandenOord:2016tk], and GTRU used in GCAE. Table \[tbl:gates\] shows that all of three gating units achieve relatively high accuracy on restaurant datasets. GTRU outperforms the other gates. It has a convolutional layer generating aspect features via ReLU activation function, which controls the magnitude of the sentiment signals according to the given aspect information. On the other hand, the sigmoid function in GTU and GLU has the upper bound $+1$, which may not be able to distill sentiment features effectively.
Visualization
=============
![The outputs of the ReLU gates in GTRU.[]{data-label="fig:vis"}](vis.pdf)
In this section, we take a concrete review sentence as an example to illustrate how the proposed gate GTRU works. It is more difficult to visualize the weights generated by the gates than the attention weights in other neural networks. The attention weight score is a global score over the words and the vector dimensions; whereas in our model, there are $N_{\text{word}} \times N_{\text{filter}} \times N_{\text{dimension}}$ gate outputs. Therefore, we train a small model with only one filter which is only three word wide. Then, for each word, we sum the $N_{\text{dimension}}$ outputs of the ReLU gates. After normalization, we plot the values on each word in Figure \[fig:vis\]. Given different aspect targets, the ReLU gates would control the magnitude of the outputs of the tanh gates.
Conclusions and Future Work
===========================
In this paper, we proposed an efficient convolutional neural network with gating mechanisms for ACSA and ATSA tasks. GTRU can effectively control the sentiment flow according to the given aspect information, and two convolutional layers model the aspect and sentiment information separately. We prove the performance improvement compared with other neural models by extensive experiments on SemEval datasets. How to leverage large-scale sentiment lexicons in neural networks would be our future work.
[^1]: The code and data is available at <https://github.com/wxue004cs/GCAE>
|
---
abstract: 'We develop a scale-invariant truncated L[é]{}vy (STL) process to describe physical systems characterized by correlated stochastic variables. The STL process exhibits L[é]{}vy stability for the probability density, and hence shows scaling properties (as observed in empirical data); it has the advantage that all moments are finite (and so accounts for the empirical scaling of the moments). To test the potential utility of the STL process, we analyze financial data.'
address: |
$^1$Department of Physics, Faculty of Science, University of Zagreb, Zagreb, Croatia\
$^2$Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215\
$^3$Cardiovascular Division, Harvard Medical School, Beth Israel Hospital, Boston, MA 02215
author:
- 'Boris Podobnik$^{1,2}$, Plamen Ch. Ivanov$^{2,3}$, Youngki Lee$^{2}$, and H. Eugene Stanley$^2$'
title: 'Scale-invariant Truncated L[é]{}vy Process'
---
[2]{}
In recent years, the L[é]{}vy process [@L37] has been proposed to describe the statistical properties of a variety of complex phenomena [@Shl93; @Sol; @Ott; @Bar; @Hay94; @Moo; @Zum94; @Orr90]. The L[é]{}vy process is characterized by “fat tails” (power law), and display scaling behavior similar to that observed in a wide range of empirical data. However, the application of the L[é]{}vy process to empirical data is limited because it is characterized by infinite second and higher moments, while empirical data have finite moments.
Truncated L[é]{}vy (TL) processes are defined to have a L[é]{}vy probability density function (PDF) in the central regime, truncated by a function decaying faster than a L[é]{}vy distribution in the tails [@Mantegna94]. The TL process is introduced to account for the finite moments observed for empirical data [@MS95]. However, the TL process (with either abrupt [@Mantegna94] or smooth [@Koponen95] truncation) has limitations when applied to empirical data. (i) The TL process is introduced for independent and identically distributed (i.i.d.) stochastic variables, while variables describing many physical systems are not i.i.d. — e.g. there are correlations. (ii) The PDF of the TL process tends to the Gaussian distribution (according to the central limit theorem), and hence does not exhibit scale invariance; PDFs for a variety of complex systems, however, are often characterized by regions of scale-invariant behavior. (iii) The time scale above which the L[é]{}vy profile becomes Gaussian depends on the size of the truncation cutoff (or the standard deviation) [@Mantegna94; @Koponen95]; to mimic the L[é]{}vy type scale invariant behavior observed for data, the TL process must be defined with a standard deviation larger than the one observed for the data \[see caption to Fig.1\].
Here we introduce a type of stochastic process which we call the scale-invariant truncated L[é]{}vy (STL) process. Stochastic variables $z$ in the STL are generated by the symmetrical probability function $f(z) = A e^{-\lambda |z|^{\beta}}
|z|^{-1-\alpha}$, where $0<\alpha<2$ [@footnote0]. The exponential prefactor [@Koponen95] ensures a smooth truncation of the L[é]{}vy distribution, where the parameter $\beta$ can take any positive value [@footnote1], $\lambda^{-1}$ is related to the size of the truncation cutoff, and $A$ is a measure of the “spread” in the central region.
From the probability function $f(z)$, we calculate the characteristic function $\phi(k) \equiv \exp[-\int^{\infty}_{-\infty}dz(1-e^{-ikz})f(z)]$ [@GK54]. The PDF ${\cal P}_{\Delta t}(z)$ is the Fourier transform of $\phi(k)$ [@GK54]: $${\cal P}(z) \equiv
\frac {1}{2\pi} \int \phi(k)
e^{ikz} dk,
\label{probability}$$ Since $f(z) \approx A|z|^{-1-\alpha}$ for small values of $z$, ${\cal
P}(z)$ has a Lévy profile in the central part. To maintain scale invariance for ${\cal P}(z)$ in the entire range including the tails [@footnote2], we introduce the scaling transformations $$A_{\Delta t} \equiv ({\Delta t})^{\epsilon} A_1,~~~~~
\lambda_{\Delta t} \equiv ({\Delta t})^{-\epsilon \beta/\alpha} \lambda_1,
\label {scale-invar.tr}$$ where $\Delta t$ is the time scale and $\epsilon $ can take any positive value. Under these transformations, the PDF ${\cal
P}(z)={\cal P}_{\Delta t}(z)$ scales as the L[é]{}vy stable distribution: $$z \equiv ({\Delta t})^{\epsilon/\alpha} z_1,~~~~~~
{\cal P}_{\Delta t}(z) \equiv
\frac{{\cal P}_1(z_1)}{({\Delta t})^{\epsilon/\alpha}}.
\label {prob.scale-invar}$$ With the transformations of Eqs. (\[scale-invar.tr\]) and (\[prob.scale-invar\]), we obtain a process with controlled dynamical properties — ${\cal P}_{\Delta t}(z)$ for any value of $\Delta t$ can be calculated from the PDF at any chosen $\Delta t$ (e.g. $\Delta t=1$) [@footnote3].
Although the PDF ${\cal P}_{\Delta t}(z)$ exhibits scale invariant properties identical to the Lévy stable distribution, the process defined by Eqs. (\[probability\]) and (\[scale-invar.tr\]) is different. While the Lévy process is defined for i.i.d. variables the STL process is characterized by correlated stochastic variables — the STL is a non-i.i.d. type process. To demonstrate this, we consider the scaling of the second moment $\sigma^2$, determined as the second derivative of $\phi(k)$ at small values of $k$ [@GK54]: $$\sigma^2_{\Delta t}=~\frac {2 A ~\Gamma ((2-\alpha)/\beta)
~\lambda ^{(\alpha -2)/\beta}}{\beta}=
({\Delta t})^{2 \epsilon/\alpha} \sigma_1^2,
\label {sigma-scale}$$ where $\sigma_1$ is the initial standard deviation for $\Delta t=1$. The second equality on the right hand side follows from the transformations of Eq. (\[scale-invar.tr\]). For an appropriate choice of $\epsilon/\alpha~ (\neq0.5$), the scaling relation (\[sigma-scale\]) indicates the presence of correlations that can be positive (or negative). In addition, the STL process exhibits scaling not only for the second moment but for all higher moments: $$< | z | ^n > \equiv \int dz~| z | ^n~{\cal P}_{\Delta t}(z)
= {\Delta t}^ {\epsilon n /\alpha} < | z_1 | ^n > .
\label{moments}$$
Hence, the STL is a process for which the PDF ${\cal P}_{\Delta
t}(z)$, the second moment $\sigma^2$, and all higher moments $< | z |^n >$ scale with the same scaling exponent $\epsilon/\alpha$.
Often with empirical data, we observe several different scaling regimes. To account for a crossover at given time scale $(\Delta
t)_{\times}$, we introduce different scale invariant transformations from the type of Eq. (\[scale-invar.tr\]) for two different regimes of time scales:
$$\lambda_{\Delta t} =
\left \{ \begin{array}{c}
(\Delta t)^{-{\epsilon}_1 \beta/\alpha} \lambda_1~~~~~~~ 1 \le \Delta t
\le (\Delta t)_{\times}
\nonumber\\
(\Delta t)^{-{\epsilon}_2 \beta/\alpha} \lambda_{\times}~~~~~~~~~~~ \Delta t >
(\Delta t)_{\times} \end{array}
\right \},
\label{scale2}$$
$$A_{\Delta t} =
\left \{ \begin{array}{c}
(\Delta t)^{\epsilon_1} A_1~~~~~~~~~~~~~1 \le \Delta t \le (\Delta t)_{\times}
\nonumber\\
(\Delta t)^{\epsilon_2} A_{\times}~~~~~~~~~~~~~~~~~~\Delta t > (\Delta t)_{\times} \end{array}
\right \}.
\label{scale3}$$
Here $\alpha$, $A_1$ and ${\lambda}_1$ are free parameters, chosen to fit ${\cal P}_{\Delta t}(z)$ at the initial time scale $\Delta t = 1$. Continuity of the PDF and the moments between the two scaling regimes is ensured by continuity in the values of $A$ and $\lambda$: from Eqs. (\[scale2\]) and (\[scale3\]) we find $A_{\times} \equiv (\Delta t)_{\times}^{{\epsilon}_1-{\epsilon}_2} A_1$ and $\lambda_{\times} \equiv (\Delta
t)_{\times}^{\beta ({\epsilon}_1-{\epsilon}_2)/\alpha} \lambda_1 $.
To exemplify the features of the STL process for a broad range of time scales, we need sufficiently large data sets. Such a large data set is the $S\&P500$ stock index over the 12 year period Jan ’84-Dec ’95. The price fluctuations $z$ of this index are the stochastic variable analyzed. In particular, we focus on the scaling behavior of several statistical characteristics: (1) the second and higher moments, (2) the probability of return to the origin ${\cal P}_{\Delta t}(0)$, and (3) the PDF ${\cal P}_{\Delta t}(z)$. For simplicity we set $\beta=1$.
We make three empirical observations. (i) Experimental results for the standard deviation as a function of $\Delta t$ show two different scaling regimes with a crossover at $(\Delta t)_{\times} \approx 30$ min [@MS95] \[Fig.1\]. The regime at small time scales is characterized by slope $0.7$, indicating the presence of positive correlations in the price fluctuations $z$ (“superdiffusive” regime). The second regime has slope $0.5$, indicating absence of correlations (“normal diffusion” regime). Therefore the fluctuations in the $S\&P500$ index cannot be described by an i.i.d. stochastic process, such as the L[é]{}vy or the TL process. (ii) The probability of return to the origin ${\cal P}_{\Delta t}(0)$, however, exhibits a L[é]{}vy type of scaling for more than three decades \[Fig.2\]. Such scaling for ${\cal P}_{\Delta t}(0)$ therefore indicates L[é]{}vy scale invariance of the central part of the probability density. (iii) The scaling exponent of ${\cal P}_{\Delta t}(0)$ is identical to the exponent of the standard deviation in the first scaling regime. However, the crossover in the scaling of the standard deviation is not followed by a change in the slope of ${\cal P}_{\Delta t}(0)$.
To account for the first empirical observations, we introduce a stochastic process with two different regimes: (a) a STL regime with $A_{\Delta t} \equiv ({\Delta t})^{\epsilon} A_1$ and $\lambda_{\Delta t} \equiv ({\Delta t})^{- \epsilon/\alpha} \lambda_1$, to account for the superdiffusive behavior $\sigma \propto (\Delta
t)^{\epsilon/\alpha}$ (Eq. \[sigma-scale\]) at short time scales $\Delta t<(\Delta t)_{\times}$ \[Fig.1\]; and (b) a regime with breakdown of scaling defined by $\lambda_{\Delta t} \equiv
\lambda_{\times}=const$ and $A_{\Delta t} \equiv ({\Delta t}) A_{\times}$ for $\Delta
t>(\Delta t)_{\times}$ to account for the normal diffusive behavior $\sigma \propto (\Delta t)^{1/2}$ (Eq. (\[sigma-scale\]) and Fig.1). This breakdown allows for a transition from a non-i.i.d. STL process to an i.i.d. TL process.
The STL process in the regime $\Delta t<(\Delta t)_{\times}$ accounts for the second empirical observation, the identical scaling exponent ($\epsilon/\alpha$) experimentally observed for both the standard deviation $\sigma$ (Eq. \[sigma-scale\]) and the probability of return to the origin ${\cal
P}_{\Delta t}(0)$ (Eq. \[prob.scale-invar\] and Fig.2). From fitting the initial probability distribution ${\cal P}_1(z)$, we obtain $\alpha=1.43$. Since empirically the standard deviation scales with exponent $\epsilon/\alpha=0.7$, we find that $\epsilon=1$ for this process.
Third, we find that the theoretical prediction for the STL process with a scaling breakdown is in good agreement with the empirical result for ${\cal P}_{\Delta t}(0)$ for more than three decades \[Fig.2\]. We note that the transition at $(\Delta t)_{\times}\approx
30$ from STL (non-i.i.d.) process to a TL (i.i.d.) process in the scaling of $\sigma$ \[Fig.1\], does not imply a sharp transition in the scaling of ${\cal P}_{\Delta t}(0)$ from a Lévy to Gaussian behavior \[Fig.2\]. The reason is that the STL scaling regime (Eq. (\[scale-invar.tr\])), ${\cal P}_{\Delta t}(0)$ exhibits Lévy scaling behavior (Eq. (\[prob.scale-invar\])) up to $(\Delta t)_{\times}\approx 30$. In this scaling regime, $\sigma$ increases superdiffusively with exponent 0.7, that is much faster than 0.5 for an i.i.d. process. At the crossover scale $(\Delta
t)_{\times}$, the standard deviation reaches the value $\sigma_{\times}=(\Delta t)_{\times}^{0.7} \sigma_1$. The value of $\sigma_{\times}=(\Delta t)_{\times}^{0.5} \sigma_{TL}$ can be also related to an i.i.d. TL process with initial standard deviation $\sigma_{TL}>\sigma_1$ \[Fig.1\]. According to the central limit theorem, an i.i.d. TL process asymptotically converges to a Gaussian process. Thus while in the short time regime (small $\Delta t$) the price fluctuation $z$ over time $\Delta t$ is a sum of correlated stochastic variables, in the asymptotic regime (large $\Delta t$), $z$ can be treated as a sum of newly-defined independent stochastic variables with standard deviation $\sigma_{TL}$. Since such a Gaussian process is defined with large initial standard deviation $\sigma_{TL}$, the transition from the L[é]{}vy to the Gaussian behavior is delayed \[Fig.2\]. The time scale $(\Delta t)_s$ of this transition can be calculated by equating the return probability ${\cal
P}_{\Delta t}(0)$ for the L[é]{}vy and Gaussian distributions [@footnote4]. We find that $(\Delta t)_s={\cal B}(\Delta
t)_{\times}$, where ${\cal B}\approx 70$ \[Fig.2\]. Such a relation is interesting, since it explicitly connects the crossover from the L[é]{}vy to Gaussian with the crossover from non-i.i.d. to i.i.d. process.
Finally, we compare the empirical distributions of the price increments $z$ of the $S\&P500$ index for different time scales $\Delta t$ with the shape of the distributions obtained analytically \[Fig.3\]. Good agreement between data and the theoretical distributions is observed both for the central part and for the tails. At small time scales, the scale-invariant behavior of ${\cal P}_{\Delta t}(z)$ is maintained in the entire range (L[é]{}vy for the central profile, and exponential in the tails) due to the scaling transformations of the STL process (Eq. \[scale-invar.tr\]). The crossover to an i.i.d. TL process at large time scales ensures a smooth transition to a Gaussian-like profile. We find that the proposed mechanism of a STL process, with breakdown, provides a reliable control of the dynamical properties of the PDF.
In our analysis, we have considered the price fluctuations of the $S\&P500$ index as the stochastic variable $z$. The choice of stochastic variable depends on the type of the stochastic process: e.g., for an [*additive*]{} process one considers increments, while for [*multiplicative*]{} processes the appropriate choice is relative increments. In finance, it is traditionally assumed that economic indicators arise from a multiplicative process, and correspondingly the preferred quantity to analyze is the rate of return or the difference in the natural logarithm of price. The additive and multiplicative processes are related for high frequency data (small $\Delta t$) and short period of analysis, so the use of price fluctuations or rates of return lead to similar results. We find that even for low frequency data (large $\Delta t$) and for long period of analysis (up to 12 years), the results for the PDF and the standard deviation remain similar for both the price fluctuations and the rates of return \[Fig.4\].
We have proposed a stochastic process that even in the presence of correlations among the stochastic variables exhibits a L[é]{}vy stability for the PDF. The STL process is characterized by identical scaling exponents for both the moments and the PDF. The STL process provides an unified dynamical picture to describe different statistical properties, and can be generalized for situations when the moments and the PDF exhibit different scaling behavior. The STL process can be utilized — as we show in the case for financial data — not only for processes with a single scaling regime but also for physical systems with different regimes of scaling behavior.
P. Lévy, [*Theorie de l’Addition des Variables Aléatories*]{} (Gauthier-Villars, Paris, 1937).
M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature [**363**]{}, 31 (1993).
T. H. Solomon, E. R. Weeks, and H. L. Swinney, Phys. Rev. Lett. [**71**]{}, 3975 (1993).
A. Ott, J.-P. Bouchaud, D. Langevin, and W. Urbach, [*Ibid.*]{} [**65**]{}, 2201 (1990).
F. Bardou, J.-P. Bouchaud, O. Emile, A Aspect, and C. Cohen-Tannoudji, [*Ibid.*]{} [**72**]{}, 203 (1994).
F. Hayot and L. Wagner, Phys. Rev. E [**49**]{}, 470 (1994); F. Hayot, Phys. Rev. A [**43**]{}, 806 (1991).
J. Moon and H. Nakanishi, Phys. Rev. A [**42**]{}, 3221 (1990).
G. Zumofen and J. Klafter, Chem. Phys. Lett. [**219**]{}, 303 (1994).
M. Orrit and J. Bernard, Phys. Rev. Lett. [**65**]{}, 2716 (1990).
R.N. Mantegna and H.E. Stanley, Phys. Rev. Lett. [**73**]{}, 2946 (1994).
R. N. Mantegna and H. E. Stanley, Nature [**376**]{}, 46 (1995).
I. Koponen, Phys. Rev. E [**52**]{}, 1197 (1995).
For $\lambda=0$ the probability function $f(z)$ corresponds to L[é]{}vy distribution.
Different types of smooth truncation can be reproduced with changing values of $\beta$: (i) for $\beta<1$ we have stretched exponential tails; (ii) for $\beta=1$ we have an exponential truncation (the case of [@Koponen95]); (iii) $\beta>1$ we have process faster than exponential; and (iv) for $\beta=2$ we have Gaussian behavior.
B. V. Gnedenko and A. N. Kolmogorov, [*Limit Distributions for Sums of Independent Random Variables*]{} (Addison-Wesley, Cambridge, MA, 1954).
In the case of TL process, L[é]{}vy-type scaling transformations for the stochastic variable $\{z\}$ and the parameter $A$ — $z \equiv ({\Delta
t})^{-\epsilon/\alpha} z_1$ and $A \equiv {\Delta t} A_1$ — ensure scale invariant behavior only for the central profile of ${\cal P}(z)$.
Note that the STL process characterized by given $\alpha$ can scale with any scaling exponent $\epsilon/\alpha$ in contrast to the Lévy stable process which scales with the scaling exponent $1/\alpha$. The parameter $\epsilon$ controls the dynamics of the process — probability distributions characterized with the same $\alpha$ can exhibit different scaling behavior for different values of $\epsilon$. E.g. for $\epsilon = 1$ and $\lambda = 0$ under the transformations of Eqs. (\[scale-invar.tr\]) the probability density ${\cal P}(z)$ scales as the L[é]{}vy stable process.
We obtain the following analytic expression: ${\cal B}=[\sqrt{2\pi}\sigma_1{\cal L}_{\Delta t=1}(0)]^{2\alpha/(2-\alpha)}$, where ${\cal L}$ is the L[é]{}vy PDF [@Koponen95].
=0.8
=0.8
=0.8
=0.82
=0.82
=0.82
|
---
author:
- Julyan Arbel
- Olivier Marchal
- Bernardo Nipoti
bibliography:
- 'biblio.bib'
title: 'On the Hurwitz zeta function with an application to the exponential-beta distribution'
---
> **Abstract**
>
> We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a connection between Hurwitz zeta function and the cumulants of the exponential-beta distribution.
Main result
===========
Let $\zeta(x,s)=\underset{k=0}{\overset{+\infty}{\sum}} (k+s)^{-x}$ be the Hurwitz zeta function [@berndt1972hurwitz] defined for $(x,s)\in (1,+\infty)\times(0,+\infty)$, and, for any $a>0$ and $b>0$, consider the function $$\label{eq:main}
x\mapsto f(x,a,b) = \left(
\zeta(x,b) - \zeta(x,a+b)
\right)^{\frac{1}{x}}, $$ defined on $[1,+\infty)$, where $f(1,a,b)$ is defined by continuity as $$\label{eq:main2}
f(1,a,b)=\sum_{k=0}^\infty\left( \frac{1}{k+b}-\frac{1}{k+a+b}\right)=\sum_{k=0}^\infty \frac{a}{(k+b)(k+a+b)}.$$ The function $f(x,a,b)$ can be alternatively written, with a geometric flavour, as $$f(x,a,b) = \left(\Vert \bm{v}_{a+b} \Vert_x^x - \Vert \bm{v}_{b} \Vert_x^x \right)^{\frac{1}{x}},$$ where, for any $s>0$, $\bm{v}_s$ is an infinite-dimensional vector whose $k^{\text{th}}$ component coincides with $(k-1+s)^{-1}$.\
The main result of the paper establishes that the function $x\mapsto f(x,a,b)$ is monotone on $[1,+\infty)$ with variations only determined by the value of $a$. More specifically,
\[thm:main\] For any $b>0$, the function $x\mapsto f(x,a,b)$ defined on $[1,+\infty)$ is increasing[^1] if $0<a<1$, decreasing if $a>1$, and constantly equal to $\frac{1}{b}$ if $a=1$.
Theorem \[thm:main\] and the derived inequalities in terms of polygamma functions (see Equation ) add to the current body of literature about inequalities and monotonicity properties of the Hurwitz zeta function [@berndt1972hurwitz; @srivastava2011two; @leping2013hilbert] and polygamma functions [@alzer1998inequalities; @alzer2001mean; @batir2005some; @qi2010complete; @guo2015sharp], respectively. The last part of the statement of Theorem \[thm:main\] is immediately verified as, when $a=1$, $f(x,a,b)$ simplifies to a telescoping series which gives $f(x,1,b)=\frac{1}{b}$ for every $x\in[1,+\infty)$. The rest of the proof is presented in Section \[sec:proof\] while Section \[sec:application\] is dedicated to an application of Theorem \[thm:main\] to the study of the so-called exponential-beta distribution [@gupta1999theory; @nadarajah2006beta], obtained by applying a log-transformation to a beta distributed random variable. More specifically, the dichotomy observed in Theorem \[thm:main\], determined by the position of $a$ with respect to 1, is shown to hold for the exponential-beta distribution at the level of (i) its cumulants (whether function is increasing or not), (ii) its dispersion (Corollary \[cor:EB-dispersion\]), (iii) the shape of its density (log-convex or log-concave, Proposition \[prop:shape-density\] and Figure \[fig:proba-interpretation\]) and (iv) its hazard function (increasing or decreasing, Proposition \[prop:hazard\]).
Proof of Theorem \[thm:main\]\[sec:proof\]
==========================================
The proof of Theorem \[thm:main\] relies on Lemma \[lem:a-larger-1\], stated below. Lemma 1 considers two sequences and establishes the monotonicity of a third one, function of the first two, whose direction depends on how the two original sequences compare with each other. The same dichotomy, in Theorem \[thm:main\], is driven by the position of the real number $a$ with respect to 1.
A technical lemma
-----------------
\[lem:a-larger-1\]Let $(s_n)_{n\geq 1}$ and $(r_n)_{n\geq 1}$ be two sequences in $(0,1)$ and define, for $N\geq 1$, $$u_N\overset{\mathrm{def}}{=}\left(1+\sum_{n=1}^N (s_n-r_n)\right)\ln\left(1+\sum_{n=1}^N (s_n-r_n)\right)-\sum_{n=1}^N(s_n\ln s_n-r_n\ln r_n).$$ We define by convention $u_0=0$. Then two cases are considered:
1. if, for any $n\geq 1$, $r_n\leq s_n$ then, for all $N\geq 0$, we have $u_{N+1}\geq u_N$, with the equality holding if and only if $s_{N+1}=r_{N+1}$;
2. if, for any $n\geq 1$, $s_{n+1}\leq r_{n+1}\leq s_n\leq r_n$ then, for all $N\geq 0$, we have $u_{N+1}\leq u_N$, with the equality holding if and only if $s_{N+1}=r_{N+1}$.
Moreover, if $\underset{n=1}{\overset{\infty}{\sum}} |s_n-r_n|<\infty$ (implying absolute convergence of the series $\underset{n=1}{\overset{\infty}{\sum}} (s_n\ln s_n-r_n\ln r_n)$) then $$u_\infty \overset{\mathrm{def}}{=}\lim_{N\to +\infty}u_N= \left(1+\sum_{n=1}^{\infty} (s_n-r_n)\right)\ln\left(1+\sum_{n=1}^{\infty} (s_n-r_n)\right)-\sum_{n=1}^{\infty}(s_n\ln s_n-r_n\ln r_n)$$ exists and satisfies $u_\infty\geq 0$ in case **1**, while $u_\infty\leq 0$ in case **2**. In both cases, $u_\infty= 0$ if and only if the two sequences $(r_n)_{n\geq 1}$ and $(s_n)_{n\geq 1}$ equal each other.
\[rem1\] Note that, in case **2**, we have $$\begin{aligned}
1+\sum_{n=1}^N (s_n-r_n)=(1-r_1)+s_{N}+\sum_{n=1}^{{N-1}}(s_n-r_{n+1})\geq (1-r_1)+s_{N}> 0,\end{aligned}$$ $$\begin{aligned}
0\leq s\ln s -r\ln r\leq s-r\,\,,\,\,\forall\,\, 0<r\leq s<1.\end{aligned}$$
**Proof of Lemma \[lem:a-larger-1\]**
**Proof for $N=0$**. We first study the case $N=0$ and define $$\begin{aligned}
h_{r_1}(s_1)=(1+s_1-r_1)\ln(1+s_1-r_1)-s_1\ln s_1+r_1\ln r_1.\end{aligned}$$ For $s_1=r_1$ we trivially have $h_{r_1}(r_1)=0$. A straightforward computation shows that $$\begin{aligned}
h_{r_1}'(s_1)=\ln(1+s_1-r_1)-\ln s_1=\ln((1-r_1)+s_1)-\ln s_1>0,\end{aligned}$$ since $r_1<1$. Hence $h_{r_1}$ is an increasing function on $(0,1)$. Since $h_{r_1}(r_1)=0$, we immediately get that $h_{r_1}$ is positive on $(r_1,1)$ and negative on $(0,r_1)$, thus proving both cases for $N=0$.
**Proof for $N\geq 1$**. We now consider the case $N\geq 1$ and define $$\begin{aligned}
h_{r_1,\ldots,r_{N+1},s_1,\ldots,s_N}(s_{N+1})=& \, u_{N+1}-u_N\\
=&\, \left(1+\sum_{n=1}^{N+1} (s_n-r_n)\right)\ln\left(1+\sum_{n=1}^{N+1} (s_n-r_n)\right)\\
&-\left(1+\sum_{n=1}^{N} (s_n-r_n)\right)\ln\left(1+\sum_{n=1}^{N} (s_n-r_n)\right)\\
&-s_{N+1}\ln s_{N+1}+r_{N+1}\ln r_{N+1}.\end{aligned}$$ We trivially get that $h_{r_1,\ldots,r_{N+1},s_1,\ldots,s_N}(r_{N+1})=0$. Moreover, we have $$\begin{aligned}
h_{r_1,\ldots,r_{N+1},s_1,\ldots,s_N}'(s_{N+1})=&\ln\left(1+\sum_{n=1}^{N+1} (s_n-r_n)\right)-\ln s_{N+1},\\
\overset{\text{(1)}}{=}&\,\,\ln\left(s_{N+1}+(1-r_{N+1})+\sum_{n=1}^{N} (s_n-r_n)\right)-\ln s_{N+1},\\
\overset{\text{(2)}}{=}&\,\,\ln\left(s_{N+1}+(1-r_{1})+\sum_{n=1}^{N} (s_n-r_{n+1})\right)-\ln s_{N+1}.\end{aligned}$$ Equality $(1)$ shows that $h_{r_1,\ldots,r_{N+1},s_1,\ldots,s_N}'$ is positive on $(r_{N+1},1)$ for conditions of case $\textbf{1}$, while equality $(2)$ shows that $h_{r_1,\ldots,r_{N+1},s_1,\ldots,s_N}'$ is positive on $(0,r_{N+1})$ for conditions of case **2**. Since $h_{r_1,\ldots,r_{N+1},s_1,\ldots,s_N}(r_{N+1})=0$ we get that $h_{r_1,\ldots,r_{N+1},s_1,\ldots,s_N}$ is positive on $(r_{N+1},1)$ for conditions of case **1**, while $h_{r_1,\ldots,r_{N+1},s_1,\ldots,s_N}$ is negative on $(0,r_{N+1})$ for conditions of case **2**. This ends the proof of monotonicity of $(u_N)_{N\geq 0}$ and conditions for strict monotonicity in both cases. Extending results from finite $N$ to $N\to \infty$ follows directly from these results and Remark \[rem1\].
Proof of Theorem \[thm:main\]
-----------------------------
We want to study the variations of $$x \mapsto f(x,a,b)=\left(\zeta(x,b)-\zeta(x,a+b)\right)^{\frac{1}{x}}$$ on $[1,\infty)$, for which it is enough, by continuity, to focus on $(1,\infty)$. Since $f$ is positive, its variations are equivalent to those of $$\begin{aligned}
F(x,a,b)&\overset{\text{def}}{=}\ln f(x,a,b)\\
&= \frac{1}{x}\ln \left(\zeta(x,b)-\zeta(x,a+b)\right)=\frac{1}{x}\ln\left( \sum_{k=0}^{\infty} \frac{1}{(k+b)^x}-\frac{1}{(k+a+b)^x}\right)\\
&=-\ln b+ \frac{1}{x} \ln\left( \sum_{k=0}^{\infty} \left(\frac{b}{k+b}\right)^x- \left(\frac{b}{k+a+b}\right)^x\right).\end{aligned}$$ A straightforward computation shows that $$\begin{aligned}
\partial_x F(x,a,b)=
\frac{H(x,a,b)}{x^2\left(\underset{k=0}{\overset{\infty}{\sum}} \left(\frac{b}{k+b}\right)^x- \left(\frac{b}{k+a+b}\right)^x\right)},$$ hence the sign of the derivative $\partial_x F(x,a,b)$ is the same as that of $H(x,a,b)$ defined by $$\begin{aligned}
\sum_{n=1}^{\infty} (s_n \ln s_n-r_n\ln r_n) -
\left(1+\sum_{n=1}^{\infty} (s_n- r_n) \right)\ln\left(1+\sum_{n=1}^{\infty} (s_n-r_n) \right),\end{aligned}$$ where, for all $n \geq 1$, we have defined $$s_n=\left(\frac{b}{n+b}\right)^x \quad \text{and} \quad r_n= \left(\frac{b}{n+a-1+b}\right)^x.$$ Note that, for any values of $a>0$ and $b>0$, we have $\underset{n=1}{\overset{\infty}{\sum}} |s_n-r_n|<\infty$ as $x>1$ implies that $\underset{n=1}{\overset{\infty}{\sum}} s_n<\infty$ and $\underset{n=1}{\overset{\infty}{\sum}} r_n<\infty$. Moreover, when $a>1$, we have $0< r_n<s_n< 1$, while when $0<a<1$, we have $0<r_{n+1}<s_n<r_n<1$ for all $n\geq 1$ and $x>1$. We can then apply Lemma \[lem:a-larger-1\] to obtain that $H(x,a,b)$, and thus $\partial_x F(x,a,b)$, is negative if $a>1$ (case **1** of Lemma \[lem:a-larger-1\]) and is positive if $0<a<1$ (case **2** of Lemma \[lem:a-larger-1\]), which concludes the proof.
Probabilistic interpretation: application to the exponential-beta distribution\[sec:application\]
=================================================================================================
The aim of this section is to identify a connection between Theorem 1 and the exponential-beta distribution. The latter arises by taking a log-transformation of a beta random variable. More specifically, let $V$ be a beta random variable with parameters $a>0$ and $b>0$, then we say that $X$ is an exponential-beta random variable with parameters $a$ and $b$ if $X = -\ln(1-V)$, and use the notation $X\sim\text{EB}(a,b)$ [see @gupta1999theory; @nadarajah2006beta]. The corresponding density function is given by $$\label{eq:dens}
g(x;a,b) = \frac{1}{B(a,b)}(1-{\mathrm{e}}^{-x})^{a-1}{\mathrm{e}}^{-bx}\mathbf{1}_{(0,+\infty)}(x),$$ where $B(a,b)$ denotes the beta function[^2]. The cumulant-generating function of $X$ can be written as $$\begin{aligned}
K(t) &\overset{\text{def}}{=}\ln {\mathbb{E}}(\exp(tX))
=\ln\Gamma(a+b)+\ln\Gamma(b-t)-\ln\Gamma(b)-\ln\Gamma(a+b-t),\end{aligned}$$ provided that $t<b$ [see Section 3 of @nadarajah2006beta]. This implies that, for any $n\geq 1$, the $n^{\text{th}}$ cumulant of $X$, denoted $\kappa_n(a,b)$, is given by $$ \kappa_n(a,b) =(-1)^n\left(\psi^{(n-1)}(b)-\psi^{(n-1)}(b+a)\right),$$ where $\psi^{(m)}$ for $m\in\mathbb{N}$ denotes the polygamma function of order $m$, which is defined as the derivative of order $m+1$ of the logarithm of the gamma function. An interesting relation across cumulants of different orders is then obtained as a straightforward application of Theorem \[thm:main\]. Before stating the result, and for the sake of compactness, we define on $\mathbb{N}\setminus\{0\}$, for any $a>0$ and $b>0$, the function $$\label{eq:cumulant-eb}
n\mapsto f_{\text{EB}}(n,a,b)=\left(\frac{\kappa_n(a,b)}{(n-1)!}\right)^{\frac{1}{n}}. $$
\[cor:EB\] For any $b>0$, the function $n\mapsto f_{\text{EB}}(n,a,b)$ defined on $\mathbb{N}\setminus \{0\}$, is increasing if $0<a<1$, decreasing if $a>1$, and constantly equal to $\frac{1}{b}$ if $a=1$.
The proof follows by observing that, when $n\in\mathbb{N}\setminus\{0\}$, $f_{\text{EB}}(n,a,b)=f(n,a,b)$, with the latter defined in and . This can be seen, when $n>1$, by applying twice the identity $\psi^{(n-1)}(s)=(-1)^{n} (n-1)! \zeta(n,s)$, and, when $n=1$, by applying twice the identity $\psi^{(0)}(s)=-\gamma+\underset{k=0}{\overset{\infty}{\sum}} \frac{s-1}{(k+1)(k+s)}$, where $\gamma$ is the Euler-Mascheroni constant, which holds for any $s>-1$ [see Identity 6.3.16 in @Abr65].\
Corollary \[cor:EB\] can alternatively be expressed as a chain of inequalities in terms of polygamma functions of different orders, which might be of independent interest. Namely, for any $b>0$ and any $0<a_1<1<a_2$, the following holds: $$\label{eq:poly}
\begin{cases}\psi^{(0)}(b+a_1)-\psi^{(0)}(b)<\ldots<
\left(\frac{\psi^{(n)}(b+a_1)-\psi^{(n)}(b)}{n!}\right)^{\frac{1}{n+1}}<\ldots <
\frac{1}{b},\\
\psi^{(0)}(b+a_2)-\psi^{(0)}(b)>\ldots>
\left(\frac{\psi^{(n)}(b+a_2)-\psi^{(n)}(b)}{n!}\right)^{\frac{1}{n+1}}>\ldots >
\frac{1}{b}.
\end{cases}$$
Corollary \[cor:EB\], as well as , highlights the critical role played by the exponential distribution with mean $\frac{1}{b}$, special case of the exponential-beta distribution recovered from by setting $a=1$. In such special instance, the cumulants simplify to $\kappa_n(1,b)=b^{-n}(n-1)!$, which makes $f_{EB}(n,1,b)=\frac{1}{b}$ for every $n\in\mathbb{N}\setminus\{0\}$. Within the exponential-beta distribution, the case $a=1$ then creates a dichotomy by identifying two subclasses of densities, namely $\{g(x;a,b)\,:\,0<a<1\}$, whose cumulants $\kappa_n(a,b)$ make $f_{\text{EB}}(n,a,b)$ an increasing function of $n$, and $\{g(x;a,b)\,:\,a>1\}$ for which $f_{\text{EB}}(n,a,b)$ a decreasing function of $n$. The left panel of Figure \[fig:proba-interpretation\] displays the function $b\mapsto f_{\text{EB}}(n,a,b)$ for some values of $n$ and $a$: it can be appreciated that, for any $b$ in the considered range, the order of the values taken by $f_{\text{EB}}(n,a,b)$ is in agreement with Corollary \[cor:EB\].\
The first two cumulants of a random variable $X$ have a simple interpretation in terms of its first two moments, namely $\kappa_1 = \mathbb{E}[X]$ and $\kappa_2 = \text{Var}[X].$ A special case of Corollary \[cor:EB\], focusing on the case $n\in\{1,2\}$, then provides an interesting result relating the dispersion of the exponential-beta distribution with its mean. Specifically,
The behaviour of the cumulants is not the only distinctive feature characterizing the two subclasses of density functions corresponding to $0<a<1$ and $a>1$. For any $b$, the value of $a$ determines the shape of the density as displayed in the right panel of Figure \[fig:proba-interpretation\] and summarized by the next proposition, whose proof is trivial and thus omitted.
\[prop:shape-density\] For any $b>0$, the exponential-beta density $g(x;a,b)$ is log-convex if $0<a<1$ and log-concave if $a>1$.
The same dichotomy within the exponential-beta distribution is further highlighted by the behaviour of the corresponding hazard function, defined for an absolutely continuous random variable $X$ as the function $x\mapsto \frac{f_X(x)}{1-F_X(x)}$, where $f_X$ and $F_X$ are, respectively, the probability density function and the cumulative distribution function of $X$.
\[prop:hazard\] For any $b>0$, the hazard function of the exponential-beta distribution with parameters $a$ and $b$ is decreasing if $a<1$, increasing if $a>1$, and constantly equal to $b$ if $a=1$.
The result follows from the log-convexity and log-concavity properties of $g(x;a,b)$ [see @barlow1975statistical].\
Finally, it is worth remarking that an analogous dichotomy holds within the class of gamma density functions with $a>0$ and $b>0$ shape and rate parameters, and that once again the exponential distribution with mean $\frac{1}{b}$, special case recovered by setting $a=1$, lays at the border between the two subclasses. The $n^{\text{th}}$ cumulant of the gamma distribution is $\kappa_n=ab^{-n}(n-1)!$, which makes the function $n\mapsto \left(\frac{\kappa_n(a,b)}{(n-1)!}\right)^{\frac{1}{n}}$, defined on $\mathbb{N}\setminus \{0\}$, increasing if $0<a<1$, decreasing if $a>1$ and constantly equal to $\frac{1}{b}$ if $a=1$. Similarly, the gamma density is log-convex if $0<a<1$ and log-concave if $a>1$ and, thus, the corresponding hazard function is decreasing if $a<1$, increasing if $a>1$ and constantly equal to $b$ if $a=1$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank [Marco Mazzola](Marco) for fruitful suggestions. O.M. would like to thank Université Lyon $1$, Université Jean Monnet and Institut Camille Jordan for material support. This work was developed in the framework of the Ulysses Program for French-Irish collaborations (43135ZK), the Grenoble Alpes Data Institute and the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) and (ANR-15-IDEX-02) operated by the French National Research Agency (ANR).
[^1]: Throughout the paper, we say that a function $f$ is increasing (resp. decreasing) if $x<y$ implies $f(x)<f(y)$ (resp. $f(x)>f(y)$), and that a quantity $A$ is positive (resp. negative) if $A>0$ (resp. $A<0$).
[^2]: The beta function is defined in this article as $$B(a,b)=\int_0^{+\infty}(1-{\mathrm{e}}^{-x})^{a-1}{\mathrm{e}}^{-bx}{\mathrm{d}}x.$$
|
**Partition Function and Open/Closed String Duality\
in Type IIA String Theory on a PP-wave**
Hyeonjoon Shin$^{1,3a}$, Katsuyuki Sugiyama$^{2b}$ and Kentaroh Yoshida$^{3c}$
$^{1}$*BK21 Physics Research Division and Institute of Basic Science,\
Sungkyunkwan University, Suwon 440-746, Korea.*\
$^{2}$*Department of Physics, Kyoto University, Kyoto 606-8501, Japan.*\
$^{3}$*Theory Division, High Energy Accelerator Research Organization (KEK),\
Tsukuba, Ibaraki 305-0801, Japan.*\
**Abstract**
We discuss partition functions of $\mathcal{N}=(4,4)$ type IIA string theory on the pp-wave background. This theory is shown to be modular invariant. The boundary states are constructed and possible D-brane instantons are classified. Then we calculate cylinder amplitudes in both closed and open string descriptions and check the open/closed string duality. Furthermore we consider general properties of modular invariant partition functions in the case of pp-waves.
[**Keywords:**]{} [pp-wave, modular invariance, boundary state, D-brane, open/closed string duality]{}
Introduction
============
Recently, superstring theories on pp-waves have been very focused upon. The maximally supersymmetric pp-wave type solution in eleven dimensions [@KG] has been known for a long time while the maximally supersymmetric type IIB pp-wave solution was found in [@BFHP1] in the recent progress. It was also pointed out in [@BFHP2] that this pp-wave background is related to the $AdS$-geometry via the Penrose limit [@P; @G]. Then, the type IIB superstring theory on the maximally supersymmetric pp-wave background was constructed [@M; @MT]. By using this pp-wave superstring theory, the study of $AdS$/CFT correspondence has greatly proceeded [@BMN]. In particular, the $AdS$/CFT correspondence has been studied at the stringy level beyond the supergravity analysis.
In the study of pp-wave backgrounds, the matrix model on the pp-wave, which was proposed by Berenstein-Maldacena-Nastase [@BMN], has been much studied. This matrix model is closely related to a supermembrane theory on the pp-wave background [@DSR; @SY1]. We have discussed the supermembrane theory and matrix model on the pp-wave from the several aspects [@SY1; @HS1; @SY2; @SY3; @NSY]. In particular, we showed the correspondence of brane charges in the supermembrane theory [@SY1] and matrix model [@HS1] in the pp-wave case as well as in flat space [@BSS].
A supermembrane in eleven dimensions is related to a string in ten dimensions via the double dimensional reduction. We constructed the type IIA pp-wave background with 24 supersymmetries and string theory on this pp-wave background [@SY4; @HS2], which is called $\mathcal{N}=(4,4)$ type IIA string theory on the pp-wave. We discussed the classification of the allowed D-branes [@SY4; @HS3] by following the work of Dabholkar and Parvizi [@DP]. After these works, the covariant classification of D-branes was done in [@HPS] where D0-branes could be studied. In addition, the spectrum of this type IIA string theory was compared to fluctuations of the linearized type IIA supergravity around the pp-wave background[@KS]. The thermodynamics of this type IIA string theory was recently studied in [@HPY] [^1].
On the other hand, as an important and interesting subject, the modular invariance of string theories on pp-waves has been studied by several authors [@BGG; @GG; @T; @S1; @S2; @HPY; @GGSS]. It has been turned out that these theories are modular invariant in spite of mass terms in the action. In this paper, motivated by the previous works, we will be interested in the $\mathcal{N}=(4,4)$ type IIA string theory on the pp-wave background obtained in [@SY4; @HS2] and study its partition function and open/closed string duality. The pp-wave background we will consider is not maximally supersymmetric but has 24 supersymmetries. Hence it is interesting to study whether the modular invariance and consistency condition between open and closed strings are satisfied or not in such less supersymmetric case. Moreover, since the number of preserved supersymmetries is nontrivial even in the case of supersymmetric D-branes, it is also interesting to construct boundary states in our model[^2].
In this paper we discuss partition functions in the $\mathcal{N}=(4,4)$ type IIA string theory on the pp-wave. We first show that our theory is modular invariant. The boundary states is constructed and then we classify the allowed D-brane instantons. They are 1/2 BPS states (preserving 12 supersymmetries) at the origin and 1/3 BPS ones (preserving 8 supersymmetries) away from the origin. This result is consistent with the classification of D-branes. Then we calculate the cylinder amplitude in the closed and open string descriptions and study the open/closed string channel duality in our theory. Finally, we discuss general properties of modular invariant partition function and classify several models.
This paper is organized as follows: In section 2 we prove the modular invariance of $\mathcal{N}=(4,4)$ type IIA string theory on the pp-wave background. The Witten index of this theory is shown to be one. Section 3 is devoted to a brief review about the supersymmetries of our theory. In section 4 we construct boundary states and classify D-brane instantons. In section 5 we calculate the cylinder amplitude in the closed string description. In section 6 the amplitude is derived in terms of open string and the open/closed string channel duality in our theory is proven. From the channel duality, the normalization factor of the boundary state is determined. In section 7, based on the result of section 2, we discuss several properties of modular invariant partition function in some general setup. Finally, section 8 is devoted to conclusions and discussions.
Modular Invariance of Type IIA String Theory
============================================
In this section we will discuss the modular invariance of type IIA string theory in the closed string description.
The action of our type IIA string in the light-cone gauge is given by $$\begin{aligned}
S_{\rm closed} &=& \frac{1}{4\pi{\alpha}'}
\int\! d\tau\!\! \int^{2\pi}_0\!\!\!d\sigma\, \Biggl[
\sum_{i=1}^8\partial_+ x^i\partial_- x^i -
\left(\frac{\mu}{3}\right)^2 \sum_{a=1}^4 (x^a)^2
- \left(\frac{\mu}{6}\right)^2\sum_{b=5}^8(x^b)^2 \Biggr] \\
&& +
\frac{i}{2\pi}
\int\! d\tau\!\! \int^{2\pi}_0\!\!\!d\sigma\, \Biggl[
\Psi^{1+}{}^{{\scriptscriptstyle}T}\partial_-\Psi^{1+}
+ \Psi^{1-}{}^{{\scriptscriptstyle}T}\partial_-\Psi^{1-}
+ \Psi^{2+}{}^{{\scriptscriptstyle}T}\partial_+\Psi^{2+}
+ \Psi^{2-}{}^{{\scriptscriptstyle}T}\partial_+\Psi^{2-} {\nonumber}\\
&&
- \frac{\mu}{3}\Psi^{1-}{}^{{\scriptscriptstyle}T}\Pi^{{\scriptscriptstyle}T}\Psi^{2+}
+ \frac{\mu}{3}\Psi^{2+}{}^{{\scriptscriptstyle}T}\Pi\Psi^{1-}
- \frac{\mu}{6}\Psi^{1+}{}^{{\scriptscriptstyle}T}\Pi^{{\scriptscriptstyle}T}\Psi^{2-}
+ \frac{\mu}{6}\Psi^{2-}{}^{{\scriptscriptstyle}T}\Pi\Psi^{1+} \Biggr]
\,, {\nonumber}\end{aligned}$$ where ${\alpha}'$ is a string tension and we have set $p^+=1$. The $\gamma^r$’s are 16 $\times$ 16 $SO(9)$ gamma matrices and we defined $\Pi \equiv \gamma^{123}$ $(\Pi^{{\scriptscriptstyle}T}\equiv \gamma^{321})$. Each of the spinors $\Psi^{i\pm}~~(i=1,2)$ has four independent components and the superscript $\pm$ represents the chirality measured by $\gamma^{1234}$: $ \gamma^{1234}\Psi^{i\pm} = \pm 1\cdot \Psi^{i\pm}\,.$ This theory has 24 supersymmetries (8 dynamical supersymmetries and 16 kinematical supersymmetries). The equations of motion are described by $$\begin{aligned}
\label{em1}
&& \partial_+ \partial_- x^a + \frac{\mu^2}{9}\,x^a = 0 \quad
(a=1,2,3,4)\,, \\
\label{em2}
&& \partial_+ \partial_- x^b + \frac{\mu^2}{36}\,x^b = 0 \quad
(b=5,6,7,8)\,, \\
\label{em3}
&& \partial_+ \Psi^{2+} + \frac{\mu}{3}\,\Pi\,\Psi^{1-}
= 0\,, \quad \partial_- \Psi^{1-} - \frac{\mu}{3}\,\Pi^{{\scriptscriptstyle}T}\,
\Psi^{2+} = 0\,, \\
&& \partial_+ \Psi^{2-} + \frac{\mu}{6}\,\Pi\,\Psi^{1+}
= 0\,, \quad
\partial_- \Psi^{1+} - \frac{\mu}{6}\,\Pi^{{\scriptscriptstyle}T}\,
\Psi^{2-} = 0\,. \label{em4} \end{aligned}$$ By solving the equations of motion (\[em1\]) and (\[em2\]), we can obtain the mode-expansions of bosonic variables represented by $$\begin{aligned}
x^a(\tau,\,\sigma) &=&
x_0^{a} \cos\left(\frac{\mu}{3}\tau\right)
+ \left(
\frac{3}{\mu}\right){\alpha}' p_0^{a}\sin\left(\frac{\mu}{3}\tau\right)
+ i\sqrt{\frac{{\alpha}'}{2}}\sum_{n\neq 0}\frac{1}{\omega_n}
\left[\alpha_n^{a}\phi_n + \bar{\alpha}_n^{a}
\tilde{\phi}_n \right]\,, \\
x^b(\tau,\,\sigma) &=&
x_0^{b} \cos\left(\frac{\mu}{6}\tau\right)
+ \left(\frac{6}{\mu}\right)
{\alpha}' p_0^{b}\sin\left(\frac{\mu}{6}\tau\right)
+ i\sqrt{\frac{{\alpha}'}{2}}\sum_{n\neq 0}\frac{1}{\omega_n'}
\left[
\alpha_n^{b}\phi_n' + \bar{\alpha}_n^b\tilde{\phi}_n'
\right]\,. \end{aligned}$$ From the equations of motion (\[em3\]) and (\[em4\]), the mode-expansions of fermionic variables are represented by $$\begin{aligned}
\Psi^{1-}(\tau,\,\sigma) &=&
\Pi^T\Psi_0\sin\left(\frac{\mu}{3}\tau\right)
-\Pi^T\widetilde{\Psi}_0\cos\left(\frac{\mu}{3}\tau\right) {\nonumber}\\
&& \qquad + \sum_{n\neq 0}c_n\left(\frac{3}{\mu}i(\omega_n - n)
\,\Pi^{{\scriptscriptstyle}T}
\Psi_n\phi_n + \widetilde{\Psi}_n\tilde{\phi}_n\right)
\,, \\
\Psi^{2+}(\tau,\,\sigma) &=&
\Psi_0 \cos\left(\frac{\mu}{3}\tau\right)
+ \widetilde{\Psi}_0\sin\left(\frac{\mu}{3}\tau\right) {\nonumber}\\
&& \qquad + \sum_{n\neq 0}
c_n\left[\Psi_n\phi_n - \frac{3}{\mu}i(\omega_n -n)
\,\Pi\widetilde{
\Psi}_n\tilde{\phi}_n \right] \,, \\
\Psi^{1+}(\tau,\,\sigma) &=&
\Pi^T\Psi_0'\sin\left(\frac{\mu}{6}\tau\right)
- \Pi\widetilde{\Psi}_0'\cos\left(\frac{\mu}{6}\tau\right) {\nonumber}\\
&& \qquad + \sum_{n\neq 0}c_n'\left(\frac{6}{\mu}i(\omega_n' - n)
\,\Pi^{{\scriptscriptstyle}T}
\Psi_n'\phi_n' + \widetilde{\Psi}_n'\tilde{\phi}_n'
\right) \,, \\
\Psi^{2-}(\tau,\,\sigma) &=&
\Psi_0' \cos\left(\frac{\mu}{6}\tau\right)
+ \widetilde{\Psi}_0'\sin\left(\frac{\mu}{6}\tau\right) {\nonumber}\\
&& \qquad + \sum_{n\neq 0}
c_n'\left[\Psi_n'\phi_n' - \frac{6}{\mu}i(\omega_n' -n)
\,\Pi\widetilde{\Psi}_n'\tilde{\phi}_n'\right] \,. \end{aligned}$$ Here we have introduced several notations: $$\begin{aligned}
&& \omega_n
\;=\; {\rm sgn}(n)\sqrt{n^2 + \left(\frac{\mu}{3}\right)^2}\,,\quad
\omega_n' \;=\;
{\rm sgn}(n)\sqrt{n^2 + \left(\frac{\mu}{6}\right)^2 }
\,, \label{omega} {\nonumber}\\
&& \phi_n
\;=\; \exp\left(-i(\omega_n\tau - n\sigma)\right)\,,\quad
\tilde{\phi}_n
\;=\; \exp\left(-i\left(\omega_n \tau + n\sigma\right)
\right)\,, {\nonumber}\\
&& \phi_n'
\;=\; \exp\left(-i(\omega_n'\tau - n\sigma)\right)\,,\quad
\tilde{\phi}_n'
\;=\; \exp\left(-i\left(\omega_n'\tau + n\sigma\right)
\right)\,, {\nonumber}\\
&& c_n \;=\; \left(1 + \left(\frac{3}{\mu}\right)^2(
\omega_n - n)^2\right)^{-1/2}\,, \quad
c_n' \;=\; \left(1 + \left(\frac{6}{\mu}\right)^2(
\omega_n' - n)^2\right)^{-1/2}
\,. {\nonumber}\end{aligned}$$ Now we shall quantize the theory by imposing (anti)commutation relations. The commutation relations for bosonic modes are given by $$\begin{aligned}
\label{com-b}
\left[x_0^i,\,p^j_0\right]
&=& i{\delta}^{ij}\,, \quad
\left[\bar{{\alpha}}^i_m, \, {\alpha}_n^j\right]
\;=\; [{\alpha}^i_m,\,\bar{{\alpha}}^j_n] \;=\; 0 \quad (i,j = 1,\ldots, 8)\,,{\nonumber}\\
\left[{\alpha}_m^a,\, \alpha_n^{a'}\right] &=&
\left[\bar{{\alpha}}_m^a,\,\bar{{\alpha}}_n^{a'}\right]
\;=\; \omega_m{\delta}_{m+n,0}\,{\delta}^{aa'}
\quad (a,\,a' = 1,2,3,4)\,, {\nonumber}\\
\left[{\alpha}_m^b,\, \alpha_n^{b'}\right] &=&
\left[\bar{{\alpha}}_m^b,\,\bar{{\alpha}}_n^{b'}\right] \;=\;
\omega_m'{\delta}_{m+n,0}\,{\delta}^{bb'}
\quad (b,\,b' = 5,6,7,8)\,, {\nonumber}\end{aligned}$$ and the anticommutation relations for fermionic modes are written as $$\begin{aligned}
\label{com-f}
\{(\Psi_m)_{{\alpha}},\, (\widetilde{\Psi}_n)_{\beta}^{{\scriptscriptstyle}T}
\} &=&
\{
(\widetilde{\Psi}_m)_{{\alpha}},\,(\Psi_n)_{\beta}^{{\scriptscriptstyle}T}
\} \; = \; 0\,, {\nonumber}\\
\{(\Psi_m)_{{\alpha}},\,(\Psi_n)_{\beta}^{{\scriptscriptstyle}T}\}
&=&
\{(\widetilde{\Psi}_m)_{{\alpha}},\,(\widetilde{\Psi}_n)_{\beta}^{{\scriptscriptstyle}T}
\} \;=\; \frac{1}{2}{\delta}_{m+n,0}\,{\delta}_{{\alpha}\beta}\,, {\nonumber}\\
\{(\Psi_m')_{{\alpha}},\, (\widetilde{\Psi}_n')_{\beta}^{{\scriptscriptstyle}T}
\} &=&
\{
(\widetilde{\Psi}_m')_{{\alpha}},\,(\Psi_n')_{\beta}^{{\scriptscriptstyle}T}
\} \; = \; 0\,, {\nonumber}\\
\{(\Psi_m')_{{\alpha}},\,(\Psi_n')_{\beta}^{{\scriptscriptstyle}T}\}
&=& \{(\widetilde{\Psi}_m')_{{\alpha}},\,(\widetilde{\Psi}_n')_{\beta}^{{\scriptscriptstyle}T}
\} \;=\; \frac{1}{2}{\delta}_{m+n,0}\,{\delta}_{{\alpha}\beta}\,. {\nonumber}\end{aligned}$$ Now let us introduce the annihilation and creation operators: $$\begin{aligned}
&&a^a_0 \equiv \sqrt{\frac{\alpha'}{2}}
\sqrt{\frac{3}{\mu}}
\left(p^a_0 - i\frac{\mu}{3\alpha'}x^a_0\right)\,,\quad
{a}^{a\dagger}_0 \equiv \sqrt{\frac{\alpha'}{2}}
\sqrt{\frac{3}{\mu}}
\left(p^a_0 + i\frac{\mu}{3\alpha'}x^a_0\right)\,\quad
(a=1,2,3,4)\,, {\nonumber}\\
&&a^a_n \equiv \frac{1}{\sqrt{\omega_n}}\alpha^a_n\,,\quad
a^{a\dagger}_n \equiv \frac{1}{\sqrt{\omega_n}}\alpha^a_{-n}\,,\quad
\bar{a}^a_n \equiv \frac{1}{\sqrt{\omega_n}}\bar{\alpha}^a_n\,,\quad
\bar{a}^{a\dagger}_n \equiv
\frac{1}{\sqrt{\omega_n}}\bar{\alpha}^a_{-n} \quad
(n>0)\,, {\nonumber}\end{aligned}$$ for the sector with mass $\mu/3$ and $$\begin{aligned}
&& a^b_0 \equiv \sqrt{\frac{\alpha'}{2}}
\sqrt{\frac{6}{\mu}}
\left(p^a_0 - i\frac{\mu}{6\alpha'}x^a_0\right)\,,\quad
{a}^{b\dagger}_0 \equiv \sqrt{\frac{\alpha'}{2}}
\sqrt{\frac{6}{\mu}}
\left(p^a_0 + i\frac{\mu}{6\alpha'}x^a_0\right) \quad (b=5,6,7,8)\,,
{\nonumber}\\
&&a^b_n=\frac{1}{\sqrt{\omega'_n}}\alpha^b_n\,, \quad
a^{b\dagger}_n=\frac{1}{\sqrt{\omega'_n}}\alpha^b_{-n}\,,\quad
\bar{a}^b_n=\frac{1}{\sqrt{\omega'_n}}\bar{\alpha}^b_n\,, \quad
\bar{a}^{b\dagger}_n=\frac{1}{\sqrt{\omega'_n}}\bar{\alpha}^b_{-n} \quad
(n>0)\,, {\nonumber}\end{aligned}$$ for the sector with mass $\mu/6$, and those for fermionic variables: $$\begin{aligned}
&& S_0=\Psi_0+i\tilde{\Psi}_0\,,\quad
S^{\dagger}_0=\Psi_0-i\tilde{\Psi}_0\,, {\nonumber}\\
&& S_n=\sqrt{2}\Psi_n\,,\quad
S^{\dagger}_n=\sqrt{2}\Psi_{-n}\,,\quad
\tilde{S}_n=\sqrt{2}\tilde{\Psi}_n\,,\quad
\tilde{S}^{\dagger}_n=\sqrt{2}\tilde{\Psi}_{-n}\,,\qquad (n>0) {\nonumber}\\
&& S'_n=\sqrt{2}\Psi'_n\,,\quad
{S'}^{\dagger}_n=\sqrt{2}\Psi'_{-n}\,,\quad
\tilde{S}'_n=\sqrt{2}\tilde{\Psi}'_n\,,\quad
\tilde{S'}^{\dagger}_n=\sqrt{2}\tilde{\Psi'}_{-n}\,.\qquad (n>0){\nonumber}\end{aligned}$$ Then the commutation relations are rewritten as $$\begin{aligned}
&&[a^a_m,a^{a'\dagger}_n]=\delta^{aa'}\delta_{m,n}\,, \quad
[\bar{a}^a_m,\bar{a}^{a'\dagger}_n]=\delta^{aa'}\delta_{m,n}\,
\qquad (a,\,a'=1,2,3,4)\,, {\nonumber}\\
&&[a^b_m,a^{b'\dagger}_n]=\delta^{bb'}\delta_{m,n}\,, \quad
[\bar{a}^b_m,\bar{a}^{b'\dagger}_n]=\delta^{bb'}\delta_{m,n}\,
\qquad (b,\,b'=5,6,7,8)\,, {\nonumber}\end{aligned}$$ and the anticommutation relations are given by $$\begin{aligned}
&&\{(S_m)_{\alpha},(S^{\dagger}_n)_{\beta}\}=
\delta_{\alpha\beta}\delta_{m,n}\,,\quad
\{(\tilde{S}_m)_{\alpha},(\tilde{S}^{\dagger}_n)_{\beta}\}
=\delta_{\alpha\beta}\delta_{m,n}\,,{\nonumber}\\
&&\{({S'}_m)_{\alpha},({S'}^{\dagger}_n)_{\beta}\}=
\delta_{\alpha\beta}\delta_{m,n}\,, \quad
\{(\tilde{S}'_m)_{\alpha},(\tilde{S'}^{\dagger}_n)_{\beta}\}
=\delta_{\alpha\beta}\delta_{m,n}\,.{\nonumber}\end{aligned}$$ By using the above (anti)commutation relations, we can represent the Hamiltonian and momentum in terms of creation and annihilation operators as follows: $$\begin{aligned}
H &=&
\sum^{\infty}_{n=-\infty}\left(\omega_n N_n +\omega'_n N'_n\right)\,, \quad
P=\sum^{\infty}_{n=-\infty}(nN_n+nN'_n)\,. \end{aligned}$$ where $N_n$ and $N_n'$ are defined by $$\begin{aligned}
&& N_0 \equiv \sum^4_{a=1}a^{a\dagger }_0a^a_0+S^{\dagger}_0S_0\,,\quad
N'_0 \equiv \sum^8_{b=5}a^{b\dagger }_0a^b_0+{S'}^{\dagger}_0S'_0\,,
{\nonumber}\\
&& N_n \equiv \sum^4_{a=1}a^{a\dagger}_na^a_n+S^{\dagger}_nS_n\,,\quad
N'_n \equiv \sum^8_{b=5}a^{b\dagger}_na^b_n+{S'}^{\dagger}_nS'_n
\qquad (n>0)\,,{\nonumber}\\
&& N_{-n} \equiv \sum^4_{a=1}\bar{a}^{a\dagger}_n\bar{a}^a_n
+\tilde{S}^{\dagger}_n\tilde{S}_n\,,\quad
N'_{-n} \equiv \sum^8_{b=5}\bar{a}^{b\dagger}_n\bar{a}^b_n
+\tilde{S'}^{\dagger}_n\tilde{S}'_n
\qquad (n>0)\,. {\nonumber}\end{aligned}$$ Now we shall introduce the Casimir Energy defined by $$\begin{aligned}
&& \Delta (\nu;a) \;\equiv\; \frac{1}{2}\sum_{n\in {\bf Z}}
\sqrt{\nu^2+(n+a)^2}-\frac{1}{2}\int^{+\infty}_{-\infty}\!dk
\,\sqrt{\nu^2+k^2}{\nonumber}\\
&=& -\frac{1}{2\pi^2}\sum_{\ell \geq 1}\cos (2\pi a\ell)
\int^{\infty}_0 \!\!\! ds\, e^{-\ell^2 s-\frac{\pi^2\nu^2}{s}}
= -\frac{1}{2}\int^{\infty}_0 \!\!\!ds\, e^{-\frac{\nu^2}{s}}
\left(\theta_3(i\pi s,a)-1\right)\,. \end{aligned}$$ Then we can express the vacuum energy for eight bosons as $$\begin{aligned}
E^0_B &=&\sum^4_{a=1}\left(\frac{1}{2}\sum_{n\in {\bf Z}}\omega_n\right)
+\sum^8_{b=5}\left(\frac{1}{2}\sum_{n\in {\bf Z}}\omega'_n\right)
\;\cong\; 4(\Delta (\mu/3 ;0)+
\Delta (\mu/6 ;0) )\,,\end{aligned}$$ and that for fermions as $$\begin{aligned}
E^0_F &=& - 4\left(
\frac{1}{2}\sum_{n\in {\bf Z}}\omega_n+
\frac{1}{2}\sum_{n\in {\bf Z}}\omega'_n
\right) \;\cong\; - 4
\left[\Delta (\mu/3 ;0)+\Delta (\mu/6 ;0)\right]\,,\end{aligned}$$ where the symbol $\cong$ means the equality after the regularization of zero-point energies and the factor 4 appears since each fermion considered here has four independent components.
Here let us evaluate the toroidal partition function: $$\begin{aligned}
&&Z= \mbox{Tr}\left[
(-1)^{\bf F}e^{-2\pi \tau_2 H+2\pi i\tau_1 P}
\right]\,,\end{aligned}$$ where $\tau_1$ and $\tau_2$ are modular parameters and ${\bf F}$ is the fermion number operator.
First we will consider a partition function for one boson with mass $\nu$. The number operator, Hamiltonian and momentum are given by $$\begin{aligned}
&&N_0=a^{\dagger}_0a_0\,,\,\,\,
N_n=a^{\dagger}_na_n\,,\,\,\,
N_{-n}=\bar{a}^{\dagger}_n\bar{a}_n \quad (n>0)\,, {\nonumber}\\
&& H = \omega_0 N_0 + \sum_{n=1}^{\infty}\left(
\omega_nN_n + \omega_nN_{-n}\right)\,,\quad
P=\sum_{n=1}^{\infty}\left( nN_n - nN_{-n}\right)\,, {\nonumber}\end{aligned}$$ and so we can obtain the partition function: $$\begin{aligned}
&&Z=\left[\Theta_{(0,0)}(\tau ,\bar{\tau};\nu)\right]^{-1/2}\,, \end{aligned}$$ where we have introduced the ‘massive’ theta function defined by $$\begin{aligned}
\Theta_{(a,b)}(\tau ,\bar{\tau};\nu)
\equiv e^{4\pi\tau_2 \Delta (\nu ;a)}
\prod_{n\in {\bf Z}}\left|
1-e^{-2\pi \tau_2 \sqrt{\nu^2 +(n+a)^2}+2\pi \tau_1 (n+a)+2\pi ib}
\right|^2\,.
\label{theta}\end{aligned}$$ Next we consider a partition function for single component fermion. The number operator, Hamiltonian and momentum are given by $$\begin{aligned}
&&N_0=S^{\dagger}_0S_0\,,\quad
N_n=S^{\dagger}_nS_n\,,\quad
N_{-n} = \tilde{S}^{\dagger}_n\tilde{S}_n \quad (n>0)\,, {\nonumber}\\
&& H = \omega_0N_0 + \sum_{n>0}\left(
\omega_nN_n + \omega_nN_{-n}\right)\,,\quad
P = \sum_{n>0}\left( nN_n - nN_{-n}\right)\,,{\nonumber}\end{aligned}$$ and hence we can obtain the partition function: $$\begin{aligned}
&&Z=\left[\Theta_{(0,0)}(\tau ,\bar{\tau};\nu)\right]^{+1/2}\,. \end{aligned}$$ If we recall the field contents of our model: $$\begin{aligned}
&&\mbox{bosons}\left\{
\begin{array}{l}
x^a ~~(a=1,2,3,4) \\
x^b~~(b=5,6,7,8)
\end{array}
\right.\,,\quad
\mbox{fermions}\left\{
\begin{array}{llll}
(S,\tilde{S}) & & (\mbox{mass}~~\frac{\mu}{3} \mbox{~~sector})\\
(S',\tilde{S}') & & (\mbox{mass}~~\frac{\mu}{6} \mbox{~~sector}) \\
\end{array}
\right.\,,\end{aligned}$$ the bosonic and fermionic partition functions $Z_B$ and $Z_F$ are given by $$\begin{aligned}
Z_B &=& \left[\Theta_{(0,0)}(\tau ,\bar{\tau};\mu/3)\right]^{-2}
\left[\Theta_{(0,0)}(\tau ,\bar{\tau};\mu/6)\right]^{-2}\,, {\nonumber}\\
Z_F&=&\left[\Theta_{(0,0)}(\tau ,\bar{\tau};\mu/3)\right]^{+2}
\left[\Theta_{(0,0)}(\tau ,\bar{\tau};\mu/6)\right]^{+2}\,, {\nonumber}\end{aligned}$$ and hence the total partition function is given by $$\begin{aligned}
&& Z=Z_B \cdot Z_F=1 \,.
\label{total}\end{aligned}$$ Thus we have shown that our theory is modular invariant at the one-loop level since the total partition function $Z$ of (\[total\]) is independent of the modular parameters $\tau$ and $\bar{\tau}$. As will be shown in more detail with some generality in the section 7, this result implies that the Witten index is one as in the cases of other string theories on pp-waves. It should be remarked that our type IIA string theory is modular invariant in the sector with mass $\mu/3$ and in that with $\mu/6$, respectively.
Supersymmetries of Type IIA String Theory
=========================================
In this section we will briefly review about $\mathcal{N}=(4,4)$ supersymmetries of the type IIA string theory on the pp-wave background [@HS2; @HS3], according to which the world-sheet variables are arranged into two supermultiplets, ($x^a,\Psi^{1-},\Psi^{2+}$) and ($x^b,\Psi^{1+},\Psi^{2-}$). Then we will rewrite the supercharges in terms of modes in order to construct the boundary states in section 4.
This type IIA string theory on the pp-wave background has 24 supersymmetries, among which 8 are dynamical and 16 are kinematical. The dynamical supersymmetry transformation laws for the multiplet ($x^a,\Psi^{1-},\Psi^{2+}$) with mass $\mu/3$ are given by $$\begin{aligned}
{\delta}x^a &=& 2i{\alpha}'\left(\Psi^{1-T}\gamma^a\epsilon^{1+}
+ \Psi^{2+T}\gamma^a\epsilon^{2-}\right)\,, \\
{\delta}\Psi^{1-} &=& \partial_+ x^a \gamma^a\epsilon^{1+}
+ \frac{\mu}{3}x^a\gamma^4\gamma^a\epsilon^{2-}\,, \quad
{\delta}\Psi^{2+} \;=\; \partial_- x^a \gamma^a\epsilon^{2-}
- \frac{\mu}{3}x^a\gamma^4\gamma^a\epsilon^{1+}\,, {\nonumber}\end{aligned}$$ and those for the multiplet ($x^b,\Psi^{1+},\Psi^{2-}$) with mass $\mu/6$, are written as $$\begin{aligned}
{\delta}x^b &=& 2i{\alpha}'\left(\Psi^{1+T}\gamma^b\epsilon^{2-}
+ \Psi^{2-T}\gamma^b\epsilon^{1+}\right)\,, \\
{\delta}\Psi^{1+} &=& \partial_- x^b \gamma^b \epsilon^{2-}
+ \frac{\mu}{6}x^b\gamma^4\gamma^b\epsilon^{1+}\,, \quad
{\delta}\Psi^{2-} \;=\; \partial_+ x^b \gamma^b \epsilon^{1+}
- \frac{\mu}{6} x^b\gamma^4\gamma^b\epsilon^{2-}\,, {\nonumber}\end{aligned}$$ where the constant spinors $\epsilon^{1+}$ and $\epsilon^{2-}$ satisfy the following chirality conditions: $$\begin{aligned}
&& \gamma^9\epsilon^{1+} = + \epsilon^{1+}\,, \quad
\gamma^{1234}\epsilon^{1+} = +\epsilon^{1+}\,, \quad
\gamma^9\epsilon^{2-} = - \epsilon^{2-}\,, \quad
\gamma^{1234}\epsilon^{2-} = - \epsilon^{2-}\,,\end{aligned}$$ respectively. As for the kinematical supersymmetry, the transformation laws are given by $\tilde{{\delta}}x^a = \tilde{\delta}x^b =0$ and $$\begin{aligned}
\tilde{\delta}\Psi^{1-}
&=& \cos\left(\frac{\mu}{3}\tau\right)\tilde{\epsilon}^{1-}
- \sin\left(\frac{\mu}{3}\tau\right)\gamma^{123}\tilde{\epsilon}^{2+}\,, \\
\tilde{\delta}\Psi^{2+}
&=& \cos\left(\frac{\mu}{3}\tau\right)\tilde{\epsilon}^{2+}
- \sin\left(\frac{\mu}{3}\tau\right)\gamma^{123}\tilde{\epsilon}^{1-}\,,{\nonumber}\\
\tilde{\delta}\Psi^{1+}
&=& \cos\left(\frac{\mu}{6}\tau\right)\tilde{\epsilon}^{1+}
- \sin\left(\frac{\mu}{6}\tau\right)\gamma^{123}\tilde{\epsilon}^{2-}\,, \\
\tilde{\delta}\Psi^{2-}
&=& \cos\left(\frac{\mu}{6}\tau\right)\tilde{\epsilon}^{2-}
- \sin\left(\frac{\mu}{6}\tau\right)\gamma^{123}\tilde{\epsilon}^{1+}\,,{\nonumber}\end{aligned}$$ where the constant spinors $\tilde{\epsilon}^{1+}$, $\tilde{\epsilon}^{1-}$, $\tilde{\epsilon}^{2+}$ and $\tilde{\epsilon}^{2-}$ satisfy the chirality conditions in terms of $\gamma^9$ and $\gamma^{1234}$ in the same way as the dynamical supersymmetry case.
By the use of Noether’s theorem, we can construct the associated supercharges. Firstly the dynamical supercharges are obtained as $$\begin{aligned}
\epsilon^{{\scriptscriptstyle}T}Q_{(\mu/3)} =
\epsilon^{1+{{\scriptscriptstyle}T}}Q^{1+} + \epsilon^{2-{{\scriptscriptstyle}T}}Q^{2-}\,, \quad
\epsilon'{}^{{\scriptscriptstyle}T}Q_{(\mu/6)} =
\epsilon^{2-{{\scriptscriptstyle}T}}Q'{}^{2-} + \epsilon^{1+{{\scriptscriptstyle}T}}Q'{}^{1+}\,.\end{aligned}$$ Here the $Q^{1+}$ and $Q^{2-}$ are the quantities defined as $$\begin{aligned}
Q^{1+} &\equiv& - \frac{i}{2\pi}\int^{2\pi}_0\!\!\!\!d\sigma\,
\left[
\partial_+ x^a \gamma^a\Psi^{1-} - \frac{\mu}{3}x^a\gamma^a\gamma^4\Psi^{2+}
\right]\,, \\
Q^{2-} &\equiv& - \frac{i}{2\pi}\int^{2\pi}_0\!\!\!\!d\sigma\,
\left[
\partial_- x^a \gamma^a\Psi^{2+} + \frac{\mu}{3}x^a\gamma^a\gamma^4\Psi^{1-}
\right]\,, \end{aligned}$$ and, for the $Q'{}^{2-}$ and $Q'{}^{1+}$, $$\begin{aligned}
Q'{}^{2-} &\equiv& - \frac{i}{2\pi}\int^{2\pi}_0\!\!\!\!d\sigma\,
\left[
\partial_+ x^b \gamma^b\Psi^{1+} - \frac{\mu}{6}x^b\gamma^b\gamma^4\Psi^{2-}
\right]\,, \\
Q'{}^{1+} &\equiv& - \frac{i}{2\pi}\int^{2\pi}_0\!\!\!\!d\sigma\,
\left[
\partial_- x^b \gamma^b\Psi^{2-} + \frac{\mu}{6}x^b\gamma^b\gamma^4\Psi^{1+}
\right]\,. \end{aligned}$$ Secondly, the kinematical supercharges are obtained as $$\begin{aligned}
\tilde{Q}_{(\mu/3)} \equiv \tilde{\epsilon}^{1-{{\scriptscriptstyle}T}}\tilde{Q}^{1-}
+ \tilde{\epsilon}^{2+{{\scriptscriptstyle}T}}\tilde{Q}^{2+}\,, \quad
\tilde{Q}_{(\mu/6)} \equiv \tilde{\epsilon}^{1+{{\scriptscriptstyle}T}}\tilde{Q}^{1+}
+ \tilde{\epsilon}^{2-{{\scriptscriptstyle}T}}\tilde{Q}^{2-}\,, \end{aligned}$$ where the $\tilde{Q}^{1-}$ and $\tilde{Q}^{2+}$ are defined by $$\begin{aligned}
\tilde{Q}^{1-} &\equiv& \frac{i}{2\pi}\int^{2\pi}_0\!\! d\sigma\,
\left[
\cos\left(\frac{\mu}{3}\tau\right)\Psi^{1-}
+ \sin\left(\frac{\mu}{3}\tau\right)\gamma^{123}\Psi^{2+}
\right]\,, \\
\tilde{Q}^{2+} &\equiv& \frac{i}{2\pi}\int^{2\pi}_0\!\! d\sigma\,
\left[
\cos\left(\frac{\mu}{3}\tau\right)\Psi^{2+}
+ \sin\left(\frac{\mu}{3}\tau\right)\gamma^{123}\Psi^{1-}
\right]\,,\end{aligned}$$ and, for the $\tilde{Q}^{1-}$ and $\tilde{Q}^{2+}$, $$\begin{aligned}
\tilde{Q}^{1+} &\equiv& \frac{i}{2\pi}\int^{2\pi}_0\!\! d\sigma\,
\left[
\cos\left(\frac{\mu}{6}\tau\right)\Psi^{1+}
+ \sin\left(\frac{\mu}{6}\tau\right)\gamma^{123}\Psi^{2-}
\right]\,, \\
\tilde{Q}^{2-} &\equiv& \frac{i}{2\pi}\int^{2\pi}_0\!\! d\sigma\,
\left[
\cos\left(\frac{\mu}{6}\tau\right)\Psi^{2-}
+ \sin\left(\frac{\mu}{6}\tau\right)\gamma^{123}\Psi^{1+}
\right]\,.\end{aligned}$$
Now we can rewrite the above supercharges in terms of creation and annihilation operators by inserting the mode-expansions of bosonic and fermionic degrees of freedom. $$\begin{aligned}
\sqrt{{\alpha}'}^{-1}Q^{1+} &=&
c_0 \sqrt{\frac{\mu}{3}} \left(a_0^{a\dagger}\gamma^a\Pi^{{\scriptscriptstyle}T}S_0
- a_0^a\gamma^a\Pi^{{\scriptscriptstyle}T}S_0^{\dagger}\right)
-i \sum_{n=1}^{\infty}c_n\sqrt{\omega_n}\left(
\bar{a}_n^{a\dagger}\gamma^a\tilde{S}_n + \bar{a}_n^a \gamma^a\tilde{S}_n^{\dagger}
\right)
{\nonumber}\\
&& \quad + \frac{1}{2}\cdot\frac{\mu}{3}\sum_{n=1}^{\infty}
\frac{1}{c_n\sqrt{\omega_n}}\left(
a_n^{a\dagger}\gamma^a\Pi^{{\scriptscriptstyle}T}S_n
- a_n^a \gamma^a\Pi^{{\scriptscriptstyle}T}S_n^{\dagger}
\right)\,, \\
\sqrt{{\alpha}'}^{-1}Q^{2-} &=& -ic_0 \sqrt{\frac{\mu}{3}}\left(
a_0^{a\dagger}\gamma^a S_0 + a_0^a\gamma^aS_0^{\dagger}
\right)
-i \sum_{n=1}^{\infty}c_n\sqrt{\omega_n}
\left(
a_n^{a\dagger}\gamma^a S_n + a_n^a\gamma^a S_n^{\dagger}
\right)
{\nonumber}\\
&& \quad
+ \frac{1}{2}\cdot\frac{\mu}{3}\sum_{n=1}^{\infty}\frac{1}{c_n\sqrt{\omega_n}}
\left(
\bar{a}^a_n\gamma^a\Pi\tilde{S}_n^{\dagger} - \bar{a}_n^{a\dagger}\gamma^a
\Pi\tilde{S}_n
\right)\,, \end{aligned}$$ and $$\begin{aligned}
\sqrt{{\alpha}'}^{-1}Q'{}^{2-} &=&
c_0' \sqrt{\frac{\mu}{6}} \left(a_0^{b\dagger}\gamma^a\Pi^{{\scriptscriptstyle}T}S_0'
- a_0^b\gamma^a\Pi^{{\scriptscriptstyle}T}S_0'{}^{\dagger}\right)
-i \sum_{n=1}^{\infty}c_n'\sqrt{\omega_n'}\left(
\bar{a}_n^{b\dagger}\gamma^b\tilde{S}'_n + \bar{a}_n^b \gamma^b
\tilde{S}_n'{}^{\dagger}
\right)
{\nonumber}\\
&& \quad + \frac{1}{2}\cdot\frac{\mu}{6}\sum_{n=1}^{\infty}
\frac{1}{c_n'\sqrt{\omega_n'}}\left(
a_n^{b\dagger}\gamma^b\Pi^{{\scriptscriptstyle}T}S'_n
- a_n^b \gamma^b\Pi^{{\scriptscriptstyle}T}S_n'{}^{\dagger}
\right)\,, \\
\sqrt{{\alpha}'}^{-1}Q'{}^{1+} &=& -ic_0' \sqrt{\frac{\mu}{6}}\left(
a_0^{b\dagger}\gamma^b S_0' + a_0^b\gamma^bS_0'{}^{\dagger}
\right)
-i \sum_{n=1}^{\infty}c_n'\sqrt{\omega_n'}
\left(
a_n^{b\dagger}\gamma^b S_n' + a_n^b\gamma^b S_n'{}^{\dagger}
\right)
{\nonumber}\\
&& \quad
+ \frac{1}{2}\cdot\frac{\mu}{6}\sum_{n=1}^{\infty}
\frac{1}{c_n'\sqrt{\omega_n'}}
\left(
\bar{a}^b_n\gamma^b\Pi\tilde{S}_n'{}^{\dagger} - \bar{a}_n^b
\gamma^b
\Pi\tilde{S}_n'
\right)\,.\end{aligned}$$ On the other hand, the kinematical supersymmetries are rewritten as $$\begin{aligned}
&& \tilde{Q}^{1-} = i\Pi\tilde{\Psi}_0 = \frac{\Pi}{2}(S_0 - S_0^{\dagger})\,,
\quad \tilde{Q}^{2+} = i\Psi_0
= \frac{i}{2}\left(S_0 + S_0^{\dagger}\right)\,, \\
&& \tilde{Q}^{1+} = i\Pi\tilde{\Psi}_0 = \frac{\Pi}{2}(S_0 - S_0^{\dagger})\,,
\quad \tilde{Q}^{2-} = i\Psi_0
= \frac{i}{2}\left(S_0 + S_0^{\dagger}\right)\,. \end{aligned}$$
In the next section, the above expressions of supercharges will be used for constructing the fermionic boundary states.
Boundary States of Type IIA String Theory
=========================================
In this section we will construct the boundary states of type IIA string theory on the pp-wave background with 24 supersymmetries. To begin with, the bosonic boundary states will be constructed. Next, we construct the fermionic boundary states and classify the allowed D-brane instantons in our theory. The resulting boundary states will be used for the calculation of amplitude in the closed string description.
Bosonic Boundary States
-----------------------
Here we will consider the bosonic part of boundary states in the Type IIA string theory. The bosonic coordinates $(x^a,x^b)~~(a=1,\ldots,4,
b=5,\ldots,8)$ are classified into $(x^{\bar{a}},x^{\bar{b}})$ (for the Neumann condition) and $(x^{\underline{a}},x^{\underline{b}})$ (for the Dirichlet condition).
The definition of bosonic boundary state $|B{\rangle}$ is given by the following boundary conditions: $$\begin{aligned}
\label{nb}
\partial_{\tau}x^{\bar{a}}|_{\tau =0} |B{\rangle}= 0\,,
\quad \partial_{\tau}x^{\bar{b}}|_{\tau =0} |B{\rangle}= 0
\quad &&(\mbox{Neumann})\,, \\
(x_0^{\underline{a}} - q_0^{\underline{a}})|_{\tau =0} | B {\rangle}= 0\,,
\quad
(x_0^{\underline{b}} - q_0^{\underline{b}})|_{\tau =0} | B {\rangle}= 0
\quad && (\mbox{Dirichlet})\,.
\label{db}\end{aligned}$$ The conditions (\[nb\]) can be rewritten as $$\begin{aligned}
(a_0^{\bar{a}} + a_0^{\bar{a}\dagger})|B{\rangle}= 0\,,
\quad (a_n^{\bar{a}} + \bar{a}_n^{\bar{a}\dagger})|B{\rangle}= 0
\quad (n>0)\,, \\
(a_0^{\bar{b}} + a_0^{\bar{b}\dagger})|B{\rangle}= 0\,,
\quad (a_n^{\bar{b}} + \bar{a}_n^{\bar{b}\dagger})|B{\rangle}= 0
\quad (n>0)\,,\end{aligned}$$ and lead to the bosonic boundary state $|B{\rangle}$ for Neumann directions described by $$\begin{aligned}
| B {\rangle}&=& \exp\left(-\frac{1}{2}\sum_{\bar{a}}a_0^{\bar{a}\dagger}
a_0^{\bar{a}\dagger} - \frac{1}{2}\sum_{\bar{b}}a_0^{\bar{b}\dagger}
a_0^{\bar{b}\dagger} - \sum_{n=1}^{\infty}
\left\{
\sum_{\bar{a}}a_n^{\bar{a}\dagger}\bar{a}_n^{\bar{a}\dagger}
+ \sum_{\bar{b}}a_n^{\bar{b}\dagger}\bar{a}_n^{\bar{b}\dagger}
\right\}\right) |0{\rangle}\,,\end{aligned}$$ where $|0{\rangle}$ is the bosonic Fock vacuum state annihilated by the operators, $a_n^{a,b}$ and $\bar{a}_n^{a,b}$.
On the other hand, the second conditions (\[db\]) can be rewritten as $$\begin{aligned}
\left(a_0^{\underline{a}} - a_0^{\underline{a}\dagger}
+ i\left(\frac{2\mu}{3{\alpha}'}\right)^{1/2}q_0^{\underline{a}}\right)
| B{\rangle}= 0\,,
\quad (a_n^{\underline{a}} - \bar{a}_n^{\underline{a}\dagger}) | B{\rangle}=0\,, \\
\left(a_0^{\underline{b}} - a_0^{\underline{b}\dagger}
+ i\left(\frac{\mu}{3{\alpha}'}\right)^{1/2}q_0^{\underline{b}}\right)
| B{\rangle}= 0\,,
\quad (a_n^{\underline{b}} - \bar{a}_n^{\underline{b}\dagger}) | B{\rangle}=0\,.\end{aligned}$$ With these boundary conditions, we can construct the boundary states for Dirichlet directions described by $$\begin{aligned}
| B {\rangle}&=& {{\rm e}}^{+\frac{1}{2}\sum_{\underline{a}}\left\{
a_0^{\underline{a}\dagger} - i\left(\frac{2\mu}{3{\alpha}'}\right)^{1/2}q_0^{\underline{a}}
\right\}^2
+ \frac{1}{2}\sum_{\underline{b}}
\left\{
a_0^{\underline{b}\dagger} -i \left(\frac{\mu}{3{\alpha}'}\right)^{1/2}q_0^{\underline{b}}
\right\}^2
}\cdot {{\rm e}}^{+\sum_{n=1}^{\infty}\left\{
\sum_{\underline{a}}a_n^{\underline{a}\dagger}\bar{a}_n^{\underline{a}\dagger}
+ \sum_{\underline{b}}a_n^{\underline{b}\dagger}
\bar{a}_n^{\underline{b}\dagger}
\right\}}|0{\rangle}\,.\end{aligned}$$ Here we shall introduce a diagonal matrix $M_{ij}=\mbox{diag}(\pm
1,\cdots,\pm 1)$ with eight components where $+1$ is assigned for Dirichlet directions and $-1$ is assigned for Neumann ones. If we will set as $q_0^{\underline{a}}=q_0^{\underline{b}}=0$, the bosonic boundary state can be rewritten as $$\begin{aligned}
\label{bvs}
| B {\rangle}&=& | B {\rangle}_{\mu/3}\otimes | B {\rangle}_{\mu/6} \,, \\
| B {\rangle}_{\mu/3} &\equiv& {{\rm e}}^{+ \frac{1}{2}M_{aa'}
a_0^{a\dagger}a_0^{a'\dagger}}\cdot
{{\rm e}}^{\sum_{n=1}^{\infty}M_{aa'}a_n^{a\dagger}\bar{a}_n^{a'\dagger}
}|0{\rangle}\,, {\nonumber}\\
| B {\rangle}_{\mu/6} &\equiv& {{\rm e}}^{+ \frac{1}{2}M_{bb'}
a_0^{b\dagger}a_0^{b'\dagger}}\cdot
{{\rm e}}^{\sum_{n=1}^{\infty}M_{bb'}a_n^{b\dagger}\bar{a}_n^{b'\dagger}
}|0{\rangle}\,. {\nonumber}\end{aligned}$$ Thus the bosonic boundary state is the product of two sectors with mass $\mu/3$ and that with $\mu/6$, and it has the $SO(4)\times SO(4)$ symmetry. We will study the fermionic boundary states in the next subsection.
Fermionic Boundary States
-------------------------
We will now consider the fermionic part of boundary states in our case. The fermionic boundary states are defined by $$\begin{aligned}
\label{orig}
&&\left(Q_{{\alpha}}^{2-} - i\eta M_{{\alpha}\beta}^{(\mu/3)}
Q_{\beta}^{1+}\right)
| B {\rangle}= 0\,, \quad \left(
Q'_{{\alpha}}{}^{2-} - i\eta M_{{\alpha}\beta}^{(\mu/6)}Q'_{\beta}{}^{1+}
\right)| B {\rangle}= 0 \,, \\
&& \left(
\tilde{Q}_{{\alpha}}^{2-} + i\eta \hat{M}_{{\alpha}\beta}^{(\mu/3)}
\tilde{Q}^{1+}_{\beta}
\right) | B {\rangle}= 0\,, \quad
\left(
\tilde{Q}^{2+} + i\eta
\hat{M}_{{\alpha}\beta}^{(\mu/6)}\tilde{Q}^{1-}_{\beta}
\right) | B {\rangle}= 0\,,
\label{kinematical}\end{aligned}$$ where matrices $M_{{\alpha}\beta}^{(\mu/3)}$, $M^{(\mu/6)}_{{\alpha}\beta}$, $\hat{M}_{{\alpha}\beta}^{(\mu/3)}$ and $\hat{M}^{(\mu/6)}_{{\alpha}\beta}$ satisfy the following relations: $$\begin{aligned}
&& M_{{\alpha}\beta}(M^{{\scriptscriptstyle}T})_{\beta\gamma} = {\delta}_{{\alpha}\gamma}\,,
\quad (M^{{\scriptscriptstyle}T})_{{\alpha}\beta}M_{\beta\gamma}
= {\delta}_{{\alpha}\gamma}\,, \\
&& \hat{M}_{{\alpha}\beta}(\hat{M}^{{\scriptscriptstyle}T})_{\beta\gamma}
= {\delta}_{{\alpha}\gamma}\,,
\quad (\hat{M}^{{\scriptscriptstyle}T})_{{\alpha}\beta}\hat{M}_{\beta\gamma}
= {\delta}_{{\alpha}\gamma}\,.\end{aligned}$$ The definition of the fermionic boundary states (\[kinematical\]) leads to the conditions written in terms of the zero-modes[^3]: $$\begin{aligned}
\label{fvs}
\left((\Psi_0)_{{\alpha}} + i\eta \hat{M}^{(\mu/3)}_{{\alpha}\beta}
(\Pi\widetilde{\Psi}_0)_{\beta}
\right) | B {\rangle}= 0\,, \quad
\left((\Psi_0')_{{\alpha}} + i\eta \hat{M}^{(\mu/6)}_{{\alpha}\beta}
(\Pi\widetilde{\Psi}_0')_{\beta}
\right) | B {\rangle}= 0\,.\end{aligned}$$ These conditions suggest us to take the following ansatz: $$\begin{aligned}
\left(
(S_n)_{{\alpha}}
+ i\eta \hat{M}^{(\mu/3)}_{{\alpha}\beta}
(\tilde{S}_n^{\dagger})_{\beta}
\right)|B{\rangle}= 0\,, \quad
\left(
(S'_n)_{{\alpha}}
+ i\eta \hat{M}^{(\mu/6)}_{{\alpha}\beta}
(\tilde{S}'_n{}^{\dagger})_{\beta}
\right) | B {\rangle}= 0\,.
\label{4.18}\end{aligned}$$ The above equations can be easily solved and the boundary state is given by $$\begin{aligned}
|B{\rangle}= {{\rm e}}^{\sum_{n=1}^{\infty}\left\{
-i\eta \hat{M}^{(\mu/3)}_{{\alpha}\beta}(S_n^{\dagger})_{{\alpha}}
(\tilde{S}_n^{\dagger})_{\beta}
-i\eta \hat{M}^{(\mu/6)}_{{\alpha}\beta}(S_n'{}^{\dagger})_{{\alpha}}
(\tilde{S}_n'{}^{\dagger})_{\beta}
\right\}
}|B{\rangle}_0\,,
\label{sol}\end{aligned}$$ where $|B{\rangle}_0$ is the fermionic vacuum state yet to be determined. By the way, this boundary state is by definition the state satisfying (\[orig\]) and (\[kinematical\]). If we now act the conditions (\[4.18\]) on (\[orig\]), we have three types of conditions that lead us to determine the structure of the matrices $M^{(\mu/3)}$, $M^{(\mu/6)}$, $\hat{M}^{(\mu/3)}$ and $\hat{M}^{(\mu/6)}$. Firstly, we obtain the conditions $$\begin{aligned}
M^{a'a}\gamma^{a'} = -M^{(\mu/3)}\gamma^a \hat{M}^{(\mu/3){{\scriptscriptstyle}T}}\,, \quad
M^{b'b}\gamma^{b'} = -M^{(\mu/6)}\gamma^b \hat{M}^{(\mu/6){{\scriptscriptstyle}T}}\,,
\label{cd1}\end{aligned}$$ which are similar to those arising in flat space [@Green]. The second type of conditions, which appears only in the pp-wave case, is $$\begin{aligned}
M^{aa'}\gamma^{a'}\Pi = - M^{(\mu/3)}\gamma^a\Pi \hat{M}^{(\mu/3)}\,, \quad
M^{bb'}\gamma^{b'}\Pi = - M^{(\mu/6)}\gamma^b\Pi \hat{M}^{(\mu/6)}\,.
\label{cd2}\end{aligned}$$ Finally, the third type of conditions we get comes from the zero-mode parts: $$\begin{aligned}
\label{zero}
&& \left\{a_0^{a\dagger}\left(
\gamma^a + \eta M^{(\mu/3)}\gamma^a\Pi
\right) S_0
+M^{aa'}a_0^{a'\dagger}\left(
\gamma^a - \eta M^{(\mu/3)}\gamma^a\Pi
\right) S_0^{\dagger}\right\} | B {\rangle}= 0\,, \\
&& \left\{a_0^{b\dagger}\left(
\gamma^b + \eta M^{(\mu/6)}\gamma^b\Pi
\right) S_0
+M^{bb'}a_0^{b'\dagger}\left(
\gamma^b - \eta M^{(\mu/6)}\gamma^b\Pi
\right) S_0^{\dagger}\right\} | B {\rangle}= 0\,.
\label{zero2}\end{aligned}$$ By the use of the definition of fermionic vacuum $S_0 | B{\rangle}_0 =0$ and the identities $a_0^a = M^{aa'} a_0^{a'\dagger}$ and $a_0^b = M^{bb'}a_0^{b'\dagger}$, we can rewrite (\[zero\]) and (\[zero2\]) as $$\begin{aligned}
\label{co}
&&\left\{
p_0^{a'}\left(
{\delta}^{aa'} - M^{aa'}
\right)
+ i \frac{\mu}{3{\alpha}'}x_0^{a'}
\left({\delta}^{aa'} + M^{aa'}\right)
\right\}
\left(
\gamma^a - \eta M^{(\mu/3)}\gamma^a\Pi
\right)
S_0^{\dagger} | B {\rangle}_0 = 0\,, \\
&&\left\{
p_0^{b'}\left(
{\delta}^{bb'} - M^{bb'}
\right)
+ i \frac{\mu}{6{\alpha}'}x_0^{b'}
\left({\delta}^{bb'} + M^{bb'}\right)
\right\}
\left(
\gamma^b - \eta M^{(\mu/6)}\gamma^b\Pi
\right)
S_0^{\dagger} | B {\rangle}_0 = 0\,.
\label{co2}\end{aligned}$$ Using the conditions (\[co\]) and (\[co2\]), we can read off supersymmetries preserved by D-brane instantons in terms of their positions. In the case that all position coordinates of a D-brane instanton $q^{r}$ for the Dirichlet directions equals zero, the D-brane instanton has 12 (4 dynamical + 8 kinematical) supersymmetries (i.e., 1/2(=12/24) BPS D-brane instanton). If the position coordinates for the Dirichlet directions are not at the origin, then the D-brane instanton has 8 (0+8) supersymmetries (i.e., 1/3(=8/24) BPS D-brane instanton). Thus all of the dynamical supersymmetries are broken. However the D-brane instantons apart from the origin are supersymmetric solutions since they have 8 kinematical supersymmetries.
As a final remark regarding the structure of the matrices, $M^{(\mu/3)}$, $M^{(\mu/6)}$, $\hat{M}^{(\mu/3)}$ and $\hat{M}^{(\mu/6)}$, we now consider the chirality condition while the above three types of conditions are almost same as in the type IIB string case [@BP]. First we can easily find that both matrices $M^{(\mu/3)}$ and $M^{(\mu/6)}$ contain the odd number of gamma matrices in order to preserve the $SO(8)$ chirality measured by $\gamma^9$. Moreover, if we consider the chirality in terms of the matrix $R = \gamma^{1234}$, we have the following conditions basically from (\[orig\]) and (\[kinematical\]): $$\begin{aligned}
\{R,\,M^{(\mu/3)}\} = 0\,, \quad
\{R,\,M^{(\mu/6)}\} = 0\,, \quad \{R,\,\hat{M}^{(\mu/3)}\} = 0\,, \quad
\{R,\,\hat{M}^{(\mu/6)}\} = 0\,.
\label{cd4}\end{aligned}$$ Then we obtain the following complete boundary state: $$\begin{aligned}
|B{\rangle}&=&
{{\rm e}}^{\sum_{n=1}^{\infty}\left\{M_{aa'}a_n^{a\dagger}\bar{a}_n^{a'\dagger}
+ M_{bb'}a_n^{b\dagger}\bar{a}_n^{b'\dagger}
-i\eta \hat{M}^{(\mu/3)}_{{\alpha}\beta}(S_n^{\dagger})_{{\alpha}}
(\tilde{S}_n^{\dagger})_{\beta}
-i\eta \hat{M}^{(\mu/6)}_{{\alpha}\beta}(S_n'{}^{\dagger})_{{\alpha}}
(\tilde{S}_n'{}^{\dagger})_{\beta}
\right\}
} | B{\rangle}_0 {\nonumber}\\
|B{\rangle}_0 &=& \left(
M_{{\scriptscriptstyle}IJ}| I {\rangle}| J {\rangle}- i \eta M_{{\alpha}\beta}|{\alpha}{\rangle}| \beta{\rangle}\right)
e^{+ \frac{1}{2}M_{aa'}
a_0^{a\dagger}a_0^{a'\dagger} + \frac{1}{2}M_{bb'}
a_0^{b\dagger}a_0^{b'\dagger}} | 0{\rangle}\,,\end{aligned}$$ where the state $|B{\rangle}_0$ is the product of the bosonic vacuum state, which is given by picking up the zero-mode parts in Eq.(\[bvs\]), and the fermionic one which is the solution of (\[fvs\]).
The remaining task is to determine the matrices $M$ and $\hat{M}$ from the conditions (\[cd1\]), (\[cd2\]), (\[co\]), (\[co2\]) and (\[cd4\]). The determined structure of the matrices leads to the classification of possible D-brane instantons. This will be done in the next subsection.
Classification of D-brane Instantons
------------------------------------
We will classify the allowed D-brane instantons by determining the matrices $M$ and $\hat{M}$. Now let us analyze each of D$p$-brane instantons.
- D0-brane instantons are expressed by the following matrices: $$\begin{aligned}
M^{(\mu/3)} = M^{(\mu/6)} = \hat{M}^{(\mu/3)} = \hat{M}^{(\mu/6)}
= \gamma^{{\scriptscriptstyle}I} ~~~(I=1,2,3)\,.\end{aligned}$$ The $x^{{\scriptscriptstyle}I}$-direction satisfies the Neumann boundary condition and other directions satisfy the Dirichlet boundary condition. When we consider $M=\gamma^1$ as an example, $x^1$ is a Neumann direction and other directions are Dirichlet ones. If we consider the D0-brane instanton at the origin $q^{2,3,4,5,6,7,8}=0$, then it is a 1/2 BPS object. If we consider the D0-brane instanton apart from the origin, then it becomes a 1/3 BPS object.
- D2-brane instantons are given by $$\begin{aligned}
&& M^{(\mu/3)} = M^{(\mu/6)} = \hat{M}^{(\mu/3)} = \hat{M}^{(\mu/6)}= \gamma^{{{\scriptscriptstyle}IJ}4}\,, \quad \\
\mbox{or}& & \quad
\quad M^{(\mu/3)} = M^{(\mu/6)} = \hat{M}^{(\mu/3)} = \hat{M}^{(\mu/6)} =
\gamma^4\gamma^{bb'}\,, {\nonumber}\end{aligned}$$ where $I$ and $J$ take values in 1,2,3, and $b$ and $b'$ run from 5 to 8. For example, if we take $M=\gamma^{124}$ then $x^{1,2,4}$-directions satisfy the Neumann condition and others are Dirichlet directions. When the D2-brane instanton sits at the origin $q^{3,5,6,7,8}=0$, it is a 1/2 BPS object. Once it goes away from the origin, it becomes 1/3 BPS.
- D4-brane instantons are described by $$\begin{aligned}
&& M^{(\mu/3)} = M^{(\mu/6)} = \hat{M}^{(\mu/3)} = \hat{M}^{(\mu/6)}
=\gamma^{123}\gamma^{bb'}\,, \quad \\
\mbox{or}& & \quad
M^{(\mu/3)} = M^{(\mu/6)}= \hat{M}^{(\mu/3)} = \hat{M}^{(\mu/6)} = \gamma^{{\scriptscriptstyle}I}\gamma^{5678}\,. {\nonumber}\end{aligned}$$ When we take $M=\gamma^{12356}$, the $x^{1,2,3,5,6}$-directions satisfy the Neumann boundary condition and others are the Dirichlet directions. When the D4-brane instanton is at the origin $q^{4,7,8}=0$, it is a 1/2 BPS object. If it is apart from the origin, it becomes 1/3 BPS.
- D6-brane instantons are given by $$\begin{aligned}
M^{(\mu/3)} = M^{(\mu/6)}= \hat{M}^{(\mu/3)} = \hat{M}^{(\mu/6)} = \gamma^{{{\scriptscriptstyle}IJ}4}\gamma^{5678}\,.\end{aligned}$$ For example, the case $M=\gamma^{1245678}$ leads to a D6-brane instanton that preserves 12 supersymmetries (i.e., 1/2 BPS) for $q^3=0$ and 8 supersymmetries (1/3 BPS) for $q^3 \neq 0$.
We should remark that the above classification of D-brane instantons is consistent with that of D-branes in the open string description [@HS3; @HPS]. The Neumann (Dirichlet) boundary condition in the closed string description is simply related by the Dirichlet (Neumann) one in the open string description, and hence this identification should hold as a matter of course. For comparison with the classification of D-branes [@HS3; @HPS], we shall summarize our result in [**Tab.**]{} \[list:tab\].
-----------------------------------------------------------------------------------
$N_N$ $M^{(\mu/3)}=M^{(\mu/6)} = \hat{M}^{(\mu/3)} =
\hat{M}^{(\mu/6)} $
------- ---------------------------------------------------------------------------
1 $\gamma^{{\scriptscriptstyle}I}$
3 $\gamma^{{{\scriptscriptstyle}IJ}4}$, $\gamma^{4}\gamma^{bb'}$
5 $\gamma^{123}\gamma^{bb'}$, $\gamma^{{\scriptscriptstyle}I}\gamma^{5678}$
7 $\gamma^{{{\scriptscriptstyle}IJ}4}\gamma^{5678}$
-----------------------------------------------------------------------------------
: List of possible D-brane instantons in our Type IIA string theory. The $N_N$ is the number of Neumann directions.[]{data-label="list:tab"}
Cylinder Amplitude in the Closed String Description
===================================================
In this section we will calculate the tree amplitude in the closed string description. The interaction energy between a pair of D-branes comes from the exchange of a closed string between two boundary states (i.e., a cylinder diagram).
The expression of the cylinder diagram (tree diagram) in the light-cone formulation can be expressed as $$\begin{aligned}
&&\mathcal{A}_{Dp_1;Dp_2}(x^+,x^-,q_1^i,q_2^j) \\
&& = \int\!\!\frac{dp^+dp^-}{2\pi i}\,{{\rm e}}^{ip^+x^- + ip^-x^+}
{\langle}Dp_1,-p^-,-p^+,q_1^i |\left(
\frac{1}{p^+(p^- + H)}
\right)
| Dp_2,p^-,p^+,q_2^j {\rangle}{\nonumber}\\
&& = \int^{+\infty}_{-\infty}\!\!dp^+\,
{{\rm e}}^{ip^+x^-} \frac{{\theta}(p^+)}{p^+}
{\langle}Dp_1,-p^+,q_1^i |
\,{{\rm e}}^{-iHx^+} | Dp_2,p^+,q_2^j{\rangle}\,, {\nonumber}\end{aligned}$$ where $H$ is the light-cone Hamiltonian of a closed string and the $|
Dp,p^+,q^i {\rangle}$ represents a boundary state of a D$p$-brane instanton located at the transverse position $q^i$ with the longitudinal momentum $p^+$. We note that the prescription given in [@BGG] has been used for obtaining the last line in the above equation. If we define the variable $t$ by $$x^+ = \pi \tau = -i\pi t$$ by performing the customary Wick rotation, the amplitude is rewritten as $$\begin{aligned}
\mathcal{A}_{Dp_1;Dp_2}(x^+,x^-,q_1^i,q_2^j) =
\int^{\infty}_0\!\!\!\! dt\,{{\rm e}}^{- \frac{x^+x^-}{\pi t}}
\tilde{\mathcal{A}}_{Dp_1;Dp_2}(t,q_1^i,q_2^j)\,, \end{aligned}$$ where $\tilde{A}_{Dp_1;Dp_2}(t,q_1,q_2)$ is the expectation value: $$\begin{aligned}
\label{tree}
\tilde{\mathcal{A}}_{Dp_1;Dp_2}(t,q_1^i,q_2^j) \equiv
{\langle}Dp_1,-p^+,q_1^i | \,{{\rm e}}^{-2\pi t (H/2) }
| Dp_2,p^+,q_2^j {\rangle}\,.\end{aligned}$$
Now the tree amplitude (\[tree\]) will be calculated by using the boundary states constructed before. We restrict ourselves to the case of identical D$p$-brane instantons for simplicity. The calculus consists of three parts: 1) vacuum energies, 2) nonzero-modes, and 3) zero-modes.
Let us concentrate only on the sector with mass $\nu\equiv\mu/3$. The other sector with mass $\nu'\equiv \mu/6$ results in the same final expression only with the difference in the value of mass parameter. When we consider the contribution of vacuum energies to the amplitude, the evaluation of vacuum energies is identical to that of the partition function in the closed string case: $$\begin{aligned}
{{\rm e}}^{2\Delta(\nu;0)} \quad (\mbox{for bosons})\,,
\quad {{\rm e}}^{-2\Delta(\nu;0)} \quad (\mbox{for fermions})\,.\end{aligned}$$ The contribution of nonzero-modes to the amplitude is also the same as that of partition function of a closed string, and so it is readily written as $$\begin{aligned}
\prod_{n=1}^{\infty}
\left(1 - q^{\omega_n}\right)^{-4} \quad (\mbox{bosons})\,, \quad
\prod_{n=1}^{\infty}
\left(1 - q^{\omega_n}\right)^{4} \quad (\mbox{fermions})\,,
\quad q\equiv e^{-2\pi t}\,.\end{aligned}$$ The contribution of bosonic zero-modes can be evaluated by using the formula: $$\begin{aligned}
{\langle}0 | {{\rm e}}^{\pm\frac{1}{2}a_0a_0}q^{\pm \frac{1}{2}m a_0^{\dagger}a_0}
{{\rm e}}^{\pm \frac{1}{2}a_0^{\dagger}a_0^{\dagger}} | 0 {\rangle}= \left(1-q^m\right)^{-1/2}\,.\end{aligned}$$ As a result, the factor $(1-q^{\nu})^{-2}$ is obtained from bosonic zero-modes. The fermionic zero-modes can be evaluated by adopting the prescription given in the appendix of [@BGG], and the resulting contribution is $(1-q^{\nu})^2$. Consequently, the contribution of zero-modes is summarized as follows: $$\begin{aligned}
(1-q^{\nu})^{-2} \quad (\mbox{for bosons})\,, \quad
(1-q^{\nu})^{2} \quad (\mbox{for fermions})\,.\end{aligned}$$
After taking account of the sector with mass $\mu/6$, the total partition function is then represented by $$\begin{aligned}
\tilde{A}_{Dp;Dp} &=& \tilde{A}_{Dp;Dp}^{B}\cdot \tilde{A}_{Dp;Dp}^{F}
\;=\; \mathcal{N}^2_{Dp}\,,
\label{5.8}\end{aligned}$$ where $\mathcal{N}_{Dp}$ is the normalization factor of boundary states, which is not determined yet. This factor can be fixed by calculating the cylinder diagram in the open string channel. This task will be done in the next section.
In the work of [@BGG], the “conformal field theory condition” $$\begin{aligned}
\label{CFT}
\tilde{\mathcal{A}}_{Dp_1;Dp_2}(t,q_1,q_2)
= \tilde{Z}_{Dp_1;Dp_2}(\tilde{t},q_1,q_2)\,, \end{aligned}$$ was analyzed in the case of the pp-wave background. This condition ensures the consistency between closed and open string channels. We now turn to the calculation of the partition function with D-branes (i.e., the cylinder diagram) in order to show that the condition (\[CFT\]) also holds in our theory.
Partition Function in the Open String Description
=================================================
In this section we will discuss the one-loop amplitude of open string, and confirm the consistency condition between open and closed string channels.
We start from the light-cone action of open string defined by $$\begin{aligned}
S_{\rm open} &=& \frac{1}{4\pi{\alpha}'}\int\!\! d\tau\!
\int^{\pi}_0\!\!\! d\sigma\,
\Biggl[
\sum^8_{i=1}
\left\{
(\partial_{\tau}x^i)^2-(\partial_{\sigma}x^i)^2
\right\}
-\left(\frac{\mu}{3}\right)^2\sum^4_{a=1}(x^a)^2
-\left(\frac{\mu}{6}\right)^2\sum^8_{b=5}(x^b)^2
\Biggr] {\nonumber}\\
&& +
\frac{i}{2\pi}\int\!\! d\tau\!
\int^{\pi}_0\!\!\! d\sigma\,
\Biggl[
\Psi^{1+{{\scriptscriptstyle}T}}\partial_{-}\Psi^{1+}+
\Psi^{1-{{\scriptscriptstyle}T}}\partial_{-}\Psi^{1-}+
\Psi^{2+{{\scriptscriptstyle}T}}\partial_{+}\Psi^{2+}+
\Psi^{2-{{\scriptscriptstyle}T}}\partial_{+}\Psi^{2-}{\nonumber}\\
&&
-\frac{\mu}{3}\Psi^{1-{{\scriptscriptstyle}T}}\Pi^{{\scriptscriptstyle}T}\Psi^{2+}
-\frac{\mu}{6}\Psi^{1+{{\scriptscriptstyle}T}}\Pi^{{\scriptscriptstyle}T}\Psi^{2-}
+\frac{\mu}{6}\Psi^{2-{{\scriptscriptstyle}T}}\Pi\Psi^{1+}
+\frac{\mu}{3}\Psi^{2+{{\scriptscriptstyle}T}}\Pi\Psi^{1-}\Biggr]\,. \end{aligned}$$
To begin with, we shall present the mode-expansion in the case of open string. The mode-expansion of D-D string is expressed as $$\begin{aligned}
x^a(\tau,\sigma) &=& q^a_0\cdot \frac{\sinh\nu (\pi -\sigma)}{\sinh (\pi\nu)}
+ q^a_1\cdot \frac{\sinh (\nu\sigma)}{\sinh (\pi\nu)} -\sqrt{2\alpha'}
\sum_{n\neq 0}\frac{1}{\omega_n}\alpha^a_ne^{-i\omega_n\tau}
\sin (n\sigma)\,, \\
x^b(\tau,\sigma) &=& q^b_0\cdot \frac{\sinh\nu' (\pi -\sigma)}{\sinh (\pi\nu')}
+ q^b_1\cdot \frac{\sinh (\nu'\sigma)}{\sinh (\pi\nu')} -\sqrt{2\alpha'}
\sum_{n\neq 0}\frac{1}{\omega'_n}\alpha^a_ne^{-i\omega'_n\tau}
\sin (n\sigma)\,, \end{aligned}$$ where the endpoints satisfy the Dirichlet conditions: $x^i(\sigma = 0) = q^i_0\,,\,\,x^i(\sigma = \pi) = q^i_1$.
The mode expansion of N-N string, whose endpoints satisfy the Neumann conditions, is written as $$\begin{aligned}
x^a(\tau,\sigma) &=&x^a_0\cos (\nu\tau)
+\frac{1}{\nu}\cdot 2\alpha' p^a_0\sin (\nu\tau)
+ i\sqrt{2\alpha'}\sum_{n\neq 0}
\frac{1}{\omega_n}
\alpha^a_n e^{-i\omega_n \tau}
\cos (n\sigma)\,, \\
x^b(\tau,\sigma) &=& x^b_0\cos (\nu'\tau)
+\frac{1}{\nu'}\cdot 2\alpha' p^b_0\sin (\nu'\tau)
+ i\sqrt{2\alpha'}\sum_{n\neq 0}
\frac{1}{\omega'_n}
\alpha^b_n e^{-i\omega'_n \tau}
\cos (n\sigma)\,.\end{aligned}$$
The mode-expansion of fermions are the same with that in the case of closed string, but we have to take account of boundary conditions at $\sigma
=0$ and $\pi$ described by $$\begin{aligned}
&&\Psi^{1-} = \Omega\Psi^{2+}\,,\quad
\Psi^{2+} = \Omega^{{\scriptscriptstyle}T}\Psi^{1-}\,,\quad
\Psi^{1+} = \Omega\Psi^{2-}\,,\quad
\Psi^{2-} = \Omega^{{\scriptscriptstyle}T}\Psi^{1+}\,,\quad {\nonumber}\\
&&\tilde{\Psi}_n = \Omega\Psi_n\,,\quad
\tilde{\Psi}_0 = - \Pi\Omega\Psi_0\,,\quad
\Psi_0 = \Pi\Omega\tilde{\Psi}_0\,,\quad \Pi\Omega\Pi\Omega = -1\,, {\nonumber}\end{aligned}$$ where $\Omega$ is the gluing matrix for fermionic modes on the boundaries.
We now introduce the creation and annihilation operators given by $$\begin{aligned}
&&a^a_n=\frac{1}{\sqrt{\omega_n}}\alpha^a_n\,,\quad
a^{a\dagger}_n=\frac{1}{\sqrt{\omega_n}}\alpha^a_{-n}\,, \quad
[a^a_m,a^{a'\dagger}_n]=\delta^{a,a'}\delta_{m,n} \quad
(m,n>0)\,,{\nonumber}\\
&&a^b_n=\frac{1}{\sqrt{\omega_n'}}\alpha^b_n\,,\quad
a^{b\dagger}_n = \frac{1}{\sqrt{\omega_n'}}\alpha^b_{-n}\,, \quad
[a^b_m,a^{b'\dagger}_n]=\delta^{b,b'}\delta_{m,n} \quad
(m,n>0)\,,{\nonumber}\\
&& a^a_0 = \sqrt{\frac{{\alpha}'}{\nu}}
\left(p^a_0-i\nu\frac{1}{2{\alpha}'}x^a_0\right)\,,\quad
a^{a\dagger}_0 = \sqrt{\frac{{\alpha}'}{\nu}}
\left(p^a_0+i\nu\frac{1}{2{\alpha}'}x^a_0\right)\,,\quad
[a^a_0,a^{a'\dagger}_0]=\delta^{a,a'}\,, {\nonumber}\\
&& a^b_0 = \sqrt{\frac{{\alpha}'}{\nu'}}
\left(p^b_0-i\nu'\frac{1}{2{\alpha}'}x^b_0\right)\,,\quad
a^{b\dagger}_0 = \sqrt{\frac{{\alpha}'}{\nu'}}
\left(p^b_0+i\nu'\frac{1}{2{\alpha}'}x^b_0\right)\,,\quad
[a^b_0,a^{b'\dagger}_0]=\delta^{b,b'}\,,{\nonumber}\end{aligned}$$ then the Hamiltonian $H_B$ for the Dirichlet directions is expressed by $$\begin{aligned}
H_B &=&\frac{1}{4\pi \alpha'}
\cdot \frac{\nu}{\sinh (\pi\nu)}
\left[
\{(q^a_0)^2+(q^a_1)^2\}\cosh (\pi\nu)-2q^a_0 q^a_1
\right] {\nonumber}\\
&& + \frac{1}{4\pi \alpha'}
\cdot \frac{\nu'}{\sinh (\pi\nu')}
\left[
\{(q^b_0)^2+(q^b_1)^2\}\cosh (\pi\nu') - 2q^b_0 q^b_1
\right] {\nonumber}\\
&& +\frac{1}{2}\sum_{n=1}^{\infty}\omega_n
(a^{a\dagger}_n a^a_n + a^{a\dagger}_n a^a_n)
+\frac{1}{2}\sum_{n=1}^{\infty}\omega'_n
(a^{b\dagger}_n a^b_n + a^{b\dagger}_n a^b_n)\,,\end{aligned}$$ and that for the Neumann directions is represented by $$\begin{aligned}
H_B &=&\frac{1}{2}\omega_0(a^{\dagger\,a}_0a^a_0+a^a_0a^{\dagger\,a}_0)
+\frac{1}{2}\sum_{n\geq 1}\omega_n
(a^{\dagger\,a}_na^a_n+a^a_na^{\dagger\,a}_n) {\nonumber}\\
&& +\frac{1}{2}\omega'_0(a^{b\dagger}_0 a^b_0+a^b_0a^{b\dagger}_0)
+\frac{1}{2}\sum_{n\geq 1}\omega'_n
(a^{b\dagger}_n a^b_n + a^b_n a^{b\dagger}_n)\,.\end{aligned}$$ The Hamiltonian of fermions $H_F$ is rewritten as $$\begin{aligned}
H_F &=& \sum_{n=1}^{\infty}(\omega_nS^{\dagger}_nS_n+{\omega'}_n
{S'}^{\dagger}_nS'_n)
-\frac{\mu}{3}i\Psi^{{\scriptscriptstyle}T}_0\Pi\Omega\Psi_0
-\frac{\mu}{6}i{\Psi'}^{{\scriptscriptstyle}T}_0\Pi\Omega {\Psi'}_0\,. \end{aligned}$$ Now we shall evaluate the Casimir energy given as follows: $$\begin{aligned}
&& \sum_{n =1}^{\infty}\frac{1}{2}\omega_n\cong
\frac{1}{2}\left(
\sum^{\infty}_{n=1}\sqrt{n^2+\nu^2}-\int^{\infty}_0\!\!dk\,\sqrt{k^2+\nu^2}
\right)
=\frac{1}{2}\left(-\frac{1}{2}\nu+\Delta (\nu ;0)\right)\,, {\nonumber}\\
&& \sum_{n =1}^{\infty}\frac{1}{2}\omega_n'\cong
\frac{1}{2}\left(
\sum^{\infty}_{n=1}\sqrt{n^2+\nu'{}^2}-\int^{\infty}_0\!\!dk\,
\sqrt{k^2+\nu'{}^2}
\right)
=\frac{1}{2}\left(-\frac{1}{2}\nu'+\Delta (\nu' ;0)\right)\,, {\nonumber}\end{aligned}$$ in terms of zero-point energy.
The zero-point energies of a single boson with the Dirichlet condition are given by $$\begin{aligned}
&&(D)\,:\,\frac{1}{2}\sum_{n\geq 1}\omega_n
\cong \frac{1}{2}\left(-\frac{1}{2}\nu +\Delta (\nu ;0)\right)
\,, \quad
\frac{1}{2}\sum_{n\geq 1}\omega'_n
\cong \frac{1}{2}\left(-\frac{1}{2}\nu' +\Delta (\nu' ;0)\right)
\,,{\nonumber}\end{aligned}$$ and those with the Neumann condition are expressed as $$\begin{aligned}
&&(N)\,:\,\frac{1}{2}\sum_{n\geq 0}\omega_n
\cong \frac{1}{2}\left(+\frac{1}{2}\nu +\Delta (\nu ;0)\right)\,, \quad
\frac{1}{2}\sum_{n\geq 0}\omega'_n
\cong \frac{1}{2}\left(+\frac{1}{2}\nu' +\Delta (\nu' ;0)\right)
\,. {\nonumber}\end{aligned}$$ As for the zero-point energies for a fermion, we have $$\begin{aligned}
E^0 = - 4\cdot \frac{1}{2} \sum_{n\geq 1}\omega_n
\cong \nu - 2\Delta (\nu ;0)
\,,\quad E'{}^0= - 4 \cdot \frac{1}{2}\sum_{n\geq 1}\omega'_n
\cong \nu' - 2\Delta (\nu' ;0)\,,{\nonumber}\end{aligned}$$ where the factor 4 in front of the summation arises since each of fermions considered here has four independent components. The contributions of zero-point energies to the partition function $Z_F$ are then ${{\rm e}}^{-2\pi t (\nu -2\Delta (\nu ;0))}=
{{\rm e}}^{-2\pi t\nu}{{\rm e}}^{4\pi t\Delta (\nu ;0)}
$ and ${{\rm e}}^{-2\pi t (\nu' -2\Delta (\nu' ;0))}=
{{\rm e}}^{-2\pi t\nu'}{{\rm e}}^{4\pi t\Delta (\nu' ;0)}\,.$
We note that we have to treat carefully the zero-mode part of the Hamiltonian $$\begin{aligned}
H_F^0 = -i\nu \Psi_0^{{\scriptscriptstyle}T}\Pi\Omega\Psi_0
- i\nu' \Psi'_0{}^{{\scriptscriptstyle}T}\Pi\Omega\Psi'_0\,. {\nonumber}\end{aligned}$$ The $(\Psi_0)_{\alpha}$ and $(\Psi'_0)_{\alpha}$ have four non-vanishing components, and hence four sets of creation and annihilation operators $S^{\pm}_1$, $S^{\pm}_2$ and $S'{}^{\pm}_1$, $S'{}^{\pm}_2$ can be constructed. Here $S^{+}_{1,2}$ ($S'{}^{+}_{1,2}$) are creation operators and $S^{-}_{1,2}$ ($S'{}^{-}_{1,2}$) are annihilation ones. Thus the Hamiltonian can be rewritten as $$\begin{aligned}
H_F^0
&=& \frac{\nu}{2} \left(S^{+}_1S^{-}_1 - S^{-}_1S^{+}_1\right)
+\frac{\nu}{2} \left(S^{+}_2S^{-}_2 - S^{-}_2S^{+}_2\right) {\nonumber}\\
&& +\frac{\nu'}{2} \left(S'{}^{+}_1S'{}^{-}_1 - S'{}^{-}_1S'{}^{+}_1\right)
+\frac{\nu'}{2} \left(S'{}^{+}_2S'{}^{-}_2 - S'{}^{-}_2S'{}^{+}_2\right)
\,,{\nonumber}\end{aligned}$$ and the associated energies are $\pm \frac{\nu}{2}$ and $\pm \frac{\nu'}{2}$. Consequently, the contribution from these zero-mode parts is evaluated as $${{\rm e}}^{2\pi \nu t} (1 - {{\rm e}}^{-2\pi \nu t})^2
+ {{\rm e}}^{2\pi \nu' t} (1 - {{\rm e}}^{-2\pi \nu' t})^2\,.$$
We now turn to the evaluation of the total partition function of the open string connecting two identical D$p$-branes. Let us first consider the partition function for bosons: $$\begin{aligned}
Z_B= {\rm Tr}\, {{\rm e}}^{-2\pi tH_B}\,,\quad q = e^{-2\pi t}\,. {\nonumber}\end{aligned}$$ We will consider the sector with the mass $\nu=\mu/3$. The bosonic partition function for the ($4-p$) Dirichlet conditions (i.e., $4-p$ = $\sharp$(D-D strings)) is given by $$\begin{aligned}
&&Z_B^{(\nu)} =\frac{1}{\displaystyle \prod_{n\geq 1}(1-q^{\omega_n})^{4-p}}
\cdot q^{(4-p)\cdot \frac{1}{2}\left(
-\frac{1}{2}\nu +\Delta (\nu ;0)
\right)}\cdot f(q^a_0,q^a_1)\,,{\nonumber}\end{aligned}$$ where the function $f(q_0^a,q_1^a)$ is defined by $$\begin{aligned}
&&f(q^a_0,q^a_1) \equiv \exp \left[
\frac{-t}{2\alpha'}
\frac{\nu}{\sinh (\pi\nu)}
\left[
\{(q^a_0)^2+(q^a_1)^2\}\cosh (\pi\nu)
-2q^a_0q^a_1\right]\right]\,. {\nonumber}\end{aligned}$$ The partition function for the $p$ Neumann directions (i.e., $p$ = $\sharp$(N-N strings)) is written as $$\begin{aligned}
Z_B^{(\nu)}=\frac{1}{\displaystyle \prod_{n\geq 0}(1-q^{\omega_n})^p}
\cdot q^{p\cdot \frac{1}{2}\left(+\frac{1}{2}\nu +\Delta (\nu ;0)\right)}
\,. {\nonumber}\end{aligned}$$ Thus, the bosonic partition function on the sector with mass $\mu/3$ is represented by $$\begin{aligned}
Z_B^{(\nu)} &=& (1 - q^{\nu})^{-p}\cdot
\frac{1}{\displaystyle \prod_{n\geq 1}(1-q^{\omega_n})^4}
\cdot f(q^a_0,q^a_1)\cdot q^{\nu (-1+\frac{1}{2}p)+2\Delta (\nu ;0)}
\,.{\nonumber}\\
&=& \left(2\sinh (\pi\nu t)\right)^{2-p}
\left[\theta_{(0,0)}(t;\nu)\right]^{-2}\cdot f(q^a_0,q^a_1)\,,{\nonumber}\end{aligned}$$ where we have utilized the theta-like function: $\theta_{(a,b)}(t;\nu)=
\sqrt{\Theta_{(a,b)}(it;-it;\nu)}$.
Next we shall evaluate the partition function for fermions with mass $\mu/3$. The fermionic partition function is given by $$\begin{aligned}
Z_F = {\rm Tr} (-1)^{\bf F}{{\rm e}}^{-2\pi tH_F}\,.{\nonumber}\end{aligned}$$ After the similar calculation to bosonic case, we obtain the fermionic partition function: $$\begin{aligned}
Z_F^{(\nu)} &=& ({{\rm e}}^{\pi\nu t}-e^{-\pi\nu t})^2 \cdot
{{\rm e}}^{-2\pi\nu t}{{\rm e}}^{4\pi t\Delta (\nu ;0)}
\prod_{n=1}^{\infty}(1 - {{\rm e}}^{-2\pi t\omega_n})^4 {\nonumber}\\
&=& \left[\theta_{(0,0)}(t;\nu)\right]^2\,.\end{aligned}$$
It is an easy task to include the sector with mass $\mu/6$, and thus the total partition function is described by $$\begin{aligned}
Z_{\rm tot} &=& Z_B^{(\nu)} Z_F^{(\nu)}
\cdot Z^{(\nu')}_B Z^{(\nu')}_F
{\nonumber}\\
&=& \left(2\sinh (\pi\nu t)\right)^{2-p_1}
\prod_{a\in D}f(q^a_0,q^a_1)\cdot
\left(2\sinh (\pi\nu' t)\right)^{2-p_2}
\prod_{b\in D}f(q^b_0,q^b_1)\,.
\label{op-pf} \end{aligned}$$ Here the numbers $p_1$ and $p_2 $ are Neumann directions in the coordinates $x^a$’s and $x^b$’s, respectively. The net number of Neumann directions is represented by $\sharp$(Neumann)$=p_1+p_2$. The $\prod_{i\in D}$ means the product in terms of Dirichlet directions $x^i$’s.
By comparing the cylinder amplitude (\[5.8\]) obtained in the last section with the resulting partition function (\[op-pf\]) in the case of $q_0 = q_1 =0$, we can determine the normalization factor of boundary states. The modular S-transformation of ‘massive’ theta function (\[theta\]) relates the parameter $\mu~(\equiv \mu_{\rm cl})$ in the closed string to that $\mu~(\equiv \mu_{\rm op})$ in the open string through the corresponding law: $\mu_{\rm op} t = \mu_{\rm cl}$ [@BGG]. Hence the normalization factor of boundary states for the D$p$-brane instanton is given by $$\mathcal{N}_{Dp} = (2\sinh(\pi\mu/3))^{(2-p_1)/2}
\cdot(2\sinh(\pi\mu/6))^{(2-p_2)/2} \quad (p=p_1+p_2)\,.$$ Thus, we have shown the open/closed string duality in the $\mathcal{N}=(4,4)$ type IIA string theory at the origin. It was already shown in [@BGG] that this duality holds in the case of the type IIB string theory on the [*maximally*]{} supersymmetric pp-wave background. Although we are in a situation of less supersymmetric case, the duality still holds at the origin.
It should be noted that the open/closed string duality holds at the origin. That is, the open/closed string duality requires that there is no dependence on transverse coordinates since the supersymmetry conditions require that both D-brane instantons should be at the origin, as discussed in the paper [@BGG]. The cylinder amplitude does not have sensible behavior once the branes are located away from the origin.
General Properties of Partition Functions of Closed String
==========================================================
We have discussed the partition function and modular invariance of the type IIA string theory on the pp-wave background above. In this consideration there are two sectors with masses $\mu/3$ and $\mu/6$, and we have found that the modular properties hold in each sector. In this section, motivated by this fact, we will discuss general properties of partition functions of closed string apart from the type IIA string theory considered above. We suggest that some characteristics of string theories on pp-waves should be fixed from the requirement of modular invariance.
In the pp-wave case, theta-like function $\Theta_{(a,b)}(\tau ,\bar{\tau},\nu)$ should appear in the closed string partition function. It contains a mass parameter $\nu$ and has peculiar properties under modular transformations $$\begin{aligned}
&&\Theta_{(a,b)}(\tau +1,\bar{\tau}+1;\nu)
=\Theta_{(a,b+a)}(\tau ,\bar{\tau};\nu)\,,\quad
\Theta_{(a,b)}(-\frac{1}{\tau},-\frac{1}{\bar{\tau}};|\tau |\nu)
=\Theta_{(b,-a)}(\tau ,\bar{\tau};\nu)\,.\nonumber\end{aligned}$$ Notably, the mass parameter $\nu$ changes into $|\tau |\nu$ under $S$-transformation $\tau \rightarrow -1/\tau$, and it gives us severe constraints in constructing modular invariant partition functions. In other words, we can make modular invariant partition functions with this clue to go upon. Now we will study a certain class of modular-invariant partition functions on the pp-wave background by using the modular properties of $\Theta_{(a,b)}(\tau ,\bar{\tau}, \nu)$.
To simplify the problem, we put the following ansatz:\
(1) There are several kinds of mass parameters $\nu$’s.\
(2) For each $\nu$, the partition function $Z_B(\tau ,\bar{\tau },\nu)$ of the boson and that of fermion $Z_F(\tau ,\bar{\tau} ,\nu)$ cancel. That is to say, $Z_B(\tau
,\bar{\tau}, \nu)\cdot Z_F(\tau , \bar{\tau}, \nu)=1 $.\
We impose the ansatz (2) because the modular $S$-transformation changes the mass parameter $\nu$ into another one $|\tau | \nu$ and it is generally difficult to construct modular invariant combinations of $Z_B$’s and $Z_F$’s. In order to avoid this complicated problem, we take the simplest ansatz $Z_B(\tau ,\bar{\tau}, \nu)\cdot Z_F(\tau , \bar{\tau},
\nu)=1 $ here. But we should emphasize that there might be other modular invariant combinations without our ansatz and we cannot say there are no other possibilities. Here we will investigate such restricted cases only and compare our results with the models proposed earlier.
Now we will classify possible models by the use of the above ansatz. In order to realize the condition (2), the degrees of freedom of bosons must be identical with those of fermions. When we consider a transverse $D$ dimensional space, the degrees of freedom of bosons are $D$ (i.e., $\sharp(\mbox{boson})=D$). On the other hand, the degrees of freedom of spinors in $D$ dimensions are evaluated as $$\begin{aligned}
\sharp(\mbox{fermion})=2^{[\frac{D}{2}]+\epsilon}\,,
\quad \epsilon =\left\{
\begin{array}{rcl}
0 & \mbox{Majorana or Weyl}\\
-1 & \mbox{Majorana and Weyl}\\
+1 & \mbox{otherwise}
\end{array}
\right.{\nonumber}\end{aligned}$$ The value of $\epsilon$ depends on what kinds of spinors we consider. From the consideration of dimensionality, we can understand that the matching of degrees of freedom between bosons and fermions happens for $D=1,2,4$ and $8$ only. For each $D=1,2,4,8$, the corresponding degree of freedom is $1,2,4,8$. That is, we have to consider four kinds of sets containing one, two, four, and eight bosons. Different sets are distinguished from mass parameters $\nu$’s. Due to these mass terms, Lorentz symmetry is broken into smaller one. Next let us classify bosonic parts based on the Lorentz symmetry.
We study superstring theories and hence the dimension of transverse space should be eight. For massless cases, the associated Lorentz symmetry is $SO(8)$ and there are eight massless bosons. However bosons have mass terms in our massive case. We set $N_a$ as the number of sets with $a(=1,2,4,8)$ bosons with the same mass parameter. Let $\nu_{a,i}$ $(i=1,2,\cdots ,N_a)$ be mass parameters for bosons and fermions in the same set. Due to mass terms, Lorentz symmetry is broken down to smaller one $$\begin{aligned}
&&SO(1)^{N_1}\otimes
SO(2)^{N_2}\otimes
SO(4)^{N_4}\otimes
SO(8)^{N_8}\subset SO(8)\,,{\nonumber}\\
&&\quad \mbox{with}\qquad N_1+2N_2+4N_4+8N_8=8\,,
\qquad N_1,N_2,N_4,N_8 \in \mathbb{Z}_{\geq 0}\,.{\nonumber}\end{aligned}$$ From this constraint, we can classify possible combinations: $$\begin{aligned}
\begin{array}{|cccc|c|}
\hline
N_1 & N_2 & N_4 & N_8 & \mbox{Symmetry}\\
\hline\hline
0 & 0 & 0 & 1 & SO(8)\\
0 & 0 & 2 & 0 & SO(4)\times SO(4)\\
0 & 2 & 1 & 0 & SO(2)\times SO(2)\times SO(4)\\
2 & 1 & 1 & 0 & SO(1)^{\otimes 2}\times SO(2)\times SO(4)\\
8-2\ell & \ell & 0 & 0 & SO(1)^{\otimes (8-2\ell)}\times SO(2)^{\otimes \ell}
\\\hline
\end{array}\nonumber\end{aligned}$$ with $\ell =0,1,2,3,4$. Our type IIA model corresponds to the second case $(N_1,N_2,N_4,N_8)=(0,0,2,0)$ in the list and symmetry is $SO(4)\times SO(4)$. The type IIB string theory on the maximally supersymmetric background corresponds to $(N_1,N_2,N_4,N_8) = (0,0,0,1)$. All of type IIB pp-wave backgrounds with the above-mentioned bosonic isometry are found (for example see [@Sakaguchi]) and we can construct superstring theories on these backgrounds.
Here we turn to partition functions based on our ansatz. Each set is labelled by the mass parameter $\nu_{a,i}$ $(i=1,2,\cdots ,N_a\,;\,a=1,2,4,8)$. The associated partition function of boson $Z_B(\tau ,\bar{\tau},\nu_{a,i})$ and that of fermion $Z_F(\tau ,\bar{\tau}, \nu_{a,i})$ are written in the massive closed string case $$\begin{aligned}
&&Z_B(\tau ,\bar{\tau},\nu_{a,i})=
\Theta_{(0,0)}(\tau ,\bar{\tau},\nu_{a,i})\,,\quad
Z_F(\tau ,\bar{\tau},\nu_{a,i})=
\Theta_{(0,0)}(\tau ,\bar{\tau},\nu_{a,i})^{-1}\,.{\nonumber}\end{aligned}$$ Then we can evaluate the total partition function $Z$ as $$\begin{aligned}
Z=\prod_{a=1,2,4,8}\prod_{i=1}^{N_a}
Z_B(\tau ,\bar{\tau},\nu_{a,i})\cdot Z_F(\tau ,\bar{\tau},\nu_{a,i})
=1\,.{\nonumber}\end{aligned}$$ It is actually modular invariant and many models proposed earlier are included in our results.
Last we explain the result $Z=1$ from the point of view of energy matching. Let $\varepsilon$ be any energy level of states in our string system. We also introduce $n_B(\varepsilon)$, $n_F(\varepsilon)$ as the number of bosonic states and that of fermionic states at each energy level $\varepsilon$ respectively. Then the associated partition function $Z$ is defined as $$\begin{aligned}
Z= {\rm Tr} (-1)^{\bf F}e^{-2\pi \tau_2 H}
=\sum_{\varepsilon}(n_B(\varepsilon)-n_F(\varepsilon))
e^{-2\pi \tau_2\varepsilon}\,.{\nonumber}\end{aligned}$$ Here ${\bf F}$ is the fermion number operator and we also take $\tau_1 =0$ for simplicity. By comparing our result $Z=1$, we understand following relations $$\begin{aligned}
&&1=Z=(n_B(\varepsilon =0)-n_F(\varepsilon =0))e^{-2\pi\tau_2 \cdot 0}
+\sum_{\varepsilon >0}
(n_B(\varepsilon )-n_F(\varepsilon ))e^{-2\pi\tau_2 \varepsilon}\,,{\nonumber}\\
&&n_B(\varepsilon =0)-n_F(\varepsilon =0)=1\,,\quad
n_B(\varepsilon )=n_F(\varepsilon )\qquad (\varepsilon >0)\,.{\nonumber}\end{aligned}$$ It shows that number of bosonic states and that of fermionic states match at each energy level $\varepsilon >0$. Then total energy of bosonic states $E_B=n_B(\varepsilon)\cdot
\varepsilon$ is equal to that of fermionic states $E_F=n_F(\varepsilon)\cdot \varepsilon$ at each energy level $\varepsilon >0$. On the other hand, there is unbalance in number between bosonic states and fermionic states for the $\varepsilon =0$ part. But the associated total energy of bosonic states $E_B^0=n_B(\varepsilon =0)\cdot 0=0$ equals to that of fermionic states $E_F^0=n_F(\varepsilon =0)\cdot 0=0$ in this $\varepsilon =0$ sector. So the partition function $Z$ is nothing but the Witten index $Z={\rm Tr}(-1)^{\bf F}$ in our massive case. This fact is already known in previous papers. Collecting these considerations, we conclude that our ansatz (2) is equivalent to a condition (Witten index)$=1$. When we impose this condition (2) on $Z$, the resulting partition function is one and does not vanish. It also ensures that total energies of states at each energy level $\varepsilon$ match between bosonic states and fermionic states.
Our ansatz satisfies sufficient conditions to construct modular invariant partition functions. But we do not know necessary condition for this problem. We think it is important to find some further extra constraints in order to construct consistent string backgrounds and classify possible strings for massive cases.
Conclusions and Discussions
===========================
We have discussed the partition function of type IIA string theory obtained from the eleven-dimensional theory through the $S^1$-compactification of a transverse direction.
The modular invariance of our type IIA string theory has been proven. This type IIA string theory is less supersymmetric but it is modular invariant by virtue of the cancellation between bosonic and fermionic degrees of freedom.
We have constructed the boundary states and classified the D-brane instantons in our theory. The resulting list of the allowed D-brane instantons is consistent with that of the allowed D-branes obtained previously in different frameworks. In addition, we have calculated the amplitude between D-branes in the closed and open string descriptions, and checked the channel duality in our theory. Furthermore, we have briefly discussed general modular properties. There are many non-maximally supersymmetric pp-wave backgrounds, but not all of them would give ‘modular invariant’ superstring theories. Thus, we believe that the modular invariance is an available clue to classify the ‘[*physical*]{}’ string theories on pp-waves.
[**Acknowledgement**]{}
K.Y thanks Makoto Sakaguchi for useful discussions and comments. The work of H.S. was supported by Korea Research Foundation Grant KRF-2001-015-DP0082. The work of K.S. is supported in part by the Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan ($\sharp$ 14740115).
[99]{}
J. Kowalski-Glikman, “Vacuum States In Supersymmetric Kaluza-Klein Theory,” Phys. Lett. B [**134**]{} (1984) 194. M. Blau, J. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, “A new maximally supersymmetric background of IIB superstring theory,” JHEP [**0201**]{} (2002) 047 \[arXiv:hep-th/0110242\]. M. Blau, J. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, “Penrose limits and maximal supersymmetry,” Class. Quant. Grav. [**19**]{} (2002) L87 \[arXiv:hep-th/0201081\]. R. Penrose, “Any spacetima has a plane wave as a limit,” Differential geometry and relativity, Reidel, Dordrecht, 1976, pp. 271-275.
R. Gueven, “Plane wave limits and T-duality,” Phys. Lett. B [**482**]{} (2000) 255 \[arXiv:hep-th/0005061\]. R. R. Metsaev, “Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background,” Nucl. Phys. B [**625**]{} (2002) 70 \[arXiv:hep-th/0112044\]. R. R. Metsaev and A. A. Tseytlin, “Exactly solvable model of superstring in plane wave Ramond-Ramond background,” Phys. Rev. D [**65**]{} (2002) 126004 \[arXiv:hep-th/0202109\]. D. Berenstein, J. M. Maldacena and H. Nastase, “Strings in flat space and pp waves from N = 4 super Yang Mills,” JHEP [**0204**]{} (2002) 013 \[arXiv:hep-th/0202021\]. K. Dasgupta, M. M. Sheikh-Jabbari and M. Van Raamsdonk, “Matrix perturbation theory for M-theory on a PP-wave,” JHEP [**0205**]{} (2002) 056 \[arXiv:hep-th/0205185\]. K. Sugiyama and K. Yoshida, “Supermembrane on the pp-wave background,” Nucl. Phys. B [**644**]{} (2002) 113 \[arXiv:hep-th/0206070\]. S. Hyun and H. Shin, “Branes from matrix theory in pp-wave background,” Phys. Lett. B [**543**]{} (2002) 115 \[arXiv:hep-th/0206090\].
K. Sugiyama and K. Yoshida, “BPS conditions of supermembrane on the pp-wave,” Phys. Lett. B [**546**]{} (2002) 143 \[arXiv:hep-th/0206132\]. K. Sugiyama and K. Yoshida, “Giant graviton and quantum stability in matrix model on PP-wave background,” Phys. Rev. D [**66**]{} (2002) 085022 \[arXiv:hep-th/0207190\]. N. Nakayama, K. Sugiyama and K. Yoshida, “Ground state of the supermembrane on a pp-wave,” arXiv:hep-th/0209081, to appear in Phys. Rev. D. T. Banks, N. Seiberg and S. H. Shenker, Nucl. Phys. B [**490**]{} (1997) 91 \[arXiv:hep-th/9612157\].
K. Sugiyama and K. Yoshida, “Type IIA string and matrix string on pp-wave,” Nucl. Phys. B [**644**]{} (2002) 128 \[arXiv:hep-th/0208029\]. S. Hyun and H. Shin, “N = (4,4) type IIA string theory on pp-wave background,” JHEP [**0210**]{} (2002) 070 \[arXiv:hep-th/0208074\]. S. Hyun and H. Shin, “Solvable N = (4,4) type IIA string theory in plane-wave background and D-branes,” Nucl. Phys. B [**654**]{} (2003) 114 \[arXiv:hep-th/0210158\]. A. Dabholkar and S. Parvizi, “Dp branes in pp-wave background,” Nucl. Phys. B [**641**]{} (2002) 223 \[arXiv:hep-th/0203231\].
S. Hyun, J. Park and H. Shin, “Covariant description of D-branes in IIA plane-wave background,” Phys. Lett. B [**559**]{} (2003) 80 \[arXiv:hep-th/0212343\]. O. K. Kwon and H. Shin, “Type IIA supergravity excitations in plane-wave background,” arXiv:hep-th/0303153. S. Hyun, J. D. Park and S. H. Yi, “Thermodynamic behavior of IIA string theory on a pp-wave,” arXiv:hep-th/0304239. Y. Sugawara, “Thermal amplitudes in DLCQ superstrings on pp-waves,” Nucl. Phys. B [**650**]{} (2003) 75 \[arXiv:hep-th/0209145\]. Y. Sugawara, “Thermal partition function of superstring on compactified pp-wave,” arXiv:hep-th/0301035. O. Bergman, M. R. Gaberdiel and M. B. Green, “D-brane interactions in type IIB plane-wave background,” JHEP [**0303**]{} (2003) 002 \[arXiv:hep-th/0205183\]. M. R. Gaberdiel and M. B. Green, “The D-instanton and other supersymmetric D-branes in IIB plane-wave string theory,” arXiv:hep-th/0211122. T. Takayanagi, “Modular invariance of strings on pp-waves with RR-flux,” JHEP [**0212**]{} (2002) 022 \[arXiv:hep-th/0206010\]. M. R. Gaberdiel, M. R. Green, S. S.-Nameki and A. Sinha, “Oblique and curved D-branes in IIB plane-wave string theory,” arXiv:hep-th/0306056.
M. B. Green and M. Gutperle, “Light-cone supersymmetry and D-branes,” Nucl. Phys. B [**476**]{} (1996) 484 \[arXiv:hep-th/9604091\]. M. Billo and I. Pesando, “Boundary states for GS superstrings in an Hpp wave background,” Phys. Lett. B [**536**]{} (2002) 121 \[arXiv:hep-th/0203028\]. M. Sakaguchi, “IIB pp-waves with extra supersymmetries,” arXiv:hep-th/0306009.
[^1]: Section 2 has some overlap with the work [@HPY]. Thermal partition function is discussed in the cases of other pp-wave strings [@S1; @S2].
[^2]: We note that, contrary to the present type IIA case, the boundary states in the type IIB string theory on the pp-wave background have been relatively much studied[@BP; @BGG; @Green; @GGSS].
[^3]: If we redefine the $\widetilde{\Psi}_0$ as $\Pi\widetilde{\Psi}_0
\rightarrow \widetilde{\Psi}_0$, then we obtain the usual expressions for zero-mode conditions. The effect from the redefinition of $\widetilde{\Psi}_0$ are absorbed into the definition of the creation and annihilation operators without the modification of anticommutation relations, and so we have no trouble for our discussion.
|
---
abstract: 'Let $G$ be an n-dimensional semisimple, compact and connected Lie group acting on both the Lie algebra $\mathfrak{g}$ of $G$ and its dual $\mathfrak{g}^*$. In this work it is shown that a nondegenerate Killing form of $G$ induces an $Ad^{*}$-equivariant isomorphism of $\mathfrak{g}$ onto $\mathfrak{g}^*$ which, in turn, induces by passage to quotients a symplectic diffeomorphism between adjoint and coadjoint orbit spaces of $G$.'
---
[**ON THE CASE WHERE ADJOINT AND COADJOINT ORBIT SPACES ARE SYMPLECTOMORPHIC**]{}
[Augustin T. Batubenge[^1] Wallace M. Haziyu ]{}
[: 20D06, 22E60, 22F30, 53D05, 57R50, 58E40.]{}\
[: Equivariant mapping, Killing form, Orbit space, Symplectomorphism. ]{}
Introduction
============
This work is concerned with morphisms of the category of symplectic spaces, so-called symplectic mappings. Of more interest among them are isomorphisms, that is, the symplectic mappings which also are diffeomorphisms between objects. They are important in that they exchange both the differentiable as well as symplectic structures. Working in this area so-called symplectic geometry is fascinating in that several studies, going back to previous centuries, constantly aimed at working out an elegant formalism of classical mechanics. For this paper, our main references among others are the book by R. Abraham and J.E. Marsden ([@Abr78]) and A. Arvanitoyeorgos ([@Arv03]). Combining the information provided in these sources as well as the constructions in the recent author’s paper (see [@BH18]), we were able to obtain the main results of this study. The work involves an important amount of background ideas on representation theory. That is, adjoint and coadjoint representations as well as the actions of Lie groups yielding orbit spaces. With high interest are those quotient spaces resulting from transitive actions, the homogeneous spaces. These are the Lie groups themselves, spheres in real as well as complex and quaternionic settings, projective spaces, Grassmann and Stiefel manifolds to cite a few. In the list, we would mention flag and generalized flag manifolds. They are an important class of homogeneous spaces which admit a complex structure, a Kähler structure and a symplectic structure as mentioned in ([@Arv03]).\
The study of coadjoint orbits was introduced by Kirillov, and the existence of a symplectic structure on these orbits is the result of Kostant and Souriau (see [@Ber01 p.52]), the fact from which we shall take steps further in this study. Briefly speaking, we consider the action of a compact, connected and semisimple Lie group $G$ on both its Lie algebra $\mathfrak{g}$ and its dual $\mathfrak{g}^*$, resulting into orbit spaces which consist of only one orbit each, and constructed a symplectic diffeomorphism between them. To this end, the paper is organized as follows. We begin by recalling the basics on homogeneous spaces. Then the notion of an adjoint orbit will follow, and we will show that it is related to flag as well as symplectic manifolds. Next, using Cartan’s criterion for semisimplicity in which case the Killing form is nondegenerate and $Ad$-invariant, we will construct an $Ad^*$-equivariant isomorphism of Lie algebras $\mathfrak{g}$ onto $\mathfrak{g}^*$ that will induce a symplectomorphism on the quotient spaces reduced to one orbit each.
Preliminaries
=============
Let $G$ be a Lie group, $H$ a subgroup, and $G/H = \lbrace aH:a\in G\rbrace$ the set of left cosets of $H$ in $G$. The map $\pi:G\rightarrow G/H$ which takes each element $a\in G$ to its coset $aH$, is called the projection map. The coset space $G/H$ is not necessarily a manifold. However, if $H$ is a closed subgroup of $G$, a manifold structure on the quotient space $G/H$ can be defined such that the projection map $\pi :G\rightarrow G/H$ is a surjective submersion. (see [@Boo75 Theorem 9.2]). Also, recall that if $\phi:G\times M\longrightarrow M;~\phi(g,p)=\phi_g(p)$ is a smooth and transitive action of $G$ on a smooth manifold $M$, the $M$ is called a homogeneous space (see [@Boo75 150]). This definition extends to the quotient space $G/H$ of the Lie group $G$ by a closed subgroup $H$ of $G$. In effect, there is a natural action $G\times G/H\rightarrow G/H$, $(g,aH)\mapsto gaH$. This action is always transitive since if $aH, bH\in G/H$, then $ba^{-1}(aH) = bH$ for all $a,b\in G$. For this reason every transitive action can be represented as a coset space $G/H$ where $H$ is a closed subgroup of $G$. In fact if $M$ is a manifold on which a Lie group $G$ acts transitively, then for any $p\in M$ let $G_{p} = \lbrace g\in G: g\cdot p = p\rbrace$ be the stabilizer of $p$, we have that $G_{p}$ is a closed subgroup of $G$ and $G/G_{p}\cong M$. Take $H = G_{p}$. Then $M\cong G/H$ as asserted. Therefore, $G/H$ is called the homogeneous space of $M$.\
Adjoint Orbits
==============
Let $G$ be a Lie group and $\mathfrak{g}\cong T_{e}G$ be its Lie algebra where $e$ is the identity element in $G$. Then the smooth action $$\Phi:G\times\mathfrak{g}\rightarrow\mathfrak{g};\quad (g,\xi)\mapsto Ad(g)\xi$$ denoted by $Ad$, is called the adjoint action of $G$ on its Lie algebra $\mathfrak{g}$.
Let $Ad:G\times\mathfrak{g}\rightarrow\mathfrak{g}$ be the adjoint action of a Lie group $G$ on its Lie algebra $\mathfrak{g}$ and let $\xi\in\mathfrak{g}$. We define the adjoint orbit of $\xi$ to be $$\begin{array}{ccc}
O_{\xi} = \lbrace Ad(g)\xi:g\in G\rbrace\subset\mathfrak{g}
\end{array}$$
That is, if $\eta\in O_{\xi}$ then there is some $g\in G$ such that $\eta = Ad(g)\xi$. The stability group also called the isotropy group of $\xi$ is given by $$\begin{array}{ccc}
G_{\xi} = \lbrace g\in G: Ad(g)\xi = \xi\rbrace.
\end{array}$$ This is a closed subgroup of $G$ (see [@Cro18 p 16]). In what follows, we show that adjoint orbits can be represented as homogeneous spaces. For a similar construction (see [@BH18 pp 127-129]). Define a map $\rho:O_{\xi}\rightarrow G/G_{\xi}$ by $\rho(\eta) = gG_{\xi}$ for $\eta\in O_{\xi}$ and $g\in G$ such that $\eta = Ad(g)\xi$. The map $\rho$ is well defined since if also $\rho(\eta) = hG_{\xi}$ for some $h\in G$ then $Ad(g)\xi = Ad(h)\xi$ which implies that $Ad(h^{-1})\circ Ad(g)\xi = \xi$. This gives $h^{-1}g\in G_{\xi}$ and $gG_{\xi} = hG_{\xi}$. The map $\rho$ is injective. For, let $\eta = Ad(g)\xi$, $\mu = Ad(h)\xi$ and suppose that $gG_{\xi} = hG_{\xi}$. Then $h^{-1}g\in G_{\xi}$ so that $Ad(h^{-1}g)\xi = Ad(h^{-1})\circ Ad(g)\xi = \xi$. This implies then that $\eta = Ad(g)\xi = Ad(h)\xi = \mu$. Clearly $\rho$ is surjective since for $g\in G$ and $\eta = Ad(g)\xi\in O_{\xi}$ gives $\rho(\eta) = gG_{\xi}$ by construction. If $\eta = Ad(h)\xi$ for some $h\in G$, then $G_{\eta} = Ad(h)G_{\xi}Ad(h^{-1})$. Thus, for all $g\in G$ we have $$\begin{array}{ccc}
G/G_{\xi}\cong G/G_{Ad(g)\xi}
\end{array}$$ induced by the map $g\mapsto hgh^{-1}$, which shows that the definition of $G/G_{\xi}$ does not depend on the choice of the element $\xi$ in its adjoint orbit. Thus, $G/G_{\xi}\cong O_{\xi}$. Now let $G/G_{\xi}\cong O_{\xi}$. From the argument above, $G$ acts transitively on $G/G_{\xi}\cong O_{\xi}$ which makes it into a homogeneous space.\
Next, let $X\in \mathfrak{g}$. Note that the vector field on $\mathfrak{g}$ corresponding to $X$, called the fundamental vector field or the infinitesimal generator of the action, is defined by $$\begin{array}{ccc}
X_{\mathfrak{g}}(\xi) = \frac{d}{dt}(Ad(\exp{tX})\xi)\mid_{t=0}
\end{array}$$ We compute the tangent space to the adjoint orbit $O_{\xi}$ at $\xi$ as follows. Let $X\in\mathfrak{g}$. Let $x(t) = \exp{tX}$ be the curve in $G$ which is tangent to $X$ at $t = 0$. Then $\xi(t) = Ad(\exp{tX})\xi$ is the curve on $O_{\xi}$ such that $\xi(0) = \xi$. Let $Y\in \mathfrak{g}$, then $\langle\xi(t) , Y\rangle = \langle Ad(\exp{tX})\xi , Y\rangle$, where $\langle\cdot , \cdot\rangle$ is the natural pairing. Differentiating with respect to $t$ at $t = 0$ we get $$\begin{array}{cll}
\langle\xi'(0) , Y\rangle &=& \frac{d}{dt}\langle Ad(\exp{tX})\xi , Y\rangle\mid_{t=0} \\
&=& \langle \frac{d}{dt}(Ad(\exp{tX})\xi)\mid_{t=0} , Y\rangle = \langle ad(X)\xi , Y\rangle.
\end{array}$$ Thus $\xi'(0) = ad(X)\xi$. Therefore, the tangent space to the orbit $O_{\xi}$ at $\xi$ is given by
$$\begin{array}{ccc}
T_{\xi}O_{\xi} = \lbrace ad(X)\xi : X\in\mathfrak{g}\rbrace
\end{array}$$
Adjoint orbits as flag manifolds
--------------------------------
The examples of adjoint orbits that will be of interest in this work are the generalized flag manifolds. These orbits are known to hold a symplectic structure. Generalized flag manifolds are homogeneous spaces which can be expressed in the form $G/C(S)$, where $G$ is a compact Lie group and\
$C(S)=\lbrace g\in G : gx = xg,~ \textrm{for~all~}x\in S\rbrace$ is the centraliser of a torus $S$ in $G$. Generalized flag manifolds just like flag manifolds are homogeneous spaces (see[@Arv03 p 70]). Here is an example in $\mathbb{C}^n$.
Let $\mathbb{C}^{n}$ be an $n$-dimensional complex space. A flag is a sequence of complex subspaces $$\begin{array}{cc}
W = V_{1}\subset V_{2}\subset\cdots\subset V_{n} = \mathbb{C}^{n}
\end{array}$$ ordered by inclusion such that $\dim V_{i} = i$ for $i = 1,\cdots,n$ and $V_i$ is a proper subset of $V_{i+1}$ for $i=1,..., n-1.$
Let $\lbrace e_{1}, e_{2},\cdots , e_{n}\rbrace$ be the canonical basis for the complex vector space $\mathbb{C}^{n}$. Then the standard flag is given by $$\begin{array}{cc}
W_{0} = Span_{\mathbb{C}}\lbrace e_{1}\rbrace \subset Span_{\mathbb{C}}\lbrace e_{1} , e_{2}\rbrace\subset\cdots\subset Span_{\mathbb{C}}\lbrace e_{1},\cdots e_{n}\rbrace = \mathbb{C}^{n}
\end{array}$$
We need to show that flag manifolds are homogeneous spaces. Let $F_{n}$ be the set of all flags in $\mathbb{C}^{n}$ and let $W_{0}$ be the standard flag above. Then the action of the Lie group $U(n)=\{A\in GL(n,\mathbb{C}):\bar{A}^TA=I\}$ on $F_{n}$ is transitive. For, consider an arbitrary flag $W = V_{1}\subset V_{2}\subset\cdots\subset V_{n} = \mathbb{C}^{n}$. Then $U(n)$ acts on $F_{n}$ by left multiplication. That is, if $S\in U(n)$ then $SW=SV_{1}\subset SV_{2}\subset\cdots\subset SV_{n} = \mathbb{C}^{n}$. Start with $v_{1}$, a unit vector in $V_{1}$ such that $V_{1} = Span_{\mathbb{C}}\lbrace v_{1}\rbrace$. Next choose a unit vector $v_{2}$ in $V_{2}$ orthogonal to $V_{1}$ such that $V_{2} = Span_{\mathbb{C}}\lbrace v_{1},v_{2}\rbrace$. Having chosen unit vectors $ v_{1},\cdots, v_{k}$ with $V_{k}=Span_{\mathbb{C}}\lbrace v_{1},\cdots,v_{k}\rbrace$, choose further a unit vector $v_{k+1}$ in $V_{k+1}$ orthogonal to $V_{k}$ such that $V_{k+1} = Span_{\mathbb{C}}\lbrace v_{1},\cdots, v_{k+1}\rbrace$. Continuing this construction we obtain a set of orthonomal unit vectors $\lbrace v_{1}, \cdots, v_{n-1}\rbrace$ such that $V_{j} = Span_{\mathbb{C}}\lbrace v_{1},\cdots, v_{j}\rbrace$. Let $v_{n}$ be a unit vector in $V_{n}$ orthogonal to $V_{n-1}$. The set $\lbrace v_{1},v_{2},\cdots, v_{n}\rbrace$ is another orthonormal basis for $\mathbb{C}^{n}$. It is now a result of linear algebra that there is $n\times n$ matrix $S=(a_{ij})$ such that $v_{i}=\displaystyle\sum_{j=1}^{n}a_{ij}e_{j}$. Then $S\in U(n)$ and $SW_{0} = W$. Thus $U(n)$ acts transitively on $F_{n}$ as earlier claimed.\
The isotropy subgroup of $W$ is $ \lbrace A\in U(n): AV_{j} = V_{j}\rbrace$. In particular, this is a set of matrices $A\in U(n)$ such that $Av_{k} = \lambda_{k}v_{k}$ for some complex number $\lambda_{k}$ with $\mid \lambda_{k}\mid = 1$ since $A\in U(n)$. Thus $\lambda_{k} = e^{i\theta_{k}}\in U(1)$. Since this must be true for each $v_{j}$, $j=1, 2,\cdots, n$, the matrix $A$ must be of the form $A = diag(e^{i\theta_{1}},\cdots, e^{i\theta_{n}})$. Thus $F_{n}=U(n)/U(1)\times\cdots\times U(1)$\
Now let $\lbrace n_{1},\cdots, n_{k}\rbrace$ be a set of positive integers such that $n_{1}+n_{2}+\cdots +n_{k}=n$. A partial flag is an element $W=V_{1}\subset \cdots\subset V_{k}$ with $\dim V_{k} = n_{1}+\cdots +n_{k}$. We can visualize this as a sum of vector spaces. For example, let $Q_{1}, Q_{2},\cdots, Q_{n}$ be a set of subspaces of $\mathbb{C}^{n}$ with $\dim Q_{1}=n_{1}$ , $\dim Q_{2} = n_{2}\cdots \dim Q_{n-1} = n-1$.\
Set $$\begin{array}{cll}
V_{1} &=& Q_{1} \\
V_{2} &=& Q_{1}\oplus Q_{2} \\
&\cdots& \\
V_{n-1} &=& Q_{1}\oplus Q_{2}\oplus\cdots\oplus Q_{n-1} \\
\end{array}$$
Then $V_{1}\subset\cdots\subset V_{n-1}$ and $\dim V_{j}=n_{1}+\cdots +n_{j}$. The flag $W=V_{1}\subset\cdots\subset V_{k}$ with $\dim V_{k}=n_{1}+\cdots +n_{k}$ is called a partial flag.\
A generalized flag manifold in $\mathbb{C}^n$ is a set $F(n_{1},\cdots,n_{k})$ of all partial flags with $n_{1}+n_{2}+\cdots +n_{k} = n$. Throughout the discussion that follows, the Lie group $G$ will be compact and connected. We chose the unitary group $U(n)$ in order to illustrate that. (see Batubenge et.al. [@BBK85])
\(i) $U(n)$ is compact.\
This is because $U(n)$ is both closed and bounded in $GL(n,\mathbb{C})$. For, $U(n)=det^{-1}(S^1)=det(U(1)),$ where we denoted by $det$ the determinant function. Next, we show that $U(n)$ is bounded. For, pick $A=(\alpha_{ij})\in U(n)$. One has $\displaystyle\sum_{j}\alpha_{ij}\cdot \beta_{jk}=\delta_{ik}$, the Kronecker delta, with $\beta_{jk}=\bar{\alpha}_{kj}$. Hence, for $i=k$ one has $\displaystyle\sum_{j}\alpha_{ij}\cdot \bar{\alpha}_{ji}=1.$ Hence, $$\displaystyle\sum_{i=1}^{n}\bigg(\displaystyle\sum_{j=1}^{n}|\alpha_{ij}|^2\bigg)=n.$$ Now, $$||A||=\displaystyle\bigg(\sum_{i,j=1}^n|\alpha_{ij}|^2\bigg)^{\frac{1}{2}}=\sqrt{n}<\sqrt{n+1}.$$ Therefore, one has $A\in B(0,\sqrt{n+1})$, where $r=\sqrt{n+1}.$ Now one has that $A\in B(0,r)$ whenever $A\in U(n)$ so that $U(n)\subset B(0,r)$, with $r=\sqrt{n+1}$. Hence, $U(n)$ is bounded. Thus, $U(n)$ is compact.
\(ii) $U(n)$ is connected
Consider the action of $U(n)$ on $\mathbb{C}^{n}$ given by $(A,X)\mapsto AX$ for all\
$A\in U(n)$ and $X\in\mathbb{C}^{n}$. We have $$\| AX\|^{2}=(\bar{AX}^{T})(AX)=\bar{X}^{T}\bar{A}^{T}AX = \bar{X}^{T}X = \|X\|^{2}.$$ Thus, this action takes sets of the form\
$\lbrace(z_{1},\cdots,z_{n}):\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}+\cdots+\mid z_{n}\mid^{2} = 1\rbrace$ into sets of the same kind. In particular, the orbit of $e_{1}$ under this action is the unit sphere $S^{2n-1}$. The stabilizer of the same element $e_{1}$ are matrices of the form $$\left(\begin{array}{ccc}
1&0\\
0&A_{1}
\end{array}\right)$$
where $A_{1}\in U(n-1)$. Thus $S^{2n-1} = U(n)/U(n-1)$. But $S^{2n-1}$ is connected which implies that $U(n)$ is connected if and only if $U(n-1)$ is connected. Since $U(1) = S^{1}$ is connected, we conclude by induction on $n$ that $U(n)$ is connected.
The Lie algebra of $U(n)$ is the space of all skew-Hermitian matrices\
$\mathfrak{u}(n)=\lbrace A\in Mat_{n\times n}(\mathbb{C}): A+\bar{A}^{T} = 0\rbrace$. We now want to determine the orbits of adjoint representation of the Lie group $G = U(n)$ on its Lie algebra $\mathfrak{g} = \mathfrak{u}(n)$.\
Let $Ad:G\times\mathfrak{g}\rightarrow\mathfrak{g}$ be the action of $G$ on its Lie algebra $\mathfrak{g}$. Let $X\in\mathfrak{g}$, then the orbit of $X$ is given by\
$$\begin{array}{cll}
O_{X} &=&\lbrace Ad_{g}X:g\in G\rbrace\\
&=& \lbrace Y\in\mathfrak{g}:Y = gXg^{-1} ~\textrm{for~some~} g\in G\rbrace
\end{array}$$
This is a set of similar matrices since the action is by conjugation. Recall that every skew Hermitian matrix is diagonalizable and that all the eigenvalues of a skew Hermitian matrix are purely imaginary. This means that $X$ is $U(n)-$ conjugate to a matrix of the form $X_{\lambda} = Diag(i\lambda_{1},i\lambda_{2},\cdots,i\lambda_{n})$ for $\lambda_{j}\in\mathbb{R},\hspace{0.4cm} j=1,\cdots,n$. Since similar matrices have same eigenvalues, without loss of generality we can describe the adjoint orbit of $X$ to be the set of all skew Hermitian matrices with eigenvalues $i\lambda_{1},i\lambda_{2},\cdots,i\lambda_{n}$. Denote this set of eigenvalues by $\lambda$ and the orbit determined by the corresponding eigenspaces by $H(\lambda)$. Note that $H(\lambda)$ is a vector space since it is a closed subgroup of a linear group $GL(n,\mathbb{C})$.\
Case 1 : All the $n$ eigenvalues are distinct\
Let $x_{j}$ be the eigenvector corresponding to the eigenvalue $i\lambda_{j}$, then we have $gx_{j} = i\lambda_{j}x_{j}$. This gives a 1-dimensional subspace $P_{j}$ of $\mathbb{C}^{n}$ which is a line in the complex plane passing through the origin.\
Assuming $\lambda_{1}<\lambda_{2}<\cdots <\lambda_{n}$. Note that the eigenvectors corresponding to distinct eigenvalues are orthogonal. Now each element in $H(\lambda)$ has same eigenvalues $i\lambda_{1},\cdots, i\lambda_{n}$, however, it is only distinguished by its corresponding eigenspaces $P_{1},\cdots, P_{n}$. Thus for each $n-$tuple $(P_{1},P_{2},\cdots, P_{n})$ of complex lines in $\mathbb{C}^{n}$ which are pairwise orthogonal, there will be an associated element $h\in H(\lambda)$ and each element $h\in H(\lambda)$ determines a family of eigenspaces $(P_{1},P_{2},\cdots, P_{n})$.\
Let $(P_{1},\cdots, P_{n})\mapsto P_{1}\subset P_{1}\oplus P_{2}\subset\cdots\subset P_{1}\oplus P_{2}\oplus\cdots\oplus P_{n}=\mathbb{C}^{n}$ and define the vector space $V_{j}$ by $V_{j} = P_{1}\oplus\cdots\oplus P_{j}$. Then $W=V_{0}\subset V_{1}\subset\cdots\subset V_{n}=\mathbb{C}^{n}$ is a flag we have already seen and the totality of such flags $F_{n} = U(n)/U(1)\times\cdots\times U(n)$ is the flag manifold described earlier. There is a bijection from $H(\lambda)$ to $F_{n}$ which associates to each element $h\in H(\lambda)$ the subspaces $V_{j} = P_{1}\oplus\cdots\oplus P_{j}$ where $P_{j}$ is the eigenspace of $h$ corresponding to the eigenvalue $i\lambda_{j}$. This shows that the adjoint orbits are diffeomorphic to flag manifolds.\
Case 2: There are $k<n$ distinct eigenvalues.\
We again order the eigenvalues $\lambda_{1}<\cdots <\lambda_{k}$. Let $n_{1}, n_{2},\cdots, n_{k}$ be their multiplicities respectively. Let $Q_{j}$ be the eigenspace corresponding to the eigenvalue $i\lambda_{j}$. We assume that $\dim Q_{i} = n_{i},\hspace{0.4cm} i=1,\cdots,k$. Then the orbit of $X$ is again determined by the eigenspaces $Q_{1},\cdots, Q_{k}$. We form an increasing sequence ordered by inclusion as before
$(Q_{1}, Q_{2},\cdots, Q_{k})\mapsto Q_{1}\subset Q_{1}\oplus Q_{2}\subset\cdots\subset Q_{1}\oplus\cdots\oplus Q_{k} = \mathbb{C}^{n}$.\
Let $F(n_{1},n_{2},\cdots, n_{k})$ be the set of all such sequences. Then the orbit of $X$ is diffeomorphic to the homogeneous space $F(n_{1},\cdots, n_{k})=U(n)/(U(n_{1})\times\cdots\times U(n_{k}))$ which as we have already seen is a generalized flag manifold. For the variation proof of this (see [@Aud04 Proposition II.1.15]).
*(**Killing form)***\
Let $\mathfrak{g}$ be any Lie algebra. The Killing form of $\mathfrak{g}$ denoted by $B$, is a bilinear form $B:\mathfrak{g}\times \mathfrak{g}\longrightarrow \mathbb{R}$ given by $$B(X,Y)=tr(ad(X)\circ ad(Y)),\rm{for~all~}X,Y\in \mathfrak{g}$$ where $tr$ refers to the usual trace of a mapping.
We shall call $B$ the Killing form of the Lie group $G$ provided $\mathfrak{g}$ is the Lie algebra of the Lie group $G$, in which case the Killing form $B$ is $Ad$-invariant. That is, $$B(X,Y)=B(Ad(g)X,Ad(g)Y )$$ for all $g\in \mathfrak{g}$. (see [@Arv03 proposition 2.10]).
We further recall that by Cartan’s criterion for semisimplicity, a finite dimensional Lie group $G$ is said to be semisimple if its Killing form is nondegenerate (see [@Arv03 p. 34]). This criterion will play a key role in the next section. We would mention that the consequences of this criterion are as follows. Let $G$ be an $n$-dimensional semisimple Lie group. If $G$ is compact then its Killing form is negative definite. Moreover, if $G$ be an $n$-dimensional connected Lie group and the Killing form of $G$ is negative definite on $\mathfrak{g}$, then $G$ is compact and semisimple.
Adjoint orbits as symplectic manifolds
--------------------------------------
We have seen that the adjoint orbits of flag manifolds are determined by the eigenspaces corresponding to a set of eigenvalues $i\lambda_{1},\cdots, i\lambda_{k}$. Denote this set of eigenvalues by $\lambda$ and the orbit determined by the corresponding eigenspaces by $H(\lambda)$. Let $G=U(n)$ be a Lie group and $\mathfrak{g}=\mathfrak{u}(n)$ its Lie algebra. First note that the dimension of orbit $H(\lambda)$ is $n^{2}-n$ which is even.\
For $X\in\mathfrak{g}$ we have seen that if $x(t) = \exp{tX}$ is a curve in $G$ tangent to $X$ at $t = 0$, then $\xi(t)=Ad_{x(t)}\xi = Ad_{\exp{tX}}\xi$ is a curve in $H(\lambda)$ passing through $\xi\in\mathfrak{u}(n)$. Then the tangent vector to this curve at $t = 0$ is given by $$\begin{array}{ccc}
\xi'(t) = \frac{d}{dt}Ad_{\exp{tX}}\xi\mid_{t=0}~\textrm{or}~\xi'(0) = ad(X)\xi = [\xi , X]
\end{array}$$
We shall now construct a symplectic 2-form on the orbit $H(\lambda)$. Let $h$ be an element of $\mathfrak{u}(n)$. Define a map $$\omega_{h}:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{R};\quad
\omega_{h}(X,Y) = B(h,[X,Y])$$
where $B$ is the Killing form of $\mathfrak{g}$, the Lie algebra of $G$.
Let $\omega_{h}$ be as defined above. Then
\(i) $\omega_{h}$ is skew symmetric bilinear form on $\mathfrak{g}=\mathfrak{u}(n)$
\(ii) $\ker\omega_{h} = \lbrace X\in\mathfrak{u}(n):[h,X]=0\rbrace$
\(iii) $\omega_{h}$ is $G$-invariant. That is, for each $g\in G$ we have\
$\omega_{Ad(g)(h)}(Ad_{g}X,Ad_{g}Y)=\omega_{h}(X,Y)$
Part (i) follows from the properties of the Lie bracket. For part (ii) (see [@Ale96 p 19]). We prove part (iii). $$\begin{array}{cll}
\omega_{Ad(g)(h)}(Ad_{g}X,Ad_{g}Y) &=& B(Ad_{g}h,[Ad_{g}X,Ad_{g}Y])\\
&=& B(Ad_{g}h,[gXg^{-1},gYg^{-1}])\\
&=& B(Ad_{g}h,\lbrace gXYg^{-1}-gYXg^{-1}\rbrace)\\
&=& B(Ad_{g}h,g[X,Y]g^{-1})\\
&=& B(Ad_{g}h,Ad_{g}[X,Y])\\
&=& B(h,[X,Y])\\
&=& \omega_{h}(X,Y)
\end{array}$$
Now for $h\in\mathfrak{u}(n)$ we consider the orbit map $$\Phi_{h}:U(n)\rightarrow\mathfrak{u}(n);\quad g\mapsto ghg^{-1}$$
That is $$\Phi_{h}:U(n)\rightarrow H(\lambda)\subset\mathfrak{u}(n)$$ Then we have $ T_{I}\Phi_{h}:\mathfrak{u}(n)\rightarrow T_{h}H(\lambda)$. But the tangent space on the orbit is generated by the vector field $ad(X)\xi = [X,\xi]$, with $X,\xi\in\mathfrak{g}$. Define a 2-form $\Omega_{h}$ on $T_{h}H(\lambda)$ by the formula $$\Omega_{h}([h,X],[h,Y]) = \omega_{h}(X,Y),\hspace{0.4cm} \textrm{for}~ X,Y\in\mathfrak{u}(n)$$
The $\Omega_{h}$ defined above is a closed and nondegenerate 2-form on the orbit $H(\lambda)$.
First note that $\Omega_{h}$ does not depend on the choice of $X,Y\in\mathfrak{u}(n)$ since if $Z\in\ker\omega_{h}$ then we have $$\begin{array}{cll}
\Omega_{h}([h, X+Z],[h, Y+Z]) &=& \omega_{h}(X+Z,Y+Z) \\
&=& B(h,[X+Z,Y+Z]) \\
&=& B(h,[X,Y]+[X,Z]+[Z,(Y+Z)]) \\
&=& B(h,[X,Y])+B(h,[X,Z])+B(h,[Z,(Y+Z)])\\
&=& \omega_{h}(X,Y)+\omega_{h}(X,Z)+\omega_{h}(Z,(Y+Z)) \\
&=& \omega_{h}(X,Y) \\
&=& \Omega_{h}([h,X],[h,Y])
\end{array}$$
Thus, $\Omega_{h}$ is well defined. It is skew-symmetric bilinear form and $G$-invariant by the construction so it is smooth. Since the Killing form $B$ is nondegenerate, $\Omega_{h}$ is nondegenerate. We only have to show that it is closed.\
From the formula (1) in Berndt R. (see ) we have $$\begin{array}{cll}
d\omega(X,Y,Z) &=& L_{X}\omega(Y,Z)-L_{Y}\omega(X,Z)+L_{Z}\omega(X,Y) \\
&+& \omega(X,[Y,Z])-\omega(Y,[X,Z])+ \omega(Z,[X,Y]),
\end{array}$$ let $X,Y,Z\in\mathfrak{u}(n)$. Then
$$\begin{array}{cll}
d\Omega_{h}([h,X],[h,Y],[h,Z]) &=& d\omega_{h}(X,Y,Z) \\
&=& \lbrace L_{X}\omega_{h}(Y,Z)-L_{Y}\omega_{h}(X,Z)+L_{Z}\omega_{h}(X,Y)\rbrace \\
&+&\lbrace\omega_{h}(X,[Y,Z])-\omega_{h}(Y,[X,Z]) +\omega_{h}(Z,[X,Y])\rbrace
\end{array}$$ We now apply the Jacobi identity to each bracket given by the braces. The second bracket gives
$$\begin{array}{cll}
\omega_{h}(X,[Y,Z]) &-&\omega_{h}(Y,[X,Z]) + \omega_{h}(Z,[X,Y]) \\
&=& B(h,[X,[Y,Z]])-B(h,[Y,[X,Z]])+B(h,[Z,[X,Y]]) \\
&=& B(h,[X,[Y,Z]]-[Y,[X,Z]]+[Z,[X,Y]])
\end{array}$$
and the term in the bracket is zero by the Jacobi identity since $\mathfrak{u}(n)$ is a Lie algebra. To deal with the first bracket we have
$$\begin{array}{cll}
L_{X}\omega_{h}(Y,Z) &=& \omega_{h}(Z,[X,Y])-\omega_{h}(Y,[X,Z]) \\
L_{Y}\omega_{h}(X,Z) &=& \omega_{h}(Z,[Y,X])-\omega_{h}(X,[Y,Z]) \\
L_{Z}\omega_{h}(X,Y) &=& \omega_{h}(Y,[Z,X])-\omega_{h}(X,[Z,Y])
\end{array}$$
Substituting into the first bracket and simplifying gives
$$\begin{array}{cll}
L_{X}\omega_{h}(Y,Z) &-&L_{Y}\omega_{h}(X,Z)+L_{Z}\omega_{h}(X,Y) \\
&=& 2\left( \omega_{h}(X,[Y,Z])+\omega_{h}(Y,[Z,X])+\omega_{h}(Z,[X,Y])\right)
\end{array}$$ which again vanishes by Jacobi identity. Thus, $d\Omega_{h} = 0$ proving that $\Omega_{h}$ is indeed closed on the orbits of the adjoint action of the Lie group $G$ on its Lie algebra $\mathfrak{g}$.
Coadjoint Orbits
================
We now describe briefly the orbits of the coadjoint action of a Lie group $G$ on the dual of its Lie algebra. There are many references to this section such as Abraham and Marsden ([@Abr78]) as well as Vilasi ([@Vil01]).\
Consider the Lie group $G$ acting on itself by left translation $L_{g}:G\rightarrow G$ given by $h\mapsto gh$ for $g\in G$. This map is a diffeomorphism. So, by lifting of diffeomorphisms, induces a symplectic action on its cotangent bundle $$\Phi:G\times T^{*}G\rightarrow T^{*}G; \quad
(g,\alpha_{h})\mapsto \Phi(g,\alpha_{h})=L_{g^{-1}}^{*}(\alpha_{h})$$
This action has a momentum mapping which is equivariant with the coadjoint action. The momentum mapping of this action is given by $$\mu:T^{*}G\rightarrow\mathfrak{g}^{*}; \quad
\mu(\alpha_{g})\xi = \alpha_{g}(\xi_{G}(g))= \alpha_{g}(R_{g})_{*{e}}\xi = (R_{g}^{*}\alpha_{g})\xi$$ for all $\xi\in\mathfrak{g}$, where $\mathfrak{g}^{*}$ is the dual to the Lie algebra of $G$..\
That is, $\mu(\alpha_{g}) = R_{g}^{*}\alpha_{g}$. Every point $\beta\in\mathfrak{g}^{*}$ is a regular value of the momentum mapping $\mu$ (see [@Vil01 p 282]). So we have for each $\beta\in\mathfrak{g}^{*}$ $$\begin{array}{cll}
\mu^{-1}(\beta) &=& \lbrace \alpha_{g}\in T^{*}G: \mu(\alpha_{g}) = \beta\rbrace\\
&=& \lbrace \alpha_{g}\in T^{*}G: R_{g}^{*}\alpha_{g}\xi = \beta\cdot\xi~\textrm{for~all~} \xi\in\mathfrak{g}\rbrace
\end{array}$$ In particular, $R_{e}^{*}\alpha_{e}\xi = \beta\cdot\xi$ implying that $\alpha_{e} = \beta$. Denote this 1-form by $\alpha_{\beta}$ so that $$\begin{array}{ccc}
\alpha_{\beta}(e) = \beta \hspace{2cm} (1)
\end{array}$$
For $g\in G$, applying the right translation $R_{g^{-1}}^{*}$ to Equation (1) gives a right-invariant 1-form on $G$ $$\begin{array}{ccc}
\alpha_{\beta}(g) = R_{g^{-1}}^{*}\beta \hspace{2cm} (2)
\end{array}$$
But now for all $g\in G$ we have $$\begin{array}{ccc}
\mu(\alpha_{\beta}(g)) = \mu(\alpha_{g}) = R_{g}^{*}R_{g^{-1}}^{*}\beta = \beta.
\end{array}$$
Thus, Equation (2) defines all and only points of $\mu^{-1}(\beta)$. Since the action is defined by $\Phi(g,\alpha_{h})= L_{g^{-1}}^{*}(\alpha_{h})$, the isotropy subgroup of $\beta$ is $$\begin{array}{ccc}
G_{\beta} = \lbrace g\in G:L_{g^{-1}}^{*}(\alpha_{\beta})= \beta\rbrace
\end{array}$$
From the map $$L_{g^{-1}}^{*}: (h,\alpha_{\beta}(h)) \longrightarrow (gh,\alpha_{\beta}(gh))$$ we see that $G_{\beta}$ acts on $\mu^{-1}(\beta)$ by left translation on the base points. This action is proper (see [@Vil01 p 283]). Since $\beta$ is also a regular value of the momentum mapping $\mu$, then $\mu^{-1}(\beta)/G_{\beta}$ is a symplectic manifold. There is a diffeomorphism
$$\mu^{-1}(\beta)/G_{\beta}\simeq G\cdot\beta = \lbrace Ad_{g^{-1}}^{*}\beta :g\in G\rbrace\subset\mathfrak{g}^{*}~~\textrm{(see \cite [p~284]{Vil01})}$$ of the reduced space $\mu^{-1}(\beta)/G_{\beta}$ onto the coadjoint orbit of $\beta\in\mathfrak{g}^{*}$. Thus the coadjoint orbit $G\cdot\beta$ is a symplectic manifold. The symplectic 2-form is given by the Kirillov-Kostant-Souriau form $$\begin{array}{ccc}
\omega_{\beta}(\nu)(\xi_{\mathfrak{g}^{*}}(\nu),\eta_{\mathfrak{g}^{*}}(\nu)) = -\nu\cdot[\xi,\eta]~ \textrm{(see \cite [pp~302-303]{Abr78})},
\end{array}$$ where $\xi, \eta\in \mathfrak{g}$ and $\nu\in\mathfrak{g}^{*}$.
If $G$ is semisimple, it is known that in this case, $H^{1}(\mathfrak{g},\mathbb{R}) = 0$. (See [@Ale96 p 19]). Thus, if $\omega$ is closed then it is exact. So, there is a 1-form $\alpha\in\mathfrak{g}^{*}$ such that $d\alpha = \omega$. The 1-form $\alpha$ satisfies $d\alpha(X,Y) = \alpha([X,Y])$.\
Thus if the Lie group $G$ is semisimple, compact and connected, then we have the relation\
$\alpha([X,Y])=d\alpha(X,Y)=\omega(X,Y)=B([\xi,X],Y)=B(\xi,[X,Y])$, where $\alpha\in\mathfrak{g}^{*}$, $\omega$ a 2-form on the homogeneous space $G/H$, $B$ the Killing form on $G/H$ and $\xi,X,Y\in\mathfrak{g}$, the Lie algebra of $G$.
Main results
============
\[th 511:th 511\] Let $Ad:G\times\mathfrak{g}\rightarrow\mathfrak{g}$ be an adjoint action of an $n$-dimensional semisimple, compact, connected Lie group $G$ on its Lie algebra $\mathfrak{g}\cong T_{e}G$. Let $\mathfrak{g}^{*}$ be the dual of $\mathfrak{g}$. Then there is an $Ad^{*}$-equivariant isomorphism $B^{\flat}:\mathfrak{g}\rightarrow\mathfrak{g}^{*}$.
Let $$B^{\flat}:\mathfrak{g}\rightarrow\mathfrak{g}^{*};\quad X\mapsto B^{\flat}(X):\mathfrak{g}\rightarrow\mathbb{R},\quad Y\mapsto B^{\flat}(X)Y:= B(X,Y)$$ where $B$ is the Killing form. Then $B^{\flat}$ is linear since of for all $X,Y,Z\in\mathfrak{g}$ and using the fact that the Killing form $B$ is bilinear, we have\
$$\begin{array}{cll}
B^{\flat}(aX+bY)Z &=& B(aX+bY, Z) \\
&=& aB(X,Z)+bB(Y,Z) \\
&=& aB^{\flat}(X)Z+bB^{\flat}(Y)Z \\
&=& (aB^{\flat}(X)+bB^{\flat}(Y))Z.
\end{array}$$ Thus $B^{\flat}(aX+bY)=aB^{\flat}(X)+bB^{\flat}(Y)$.\
First, $B^{\flat}$ is injective. For, let $B^{\flat}(X) = B^{\flat}(Y)$. Then for all $Z\in\mathfrak{g}$ one has $$B^{\flat}(X)Z = B^{\flat}(Y)Z\Rightarrow B(X,Z)=B(Y,Z)\Rightarrow B(X-Y,Z)=0$$ and since the Killing form is nondegenerate we get $X = Y$. Next, $B^{\flat}$ is surjective since, first we note that $G$ is finite dimensional Lie group and $B^{\flat}$ is injective, thus $\ker B^{\flat} = \lbrace 0\rbrace$ implying that $\dim\ker B^{\flat} = 0$. But $\dim\ker B^{\flat}+\textrm{Rank} B^{\flat} = \dim\mathfrak{g}$, so we must have $\dim\mathfrak{g}^{*} = \dim \textrm{Im}B^{\flat} = \textrm{Rank} B^{\flat} = \dim\mathfrak{g}$. This shows that the map $B^{\flat}$ is surjective.
We now show that $B^{\flat}:\mathfrak{g}\rightarrow\mathfrak{g}^{*}$ is equivariant with respect to the adjoint action of $G$ on $\mathfrak{g}$ and the coadjoint action of $G$ on $\mathfrak{g}^{*}$. Define a map $$u:G\times\mathfrak{g}\rightarrow G\times\mathfrak{g}^{*};\quad (g,X)\mapsto (g,B^{\flat}X),$$ where $X\in\mathfrak{g}, g\in G$. That is, $u = Id_{G}\times B^{\flat}$. Then the following diagram commutes $$\xymatrixcolsep{4pc}\xymatrixrowsep{4pc}
\xymatrix{
G\times \mathfrak{g} \ar[d]_{Ad} \ar[r]^-{u} & G\times \mathfrak{g}^{*} \ar[d]^{Ad^{*}}\\
\mathfrak{g} \ar[r]^{B^{\flat}} & \mathfrak{g}^{*}}$$
Let $(g,X)\in G\times \mathfrak{g}$. Then for all $Y\in\mathfrak{g}$ we have\
$B^{\flat}(Ad_{g}X)Y = B(Ad_{g}X,Y) = B(Ad_{g^{-1}}\circ Ad_{g}X, Ad_{g^{-1}}Y)\\
= B(X, Ad_{g^{-1}}Y) = Ad_{g}^{*}B^{\flat}(X)(Y)$. The second and the third equalities is because the Killing form $B$ is Ad-invariant. That is, $$B^{\flat}(Ad_{g}X) = Ad_{g}^{*}B^{\flat}X.$$
Thus $B^{\flat}\circ Ad = Ad^{*}\circ B^{\flat}$ and $B^{\flat}$ is equivariant.\
Let $\pi_{\mathfrak{g}}:\mathfrak{g}\rightarrow \mathfrak{g}/G$ and $\pi_{\mathfrak{g}^{*}}:\mathfrak{g}^{*}\rightarrow \mathfrak{g}^{*}/G$ be the projection maps into the respective orbit spaces. Then, (see [@Mei03 p 10]) there is at most one manifold structure on $\mathfrak{g}/G$ respectively on $(\mathfrak{g}^{*}/G)$ such that $\pi_{\mathfrak{g}}$ respectively $(\pi_{\mathfrak{g}^{*}})$ are submersions. In fact note for example that the rank of $d\pi_{\mathfrak{g}}$ is equal to the dimension of its image and since $\dim\mathfrak{g}/G\leq\dim\mathfrak{g}$ then $\pi_{\mathfrak{g}}$ is a submersion. Since $B^{\flat}:\mathfrak{g}\rightarrow \mathfrak{g}^{*}$ is equivariant and $\pi_{\mathfrak{g}}$ and $\pi_{\mathfrak{g}^{*}}$ are submersions, the criterion of passage to quotients (see [@Abr78 p 264]) implies that it induces a smooth map $\hat{B^{\flat}}:\mathfrak{g}/G\rightarrow\mathfrak{g}^{*}/G$, $\hat{B^{\flat}}[X] = [\alpha]:=[B^{\flat}(X)]$, where $[X]$ is adjoint orbit through $X$ and $[\alpha]:=[B^{\flat}(X)]$ the corresponding coadjoint orbit through $B^{\flat}(X)=\alpha$. This gives the following diagram $$\xymatrixcolsep{4pc}\xymatrixrowsep{4pc}
\xymatrix{
G\times\mathfrak{g}\ar[d]_{Ad} \ar[r]^-{u} &G\times\mathfrak{g}^{*}\ar[d]^{Ad^{*}}\\
\mathfrak{g} \ar[d]_{\pi_{\mathfrak{g}}} \ar[r]^-{B^{\flat}} &\mathfrak{g}^{*}\ar[d]^{\pi_{\mathfrak{g}^{*}}}\\
\mathfrak{g}/G \ar[r]^{\hat{B^{\flat}}} &\mathfrak{g}^{*}/G}$$
Let $G$ be a compact, connected semisimple Lie group. Let $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^{*}$ the dual of $\mathfrak{g}$. Let $B^{\flat}$ be as in Theorem 5.0.1 and let $\hat{B^{\flat}}:\mathfrak{g}/G\rightarrow \mathfrak{g}^{*}/G$ be the map induced by passage to quotients as described above between adjoint and coadjoint orbit spaces. Then the map $\hat{B^{\flat}}$ is a local symplectomorphism.
The map $\hat{B^{\flat}}$ is well defined since if $\hat{B^{\flat}}([X])= [B^{\flat}(X)]$ and $\hat{B^{\flat}}([X])=[B^{\flat}(Y)]$, then $X$ and $Y$ belong to the same orbit $[X]$ so that there is some $g\in G$ such that $Y = gXg^{-1}$. Let $\alpha = B^{\flat}(X)$ and $\beta = B^{\flat}(Y)$. Then $\beta = B^{\flat}(Y) = B^{\flat}(gXg^{-1})=gB^{\flat}(X)g^{-1}=g\alpha g^{-1}$. This shows that $\alpha$ and $\beta$ belong to the same orbit. Therefore, $[B^{\flat}(X)]= [B^{\flat}(Y)]$ so that $\hat{B^{\flat}}$ is well defined.\
To show that $\hat{B^{\flat}}$ is injective we first have to show that the following diagram commutes.
$$\xymatrixcolsep{4pc}\xymatrixrowsep{4pc}
\xymatrix{
\mathfrak{g} \ar[d]_{\pi_{\mathfrak{g}}} \ar[r]^-{B^{\flat}} & \mathfrak{g}^{*} \ar[d]^{\pi_{\mathfrak{g}^{*}}}\\
\mathfrak{g}/G \ar[r]^{\hat{B^{\flat}}} & \mathfrak{g}^{*}/G}$$
The commuting of this diagram is now a consequence of the fact that $B^{\flat}$ is both an isomorphism and is equivariant with respect to the adjoint action and the coadjoint action. That is, $B^{\flat}\circ Ad_{g}(X) = Ad_{g}^{*}\circ B^{\flat}(X)$ for all $X\in\mathfrak{g}$ and for all $g\in G$. If we fix $X\in\mathfrak{g}$ and let $g$ run through all the elements of $G$ then on the left we get all the elements in the orbit through $X$ while on the right we get all the elements in the orbit through $B^{\flat}(X) = \alpha$. Consequently, we must have $\hat{B^{\flat}}\circ\pi_{\mathfrak{g}}(X)=\pi_{\mathfrak{g}^{*}}\circ B^{\flat}(X)$ for all $X\in\mathfrak{g}$.
We can now show that $\hat{B^{\flat}}$ is injective. The commuting of the above diagram says that $\hat{B^{\flat}}\circ\pi_{\mathfrak{g}} = \pi_{\mathfrak{g}^{*}}\circ B^{\flat}$. Suppose $\hat{B^{\flat}}([X]) = \hat{B^{\flat}}([Y])$, then $\pi_{\mathfrak{g}^{*}}\circ B^{\flat}(X) = \pi_{\mathfrak{g}^{*}}\circ B^{\flat}(Y)$. This implies that there is a $g\in G$ such that $B^{\flat}(Y)=gB^{\flat}(X)g^{-1}$. Then for all $Z\in\mathfrak{g}$ we have $B^{\flat}(Y)Z = gB^{\flat}(X)Z)g^{-1}\Rightarrow B(Y,Z) = gB(X,Z)g^{-1}\Rightarrow B(Y,Z)=B(X,Z)\Rightarrow Y=X$ so that $[X] = [Y]$ and $\hat{B^{\flat}}$ is injective. From the relation $\hat{B^{\flat}}\circ\pi_{\mathfrak{g}} = \pi_{\mathfrak{g}^{*}}\circ B^{\flat}$, the right hand side is a composition of smooth map and on the left $\pi_{\mathfrak{g}}$ is smooth, this then implies that $\hat{B^{\flat}}$ must be a smooth map.\
To show that $\hat{B^{\flat}}$ is a surjective map consider the following commutative diagram:
$$\xymatrixcolsep{4pc}\xymatrixrowsep{4pc}
\xymatrix{
\mathfrak{g} \ar[d]_{\pi_{\mathfrak{g}}} \ar[rd]^{\varphi} \ar[r]^-{B^{\flat}} & \mathfrak{g}^{*} \ar[d]^{\pi_{\mathfrak{g}^{*}}}\\
\mathfrak{g}/G \ar[r]^{\hat{B^{\flat}}} & \mathfrak{g}^{*}/G}$$
We have $\varphi = \pi_{\mathfrak{g}^{*}}\circ B^{\flat}$. But the right hand side is surjective since $B^{\flat}$ is an isomorphism hence bijective and $\pi_{\mathfrak{g}^{*}}$ is the projection which is surjective, this shows that $\varphi:\mathfrak{g}\rightarrow\mathfrak{g}^{*}/G$, $X\mapsto [B^{\flat}(X)]$ is surjective. But $\hat{B^{\flat}}$ is the factorization of $\varphi$ through $\mathfrak{g}/G$,(see also [@Ton64 pp 15-16]), that is, $\varphi = \hat{B^{\flat}}\circ \pi_{\mathfrak{g}}$, therefore, for any $ [B^{\flat}(X)]\in \mathfrak{g}^{*}/G$ there is $X\in\mathfrak{g}$ such that $\varphi(X)=[B^{\flat}(X)]$. This gives $\varphi(X)=\hat{B^{\flat}}(\pi_{\mathfrak{g}}(X)) = \hat{B^{\flat}}([X]) = [B^{\flat}(X)]$. Thus for each $[B^{\flat}(X)]\in\mathfrak{g}^{*}/G$ there is $[X]\in\mathfrak{g}/G$ such that $\hat{B^{\flat}}([X]) = [B^{\flat}(X)]$ which shows that $\hat{B^{\flat}}$ is bijective so that its inverse $(\hat{B^{\flat}})^{-1}$ exists. We must show that the inverse is smooth. But now $(\hat{B^{\flat}})^{-1}\circ\pi_{\mathfrak{g}^{*}}\circ B^{\flat} = \pi_{\mathfrak{g}}$ and since $\pi_{\mathfrak{g}}$ is smooth and the other two maps on the left are smooth, this forces $(\hat{B^{\flat}})^{-1}$ to be smooth. Therefore, $\hat{B^{\flat}}$ is a diffeomorphism. We shall now write $O_{X}$ for the orbit $[X]$ and $O_{B^{\flat}(X)}$ for the orbit $[B^{\flat}(X)]$.\
Let $O_{X}$ be the adjoint orbit through $X\in\mathfrak{g}$. Define a set map on $O_{X}$ as follows: Since each element in $O_{X}$ is of the form $gX$ for some $g\in G$, for any two points $y=hX$ and $z=gX$ in $O_{X}$ let $$f_{X}:O_{X}\rightarrow O_{X},\quad y\mapsto z;\quad f_{X}(y)=(gh^{-1})y=z.$$
Then $f_{X}$ maps all points of $O_{X}$ into points of $O_{X}$. Since $G$ is a group and $gh^{-1}$ is smooth for all $g,h\in G$, the map $f_{X}$ is smooth with smooth inverse $f_{X}^{-1}=hg^{-1}$.\
In a similar way define a set map $k_{\alpha}$ on the coadjoint orbit $O_{B^{\flat}(X)}= O_{\alpha}$ corresponding to the adjoint orbit $O_{X}$. That is, $$k_{\alpha}:O_{\alpha}\rightarrow O_{\alpha},\quad \beta\mapsto\gamma;\quad k_{\alpha}(\beta)=(rs^{-1})\beta=\gamma,$$
where $\alpha = B^{\flat}(X), \beta = s\alpha, \gamma = r\alpha$ and $r,s\in G$. Let $\hat{B^{\flat}}_{X}$ be the restriction of $\hat{B^{\flat}}$ to a small neighborhood of the point $O_{X}$. Then $$\begin{array}{ccc}
k_{\alpha}\circ \hat{B_{X}^{\flat}}\circ f_{X}^{-1}:O_{X}\rightarrow O_{B^{\flat}(X)} = O_{\alpha}\hspace{2cm}(1)
\end{array}$$ maps points of $O_{X}$ into points of $O_{B^{\flat}(X)}=O_{\alpha}$ and it is smooth since it is a composition of smooth maps. It is known that the coadjoint orbit is symplectic. Let $\hat{\omega}$ be the Kirillov-Kostant-Souriau form on the coadjoint orbit $O_{B^{\flat}(X)}=O_{\alpha}$ which is known to be symplectic. Then for all $Y,Z\in \mathfrak{g}$ and $r,s\in G$ we have: $$\begin{array}{cll}
k_{\alpha}^{*}\hat{\omega}(Y,Z) &=& \hat{\omega}(k_{\alpha{*}}Y,k_{\alpha{*}}Z)\\
&=& \hat{\omega}\left((rs^{-1})_{*}Y,(rs^{-1})_{*}Z\right)\\
&=& \hat{\omega}\left(r_{*}(s_{*}^{-1}Y),r_{*}(s_{*}^{-1}Z)\right)\\
&=& \hat{\omega}(r_{*}Y,r_{*}Z)\\
&=& \hat{\omega}(Y,Z)
\end{array}$$ since $Y,Z\in\mathfrak{g}$ are left invariant. Thus $k_{\alpha}^{*}\hat{\omega} = \hat{\omega}$. By similar calculations, for any 2-form $\hat{\Omega}$ on the adjoint orbit $O_{X}$ we must have $f_{X}^{*}\hat{\Omega} = \hat{\Omega}$.
Consider now the pull back of the form $\hat{\omega}$ by the map in (1), $\left(k_{\alpha}\circ \hat{B_{X}^{\flat}}\circ f_{X}^{-1}\right)^{*}\hat{\omega}$. We have $$\begin{array}{cll}
\left(k_{\alpha}\circ \hat{B_{X}^{\flat}}\circ f_{X}^{-1}\right)^{*}\hat{\omega} &=& (f_{X}^{-1})^{*}\circ (\hat{B_{X}^{\flat}})^{*}\circ k_{\alpha}^{*}\hat{\omega}\\
&=& (f_{X}^{-1})^{*}\circ (\hat{B_{X}^{\flat}})^{*}\hat{\omega}
\end{array}$$
But $\hat{B_{X}^{\flat}}$ is a smooth map so that it pulls back a 2-form into a 2-form. Thus $(\hat{B_{X}^{\flat}})^{*}\hat{\omega}$ is a 2-form. We now check if the 2-form $(\hat{B_{X}^{\flat}})^{*}\hat{\omega}$ is symplectic, that is, if it is closed and nondegenerate. Since a pull back commutes with exterior derivative we have $d\hat{B_{X}^{\flat{*}}}\hat{\omega} = (\hat{B_{X}^{\flat}})^{*}d\hat{\omega} = 0$ since $\hat{\omega}$ is closed. Thus the 2-form $(\hat{B_{X}^{\flat}})^{*}\hat{\omega}$ is closed. For non degeneracy, if $(\hat{B_{X}^{\flat}})^{*}\hat{\omega}(Y,Z)=0$ for all $Z\in\mathfrak{g}$ then $\hat{\omega}(d\hat{B^{\flat}}_{X}(Y),d\hat{B^{\flat}}_{X}(Z))=0$ for all $Z\in\mathfrak{g}$. Since $\hat{\omega}$ is symplectic, $\hat{\omega}(d\hat{B^{\flat}}_{X}(Y),d\hat{B^{\flat}}_{X}(Z))=0$ for all $Z\in\mathfrak{g}$ implies that $d\hat{B^{\flat}}_{X}(Y) = 0$. But $d\hat{B^{\flat}}$ is a linear isomorphism so that $d\hat{B^{\flat}}_{X}(Y)=0\Rightarrow Y\in\ker{d\hat{B^{\flat}}} = \lbrace 0\rbrace$ which gives $Y=0$. Thus $(\hat{B_{X}^{\flat}})^{*}\hat{\omega}(Y,Z)=0$ for all $Z\in\mathfrak{g}$ implies that $Y=0$ and $(\hat{B^{\flat}})_{X}^{*}\hat{\omega}$ is nondegenerate. This proves that $\hat{B^{\flat}}$ is a symplectic map orbitwise. So $\hat{B^{\flat}}$ can be used to pull back a symplectic form on a coadjoint orbit space to a symplectic form on an adjoint orbit space. Since the action is transitive by assumption, the orbit spaces reduce to only one each. In this case, we have proved that they are symplectomorphic spaces. More details will appear elsewhere.
Acknowledgements
================
Augustin Batubenge is grateful to Professor François Lalonde for his financial support and for hosting him as an invited researcher in the Canada chair of mathematics during the time of writing this paper at the University of Montréal from 2017 to 2019.\
Wallace Haziyu acknowledges the financial support from the International Science Program, ISP, through East African Universities Mathematics Project, EAUMP and more particularly to Professor Lief Abrahamson for his significant input in funding his research.
[99]{} R. Abraham and J. E. Marsden. *Foundations of Mechanics*, Second Edition. Addison-Wesley Publishing Company, Inc., New York 1978. D.V. Alekseevsky. *Flag Manifolds*. 11 Yugoslav Geometrical Seminar, Divčibare 10-17 Oct. 1996, 3-35. A. Arvanitoyeorgos. *An Introduction to Lie Groups and the Geometry of Homogeneous Spaces*,(Vol. 22). American Mathematical Society, Rhode Island 2003. M. Audin. *Torus Action on Symplectic Manifolds*. [Second Revised Version]{}. Birkhäuser Verlag, Berlin 2004. T. Batubenge, T. Bukasa, M. Kasongo. *Une Structure Fibrée sur le Groupe Unitaire $U(n)$*. Revue de Pédagogie Appliquée, Vol.3, No.2; Presses Universitaires du Zaïre, Kinshasa 1985. A. Batubenge and W. Haziyu. Symplectic Affine Action and Momentum with Cocycle. In *Mathematical Structures and Applications* by Toka D. and Toni B., Springer Nature, Switzerland 2018.
R. Berndt. *An Introduction to Symplectic Geometry*. Graduate Studies in Mathematics, (Vol. 26), AMS, Providence, Rhode Island 2001.
W. M. Boothby. *An Introduction to Differentiable Manifolds and Riemannian Geometry*. Revised Second Ed., Academic Press Inc, San Diego 2003.
P. Crooks. *Complex Adjoint Orbits in Lie Theory and Geometry*, Expo. Math.(2018),https://doi.org/10.1016/j.exmath.2017.12.001.
E. Meinrenken. *Group Actions on Manifolds*, Lecture Notes. http://www.math.toronto.edu-mein-teaching-action.pdf (2003).
Philippe Tondeur. *Introduction to Lie Groups and Transformation Groups*, (Second Edition). Springer-Verlag, Berlin 1964.
G. Vilasi. *Hamiltonian Dynamics*. World Scientific Publishing Co. Pte Ltd, Singapore 2001.
F. W. Warner. *Foundations of Differentiable Manifolds and Lie Groups*. Springer-Verlag, New York 1983.
[**Authors**]{}
- Augustin Tshidibi Batubenge\
Department of Mathematics and Statistics Université de Montréal and University of Zambia\
email: a.batubenge@gmail.com\
- Wallace Mulenga Haziyu\
Department of Mathematics and Statistics\
University of Zambia\
P.O. Box 32379 Lusaka, Zambia\
email: whaziyu@unza.zm
[^1]: Corresponding author
|
---
abstract: 'This paper deals with model order selection in context of correlated noise. More precisely, one considers sources embedded in an additive Complex Elliptically Symmetric (CES) noise, with unknown parameters. The main difficultly for estimating the model order lies into the noise correlation, namely the scatter matrix of the corresponding CES distribution. In this work, to tackle that problem, one adopts a two-step approach: first, we develop two different methods based on a Toeplitz-structured model for estimating this unknown scatter matrix and for whitening the correlated noise. Then, we apply Maronna’s $M$-estimators on the whitened signal to estimate the covariance matrix of the “decorrelated” signal in order to estimate the model order. The proposed methodology is based both on robust estimation theory as well as large Random Matrix Theory, and original results are derived, proving the efficiency of this methodology. Indeed, the main theoretical contribution is to derive consistent robust estimators for the covariance matrix of the signal-plus-correlated noise in a large dimensional regime and to propose efficient methodology to estimate the rank of signal subspace. Finally, as shown in the analysis, these results show a great improvement compared to the state-of-the-art, on both simulated and real hyperspectral images.'
author:
- |
\
\
\
bibliography:
- 'bibliographieeug.bib'
title: Robust Model Order Selection in Large Dimensional Elliptically Symmetric Noise
---
Model order selection, RMT, correlated noise, CES distribution, robust estimation.
Introduction
============
order selection is a challenging issue in signal processing for example in wireless communication [@Julia13], array processing [@Nadler10], or other related problems [@Ottersten92], [@Nadler11]. Classically, for a white noise, statistical methods such as the one based on the application of the information theoretic criteria for model order selection, allow to estimate the model order thanks to eigenvalues and eigenvectors of the covariance matrix of the signal. This is the case of the Akaike Information Criterion (AIC) [@Akaike74] or the Minimum Description Length (MDL) [@Rissanen78; @schwarz78]. Other examples are the problem of source localization [@schmidt86], where the estimation of the signal subspace is done by the estimation of the eigenvalues of the covariance matrix, channel identification [@Meraim97], waveform estimation [@Liu96] and many other parametric estimation problems. Though, all these methods are no more relevant for large dimensional and correlated data. Even if particular cases have been studied for correlated signals as in [@Bai98] or [@Silverstein95], these methods can not be generalized for all kind of signals and a whitening step, when possible, can not be systematically set up [@Cawse11]. Moreover, the commonly used statistical model for this problem has not the same matrix properties when the data are large and when they are not: the covariance matrix is not correctly apprehended and the methods fail to estimate the model order, for example in [@Combernoux14], in [@Nadler9] or in [@Farsi15]. In the field of model order selection for large dimensional regime, that is when the number of snapshots $N$ and the dimension of the signal $m$ tend to infinity with a constant positive ratio, and for white or whitened noise, the Random Matrix Theory (RMT) proposes methods to estimate the model order selection relying on the study of the largest eigenvalues distribution of the covariance matrix [@Couillet13]. The RMT introduces new methodologies which correctly handle the statistical properties of large matrices thanks to a statistical and probability approach: see [@Couillet11] for a review of this theory, [@Kritchman9] for a general detection algorithm, [@Hachem13] for an adapted MUSIC detection algorithm, [@Pascal16] for applications to radar detection and [@Cawse10] for an application on hyperspectral imaging. When the noise is spatially correlated, it is still possible to estimate the model order for example by evaluating the distance between the eigenvalues of the covariance matrix [@Vinogradova13]. Nevertheless, these methods require a threshold that has no explicit expression and can be fastidious to obtain [@Terreaux15]. In addition to the problem of the large dimension and the correlation, another recurrent problem in signal processing is the non-Gaussianity of the noise. To be less dependent of the noise statistic, that is for the model order selection not to be degraded with a noise more or less sightly different than targeted, robust methods for model order selection have been developed [@Breloy16] in hyperspectral imaging [@Halimi16]. Nevertheless, these methods depend on unknown parameters [@Julia13] or are not adapted for large data. Recent results in RMT enable to correctly estimate the covariance matrix for textured signals [@Couillet15b]. But the correlation matrix is assume to be known and the signal is whitened before processed.\
In this works, one considers a Complex Elliptically Symmetric (CES) noise. The CES distributions modelling is often exploited in signal processing, because of its flexibility, that is the ability to model a large panel of random signals. The signal can be split in two parts: a texture and a speckle. They are rather often used in various fields, as in [@Ovarlez11] for hyperspectral imaging, or [@Gini97] for radar clutter echoes modelling. This article deals with large dimansional non-Gaussian data, and proposes a robust method to estimate the model order. The robustness of our method comes from the robust estimation of the covariance matrix, with a Maronna $M$-estimator [@Maronna76] which assigns different weights according to the Mahalanobis distance between the signals received by the different sensors. It is a generalization of [@Couillet15b] and [@Vinogradova14] to the case of left hand side correlation (with an unknown covariance matrix). Moreover, this article proposes a new algorithm to estimate the model order.\
In a first part, an estimator for the correlation matrix is presented: the toeplitzified Sample Covariance Matrix (SCM), that is, the SCM enforced to be of Toeplitz form [@Vallet14]. Indeed, as the covariance matrix is supposed to be Toeplitz, the SCM is toeplitzified as in [@Couillet15c] to enhance the estimation. The data are then whitened with this Toeplitz matrix and a robust Maronna $M$-estimator of the covariance matrix is then used after the data whitening. This robust estimation is studied and a threshold on its eigenvalues can be derived to select the model order. A second part presents the same procedure for the toeplitzified Fixed-Point (FP) estimator [@Tyler87] and [@Pascal8]. The third part presents some simulations on both simulated and real hyperspectral images. Proofs of the main results are postponed in the appendices.\
*Notations*: Matrices are in bold and capital, vectors in bold. Let $\mathbf{X}$ be a square matrix of size $s\times s$, $(\lambda)_{i}(\mathbf{X})$, $i \in \rrbracket 1,...,s \llbracket$, are the eigenvalues of $\mathbf{X}$. $Tr(\mathbf{X})$ is the trace of the matrix $\mathbf{X}$. $\left\Vert \mathbf{X} \right\Vert$ stands for the spectral norm. Let $\mathbf{A}$ be a matrix, $\mathbf{A}^T$ is the transpose of $\mathbf{A}$ and $\mathbf{A}^H$ the Hermitian transpose of $\mathbf{A}$. $\mathbf{I}_n $ is the $n \times n$ identity matrix. For any $m-$vector $\mathbf{x}$, $\mathcal{L} : \, \mathbf{x} \mapsto \mathcal{L}(\mathbf{x})$ is the $m \times m$ matrix defined as the Toeplitz operator : $\left( [\mathcal{L} (\mathbf{x})]_{i,j} \right) _{i\leq j} = x_{i-j}$ and $\left( [\mathcal{L} (\mathbf{x})]_{i,j} \right)_{i>j} = x_{i-j}^\ast$. For any matrix $\mathbf{A}$ of size $m \times m$, $\mathcal{T}(\mathbf{A})$ represents the matrix $\mathcal{L}(\check{\mathbf{a}})$ where $\check{\mathbf{a}}$ is a vector for which each component $\check{\mathbf{a}}_{i , \, 0<i<m-1}$ contains the sum of the $i-$th diagonal of $\mathbf{A}$ divided by $m$. For $x \in \mathds{R}$, $\delta_x$ is the Dirac measure at $x$. For any complex $z$, $z^{\star}$ is the conjugate of $z$. The notation *dist* stands for the distance associated to the $L_1$ norm. $\mathrm{supp}$ is the support of a set. Eventually, $\mathcal{R}e$ and $\mathcal{I}m$ stand respectively for the real and the imaginary part for a complex number. The notation $\overset{a.s.}{\longrightarrow}$ means “tends to almost surely”.
Model and Assumptions {#sec::2}
=====================
This section introduces the model as well as the general assumptions needed to derive the results. Let us consider the following general sources-plus-noise model. Let $\mathbf{Y} = [\mathbf{y}_0, \ldots, \mathbf{y}_{N-1}] $ be a matrix of size $m$ $\times$ $N$, containing $N$ observations $\left\{\mathbf{y}_i\right\}_{i\in \llbracket 0,N-1\rrbracket}$ of size $m$, constituted of $p$ mixed sources corrupted with an additive noise: $$\mathbf{y}_i = \displaystyle \sum_{j=1}^p s_{i,j} \, \mathbf{m}_j + \sqrt{\tau_i} \,\mathbf{C}^{1/2} \, \mathbf{x}_i\, , \, \hspace{0.3cm} i\in \llbracket 0,N-1 \rrbracket \, ,
\label{modele}$$ which can be rewritten as $$\mathbf{Y} = \mathbf{MS} + \mathbf{C}^{1/2}\,\mathbf{ X}\,\mathbf{ T}^{1/2} \, ,
\label{modele2}$$ where the $\left\{\tau_i\right\}_{i\in \llbracket 0, N-1 \rrbracket}$ are positive random variables, and $\mathbf{T}$ is the $N \times N$-diagonal matrix containing the $\left\{\tau_i\right\}_{i\in \llbracket 0, N-1 \rrbracket}$. Moreover, the $m \times p$ matrix $\mathbf{M}$ with elements $M_{i,j} = (\mathbf{M})_{i,j} = (\mathbf{m}_j)_i $ is referred to as the mixing matrix and contains the $p$ vectors of the sources. In this work, the additive noise is modelled thanks to the general family of Complex Elliptically Symmetric (CES) distributions [@Kelker70; @Yao73] (see also [@Ollila12] for more details on CES as well as their use in signal processing). Thus, each component of the noise is characterized by a random vector $\mathbf{x}_i$ uniformly distributed on a sphere times an independent positive random scalar $\tau_i$ with unspecified probability distribution function. The left hand side spectral correlation is handled by the scatter matrix $\mathbf C$.
Each element $s_{i,j}$ of the $p \times N$ matrix $\mathbf{S}$ corresponds to the power variation of each source in the received vector. This matrix can be written $\mathbf{S} = \boldsymbol{\delta}^H \, \mathbf{\Gamma}^{1/2}$ where $\boldsymbol{\delta}$ is a $N \times p$ random matrix, independent of $\mathbf{X}$, whose elements are normally distributed with zero-mean and unit variance. $\mathbf{\Gamma}$ is a $N \times N$ Hermitian covariance matrix. Eventually, $\mathbf{C} = \mathcal{L} \left( [c_{0},\ldots,c_{m-1}]^T\right)$ is a $m \times m$ Hermitian nonnegative definite Toeplitz matrix: $$\mathbf{C}=
\begin{pmatrix}
c_0 & c_1 & ... & c_{m-1} \\
c_{1}^{\star} & c_0 &... & c_{m-2} \\
...\\
c_{m-1}^{\star} & c_{m-2}^{\star} & ... & c_{0}
\end{pmatrix} \, .$$
In the sequel, we will consider the following assumptions:\
[**Assumption 1**]{}: One assumes the usual random matrix regime, *i.e.*: $N \rightarrow \infty$, $m \rightarrow \infty $ and $c_N = \displaystyle \frac{m}{N} \rightarrow c > 0$,\
[**Assumption 2**]{}: for matrices $\mathbf{C}$, $\mathbf{T}$ and $\mathbf{X}$ of equation , one has:
- $\mathrm{dist}(\lambda_i(\mathbf{C}),\mathrm{supp}(\nu)) \rightarrow 0$ with $\nu$ the limit of $ \frac{1}{N} \,\sum \delta_{\lambda_i(\mathbf{C})}$ when $N \rightarrow \infty$,
- $\left\{c_k\right\}_{k\in \llbracket 0, m-1 \rrbracket}$ are absolutely summable coefficients, such that $c_0 \neq 0$,
- The random measure $\mu_N = \displaystyle \frac{1}{N} \displaystyle \sum_{i=1}^{N} \delta_{\tau_i} $ satisfies $\displaystyle\int \tau \, \mu_N(d\tau) \overset{a.s.}{\longrightarrow} 1$,
- $\mathbf{X}$ is a white noise, with independent and identically distributed entries with zero-mean and with unit variance,
- $\left\Vert \mathbf{\Gamma} \right\Vert < \infty $ and $\left\Vert \mathbf{M} \right\Vert < \infty$.
[**Assumption 3**]{}:\
$\bullet$ In each column of $\mathbf{M}$, the coefficients are absolutely summable that is, for all fixed $j$, $\displaystyle \sum_{i=1}^m \vert M_{i,j} \vert < \infty$. This is a common assumption in several applications and especially in hyperspectral imaging.\
$\bullet$ $\mathbf{\Gamma}$ has coefficients absolutely summable.\
[**Assumption 4**]{}: Let $[\mathbf{Y}]_{i,j} = [\mathbf{y}_i]_j$, then the coefficients $[\mathbf{y}_i]_j$ are absolutely summable, that is, for a fixed $i$, $\displaystyle \sum_{j} \left\vert [\mathbf{y}_i]_j \right\vert$ exists.
Model Order Selection: a Gaussian Approach {#sec::3}
==========================================
In this section, the consistency of the SCM is used to whiten the signal and to estimate the model order thanks to a Maronna $M$-estimator. The step which consists in directly evaluating the model order with a Maronna $M$-estimator has been already studied in [@Couillet15b] for the special case of [*spiked*]{} model with CES white noise. In this work, one considers the more challenging problem of correlated CES noise.
Whitening Step
--------------
The noise being correlated, one proposes here to whiten it using the Toeplitz structure of the noise covariance matrix. As a reminder, the model under consideration is the following: $$\mathbf{Y} = \mathbf{M} \, \boldsymbol{\delta}^H \, \mathbf{\Gamma}^{1/2} + \mathbf{C}^{1/2}\,\mathbf{ X}\,\mathbf{ T}^{1/2}\, .
\label{modele3}$$
Let $\mathbf{Y}$ be written as $\mathbf{Y} = [\mathbf{y}_0, ... , \mathbf{y}_{N-1}]$ where $\mathbf{y}_j = \left(y_{0,j}, y_{1,j}, \ldots,y_{m-1,j}\right)^T$, $j\in \llbracket 0, N-1 \rrbracket$ and let $\widetilde{\mathbf{C}}_{SCM}$ be a biased Toeplitz estimation of the covariance matrix $\mathbf{C}$ such that : $$\left[ \tilde{\mathbf{C}}_{SCM} \right]_{i,j} = \left[ \mathcal{L} (\tilde{\mathbf{c}}_{SCM})\right]_{i,j}
\label{eq1}$$
with $ \tilde{c}_{SCM,k} = \frac{1}{m\,N} \, \displaystyle \sum_{i=0}^{m-1} \sum_{j=0}^{N-1} y_{i,j} \, y_{i+k,j}^\star \, \mathds{1}_{0 \leq i+k < m}$
It can be equivalently stated as $\widehat{\mathbf{C}}_{SCM} = \displaystyle\frac{1}{N} \mathbf{Y} \, \mathbf{Y}^H$ and $\widetilde{\mathbf{C}}_{SCM} = \mathcal{T}(\widehat{\mathbf{C}}_{SCM})$. The following theorem establishes the consistency of $\widetilde{\mathbf{C}}_{SCM}$.
Under above assumptions, one has the following result: $$\left\Vert \widetilde{\mathbf{C}}_{SCM} - \mathds{E}[\tau] \,\mathbf{C} \right\Vert \overset{a.s.}{\longrightarrow} 0 \, .
\label{eqth1}$$ The covariance matrix defined by $\check{\mathbf{C}}_{SCM} = \displaystyle \frac{1}{\mathds{E}(\tau)} \, \widetilde{\mathbf{C}}_{SCM}$ characterizes the biased Toeplitz estimation of $\mathbf{C}$.
The complete proof is in Appendix A.
This estimator is then used to whiten the samples: $$\check{\mathbf{Y}}_{wSCM} = \check{\mathbf{C}}_{SCM}^{-1/2} \,\mathbf{M} \, \boldsymbol{\delta}^H \mathbf{\Gamma}^{1/2} + \check{\mathbf{C}}_{SCM}^{-1/2} \, \mathbf{C}^{1/2}\,\mathbf{X} \,\mathbf{ T}^{1/2}\, .
\label{scm_blanc}$$ In practice, $\mathds{E}(\tau)$ can be empirically estimated or is supposed to be equal to $1$.
Estimation of the covariance matrix
-----------------------------------
Once the signal $\mathbf{Y}$ has been whitened, a robust estimation of the (unobservable) covariance matrix $\mathds{E}\left[ \mathbf{X}\,\mathbf{X}^H \right]$ can be performed through the samples $\check{\mathbf{Y}}_{wSCM}$. This estimation is said to be robust in the sense that it can annihilate the high values of the texture $\tau$, which can alter the structure quality of the estimated covariance matrix. The chosen estimator is a Maronna’s $M$-estimator [@Maronna76], which gives good performances for CES signals. This robust estimation of the scatter matrix is therefore a fixed-point estimator noted $\check{\mathbf{\Sigma}}_{SCM}$ and defined through $\check{\mathbf{Y}}_{wSCM} = [\check{\mathbf{y}}_{wS0}, ...,\check{\mathbf{y}}_{wS \, N-1} ]$ as the unique solution of the following equation: $$\mathbf{\Sigma} = \frac{1}{N} \displaystyle \sum_{i=0}^{N-1} u\left(\frac{1}{m} \, \check{\mathbf{y}}_{wSi}^H \,\mathbf{\Sigma}^{-1} \,\check{\mathbf{y}}_{wSi} \right)\, \check{\mathbf{y}}_{wSi} \check{\mathbf{y}}_{wSi}^H \, ,
\label{eqsigma1}$$ under
- $u$: \[0, $+\infty$) $\mapsto$ (0, $+\infty$) nonnegative, continuous et non-increasing function derived thanks to the probability distribution function of the CES (for the complete calculus, see [@Mahot12]),
- $\phi: x \mapsto x \, u(x)$ increasing and bounded, with $ \lim\limits_{\substack{x \to \infty}} \phi(x) = \Phi_{\infty} > 1$,
- $\underset{N \longrightarrow \infty}{\text{lim}} \, c_N < \,\Phi_{\infty}^{-1}$.
Next step consists in evaluating the rank of the signal subspace from this matrix.
Model order selection
---------------------
The mean idea is to study the eigenvalues distribution of this Maronna $M$-estimator to find the model order or the number of sources. Indeed, in a non-RMT regime, that is if Assumption 1 is not satisfied, and in the case of a white Gaussian noise, it is possible to set a threshold such that no eigenvalues of the noise can be found upon. If eigenvalues are found beyond this threshold, they are due to sources. Here, under Assumption 1 and thanks to [@Bai98] in the case of a white Gaussian noise plus an additive signal, no eigenvalues outside the support of the Marchenko-Pastur law can belong to the noise. However, due to the presence of the texture matrix $\mathbf{T}$, some eigenvalues could exist upon the right edge of the Marchenko-Pastur distribution support. A more precise threshold can then be derived to ensure that no eigenvalue found upon are due to the noise. However, it does not ensure that all the sources eigenvalues will be located beyond this threshold. Indeed, this depends of the sources Signal to Noise Ratio (SNR).\
The proposed estimator $\check{\mathbf{\Sigma}}_{SCM}$ has so to be analysed for CES distribution. However, some characteristics such as its eigenvalues distribution can not be easily and theoretically studied when both $m$ and $N \rightarrow \infty $ as the term $ u\left(\displaystyle \frac{1}{m} \, \mathbf{\check{y}}_{wSi}^H\, \check{\mathbf{\Sigma}}_{SCM}^{-1} \,\mathbf{\check{y}}_{wSi} \right)$ is not independent on $\mathbf{\check{y}}_{wSi}$. To fill this gap, the following white model [@Couillet15b] is considered : $$\mathbf{Y}_w = [\mathbf{y}_{w0}, \ldots, \mathbf{y}_{w\,N-1} ] = \mathbf{C}^{-1/2} \,\mathbf{M} \, \boldsymbol{\delta}^H \mathbf{\Gamma}^{1/2} + \mathbf{X}\, \mathbf{T}^{1/2} \, .
\label{scm_normal}$$
Notice that the difference between models and lies in the empirical whitening. Then,
$$\hat{\mathbf{S}} \overset{\triangle}{=} \frac{1}{N} \displaystyle \sum_{i=0}^{N-1} v\left( \tau_i \, \gamma \right) \, \mathbf{y}_{wi} \, \mathbf{y}_{wi}^{H} \, ,
\label{matrixS}$$
which can be rewritten as $\hat{\mathbf{S}} = \mathbf{Y}_w \, \mathbf{D}_{\nu} \, \mathbf{Y}_w^H$ where $\mathbf{D}_{\nu}$ a diagonal matrix containing the $\left\{v(\tau_i \,\gamma)\right\}_i$, where:
- $g: x \mapsto \displaystyle\frac{x}{1-c \, \phi(x)}$,
- $v: x \mapsto u \circ g^{-1}(x)$, $\psi: x \mapsto x\, v(x)$, with $\displaystyle\lim_{x\rightarrow \infty} \psi(x) = \displaystyle \frac{\Phi_{\infty}}{1 - c \, \Phi_{\infty}}$,
- $\gamma$ is the unique solution, if defined, of the equation in $\gamma$: $1 = \displaystyle \frac{1}{N} \, \sum_{i=1}^N \frac{\psi\left( \tau_i \, \gamma \right)}{1+ c \, \psi( \tau_i \,\gamma)}$.
Moreover, it is proved in [@Couillet15b] that: $$\left\Vert \hat{\mathbf{\Sigma}} - \hat{\mathbf{S}} \right\Vert \overset{a.s.}{\longrightarrow} 0\, .
\label{eqcouillet2}$$
where $\hat{\mathbf{\Sigma}}$ is the unique solution (if it exists) of: $$\mathbf{\Sigma} = \frac{1}{N} \displaystyle \sum_{i=0}^{N-1} u\left(\frac{1}{m} \, \mathbf{y}_{wi}^H \,\mathbf{\Sigma}^{-1} \,\mathbf{y}_{wi}\right)\, \mathbf{y}_{wi} \, \mathbf{y}_{wi}^H\, .$$
The distribution of the eigenvalues of $\hat{\mathbf{S}}$ can hence be more efficiently studied, the terms $\left[ v\left( \tau_i \, \gamma\right) \right]_{i\in \llbracket 0, \, N-1 \rrbracket}$ being independent of the $\left\{\mathbf{x}_i\right\}_i$. The goal being to study $\check{\mathbf{\Sigma}}_{SCM}$ which is the unique solution of , the following theorem enables to establish the relationship between $\check{\mathbf{\Sigma}}_{SCM}$ and $\hat{\mathbf{S}}$ thanks to .
With previous definitions, one has the following convergence: $$\left\Vert \check{\mathbf{\Sigma}}_{SCM} - \hat{\mathbf{S}} \right\Vert \overset{a.s.}{\longrightarrow} 0\, .
\label{eqcouilletmod}$$
The proof is provided in Appendix B.
As the eigenvalues distribution of $\hat{\mathbf{S}}$ can be theoretically analysed when $N$, $m \rightarrow \infty$, it can characterize also those of $\check{\mathbf{\Sigma}}_{SCM}$ thanks to . Under the hypothesis that there is no source present in the signal, it is possible to set a threshold similarly to [@Couillet15b]. Indeed, in this case: $$\begin{aligned}
\left\Vert \hat{\mathbf{S}} \right\Vert & = & \left\Vert \frac{1}{N} \, \displaystyle \sum_{i=0}^{N-1} \tau_i \, v(\tau_i \, \gamma) \, \mathbf{x}_i \, \mathbf{x}_i^H \right\Vert = \left\Vert \frac{1}{N} \displaystyle \sum_{i=0}^{N-1} \frac{1}{\gamma} \, \psi{(\tau_i \,\gamma)} \, \mathbf{x}_i \, \mathbf{x}_i^H \right\Vert \nonumber\, , \\
& \leq & \frac{\Phi_{\infty}}{\gamma \, \left(1-c \,\Phi_{\infty}\right)} \, \left \Vert \, \frac{1}{N} \, \displaystyle \sum_{i=0}^{N-1} \mathbf{x}_i \, \mathbf{x}_i^H \right\Vert \nonumber \, .\end{aligned}$$
Thanks to [@Bai98], and the bounds of the Marchenko Pastur distribution support, this inequality becomes $$\left\Vert \hat{\mathbf{S}} \right\Vert \leq t\, ,
\label{Seuil}$$ where the threshold $t$ is defined for the covariance matrix $\check{\mathbf{\Sigma}}_{SCM}$ by: $$t = \frac{\Phi_{\infty} \, \left(1+\sqrt{c}\right)^2}{\gamma \, \left(1-c \, \Phi_{\infty}\right)}\, .
\label{eqseuil}$$ Then, if the signal contains sources of sufficiently high SNR, eigenvalues might be found upon this threshold $t$ and all these eigenvalues correspond to sources. Let $\left\{ \lambda_i(\check{\mathbf{\Sigma}}_{SCM})\right\}_{i\in\llbracket 1,N\rrbracket}$ be the sorted eigenvalues of $\check{\mathbf{\Sigma}}_{SCM}$ when sources are present in the samples. As all sources are assumed to be independent, the estimated number of sources $\hat{p}$ which corresponds to the rank of the signal subspace is then given by $\hat{p} = \displaystyle\min_k (\lambda_k > t)$, if $p << \min{(N,m)}$.
Results
-------
This section is devoted to the presentation of some simulations relative to the estimation of the covariance matrix. Samples are considered here sources-free. The parameters are set to $c = 0.45$, $m=900$ and $N=2000$. Thus, $\mathbf{Y} = \mathbf{C}^{1/2}\,\mathbf{ X}\,\mathbf{ T}^{1/2}$ with $\mathbf{C} = \mathcal{L} \, \left(\left(\rho^0, \rho^1,\ldots ,\rho^{m-1}\right)^T\right)$ where $\rho = 0.7$ and $\mathbf{X}$ is a zero-mean complex Gaussian noise with identity covariance matrix. The texture matrix $\mathbf{T}$ is a diagonal $N\times N$-matrix containing the $\left\{\tau_i\right\}_{i\in \llbracket 0,N-1\rrbracket}$ on its diagonal where $\left\{\tau_i\right\}_i$ are i.i.d. inverse gamma distributed with mean equal to $1$ and with shape parameter equal to $10$. The function $u$ is here defined as $u : x \mapsto \displaystyle \frac{1+\alpha}{x+\alpha} $ where $\alpha$ is a fixed parameter equal to $0.1$.
Figure \[fig2\] shows the eigenvalues of the estimated covariance matrix $\check{\mathbf{\Sigma}}_{SCM}$ when samples $\mathbf{Y}$ have been whitened by $\check{\mathbf{C}}_{SCM}^{-1/2}$. On the figure \[fig3\], the signal $\mathbf{Y}$ has not been whitened. The green histogram corresponds to the eigenvalues distribution of $\hat{\mathbf{S}}$ whose histogram is expected to coincide with the distribution of the eigenvalues of $\check{\mathbf{\Sigma}}_{SCM}$ as the equation indicates. Moreover, the threshold $t = \displaystyle \frac{(1+\alpha) \, (1+\sqrt{c})^2}{\gamma \, (1-c\, (1+\alpha))}$ given by has been estimated and drawn in red, in order to confirm that the eigenvalues are all smallest than the threshold.
As the eigenvalues distribution of $\check{\mathbf{\Sigma}}_{SCM}$ are closed to those of $\hat{\mathbf{S}}$, the fixed-point estimator correctly annihilates the influence of the textures $\tau_i$’s and the whitening balances the matrix of correlation. On Figure \[fig2\], we can observe that the eigenvalues do not exceed the upper bound $t$. When the signal has not been whitened, this threshold $t$ does not theoretically correspond. Indeed, in Figure \[fig3\], the threshold is found to be smaller than the largest eigenvalues of the estimated covariance matrix. These figures illustrate first the results of Theorem 2 and show the importance of the whitening process.\
Figure \[fig4\] presents the eigenvalues distributions of $\hat{\mathbf{S}}$ and $\check{\mathbf{C}}_{SCM}$ for samples distributed according to a different CES distribution. Here, the texture $\mathbf{T}$ is a diagonal matrix containing the $\left\{\tau_i\right\}_{i\in \llbracket 0,, N-1\rrbracket}$ on its diagonal where each $\tau_i$ is independent and identically distributed and follows a distribution equal to $t^2$ where $t$ is a Student-t distributed random variable with parameter $100$ and $\alpha = 0.1$. The eigenvalues are not so close than the eigenvalues of $\hat{\mathbf{S}}$ and are found to get closer to the threshold $t$. If the distribution of $\tau$ is getting away to the one for which the function $u$ has been calculated, the method seems so to be less reliable. To fill this gap, we propose to enhance the proposed SCM-based method for the whitening through robust $M$-estimators-based method.
Model Order Selection: a robust method approach
===============================================
This section aims at developing a robust estimator based technique to whiten the signal instead of the previous SCM-based one. This section follows the same steps than in the previous section but by using a $M$-estimator in the whitening process.
Whitening Step
--------------
Let $\widetilde{\mathbf{C}}_{FP}$ be an biased estimator of the covariance matrix $\mathbf{C}$ such that $ \widetilde{\mathbf{C}}_{FP} = \mathcal{T} (\widehat{\mathbf{C}}_{FP}) $ where $\widehat{\mathbf{C}}_{FP}$ is the unique solution to the Maronna’s $M$-estimator [@Maronna76]: $$\mathbf{Z} = \frac{1}{N} \, \displaystyle \sum_{i=0}^{N-1} u\left(\frac{1}{m} \,\mathbf{y}_i^{H} \, \mathbf{Z}^{-1} \, \mathbf{y}_i \right) \, \mathbf{y}_i \, \mathbf{y}_i^{H}\, .$$ As in the previous section, $u(.)$ is a function derived thanks to the probability distribution function of the CES noise: $u$: \[0, $+\infty$) $\mapsto$ (0, $+\infty$) nonnegative, continuous and non-increasing. The following theorem stands for $\widetilde{\mathbf{C}}_{FP}$:
Let $\widetilde{\mathbf{C}}_{FP}$ be a fixed-point estimator of the covariance matrix $\mathbf{C}$ as defined above, the following result holds: $$\left\Vert \widetilde{\mathbf{C}}_{FP} - \mathds{E}\left[v(\tau \,\gamma)\,\tau\right] \,\mathbf{C} \right\Vert \overset{a.s.}{\longrightarrow} 0 \, ,
\label{eqth2}$$ where:
- $\phi: x \mapsto x \,u(x)$ increasing and bounded, with $ \lim\limits_{\substack{x \to \infty}} \phi(x) = \Phi_{\infty} > 1$,
- $\underset{N \rightarrow \infty}{\text{lim}} \, c_N < \, \Phi_{\infty}^{-1}$,
- $g: x \mapsto \displaystyle \frac{x}{1-c \,\phi(x)}$,
- $v: x \mapsto u \circ g^{-1}(x)$, $\psi: x \mapsto x\, v(x)$,
- $\gamma$ is the unique solution, if defined, of: $1 = \displaystyle \frac{1}{N} \, \sum_{i=1}^N \frac{\psi\left( \tau_i \, \gamma \right)}{1+ c \, \psi( \tau_i \,\gamma)}$.
The covariance matrix $\check{\mathbf{C}}_{FP} =\displaystyle \frac{ \tilde{\mathbf{C}}_{FP} }{\mathds{E} \left[v(\tau \,\gamma) \, \tau\right]}$ characterizes the estimator of the true covariance matrix $\mathbf{C}$.
The proof, inspired by [@Vinogradova14] and [@Loubaton16], is provided in Appendix B.
*Remark*: [@Couillet13] proves that $\tilde{\mathbf{C}}_{SCM} = \phi^{-1}(1) \,\tilde{\mathbf{C}}_{FP}$. When the function $u$ is well chosen, it is possible to have $\phi^{-1}(1) = 1$ and $\tilde{\mathbf{C}}_{SCM} = \tilde{\mathbf{C}}_{FP} $, as it will be the case for the $u$ chosen in the following sections. But even in this case, $\check{\mathbf{C}}_{SCM}$ and $\check{\mathbf{C}}_{FP} $ differ up to a scale factor as $\check{\mathbf{C}}_{SCM} = \displaystyle \frac{\mathds{E} \left[v(\tau \gamma) \,\tau\right]}{\mathds{E} (\tau)} \, \check{\mathbf{C}}_{FP} $.\
As in the previous section, the samples $\mathbf{Y}$ can then be whitened thanks to $ \check{\mathbf{C}}_{FP}^{-1/2}$. Let $\check{\mathbf{Y}}_{wFP} = [\check{\mathbf{y}}_{wF0} ,... \check{\mathbf{y}}_{wF \, N-1}]$ be the whitened samples: $$\check{\mathbf{Y}}_{wFP} = \check{\mathbf{C}}_{FP}^{-1/2} \,\mathbf{M} \, \boldsymbol{\delta}^H \, \mathbf{\Gamma}^{1/2} + \check{\mathbf{C}}_{FP}^{-1/2} \, \mathbf{C}^{1/2}\, \mathbf{X}\, \mathbf{T}^{1/2}\, .$$
The parameter $\mathds{E}[\tau]$ can be in practice evaluated with the empirical estimator of the mean, or, as in the previous section, be considered as equal to one. The quantity $\mathds{E}[v(\tau \,\gamma)\,\tau]$ can be also evaluated through an estimate $\hat{\gamma}$ of $\gamma$ as explained in the Results section.
Robust estimation of the covariance matrix and model order selection
--------------------------------------------------------------------
The robust estimation of the covariance matrix and the model order selection are done as previously. The robust estimator of the scatter matrix of the whitened signal $\check{\mathbf{Y}}_{wFP}$ is a fixed-point estimator denoted by $\check{\mathbf{\Sigma}}_{FP}$ and defined as the unique solution of the equation: $$\mathbf{\Sigma} = \frac{1}{N} \displaystyle \sum_{i=0}^{N-1} u\left(\frac{1}{m} \, \check{\mathbf{y}}_{wFi}^H \,\mathbf{\Sigma}^{-1} \,\check{\mathbf{y}}_{wFi} \right)\, \check{\mathbf{y}}_{wFi} \, \check{\mathbf{y}}_{wFi}^H\, .
\label{eqsigma2}$$
Thus, $\check{\mathbf{\Sigma}}_{FP}$ is a robust estimator of the covariance matrix of the whitened signal.\
The equation is still effective when replacing $\check{\mathbf{\Sigma}}_{SCM}$ by $\check{\mathbf{\Sigma}}_{FP}$. Indeed, Theorem 2 can be adapted as follows:
The following convergence holds: \[theorem2\] $$\left\Vert \check{\mathbf{\Sigma}}_{FP} - \hat{\mathbf{S}} \right\Vert \overset{a.s.}{\longrightarrow} 0 \, .
\label{eqth2mod}$$
The proof is the same as in Theorem 2 and is provided in Appendix B.
The same threshold $t$ given by equation can be used on the eigenvalues of $\check{\mathbf{\Sigma}}_{FP}$ to estimate $p$. The final corresponding algorithm is presented below.\
Results
-------
As in the previous section, it seems interesting to analyse the eigenvalues distributions of $\hat{\mathbf{S}}$ and $\check{\mathbf{\Sigma}}_{FP}$. For the next simulations, source-free samples are considered and the parameters are set to $c = 0.45$, $m=900$ and $N=2000$. The function $u$ chosen for the FP and Maronna $M$-estimators is the same function as before with $\alpha=0.1$.\
Figure \[fig6\] presents the eigenvalues distribution of the covariance matrices $\hat{\mathbf{S}}$ and $\check{\mathbf{\Sigma}}_{FP}$ when the signal has been whitened by $\check{\mathbf{C}}_{FP}$. One can notice that the results are the same as Figure \[fig2\]: for $N$ large, the distribution of eigenvalues is almost the same as those of $\mathbf{\hat{S}}$. However, as the rate of convergence of is faster than in , it is more interesting to consider the robust method. Moreover, if a robust estimator is not used after the whitening process, the eigenvalues distribution will not follow those of $\hat{\mathbf{S}}$ and will exceed the threshold $t$.
For robustness analysis (not the same texture distribution for the $u$ function and the observed samples), Figure \[fig8\] shows quite good results when $\mathbf{T}$ is a diagonal matrix containing the $\{\tau_i\}_{i\in \llbracket0,N-1\rrbracket}$ on its diagonal where $\{\tau_i\}_{i\in \llbracket 0, N-1\rrbracket}$ are i.i.d. and follow a distribution equal to $t^2$ with $t$ a Student-t random variable with parameter $100$ and $\alpha = 0.1$.
Figure \[fig9\_bis\] presents the same histograms as in Figure \[fig6\] for a single source of SNR equal to $10~$dB present in the samples. One can observe that only single eigenvalue exceeds the threshold and that the noise eigenvalues distribution of $\check{\mathbf{\Sigma}}_{FP}$ fits well those of $\hat{\mathbf{S}}$.
The results are better for this robust method than in the previous section (e.g. figure \[fig4\]). Indeed, the robust method provides robustness with respect to the distribution of $\tau$: if the distribution of the texture differs to those for which the function $u$ has been computed, the method is still reliable, this can be explained by the robustness of the covariance matrix estimation. As for the non-whitening case, the eigenvalues get over the threshold and no conclusion or model order can be deducted. These results have so extended the paper of [@Vinogradova14] to the left hand side correlated noise case. The L2-norm of the estimated covariance matrix compared to the SCM tends to zero when $N$ and $m$ tends to infinity with a constant ratio $c$. As a lot of estimation methods for the rank of the signal subspace are based on the estimation of the eigenvalues of the covariance matrix, this new estimator improves the consistency for resolution of this problem.
Results and Comparisons
========================
In this section some results of order selection are presented, on both simulated and real hyperspectral images. The simulations are based on $\check{\mathbf{\Sigma}}_{SCM}$ and $\check{\mathbf{\Sigma}}_{FP}$.
Estimation of the model order
-----------------------------
In order to test the proposed method, we simulate hyperspectral images, before dealing with real images. As a reminder, we first whiten the received signal thanks to a Toeplitz matrix coming from the SCM or a Fixed-Point estimator. Thus, a $M$-estimator is used to estimate the scatter matrix of the whitened signal. The distribution of its eigenvalues is then studied: a threshold is applied to count how many eigenvalues are higher than this threshold, providing the estimated model order $\hat{p}$.\
For simulated and correlated ($\rho=0.7$) CES noise, the $\left\{\tau_i\right\}_{i\in \llbracket 0, N-1 \rrbracket}$ are inverse gamma distributed with parameter $\nu=0.1$. On Figure \[fig12\] ($m=400$ and $N=2000$), $p=4$ sources are added in the observations with a SNR varying from $-15$ to $20 $dB. For this figure, the number of sources $\hat{p}$ (average on 4 trials) is estimated through three methods: AIC, the non-whitened signal and the two proposed methods: when the signal is whitened with the Toeplitz version of the SCM and the one of the FP. The proposed method starts to find sources from a SNR equal to $-5$dB. The FP method seems to better evaluate the number of sources. For a greater SNR, whereas it systematically gives the correct number of sources, the other methods overestimate it. On Figure \[fig13\] the same simulation is done for $p=4$ but with the $\left\{\tau_i\right\}_i$ following a distribution equal to $t^2$ where $t$ is a Student-t random variable, as before. On Figure \[fig13\], one notice that the proposed estimators still present better performance than the others, and allow to find sources with SNR greater than 0 dB.
Now, we compare the results obtained with three different methods on several real hyperspectral images found in public access: [*Indian Pines*]{}, [*SalinasA*]{} from AVIRIS database and [*PaviaU*]{} from ROSIS database [@siteinternet]. Let $M1$ be the proposed method with a whitening made with the SCM estimator, $M2$ be the proposed method with a whitening made with a Fixed-Point estimator, $M3$ be the method consisting in thresholding the eigenvalues of the Fixed-Point estimator without the whitening step, and the usual AIC method. For the function $u(.)$ corresponding to the Student-t distribution, we choose $\nu=0.1$ for the whitening process if it is done by a fixed-point estimator, and zero for the estimation process. As we do not have any access to the true distribution of the noise, an empirical estimator of $\gamma$ is used, $\hat{\gamma} = \displaystyle \frac{1}{N} \, \sum_{i=1}^{N} \frac{1}{m} \, \mathbf{y}_i^{H} \, \check{\mathbf{\Sigma}}^{-1}_{(i)} \, \mathbf{y}_i$, where $\check{\mathbf{\Sigma}}_{(i)} = \check{\mathbf{\Sigma}} - \displaystyle\frac{1}{N} \, u\left(\frac{1}{m} \,\mathbf{y}_i^{H} \, \check{\mathbf{\Sigma}}^{-1} \, \mathbf{y}_i\right) \, \mathbf{y}_i \, \mathbf{y}_i^H$. Then [@Couillet15b] shows that $\gamma -\hat{\gamma} \overset{a.s.}{\longrightarrow} 0$. Moreover, as the distribution of $\tau$ is unknown, we choose to consider that $\mathds{E} \left[\tau\right] $ and $\mathds{E}\left[v(\tau \, \gamma) \, \tau \right]$ are equal to 1. Further works can be carried out to estimate correctly these unknown quantities. However, we can reasonably assume than $\mathds{E}\left[v(\tau \, \gamma) \, \tau \right]$ and $\mathds{E} \left[\tau\right] $ are not to large and that the estimation error will not impact the results a lot. The results are summarized in table \[table-Tableau1\]. On each image, the result tends to be better than those of classical methods.
Images Indian Pines SalinasA PaviaU Cars
--------------- -------------- ---------- -------- ------
$p$ 16 9 9 6
$\hat{p}$ M1 11 9 1 3
$\hat{p}$ M2 12 9 1 3
$\hat{p}$ M3 220 204 103 1
$\hat{p}$ AIC 219 203 102 143
: Estimated $p$ for different hyperspectral images.
\[table-Tableau1\]
Conclusion
==========
The model order selection for large dimensional data and for sources embedded in correlated CES noise is tackled in this article. Two Toeplitz-based covariance matrix estimators are first introduced, and their consistency has been proved. As for the CES texture, it is handled with any $M$-estimator, which can then be used to estimate the correct structure of the scatter matrix built on whitened observations. The Random Matrix Theory provides tools to correctly estimate the model order. Results obtained on real and simulated hyperspectral images are promising. Moreover, the proposed method can be applied on a lot of other kind of model order selection problems such as radar clutter rank estimation, sources localization or any hyperspectral problems such as anomaly detection or linear or non-linear unmixing techniques.
Proofs of Theorem 1 and Theorem 3 {#sec::4}
=================================
The proofs of Theorem 1 and Theorem 3 are inspired by [@Vinogradova14]. For these theorems, we will use the lemma 4.1 in [@Gray6], that is, for $\mathbf{T} = \mathcal{L}\left(\left(t_0, \ldots, t_{m-1}\right)^T\right)$ a Toeplitz Hermitian $m \times m$-matrix with $\left\{t_{k}\right\}_{k\in \llbracket0,m-1\rrbracket}$ absolutely summable ($t_{-k} = t_k^\ast$), we can define the function $f(.)$ such that for any $\lambda \in[0,2\,\pi]$, $f(\lambda) = \displaystyle \sum_{k=1-m}^{m-1} t_k \, e^{i \lambda k} $ and $M_f$ characterizes its essential supremum: $$\Vert \mathbf{T} \Vert \leq M_f = \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \displaystyle \sum_{k=1-m}^{m-1} t_k \, e^{i \lambda k} \right\vert \, .
\label{eqGray}$$
Proof of Theorem 1
------------------
As in the main body of this article, let $\mathbf{Y} = \mathbf{M} \, \boldsymbol{\delta}^H \, \mathbf{\Gamma}^{1/2} + \mathbf{C}^{1/2}\,\mathbf{ X}\,\mathbf{ T}^{1/2}$ and let $\mathcal{T}$ be the Toeplitz operator as defined in the introduction and for any $m$-vector $\mathbf{x}$, $\left( \left[\mathcal{L} (\mathbf{x})\right]_{i,j} \right) _{i\leq j} = x_{i-j}$ and $\left( \left[\mathcal{L} (\mathbf{x})\right]_{i,j} \right)_{i>j} = x_{i-j}^\ast$, of size $m \times m$. Under Assumption 1, Assumption 2, Assumption 3 and as $\mathcal{T} \left(\displaystyle \frac{1}{N} \, \mathbf{Y}\, \mathbf{Y}^H\right) $ and $\mathds{E}[\tau] \, \mathbf{C}$ are Toepltiz matrices, one can write, thanks to : $$\left\Vert \mathcal{T} \left(\frac{1}{N} \, \mathbf{Y}\, \mathbf{Y}^H\right) - \mathds{E}[\tau]\, \mathbf{C} \right\Vert \leq \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m (\lambda) - \mathds{E}[\tau] \,\gamma_m (\lambda) \right\vert \, ,
\label{eqproof1}$$ where $\mathbf{\gamma}_m (\lambda) = \displaystyle \sum_{k =1-m}^{m-1} c_{k,m} \, e^{i\,k \,\lambda}$ with $c_{-k} = c_k^{\star}$ and $\hat{\mathbf{\gamma}}_m (\lambda) = \displaystyle \sum_{k =1-m}^{m-1} \check{c}_{k,m} \, e^{i\,k \,\lambda}$ with $\check{c}_{-k} = \check{c}_k^{\star}$.\
The following lemma is essential for the development of the proof:
\[lemma1\] The quantity $\hat{\mathbf{\gamma}}_m (\lambda)$ can be rewritten as: $$\hat{\mathbf{\gamma}}_m (\lambda) = \mathbf{d}_m^H (\lambda) \, \frac{\mathbf{Y}\, \mathbf{Y}^H}{N} \, \mathbf{d}_ m (\lambda) \, ,
\label{eq1th1lemme1}$$ with $\mathbf{d}_m (\lambda) =\displaystyle \frac{1}{\sqrt{m}} \, \left(1, e^{-i \,\lambda}, \ldots, e^{-i\,(m-1) \,\lambda} \right)^T$.
The proof draws his inspiration from the one of Appendix A1 in [@Vinogradova14]. Equation can be rewritten as: $$\begin{aligned}
& \mathbf{d}_m^H (\lambda) & \,\displaystyle\frac{\mathbf{Y} \,\mathbf{Y}^H}{N} \,\mathbf{d}_m (\lambda) = \frac{1}{m\,N} \displaystyle \sum_{l,l' =0}^{m-1} e^{-i\,(l'-l)\,\lambda} \, \left[\mathbf{Y} \,\mathbf{Y}^H \right]_{l,l'} \, ,\nonumber \\
&= &\displaystyle \sum_{k= 1-m}^{m-1} e^{-i\,k \,\lambda} \,\frac{1}{m \,N} \displaystyle \sum_{i=0}^{m-1} \displaystyle \sum_{j=0}^{N-1} y_{i,j} \, y_{i+k,j}^\star \mathds{1}_{0 \leq i+k\leq m } \, ,\nonumber \\
& = & \displaystyle \sum_{k= 1-m}^{m-1} \check{c}_k \, e^{-i\,k \,\lambda} \nonumber \, .\end{aligned}$$
Thereby, we have:
$$\begin{aligned}
& \hat{\mathbf{\gamma}}&_m (\lambda) = \mathbf{d}_m^H (\lambda) \, \frac{\mathbf{Y}\, \mathbf{Y}^H}{N} \, \mathbf{d}_m (\lambda)\, , \nonumber \\
&=& \mathbf{d}_m^H (\lambda)\, \frac{\mathbf{M} \boldsymbol{\delta}^H \, \mathbf{\Gamma} \, \boldsymbol{\delta} \, \mathbf{M}^H}{N} \, \mathbf{d}_m (\lambda) \nonumber \\
&+& \mathbf{d}_m^H (\lambda) \, \frac{\mathbf{C}^{1/2}\, \mathbf{X} \, \mathbf{T}^{1/2} \, \mathbf{\Gamma}^{1/2} \, \boldsymbol{\delta} \, \mathbf{M}^H}{N} \, \mathbf{d}_m (\lambda) \nonumber \\
&+& \mathbf{d}_m^H (\lambda) \, \frac{\mathbf{M} \, \boldsymbol{\delta}^H \mathbf{\Gamma}^{1/2} \, \mathbf{T}^{1/2} \, \mathbf{X}^H \, \mathbf{C}^{1/2}}{N} \, \mathbf{d}_m (\lambda) \nonumber \\
&+& \mathbf{d}_m^H (\lambda) \, \frac{\mathbf{C}^{1/2} \, \mathbf{X} \, \mathbf{T} \, \mathbf{X}^H \, \mathbf{C}^{1/2}}{N} \, \mathbf{d}_m (\lambda) \, \label{eqcomp} . \end{aligned}$$
And we note : $ \hat{\mathbf{\gamma}}_m^{sign} (\lambda) = \mathbf{d}_m^H (\lambda)\, \frac{\mathbf{M} \boldsymbol{\delta}^H \, \mathbf{\Gamma} \, \boldsymbol{\delta} \, \mathbf{M}^H}{N} \, \mathbf{d}_m (\lambda)$,\
$\hat{\mathbf{\gamma}}_m^{cross} (\lambda) =$\
$\mathbf{d}_m^H (\lambda) \, \frac{\mathbf{C}^{1/2}\, \mathbf{X} \, \mathbf{T}^{1/2} \, \mathbf{\Gamma}^{1/2} \, \boldsymbol{\delta} \, \mathbf{M}^H + \mathbf{M} \, \boldsymbol{\delta}^H \mathbf{\Gamma}^{1/2} \, \mathbf{T}^{1/2} \, \mathbf{X}^H \, \mathbf{C}^{1/2}}{N} \, \mathbf{d}_m (\lambda)$\
$\hat{\mathbf{\gamma}}_m^{noise} (\lambda) = \mathbf{d}_m^H (\lambda) \, \frac{\mathbf{C}^{1/2} \, \mathbf{X} \, \mathbf{T} \, \mathbf{X}^H \, \mathbf{C}^{1/2}}{N} \, \mathbf{d}_m (\lambda) $ .
And the equation becomes: $$\begin{aligned}
&& \left\Vert \mathcal{T} \left(\frac{1}{N} \, \mathbf{Y}\, \mathbf{Y}^H\right) - \mathds{E}[\tau]\, \mathbf{C} \right\Vert \leq \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \vert \hat{\gamma}_m^{noise} (\lambda) \nonumber \\
&& + \hat{\gamma}_m^{sign} (\lambda) + \hat{\gamma}_m^{cross} (\lambda) - \mathds{E}[\tau] \,\gamma_m (\lambda) \vert \, .
\label{eqproof3}\end{aligned}$$
This leads to: $$\begin{aligned}
&& \left\Vert \mathcal{T} \left(\frac{1}{N} \mathbf{Y} \, \mathbf{Y}^H\right) - \mathds{E}[\tau]\, \mathbf{C} \right\Vert \nonumber \\
&& \leq \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m^{noise} (\lambda) - \mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda) \right] \right\vert \nonumber \\
&& + \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda)\right] - \mathds{E}(\tau) \, \gamma_m (\lambda) \right\vert \nonumber \\
&& + \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m^{sign} (\lambda)\right\vert + \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m^{cross} (\lambda) \right\vert \, .
\label{eqproof5}\end{aligned}$$
We will now analyse each term of .\
### Analysis of $\underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda)\right] - \mathds{E}(\tau) \, \gamma_m (\lambda) \right\vert$
We first need the following lemma:
\[lemma2\] $$\mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda) \right] = \mathds{E}[\tau] \, \mathbf{d}_m^H (\lambda) \, \mathbf{C} \, \mathbf{d}_m (\lambda) = \mathds{E}[\tau] \, \gamma_m (\lambda)\, .
\label{eq2th1lemme1}$$
The equation gives $\mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda)\right] = \mathbf{d}_m^H(\lambda) \, \mathds{E} \left[ \displaystyle \frac{\mathbf{C}^{1/2} \mathbf{X} \, \mathbf{T} \, \mathbf{X}^H \mathbf{C}^{1/2}}{N} \right]\, \mathbf{d}_m (\lambda)$. Let $\mathbf{V} = \mathbf{C}^{1/2} \mathbf{X}$ and $\left(\mathbf{V}\right)_{i,j} = v_{i,j}$. We obtain $\mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda)\right] = \mathbf{d}_m^H(\lambda) \, \mathds{E} \left[ \displaystyle \frac{ \mathbf{V} \, \mathbf{T} \, \mathbf{V}^H }{N} \right]\, \mathbf{d}_m (\lambda)$. As $ \left( \mathds{E} \left[\mathbf{V} \,\mathbf{T} \,\mathbf{V}^H \right]\right)_{i j} = \displaystyle \sum_{k=1}^N \mathds{E}[\tau]\, \mathds{E} \left[v_{j,k}^{\ast} \, v_{ik}\right]$ and $c_{k'} = \mathds{E} \left[v_{p, n} \,v_{p+k',n}^{\ast}\right]$, we have $\left( \mathds{E} \left[\mathbf{V} \, \mathbf{T} \, \mathbf{V}^H\right] \right)_{i j} = \displaystyle \sum_{k=1}^N \mathds{E}[\tau]\, c_{j-i} = N \,\mathds{E}[\tau] \, c_{j-i}$. This leads to $$\mathbf{\mathds{E}} \left[\hat{\gamma}_m^{noise} (\lambda)\right] = \frac{N\, \mathds{E}(\tau) }{N} \,\mathbf{d}_m^H (\lambda) \, \mathbf{C} \, \mathbf{d}_m (\lambda) = \mathds{E}[\tau] \, \gamma_m (\lambda) \, .$$
Thereby, the second term of leads to: $$\begin{aligned}
& \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \mathds{E} \left[ \hat{\gamma}_m^{noise} (\lambda) \right] - \mathds{E}(\tau) \,\gamma_m(\lambda) \right \vert \\
& = \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \mid \mathds{E}(\tau) \,\gamma_m(\lambda) - \mathds{E}(\tau) \, \gamma_m (\lambda) \mid = 0\, .\end{aligned}$$ The second term is equal to zero.
### Analysis of $\underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m^{noise} (\lambda) - \mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda) \right] \right\vert $
As in [@Vinogradova14], the method consists in proving for a $\lambda_i \in \left[ 0,2 \pi \right) $ and a real $ x>0 $ that $ \mathds{P} \left[ \left\vert \hat{\gamma}_m^{noise} (\lambda_i) - \mathds{E} \left[ \gamma_m^{noise} (\lambda_i) \right] \right\vert > x \right] \rightarrow 0$. After that, it remains to prove that $ \mathds{P} \left[ \underset{\lambda \in \left[ \lambda_i , \lambda_{i+1} \right) }{\sup} \left\vert \gamma_m^{noise} (\lambda) - \gamma_m^{noise} (\lambda_i) \right\vert > x \right] \rightarrow 0 $ and that $ \mathds{P} \left[ \underset{\lambda \in \left[ \lambda_i , \lambda_{i+1} \right) }{\sup} \left\vert \mathds{E} \gamma_m^{noise} (\lambda_i) - \mathds{E} \gamma_m^{noise} (\lambda) \right\vert > x \right] \rightarrow 0$. Let $\lfloor\cdot\rfloor $ be the floor function, choosing a $\beta > 2 $, $\mathcal{I} = \left[ 0, \ldots, \lfloor N^{\beta} \rfloor -1 \right] $, $\lambda_i = 2\, \pi \,\frac{i}{\lfloor N^{\beta}\rfloor }$, $i \in \mathcal{I}$: $$\begin{aligned}
&& \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m^{noise} (\lambda) - \mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda)\right] \right\vert \nonumber \\
&& \leqslant \max_{i \in \mathcal{I}} \underset{\lambda \in \left[ \lambda_i \lambda_{i+1} \right] }{\sup} \left\vert \hat{\gamma}_m^{noise} (\lambda) - \hat{\gamma}_m^{noise} (\lambda_i) \right\vert \nonumber \\
& & + \max_{i \in \mathcal{I}} \left\vert \hat{\gamma}_m^{noise} (\lambda_i) - \mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda_i)\right] \right\vert \nonumber \\
& & + \max_{i \in \mathcal{I}} \underset{\lambda \in \left[ \lambda_i \lambda_{i+1} \right] }{\sup} \left\vert \mathds{E} \left[ \hat{\gamma}_m^{noise} (\lambda_i)\right] - \mathds{E} \left[ \hat{\gamma}_m^{noise} (\lambda)\right] \right\vert \nonumber \, ,\\
& \overset{\triangle}{=} & \chi_1 + \chi_2 +\chi_3\, .
\label{eqgamma}\end{aligned}$$
The idea of the proof in [@Vinogradova14] is then to provide concentration inequalities for the term $\chi_1$ and $\chi_2$ (random terms) and a bound on $\chi_3$. The only difference with [@Vinogradova14] is the presence of the matrix $\mathbf{T}$ in $\hat{\gamma}_m^{noise}(\lambda)$ and the left side correlation of the noise. Let note $ \Vert \boldsymbol{\gamma} \Vert_{\infty}$ the sup norm of the function $\boldsymbol{\gamma}: \lambda \longrightarrow \displaystyle \sum_{k=-\infty}^{\infty}c_k \, e^{-ik\lambda}$ for $\lambda \in [0 \, 2\pi )$. The convergence of the first term $\chi_1$ is proposed in the following lemma.
\[lemma3\] A constant $A > 0$ can be found
such that, for any $x>0$ and $N$ large enough, $$\mathds{P} \left[ \chi_1 > x \right] \leq \exp \left( - \frac{c\,N^2}{\left\Vert\mathbf{T} \right\Vert_{\infty}} \, \left( \frac{x\,N^{\beta-2}}{A \, \left\Vert \boldsymbol{\gamma} \right\Vert_{\infty}} - \log\left( \frac{x\,N^{\beta-2}}{A \, \left\Vert \boldsymbol{\gamma} \right\Vert_{\infty}}\right) -1 \right) \right) \, .$$
As already mentioned, the proof is the same as in [@Vinogradova14] except for two points: the presence of the matrix $\mathbf{T}$ and the left side correlation of the noise instead of right side in [@Vinogradova14]. The inequality: $\left\Vert \mathbf{V}_N \,\mathbf{T} \,\mathbf{V}^H_N \right\Vert \leqslant \left\Vert \mathbf{T} \right\Vert_{\infty} \, \left\Vert \mathbf{V}_N \,\mathbf{V}^H_N \right\Vert \leq \left\Vert \mathbf{T} \right\Vert_{\infty} \, \left\Vert \mathbf{C} \right\Vert \, \left\Vert \mathbf{X\, X}^H \right\Vert$ enables to write: $$\begin{aligned}
& \left\vert \hat{\gamma}_m^{noise} (\lambda) - \hat{\gamma}_m^{noise} (\lambda_i) \right\vert \\
&= \left\vert \mathbf{d}_m^H(\lambda) \, \frac{\mathbf{V} \, \mathbf{T} \, \mathbf{V}^H}{N} \, \mathbf{d}_m(\lambda) - \mathbf{d}_m^H(\lambda_i) \, \frac{\mathbf{V} \, \mathbf{T} \, \mathbf{V}^H}{N} \, \mathbf{d}_m(\lambda_i) \right\vert\, , \\
&\leq \frac{2}{N} \, \left\vert \mathbf{d}_m(\lambda) - \mathbf{d}_m(\lambda_i) \right\vert \, \left\Vert \mathbf{C} \right\Vert \, \left\Vert \mathbf{T} \right\Vert_\infty \, \left\Vert \mathbf{X\, X}^H \right\Vert \, .\end{aligned}$$ And then the end of the proof is exactly the same as those of the Lemma 4 in [@Vinogradova14] replacing $c$ by $\displaystyle\frac{c}{\left\Vert \mathbf{T} \right\Vert_\infty}$ in the exponential. The left correlation is without consequences on the proof.
The convergence of the second term $\chi_2$ is proposed in the following lemma.
\[lemma4\] $$\mathds{P} \left[ \chi_2 > x \right] \leq 2\,N^{\beta}\, \exp \left( - \frac{c\,N}{\left\Vert \mathbf{T} \right\Vert_{\infty} } \, \left( \frac{x}{\left\Vert \boldsymbol{\gamma} \right\Vert_{\infty}} - \log\left(\frac{x}{ \left\Vert \boldsymbol{\gamma} \right\Vert_{\infty}} + 1 \right) \right) \right) \, .$$
The proof is the same as those of the Lemma 5 in [@Vinogradova14], with the $\displaystyle\frac{c}{\left\Vert \mathbf{T} \right\Vert_\infty}$ on the denominator.
The convergence of the third term $\chi_3$ is proposed in the following lemma.
\[lemma5\] $$\chi_3 \leq A \left\Vert \boldsymbol{\gamma} \right\Vert_{\infty} \, N^{- \beta + 1}\, .$$
The proof is the same as those of the lemma 6 of [@Vinogradova14], still with the $\displaystyle\frac{c}{\Vert \mathbf{T} \Vert_\infty}$ on the denominator.
These inequalities proves that $\mathds{P} \left[ \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m^{noise} (\lambda) - \mathds{E} \left[\hat{\gamma}_m^{noise} (\lambda) \right] \right\vert > x \right] \overset{a.s.}{\longrightarrow} 0$ for any $x$ positive real and with a $e^{-N^2}$ rate of decrease.\
### Analysis of $\underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m^{cross} (\lambda) \right\vert$
To prove the convergence of the last term of , let us recall that $$\begin{aligned}
\hat{\gamma}_m^{cross} (\lambda) & = \mathbf{d}_m^H (\lambda) \, \frac{\mathbf{C}^{1/2}\, \mathbf{X} \, \mathbf{T}^{1/2} \, \mathbf{\Gamma}^{1/2} \, \boldsymbol{\delta} \, \mathbf{M}^H}{N} \, \mathbf{d}_ m (\lambda)\\
& + \mathbf{d}_m^H (\lambda) \frac{\mathbf{M} \, \boldsymbol{\delta}^H \mathbf{\Gamma}^{1/2} \, \mathbf{T}^{1/2} \, \mathbf{X}^H \, \mathbf{C}^{1/2}}{N} \, \mathbf{d}_ m (\lambda) \, .\end{aligned}$$ Let $\mathds{I}_m$ be a $m \times m$ matrix containing 1 everywhere and $\mathbf{D}_m(\lambda)$ be the matrix containing the elements of $\mathbf{d}_m(\lambda)$ on its diagonal. It can be easily verified that, for any matrix $\mathbf{A}$, $\mathbf{d}_m^H(\lambda)\, \mathbf{A} \, \mathbf{d}_m(\lambda) = \mathrm{Tr}\left(\mathbf{D}_m^H(\lambda) \, \mathbf{A} \, \mathbf{D}_m(\lambda) \, \mathds{I}_m\right)$. We obtain: $$\begin{aligned}
& \hat{\gamma}_m^{cross} (\lambda) = \\
& 2 \, \mathcal{R}e \left[ \frac{1}{N} \, \mathrm{Tr} \left(\mathbf{X} \, \mathbf{T}^{1/2} \,\boldsymbol{\Gamma}^{1/2} \, \boldsymbol{\delta} \, \mathbf{M}^H \, \mathbf{D}_m (\lambda) \, \mathds{I}_{m} \, \mathbf{D}_m^H (\lambda) \, \mathbf{C}^{1/2} \right) \right] \, .\end{aligned}$$ For readability, let $\mathbf{E}(\lambda) = \mathbf{D}_m(\lambda) \, \mathds{I}_{m} \, \mathbf{D}_m^H(\lambda)$ defined as:
$$\mathbf{E}(\lambda)=
\begin{pmatrix}
1 & e^{i \lambda} & \ldots & e^{i(m-1)\lambda} \\
e^{-i \lambda} & 1 & \ldots & e^{i(m-2) \lambda} \\
e^{-i(m-1)\lambda} & \ldots & \ldots & 1
\end{pmatrix} \, ,$$
let $\mathbf{G}(\lambda) = \mathbf{M}^H \, \mathbf{E}(\lambda) \,\mathbf{C}^{1/2} \,\mathbf{X}$ and $\mathbf{J} = \mathbf{T}^{1/2} \, \boldsymbol{\Gamma}^{1/2} \, \boldsymbol{\delta}$, two matrices respectively of size $p \times N$ and $N \times p$. Moreover, let $\mathbf{g}(\lambda) = \left[g_1(\lambda), \ldots, g_{N\, p}(\lambda) \right]^T = \mathrm{vec}(\mathbf{G}(\lambda))$ and $\mathbf{j}= \left[j_1, \ldots, j_{N\, p} \right]^T= \mathrm{vec}(\mathbf{J})$. We obtain: $$\begin{aligned}
\hat{\gamma}_m^{cross} (\lambda)
&= \frac{2}{N} \, \mathcal{R}e \left( \mathrm{vec}^T(\mathbf{G}(\lambda)) \, \mathrm{vec}(\mathbf{J}) \right)\, , \\
&= \frac{2}{N} \, \displaystyle \sum_{k=1}^{N \, p} \mathcal{R}e (g_k(\lambda) ) \, \mathcal{R}e (j_k) - \mathcal{I}m (g_k(\lambda)) \, \mathcal{I}m (j_k)\, .\end{aligned}$$
This expression can be transformed by introducing $\mathbf{A} = \mathbf{M}^H \, \mathbf{E} \, \mathbf{C}^{1/2} \otimes \mathbf{I}_N$, $\mathbf{B} = \mathbf{T}^{1/2} \, \boldsymbol{\Gamma}^{1/2} \otimes \mathbf{I}_p$, $\tilde{\mathbf{g}}(\lambda) = \mathbf{A}^{-1}\, \mathbf{g}(\lambda)$ $\tilde{\mathbf{j}} = \mathbf{B}^{-1} \, \mathbf{j}$, $a_k = \displaystyle \sum_{l=1}^{N \, p} \left( \mathbf{A}^{T}\right)_{l,k}$ and $b_k = \displaystyle \sum_{s=1}^{N \, p} \left(\mathbf{B}\right)_{s,k}$: $$\begin{aligned}
&\hat{\gamma}_m^{cross} (\lambda) \\
& =\frac{2}{N} \, \mathcal{R}e \left( \left(\mathbf{A}^{-1}\, \mathbf{g}(\lambda)\right)^{T} \, \mathbf{A}^{T} \, \mathbf{B} \, \left( \mathbf{B}^{-1} \, \mathbf{j}\right) \right) \, \\
& = \displaystyle \frac{2}{N} \, \sum_{k=1}^{N \, p} \mathcal{R}e \left( \displaystyle \sum_{l=1}^{N \, p} \left(\mathbf{A}^{T}\right)_{l,k} \, \tilde{\mathbf{g}}_k (\lambda) \right) \, \mathcal{R}e \left( \displaystyle \sum_{s=1}^{N \, p} \left(\mathbf{B}\right)_{s,k} \, \tilde{\mathbf{j}}_k \right) \\
&- \mathcal{I}m \left( \displaystyle \sum_{l=1}^{N \, p} \left(\mathbf{A}^{T}\right)_{l,k} \, \tilde{\mathbf{g}}_k (\lambda) \right) \, \mathcal{I}m \left( \displaystyle \sum_{s=1}^{N \, p} \left( \mathbf{B}\right)_{s,k} \tilde{\mathbf{j}}_k \right) \, , \\
& = \displaystyle \frac{2}{N} \, \sum_{k=1}^{N \, p} \, \mathcal{R}e \left( a_k \, \tilde{g}_k (\lambda) \right) \, \mathcal{R}e \left( b_k \, \tilde{j}_k \right) \\
&- \mathcal{I}m \left( a_k \, \tilde{g}_k (\lambda) \right) \, \mathcal{I}m \left( b_k \, \tilde{j}_k \right) \, . \end{aligned}$$ The variables $a_k \, \tilde{g}_k (\lambda)$ and $b_k \, \tilde{j}_k$ are two independent complex Gaussian variables with variances respectively equal to $\left|\tilde{a}_k(\lambda)\right|^2$ and $\left|b_k\right|^2$. We can apply the following lemma:
\[lemma6\] Let $x$ and $y$ be two independent Gaussian $\mathcal{N}(0,1)$
*scalar random variables, then for any $\tau \in (-1 \,\, 1)$, then $\mathds{E}\left[\exp{\left(\tau \, x \, y\right)}\right]= (1- \tau^2)^{-1/2}$.*
The proof is derived in [@Vinogradova14] through lemma 13.
Let $\nu >0$ a real such that : $\nu^{-1} > \underset{k\in \llbracket1, N p\rrbracket, \lambda \in \left[ 0,2 \pi \right) }{\sup} \left(\left\vert \tilde{a}_k(\lambda) \right\vert^2 \, \left\vert b_k \right\vert^2\right)$. Then, for a fixed $\lambda \in [0 \, \,2\pi )$, from Lemma \[lemma6\] and from the Markov Inequality: $$\begin{aligned}
& \mathds{P} \left[ \hat{\gamma}_m^{cross} (\lambda) > x \mid \mathbf{T} \right] \\
&= \mathds{P} \left[ \exp{ \left( N \,\nu \,\hat{\gamma}_m^{cross} (\lambda) \right)} > \exp{ \left(N \,\nu \,x\right) } \, \mid \mathbf{T} \right] \\
& \leq \exp{\left(-N \,\nu \,x\right)} \, \mathds{E} \left[ \exp \left(2 \,\nu \displaystyle \sum_{k=1}^{N \, p} \, \left[ \mathcal{R}e \left( a_k \tilde{g}_k (\lambda) \right) \, \mathcal{R}e \left( b_k \tilde{j}_k \right) \right. \right. \right. \\
& \left. \left. \left. - \mathcal{I}m \left( a_k \, \tilde{g}_k (\lambda) \right) \, \mathcal{I}m \left( b_k \, \tilde{j}_k \right) \right] \right) \right] \\
&\leq \exp{ \left(-N \, \nu \, x\right)} \displaystyle \prod_{k=1}^{Np} \left( 1 - 4 \, \nu^2 \, \frac{\vert \tilde{a}_k(\lambda) \vert^2}{2} \, \frac{\vert b_k \vert^2}{2} \right)^{-1/2} \, \\
& \left( 1 - 4 \,\nu^2 \, \frac{\vert \tilde{a}_k(\lambda) \vert^2}{2} \, \frac{\vert b_k \vert^2}{2} \right)^{-1/2} \\
&\leq \exp{\left(-N \nu x - \displaystyle \sum_{k=1}^{N\, p} \, \log{\left( 1 - \nu^2 \, \vert \tilde{a}_k(\lambda) \vert^2 \, \vert b_k \vert^2 \right)^{-1} }\right)}\end{aligned}$$
Moreover, since the $\Gamma_{i,j}$ are absolutely summable (Assumption 3), it exists a constant $K$ such that: $$\left\vert b_k \right\vert ^2 = \left\vert \displaystyle \sum_{l=1}^{N p} \sqrt{\tau_l} \, \Gamma_{l,k}^{1/2} \right\vert^2 \leq K \displaystyle \sum_{l=1}^{N p} \tau_l \, .$$ Furthermore, since $\displaystyle\frac{1}{N} \, \displaystyle \sum_{l=1}^{N p} \tau_l \underset{N \longrightarrow \infty}{\longrightarrow} \mathds{E} (\tau_i) = 1$, we obtain $\vert b_k \vert ^2 \leq N \, K$. To deal with $\vert \tilde{a}_k(\lambda) \vert$, let $K_1$ and $K_2$ be some constants and remind that, for a fixed $j$, the $\left\{c_{i,j}\right\}_i$ and the $\left\{M_{i,j}\right\}_i$ are absolutely summable: $$\begin{aligned}
\left\vert a_k(\lambda) \right\vert &= \frac{1}{m} \, \left\vert \displaystyle \sum_{s=1}^p \sum_{l,j =1}^m c_{l,k} \, M_{j,s}^{\star} \, e^{i(l-j)\,\lambda} \right\vert \, ,\\
&\leq \frac{1}{m} \displaystyle \sum_{l=1}^m \left\vert c_{l,k}\right\vert \, m \, \sum_{s,j =1}^{p,m} \left\vert M_{j,s}^{\star} \right\vert\, ,\\
&\leq p \, K_2 \, \underset{s}{\max} \left(\displaystyle \sum_{j =1}^m \left\vert M_{j,s}^{\star} \right\vert\right) = p\, K_1 \, .\end{aligned}$$
We obtain $\nu ^2 \, \left\vert \tilde{a}_k(\lambda) \right\vert^2 \, \left\vert b_k\right\vert^2 \leq \nu^2 \, N^2 \, p^2 \, K \, K_1^2$ with $p \ll N $. Let $q$ and $\epsilon$ be two positive reals small enough and such that: $$\nu^2 = \left( \frac{q}{N^{1/2 + \epsilon}} \right)^2 < \frac{K\, K_1^2}{N}\, .$$ Then $\underset{N \longrightarrow \infty}{\lim} \nu^2 \left\vert a_k(\lambda) \right\vert^2 \, \left\vert b_k \right\vert^2 = 0$ and $\log \left( 1 - \nu^2 \, \left\vert \tilde{a}_k(\lambda) \right\vert^2 \, \left\vert b_k \right\vert^2 \right)^{-1} \sim \nu^2 \left\vert \tilde{a}_k(\lambda) \right\vert^2 \, \left\vert b_k \right\vert^2 $. Thereby, with $A$ defining a constant, it can be obtained: $$\mathds{P} \left[ \hat{\gamma}_m^{cross} (\lambda) > x \mid \mathbf{T} \right] \leq \exp{\left(-N^{1/2 - \epsilon} \, q \, x - A\right)}\, .$$ Then, integrating with respect to any density $p_\mathbf{T}(.)$ of $\mathbf{T}$ leads to: $$\begin{aligned}
&\mathds{P} \left[ \hat{\gamma}_m^{cross} (\lambda) > x \right] = \int \mathds{P} \left[ \hat{\gamma}_m^{cross} (\lambda) > x \mid \mathbf{T} \right] \, p_\mathbf{T}(\mathbf{T}) \, d\mathbf{T} \\
& \leq \exp{\left(-N^{1/2 - \epsilon} \, q \, x - A\right)} \, .\end{aligned}$$
This proves that, for any $\lambda_i$, $\mathds{P} \left[ \hat{\gamma}_m^{cross} (\lambda_i) > x \right] \underset{N \rightarrow \infty}{\rightarrow} 0$.\
It remains now to prove that $\underset{i \in \mathcal{I}}{\max} \underset{\lambda \in [\lambda_i \, \, \lambda_{i+1}] }{\sup} \left\vert \hat{\gamma}_m^{cross} (\lambda) - \hat{\gamma}_m^{cross} (\lambda_i)\right\vert \overset{a.s.}{\longrightarrow} 0$. This will be left to the reader as it follows the same proof as for $\chi_1$ of . We have so $\mathds{P} \left[ \underset{\lambda \in [0 \, 2\pi)}{\sup} \hat{\gamma}_m^{cross} (\lambda) > x \right] \underset{N \longrightarrow \infty}{\longrightarrow} 0 $.
### Analysis of $\underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \hat{\gamma}_m^{sign} (\lambda) \right\vert$
The proof of convergence of this quantity follows the same principles. We have: $$\hat{\gamma}_m^{sign} (\lambda) = \mathbf{d}_m^H (\lambda) \, \frac{\mathbf{M} \, \boldsymbol{\delta}^H \, \boldsymbol{\Gamma} \, \boldsymbol{\delta}\, \mathbf{M}^H}{N} \, \mathbf{d}_ m (\lambda) \, .$$
As previously, let $\mathds{I}_{m}$ be a $m \times m$ matrix containing 1 everywhere and let $\mathbf{E}(\lambda)= \mathbf{D}_m(\lambda) \, \mathds{I}_m \, \mathbf{D}_m^H(\lambda)$. Then: $$\hat{\gamma}_m^{sign} (\lambda) = 2 \, \mathcal{R}e \left[ \frac{1}{N} \, \mathrm{Tr}\left( \mathbf{M} \, \boldsymbol{\delta} \, \boldsymbol{\Gamma} \, \boldsymbol{\delta}^H \, \mathbf{M}^H \mathbf{E}\right) \right]\, .$$ Let $\mathbf{A}(\lambda) = \mathbf{M}^H \, \mathbf{E} \, \mathbf{M} \, \boldsymbol{\delta}$ and $\mathbf{B} = \boldsymbol{\Gamma} \, \boldsymbol{\delta}^H$ be two matrix respectively of size $p \times N$ and $N \times p$. Defining $\mathbf{a}(\lambda) = \mathrm{vec}(\mathbf{A}(\lambda))$ and $\mathbf{b}= \mathrm{vec}(\mathbf{B})$, we have: $$\begin{aligned}
\hat{\gamma}_m^{sign} (\lambda) &= \frac{2}{N} \mathcal{R}e \left( \mathrm{vec}^T(\mathbf{A}(\lambda)) \, \mathrm{vec}(\mathbf{B}) \right)\, , \\
&= \displaystyle \frac{2}{N} \, \mathcal{R}e \left( \mathbf{a}^T(\lambda) \, \left(\mathbf{M}^H \, \mathbf{E} \, \mathbf{M} \otimes \mathbf{I}_N\right) ^{-T} \right. \\
& \left. \left(\mathbf{M}^H \,\mathbf{E} \,\mathbf{M} \otimes \mathbf{I}_N \right)^T \, \left( \boldsymbol{\Gamma} \otimes \mathbf{I}_p \right) \, \left(\boldsymbol{\Gamma} \otimes \mathbf{I}_p\right)^{-1} \, \mathbf{j} \right)\, , \\
&=\displaystyle \frac{2}{N} \, \sum_{k=1}^{N \, p} \mathcal{R}e (a_k(\lambda) ) \, \mathcal{R}e (b_k) - \mathcal{I}m (a_k(\lambda)) \, \mathcal{I}m (b_k)\, .\end{aligned}$$
Let us define $ \mathbf{C}(\lambda) = \mathbf{M}^H \, \mathbf{E} \, \mathbf{M} \otimes \mathbf{I}_N$, $\mathbf{D} = \boldsymbol{\Gamma} \otimes \mathbf{I}_p$, $\tilde{\mathbf{a}}(\lambda) = \mathbf{C}^{-1}(\lambda) \, \mathbf{a}(\lambda)$, $\tilde {\mathbf{b}} = \mathbf{D}^{-1} \, \mathbf{b}$, $ c_k = \displaystyle \sum_{l=1}^{N \, p} \left(\mathbf{C}^{T}(\lambda)\right)_{l,k}$ and $d_k = \displaystyle \sum_{s=1}^{N \, p} \left(\mathbf{D}\right)_{s,k}$. Using Lemma \[lemma6\] and the Markov inequality, it can be shown that, for any fixed $\lambda \in [0 \, \,2\pi )$ and a constant $\mu$ such that $0 < \mu < \left( \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \Vert \mathbf{C}(\lambda) \Vert \sup \Vert \mathbf{D} \Vert \right)^{-1}$: $$\begin{aligned}
&\mathds{P} \left[ \hat{\gamma}_m^{sign} (\lambda) > x \right] \\
&\leq \exp{\left(-N \, \nu \, x - \displaystyle \sum_{k=1}^{N\, p} \log \left( 1 - \mu^2 \, \left\vert c_k(\lambda) \right\vert^2 \, \left\vert d_k \right\vert^2 \right)^{-1}\right) } \, .\end{aligned}$$ As the matrix $\mathbf{\Gamma}$ is absolutely summable, then, for all $k$, $\left\vert d_k \right\vert^2 \leq K$ where $K$ is a constant. Now, for all $k$, we have $$\begin{aligned}
\vert c_k(\lambda) \vert &= \left\vert \displaystyle \sum_{s=1}^{p} \left[ \sum_{l=1}^{m} M_{l,s} \sum_{j=1}^{m} M_{j,k} \, e^{i\,(j-l) \lambda} \right] \right\vert \, ,\\
&\leq \displaystyle \sum_{s=1}^{p} \left[ \sum_{l=1}^{m} \left\vert M_{l,s} \right\vert \sum_{j=1}^{m} \left\vert M_{j,k} \right\vert \right] \, .\end{aligned}$$
The columns of $\mathbf{M}$ are absolutely summable. As $p$ is fixed and $p \ll N$, with $K$ a constant, we have $\vert c_k(\lambda) \vert \leq K$. The coefficients of the matrix $\boldsymbol{\Gamma}$ being absolutely summable, for all $k$, we have find a constant $K_1$ such that $\vert d_k \, \vert \leq K_1$ . By defining $w$ as a constant small enough and $\mu = \displaystyle \frac{w}{\sqrt{N}}$ such that $\mu ^2 \, \vert c_k \, \vert^2 \, \vert d_k \vert^2 \underset{N \longrightarrow \infty}{\longrightarrow} 0 $, then, for all $x > 0$ and $A$ a constant, we have the following inequality: $$\mathds{P} [\hat{\gamma}_m^{sign} (\lambda) > x] \leq \exp{\left(- N^{1/2} \, w \, x - A\right)} \, .$$
As for $\gamma_m^{cross} (\lambda)$, it remains to prove than $\underset{i \in \mathcal{I}}{\max} \underset{\lambda \in [\lambda_i \, \, \lambda_{i+1}] }{\sup} \vert \hat{\gamma}_m^{sign} (\lambda) - \hat{\gamma}_m^{sign} (\lambda_i)\vert \overset{a.s.}{\longrightarrow} 0$ and this will left to the reader as it is the same as the proof of $\chi_1$. We have proven than $\mathds{P} \left[ \underset{\lambda \in [0 \, 2\pi)}{\sup} \hat{\gamma}_m^{sign} (\lambda) > x \right] \underset{N \longrightarrow \infty}{\longrightarrow} 0 $. As the right term of tends to zero when $N$ is tends to infinity, the proof of Theorem 1 is completed.
Proof of Theorem 3
-------------------
The proof follow the same idea. With the notation $\check{\mathbf{C}}_{FP} = \mathcal{T} (\hat{\mathbf{C}}_{FP})$ where $\mathcal{T}$ is the Toeplitz operator defined in the introduction, the equation to prove becomes: $$\left\Vert \mathcal{T} (\hat{\mathbf{C}}_{FP}) - \mathds{E}\left[v(\tau \gamma) \, \tau\right] \,\mathbf{C} \right\Vert \overset{a.s.}{\longrightarrow} 0 \, .
\label{eqprooof2}$$
This equation can be split as: $$\begin{aligned}
&\left\Vert \mathcal{T} \left( \hat{\mathbf{C}}_{FP} \right) - \mathds{E}\left[v(\tau \gamma) \,\tau\right] \, \mathbf{C} \right\Vert \\
& \leq \left\Vert \mathcal{T} \left(\hat{\mathbf{C}}_{FP}- \hat{\mathbf{S}}\right) \right\Vert + \left\Vert \mathcal{T} \left( \hat{\mathbf{S}} \right) - \mathds{E}\left[v(\tau \,\gamma) \,\tau\right]\, \mathbf{C} \right\Vert \, .\end{aligned}$$
Let us considering the following notations:
- $\hat{\mathbf{S}}$ the matrix such as $ \left\Vert \mathbf{\check{\Sigma}} - \hat{\mathbf{S}} \right\Vert \overset{a.s.}{\longrightarrow} 0$, as Theorem 3 has stated. As a reminder, $\hat{\mathbf{S}}$ is the matrix defined by: $$\hat{\mathbf{S}} = \frac{1}{N} \displaystyle \sum_{i=1}^{N} v\left(\tau_i \,\gamma\right) \, \mathbf{y}_{wi} \, \mathbf{y}_{wi}^{H} \, ,$$ where $\gamma$ is the unique solution (if defined) of: $$1 = \frac{1}{N} \displaystyle \sum_{i=1}^{N} \frac{\psi(\tau_i \, \gamma)}{1+ c \,\psi(\tau_i \,\gamma)}\, ,$$ where $g : x \mapsto \displaystyle \frac{x}{1-c\,\phi(x)}$, $v : x \mapsto u \, o \, g^{-1} (x)$ and $\psi:x\mapsto x \, v(x)$.
- If $\mathbf{A} = \mathcal{T}\left(\left(a_0, \ldots, a_{m-1}\right)^T\right)$ is a Toeplitz matrix ($a_{-k} = a_k^\ast$), we can define the spectral density as: $$\gamma^{\mathbf{A}} (\lambda) \overset{\Delta}{=} \displaystyle \sum_{k=1-m}^{m-1} a_k \,e^{i\,k\,\lambda} \, .$$ Finally, we denote by $\hat{\gamma}^{\mathbf{A}}(\lambda)$ the estimated spectral density of Toeplitz matrix $\mathbf{A}$.
To prove the consistency, we will decompose, as for Theorem 1, the equation in two parts. As matrices $\mathcal{T} \left(\hat{\mathbf{C}}_{FP}\right)$ and $\mathbf{C}$ are Toeplitz, it follows through : $$\begin{aligned}
&& \left\Vert \mathcal{T} \left(\hat{\mathbf{C}}_{FP}\right) - \mathds{E}[v(\tau \,\gamma) \, \tau] \, \mathbf{C} \right\Vert \nonumber \\
& \leq & \underset{\lambda \in \left[ 0, 2 \,\pi \right) }{\sup} \left\vert \hat{\gamma}^{\hat{\mathbf{S}}} (\lambda) - \gamma^{\mathds{E}[v(\tau \, \gamma) \,\tau] \, \mathbf{C}}(\lambda) \right\vert + \left\Vert \mathcal{T} \left( \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right) \right\Vert \, \nonumber \,\\
& \leq & \chi_1 + \chi_2 \label{eqpreuveth21} ,\end{aligned}$$ where $\chi_1 = \underset{\lambda \in \left[ 0,2 \, \pi \right) }{\sup} \left\vert \hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda) - \gamma^{ \mathds{E}[v(\tau \,\gamma) \,\tau]\, \mathbf{C}}(\lambda) \right\vert $ and $\chi_2 = \left\Vert \mathcal{T} ( \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} ) \right\Vert$.\
### Part 1: convergence of $\chi_1 = \underset{\lambda \in \left[ 0,2 \, \pi \right) }{\sup} \left\vert \hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda) - \gamma^{ \mathds{E}[v(\tau \,\gamma) \,\tau]\, \mathbf{C}}(\lambda) \right\vert $
We will split $\chi_1$ into two sub-terms: $$\begin{aligned}
&& \underset{\lambda \in \left[ 0, 2 \,\pi \right) }{\sup} \left\vert \hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda) - \gamma^{ \mathds{E}[v(\tau \,\gamma) \,\tau] \, \mathbf{C}}(\lambda) \right\vert \\
& \leq & \underset{\lambda \in \left[ 0,2 \,\pi \right) }{\sup} \left\vert \hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda) - \mathds{E} \left[\hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda)\right] \right\vert \\
&+& \underset{\lambda \in \left[ 0,2 \,\pi \right) }{\sup} \left\vert \mathds{E} \left[\hat{\gamma}^{\hat{\mathbf{S}}} (\lambda)\right] - \gamma^{ \mathds{E}[v(\tau \,\gamma) \,\tau] \, \mathbf{C}}(\lambda) \right\vert \, , \\
& \leq & \chi_{11} + \chi_{12} \, ,
\end{aligned}$$ where $\chi_{11} = \underset{\lambda \in \left[ 0,2\, \pi \right) }{\sup} \left\vert \hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda) - \mathds{E} \left[\hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda)\right] \right\vert $ and $\chi_{12} = \underset{\lambda \in \left[ 0,2 \, \pi \right) }{\sup} \left\vert \mathds{E} \left[\hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda)\right] - \gamma^{ \mathds{E}\left[v(\tau \,\gamma) \, \tau\right] \, \mathbf{C}}(\lambda) \right\vert$.\
### Part 1.1: convergence of $\chi_{11}$ {#part-1.1-convergence-of-chi_11 .unnumbered}
We will need the following lemma:
\[lemma7\] $$\hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda) = \mathbf{d}_m^H (\lambda) \, \hat{\mathbf{S}} \, \mathbf{d}_m (\lambda)\, ,
\label{eq1th2lemme5}$$ and:
$$\mathds{E} \left[\hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda) \right] = \mathds{E}\left[v(\tau \, \gamma) \, \tau\right] \, \mathbf{d}_m^H (\lambda) \, \mathbf{I}_m \, \mathbf{d}_m (\lambda)\, ,
\label{eq2th1lemme5}$$
where $\mathbf{d}_m(\lambda) =\displaystyle \frac{1}{\sqrt{m}} \left[1, e^{-i \,\lambda}, \ldots, e^{-i\,(m-1)\,\lambda}\right]^T$.
This is the same idea than for Lemma \[lemma1\]. First, we can write: $$\begin{aligned}
&& \hat{\gamma}^{\hat{\mathbf{S}}}(\lambda) = \displaystyle \sum_{k=1-m}^{m-1} \check{s}_k \, e^{i\,k \,\lambda} \, , \nonumber \\
& = & \frac{1}{m\,N} \displaystyle \sum_{k=1-m}^{m-1} e^{i\,k \,\lambda} \displaystyle \sum_{j=0}^{m-1} \sum_{n=0}^{N-1} \hat{s}_{j,n} \, \hat{s}^{\star}_{j+k,n} \, \mathds{1}_{0 \leq j+k< m} \, ,\nonumber \\
& = & \displaystyle \frac{1}{m\,N} \displaystyle \sum_{l,l'=0}^{m-1} e^{-i\,(l'-l) \,\lambda} \displaystyle \sum_{n=0}^{N-1} \hat{s}_{l,n} \, \hat{s}^{\star}_{l',n}\nonumber = \mathbf{d}_m^H (\lambda)\, \hat{\mathbf{S}} \, \mathbf{d}_m (\lambda) \, .\end{aligned}$$
The first part of the Lemma is then proven. Concerning $\mathds{E} \left[\hat{\gamma}^{\hat{\mathbf{S}}} (\lambda) \right]$, we can define $\mathbf{D}$ as the diagonal matrix containing the $\left\{v(\tau_i \,\gamma)\right\}_{i\in \llbracket 0, N-1\rrbracket}$. We obtain: $$\begin{aligned}
\mathds{E} \left[ \hat{\gamma}^{ \hat{\mathbf{S}}} (\lambda) \right] &= \mathbf{d}_m^H (\lambda) \, \mathds{E}\left[\hat{\mathbf{S}}\right] \, \mathbf{d}_m (\lambda) \, , \\
&= \mathbf{d}_m^H(\lambda) \, \mathds{E} \left[ \frac{\mathbf{Y}_w \, \mathbf{D}\, \mathbf{Y}_w^H}{N} \right] \, \mathbf{d}_m (\lambda) \, .\end{aligned}$$
Then expliciting each element of $\mathds{E}\left[\mathbf{Y}_w \, \mathbf{D} \, \mathbf{Y}_w^H\right]$ leads to: $$\begin{aligned}
& \left( \mathds{E}\left[ \mathbf{Y}_w \, \mathbf{D} \, \mathbf{Y}_w^H \right] \right) _{i,j} = \mathds{E} \left[ \displaystyle \sum_{n=0}^{N-1} v(\tau_n \,\gamma) \, y_{w \,i,n} \, y^{\star}_{w \, j,n} \right]\, , \\
&= \displaystyle \sum_{n=0}^{N-1} \mathds{E} \left[ v(\tau_n \, \gamma) \, \tau_n\right] = N \, \mathds{E}\left[ v(\tau_n \, \gamma) \, \tau_n\right]\, .\end{aligned}$$
We obtain the following result: $\mathds{E}\left[\hat{\gamma}^{\hat{\mathbf{S}}} (\lambda)\right] = \mathds{E} \left[v(\tau \,\gamma) \,\tau\right] \, \mathbf{d}_m^H(\lambda) \,\mathbf{I}_m \, \mathbf{d}_m(\lambda)$.
The rest of the proof for $\chi_{11}$ is the same as for Theorem 1 $\hat{\gamma}^{noise}$, but with $\mathbf{T}$ containing the $\left\{\tau_i\right\}_{i}$ on its diagonal, we will have $\left\Vert \mathbf{\mathbf{T}} \right\Vert_{\infty} \, \left\Vert \mathbf{\mathbf{D}} \right\Vert_{\infty}$ instead of $\left\Vert \mathbf{\mathbf{T}} \right\Vert_{\infty}$. We obtain so $\chi_{11} \overset{a.s.}{\longrightarrow} 0$ as $m \longrightarrow \infty$.\
### Part 1.2: convergence of $\chi_{12}$ {#part-1.2-convergence-of-chi_12 .unnumbered}
Lemma \[lemma7\] and give us $$\mathds{E}\left[\hat{\gamma}^{\hat{\mathbf{S}}} (\lambda)\right]= \mathds{E} \left[v(\tau \,\gamma) \,\tau \right] \, \mathbf{d}_m^H(\lambda) \, \mathbf{C} \, \mathbf{d}_m (\lambda) \, .$$ and $\mathds{E}\left[v(\tau \,\gamma) \,\tau\right] \, \gamma^\mathbf{C}(\lambda) = \mathds{E} \left[v(\tau \,\gamma) \,\tau\right] \, \mathbf{d}_m^H(\lambda) \, \mathbf{C} \, \mathbf{d}_m (\lambda)$. This yields $\chi_{12} = 0$.\
### Part 2: convergence of $\chi_2 = \left\Vert \mathcal{T} \left( \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}}\right) \right\Vert$
It is proven, in [@Couillet15a] that $\left\Vert \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right\Vert \overset{a.s.}{\longrightarrow} 0$. Let $\mathbf{J}$ be a matrix such that $\left(\mathbf{J}\right)_{j-i = 1} = 1$ and $0$ elsewhere. $\mathbf{J}^k$ contains 1 only on the $k^{th}$ diagonal. As before, thanks to , we have: $$\left\Vert \mathcal{T} \left( \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right) \right\Vert \leq \underset{\lambda \in \left[ 0,2 \pi \right) }{\sup} \left\vert \displaystyle \sum_{k=1-m}^{m-1} \left( \check{fp}_k - \check{s}_k \right) \, e^{i\,k\, \lambda} \right\vert \, .$$ Let us define $\mathcal{T} \left( \hat{\mathbf{C}}_{FP} \right) = \mathcal{L}\left(\left(\check{fp}_0, \ldots, \check{fp}_{m-1}\right)^T\right)$ with $\check{fp}_{-k} = \check{fp}_k^\ast$ and $\mathcal{T} \left( \hat{\mathbf{S}}\right) = \mathcal{L}\left(\left(\check{s}_0, \ldots, \check{s}_{m-1}\right)^T\right)$ with $\check{s}_{-k} = \check{s}_k^\ast$. We have: $$\begin{aligned}
& \underset{\lambda \in \left[ 0,2 \,\pi \right) }{\sup} \left\vert \displaystyle \sum_{k=1-m}^{m-1} \left( \check{fp}_k - \check{s}_k \right) \, e^{i\, k\, \lambda} \right\vert \\
&= \underset{\lambda \in \left[ 0, 2 \,\pi \right) }{\sup} \left\vert \displaystyle \sum_{k=1-m}^{m-1} \frac{1}{m} \sum_{p-1}^m \left(\check{fp}_k - \check{s}_k \right) \, e^{i\, k \, \lambda} \, \mathds{1}_{0 \leq p+k \leq m} \right\vert \, ,
\\ &= \underset{\lambda \in \left[ 0,2 \,\pi \right) }{\sup} \left\vert \mathrm{Tr} \left( \left(\hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}}\right) \, \frac{1}{m} \displaystyle \sum_{k=1-m}^{m-1} \left(\mathbf{J}^T\right)^k \, e^{i \, k\, \lambda} \right) \right\vert \, .
\end{aligned}$$
Moreover $\displaystyle \frac{1}{m} \displaystyle \sum_{k=1-m}^{m-1} \left(\mathbf{J}^T\right)^k \, e^{i \, k\, \lambda} = \mathbf{d}_m (\lambda) \, \mathbf{d}_m^{H} (\lambda)$. This leads to: $$\begin{aligned}
&\left\Vert \mathcal{T} \left(\hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right) \right\Vert \\
&\leq \underset{\lambda \in \left[ 0, 2 \,\pi \right) }{\sup} \left\vert \mbox{Tr} \left( \left(\hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}}\right) \, \mathbf{d}_m(\lambda) \, \mathbf{d}_m^{H} (\lambda) \right) \right\vert \\
&= \underset{\lambda \in \left[ 0, 2 \, \pi \right) }{\sup} \left\vert \mathbf{d}_m^{H} (\lambda) \, \left(\hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right) \, \mathbf{d}_m(\lambda) \right\vert \, .\end{aligned}$$
For any vector $\mathbf{x}$, the last equation becomes: $$\begin{aligned}
&& \underset{\lambda \in \left[ 0,2 \,\pi \right) }{\sup} \left\vert \mathbf{d}_m^{H} (\lambda) \, \left(\hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right) \, \mathbf{d}_m(\lambda) \right\vert \\
& \leq & \underset{\left\Vert \mathbf{x} \right\Vert_2 = 1}{\sup} \left\vert \mathbf{x}^H \, \left( \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}}\right) \, \mathbf{x} \right\vert \, , \\
& \leq &\underset{\left\Vert \mathbf{x} \right\Vert_2 = 1}{\sup} \left\Vert \left( \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}}\right) \, \mathbf{x} \right\Vert_2 \leq \left\Vert \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}}\right\Vert \, .\end{aligned}$$
Finally, we obtain: $$\left\Vert \mathcal{T} \left( \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right) \right\Vert \leq \left\Vert \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right\Vert \, .$$
As $\left\Vert \hat{\mathbf{C}}_{FP} - \hat{\mathbf{S}} \right\Vert \overset{a.s.}{\longrightarrow} 0$ then $\chi_2 \overset{a.s.}{\longrightarrow} 0$ and the proof of Theorem 3 is completed.
Proof of Theorem 2
==================
As the proof is the same for $\check{\mathbf{\Sigma}}_{SCM}$ and $\check{\mathbf{\Sigma}}_{FP}$, let $\check{\mathbf{\Sigma}}$ denote one or the other of these matrices.\
From the equations and , as $\mathbf{\check{y}}_{wi} = \mathbf{\check{C}}^{-1/2} \, \mathbf{y}_i$, $\check{\mathbf{\Sigma}}$ is the unique solution of: $$\begin{aligned}
\mathbf{\Sigma} = \frac{1}{N} \displaystyle \sum_{i=0}^{N-1} & u \left( \frac{1}{m} \, \mathbf{y}_i^H \, \mathbf{\check{C}}^{-1/2} \, \mathbf{\Sigma}^{-1} \, \mathbf{\check{C}}^{-1/2} \,\mathbf{y}_i \right)\, \times \\
& \mathbf{\check{C}}^{-1/2} \,\mathbf{y}_i \, \mathbf{y}_i^H \, \mathbf{\check{C}}^{-1/2} \, .\end{aligned}$$
Rewriting this equation with the $\left\{\mathbf{\check{y}}_{wi}\right\}_i$ $$\begin{aligned}
& \mathbf{C}^{-1/2} \mathbf{\check{C}}^{1/2} \, \mathbf{\Sigma} \, \mathbf{\check{C}}^{1/2} \, \mathbf{C}^{-1/2} \\
& = \displaystyle \frac{1}{N} \, \sum_{i=0}^{N-1} u \left( \frac{1}{m} \, \mathbf{\check{y}}_{wi}^H \, \left( \mathbf{C}^{-1/2} \, \mathbf{\check{C}}^{1/2} \, \mathbf{\Sigma}\, \mathbf{\check{C}}^{1/2} \, \mathbf{C}^{-1/2} \, \right)^{-1} \right. \times \\
& \left. \mathbf{\check{y}}_{wi} \right) \, \mathbf{\check{y}}_{wi} \, \mathbf{\check{y}}_{wi}^H \, ,\end{aligned}$$ we obtain the following relationship between $\check{\mathbf{\Sigma}}$ and $\hat{\mathbf{\Sigma}}$: $$\check{\mathbf{\Sigma}} = \mathbf{\check{C}}^{-1/2} \, \mathbf{C}^{1/2} \, \hat{\mathbf{\Sigma}} \, \mathbf{C}^{1/2} \, \mathbf{\check{C}}^{-1/2} \, .
\label{Sigmacheck}$$
Then, equation can be rewritten as $$\left\Vert \check{\mathbf{\Sigma}} - \hat{\mathbf{S}}\right\Vert \leq \left\Vert \check{\mathbf{\Sigma}} - \hat{\mathbf{\Sigma}}\right\Vert \, + \left\Vert \hat{\mathbf{\Sigma}} - \hat{\mathbf{S}}\right\Vert\, .
\label{inequality}$$
Concerning the second term of the right hand side of , it is proven in [@Couillet15b] that the matrix $\hat{\mathbf{S}}$ given by is such that $$\left\Vert \hat{\mathbf{\Sigma}} - \hat{\mathbf{S}} \right\Vert \overset{a.s.}{\longrightarrow} 0 \, .
\label{eqcouillet}$$
With , the first term of right hand side of can be rewritten as: $$\begin{aligned}
&& \left\Vert \check{\mathbf{\Sigma}}- \mathbf{\hat{\Sigma}} \right\Vert \le \left\Vert \mathbf{\check{C}}^{-1/2} \, \mathbf{C}^{1/2} \, \mathbf{\hat{\Sigma}} \, \mathbf{C}^{1/2} \, \mathbf{\check{C}}^{-1/2} - \mathbf{\hat{\Sigma}} \, \mathbf{C}^{1/2} \, \mathbf{\check{C}}^{-1/2}\right\Vert \nonumber \\
&& + \left\Vert \mathbf{\hat{\Sigma}} \, \mathbf{C}^{1/2} \, \mathbf{\check{C}}^{-1/2} - \mathbf{\hat{\Sigma}} \right\Vert \, .\end{aligned}$$ After left and right factorizations, we obtain: $$\begin{aligned}
\left\Vert \check{\mathbf{\Sigma}}- \mathbf{\hat{\Sigma}} \right\Vert \le \left\Vert \mathbf{\check{C}}^{-1/2} \, \mathbf{C}^{1/2} -\mathbf{I}_m\right \Vert \left\Vert \mathbf{\hat{\Sigma}} \right\Vert \left(\left\Vert \mathbf{C}^{1/2} \, \mathbf{\check{C}}^{-1/2} \right\Vert +1\right) \, . \nonumber\end{aligned}$$ As $\Vert \mathbf{C}\Vert $ has a bounded support, $\Vert \check{\mathbf{C}} \Vert $ is bounded too since its eigenvalues support converges almost surely toward the true distribution. Moreover, Theorem 1 and Theorem 2 have proved the consistency $ \left\Vert \mathbf{C} -\check{\mathbf{C}} \right\Vert \overset{a.s.}{\longrightarrow} 0$. This ensures the proof.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank the DGA for its financial support.
|
---
abstract: 'The quartic confining potential has emerged as a key ingredient to obtain fast rotating vortices in BEC as well as observation of quantum phase transitions in optical lattices. We calculate the critical temperature $T_c$ of bosons at which normal to BEC transition occurs for the quartic confining potential. Further more, we evaluate the effect of finite particle number on $T_c$ and find that $\Delta T_c/T_c$ is larger in quartic potential as compared to quadratic potential for number of particles $ < 10^5$. Interestingly, the situation is reversed if the number of particles is $\gtrsim10^5$.'
author:
- 'S. Gautam'
- 'D. Angom'
title: 'Critical Temperature for Bose-Einstein condensation in quartic potentials'
---
Introduction
============
Particles in anharmonic potentials is a well studied example of chaotic system. A simple but good model of a system which exhibits parametric transitions from regular to chaos and vice-versa is the quartic oscillator. The parameters which control the transitions are the coupling strengths of the cross terms. The quartic oscillator besides being a case study of chaotic system, plays a very important role in the guise of Higg’s field, through which all the known fundamental particles acquire finite masses. The same potential also appears in optical lattices, where quantum phase transition from superfluid to Mott insulator has been observed [@marcus-02]. Optical lattices are regular intensity patterns of light created with counter propagating laser beams. The laser beams have a Gaussian profile and create an overlapping anharmonic potential across the optical lattice. Recent theoretical studies show that realization of the quantum phase transition can be more robust with a quartic potential [@olivier-06]. Besides the quantum phase transitions, introducing a quartic confining potential stabilizes fast rotating vortices in Bose-Einstein condensates [@fetter-01; @ghosh-04; @kim-05; @ionut-05; @fetter-05; @bargi-06]. In a recent work [@kling-07], the condensation temperature and thermodynamic properties of a rotating ideal Bose gas in an anharmonic trap has been studied. However, in the experimental realization of the quadratic plus quartic confining potential, the observation of the fast rotating vortices eludes an unambiguous detection [@vincent-04].
In this paper we calculate the critical temperature $T_c$ at which bosons confined by a quartic potential condense. BEC in such a potential was studied for low-dimensional systems [@vanderlei-91]. This calculation requires the density of states. In our calculations we use the semiclassical expression of the density of states, which is valid at higher energies. It is however not appropriate to study low lying states. In particular, the energy of the ground state is essential to estimate $T_c$ for finite number of particles. To estimate the correction to $T_c$ in the finite particle case, we calculate the ground state energy analytically [@mathews]. The calculation is based on a method which optimizes the matrix elements of the quartic potential Hamiltonian in the harmonic oscillator basis.
Our calculations show that $T_c$ in quartic confining potential is higher than quadratic potential. This is perhaps to be expected, since $T_c$ varies as $N^{1/\alpha}$ when the density of states is proportional to $\epsilon^{\alpha -1}$. In case of 3D harmonic oscillator potential $\alpha=3$ whereas it is 9/4 in the case of 3D quartic potential. Hence, in the quartic potential case $T_c$ varies as $N^{4/9}$ compared to $N^{1/3}$ in the case of harmonic potential. From our calculations, it is evident that the cross terms increase $T_c$ in 2D as well as 3D quartic potential. We find that $T_c$ rises by factor of 1.2 and 1.1 in the 3D and 2D potentials respectively. However, the experimental realizations of the trapping potentials, which are created from laser beams, are more appropriately described without the cross terms.
$T_c$ for 3D quartic potential trap
===================================
The general form of the quartic oscillator potential is $\lambda(\bm{r}.\bm{r})^2 $. It is homogeneous and has cross terms in the Cartesian coordinate representation which couple motions along different axes. Potentials of this form occur in optical lattices, where counter propagating lasers create undulating patterns of standing radiation field. In one dimension, a pair of counter propagating Gaussian laser beams along $z$-axis of intensity profile $I_0\exp(-2r^2/w)$ creates an array of periodic intensity minima and maxima. These are located along the $z$-axis. Depending on the detuning of the laser, the atoms are attracted to the intensity minima or maxima. Usually the wavelength of the laser $\lambda$ is much smaller than the beam width $w$. To a very good approximation, the potential across a surface normal to the laser beam $$\label{eq.a.a}
V(r) = I_0\left (\frac{-2r^2}{w} + \frac{4r^4}{w^2}\right ).$$ Tuning the parameters of the laser beams, it possible to retain only the quartic term. For simplicity, neglecting the cross terms, in three dimension $$\label{eq.a.b}
V(x, y, z)=\lambda(x^4+y^4+z^4).$$ The eigen energies of the corresponding Hamiltonian is the sum of eigen values corresponding to each dimension. For the one dimensional quartic oscillator, eigen values can be calculated by minimizing the expectation of the Hamiltonian in the basis states of harmonic oscillator of appropriately chosen frequency[@mathews]. Generalizing the result to three dimensional case, the eigen energy $$\label{eq.a.c}
\epsilon(n_1,n_2,n_3)= 1.389\sum_{i=1}^3(n_i+\frac{1}{2})^\frac{4}{3}
\left(\frac{\lambda{\hbar}^4}{m^2}\right)^\frac{1}{3}.$$ Since $1.389(\lambda\hbar^4/m^2)^{1/3}$ has the dimensions of energy we can represent this factor by $\hbar\omega$, then $$\label{eq.a.d}
\epsilon(n_1,n_2,n_3)=\sum_{i=1}^3(n_i+\frac{1}{2})^\frac{4}{3}\hbar\omega .$$ We now determine the number of states $G(\epsilon)$ with energy less than a given value $\epsilon$. For energies large compared to $\hbar\omega$, we may treat $n_i$’s as continuous variables and neglect the ground state energy. To calculate $G(\epsilon)$, we introduce a coordinate system in terms of the three variables $\epsilon_i=n_i^{4/3}\hbar\omega$. In this coordinate system $\epsilon=\epsilon_1+\epsilon_2+\epsilon_3$ defines a surface of constant energy $\epsilon$. Then $G(\epsilon)$ is proportional to the volume in the first octant bounded by the surface $$\label{eq.a.e}
G(\epsilon)=\frac{27}{64(\hbar \omega)^{9/4}}\int_0^{\epsilon}
{\epsilon_1}^{-1/4}d{\epsilon_1} \int_0^{\epsilon- \epsilon_1}
\!\!\!\!\!\!\!\!\!\!{\epsilon_2}^{-1/4}d\epsilon_2
\int_0^{\epsilon-\epsilon_1-\epsilon_2}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\epsilon_3^{-1/4}d{\epsilon_3}.$$ To evaluate the integral we use the relation $\int_0^u{x}^{\nu-1}( u-x) ^{\mu-1} =u^{\mu+\nu-1}B(\mu,\nu)$, where $B(\mu,\nu)= \Gamma(\mu)\Gamma(\nu)/\Gamma(\mu+\nu)$ [@grad]. Then the density of states $$\label{eq.a.f}
g(\epsilon)=\frac{dG(\epsilon)}{d\epsilon}=0.6852\frac{\epsilon^{5/4}}
{(\hbar\omega)^{9/4}}.$$ This expression for density of states is used to calculate $T_c$. For bosons, the total number of particles occupying the excited states is $$\label{eq.a.g}
N_{\rm exc}=\int_0^{\infty}\frac{g(\epsilon)}{e^{(\epsilon-\mu)/kT}-1}
d\epsilon.$$ At critical temperature $\mu \rightarrow 0$ in the case of bosons and $N_{\rm exc}$ is equal to the total number of bosons $N$. Evaluating the integral gives $T_c$ in terms of the number of bosons $$\label{eq.a.h}
kT_c=\frac{N^{4/9}\hbar\omega}{\left[0.6852\Gamma(9/4)
\zeta(9/4)\right] ^{4/9}}.$$ The corresponding expression for three dimensional isotropic harmonic oscillator potential is [@pethick] $$\label{eq.c.c}
kT_c=\frac{\hbar \omega_o N^{1/3}}{[\zeta(3)]^{1/3}}.$$ From eq. (\[eq.a.h\]) and eq. (\[eq.c.c\]), the ratio of the critical temperatures in the two potentials is $$\begin{aligned}
\label{eq.a.i}
\frac{( T_c)_{\rm quartic}}{( T_c)_{\rm harmonic}} &= &
\frac{N^{4/9}\zeta(3)^{1/3}}{N^{1/3}\left[ 0.6852\Gamma(9/4)\zeta(9/4)
\right]^{4/9}} \nonumber \\
&=&1.006N^{1/9}.\end{aligned}$$ Where we have taken $\hbar\omega=\hbar\omega_o$ to obtain the ratio. The ratio is proportional to $N^{1/9}$, which means that for $10^7$ atoms, $T_c$ in the case of 3D quartic potential trap is approximately six times higher than that of the 3D isotropic harmonic trap.
Effect of finite particle number
================================
The expression of $T_c$ in the previous section is with the approximation that the ground state energy is zero, which is a valid approximation when the system has large number of bosons. For finite number of bosons, the zero point energy causes a change in the value of $T_c$. For the 3D quartic potential, the ground state energy is [@mathews] $$\label{eq.a.j}
\epsilon_{\rm min}=2.41\left( \frac{\lambda\hbar^4}{m^2}\right)^{1/3}.$$ This should be equal to the change in the chemical potential at the critical temperature, that is $\Delta\mu=\epsilon_{\rm min}$. As the total number of bosons is fixed [@pethick] $$\label{eq.a.k}
dN=\left( \frac{\partial N}{\partial T}\right) _{\mu}dT+
\left( \frac{\partial N}{\partial{\mu}}\right)_T d\mu=0.$$ This implies $$\label{eq.a.l}
\left( \frac{\partial \mu}{\partial T}\right)_N= -\left( \frac{\partial N}
{\partial T}\right) \left(\frac{\partial N}{\partial \mu} \right)^{-1}_T.$$ Using the expression of $N$ at temperatures slightly above $T_c$ $$\label{eq.a.m}
N=C_{\alpha}\int_0^{\infty}\frac{\epsilon^{\alpha-1}}{e^{(\epsilon-\mu)/kT}
-1}d\epsilon.$$ This relation is obtained by substituting the general expression for the density of states i.e. $g(\epsilon)=C_\alpha \epsilon^{\alpha-1}$ in eq. (\[eq.a.g\]). Here $C_\alpha$ is a constant whose value depends on the form of the trapping potential. Then from eq. (\[eq.a.l\]) and eq. (\[eq.a.m\]) we get $$\label{eq.a.n}
\left( \frac{\partial \mu}{\partial T}\right)_N=-\alpha
\frac{\zeta(\alpha)}{\zeta(\alpha-1)}k.$$ In this expression $\alpha$ should be greater than 2, otherwise the relation is not valid since $\zeta(1)$ diverges. Using this expression, the change in the critical temperature due to the finite particle number is $$\label{eq.a.o}
\Delta T_c=-\frac{\zeta(\alpha-1)}{\alpha\zeta(\alpha)k}\Delta\mu.$$ In the case of 3D quartic potential $\alpha$ is equal to $9/4$. Then $$\begin{aligned}
\label{eq.a.p}
\Delta T_c & = & \frac{-4\zeta(5/4)}{9\zeta(9/4)k}\Delta\epsilon_{\rm min}
\nonumber \\
& = &\frac{-1.071\zeta(5/4)}{\zeta(9/4)k}
\left(\frac{\lambda\hbar^4}{m^2} \right)^{1/3}.\end{aligned}$$ A relative measure of the effect of zero point energy on the critical temperature is the fractional change of the critical temperature. It is the ratio between $\Delta T_c$ and $T_c$, for the present case $$\label{eq.a.q}
\frac{\Delta T_c}{T_c}=\frac{-0.9054\zeta(5/4)}{\zeta(9/4)}
\left( \frac{\lambda\hbar^4}{m^2}\right)^{1/3}
\frac{\left( \Gamma(9/4)\zeta(9/4)
\right)^{4/9}}{\hbar\omega N^{4/9}}.$$ Noting that $\left( \lambda\hbar^4/m^2\right)^{1/3} $ is equivalent to $\hbar\omega/1.389$ we get $$\label{eq.a.r}
\frac{\Delta T_c}{T_c}=\frac{-0.6891\zeta(1.25)N^{-4/9}}
{\zeta(2.25)^{5/9}}$$ $$\label{eq.a.s}
=-2.56N^{-4/9}.$$ For the 3D isotropic harmonic potential, the fractional change of the critical temperature is $$\label{eq.c.a}
\frac{\Delta T_c}{T_c}=-0.73N^{-1/3}.$$ If we compare eq. (\[eq.a.s\]) and eq. (\[eq.c.a\]), we find that the percentage decrease in $T_c$ is larger in the case of 3D quartic potential trap for number of particles $\lessapprox80,000$. But the scenario is reversed for number of particles $>80,000$. This is also evident from fig. (\[fig.2\]) where cross over point corresponds to the number of particles $\thickapprox80,000$.
Effect of cross terms
=====================
3D case
-------
Consider the general form of the quartic potential, as mentioned earlier $$\label{eq.a.t}
V(\bm{r})=\lambda(\bm{r}.\bm{r})^2.$$ In optical traps, it is possible to create confining potentials which are approximately close to this form but a truly spherically symmetric one is not realizable. The difficulty is in producing the cross terms of the potential, for example, terms like $x^2y^2$ in Cartesian coordinate representation. The absence and presence of the cross terms in quadratic and quartic potentials respectively introduce a key difference between the dynamics in the two potentials. In absence of the cross terms like $xy$, in quadratic potential, a perturbation to the dynamics of a particle along an axis remains confined along that axis. In contrast, it propagates to other axes in the case of quartic potential. For condensates in traps, an important parameter which reflects the effects of these terms is the critical temperature.
Semiclassically, total number of states available to the system can be obtained by dividing the total phase space volume by $h^3$, the volume of a single state $$\begin{aligned}
\label{eq.a.u}
G(\epsilon) & = &\frac{1}{h^3}\int{d\mathbf{x}}\int{d\mathbf{p}}, \\
\label{eq.a.v}
& = &\frac{16\pi^2}{h^3}\int_0^{r^*}r^2dr\int_0^{p^*}p^2dp.\end{aligned}$$ In the above equation $r^*$ and $p^*$ are radial coordinate and momentum corresponding to the classical turning point respectively. Transforming the variable of integration from $p$ to $\epsilon$ (using the relation $p^2/{2m}=\epsilon-V(\bm{r})$) we get $$\label{eq.a.w}
G(\epsilon)=\frac{16\pi^2m}{h^3}\int_0^{r^*}r^2dr
\int_0^{\epsilon^*}\sqrt{2m(\epsilon-V(\bm{r}))}d\epsilon.$$ Thus the density of states is $$\label{eq.a.x} g(\epsilon)=\frac{16\pi^2m}{h^3}\int_0^{r^*}r^2
\sqrt{2m\left( \epsilon-\lambda r^4\right) }dr.$$ Substituting $r^4=x$ we can transform the integral into a form which can be evaluated analytically [@grad] $$\begin{aligned}
\label{eq.a.y}
g(\epsilon)&=&\frac{4\pi^2m^{3/2}\sqrt{2\lambda}}{h^3}
\int_0^{\epsilon/\lambda}\left( \sqrt{\frac{\epsilon}
{\lambda}-x}\right) x^{-1/4}dx \\
\label{eq.a.z}
&=&\frac{4\sqrt{2}\pi^2m^{3/2}\Gamma( 3/2)\Gamma( 3/4)\epsilon^{5/4}}
{h^3\lambda^{3/4}\Gamma(9/4) }.\end{aligned}$$ Using this expression for the density of states in eq. (\[eq.a.g\]) we get $$\label{eq.b.a}
kT_c=\frac{N^{4/9}}{\left[ \Gamma(9/4)\zeta(9/4)\right]^{4/9} }
\left[ \frac{h^3\lambda^{3/4}\Gamma(9/4)}
{\Gamma(3/2)\Gamma(3/4)4\sqrt{2}\pi^2m^{3/2}}\right]^{4/9}.$$ Comparing with eq. (\[eq.a.h\]) we can obtain the ratio of $T_c$ in the two cases, with and without the cross terms, for the 3D isotropic quartic potential. It is found that $T_c$ with the cross terms is 1.2 times higher. This rise in $T_c$ can be attributed to the contribution from the cross terms which were neglected while deriving eq. (\[eq.a.h\]). The reason for the difference is, when $g(\epsilon)= C_\alpha \epsilon^{\alpha-1}$ then $T_c$ varies as $1/{C_\alpha}^{1/\alpha}$. Hence, the lower $T_c$ in 3D quartic potentials without the cross terms is due to the higher value of $C_\alpha$.
2D case
-------
In the 2D case, neglecting the cross terms, the potential is of the form $$\label{eq.b.b}
V(x,y)=\lambda(x^4+y^4).$$ Using the same approach as adopted in the 3D case, the total number of states available to the system is $$\begin{aligned}
\label{eq.b.c}
G(\epsilon)&=&\frac{9}{16(\hbar\omega)^{3/2}}
\int_0^\epsilon{\epsilon_1}^{-1/4}d \epsilon_1
\int_0^{\epsilon-\epsilon_1}\epsilon_2^{-1/4} d\epsilon_2 \\
\label{eq.b.d}
&=&\frac{3\Gamma(3/4)\Gamma(7/4)}{4(\hbar\omega)^{3/2}\Gamma(5/2)}
\epsilon^{3/2}.\end{aligned}$$ Thus the density of states $$\label{eq.b.e}
g(\epsilon)=\frac{0.9531\epsilon^{1/2}}{(\hbar\omega)^{3/2}}.$$ Substituting this expression of $g(\epsilon)$ in eq. (\[eq.a.g\]) we get $$\label{eq.b.f}
kT_c=\frac{{\hbar\omega}N^{2/3}}{\left[ \zeta(3/2)\Gamma(3/2)
0.9531\right]^{2/3}} .$$ The corresponding expression when cross terms are considered is [@vanderlei-91] $$\label{eq.b.g}
kT_c=\left[ \frac{Nh^2\sqrt {\lambda}}{2\pi^2m\Gamma(3/2)\zeta(3/2)}
\right]^{2/3}.$$ Comparing eq. (\[eq.b.f\]) and eq. (\[eq.b.g\]) we find that $T_c$ with the cross terms in eq. (\[eq.b.g\]) is approx. 1.12 times higher than $T_c$ without the cross terms in eq. (\[eq.b.f\]). Thus the cross terms increase $T_c$ in 2D as well as 3D case.
Conclusions
===========
Our calculations show that $T_c$ in the case of the 3D quartic potential trap is higher than that of the isotropic harmonic potential trap. This is due to the form of the density of states $g(\epsilon)$, which varies as $\epsilon^{5/4}$ and $\epsilon^2$ in 3D isotropic quartic and quadratic trapping potentials respectively. This implies lower density of states in quartic oscillator potential compared to isotropic harmonic oscillator potential. However, more interesting is the effect of the cross terms. In the 3D isotropic harmonic potential trap the cross terms are absent, which is not the case for the 3D quartic potential trap. The cross terms tend to decrease the density of states and raise $T_c$. These terms increase $T_c$ by factor of 1.2 and 1.1 in the 3D and 2D quartic trap potentials respectively as compared to the case without the cross terms. Experimentally, in optical traps, the potentials without the cross terms are more appropriate. We find that the effect of finite particle number is more pronounced in the 3D quartic potential when the number of particles is $< 10^5$. The situation is reversed when the number of particles is $\gtrsim10^5$. The cause of the reversal lies in the form of the fractional change $\Delta T_c/T_c$ for the two potentials. The ratio of the fractional change between 3D isotropic quartic potential to harmonic potential is $3.51/N^{1/9}$. It is $\thickapprox 1$ for $N\thickapprox 10^5$, $>1$ for $N<10^5$ and $<1$ for $ N\gtrsim10^5$. Thus, when $N < 10^5$ the constant factor is dominant in eq. (\[eq.a.s\]) and is responsible for the larger value of $\Delta T_c/T_c$ in quartic potential. But when $N\gtrsim 10^5$, $N^{-4/9}$ dominates and $\Delta T_c/T_c$ of the quartic potential is lower than that of the harmonic potential.
[99]{} M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch and I. Bloch, Nature [**415**]{}, 39 (2002). O. Gygi, H. G. Katzgraber, M. Troyer, S. Wessel and G. G. Batrouni, Phys. Rev. A [**73**]{}, 063606 (2006). A. L. Fetter, Phys. Rev. A. [**64**]{}, 063608 (2001). T. K. Ghosh, Phys. Rev. A. [**69**]{}, 043606 (2004). J. Kim and A. L. Fetter, Phys. Rev. A. [**72**]{}, 023619 (2005). I. Danaila, Phys. Rev. A. [**72**]{}, 013605 (2005). A. L. Fetter, B. Jackson and S. Stringari, Phys. Rev. A. [**71**]{}, 013605 (2005). S. Bargi, G. M. Kavoulakis and S. M. Reimann, Phys. Rev. A. [**73**]{}, 033613 (2006). S. Kling and A. Pelster, Phys. Rev. A [**76**]{}, 023609 (2007). V. Bretin, S. Stock, Y. Seurin and J. Dalibard, Phys. Rev. Lett. [**92**]{}, 050403 (2004). V. Bagnato and D. Kleppner, Phys. Rev. A. [**44**]{}, 7439 (1991). P. M. Mathews, M. Seetharaman and S. Raghavan, J. Phys. A [**15**]{}, 103 (1981). I. S. Gradshteyn and I. M. Ryzhik, [*Table of Integral Series And Products*]{}. Academic Press, sixth edition, 2001. C. J. Pethick and H. Smith, [*Bose-Einstein Condensation in Dilute Gases*]{} Cambridge University Press, 2002.
|
---
abstract: 'We study the radius of convergence of a differential equation on a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Several properties of this function are known: F. Baldassarri proved that it is continuous (see [@ContinuityCurves]) and the authors showed that it factorizes by the retraction through a locally finite graph (see [@finiteness] and [@finiteness2]). Here, assuming that the curve has no boundary or that the differential equation is overconvergent, we provide a shorter proof of both results by using potential theory on Berkovich curves.'
address:
- 'Institut de recherche mathématique avancée, 7, rue René Descartes, 67084 Strasbourg, France'
- 'Département de Mathématiques, Université de Montpellier II, CC051, Place Eugène Bataillon, 34095, Montpellier Cedex 5, France'
author:
- Jérôme Poineau
- Andrea Pulita
bibliography:
- 'biblio.bib'
nocite: '\nocite{}'
title: 'Continuity and finiteness of the radius of convergence of a $p$-adic differential equation *via* potential theory'
---
[^1]
Introduction
============
Let $K$ be a non-archimedean complete valued field of characteristic 0. Let $X$ be a quasi-smooth $K$-analytic curve, in the sense of Berkovich theory. Let ${\ensuremath{\mathscr{F}}}$ be a locally free ${\ensuremath{\mathscr{O}}}_{X}$-module of finite type endowed with an integrable connection $\nabla$.
By the implicit function theorem, in the neighbourhood of any $K$-rational point $x$, the curve is isomorphic to a disc, and it makes sense to consider the radius ${\ensuremath{\mathcal{R}}}(x)$ of the biggest disc on which $({\ensuremath{\mathscr{F}}},\nabla)$ is trivial. Extending the scalars, we may find a rational point above any given point and extend the definition to the whole curve $X$.
The radius of convergence is a function that has been actively investigated and its general behaviour is now rather well understood. The main features we are interested in are its continuity and a finiteness property (the fact that the map is controlled by its behaviour on a locally finite graph). Both are known. In the nineties already, G. Christol and B. Dwork proved the continuity of the radius of convergence over the skeleton of an annulus (see [@ChristolDwork]). This result was later extended to a continuity statement on affinoid domains of the affine line by F. Baldassarri and L. Di Vizio (see [@ContinuityBDV]), then on curves by F. Baldassarri (see [@ContinuityCurves]). As for the finiteness property, it was recently proven by the authors (see [@finiteness] and [@finiteness2]). The proofs of all these results are quite long and involved, which led us to believe that it was worth finding shorter ones, even to the expense of restricting the setting: here, we assume that the curve is boundary-free (*e.g.* the analytification of an algebraic curve) or that the connection is overconvergent. We have tried to make our paper as self-contained as possible.
Our techniques rely, for the main part, on potential theory on Berkovich curves, as developed by A. Thuillier in his thesis [@TheseThuillier]. We will provide a reminder in section \[sec:potential\] and then use freely the notions of Laplacian operator, harmonic and sub-harmonic functions, etc. As regards differential equations, we will only need to know how the radius of convergence behaves on an interval inside a disc or an annulus: it is continuous, concave, piecewise $\log$-linear with slopes that are rational numbers whose denominators are bounded by the rank of ${\ensuremath{\mathscr{F}}}$ (see theorem \[thm:Kedlaya\] and the discussion that precedes for attribution of the results).
Let us say a few words about the strategy of the proof. In the case where $X$ is an analytic domain of the affine line, with coordinate $t$, the ring ${\ensuremath{\mathscr{O}}}(X)$ may be endowed with the usual derivation $d=\mathrm{d}/\mathrm{d}t$. If the sheaf ${\ensuremath{\mathscr{F}}}$ is free on $X$, then, together with its connection $\nabla$, it comes from a differential module $(M,D)$. The radius of convergence of $(M,D)$ may then be computed as the radius of convergence of a power series, hence by the usual formula $\liminf_{n} (|f_{n}|^{-1/n})$, for some $f_{n} \in {\ensuremath{\mathscr{O}}}(X)$. It is here that potential theory enters the picture: by general arguments, the logarithm of the resulting function, or more precisely its lower semicontinuous envelope, is super-harmonic.
Unfortunately, this proof cannot be directly adapted to the case of a general smooth curve, even locally, for lack of a canonical derivation. Still, using geometric arguments, around any point $x$ of type 2, we manage to find a suitable derivation and use it to show that the logarithm of the radius of convergence of $({\ensuremath{\mathscr{F}}},\nabla)$ coincides with a super-harmonic function on some affinoid domain $Y$ of $X$ containing $x$ (*i.e.* outside a finite number of branches emanating from $x$). This will be sufficient for our purposes.
The rest of the proof relies on general properties of super-harmonic functions. We will use in a crucial way the fact that their Laplacians are Radon measures. More precisely, together with the fact that the non-zero slopes of the logarithm of the radius of convergence are bounded below in absolute value, this property implies that, around any point of type 2 of $X$, there may only be a finite number of directions along which this radius is not constant. Using the same strategy as in [@finiteness], we will deduce that it is locally constant outside a locally finite subgraph $\Gamma$ of $X$.
To prove that the radius of convergence is continuous, it is now enough to show that it is continuous on $\Gamma$. This will follow from another general property of super-harmonic functions: their restrictions to segments are continuous at points of type 2, 3 and 4.
Let us finally point out that the restriction to boundary-free curves (or overconvergent connections) is due to an inherent limitation of the methods of potential theory. Indeed, points that lie at the boundary of the space behave as if some directions out of them were missing and, at those points, the Laplacian of a function (which is a weighted sum of all outer derivatives) carries too little information for us to use.
**Acknowledgments**
We are very grateful to Amaury Thuillier for his comments and helpful discussions on the subject.
\[setting\] For the rest of the article, we fix the following: $K$ is a complete non-archimedean valued field of characteristic 0, $X$ is a quasi-smooth $K$-analytic curve[^2] endowed with a weak triangulation $S$, ${\ensuremath{\mathscr{F}}}$ is a locally free ${\ensuremath{\mathscr{O}}}_{X}$-module of finite type endowed with an integrable connection $\nabla$.
Definitions
===========
To define the radius of convergence of $({\ensuremath{\mathscr{F}}},\nabla)$, one needs to understand precisely the geometry of the curve $X$. We find it convenient to use A. Ducros’s notion of triangulation (see [@RSSen]), which enables to cut the curve into simple pieces. Our definition of radius of convergence actually depends on the choice of such a (weak) triangulation $S$ on $X$. We will not carry out the construction in every detail but content ourselves with the basic definitions and properties. We refer to section 2 of our previous paper [@finiteness2] for a more thorough exposition.
Triangulations
--------------
Our reference for this part is A. Ducros’s manuscript [@RSSen] and especially chapter 4.
Let us first recall that a connected analytic space is called a virtual open disc (resp. annulus) if it becomes isomorphic to a union of open discs (resp. annuli) over an algebraically closed valued field.
For any subset $Y$ of $X$, we denote $Y_{[2,3]}$ its subset of points of type 2 or 3.
\[defi:triangulation\] A locally finite subset $S$ of $X_{[2,3]}$ is said to be a weak triangulation of $X$ if any connected component of $X\setminus S$ is a virtual open disc or annulus.
Note that a weak triangulation may be empty (*e.g.* in the case of a disc or an annulus).
It is possible to associate a skeleton $\Gamma_{S}$ to a weak triangulation $S$ by considering the union of the skeletons of the connected components of $X\setminus S$ that are virtual annuli. It is a locally finite subgraph of $X_{[2,3]}$.
One of the main results of A. Ducros’s manuscript [@RSSen] is the existence of a triangulation[^3] on any quasi-smooth curve. For the rest of the article, we assume that $X$ is endowed with a weak triangulation $S$.
For any complete valued extension $L$ of $K$, the weak triangulation $S$ may be canonically extended to a weak triangulation $S_{L}$ of $X_{L}$.
Distances
---------
In the following, we will need to measure distances on the curve $X$, or at least on some segments inside it. We will explain quickly how this may be done by recalling the rough lines of A. Ducros’s notion of gauge (“toise” in French, see [@RSSen 1.6.1]). This construction is not new and several equivalent ones may be found in the literature (see [@BR section 2.7] in the case of the line over an algebraically closed field or [@TheseThuillier section 2.2] in the general case, for instance).
We first introduce notations for discs and annuli that will also be useful in the rest of the text.
Let ${\ensuremath{\mathbf{A}^{1,\mathrm{an}}_{K}}}$ be the affine analytic line with coordinate $t$. Let $L$ be a complete valued extension of $K$ and $c\in L$. For $R>0$, we set $$D_{L}^+(c,R) = \big\{x\in {\ensuremath{\mathbf{A}^{1,\mathrm{an}}_{L}}}\, \big|\, |(t-c)(x)|\le R\big\}$$ and $$D_{L}^-(c,R) = \big\{x\in {\ensuremath{\mathbf{A}^{1,\mathrm{an}}_{L}}}\, \big|\, |(t-c)(x)|<R\big\}.$$ For $R_{1},R_{2}$ such that $0 < R_{1} \le R_{2}$, we set $$C_{L}^+(c;R_{1},R_{2}) = \big\{x\in {\ensuremath{\mathbf{A}^{1,\mathrm{an}}_{L}}}\, \big|\, R_{1}\le |(t-c)(x)|\le R_{2}\big\}.$$ For $R_{1},R_{2}$ such that $0 < R_{1} < R_{2}$, we set $$C_{L}^-(c;R_{1},R_{2}) = \big\{x\in {\ensuremath{\mathbf{A}^{1,\mathrm{an}}_{L}}}\, \big|\, R_{1} < |(t-c)(x)| < R_{2}\big\}.$$
\[defi:modulus\] The modulus of the closed annulus $C_{L}^+(c;R_{1},R_{2})$ is defined by $$\textrm{Mod}(C_{L}^+(c;R_{1},R_{2})) = \frac{R_{2}}{R_{1}}.$$ It is independent of the coordinate $t$ on the annulus.
The modulus is the basic tool that helps define distances on curves. To explain the idea in a simple case, let us assume, for a short moment, that $K$ is algebraically closed and that $X$ is the affine analytic line ${\ensuremath{\mathbf{A}^{1,\mathrm{an}}_{K}}}$. In this case, any segment $I \subset X_{[2,3]}$ is the skeleton of a closed annulus $I^\sharp$ and we may set $\ell(I) = \log(\textrm{Mod}(I^\sharp))$. This defines a gauge on $X_{[2,3]}$.
Since moduli of annuli are invariant under Galois action, we may consider virtual annuli instead of annuli and relax the hypothesis that the field $K$ be algebraically closed. The method may actually be extended to curves, by cutting the segments into a finite number of pieces whose interiors lie inside the affine line. Using this kind of arguments, A. Ducros shows that there exists a canonical gauge $\ell$ on $X_{[2,3]}$ in the general case (see [@RSSen proposition 3.4.19]).
In what follows, every time we need to measure a segment (to speak of linear or $\log$-linear maps, to compute derivatives, etc.), we will use this canonical gauge.
Radius of convergence {#sec:radius}
---------------------
We are now ready to define the radius of convergence of $({\ensuremath{\mathscr{F}}},\nabla)$.
\[def:DxS\] Let $x \in X$. Let $L$ be a complete valued extension of $K$ such that $X_{L}$ contains an $L$-rational point ${\ensuremath{\tilde{x}}}$ over $x$. We denote $D({\ensuremath{\tilde{x}}},S_{L})$ the biggest open disc centred at ${\ensuremath{\tilde{x}}}$ that is contained in $X_{L}\setminus S_{L}$, *i.e.* the connected component of $X_{L}\setminus \Gamma_{S_{L}}$ that contains ${\ensuremath{\tilde{x}}}$.
\[def:radius\] Let $x$ be a point in $X$ and $L$ be a complete valued extension of $K$ such that $X_{L}$ contains an $L$-rational point ${\ensuremath{\tilde{x}}}$ over $x$. Let us consider the pull-back $({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})$ of $({\ensuremath{\mathscr{F}}},\nabla)$ on $D({\ensuremath{\tilde{x}}},S_{L}) \simeq D_{L}^-(0,R)$. We denote ${\ensuremath{\mathcal{R}}}'_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))$ the radius of the biggest open disc centred at $0$ on which $({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})$ is trivial and ${\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla)) = {\ensuremath{\mathcal{R}}}'_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))/R$.[^4]
The definition is independent of the choices of $L$ and ${\ensuremath{\tilde{x}}}$ and invariant by extension of the base field $K$.
For any complete valued extension $L$ of $K$, we denote by $\pi_{L} : X_{L} \to X$ the natural projection.
\[lem:basechange\] Let $L$ be a complete valued extension of $K$. For any $x\in X_{L}$, we have $${\ensuremath{\mathcal{R}}}_{S_{L}}(x,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla)) = {\ensuremath{\mathcal{R}}}_{S}(\pi_{L}(x),({\ensuremath{\mathscr{F}}},\nabla)).$$
Let us now explain how the function behaves with respect to changing triangulations. Let $S'$ be a weak triangulation of $X$ that contains $S$. Let $x \in X$. Let $L$ be a complete valued extension of $K$ such that $X_{L}$ contains an $L$-rational point ${\ensuremath{\tilde{x}}}$ over $x$. Inside $X_{L}$, the disc $D({\ensuremath{\tilde{x}}},S'_{L})$ is included in $D({\ensuremath{\tilde{x}}},S_{L}) \simeq D_{L}^-(0,R)$. Let $R'$ be its radius as a sub-disc of $D_{L}^-(0,R)$ and set $\rho_{S',S}(x) = R'/R \in (0,1]$. It is also the modulus of the semi-open annulus $D({\ensuremath{\tilde{x}}},S_{L}) \setminus D({\ensuremath{\tilde{x}}},S'_{L})$. Remark that the map $\rho_{S',S}$ is constant and equal to 1 on $S$, and even $\Gamma_{S}$. It is now easy to check that $$\label{eq:rhoS'S}
{\ensuremath{\mathcal{R}}}_{S'}(x,({\ensuremath{\mathscr{F}}},\nabla)) = \min \left(\frac{{\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))}{\rho_{S',S}(x)},1\right).$$
It is possible to describe in a very concrete way the behaviour of the function $\rho_{S',S}$ on discs and annuli, hence on the whole $X$. We deduce the following result.
\[lem:rhoS’S\] The map $x\in X \mapsto \rho_{S',S}(x)$ is continuous on $X$, locally constant outside the skeleton $\Gamma_{S'}$ and piecewise $\log$-linear on $\Gamma_{S'}$ with slopes $0$ or $\pm 1$.
Analytic domains of the affine line {#sec:affineline}
-----------------------------------
Assume that $X$ is an analytic domain of the affine line ${\ensuremath{\mathbf{A}}}^{1,{\textrm{an}}}_{K}$. The choice of a coordinate $t$ on ${\ensuremath{\mathbf{A}}}^{1,{\textrm{an}}}_{K}$ provides a global coordinate on $X$ and it seems natural to use it in order to measure the radii of convergence. We will call “embedded” the radii we define in this setting.
Let us first give a definition that does not refer to any triangulation.
\[def:multiradiusemb\] Let $x$ be a point of $X$ and $L$ be a complete valued extension of $K$ such that $X_{L}$ contains an $L$-rational point ${\ensuremath{\tilde{x}}}$ over $x$. Let $D({\ensuremath{\tilde{x}}},X_{L})$ be the biggest open disc centred at ${\ensuremath{\tilde{x}}}$ that is contained in $X_{L}$.
Let us consider the pull-back $({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})$ of $({\ensuremath{\mathscr{F}}},\nabla)$ on $D({\ensuremath{\tilde{x}}},X_{L})$. We denote ${\ensuremath{\mathcal{R}}}^\mathrm{emb}(x,({\ensuremath{\mathscr{F}}},\nabla))$ the radius of the biggest open disc centered at ${\ensuremath{\tilde{x}}}$, measured using the coordinate $t$ on ${\ensuremath{\mathbf{A}}}^{1,{\textrm{an}}}_{L}$, on which $({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})$ is trivial.
The definition of ${\ensuremath{\mathcal{R}}}^\mathrm{emb}(x,({\ensuremath{\mathscr{F}}},\nabla))$ only depends on the point $x$ and not on $L$ or ${\ensuremath{\tilde{x}}}$.
Let us now state a second definition that takes into account the weak triangulation $S$ of $X$.
\[def:multiradiusembT\] Let $x$ be a point of $X$ and $L$ be a complete valued extension of $K$ such that $X_{L}$ contains an $L$-rational point ${\ensuremath{\tilde{x}}}$ over $x$. As in definition \[def:DxS\], consider $D({\ensuremath{\tilde{x}}},S_{L})$, the biggest open disc centred at ${\ensuremath{\tilde{x}}}$ that is contained in $X_{L}\setminus S_{L}$. We denote $\rho_{S}(x)$ its radius, measured using the coordinate $t$ on ${\ensuremath{\mathbf{A}}}^{1,{\textrm{an}}}_{L}$.
Let us consider the pull-back $({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})$ of $({\ensuremath{\mathscr{F}}},\nabla)$ on $D({\ensuremath{\tilde{x}}},S_{L})$. We denote ${\ensuremath{\mathcal{R}}}^\mathrm{emb}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))$ the radius of the biggest open disc centered at ${\ensuremath{\tilde{x}}}$, measured using the coordinate $t$ on ${\ensuremath{\mathbf{A}}}^{1,{\textrm{an}}}_{L}$, on which $({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})$ is trivial.
Once again, the definitions of $\rho_{S}(x)$ and ${\ensuremath{\mathcal{R}}}^\mathrm{emb}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))$ are independent of the choices of $L$ and ${\ensuremath{\tilde{x}}}$.
The radii we have just defined may easily be linked to the one we introduced in definition \[def:radius\]. For the second radius, we have $$\label{eq:radiusembT}
{\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla)) = \frac{{\ensuremath{\mathcal{R}}}^\textrm{emb}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))}{\rho_{S}(x)}.$$
Assume that $X$ is not the affine line and let $S_{0}$ be its smallest weak triangulation. We have $$\label{eq:radiusembT0}
{\ensuremath{\mathcal{R}}}_{S_{0}}(x,({\ensuremath{\mathscr{F}}},\nabla)) = \frac{{\ensuremath{\mathcal{R}}}^\textrm{emb}_{S_{0}}(x,({\ensuremath{\mathscr{F}}},\nabla))}{\rho_{S_{0}}(x)} = \frac{{\ensuremath{\mathcal{R}}}^\textrm{emb}(x,({\ensuremath{\mathscr{F}}},\nabla))}{\rho_{S_{0}}(x)}.$$
The different radii satisfy similar properties thanks to the following result.
\[lem:rho\] The map $x\in X \mapsto \rho_{S}(x)$ is continuous on $X$, locally constant outside the skeleton $\Gamma_{S}$ and piecewise $\log$-linear on $\Gamma_{S}$ with slopes $0$ or $\pm 1$.
Computation in coordinates
--------------------------
We now present a concrete way to compute the radius of convergence. Consider an open disc $D = D^-(0,R)$ endowed with the empty weak triangulation and choose a coordinate $t$ on it. Endow ${\ensuremath{\mathscr{O}}}(D)$ with the usual derivation $d = \mathrm{d}/\mathrm{d}t$. Assume that ${\ensuremath{\mathscr{F}}}$ is free of rank $m$ on $D$. In this case, the connection $\nabla$ on $D$ may be given by a matrix $G \in M_{m}({\ensuremath{\mathscr{O}}}(D))$.
Let $x\in D(K)$. We can compute the Taylor series of a fundamental solution matrix in the neighbourhood of the point $x$: $$\sum_{n\ge 0} \frac{G_{n}(x)}{n!}\, (t-t(x))^n,$$ where $G_{0} = \textrm{Id}$, $G_{1}=G$ and, for every $n\ge 1$, $G_{n+1} = d(G_{n}) + G_{n}\, G$. Then the radius of convergence at $x$ may be computed by the following formula: $$\label{eq:formularadius1}
{\ensuremath{\mathcal{R}}}_{\emptyset}(x,({\ensuremath{\mathscr{F}}},\nabla)) = \min\left( \frac{1}{R}\, \liminf_{n\ge 1} \left(\left|\frac{G_{n}(x)}{n!}\right|^{-\frac{1}{n}}\right),1\right).$$ Since the matrices $G_{n}$ stay the same if we enlarge the field $K$, the formula actually holds for any point $x$ of $D$.
Even more generally, assume that $X$ is an analytic domain on the affine line different from the affine line and that ${\ensuremath{\mathscr{F}}}$ is free on it. Choose a coordinate $t$ on ${\ensuremath{\mathbf{A}^{1,\mathrm{an}}_{K}}}$ and endow ${\ensuremath{\mathscr{O}}}(X)$ with the usual derivation $d = \mathrm{d}/\mathrm{d}t$. We may define a sequence $(G_{n})_{n\ge 0}$ of matrices as above and check that, for any $x\in X$, we have $$\label{eq:RdT0}
{\ensuremath{\mathcal{R}}}_{S_{0}}(x,({\ensuremath{\mathscr{F}}},\nabla)) = \min\left( \frac{1}{\rho_{S_{0}}(x)}\, \liminf_{n\ge 1} \left(\left|\frac{G_{n}(x)}{n!}\right|^{-\frac{1}{n}}\right),1\right),$$ where $S_{0}$ is the smallest weak triangulation of $X$.
Let us now return to the case of the open disc $D = D^-(0,R)$ as above. In general, the sheaf ${\ensuremath{\mathscr{F}}}$ need not be free on it (unless $K$ is maximally complete, see [@Lazard]). On the other hand, it is free on any disc $D^+(0,r)$, with $r<R$, since ${\ensuremath{\mathscr{O}}}(D^+(0,r))$ is principal. For any $r\in [0,R)$ and any $n\ge 0$, we denote $G_{r,n}$ the matrix associated to the restriction of $\nabla^n$ to $D^+(0,r)$. We set $$\label{eq:Rr}
{\ensuremath{\mathcal{R}}}_{\emptyset,r}(x,({\ensuremath{\mathscr{F}}},\nabla)) = \min\left(\frac{1}{R}\, \liminf_{n\ge 1} \left(\left|\frac{G_{r,n}(x)}{n!}\right|^{-\frac{1}{n}}\right),\frac{r}{R}\right).$$ Then, it is easy to check that the map $r\in [0,R) \mapsto {\ensuremath{\mathcal{R}}}_{\emptyset,r}(x,({\ensuremath{\mathscr{F}}},\nabla))$ is non-decreasing and that, for any $x\in D$, we have $$\label{eq:R1}
{\ensuremath{\mathcal{R}}}_{\emptyset}(x,({\ensuremath{\mathscr{F}}},\nabla)) = \lim_{r\to R^-} {\ensuremath{\mathcal{R}}}_{\emptyset,r}(x,({\ensuremath{\mathscr{F}}},\nabla)).$$
Let us now finally turn back to the general case where $X$ is a quasi-smooth curve. Let $x\in X$. Let $L$ be a complete valued extension of $K$ such that $X_{L}$ contains an $L$-rational point ${\ensuremath{\tilde{x}}}$ over $x$. The radius of convergence ${\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))$ is computed by pulling back $({\ensuremath{\mathscr{F}}},\nabla)$ to $({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})$ on $D({\ensuremath{\tilde{x}}},S_{L})$. It follows from the definition that $${\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla)) = {\ensuremath{\mathcal{R}}}_{\emptyset}(x,({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})),$$ where the latter radius is computed on $D({\ensuremath{\tilde{x}}},S_{L})$ endowed with the empty weak triangulation. Hence the formulas of the preceding paragraph may be used in the general case.
The following result may now be easily proven.
\[thm:Cauchypadique\] For any $x\in X(K)$, we have ${\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla)) >0$. In particular, the function ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is constant in the neighbourhood of any rational point of $X$.
We may use the setting of formula (\[eq:R1\]). Let $r\in (0,R)$. We denote $\|.\|$ the norm on $D^+(0,r)$. It is enough to show that the sequence $\big((\|G_{r,n}\| / |n !|)^{1/n}\big)_{n\ge 1}$ is bounded. It is well known that the sequence $(|n!|^{1/n})_{n\ge 1}$ converges, hence it is enough to show that $(\|G_{r,n}\|^{1/n})_{n\ge 1}$ is bounded. For any $n\ge 1$, we have $G_{r,n+1} = d(G_{r,n}) + G_{r,n}\, G_{r,1}$, hence $\|G_{r,n+1}\| \le \max(\|d\|, \|G_{r,1}\|) \, \|G_{r,n}\|$. We deduce that, for any $n\ge 1$, we have $\|G_{r,n}\|^{1/n} \le \max(\|d\|, \|G_{r,1}\|)$.
Let us now try to carry out computations similar to those of formula (\[eq:formularadius1\]) around an arbitrary point of $X$. Let $U$ be an analytic domain of $X$ on which the sheaves ${\ensuremath{\mathscr{F}}}$ and $\Omega_{X}$ are free (such an analytic domain exists in the neighbourhood of any point). Let $d$ be a derivation on ${\ensuremath{\mathscr{O}}}(U)$. Let $G$ be the matrix associated to the connection $({\ensuremath{\mathscr{F}}},\nabla)$. For every $x\in U$, we can now define $$\label{eq:radiusd}
{\ensuremath{\mathcal{R}}}^d(x,({\ensuremath{\mathscr{F}}},\nabla)) = \liminf_{n\ge 1} \left(\left|\frac{G_{n}(x)}{n!}\right|^{-\frac{1}{n}}\right).$$ However, it is not clear how this relates to the radius of convergence ${\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))$. We will study this question later (see corollary \[cor:Rdbranch\] and remark \[rem:Rdx\]).
The result
==========
Now that we have made precise the meaning of radius of convergence of $({\ensuremath{\mathscr{F}}},\nabla)$ at any point of the curve $X$ (see definition \[def:radius\]), we start investigating its properties.
Statement
---------
Let us state precisely the result we are interested in. To this end, we need to extend the notion of $\log$-linearity beyond $X_{[2,3]}$.
\[defi:loglinear\] Let $J$ be a segment of $X$. A map $f : J \to {\ensuremath{\mathbf{R}}}$ is said to be linear if it is continuous and linear on the interior $\mathring{J}$ of $J$.
A map $f : \Gamma\to {\ensuremath{\mathbf{R}}}$ on a locally finite subgraph $\Gamma$ of $X$ is said to be piecewise linear if $\Gamma$ may be covered by a locally finite family ${\ensuremath{\mathscr{J}}}$ of segments such that, for any $J \in {\ensuremath{\mathscr{J}}}$, the restriction of the map $f$ to $\mathring{J}$ is linear.
A map with values in ${\ensuremath{\mathbf{R}}}_{+}^*$ is said to be $\log$-linear if its logarithm is linear.
\[thm:continuousandfinite\] The map $$x \in X \mapsto {\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla)) \in {\ensuremath{\mathbf{R}}}_{+}^*$$ satisfies the following properties:
1. it is continuous;
2. it is locally constant outside a locally finite subgraph $\Gamma$ of $X$;
3. its restriction to $\Gamma$ is piecewise $\log$-linear and, for any connected subgraph $\Gamma_{c}$ of $\Gamma$, its slopes on $\Gamma_{c}$ are rational numbers of the form $\pm m/i$, with $m\in{\ensuremath{\mathbf{Z}}}$, .
\[rem:factorisation\] Let us enlarge $\Gamma$ to a locally finite subgraph $\Gamma'$ of $X$ such that
1. $\Gamma'$ contains $\Gamma_{S}$ ;
2. $\Gamma'$ meets every connected component of $X$ ;
3. for any connected component $V$ of $X\setminus S$, the graph $\Gamma'\cap V$ is convex.
In this case, there is a natural continuous retraction $X \to \Gamma'$ and the map ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ factorizes by it.
In the following, we will give a new proof of the theorem assuming that $X$ is boundary-free or that $({\ensuremath{\mathscr{F}}},\nabla)$ is overconvergent.
As we mentioned in the introduction, the contents of theorem \[thm:continuousandfinite\] already appeared in the literature:
1. The continuity property on the skeleton of an annulus is due to G. Christol and B. Dwork (see [@ChristolDwork théorème 2.5]). It has been extended to affinoid domains of the affine line by F. Baldassarri and L. Di Vizio (see [@ContinuityBDV]) and to general curves by F. Baldassarri (see [@ContinuityCurves]). His setting is actually slightly less general than ours, but his result extends easily. The second author also proved the continuity of all the radii of convergence (*i.e.* all the slopes of the Newton polygon) on affinoid domains of the affine line by another method (see [@finiteness]). It has been extended to curves by both authors (see [@finiteness2]).
2. The local constancy outside a locally finite subgraph has been proven for all the radii of convergence on an affinoid domain of the affine line by the second author (see [@finiteness]) and then extended to general curves by both authors (see [@finiteness2]).
3. The last property has been proven by É. Pons (in a weaker form) for the skeleton of an annulus (see [@PonsPadova théorème 2.2]) and by F. Baldassarri for an interval inside the skeleton of a curve (see [@ContinuityCurves corollary 6.0.6]). In the case of the skeleton of an annulus again, K. Kedlaya extended it to the other radii of convergence (see [@pde theorem 11.3.2]).
In the sequel, we will use the fact that the result is already known for the restriction of the radius to intervals inside discs and annuli. For future reference, let us write it down explicitly.
\[thm:Kedlaya\] Assume that $X$ is an open or closed annulus (possibly a disc). Fix a coordinate $t$ on $X$ and let $d = \textrm{d}/\textrm{d}t$ be the usual derivation on ${\ensuremath{\mathscr{O}}}(X)$. Assume that $({\ensuremath{\mathscr{F}}},\nabla)$ comes from a global differential module $(M,D)$ on $({\ensuremath{\mathscr{O}}}(X),d)$.
If $X$ is an annulus, let $J$ be its skeleton. If $X$ is a disc, pick a point $c \in X(K)$ and let $J$ be the interval in $X$ that joins $c$ to its boundary.
Then, the restriction of the map ${\ensuremath{\mathcal{R}}}^\mathrm{emb}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ to $J$ is concave, continuous and piecewise $\log$-linear with slopes that are rational numbers of the form $m/i$, with $m\in {\ensuremath{\mathbf{Z}}}$ and $1\le i \le \textrm{rk}\, ({\ensuremath{\mathscr{F}}})$.
In most of the literature (see [@pde theorem 11.3.2], for instance), the result is actually stated with the generic radius of convergence. Since the relation between the two radii is well understood (see [@pde proposition 9.7.5] or [@finiteness section 3.3]), this actually causes no harm.
Potential theory {#sec:potential}
----------------
In this section, we assume that the absolute value of $K$ is non-trivial and that $X$ is strictly $K$-analytic and boundary-free. We briefly introduce non-archimedean potential theory, as developed in Amaury Thuillier’s manuscript [@TheseThuillier]. This will be our main tool in the proof of theorem \[thm:continuousandfinite\].
Let us recall that any $K$-analytic space $Y$ in the sense of V. Berkovich has a boundary $\partial Y$ and an interior $\textrm{Int}(Y) = Y \setminus \partial Y$ (see [@rouge], section 2.5 for the affinoid case and the discussion before proposition 3.1.3 for the general one). For instance, the closed disc $D^+(c,R)$, with $R>0$ has boundary $\{\eta_{c,R}\}$ and the closed annulus $C^+(c;R_{1},R_{2})$, with $0<R_{1}\le R_{2}$ has boundary $\{\eta_{c,R_{1}},\eta_{c,R_{2}}\}$. Any open disc or annulus and, more generally, any open subset of the affine line, any open subset of the analytification of an algebraic variety is boundary-free. In this section, $X$ is assumed to be boundary-free and all of its open subsets will also be.
Let us now turn to potential theory. Topologically speaking, Berkovich analytic curves can be reconstructed from their finite subgraphs and we will first explain what it looks like on the latter. On a finite metrized graph $\Gamma$, one may define
- smooth functions: they are the continuous piecewise linear functions;
- the Laplacian of a smooth function $f$: it is the finite measure $$\label{eq:ddc}
\textrm{dd}^c(f) = \sum_{p\in \Gamma} \big(\sum_{\vec v \in T_{p}\,\Gamma} m_{\vec v}\, \mathrm{d}_{\vec v}f(p)\big) \delta_{p},$$ where $T_{p}\,\Gamma$ denotes the set of directions out of $p$, $m_{\vec v}$ is a weight, $\mathrm{d}_{\vec v}f(p)$ denotes the outer derivative of $f$ at $p$ and $\delta_{p}$ denotes the Dirac measure at $p$.
One may push further this line of thought and define harmonic functions (those for which ), super-harmonic functions (those for which $\textrm{dd}^c(f)\le 0$) and sub-harmonic functions.
To give a rough idea of what is going on, let us give a few examples. Saying that a smooth function $f$ on a segment $[a,b]$ is harmonic on $(a,b)$ is equivalent to saying that its slope never changes. We deduce that such a function $f$ is linear on $[a,b]$, hence determined by its values $f(a)$ and $f(b)$ at the boundary. Conversely, any prescribed values at the boundary may be realized by a smooth function on $[a,b]$ which is harmonic on $(a,b)$. The analogues of both statements hold for arbitrary finite graphs (see [@TheseThuillier proposition 1.2.15]). To put it in other words, in this context, the Dirichlet problem admits a unique solution.
The same kind of arguments show that smooth functions on segments that are super-harmonic in the interior correspond to concave functions.
Building on those ideas in the case of graphs, A. Thuillier managed to develop a full-fledged potential theory on Berkovich analytic curves which is quite similar to the complex one. We briefly review here the definitions he introduces (see [@TheseThuillier sections 2 and 3]).
The first step is to extend the notion of smooth and harmonic functions. Let ${\ensuremath{\mathfrak{Y}}}$ be a semi-stable formal scheme whose generic fiber $Y$ identifies to an affinoid domain of $X$. In this situation, one defines a skeleton $\Gamma({\ensuremath{\mathfrak{Y}}})$[^5], which is a finite subgraph of $Y$ that contains $\partial Y$, and a retraction $\tau_{{\ensuremath{\mathfrak{Y}}}} : Y \to \Gamma({\ensuremath{\mathfrak{Y}}})$. Formula (\[eq:ddc\]) may now be made more precise by choosing for the weight $m_{\vec v}$ the residual degree of the field on which the direction corresponding to $\vec v$ is defined. In particular, if $K$ is algebraically closed, it is always 1. Let $H(Y)$ be the pull-back by $\tau_{{\ensuremath{\mathfrak{Y}}}}$ of the set of smooth functions on $\Gamma({\ensuremath{\mathfrak{Y}}})$ that are harmonic outside $\partial Y$.
If $Y$ is an arbitrary strictly $K$-affinoid domain of $X$, by the semi-stable reduction theorem, one may carry out the previous construction after passing to a finite Galois extension $K'/K$ and then consider the invariants under $\mathrm{Gal}(K'/K)$. The resulting set $H(Y) \subset {\ensuremath{\mathscr{C}}}^0(Y,{\ensuremath{\mathbf{R}}})$, the set of harmonic functions on $Y$, depends only on $Y$.
For any open subset $U$ of $X$, one may now set ${\ensuremath{\mathcal{H}}}_{X}(U) = \varprojlim H(Y)$, where the limit is taken over the strictly $K$-affinoid domains $Y$ of $U$. This defines the sheaf ${\ensuremath{\mathcal{H}}}_{X}$ of harmonic functions on $X$.
If $Y$ is a $K$-affinoid domain of $X$, we extend the previous definition of $H$ by setting $H(Y) = \Gamma(Y\setminus \partial Y,{\ensuremath{\mathcal{H}}}_{X})\, \cap\, {\ensuremath{\mathscr{C}}}^0(Y,{\ensuremath{\mathbf{R}}})$. In this general case, the Dirichlet problem also admits a unique solution: the restriction map $H(Y) \to {\textrm{Hom}}(\partial Y,{\ensuremath{\mathbf{R}}})$ is bijective.
Let $U$ be an open subset of $X$. The ${\ensuremath{\mathbf{R}}}$-vector space $A^0(U)$ of smooth functions on $U$ consists of the continuous functions $f \in {\ensuremath{\mathscr{C}}}^0(U,{\ensuremath{\mathbf{R}}})$ for which there exists a locally finite covering of $U$ by $K$-affinoid domains $Y$ such that $f_{|Y} \in H(Y)$. If we denote by $A^1(U)$ the ${\ensuremath{\mathbf{R}}}$-vector space of real measures on $U$ whose support is a locally finite subset of $U_{[2,3]}$ (which is denoted by $I(U)$ in [@TheseThuillier]), we may naturally extend the Laplacian operator defined by formula (\[eq:ddc\]) to a map $$\textrm{dd}^c : A^0(U) \to A^1(U).$$ As we expect, its kernel is nothing but ${\ensuremath{\mathcal{H}}}_{X}(U)$.
This operator sends $A_{c}^0(U)$ to $A_{c}^1(U)$, where the subscript $c$ indicates a compactness condition on the support, and induces a map between their duals $$\textrm{dd}^c : D^0(U) \to D^1(U).$$ It may be useful to remark that the set $D^0(U)$ of currents of degree 0 is naturally isomorphic to ${\textrm{Hom}}(U_{[2,3]},{\ensuremath{\mathbf{R}}})$.
In a more restricting setting, we would also like to mention the book [@BR] by M. Baker and R. Rumely where a potential theory on the line over an algebraically closed field is developed. Of course, it is equivalent to A. Thuillier’s.
In this text, we will be especially interested in super-harmonic functions (see [@TheseThuillier sections 3.1.2 and 3.4] and [@BR chapter 8]). We will recall the results we need. Let us begin with the definition (see [@TheseThuillier définition 3.1.5]).
Let $U$ be an open subset of $X$. We say that a map $u : U \to {\ensuremath{\mathbf{R}}}\cup\{+\infty\}$ is pre-super-harmonic if, for any strictly $k$-affinoid $Y$ of $U$ and any harmonic function $h$ on $Y$, the following condition holds: $$(u_{|\partial Y} \ge h_{|\partial Y}) \implies (u_{|Y} \ge h).$$ The map $u$ is said to be super-harmonic if, moreover, it is lower semicontinuous and identically equal to $+\infty$ on no connected component of $U$.
Super-harmonic functions may be characterised by a non-positivity property of their Laplacians (see [@TheseThuillier proposition 3.4.4 and théorème 3.4.12] or [@BR theorem 8.19]).
\[thm:Radon\] Let $U$ be an open subset of $X$.
1. Let $f\in A^0(U)$. The smooth function $f$ is super-harmonic if, and only if, $\mathrm{d}\mathrm{d}^c(f) \le 0$.
2. Let $T\in D^0(U)$. The current $T$ is a super-harmonic function if, and only if, (as a current of degree 1). In this case, $\mathrm{d}\mathrm{d}^c(T)$ is a non-positive Radon measure.
Let us introduce the basic examples of harmonic and super-harmonic functions (see [@TheseThuillier propositions 2.3.20 and 3.1.6]).
Let $U$ be an open subset of $X$. Let $f\in {\ensuremath{\mathscr{O}}}(U)$.
1. If $f$ is invertible on $U$, then $-\log(|f|)$ is harmonic on $U$.
2. If $f$ vanishes identically on no connected component of $U$, then $-\log(|f|)$ is super-harmonic on $U$.
Since we will use it later, let us mention that we already encountered an example of harmonic function in section \[sec:affineline\].
\[lem:rhoh\] Let $C$ be an open disc or annulus endowed with the empty weak triangulation. The map $\log(\rho_{\emptyset})$ is harmonic on $C$.
Assume that $C$ is the open disc $D^-(c,R)$. Then $\rho_{\emptyset} \equiv R$ and the result is obvious.
Assume that $C$ is the open annulus $C^-(c;R_{1},R_{2})$ with coordinate $t$. Then, for every , we have $\rho_{\emptyset}(x) = |(t-c)(x)|$ and the result follows from the proposition.
In general, it is easy to check that a map of the form $\rho_{S}$ on an arbitrary analytic domain of the affine line is super-harmonic, but not necessarily harmonic.
We now state some properties of super-harmonic functions. It is well-known that a concave map on a segment is left and right-differentiable in the interior of this segment. The next proposition is the analogue of this fact for more general finite 1-dimensional graphs. It appears in [@BR proposition 8.24] in the case of the line and may be generalised to the case of curves.
\[prop:shcontinuous\] Let $U$ be a connected open subset of $X$ and $u : U \to {\ensuremath{\mathbf{R}}}\cup\{+\infty\}$ be a pre-super-harmonic function on $U$ that is not identically equal to $+\infty$. Let $\Gamma$ be a subgraph of $U$. For any point $p\in \Gamma$ of type 2, 3 or 4 and any direction $\vec{v} \in T_{p}\, \Gamma$, the directional derivative $\mathrm{d}_{\vec v}f(p)$ exists and is finite. In particular, the restriction of $f$ to $\Gamma$ is continuous at any point of type 2, 3 or 4.
Let us quote the other properties of super-harmonic functions that we will need. They come from [@TheseThuillier proposition 3.1.8], except for the last property, which may be deduced from the preceding proposition (see also [@BR proposition 8.26]).
Recall that, for any topological space $U$, the lower semicontinuous regularization $f^*$ of a map $f : U \to (-\infty,+\infty]$ which is locally bounded below is defined by $$\forall x\in U, f^*(x) = \liminf_{y\to x} f(y).$$
\[prop:propsh\]
1. Let $U$ be an open subset of $X$. If $f$ and $g$ are super-harmonic functions on $U$, then $\min(f,g)$ is super-harmonic on $U$ and, for any $\lambda,\mu \ge 0$, $\lambda f +\mu g$ is super-harmonic of $U$.
2. Let $U$ be a connected open subset of $X$. Let $(f_{n})_{n\ge 0}$ be a sequence of super-harmonic functions on $U$ which is locally bounded below. Put $f = \liminf_{n} (f_{n})$. Then either $f$ is identically equal to $+\infty$ on $U$ or the lower semicontinuous regularisation $f^*$ of $f$ is super-harmonic. Furthermore, for every $x\in U\setminus U(K)$, we have $f^*(x)=f(x)$.
\[cor:Rdsh\] Let $U$ be a connected open subset of $X$ on which the sheaf ${\ensuremath{\mathscr{F}}}$ is free and $d$ be a derivation on ${\ensuremath{\mathscr{O}}}(U)$. The map $\log({\ensuremath{\mathcal{R}}}^{d})^*$ is either identically equal to $+\infty$ or super-harmonic on $U$.
With the notations of formula (\[eq:radiusd\]), for every $x$ in $U$, we have $$\log({\ensuremath{\mathcal{R}}}^{d})(x,({\ensuremath{\mathscr{F}}},\nabla)) = \liminf_{n\ge 1} \left(-\frac{1}{n}\, \log\left(\left|\frac{G_{n}(x)}{n!}\right|\right)\right).$$ We only need to check that the above sequence is locally bounded from below. Let $V$ be a compact subset of $U$. Arguing as in the proof of theorem \[thm:Cauchypadique\], we show that the sequence $\big((\|G_{n}\|_{V} / |n !|)^{1/n}\big)_{n\ge 1}$ is bounded.
\[cor:logRT0sh\] Assume that $X$ is an open disc or annulus endowed with the empty weak triangulation on which ${\ensuremath{\mathscr{F}}}$ is free. Then the map $\log({\ensuremath{\mathcal{R}}}_{\emptyset})$ is super-harmonic on $X$.
Set the notations as in formula (\[eq:RdT0\]). We have $$\log({\ensuremath{\mathcal{R}}}_{\emptyset}) = \min (\log({\ensuremath{\mathcal{R}}}^d) - \log(\rho_{\emptyset}),0).$$
By proposition \[prop:propsh\], *ii*), and lemma \[lem:rhoh\], the map $\min(\log({\ensuremath{\mathcal{R}}}^d) - \log(\rho_{\emptyset}),0)$ may only fail to be lower semicontinuous at rational points. By theorem \[thm:Cauchypadique\], it is actually constant in the neighbourhood of rational points, hence lower semicontinuous. Thus, we have $$\log({\ensuremath{\mathcal{R}}}_{\emptyset}) = \min (\log({\ensuremath{\mathcal{R}}}^d)^* - \log(\rho_{\emptyset}),0).$$ By the preceding corollary, lemma \[lem:rhoh\] and proposition \[prop:propsh\], *i)*, it is super-harmonic.
Proof of theorem \[thm:continuousandfinite\] in the boundary-free case
----------------------------------------------------------------------
Let us begin with some reductions. Thanks to lemma \[lem:basechange\], we may extend the base field and assume that $K$ is algebraically closed, non-trivially valued and maximally complete and that $X$ is strictly $K$-affinoid. We will do so in the rest of the section.
We begin with the simple case of a differential module on an open disc or annulus. We will follow the strategy that was implemented by the second author in [@finiteness proof of theorem 2.14].
For $c\in K$ and $R>0$, we denote $\eta_{c,R}$ the unique point of the Shilov boundary of the disc $D^+(c,R)$. We set $\eta_{R} = \eta_{0,R}$.
\[lem:borddisque\] Assume that $X$ is a closed disc $D^+(c,R)$ endowed with the smallest weak triangulation $S_{0} = \{\eta_{c,R}\}$. Let $J$ be the segment $[c,\eta_{c,R}]$. The map ${\ensuremath{\mathcal{R}}}_{S_{0}}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is piecewise $\log$-linear on $J$. All its slopes are non-positive. If the last slope is zero, then it is constant on the open disc $D^-(c,R)$.
On the closed disc $D^+(c,R)$, we have ${\ensuremath{\mathcal{R}}}^\mathrm{emb}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)) = R\, {\ensuremath{\mathcal{R}}}_{S_{0}}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$. Hence theorem \[thm:Kedlaya\] also holds for the map ${\ensuremath{\mathcal{R}}}_{S_{0}}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$, which proves the first statement.
By theorem \[thm:Cauchypadique\], the map ${\ensuremath{\mathcal{R}}}_{S_{0}}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is constant in the neighbourhood of $c$. Hence, its first slope on $J$ is 0. Since it is concave, all its slopes are non-positive.
Now, assume that the last slope of the map ${\ensuremath{\mathcal{R}}}_{S_{0}}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ on the segment $J$ is 0. The previous argument shows that it is indeed constant on $J$. Let $y \in D^-(c,R)$. Since $K$ is maximally complete, there exists $d \in D^-(c,R)(K)$ and $r \in [0,R)$ such that $y = \eta_{d,r}$ (see [@rouge section 1.4.4]). The segment $[d,\eta_{d,R}] = [d,\eta_{c,R}]$ meets $J$ in a neighbourhood of its end. Hence the last slope of the map on this segment is 0 too. Using the same arguments as before, we show that the map is actually constant on the whole segment. Hence ${\ensuremath{\mathcal{R}}}_{S_{0}}(y,({\ensuremath{\mathscr{F}}},\nabla)) = {\ensuremath{\mathcal{R}}}_{S_{0}}(\eta_{c,R},({\ensuremath{\mathscr{F}}},\nabla))$.
We now deal with the case of an open disc or annulus. Recall that we assumed that $K$ is maximally complete. Hence, by [@Lazard] (which actually deals with the case of a disc, but the result for an annulus follows), any locally free sheaf on such a space is actually free.
We say that a property holds for almost every element of a set $E$ if it holds for every element of $E$ with a finite number of exceptions.
\[lem:dirfinite\] Let $C = C^-(0;R_{1},R_{2})$ be an open annulus. Let $x\in (\eta_{R_{1}},\eta_{R_{2}})$. Let $\vec{v}_{1}$ and $\vec{v}_{2}$ be the directions out of $x$ towards $\eta_{R_{1}}$ and $\eta_{R_{2}}$ respectively. Let $f$ be a super-harmonic function on $C$. Assume that, for every direction $\vec v$ out of $x$, there exists a point $x_{\vec{v}}$ in the direction of $\vec v$ such that $f$ is linear on the segment $[x,x_{\vec v}]$, with slope $p_{\vec{v}}$. Assume moreover that there exists $m>0$ such that, for any direction $\vec{v}$ different from $\vec{v}_{1}$ and $\vec{v}_{2}$, we have $p_{\vec{v}} \in \{0\} \cup [m,+\infty)$.
Then, for almost every direction out of $x$, the slope of the map $f$ is zero.
Let $E$ be the set of directions out of $x$, different from $\vec{v}_{1}$ and $\vec{v}_{2}$, on which the slope of the map $f$ is not zero (hence at least $m$). Let $F$ be a finite subset of $E$. Let $r$ denote its cardinality. We may assume that there exists $d$ such that, for any $\vec{v} \in F\cup\{\vec{v}_{1},\vec{v}_{2}\}$, the point $x_{\vec{v}}$ lies at distance $d$ from the point $x$.
Let us consider the continuous function $g : C \to {\ensuremath{\mathbf{R}}}$ such that, for any $\vec{v} \in F\cup\{\vec{v}_{1},\vec{v}_{2}\}$, the restriction of $g$ to $[x,x_{\vec{v}}]$ is the linear map equal to 1 at $x$ and 0 at $x_{\vec{v}}$ and $g$ is contant outside the complement of those segments. The function $g$ is smooth with compact support. By definition of the Laplacian operator, we have $$\begin{aligned}
{\langle}\mathrm{dd}^c(f), g{\rangle}&= {\langle}f, \mathrm{dd}^c(g) {\rangle}\\
&= \bigg\langle f, - \frac{r+2}{d}\, \delta_{x} + \sum_{\vec{v} \in F\cup\{\vec{v}_{1},\vec{v}_{2}\}} \frac{1}{d}\, \delta_{x_{\vec{v}}} \bigg\rangle\\
&= \frac{1}{d} \bigg(-(r+2) f(x) + \sum_{\vec{v} \in F\cup\{\vec{v}_{1},\vec{v}_{2}\}} (f(x) + d\, p_{\vec{v}}) \bigg)\\
&= \sum_{\vec{v} \in F\cup\{\vec{v}_{1},\vec{v}_{2}\}} p_{\vec{v}}\\
&\ge p_{\vec{v}_{1}} + p_{\vec{v}_{2}} + rm.\end{aligned}$$ Since $f$ is super-harmonic, by theorem \[thm:Radon\], the current $\mathrm{dd}^c(f)$ is non-positive, which implies that $r \le -(p_{\vec{v}_{1}} + p_{\vec{v}_{2}})/m$. We deduce that the set $E$ is finite.
It is possible to give another proof of the result using M. Baker and R. Rumely’s theory. They actually show that a super-harmonic function $f : U \to {\ensuremath{\mathbf{R}}}$ is locally of bounded differential variation (see [@BR theorem 8.19]): for any $x\in U$, there exists an open neighbourhood $V$ and a constant $B$ such that, for any finite subgraph $\Gamma$ of $V$ that contains no points of type 1, we have $|\mathrm{dd}^c(f)(\Gamma)| \le B$. With the notations of the proof of the lemma, this also implies that $r$ is bounded.
\[prop:finiteopendisc\] Assume that $X$ is an open disc or annulus endowed with the empty weak triangulation. Then theorem \[thm:continuousandfinite\] holds.
Assume that $X$ is the open annulus $C^-(0;R_{1},R_{2})$. This restriction is only a matter of notation and the case of a disc may be handled in the same way.
By theorem \[thm:Kedlaya\], the map $\log({\ensuremath{\mathcal{R}}}^\mathrm{emb}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)))$ is continuous and piecewise linear on $J = (\eta_{R_{1}},\eta_{R_{2}})$. We also know that the (archimedean) absolute values of its non-zero slopes in the directions out of a point of $J$ are uniformly bounded below by a positive constant (the inverse of the rank of ${\ensuremath{\mathscr{F}}}$). Moreover, by lemma \[lem:borddisque\], the slopes that do not correspond to the directions $\vec{v}_{1}$ and $\vec{v}_{2}$ towards $\eta_{R_{1}}$ and $\eta_{R_{2}}$ respectively are non-negative (a minus sign appeared since we now compute the slopes in the other direction). By formula (\[eq:radiusembT0\]) and the explicit description of $\rho_{\emptyset}$ (constant out of $J$, log-linear with slope 1 on $J$; see the proof of lemma \[lem:rhoh\]), the map $LR = \log({\ensuremath{\mathcal{R}}}_{\emptyset}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)))$ satisfies the same properties.
By corollary \[cor:logRT0sh\], the map $LR$ is super-harmonic. Let $x$ be a point of $J$. By lemma \[lem:dirfinite\], there may only be a finite number of directions out of $x$ in which the slopes of $LR$ are non-zero. Recall that, by lemma \[lem:borddisque\], a zero slope correspond to an open disc on which the map $LR$ is constant. In particular, the map $LR$ is locally a smooth function, which enables to compute its Laplacian by formula (\[eq:ddc\]).
Let $B_{J}$ be the subset of break-points of $J$, *i.e.* the set of points $x$ on $J$ at which the slope of the map $LR$ on $J$ changes. We want to prove that every connected component of $C\setminus J$ on which the map $LR$ is not constant branches at a point of $B_{J}$. Let $x\in J\setminus B_{J}$. The map $LR$ has non-negative slopes at $x$ in the directions outside $J$ and the slopes in the directions $\vec{v}_{1}$ and $\vec{v}_{2}$ balance out. On the other hand, by super-harmonicity, the Laplacian of $LR$ at the point $x$, *i.e.* the sum of all the slopes out of $x$, is a non-positive real number, which forces all the slopes in direction different from $\vec{v}_{1}$ and $\vec{v}_{2}$ to be zero.
Let $D$ be a connected component of $X\setminus J$, necessarily an open disc, and let $\eta_{D}$ be its boundary point. Let $x \in D(K)$ and set $J_{D} = [x,\eta_{D})$. Remark that the map $LR$ may only have finitely many different slopes on $J_{D}$ in the neighbourhood of $\eta_{D}$. Indeed, on $J_{D}$, it is concave, piecewise linear with slopes that are rational numbers with bounded denominators, and by proposition \[prop:shcontinuous\], it admits a finite derivative at the point $\eta_{D}$ in the direction of $D$. In particular, the number of break-points of $LR$ on $J_{D}$ is finite.
We now use the argument of the third and fourth paragraphs repeatedly for any connected component of $X\setminus J$ on which $LR$ is not constant. We need only repeat the process a finite number of times, otherwise we would find an infinite number of breaks on some segment $[\eta_{c,R'_{1}},\eta_{c,R'_{2}}]$ inside a closed sub-disc, which would contradict theorem \[thm:Kedlaya\]. This proves that the map $LR$ is locally constant outside a finite graph $\Gamma$, which is a finite union of segments of the form $[\eta_{d,R''_{1}},\eta_{d,R''_{2}}]$.
To prove property *i)* of theorem \[thm:continuousandfinite\], *i.e.* that ${\ensuremath{\mathcal{R}}}_{\emptyset}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is continuous, it is now enough to prove that it is continuous on $\Gamma$, hence on a segment of the form $[\eta_{d,R''_{1}},\eta_{d,R''_{2}}]$. This follows from theorem \[thm:Kedlaya\]. Property *iii)* also follows from theorem \[thm:Kedlaya\].
The following result may be proved by the same arguments.
\[cor:finitegraph\] Fix the setting as in proposition \[prop:finiteopendisc\].
Assume that $X = D^-(0,R)$ and let $J=[0,\eta_{R}]$. Assume that the restriction of the map ${\ensuremath{\mathcal{R}}}^\mathrm{emb}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ to $J$ is piecewise $\log$-linear with a finite number of slopes. Then the graph $\Gamma$ of theorem \[thm:continuousandfinite\] may be chosen finite in the direction of $\eta_{R}$: there exists a finite subgraph $\bar\Gamma$ of the closure $\overline{D^-(0,R)} = D^-(0,R) \cup \{\eta_{R}\}$ of $D^-(0,R)$ such that $\bar\Gamma \cap D^-(0,R) = \Gamma \cap D^-(0,R)$.
Assume that $C = C^-(0;R_{1},R_{2})$ and let $J=[\eta_{R_{1}},\eta_{R_{2}}]$. Let $R'\in (R_{1},R_{2})$. Assume that the restriction of the map ${\ensuremath{\mathcal{R}}}^\mathrm{emb}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ to $[\eta_{R'},\eta_{R_{2}})$ is piecewise $\log$-linear with a finite number of slopes. Then the graph $\Gamma$ of theorem \[thm:continuousandfinite\] may be chosen finite in the direction of $\eta_{R_{2}}$: there exists a finite subgraph $\bar\Gamma$ of $\overline{C^-(0;R',R_{2})} = C^-(0;R',R_{2}) \cup \{\eta_{R'},\eta_{R_{2}}\}$ such that $\bar\Gamma \cap C^-(0;R',R_{2}) = \Gamma \cap C^-(0;R',R_{2})$.
The same statement holds if we consider the other end of the annulus.
The main issue is now to understand the behaviour of the radius of convergence at a point of the triangulation. We will first consider points of type 2 and study the local structure of the curve $X$ in the neighbourhood of those points. The main result we use is adapted from A. Ducros’s manuscript [@RSSen] (see the proof of théorème 3.4.1 and also [@finiteness2 theorem 3.2.1]).
Let us recall a few definitions from [@RSSen]. A branch roughly corresponds to a direction out of a point (see [@RSSen section 1.7] for a precise definition). A section of a branch out of a point $x$ is a connected open subset $U$ that lies in the prescribed direction and such that $x$ belongs to the closure $\bar{U}$ of $U$ but not to $U$ itself.
\[thm:bonvois\] Let $x$ be a point of $X$ of type 2 and $b$ a branch out of $x$. There exists an affinoid neighbourhood $Y$ of $x$ in $X$, an affinoid domain $W$ of ${\ensuremath{\mathbf{P}}}^{1,\textrm{an}}_{K}$ and a finite étale map such that
1. $\psi^{-1}(\psi(x))=\{x\}$;
2. almost every connected component of $Y\setminus\{x\}$ is an open unit disc with boundary $\{x\}$;
3. almost every connected component of $W\setminus\{\psi(x)\}$ is an open unit disc with boundary $\{\psi(x)\}$;
4. for almost every connected component $V$ of $Y\setminus\{x\}$, the induced morphism $V \to \psi(V)$ is an isomorphism;
5. the map $\psi$ induces an isomorphism between a section of $b$ and a section of $\psi(b)$.
\[cor:Rdbranch\] Let $x$ be a point of $S$ of type 2 and $b$ a branch out of $x$. There exists an affinoid neighbourhood $Y$ of $x$ in $X$, a derivation $d$ on ${\ensuremath{\mathscr{O}}}(Y)$ and a constant $R>0$ such that
1. the sheaf ${\ensuremath{\mathscr{F}}}$ is free on $Y$;
2. almost every connected component of $Y\setminus\{x\}$ is an open unit disc on which ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)) = \min({\ensuremath{\mathcal{R}}}^d(\cdot,({\ensuremath{\mathscr{F}}},\nabla))/R, 1) = \min({\ensuremath{\mathcal{R}}}^d(\cdot,({\ensuremath{\mathscr{F}}},\nabla))^*/R, 1)$;
3. there exists a section $U$ of $b$ which is isomorphic to a semi-open annulus with boundary $x$ and on which ${\ensuremath{\mathcal{R}}}_{\emptyset}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)_{|U}) = \min({\ensuremath{\mathcal{R}}}^d(\cdot,({\ensuremath{\mathscr{F}}},\nabla))/R, 1) = \min({\ensuremath{\mathcal{R}}}^d(\cdot,({\ensuremath{\mathscr{F}}},\nabla))^*/R, 1)$.
Consider a morphism $\psi$ as in the previous theorem. Denote $U$ the section of $b$ that is mentioned there. Up to shrinking $Y$ and $W$, we may assume that $U$ is a semi-open annulus with boundary $x$ and that it is the connected component of $Y\setminus\{x\}$ that lies in the direction associated to $b$. We may also assume that ${\ensuremath{\mathscr{F}}}$ is free on $Y$.
Let $t$ denote a coordinate on $W \subset {\ensuremath{\mathbf{P}}}^{1,\textrm{an}}_{K}$ and consider the derivation $\mathrm{d}/\mathrm{d}t : \Omega^1_{W} \to {\ensuremath{\mathscr{O}}}_{W}$. Since $\psi$ is étale, it induces a derivation $$d : \Omega^1_{Y} \simeq \psi^*\Omega^1_{W} \to \psi^*{\ensuremath{\mathscr{O}}}_{W} \to {\ensuremath{\mathscr{O}}}_{Y}$$ on $Y$. By formula (\[eq:formularadius1\]), it satisfies the properties of the statement with $R=|\psi(x)|$, except for the last equalities in the last two items, which is a lower semicontinuity issue.
By proposition \[prop:propsh\], *ii*), the map $\min({\ensuremath{\mathcal{R}}}^d(\cdot,({\ensuremath{\mathscr{F}}},\nabla))/R, 1)$ may only fail to be lower semicontinuous at rational points. If it is equal to ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ or ${\ensuremath{\mathcal{R}}}_{\emptyset}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)_{|U})$ in the neighbourhood of such a point, by theorem \[thm:Cauchypadique\], there exists a possibly smaller neighbourhood on which it is actually constant, hence lower semicontinuous.
To be able to use the last point of the previous result, we will need to be able to compare the radius ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ restricted to some annulus $C$ in $X$ to the radius of the restriction ${\ensuremath{\mathcal{R}}}_{\emptyset}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)_{|C})$. This relies on formula (\[eq:rhoS’S\]).
\[lem:restriction\] Let $x$ be a point of $S$. Let $C$ be an open disc or annulus inside $X$ such that $\bar{C}\cap S = \{x\}$, where $\bar{C}$ denotes the closure of $C$ in $X$.
a) Assume that $C$ is an open disc. Then, for any $y\in C$, we have $${\ensuremath{\mathcal{R}}}_{\emptyset}(y,({\ensuremath{\mathscr{F}}},\nabla)_{|C}) = {\ensuremath{\mathcal{R}}}_{S}(y,({\ensuremath{\mathscr{F}}},\nabla)).$$
b) Assume that $C$ is an open annulus such that $\Gamma_{S}$ meets the skeleton of $C$. Then, for any $y\in C$, we have $${\ensuremath{\mathcal{R}}}_{\emptyset}(y,({\ensuremath{\mathscr{F}}},\nabla)_{|C}) = {\ensuremath{\mathcal{R}}}_{S}(y,({\ensuremath{\mathscr{F}}},\nabla)).$$
c) Assume that $C$ is an open annulus such that $\Gamma_{S}$ does not meet the skeleton of $C$. Identify $C$ with an annulus $C^-(0;R_{1},R_{2})$, with coordinate $t$, in such a way that $\lim_{R\to R_{2}^-} \eta_{R} = x$. Then, for any $y\in C$, we have $${\ensuremath{\mathcal{R}}}_{\emptyset}(y,({\ensuremath{\mathscr{F}}},\nabla)_{|C}) = \min\left(\frac{R_{2}}{|t(y)|}\, {\ensuremath{\mathcal{R}}}_{S}(y,({\ensuremath{\mathscr{F}}},\nabla)), 1\right).$$
The following result will now allow us to go from one radius to the other to prove the properties we want. It is based on the fact that radii cannot exceed 1.
\[lem:change\] Let $x$ be a point of $S$. Let $C$ be an open disc or annulus inside $X$ such that $\bar{C}\cap S = \{x\}$.
a) Let $y \in C$. The restriction of ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ to $[y,x)$ admits a limit at $x$ if, and only if, the restriction of ${\ensuremath{\mathcal{R}}}_{\emptyset}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)_{|C})$ to $[y,x)$ admits a limit at $x$. Moreover, in this case, the limits coincide.
b) Let $\Gamma$ be a subgraph of $C$. If $C$ is an annulus and not a disc, assume that $\Gamma$ contains its skeleton. The restriction of ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ to $C$ is locally constant outside $\Gamma$ if, and only if, the restriction of ${\ensuremath{\mathcal{R}}}_{\emptyset}(\cdot,({\ensuremath{\mathscr{F}}},\nabla)_{|C})$ to $C$ is locally constant outside $\Gamma$.
Let us now go back to the results we want to prove. We are ready to adapt the proofs of lemma \[lem:dirfinite\] and proposition \[prop:finiteopendisc\] in the case of a general curve.
\[cor:locfinite\] Let $x$ be a point of $S \cap \mathrm{Int}(X)$ of type 2 and let $C$ be the connected component of $X$ containing $x$. The map ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is constant on almost every connected component of $C\setminus\{x\}$.
Let $U$ be an open neighbourhood of $x$ in $C$ on which ${\ensuremath{\mathscr{F}}}$ and $\Omega_{X}$ are free. Almost every connected component of $C\setminus\{x\}$ is a disc that is entirely contained in $U$. By lemma \[lem:borddisque\], it is enough to prove that, for almost every direction out of $x$, the slope of ${\ensuremath{\mathcal{R}}}_{S}$ at $x$ in the corresponding direction is $0$. Thanks to corollary \[cor:Rdbranch\], we may prove the result for the map $\min(\log({\ensuremath{\mathcal{R}}}^d)^*,\log(R))$.
By corollary \[cor:Rdsh\], the map $\min(\log({\ensuremath{\mathcal{R}}}^d)^*,\log(R))$ is super-harmonic on $U$ and, by theorem \[thm:Radon\], the current $\textrm{dd}^c \min(\log({\ensuremath{\mathcal{R}}}^d)^*,\log(R))$ is a non-positive Radon measure. In particular, it defines a continuous linear form on the Fréchet space ${\ensuremath{\mathscr{C}}}^0(U,{\ensuremath{\mathbf{R}}})$: for any compact subset $V$ of $U$, there exists $C_{V}\in {\ensuremath{\mathbf{R}}}$ such that, for $f\in{\ensuremath{\mathscr{C}}}^0(U,{\ensuremath{\mathbf{R}}})$ supported on $V$, we have $$\left|\int_{U} f\, \textrm{dd}^c \min(\log({\ensuremath{\mathcal{R}}}^d)^*,\log(R))\right| \le C_{V} \sup_{z\in V}(|f(z)|).$$ By theorem \[thm:Kedlaya\] and corollary \[cor:Rdbranch\], the absolute values of the non-zero slopes of $\min(\log({\ensuremath{\mathcal{R}}}^d)^*,\log(R))$ in almost every direction out of $x$ are uniformly bounded below by a positive constant. Hence only a finite number of theses slopes may be different from $0$.
\[cor:finiteopendiscx\] Let $x$ be a point of $S \cap \mathrm{Int}(X)$, $b$ a branch out of $x$ and $C$ an open annulus which is a section of $b$. By proposition \[prop:finiteopendisc\], there exists a locally finite graph $\Gamma_{C}$ outside which the map ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))_{|C}$ is constant. The graph $\Gamma_{C}$ may actually be chosen finite in the neighbourhood of $x$.
If $x$ is a point of type 3 of the triangulation, by [@RSSen théorème 3.3.5], it has a neighbourhood $U$ which is isomorphic to an open annulus, and the result follows.
We may assume that $x$ is of type 2, that $C = C^-(0;R_{1},R_{2})$ and that . By corollary \[cor:Rdbranch\] applied with the branch $b$ and lemma \[lem:change\], we may prove the result for the map $\min(\log({\ensuremath{\mathcal{R}}}^d)^*,\log(R))$. Let $R' \in (R_{1},R_{2})$. By corollary \[cor:finitegraph\], it is enough to prove that the restriction to $I = [\eta_{R'},\eta_{R_{2}})$ of this map has a finite number of slopes. By theorem \[thm:Kedlaya\], it is concave, piecewise $\log$-linear and all its slopes are of the form $m/i$ with $m\in{\ensuremath{\mathbf{Z}}}$ and $1\le i\le \mathrm{rk}({\ensuremath{\mathscr{F}}}_{|C})$. By corollary \[cor:Rdsh\], it is also super-harmonic and, by proposition \[prop:shcontinuous\], it has a finite derivative at the point $x$ in the direction of $C$. By concavity, the slopes of the map on $I$ are bounded below, hence there may only be a finite number of them.
\[cor:boundaryfreefinite\] Assume that $X$ is boundary-free. Then, there exists a locally finite subgraph $\Gamma$ of $X$ outside which the map ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is locally constant.
We may use corollary \[cor:locfinite\] for every point of type 2 of the triangulation $S$. If $x$ is a point of type 3 of the triangulation, by [@RSSen théorème 3.3.5], it has a neighbourhood $U$ which is isomorphic to an open annulus. In this case, $U\setminus\{x\}$ is a disjoint union of two annuli.
We are left with a locally finite family ${\ensuremath{\mathscr{E}}}$ of open discs and annuli on which the function ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is not constant. On each member $E$ of ${\ensuremath{\mathscr{E}}}$, we may apply proposition \[prop:finiteopendisc\] to find a locally finite graph $\Gamma_{E}$ outside which the map ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))_{|E}$ is constant. We check that $\Gamma = \Gamma_{S} \cup \bigcup_{E\in {\ensuremath{\mathscr{E}}}} \Gamma_{E}$ is a graph and that the map ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is locally constant on its complement.
To conclude, we need to prove that the graph $\Gamma$ is indeed locally finite. It is enough to prove that, for each member $E$ of ${\ensuremath{\mathscr{E}}}$ and each point $x\in \partial E$, the graph $\Gamma_{E}$ is finite in the neighbourhood of $E$. This is the content of corollary \[cor:finiteopendiscx\].
Let us finally deal with the continuity of the radius of convergence. Once again, it will follow from corollary \[cor:Rdbranch\] and the properties of super-harmonic functions.
\[cor:continuousinterval\] Let $x$ be a point of $S \cap \mathrm{Int}(X)$ of type 2 and $I$ be an interval with end-point $x$. The restriction of ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ to $I$ is continuous at $x$.
Let us first explain the strategy of the proof before going into the technical details, which we fear may appear messy. The idea is to use corollary \[cor:Rdbranch\] to express the radius ${\ensuremath{\mathcal{R}}}_{S}$ as a radius of the form ${\ensuremath{\mathcal{R}}}^{d}$, for some derivation $d$, and then use the continuity property of super-harmonic functions (see proposition \[prop:shcontinuous\]). We will use corollary \[cor:Rdbranch\] twice: first, after a base change to a field $L$ containing ${\ensuremath{\mathscr{H}}}(x)$, with a branch inside $\pi_{L}^{-1}(x)$ (we find a first limit $\ell_{1}$ at $x$, which is ${\ensuremath{\mathcal{R}}}_{S}(x)$) and second with the branch associated to $I$ (we find a second limit $\ell_{2}$ at $x$, which is the limit of ${\ensuremath{\mathcal{R}}}_{S}$ at $x$ along $I$). We will conclude that the two limits are the same by a kind of genericity argument: on almost every branch out of $x$, the two ${\ensuremath{\mathcal{R}}}^{d}$’s are equal to ${\ensuremath{\mathcal{R}}}_{S}$ (up to a multiplicative constant), hence they coincide and so do their limits at $x$.
Let us now provide the details of the proof. We may assume that $I$ is non-trivial. Let $b$ be the branch out of $x$ defined by $I$. Let $L$ be an algebraically closed complete valued extension of ${\ensuremath{\mathscr{H}}}(x)$. Let $y$ be a point of $S_{L}$ over $x$, $I_{0}$ be a non-trivial interval with end-point $y$ which is contained in $\pi^{-1}_{L}(x)$ and $b_{0}$ be the branch out of $y$ defined by $I_{0}$. Let us use corollary \[cor:Rdbranch\] with $y$ and $b_{0}$ to find some $Y_{0}$, $d_{0}$, $R_{0}$ and $U_{0}$.
By proposition \[prop:shcontinuous\], ${\ensuremath{\mathcal{R}}}^{d_{0}}(z,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla))$ tends to ${\ensuremath{\mathcal{R}}}^{d_{0}}(y,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla))$ when $z$ tends to $y$ along $I_{0}$. By lemma \[lem:basechange\], for any $z$ in a section of $c$, we have ${\ensuremath{\mathcal{R}}}_{S_{L}}(z,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla)) = {\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))$. Hence, using lemma \[lem:change\], case a, we have $$\begin{aligned}
\min ( {\ensuremath{\mathcal{R}}}^{d_{0}}(y,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla)), R_{0} ) &= \lim_{z \xrightarrow[I_{0}]{} y} \min( {\ensuremath{\mathcal{R}}}^{d_{0}}(z,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla)) , R_{0})\\
&= \lim_{z \xrightarrow[I_{0}]{} y} R_{0}\, {\ensuremath{\mathcal{R}}}_{\emptyset}(z,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla)_{|U_{0}})\\
&= \lim_{z \xrightarrow[I_{0}]{} y} R_{0}\, {\ensuremath{\mathcal{R}}}_{S_{L}}(z,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla))\\
& = R_{0}\, {\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla)).\end{aligned}$$
Now, let us use corollary \[cor:Rdbranch\] with $x$ and $b$ to find some $Y$, $d$, $R$ and $U$. We may assume that $I \subset U$. By proposition \[prop:shcontinuous\] and lemma \[lem:change\], we have $$\begin{aligned}
\lim_{z \xrightarrow[I]{} x} {\ensuremath{\mathcal{R}}}_{S}(z,({\ensuremath{\mathscr{F}}},\nabla)) &= \lim_{z \xrightarrow[I]{} x} {\ensuremath{\mathcal{R}}}_{\emptyset}(z,({\ensuremath{\mathscr{F}}},\nabla)_{|U})\\
&=\lim_{z \xrightarrow[I]{} x} \min ( {\ensuremath{\mathcal{R}}}^{d}(z,({\ensuremath{\mathscr{F}}},\nabla))/R, 1)\\
& = \min ( {\ensuremath{\mathcal{R}}}^{d}(x,({\ensuremath{\mathscr{F}}},\nabla))/R, 1).\end{aligned}$$
Let us now extend $d$ to the derivation $d_{L} = d \otimes \mathrm{Id}$ on ${\ensuremath{\mathscr{O}}}(Y_{L}) = {\ensuremath{\mathscr{O}}}(Y){\ensuremath{\hat{\otimes}}}_{K} L$. The conclusion of corollary \[cor:Rdbranch\], forgetting point *(iii)* that we will not need, still holds with $y$, $Y_{L}$, $d_{L}$ and $R$. We have $${\ensuremath{\mathcal{R}}}^{d_{L}}(y,({\ensuremath{\mathscr{F}}},\nabla)) = {\ensuremath{\mathcal{R}}}^{d}(x,({\ensuremath{\mathscr{F}}},\nabla)).$$
Let us now remark that there exists an open subset $V$ of $X_{L}$ that is both a connected component of $Y_{0}\setminus\{y\}$ and $Y_{L}\setminus\{y\}$ and that satisfies condition *ii*) of corollary \[cor:Rdbranch\] for $d_{0}$ (with $R_{0}$) and $d_{L}$ (with $R$). In fact, this is the case for almost every connected component of $C\setminus\{y\}$, where $C$ denotes the connected component of $X_{L}$ that contains $y$. Let $J$ be a non-trivial interval with end-point $y$ which is contained in $V$.
By proposition \[prop:shcontinuous\], again, we have $$\begin{aligned}
\min ( {\ensuremath{\mathcal{R}}}^{d_{0}}(y,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla))/R_{0},1) &= \lim_{z \xrightarrow[J]{} y} \min ({\ensuremath{\mathcal{R}}}^{d_{0}}(z,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla))/R_{0},1)\\
&= \lim_{z \xrightarrow[J]{} y} {\ensuremath{\mathcal{R}}}_{S}(z,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla))\\
&= \lim_{z \xrightarrow[J]{} y} \min ({\ensuremath{\mathcal{R}}}^{d_{L}}(z,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla))/R,1)\\
&= \min( {\ensuremath{\mathcal{R}}}^{d_{L}}(y,\pi_{L}^*({\ensuremath{\mathscr{F}}},\nabla))/R,1).\end{aligned}$$
The result follows.
\[rem:Rdx\] Under the conditions of corollary \[cor:Rdbranch\], the preceding proof shows that ${\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla)) = \min({\ensuremath{\mathcal{R}}}^d(x,({\ensuremath{\mathscr{F}}},\nabla))/R, 1)$.
Let us point out that we only used the fact that the point $x$ lies in the interior of $X$ to ensure that the map ${\ensuremath{\mathcal{R}}}^d$ is continuous at $x$ along any interval. As a consequence, equality holds for a point $x$ inside the boundary of $X$, as soon as the continuity property is satisfied. It could be deduced from [@ContinuityBDV theorem 4.11] or [@ContinuityCurves section 5.2], for instance.
Theorem \[thm:continuousandfinite\] holds if $X$ is boundary-free.
By corollary \[cor:boundaryfreefinite\], there exists a locally finite subgraph $\Gamma$ of $X$ outside which the map ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ is locally constant. This is the content of property *(ii)*.
It is now enough to prove the continuity of the restriction of ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ to $\Gamma$. Let $x \in \Gamma$. If $x$ does not belong to the triangulation $S$, it belongs to an open disc or annulus and the continuity at $x$ follows from theorem \[thm:Kedlaya\]. If $x$ is a point of $S$ of type 3, by [@RSSen théorème 3.3.5], it has a neighbourhood $U$ which is isomorphic to an open annulus, and continuity at $x$ holds, by the same argument. If $x$ is a point of $S$ of type 2, it follows from corollary \[cor:continuousinterval\].
Finally, to prove property *(iii)*, it is enough to consider the restriction of ${\ensuremath{\mathcal{R}}}_{S}(\cdot,({\ensuremath{\mathscr{F}}},\nabla))$ to $\Gamma \cap X\setminus S$, and we may now conclude by theorem \[thm:Kedlaya\] again.
Let us recall that $({\ensuremath{\mathscr{F}}},\nabla)$ is said to be overconvergent if there exist a strictly $K$-analytic curve $X_{0}$ and a locally free ${\ensuremath{\mathscr{O}}}_{X_{0}}$-module of finite type ${\ensuremath{\mathscr{F}}}_{0}$ endowed with an integrable connection $\nabla_{0}$ such that
1. $X$ embeds in $\textrm{Int}(X_{0})$ ;
2. $({\ensuremath{\mathscr{F}}}_{0},\nabla_{0})$ restricts to $({\ensuremath{\mathscr{F}}},\nabla)$ on $X$.
Here, since $X$ is quasi-smooth, the analytic curve $X_{0}$ may be assumed to be quasi-smooth too.
Theorem \[thm:continuousandfinite\] holds if $({\ensuremath{\mathscr{F}}},\nabla)$ is overconvergent.
[^1]: The research for this article was partially supported by the ANR projects Berko and CETHop.
[^2]: Quasi-smooth means that $\Omega_{X}$ is locally free, see [@RSSen 2.1.8]. This corresponds to the notion called “rig-smooth” in the rigid analytic setting.
[^3]: A. Ducros’s definition is actually stronger than ours since he requires the connected components of $X\setminus S$ to be relatively compact. Any triangulation is a weak triangulation.
[^4]: If ${\ensuremath{\tilde{D}}}$ denotes the biggest open disc centred at ${\ensuremath{\tilde{x}}}$ on which $({\ensuremath{\tilde{{\ensuremath{\mathscr{F}}}}}},{\ensuremath{\tilde{\nabla}}})$ is trivial, then ${\ensuremath{\mathcal{R}}}_{S}(x,({\ensuremath{\mathscr{F}}},\nabla))$ may also be defined as the modulus of the annulus $D({\ensuremath{\tilde{x}}},S_{L}) \setminus {\ensuremath{\tilde{D}}}$ (with an obvious generalisation of definition \[defi:modulus\]).
[^5]: This skeleton is actually denote $S({\ensuremath{\mathfrak{Y}}})$ in [@TheseThuillier]. We changed the notation to avoid the confusion with a triangulation.
|
---
abstract: |
The gauged Witten equation was essentially introduced by Witten in his formulation of gauged linear $\sigma$-model (GLSM) in [@Witten_LGCY]. GLSM is a physics theory which explains the so-called Landau-Ginzburg/Calabi-Yau correspondence. This is the first paper in a series towards a mathematical construction of GLSM. In this paper we study some analytical properties of the gauged Witten equation for a Lagrange multiplier type superpotential. It contains the asymptotic property of finite energy solutions, the linear Fredholm property, the uniform $C^0$-bound, and the compactness of the moduli space of solutions over a fixed smooth $r$-spin curve with uniform energy bound.
[*Keywords*]{}: gauged linear $\sigma$-model, gauged Witten equation, moduli space, compactness
[*Mathematics Subject Classification 2010*]{}: Primary 58J05, Secondary 53D45
address:
- |
Department of Mathematics\
Princeton University\
Fine Hall, Washington Road\
Princeton, NJ 08544 USA
- |
Department of Mathematics\
University of California, Irvine\
Irvine, CA 92697 USA
author:
- Gang Tian
- Guangbo Xu
bibliography:
- 'symplectic\_ref.bib'
title: Analysis of gauged Witten equation
---
\[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Conjecture]{}
\[thm\][Definition]{}
\[thm\][Remark]{} \[thm\][Hypothesis]{} \[thm\][Example]{}
Introduction
============
In this paper we study a system of elliptic partial differential equations over a Riemann surface, called the gauged Witten equation, which originated from physicists’ study of superstring theory. This is the first piece of work in a series which aims at a rigorous construction of Witten’s gauged linear $\sigma$-model ([@Witten_LGCY]), and which, from a mathematical point of view, generalizes both the theory of symplectic vortex equation (see [@Cieliebak_Gaio_Salamon_2000], [@Mundet_thesis], [@Mundet_2003]) and the theory of Witten equation (see [@FJR1; @FJR3; @FJR2]). It is also a new member of the collection of mathematical works related to quantum field theory, which has greatly influenced mathematics in the past few decades. Therefore we would like to explain our motivation from a historical perspective, and many related works will be recalled.
Two celebrated members of this collection are Gromov-Witten theory and gauge theory. Gromov-Witten theory, inspired by Gromov’s work on $J$-holomorphic curves [@Gromov_1985] and Witten’s interpretation [@Witten_sigma_model], has been constructed rigorously by mathematicians ([@Ruan_Tian], [@Ruan_96], [@Li_Tian_2], [@Li_Tian], [@Fukaya_Ono] etc.). The field-theoretic correlation function, called the Gromov-Witten invariant, has become a fundamental tool in symplectic topology as well as in algebraic geometry. On the other hand, for last few decades, a lot of exciting results about gauge theory have been proven, notably, Atiyah-Bott’s famous work [@Atiyah_Bott], Uhlenbeck’s compactness theorem [@Uhlenbeck_82], Taubes’ construction of self-dual connections ([@Taubes_SDYM]) and Donaldson theory on differentiable 4-manifolds ([@Donaldson_Kronheimer]).
The coupling of gauge theory and $\sigma$-model is fundamental in physics, which also has been adapted by mathematicians. Many interesting examples came from “dimensional reduction” of four-dimensional gauge theory to dimension 2, where conformal invariance brings richer structures. For example, one considers a rank $n$ Hermitian vector bundle $E$ over a Riemann surface $\Sigma$, and consider the (linear) vortex equation on a pair $(A, u)$: $$\begin{aligned}
\label{equation11}
\left\{ \begin{array}{ccc}
D_A^{0,1} u & = & 0,\\
{\bm i} * F_A + \left( u\otimes u^* - \tau {\rm Id}_E\right) & = & 0.
\end{array} \right.\end{aligned}$$ Here $A$ (gauge field) is a unitary connection on $E$, $u$ (matter field) is a smooth section of $E$, and $\tau$ is a constant parameter (see [@Bradlow_stable_pairs]). The vortex equation carries a new feature: the moduli space undergoes a birational-like transformation when $\tau$ varies (see for example [@Bradlow_birational]). In the language of algebraic geometry, this is called the variation of GIT quotient. Interesting results have been proved by utilizing this feature (cf. [@Thaddeus_pairs]), which share similar spirit of the Landau-Ginzburg/Calabi-Yau correspondence we will soon review.
Observing that the term $(u\otimes u^* - \tau {\rm Id}_E)$ is of the form of a moment map of the $U(n)$-action on ${{\mathbb}C}^n$, the vortex equation can be generalized to a symplectic manifold $X$ with a Hamiltonian $G$-action. This was firstly studied by Mundet in his thesis (cf. [@Mundet_thesis], [@Mundet_2003]) and Cieliebak-Gaio-Salamon ([@Cieliebak_Gaio_Salamon_2000]). The equation is now called the [**symplectic vortex equation**]{}. Using the moduli space of solutions to the symplectic vortex equation, certain invariants of Hamiltonian $G$-manifolds, called the gauged (or Hamiltonian) Gromov-Witten invariants can be defined (see [@Mundet_2003], [@Cieliebak_Gaio_Mundet_Salamon_2002], [@Mundet_Tian_Draft] etc.). On the other hand, such invariants are closely related to the Gromov-Witten invariants of the symplectic quotient of $X$: in the “adiabatic limit” the symplectic vortex equation reduces to $J$-holomorphic curves in the symplectic quotient (see [@Gaio_Salamon_2005]). Therefore, the gauged Gromov-Witten invariants also relate the Gromov-Witten invariants of different symplectic/GIT quotients (cf. [@Woodward_1; @Woodward_2; @Woodward_3], [@Gonzalez_Woodward_wall_crossing] for the algebraic case).
Another important ingredient in field theory is the potential energy. Via localization, such field theories are closely related to the geometry of the “singularity” of the potential function. If the potential is a holomorphic function on a complex manifold, then such a theory is usually referred to as a Landau-Ginzburg theory. LG theories are naturally related to the study of singularities in topology and algebraic geometry.
In [@Witten_spin], Witten proposed an elliptic equation associated to a quasi-homogeneous polynomial $W$ in $n$ complex variables (now called [**Witten equation**]{}), which was motivated from physicists’ study of matrix models of two dimensional quantum gravity. His equation takes a simple form as a “complex gradient flow equation”: $$\begin{aligned}
\label{equation12}
{\partial u_i \over \partial {\overline}{z}} + {\overline}{ \partial_i W} (u_1, \ldots, u_n) = 0.\end{aligned}$$ In particular for polynomials which define the simple singularities (which have the famous A-D-E classification), Witten conjectured that certain generating functions satisfy the generalized KdV hierarchies. This generalized his earlier conjecture about $A_1$-singularity and KdV hierarchy ([@Witten_conjecture]), which was proved by Kontsevich [@Kontsevich_92] (and later by Mirzakhani [@Mirzakhani]). For higher type $A$ singularities, generalized Witten’s conjecture were proved by various people (Jarvis-Kimura-Vaintrob [@JKV], Lee [@YPLee], and Faber-Shadrin-Zvonkine [@FSZ]) using algebro-geometric method; while for type $D$ and type $E$ singularities, algebraic method seemed to be insufficient. In a series of papers ([@FJR1; @FJR3; @FJR2]), Fan-Jarvis-Ruan used analytic method to study the moduli space of Witten equation (\[equation12\]) for general nondegenerate quasi-homogeneous polynomials, and proved generalized Witten’s conjecture for $D_n$-singularities for even $n \geq 6$ and all type $E$-singularities. This much more systematic approach is referred to as the Landau-Ginzbug A-model theory, which can be viewed as a quantum theory about singularities.
Around 1990s, physicists discovered a correspondence between the “Landau-Ginzburg model” and the nonlinear $\sigma$-model of Calabi-Yau hypersurfaces ([@GVW], [@Martinec], [@VafaW]). It becomes a famous mathematical conjecture, often referred to as the Landau-Ginzburg/Calabi-Yau correspondence (LG/CY for short). The conjecture can be vaguely stated as follows.
The Landau-Ginzburg theory of a quasi-homogeneous superpotential $W$ of Calabi-Yau type is isomorphic to the nonlinear $\sigma$-model of the weighted projective hypersurface defined by $W$ in a certain sense.
This conjecture is certainly one of the most important problems in studying mathematical aspects of 2-dimensional quantum field theories. It has many deep implications, e.g., simpler method of calculating Gromov-Witten invariants of Calabi-Yau manifolds and applications to mirror symmetry, etc..
Witten ([@Witten_LGCY]) observed that this correspondence can be explained as a phase transition via the variation of the Fayet-Iliopoulos D-term (something similar to the $\tau$ in (\[equation11\])) in certain “master theory”. This master theory, usually referred to as the [**gauged linear $\sigma$-model**]{} (GLSM), flows in low energy to the LG and CY models respectively in different phases. Let us illustrate Witten’s idea through the following important example.
More precisely, if $Q$ is a quintic polynomial in variables $x = (x_1, \ldots, x_5)$, then Witten proposed to study (\[equation12\]) for $W(x, p) = pQ(x)$. Moreover, on the $(x, p)$-space there is an $S^1$-action with weight $(1, 1, 1, 1, 1, -5)$ under which $W$ is invariant. Then the equation carries a gauge invariance with respect to this action. Moreover, this action is Hamiltonian with moment map proportional to $$\begin{aligned}
\label{equation13}
\mu(x, p) = - 5 |p|^2 + \sum_{i=1}^5 |x_i|^2 + \tau.\end{aligned}$$ For $\tau>0$, the “classical vacuum” is $\left( {\rm Crit} W \cap \mu^{-1}(0)\right) / S^1$, which is the same as the singularity defined by $Q$; for $\tau<0$, the classical vacuum $\left( {\rm Crit} W \cap \mu^{-1}(0) \right) /S^1$ is the quintic hypersurface in ${{\mathbb}P}^4$ defined by $Q$. The variation of $\tau$ parametrizes the phase transition therefore the two theories are related.
GLSM has been extensively used by physicists in their research, for example, in the study of mirror symmetry (cf. [@Hori_Vafa]). Mathematicians have been also thinking about its mathematical formulation and applications: How to construct them mathematically? How can it be applied to studying mirror symmetry? For instance, in [@Chang_Li] and [@Chang_Li_Li], J. Li [*et al.*]{} studied the Gromov-Witten theory of a quintic hypersurface and the narrow case of Landau-Ginzburg theory by using cosection localization, which they believe to lead to an algebraic approach to GLSM and LG/CY correspondence. Fan-Jarvis-Ruan also have a project towards GLSM.
The purpose of our series of papers is to establish a mathematical theory of GLSM, at least, in some simple cases. Our approach is symplecto-geometric and uses geometric analysis. We will start our series by solving some serious technical problems, among which the most crucial one is the compactness of its moduli space. There are well-known difficulties we need to overcome in solving these problems. Our framework also includes the gauged Gromov-Witten theory as a special case where the superpotential is zero. We hope, via certain adiabatic limits, our construction can relates the work of Fan-Jarvis-Ruan on one side and the nonlinear $\sigma$-model on the other side, so it can give a good mathematical understanding of the LG/CY correspondence.
Main results of this paper
--------------------------
Now we briefly describe our main set-up and result of this first paper in our series. Let $(X, \omega, J)$ be a noncompact Kähler manifold (with “bounded geometry” at infinity), $Q: X \to {{\mathbb}C}$ be a nonzero holomorphic function which is homogeneous with respect to a ${{\mathbb}C}^*$-action on $X$. Consider ${\widetilde}{X} = X \times {{\mathbb}C}$ and the function $W: {\widetilde}{X} \to {{\mathbb}C}$ given by $W (x, p) = p Q(x)$. $W$ is invariant under another ${{\mathbb}C}^*$-action on ${\widetilde}{X}$. Let $G = S^1 \times S^1$, and there is a moment map $\mu: {\widetilde}{X} \to ({\rm Lie} G)^*$ for the $G$-action.
Let $\Sigma$ be a compact Riemann surface with punctures. The gauged Witten equation is roughly a union of the Witten equation and the vortex equation, which reads $$\begin{aligned}
\label{equation14}
\left\{ \begin{array}{ccc} {\overline}\partial_A u + \nabla W(u) & = & 0; \\
* F_A + \mu(u) & = & 0.
\end{array} \right.\end{aligned}$$ The variables of this system are $A$ and $u$, where $A$ is a connection on a $G$-bundle $P \to \Sigma$ and $u$ is a section of the associated bundle $P \times_G {\widetilde}{X}$. In fact, such a system can be defined for a quite general class of superpotentials $W$ on a general Kähler manifold, which is not necessarily a Lagrange multiplier type one. But in this paper we only consider a special class, because of difficulties in proving compactness for general superpotentials.
The gauged Witten equation generalizes both the symplectic vortex equation (\[equation11\]) and the Witten equation (\[equation12\]). It is also the classical equation of motion with respect to the following energy functional. For each pair $(A, u)$, using the superpotential $W$, its energy is defined as $$\begin{aligned}
\label{equation15}
E(A, u) = {1\over 2} \Big( \big\|d_A u \big|_{L^2(\Sigma)}^2 + \big\| F_A \big\|_{L^2(\Sigma)}^2 + \big\| \mu(u) \big\|_{L^2(\Sigma)}^2 \Big) + \big\|\nabla W(u) \big\|_{L^2(\Sigma)}^2.\end{aligned}$$
(\[equation14\]) is not Fredholm in general because $W$ may have degenerate critical points. On a cylindrical end $[0, +\infty)\times S^1$ of the punctured surface with cylindrical coordinates $(s, t)$, the unperturbed equation is essentially the following Floer-type equation $$\begin{aligned}
\label{equation16}
{\partial u \over \partial s} + J {\partial u \over \partial t} + \nabla W (u) = 0.\end{aligned}$$ To have a Fredholm operator we have to modify $W$ on cylindrical ends. In Section \[section2\] we discuss the perturbation of the equation on the cylindrical ends at the “broad” punctures of $\Sigma$, so that after perturbation, $W$ becomes a holomorphic Morse function. After perturbation, (\[equation14\]) gives a nonlinear Fredholm operator. In Section \[section3\] we prove
Every bounded solution (see Definition \[defn41\]) to the perturbed version of (\[equation14\]) converges to a critical point of the perturbed $W$ at each cylindrical end, and the energy density decays exponentially. Modulo gauge transformation, the linearization of the left-hand-side of the perturbed version of (\[equation14\]) is a linear Fredholm operator between certain Sobolev spaces (see Theorem \[thmx52\]). Its Fredholm index is given by (\[equationx58\]).
There are certain difficulties in formulating this problem properly: First, to write down the Witten equation (\[equation12\]) for a superpotential $W$ on a general Riemann surface which has no global holomorphic coordinate, one needs an extra structure (the $W$-structure) on the Riemann surface. For example, if $W$ is a generic homogeneous polynomial of degree $r$, then a natural choice of such a structure is an $r$-spin structure, i.e., an $r$-th root of the canonical bundle of the Riemann surface. (see [@FJR2] for a comprehensive study of $W$-structures and their moduli.) Based on Fan-Jarvis-Ruan’s work, we realized that the purpose of having a $W$-structure is to lift the superpotential to the fibre bundle. For GLSM, $W$ is invariant under the action copy of ${{\mathbb}C}^*$. Therefore we have to make the $W$-structure consistent with another line bundle so that $W$ can be lifted and we can write (\[equation12\]) globally on a Riemann surface.
Another difficulty is how to set up a proper perturbation scheme of the gauged Witten equation (\[equation14\]). In Fan-Jarvis-Ruan’s framework, $W$ is a nondegenerate quasi-homogeneous polynomial and the perturbation in [@FJR3] was done by adding a small generic holomorphic function $\epsilon f$ to $W$ so that $W+ \epsilon f$ becomes a holomorphic Morse function. Using a cut-off function one can extend the perturbation to the whole Riemann surface. On the other hand, the beautiful Picard-Lefschetz theory about isolated hypersurface singularities guarantees that generic perturbations can give topological information about the singularity. For general $W$ with non-isolated critical points, there is no Picard-Lefschetz theory and we don’t know if generic perturbations can unwrap interesting topology. This is one reason why we restrict to the case of superpotentials of Lagrange multiplier type (i.e., $W = pQ$). In this case we perturb $pQ(x)$ to $p(Q(x) -a) + F(x)$, so that the topology of the regular hypersurface $Q^{-1}(a)$ will be relevant, and objects like vanishing cycles appear again.
Difficulties in proving compactness
-----------------------------------
The most important technical result of the current paper is the compactness of solutions to the perturbed gauged Witten equation. The theorem reads (see Theorem \[thm65\])
Let $\vec{{\mathcal}C}$ be a rigidified $r$-spin curve (see Definition \[defn210\]). Then any sequence $(A^{(i)}, u^{(i)})$ of stable solutions to the perturbed gauged Witten equation on $\vec{{\mathcal}C}$ with $\sup_i E (A^{(i)}, u^{(i)} ) < \infty$, modulo gauge transformation, has a convergent subsequence with respect to the natural Gromov-type topology.
Its proof occupies the last three sections (Section \[section6\]–\[section8\]) of the paper. Moreover, in order to use the compactness theorem, we need to prove that the energy of solutions with fixed homology class is uniformly bounded (Theorem \[thm44\]). This requires a delicate control on the contribution of the perturbation term, for which we have to include a non-local parameter in the perturbation term (Definition \[defn215\]) and impose a few more properties (see Hypothesis \[hyp28\]).
The main issue in proving compactness is to establish a uniform $C^0$-bound on solutions. Since the target space is noncompact, this is not automatic and usually one has to assume conditions about the geometry of the target space at infinity. For example, in Gromov-Witten theory one can assume the existence of a plurisubharmonic function on the manifold; in the case of symplectic vortex equation, there is also an analogous, $G$-equivariant version of this convexity assumption (see [@Cieliebak_Gaio_Mundet_Salamon_2002 Section 2.5]). The uniform bound then follows from a strong maximal principle argument.
In our situation, if the equation is unperturbed, the solutions are holomorphic and they are special solutions to the symplectic vortex equation. So one can prove the $C^0$-bound in the same way as in [@Cieliebak_Gaio_Mundet_Salamon_2002]. The difficulty lies in the perturbed case, where the perturbation term disturbs the control. Even worse, in our case, the gradient $\nabla W$ is not a proper map, so $\nabla W(u)$ cannot control $u$ (such a control [@FJR1 Theorem 5.8] is a crucial technical ingredient in the compactness theorem of Fan-Jarvis-Ruan).
We take a different route. We prove that for a sequence of solutions $(A_i, u_i)$ with uniform energy bound, if $u_i$ blows up near some point on the Riemann surface, then there must be an energy concentration (Corollary \[cor72\]). Such a quantization property implies that the sequence are uniformly bounded up to blowing up at finitely many points. Then we argue that the blowing up contradicts with a local maximal principle.
The establishment of this energy quantization property is lengthy due to the complicated behavior of the superpotential $W$ at infinity. The critical point set ${\rm Crit} W$ is a stratified space, and near infinity of the target space ${\widetilde}{X}$, ${\rm Crit} W$ has components of different nature. If the blow-up of solutions happens away from ${\rm Crit} W$, then the energy quantization is easy to achieve; if the blow-up happens near ${\rm Crit} W$, then in general, we can prove the energy concentration only when it is near a component of ${\rm Crit} W$ of Bott type. However, since in our main example $W$ is the Lagrange multiplier of a homogeneous polynomial, whose critical point set has necessarily a degenerate component, considering only Bott type critical loci is not enough. For the degenerate component, we have to use the special structure of the Lagrange multiplier; this is another (and a more important) technical reason why we have to restrict to such type of superpotentials. On the other hand, this part of argument is purely local and it may shed some light on more general cases.
Once $C^0$-bound is established, the remaining part of the proof of the compactness problem is straightforward. In this paper we assume that the target space is aspherical so that we can rule out sphere bubbles. On the other hand, on the cylindrical ends the solutions may undergo a Morse-Floer type degeneration, similar to the situation of [@FJR3 Section 4]. In this situation we have to consider “solitons”, which are solutions to (\[equation16\]) on the cylinder ${{\mathbb}R} \times S^1$ with $W$ properly perturbed. A stable solution to the perturbed gauged Witten equation is the concatenation of a usual solution with (broken) solitons attached to the cylindrical ends. The construction of a stable solution in a subsequence limit follows from standard arguments.
In this paper we only consider the compactification of the moduli space for a fixed complex structure on the Riemann surface $\Sigma$. The compactification with degenerating complex structures will be much more complicated because the variations of holonomies on the forming nodes can give extra pieces of the limiting stable objects like the situation of [@Mundet_Tian_2009], and it awaits further consideration.
A formal definition of the GLSM correlation functions
-----------------------------------------------------
Our main goal of this series of papers is to define the correlation functions of the gauged linear $\sigma$-model. For this purpose we have to work out the transversality problem of the moduli space and prove that the correlation functions are independent of many choices we made in defining them. The details of constructing the virtual cycle and proving its properties will be given in a forthcoming paper [@Tian_Xu_3]. Assuming the existence of virtual cycle, we announced the definition of the correlation function in [@Tian_Xu_2], which we sketch here.
The correlation function can be defined for general Lagrange multiplier type superpotentials with appropriate assumptions on the pair $(X, Q)$ (see [@Tian_Xu_2]). For simplicity we sketch it for the case of (the Lagrange multiplier of) a quintic polynomial in 5 variables. Let $Q: {{\mathbb}C}^5 \to {{\mathbb}C}$ be a nondegenerate quintic polynomial and $W = pQ: {{\mathbb}C}^6 \to {{\mathbb}C}$ be the superpotential of GLSM. The [**state space**]{} is the direct sum of the narrow sectors and the broad sector. For $\upgamma^{(k)} = \exp \left( {2k\pi {\bm i} \over 5} \right) \in {{\mathbb}Z}_5$ for $k = 1, 2, 3, 4$, the $\upgamma^{(k)}$-sector (which is narrow) of the state space ${{\mathscr}H}_k$ is a one-dimensional rational vector space, generated by one vector $\alpha_k$ of degree $2k -2$. For for $\upgamma^{(0)} = 1$, the broad sector ${{\mathscr}H}_0$ has pure degree $5$, and is isomorphic to the cohomology group $$\begin{aligned}
{{\mathscr}H}_0 = H^3 \big( {\overline}{X}_Q; {{\mathbb}Q} \big).\end{aligned}$$ Here ${\overline}{X}_Q \subset {{\mathbb}P}^4$ is the quintic hypersurface defined by $Q$. For each $a \in {{\mathbb}C}^*$, ${{\mathscr}H}_0$ can be identified canonically with the ${{\mathbb}Z}_5$-invariant part of the cohomology $H^4 \big( Q^{-1}(a); {{\mathbb}Q} \big)$. A perfect pairing can be defined on ${{\mathscr}H}_0$ so it is also identified with ${{\mathscr}H}_4 \big( Q^{-1}(a); {{\mathbb}Q} \big)^{{{\mathbb}Z}_5}$, i.e., the invariant part of the space of vanishing cycles.
We denote by ${{\mathscr}H}_{\rm GLSM}$ the direct sum of broad and narrow sectors. The correlation function is the collection of multi-linear maps $$\begin{aligned}
\label{equation17}
\left\langle \ \cdots\ \right\rangle_{g, n}^d: \big( {{\mathscr}H}_{\rm GLSM} \big)^{\otimes n} \to {{\mathbb}Q},\ g, n, d \in {{\mathbb}Z}, g \geq 0,\ 2g-2+n >0.\end{aligned}$$ To define the correlation function, we need to do certain virtual integration on the moduli space of solutions to the perturbed gauged Witten equation. Here for simplicity, we omit the discussion about gravitational descendents.
In this simplified situation, the topological data we need to fix is the degree of the additional $S^1$-bundle $P_1$. This corresponds to the degrees of holomorphic curves in the quintic 3-fold ${\overline}{X}_Q$. For each rigidified $5$-spin curve, the perturbation data at broad punctures is given by the choice of $a \in {{\mathbb}C}^*$ and a linear function $F(x_1, \ldots, x_5)$. Then denote by ${{\mathcal}W}_{g, n}^d$ the moduli space of solutions to the perturbed gauged Witten equation over a genus $g$, $n$-marked rigidified $r$-spin curve, of degree $d$. The moduli space can be subdivided as the disjoint union of moduli spaces $$\begin{aligned}
{{\mathcal}W}_{g, n}^d \big( \vec{\upgamma}, \vec{\upkappa} \big).\end{aligned}$$ Here $\upgamma = ( \upgamma_1, \ldots, \upgamma_n ) \in ( {{\mathbb}Z}_5 )^n$ describes the monodromies of the $r$-spin structure at the $n$ punctures; $\vec{\upkappa} = (\upkappa_{i_1}, \ldots, \upkappa_{i_b} )$ describes the asymptotics at the broad punctures where each $\upkappa_{i_\alpha}$ is a critical point of the Lagrange multiplier ${\widetilde}{W} = p(Q -a) + F$, or equivalently of the function $F_a:= F|_{Q^{-1}(a)}$. We assume that each ${{\mathcal}W}_{g, n}^d \left( \vec{\upgamma}, \vec{\upkappa} \right)$ has a good compactification, over which we have a well-defined virtual cycle. Then, we can define the virtual counting $$\begin{aligned}
\# {{\mathcal}W}_{g, n}^d \left( \vec{\upgamma}, \vec{\upkappa} \right) \in {{\mathbb}Q}\end{aligned}$$ which is zero if the virtual dimension of ${{\mathcal}W}_{g, n}^d \left( \vec{\upgamma}, \vec{\upkappa} \right)$ is not zero. The correlation function will just be a linear combination of the virtual numbers.
For each critical point $\upkappa$ of $F_a$, its unstable submanifold with respect to the flow of the real part of $F_a$ is a 4-dimensional cycle in $Q^{-1}(a)$ relative to infinity, denoted by $[\upkappa] \in H_4 ( Q^{-1}(a), \infty)$. We define the correlation $$\begin{aligned}
\label{equation18}
\big\langle \theta_1, \ldots, \theta_n \big\rangle_{g, n}^d := \sum_{\vec{\upgamma}} \sum_{ \vec{\upkappa}} \# {{\mathcal}W}_{g, n}^d \big( \vec{\upgamma}, \vec{\upkappa} \big) \big( \theta_{i_1}^* \cap [\upkappa_{i_1}] \big) \cdots \big( \theta_{i_b}^* \cap [\upkappa_{i_b}] \big).\end{aligned}$$ Here we assume that each $\theta_i\in {{\mathscr}H}_{\rm GLSM}$ is homogeneous, i.e., coming from a single sector and if $\theta_i$ is a narrow state, then $\theta_i$ is the generator of the corresponding narrow sector. The first summation in (\[equation18\]) runs over all possible combination of monodromies $\vec{\upgamma} =(\upgamma_1, \ldots, \upgamma_n )$ of an $r$-spin structure over a genus $g$, $n$-marked stable curve, such that if $\theta_i \in {{\mathscr}H}_k$, then $\upgamma_i = \upgamma^{(k)}$; the second summation runs over all combinations of critical points $\vec{\upkappa} = ( \upkappa_{i_1}, \ldots, \upkappa_{i_b})$ of ${\widetilde}{W}$; $\theta_{i_1}^*, \ldots, \theta_{i_b}^* \in H_4 \big( Q^{-1}(a); {{\mathbb}Q} \big)^{{{\mathbb}Z}_5})$ are the duals of the broad states $\theta_{i_1}, \ldots, \theta_{i_b}$. The correlator (\[equation17\]) is then defined by extending (\[equation18\]) linearly.
Organization of the paper
-------------------------
In Section \[section2\], we give the basic set-up of the gauged Witten equation, including the basic assumptions, and how to perturb the equation. In Section \[section3\], we consider the asymptotic behavior of bounded solutions to the perturbed gauged Witten equation. In Section \[section4\] we study the linear Fredholm property of the perturbed Witten equation and compute the index of the linearized operator.
In Section \[section6\]–\[section8\], we consider the compactification of the moduli space when the complex structure of the Riemann surface $\Sigma$ is fixed. In Section \[section6\] we first define the stable objects which are possible geometric limits of a sequence of solutions, and then state the compactness theorem. In Section \[section7\] we establish the energy quantization about blowing-up of solutions. In Section \[section8\] we establish the uniform $C^0$-bound and prove the compactness theorem.
In Appendix \[appendixa\] we provide some basic analytical results which are used in this paper. In Appendix \[appenxib\] we include some basic facts about equivariant topology.
Acknowledgements
----------------
We would like to thank Simons Center for Geometry and Physics for hospitality during our visit in 2013. We thank Kentaro Hori, David Morrison, Edward Witten for useful conversations on GLSM. The second author would like to thank Chris Woodward for helpful discussions. The revision of this paper were partially made during the second author’s visit to Institute for Advanced Study and he would like to thank Helmut Hofer for hospitality.
The gauged Witten equation and perturbations {#section2}
============================================
The target space {#subsection21}
----------------
Let $(X, \omega, J)$ be a Kähler manifold and $Q: X \to {{\mathbb}C}$ is a holomorphic function, with a single critical point $\star \in X$. We assume that there exists a Hamiltonian $S^1$-action with moment map $\mu_0 : X \to {\bm i}{{\mathbb}R}$. Here we identify ${\bm i} {{\mathbb}R} \simeq {\rm Lie} S^1$ with its dual space by the standard metric on ${{\mathbb}R}$. Then for the generator ${\bm i}$ of ${\rm Lie} S^1$, we denote its infinitesimal action by ${{\mathcal}X}_0 \in \Gamma (TX)$.
We suppose that the $S^1$-action extends to a holomorphic ${{\mathbb}C}^*$-action. We also assume that $Q$ is homogeneous of degree $r$, $r>1$ with respect to this ${{\mathbb}C}^*$-action. This means for $x \in X$ and $\xi \in {{\mathbb}C}^*$, $$\begin{aligned}
Q(\xi x) = \xi^r Q(x).\end{aligned}$$ Let $X_Q:= Q^{-1}(0)$, which is smooth away from $\star$. For any $\upgamma \in {{\mathbb}Z}_r$, let $X_\upgamma \subset X$ be the fixed point set of $\upgamma$ and ${\widetilde}{X}_\upgamma= X_{\upgamma}\times {{\mathbb}C}$.
The GLSM target space is the product ${\widetilde}{X} = X \times {{\mathbb}C}$, whose coordinates are denoted by $(x, p)$. The factor ${{\mathbb}C}$ has the standard Kähler structure so that it induces a product Kähler structure, which, for simplicity, is still denoted by $(\omega, J)$ on ${\widetilde}{X}$. We lift the ${{\mathbb}C}^*$-action on $X$ trivially to ${\widetilde}{X}$. The [**superpotential**]{} is the holomorphic function $$\begin{aligned}
W: {\widetilde}{X} \to {{\mathbb}C},\ W(x, p) = p Q(x).\end{aligned}$$ $W$ is also of degree $r$ with respect to the ${{\mathbb}C}^*$-action because $W(\xi (x,p)) = W(\xi x, p) = \xi^r W(x, p)$.
On the other hand, let $G_1 = S^1$ and we consider the $G_1^{{\mathbb}C} = {{\mathbb}C}^*$-action on ${\widetilde}{X}$, given by $$\begin{aligned}
\zeta (x, p) = ( \zeta x, \zeta^{-r} p).\end{aligned}$$ $W$ is then $G_1^{{\mathbb}C}$-invariant. We use $G_0$ (resp. $G_0^{{\mathbb}C}$) to denote the copy of $S^1$ (resp. ${{\mathbb}C}^*$) which acts on $X$ and denote $G = G_0 \times G_1$ (resp. $G^{{\mathbb}C} = G_0^{{\mathbb}C} \times G_1^{{\mathbb}C}$). Then the $G$-action on ${\widetilde}{X}$ is Hamiltonian, with a moment map $$\begin{aligned}
\mu (x, p) = \Big( \mu_0 (x), \mu_0 (x) + {{\bm i}r \over 2}|p|^2 - \tau \Big).\end{aligned}$$ Here $\tau \in {\bm i} {{\mathbb}R}$ is a constant, which we fix from now on. We denote $$\begin{aligned}
\mu_1(x, p) = \mu_0(x) + {{\bm i} r\over 2}|p|^2 - \tau\end{aligned}$$ which is a moment map for the $G_1$-action. Let ${{\mathfrak}g}_i$ be the Lie algebra of $G_i$ for $i = 0, 1$ and ${{\mathfrak}g} = {{\mathfrak}g}_0 \oplus {{\mathfrak}g}_1$. For any $\xi = (\xi_0, \xi_1) \in {{\mathfrak}g}$, we denote by ${{\mathcal}X}_\xi = {{\mathcal}X}_{\xi_0} + {{\mathcal}X}_{\xi_1}\in \Gamma (T{\widetilde}{X})$ the infinitesimal action of $\xi$.
We make the following assumptions on the structures, which are all satisfied by the typical example of nondegenerate homogeneous polynomials on ${{\mathbb}C}^n$ of degree at least 2.
1. $(X, \omega)$ is symplectically aspherical.
2. The Riemannian curvature of $X$ is uniformly bounded; the complex structure $J$ is uniformly continuous on $X$ with respect to the Kähler metric in the sense of Definition \[defna1\].
3. The moment map $\mu_0$ is proper and there exists $c >0$ such that for any $x \in X$, $$\begin{aligned}
{1\over c} {\bm i} \mu_0(x) - c \leq \big| {{\mathcal}X}_0(x) \big|^2 \leq c {\bm i} \mu_0(x) + c.\end{aligned}$$
4. As a real quadratic form on $TX$, we have $$\begin{aligned}
0 \leq \nabla^2 \big( {\bm i}\mu_0 \big) \leq r. \end{aligned}$$
\[hyp21\]
([**X1**]{}) is imposed in order to simplify the proof of compactness; it can be removed. ([**X2**]{}) is a bounded geometry at infinity assumption of $X$, which is used to prove the uniform $C^0$-bound of solutions (see Section \[section7\]-\[section8\]). The precise upper bound of ([**X4**]{}) seems to be too strong but it is satisfied by all quasi-homogeneous polynomials on ${{\mathbb}C}^n$ of positive degrees. The condition ([**X3**]{}) implies certain convexity about the geometry of ${\widetilde}{X}$ near infinity.
\[lemma23\] For each $h \in {\bm i}{{\mathfrak}g}$, we denote $$\begin{aligned}
\label{equation21}
\big( |h|_{{\widetilde}{X}} \big)^2 = \big\| (e^h)^* \omega \|_{L^\infty({\widetilde}{X})}.\end{aligned}$$ There is a constant $c>0$ such that for any $h \in {\bm i} {{\mathfrak}g}$, $$\begin{aligned}
\big( |h|_{{\widetilde}{X}} \big)^2 \leq c|h|.\end{aligned}$$
For any two tangent vector fields $Y, Z$, we have $$\begin{aligned}
\begin{split}
\big({{\mathcal}L}_{J{{\mathcal}X}} \omega \big)(Y, Z) = &\ Y \omega (J{{\mathcal}X}, Z) - Z \omega( J{{\mathcal}X}, Y) - \omega( J{{\mathcal}X}, [Y, Z]) \\
= &\ Z \langle {{\mathcal}X}, Y \rangle - Y \langle {{\mathcal}X}, Z \rangle + \langle {{\mathcal}X}, [Y, Z]\rangle \\
= &\ \langle Y, \nabla_Z {{\mathcal}X}\rangle - \langle Z, \nabla_Y {{\mathcal}X} \rangle \\
= &\ 2 \langle Y, \nabla_Z {{\mathcal}X}\rangle.
\end{split}\end{aligned}$$ The last inequality follows from the fact that ${{\mathcal}X}$ is Killing. Notice that $|\nabla {{\mathcal}X}| = |\nabla^2\mu|$. Then by ([**X4**]{}) of Hypothesis \[hyp21\], $\big| {{\mathcal}L}_{J{{\mathcal}X}} \omega \big|$ is uniformly bounded throughout ${\widetilde}{X}$. Then the lemma follows from the fact that $J{{\mathcal}X}$ is the infinitesimal ${\bm i} {{\mathfrak}g}$-action.
For any $b \in (0,1)$, define ${{\mathcal}F}_{b}: {\widetilde}{X} \to {{\mathbb}R}$ by $$\begin{aligned}
{{\mathcal}F}_b(x, p):= \mu_0(x) \cdot \Big({ {\bm i} (1- b) \over r} \Big) + {b \over 2} |p|^2 = \mu \cdot \Big( - {{\bm i} \over r}, {{\bm i}b \over r} \Big).\end{aligned}$$
\[lemma24\] For any $b \in (0, 1)$, ${{\mathcal}F}_b: {\widetilde}{X} \to {{\mathbb}R}$ is a proper function and is bounded from below. Moreover, there exist a constant $c_0 >0$, a choice of $b_0 \in (0, 1)$ and $\lambda_0 >0$ such that $$\begin{aligned}
\label{equation22}
\big\langle \nabla {{\mathcal}F}_{b_0}, J{{\mathcal}X}_{(\lambda_0 \mu_0, \mu_1)} \big\rangle \geq {1\over c_0} \big| \mu(u) \big|^2 - c_0.\end{aligned}$$
The properness and the fact that ${{\mathcal}F}_b$ is bounded from below follow immediately from ([**X3**]{}) of Hypothesis \[hyp21\]. On the other hand, denoting $\rho = |p|$, we have $$\begin{aligned}
\begin{split}
J{{\mathcal}X}_{\mu_0} (x, p) = &\ \Big( -{\bm i} \mu_0(x) J{{\mathcal}X}_0 (x) , 0 \Big),\\
J{{\mathcal}X}_{\mu_1} (x, p) = &\ \Big( \big( - {\bm i} \mu_0(x) + {r \over 2} |p|^2 + {\bm i} \tau \big) J {{\mathcal}X}_0(x), r \big( -{\bm i} \mu_0(x) + {r\over 2}|p|^2 + {\bm i} \tau \big) \rho {\partial \over \partial \rho} \Big).
\end{split}\end{aligned}$$ Then by ([**X3**]{}) of Hypothesis \[hyp21\], we have $$\begin{aligned}
\begin{split}
\Big\langle \nabla {{\mathcal}F}_b, J{{\mathcal}X}_{\mu_0} \Big\rangle = &\ \Big\langle - \big({ 1- b \over r} \big)J{{\mathcal}X}_0, - {\bm i} \mu_0 J {{\mathcal}X}_0 \Big\rangle\\
= &\ \big({1-b \over r} \big) {\bm i}\mu_0 |{{\mathcal}X}_0|^2\\
\geq &\ \big( {1-b \over c r} \big) |{{\mathcal}X}_0|^4 - \big({1-b \over r} \big) |{{\mathcal}X}_0|^2;\\
\Big\langle \nabla {{\mathcal}F}_b, J{{\mathcal}X}_{\mu_1} \Big\rangle = &\ \Big\langle - \big({1- b \over r} \big) J {{\mathcal}X}_0 , \big( - {\bm i} \mu_0(x) + {r \over 2} |p|^2 + {\bm i} \tau \big) J {{\mathcal}X}_0(x) \Big\rangle\\
&\ + \Big\langle b \rho{\partial \over \partial \rho}, r \big( -{\bm i} \mu_0(x) + {r\over 2}|p|^2 + {\bm i} \tau \big) \rho {\partial \over \partial \rho} \Big\rangle\\
= &\ \big({1-b \over r}\big) \big( {\bm i} \mu_0 - {\bm i} \tau \big) |{{\mathcal}X}_0|^2 + r b |p|^4 - \big( {1-b \over 2} \big) |p|^2 |{{\mathcal}X}_0|^2 \\
&\ - r b {\bm i}\mu_0 |p|^2 + br {\bm i} \tau |p|^2\\
\geq &\ \big({1-b \over c r} \big) |{{\mathcal}X}_0|^4 + rb |p|^4 - \Big( \big({1- b \over 2}\big) + rb c \Big) |p|^2 |{{\mathcal}X}_0|^2 \\
&\ - \big({1- b \over r} \big) ( 1 + {\bm i} \tau) |{{\mathcal}X}_0|^2 + br ( {\bm i} \tau - c^2) |p|^2 .
\end{split}\end{aligned}$$ It suffices to choose $\lambda$ and $b$ so that the quartic part of $\langle \nabla {{\mathcal}F}_b, J{{\mathcal}X}_{(\lambda \mu_0, \mu_1)} \rangle$ is a positive definite form in $|{{\mathcal}X}_0|^2$ and $|p|^2$, i.e., to guarantee that the quadratic form $$\begin{aligned}
( 1+ \lambda) \big( {1- b \over c r} \big) A^2 - \Big( \big({1- b\over 2} \big) + rb c \Big) AB + rb B^2\end{aligned}$$ is positive definite. This is equivalent to $$\begin{aligned}
\Big( \big({1- b\over 2} \big) + rb c \Big) < 4 b ( 1+ \lambda) \big( {1- b \over c } \big).\end{aligned}$$ It holds for certain $b = b_0 \in (0, 1)$ and $\lambda = \lambda_0 >0$. Then $c_0>0$ exists.
We fix $b_0$ and $\lambda_0 $ and denote ${{\mathcal}F}_{b_0}$ by ${{\mathcal}F}$. We use $\lambda_0 $ to define a metric on ${{\mathfrak}g}$ as $$\begin{aligned}
\label{equation23}
\big| (\xi_0, \xi_1) \big|^2 = \lambda_0^{-1} \big| \xi_0 \big|^2 + \big| \xi_1 \big|^2.\end{aligned}$$ This metric induces an identification ${{\mathfrak}g} \simeq {{\mathfrak}g}^*$ and $(\lambda_0 \mu_0, \mu_1)$ can be viewed as the dual of the moment map with respect to this metric. Then (\[equation22\]) can be rewritten as $$\begin{aligned}
\label{equation24}
\big\langle \nabla {{\mathcal}F}, J {{\mathcal}X}_{\mu^*} \big\rangle \geq {1\over c_0} \big| \mu \big|^2 - c_0.\end{aligned}$$
Now we give the assumptions on the function $Q$.
\[hyp25\]
1. There is a constant $c_Q >1 $ and a $G_0$-invariant compact subset $K_Q \subset X$ such that $$\begin{aligned}
x \notin K_Q \Longrightarrow {1\over c_Q} \left| \nabla^3 Q(x) \right| \leq \left| \nabla^2 Q(x) \right| \leq c_Q \left| \nabla Q(x) \right|.\end{aligned}$$ Moreover, for every $\delta>0$, there exists $c_Q(\delta)>0$ such that $$\begin{aligned}
d(x, X_Q) \geq \delta, x \notin K_Q \Longrightarrow |\nabla Q(x)| \leq c_Q(\delta) |Q(x)|.\end{aligned}$$
2. For every $\upgamma \in {{\mathbb}Z}_r$, it is easy to see that $dQ$ vanishes along the normal bundle $N_{\upgamma} \to X_{\upgamma}$. We assume that the Hessian $\nabla^2 Q$ vanishes along $N_{\upgamma}$.
The condition ([**Q2**]{}) is not essential but it helps reduce the technicality in proving the asymptotic property of solutions in Section \[section3\].
$\upgamma \in {{\mathbb}Z}_r$ is called [**broad**]{} (resp. [**narrow**]{}) if $X_\upgamma \neq \{\star\}$ (resp. $X_\upgamma = \{\star\}$).
\[hyp28\]For any broad $\upgamma \in {{\mathbb}Z}_r$, there are a function $F_\upgamma: X \to {{\mathbb}C}$ and $a_\upgamma \in {{\mathbb}C}^*$ satisfying the following conditions.
1. $F_\upgamma$ can be written as $$\begin{aligned}
F_\upgamma = \sum_{l=1}^{s-1} F_{\upgamma;l},\ (2 \leq s \leq r)\end{aligned}$$ where $F_{\upgamma;l}: X \to {{\mathbb}C}$ is a holomorphic function of degree $l$ with respect to the $G_0^{{\mathbb}C}$-action on $X$. The pull-back of $F_{\upgamma;l}$ to ${\widetilde}{X}$ is still denoted by $F_{\upgamma;l}$.
2. Each $F_{\upgamma,l}$ is $\upgamma$-invariant. It is easy to see that $dF_{\upgamma;l}$ vanishes along the normal bundle $N_\upgamma \to X_\upgamma$. We require that for every $l$, the Hessian $\nabla^2 F_{\upgamma;l}$ vanishes along $N_\upgamma$.
3. For $j = 0, 1, \ldots$, there exist constants $c^{(j)} >0$ such that for $l = 1, \ldots, s-1$ and $x \in X$, $$\begin{aligned}
\Big| F_{\upgamma;l} (x) \Big| \leq c^{(0)} \Big( 1 + \big| \mu_0 (x) \big| \Big)^{1\over 2},\ \Big| \nabla^j F_{\upgamma;l} \Big| \leq c^{(j)}, j \geq 1.\end{aligned}$$
4. The restriction of $F_\upgamma$ to $Q^{-1}(a_\upgamma) \cap X_{\upgamma}$ is a holomorphic Morse function. This is equivalent to saying that the Lagrange multiplier $p(Q_\upgamma - a_\upgamma ) + F_\upgamma$ is a holomorphic Morse function on ${\widetilde}{X}_\upgamma$.
5. The perturbation has no critical point at infinity, in the following sense. There exist a compact subset ${\widetilde}{K}_\upgamma \subset {\widetilde}{X}$ and a constant $c_\upgamma > 0$ such that $$\begin{aligned}
(x, p) \notin {\widetilde}{K}_\upgamma \Longrightarrow \Big| \nabla \big( W - a_\upgamma p + F_\upgamma \big)(x, p) \Big| \geq c_\upgamma.\end{aligned}$$
The conditions ([**P1**]{})–([**P5**]{}) are modelled on linear functions on ${{\mathbb}C}^n$, when $Q$ is a nondegenerate quasi-homogeneous polynomial. Like ([**Q2**]{}), the second part of the condition ([**P2**]{}) is not essential and can be removed.
Now for each broad $\upgamma$, we fix the choice of the perturbation data $(a_\upgamma, F_\upgamma)$. We denote $F_{\upgamma; s} = - a_\upgamma p: {\widetilde}{X} \to {{\mathbb}C}$. We introduced $$\begin{aligned}
W_\upgamma':= -a_\upgamma p + \sum_{l=1}^{s-1} F_{\upgamma; l} =: \sum_{l=1}^{s} F_{\upgamma; l}.\end{aligned}$$ We also denote $F_0 (x, p) = W(x, p) = pQ(x)$. For notational purpose, if $\upgamma$ is narrow, we take $W_\upgamma' = \sum_{l=1}^s F_{\upgamma; l}$ to be the sum of $s$ zero functions. Then we denote $$\begin{aligned}
\label{equation25}
{\widetilde}{W}_\upgamma = W + W_\upgamma' = \sum_{l=0}^s F_{\upgamma; l}.\end{aligned}$$
For each $t\in G_0^{{\mathbb}C} = {{\mathbb}C}^*$, we denote $$\begin{aligned}
{\widetilde}{W}_\upgamma^{(t)}(x, p) = t^r {\widetilde}{W}_\upgamma ( t^{-1} x, p) = W(x, p) - t^r a_\upgamma p + \sum_{l=1}^{s-1} t^{r- l} F_{\upgamma; l}(x) =: \sum_{l=0}^s F_{\upgamma; l}^{(t)}.\end{aligned}$$ Most of the time we will consider the case $t \in {{\mathbb}R}_+$ and use $\delta$ instead of $t$. We have $$\begin{aligned}
\label{equation26}
(x, p)\in {\rm Crit} \big( {\widetilde}{W}_\upgamma |_{{\widetilde}{X}_\upgamma} \big) \Longleftrightarrow (tx, p) \in {\rm Crit} \big( {\widetilde}{W}_\upgamma^{(t)}|_{{\widetilde}{X}_\upgamma} \big).\end{aligned}$$
For $l=1, \ldots, s-1$, we denote by $\rho_l: G^{{\mathbb}C} \simeq {{\mathbb}C}^* \times {{\mathbb}C}^* \to {{\mathbb}C}^*$ the character $(\xi_0, \xi_1) \mapsto (\xi_0^l, \xi_1^l)$; we denote by $\rho_{s}: G^{{\mathbb}C} \to {{\mathbb}C}^*$ the character which is trivial on the first ${{\mathbb}C}^*$-factor and is $\xi\mapsto \xi^{-r}$ on the second; we denote by $\rho_0: G^{{\mathbb}C} \to {{\mathbb}C}^*$ the character which is $\xi \mapsto \xi^r$ on the first ${{\mathbb}C}^*$-factor and is trivial on the second. Then each $F_{\upgamma;l}: {\widetilde}{X} \to {{\mathbb}C}$ above is $\rho_l$-equivariant for $l=0, \ldots, s$.
The domain {#subsection22}
----------
### Rigidified $r$-spin curves {#rigidified-r-spin-curves .unnumbered}
We recall the notion of rigidified $r$-spin curves following [@FJR2 Section 2.1].
Let $\Sigma$ be a compact Riemann surface and ${\bm z}= \{z_1, \ldots, z_k\}$ is a finite subset of punctures (marked points). We denote $\Sigma^*:= \Sigma \setminus {\bm z}$. We can attach orbifold charts near each puncture to obtain an [**orbicurve**]{} ${{\mathcal}C}$. Suppose the local group of orbifold chart near each $z_j$ is $\Gamma_j$, which is canonically isomorphic to a cyclic group ${{\mathbb}Z}_{r_j}$. Then $\Sigma$ can be viewed as the “desingularization” of ${{\mathcal}C}$, also denoted by $|{{\mathcal}C}|$. There is a projection $\pi_{{\mathcal}C}: {{\mathcal}C} \to \Sigma$. The orbicurve ${{\mathcal}C}$ has the log-canonical bundle ${{\mathcal}K}_{\log}\simeq \pi_{{\mathcal}C}^* K_{\log}$, where $K_{\log}\to \Sigma$ is the bundle $$\begin{aligned}
K_{\log} = K_\Sigma \otimes {{\mathcal}O}(z_1) \otimes \cdots \otimes {{\mathcal}O}(z_k).\end{aligned}$$
\[defn210\] Fix $r \in {{\mathbb}Z}$, $r \geq 3$. An [**$r$-spin curve**]{} is a triple $({{\mathcal}C}, {{\mathcal}L}, \upvarphi)$ where ${{\mathcal}C}$ is an orbicurve, ${{\mathcal}L}\to {{\mathcal}C}$ is an orbibundle, and $$\begin{aligned}
\upvarphi: {{\mathcal}L}^{\otimes r} \to {{\mathcal}K}_{\log}\end{aligned}$$ is an isomorphism of orbibundles.
A [**rigidification**]{} of the $r$-spin structure $({{\mathcal}L}, \upvarphi)$ at $z_j$ is a choice of an element $e_j$ of ${{\mathcal}L}|_{z_j}$ such that $$\begin{aligned}
\upvarphi( e_j^{\otimes r}) = {dw \over w}.\end{aligned}$$
We denote a rigidification at $z_j$ by a map $\upphi_j: {{\mathbb}C}/ \Gamma_j \to {{\mathcal}L}|_{z_j}$. For a choice of rigidification $\upphi_j$ for each $j$, we call the tuple $\vec{{{\mathcal}C}}:= ({{\mathcal}C}, {{\mathcal}L}, \upvarphi; {\bm \upphi}):= ({{\mathcal}C}, {{\mathcal}L}, \upvarphi; \upphi_0, \ldots, \upphi_k)$ a [**rigidified $r$-spin curve**]{}.
In this paper, from now on, we fix a rigidified $r$-spin curve $\vec{{{\mathcal}C}}= ({{\mathcal}C}, {{\mathcal}L}, \upvarphi; {\bm \upphi})$.
It is more convenient to look at rigidifications on the smooth curve $\Sigma$. Indeed, at each marked point $z_j$, the orbibundle ${{\mathcal}L}$ has its local monodromy, which is a representation $\Gamma_j \to S^1$. As a convention, we always assume that this representation is faithful. Then since ${{\mathcal}K}_{\log}$ always has trivial monodromy, we can view $\Gamma_j$ as a subgroup of ${{\mathbb}Z}_r$. So the generator of $\Gamma_j$ can be written as $\exp \left( {2\pi{\bm i} m_j \over r} \right)$, with $m_j \in \{0, 1, \ldots, r-1\}$. Then the $r$-spin structure induces an isomorphism $$\begin{aligned}
|\upvarphi|: |{{\mathcal}L}|^{\otimes r} \to K_{\log} \otimes {{\mathcal}O}\Big( - \sum_{j=0}^k m_j z_j \Big)\end{aligned}$$ as usual line bundles over $\Sigma$, where $|{{\mathcal}L}| \to \Sigma$ is the desingularization of ${{\mathcal}L}$. Therefore, for any choice of local coordinate $w$ around $z_j$, a rigidification induces a choice of local frame $e_j$ of $|{{\mathcal}L}|$ near $z_j$ such that $$\begin{aligned}
\label{equation27}
|\upvarphi|(e_j^{\otimes r}) = w^{m_j} {dw \over w}.\end{aligned}$$ We denote $\lambda_j = {\bm i} m_j/ r$ (resp. $\upgamma_j = \exp (2\pi \lambda_j)$) and call it the [**residue**]{} (resp. [**monodromy**]{}) of the $r$-spin structure at $z_j$. We define the type of the punctures.
A puncture $z_j$ is called [**narrow**]{} (resp. [**broad**]{}) if $\upgamma_j\in {{\mathbb}Z}_r$ is narrow (resp. broad).
We take a smooth area form $\nu$ on the closed Riemann surface $\Sigma$. Then together with the complex structure, it determines a Riemannian metric, to which we will refer as the “smooth metric”. On the other hand, for each $z_j$, we fix a holomorphic coordinate patch $w: B_1 \to \Sigma$ with $w(0) = z_j$ and use the $\log$ function to identify the punctured $U_j = w (B_1) \setminus \{z_j\}$ with the cylinder $\Theta_+:= [0, +\infty) \times S^1$. The latter has coordinates $s + {\bm i} t = - \log w$. We can choose a different area form $\nu'$ such that $\nu = \sigma \nu'$ where the conformal factor $\sigma: \Sigma^* \to {{\mathbb}R}_+$ is a smooth function whose restriction to each $\Theta_+$ is equal to $e^{-2s}$. The metric determined by $\nu'$ and the complex structure is called the “cylindrical metric” on $\Sigma^*$.
From now on, for each puncture $z_j$, we fix the coordinate $w$ centered at $z_j$, the cylindrical end $U_j$, and its identification with $\Theta_+$. For any $S \geq 0$, we denote by $U_j(S) \subset \Sigma^*$ the subset identified with $[S, +\infty) \times S^1$.
The cylindrical metric has injectivity radius bounded from below. We choose an $r^* \in (0, 1]$ such that for every point $q \in \Sigma^*$, there exists a holomorphic coordinate $$\begin{aligned}
\label{equation28}
z_q = s + {\bm i} t: B_{r^*}(q) \to B_{r^*} \subset {{\mathbb}C}.\end{aligned}$$ such that $z_q(0) = q$. Here $B_{r^*}(q)$ is the $r^*$-neighborhood of $q$ with respect to the cylindrical metric. Then, for each such neighborhood $B_{r^*}(q)$, the area form $\Omega$ can be written as $$\begin{aligned}
\nu = \sigma_q (z_q) {{\bm i}\over 2} dz_q \wedge d{\overline}{z_q}.\end{aligned}$$ Then by shrinking $r^*$ properly, we have $$\begin{aligned}
\label{equation29}
\sup_{p \in \Sigma^*} \sup_{z_q \in B_{r^*}} \sigma_q (z_q) < \infty,\ \forall q\ \sup_{B_{r^*}} \sigma_q \leq 2 \inf_{B_{r^*}} \sigma_q.\end{aligned}$$
We require that, if $B_{r^*}(q) \subset U_j$, then $z_q$ is the restriction of the cylindrical coordinate $ s+ {\bm i} t$ (after proper translation) to $B_{r^*}(q)$.
### Adapted Hermitian metrics {#adapted-hermitian-metrics .unnumbered}
We define the weighted Sobolev space $W_\delta^{k, p}$ to be the Banach space completion of $C^\infty_0(\Sigma^*)$ with respect to the norm $$\begin{aligned}
\big\| f \big\|_{W_\delta^{k, p}(\Sigma^*)} := \big\| \sigma^{- {\delta \over 2}} f \big\|_{W^{k, p}(\Sigma^*)} \end{aligned}$$ where the latter Sobolev norm is taken with respect to the cylindrical metric on $\Sigma^*$. It is similar to define the Sobolev space $W_\delta^{k, p}(U_j)$ for each cylindrical end $U_j$. We denote by ${{\mathpzc}U}_\delta^{2, p}$ the space of $W_{loc}^{2, p}$-maps $g: \Sigma^* \to S^1$ such that for each $j$, $$\begin{aligned}
g|_{U_j} = \exp \big( {\bm i}\xi_j \big),\ \xi_j \in W_\delta^{2, p}(U_j).\end{aligned}$$
From now on we fix $p>2$.
A $W_{loc}^{2,p}$-Hermitian metric $H$ on $|{{\mathcal}L}||_{\Sigma^*}$ is called [**adapted**]{} if there is $\delta>0$ such that $$\begin{aligned}
\log \big( |w|^{- {m_j \over r}} | e_j |_H \big) \in W_\delta^{2, p}(U_j).\end{aligned}$$ Here $m_j$ and $e_j$ are the ones in (\[equation27\]).
Let ${{\mathpzc}H}$ be the space of all adapted metrics on $|{{\mathcal}L}||_{\Sigma^*}$. There is an ${{\mathbb}R}_+$-action on ${{\mathpzc}H}$ by rescaling a metric. For any $H \in {{\mathpzc}H}$, denote by $P_0 (H)$ the $S^1$-frame bundle of $|{{\mathcal}L}|$ with respect to $H$. Then if $H$ is an adapted Hermitian metric, the Chern connection $A_0 (H)$ of $H$ will be a unitary connection on $P_0 (H)$ such that near each puncture, with respect to the trivialization determined by (\[equation27\]), it can be written as $$\begin{aligned}
A_0 (H) = d + \lambda_j d t + \alpha_j\end{aligned}$$ and $\alpha_j$ is a purely imaginary valued 1-form on $U_j$ of class $W_\delta^{1, p}$ for some $\delta>0$. Note that the map $H \mapsto A_0 (H)$ is not injective but is constant on each ${{\mathbb}R}_+$-orbit of ${{\mathpzc}H}$.
Now choose (arbitrarily) a smooth element $H_0 \in {{\mathpzc}H}$ as a reference, and consider the subset ${{\mathpzc}H}_+ \subset {{\mathpzc}H}$ consisting of metrics of the form $e^{2h_0} H_0$ with $h_0 \in W_\delta^{2, p}$ for some $\delta>0$. Then the map $H \mapsto A_0 (H)$ is injective on ${{\mathpzc}H}_+$. We define $$\begin{aligned}
{{\mathpzc}U} =\bigcup_{\delta>0} {{\mathpzc}U}_\delta^{2, p},\ {{\mathpzc}A}_0 = \left\{ g^* A_0 (H)\ |\ H \in {{\mathpzc}H}_+,\ g\in {{\mathpzc}U} \right\}.\end{aligned}$$ Then every element of ${{\mathpzc}A}_0$ has a unique expression as $g^* A_0 (H)$ for $g \in {{\mathpzc}U}_{loc}^{2, p}$ and $H \in {{\mathpzc}H}_+$.
It is necessary for us to remove the ${{\mathbb}R}_+$-action because it will cause trouble in proving compactness. On the other hand, we can release the restriction of ${{\mathpzc}H}_+$ such that we can vary the value of $H$ at the punctures. However, those variations only form a finite dimensional degree of freedom, so they don’t affect compactness and don’t essentially change Fredholm property.
Denote $P_0=P_0(H_0)$, which is the unit circle bundle of $|{{\mathcal}L}|$ with respect to the reference $H_0$. For any $H = e^{2h_0} H_0 \in {{\mathpzc}H}_+$, there is a canonical isomorphism between $P_0 (H)$ and $P_0 (H_0)$, given by $v \mapsto e^{h_0} v$. Then any connection in ${{\mathpzc}A}_0$ is transformed to a $S^1$-connection on $P_0$. We still denote this set of connections by ${{\mathpzc}A}_0$. In particular, for every $A_0 \in {{\mathpzc}A}_0$, the holomorphic line bundle structure of $P_0 \times_{S^1} {{\mathbb}C}$ determined by (the $(0,1)$-part of) $A_0$ is isomorphic to the holomorphic line bundle $|{{\mathcal}L}|$.
Now we will choose a trivialization of $P_0$ on each $B_{r^*}(q)$ as well as on each $U_j$. On each $B_{r^*}(q)$, there is a local holomorphic section $e_q$ of $|{{\mathcal}L}|$ such that $$\begin{aligned}
\label{equation210}
|\upvarphi|(e_q^{\otimes r}) = dz_q.\end{aligned}$$ Here $z_q$ is the fixed one in (\[equation28\]) and $e_q$ is unique up to a ${{\mathbb}Z}_r$-action and we just choose one of them. Then we trivialize $P_0$ over $B_{r^*}(q)$ by the local unitary frame $$\begin{aligned}
\epsilon_q:= { e_q \over \|e_q\|_{H_0}}.\end{aligned}$$ This trivialization is denoted by $$\begin{aligned}
\phi_{q, 0}: B_{r^*}(q)\times S^1 \to P_0 |_{B_{r^*}(q)}.\end{aligned}$$ On the other hand, on each cylindrical end $U_j$, there is a local holomorphic section $e_j$ of $|{{\mathcal}L}|$ such that $$\begin{aligned}
\label{equation211}
|\upvarphi| (e_j^{\otimes r}) = w^{m_j} {dw \over w}.\end{aligned}$$ $e_j$ is unique up to a ${{\mathbb}Z}_r$-action. Then we trivialize $P_0 |_{U_j}$ by the local unitary frame $$\begin{aligned}
\epsilon_j:= { e_j \over \| e_j\|_{H_0} }.\end{aligned}$$ This trivialization is denoted by $$\begin{aligned}
\phi_{j, 0}: U_j \times S^1 \to P_0 |_{U_j}.\end{aligned}$$
Now for each $A_0 \in {{\mathpzc}A}_0$ and each $B_{r^*}(q)$ (resp. $U_j$), we define a function $h_0(A_0): B_{r^*}(q) \to {\rm Lie} {{\mathbb}C}^*$ (resp. $h_0 (A_0): U_j \to {{\mathbb}C}$) as follows. If $A_0 = g^* A_0(H)$ for $H \in {{\mathpzc}H}_+$ and $g \in {{\mathpzc}U}$, then for each $B_{r^*}(q)$ (resp. $U_j$), there is a unique ${\bm i} {{\mathbb}R}$-valued function $h_0':= h_0' (A_0)$ on $B_{r^*}(q)$ (resp. $U_j$) such that $$\begin{aligned}
\label{equation212}
e^{h_0'} = g|_{B_{r^*}(q)},\ -{\bm i}h_0'(q) \in [0, 2\pi),\ \Big( {\rm resp.}\ e^{h_0'} = g|_{U_j},\ \lim_{z \to z_j} h_0' (z) = 0 \Big).\end{aligned}$$ On the other hand, we define $$\begin{aligned}
\label{equation213}
h_0'':= h_0''(A_0) = {\bm i} \log \| e_q \|_{H}\ \Big( {\rm resp.}\ h_0''= {\bm i} \log \Big( \|e_j\|_{H} - |w|^{m_j \over r} \Big) \Big),\end{aligned}$$ where $w = e^{-z}$ is the coordinate centered at $z_j$; then on either $B_{r^*}(q)$ or $U_j$, define $$\begin{aligned}
\label{equation214}
h_0:= h_0(A_0):= h_0' + {\bm i} h_0''.\end{aligned}$$
By the definition of the Chern connection and that of gauge transformation, on each $B_{r^*}(q)$, with respect to the trivialization $\phi_{q, 0}$ of $S|_{B_{r^*}(q)}$, if $A_0 \in {{\mathpzc}A}_0$ is written as $A_0 = d + \phi_0 ds + \psi_0 dt$ for $\phi_0, \psi_0: B_{r^*}(q) \to {\bm i} {{\mathbb}R}$, then $$\begin{aligned}
\phi_0 = \partial_s h_0'- \partial_t h_0'',\ \psi_0 = \partial_s h_0'' + \partial_t h_0'.\end{aligned}$$ Similarly, if on $U_j$, $A_0 = \phi_0 ds + \psi_0 dt$, then $$\begin{aligned}
\phi_0 = \partial_s h_0'- \partial_t h_0'',\ \psi_0 - \lambda_j = \partial_s h_0'' + \partial_t h_0'.\end{aligned}$$ In either case, the curvature form of $A_0$ is equal to $\Delta h_0'' ds dt$.
### The $G_1$-bundle and connections {#the-g_1-bundle-and-connections .unnumbered}
We used $G_1$ to denote another copy of the group $S^1$ to distinguish from the structure group of $P_0$. We fix an arbitrary smooth $G_1$-bundle $P_1 \to \Sigma$. We denote its restriction to $\Sigma^*$ still by $P_1$ and denote by $$\begin{aligned}
P = P_0 \times_{\Sigma^*} P_1 \to \Sigma^*\end{aligned}$$ the fibre product, which is a $G = G_0\times G_1 = S^1 \times S^1$-bundle over $\Sigma^*$. For each coordinate patch $B_{r^*}(q) \subset \Sigma^*$, we fix a trivialization $\phi_{q,1}: U_q \times G_1 \to P_1|_{U_q}$ arbitrarily. For each cylindrical end $U_j$ we can also take a trivialization $\phi_{j,1}: U_j \times G_1 \to P_1|_{U_j}$ which is the restriction of a local trivialization of $P_1$ near $z_j$. Together with the trivializations $\phi_{q, +}$ (resp. $\phi_{j, +}$), this gives a trivialization $\phi_q = (\phi_{q, 0}, \phi_{q, 1}): B_{r^*} (q) \times G \to P|_{B_{r^*}(q)}$ (resp. $\phi_j= (\phi_{j, 0}, \phi_{j, 1}): U_j \times G \to P|_{U_j}$).
We denote ${{\mathpzc}A}_1$ to be the space of $W_{loc}^{1, p}$-connections on $P_1|_{\Sigma^*}$ such that for each cylindrical end $U_j$, with respect to the trivialization of $P_1|_{U_j}$ induced from $\phi_{j, 1}$, any $A_1 \in {{\mathpzc}A}_1$ can be written as $A_1 = d + \alpha_1$ where $\alpha_1$ is a ${{\mathfrak}g}_1$-valued 1-form on $U_j$ of class $W_\delta^{1, p}$ for some $\delta>0$ (with respect to the cylindrical metric).
Now consider ${{\mathpzc}A} = {{\mathpzc}A}_0 \times {{\mathpzc}A}_1$. This is a set of $G$-connections on $P$. For any $\delta>0$, denote by ${{\mathpzc}G}_{1,\delta}^{2,p}$ the group of $G_1$-gauge transformations on $\Sigma^*$ of class $W_{\delta}^{2, p}$ and denote $$\begin{aligned}
{{\mathpzc}G}_1 = \bigcup_{\delta>0} {{\mathpzc}G}_{1, \delta}^{2, p}, \ {{\mathpzc}G} = {{\mathpzc}U} \times {{\mathpzc}G}_1.\end{aligned}$$ Then ${{\mathpzc}G} = {{\mathpzc}U} \times {{\mathpzc}G}_1$ acts on ${{\mathpzc}A}$ naturally.
We would like to define functions similar to $h_0 (A_0)$ given by (\[equation212\])–(\[equation214\]). On $B_{r^*}(q)$, with respect to the trivialization $\phi_{q, 1}$, a $G_1$-connection $A_1 \in {{\mathpzc}A}_1$ can be written as $$\begin{aligned}
A_1 = d + \phi_1 ds + \psi_1 dt,\ \phi_1, \psi_1: B_{r^*}(q) \to {{\mathfrak}g}_1,\end{aligned}$$ where $s + {\bm i} t = z$ is the local coordinate. Then we define a function $h_1 = h_1' + {\bm i} h_1'':= h_1(A_1) = h_1'(A_1) + {\bm i} h_1(A_1)'': B_{r^*}(q) \to {{\mathfrak}g}^{{\mathbb}C}$ by the Cauchy integral formula $$\begin{aligned}
h_1 (A_1)(z) = {1\over 4 \pi {\bm i}} \iint_{B_{r^*}(q)} \left( { \phi_1 + {\bm i} \psi_1 \over \zeta - z} - {\phi_1 + {\bm i} \psi_1 \over \zeta} \right) d\zeta d{\overline}{\zeta}.\end{aligned}$$ Similarly, for $U_j$, we write $A_1$ as $$\begin{aligned}
A_1 =d + \phi_1 ds + \psi_1 dt= d + \vartheta dx + \varsigma dy\end{aligned}$$ where $w = x + {\bm i} y= e^{-z}$ is the smooth coordinate near $z_j$. Then we define $$\begin{aligned}
h_1 (A_1)(z) = {1\over 4 \pi {\bm i}} \iint_{U_j} \left( {\vartheta + {\bm i} \varsigma \over w-z} - {\vartheta + {\bm i} \varsigma \over w }\right) dw d{\overline}{w}.\end{aligned}$$ Since $\phi_1 ds + \psi_1 dt$ is of class $W_\delta^{1, p}$ on $U_j$, we see that $$\begin{aligned}
\Big| \iint_{U_j} { \vartheta + {\bm i} \varsigma \over w} dw d{\overline}{w} \Big| \leq \iint_{U_j}| \phi_1 + {\bm i} \psi_1 | ds dt \leq \big\|\phi_1 + {\bm i} \psi_1 \big\|_{L_\delta^p(U_j) } \big\| e^{- \delta s} \big\|_{L^{p \over p-1}(U_j)} < \infty.\end{aligned}$$ Therefore $h_1$ is well-defined on $U_j$ and $\displaystyle \lim_{s \to +\infty} h_1(s, t) = 0$. On either $B_{r^*}(q)$ or $U_j$, we have $$\begin{aligned}
\label{equation215}
\phi_1 = \partial_s h_1' - \partial_t h_1'',\ \psi_0 = \partial_s h_1'' + \partial_t h_1'.\end{aligned}$$ In particular, the curvature of $A_1$ is $F_{A_1} = \Delta h_1 (A_1)'' ds dt$.
Now for a connection $A = (A_0, A_1)\in {{\mathpzc}A}$, for $U$ being either $B_{r^*}(q)$ or $U_j$, we define $$\begin{aligned}
\label{equation216}
h_A:= (h_0, h_1)= (h_0(A_0), h_1(A_1)): U \to {{\mathfrak}g}^{{\mathbb}C}.\end{aligned}$$ This family of functions are useful when we do local analysis.
### The fibre bundle {#the-fibre-bundle .unnumbered}
Since $G$ acts on ${\widetilde}{X}$, we have the associated fibre bundle $$\begin{aligned}
\pi: Y:= P\times_G {\widetilde}{X} \to \Sigma^*.\end{aligned}$$ The vertical tangent bundle $T^\bot Y \subset TY$ consists of vectors tangent to a fibre. Then since the $G$-action is Hamiltonian and preserves $J$, the Kähler structure on ${\widetilde}{X}$ induces a Hermitian structure on $T^\bot Y$. On the other hand, for any continuous connection $A$, the tangent bundle $TY$ splits as the direct sum of $T^\bot Y$ and the horizontal tangent bundle. The horizontal bundle is isomorphic to $\pi^* T\Sigma^*$, therefore the connection induces an almost complex structure on $Y$. Since ${\widetilde}{X}$ is Kähler, this almost complex structure is integrable and $Y$ becomes a holomorphic fibre bundle over $\Sigma^*$.
We will consider sections of $Y$. A general smooth section is denoted by $u \in \Gamma(Y)$; more generally, we will consider sections $u \in \Gamma_{loc}^{1, p}(Y)$ of class $W^{1, p}_{loc}$. The group ${{\mathpzc}G}$ also acts on the space of sections.
The trivialization $\phi_q: B_{r^*}(q) \times G \to P|_{B_{r^*}(q)}$ (resp. $\phi_j: U_j \times G \to P|_{U_j}$) induces a corresponding local trivialization of $Y$, which is denoted by the same symbol.
The superpotential and gauged Witten equation
---------------------------------------------
### The lift of the superpotential {#the-lift-of-the-superpotential .unnumbered}
Using the $r$-spin structure $\upvarphi: {{\mathcal}L}^{\otimes r} \to {{\mathcal}K}_{\log}$ we can lift the potential function $W$ to the total space $Y$. More precisely, for each $B_{r^*}(q) \subset \Sigma^*$, let $(z_q, e_q)$ satisfy (\[equation210\]). Let $\epsilon_{q, 1}$ be an arbitrary local frame of $P_1|_{B_{r^*}(q)}$. Then a point of $Y|_{B_{r^*}(q)}$ can be represented by $[e_q, \epsilon_{q, 1}, x]$ with the equivalence relation $$\begin{aligned}
[g_0 e_q, g_1 \epsilon_{q, 1}, x] = [e_q, \epsilon_{q,1}, g_0 g_1 x],\ \forall x\in {\widetilde}{X},\ g_0 \in G_0^{{\mathbb}C},\ g_1 \in G_1.\end{aligned}$$ Then we define $$\begin{aligned}
{{\mathcal}W}_{H_0} = \left( [e_q, \epsilon_{0, q}, x] \right) = W(x) dz_q.\end{aligned}$$ Then with respect to the unitary frame $\epsilon_q :=e_q/ \|e_q\|_{H_0}$ of $P_0$, we have $$\begin{aligned}
{{\mathcal}W}_{H_0} \big( [\epsilon_q, \epsilon_{q, 0}, x] \big) = {{\mathcal}W}_{H_0} \big( [e_q, \epsilon_{q,0}, \|e_q\|_{H_0}^{-1} x] \big)= \left\| e_q \right\|_{H_0}^{-r} W(x) dz_q. \end{aligned}$$ Then it is easy to see that the above definition is independent of the choice of the pair $(z_q, e_q)$ satisfying (\[equation210\]) and the choice of the frame $\epsilon_{q, 0}$, so ${{\mathcal}W}_{H_0}$ is a well-defined section of the bundle $\pi^* K_\Sigma \to Y$. Moreover since $W$ is holomorphic we see that ${{\mathcal}W}_{H_0}$ is actually holomorphic with respect to the holomorphic structure on $Y$ induced from the $S^1$-connection $A_0 (H_0)$ and any $G_1$-connection $A_1$.
Now let $H = e^{2h_0} H_0 \in {{\mathpzc}H}_+$. Then we define $$\begin{aligned}
{{\mathcal}W}_{H} = e^{-r h_0} {{\mathcal}W}_{H_0} \in \Gamma \left( Y, \pi^* K_{\Sigma^*} \right).\end{aligned}$$ We see it is holomorphic with respect to the holomorphic structure on $Y$ induced from $A_0 (H)$ and any $G_1$-connection $A_1$. Moreover, for any connection $A = (A_0, A_1) \in {{\mathpzc}A}$, we can express $A_0$ uniquely as $g_0^* A_0 (H)$ for some $g \in {{\mathpzc}U}$ and $H\in {{\mathpzc}H}_+$. Then we define $$\begin{aligned}
{{\mathcal}W}_A(y) = {{\mathcal}W}_{H} ( g y).\end{aligned}$$ Again, this is a section of $\pi^* K_\Sigma$ which is holomorphic with respect to the holomorphic structure on $Y$ induced from $A$. By the $G_1$-invariance of $W$, we also see that for any $g \in {{\mathpzc}G}$, we have $$\begin{aligned}
\label{equation217}
{{\mathcal}W}_{g^* A}(y) = {{\mathcal}W}_A (gy).\end{aligned}$$ On the other hand, using the trivialization $\phi_q: B_{r^*}(q) \times {\widetilde}{X} \to Y|_{B_{r^*}(q)}$, we have $$\begin{aligned}
{{\mathcal}W}_A \circ \phi_q(z, x) = e^{\rho_0(h_A(z))} W(x),\end{aligned}$$ where $\rho_0: G^{{\mathbb}C} \to {{\mathbb}C}^*$ is the character defined at the end of Subsection \[subsection21\]. Similarly, the trivialization $\phi_j: U_j \times G \to P|_{U_j}$ induces a trivialization $\phi_j: U_j \times X \to Y|_{U_j}$, and $$\begin{aligned}
{{\mathcal}W}_A \circ \phi_j(z, x) = e^{\rho_0(h_A(z))} W(e^{\lambda_j t} x) dz = e^{\rho_0(h_A(z)) + r\lambda_j t} W(x) dz.\end{aligned}$$
### The gauged Witten equation {#the-gauged-witten-equation .unnumbered}
The vertical differential of ${{\mathcal}W}_A$ is a section $$\begin{aligned}
d{{\mathcal}W}_A \in \Gamma \Big( Y, \pi^* K_{\Sigma^*} \otimes \big( T^\bot Y \big)^* \Big).\end{aligned}$$ The vertical Hermitian metric on $T^\bot Y$ induces a conjugate linear isomorphism $T^\bot Y \simeq \left( T^\bot Y \right)^*$. On the other hand, the complex structure on $\Sigma^*$ induces a conjugate linear isomorphism $K_{\Sigma^*} \simeq \Lambda^{0,1}T^* \Sigma^*$. Therefore we have a conjugate linear isomorphism $$\begin{aligned}
\pi^* K_{\Sigma^*} \otimes \big( T^\bot Y \big)^* \simeq \pi^* \Lambda^{0,1}_{\Sigma^*} \otimes T^\bot Y.\end{aligned}$$ The image of $d{{\mathcal}W}_A$ under this map is called the [**vertical gradient**]{} of ${{\mathcal}W}_A$, denoted by $$\begin{aligned}
\nabla {{\mathcal}W}_A \in \Gamma \Big( Y, \pi^* \Lambda^{0,1}_{\Sigma^*} \otimes T^\bot Y \Big).\end{aligned}$$
Now we can write down [**the gauged Witten equation**]{}. It is the following system on the pair $(A, u)$, where $A \in {{\mathpzc}A}$ and $u \in \Gamma_{loc}^{1, p}(Y)$: $$\begin{aligned}
\label{equation218}
\left\{ \begin{array}{ccc}
{\overline}\partial_A u + \nabla {{\mathcal}W}_A(u) & = & 0;\\
* F_A + \mu^* (u) & = & 0.
\end{array} \right.\end{aligned}$$ Each term in the system is defined as follows: the connection $A$ induces a continuous splitting $TY \simeq T^\bot Y \oplus \pi^* T \Sigma^*$ and $d_A u \in W^{1, p}_{loc}(T^* \Sigma^* \otimes u^* T^\bot Y )$ is the covariant derivative of $u$; the $G$-invariant complex structure $J$ induces a complex structure on $T^\bot Y$ and ${\overline}\partial_A u$ is the $(0, 1)$-part of $d_A u$ with respect to this complex structure. $\nabla {{\mathcal}W}_A (u)$ is the pull-back of $\nabla {{\mathcal}W}_A$ by $u$, which lies in the same vector space as ${\overline}\partial_A u$. $F_A\in \Omega^2(\Sigma^*) \otimes {{\mathfrak}g}$ is the curvature form of $A$, $*: \Omega^2(\Sigma^*) \to \Omega^0(\Sigma^*)$ is the Hodge-star operator with respect to the smooth metric on $\Sigma$; the moment map $\mu$ lifts to a ${{\mathfrak}g}$-valued function on $Y$ and $\mu^*(u)$ is the dual of $\mu(u)$ with respect to the metric defined by (\[equation23\]).
By (\[equation217\]) and the fact that the $G$-action is Hamiltonian, the gauged Witten equation is ${{\mathpzc}G}$-invariant, in the sense that for any $(A, u)\in {{\mathpzc}A} \times \Gamma_{loc}^{1, p}(Y)$ and any $g\in {{\mathpzc}G}$, we have $$\begin{aligned}
\label{equation219}
\begin{split}
{\overline}\partial_{g^*A} (g^* u) + \nabla {{\mathcal}W}_{g^* A}( g^* u) = &\ \left( g^{-1}\right)_* \left( {\overline}\partial_A u + \nabla {{\mathcal}W}_A (u) \right),\\
* F_{g^* A} + \mu^* ( g^* u) = &\ {\rm Ad}_g^{-1} \left( * F_A + \mu^*(u) \right).
\end{split}\end{aligned}$$
In this paper, all vector fields are regarded as real vector fields. So for a holomorphic function $F: X \to {{\mathbb}C}$, its gradient $\nabla F$ is the gradient of the real part of $F$.
Perturbation
------------
The function $W: {\widetilde}{X} \to {{\mathbb}C}$ has highly degenerate critical points. The degeneracy will cause the problem that the linearized equation doesn’t give a Fredholm operator, in the presence of broad punctures. The usual way to deal with this situation is to perturb the potential ${{\mathcal}W}_A$ near the broad punctures, which is already adopted in the study of Witten equation in [@FJR3]. We will use the functions $F_\upgamma$ given in Hypothesis \[hyp28\] to perturb the superpotential.
### A bounding functional on ${{\mathpzc}A}$ {#a-bounding-functional-on-mathpzca .unnumbered}
In this subsection we would like to construct a [*smooth*]{} functionals on ${{\mathpzc}A}$ which can control certain Sobolev norms. The purpose of having such bounding functionals is to give uniform energy bound on solutions with fixed topological type (see the proof of Theorem \[thm44\]).
\[defn215\] For each $A = (A_0, A_1) \in {{\mathpzc}A} = {{\mathpzc}A}_0 \times {{\mathpzc}A}_1$ and for each broad puncture $z_j$, we define $$\begin{aligned}
\label{equation220}
m_{j,A} = \sum_{l=1}^{s} \big\| e^{\rho_l (h_{j,A}) } \big\|_{L^2(U_j \setminus U_j (2))} + 1,\ \delta_{j, A} = \big( m_{j, A}\big)^{-1}.\end{aligned}$$ Here $h_{j, A}: U_j \to {{\mathfrak}g}^{{\mathbb}C}$ is the function defined by (\[equation216\]). If $z_j$ is narrow, we define $\delta_{j, A} = 1$.
$\delta_{j, A}$ only depends on the gauge equivalence class of $A$ because a gauge transformation only changes the real part of $h_{j,A}$. Moreover, the function $A \mapsto \delta_{j,A}$ is smooth in $A \in {{\mathpzc}A}$. Indeed, the map $A \mapsto h_{j,A}$ is smooth; it follows with the restriction to $U_j \setminus U_j(2)$, and a Sobolev embedding $L_1^p \to C^0$, which are both linear, hence smooth. Now $C^0(U_j \setminus U_j(2))$ is a Banach algebra, so the exponential map is smooth. It is followed by taking the $L^2$-norm of a nonzero continuous function, which is smooth.
### The perturbed gauged Witten equation {#the-perturbed-gauged-witten-equation .unnumbered}
For each broad puncture $z_j$, we can lift $W_{\upgamma_j}'= \sum_{l=1}^s F_{\upgamma_j,l}$ to $Y|_{U_j}$. The trivialization $\phi_j$ gives the local frame $\epsilon_j$ of $P|_{U_j}$. We define $$\begin{aligned}
\begin{array}{cccc}
\displaystyle {{\mathcal}W}_{j,A}': & Y|_{U_j} & \to & ( T^* U_j )^{1,0}\\[0.2cm]
\displaystyle & \big( [ \epsilon_j, x ] \big) & \mapsto & \displaystyle \Big( \sum_{l=1}^{s} e^{\rho_l(h_{j,A} + \lambda_j t)} F_{\upgamma_j,l}^{(\delta_j)} ( x) \Big) {dw \over w}.
\end{array}
\end{aligned}$$ Indeed, $e^{h_{j,A} + \lambda_j t} \epsilon_j$ gives a local frame of $P^{{\mathbb}C}|_{U_j}$ which is holomorphic with respect to $A$, and we have $$\begin{aligned}
{{\mathcal}W}_{j,A}'\Big( [e^{h_{j,A} + \lambda_j t} \epsilon_j, x ] \Big) =\Big( \sum_{l=1}^s F_{\upgamma_j,l}^{(\delta_j)} (x) \Big) {dw \over w}.\end{aligned}$$ This expression shows that ${{\mathcal}W}_{j,A}'$ is holomorphic with respect to the connection $A$.
For each broad puncture $z_j$, fix a cut-off function $\beta_j$ supported in $U_j$ and $\beta|_{U_0(2)} \equiv 1$ such that with respect to the cylindrical metric, $$\begin{aligned}
\big| \nabla \beta_j \big| \leq 1,\ \big| \nabla^2 \beta_j \big| \leq 1\end{aligned}$$ Denote $\beta = \sum_{z_j\ {\rm broad}} \beta_j$. We define $$\begin{aligned}
{\widetilde}{{\mathcal}W}_A = {{\mathcal}W}_A + \sum_{z_j\ {\rm broad}} \beta_j {{\mathcal}W}_{j,A}'.\end{aligned}$$ ${\widetilde}{{\mathcal}W}_A$ is only vertically holomorphic and is holomorphic outside the support of $d\beta$. Then the [**perturbed gauged Witten equation**]{} is $$\begin{aligned}
\label{equation221}
\left\{ \begin{array}{ccc}
{\overline}\partial_A u + \nabla {\widetilde}{{\mathcal}W}_A(u) & = & 0;\\
* F_A + \mu^* (u) & = & 0.
\end{array}\right.\end{aligned}$$ Similar to the unperturbed case, the perturbed gauged Witten equation is gauged invariant in a similar sense as (\[equation219\]).
Energy
------
For $(A, u)\in {{\mathpzc}A} \times \Gamma_{loc}^{1,p}(Y)$, we define its energy as $$\begin{aligned}
E (A, u) = {1\over 2} \Big( \big\| d_A u \big\|_{L^2(\Sigma^*)}^2 + {1\over 2} \big\| F_A \big\|_{L^2(\Sigma^*)}^2 + \big\| \mu(u) \big\|_{L^2(\Sigma^*)}^2\Big) + \big\| \nabla {\widetilde}{{\mathcal}W}_A (u) \big\|_{L^2(\Sigma^*)}^2.\end{aligned}$$ Here the Sobolev norms are taken with respect to the smooth metric on $\Sigma$. The sum of the first two terms is sometimes referred to as the kinetic energy, and the sum of the last two terms is sometimes referred to as the potential energy.
This energy functional generalizes the Yang-Mills-Higgs functional used in gauged Gromov-Witten theory, which can be viewed as a special case of our setting where $W = 0$. One can compare it with the bosonic part of the supersymmetric action in [@Witten_LGCY].
Regularity
----------
\[prop216\] Suppose $(A, u) \in {{\mathpzc}A} \times \Gamma_{loc}^{1, p}(Y)$ is a solution to (\[equation221\]). Then there exists a gauge transformation $g\in {{\mathpzc}G}$ such that $g^* (A, u)$ is smooth.
Suppose $A \in {{\mathpzc}A}_\delta^{1, p}$ for some $\delta>0$. Let $d^*$ be the dual of $d$ with respect to the cylindrical metric and $\Delta = - d^* d$. Then $\Delta: W_\delta^{2, p}(\Sigma^*) \otimes {{\mathfrak}g} \to L_\delta^p(\Sigma^*) \otimes {{\mathfrak}g}$ is Fredholm. Therefore we can find a smooth element $A' \in {{\mathpzc}A}$ such that $d^* (A - A')\in L_\delta^p(\Sigma^*) \otimes {{\mathfrak}g}$ lies in the range of $\Delta$. Choose $h \in \Delta^{-1} \big( d^* (A - A')\big)$ and denote $g = \exp h \in {{\mathpzc}G}$. Then $$\begin{aligned}
d^* \big( g^* A - A' \big) = - \Delta h + d^* (A - A') = 0.\end{aligned}$$ This means that $g^* A$ is in Coulomb gauge relative to $A'$. Let $\alpha = g^* A - A'$. Then $$\begin{aligned}
{\overline}\partial_{A'} u = - \nabla {\widetilde}{{\mathcal}W}_A (u) - \big({{\mathcal}X}_{\alpha} (u) \big)^{0,1},\ d\alpha = - \mu^* (u) \nu - F_{A'}.\end{aligned}$$ Apply the standard elliptic bootstrapping argument to the pair $(\alpha, u)$ we see that $(\alpha, u)$ is indeed smooth.
So it suffices to consider smooth solutions to the perturbed gauged Witten equation.
Local and cylindrical models of gauged Witten equation
======================================================
Let $r >0$. The parameters of [**local models of gauged Witten equation**]{} over $B_r$ are triples $(\beta, \sigma, \delta)$, where: $\beta: B_r \to [0,1]$, $\sigma: B_r \to [0, +\infty)$ are smooth functions satisfying $$\begin{aligned}
|d\beta| \leq 1,\ \sigma \leq C(\sigma), \ \sigma^+:= \sup_{B_r} \sigma \leq 2 \sigma^-:= 2\inf_{B_r} \sigma\end{aligned}$$ and $\delta \in (0, 1]$ is a constant. A solution to the local model with parameter $(\beta, \sigma, \delta)$ is a pair ${\bm u} = (u, h = h' + {\bm i} h'') \in C^\infty( B_r, {\widetilde}{X} \times {{\mathfrak}g}^{{\mathbb}C})$ solving the equation $$\begin{aligned}
\label{equation31}
\partial_s u + {{\mathcal}X}_\phi(u) + J \big( \partial_t u + {{\mathcal}X}_\psi (u) \big) + 2 \nabla {\widetilde}{W}_h^{(\delta)}(u) = 0,\ \Delta h'' + \sigma \mu^* (u) = 0.\end{aligned}$$ Here $(\phi, \psi): B_r \to {{\mathfrak}g} \times {{\mathfrak}g}$ is given by $\phi + {\bm i} \psi = 2(\partial h / \partial {\overline}{z})$, $$\begin{aligned}
{\widetilde}{W}_h^{(\delta)}(z, x) = e^{\rho_0(h(z))} F_0(x) + \beta(z) \sum_{l=1}^s e^{\rho_l(h(z))} F_l^{(\delta)}(x).\end{aligned}$$ Here ${\widetilde}{W} = \sum_{l=0}^s F_l$ is equal to the function ${\widetilde}{W}_\upgamma$ we specified in (\[equation25\]) for some $\upgamma \in {{\mathbb}Z}_r$.
Let $\Theta_+ = [0, +\infty) \times S^1$ and let $\sigma: \Theta_+ \to {{\mathbb}R}_+ \cup \{0\}$ be a smooth function satisfying $$\begin{aligned}
\label{equation32}
\big| \nabla^j \sigma(s, t) \big| \leq C^{(j)}(\sigma) e^{-2s},\ j = 0, 1, \ldots.\end{aligned}$$
Let $\lambda \in {\bm i} ({{\mathbb}Z}/r \cap [0, 1)) \subset {{\mathfrak}g}_0$ and $\upgamma = \exp (2\pi \lambda)$. The parameters of [**cylindrical models of gauged Witten equation with residue $\lambda$**]{} (called $\lambda$-cylindrical model for short) is a pair $(\sigma, \delta)$, where: $\sigma: \Theta_+ \to {{\mathbb}R}_+ \cup\{0\}$ is a smooth function satisfying (\[equation32\]) and $\delta\in (0, 1]$ is a constant, such that if $\upgamma$ is narrow, then $\delta = 1$.
A smooth solution to a $\lambda$-cylindrical model with parameter $(\sigma, \delta)$ is a map ${\bm u} = (u, h) \in C^\infty( \Theta_+, {\widetilde}{X} \times {{\mathfrak}g}^{{\mathbb}C})$ which solves $$\begin{aligned}
\label{equation33}
\partial_s u + {{\mathcal}X}_\phi(u) + J \big( \partial_t u + {{\mathcal}X}_{\psi} \big) + 2 {\widetilde}{W}_{h, \lambda}^{(\delta)}(u) = 0,\ \Delta h'' + \sigma \mu^*(u) = 0.\end{aligned}$$ and which satisfies $$\begin{aligned}
\label{equation34}
\lim_{s \to +\infty} h(s, t) = 0,\ \phi \in W_\delta^{1, p}(\Theta, {{\mathfrak}g})\ {\rm for\ some\ }\delta>0.\end{aligned}$$ Here $(\phi, \psi): \Theta_+ \to {{\mathfrak}g} \times {{\mathfrak}g}$ is given by $\phi + {\bm i}(\psi - \lambda) = 2( \partial h / \partial {\overline}{z})$ and $$\begin{aligned}
\label{equation35}
{\widetilde}{W}_{h, \lambda}^{(\delta)}(z, x) = \sum_{l=0}^s e^{\rho_l(h(z) + \lambda t)} F_{\upgamma;l}^{(\delta)}(x)\end{aligned}$$ where the function $\sum_{l=0}^s F_{\upgamma; l} ={\widetilde}{W}_\upgamma$ is the one we specified in (\[equation25\]).
For either the local model or the cylindrical model, we have the coordinate $z = s + {\bm i} t$. If ${\bm u} = (u, h)$ is a solution to either type of model, we abbreviate $A = \phi ds + \psi dt$, $d_A u = (\partial_s u + {{\mathcal}X}_\phi(u)) ds + (\partial_t u + {{\mathcal}X}_{\psi}(u)) dt$ and ${\widetilde}{{\mathcal}W}_A = {\widetilde}{W}_h^{(\delta)}$. The energy density of ${\bm u}$ is $$\begin{aligned}
e({\bm u}) = {1\over 2} \big| d_A u \big|^2 + \big| \nabla {\widetilde}{{\mathcal}W}_A(u) \big|^2 + \big| \sqrt{\sigma} \mu(u) \big|^2.\end{aligned}$$ The total energy $E({\bm u})$ is the integral of $e({\bm u})$ over the domain (either $B_r$ or $\Theta_+$).
Suppose $(A, u)$ is a solution to the perturbed gauged Witten equation (\[equation221\]) over the rigidified $r$-spin curve $\vec{{\mathcal}C}$. Then for any $q \in \Sigma^*$, the restriction of $(A, u)$ to $B_r(q)$ ($r \leq r^*$) gives a solution to a corresponding local model, via the local coordinate on $B_r(q)$ and the local trivialization of $P|_{B_r(q)}$. Moreover, if the puncture $z_j$ has residue $\lambda_j$, then the restriction of $(A, u)$ to any $U_j(S)\simeq \Theta_+$ ($S \geq 1$), via the trivialization $\phi_j$, gives a solution to a solution to a $\lambda_j$-cylindrical model.
On the other hand, suppose $(u, h)$ is a solution to a $\lambda$-cylindrical model with parameter $(\sigma, \delta)$. Then on any disk $B_r \subset \Theta_+$, the function $\lambda t: B_r \to {{\mathfrak}g}$ is single-valued and the pair $(u|_{B_r}, h|_{B_r} + \lambda t)$ is a solution to the local model over $B_r$ with parameter $(\beta = 1, \sigma|_{B_r}, \delta)$.
Suppose ${\bm u} = (u, h)$ is a solution to a local model over $B_r$ and $f: U \to {{\mathfrak}g}$ is a smooth function. Let $g = \exp f$. Then $g^*{\bm u} := (g^{-1} u, h + f)$ is another solution to the original model. We simply say that ${\bm u}$ and $g^* {\bm u}$ are gauge equivalent.
Digress: holomorphic 1-forms {#subsection31}
----------------------------
To carry out local calculations, we make a digress on properties of holomorphic 1-forms on Kähler manifolds. These properties apply to the case of holomorphic functions directly. Within this subsection, ${\widetilde}{X}$ is a general Kähler manifold and $G$ is a compact Lie group. Assume that there is a Hamiltonian $G$-action on ${\widetilde}{X}$, which extends to a holomorphic $G^{{\mathbb}C}$-action.
We say a holomorphic 1-form $\alpha$ on ${\widetilde}{X}$ is homogeneous with respect to a character $\rho: G^{{\mathbb}C} \to {{\mathbb}C}^*$ if for any $g\in G^{{\mathbb}C}$, $g^* \alpha = \rho(g) \alpha$. In particular, if $f$ is a homogeneous function, then $df$ is homogeneous with respect to the same character. On the other hand, for a holomorphic 1-form, $\alpha$, we define its metric dual $\alpha^*$ to be the [*real*]{} vector field satisfying that for any real vector field $Z$, $$\begin{aligned}
\langle \alpha^*, Z\rangle = {\rm Re} \left( \alpha(Z)\right).\end{aligned}$$
\[lemma33\] If $\alpha$ is a holomorphic 1-form which is homogeneous with respect to $\rho: G^{{\mathbb}C} \to {{\mathbb}C}^*$, then for any $\xi \in {{\mathfrak}g}$, $$\begin{aligned}
\label{equation37}
\left[ \alpha^*, {{\mathcal}X}_\xi \right] = \rho(\xi) \alpha^*.\end{aligned}$$ Moreover, for any real vector field $Z$, $$\begin{aligned}
\label{equation38}
\left[ \nabla_Z \alpha^*, {{\mathcal}X}_\xi \right] = \rho(\xi) \nabla_Z \alpha^* + \nabla_{[Z, {{\mathcal}X}_\xi]} \alpha^*.\end{aligned}$$
The homogeneity of $\alpha$ implies that ${{\mathcal}L}_{{{\mathcal}X}_\xi} \alpha = \rho(\xi) \alpha$. Therefore, for any $G$-invariant vector field $Z$, since ${{\mathcal}X}_\xi$ is Killing, we have $$\begin{aligned}
\big\langle Z, [ \alpha^*, {{\mathcal}X}_\xi ] \big\rangle = - {{\mathcal}L}_{{{\mathcal}X}_\xi} \big( {\rm Re} ( \alpha(Z)) \big) = - \big({\rm Re} {{\mathcal}L}_{{{\mathcal}X}_\xi} \alpha \big)(Z) = - {\rm Re} \big( \rho(\xi) \alpha \big) (Z) = \big\langle Z, \rho (\xi) \alpha^* \big\rangle.\end{aligned}$$ Therefore $[ \alpha^*, {{\mathcal}X}_\xi ] = \rho(\xi) \alpha^*$. To prove the second equality, we may assume that $Z$ is $G$-invariant. Then take another $G$-invariant vector field $Z'$, we see $$\begin{aligned}
\begin{split}
\big\langle Z', [ \nabla_Z \alpha^* , {{\mathcal}X}_\xi ] \big\rangle = &\ - {{\mathcal}X}_\xi \big\langle Z', \nabla_Z \alpha^* \big\rangle \\
= &\ - {{\mathcal}X}_\xi Z \big\langle Z, \alpha^* \big\rangle + {{\mathcal}X}_\xi \big\langle \nabla_Z Z', \alpha^* \big\rangle \\
= &\ -Z {{\mathcal}X}_\xi \big\langle Z', \alpha^* \big\rangle + {{\mathcal}X}_\xi \big\langle \nabla_Z Z', \alpha^* \big\rangle \\
= &\ Z \big\langle Z', \rho (\xi) \alpha^* \big\rangle - \big\langle \nabla_Z Z', \rho (\xi) \alpha^* \big\rangle \\
= &\ \big\langle Z', \nabla_Z (\rho (\xi)) \alpha^* \big\rangle \\
= &\ \big\langle Z', \rho (\xi) \nabla_Z \alpha^* \big\rangle.
\end{split}\end{aligned}$$ Therefore (\[equation38\]) holds.
\[lemma34\] Suppose $\alpha$ is a holomorphic 1-form and $\alpha^*$ is the real vector field defined by $\langle \alpha^*, Z \rangle = {\rm Re} (\alpha(Z))$. Then we have $$\begin{aligned}
\nabla_{JZ} \alpha^* = - J \nabla_Z \alpha^*.\end{aligned}$$
It suffices to prove for any tangent vector $V$, we have $$\begin{aligned}
\langle \nabla_{JZ} \alpha^*, V\rangle = \langle \nabla_Z \alpha^*, J V \rangle.\end{aligned}$$ Indeed, the equality is bilinear in $Z$ and $V$. So it suffices to consider the case when $[Z,V] = [JZ, V] = 0$. In this case we have $$\begin{aligned}
\begin{split}
\big\langle \nabla_{JZ} \alpha^*, V \big\rangle = &\ JZ \big\langle \alpha^*, V \big\rangle - \big\langle \alpha^*, \nabla_{JZ} V \big\rangle\\
=&\ {\rm Re} \big( JZ \alpha(V) - \alpha (J \nabla_Z V) \big) \\
= &\ {\rm Re} \big( d\alpha (JZ, V) + V \alpha(JZ) - {\bm i} \alpha(\nabla_Z V) \big) \\
= &\ {\rm Re} \big( {\bm i} d\alpha(Z, V) - {\bm i} V \alpha(Z) - {\bm i} \alpha( \nabla_Z V) \big) \\
= &\ {\rm Re} \big( Z \alpha(JZ) - \alpha (\nabla_Z (JV))\big) \\
= &\ Z \big\langle \alpha^*, JV \big\rangle - \big\langle \alpha^*, \nabla_Z (J V) \big\rangle \\
= &\ \big\langle \nabla_Z \alpha^*, JV \big\rangle.
\end{split}\end{aligned}$$ Here in the third and fourth equalities we used the fact that $\alpha$ is holomorphic.
Local calculations {#subsection32}
------------------
Consider an arbitrary smooth map $(u, \phi, \psi): U \to {\widetilde}{X}\times {{\mathfrak}g} \times {{\mathfrak}g}$ and denote $A = \phi ds + \psi dt$. Here $U$ is a region which is either $B_r$ or $\Theta_+$, having a coordinate $z = s + {\bm i} t$. We recall certain differential operators naturally associated to the triple $(u, \phi, \psi)$ (cf. [@Cieliebak_Gaio_Mundet_Salamon_2002] and [@Gaio_Salamon_2005] for more comprehensive treatment of such operators). For any $\xi \in \Gamma (U, u^* T{\widetilde}{X})$, we define $$\begin{aligned}
D_{A, s} \xi = \nabla_s \xi + \nabla_\xi {{\mathcal}X}_\phi,\ D_{A, t} \xi = \nabla_t \xi + \nabla_\xi {{\mathcal}X}_\psi.\end{aligned}$$ We list some of their properties whose proofs can be found in [@Cieliebak_Gaio_Mundet_Salamon_2002 Section 2.4] and [@Gaio_Salamon_2005 Section 4].
\(I) For $\xi_1, \xi_2 \in \Gamma( U, u^* T{\widetilde}{X} )$, let $\langle \xi_1, \xi_2 \rangle$ be the real inner product on $T{\widetilde}{X}$. Then $$\begin{aligned}
\label{equation39}
\begin{split}
\partial_s \langle \xi_1, \xi_2 \rangle =&\ \langle D_{A, s} \xi_1, \xi_2 \rangle + \langle \xi_1, D_{A, s} \xi_2 \rangle,\\
\partial_t \langle \xi_1, \xi_2 \rangle =&\ \langle D_{A, t} \xi_1, \xi_2 \rangle + \langle \xi_1, D_{A, t} \xi_2 \rangle.
\end{split}\end{aligned}$$
\(II) Since $J$ is integrable and $G$-invariant, we have $$\begin{aligned}
\label{equation310}
[ D_{A, s}, J ] = [ D_{A, t}, J ] = 0.\end{aligned}$$
\(III) Let $R$ be the curvature tensor of ${\widetilde}{X}$ and $F_A = \partial_s \psi - \partial_t \phi$. Denote $v_s = \partial_s u + {{\mathcal}X}_\phi$, $v_t = \partial_t u + {{\mathcal}X}_\psi$, then for $\xi \in \Gamma ( U, u^* T{\widetilde}{X} )$, we have $$\begin{aligned}
\label{equation311}
D_{A, s} v_t - D_{A, t} v_s = {{\mathcal}X}_{F_A};\end{aligned}$$ $$\begin{aligned}
\label{equation312}
[ D_{A, s}, D_{A, t} ] \xi = R(v_s, v_t) \xi + \nabla_\xi {{\mathcal}X}_{F_A}.\end{aligned}$$
Let $h = h'+ {\bm i} h'': U \to {{\mathfrak}g}^{{\mathbb}C}$ be a smooth function such that $\phi + {\bm i} \psi = 2 ( \partial h/ \partial {\overline}{z})$. As an application of Lemma \[lemma33\] and \[lemma34\], we have
\[lemma35\] Let $F: {\widetilde}{X} \to {{\mathbb}C}$ be a homogeneous function with respect to a character $\rho: G^{{\mathbb}C} \to {{\mathbb}C}^*$. Then we have $$\begin{aligned}
\begin{split}
D_{A, s} e^{{\overline}{\rho(h)}} \nabla F(u) = &\ e^{{\overline}{\rho(h)}} \big( \nabla_{v_s} \nabla F + 2 \rho ( {\bm i} \partial h'' / \partial {\overline}{z} ) \nabla F \big),\\
D_{A, t} e^{{\overline}{\rho(h)}} \nabla F(u) = &\ e^{{\overline}{\rho(h)}} \big( \nabla_{v_t} \nabla F + 2 \rho ( \partial h''/ \partial {\overline}{z}) \nabla F \big).
\end{split}\end{aligned}$$
We have $$\begin{aligned}
\begin{split}
D_{A,s} e^{{\overline}{\rho(h)}} \nabla F(u) = &\ e^{{\overline}{\rho(h)}} \left( {\overline}{\rho (\partial_s h)} \nabla F + \nabla_s \nabla F + \nabla_{\nabla F } {{\mathcal}X}_{\phi} \right)\\
= &\ e^{{\overline}{\rho(h)}} \left( {\overline}{\rho ( \partial_s h)} \nabla F + \nabla_{v_s} \nabla F + [\nabla F, {{\mathcal}X}_\phi] \right)\\
= &\ e^{{\overline}{\rho(h)}} \left( \rho ( - \partial_s h' + {\bm i} \partial_s h'') \nabla F + \nabla_{v_s} \nabla F + \rho ( \partial_s h' - \partial_t h'') \nabla F \right)\\
= &\ \nabla_{v_s} e^{{\overline}{\rho(h)}} \nabla F + \rho ( {\bm i} \partial_s h'' - \partial_t h'' ) e^{{\overline}{\rho(h)}} \nabla F.
\end{split}\end{aligned}$$ The third equality follows from Lemma \[lemma33\]. The formula for $D_{A, t} e^{{\overline}{\rho(h)}} \nabla F(u)$ follows in the same way.
On the other hand, consider a vector field $Z$ along $u: U \to {\widetilde}{X}$. We have $$\begin{aligned}
\label{equation313}
\begin{split}
&\ D_{A, s} \Big( e^{{\overline}{\rho(h)}} \nabla_Z \nabla F \Big) \\
= &\ e^{{\overline}{\rho(h)}} \Big( {\overline}{\rho(\partial_s h)} \nabla_Z \nabla F + \nabla_s \nabla_Z \nabla F + \nabla_{\nabla_Z \nabla F} {{\mathcal}X}_\phi \Big)\\
= &\ e^{{\overline}{\rho(h)}} \Big( {\overline}{\rho(\partial_s h) } \nabla_Z \nabla F + \big[ \nabla_Z \nabla F, {{\mathcal}X}_\phi \big] + \big( \nabla_s + \nabla_{{{\mathcal}X}_\phi} \big) \nabla_Z \nabla F \Big)\\
= &\ e^{{\overline}{\rho(h)}} \Big( 2 \rho ({\bm i} \partial h''/ \partial {\overline}{z} ) \nabla_Z \nabla F + \nabla_{[Z, {{\mathcal}X}_\phi]} \nabla F + \big( \nabla_s + \nabla_{{{\mathcal}X}_\phi} \big) \nabla_Z \nabla F \Big)\\
= &\ e^{{\overline}{\rho (h)}} \Big( 2 \rho ({\bm i} \partial h''/ \partial {\overline}{z} ) \nabla_X \nabla F + \nabla_{\nabla_s Z + \nabla_{{{\mathcal}X}_\phi} Z + [Z, {{\mathcal}X}_\phi] } \nabla F \Big) \\
&\ + e^{{\overline}{\rho (h)}} \Big( \big( \nabla_s + \nabla_{{{\mathcal}X}_\phi} \big) \nabla_Z \nabla F - \nabla_{\nabla_s Z + \nabla_{{{\mathcal}X}_\phi} Z} \nabla F \Big) \\
= &\ e^{{\overline}{\rho(h)}} \Big( 2 \rho ({\bm i} \partial h''/ \partial {\overline}{z} ) \nabla_Z \nabla F + \nabla_{D_{A, s} Z} \nabla F + G_F (v_s, Z) \Big)
\end{split}
\end{aligned}$$ where the tensor $G_F$ is the third derivative of $F$, given by $$\begin{aligned}
G_F (V, Z) = \nabla_V(\nabla_Z \nabla F) - \nabla_{\nabla_V Z} \nabla F.\end{aligned}$$ In deriving the third equality we used the second part of Lemma \[lemma33\]. Similar to (\[equation313\]), $$\begin{aligned}
\label{equation314}
D_{A, t} \Big( e^{{\overline}{\rho(h)}} \nabla_Z \nabla F \Big) = e^{{\overline}{\rho(h)}} \Big( 2 \rho( \partial h''/ \partial {\overline}{z}) \nabla_Z \nabla F + \nabla_{D_{A, t} Z} \nabla F + G_F (v_t, Z) \Big). \end{aligned}$$
Now we denote $$\begin{aligned}
D_A^{1, 0} = \big( D_{A, s} - J D_{A, t} \big)/2,\ D_A^{0,1} = \big( D_{A, s} + J D_{A, t} \big)/2.\end{aligned}$$ Then Lemma \[lemma35\] and Lemma \[lemma34\] imply that $$\begin{aligned}
\label{equation315}
\begin{split}
D_A^{1,0} e^{{\overline}{\rho(h)}} \nabla F(u) = &\ \nabla_{{\overline}\partial_A u} e^{{\overline}{\rho(h)}} \nabla F(u),\\
D_A^{0,1} e^{{\overline}{\rho(h)}} \nabla F(u) = &\ \nabla_{\partial_A u } e^{{\overline}{\rho(h)}} \nabla F(u) + 2 e^{{\overline}{\rho(h)}} \rho( {\bm i} \partial h''/ \partial {\overline}{z} ) \nabla F(u).
\end{split}\end{aligned}$$
Suppose $(\beta, \sigma, \delta)$ parametrizes a local model. For any smooth $(u, h): B_r \to {\widetilde}{X} \times {{\mathfrak}g}^{{\mathbb}C}$, abbreviate the inhomogeneous term in the first equation of (\[equation31\]) as $$\begin{aligned}
\nabla {\widetilde}{{\mathcal}W}_A= e^{{\overline}{\rho_0(h(z))}} \nabla W(x) + \beta(z) \sum_{l=1}^s e^{{\overline}{\rho_l (h(z))}}\nabla F_l^{(\delta)} (x) = {{\mathcal}W}_A + \beta {{\mathcal}W}_A'.\end{aligned}$$ Then by (\[equation315\]), we have $$\begin{aligned}
\label{equation317}
\begin{split}
D_A^{1,0} \nabla {\widetilde}{{\mathcal}W}_A(u) = &\ \nabla_{{\overline}\partial_A u} \nabla {\widetilde}{{\mathcal}W}_A(u) + (\partial \beta / \partial z) \nabla {{\mathcal}W}_A'(u);
\end{split}\\
\begin{split}\label{equation318}
D_A^{0,1} \nabla {\widetilde}{{\mathcal}W}_A (u) = &\ \nabla_{\partial_A u} \nabla {\widetilde}{{\mathcal}W}_A (u) + ( \partial \beta / \partial {\overline}{z}) \nabla {{\mathcal}W}_A'(u) \\
+\ 2 e^{{\overline}{\rho_0(h)}} \rho_0( {\bm i} \partial h''/\partial {\overline}{z} ) \nabla F_0^{(\delta)}(u) & + 2 \beta \sum_{l=1}^s e^{{\overline}{\rho_l(h)}} \rho_l ( {\bm i} \partial h''/ \partial {\overline}{z}) \nabla F_l^{(\delta) } (u) .
\end{split}\end{aligned}$$
Moreover, for $l = 0, 1, \ldots, s$, we define $$\begin{aligned}
\label{equation319}
\begin{split}
H_{A, s}^{(l)}(u, d_A u, Z) = &\ e^{{\overline}{\rho_l (h)}} \Big( 2 \rho_l ( {\bm i} \partial h''/\partial {\overline}{z}) \nabla_Z \nabla F_l^{(\delta)} + G_{F_l^{(\delta)} } (v_s, Z) \Big),\\
H_{A, t}^{(l)}(u, d_A u, Z) = &\ e^{{\overline}{\rho_l (h)}}\Big( 2 \rho_l (\partial h''/ \partial {\overline}{z}) \nabla_Z \nabla F_l^{(\delta)} + G_{F_l^{(\delta)}} (v_t, Z) \Big).
\end{split}\end{aligned}$$ We define $$\begin{aligned}
\label{equation320}
\begin{split}
{\widetilde}{H}_{A, s} = H_{A, s}^{(0)} + \beta \sum_{l =1}^s H_{A, s}^{(l)}&, \ {\widetilde}{H}_{A, t} = H_{A, t} + \beta \sum_{l =1}^s H_{A, t}^{(l)},\\
{\widetilde}{H}_A^{0,1} = {1\over 2} &\big( {\widetilde}{H}_{A, s} + J {\widetilde}{H}_{A, t}\big).
\end{split}\end{aligned}$$ Then by (\[equation313\]) and (\[equation314\]) we have $$\begin{aligned}
\label{equation321}
\begin{split}
&\ D_A^{0,1} \nabla_Z \nabla {\widetilde}{{\mathcal}W}_A (u)\\
= &\ D_A^{0,1} \big( e^{{\overline}{\rho(h)}} \nabla_Z \nabla F_0(u) \big) + (\partial \beta / \partial {\overline}{z}) \nabla_Z \nabla {{\mathcal}W}_A'(u) + \beta D_A^{0,1} \sum_{l=1}^s e^{{\overline}{\rho_l(h)}} \nabla_Z \nabla F_l^{(\delta)}(u) \\
= &\ {\widetilde}{H}_A^{0,1}(u, d_A u, Z) + (\partial \beta /\partial {\overline}{z}) \nabla_Z \nabla {{\mathcal}W}_A'(u) + \nabla_{D_A^{1,0} Z} \nabla {\widetilde}{{\mathcal}W}_A(u).
\end{split}\end{aligned}$$
Asymptotic behavior {#section3}
===================
In this section we consider the asymptotic behavior of solutions to the gauged Witten equation. It suffices to consider the equation over cylindrical ends of $\Sigma^*$ and hence we can use cylindrical models introduced in the last section.
Within this section, we fix $\lambda \in {\bm i} \big( {{\mathbb}Z}/r \cap [0, 1) \big)$ and denote $\upgamma = \exp(2\pi \lambda)$.
\[defn41\] A solution ${\bm u}$ to a cylindrical model is called [**bounded**]{}, if $E({\bm u}) < \infty$ and there is a compact subset ${\widetilde}{K} \subset {\widetilde}{X}$ such that $u(\Theta_+) \subset {\widetilde}{K}$.
A solution $(A, u)$ to the perturbed gauged Witten equation over the rigidified $r$-spin curve $\vec{{\mathcal}C}$ is called [**bounded**]{} if its restriction to any of its cylindrical ends gives a bounded solution to the corresponding cylindrical model.
Our main theorems of this section are
\[thm42\] Suppose ${\bm u} = (u, h)$ is a bounded solution to a $\lambda$-cylindrical model with parameters $(\sigma, \delta)$. Then there is a point $\upkappa \in {\widetilde}{X}_\upgamma$ such that $$\begin{aligned}
\lim_{s \to +\infty} e({\bm u})(s, t) = 0, \lim_{s \to +\infty} e^{\lambda t} u(s, t) = \upkappa.\end{aligned}$$ both uniformly for $t\in S^1$.
\[thm43\] For every $G$-invariant compact subset ${\widetilde}{K} \subset {\widetilde}{X}$ and every ${\underline}\delta\in (0, 1]$, there are constants $\epsilon ({\widetilde}{K}, {\underline}\delta ), c ({\widetilde}{K}, {\underline}\delta), \tau ({\underline}\delta) > 0$ satisfying the following conditions. Suppose ${\bm u} = (u, h)$ is a bounded solution to a $\lambda$-cylindrical model parametrized by $(\sigma, \delta)$ such that $u (\Theta_+) \subset {\widetilde}{K}$ and if $\upgamma$ is broad, then $\delta \geq {\underline}\delta$. Then $$\begin{aligned}
\big\| e({\bm u}) \big\|_{L^\infty(\Theta_+)} \leq \epsilon ({\widetilde}{K}, {\underline}\delta ) \Longrightarrow e({\bm u})(s, t) \leq c ({\widetilde}{K}, {\underline}\delta ) e^{-\tau ({\underline}\delta) s}.\end{aligned}$$
The proof of Theorem \[thm42\] and Theorem \[thm43\] are given in Subsection \[subsection41\].
It follows from Theorem \[thm42\] that we can define the evaluations of a bounded solutions the perturbed gauged Witten equation at the punctures. Indeed, let $(A, u)$ be a bounded solution to the perturbed gauged Witten equation over $\vec{{{\mathcal}C}}$. Restrict $(A, u)$ to the cylindrical end near $z_j$ with residue $\lambda_j$, we obtain a bounded solution to a $\lambda_j$-cylindrical model. Then by Theorem \[thm42\], we have the well-defined limit $$\begin{aligned}
\lim_{z \to z_j} e^{-\lambda_j t} \phi_j^{-1} u(z) = \upkappa_j \in {\widetilde}{X}_{\upgamma_j}.\end{aligned}$$ We denote $ev_j(A, u) = \upkappa_j$ but indeed, the evaluation of the solution $(A, u)$ is a point on the fibre of $Y$ at $z_j$. On the other hand, for each $j =0, \ldots, k$, we have a solution to a $\lambda_j$-cylindrical model with parameters $(\sigma_j, \delta_j)$, where $\sigma_j ds dt $ is the restriction of the area form $\nu$ onto $U_j$ and $\delta_j = \delta_{j, A}$. Then the residue of $(A, u)$ at $z_j$ is $$\begin{aligned}
{\rm Res}_j (A, u) = \sum_{l=0}^s F_{\upgamma_j; l}^{(\delta_j)}( \upkappa_j) \in {{\mathbb}C}.\end{aligned}$$ Indeed, the residue ${\rm Res}_j(A, u)$ is nonzero only if $z_j$ is a broad puncture of $\vec{{\mathcal}C}$. A corollary to Theorem \[thm42\] is the following uniform energy bound.
\[thm44\] If $(A, u)$ is a bounded solution to the perturbed gauged Witten equation, then $u$ extends to a continuous orbifold section ${{\mathcal}U}$ of ${{\mathcal}Y} \to {{\mathcal}C}$, which defines a rational homology class $$\begin{aligned}
\big[ A, u \big] \in H^G_2 \big( {\widetilde}{X}; {{\mathbb}Z}[r^{-1}] \big).\end{aligned}$$ (See Appendix \[appenxib\] for the precise meanings.) We have, $$\begin{aligned}
E(A, u) = \big\langle \big[ \omega - \mu \big], \big[ A, u \big] \big\rangle + {\rm Re} \Big( \int_\Sigma {\bm i}{{\mathcal}W}_A'(u) \wedge {\overline}\partial \beta - 4 \pi \sum_{j=0}^k {\rm Res}_j (A, u)\Big).\end{aligned}$$ Here $\big[ \omega - \mu \big] \in H^2_G \big( {\widetilde}{X}; {{\mathbb}R} \big)$ is the equivariant cohomology class represented by the equivariant symplectic form $\omega - \mu$. Moreover, there is a constant $E$ depending only on the class $\big[ A, u \big]$ such that $E(A, u) \leq E$.
Its proof is given in Subsection \[subsection44\].
Proof of Theorem \[thm42\] and \[thm43\] {#subsection41}
----------------------------------------
### Decay of energy density {#decay-of-energy-density .unnumbered}
We first prove the first half of Theorem \[thm42\].
\[prop45\] For any bounded solution $(u, h)$ to a $\lambda$-cylindrical model, we have $$\begin{aligned}
\lim_{s \to +\infty} \big| \partial_s u + {{\mathcal}X}_\phi(u) \big| = \lim_{s \to +\infty} \big| \partial_t u + {{\mathcal}X}_\psi (u) \big| = \lim_{s \to +\infty} \big| \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)}(u) \big| = 0.\end{aligned}$$ In particular, $$\begin{aligned}
\big\| e({\bm u}) \big\|_{L^\infty(\Theta_+)} < +\infty.\end{aligned}$$
We abbreviate $v_s = \partial_s u + {{\mathcal}X}_\phi(u)$, $v_t = \partial_t u + {{\mathcal}X}_\psi(u)$. The proof is based on estimating $\Delta |v_s|^2$ and $\Delta |v_t|^2$. For any $z \in {\rm Int} \Theta_+$, choose a small disk $B_r(z) \subset \Theta_+$. Then the function $\lambda t$ is single-valued on $B_r(z)$ and the restriction of $(u, h+\lambda t)$ to $B_r(z)$ gives a solution to the local model parametrized by $(\beta = 1, \sigma|_{B_r(z)}, \delta)$. Replacing $h$ by $h + \lambda t$, and using the notations introduced in Subsection \[subsection32\], by (\[equation311\]) and (\[equation312\]), we have $$\begin{aligned}
\label{equation41}
\begin{split}
&\ \big( D_{A, s}^2 + D_{A, t}^2 \big) v_s\\
= &\ D_{A, s} \big( D_{A, s} v_s + D_{A, t} v_t \big) - \big[ D_{A, s}, D_{A, t} \big] v_t - D_{A, t} \big( D_{A, s} v_t - D_{A, t} v_s \big)\\
= &\ D_{A, s} \Big( D_{A, s} \big( -J v_t - 2 \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)} \big) + D_{A, t} \big( J v_s + 2 J \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)} \big) \Big) \\
&\ - R(v_s, v_t) v_t - \nabla_{v_t} {{\mathcal}X}_{F_A} - D_{A, t} {{\mathcal}X}_{F_A}\\
= &\ - J D_{A, s} {{\mathcal}X}_{F_A} - D_{A, t} {{\mathcal}X}_{F_A} - \nabla_{v_t} {{\mathcal}X}_{F_A} - R(v_s, v_t) v_t - 4 D_{A, s} D_A^{1, 0} \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)}\\
= &\ J D_{A, s} \big( \sigma {{\mathcal}X}_{\mu^*}\big) + D_{A, t} \big(\sigma {{\mathcal}X}_{\mu^*}\big) + \sigma \nabla_{v_t} {{\mathcal}X}_{\mu^*} - R(v_s, v_t) v_t - 4 D_{A, s} D_A^{1, 0} \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)}.
\end{split}\end{aligned}$$ By the definition of $D_{A, s}$, $D_{A, t}$, the invariance of $\mu$ and the boundedness of ${\bm u}$, there is a constant $C ({\bm u})>0$ such that $$\begin{aligned}
\begin{split}
&\ \big| JD_{A, s} \big( \sigma {{\mathcal}X}_{\mu^*} \big) + D_{A, t} \big( \sigma {{\mathcal}X}_{\mu^*} \big) \big| \\
\leq &\ 2 \big| {\overline}\partial \sigma \big| \big| {{\mathcal}X}_{\mu^*} \big| + \big| \sigma \big| \big( \big| {{\mathcal}X}_{d \mu^* \cdot v_s}\big| + \big| {{\mathcal}X}_{d \mu^*\cdot v_t} \big| + \big| \nabla {{\mathcal}X}_{\mu^*} \big| \big| d_A u \big| \big)\\
\leq &\ C ({\bm u}) \big( 1 + \big| d_A u \big| \big).
\end{split}\end{aligned}$$ On the other hand, by (\[equation317\]) and (\[equation313\]), we have $$\begin{aligned}
\label{equation43}
D_{A, s} D_A^{1, 0} \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)} = &\ D_{A, s} \nabla_{{\overline}\partial_A u} \nabla {\widetilde}{W}_h^{(\delta)} =\nabla_{D_{A, s} {\overline}\partial_A u } \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)} + {\widetilde}{H}_{A, s} (u, d_A u, {\overline}\partial_A u) .\end{aligned}$$ Here ${\widetilde}{H}$ is the tensor field defined by (\[equation320\]). We see that in the expression of $H_{A,s}^{(l )}$ in (\[equation319\]), the tensor field $G_{F_l^{(\delta)}}$ and the Hessian of $F_l^{(\delta)}$ are uniformly bounded because $u(\Theta_+)$ is contained in a compact subset of ${\widetilde}{X}$. Moreover, the equation $\Delta h'' = -\sigma \mu^* (u)$ and the condition $\lim_{s \to +\infty} h = 0$ imply the uniform bound on $dh''$ and $e^{{\overline}{\rho_l (h)}}$. Therefore, abusing $C({\bm u})>0$, we have $$\begin{aligned}
\label{equation44}
\begin{split}
\big| {\widetilde}{H}_{A, s} (u, d_A u, {\overline}\partial_A u ) \big| \leq &\ C ({\bm u}) \big( 1 + \big| d_A u \big| \big) \big| {\overline}\partial_A u \big|;\\
\big| \nabla_{D_{A, s} {\overline}\partial_A u} \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)} \big| \leq &\ C({\bm u}) \big( \big| D_{A, s} v_s \big| + \big| D_{A, s} v_t \big|\big).
\end{split}\end{aligned}$$ By (\[equation41\])-(\[equation44\]) and abusing $C( {\bm u})$, we obtain $$\begin{aligned}
\begin{split}
{1\over 2} \Delta|v_s|^2 = &\ \big\langle (D_{A, s}^2 + D_{A, t}^2 ) v_s, v_s \big\rangle + \big| D_{A, s} v_s \big|^2+ \big| D_{A, t} v_s \big|^2 \\
= & \big\langle - J D_A^{1,0} {{\mathcal}X}_{F_A} - \nabla_{v_t} {{\mathcal}X}_{F_A} - R(v_s, v_t) v_t , v_s \big\rangle \\
& -4 \big\langle D_{A, s} D_A^{1, 0} \nabla {\widetilde}{W}_h^{(\delta)}, v_s \big\rangle +\big| D_A v_s\big|^2\\
\geq & - C( {\bm u}) \big( 1 + |v_s|^4 + |v_t|^4 \big) + \big| D_A v_s \big|^2 - C({\bm u}) \big(\big| D_{A, s} v_s \big| + \big| D_{A, s} v_t \big| \big) \big| v_s \big|
\end{split}\end{aligned}$$ In the same way, we have $$\begin{aligned}
\Delta |v_t|^2 \geq - C( {\bm u}) \big( 1+ |v_s|^4 + |v_t|^4 \big) + \big| D_A v_t \big|^2 - C({\bm u}) \big( \big| D_{A, t} v_s \big| + \big| D_{A, t} v_t \big| \big) |v_t|.\end{aligned}$$ Therefore, abusing $C({\bm u})$ again, we have $$\begin{aligned}
\Delta \big( |v_s|^2 + |v_t|^2 \big) \geq -C({\bm u}) - C({\bm u}) \big( |v_s|^2 + |v_t|^2 \big)^2.\end{aligned}$$ Then by the mean value estimate (Lemma \[lemmaa5\]), there exist positive numbers $\epsilon, L >0$ depending on $C({\bm u})$, such that for any $z\in \Theta_+$ and $B_r(z) \subset \Theta_+$, we have $$\begin{aligned}
\int_{B_r(z)} \big( |v_s|^2 + |v_t|^2 \big) \leq \epsilon \Longrightarrow |v_s(z)|^2 + |v_t(z)|^2 \leq L \Big( r^2 + {1\over r^2} \int_{B_r(z)} \big(|v_s|^2 + |v_t|^2 \big) \Big).\end{aligned}$$ Since the energy of the solution is finite, this estimate implies that $$\begin{aligned}
\lim_{s \to +\infty} \left(| v_s(s, t)|^2 + |v_t(s, t)|^2 \right)= 0.\end{aligned}$$ The equation (\[equation33\]) implies $\displaystyle \lim_{s \to +\infty} \big| \nabla {\widetilde}{W}_{h, \lambda}^{(\delta)} (u(s, t)) \big|= 0$.
### Temporal gauge {#temporal-gauge .unnumbered}
Suppose ${\bm u} = (u, h)$ is a bounded solution to a $\lambda$-cylindrical model with parameters $(\sigma, \delta)$ and $u(\Theta_+) \subset {\widetilde}{K}$. Then we can transform it into temporal gauge as follows. Define $$\begin{aligned}
\label{equation45}
f(s, t) = \int_s^{+\infty} \phi(v, t) dv,\ g(s, t) = \exp f(s, t).\end{aligned}$$ By (\[equation34\]) $f$ is finite and has limit $0$ as $s \to +\infty$. Denote $u'(s, t) = g^{-1}(s, t) u(s, t),\ h' = h + f$. We call $(u', h')$ a temporal gauge solution.
For $l = 1, \ldots, s$, we abbreviate $F_{\upgamma; l}$ by $F_l$. We denote $$\begin{aligned}
{\widetilde}{W}^{(\delta)} := {\widetilde}{W}^{(\delta)} (x) := \sum_{l=0}^s F_l^{(\delta)}(x),\ {\widetilde}{W}_\lambda^{(\delta)} (z, x) = {\widetilde}{W}_\lambda^{(\delta)} (e^{\lambda t} x),\end{aligned}$$ Note that although $e^{\lambda t}x$ is multi-valued, ${\widetilde}{W}_\lambda^{(\delta)}$ is single-valued. We denote $$\begin{aligned}
R_h^{(\delta)} (z, x) = {\widetilde}{W}_{h, \lambda}^{(\delta)} (z, x) - {\widetilde}{W}_\lambda^{(\delta)}(z, x).\end{aligned}$$ Then by the expression (\[equation35\]), it is easy to see that, for every $l_1, l_2 \geq 0$, there is a constant $C^{l_1, l_2}({\widetilde}{K}) >0$ only depending on the compact subset ${\widetilde}{K}$ such that $$\begin{aligned}
\label{equation46}
\sup_{x \in {\widetilde}{K}} e^{2s} \big| \nabla_z^{(l_1)} \nabla_x^{(l_2)} R_h^{(\delta)} (z, x) \big| \leq C^{l_1, l_2}({\widetilde}{K}) \big| \nabla^{(l_1)} h \big|.\end{aligned}$$ Here $\nabla^{(l)}_z$ (resp. $\nabla^{(l)}_x$) means the derivative in the $z$-direction (resp. $x$-direction) of order $l$. The norm is taken with respect to the cylindrical metric.
By elliptic regularity and the boundedness of the solution, it is easy to prove
\[lemma46\] For any real number $M>0$ and any natural number $l$, there exists a constant $C^l({\widetilde}{K}, M) >0$ satisfying the following condition. If $(u, h)$ is a smooth bounded temporal gauge solution to a cylindrical model and $u(\Theta_+) \subset {\widetilde}{K}$, $\big\| e({\bm u}) \big\|_{L^\infty} \leq M$, then $$\begin{aligned}
\big\| h \big\|_{C^l(\Theta_+)} + \big\| du \big\|_{C^l(\Theta_+)} \leq C^l({\widetilde}{K},M). \end{aligned}$$
In radial gauge, $\partial_s \psi = - \sigma \mu^* (u)$. Then by (\[equation32\]), $\psi$ has a uniform $C^0$-bound. Moreover, the radial gauge condition implies that $$\begin{aligned}
\partial_s h = \partial_s h' + {\bm i} \partial_s h'' = \partial_t h'' + {\bm i} \partial_s h'',\ \partial_t h = \partial_t h' + {\bm i} \partial_t h'' = \psi -\lambda - \partial_s h'' + {\bm i} \partial_t h'',\end{aligned}$$ which is bounded by $\psi$ and $dh''$, while $dh''$ can be bounded via elliptic estimate by $\Delta h'' = F_A = - \sigma \mu^* (u)$. The uniform bound on $h$ follows from the fact that $\lim_{s \to +\infty} h = 0$. On the other hand, the bound on energy density implies uniform gradient bound on $u$. Therefore, using (\[equation46\]) to bound the inhomogeneous term, by elliptic bootstrapping for Cauchy-Riemann equations and Sobolev embedding we obtain the uniform bounds on all derivatives of $u$.
To proceed with the proof of exponential convergence, we need the following result.
\[lemma47\] For any natural number $l$ and any real number $M>0$, there is a constant $C^l({\widetilde}{K}, M) >0$ such that if $(u, h)$ is a solution to a cylindrical model and $\big\| e({\bm u}) \big\|_{L^\infty} \leq M$, then for any $s \geq 0$, we have $$\begin{aligned}
\label{equation47}
\big\| \partial_s \psi \big\|_{C^l([s, +\infty)\times S^1)} \leq C^l({\widetilde}{K}, M) e^{-2s}.\end{aligned}$$ $$\begin{aligned}
\big\| \partial_s u \big\|_{C^l([s, +\infty) \times S^1)} \leq C^l({\widetilde}{K}, M) \Big( e^{-2s} + \big\|\partial_s u \big\|_{C^0([s-1, +\infty) \times S^1)} \Big).\end{aligned}$$
It is easy to see that (\[equation47\]) follows from the vortex equation $\partial_s \psi = - \sigma \mu^* (u)$ and the uniform bound on all derivatives of $u$, which is provided by Lemma \[lemma46\]. On the other hand, apply $\nabla_s$ to the (\[equation33\]), we obtain $$\begin{aligned}
2 \nabla^{0,1} \partial_s u = - J \nabla {{\mathcal}X}_{\psi} (\partial_s u ) - J {{\mathcal}X}_{F_A} (u) - 2 \nabla^2 {\widetilde}{W}_h^{(\delta)}(\partial_s u) - 2 (\nabla_s \nabla R_h)(u).\end{aligned}$$ Here $2\nabla^{0,1} = \nabla_s + J \nabla_t$. Fix $p > 2$. Lemma \[lemma46\] implies uniform bounds on $\psi$, $F_A$. Then by (\[equation46\]) and elliptic estimate, there is a constant $b_1({\widetilde}{K}, M)>0$ such that $$\begin{aligned}
\big\| \partial_s u \big\|_{W^{1, p}([s, s+1]\times S^1)} \leq b_1({\widetilde}{K}, M) \Big( e^{-2s} + \big\|\partial_s u \big\|_{C^0([s-1, +\infty)\times S^1)} \Big).\end{aligned}$$ By elliptic bootstrapping we can replace the $W^{1, p}$-norm by the $W^{k, p}$-norm and the constant $b_1({\widetilde}{K}, M)$ by some $b_k({\widetilde}{K}, M)$. Indeed, if it is true for $k\geq 1$, then we see, the term $J \nabla {{\mathcal}X}_\psi(\partial_s u)$ and the term $\nabla^2 {\widetilde}{W}_h^{(\delta)} (\partial_s u)$ are linear in $\partial_s u$ and all derivatives of $J \nabla {{\mathcal}X}_\psi$ and $\nabla^2 {\widetilde}{W}_h^{(\delta)}$ are uniformly bounded by Lemma \[lemma46\]; the term $J {{\mathcal}X}_{F_A} (u)$ is linear in $F_A = \partial_s \psi ds dt$ and all derivatives of $u$ are uniformly bounded by Lemma \[lemma46\]; finally, all derivatives of $\nabla_s R_h$ are uniformly exponentially decay by (\[equation46\]). Therefore, (\[equation47\]), elliptic estimate and induction hypothesis imply that there is $b_{k+1}({\widetilde}{K}, M) >0$ such that $$\begin{aligned}
\big\| \partial_s u \big\|_{W^{k+1, p}([s, s+1]\times S^1)} \leq b_{k+1}({\widetilde}{K}, M) \Big( e^{-2s} + \big\| \partial_s u \big\|_{C^0([s-1, +\infty)\times S^1)} \Big).\end{aligned}$$ The bound on $C^l$-norm is obtained by Sobolev embedding.
### Exponential decay {#exponential-decay .unnumbered}
Let ${\widetilde}{N}_\upgamma \to {\widetilde}{X}_\upgamma$ be the normal bundle. Let $D >0$ be a small number and let ${\widetilde}{N}_\upgamma^D \cap {\widetilde}{K}$ be the $D$-neighborhood of ${\widetilde}{X}_{\upgamma}\cap {\widetilde}{K}$. There is a small $D_0>0$ such that the exponential map identifies ${\widetilde}{N}_\upgamma^{D_0}\cap {\widetilde}{K}$ with a neighborhood of the zero section of ${\widetilde}{N}_\upgamma \cap {\widetilde}{K}$. A point in this neighborhood is denoted either by $\exp_{{\overline}{x}} \xi$ or $({\overline}{x}, \xi)$, for ${\overline}{x}\in {\widetilde}{X}_\upgamma$ and $\xi \in {\widetilde}{N}_{\upgamma}|_{{\overline}{x}}$.
Now we state the result about the exponential decay of the normal component, which will be proved in Subsection \[subsection42\]. The derivative of ${{\mathcal}X}_\lambda$ in the direction of ${\widetilde}{N}_\upgamma$ defines a skew-adjoint map $d{{\mathcal}X}_\lambda^N: {\widetilde}{N}_\upgamma \to {\widetilde}{N}_\upgamma$, whose spectra are locally constant and are disjoint from ${\bm i} {{\mathbb}Z}$. We define $$\begin{aligned}
\tau_0 = \tau_0(\lambda):= d \big( {\bm i} {{\mathbb}Z}, {\rm Spec} ( d{{\mathcal}X}_\lambda^N ) \big) \in (0,1).\end{aligned}$$
\[prop48\] For every $G$-invariant compact subset ${\widetilde}{K} \subset {\widetilde}{X}$, there exist a constant $\epsilon_1 = \epsilon_1 ({\widetilde}{K})>0$ and for every $l$, a constant $C^l({\widetilde}{K}) >0$ satisfying the following conditions. Suppose $(u, h)$ is a smooth temporal gauge solution to a $\lambda$-cylindrical model, and $u(\Theta_+) \subset {\widetilde}{K}$. Suppose $$\begin{aligned}
\label{equation48}
\big\| e({\bm u}) \big\|_{L^\infty(\Theta_+)} \leq (\epsilon_1 )^2.\end{aligned}$$ Then $u(\Theta_+) \subset {\widetilde}{N}_\upgamma^{D_0} \cap {\widetilde}{K}$. Moreover, if we write $u = \exp_{{\overline}{u}} \xi$ where ${\overline}{u}: \Theta_+ \to {\widetilde}{X}_\upgamma$ and $\xi \in \Gamma \left( {\overline}{u}^* {\widetilde}{N}_\upgamma\right)$, then for each $l$ and every $s \geq 0$, we have $$\begin{aligned}
\label{equation49}
\| \xi \|_{C^l([s, +\infty) \times S^1)} \leq C^l({\widetilde}{K}) e^{- {1 \over 2} \tau_0 s}.\end{aligned}$$
In the broad case, ${\widetilde}{W}^{(\delta)}: {\widetilde}{X}_\upgamma \to {{\mathbb}C}$ is a holomorphic Morse function having finitely many critical points. Then for any ${\underline}\delta > 0$, there exists $\tau_1 = \tau_1( {\underline}\delta )>0$ such that for any $\delta \in [ {\underline}\delta, 1]$, for any critical point $\upgamma$ of ${\widetilde}{W}^{(\delta)}|_{{\widetilde}{X}_\upgamma}$, each eigenvalue of the Hessian of ${\widetilde}{W}^{(\delta)}|_{{\widetilde}{X}_\upgamma}$ has absolute value no less than $\tau_1$.
The following two propositions will be proved in Subsection \[subsection43\].
\[prop49\] Suppose $\upgamma$ is broad. Then for every $G$-invariant compact subset ${\widetilde}{K}\subset {\widetilde}{X}$ and every ${\underline}\delta>0$, there are constants $\epsilon_2 = \epsilon_2( {\widetilde}{K}, {\underline}\delta )>0, C_2 = C_2({\widetilde}{K}, {\underline}\delta ) >0$ satisfying the following conditions. Suppose $(u, h)$ is a bounded smooth temporal gauge solution to a $\lambda$-cylindrical model with parameters $(\sigma, \delta)$ such that $\delta \geq {\underline}\delta$. Suppose $$\begin{aligned}
\big\| e({\bm u}) \big\|_{L^\infty(\Theta_+)} \leq ( \epsilon_2 )^2.\end{aligned}$$ Then $u(\Theta_+) \subset {\widetilde}{N}_\upgamma^{D_0} \cap {\widetilde}{K}$ and there is a unique critical point $\upkappa$ of ${\widetilde}{W}^{(\delta)} |_{{\widetilde}{X}_\upgamma}$ such that for all $(s, t)\in \Theta_+$, $$\begin{aligned}
d \big( e^{\lambda t} {\overline}{u}(s, t), \upkappa \big) \leq C_2 e^{- {1\over 2} \min\{ \tau_0, \tau_1 \} s}.\end{aligned}$$
\[prop410\] Suppose $\upgamma$ is narrow. Then for every $G$-invariant compact subset ${\widetilde}{K}\subset X$, there are constants $\epsilon_3 = \epsilon_3({\widetilde}{K} ) >0$, $C_3 = C_3 ({\widetilde}{K} ) >0$ satisfying the following conditions. Suppose $(u, h)$ is a bounded smooth temporal gauge solution to a $\lambda$-cylindrical model. Suppose $$\begin{aligned}
\big\| e({\bm u}) \big\|_{L^\infty(\Theta_+)} \leq ( \epsilon_3 )^2.\end{aligned}$$ Then $u(\Theta_+) \subset {\widetilde}{N}_\upgamma^{D_0} \cap {\widetilde}{K}$ and there is a point $\upkappa\in {\widetilde}{X}_{\upgamma}$ such that for all $(s, t) \in \Theta_+$, $$\begin{aligned}
d\big( e^{\lambda t} {\overline}{u}(s, t), \upkappa \big) \leq C_3 e^{-{1\over 2} \tau_0 s}.\end{aligned}$$
It is easy to see that Proposition \[prop45\], \[prop49\] and \[prop410\] imply Theorem \[thm42\]. On the other hand, we write $u = \exp_\upkappa \xi$. By Proposition \[prop48\], \[prop49\] and \[prop410\], $\xi$ decays exponentially. Then Theorem \[thm43\] follows from the elliptic estimates for a Cauchy-Riemann equation in $\xi$. The choices of the constants in Theorem \[thm43\] are obvious.
Proof of Proposition \[prop48\] {#subsection42}
-------------------------------
\[lemma411\] For any compact $G$-invariant subset ${\widetilde}{K}\subset {\widetilde}{X}$ and $D >0$, there is an $\epsilon_4 = \epsilon_4 ({\widetilde}{K}, D) >0$ such that if a $C^1$-loop $(x, \eta): S^1 \to {\widetilde}{K} \times {{\mathfrak}g}$ satisfies $$\begin{aligned}
\label{equation410}
\sup_{t \in S^1}\Big( \big| x'(t) + {{\mathcal}X}_\eta (x(t)) \big| + \sup_{t \in S^1} \big| \eta(t) -\lambda \big| \Big) \leq \epsilon_4,\end{aligned}$$ then $x(S^1) \subset {\widetilde}{N}_{\upgamma}^D$.
Define $(g, y): [0,2\pi] \to G \times {\widetilde}{X}$ by $$\begin{aligned}
g(t) = \exp \Big( \int_0^t \eta(\tau) d\tau \Big),\ y(t) = g(t) x(0).\end{aligned}$$ Then $y'(t) = g(t)_* \left( x'(t) + {{\mathcal}X}_{\eta(t)} (x(t)) \right)$ and (\[equation410\]) implies $d(y(2\pi), y(0)) \leq 2\pi \epsilon_4$. Then $$\begin{gathered}
d ( \upgamma x(0), x(0) ) \leq d ( \upgamma x(0), y(2\pi) ) + d ( y(2\pi), x(2\pi) )\\
= d \Big( \exp ( 2\pi \lambda) x(0), \exp \Big( \int_0^{2\pi} \eta(\tau) d\tau \Big) x(0) \Big) + d ( y(2\pi), y(0) ) \\
\leq d \Big( \upgamma x(0), \exp \Big( \int_0^{2\pi} \eta(\tau) d\tau \Big) x(0) \Big) + 2\pi \epsilon_4.\end{gathered}$$ (\[equation410\]) also implies that $\big| 2\pi \lambda - \int_0^{2\pi} \eta(\tau) d\tau \big| \leq 2\pi \epsilon_4$. Then since $x(0)$ is in a compact subset, for $\epsilon_4$ small enough, $\upgamma x(0)$ is sufficiently close to $x(0)$ so that $x(0) \in {\widetilde}{N}_{\upgamma}^{{1\over 2} D} \cap {\widetilde}{K}$. Then since $|y'(t)|$ is very small, $y([0, 2\pi])$ is contained ${\widetilde}{N}_\upgamma^D \cap {\widetilde}{K}$ for $\epsilon_4$ small enough.
Now let ${\bm u} = (u, h)$ is a bounded smooth temporal gauge solution to a $\lambda$-cylindrical model with $u(\Theta_+) \subset {\widetilde}{K}$. Take $\epsilon_1 = \epsilon_1 ({\widetilde}{K}, \upgamma) > 0$ undetermined. Then if $\big\| e({\bm u}) \big\|_{L^\infty} \leq (\epsilon_1)^2$, we have $$\begin{aligned}
\big| \partial_t u(s, t) + {{\mathcal}X}_{\psi(s, t)} (u(s, t)) \big| \leq \epsilon_1,\end{aligned}$$ $$\begin{aligned}
\big| \psi(s, t) - \lambda \big| \leq \int_s^{+\infty} \big| \sigma(\rho, t) \mu^* (u) \big| d\rho \leq \sqrt{ \epsilon_1} \int_s^\infty \sqrt{\sigma(\rho, t)} d\rho \leq \sqrt{ \epsilon_1 C^{(0)}(\sigma)} e^{-s}.\end{aligned}$$ Here $C^{(0)}(\sigma)$ is the one in (\[equation32\]). Then we can choose $\epsilon_1$ sufficiently small so that by Lemma \[lemma411\], the first claim of Proposition \[prop48\] is satisfied, i.e., $u ( \Theta_+ ) \subset {\widetilde}{N}_{\upgamma}^{D_0}$. Then we can use the exponential map to write $u = \exp_{{\overline}{u}} \xi$ for ${\overline}{u}: \Theta_+ \to {\widetilde}{X}_{\upgamma}$ and $\xi \in \Gamma \left( {\overline}{u}^* {\widetilde}{N}_\upgamma \right)$.
Now we consider the equation that the normal component $\xi$ of $u$ should satisfy. Let $\pi: {\widetilde}{N}_\upgamma \to {\widetilde}{X}_\upgamma$, $\pi(\exp_{{\overline}{x}}\xi) = {\overline}{x}$ be the projection. The exponential map induces a bundle isomorphism $$\begin{aligned}
\label{equation411}
T{\widetilde}{X}|_{{\widetilde}{N}_\upgamma^{D_0}} \simeq \pi^* T{\widetilde}{X}_\upgamma \oplus \pi^* {\widetilde}{N}_\upgamma.\end{aligned}$$ For any $V \in T {\widetilde}{X}|_{{\widetilde}{N}_\upgamma^{D_0}}$, we denote by $V^T$ the tangential component and $V^N$ the normal component, with respect to the above decomposition. This decomposition respects the $G$-action, i.e. for any $g \in G$ and $({\overline}{x}, \xi) \in {\widetilde}{N}_\upgamma^{D_0}$, $g ({\overline}{x}, \xi) = (g {\overline}{x}, g \xi)$. Therefore, $$\begin{aligned}
{{\mathcal}X}_\lambda (\exp_{{\overline}{x}} \xi) = \left( {{\mathcal}X}_\lambda({\overline}{x}) , {{\mathcal}X}_\lambda^N( {\overline}{x}, \xi) \right)\end{aligned}$$ where the second component is linear in $\xi$. However, the decomposition (\[equation411\]) may not respect the complex structure and we can write the complex structure as $$\begin{aligned}
J ({\overline}{x}, \xi) = \left( \begin{array}{cc} J^T({\overline}{x}) & 0\\
0 & J^N({\overline}{x})
\end{array} \right) + R_J({\overline}{x}, \xi),\end{aligned}$$ where $R_J$ depends smoothly on $({\overline}{x}, \xi)$ and there is a constant $C_J({\widetilde}{K})>0$ depending on the compact set ${\widetilde}{K}$ such that for $({\overline}{x},\xi) \in {\widetilde}{K}$, we have $$\begin{aligned}
\label{equation412}
| R_J ({\overline}{x}, \xi) | \leq C_J({\widetilde}{K}) | \xi |.\end{aligned}$$ Lastly, by ([**Q2**]{}) of Hypothesis \[hyp25\] and ([**P2**]{}) of Hypothesis \[hyp28\], the Hessian of ${\widetilde}{W}_h^{(\delta)}$ vanishes along the normal bundle ${\widetilde}{N}_\upgamma$. By the uniform bound on $h$ (Lemma \[lemma46\]), there is a constant $c^N({\widetilde}{K})$ depending only on ${\widetilde}{K}$ such that $$\begin{aligned}
\label{equation413}
\big| \big( \nabla {\widetilde}{W}_h^{(\delta)} ({\overline}{x}, \xi) \big)^N \big| \leq c^N({\widetilde}{K}) | \xi |^2.\end{aligned}$$
Use the above notations, the normal component of (\[equation33\]) can be written as $$\begin{gathered}
\label{equation414}
\nabla_s \xi + J^N({\overline}{u}) ( \nabla_t \xi + {{\mathcal}X}_\lambda(\xi) ) \\
= - \big( R_J ({\overline}{u}, \xi) \left( \partial_t u + {{\mathcal}X}_{ \lambda}(u) \right) \big)^N - \big( J {{\mathcal}X}_{\psi-\lambda} (u) \big)^N - 2 \big( \nabla {\widetilde}{W}_h^{(\delta)} ({\overline}{u}, \xi) \big)^N.\end{gathered}$$
\[lemma412\] Denote the right hand side of (\[equation414\]) by $R(s, t)$. There exists $c_1 >0$ and for any $\rho>0$, there are constants $\varepsilon_1 = \varepsilon_1(\rho)>0$ and $S_1 = S_1(\rho)>0$ such that if $\big\| e({\bm u}) \big\|_{L^\infty} \leq (\varepsilon_1)^2$, then for $s \geq S_1$, we have $$\begin{aligned}
| R(s, t) | \leq \rho |\xi| ,\ | \nabla_s R(s, t) | \leq \rho^2 |\xi| + \rho |\nabla_s \xi|,\ | \nabla_t R | \leq c_1 ( |\xi| + |\nabla_t \xi| ).\end{aligned}$$
We will estimate each term in the expression of $R(s, t)$ and all constants appeared below will depend on ${\widetilde}{K}$.
First, by the vortex equation, there is a constant $a_1>0$ such that $| \psi - \lambda | + |\partial_s \psi | \leq a_1 e^{-2s}$. Moreover, by Lemma \[lemma46\], there is a constant $a_2>0$ such that $|\partial_t \psi | \leq a_2$. Then in the expression of $R$, the contribution from $(J{{\mathcal}X}_{\psi - \lambda}(u))^N$ can be bounded in the desired way since it is linear in $\xi$. Moreover, by (\[equation413\]), Lemma \[lemma46\] and (\[equation46\]), the contribution of $\big( \nabla {\widetilde}{W}_h^{(\delta)}({\overline}{u}, \xi) \big)^N$ can be controlled in the desired way.
On the other hand, by (\[equation412\]), we have $$\begin{gathered}
\label{equation415}
\big| \big( R_J({\overline}{u}, \xi) \left( \partial_t u + {{\mathcal}X}_\lambda(u)\right) \big)^N \big| \leq \big| R_J({\overline}{u}, \xi) \big| \big| \partial_t u + {{\mathcal}X}_\lambda(u) \big|\\
\leq C_J({\widetilde}{K}) \big| \xi \big| \big( \big| \partial_t u + {{\mathcal}X}_\psi(u) \big| + \big| {{\mathcal}X}_{\psi - \lambda}(u) \big| \big) \leq C_J({\widetilde}{K}) \big( \varepsilon_1 + a_1 e^{-2s} \big) |\xi|;\end{gathered}$$ applying $\nabla_s$, we have that there is a constant $a_3>0$ such that $$\begin{gathered}
\label{equation416}
\big| \nabla_s \big( R_J ({\overline}{u}, \xi) (\partial_t u + {{\mathcal}X}_\lambda(u)) \big)^N \big| \\
\leq a_3 \big( \big| \partial_s {\overline}{u} \big| \big| \xi \big| + \big| \nabla_s \xi \big| \big) \big| \big(\partial_t u + {{\mathcal}X}_\lambda(u) \big)^N \big| + a_3 \big| \xi \big| \big( \big| \nabla_s \partial_t u \big| + \big|\nabla_s {{\mathcal}X}_\lambda(u) \big| \big).\end{gathered}$$ Then by choosing $\varepsilon_1$ sufficiently small, $S_1$ sufficiently large, and using Lemma \[lemma47\] to control $\nabla_t \partial_s u$, we see that for $s \geq S_1$, we have $$\begin{aligned}
\left| \nabla_s \left( R_J({\overline}{u}, \xi) (\partial_t u + {{\mathcal}X}_\lambda(u)) \right)^N \right| \leq \rho^2 |\xi| + \rho|\nabla_s \xi|.\end{aligned}$$ Applying $\nabla_t$ to $R_J(\partial_t u + {{\mathcal}X}_\lambda(u))^N$ and using Lemma \[lemma46\], we see there are constant $a_4, a_5>0$ such that $$\begin{gathered}
\left| \nabla_t R_J({\overline}{u}, \xi) (\partial_t u + {{\mathcal}X}_\lambda(u))^N \right| \\
\leq a_4 \left( |\partial_t {\overline}{u}| |\xi| + |\partial_t \xi|\right) \left| (\partial_t u + {{\mathcal}X}_\lambda(u))^N \right| + a_4 |\xi| \left( |\nabla_t \partial_t u | + |\nabla_t {{\mathcal}X}_\lambda(u) |\right)\\
\leq a_5 |\xi| + a_5 |\nabla_t \xi|.\end{gathered}$$ So the lemma is proven.
\[lemma413\] There exist $c_2>0$ and $\varepsilon_2>0$ depending only on ${\widetilde}{K}$ that satisfy the following conditions. If $u(\Theta_+) \subset {\widetilde}{N}_{\upgamma}^{D_0} \cap {\widetilde}{K}$ and $\big\| e({\bm u}) \big\|_{L^\infty} \leq (\varepsilon_2)^2$, then $$\begin{aligned}
\label{equation417}
\| \xi \|_{L^2(\{s\}\times S^1)} \leq c_2 e^{-{1\over 2} \tau_0 s}.\end{aligned}$$
Let $s \geq S_1$ where $S_1$ is the one in Lemma \[lemma412\]. Let ${\overline}{u}_s: S^1 \to {\widetilde}{X}_\upgamma$ be the restriction of ${\overline}{u}$ to $\{s\}\times S^1$. We denote by ${{\mathcal}L}(s): L^2( {\overline}{u}_s^* {\widetilde}{N}_\upgamma) \to L^2( {\overline}{u}_s^* {\widetilde}{N}_\upgamma)$ the following self-adjoint operator $$\begin{aligned}
{{\mathcal}L}(s) \eta = J^N({\overline}{u}_s) \left( \nabla_t \eta + {{\mathcal}X}_\lambda(\eta)\right).\end{aligned}$$ We claim that for all $s \geq S_1$, ${{\mathcal}L}(s)$ is coercive in the sense that $$\begin{aligned}
\label{equation418}
\big\| {{\mathcal}L}(s) \eta \big\|_{L^2(S^1)}^2 \geq (\tau_0)^2 \big\| \eta \big\|_{L^2(S^1)}^2.\end{aligned}$$ Indeed, any $\eta \in L^2(S^1)$ can be written as Fourier series $\eta = \sum_{k \in {{\mathbb}Z}} \eta_k e^{{\bm i} k t}$. Then $$\begin{aligned}
\big\| {{\mathcal}L}(s) \eta \big\|_{L^2} = \Big\| \sum_{k\in {{\mathbb}Z}} e^{{\bm i} k t} ( - k \eta_k + {\bm i} {{\mathcal}X}_\lambda( \eta_k) ) \Big\|_{L^2} = \Big( \sum_{k\in {{\mathbb}Z}} | - k \eta_k + {\bm i} {{\mathcal}X}_\lambda( \eta_k) |^2 \Big)^{1\over 2} \geq \tau_0 \big\| \eta \big\|_{L^2}.\end{aligned}$$
On the other hand, since the covariant derivative on ${\widetilde}{N}_{\upgamma}$ preserves the complex structure $J^N$, we have $$\begin{aligned}
{{\mathcal}L}'(s) = J^N \left( [\nabla_s, \nabla_t] + \nabla_s {{\mathcal}X}_\lambda \right)= J^N \left( {\sf R}^N( \partial_s {\overline}{u}, \partial_t {\overline}{u}) + \nabla_s {{\mathcal}X}_\lambda \right).\end{aligned}$$ Here ${\sf R}^N$ is the curvature tensor in the normal bundle. Then ${{\mathcal}L}'(s)$ is a family of bounded operators of $L^2$, and there exists a constant $a_6>0$ depending on ${\widetilde}{K}$ such that $$\begin{aligned}
\label{equation419}
\big\|{{\mathcal}L}'(s) \big\| \leq a_6 \big\| \partial_s {\overline}{u} \big\|_{L^\infty} \leq a_6 \varepsilon_2.\end{aligned}$$ Here we used the fact of Lemma \[lemma46\] that $\|du\|$ is uniformly bounded. Then applying $\nabla_s$ to (\[equation414\]), we obtain $$\begin{aligned}
\nabla_s^2 \xi + {{\mathcal}L}(s) \nabla_s \xi + {{\mathcal}L}'(s) \xi = \nabla_s R(s, t).\end{aligned}$$ Denote $v(s) = \| \xi(s, \cdot) \|_{L^2(S^1)}^2$. We claim that there exist $\rho>0$, $S_2 \geq 0$ and $\varepsilon_2>0$ such that $$\begin{aligned}
\label{equation420}
s\geq S_2 \Longrightarrow v''(s) \geq \left\| {{\mathcal}L}(s) \xi \right\|_{L^2(S^1)}^2.\end{aligned}$$ Indeed, for any $\rho>0$, for $s\geq S_1$ where $S_1$ is the one in Lemma \[lemma412\], we have $$\begin{aligned}
\begin{split}
{1\over 2} v''(s) = &\ \big\langle \nabla_s^2 \xi , \xi \big\rangle + \big\| \nabla_s \xi \big\|^2\\
= &\ \big\langle \nabla_s R - {{\mathcal}L} (s) \nabla_s \xi - {{\mathcal}L}'(s) \xi , \xi \big\rangle + \big\| \nabla_s \xi \big\|^2\\
= &\ \big\langle \nabla_s R - {{\mathcal}L}'(s) \xi + {{\mathcal}L}(s) ( {{\mathcal}L}(s) \xi - R), \xi \big\rangle + \big\| \nabla_s \xi \big\|^2\\
= &\ \big\| {{\mathcal}L}(s) \xi \big\|^2 + \big\langle -R, {{\mathcal}L}(s) \xi \big\rangle + \big\langle \nabla_s R - {{\mathcal}L}'(s) \xi, \xi \big\rangle + \big\| \nabla_s \xi \big\|^2 \\
\geq &\ \big\| {{\mathcal}L}(s) \xi \big\|^2 - \big\| R \big\| \big\| {{\mathcal}L}(s) \xi \big\| - \big\| \nabla_s R \big\| \big\| \xi \big\| - \big\| {{\mathcal}L}'(s) \xi \big\| \big\| \xi\big\| + \big\| \nabla_s \xi \big\|^2 \\
\geq &\ \big\| {{\mathcal}L}(s) \xi \big\|^2 - {1\over 4} \big\|{{\mathcal}L}(s) \xi \big\|^2 - 2 \rho^2 \big\| \xi \big\|^2 - \rho \big\| \xi \big\| \big\| \nabla_s \xi \big\| - a_6 \varepsilon_2 \big\| \xi\big\|^2 + \big\| \nabla_s \xi \big\|^2 \\
\geq &\ {3\over 4} \big\| {{\mathcal}L}(s) \xi \big\|^2 - { 9 \over 4} \rho^2 \big\| \xi \big\|^2 - a_6 \varepsilon_2 \big\| \xi \big\|^2.
\end{split}\end{aligned}$$ Here norms and inner products are the ones in the $L^2$ space, and we used (\[equation419\]) and the estimates of Lemma \[lemma412\]. We choose $\rho$, $\varepsilon_2$, $S_2$ so that $$\begin{aligned}
{9 \over 4} \rho^2 \leq {1 \over 8} (\tau_0)^2,\ \varepsilon_2 \leq \min \big\{ \varepsilon_1(\rho), (\tau_0)^2/ 8 a_1 \big\},\ S_2 \geq S_1(\rho).\end{aligned}$$ Then for $s \geq S_2$, (\[equation420\]) holds, and by (\[equation418\]), $v''(s) \geq (\tau_0)^2 v(s)$. Thus the function $$\begin{aligned}
e^{-\tau_0 s } \big( v'(s) + \tau_0 v(s) \big) \end{aligned}$$ is non-decreasing on $[S_2, +\infty)$. Since $\displaystyle \lim_{s \to \infty} v(s) = 0$, we see that for $s\geq S_2$, $$\begin{aligned}
v'(s) + \tau_0 v(s) \leq 0 \Longleftrightarrow {d\over ds} \big( e^{\tau_0 s} v(s) \big) \leq 0.\end{aligned}$$ Therefore $ v(s) \leq e^{- \tau_0 s} \big( v(S_2) e^{ \tau_0 S_2} \big)$. Moreover, since $v(s)$ is uniformly bounded for all $s \geq 0$, there is $c_2>0$ such that (\[equation417\]) holds.
Then in the above situation, there is $c_3 = c_3({\widetilde}{K}) > 0$ such that $$\begin{aligned}
\label{equation421}
\| \xi \|_{L^2([s-1, s+1]\times S^1)} \leq c_3 e^{-{1\over 2} \tau_0 s}\end{aligned}$$ To derive pointwise estimate, we apply $\nabla_s - J^N \nabla_t$ to (\[equation414\]). Then we obtain $$\begin{aligned}
\label{equation422}
\Delta \xi = (\nabla_s - J^N \nabla_t )(\nabla_s + J^N \nabla_t )\xi = \left( \nabla_s - J^N \nabla_t \right) \left( R - J {{\mathcal}X}_\lambda(\xi) \right).\end{aligned}$$ Choose $z_0 = (s_0, t_0) \in [1, +\infty) \times S^1$. Then by the uniform bound on derivatives of $u$ and Lemma \[lemma412\], we see there is a constant $c_4 = c_4({\widetilde}{K}) >0$ such that $$\begin{aligned}
{1\over 2} \Delta |\xi|^2 \geq \langle \Delta \xi , \xi \rangle \geq - c_4 |\xi|^2 \geq - {1\over 2} \Big( {\pi \over 16 c_3^2} e^{\tau_0 s_0}|\xi|^4 + {16 c_4^2 c_3^2 \over \pi} e^{- \tau_0 s_0 } \Big).\end{aligned}$$ Denote $$\begin{aligned}
A = {16 c_4^2 c_3^2 \over \pi} e^{- \tau_0 s_0 },\ B = {\pi \over 16 c_3^2} e^{ \tau_0 s_0}.\end{aligned}$$ By (\[equation421\]), $\int_{B_r(z_0 )} |\xi|^2 \leq \pi/ 16 B$. Then by the mean value estimate (Lemma \[lemmaa5\]) for the differential inequality $\Delta u \geq - A - Bu^2$, for $r = 1$, we have $$\begin{aligned}
\label{equation424}
|\xi(z_0 )|^2 \leq { 8 \over \pi} \int_{B_r(z_0 )} |\xi|^2 + {A\over 4} = \Big( {8 c_3^2 \over \pi} + {4 c_4^2 c_3^2 \over \pi} \Big) e^{ - \tau_0 s_0 }=: c_5 e^{- \tau_0 s}.\end{aligned}$$ For $l \geq 1$ the estimate (\[equation49\]) follows from elliptic estimate.
Proof of Proposition \[prop49\] and \[prop410\] {#subsection43}
-----------------------------------------------
In this subsection we use the symbols $c_1, c_2, \ldots$ abusively, which could be different from the ones in the last subsection. We also use $\nabla$ to denote the Levi-Civita connection on ${\widetilde}{X}_\upgamma$.
Now suppose ${\bm u} = (u, h)$ is a temporal gauge solution to a $\lambda$-cylindrical model, satisfying (\[equation48\]). Then $u(\Theta_+) \subset {\widetilde}{N}_\upgamma^{D_0} \cap {\widetilde}{K}$ and $u$ can be written as $u = \exp_{{\overline}{u}}\xi$. Then with respect to the decomposition (\[equation411\]), the tangential direction of $ \nabla {\widetilde}{W}_\lambda^{(\delta)}(u)$ is $$\begin{aligned}
\big( \nabla {\widetilde}{W}_\lambda^{(\delta)} \big)^T = \nabla {\widetilde}{W}_\lambda^{(\delta)} ({\overline}{u}) + R_\lambda^{(\delta), T} ({\overline}{u}, \xi),\end{aligned}$$ where the remainder $R_\lambda^{(\delta), T}( {\overline}{u}, \xi)$ has norm less than a constant multiple of $|\xi |$. Then if we project the first equation of (\[equation33\]) to the tangential direction, we have $$\begin{gathered}
\label{equation425}
\partial_s {\overline}{u} + J^T \big( \partial_t {\overline}{u} + {{\mathcal}X}_{\lambda}( {\overline}{u} ) \big) + 2 \nabla {\widetilde}{W}_\lambda^{(\delta)} ({\overline}{u}) \\
= - \big( J {{\mathcal}X}_{\psi - \lambda}(u) \big)^T - \big( R_J ( \partial_t u + {{\mathcal}X}_\psi(u) ) \big)^T - 2 \big( R_h^{(\delta)} (u)\big)^T - 2 R_\lambda^{(\delta), T} ({\overline}{u}, \xi).\end{gathered}$$ Denote the right hand side by $R_0^T$.
\[lemma414\] There is a constant $c_1>0$ depending only on ${\widetilde}{K}$ such that for $$\begin{aligned}
\big| R_0^T ( {\overline}{u}, \xi) \big| + \big| \nabla_s R_0^T ({\overline}{u}, \xi) \big| + \big| \nabla_t R_0^T ({\overline}{u}, \xi) \big| \leq c_1 e^{-{1\over 2} \tau_0 s}.\end{aligned}$$
On the right hand side of (\[equation425\]), $(J{{\mathcal}X}_{\psi- \lambda})^T$ and $\big( R_h^{(\delta)} (u)\big)^T$ decay like $e^{-2s}$ (with all derivatives) which is faster than $e^{-{1\over 2} \tau_0 s}$. The other two terms together with their derivatives can be controlled by $|\xi|$, which decays like $e^{-{1\over 2} \tau_0 s}$ (with all derivatives) by Proposition \[prop48\].
Since the image of ${\overline}{u}$ is contained in ${\widetilde}{X}_\upgamma$, the map ${\overline}{v}:= e^{\lambda t} {\overline}{u}(s, t)$ is still a smooth map from $\Theta_+$ to ${\widetilde}{X}_\upgamma$, which satisfies $$\begin{aligned}
\label{equation426}
\partial_s {\overline}{v} + J^T({\overline}{v}) \partial_t {\overline}{v} + \nabla {\widetilde}{W}^{(\delta)} ({\overline}{v}) = e^{-\lambda t} R_0^T. \end{aligned}$$
### The broad case {#the-broad-case .unnumbered}
Now we suppose that $\upgamma = \exp(2\pi \lambda)$ is broad and $\delta \geq {\underline}\delta_0>0$. Then there exists $D_1 = D_1 ({\underline}\delta)>0$ such that for all $\delta \in [{\underline}\delta, 1]$ and any two distinct critical points $\upkappa, \upkappa'$ of ${\overline}{W}^{(\delta)}: {\widetilde}{X}_\upgamma \to {{\mathbb}C}$ as distance bigger than $2D_1$. Therefore we can take $\epsilon_2$ small enough so that for each such solution, there is a unique critical point $\upkappa$ of ${\widetilde}{W}^{(\delta)}$ such that ${\overline}{u}(\Theta_+)$ is contained in the $D_1$-neighborhood of $\upkappa$. We also assume that $D_1$ is smaller than the injectivity radius of ${\widetilde}{X}_\upgamma \cap {\widetilde}{K}$. Then we can write ${\overline}{v} = \exp_{\upkappa} \eta$ for $\eta \in \Gamma ( \Theta_+, T_\upkappa {\widetilde}{X}_\upgamma )$.
The derivative of $\exp_{\upkappa}$ induces a smooth family of isomorphisms $E_2(\eta): T_{\upkappa} {\widetilde}{X}_\upgamma \to T_{\exp_{\upkappa} \eta} {\widetilde}{X}_\upgamma$. Then for the tangential part $J^T$ of the complex structure $J$, we have $$\begin{aligned}
J^T(\exp_{\upkappa} \eta) E_2(\eta) = E_2(\eta) J^T( \upkappa) + B_J^T(\eta).\end{aligned}$$ $B_J^T$ depends smoothly on $\eta$ and there is a constant $c_{\upkappa}>0$ such that $| B_J^T (\eta) | \leq c_\upkappa |\eta|$. On the other hand, let ${\sf A}_\upkappa^{(\delta)}: T_\upkappa {\widetilde}{X}_\upgamma \to T_\upkappa {\widetilde}{X}_\upgamma$ be the Hessian of ${\widetilde}{W}^{(\delta)} |_{{\widetilde}{X}_\upgamma}$ at $\upkappa$. We can write $$\begin{aligned}
\nabla {\widetilde}{W}^{(\delta)} ( \exp_\upkappa \eta) = E_2(\eta) ( {\sf A}_{\upkappa}^{(\delta)} \eta ) + R_1^T (\eta).\end{aligned}$$ $R_1^T$ depends smoothly on $\eta$ and we may assume $| R_1^T (\eta) | \leq c_\upkappa |\eta|^2$ for the same $c_\upkappa$. This $c_\upkappa$ can be taken uniformly for all $\delta \in [{\underline}\delta, 1]$.
Therefore, (\[equation426\]) can be written as $$\begin{aligned}
\label{equation427}
E_2 \left( \partial_s \eta + J^T(\upkappa)\left( \partial_t \eta\right) + {\sf A}_{\upkappa} \eta \right)= e^{-\lambda t} R_0^T + R_1^T.\end{aligned}$$
The following lemma can be proved in a similar way as proving Lemma \[lemma412\]. We leave the proof to the reader.
\[lemma415\] There exists $c_3 = c_3 ( {\widetilde}{K}, {\underline}\delta, \upgamma) >0$ and for any $\rho>0$, there are constants $\varepsilon_3 = \varepsilon_3(\rho)>0$ and $S_3 = S_3(\rho)>0$ (which also depend on ${\widetilde}{K}, {\underline}\delta, \upgamma$) satisfying the following condition. If $\big\| e({\bm u}) \big\|_{L^\infty(\Theta_+)} \leq (\varepsilon_3)^2$, then $$\begin{aligned}
\big| R_1^T(s, t) \big| \leq \rho \big| \eta \big| ,\ \big| \nabla_s R_1^T (s, t) \big| \leq \rho^2 \big| \eta \big| + \rho \big| \nabla_s \eta \big|;\end{aligned}$$ $$\begin{aligned}
\big| \nabla_t R_1^T \big| \leq c_3 \big( |\eta | + |\nabla_t \eta | \big).\end{aligned}$$
Let ${{\mathcal}L}_\upkappa^{(\delta)}: L^2(S^1, T_\upkappa {\widetilde}{X}_\upgamma) \to L^2(S^1, T_\upkappa {\widetilde}{X}_\upgamma)$ be the operator $$\begin{aligned}
{{\mathcal}L}_\upkappa(\eta) = J^T(\upkappa) \partial_t \eta + {\sf A}_\upkappa^{(\delta)} \eta.\end{aligned}$$ In the same way as proving (\[equation418\]) we can show that it is self-adjoint and coercive, i.e., $$\begin{aligned}
\label{equation428}
\big\| {{\mathcal}L}_\upkappa^{(\delta)} (\eta ) \big\|^2 \geq (\tau_1)^2 \big\| \eta \big\|^2.\end{aligned}$$
Denoting $R^T = E_2( \upkappa, \eta )^{-1} \left( e^{- \lambda t} R_0^T + R_1^T \right)$, (\[equation427\]) implies that $$\begin{aligned}
\partial_s^2 \eta = \partial_s \big( R^T - {{\mathcal}L}_\upkappa^{(\delta)} (\eta) \big) = \partial_s R^T - {{\mathcal}L}_\upkappa^{(\delta)} ( R^T) + \big( {{\mathcal}L}_\upkappa^{(\delta)} \big)^2 (\eta).\end{aligned}$$ Then we denote $v(s) = \big\| \eta(s, \cdot) \big\|_{L^2(S^1)}^2$. Then by Lemma \[lemma414\] and Lemma \[lemma415\], we have $$\begin{gathered}
{1\over 2} v''(s) = \big\langle \partial_s^2 \eta, \eta \big\rangle + \big\| \partial_s \eta \big\|^2 = \big\langle \partial_s R^T - {{\mathcal}L}_\upkappa^{(\delta)} (R^T) +\big( {{\mathcal}L}_\upkappa^{(\delta)} \big)^2 (\eta) , \eta \big\rangle + \big\| \partial_s \eta \big\|^2 \\
\geq - \rho^2 \big\| \eta \big\|^2 - c_1 e^{-{1\over 2} \tau_0 s} \big\| \eta \big\| - \rho \big\| \partial_s \eta \big\| \big\| \eta \big\| - \big\|{{\mathcal}L}_\upkappa^{(\delta)} \eta \big\| \big( \rho \big\| \eta \big\| + c_1 e^{-{1\over 2} \tau_0 s} \big) + \big\|{{\mathcal}L}_\upkappa^{(\delta)} \eta \big\|^2 + \big\|\partial_s \eta \big\|^2 \\
\geq {5 \over 8} \big\| {{\mathcal}L}_\upkappa^{(\delta)} \eta \big\|^2 - {9 \over 4} \rho^2 \big\| \eta \big\|^2 + 2 \big( c_1 e^{-{1\over 2} \tau_0 s} \big)^2 - c_1 e^{-{1\over 2} \tau_0 s} \big\| \eta \big\|.\end{gathered}$$ Then by (\[equation428\]) and Lemma \[lemma415\], there are $\rho_4 >0$, $S_4 \geq S_3(\rho_4) > 0$ and $c_4 >0$ such that if $\big\| e({\bm u}) \big\|_{L^\infty(\Theta_+)} \leq (\varepsilon_3(\rho_4 ))^2$, then for $s \geq S_4$, we have $$\begin{aligned}
\label{equation429}
v''(s) \geq \big\|{{\mathcal}L}_\upkappa^{(\delta)} \eta \big\|^2 - c_4 e^{- \tau_0 s}\geq (\tau_1)^2 \big\| \eta \big\|^2 - c_4 e^{- \tau_0 s}.\end{aligned}$$ This implies that the function $$\begin{aligned}
e^{- \tau_1 s} \Big( v'(s) + \tau_1 v(s) - { c_4 \over \tau_0 + \tau_1} e^{- \tau_0 s} \Big)\end{aligned}$$ is non-decreasing on $[S_4, +\infty)$. Then by the fact that $\lim_{s \to \infty} v(s) = 0$, we see for $s \geq S_4$, $$\begin{aligned}
v'(s) + \tau_1 v(s) - { c_4 \over \tau_0 +\tau_1} e^{-\tau_0 s}\leq 0.\end{aligned}$$ We can assume that $\tau_0 \neq \tau_1$; otherwise we can slightly improve (\[equation418\]) or (\[equation428\]) so that the $\tau_0$ and $\tau_1$ appeared there are different. Therefore $$\begin{aligned}
{d\over ds} \Big( e^{\tau_1 s} \big( v(s) + { c_4 \over (\tau_0 - \tau_1 )(\tau_0 + \tau_1)} e^{ - \tau_0 s} \big) \Big) \leq 0.\end{aligned}$$ Therefore we see there is a constant $c_5 >0$ such that for $s \geq S_4$, $$\begin{gathered}
\label{equation430}
v(s) \leq { c_4 \over (\tau_1 - \tau_0) (\tau_1 + \tau_0)} e^{-\tau_0 s} + e^{- \tau_1 s} \Big( v(S_4) + {c_4 \over ( \tau_1 - \tau_0 ) (\tau_1 + \tau_0 )} e^{-\tau_0 S_4 } \Big) \\
\leq c_5 e^{- \min\{\tau_0, \tau_1\} s}.\end{gathered}$$
To derive pointwise estimate we can use the similar method as in did in (\[equation422\])–(\[equation424\]). Indeed, apply $\partial_s - J^T(\upkappa) \partial_t$ to (\[equation427\]), we obtain $$\begin{aligned}
\Delta \eta = (\partial_s - J^T(\upkappa)\partial_t ) R^T - ( \partial_s - J^T(\upkappa)\partial_t ) {\sf A}_{\upkappa}^{(\delta)} \eta.\end{aligned}$$ Therefore by Lemma \[lemma414\] and Lemma \[lemma415\], there is a constant $C$ such that $$\begin{aligned}
{1\over 2} \Delta |\eta|^2 = \langle \Delta \eta, \eta \rangle + | d\eta |^2 \geq -C ( e^{-\tau_0 s} + |\eta|^2 ).\end{aligned}$$ Denoting $\tau = \min\{ \tau_0, \tau_1\}$, then there is another constant $C'>0$ such that $$\begin{aligned}
\Delta |\eta|^2 \geq - C' ( e^{-\tau s} + e^{\tau s} |\eta|^4 ).\end{aligned}$$ This allows us to derive a similar mean value estimate as did in (\[equation422\])–(\[equation424\]) and therefore Proposition \[prop49\] is proven.
### The narrow case {#the-narrow-case .unnumbered}
Now we assume $\upgamma = \exp(2\pi \lambda)$ is narrow.
For the compact set ${\widetilde}{X}_\upgamma \cap {\widetilde}{K}$, there is a constant $D_2 > 0$ satisfying the following condition. For any smooth loop $x: S^1 \to {\widetilde}{X}_{\upgamma}\cap {\widetilde}{K}$, if ${\rm diam} ( x(S^1) ) \leq D_2$, then we can define the [**center of mass**]{}, which is a unique point $\alpha \in {\widetilde}{X}_{\upgamma}$ such that there is a function $\eta: S^1 \to T_\alpha {\widetilde}{X}_\upgamma$ such that $$\begin{aligned}
x(t) = \exp_\alpha \eta(t), \ \int_{S^1} \eta(t) dt = 0.\end{aligned}$$ Therefore, it is easy to see that there is a constant $\varepsilon_4>0$ such that if $\big\| e({\bm u}) \big\|_{L^\infty} \leq (\varepsilon_4)^2$, then $| \partial_t {\overline}{u} + {{\mathcal}X}_\lambda({\overline}{u}) |$ is small enough and hence the diameter of the loop ${\overline}{v}(s, \cdot)$ is smaller than $D_2$. Then the [**center of mass**]{} of ${\overline}{v}(s, \cdot)$ is smooth curve $\alpha: [0, +\infty) \to {\widetilde}{X}_\upgamma$. We regard $\alpha$ as a map $\alpha: \Theta_+ \to {\widetilde}{X}_\upgamma$ which is independent of the $t$-variable. Then there is a section $\eta \in \Gamma ( \Theta_+, \alpha^* T {\widetilde}{X}_\upgamma )$ so that $$\begin{aligned}
{\overline}{v}(s, t) = \exp_{\alpha(s)} \eta(s, t),\ \int_{S^1} \eta(s, t) dt = 0.\end{aligned}$$
Let $E_1, E_2$ be the components of the derivative of the exponential map of ${\widetilde}{X}_{\upgamma}$, i.e., $$\begin{aligned}
d \exp_x V = E_1(x, V) dx + E_2(x, V) \nabla V,\ x \in {\widetilde}{X}_\upgamma,\ V \in T_x {\widetilde}{X}_\upgamma.\end{aligned}$$ Then using the center of mass, we rewrite (\[equation426\]) as $$\begin{aligned}
\label{equation431}
E_1( \alpha, \eta) \alpha'(s) + E_2( \alpha, \eta) \nabla_s \eta + J^T({\overline}{v}) \left( E_2(\alpha, \eta) \partial_t \eta \right) = e^{-\lambda t} R_0^T.\end{aligned}$$ Moreover, there exists a linear map $R_J^T(s, t) : T_{\alpha(s)} {\widetilde}{X}_\upgamma \to T_{{\overline}{v}(s, t)} {\widetilde}{X}_\upgamma$ such that $$\begin{aligned}
\label{equation432}
E_2(\alpha, \eta)^{-1} J^T( {\overline}{v}) E_2 (\alpha, \eta) - J^T (\alpha) = R_J^T,\end{aligned}$$ We denote $R_1^T = R_J^T (\partial_t \eta)$, $R_2^T = E_2^{-1} E_1 \alpha'(s) - \alpha'(s)$ and $R^T = R_1^T + R_2^T$. Then (\[equation431\]) can be rewritten as $$\begin{aligned}
\label{equation433}
\alpha'(s) + \nabla_s \eta + J^T(\alpha) \partial_t \eta = E_2^{-1} \big( e^{-\lambda t} R_0^T \big) + R^T.\end{aligned}$$
\[lemma416\] There exists $c_6 >0$ and for any $\rho>0$, there are constants $\varepsilon_6 = \varepsilon_6(\rho)>0$ and $S_6 = S_6 (\rho)>0$ satisfying the following conditions. If $\big\| e({\bm u}) \big\|_{L^\infty} \leq (\varepsilon_6 )^2$, then for $s \geq S_6$, we have $$\begin{aligned}
\big| R^T(s, t) \big| \leq \rho \big| \eta \big| ,\ \big| \nabla_s R^T (s, t) \big| \leq \rho^2 \big| \eta \big| + \rho \big| \nabla_s \eta \big|;\end{aligned}$$ $$\begin{aligned}
\big| \nabla_t R^T \big| \leq c_6 \big( \big| \eta \big| + \big| \nabla_t \eta \big| \big).\end{aligned}$$
It can be proved in a similar way as Lemma \[lemma412\]. The proof is left to the reader.
\[lemma417\] There exist $c_7>0$ and $\varepsilon_7>0$ depending only on ${\widetilde}{K}$ that satisfy the following condition. If $\big\| e({\bm u}) \big\|_{L^\infty} \leq (\varepsilon_7)^2$, then $$\begin{aligned}
\label{equation434}
\big\| \eta(s, \cdot) \big\|_{L^2(S^1)} \leq c_7 e^{-{1\over 2} \tau_0 s}.\end{aligned}$$
Denote by $R(s, t)$ the right hand side of (\[equation433\]) and let ${\overline}{H}_s \subset L^2(T_{\alpha(s)} T {\widetilde}{X}_{\upgamma})$ be the subspace of functions with zero average on $S^1$. Then $\nabla_s$ preserves this subspace. Projecting (\[equation433\]) onto ${\overline}{H}_s$, $\alpha'(s)$ is killed and we have $$\begin{aligned}
\nabla_s \eta + J^T \partial_t \eta = {\overline}{R}(s, t)\end{aligned}$$ where ${\overline}{R}(s, \cdot) \in {\overline}{H}_s$ is the image of $R(s, \cdot)$ under the projection.
The operator $J \partial_t$ is coercive on ${\overline}{H}_s$, satisfying $$\begin{aligned}
\| J \partial_t \eta \|_{L^2}^2 \geq \|\eta\|_{L^2}^2.\end{aligned}$$ Notice that $\tau_0 < 1$. Then (\[equation434\]) can be derived in the same way as deriving (\[equation430\]).
Applying the mean value estimate as did in (\[equation422\])–(\[equation424\]), we can prove that $$\begin{aligned}
| \eta(s, t) | \leq c_7 e^{-{1\over 2} \tau_0 s}.\end{aligned}$$ Then it implies that $\left| R(s, t) \right| \leq c_7 e^{- {1\over 2}\tau_0 s}$ with $c_7$ abusively used. Taking $L^2$-paring of (\[equation433\]) with $\alpha'(s)$, one has $$\begin{aligned}
\left| \alpha'(s) \right| \leq c_7 e^{- {1\over 2} \tau_0 s}.\end{aligned}$$ Then it implies that there exists $\upkappa \in {\widetilde}{X}_\upgamma$ such that $$\begin{aligned}
\lim_{s \to \infty} \alpha(s) = \lim_{s \to \infty} {\overline}{v}(s, t) = \upkappa. \end{aligned}$$ Therefore Proposition \[prop410\] is proven.
Proof of Theorem \[thm44\] {#subsection44}
--------------------------
Let $(A, u)$ be a bounded solution to the perturbed gauged Witten equation over $\vec{{\mathcal}C}$. By Theorem \[thm42\] and \[thm43\], $u$ extends to a continuous orbifold section ${{\mathcal}U}: {{\mathcal}C} \to {{\mathcal}Y}$.
Now we compute the energy of $(A, u)$. For the fibration $Y \to \Sigma$ and any $G$-connection $A$ on $P$, we have the minimal coupling form $\omega_A \in \Omega^2(Y)$. For any smooth section $u: \Sigma^* \to Y$, the following identity is well-known (see for example, the proof of [@Cieliebak_Gaio_Mundet_Salamon_2002 Proposition 2.2]). $$\begin{aligned}
{1\over 2} \big\| d_A u \big\|^2 \nu = u^* \omega_A + \big\| {\overline}\partial_A u \big\|^2 \nu + \mu (u) \cdot F_A.\end{aligned}$$ Then by the definition of the kinetic energy we have: $$\begin{aligned}
\label{equation435}
E_K (A, u)= \big\| {\overline}\partial_A u \big\|_{L^2(\Sigma^*)}^2 + \int_\Sigma u^*\omega_A + {1\over 2} \big\| * F_A + \mu^* (u) \big\|_{L^2(\Sigma^*)}^2 - {1\over 2}\big\| \mu(u) \big\|_{L^2(\Sigma^*) }^2.\end{aligned}$$ Since $u$ extends to ${{\mathcal}U}$, we have $$\begin{aligned}
\label{equation436}
\int_{\Sigma^*} u^* \omega_A = \big\langle \big[ {{\mathcal}U} \big], \big[ \omega - \mu \big] \big\rangle = \big\langle \big[ A, u\big], \big[\omega - \mu\big] \big\rangle.\end{aligned}$$ On the other hand, let $\langle \cdot, \cdot \rangle$ be the real part of the Hermitian pairing on $T{\widetilde}{X}$. Then $$\begin{aligned}
\label{equation437}
\begin{split}
& \big| {\overline}\partial_A u \big|^2 + \big| \nabla {\widetilde}{{\mathcal}W}_A (u) \big|^2 - \big| {\overline}\partial_A u + \nabla {\widetilde}{{\mathcal}W}_A (u) \big|^2 \\
=& - 2 \big\langle {\overline}\partial_A u , \nabla {\widetilde}{{\mathcal}W}_A (u) \big\rangle = 2 {\rm Im} * \big( d {\widetilde}{{\mathcal}W}_A (u) \wedge {\overline}\partial_A u \big) \\
=& - 2 {\rm Im} * \big( {\overline}\partial {\widetilde}{{\mathcal}W}_A(u) \big) +2 {\rm Im} * \big( {\overline}\partial \beta \wedge {{\mathcal}W}_A'(u) \big).
\end{split}\end{aligned}$$ Now we identify each cylindrical end $U_j(2)$ with $\Theta_+$ and identify the restriction of $(A, u)$ to $U_j(2)$ with a solution ${\bm u}_j = (u_j, h_j)$ to a $(\lambda_j, \delta_j)$-cylindrical model, where $\delta_j = \delta_{j, A}$. Then by Stokes formula, Theorem \[thm42\] and \[thm43\], we have $$\begin{aligned}
\label{equation438}
\begin{split}
- 2{\rm Im} \int_{\Sigma^*} {\overline}\partial {\widetilde}{{\mathcal}W}_A(u) = &\ - 2 \int_{\Sigma^*} d \big( {\rm Im} {\widetilde}{{\mathcal}W}_A(u) \big) \\
= &\ -2 {\rm Re} \Big[ \sum_{j=1}^k \lim_{s \to +\infty} \int_{\{s \} \times S^1} {\widetilde}{W}_{h_j, \lambda_j}^{(\delta_j)}(u_j (s, \cdot)) dt \Big] \\
= &\ - 4\pi {\rm Re} \Big[ \sum_{j=0}^k {\widetilde}{W}_{\upgamma_j}^{(\delta_j)} (\upkappa_j) \Big]\\
= &\ -4 \pi {\rm Re} \Big[ \sum_{j=0}^k {\rm Res}_j(A, u)\Big].
\end{split}\end{aligned}$$ Then by (\[equation435\])-(\[equation438\]), $$\begin{aligned}
\label{equation439}
E(A, u) = \big\langle \big[A, u\big] , \big[ \omega, - \mu\big] \big\rangle - {\rm Re} \Big[ 4\pi \sum_{j=1}^k {\rm Res}_j(A, u) + \int_{\Sigma^*} {\bm i} {\overline}\partial \beta \wedge {{\mathcal}W}_A'(u) \Big].\end{aligned}$$ Therefore the first part of Theorem \[thm44\] is proved.
To prove the second part, i.e., the uniform bound on the energy, we have to control the residues ${\rm Res}_j(A, u)$ and the integral of ${\overline}\partial\beta \wedge {{\mathcal}W}_A'(u)$. Indeed, if $z_j$ is narrow, ${\rm Res}_j(A, u) = 0$; if $z_j$ is broad, the limit $\upkappa_j = (x_j, p_j) \in X \times {{\mathbb}C}$ is a critical point of the function $$\begin{aligned}
{\widetilde}{W}^{(\delta_j)}_{\upgamma_j} = \sum_{l=0}^s F_{\upgamma_j; l}^{(\delta_j)}.\end{aligned}$$ By (\[equation26\]), $( \delta_j^{-1} x_j, p_j)$ is a critical point of the function ${\widetilde}{W}_{\upgamma_j}|_{{\widetilde}{X}_{\upgamma_j}}$, which is independent of $(A, u)$. Therefore $$\begin{aligned}
\label{equation440}
{\widetilde}{W}^{(\delta_j)}_{\upgamma_j} ( \upkappa_j) = \sum_{l=0}^s F_{\upgamma_j;l}^{(\delta_j)} (x_j, p_j) = \sum_{l=1}^{s-1} (\delta_j)^{r- l} F_{\upgamma_j; l} ( x_j ) = (\delta_j)^r F_{\upgamma_j; l}( \delta_j^{-1} x_j) .\end{aligned}$$ So the sum of the residues in (\[equation439\]) is uniformly bounded. On the other hand, for each broad $z_j$, denote $C_j:= U_j\setminus U_j(2)$, which contains the support of $d\beta_j$. We have $$\begin{aligned}
\label{equation441}
\begin{split}
\Big| \int_{\Sigma^*} {\bm i} {\overline}\partial \beta \wedge {{\mathcal}W}_A'(u) \Big| \leq &\ \sum_{z_j\ {\rm broad}} \Big| \int_{C_j} {\overline}\partial \beta_j \wedge \sum_{l =1}^s e^{\rho_l (h + \lambda_j t) } F_{\upgamma_j; l}^{(\delta_j)} (u_j) dz \Big| \\
\leq &\ \sum_{z_j\ {\rm broad}} \delta_j \int_{C_j} \Big( \sum_{l =1}^{s} \big| e^{\rho_l (h)} \big| \big| F_{\upgamma_j; l} (u_j) \big| \Big) ds dt \\
\leq &\ c^{(0)} \sum_{z_j\ {\rm broad}} \delta_j \int_{N_j} \Big( \sum_{l =1}^{s} \big| e^{\rho_l (h)} \big| \Big) \sqrt{ 1 + \big| \mu (u_j) \big|} ds dt\\
\leq &\ c^{(0)} \sum_{z_j\ {\rm broad}} \delta_j \Big( \sum_{l =1}^{s} \big\| e^{\rho_l (h)} \big\|_{L^2(N_j)} \Big) \big\| \sqrt{ 1 + \big| \mu (u_j) \big|} \big\|_{L^2(N_j) } \\
\leq & \ c \Big( 1 + \big\| \sqrt{\sigma} \mu (u) \big\|_{L^2(\Sigma^*)} \Big) \\[0.2cm]
\leq &\ c \Big( 1 + \sqrt{ E (A, u) } \Big).
\end{split}\end{aligned}$$ Here the second line follows from the fact that each $F_{\upgamma_j;l}^{(\delta_j)}$ has at least one $\delta_j$ factor; the third line uses ([**P3**]{}) of Hypothesis \[hyp28\]; the fifth line uses the definition of $\delta_j$; $c>0$ is a constant depending on $c^{(0)}$. It follows from (\[equation439\])–(\[equation441\]) that for some constant $c'$ independent of $(A,u)$, we have $$\begin{aligned}
E (A, u) \leq \big\langle \big[ A, u \big], \big[ \omega - \mu \big] \big\rangle + c' + c'\sqrt{ E (A, u)}.\end{aligned}$$ It implies a uniform bound on $E(A, u)$ in terms of $\big[ A, u \big]$, so Theorem \[thm44\] is proved.
In the narrow case, i.e., when all punctures are narrow, we don’t have to perturb the equation. Then by (\[equation437\]), the $L^2$-norm of ${\overline}\partial_A u$ is zero because all the residues are zero. So any solution of the gauged Witten equation is also a solution to the symplectic vortex equation, whose image is contained in ${\rm Crit} W$. So analysis in the narrow case are much easier than the broad case (when there is at least one broad puncture).
Linear Fredholm Theory {#section4}
======================
In this section we consider the linearized operator of the perturbed gauged Witten equation modulo gauge transformations. This section is more or less independent of the other sections of this paper and can be treated under a much more general set-up, for example, the Lie group $G$ could be nonabelian, and the superpotential ${\widetilde}{{\mathcal}W}_A$ need not be holomorphic. The condition on narrowness and broadness of punctures can also be more flexible. However for simplicity we still use the set-up of given in Section \[section2\].
Banach manifolds, Banach bundles and sections
---------------------------------------------
Let $\vec{{\mathcal}C}$ be a rigidified $r$-spin curve with underlying Riemann surface $\Sigma$. Let ${\bm z} =\{ z_1, \ldots, z_k\}$ be the set of punctures and the monodromy of $\vec{{\mathcal}C}$ at $z_j$ is $\upgamma_j$. The corresponding punctured Riemann surface $\Sigma^*$ is equipped with the cylindrical metric, which is used to define the weighted Sobolev spaces. Let $\tau>0$. We use $W_\tau^{k, p}(\Sigma^*, E)$ to denote the space of sections of a vector bundle $E$ over $\Sigma^*$, of class $W_\tau^{k, p}$, with respect to some fixed choice of connection on $E$. We will omit the domain $\Sigma^*$ in this section and abbreviate the space by $W_\tau^{k, p}(E)$.
For each $\upgamma \in {{\mathbb}Z}_r$, we have the function ${\widetilde}{W}_\upgamma: {\widetilde}{X} \to {{\mathbb}C}$ as introduced in (\[equation25\]) such that if $\upgamma$ is broad, then ${\widetilde}{W}_\upgamma |_{{\widetilde}{X}_\upgamma}$ is a holomorphic Morse function with finitely many critical points $\upkappa_\upgamma^{(\iota)},\ \iota = 1,\ldots, m_\upgamma$. For $\delta \in (0, 1]$, $\upkappa_{\upgamma; \delta}^{(\iota)} := \delta \upkappa_\upgamma^{(\iota)}$ is a critical point of ${\widetilde}{W}_\upgamma^{(\delta)}|_{{\widetilde}{X}_\upgamma}$.
We abbreviate ${\widetilde}{X}_j = {\widetilde}{X}_{\upgamma_j}$. Now for each $A\in {{\mathpzc}A}$, we have defined $\delta_{j,A}$ in (\[equation220\]). Then for each $A \in {{\mathpzc}A}$, denote $\upkappa_{j, A}^{(\iota)} = \upkappa_{\upgamma_j; \delta_{j,A}}^{(\iota)}$.
For any $x_j \in {\widetilde}{X}_j$, define ${\widetilde}{x}_j: S^1 \to {\widetilde}{X}_j$ by ${\widetilde}{x}_j(t) = e^{- \lambda_j t} \upkappa_j$. The pull-back of ${\widetilde}{x}_j$ to any cylinder $[a, b]\times S^1$ via the projection $[a, b] \times S^1 \to S^1$ is still denoted by ${\widetilde}{x}_j$. When $z_j$ is broad, denote ${\widetilde}\upkappa^{(\iota)}_{j,A}= e^{-\lambda_j t} \upkappa^{(\iota)}_{j,A}$ for all $A \in {{\mathpzc}A}$.
### The Banach manifold {#the-banach-manifold .unnumbered}
Choose ${\underline}\delta \in (0, 1]$ and $\tau \in (0, \tau({\underline}\delta)/2 )$ where $\tau({\underline}\delta)>0$ is the one of Theorem \[thm43\]. Choose $B \in H_2^G\big( {\widetilde}{X}; {{\mathbb}Z}[r^{-1}]\big)$. For each broad puncture $z_j$, choose $\iota_j \in \{1, \ldots, m_{\upgamma_j}\}$ and denote $$\begin{aligned}
\vec\upkappa = \big( \upkappa_{\upgamma_j}^{(\iota_j)}\big)_{z_j\ {\rm broad}}.\end{aligned}$$
Consider the space $$\begin{aligned}
{{\mathpzc}B} = {{\mathpzc}B}_{\tau, {\underline}\delta}\big( B, \vec\upkappa\big) \subset {{\mathpzc}A}_\tau^{1, p} \times W_{loc}^{1, p} \big( \Sigma^*, Y \big),\end{aligned}$$ consisting of pairs $(A, u)\in {{\mathpzc}A}_\tau^{1, p}\times W_{loc}^{1, p}(\Sigma^*, Y)$ such that $\delta_{j,A} \in ({\underline}\delta, 1]$ and such that there is an $S>0$ satisfying the following conditions.
\(I) For each broad $z_j$, there is a section ${\widetilde}\eta_j \in W_\tau^{1, p} \big( \Theta_+(S), \big( {\widetilde}\upkappa_{j,A}^{(\iota_j)} \big)^* T{\widetilde}{X} \big)$ such that $$\begin{aligned}
\label{equation51}
u\circ \phi_j|_{\Theta_+(S)} = \exp_{{\widetilde}\upkappa_{j, A}^{(\iota_j)}} {\widetilde}\eta_j.\end{aligned}$$
\(II) For each narrow $z_j$, there is $(\star, p_j) \in {\widetilde}{X}_j$ and ${\widetilde}\eta_j \in W_\tau^{1, p} \big( \Theta_+(S), T_{p_j} {\widetilde}{X} \big)$ such that $$\begin{aligned}
\label{equation52}
u\circ \phi_j |_{\Theta_+(S)} = \exp_{p_j} {\widetilde}\eta_j.\end{aligned}$$
\(III) The above two conditions implies that $u$ extends to an orbifold section over $\vec{{\mathcal}C}$ and we require that $\big[ A, u\big] = B$.
From now on within this section, $\vec\upkappa$ is fixed and we abbreviate $\upkappa_j = \upkappa_{\upgamma_j}^{(\iota_j)}$, $\upkappa_{j, A} = \upkappa_{j, A}^{(\iota_j)}$.
The space of connections ${{\mathpzc}A}_\tau^{1, p}$ doesn’t contain all $W_\tau^{1, p}$-connections on $P$ but it gives a constrain on the holomorphic structure. By the definition of ${{\mathpzc}A}_\tau^{1, p}$, ${{\mathpzc}A}_\tau^{1, p}$ is an affine space modeled on the vector space $$\begin{aligned}
T{{\mathpzc}A}_\tau^{1, p} \simeq \Big\{ \alpha = ( \alpha_0, \alpha_1) \in W_\tau^{1, p} \big({{\mathfrak}g}_0 \oplus {{\mathfrak}g}_1 \big) \ |\ \alpha_0 = * d h + d f,\ f, h \in W_\tau^{2, p}({{\mathfrak}g}_0) \Big\}.\end{aligned}$$ Here $* d h$ is the infinitesimal change of $A$ with respect to the infinitesimal change of the Hermitian metric, and $d f$ is the infinitesimal gauge transformation. Then we have
${{\mathpzc}B}$ carries a Banach manifold structure, whose tangent space at ${{\mathpzc}X}= (A, u) \in {{\mathpzc}B}$ is isomorphic to $$\begin{aligned}
\label{equation53}
T_{{\mathpzc}X} {{\mathpzc}B}\simeq T{{\mathpzc}A}_\tau^{1, p} \oplus W_\tau^{1, p} \big( u^* T^\bot Y \big) \oplus {{\mathbb}C}^{k_n}.\end{aligned}$$ Here $k_n$ is equal to the number of narrow punctures.
Moreover, the group ${{\mathpzc}G}_\tau = {{\mathpzc}G} \cap {{\mathpzc}G}_\tau^{2, p}$ acts smoothly on ${{\mathpzc}B}$ such that the isomorphism (\[equation53\]) is equivariant in a natural way.
We define an exponential map for ${{\mathpzc}X} = (A, u) \in {{\mathpzc}B}$ and $$\begin{aligned}
\big( \alpha, \xi, {\bm \zeta} \big) \in T{{\mathpzc}A}_\tau^{1, p} \oplus W_\tau^{1, p}\big( u^* T^\bot Y \big) \oplus {{\mathbb}C}^{k_n}\end{aligned}$$ with sufficiently small norm. This will give a local chart of the Banach manifold structure. Let $S$ and ${\widetilde}\eta_j$ be the same as in (\[equation51\]) and (\[equation52\]).
For a broad puncture $z_j$, since the map $A \mapsto \upkappa_{j,A}$ is smooth, for $\|\alpha\|_{W^{1, p}_\tau}$ sufficiently small, there is a unique $\xi_j(\alpha) \in T_{\upkappa_{j, A}} {\widetilde}{X}_j$ such that $$\begin{aligned}
\upkappa_{j, A + \alpha} = \exp_{\upkappa_{j,A}} \xi_j(\alpha).\end{aligned}$$ Denote ${\widetilde}\xi_j(\alpha):= e^{-\lambda t} \xi_j(\alpha)$, which is along the map ${\widetilde}\upkappa_{j,A}$. Then we can extend ${\widetilde}\xi_j$ to a vector field along $( u \circ \phi_0 )|_{\Theta_+(S)}$, by the parallel transport of ${\widetilde}\xi_j$ along the family of geodesics $$\begin{aligned}
x_{s, t}(\epsilon) = \exp_{{\widetilde}\upkappa_{j, A}} \epsilon {\widetilde}\eta_j (s, t),\ (s, t) \in \Theta_+,\ \epsilon \in [0,1].\end{aligned}$$ Denote the vector field still by ${\widetilde}\xi_j$. Then choose a cut-off function $\beta_j': \Sigma^* \to [0,1]$ vanishing outside $\Theta_+(S+1)$ and being identically $1$ on $\Theta_+(S+2)$. Then $\beta_j' {\widetilde}\xi_j$ extends to a vertical tangent vector field along $u$, denoted by the same symbol.
For a narrow puncture $z_j$, any $\zeta_j \in {{\mathbb}C} \simeq T_{(\star, p_j)} {\widetilde}{X}_j$ can be viewed as a vector field $u\circ \phi_j$. We choose a cut-off function $\beta_j'$ similarly as the above, and denote ${\widetilde}\zeta_j = \beta_j' \zeta_j \in \in W_{loc}^{1, p} \big( \Sigma^*, u^* T^\bot Y \big)$.
We define $$\begin{aligned}
V(\alpha, \xi, {\bm \zeta}):= \xi + \sum_{z_j\ {\rm broad}} \beta_j' {\widetilde}\xi_j + \sum_{z_j\ {\rm narrow}} \beta_j' {\widetilde}\zeta_j.\end{aligned}$$ Then the section $\exp_u V(\alpha, \xi, {\bm \zeta})$ represent the same homology class $\big[ A, u\big]$. Moreover, if $\big\|\alpha\big\|_{W_\tau^{1, p}}$ is small enough, then $\delta_{j; A + \alpha} > {\underline}\delta$. Therefore the object $(A', u') = ( A + \alpha, u') \in {{\mathpzc}B}$. This gives local charts of ${{\mathpzc}B}$. The assertion about the ${{\mathpzc}G}_\tau$-action is easy to check.
There exist $\tau>0$ and ${\underline}\delta\in (0, 1]$ such that for any bounded solution $(A, u)$ to the perturbed gauged Witten equation over $\vec{{\mathcal}C}$, if $\big[ A, u \big] = B$ and for each broad puncture $z_j$, $ev_j(A, u) = \upkappa_{j, A}$, then there is a gauge transformation $g\in {{\mathpzc}G}$ such that $g^*(A, u) \in {{\mathpzc}B}_{\tau, {\underline}\delta}(B, \vec{\upkappa})$.
By Theorem \[thm44\], such solutions have a uniform energy bound $E(B)$. Therefore there exists ${\underline}\delta\in (0, 1]$ such that for any such solution $(A, u)$ and any broad puncture $z_j$, $\delta_{j, A}\geq {\underline}\delta$. There exists $g\in {{\mathpzc}G}$ such that on each cylindrical end $g^*A$ is in temporal gauge. Then by Theorem \[thm42\], the vortex equation, the decay of the area form $\nu$, $A\in {{\mathpzc}A}_\tau^{1, p}$ for any $\tau < 2$. On the other hand, Theorem \[thm43\] implies that $u$ converges exponentially fast to its limits at punctures. Therefore, there exists $\tau>0$ such that $g^* (A, u) \in {{\mathpzc}B}_{\tau, {\underline}\delta}(B, \vec{\upkappa})$.
### The Banach bundle and the section {#the-banach-bundle-and-the-section .unnumbered}
Now we consider the ${{\mathpzc}G}_\tau$-equivariant Banach vector bundle ${{\mathpzc}E}\to {{\mathpzc}B}$ whose fibre over $(A, u) \in {{\mathpzc}B}$ is $$\begin{aligned}
\label{equation54}
{{\mathpzc}E}|_{(A, u)} = L_\tau^p \big( \Lambda^{0,1}T^* \Sigma^*\otimes u^* T^\bot Y \big) \oplus L_\tau^p ({{\mathfrak}g} ). \end{aligned}$$ The fact that ${{\mathpzc}E}$ carries a smooth Banach bundle structure over ${{\mathpzc}B}$ is the same as many classical cases (for example, Gromov-Witten theory, see [@McDuff_Salamon_2004 Section 3]); a local trivialization of ${{\mathpzc}E}$ can be obtained by using parallel transport. The ${{\mathpzc}G}_\tau$-action also lifts naturally to a linear action on ${{\mathpzc}E}$, making ${{\mathpzc}E}$ a ${{\mathpzc}G}_\tau$-equivariant bundle.
The perturbed gauged Witten equation gives a smooth section of ${{\mathpzc}E}\to {{\mathpzc}B}$. More precisely, for $(A, u) \in {{\mathpzc}B}$, the left-hand-side of (\[equation221\]) defines $$\begin{aligned}
{{\mathpzc}W}(A, u) \in {{\mathpzc}E}|_{(A, u)}.\end{aligned}$$ The smoothness of ${{\mathpzc}W}$ relies on the smoothness of the function $A \mapsto \delta_{j,A}$ (see Definition \[defn215\]). It is also an equivariant section by the gauge invariance property of the perturbed gauged Witten equation. Moreover, for every ${{\mathpzc}X} \in {{\mathpzc}B}$, $g\in {{\mathpzc}G}_\tau$ and ${{\mathpzc}X}'= g^* {{\mathpzc}X}$, $g$ induces an isomorphism $$\begin{aligned}
g^*: \left( T_{{\mathpzc}X} {{\mathpzc}B}, E|_{{{\mathpzc}X}} \right) \to \left( T_{{{\mathpzc}X}'} {{\mathpzc}B}, E|_{{{\mathpzc}X}'} \right).\end{aligned}$$ This makes the isomorphism (\[equation53\]) and (\[equation54\]) both transform naturally.
### The deformation complex and the index formula {#the-deformation-complex-and-the-index-formula .unnumbered}
The linearization of ${{\mathpzc}W}$ at ${{\mathpzc}X}\in {{\mathpzc}B}$ is a bounded linear map $$\begin{aligned}
d{{\mathpzc}W}_{{\mathpzc}X}: T_{{\mathpzc}X} {{\mathpzc}B} \to {{\mathpzc}E}|_{{{\mathpzc}X}}.\end{aligned}$$ With respect to (\[equation53\]) and (\[equation54\]), it reads, $$\begin{aligned}
\label{equation55}
d{{\mathpzc}W}_{{\mathpzc}X}(\alpha, \xi, {\bm \zeta}) = \big( {{\mathpzc}D}_{{\mathpzc}X} V(\alpha, \xi, {\bm \zeta}) + \delta_\alpha \nabla {\widetilde}{{\mathcal}W}_A(u),\ *_c d\alpha + \sigma d\mu^* (u) V(\alpha, \xi, {\bm \zeta} ) \big).\end{aligned}$$ Here ${{\mathpzc}D}_{{\mathpzc}X}: W_{loc}^{1, p} \big( u^* T^\bot Y \big) \to L_{loc}^p \big( \Lambda^{0,1} T^* \Sigma^* \otimes u^* T^\bot Y \big)$ is the linearization of ${\overline}\partial_A u + \nabla {\widetilde}{{\mathcal}W}_A (u)$ in the direction of $\xi$, which reads $$\begin{aligned}
{{\mathpzc}D}_{{\mathpzc}X} (V) = {\overline}\partial_A V + \nabla_V \nabla {\widetilde}{{\mathcal}W}_A (u);\end{aligned}$$ and $$\begin{aligned}
\delta_\alpha \nabla {\widetilde}{{\mathcal}W}_A(u) = {d\over dt}|_{t=0} \nabla {{\mathcal}W}_{A + t\alpha}(u).\end{aligned}$$ The second component of (\[equation55\]) is the linearization of the left-hand-side of the vortex equation $*_c ( F_A + \mu^* (u) \nu)$, where $\nu$ is the smooth area form and $*_c$ is the Hodge-star of of the cylindrical metric; we use this modification because otherwise the $* d$ is not uniformly elliptic with respect to the cylindrical coordinates.
On the other hand, the linearization of infinitesimal gauge transformation at ${{\mathpzc}X}\in {{\mathpzc}B}$ is a linear operator $$\begin{aligned}
\begin{array}{cccc}
d{{\mathpzc}G}_{{\mathpzc}X}: & {\rm Lie} {{\mathpzc}G}_\tau:= W_\tau^{2, p} ({{\mathfrak}g}) & \to & \big( W_\tau^{1, p} ( T^* \Sigma \otimes {{\mathfrak}g} ) \big) \oplus W_\tau^{1, p} \big( u^* T^{\bot}Y \big)\\
& \xi & \mapsto & \big( d \xi, - {{\mathcal}X}_\xi \big).
\end{array}\end{aligned}$$ Then the [**deformation complex**]{} at ${{\mathpzc}X}$ is the following complex of Banach spaces $$\begin{aligned}
\label{equation56}
{{\mathpzc}C}_{{\mathpzc}X}: \begin{CD}
{\rm Lie} {{\mathpzc}G}_\tau @> d{{\mathpzc}G}_{{\mathpzc}X} > > T_{{\mathpzc}X}{{\mathpzc}B} @> d{{\mathpzc}W}_{{\mathpzc}X} >> {{\mathpzc}E}|_{{\mathpzc}X}.
\end{CD}\end{aligned}$$ For any ${{\mathpzc}X}\in {{\mathpzc}B}$, we abbreviate $A_1 = {\rm Lie} {{\mathpzc}G}_\tau $, $A_2 = T_{{\mathpzc}X} {{\mathpzc}B}$ and $A_3 = {{\mathpzc}E}|_{{{\mathpzc}X}}$.
To state the index formula, we need to introduced some notations. For each broad puncture $z_j$, we define $b_j(\vec{{\mathcal}C}) = {\rm dim}_{{\mathbb}C} {\widetilde}{X}_j$. We define $$\begin{aligned}
b(\vec{{\mathcal}C}) = \sum_{z_j \ {\rm broad}} b_j(\vec{{\mathcal}C}).\end{aligned}$$ On the other hand, for each $j$, the normal bundle ${\widetilde}{N}_j:= {\widetilde}{N}_j \to {\widetilde}{X}_j$ splits as $$\begin{aligned}
{\widetilde}{N}_j = \bigoplus_{i=1}^{{\rm codim} {\widetilde}{X}_j} {\widetilde}{N}_j^{(i)}\end{aligned}$$ where each line bundles ${\widetilde}{N}^{(i)}_j$ has an associated weight $\nu^{(i)}_j \in {{\mathbb}Z}$ such that $(\upgamma_j)^{\nu^{(i)}_j} \neq 1$. We define $$\begin{aligned}
n_j(\vec{{\mathcal}C}) = - {\bm i} \sum_i \big( \nu^{(i)}_j \lambda_j - \big\lfloor \nu^{(i)}_j \lambda_j \big\rfloor \big) \in {{\mathbb}Q}_{\geq 0},\ n(\vec{{\mathcal}C}) = \sum_{j=1}^k n_j(\vec{{\mathcal}C}).\end{aligned}$$ Here $\lfloor a\rfloor\in {{\mathbb}Z}$ is the greatest integer which is no greater than $a\in {{\mathbb}R}$.
Our main theorem of this section is the following.
\[thmx52\] For any bounded solution ${{\mathpzc}X} \in {{\mathpzc}B}$ to the perturbed gauged Witten equation, the deformation complex (\[equation56\]) is Fredholm. That means, the image of $d{{\mathpzc}G}_{{\mathpzc}X}$ is a closed subspace of $T_{{\mathpzc}X}{{\mathpzc}B}$ and has finite codimension in ${\rm ker} \left( d{{\mathpzc}W}_{{\mathpzc}X} \right)$. Moreover, in this case, the Euler characteristic of ${{\mathpzc}C}_{{\mathpzc}X}$ is given by the formula $$\begin{aligned}
\label{equationx58}
\chi ({{\mathpzc}C}_{{\mathpzc}X} ) := (2-2g) {\rm dim}_{{\mathbb}C} X + 2 c_1^G \cdot \big[ {{\mathpzc}X} \big] - b(\vec{{\mathcal}C}) - 2 n(\vec{{\mathcal}C}).\end{aligned}$$ Here $c_1^G$ is the equivariant first Chern class of $T{\widetilde}{X}$, and $[{{\mathpzc}X}]\in H_2^G \big( X; {{\mathbb}Z}[r^{-1}] \big)$ is the homology class of ${{\mathpzc}X}$.
The theorem follows immediately from the following two propositions.
\[prop54\] For any ${{\mathpzc}X} \in {{\mathpzc}B}$, the complex ${{\mathpzc}C}_{{\mathpzc}X}$ is Fredholm if and only if the operator ${{\mathpzc}D}_{{\mathpzc}X}: W_\tau^{1, p}\big( u^* T^\bot Y\big) \to L_\tau^p \big( \Lambda^{0,1} T^* \Sigma \otimes u^* T^\bot Y \big)$ is Fredholm. In that case, $$\begin{aligned}
\chi \big( {{\mathpzc}C}_{{\mathpzc}X} \big) = {\rm ind} \big( {{\mathpzc}D}_{{\mathpzc}X} \big) - 2 (1-g).\end{aligned}$$
\[prop55\] For $\tau\in (0, 1)$ sufficiently small, the operator ${{\mathpzc}D}_X: W_\tau^{1, p}\big( u^* T^\bot Y\big) \to L_\tau^p \big( \Lambda^{0,1} T^* \Sigma \otimes u^* T^\bot Y \big)$ is Fredholm and $$\begin{aligned}
{\rm ind} \big( D_{{\mathpzc}X} \big) = (2- 2g) {\rm dim}_{{\mathbb}C} {\widetilde}{X} + 2 c_1^G \cdot \big[ {{\mathpzc}X} \big] - b(\vec{{\mathcal}C}) - 2n(\vec{{\mathcal}C}) - 2k_n.\end{aligned}$$
Proof of Proposition \[prop54\]
-------------------------------
The following results about Fredholm property of complexes of Banach spaces are standard.
\[lemma56\] Suppose that $\begin{CD}A_1 @> d_1 >> A_2 @ > d_2 >> A_3\end{CD}$ is a complex of Banach spaces, and assume that there exists another Banach space $B$ and an operator $\delta_1: A_2 \to B$ such that the operator $\delta_1 d_1: A_1 \to B$ and $F= (\delta_1, d_2): A_2 \to B \oplus A_3$ are both Fredholm. Then the cohomology of the original complex is finite dimensional, and its Euler characteristic is $$\begin{aligned}
\chi = {\rm ind} (F) - {\rm ind} (\delta_1 d_1).\end{aligned}$$
\[lemma57\] Suppose $D: A_1 \oplus A_2 \to B_1 \oplus B_2$ is a bounded operator, which is written in the matrix form as $$\begin{aligned}
D = \left( \begin{array}{cc} D_1 & \alpha_1 \\
0 & D_2 \end{array}\right)。\end{aligned}$$ If $D_1: A_1 \to B_1$, $D_2: A_2 \to B_2$ are both Fredholm, then $D$ is Fredholm and $$\begin{aligned}
{\rm ind} (D) = {\rm ind} (D_1) + {\rm ind} (D_2).\end{aligned}$$
The operator $(\alpha, \xi, {\bm \zeta})\mapsto \sigma d\mu^* (u) V(\alpha, \xi, {\bm \zeta})$ inside (\[equation55\]) is a compact operator because it is of zero-th order and $\sigma$ converges to zero at punctures. Therefore it can be ignored when considering Fredholm properties. So we denote by $$\begin{aligned}
{\widetilde}{{\mathpzc}D}_{{\mathpzc}X}: W_\tau^{1, p} \big( T^* \Sigma \otimes {{\mathfrak}g} \big) \oplus W_\tau^{1, p} \big( u^* T^\bot Y \big) \to L_\tau^p \big( \Lambda^{0,1}T^* \Sigma^* \otimes u^* T^\bot Y \big) \oplus L_\tau^p ( {{\mathfrak}g})\end{aligned}$$ the operator defined by $$\begin{aligned}
{\widetilde}{{\mathpzc}D}_{{\mathpzc}X}\big( \alpha, \xi, {\bm \zeta} \big) = \big( {{\mathpzc}D}_{{\mathpzc}X} (V(\alpha, \xi, {\bm \zeta})) + \delta_\alpha \nabla {\widetilde}{{\mathcal}W}_A(u) ,\ *_c d \alpha \big).\end{aligned}$$ Then since $G$ is abelian, the modified sequence $$\begin{aligned}
{{\mathpzc}C}_{{\mathpzc}X}': \begin{CD} A_1 @> d{{\mathpzc}G}_{{\mathpzc}X} >> A_2 @> {\widetilde}{{\mathpzc}D}_{{\mathpzc}X} >> A_3\end{CD} \end{aligned}$$ is still a chain complex. It has the same Euler characteristic as ${{\mathpzc}C}_{{\mathpzc}X}$ when one of them is Fredholm.
Now we define $$\begin{aligned}
\begin{array}{cccc}
\delta_1 : & A_2 &\to & L_\tau^p({{\mathfrak}g})\\
& \big( \alpha , \xi, {\bm \zeta}\big) & \mapsto & - *_c d *_c \alpha.
\end{array}\end{aligned}$$ Then for $h \in A_1$, we have $\delta_1 d{{\mathpzc}G}_{{\mathpzc}X}(h) = \Delta h$, where $\Delta: A_1 \to L_\tau^p( {{\mathfrak}g})$ is the positive-definite Laplacian with respect to the cylindrical metric. Then by Lemma \[lemma56\], we see that ${{\mathpzc}C}_{{\mathpzc}X}'$ is Fredholm if and only if both $\Delta$ and ${{\mathpzc}I} = (\delta_1, d{{\mathpzc}W}_{{\mathpzc}X}): A_2\to A_3 \oplus L_\tau^p( {{\mathfrak}g})$ are Fredholm operators. Indeed, since $\tau \in (0, 1)$, $\Delta$ is Fredholm and $$\begin{aligned}
{\rm ind} \Delta = -k {\rm dim}G = - \# \{{\rm punctures}\}\cdot {\rm dim} G = -2k.\end{aligned}$$ Therefore, if $\delta \in (0, 1)$ and ${{\mathpzc}I}$ is Fredholm, then by Lemma \[lemma56\], we have $$\begin{aligned}
\label{equation59}
\chi\big( {{\mathpzc}C}_{{\mathpzc}X}' \big) = {\rm ind} \big({{\mathpzc}I} \big) - {\rm ind} \Delta = {\rm ind} \big( {{\mathpzc}I} \big) + 2k.\end{aligned}$$
Now we look at the operator ${{\mathpzc}I}$, which is $$\begin{aligned}
{{\mathpzc}I} \left( \begin{array}{c}
\alpha \\ \xi \\ {\bm \zeta}
\end{array}\right) \mapsto \left( \begin{array}{c} {\widetilde}{{\mathpzc}D}_{{\mathpzc}X} (\alpha, \xi, {\bm \zeta}) \\ - *_c d *_c a
\end{array} \right)= \left( \begin{array}{c} {{\mathpzc}D}_{{\mathpzc}X} (V(\alpha, \xi, {\bm \zeta})) + \delta_\alpha \nabla {\widetilde}{{\mathcal}W}_A(u) \\ *_c d \alpha \\ - *_c d *_c \alpha
\end{array} \right).\end{aligned}$$ We claim that when $\tau \in (0,1)$, the operator $\alpha \mapsto ( *_c d\alpha, \ - *_c d *_c \alpha)$ is Fredholm and has index equal to $-2k - 2(1-g)$. By Lemma \[lemma57\] and (\[equation59\]), the proposition follows from this claim.
To prove the claim, notice that the ${{\mathfrak}g}_0$-component of $\alpha$, denoted by $\alpha_0$, is mapped by $$\begin{aligned}
\alpha_0 = *_c d h + d f \mapsto ( *_c d *_c dh,\ - *_c d *_c df ) = ( - \Delta h, \Delta f).\end{aligned}$$ It is Fredholm and has index $-2k$. On the other hand, for the ${{\mathfrak}g}_1$ component of $\alpha$, denoted by $\alpha_1$, we define ${{\mathbb}R}$-linear isomorphisms $\iota_1: \Lambda^{0,1} T^* \Sigma^* \otimes_{{{\mathbb}C}} {{\mathfrak}g}_1^{{\mathbb}C} \to T^*\Sigma^* \otimes_{{\mathbb}R} {{\mathfrak}g}_1$ by $b\mapsto (b + {\overline}{b})$ and $\iota_2: {{\mathfrak}g}_1 \oplus {{\mathfrak}g}_1 \to {{\mathfrak}g}_1^{{\mathbb}C}$ by $\iota(a_1, a_2) = a_1 + {\bm i} a_2$. Then we have $$\begin{aligned}
\begin{split}
\iota_2 \big( *_c d, - *_c d *_c \big) \iota_1 \theta = & *_c d( \theta + {\overline}\theta) - {\bm i} *_c d *_c ( \theta + {\overline}\theta )\\
= & *_c (\partial \theta + {\overline}\partial {\overline}\theta ) + {\bm i} ( {\overline}\partial^* \theta + \partial^* {\overline}\theta )\\
= &\ {\bm i} {\overline}\partial^* \theta - {\bm i} \partial^* {\overline}\theta + {\bm i} {\overline}\partial^* \theta + {\bm i} \partial^* {\overline}\theta \\
= &\ 2 {\bm i} {\overline}\partial^* \theta.
\end{split}\end{aligned}$$ Here $\partial^*$ and ${\overline}\partial^*$ are the adjoint of $\partial$ and ${\overline}\partial^*$ with respect to the cylindrical metric, respectively; the third equality follows from the Kähler identities on $\Sigma^*$. Therefore we see that the operator $\alpha_1 \mapsto (*_c d\alpha_1, -*_c d *_c \alpha_1)$ is Fredholm if and only if the operator $$\begin{aligned}
{\overline}\partial^*: W_\tau^{1, p} \big( \Lambda^{0,1} T^* \Sigma^* \big) \to L_\tau^p\otimes {{\mathbb}C}\end{aligned}$$ is Fredholm. When $\tau \in (0, 1)$, it is the case and $$\begin{aligned}
{\rm ind}_{{\mathbb}R} \big( {\overline}\partial^* \big) = -2 (1-g).\end{aligned}$$
Proof of Proposition \[prop55\]
-------------------------------
The proof of Proposition \[prop55\] is a generalization of the computation of Fredholm indices in [@Mundet_Tian_Draft] and [@FJR3 Section 5.1].
### Riemann-Roch for orbifold line bundles {#riemann-roch-for-orbifold-line-bundles .unnumbered}
We consider a smooth Hermitian line bundle $L \to \Sigma^*$ together with a meromorphic unitary connection $A$. Suppose for each marked point $z_j$, over the cylindrical ends $U_j\simeq \Theta_+$, we choose a unitary trivialization $\xi_j: U_j \times {{\mathbb}C} \to L|_{U_j}$ so that the connection form is $$\begin{aligned}
A = d + \alpha + \lambda_j dt\end{aligned}$$ where $\alpha \in \Omega^1(\Theta_+, {\bm i} {{\mathbb}R})$ extends to a continuous 1-form over the marked point and $\lambda_j \in {\bm i} {{\mathbb}R}$ (the residue) is a constant. $\lambda_j$ only depends on the homotopy class of the local trivialization $\xi_j$, and for different trivializations, the residues differ by an integer multiple of ${\bm i}$. $\exp (2\pi \lambda_j) \in U(1)$ is called the monodromy of the connection.
We assume that for every $z_j$, $\lambda_j \in {\bm i} {{\mathbb}Z}/ r$. Then we can define an “orbifold completion” ${{\mathcal}L} \to {{\mathcal}C}$ of $L \to \Sigma^*$, where ${{\mathcal}C}$ is an orbicurve obtained by adding orbifold charts near $z_j$ to $\Sigma^*$, and ${{\mathcal}L}$ is an orbifold line bundle. The orbifold degree of ${{\mathcal}L}$ is defined as follows. The trivializations ${\bm \xi}= (\xi_j)_{j=1}^k$ defines a smooth line bundle $L({\bm \xi}) \to \Sigma$. We define $$\begin{aligned}
{\rm deg}^{orb} {{\mathcal}L} = {\rm deg} L({\bm \xi}) - {\bm i} \sum_{j=1}^k \lambda_j \in {{\mathbb}Z}/r.\end{aligned}$$ We also define $$\begin{aligned}
\lfloor {{\mathcal}L} \rfloor = {\rm deg} L({\bm \xi}) + \sum_{j=1}^k \lfloor -{\bm i} \lambda_j \rfloor \in {{\mathbb}Z}.\end{aligned}$$ Both ${\rm deg}^{orb} {{\mathcal}L}$ and $\lfloor {{\mathcal}L} \rfloor$ are independent of the choice of ${\bm \xi}$.
Consider a class of real linear Cauchy-Riemann operators $$\begin{aligned}
D: \Omega^0 (L) \to \Omega^{0,1}(L).\end{aligned}$$ Their Fredholm properties essentially only depends on their behavior near the punctures.
\[defn58\] Let $L \to \Theta_+$ be a Hermitian line bundle and $D: \Omega^0(\Theta_+, L) \to \Omega^{0,1}(\Theta_+, L)$ is a real linear, first-order differential operator. $D$ is called [**admissible**]{} if the following conditions are satisfied
1. $D - {\overline}\partial_A$ is a zero-th order operator for some meromorphic unitary connection $A$ on $L$.
2. If the monodromy of $A$ at the infinity of $\Theta_+$ is not 1, then $D = {\overline}\partial_A$. In this case we say that $D$ is of type I (at the puncture at infinity).
3. If the monodromy of $A$ at the infinity of $\Theta_+$ is 1, then there exists a trivialization $\xi: \Theta_+ \times {{\mathbb}C} \to L$ such that with respect to this trivialization, either $Df = {\overline}\partial f + \tau {\overline}{f}$ for some $\tau >0$, or $Df = {\overline}\partial f$. In the first case we say that $D$ is of type ${\rm II}_1$ and in the second case we say that $D$ is of type ${\rm II}_2$.
If $L\to \Sigma^*$ is a Hermitian line bundle and $D: \Omega^0(\Sigma^*, L) \to \Omega^{0,1}(\Sigma^*, L)$ is a real linear first-order differential operator, then we say that $D$ is admissible if its restriction to each cylindrical end $U_j \simeq \Theta_+$ is admissible in the above sense. If the restriction of $D$ to $U_j$ is of one of the three types defined above, we say that $z_j$ is a puncture of that type. We define $b({{\mathcal}L}, D) \in {{\mathbb}Z}$ be the number of type ${\rm II}_1$ punctures plus twice of the number of type ${\rm II}_2$ punctures.
We have the following index formula
\[prop57\] Suppose $D: \Omega^0(\Sigma^*, L) \to \Omega^{0,1}(\Sigma^*, L)$ is admissible. Then there exists $\tau_0>0$ such that for $\tau \in (0, \tau_0)$, the operator $D$ defines a Fredholm operator $$\begin{aligned}
D: W_\tau^{1, p} (L ) \to L_\tau^p (\Lambda^{0,1} T^* \Sigma^* \otimes L ).\end{aligned}$$ Moreover, its (real) index is given by $$\begin{aligned}
{\rm ind} ( D ) = 2-2 g- b({{\mathcal}L}, D) + 2 \lfloor {{\mathcal}L} \rfloor. \end{aligned}$$
We can use the index gluing formula (about Cauchy-Riemann operators with totally real boundary conditions, see [@McDuff_Salamon_2004 Appendix C]) to reduce the proof to a simple case. More precisely, we can cut the Riemann surface $\Sigma$ into the union of pair-of-pants, disks and cylinders, glued along common boundaries. Then the index of $D$ is the sum of the indices of Cauchy-Riemann operators $D_i$ on the $i$-th component, with totally real boundary conditions. If the component doesn’t contains an original puncture, then its index formula is known. The only unknown case can be deduced from the case of an operator $D_0$ on the trivial line bundle on the sphere with only one puncture, where the puncture is either of type ${\rm II}_1$ or ${\rm II}_2$ (type I case is well-known).
In such a case $\lfloor {{\mathcal}L} \rfloor = 0$. If the puncture is of type ${\rm II}_1$, then using the cylindrical coordinates near the puncture, $D_0$ can be written as (up to a compact operator) $$\begin{aligned}
D_0 = {1\over 2} {\partial \over \partial s} + {1\over 2} {\bm i} {\partial\over \partial t} + \left( \begin{array}{cc} \tau & 0 \\ 0 & -\tau \end{array} \right).\end{aligned}$$ If we denote $S = \left( \begin{array}{cc} \tau & 0 \\ 0 & -\tau \end{array}\right)$, then the symplectic path $\left\{ e^{{\bm i} S t}\right\}_{t\geq 0}$ has eigenvalues $e^{\tau t}$ and $e^{-\tau t}$ which are not on the unit circle for $t >0$. Therefore, the Conley-Zehnder index of this path is zero. By the index formula for Cauchy-Riemann operators of this type, for $\delta_0>0$ small enough, $D_0$ is Fredholm and $$\begin{aligned}
{\rm ind} (D_0) = 1 = 2 - 2g(S^2) - 1 = 2 - 2g - b({{\mathcal}L}, D_0).\end{aligned}$$
If the puncture is of type ${\rm II}_2$, then $D_0$ is the same as a complex Cauchy-Riemann operator (up to a compact operator) with one point constrain. Therefore $$\begin{aligned}
{\rm ind} (D_0) = 2 ( 1- g(S^2) ) - 2 = 2 - 2 g(S^2) - b({{\mathcal}L}, D_0).\end{aligned}$$
### A splitting of $u^* T^\bot Y$ {#a-splitting-of-u-tbot-y .unnumbered}
For the fixed solution ${{\mathpzc}X}= (A, u)\in {{\mathpzc}B}$, denote $E := u^* T^\bot Y \to \Sigma^*$. Remember that the principal $G$-bundle extends to an orbifold $G$-bundle ${{\mathcal}P} \to {{\mathcal}C}$. Moreover, the section $u$ extends to an orbifold section ${{\mathcal}U}: {{\mathcal}C} \to {{\mathcal}Y}$. Similarly, we can show that $E$ extends to an orbifold vector bundle ${{\mathcal}E}\to {{\mathcal}C}$.
Now we consider the linearization ${{\mathpzc}D}_{{\mathpzc}X}$. The idea of computing ${\rm ind}\left( {{\mathpzc}D}_{{\mathpzc}X}\right)$ is that near each puncture, we can split $E$ as direct sums of line bundles, and, up to compact operators, the restriction of ${{\mathpzc}D}_{{\mathpzc}X}$ to each cylindrical end is the direct sums of admissible operators. Moreover, we can extend the splittings over $\Sigma^*$, i.e., we have a decomposition $$\begin{aligned}
E = \bigoplus_{i=1}^n L^{(i)}.\end{aligned}$$ Then we can show that, on each $L^{(i)}$, there is an operator $D^{(i)}$ which is an admissible Cauchy-Riemann operator on $L^{(i)}$ such that ${{\mathpzc}D}_{{\mathpzc}X}- \displaystyle \oplus_{i=1}^n D^{(i)}$ is compact. We carry out this idea in the following steps. Similar procedures are used in [@Mundet_Tian_Draft].
[**Step 1.**]{} First we examine the operator ${{\mathpzc}D}_{{\mathpzc}X}$ around each puncture $z_j$, with monodromy $\upgamma_j \in {{\mathbb}Z}_r$. With respect to the trivialization $\phi_j$, $u$ is identified with a map $u_j: \Theta_+ \to X$ and the connection is identified with a 1-form $\phi ds + \psi dt + \lambda_j dt$ for $\phi, \psi: \Theta_+ \to {\mathfrak}g$. Since $\displaystyle \lim_{s \to +\infty} u_j(s, t) = v_j(t):=e^{ - \lambda_j t} \upkappa_j$. For the purpose of studying Fredholm properties of ${{\mathpzc}D}_{{\mathpzc}X}$, we can deform ${{\mathpzc}X} = (A, u)$ such that over $\Theta_+$, $u_j(s, t) = e^{ - \lambda_j t} \upkappa_j$, and $A = d + \lambda_j dt$. Then, after this modification, we have $$\begin{aligned}
{{\mathpzc}D}_{{\mathpzc}X} \xi = {\overline}\partial \xi + {1\over 2} \nabla_\xi {{\mathcal}X}_{\lambda_j}(u_j) + \sum_{l=0}^s e^{- \rho_l(\lambda_j t)}\nabla_\xi \nabla F_l^{(\delta_{j, A})}(u_j).\end{aligned}$$ Denote $W_j = \sum_{l=0}^s F_{j; l}^{(\delta_{j, A})}$.
[**Step 2.**]{} Now we see that on $U_j$ we have an $S^1$-equivariant splitting $v_j^* T {\widetilde}{X} \simeq v_j^* T {\widetilde}{X}_j \oplus v_j^* {\widetilde}{N}_j$. Moreover, since $dF_l$ vanishes along the normal bundle ${\widetilde}{N}_j$, the operator ${{\mathpzc}D}_{{\mathpzc}X}$ splits over $U_j$ as the direct sum of two operators $$\begin{aligned}
\begin{array}{cccc}
{{\mathpzc}D}_j^T: & \Gamma \big( \Theta_+, v_j^* T{\widetilde}{X}_j \big) &\to & \Omega^{0,1} \big( \Theta_+, v_j^* T{\widetilde}{X}_j \big),\\
{{\mathpzc}D}_j^N: & \Gamma \big( \Theta_+, v_j^* {\widetilde}{N}_j \big) & \to & \Omega^{0,1} \big( \Theta_+, v_j^* {\widetilde}{N}_j \big).
\end{array}\end{aligned}$$
[**Step 3.**]{} We consider the tangential part ${{\mathpzc}D}_j^T$. If $z_j$ is narrow, then $W_j|_{{\widetilde}{X}_j} \equiv 0$. In this case ${{\mathpzc}D}_j^T$ is the same as a usual homogeneous Cauchy-Riemann operator. We trivialize $v_j^* T{\widetilde}{X}_j$ over $\Theta_+$ so that we can write $$\begin{aligned}
v_j^* T{\widetilde}{X}_j \simeq \bigoplus_{\nu=1}^{b_j} L^{(\nu)}\end{aligned}$$ and the restriction of ${{\mathpzc}D}_j^T$ to $U_j$ is the direct sum of $D_j^{(\nu)}: \Omega^0( U_j, L^{(\nu)}) \to \Omega^{0,1}(U_j, L^{(\nu)})$. Here each $D_j^{(\nu)}$ is of type ${\rm II}_2$ in the sense of Definition \[defn58\].
If $z_j$ is broad, then $W_j|_{{\widetilde}{X}_j}$ is a holomorphic Morse function. The Hessian of ${\widetilde}{W}_j$ at $\upkappa_{j, A}$ is a real quadratic form $H_j$ on $T_{\upkappa_j} {\widetilde}{X}_j$ satisfying $H_j( \cdot, \cdot) = -H_j(J \cdot, J \cdot)$. Then we have decomposition of $T_{\upkappa_j} {\widetilde}{X}_j$ into complex lines $$\begin{aligned}
T_{\upkappa_j} {\widetilde}{X}_j \simeq \bigoplus_{\nu =1}^{b_j} Z^{(\nu)}\end{aligned}$$ with respect to which the Hessian is diagonalized. On each $Z^{\nu}$, $H_j$ has eigenvalues $\pm b_\nu$ for some $b_\nu > 0$. The path of diffeomorphisms $e^{\lambda_j t}$ induces a trivialization of $v_j^* T{\widetilde}{X}_j$ along $S^1$. Therefore we have a trivialization $U_j \times T_{\upkappa_j} {\widetilde}{X}_j \to v_j^* T{\widetilde}{X}_j$, which is well-defined since ${\widetilde}{X}_j$ is fixed by $\upgamma_j$. With respect to this trivialization, ${{\mathpzc}D}_j^T$ splits as the direct sum of operators $$\begin{aligned}
D_j^{(\nu)}: \Omega^0 \big( \Theta_+, L_j^{(\nu)} \big) \to \Omega^{0,1} \big( \Theta_+, L_j^{(\nu)} \big),\ \nu =1, \ldots, b_j.\end{aligned}$$ Each $D_j^{(\nu)}$ is of type ${\rm II}_1$ in the sense of Definition \[defn58\].
[**Step 4.**]{} Now we consider the normal component ${{\mathpzc}D}_j^N$. By ([**P2**]{}) of Hypothesis \[hyp25\] and ([**Q2**]{}) of Hypothesis \[hyp28\], the Hessian of $W_j$ vanishes in the normal direction. Therefore, $$\begin{aligned}
{{\mathpzc}D}_j^N \xi = {\overline}\partial \xi + {1\over 2} \nabla_\xi {{\mathcal}X}_{\lambda_j}(u_j).\end{aligned}$$ On the other hand, we have the splitting of normal bundles $$\begin{aligned}
{\widetilde}{N}_j \simeq \bigoplus_{i=b_j+1}^n {\widetilde}{N}^{(i)}_j,\end{aligned}$$ where each ${\widetilde}{N}^{(i)}$ is an $S^1$-equivariant line bundle over ${\widetilde}{X}_j$. If we denote $L^{(i)}_j= v_j^* {\widetilde}{N}^{(i)}_j$, then ${{\mathpzc}D}_j^N$ splits as the direct sum of Cauchy-Riemann operators $D_j^{(i)}: \Omega^0(\Theta_+, L^{(i)}_j)\to \Omega^{0,1}(\Theta_+, L^{(i)}_j)$. Each $D_j^{(i)}$ is of type I in the sense of Definition \[defn58\].
[**Step 5.**]{} So far, for each cylindrical end, we have constructed a splitting $$\begin{aligned}
\label{equation510}
E|_{U_j} = v_j^* T {\widetilde}{X} \simeq \bigoplus_{i=1}^n L^{(i)}_j\end{aligned}$$ and differential operators $$\begin{aligned}
{{\mathpzc}D}^{(i)}_j : \Omega^0 \big( \Theta_+, L^{(i)}_j \big) \to \Omega^{0,1} \big( \Theta_+, L^{(i)}_j \big)\end{aligned}$$ such that ${{\mathpzc}D}_{{\mathpzc}X} - \bigoplus_{i=1}^n {{\mathpzc}D}^{(i)}_j$ is a compact operator. We claim that the union of the splittings over $\cup_{j=1}^k U_j$ can be extended to whole $\Sigma^*$.
Indeed, over $\Sigma^* \setminus \cup_{j=1}^k U_j$ the bundle $E$ is trivial. Choosing a trivialization, the splitting (\[equation510\]) induces a smooth map from $\partial \left( \Sigma^* \setminus \cup_{j=1}^l U_j \right)$ to the flag manifold ${\rm Flag}({{\mathbb}C}^n)$. Since ${\rm Flag}({{\mathbb}C}^n)$ is simply-connected, this map can be smoothly extended to $\Sigma^* \setminus \cup_{j=1}^l U_j$, which means we extend the splitting (\[equation510\]) to the interior.
Then we obtained a splitting of $E$ as direct sum of line bundles $L^{(i)} \to \Sigma^*$ for $i = 1, \ldots, n$. The differential operators ${{\mathpzc}D}^{(i)}_j$ on $L^{(i)}|_{U_j} = L_j^{(i)}$ can be extended smoothly to ${{\mathpzc}D}^{(i)}: \Omega^0 (L^{(i)}) \to \Omega^{0,1}(L^{(i)})$, while the ambiguities of the extensions are compact operators. By our construction in previous steps, ${{\mathpzc}D}^{(i)}$ is admissible in the sense of Definition \[defn58\]. Apply Proposition \[prop57\] to each ${{\mathpzc}D}^{(i)}$, we see that there exists $\delta_0>0$ such that for all $\delta \in (0, \delta)$, each ${{\mathpzc}D}^{(i)}$ induces a Fredholm operator $$\begin{aligned}
{{\mathpzc}D}^{(i)}: W_\tau^{1, p} \big( L^{(i)} \big) \to L_\tau^p \big( \Lambda^{0,1}\otimes L^{(i)} \big).\end{aligned}$$ Moreover, each $L^{(i)}$ extends to an orbi-bundle ${{\mathcal}L}^{(i)} \to {{\mathcal}C}$ and $$\begin{aligned}
\label{equation511}
\begin{split}
{\rm ind} \big( {{\mathpzc}D}_{{\mathpzc}X} \big) = &\ \sum_{i=1}^n {\rm ind} \big( {{\mathpzc}D}^{(i)} \big) \\
= &\ \sum_{i=1}^n \big( 2- 2g - b ( {{\mathcal}L}^{(i)}, {{\mathpzc}D}^{(i)} ) + 2 \big\lfloor {{\mathcal}L}^{(i)} \big\rfloor \big) \\
= &\ (2- 2g) {\rm dim}_{{\mathbb}C} {\widetilde}{X} - b(\vec{{\mathcal}C}) + 2 \sum_{i=1}^n \big\lfloor {{\mathcal}L}^{(i)} \big\rfloor.
\end{split}\end{aligned}$$ Proposition \[prop55\] follows by noticing that the sum of all $\lfloor {{\mathcal}L}^{(i)} \rfloor$ is equal to $$\begin{aligned}
c_1^G \cdot \big[ {{\mathpzc}X} \big] - \sum_{z_j\ {\rm narrow}} \big( n_j(\vec{{\mathcal}C}) + 1 \big).\end{aligned}$$
Stable solutions and the compactness theorem {#section6}
============================================
From this section on we start to consider the compactification of the moduli space of the perturbed gauged Witten equation.
Solitons
--------
Let $\delta \in (0, 1]$, $\lambda \in {\bm i}[0, 1) \cap ({\bm i} {{\mathbb}Z}/r)$ and $\upgamma = \exp (2\pi \lambda)$. We have the function ${\widetilde}{W}_\upgamma^{(\delta)}: {\widetilde}{X} \to {{\mathbb}C}$ introduced in (\[equation25\]) and ${\widetilde}{W}_{\lambda}^{(\delta)}: \Theta \times {\widetilde}{X} \to {{\mathbb}C}$ given by $$\begin{aligned}
{\widetilde}{W}_{\lambda}^{(\delta)} (s, t, x) = \sum_{l=0}^s e^{\rho_l(\lambda t)} F_{\upgamma; l}^{(\delta)} (x).\end{aligned}$$ Consider the equation for a map $u: \Theta \to {\widetilde}{X}$ $$\begin{aligned}
\label{equation62}
\partial_s u + J \big( \partial_t u + {{\mathcal}X}_\lambda(u) \big) + 2 \nabla {\widetilde}{W}_{\lambda}^{(\delta)} (u) = 0.\end{aligned}$$ The energy of a solution $u$ is defined as $$\begin{aligned}
E(u) = {1\over 2} \big\| \partial_s u \big\|_{L^2(\Theta)}^2 + {1\over 2} \big\| \partial_t u + {{\mathcal}X}_{\lambda}(u) \big\|_{L^2(\Theta)}^2 + \big\| \nabla {\widetilde}{W}_\lambda^{(\delta)} (u) \big\|_{L^2(\Theta)}^2.\end{aligned}$$ A solution $u$ to (\[equation62\]) whose energy is finite and whose image has compact closure is called a $(\lambda, \delta)$-soliton, or simply a soliton. A soliton having nonzero energy is called nontrivial, otherwise it is called trivial.
We see that the data $(\lambda, \delta)$ naturally gives a $\pm\lambda$-cylindrical model of the perturbed gauged Witten equation with parameters $(\sigma = 0, \delta)$ (on the positive part $\Theta_+$ and the negative part $\Theta_-$, respectively). If $u$ is a soliton, then the restriction of $(u, 0)$ to $\Theta_\pm$ is a bounded solution to the corresponding cylindrical model. Then by Theorem \[thm42\], for any $(\lambda, \delta)$-soliton, there exist $\upkappa_\pm \in {\widetilde}{X}_\upgamma$ such that $$\begin{aligned}
\lim_{s\to \pm \infty} e^{\lambda t} u(s, t) = \upkappa_\pm.\end{aligned}$$ We define the [**evaluation**]{} of the soliton by $u_\pm = \upkappa_\pm$.
\[lemma62\] If $\upgamma = \exp(2\pi \lambda)$ is narrow, then every $(\lambda, \delta)$-soliton is trivial.
Abbreviate ${\widetilde}{W}_\lambda = {\widetilde}{W}_{\lambda}^{(\delta)}$ and ${\widetilde}{W} = {\widetilde}{W}_\upgamma^{(\delta)}$. Define $v: \Theta \to X$ by $v(s, t) = e^{r \lambda t} u( rs, rt)$. Then $$\begin{aligned}
\begin{split}
\partial_s v(s, t) + J \partial_t v (s, t) = &\ m ( e^{r \lambda t} )_* \big( \partial_s u (rs, rt) + J (\partial_t u (rs, rt) + {{\mathcal}X}_\lambda(u(rs, rt)) ) \big) \\
= &\ -2 m (e^{r\lambda t})_* \nabla {\widetilde}{W}_\lambda(rs, rt, u(rs, rt)) = -2m \nabla {\widetilde}{W}( v(s, t)) .
\end{split}\end{aligned}$$ Moreover, $v(s, \cdot)$ converges uniformly to $\upkappa_\pm$ as $s\to \pm\infty$. Therefore $$\begin{aligned}
\label{equation63}
\begin{split}
2 m \big\| \nabla {\widetilde}{W} (v) \big\|_{L^2(\Theta)}^2 = &\ - \int_\Theta \big\langle \partial_s v + J \big( \partial_t v \big) , \nabla {\widetilde}{W} (v) \big\rangle dsdt\\
= &\ - \int_\Theta d {\widetilde}{W} \cdot \big( \partial_s v + J \big( \partial_t v \big) \big) ds dt \\
= &\ -2 \int_\Theta {\partial {\widetilde}{W} (v) \over \partial {\overline}{z}} ds dt \\
= &\ 2\pi \big( {\widetilde}{W}(\upkappa_-) - {\widetilde}{W} (\upkappa_+) \big).
\end{split}\end{aligned}$$ Since $\upgamma$ is narrow, ${\widetilde}{W} (\upkappa_-) = {\widetilde}{W}(\upkappa_+) = 0$. Then $v$ is holomorphic. Since $v$ is a multiple cover of $u$, $v$ has finite energy. By removal of singularity, $v$ extends to a holomorphic sphere. By ([**X1**]{}) of Hypothesis \[hyp21\], $({\widetilde}{X}, \omega)$ is aspherical, $v \equiv \upkappa_\pm$ is a constant. So $u$ has zero energy.
On the other hand, if $\upgamma$ is broad, then similar to (\[equation63\]), $$\begin{aligned}
\big\| \nabla {\widetilde}{W}_{\lambda}^{(\delta)} (u) \big\|_{L^2(\Theta)}^2 = 2\pi \big( {\widetilde}{W}_\upgamma^{(\delta)}(\upkappa_-) - {\widetilde}{W}_\upgamma^{(\delta)} (\upkappa_+) \big).\end{aligned}$$ In particular, this implies that ${\rm Im} {\widetilde}{W}_\upgamma^{(\delta)} (\upkappa_-) = {\rm Im}{\widetilde}{W}_\upgamma^{(\delta)} (\upkappa_+)$.
A stable $(\lambda, \delta)$-soliton is a finite sequence $$\begin{aligned}
{\bm u} = ( u_1, \ldots, u_\nu )\end{aligned}$$ where for each $\alpha = 1, \ldots, \nu$, $u_\alpha$ is a [*nontrivial*]{} $(\lambda, \delta)$-soliton such that $$\begin{aligned}
( u_\alpha )_+ = ( u_{\alpha+1})_- \in {\widetilde}{X}_{\upgamma},\ \alpha = 1, \ldots, \nu-1.\end{aligned}$$
Stable solutions and convergence
--------------------------------
Let $\vec{{\mathcal}C}$ be a rigidified $r$-spin curve with punctures $z_1, \ldots, z_k$.
A [**stable solution**]{} to the perturbed gauged Witten equation over $\vec{{\mathcal}C}$ is a triple $$\begin{aligned}
\big( A, u, \{ {\bm u}_{j}\}_{z_j\ {\rm broad}} \big)\end{aligned}$$ where
1. $(A, u)$ is a bounded solution to the perturbed gauged Witten equation on $\vec{{{\mathcal}C}}$.
2. For each broad puncture $z_j$, ${\bm u}_j = ( u_{j;1}, \ldots, u_{j; \nu_j})$ is a stable $(\lambda_j, \delta_j)$-soliton, where $\lambda_j$ is the residue of the $r$-spin structure at $z_j$ and $\delta_j = \delta_{j, A} \in (0, 1]$ introduced in (\[equation220\]).
3. If $\nu_j \geq 1$, then $$\begin{aligned}
ev_j (A, u) = ( u_{j; 1} )_- \in {\rm Crit} \Big( {\widetilde}{W}^{(\delta_j)}_{\upgamma_j} |_{{\widetilde}{X}_{\upgamma_j}} \Big).\end{aligned}$$
Now we can define the topology in the space of stable solutions. At the “tails”, the convergence of the stable solitons are just an $A$-parametrized version of convergence of stable solutions to the corresponding Floer type equation (\[equation62\]), because in $\nabla {\widetilde}{W}_{\lambda_i}^{(\delta_j)}$, the parameter $\delta_j$ depends on $A$. Therefore it suffices to define the convergence of a sequence of usual solutions over $\vec{{\mathcal}C}$ to a stable solution. The definition in the rest of cases can be easily written down and we omit it.
\[defn64\] Suppose $( A^{(i)}, u^{(i)} )$ is a sequence of solutions to the gauged Witten equation over a fixed rigidified $r$-spin curve $\vec{{{\mathcal}C}}$ with underlying punctured Riemann surface $\Sigma^*$. We say that the sequence converges to a stable solution $( (A, u), \{{\bm u}_j\}_{z_j\ {\rm broad}} )$ if the following conditions are satisfied.
1. $( A^{(i)}, u^{(i)} )$ converges to $(A, u)$ in $W^{1, p}_{loc}$-topology.
2. For each broad puncture $z_j$, If ${\bm u}_j = (u_{j;1}, \ldots, u_{j; \nu_j} )$ and $\nu_j \geq 1$, then the following conditions are satisfied.
- For $\alpha = 1, \ldots, \nu_j$, there are sequences $s_{\alpha}^{(i)}>0$ such that $$\begin{aligned}
\lim_{i \to +\infty} s_{\alpha}^{(i)} = +\infty,\ \alpha> \alpha' \Longrightarrow \lim_{i \to +\infty} s_{\alpha}^{(i)} - s_{\alpha'}^{(i)}= +\infty.\end{aligned}$$
- Let $(u^{(i)}, h^{(i)})$ be the sequence of solutions to the cylindrical model obtained by restricting $(A^{(i)}, u^{(i)})$ to $U_j(2)$. Then for each $\alpha$, the sequence $u^{(i)} ( s_{\alpha}^{(i)} + \cdot, \cdot )$ converges to $u_{j;\alpha}$ uniformly on any compact subset of $\Theta$.
- We have $$\begin{aligned}
\lim_{s \to +\infty} \limsup_{i \to \infty} E \big( A^{(i)}, u^{(i)}; U_j (s_{\nu_j}^{(i)} + s) \big) = 0.\end{aligned}$$
3. For each broad puncture $z_j$, if $\nu_j = 0$, then $$\begin{aligned}
\lim_{s \to +\infty} \limsup_{i \to \infty} E \big( A^{(i)}, u^{(i)}; U_j (s) \big) = 0.\end{aligned}$$
Now we state the compactness theorem.
\[thm65\] If $(A^{(i)}, u^{(i)} ) \in {{\mathpzc}A} \times \Gamma(Y)$ is a sequence of smooth bounded solutions to the gauged Witten equation (\[equation221\]) with $$\begin{aligned}
\sup_i E ( A^{(i)}, u^{(i)} ) < \infty,\end{aligned}$$ then there is a subsequence (still indexed by $i$), a stable solution $( (A, u), \{{\bm u}_j\}_{z_j\ {\rm broad}})$, and a sequence of smooth gauge transformations $g^{(i)}\in {{\mathpzc}G}$ such that $( g^{(i)} )^* ( A^{(i)}, u^{(i)} )$ converges to $( (A, u), \{{\bm u}_j\}_{z_j\ {\rm broad}} )$ in the sense of Definition \[defn64\].
Energy quantization in blowing up {#section7}
=================================
Now we start to prove the compactness theorem of the moduli space of gauged Witten equation. The first main concern is about the uniform $C^0$-bound on the solutions, which is the prerequisite of applying all the bubbling analysis.
We first summarize the methods of achieving $C^0$-bound in relevant situations when the target space is noncompact. In gauged Gromov-Witten theory, one can achieve the $C^0$-bound by imposing the equivariant convexity assumption on the target space (see [@Cieliebak_Gaio_Mundet_Salamon_2002 Page 555]). This is an assumption generalizing the convexity condition in [@Eliashberg_Gromov]. In Fan-Jarvis-Ruan’s LG A-model theory, for a quasi-homogeneous superpotential $f$ on ${{\mathbb}C}^N$, the crucial condition for $C^0$-control is a growth estimate of $df$ ([@FJR1 Theorem 5.8]), deduced from the nondegeneracy of the singularity.
In this paper we also imposed a convexity condition at infinity (([**X4**]{}) of Hypothesis \[hyp21\]), a more concrete form of the assumption used in [@Cieliebak_Gaio_Mundet_Salamon_2002]. However, since we have to perturb the equation, the solutions are no longer holomorphic and there are error terms in the estimate (see Proposition \[prop82\]). So [*a priori*]{}, there could exist a sequence of bounded solutions which escape to infinity in the limit near broad punctures. One way to overcome this trouble is to establish an energy quantization property for the [*a priori*]{} blow-up of $C^0$-norm (Theorem \[thm71\]). Then a $C^0$-bound follows from a local maximal principle argument.
One of the difficulty in establishing the energy quantization comes from the fact that the inhomogeneous term of the Witten equation is not bounded and not proper. The $\epsilon$-regularity argument only applies in a scale comparable to $\big| \nabla {\widetilde}{{\mathcal}W}_A \big|^{-1}$. Moreover, ${\rm Crit}W$ is the union of two parts, $$\begin{aligned}
{\widetilde}{X}_B:= \big\{ (x, p)\ |\ Q(x) = 0 \big\},\ {\widetilde}{X}_S : = \big\{ (\star, p) \ |\ p \in {{\mathbb}C} \big\}.\end{aligned}$$ If the blow-up happens in the region away from ${\widetilde}{X}_B$ and ${\widetilde}{X}_S$, then it is easy to establish the energy quantization (Proposition \[prop75\]); if the blow-up happens near ${\widetilde}{X}_B$ and ${\widetilde}{X}_S$, then the magnitude of $\nabla {\widetilde}{{\mathcal}W}_A$ can change dramatically and we have to use different arguments (Proposition \[prop74\] and Proposition \[prop76\]).
We remark that one should be able to generalize the results of this section to the case of complete intersections, i.e., the superpotential is of the form $p_1 Q_1 + p_2 Q_2 + \ldots + p_k Q_k$ on a manifold $X \times {{\mathbb}C}^k$, where $Q_i: X \to {{\mathbb}C}$ are homogeneous functions and $p_1, \ldots, p_k$ are the complex variables of the ${{\mathbb}C}^k$-factor.
The main technical result of this section is stated in terms of local models.
\[thm71\] For each $H>0$, there exists $\epsilon_0 = \epsilon_0(H) >0$ satisfying the following condition. Suppose we have a sequence $(\beta_i, \sigma_i, \delta_i)$ of parameters of local models over $B_{r^*}$ and a corresponding sequence of solutions $(u_i, h_i)$. Suppose $$\begin{aligned}
\label{equation71}
\lim_{i \to \infty} \big| \mu(u_i(0)) \big| = +\infty,\ \big\|h_i \big\|_{L^\infty(B_{r^*})} \leq H.\end{aligned}$$ Then there exists a subsequence (still indexed by $i$) such that one of the following conditions holds.
1. We have $\displaystyle \lim_{r \to 0}\lim_{i \to \infty} E (u_i, h_i; B_r) \geq \epsilon_0$.
2. We have $\displaystyle \lim_{i \to \infty} \sigma_i = 0$ (uniformly on $B_{r^*}$) and there exists $r_0 >0$ (depending on the subsequence) such that $$\begin{aligned}
\label{equation72}
\lim_{i \to \infty} \inf_{B_{r_0} } \big| \mu(u_i) \big| = +\infty.\end{aligned}$$
The proof is given in Subsection \[subsection71\]
\[cor72\] For every $E>0$, there exists $\epsilon_E >0$ satisfying the following conditions. Suppose $(A_i, u_i)$ is a sequence of solutions to the perturbed gauged Witten equation over $\vec{{\mathcal}C}$ such that $E (A_i, u_i ) \leq E$. Then there exist a subsequence (still indexed by $i$), and sequences of points $$\begin{aligned}
\{z_i^\alpha\}_{1 \leq \alpha \leq l},\ \{ z_i^{j, \beta}\}_{z_j\ {\rm broad},\ 1\leq \beta \leq l_j}\end{aligned}$$ satisfying the following conditions (here $d$ (resp. ${\widetilde}{d}$) is the distance of the cylindrical metric on $\Sigma^*$ (resp. smooth metric on $\Sigma$)).
1. For each $\alpha = 1, \ldots, l$, $\lim_{i \to \infty} z_i^\alpha = z^\alpha \in \Sigma^*$ and all $z^\alpha$’s are distinct.
2. For each $\alpha = 1, \ldots, l$, we have $$\begin{aligned}
\lim_{i \to \infty} \big| \mu(u_i(z_i^\alpha)) \big| = +\infty,\ \lim_{r \to 0} \lim_{i \to \infty} E ( A_i, u_i; B_r(z_i^\alpha)) \geq \epsilon_E .\end{aligned}$$
3. For each broad puncture $z_j$ and $\beta = 1, \ldots, l_j$, $z_i^{j, \beta} \in U_j$ and
- For each $\beta$, $\displaystyle \lim_{i \to \infty} {\widetilde}{d} \big( z_i^{j, \beta}, z_j\big) = 0$;
- For any $\beta_1 \neq \beta_2$, we have $\displaystyle \liminf_{i \to \infty} d \big( z_i^{j,\beta} , z_i^{j,\beta'} \big) >0$.
- For any $\beta$, $\displaystyle \lim_{i \to \infty} \big| \mu \big(u_i \big( z_i^{j,\beta} \big) \big) \big| = \infty$ and $\displaystyle \lim_{r \to \infty} \lim_{i \to \infty} E \big( A_i, u_i ; B_r(z_i^{j,\beta}) \big) \geq \epsilon_E$.
4. For any sequence $z_i$ of points in $\Sigma^*$, if $\displaystyle \liminf_{i \to \infty} d(z_i, z_i^\alpha) > 0$ for all $\alpha = 1, \ldots, l$, $\displaystyle \liminf_{i \to \infty} d( z_i, z_i^{j, \beta})>0$ for all broad punctures $z_j$ and $\beta = 1, \ldots, l_j$, $\displaystyle \liminf_{i \to \infty} {\widetilde}{d}(z_i, z_j)>0$ for all narrow punctures $z_j$, then $$\begin{aligned}
\limsup_{i \to \infty} \big| \mu(u_i(z_i)) \big| < \infty.\end{aligned}$$
For each $q \in \Sigma^*$, we can restrict $(A_i, u_i)$ to $B_{r^*} (q)$ to obtain a solution $(u_i, h_i)$ to a local model. By the equation $\Delta h_i ds dt = F_{A_i}$, the elliptic estimate and Sobolev inequality, we see that there exists $H(E)>0$ such that for all $i$ and all $q\in \Sigma^*$, $$\begin{aligned}
\label{equation73}
\sup_q \sup_i \big\| h_i \big\|_{B_{r^*}(q)} \leq H(E).\end{aligned}$$ Abbreviate $\epsilon_0 = \epsilon_0(H(E))$.
We construct the sequences $z_i^\alpha$ by an induction argument. We take an exhausting sequence of compact subsets of $\Sigma^*$, denoted by $K^{(l)}$, $l = 1, 2, \ldots$. We consider $$\begin{aligned}
\limsup_{i \to \infty} \big\| \mu (u_i) \big\|_{L^\infty(K^{(l)})}.\end{aligned}$$ If it is finite, then we move on to $K^{(l+1)}$. If it is infinite, then there exist a subsequence (still indexed by $i$) and a sequence of points $q_i \in K^{(l)}$ which converges to some $q \in K^{(l)}$, such that $\displaystyle \lim_{i \to \infty} \big| \mu (u_i(q_i)) \big| = + \infty$. Then we can apply Theorem \[thm71\] to the sequence $(u_i, h_i)$ which is the solution to the local model obtain by restricting $(A_i, u_i)$ to $B_{r^*}(q_i)$. Then there is a subsequence (still indexed by $i$) such that $$\begin{aligned}
\lim_{r \to 0} \lim_{i \to \infty} E ( A_i, u_i; B_r(q_i) ) \geq \epsilon_0.\end{aligned}$$ (Here the second case of Theorem \[thm71\] doesn’t happen because the area form is uniformly bounded from below near $q$.)
Now we replace $\Sigma^*$ by $\Sigma^* \setminus \{q\}$, and retake an exhausting sequence of compact subsets $\{K^{(l)} \}$ of $\Sigma^* \setminus \{q\}$. We restart the induction process. It is easy to see that the induction process stops until we find a finite subset $Z = \{ z_1, \ldots, z_l\}$, a subsequence (still indexed by $i$) and sequences $z_\alpha^i$ for which (1) and (2) are satisfied for $\epsilon_E = \epsilon_0$, because the total energy of $(A_i, u_i)$ is uniformly bounded.
Now we consider the possible blowing up at a broad puncture $z_j$. Take $S>0$ sufficiently large so that $U_j(S)\cap Z = \emptyset$. Therefore for each $K>0$, we have $$\begin{aligned}
\limsup_{i \to \infty} \big\| \mu (u_i) \big\|_{L^\infty([S, S+K]\times S^1)} < \infty.\end{aligned}$$ Now suppose there exist a subsequence (still indexed by $i$) and a sequence of points $z_i= (s_i, t_i)\in U_j(S)$ such that with $$\begin{aligned}
\lim_{i \to \infty} s_i = +\infty,\ \lim_{i \to \infty} \big| \mu (u_i(z_i)) \big| = +\infty.\end{aligned}$$ We claim that $$\begin{aligned}
\label{equation74}
\lim_{r \to 0} \limsup_{i \to \infty} E \left( A_i, u_i; B_r(z_i) \right) \geq \epsilon_0.\end{aligned}$$ Suppose it is not true, then consider the subset $$\begin{aligned}
\Theta^*:= \Big\{ z\in \Theta = {{\mathbb}R} \times S^1 \ |\ \limsup_{i \to \infty} \left| \mu ( z_i + z ) \right| = \infty \Big\}.\end{aligned}$$ Theorem \[thm71\] implies that $\Theta^*$ has nonzero measure. There are two possibilities.
[**(I)**]{} Suppose the boundary of $\Theta^*$ is a finite set, then $\Theta^*$ has infinite area. On the other hand, let $h_i = h_{A_i}: U_j \to {{\mathfrak}g}^{{\mathbb}C}$ be the function defined by (\[equation216\]). In the same way as deriving (\[equation73\]), we may assume that $$\begin{aligned}
\sup_i \big\| h_i \big\|_{L^\infty(U_j)} \leq H(E). \end{aligned}$$ By the definition of $\delta_{j, i} = \delta_{j, A_i}$ (see (\[equation220\])), we have $\inf_i \delta_{j,i} =: \underline\delta > 0$. Then take $$\begin{aligned}
M(E):= \sup_{|h|\leq H(E),\ {\underline}\delta \leq \delta \leq 1} |h - \log \delta |_{{\widetilde}{X}},\end{aligned}$$ which, by Lemma \[lemma23\], is finite. Let $L$ of $\Theta_+$ be a compact subset. Then for $i$ sufficiently large, for any $z \in z_i + L$, $e^{h_i(z)} \delta_{j, i}^{-1} u_i(z) \notin {\widetilde}{K}_{\upgamma_j}$ where ${\widetilde}{K}_{\upgamma_j} \subset {\widetilde}{X}$ is the compact subset in ([**P5**]{}) of Hypothesis \[hyp28\]. Then by the definition of $|h|_{{\widetilde}{X}}$, we have $$\begin{aligned}
\begin{split}
\Big| \nabla {\widetilde}{{\mathcal}W}_{A_i} (u_i (z)) \Big| = &\ \Big| \sum_{l=0}^s e^{{\overline}{\rho_l(h_i(z))}} \nabla F_l^{(\delta_{j,i})}(u_i(z)) \Big|\\
= &\ \Big| ( e^{h_i(z)})^* \Big( d {\widetilde}{W}_{\upgamma_j}^{(\delta_{j, i})} ( e^{h_i(z)} u_i(z))\Big) \Big|\\
= &\ \big(\delta_{j, i}\big)^r \Big| ( e^{h_i(z)} \delta_{j, i}^{-1} )^* \Big( d {\widetilde}{W}_{\upgamma_j} ( e^{h_i(z)} \delta_{j, i}^{-1} u_i(z) ) \Big) \Big| \\
\geq &\ {\underline}\delta^r M(E)^{-1} c_{\upgamma_j}.
\end{split}\end{aligned}$$ Here $c_{\upgamma_j} > 0$ is the one in ([**P5**]{}) of Hypothesis \[hyp28\]. Therefore $$\begin{aligned}
E( A_i, u_i ) \geq E ( A_i, u_i; z_i + L ) \geq \int_{z_i + L} \Big| \nabla {\widetilde}{{\mathcal}W}_{A_i}(u_i) \Big|^2 ds dt \geq {\underline}\delta^r M(E)^{-1} c_{\upgamma_j} {\rm Area}(L).\end{aligned}$$ This contradicts with the energy bound because ${\rm Area} (L)$ can be arbitrarily large.
[**(II)**]{} Suppose the boundary of $\Theta^*$ is an infinite set, then choose an integer $m > E/ \epsilon_0$ and $m$ distinct points $w_1, \ldots, w_m$ of $z \in \partial \Theta^*$. Then by the definition of $\Theta^*$, there exists a subsequence (still indexed by $i$) and sequences of points $w_{l, i}$, $l = 1, \ldots, m$ such that $$\begin{aligned}
\lim_{i \to \infty} w_{l, i} = w_l,\ \lim_{i \to \infty} \big| \mu(u_i( z_i + w_{l, i})) \big| = +\infty.\end{aligned}$$ On the other hand, use the trivialization of the $G$-bundle over $U_j$, the restriction of $(A_i, u_i)$ to the disk $B_{r^*}(z_i + w_{l, i})$ is a sequence of solutions to some sequence of local models over $B_{r^*}$, which satisfy the hypothesis of Theorem \[thm71\]. However, the second implication of Theorem \[thm71\] doesn’t holds because $w_{l, i}$ converges to a point on the boundary of $\Theta^*$. Therefore $$\begin{aligned}
\lim_{r \to 0} \lim_{i \to \infty} E(A_i, u_i; B_r(z_i + w_{l, i})) \geq \epsilon_0.\end{aligned}$$ By the choice of $m$, this contradicts with the energy bound of $(A_i, u_i)$.
Therefore (\[equation74\]) is true. Moreover, the set $\Theta^*$ must be finite because for any $z\in \Theta^*$, we can prove (\[equation74\]) is true with $z_i$ replaced by $z_i + z$. The energy bound implies that such points are only of finitely many. Then we do an induction to construct a subsequence (still indexed by $i$) and sequences $z_i^{j,\beta}= (s_i^{j,\beta}, t_i^{j,\beta})$, $\beta = 1,2, \ldots$ such that $$\begin{aligned}
\lim_{i \to \infty} s_i^{j,\beta} = +\infty,\ \forall \beta \neq \beta',\ \lim_{i \to \infty} d \big( z_i^{j,\beta} , z_i^{j,\beta'} \big) > 0\end{aligned}$$ and $$\begin{aligned}
\lim_{i \to \infty} \big| \mu(u_i (z_i^{j,\beta})) \big| = +\infty,\ \lim_{r \to \infty} \lim_{i \to \infty} E ( A_i, u_i; B_r (z_i^{j,\beta}) ) \geq \epsilon_0.\end{aligned}$$ Since the energy is uniformly bounded, the induction process stops at finite time. Therefore, whenever the induction stops, the sequences $z_i^{j,\beta}$ satisfy the conditions listed in (3) of this corollary \[cor72\] for $\epsilon_E = \epsilon_0$. Item (4) of this corollary is obvious from our construction.
Proof of Theorem \[thm71\] {#subsection71}
--------------------------
The proof of Theorem \[thm71\] follows from Lemma \[lemma73\], Proposition \[prop74\]–\[prop76\] below. First we introduce some notations. For any solution ${\bm u} = (u, h)$ to a local model over $B_r$, parametrized by $(\sigma, \beta, \delta)$, we have the following density functions of $(u, h)$ which are comparable to the square root of the potential density functions. $$\begin{aligned}
\begin{array}{lcll}
{{\mathfrak}p}'(z) & := & {{\mathfrak}p}'({\bm u})(z) & =\ \big| \nabla {\widetilde}{W}_h^{(\delta)} (u(z)) \big|,\\
{{\mathfrak}p}''(z) & := & {{\mathfrak}p}''({\bm u})(z) & =\ \sqrt{\sigma(z)} \big|\mu(u(z)) \big|,\\
{{\mathfrak}p}(z) & := & {{\mathfrak}p}( {\bm u} )(z) & =\ {{\mathfrak}p}'(z) + {{\mathfrak}p}''(z).
\end{array}\end{aligned}$$
\[lemma73\] There exist $\epsilon_1>0$, $C_1>0$, $r_1 \in (0, r^*]$, $\lambda_1 \in (0,{1\over 2}]$ and for $p>2$, $c_{(p)}>0$, satisfying the following conditions. Suppose $r\in (0, r_1]$ and $( u, h)$ is a solution to the local model on $B_r$ parametrized by $(\beta, \sigma, \delta)$. If $$\begin{aligned}
\label{equation76}
r\sup_{z\in B_r} {{\mathfrak}p}(z) \leq 1,\ E(u, h; B_r) \leq \epsilon_1,\end{aligned}$$ then there exists a gauge transformation $g: B_r \to G$ such that if we denote by $(u', h') = g^* (u, h)$ and $\phi' + {\bm i} \psi' = 2( \partial h'/ \partial {\overline}{z})$, then for every $\lambda \in (0, \lambda_1]$, we have $$\begin{aligned}
\label{equation77}
\begin{split}
{\rm diam} \big( u' (B_{\lambda r}) \big) \leq &\ C_1 \Big( \big\| d_A u \big\|_{L^2(B_r)} + \sqrt{r} + \lambda \Big), \\
\big\| d_A u \big\|_{L^p(B_{\lambda r})} \leq &\ c_{(p)} (\lambda r)^{{2\over p} - 1} \Big( \big\| d_A u \big\|_{L^2(B_r)} + \sqrt{r} + \lambda \Big).
\end{split}\end{aligned}$$
Let $K_Q \subset X$ be the subset in ([**Q1**]{}) of Hypothesis \[hyp25\] and ${\widetilde}{K}_0 = \big\{ (x, p) \in {\widetilde}{X} \ |\ x \in K_Q,\ |p| \leq 1 \big\}$. For any $D>0$, denote $$\begin{aligned}
{\widetilde}{X}_B^D = \big\{ x \in {\widetilde}{X} \setminus {\widetilde}{K}_0 \ |\ d(x, {\widetilde}{X}_B) \leq D \big\},\ {\widetilde}{X}_S^D = \big\{ x \in {\widetilde}{X} \setminus {\widetilde}{K}_0\ |\ d(x, {\widetilde}{X}_S) \leq D \big\}.\end{aligned}$$
\[prop74\] For each $H>0$ there exist $\epsilon_2 = \epsilon_2(H)>0$, $M_2 = M_2(H)>0$ and $D = D(H)>0$ satisfying the following condition.
Suppose $r \in (0, r^*]$ and $(u, h)$ is a solution to the local model with parameter $(\beta, \sigma, \delta)$ on $B_r$ with finite energy. If $u(0) \in {\widetilde}{X}_B^D$ and $$\begin{aligned}
\label{equation79}
\big\| h \big\|_{L^\infty(B_r)} \leq H,\ {{\mathfrak}p}(0) \geq {1\over 2} \sup_{B_r} {{\mathfrak}p} \geq M_2,\ {{\mathfrak}p}'(0) \geq {{\mathfrak}p}''(0),\ r {{\mathfrak}p}(0) = \lambda \in (0, 1],\end{aligned}$$ then we have $$\begin{aligned}
E( u, h; B_r ) \geq \epsilon_2 \lambda^2.\end{aligned}$$
The next proposition considers the case that the image of $u$ is away from ${\widetilde}{X}_S \cup {\widetilde}{X}_B$.
\[prop75\] There exist $\epsilon_3>0$ and for each $H>0$, a constant $M_3 = M_3(H) > 0$ satisfying the following condition. Suppose $(u, h)$ is a solution the local model with parameter $(\beta, \sigma, \delta )$ on $B_r$ with $r \in (0, r^*]$, such that $$\begin{aligned}
\label{equation710}
\big\| h\big\|_{L^\infty(B_r)} \leq H,\ {{\mathfrak}p}(0) \geq {1\over 2} \sup_{ z\in B_r } {{\mathfrak}p}(z)\geq M_3,\ r {{\mathfrak}p}(0) = \lambda \in (0, 1];\end{aligned}$$ and such that either of the following two conditions are satisfied: $$\begin{aligned}
\label{equation711}
{\rm (I)}\ u(0) \notin {\widetilde}{X}_B^{D} \cup {\widetilde}{X}_S^{D}, \hspace{1cm} {\rm (II)}\ {{\mathfrak}p}''(0) \geq {{\mathfrak}p}'(0).\end{aligned}$$ Here $D = D(H) >0$ is the one of Proposition \[prop74\]. Then $$\begin{aligned}
E ( u, h; B_r ) \geq \epsilon_3 \lambda^2. \end{aligned}$$
The remaining case is that the blow up happens near ${\widetilde}{X}_S= \{ (\star, p)\ |\ p \in {{\mathbb}C}\}$. The function $W$ is degenerate along the normal direction of ${\widetilde}{X}_S$ and we couldn’t find a straightforward argument to deal with this situation. Instead we have the following proposition, from whose proof one can see that it is a corollary to the above two propositions.
\[prop76\] For each $H>0$ there exists $\epsilon_4 = \epsilon_4(H) >0$ satisfying the following condition.
Suppose $(\beta_i, \sigma_i, \delta_i)$ is a sequence of parameters of local models of gauged Witten equation over $B_r$ with $r \in (0, r^*]$ and suppose $(u_i, h_i)$ are a sequence of corresponding solutions. Suppose $$\begin{aligned}
\lim_{i \to \infty} {{\mathfrak}p}_i(0) = \infty,\ \big\| h_i \big\|_{L^\infty(B_r)} \leq H\end{aligned}$$ and $$\begin{aligned}
{{\mathfrak}p}_i'(0) \geq {{\mathfrak}p}_i''(0),\ {{\mathfrak}p}_i(0) \geq {1\over 2} \sup_{B_{r_i}} {{\mathfrak}p}_i, u_i(0) \in {\widetilde}{X}_S^D.\end{aligned}$$ Here $r_i = {{\mathfrak}p}_i(0)^{-1}$ and $D = D(H)$ is the one of Proposition \[prop74\]. Then there exists a subsequence (still indexed by $i$) such that one of the following conditions holds
1. We have $$\begin{aligned}
\label{equation712}
\lim_{r \to \infty} \lim_{i \to \infty} E ( u_i, h_i; B_r ) \geq \epsilon_4.\end{aligned}$$
2. $\displaystyle \lim_{i \to \infty} \sigma_i = 0$ uniformly on $B_{r^*}$ and there exists $\tau>0$ (which may depend on the subsequence) such that $$\begin{aligned}
\label{equation713}
\lim_{i \to \infty} \inf_{B_\tau} \big| \mu( u_i) \big| = + \infty.\end{aligned}$$
First, if there exists a subsequence (still indexed by $i$) and a sequence $z_i \to 0$ such that $\lim_{i \to \infty} {{\mathfrak}p}_i(z_i) = \infty$, then the conclusion holds according to Proposition \[prop74\], \[prop75\] and \[prop76\]. Indeed, let $r_i:= {{\mathfrak}p}_i(0)^{-1}$ which converges to zero. Apply Hofer’s lemma (Lemma \[lemmaa7\]) to the function ${{\mathfrak}p}_i: B_{r_i}(z_i) \to {{\mathbb}R}$. Then there exist a point $y_i \in B_{r_i}(z_i)$ and $\delta_i \in (0, r_i/ 2]$ such that $$\begin{aligned}
{{\mathfrak}p}_i(y_i) \geq {1\over 2} \sup_{B_{\delta_i}(y_i)}{{\mathfrak}p}_i,\ \delta_i {{\mathfrak}p}_i(y_i) \geq {r_i \over 2} {{\mathfrak}p}_i(z_i) = {1\over 2}.\end{aligned}$$ By taking a subsequence, we may assume that either ${{\mathfrak}p}_i''(y_i) \geq {{\mathfrak}p}_i'(y_i)$, or ${{\mathfrak}p}_i'(y_i) \geq {{\mathfrak}p}_i''(y_i)$. In the former case, for $i$ large enough, (\[equation710\]) and (II) of (\[equation711\]) are satisfied by $(u_i, h_i)$ over $B_{\delta_i}(y_i)$. Then by Proposition \[prop75\], for $i$ large enough, we have $$\begin{aligned}
E(u_i, h_i; B_{\delta_i}(y_i)) \geq \epsilon_3 \big( \delta_i {{\mathfrak}p}_i(y_i) \big)^2 \geq {\epsilon_3 \over 4}.\end{aligned}$$ In the latter case, by taking a subsequence, we have three distinct possibilities.
1. For all $i$, $u_i(y_i) \in {\widetilde}{X}_B^D$. Then by Proposition \[prop74\], for $i$ large enough (so that (\[equation79\]) is satisfied by $(u_i, h_i)$ over $B_{\delta_i}(y_i)$), we have $$\begin{aligned}
E(u_i, h_i; B_{\delta_i}(y_i)) \geq \epsilon_2 \big( \delta_i {{\mathfrak}p}_i(y_i) \big) \geq {\epsilon_2 \over 4}.\end{aligned}$$
2. For all $i$, $u_i(y_i) \notin {\widetilde}{X}_B^D \cup {\widetilde}{X}_S^D$, Then by Proposition \[prop75\], for $i$ large enough (so that (\[equation710\]) and (I) of (\[equation711\]) are satisfied by $(u_i, h_i)$ over $B_{\delta_i}(y_i)$), we have $$\begin{aligned}
E(u_i, h_i; B_{\delta_i}(y_i)) \geq \epsilon_3 \big( \delta_i {{\mathfrak}p}_i(y_i) \big)^2 \geq {\epsilon_3 \over 4}.\end{aligned}$$
3. For all $i$, $u_i(y_i) \in {\widetilde}{X}_S^D$. Then since $\lim_{i \to \infty} y_i = 0$, we can choose $r'$ small enough so that $B_{r'}(y_i) \subset B_r$. Then we can apply Proposition \[prop76\] to $(u_i, h_i)$ restricted to $B_{r'}(y_i)$. It implies that by taking a further subsequence, we have either $$\begin{aligned}
\lim_{r \to 0}\lim_{i \to \infty} E(u_i, h_i; B_r(y_i)) \geq \epsilon_4,\end{aligned}$$ or $\displaystyle \lim_{i \to \infty} \sigma_i = 0$ uniformly and there is $\tau >0$ (depending on the subsequence) such that $$\begin{aligned}
\lim_{i \to \infty} \inf_{B_{\tau} (y_i)} \big| \mu(u_i) \big| = \infty.\end{aligned}$$
Since $y_i \to 0$, this implies the conclusion for $\epsilon_0 = \min\{ \epsilon_2/4, \epsilon_3/4, \epsilon_4 \}$ and $\tau_0 = \tau$.
It remains to consider the case that ${{\mathfrak}p}_i$ doesn’t blow up at $0$. Then we can assume that there exist a subsequence (still indexed by $i$) and $\tau >0$ (which depends on the subsequence) such that $$\begin{aligned}
\limsup_{i \to \infty} \big\|{{\mathfrak}p}_i \big\|_{L^\infty(B_\tau)} = M < \infty. \end{aligned}$$ Then we can take $\tau$ smaller than both the $r_1$ of Lemma \[lemma73\] and ${1\over M}$. By taking a subsequence, we can assume for all $i$, either $E(u_i, h_i; B_\tau ) > \epsilon_0$ or $E(u_i, h_i; B_\tau ) \leq \epsilon_0$. In the former case the current lemma is proven; in the latter case, (\[equation76\]) is satisfied and by Lemma \[lemma73\] and (\[equation77\]), $(u_i, h_i)$ is gauged equivalent to some $(u_i', h_i')$ such that $$\begin{aligned}
{\rm diam} (u' (B_{\lambda_0 \tau })) \leq C,\end{aligned}$$ where $\lambda_0 \in (0, {1 \over 2}]$ is the one in Lemma \[lemma73\] and $C$ is a constant, independent of the sequence. Thus this implies $u_i(B_{\lambda_0 \tau })$ escape to infinity uniformly. Thus (\[equation72\]) is true and necessarily $\sigma_i$ should converges to zero uniformly.
Proof of Lemma \[lemma73\]
--------------------------
Let $\phi, \psi: B_r \to {{\mathfrak}g}$ be the functions defined by $\phi + {\bm i} \psi = 2(\partial h / \partial {\overline}{z})$. We pull back $(u, h)$ and $\phi ds + \psi dt$ via the rescaling $B_1 \to B_r$ given by $w \mapsto z= rw$, which is denoted by $( u_r, h_r)$,and $\phi_r ds + \psi_r dt$. Denote $A_r= d + \phi_r ds + \psi_r dt$. Then $$\begin{aligned}
\partial \psi_r - \partial_t \phi_r + r^2 \sigma \mu^* (u_r) = 0.\end{aligned}$$ Hence by (\[equation76\]) we have $$\begin{aligned}
\big\| F_{A_r} \big\|_{L^\infty(B_1)} = r^2 \big\| \sigma \mu (u_r) \big\|_{L^\infty(B_r)} \leq r^2 \Big( \sup_{B_r} \sqrt{\sigma} \Big) \Big( \sup_{B_r} {{\mathfrak}p}'' \Big)\leq r \sqrt{\sigma^+}.\end{aligned}$$ There exists $f: B_1 \to {{\mathfrak}g}$ solving the Neumann boundary value problem $$\begin{aligned}
\Delta f = d^* (\phi_r ds + \psi_r dt),\ -{\partial f \over \partial {\bm n}}d \theta = (\phi dt - \psi ds)|_{\partial B_1},\ f(0) = 0.\end{aligned}$$ It means that the gauge transformation $g = e^f$ will turn $A_r$ into Coulomb gauge on $B_1$. Denote $g^* A_r = d + \phi_r' ds + \psi_r' dt$, then there is a universal constant $c >0$ such that $$\begin{aligned}
\big\| \phi_r' \big\|_{L^\infty(B_1)} + \big\| \psi_r' \big\|_{L^\infty(B_1)} \leq c \big\| F_{A_r} \big\|_{L^\infty(B_1)} \leq c r \sqrt{\sigma^+}.\end{aligned}$$ Denote $u_r' = g^{-1} u_r$, $h_r' = h_r + f$, $u' (z) = u_r'(z/r)$. Then in this new gauge, $$\begin{aligned}
\label{equation714}
\partial_s u_r' + {{\mathcal}X}_{\phi_r'} (u_r') + J ( \partial_t u_r' + {{\mathcal}X}_{\psi_r'} (u_r') ) + 2 r \nabla {\widetilde}{W}_{h_r'}^{(\delta)} (u_r')= 0.\end{aligned}$$ By ([**X3**]{}) of Hypothesis \[hyp21\], there exists a constant $C_1>0$ (which is abusively used in this proof) such that $$\begin{aligned}
\label{equation715}
\begin{split}
&\ \big\| {{\mathcal}X}_{\phi_r'}( u_r') \big\|_{L^2(B_1)} + \big\| {{\mathcal}X}_{\psi_r'}( u_r') \big\|_{L^2(B_1)} \\[0.1cm]
\leq &\ \sqrt{\pi} \Big( \big\| {{\mathcal}X}_{\phi_r'}(u_r') \big\|_{L^\infty(B_1)} + \big\| {{\mathcal}X}_{\psi_r'}(u_r') \big\|_{L^\infty(B_1)} \Big) \\
\leq &\ C_1 r \sqrt{\sigma^+} \Big( 1 + \sup_{B_r} \sqrt{|\mu(u)|} \Big)\\
\leq &\ C_1 r \sqrt{\sigma^+} \Big( 1 + (\sigma^-)^{-{1\over 4}} \sup_{B_r} \sqrt{{{\mathfrak}p}} \Big)\\
\leq &\ C_1 (\sigma^+)^{1\over 4} \sqrt{r}.
\end{split}\end{aligned}$$ Since $\sigma^+$ is bounded from above, we can take $r_1$ sufficiently small so that $$\begin{aligned}
\label{equation716}
r \leq r_1 \Longrightarrow C_1 (\sigma^+)^{1\over 4} \sqrt{r} \leq {1\over 2} {\bm \epsilon_2}.\end{aligned}$$ Here ${\bm \epsilon_2}$ is the one in Lemma \[lemmaa4\]. We can also assume that $E ( u, h; B_r ) \leq \epsilon_1 \leq \big( {1\over 4} {\bm \epsilon_2} \big)^2$. Then by (\[equation715\]) and (\[equation716\]), we have $$\begin{aligned}
\begin{split}
\big\| d u_r' \big\|_{L^2(B_1)} \leq &\ \big\| \partial_s u_r' + {{\mathcal}X}_{\phi_r'}(u_r' ) \big\|_{L^2(B_1)} + \big\| \partial_t u_r' + {{\mathcal}X}_{\psi_r'} (u_r' ) \big\|_{L^2(B_1)} \\
&\ + \big\| {{\mathcal}X}_{\phi_r' }( u_r' ) \big\|_{L^2(B_1)} + \big\| {{\mathcal}X}_{\psi_r'} ( u_r' ) \big\|_{L^2(B_1)} \\
\leq &\ 2 \sqrt{ E ( u, h; B_r ) } + C_1 (\sigma^+)^{1\over 4} \sqrt{r} \leq {\bm \epsilon}_2.
\end{split}\end{aligned}$$ On the other hand, by (\[equation715\]) and (\[equation716\]), we have $$\begin{gathered}
\Big\| {{\mathcal}X}_{\phi_r'} ( u_r' ) + J {{\mathcal}X}_{\psi_r' } ( u_r' ) + 2 r \nabla {\widetilde}{W}_{h_r'}^{(\delta)} (u_r') \Big\|_{L^\infty(B_1)} \\
\leq \big\| {{\mathcal}X}_{\phi_r'}( u_r' ) \big\|_{L^\infty(B_1)} + \big\| {{\mathcal}X}_{\psi_r'}( u_r' ) \big\|_{L^\infty(B_1)} + 2 r \big\| \nabla {\widetilde}{W}_{h_r'}^{(\delta)} ( u_r') \big\|_{L^\infty(B_1)} \leq 3.\end{gathered}$$
Take $\lambda_1 = {1\over 6} {\bm \epsilon_2}{\bm \epsilon_p}$. For $\lambda \in (0, \lambda_1]$, the restriction of $u_r'$ to $B_{2\lambda}$ satisfies the assumptions of Lemma \[lemmaa4\]. Thus there exists $c_{(p)}>0$ (which is abusively used below) such that $$\begin{aligned}
\begin{split}
&\ \big\| d u' \big\|_{L^p(B_{\lambda r})} \\
= &\ r^{{2\over p} - 1} \big\| d u_r' \big\|_{L^p(B_\lambda)} \\
\leq &\ c_{(p)} (\lambda r)^{{2\over p }-1} \Big( \big\| d u_r' \big\|_{L^2(B_{2\lambda_1 })} + 6 \lambda \Big) \\
\leq &\ c_{(p)} (\lambda r)^{{2\over p} - 1} \Big( \big\| d_A u \big\|_{L^2(B_r)} + \big\| {{\mathcal}X}_{\phi_r'}(u_r') \big\|_{L^2(B_{2\lambda_1})} + \big\| {{\mathcal}X}_{\psi_r'}( u_r' ) \big\|_{L^2(B_{2\lambda_1})} + 6\lambda \Big)\\
\leq &\ c_{(p)} (\lambda r)^{{2\over p}- 1} \Big( \big\| d_A u \big\|_{L^2(B_r)} + \sqrt{r} + \lambda \Big).
\end{split}\end{aligned}$$ We used (\[equation710\]) to derive the last inequality. Moreover, there exists $C_1 >0$ such that $$\begin{aligned}
{\rm diam} \big( u' (B_{\lambda r}) \big)= {\rm diam} \big( u_r' (B_{\lambda}) \big) \leq C_1 \Big( \big\| d_A u \big\|_{L^2(B_r)} + \sqrt{r} + \lambda \Big). \end{aligned}$$ Lastly, we may assume that for the same $c_{(p)}>0$, we have $$\begin{aligned}
\begin{split}
\big\| d_A u \big\|_{L^p(B_{\lambda r})} \leq &\ \big\| d u' \big\|_{L^p(B_{\lambda r})} + \big\| {{\mathcal}X}_{\phi'}(u' ) \big\|_{L^p(B_{\lambda r})} + \big\| {{\mathcal}X}_{\psi'}(u' ) \big\|_{L^p(B_{\lambda r})} \\
\leq &\ \big\| d u' \big\|_{L^p(B_{\lambda r})} + r^{-1} \Big( \big\| {{\mathcal}X}_{\phi_r'}(u_r') \big\|_{L^\infty} + \big\| {{\mathcal}X}_{\psi_r'}(u_r' ) \big\|_{L^\infty} \Big) \big( \pi \lambda^2 r^2 \big)^{1\over p}\\
\leq &\ c_{(p)} (\lambda r)^{{2\over p} -1} \Big( \big\| d_A u \big\|_{L^2(B_r)} + \sqrt{r}+ \lambda \Big).
\end{split}\end{aligned}$$ Thus Lemma \[lemma73\] is proved.
Proof of Proposition \[prop74\]
-------------------------------
We assume first that $D \leq 1/2$. By Lemma \[lemma73\], we know that there exist ${\widetilde}\epsilon_1 >0$, ${\widetilde}{r}_1>0$ and ${\widetilde}\lambda_1 \in (0, {1\over 2}]$ such that if $E ( u, h; B_{{\widetilde}{r}} ) \leq {\widetilde}\epsilon_1$, ${\widetilde}{r} \leq {\widetilde}{r}_1$, then up to gauge transformation, ${\rm diam} \big( u(B_{ {\widetilde}\lambda_1 {\widetilde}{r}}) \big) \leq 1/2$. Then we can assume that $M_2$ is big enough such that $r \leq \lambda M_2^{-1} \leq {\widetilde}{r}_1$. Then $u\big( B_{{\widetilde}\lambda_1 r} \big) \subset {\widetilde}{X}_B^1$. It suffices to prove that there exists $\epsilon_2 > 0$ such that $$\begin{aligned}
\label{equation718}
E (u, h; B_{{\widetilde}\lambda_1 r} ) \geq \epsilon_2 \lambda^2.\end{aligned}$$
Denote ${\overline}\partial_A u = {1\over 2} \big( \partial_s u + {{\mathcal}X}_\phi(u) + J \partial_t u + J {{\mathcal}X}_\psi(u) \big)$. Let $\pi_N: T{\widetilde}{X}_B^1 \to T{\widetilde}{X}_B^1$ be the orthogonal projection onto the distribution spanned over ${{\mathbb}C}$ by $\partial / \partial p$ and $\nabla Q$ and abbreviate $\pi_N({\overline}\partial_A u(z)) = V_N(z)$. To prove (\[equation718\]), we need the following estimate, which follows from straightforward but technical calculations and estimates.
\[lemma77\] For each $H>0$, there exist $c > 0$, ${\widetilde}\lambda_2 \in (0, {1\over 2}]$, $M_2 >0$, ${\widetilde}\epsilon_2 > 0$ and $D > 0$ such that if $(u, h)$ satisfies (\[equation79\]) with this $H$, this $M_2$ and $u(0) \in {\widetilde}{X}_B^D$, and $E(u, h) \leq {\widetilde}\epsilon_2$, then $u \big( B_{{\widetilde}\lambda_2 r} \big) \subset {\widetilde}{X}_B^{2 D}$ and over $B_{{\widetilde}\lambda_2 r}$ we have $$\begin{aligned}
\label{equation719}
\Delta \big| V_N \big|^2 \geq - c \big| V_N(0) \big) \big|^2 \Big( 1+ \big| d_A u \big|^2 + \big| d h'' \big|^2 + \big| F_A \big| \Big).\end{aligned}$$
The proof is given in Subsection \[subsection76\].
Now take any ${\widetilde}\lambda \in (0, {\widetilde}\lambda_2]$. We apply the local maximal principle [@Gilbarg_Trudinger Theorem 9.20] for $p =1$, $n=2$, $R = {\widetilde}\lambda r$, $u = \left| {\overline}\partial_A u \right|^2$ and $f$ equal to the right hand side of (\[equation719\]), we see that there exists a constant $c' > 0$ (independent of ${\widetilde}\lambda$ and $r$) such that $$\begin{gathered}
{1\over c'} \big| V_N(0) \big|^2 \leq {1\over ({\widetilde}\lambda r)^2 } \int_{B_{{\widetilde}\lambda r}} \big| V_N \big|^2 dsdt \\
+ c {\widetilde}\lambda r \big| V_N(0) \big|^2 \Big( {\widetilde}\lambda r+ \big\| d_A u \big\|_{L^4(B_{{\widetilde}\lambda r})}^2 + \big\| dh'' \big\|_{L^4(B_{{\widetilde}\lambda r})}^2 + \big\| F_A \big\|_{L^2(B_{{\widetilde}\lambda r})}\Big).\end{gathered}$$ Therefore, we see $$\begin{gathered}
\label{equation720}
E ( u, h; B_{{\widetilde}\lambda r} ) \geq {1\over 2} \int_{B_{{\widetilde}\lambda r}} \big| V_N \big|^2 ds dt \geq { ({\widetilde}\lambda r)^2 \over 2 c'} \big| V_N(0) \big|^2 \\
- { ( {\widetilde}\lambda r )^3 c \over 2} \big| V_N(0) \big|^2 \Big( {\widetilde}\lambda r + \big\| d_A u \big\|_{L^4(B_{{\widetilde}\lambda r})}^2 + \big\| dh'' \big\|_{L^4(B_{{\widetilde}\lambda r})}^2 + \big\| F_A \big\|_{L^2(B_{{\widetilde}\lambda r})} \Big).\end{gathered}$$ To proceed, we need
\[lemma78\] There exist ${\widetilde}\epsilon_2' >0$, ${\widetilde}\lambda_2' \in (0, {1\over 2}{\widetilde}\lambda_2]$ such that if $E (u, h; B_{{\widetilde}\lambda_2 r} ) \leq {\widetilde}\epsilon_2'$, then $$\begin{aligned}
\label{equation721}
{\widetilde}\lambda_2' r \Big( {\widetilde}\lambda_2' r + \big\| d_A u \big\|_{L^4(B_{{\widetilde}\lambda_2' r})}^2 + \big\| dh'' \big\|_{L^4(B_{{\widetilde}\lambda_2' r})}^2 + \big\| F_A \big\|_{L^2(B_{{\widetilde}\lambda_2' r})} \Big) \leq {1\over 2 c c'}.\end{aligned}$$
We can ignore the terms ${\widetilde}\lambda_2' r$ and $\|F_A\|_{L^2(B_{{\widetilde}\lambda_2' r})}$ in (\[equation721\]) because they are easily bounded by the radius and the energy on $B_{{\widetilde}\lambda_2' r}$. On the other hand, by Lemma \[lemma73\] and (\[equation77\]), if $E ( u,h; B_r )$ is sufficiently small, then for $p = 4$, we have $$\begin{aligned}
{\widetilde}\lambda_2' r \big\| d_A u \big\|_{L^4(B_{{\widetilde}\lambda_2' r})}^2 \leq c_{(4)}^2 \Big( \big\| d_A u \big\|_{L^2(B_r)} + \sqrt{r} + {\widetilde}\lambda_2' \Big)^2.\end{aligned}$$ On the other hand, define $h_r''(z) = h'' (rz)$ for $z \in B_1$. Then by the elliptic estimate over $B_1$, for some universal constant $c'' > 0$, $$\begin{gathered}
{\widetilde}\lambda_2' r \big\| dh'' \big\|_{L^4(B_{{\widetilde}\lambda_2' r})}^2 = {\widetilde}\lambda_2' \big\| d h_r'' \big\|_{L^4(B_{{\widetilde}\lambda_2'})}^2 \leq {\widetilde}\lambda_2' c'' \big\| \Delta h_r'' \big\|_{L^2(B_1)}^2 \\
= c'' {\widetilde}\lambda_2' r^2 \big\| \Delta h'' \big\|_{L^2(B_r)}^2 = c'' {\widetilde}\lambda_2' r^2 \big\| F_A \big\|_{L^2(B_r)}^2.\end{gathered}$$ Therefore it is easy to see Lemma \[lemma78\] is true.
Then by (\[equation720\]) and Lemma \[lemma78\], we see that if $E( u, h; B_r ) \leq {\widetilde}\epsilon_2'$, then $$\begin{aligned}
E(u, h; B_r) \geq E ( u, h; B_{{\widetilde}\lambda_2' r} ) \geq { ( {\widetilde}\lambda_2' r )^2 \over 4 c_2} \big| V_N(0) \big|^2 \geq {({\widetilde}\lambda_2' r)^2 \over 4c_2} {{{\mathfrak}p}(0)^2 \over 16} = { ({\widetilde}\lambda_2')^2 \over 64 c_2} \lambda^2.\end{aligned}$$ Here the third inequality uses Lemma \[lemma712\]. Therefore Proposition \[prop74\] holds for $$\begin{aligned}
\epsilon_2 = \min \Big\{ {\widetilde}\epsilon_1, {\widetilde}\epsilon_2, {\widetilde}\epsilon_2', {( {\widetilde}\lambda_2')^2 \over 64 c_2 } \Big\}\end{aligned}$$ and the $M_2$ and $D$ which we already specified.
Proof of Proposition \[prop75\]
-------------------------------
Suppose $(u, h)$ satisfies (\[equation710\]) with $M_3$ undetermined. By Lemma \[lemma73\], there exist ${\widetilde}\epsilon>0$, ${\widetilde}\lambda \in (0, 1/2]$ and ${\widetilde}{M}>0$ such that if $E( u, h; B_r ) \leq {\widetilde}\epsilon$ and ${{\mathfrak}p} (0) \geq {\widetilde}{M}$ (so $r$ is small enough), then $( u,h)$ is gauge equivalent to a triple $( u', h')$ such that ${\rm diam} \big( u'(B_{{\widetilde}\lambda r}) \big) \leq 1/4$. Take $M_3 \geq {\widetilde}{M}$. So without loss of generality we may assume that ${\rm diam} \big( u ( B_{ {\widetilde}\lambda r} ) \big) \leq 1/4$. Moreover, by the equation $\Delta h'' + \sigma \mu^*(u) = 0$, we can take ${\widetilde}\epsilon>0$ small enough such that $$\begin{aligned}
\label{equation722}
\sup_{B_{{\widetilde}\lambda r}} \big| \rho_0(h'') \big| - \inf_{B_{{\widetilde}\lambda r}} \big| \rho_0( h'') \big| \leq \log 2.\end{aligned}$$
\(I) Assume that ${{\mathfrak}p}''(0) \geq {{\mathfrak}p}'(0)$. Then $$\begin{aligned}
\big| \mu(0) \big| \geq \big(\sqrt{ \sigma^+}\big)^{-1} {{\mathfrak}p}''(0) \geq {M_3 \over 2 \sqrt{\sigma^+}}.\end{aligned}$$ By the properness of $\mu$ and ([**X3**]{}) of Hypothesis \[hyp21\], we may take $M_3$ big enough so that for any $z \in B_{{\widetilde}\lambda r}$ ($u(z)$ is at most $1/4$ away from $u(0)$), we have $$\begin{aligned}
\big| \nabla \mu(u(z))) \big| \leq \big| \mu(u(z)) \big|.\end{aligned}$$ Therefore $$\begin{aligned}
\big| \mu(u(z)) - \mu(u(0)) \big| \leq \sup_{t \in [0,1]} \big| \nabla \mu(u(tz))\big| \cdot d (u(z), u(0)) \leq {1\over 4} \sup_{B_{{\widetilde}\lambda r}} \big| \mu(u) \big|.\end{aligned}$$ Therefore by the triangle inequality, we have $$\begin{aligned}
\sup_{B_{{\widetilde}\lambda r}} \big| \mu(u) \big| \leq {4 \over 3}\big| \mu(u(0)) \big|.\end{aligned}$$ Then we have $$\begin{aligned}
\big| \mu(u(z)) \big| \geq \big| \mu(u(0)) \big| - \big| \mu(u(0)) - \mu(u(z)) \big| \geq {1\over 2} \big| \mu(u(0)) \big|.\end{aligned}$$ Therefore by the property of $\sigma$, we have $$\begin{aligned}
{{\mathfrak}p}''(z)^2 = \sigma(z) \big| \mu (u(z)) \big|^2 \geq {1\over 8} \sigma(0) \big| \mu (u(0)) \big|^2 = {1\over 8} {{\mathfrak}p}''(0)^2.\end{aligned}$$ Therefore $$\begin{aligned}
E (u,h; B_r ) \geq \int_{B_{ {\widetilde}\lambda r}} {{\mathfrak}p}''(z)^2 \geq {\pi \over 8} \big( {\widetilde}\lambda r {{\mathfrak}p}''(0) \big)^2 \geq \Big( {\pi \over 32} ({\widetilde}\lambda)^2 \Big) \lambda^2.\end{aligned}$$
\(II) Assume that ${{\mathfrak}p}'(0) \geq {{\mathfrak}p}''(0)$ and $u(0) \notin {\widetilde}{X}_B^D \cup {\widetilde}{X}_S^D$. For convenience we introduce $$\begin{aligned}
{{\mathfrak}p}_0'(z) = \Big| e^{{\overline}{\rho_0(h)}} \nabla F_0(u(z)) \Big|,\ {{\mathfrak}p}_+'(z) = \Big| \beta(z) \sum_{l=1}^s e^{{\overline}{\rho_l(h)}} \nabla F_l^{(\delta)}(u(z)) \Big|.\end{aligned}$$
Let $K_Q\subset X$ be the compact subset of ([**Q1)**]{} of Hypothesis \[hyp25\] and ${\widetilde}{K} = K_Q \times B_1\subset {\widetilde}{X}$. We claim that there exists a constant $\alpha= \alpha(D) \in (0, 1)$ such that if $$\begin{aligned}
\label{equation723}
(x, p) \notin {\widetilde}{K} \cup {\widetilde}{X}_B^{D\over 2} \cup {\widetilde}{X}_S^{D\over 2} \Longrightarrow \alpha \big| \nabla^2 W(x, p) \big| \leq \big| \nabla W(x, p) \big|.\end{aligned}$$ To justify our claim, with respect to the product structure ${\widetilde}{X} = X \times {{\mathbb}C}$, we write $$\begin{aligned}
\nabla W(x, p) = \big( {\overline}{p} \nabla Q(x),\ {\overline}{Q(x)} \big),\ \nabla^2 W(x, p) = \left( \begin{array}{cc} {\overline}{p} \nabla^2 Q(x) & \nabla Q(x)\\
\langle \cdot, \nabla Q(x)\rangle & 0
\end{array} \right).\end{aligned}$$ It suffices to bound each component of $\nabla^2 W$ by components of $\nabla W$. Indeed, if $(x, p) \notin {\widetilde}{K} \cup {\widetilde}{X}_B^{D\over 2} \cup {\widetilde}{X}_S^{D\over 2}$, then either $|p| \geq 1$ and $d(x, \star) \geq D/2$ or $|p| \leq 1$, $x \notin K_Q$ and $d(x, X_Q) \geq D/2$. In the former case, by ([**Q1**]{}) of Hypothesis \[hyp25\], there exists $\alpha_1(D)>0$ such that $\alpha_1(D) |\nabla^2 Q(x)| \leq |\nabla Q(x)|$; so ${\overline}{p}\nabla^2 Q$ and $\nabla Q$ can be controlled by ${\overline}{p}\nabla Q$. In the latter case, there exists $\alpha_2(D)>0$ such that $\alpha_2(D) |\nabla Q(x)| \leq |Q(x)|$. Then $\nabla Q$ and ${\overline}{p} \nabla^2 Q$ can be controlled by $Q$ (by ([**Q1**]{}) of Hypothesis \[hyp25\]). The claim is proved.
On the other hand, by Lemma \[lemma710\] below, we have ${{\mathfrak}p}_+'(z) \leq c_P(H)$. Therefore $$\begin{aligned}
M_3 \leq {{\mathfrak}p}(0) \leq 2 {{\mathfrak}p}'(0) \leq 2 {{\mathfrak}p}_0'(0) + 2 {{\mathfrak}p}_+'(0) \leq 2 {{\mathfrak}p}_0'(0) + 2 c_P (H).\end{aligned}$$ Taking $M_3 = M_3(H) \geq 4 c_P (H)$, we have $$\begin{aligned}
\label{equation724}
{{\mathfrak}p}_0'(0) \geq M_3/4,\ {{\mathfrak}p}_0'(0) \geq {{\mathfrak}p}_+'(0).\end{aligned}$$ By the expression of ${{\mathfrak}p}_0'$ and the fact that $|h| \leq H$, we may take $M_3$ big enough such that $$\begin{aligned}
u(B_{{\widetilde}\lambda r}) \cap {\widetilde}{K} = \emptyset. \end{aligned}$$ On the other hand, by Lemma \[lemma73\] we may take $M_3$ big enough and ${\widetilde}\lambda$ small enough such that $$\begin{aligned}
{\rm diam}\big( u( B_{{\widetilde}\lambda r})\big) \leq \min \big\{ \alpha/4, D/2\big\}.\end{aligned}$$ Then $u(B_{{\widetilde}\lambda r}) \subset {\widetilde}{X} \setminus ( {\widetilde}{K} \cup {\widetilde}{X}_B^{D \over 2} \cup {\widetilde}{X}_S^{D\over 2})$. Then by (\[equation723\]), for any $z \in B_{{\widetilde}\lambda r}$, we have $$\begin{aligned}
\Big| \big| \nabla W(u(z)) \big| - \big| \nabla W(u(0))\big|\Big| \leq \sup_{t \in [0,1]} \big| \nabla^2 W(u(tz)) \big| {\alpha \over 4} \leq {1 \over 4} \sup_{B_{{\widetilde}\lambda r}} \big| \nabla W(u)\big|. \end{aligned}$$ Then by triangle inequality, $$\begin{aligned}
\sup_{B_{{\widetilde}\lambda r}} \big|\nabla W(u) \big| \leq {4 \over 3} \big| \nabla W(u(0)) \big|.\end{aligned}$$ Therefore $$\begin{aligned}
\big| \nabla W(u(z)) \big| = \big| \nabla W(u(0)) \big| - \Big( \big| \nabla W(u(0)) \big| - \big| \nabla W(u(z)) \big| \Big) \geq {1\over 2} \big| \nabla W(u(0)) \big|.\end{aligned}$$ Then by (\[equation722\]), we have ${{\mathfrak}p}_0'(z) \geq {1\over 4} {{\mathfrak}p}_0'(0)$; then by (\[equation724\]) we have ${{\mathfrak}p}'(0) \geq {{\mathfrak}p}'(0)/8$ and $$\begin{aligned}
E ( u, h; B_r ) \geq \int_{B_{ {\widetilde}\lambda r}} {{\mathfrak}p}'(z)^2 \geq { \pi \over 64} \big( {\widetilde}\lambda r {{\mathfrak}p}'(0)\big)^2 \geq {\pi ({\widetilde}\lambda)^2 \over 256}\lambda^2.\end{aligned}$$
Therefore we see that $\epsilon_3 = \min \big\{ {\widetilde}\epsilon, 2^{-8} \pi ({\widetilde}\lambda)^2 \big\}$ and $M_3$ big enough satisfy the condition stated in this proposition.
Proof of Proposition \[prop76\]
-------------------------------
Suppose that the sequence $( u_i, h_i)$ satisfies the hypothesis of Proposition \[prop76\]. Since $\| h_i\|_{L^\infty(B_r)}$ is uniformly bounded, ${{\mathfrak}p}_i(0) \to +\infty$ implies that $\big| \mu (u_i(0)) \big|\to +\infty$. We write the map as $u_i (z) = ( {\overline}{u}_i(z), p_i(z))$ with respect to the decomposition ${\widetilde}{X} = X \times {{\mathbb}C}$. Then the condition $u_i(0) \in {\widetilde}{X}_S^D$ implies that $$\begin{aligned}
\lim_{i \to \infty} \big| p_i(0) \big| = +\infty.\end{aligned}$$
Projecting the Witten equation onto the ${{\mathbb}C}$-factor, we have $$\begin{aligned}
\label{equation725}
{\partial \over \partial {\overline}{z}} \left( e^{\rho_s (h_i)} p_i \right) = - e^{\rho_s(h_i)} \left( e^{{\overline}{\rho_0(h_i)}} {\overline}{Q}(\underline{u}_i) + \beta e^{{\overline}{\rho_s(h_i)}}\delta_i^r {\overline}{a} \right).\end{aligned}$$ Denote ${{\mathfrak}s}_i(z)$ the norm of the inhomogeneous term of (\[equation725\]) at $z$. By taking a subsequence, there are two possibilities.
\(I) There exist $\rho>0$ and $M>0$ (which may depend on the subsequence) such that for every $i$, $$\begin{aligned}
\sup_{z\in B_\rho} {{\mathfrak}s}_i(z) \leq M.\end{aligned}$$ Then by the standard diameter estimate for Cauchy-Riemann equations (with target ${{\mathbb}C}$), we see that $p_i$ diverges to infinity uniformly on $B_{\rho \over 2}$. Therefore (\[equation713\]) holds and $\displaystyle \lim_{i \to \infty} \sigma_i = 0$. Otherwise $$\begin{aligned}
E ( u_i, h_i ; B_r ) \geq \int_{B_{\rho \over 2}} \sigma_i \big| \mu (u_i) \big|^2 dsdt \to \infty.\end{aligned}$$ This contradicts with the uniform bound on energies of $(u_i, h_i)$.
\(II) There exists a sequence $y_i \in B_{r^*}$, $y_i \to 0$ such that $$\begin{aligned}
\label{equation726}
\lim_{i \to \infty} {{\mathfrak}s}_i(y_i) = \infty. \end{aligned}$$ Denote $\tau_i = {{\mathfrak}s}_i(y_i)^{-1}$. Applying Hofer’s lemma (Lemma \[lemmaa7\]) to the function ${{\mathfrak}s}_i$ on $B_{{\tau_i \over 2}}(y_i)$, we see that there exist $z_i \in B_{\tau_i \over 2}(y_i)$ and $\rho_i \in (0, {\tau_i \over 4}]$ such that $$\begin{aligned}
\label{equation727}
{{\mathfrak}s}_i(z_i) \geq {1\over 2} \sup_{B_{\rho_i}(z_i)} {{\mathfrak}s}_i,\ {{\mathfrak}s}_i(z_i) \rho_i = {1\over 4}.\end{aligned}$$ Now, equation (\[equation725\]) and Lemma \[lemmaa4\] imply that for some $\lambda_0$ small enough up to gauge transformation, $$\begin{aligned}
\label{equation728}
{\rm diam} \left( p_i \left( u_i (B_{\lambda_0 \rho_i}(z_i)) \right) \right) \leq 1.\end{aligned}$$ (Here the target is ${{\mathbb}C}$, so we don’t need the condition that the energy of $u_i|_{B_{\rho_i}(z_i)}$ is small.) Then applying Hofer’s lemma to the function ${{\mathfrak}p}_i$ on $B_{\lambda_0 \rho_i}(z_i)$, we obtain $w_i \in B_{{1\over 2} \lambda_0 \rho_i}(z_i)$, $\kappa_i \in (0, {1\over 4} \lambda_0 \rho_i]$ such that $$\begin{aligned}
\label{equation729}
{{\mathfrak}p}_i(w_i) \geq {1\over 2} \sup_{B_{\kappa_i}(w_i)} {{\mathfrak}p}_i,\ \kappa_i {{\mathfrak}p}_i(w_i) \geq { \lambda_0 \rho_i {{\mathfrak}p}_i (z_i) \over 4 } \geq {\lambda_0 \over 16}.\end{aligned}$$ Notices that $$\begin{aligned}
\label{equation730}
{{\mathfrak}p}_i(w_i) \geq {{\mathfrak}p}_i(z_i) \geq {{\mathfrak}s}_i(z_i) \geq {{\mathfrak}s}_i(y_i) \to +\infty.\end{aligned}$$
[*Claim.*]{} For sufficiently large $i$, we have that either $u_i(w_i) \notin {\widetilde}{X}_S^{D}$ or ${{\mathfrak}p}_i''(w_i) \geq {{\mathfrak}p}_i'(w_i)$.
Suppose that there is a subsequence (still indexed by $i$) such that $u_i(w_i) \in {\widetilde}{X}_S^D$ and ${{\mathfrak}p}_i'(w_i) \geq {{\mathfrak}p}_i''(w_i)$. We write $\nabla {\widetilde}{{\mathcal}W}_{h_i}^{(\delta_i)}(u_i)$ as $$\begin{aligned}
\nabla {\widetilde}{{\mathcal}W}_{h_i}^{(\delta_i)} (u_i) = e^{{\overline}{\rho_0(h_i)}} \left(\begin{array}{c} {\overline}{Q}(u_i) \\ {\overline}{p_i} \nabla Q(u_i) \end{array} \right) + \beta \nabla {{\mathcal}W}_{h_i}^{(\delta_i)'}(u_i).\end{aligned}$$
Then $u_i(w_i) \in {\widetilde}{X}_S^D$ implies that ${\overline}{Q}(u_i(w_i))$ and $\nabla Q(u_i(w_i))$ are both bounded. Then by (\[equation730\]) and the uniformly bound on $h_i$, we have $$\begin{aligned}
{{\mathfrak}p}_i'(w_i) \geq {{\mathfrak}p}_i''(w_i) \Longrightarrow {{\mathfrak}p}_i'(w_i) \to +\infty \Longrightarrow \big| p_i(w_i) \big| \to +\infty\end{aligned}$$ Then (\[equation728\]) implies that $\big| p_i (u_i(z_i)) \big| \geq \big| p_i (u_i(w_i)) \big|/2$. However, (\[equation726\]) and (\[equation727\]) imply that $\big| Q(u_i(z_i)) \big| \to +\infty$ and hence $\big| \nabla Q (u_i(z_i)) \big| + \infty$. Therefore $$\begin{aligned}
\Big| {\overline}{p}_i(z_i) \nabla Q(u_i(z_i)) \Big| >> \Big| {\overline}{p}_i(w_i) \nabla Q(u_i(w_i)) \Big|,\end{aligned}$$ which contradicts with (\[equation729\]). Therefore the claim holds.
Now applying Proposition \[prop74\] (if $u_i(w_i) \in {\widetilde}{X}_B^D$ and ${{\mathfrak}p}_i''(w_i) < {{\mathfrak}p}_i'(w_i)$) or Proposition \[prop75\] (if $u_i(w_i) \notin {\widetilde}{X}_B^D$ or ${{\mathfrak}p}_i''(w_i) \geq {{\mathfrak}p}_i'(w_i)$) to the disk $B_{\kappa_i}(w_i)$, with the condition (\[equation729\]), we see that for any $r>0$ and sufficiently large $i$, $$\begin{aligned}
E ( u_i, h_i; B_r ) \geq E ( u_i, h_i; B_{\kappa_i} (w_i) ) \geq 2^{-8} \min\{\epsilon_2, \epsilon_3 \} \lambda_0^2.\end{aligned}$$ (\[equation712\]) holds for $\epsilon_4 = 2^{-8} \min \{ \epsilon_2, \epsilon_3 \} \lambda_0^2$.
Proof of Lemma \[lemma77\] {#subsection76}
--------------------------
Let $NB\subset T{\widetilde}{X}_B^1$ be the distribution spanned over ${{\mathbb}C}$ by $\partial / \partial p$ and $\nabla Q$ and let $TB$ be its orthogonal complement. Let $\pi_T = {\rm Id} - \pi_N$.
\[lemma79\] $\pi_T$ and $\pi_N$ are $G$-invariant tensor fields and
1. For any $Z \in T{\widetilde}{X}_B^1$, $$\begin{aligned}
\label{equation731}
\nabla_{JZ} \pi_T = -J \nabla_Z \pi_T,\ \nabla_{JZ} \pi_N = - J \nabla_Z \pi_N.\end{aligned}$$
2. There exists $c_Q>0$ (which we can assume to coincide with the one of ([**Q1**]{}) of Hypothesis \[hyp28\]) such that in ${\widetilde}{X}_B^1$, $$\begin{aligned}
\label{equation732}
\big| \nabla \pi_T \big| \leq c_Q,\ \big| \nabla \pi_N \big| \leq c_Q,\ \big| \nabla^2 \pi_T \big| \leq c_Q,\ \big| \nabla^2 \pi_N \big| \leq c_Q.\end{aligned}$$
The distribution $NB$ and the metric are both $G$-invariant so $\pi_N$ and $\pi_T$ are $G$-invariant.
It is easy to see that with respect to the decomposition $T{\widetilde}{X}_B^1 \simeq TB \oplus NB$, for any tangent vector $Z$, we can write It is easy to see that with respect to the decomposition $T{\widetilde}{X}_B^1 \simeq TB \oplus NB$, for any tangent vector $Z$, we can write $$\begin{aligned}
\nabla_Z \pi_T = - \nabla_Z \pi_N = \left( \begin{array}{cc} 0 & F_Z \\
F^*_Z & 0
\end{array} \right),\ F_Z: NB \to TB.\end{aligned}$$ Moreover, the restriction of $F_Z$ to the ${\partial \over \partial p}$-direction is zero. Now we have $$\begin{aligned}
F_{JZ} \left(\nabla Q\right) = - \pi_T \nabla_{JZ} \nabla Q = \pi_T \left( J \nabla_Z \nabla Q \right) = J \pi_T \nabla_Z \nabla Q = - J F_Z (\nabla Q).\end{aligned}$$ Here the second equality follows from Lemma \[lemma34\]. Since $T{\widetilde}{X}_B^1 \simeq TB \oplus TN$ is $J$-linear, we see $F^*_{JZ} = - J F^*_Z$. Therefore (\[equation731\]) is proven.
To estimate $\nabla \pi_T$, we see that by ([**Q1)**]{} of Hypothesis \[hyp25\], for any $Z$, we have $$\begin{aligned}
\label{equation733}
\left| F_Z (\nabla Q ) \right| = \left| \pi_T \nabla_Z \nabla Q \right| \leq c_Q \left| \nabla Q\right||Z|.\end{aligned}$$
Now we consider the second derivative of $\pi_T$. In ${\widetilde}{X}_B^1$, we can write the Levi-Civita connection as $$\begin{aligned}
\nabla = \left( \begin{array}{cc} \nabla^T & - F \\
F^* & \nabla^N
\end{array} \right).\end{aligned}$$ Then $$\begin{aligned}
\label{equation734}
\nabla^2 \pi_T = \left[ \nabla, \nabla \pi_T \right] = \left( \begin{array}{cc} - 2 FF^* & \nabla^T F - F \nabla^N\\
\nabla^N F^* - F^* \nabla^T & 2 F^* F
\end{array} \right).\end{aligned}$$ Therefore it suffices to consider the two off-diagonal terms, which are adjoint to each other. Consider the upper-right one. Take tangent vectors $Z_1, Z_2$ with $\nabla_{Z_1} Z_2$ vanishes at a point. Then at that point, using ([**Q1**]{}) of Hypothesis \[hyp25\], we see $$\begin{aligned}
\begin{split}
&\ \big| \big( \nabla^T_{Z_1} F_{Z_2} - F_{Z_2} \nabla_{Z_1}^N \big) \nabla Q \big| \\
\leq &\ \big| \nabla_{Z_1}^T \pi_T \nabla_{Z_2}\nabla Q \big| + \big| F_{Z_2} \pi_N \nabla_{Z_1} \nabla Q\big| \\
\leq &\ \big| \nabla_{Z_1} \pi_T \nabla_{Z_2} \nabla Q \big| + \big| F_{Z_2} \pi_N \nabla_{Z_1} \nabla Q \big| \\
\leq &\ \big| F_{Z_1} \nabla_{Z_2} \nabla Q \big| + \big| \nabla_{Z_1} \nabla_{Z_2} \nabla Q \big| + \big| F_{Z_2} \pi_N \nabla_{Z_1} \nabla Q \big| \\
\leq &\ c_Q |Z_1||Z_2| \big|\nabla Q \big|.
\end{split}\end{aligned}$$ By (\[equation733\]), (\[equation734\]) and above we see (\[equation732\]) holds.
\[lemma710\] For any $H>0$, there exists $c_P = c_P(H)>0$ such that if $|h| \leq H$ and $\delta \leq 1$, then for any $x \in {\widetilde}{X}$, $$\begin{aligned}
\label{equation735}
\sum_{l=1}^s \big| e^{{\overline}{\rho_l(h)}} \nabla^{(j)} F_l^{(\delta)} (x) \big| \leq c_P\ (j=1, 2, 3).\end{aligned}$$ Moreover, if $(u, h)$ is a solution to a local model over $B_r$ such that $\big\| h\big\|_{L^\infty(B_r)} \leq H$ and $u(B_r) \subset {\widetilde}{X}_B^1$, then for any $z \in B_r$, $$\begin{aligned}
\label{equation736}
\big| \pi_T( {\overline}\partial_A u (z)) \big| \leq c_P.\end{aligned}$$
The first estimate follows from the hypothesis $|h(z)|\leq H$ and ([**P3**]{}) of Hypothesis \[hyp28\]. By the equation ${\overline}\partial_A u + {\widetilde}{W}(u) = 0$ we have $$\begin{aligned}
\pi_T ({\overline}\partial_A u) = - \pi_T( \nabla {\widetilde}{W} (u)) = - \pi_T ( \nabla W' (u)) = - \beta \pi_T \big( \sum_{l=1}^s e^{{\overline}{\rho_l(h)}} \nabla F_l^{(\delta)}(u) \big).\end{aligned}$$ Then (\[equation736\]) follows from (\[equation735\]) with $j=1$.
Now we consider the Hessian of ${\widetilde}{W}_{h}$ in ${\widetilde}{X}_B^1$. With respect to the decomposition $NB \oplus TB$, we write $$\begin{aligned}
\nabla^2 W_h = \left( \begin{array}{cc} E_1 & E_2 \\ E_3 & E_4 \end{array} \right),\ \nabla^2 {\widetilde}{W}_h = {\widetilde}{E} = \left( \begin{array}{cc} {\widetilde}{E}_1 & {\widetilde}{E}_2 \\ {\widetilde}{E}_3 & {\widetilde}{E}_4 \end{array} \right).\end{aligned}$$ On the other hand, with respect to the splitting $T {\widetilde}{X} = {{\mathbb}C} \oplus TX$, we have $$\begin{aligned}
\label{equation737}
\nabla^2 W_h = e^{{\overline}{\rho_0(h)}} \left( \begin{array}{cc} 0 & \left(\nabla Q \right)^*\\
\nabla Q & {\overline}{p} \nabla^2 Q
\end{array} \right).\end{aligned}$$
\[lemma711\]For any $H>0$, there exist $c_H>0$, $D_1 = D_1(H)>0$ and a compact subset ${\widetilde}{K}_1:= {\widetilde}{K}_1(H) \subset {\widetilde}{X}$ such that if $|h|\leq H$ and $x \in {\widetilde}{X}_B^{D_1} \setminus {\widetilde}{K}_1$, then $$\begin{aligned}
\big| {\widetilde}{E}_i(z, x) \big| \leq c_H \big( 1 + \big| \nabla {\widetilde}{W}_h (z, x) \big| \big),\ \big| {\widetilde}{E}_i(z, x) \big| \leq {1\over 6} \big| {\widetilde}{E}_1(z, x) \big|.\end{aligned}$$ In particular, $$\begin{aligned}
\big| {\widetilde}{E}_1(z, x) \big| \geq {1\over 2} \big| {\widetilde}{E}(z, x) \big|.\end{aligned}$$
By the definition of ${\widetilde}{E}_i$ and (\[equation737\]), we see that for $i = 2, 3, 4$, $$\begin{aligned}
\begin{split}
\big| {\widetilde}{E}_i \big| \leq &\ \big| e^{{\overline}{\rho_0(h)}} {\overline}{p} \nabla^2 Q \big| + \big| \nabla^2 W_h' \big| \\
\leq &\ c_Q \big| e^{{\overline}{\rho_0(h)}} {\overline}{p} \nabla Q \big| + c_P(H)\\
\leq &\ c_Q \big| \nabla {\widetilde}{W}_h \big| + c_Q c_P(H) + c_P(H).
\end{split}\end{aligned}$$ Therefore the first inequality holds by choosing $c_H$ properly. On the other hand, if $x \in {\widetilde}{X}_B^D$ and $D \leq 1/2 c_Q$, then by (\[equation737\]), $$\begin{aligned}
\begin{split}
\big| {\widetilde}{E}_1 \big| \geq &\ \big| E_1 \big| - \big| \nabla^2 W_h' \big|\\
\geq &\ \big| e^{{\overline}{\rho_0(h)}} \nabla Q \big| - \big| e^{{\overline}{\rho_0(h)}} {\overline}{p} \nabla^2 Q \big| - c_P(H)\\
\geq &\ \big| e^{{\overline}{\rho_0(h)}} \nabla Q \big| - c_Q D \big| e^{{\overline}{\rho_0(h)}} \nabla Q \big| - c_P(H)\\
\geq &\ {1\over 2} \big| e^{{\overline}{\rho_0(h)}} \nabla Q \big| - c_P(H).
\end{split}\end{aligned}$$ Moreover, since $|h|\leq H$, we can take ${\widetilde}{K}(H)$ sufficiently big such that if $x \notin {\widetilde}{K}(H)$, then $$\begin{aligned}
\big| e^{{\overline}{\rho_0(h)}} \nabla Q(x) \big| \geq e^{-|\rho_0(H)|} \big| \nabla Q(x)\big| \geq 50 c_P(H).\end{aligned}$$ We take $D \leq 1/ 24c_Q$. Then for $i = 2, 3, 4$, we have $$\begin{aligned}
\begin{split}
\big| {\widetilde}{E}_i\big| \leq &\ c_Q \big| e^{{\overline}{\rho_0(h)}} {\overline}{p} \nabla Q \big| + c_P(H)\\
\leq &\ c_Q D \big| e^{{\overline}{\rho_0(h)}} \nabla Q \big| + c_P(H)\\
\leq &\ 2 c_Q D \big| {\widetilde}{E}_1 \big| + 2 c_Q c_P(H) D + c_P(H)\\
\leq &\ 2 c_Q D \big| {\widetilde}{E}_1 \big| + 2c_P(H)\\
\leq &\ {1\over 6} \big| {\widetilde}{E}_1 \big|.
\end{split}\end{aligned}$$
\[lemma712\] For any $H>0$, there exists $M_2(H)>0$ such that if $(u, h)$ is a solution to a local model over $B_r$ satisfying (\[equation79\]) and $\big\| h\big\|_{L^\infty(B_r)} \leq H$, $u(B_\rho) \subset {\widetilde}{X}_B^1$. Then for any $z \in B_\rho$, we have $$\begin{aligned}
\label{equation738}
\big| {\overline}\partial_A u(z) \big| \leq 4 \big| {\overline}\partial_A u(0) \big| \leq 8 \big| \pi_N ( {\overline}\partial_A u (0) ) \big|\end{aligned}$$
Indeed, by (\[equation79\]), $$\begin{aligned}
\big| {\overline}\partial_A u(z)\big| = {{\mathfrak}p}'(z) \leq {{\mathfrak}p}(z) \leq 2 {{\mathfrak}p}(0) \leq 4 {{\mathfrak}p}'(0) = 4 \big| {\overline}\partial_A u(0) \big|.\end{aligned}$$ On the other hand, by (\[equation79\]) and (\[equation735\]), we have $$\begin{aligned}
M_2 (H) /2 \leq \big| {\overline}\partial_A u(0) \big| \leq \big| \pi_N({\overline}\partial_A u(0))\big| + c_P.\end{aligned}$$ So the second inequality of (\[equation738\]) holds if $M_2 (H)$ is big enough.
Let $D = D_1 (H)/2$ where $D_1(H)$ comes from Lemma \[lemma711\]. Then by Lemma \[lemma73\], for certain $M_2 > 0$ and ${\widetilde}{\lambda}_2>0$, if $(u, h)$ satisfies (\[equation79\]) for this $M_2$ and $u(0) \in {\widetilde}{X}_B^D$, then ${\rm diam} \big( u(B_{{\widetilde}\lambda_2 r}) \big) \leq D$. Then we can take $M_2 = M_2(H)>0$ big enough such that $u(B_{{\widetilde}\lambda_2 r}) \subset {\widetilde}{X}_B^{2D} \setminus {\widetilde}{K}_1 (H)$ and (\[equation738\]) is satisfied. Then we can apply the estimates obtained in Lemma \[lemma710\], \[lemma711\], \[lemma712\].
Abbreviate ${\overline}\partial_A u = V$ and $\pi_N({\overline}\partial_A u) = V_N$. Then we have $$\begin{aligned}
\label{equation739}
\begin{split}
{1\over 2} \Delta \big( \big| V_N \big|^2 \big) = &\ {1\over 2} \partial_s^2 \big\langle V_N, V_N \big\rangle + {1\over 2} \partial_t^2 \big\langle V_N , V_N \big\rangle \\
= &\ \partial_s \big\langle D_{A, s} V_N, V_N \big\rangle + \partial_t u \big\langle D_{A, t} V_N, V_N \big\rangle\\
= &\ \big| D_{A, s} V_N \big|^2 + \big| D_{A, t} V_N \big|^2 + \big\langle \big( D_{A, s}^2 + D_{A, t}^2 \big) V_N, V_N\big\rangle.
\end{split}\end{aligned}$$ Then we have $$\begin{aligned}
\begin{split}
&\ \big( D_{A, s}^2 + D_{A,t}^2 \big) V_N \\
= &\ D_{A, s} \big( D_{A, s} - J D_{A, t} \big) V_N + J D_{A, t} \big( D_{A, s} - J D_{A, t} \big) V_N + J \big[ D_{A, s}, D_{A, t} \big] V_N\\
= &\ 4 D_A^{0,1} D_A^{1, 0} V_N + J \big[ D_{A, s}, D_{A, t} \big] V_N \\
= &\ 4 D_A^{0,1} D_A^{1,0} V_N + J R(v_s, v_t) V_N + J \nabla_{V_N} {{\mathcal}X}_{F_A}.
\end{split}\end{aligned}$$ By ([**X2**]{}) and ([**X4**]{}) of Hypothesis \[hyp21\], there exist $c_R, c_\mu >0$ such that $$\begin{aligned}
\Big| \big\langle J R(v_s, v_t) V_N, V_N \big\rangle \Big| \leq c_R \big| V_N\big|^2 \big|d_A u \big|^2.\end{aligned}$$ $$\begin{aligned}
\Big| \big\langle J \nabla_{V_N} {{\mathcal}X}_{F_A}, V_N \big\rangle \Big| \leq \big| V_N \big|^2 \big| F_A \big| \big| \nabla^2 \mu \big| \leq c_\mu \big| F_A \big| \big| V_N \big|^2.\end{aligned}$$
Abbreviate $\partial = \partial / \partial z$, ${\overline}\partial = \partial / \partial {\overline}{z}$, $\partial {\overline}\partial = \partial^2 / \partial z \partial {\overline}{z}$. Then We have $$\begin{aligned}
\label{equation743}
\begin{split}
D_A^{0,1} D_A^{1,0} V_N = &\ - D_A^{0,1} D_A^{1, 0} \pi_N \big( \nabla {\widetilde}{W}_h \big) \\
= &\ - D_A^{0,1} \Big( \big( \nabla_V \pi_N \big) \nabla {\widetilde}{W}_h + \pi_N \big( \nabla_V \nabla {\widetilde}{W}_h + \partial \beta \nabla W_h' (u) \big) \Big) \\
= &\ \big[ D_A^{0,1}, \nabla_{ \nabla {\widetilde}{W}_h} \pi_N \big] \nabla {\widetilde}{W}_h + \big( \nabla_{ \nabla {\widetilde}{W}_h} \pi_N \big) D_A^{0,1} \nabla {\widetilde}{W}_h \\
&\ - \big( \nabla_{\partial_A u} \pi_N \big) \big( \nabla_V \nabla {\widetilde}{W}_h + \partial \beta \nabla W_h' (u) \big) \\
&\ + \pi_N \Big( D_A^{0,1} \nabla_{\nabla {\widetilde}{W}_h} \nabla {\widetilde}{W}_h - \partial {\overline}\partial \beta \nabla W_h'(u) - \partial \beta D_A^{0,1} \nabla W_h'(u) \Big).
\end{split}\end{aligned}$$ We estimate the above expression term by term.
\(I) By (1) of Lemma \[lemma79\], for any tangent vector field $Z_1$ and $Z_2$, we have $$\begin{aligned}
\label{equation744}
\begin{split}
\big[ D_{A, s}, \nabla_{Z_1} \pi_N \big] (Z_2) = &\ \big( \nabla^2_{v_s, Z_1} \pi_N \big) (Z_2) + \big( \nabla_{D_{A, s} Z_1} \pi_N \big) (Z_2),\\
\big[ D_{A, t}, \nabla_{Z_1} \pi_N \big] (Z_2) = &\ \big( \nabla^2_{v_t, Z_1} \pi_N \big) (Z_2) + \big( \nabla_{D_{A, t} Z_1} \pi_N \big) (Z_2).\\
\end{split}\end{aligned}$$ Therefore, $$\begin{aligned}
\begin{split}
&\ \Big| \big[ D_A^{0,1}, \nabla_{ \nabla {\widetilde}{W}_h} \pi_N \big] \nabla {\widetilde}{W}_h \Big|\\
\leq &\ \Big( \big| \nabla_{D_A^{1,0} \nabla {\widetilde}{W}_h } \pi_N \big| + \big|\nabla^2_{v_s, \nabla {\widetilde}{W}_h} \pi_N \big| + \big|\nabla^2_{v_t, \nabla {\widetilde}{W}_h} \pi_N \big| \Big) \big| V \big| \\
\leq &\ c_Q \Big( \big| D_A^{1,0} \nabla {\widetilde}{W}_h \big| + \big| d_A u \big| \big| \nabla {\widetilde}{W}_h \big|\Big) \big| V \big| \\
\leq &\ c_Q \Big( c_P + \big| \nabla_V \nabla {\widetilde}{W}_h \big| + \big| \nabla {\widetilde}{W}_h \big| \big|d_A u \big| \Big) \big| V \big| \\
\leq &\ c_Q \Big( \big| {\widetilde}{E}_1 \big| \big| V_N \big| + \big| {\widetilde}{E}_2 \big| \big| \pi_T(V) \big| + c_P + \big| \nabla {\widetilde}{W}_h \big| \big|d_A u \big| \Big) \big| V \big|\\
\leq &\ c_Q \Big( \big| {\widetilde}{E} \big| \big| V_N \big| + c_P \big| {\widetilde}{E} \big| + c_P + \big| d_A u \big|^2 \Big) \big| V\big|.
\end{split}\end{aligned}$$ We briefly explain how we obtain this estimate. To derive the first inequality, we used (\[equation744\]) and (2) of Lemma \[lemma79\]; to derive the second inequality we used (2) of Lemma \[lemma79\]; to derive the third inequality we used the expression of $D_A^{1,0} \nabla {\widetilde}{W}_h$ in (\[equation317\]). Then $$\begin{aligned}
\begin{split}
&\ \big\langle \big[ D_A^{0,1}, \nabla_{ \nabla {\widetilde}{W}_h} \pi_N \big] \nabla {\widetilde}{W}_h, V_N \big\rangle\\
\geq &\ - c_Q \big| {\widetilde}{E} \big| \big| V \big| \big| V_N \big|^2 - c_Q c_P \big| {\widetilde}{E} \big| \big| V \big| \big| V_N \big| - c_Q \big( c_P + \big| d_A u \big|^2 \big) \big| V \big| \big| V_N \big|\\
\geq &\ - {1\over 64} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - (4 c_Q)^2 \big| \nabla {\widetilde}{W}_h \big|^2 \big| V_N \big|^2\\
&\ - {1\over 64} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - (4 c_Q c_P)^2 \big| V \big|^2 - c_Q \big( c_P + \big| d_A u\big|^2 \big) \big| V \big|^2\\
\geq &\ -{1\over 32} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - c_{P, Q}^{(1)} \big( 1 + \big| d_A u \big|^2 \big) \big| V_N(0) \big|^2.
\end{split}\end{aligned}$$ Here $c_{P, Q}^{(1)}>0$ depends on $c_P$ and $c_Q$ and the last inequality uses Lemma \[lemma712\].
\(II) For the second summand of (\[equation743\]), we have $$\begin{aligned}
\begin{split}
&\ \big| \big( \nabla_{\nabla {\widetilde}{W}_h} \pi_N \big)\big( D_A^{0,1} \nabla {\widetilde}{W}_h \big) \big|\\
\leq &\ c_Q \big| V \big| \big| D_A^{0,1} \nabla {\widetilde}{W}_h \big|\\
\leq &\ c_Q \big| V \big| \Big( c_P + \big| \partial_A u \big| \big| \nabla^2 {\widetilde}{W}_h \big| + \big| dh''\big| \big( c_P + \big| \nabla W_h \big| \big) \Big)\\
\leq &\ c_Q \big| V \big| \big| {\widetilde}{E} \big| \big| \partial_A u \big| + c_Q \big| V \big| \big( c_P + 2 c_P \big| dh'' \big| + \big| dh'' \big| \big| V \big| \big)\\
\leq &\ c_Q \big| {\widetilde}{E} \big| \big| V \big| \big| d_A u \big| + c_Q \big| V \big| \big( c_P + (c_P)^2 + 2 \big| dh'' \big|^2 + \big| d_A u \big|^2 \big).
\end{split}\end{aligned}$$ Here the first inequality uses (2) of Lemma \[lemma79\]; the second one uses the expression of $D_A^{0,1} \nabla {\widetilde}{W}_h$ in (\[equation318\]) and the bound on perturbation terms given by Lemma \[lemma710\]. Then $$\begin{aligned}
\begin{split}
&\ \big\langle \big( \nabla_{\nabla {\widetilde}{W}_h} \pi_N \big)\big( D_A^{0,1} \nabla {\widetilde}{W}_h \big), V_N \big\rangle\\
\geq & - c_Q \big| {\widetilde}{E} \big| \big| V_N\big| \big| V\big|\big| \partial_A u\big| - c_Q \big( c_P + (c_P)^2 + 2 \big| dh''\big|^2 + \big| d_A u\big|^2 \big) \big| V \big|^2\\
\geq & - {1\over 64} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - (4 c_Q)^2 \big| d_A u \big|^2 \big| V\big|^2 - c_Q \big( c_P + (c_P)^2 + 2 \big| dh''\big|^2 + \big| d_A u\big|^2 \big) \big| V \big|^2\\
\geq & - {1\over 64} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - c_{P, Q}^{(2)} \big( 1 + \big| dh'' \big|^2 + \big| d_A u \big|^2 \big) \big| V_N(0)\big|^2.
\end{split}\end{aligned}$$ Here $c_{P,Q}^{(2)}>0$ depends on $c_P$ and $c_Q$ and the last inequality uses Lemma \[lemma712\].
\(III) For the third summand of (\[equation743\]), we have $$\begin{aligned}
\begin{split}
&\ - \big\langle \big( \nabla_{\partial_A u } \pi_N \big) \big( \nabla_V \nabla {\widetilde}{W}_h + (\partial \beta) \nabla W_h' \big), V_N \big\rangle\\
\geq &\ -c_Q \big| \partial_A u \big| \big| \nabla_V \nabla {\widetilde}{W}_h \big| \big| V_N\big| - c_P c_Q \big| \partial_A u \big| \big| V_N\big|\\
\geq &\ - c_Q \big| {\widetilde}{E}\big| \big| V_N \big| \big| d_A u \big| \big| V \big| - c_P c_Q \big| d_A u \big| \big| V \big| \\
\geq &\ - {1 \over 64} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - ( 4 c_Q)^2 \big| d_A u \big|^2 \big| V \big|^2 - c_P c_Q \big| d_A u \big| \big| V \big|\\
\geq &\ - {1 \over 64} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - c_{P, Q}^{(3)} \big( 1 + \big| d_A u \big|^2 \big) \big| V_N(0) \big|^2.
\end{split}\end{aligned}$$ Here the first inequality we used (2) of Lemma \[lemma79\] and the bound on $\nabla W_h'$ given by Lemma \[lemma710\]; $c_{P, Q}^{(3)}>0$ depends on $c_P$ and $c_Q$ and the last inequality uses Lemma \[lemma712\].
\(IV) For the fourth summand of (\[equation743\]), we have $$\begin{aligned}
\begin{split}
&\ - \big\langle \pi_N \big( (\partial {\overline}\partial \beta) \nabla W_h' + (\partial \beta) D_A^{0,1} \nabla W_h' \big), V_N \big\rangle\\
\geq &\ -c_P \big| V_N\big| - \big| \partial_A u \big| \big| \nabla^2 W_h' \big|\big| V_N \big| - c_P \big| dh'' \big| \big| V_N \big|\\
\geq &\ - c_{P, Q}^{(4)} \big( 1 + \big| dh'' \big|^2 \big) \big| V_N(0) \big|^2.
\end{split}\end{aligned}$$ Here we used the bounds on $\nabla W_h'$ and $\nabla^2 W_h'$ given by Lemma \[lemma710\], and an expression of $D_A^{0,1} \nabla W_h'$ similar to (\[equation318\]). The last inequality uses Lemma \[lemma712\].
\(V) Lastly, the dominating part of (\[equation743\]) is estimated as follows.
\[lemma713\] There exist a constant $c_H^{(5)} > 0$, a compact subset ${\widetilde}{K}_2 ={\widetilde}{K}_2(H)\subset {\widetilde}{X}$ which depends on $H>0$ such that if a solution $(u, h)$ to a local model over $B_\rho$ (for some $\rho>0$) satisfies (\[equation738\]) and $$\begin{aligned}
\big\| h \big\|_{L^\infty(B_\rho)} \leq H,\ u(B_\rho) \subset {\widetilde}{X}_B^{D_1} \setminus {\widetilde}{K}_2,\end{aligned}$$ then $$\begin{aligned}
\label{equation750}
\big\langle \pi_N \big( D_A^{0,1} \nabla_{\nabla {\widetilde}{W}_h} \nabla {\widetilde}{W}_h \big), V_N \big\rangle \geq {1\over 16} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - c_H^{(5)} \big( 1 + \big| d_A u \big|^2 + \big| dh'' \big|^2 \big)\big| V_N(0)\big|^2.\end{aligned}$$
This lemma is proved at the very end of this section.
Without loss of generality, we may take $M_2(H)$ big enough so that $u(B_{{\widetilde}\lambda_2 r}) \subset {\widetilde}{X}_B^{2D} \setminus {\widetilde}{K}_2$. Then by (\[equation739\])–(\[equation750\]), we see that there is a constant $c(H)>0$ $$\begin{aligned}
\begin{split}
{1\over 2} \Delta \big| V_N \big|^2 \geq &\ - \big| V_N \big|^2 \big( c_R \big| d_A u \big|^2 + c_\mu \big| F_A \big| \big) \\
- & {1\over 32} \big| {\widetilde}{E} \big|^2 \big| V_N\big|^2 - c_{P, Q}^{(1)} \big| V_N(0) \big|^2 \big( 1+ \big| d_A u \big|^2 \big)\\
- & {1 \over 64} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - c_{P, Q}^{(2)} \big| V_N(0) \big|^2 \big( 1 + \big| dh'' \big|^2 + \big| d_A u \big|^2 \big)\\
- & {1\over 64} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - c_{P, Q}^{(3)} \big| V_N(0) \big|^2 \big( 1+ \big| d_A u \big| \big)^2 \\
- & c_{P, Q}^{(4)} \big| V_N(0) \big|^2 \big( 1 + \big| dh'' \big|^2 \big)\\
+ & {1\over 16} \big| {\widetilde}{E}\big|^2 \big| V_N \big|^2 - c_{P, Q}^{(5)} \big| V_N(0) \big|^2 \big( 1 + \big| dh'' \big|^2 + \big| d_A u \big|^2 \big)\\
\geq &\ - c(H) \big| V_N(0) \big|^2 \big( 1 + \big| dh'' \big|^2 + \big| d_A u \big|^2 + \big| F_A \big| \big).
\end{split}\end{aligned}$$
### Proof of Lemma \[lemma713\] {#proof-of-lemma-lemma713 .unnumbered}
We need the following estimate on the tensor field ${\widetilde}{H}_A$ defined locally by (\[equation319\])–(\[equation320\]).
\[lemma714\] For each $H>0$, there exist $c_H>0$ and a compact subset ${\widetilde}{K}_2(H)\subset {\widetilde}{X}$ satisfying the following conditions.
Let $(\beta, \sigma, \delta)$ be a parameter of a local model on $B_\rho$ and $(u, h) \in C^\infty( B_\rho, {\widetilde}{X}\times {{\mathfrak}g} \times {{\mathfrak}g})$. Suppose $$\begin{aligned}
\big\| h \big\|_{L^\infty(B_r)} \leq H,\ u(B_\rho) \subset {\widetilde}{X} \setminus {\widetilde}{K}_2(H).\end{aligned}$$ Then for any smooth vector field $Z$ along $u$, we have $$\begin{aligned}
\big| {\widetilde}{H}_A (u, d_A u, Z) \big| \leq c_H \big( \big| d_A u \big| + \big| d h'' \big| \big) \big| \nabla^2 {\widetilde}{W}_h \big| \big| Z \big|.\end{aligned}$$
By (\[equation319\]) and Lemma \[lemma710\], it is easy to see that $$\begin{aligned}
\label{equation751}
\big| \beta \sum_{l =1}^s H_A^{(l)}(u, d_A u, Z) \big| \leq c_P \big( \big| d_A u \big| + \big| dh'' \big| \big) \big| Z \big|.\end{aligned}$$ Therefore we only have to consider $H_A^{(0)}$. By the expression of $H_A^{(0)}$, we see $$\begin{aligned}
\begin{split}
\big| e^{{\overline}{\rho_0(h)}} \rho_0( {\bm i }\partial_s h'' - \partial_t h'' ) \nabla_Z \nabla F_0 \big| \leq &\ r e^{rH} \big|d h''\big| \big| \nabla^2 W_h \big| \big| Z \big|;\\
\big| e^{{\overline}{\rho_0(h)}} \rho_0(\partial_s h'' + {\bm i} \partial_t h'' ) \nabla_Z \nabla F_0 \big| \leq &\ r e^{rH} \big|d h'' \big| \big| \nabla^2 W_h \big| \big| Z \big|.
\end{split}\end{aligned}$$ On the other hand, the term $e^{{\overline}{\rho_0(h)}} G_{F_0}(v_s, Z)$ (resp. $e^{{\overline}{\rho_0 (h)}} G_{F_0} (v_t, Z)$) is bounded by $|v_s|$ (resp. $|v_t|$) times the third order derivative of $W_h$ in the vertical direction. By ([**Q1**]{}) in Hypothesis \[hyp25\], we have (up to certain universal or dimensional constants) $$\begin{aligned}
\big| \nabla^3 F_0 \big| \leq \big| p \nabla^3 Q \big| + \big| \nabla^2 Q \big| \leq c_Q \big| p \nabla^2 Q \big| + c_Q \big| \nabla Q \big| \leq c_Q \big| \nabla^2 F_0 \big|.\end{aligned}$$ Then we have $$\begin{aligned}
\label{equation753}
\big| e^{{\overline}{\rho_0 (h)}} G_{F_0} (v_s, Z) \big| \leq c_Q \big| \nabla^2 W_h \big| \big| v_s \big| \big| Z \big|,\ \big| e^{{\overline}{\rho_0 (h)}} G_{F_0} (v_t, Z) \big| \leq c_Q \big| \nabla^2 W_h \big| \big| v_t \big| \big| Z\big|.\end{aligned}$$ Since $\big| \nabla^2 W_h \big| \geq \big| e^{{\overline}{\rho_0(h)}} \nabla Q \big|$ and $\big| \nabla^2 W_h' \big| \leq c_P(H)$, there exists ${\widetilde}{K}_2(H)$ such that outside ${\widetilde}{K}_2(H)$, $1 \leq \big| \nabla^2 W_h \big| \leq 2 \big| \nabla^2 {\widetilde}{W}_h \big|$. Then the lemma follows from (\[equation751\])–(\[equation753\]).
Now we can prove Lemma \[lemma713\]. By the definition of ${\widetilde}{H}_A$ and (\[equation321\]) we have $$\begin{aligned}
\label{equation754}
\begin{split}
&\ D_A^{0,1} \nabla_{\nabla {\widetilde}{W}_h} \nabla {\widetilde}{W}_h \\
= &\ \nabla_{D_A^{1,0} \nabla {\widetilde}{W}_h} \nabla {\widetilde}{{\mathcal}W}_A + {\overline}\partial \beta \nabla_{\nabla {\widetilde}{W}_h} \nabla W_h' + {\widetilde}{H}_A^{0,1} \big( u, d_A u, \nabla {\widetilde}{W}_h \big)\\
= &\ \nabla_{\nabla_{{\overline}\partial_A u} \nabla {\widetilde}{W}_h} \nabla {\widetilde}{W}_h + {\overline}\partial \beta \nabla_{\nabla {\widetilde}{W}_h} \nabla W_h'+ \nabla_{ \partial \beta \nabla {{\mathcal}W}'_A } \nabla {\widetilde}{W}_h + {\widetilde}{H}^{0,1}_A \big( u, d_A u, \nabla {\widetilde}{W}_h \big).
\end{split}\end{aligned}$$ The summands of (\[equation754\]) can be estimated as follows.
\(I) We have $$\begin{aligned}
\begin{split}
&\ \big\langle \nabla_{\nabla_{{\overline}\partial_A u} \nabla {\widetilde}{W}_h} \nabla {\widetilde}{W}_h, V_N \big\rangle \\
= &\ \big\langle \nabla_{V_N} \nabla {\widetilde}{W}_h, \nabla_{{\overline}\partial_A u} \nabla {\widetilde}{W}_h \big\rangle \\
= &\ \big| {\widetilde}{E}_1 (V_N) \big|^2 + \big| {\widetilde}{E}_2 (V_N)\big|^2 + \big\langle {\widetilde}{E}_1 (V_N), {\widetilde}{E}_3 (\pi_T (V)) \big\rangle + \big\langle {\widetilde}{E}_2 (V_N), {\widetilde}{E}_4 (\pi_T (V)) \big\rangle\\
\geq &\ {3\over 4} \big| {\widetilde}{E}_1 (V_N) \big|^2 - \big| {\widetilde}{E}_3 (\pi_T (V)) \big|^2 - {1\over 4} \big| {\widetilde}{E}_4 ( \pi_T (V)) \big|^2\\
\geq &\ {3\over 16} \big| {\widetilde}{E} \big| \big|V_N \big|^2 - \big( c_P c_H \big( 1 + \big| V \big| \big)\big)^2.
\end{split}\end{aligned}$$ To derive the last inequality we used Lemma \[lemma711\].
\(II) The terms in (\[equation754\]) containing the cut-off function $\beta$ can be controlled as follows. $$\begin{aligned}
\begin{split}
\big\langle {\overline}\partial \beta \nabla_{\nabla {\widetilde}{W}_h} \nabla W_h', V_N \big\rangle \geq &\ - c_P \big| \nabla {\widetilde}{W}_h \big| \big| V_N \big| \geq - c_P \big| V_N \big| \big| d_A u \big|;\\
\big\langle \nabla_{\partial \beta \nabla W_h'} \nabla {\widetilde}{W}_h, V_N \big\rangle = &\ \big\langle \nabla_{V_N} \nabla {\widetilde}{W}_h, (\partial \beta) \nabla W_h' \big\rangle\\
\geq &\ - c_P \big| {\widetilde}{E}_1 (V_N) \big| - c_P \big| {\widetilde}{E}_2 (V_N) \big| \\
\geq &\ - 2 c_P \big| {\widetilde}{E} \big| \big| V_N \big|\\
\geq &\ - {1\over 16} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - (4c_P)^2.
\end{split}\end{aligned}$$ Here the first estimate follows from the uniform bound on $\nabla^2 W_h'$ and the second estimate follows from Lemma \[lemma710\] and \[lemma711\].
\(III) By Lemma \[lemma714\], we have $$\begin{aligned}
\label{equation757}
\begin{split}
&\ \big\langle {\widetilde}{H}_A^{0,1} \big( u, d_A u, \nabla {\widetilde}{W}_h \big) , V_N \big\rangle \\
\geq &\ - c_{P, Q}^{(6)} \big( \big|d_A u \big| + \big| d h'' \big| \big) \big| {\widetilde}{E} \big| \big| \nabla {\widetilde}{W}_h \big| \big| V_N \big|\\
\geq &\ -{1 \over 16} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - (4 c_{P, Q}^{(6)} )^2 \big| V \big|^2 \big( \big| d_A u \big|^2 + \big|dh''\big|^2 \big).
\end{split}\end{aligned}$$
Then by (\[equation754\])–(\[equation757\]) and redefining $c_{P, Q}^{(6)} > 0$ properly, we have $$\begin{aligned}
\big\langle D_A^{0,1} \big( \nabla_{\nabla {\widetilde}{W}_h} \nabla {\widetilde}{W}_h \big), V_N \big\rangle \geq {1\over 16} \big| {\widetilde}{E} \big|^2 \big| V_N \big|^2 - c_{P, Q}^{(6)} \big| V_N(0) \big|^2 \big( 1 + \big| d_A u \big|^2 + \big| dh'' \big|^2 \big).\end{aligned}$$ So Lemma \[lemma713\] is proved.
Proof of the compactness theorem {#section8}
================================
The uniform $C^0$-bound
-----------------------
In this subsection, we show that the “bubbling at infinity” won’t happen and solutions to the gauged Witten equation are uniformly bounded everywhere. The argument is based on a maximal principle near the point where the bubbling may happen [*a priori*]{}. Similar estimates appear [@Cieliebak_Gaio_Salamon_2000 Page 859], [@Cieliebak_Gaio_Mundet_Salamon_2002 556] and [@FJR1 Page 780].
Let ${{\mathcal}F}= {{\mathcal}F}_{b_0}: {\widetilde}{X} \to {{\mathbb}R}$ be the $G$-invariant function in Lemma \[lemma24\].
\[lemma81\] Let ${{\mathcal}H}$ be the Hessian of ${{\mathcal}F}$. Then we have $$\begin{aligned}
\label{equation81}
{{\mathcal}H}(J\cdot, J\cdot) = {{\mathcal}H}(\cdot, \cdot).\end{aligned}$$ Moreover, as a quadratic form on $TX$, we have $$\begin{aligned}
\label{equation82}
0 \leq {{\mathcal}H} \leq 1.\end{aligned}$$
Since $J$ is integrable, for any tangent vector $V$, we have $$\begin{gathered}
{{\mathcal}H} (JV, JV) = JV (JV {{\mathcal}F}) - \left(\nabla_{JV} JV\right) {{\mathcal}F} = JV \omega( {{\mathcal}X}_\upxi, JV) - \omega({{\mathcal}X}_\xi, J \nabla_{JV} V ) \\
=JV \langle {{\mathcal}X}_\upxi, V \rangle - \langle {{\mathcal}X}_\upxi, \nabla_{JV} V \rangle = \langle \nabla_{JV} {{\mathcal}X}_\upxi, V \rangle.\end{gathered}$$ Replacing $JV$ by $V$, we have $$\begin{aligned}
{{\mathcal}H} (V, V) = \left\langle \nabla_{ J(-JV)} {{\mathcal}X}_\upxi, -JV\right\rangle = \left\langle \nabla_V {{\mathcal}X}_\upxi, -JV \right\rangle = \left\langle \nabla_{JV} {{\mathcal}X}_\upxi, V \right\rangle.\end{aligned}$$ The last equality is true because ${{\mathcal}X}_\upxi$ is Killing. Therefore (\[equation81\]) holds. On the other hand, (\[equation82\]) follows from ([**X4**]{}) of Hypothesis \[hyp21\] and the definition of ${{\mathcal}F}$.
Since ${{\mathcal}F}$ is $G$-invariant, it lifts to a function ${{\mathcal}F}: Y \to {{\mathbb}R}$. We have
\[prop82\] For each $E>0$, there exist $c(E)>0$ such that for any solution $(A, u)$ to the perturbed gauged Witten equation with $E(A, u)\leq E$, we have $$\begin{aligned}
\Delta_c {{\mathcal}F}(u) \geq {\sigma \over 2c_0} \big| \mu(u)\big|^2 - c(E).\end{aligned}$$ Here $\sigma: \Sigma^* \to {{\mathbb}R}_+$ is the ratio of the smooth metric over the cylindrical metric of $\Sigma^*$ and $c_0>0$ is the constant in Lemma \[lemma24\].
Near any $q \in \Sigma^*$, we use the local model of the perturbed gauged Witten equation so $(A, u)$ gives a solution $(u, h): B_{r^*} \to {\widetilde}{X} \times {{\mathfrak}g} \times {{\mathfrak}g}$ to a local model parametrized by $(\beta, \sigma,\delta)$. Moreover, there exists $H = H(E)$ such that $\big\| h \big\|_{L^\infty(B_{r^*})} \leq H(E)$.
Over $B_{r^*}(q)$, the cylindrical area form can be expressed as $\tau ds dt$ where $\tau: B_{r^*} \to {{\mathbb}R}$, which is uniformly bounded from above and uniformly bounded away from zero. Then by the equation, we have $$\begin{aligned}
\label{equation83}
\begin{split}
\tau \Delta_c {{\mathcal}F}(u) = &\ \partial_s \big\langle \nabla {{\mathcal}F}, v_s \big\rangle + \partial_t \big\langle \nabla{{\mathcal}F} , v_t \big\rangle \\
= &\ \partial_s \big\langle \nabla {{\mathcal}F}, - J v_t -2 \nabla {\widetilde}{{\mathcal}W}_A(u) \big\rangle + \partial_t \big\langle \nabla {{\mathcal}F}, J v_s+ 2 J \nabla {\widetilde}{{\mathcal}W}_A (u) \big\rangle\\
= &\ - 2 \partial_s \big\langle \nabla {{\mathcal}F}, \nabla {\widetilde}{{\mathcal}W}_A(u) \big\rangle + 2 \partial_t \big\langle \nabla {{\mathcal}F}, J \nabla {\widetilde}{{\mathcal}W}_A(u) \big\rangle\\
&\ + \partial_s \big\langle \nabla {{\mathcal}F}, - J v_t \big\rangle + \partial_t \big\langle \nabla F , J v_s \big\rangle.
\end{split}\end{aligned}$$ Now by Lemma \[lemma81\] and (\[equation311\]) we have $$\begin{aligned}
\label{equation84}
\begin{split}
&\ \partial_s \big\langle \nabla {{\mathcal}F}, - J v_t \big\rangle + \partial_t \big\langle \nabla{{\mathcal}F}, J v_s \big\rangle \\
= &\ {{\mathcal}H} \big( v_s , - J v_t \big) + {{\mathcal}H} \big( v_t , J v_s \big) + \big\langle \nabla {{\mathcal}F}, - J D_{A, s} v_t + J D_{A, t} v_s \big\rangle\\
= &\ 2 {{\mathcal}H} \big( \partial_A u, \partial_A u \big) - 2 {{\mathcal}H} \big( J {\overline}\partial_A u, J {\overline}\partial_A u \big) + \big\langle \nabla {{\mathcal}F}, - J {{\mathcal}X}_{F_A} \big\rangle\\
= &\ 2 {{\mathcal}H} \big( \partial_A u , \partial_A u \big) - 2 {{\mathcal}H} \big( {\overline}\partial_A u, {\overline}\partial_A u \big) + \sigma \big\langle \nabla {{\mathcal}F}, J {{\mathcal}X}_{\mu^*} \big\rangle.
\end{split}\end{aligned}$$ In the second identity we used (\[equation311\]) and in the third equality we used the vortex equation $F_A + \sigma \mu^*(u) ds dt = 0$.
On the other hand, denote ${{\mathcal}W}_{A, b}' = d {{\mathcal}W}_A' \cdot \nabla {{\mathcal}F} $ where $d$ denotes the differential in the vertical direction. Then $$\begin{aligned}
\label{equation85}
\begin{split}
&\ \partial_s \big\langle \nabla {{\mathcal}F}, -2\nabla {\widetilde}{{\mathcal}W}_A \big\rangle + \partial_t \big\langle \nabla {{\mathcal}F}, + 2 J \nabla {\widetilde}{{\mathcal}W}_A \big\rangle \\
= &\ -4 {\rm Re} \Big[ {\partial \over \partial {\overline}{z}} \big\langle\big\langle \nabla {{\mathcal}F}, \nabla {\widetilde}{{\mathcal}W}_A(u) \big\rangle\big\rangle \Big] \\
= &\ -4 {\rm Re} \Big[ {\partial \over \partial {\overline}{z}} {{\mathcal}W}_A (u) + {\partial \over \partial {\overline}{z}} (\beta {{\mathcal}W}_{A, b}'(u)) \Big] \\
= &\ -4 {\rm Re} \Big[ d {{\mathcal}W}_{A} \cdot {\overline}\partial_A u + \beta d {{\mathcal}W}_{A,b}' \cdot {\overline}\partial_A u + {\partial \beta \over \partial {\overline}{z}} {{\mathcal}W}_{A,b}'(u) \Big] \\
= &\ - 4 {\rm Re} \Big[ d {\widetilde}{{\mathcal}W}_A \cdot {\overline}\partial_A u + \beta d \big( {{\mathcal}W}_{A, b}' - {{\mathcal}W}_A' \big) \cdot {\overline}\partial_A u + {\partial \beta\over \partial {\overline}{z}} {{\mathcal}W}_{A, b}'(u) \Big] \\
= &\ 4 \big| {\overline}\partial_A u \big|^2 - 4 {\rm Re} \Big[ \beta d \big( {{\mathcal}W}_{A, b}' - {{\mathcal}W}_A' \big) \cdot {\overline}\partial_A u + {\partial \beta \over \partial {\overline}{z}} {{\mathcal}W}_{A,b}'(u) \Big].
\end{split}\end{aligned}$$ Then (\[equation83\])–(\[equation85\]) imply that $$\begin{aligned}
\label{equation86}
\begin{split}
\tau \Delta_c {{\mathcal}F}(u) = &\ 2{{\mathcal}H} \big( \partial_A u, \partial_A u \big) - 2{{\mathcal}H}\big( {\overline}\partial_A u , {\overline}\partial_A u \big) + \sigma \big\langle \nabla {{\mathcal}F}, J{{\mathcal}X}_{\mu^*} \big\rangle + 4 \big| {\overline}\partial_A u \big|^2\\
&\ - 4 {\rm Re} \Big[ \beta d \big( {{\mathcal}W}_{A, b}' - {{\mathcal}W}_A' \big) \cdot {\overline}\partial_A u + {\partial \beta \over \partial {\overline}{z}} {{\mathcal}W}_{A, b}'(u) \Big]\\
\geq &\ 2 {{\mathcal}H} \big( \partial_A u, \partial_A u \big) - 2{{\mathcal}H} \big( {\overline}\partial_A u , {\overline}\partial_A u \big) + \sigma \big\langle \nabla {{\mathcal}F}, J {{\mathcal}X}_{\mu^*} \big\rangle + 2 \left| {\overline}\partial_A u \right|^2 \\
&\ - 2 |\beta|^2 \big| d {{\mathcal}W}_A'(u) - d {{\mathcal}W}_{A, b}'(u) \big|^2 - 4 {\rm Re} \Big[ { \partial \beta \over \partial {\overline}{z}} {{\mathcal}W}_{A, b}'(u) \Big]\\
\geq &\ - 2 |\beta|^2 \big| d {{\mathcal}W}_A'(u) - d {{\mathcal}W}_{A, b}'(u) \big|^2+ \sigma \big\langle \nabla {{\mathcal}F}, J {{\mathcal}X}_{\mu^*} \big\rangle - 4 {\rm Re} \Big[ {\partial \beta \over \partial {\overline}{z}} W_{A, b}'(u) \Big].
\end{split}\end{aligned}$$ Here the second inequality follows from (\[equation82\]). Moreover, by Lemma \[lemma710\] and ([**P3**]{}) of Hypothesis \[hyp28\] there exists $c(E)>0$ depending on $E$ such that $$\begin{aligned}
\big| d {{\mathcal}W}_A'(u) - d {{\mathcal}W}_{A, b}'(u) \big|^2 \leq c(E),\ \big| {{\mathcal}W}_{A, b}'(u) \big| \leq c(E) \sqrt{ 1+ \big|\mu(u) \big|}.\end{aligned}$$ Then with the symbol $c(E)$ abusively used, we have $$\begin{aligned}
\begin{split}
\tau \Delta_c {{\mathcal}F}(u) \geq &\ \sigma \big\langle \nabla {{\mathcal}F}, J{{\mathcal}X}_{\mu^*} \big\rangle - c(E) |\beta|^2 - c(E) \big| d\beta \big| \sqrt{ 1+ \big| \mu(u) \big| }\\
\geq &\ {\sigma \over c_0} \big| \mu(u) \big|^2 - c(E)|\beta|^2 - c(E) \big|d\beta\big| \big| \mu(u) \big|\\
\geq &\ {\sigma \over 2c_0} \big| \mu(u) \big|^2 - c(E)|\beta|^2 - { c(E) |d\beta|^2 \over 2 \sigma}\\
\geq &\ {\sigma \over 2c_0} \big| \mu(u) \big|^2 - c(E).
\end{split}\end{aligned}$$ Here the last inequality follows from the fact that $d\beta$ is controlled by $\sigma$.
Now we can prove the uniform bound on the section.
\[thm83\] For every $E>0$, there exists $K(E) > 0$ such that for every solution $(A, u)$ to the perturbed gauged Witten equation with $E(A, u) \leq E$, we have $$\begin{aligned}
\big\| {{\mathcal}F}(u) \big\|_{L^\infty(\Sigma^*)} \leq K(E).\end{aligned}$$
For any bounded solution $(A, u)$ to the perturbed gauged Witten equation over $\vec{{\mathcal}C}$, the function ${{\mathcal}F}(u)$ extends continuously to $\Sigma$, thanks to Theorem \[thm42\]. Moreover, the value of ${{\mathcal}F}(u)$ at every broad punctures is uniformly bounded because the limit at each broad puncture $z_j$ lies in a uniformly bounded subset of ${\widetilde}{X}_{\upgamma_j}$.
Suppose the statement is not true, then there exists a sequence $(A^{(i)}, u^{(i)})$ of solutions to the perturbed gauged Witten equation over $\vec{{\mathcal}C}$ with $E(A^{(i)}, u^{(i)})\leq E$ such that $$\begin{aligned}
\label{equation88}
\lim_{i \to \infty} \big\| {{\mathcal}F}(u^{(i)}) \big\|_{L^\infty(\Sigma)} = +\infty.\end{aligned}$$ By Corollary \[cor72\], there is a subsequene of the sequence $(A^{(i)}, u^{(i)})$ (still indexed by $i$), and sequences of points $\{z^{(i)}_\beta\}_{1\leq \beta\leq m}$ contained in $\Sigma^*$ which satisfy (3) of Corollary \[cor72\]. In particular, for each $\beta$ and each small $r>0$, the restriction of ${{\mathcal}F}(u^{(i)})$ to $\partial B_r(z_i)$ is uniformly bounded. Then, apply the mean value estimate to ${{\mathcal}F}(u^{(i)} )$ restricted to $B_r(z_i)$, with the first differential inequality in Proposition \[prop82\], we see that $$\begin{aligned}
{{\mathcal}F}\big( u^{(i)}(z_\beta^{(i)}) \big) \leq \max_{ \partial B_r(z_\beta^{(i)} )} {{\mathcal}F}\big( u^{(i)} \big) + { C(E) \over 8 \pi}r.\end{aligned}$$ This contradicts with the divergence of ${{\mathcal}F}(u^{(i)} (z_\beta^{(i)}))$. Therefore, $m=0$; by Corollary \[cor72\], it means on any compact subset of the complement of narrow punctures, ${{\mathcal}F}(u^{(i)} )$ is uniformly bounded.
On the other hand, for any bounded solution $(A, u)$, any $r>0$ sufficiently small and any narrow puncture $z_j$, we define $$\begin{aligned}
K_j(A, u) = \sup_{\partial {\widetilde}{B}_r(z_j)} {{\mathcal}F}(u).\end{aligned}$$ Here ${\widetilde}{B}_r(z_j)$ is the radius $r$ disk around $z_j$ with respect to the smooth metric. We also take $$\begin{aligned}
K' = \sup \Big\{ {{\mathcal}F}(x)\ |\ x\in {\widetilde}{X},\ \big| \mu(x) \big| \leq c_0 \Big\}\end{aligned}$$ where $c_0$ is the one in (\[equation24\]). Since $\mu$ is proper, $K'(E)$ is finite. We claim that for each narrow puncture $z_j$, $$\begin{aligned}
\label{equation89}
\sup_{B_r(z_j)} {{\mathcal}F}(u) \leq \max \big\{ K_j(A, u), K' \big\}. \end{aligned}$$ Then it leads to a contradiction with (\[equation88\]).
Indeed, take $w \in B_r(z_j)$ with ${{\mathcal}F}(u(w)) = \sup_{B_r(z_j)} {{\mathcal}F}(u) > \max \{ K_j(A, u), K'\}$. If $w\neq z_j$, then near $w$ we have $$\begin{aligned}
\Delta_c {{\mathcal}F}(u) \geq \sigma \big\langle \nabla {{\mathcal}F}(u), J{{\mathcal}X}_{\mu^*(u)} \big\rangle \geq {\sigma \over c_0} \big( \big|\mu(u) \big|^2 - c_0^2 \big) > 0.\end{aligned}$$ So ${{\mathcal}F}(u)$ is subharmonic near $w$ and hence ${{\mathcal}F}(u)$ is constant on $B_r(z_j)$. It contradicts with the definition of $K_j(A, u)$. On the other hand, if $w = z_j$, choose $\tau\in (0, r)$ such that $\inf_{B_\tau(z_j)} {{\mathcal}F}(u) \geq K'$. Let ${\widetilde}\Delta$ be the Laplacian with respect to the smooth metric on ${\widetilde}{B}_r(z_j)$. Then for any $\kappa \in (0, \tau)$, by the divergence formula, we have $$\begin{aligned}
\label{equation810}
\int_{{\widetilde}{B}_r(z_j) \setminus {\widetilde}{B}_\kappa(z_j)} {\widetilde}\Delta {{\mathcal}F}(u) = \int_{\partial {\widetilde}{B}_r(z_j)} {\partial \over \partial \rho} {{\mathcal}F}(u) - \int_{\partial {\widetilde}{B}_\kappa (z_j)} {\partial \over \partial \rho} {{\mathcal}F}(u).\end{aligned}$$ Here $\rho$ is the radial coordinate on ${\widetilde}{B}_r(z_j)$. By the exponential convergence of $u$ near each puncture (Theorem \[thm43\]), there is an $\alpha>0$ such that $| \partial_\rho {{\mathcal}F}(u) | \leq \rho^{\alpha -1}$. Therefore let $\kappa$ go to zero in (\[equation810\]), we see $$\begin{aligned}
\int_{\partial {\widetilde}{B}_\tau (z_j)} {\partial \over \partial \rho} {{\mathcal}F}(u) = \int_{{\widetilde}{B}_\tau (z_j)} {\widetilde}\Delta {{\mathcal}F}(u) \geq \int_{{\widetilde}{B}_r(z_j)} \big\langle \nabla {{\mathcal}F}, J{{\mathcal}X}_{\mu^*} \big\rangle \geq {1\over c_0} \int_{{\widetilde}{B}_\tau(z_j)} \big( \big| \mu(u)\big|^2 -c_0^2 \big) \geq 0.\end{aligned}$$ Then since ${{\mathcal}F}(u)$ attains maximum at $z_j$, ${{\mathcal}F}(u)$ is a constant on $B_r(z_j)$. This constant is bigger than $K_j(A, u)$, which is a contradiction. Therefore (\[equation89\]) holds.
Proof of Theorem \[thm65\]
--------------------------
\[prop84\] For every $E>0$, with abuse of notation, there exists $K(E) > 0$ such that for every solution $(A, u)$ to the perturbed gauged Witten equation (\[equation221\]) with $E(A, u) \leq E$, we have $$\begin{aligned}
\big\| d_A u \big\|_{L^\infty(\Sigma^*)} \leq K(E).\end{aligned}$$
The uniform bound on the section $u$ implies that the inhomogeneous term of the Witten equation is uniformly bounded. Then it is a standard argument to extract a subsequence from any sequence of solutions with energy uniform bound, such that the subsequence bubbles off a non-constant holomorphic sphere. However, since the target space ${\widetilde}{X}$ is symplectically aspherical, this is impossible.
Now we can prove Theorem \[thm65\]. Suppose $(A^{(i)}, u^{(i)})$ is a sequence of smooth solutions to the perturbed gauged Witten equation over $\vec{{\mathcal}C}$ with $$\begin{aligned}
\sup_i E\big( A^{(i)}, u^{(i)} \big) = E < \infty.\end{aligned}$$ As we did in the proof of Proposition \[prop216\], there is a sequence of (smooth) gauge transformations $g^{(i)}\in {{\mathpzc}G}$ such that $(g^{(i)})^* A^{(i)}$ is in Coulomb gauge, relative to some reference connection $A_0$. We replace $A^{(i)}$ by $(g^{(i)})^* A^{(i)}$. On the other hand, by Theorem \[thm83\], there is a $G$-invariant compact subset ${\widetilde}{K} \subset {\widetilde}{X}$ such that the images of $u^{(i)}$ are contained in $ P\times_G {\widetilde}{K}$. So by the equation $* F_{A_i} + \mu^* (u_i) = 0$, the curvature form has uniformly bounded $L^\infty$-norm. Therefore elliptic estimate shows that $A^{(i)}$ converges to some $A \in {{\mathpzc}A}$ in weak $W^{1, p}$-topology. In particular, the monodromy and residue of $A$ at each $z_j$ is the same as that of each $A^{(i)}$. The weak convergence implies that $A$ is also in Coulomb gauge relative to $A_0$.
Therefore, by the continuous dependence of ${\widetilde}{{\mathcal}W}_A$ on $A \in {{\mathpzc}A}$, for any compact subset $\Sigma_c \subset \Sigma^*$ and any $G$-invariant compact subset ${\widetilde}{K}$, $ \lim_{i \to \infty} {\widetilde}{{\mathcal}W}_{A^{(i)}} = {\widetilde}{{\mathcal}W}_A$ uniformly on $(P|_{\Sigma_c}) \times_G {\widetilde}{K}$. By the basic compactness about inhomogeneous Cauchy-Riemann equation with Proposition \[prop84\], there is a subsequence (still indexed by $i$) and a section $u \in \Gamma_{loc}^{1, p}(Y)$ such that $u_i$ converges to $u$ in $W_{loc}^{1, p}$-topology. Moreover, the pair $(A, u)$ satisfies the perturbed gauged Witten equation on $\Sigma^*$. By Proposition \[prop216\] and its proof, the Coulomb gauge condition on $A$ implies that $(A, u)$ is smooth. Bootstrapping shows that the convergence of $(A^{(i)}, u^{(i)})$ to $(A, u)$ is uniform on any compact subset of $\Sigma^*$, together with all derivatives.
Now near each puncture $z_j$, we consider the corresponding cylindrical models on a cylindrical end. By Theorem \[thm42\], for each $z_j$, there exists $\upkappa_j \in {\widetilde}{X}_{\upgamma_j}$, such that $$\begin{aligned}
\lim_{z \to z_j} e^{\lambda_j t} u(z) = \upkappa_j.\end{aligned}$$
It is possible that a sequence of solutions degenerate to a stable solution with a sequence of solitons “attached” at the broad punctures. The situation is almost the same as the situation in Floer theory where a sequence of connecting orbits degenerate to a stable connecting orbits in the limit.
For each broad puncture $z_j$, identify $U_j \simeq \Theta_+$, we express $(A^{(i)}, u^{(i)})$ as a solution ${\bm u}^{(i)} = (u^{(i)}, h^{(i)})$ to a cylindrical model where the connection form is $\phi^{(i)} ds + \psi^{(i)} dt$. We also write $A = d + \phi ds + \psi dt$. Since $A^{(i)}$ converges to $A$ uniformly on any compact subset of $\Sigma^*$, $(\phi^{(i)}, \psi^{(i)})$ converges to $(\phi, \psi)$ uniformly on any compact subset. We define $$\begin{aligned}
f^{(i)}(s, t) = \int_s^{+\infty} \phi^{(i)}(v, t) dv,\ f(s, t) = \int_s^{+\infty} \phi(v, t) dv. \end{aligned}$$ By the symplectic vortex equation we see that $f^{(i)}, f \in W_\delta^{2, p}(\Theta_+)$ for some $\delta>0$. Denote $\cancel{g}^{(i)} = \exp f^{(i)}$, $\cancel{g} = \exp f$. Therefore we can extend $\cancel{g}^{(i)}$ and $\cancel{g}$ to gauge transformations in ${{\mathpzc}G}$ so that $\cancel{g}^{(i)}$ converges to $\cancel{g}$ uniformly on any compact subset of $\Sigma^*$. Therefore, we can absorb $\cancel{g}^{(i)}$ into $g^{(i)}$, and without loss of generality, we may assume that ${\bm u}^{(i)}$ is in temporal gauge.
Now if there is a subsequence (still indexed by $i$) and $S_0>0$ such that $$\begin{aligned}
\sup_{[S_0, +\infty)\times S^1} e \big( {\bm u}^{(i)} \big) \leq \epsilon_2,\end{aligned}$$ where $\epsilon_2 = \epsilon_2 ( {\widetilde}{K}, {\underline}\delta)$ is the one in Theorem \[thm43\]. Then by Theorem \[thm43\] we see that $$\begin{aligned}
\lim_{S \to +\infty} \lim_{i \to \infty} E \big( {\bm u}^{(i)}; \Theta_+(S) \big) = 0.\end{aligned}$$ Therefore in this situation, Theorem \[thm65\] holds.
On the other hand, suppose there is a sequence $(s^{(i)}, t^{(i)})\in \Theta_+$ such that $$\begin{aligned}
\label{equation813}
\lim_{i \to \infty} s^{(i)} \to +\infty,\ e \big( {\bm u}^{(i)}\big) \big( s^{(i)}, t^{(i)} \big) > \epsilon_2.\end{aligned}$$ Then we can extract a subsequence (still indexed by $i$) such that the sequence $$\begin{aligned}
\big( u^{(i)} \big( s + s^{(i)}, t \big), h^{(i)} \big( s + s^{(i)}, t \big) \big)\end{aligned}$$ converges uniformly on compact subsets of $\Theta$ with all derivatives to $(u, 0)$, where $u: \Theta \to {\widetilde}{X}_{\upgamma_j}$ is a soliton with nonzero energy. Indeed, $(u^{(i)}, h^{(i)})$ satisfies the equation $$\begin{aligned}
\partial_s u^{(i)} + J\big( \partial_t u^{(i)} + {{\mathcal}X}_{\psi^{(i)}}(u^{(i)}) \big) + \nabla {\widetilde}{W}_{h^{(i)}, \lambda_j}^{(\delta_j^{(i)})}(u^{(i)}) = 0,\ \Delta (h^{(i)})'' + \sigma \mu^* (u^{(i)}) = 0.\end{aligned}$$ Here $\delta_j^{(i)} = \delta_{j, A^{(i)}} \in (0,1]$. Since all derivatives of $u^{(i)}$ are uniformly bounded, the second equation implies that a subsequence of $h^{(i)}(s + s^{(i)}, t)$ (still indexed by $i$) converges uniformly on any compact subset of $\Theta$ to $0$, together all derivatives. Then $\psi^{(i)}(s + s^{(i)}, t)$ converges uniformly on any compact subset of $\Theta$ to the constant $\lambda$ together will all derivatives. Moreover, $\delta_j^{(i)}$ converges to $\delta_j := \delta_{j, A}$. Therefore the standard argument shows that $u^{(i)}$ converges uniformly with all derivatives to a $(\lambda_j, \delta_j)$-soliton. (\[equation813\]) implies that this soliton must be nontrivial.
More generally, using the same trick as in [@Mundet_Tian_2009 Section 8.5], we define an [**$\epsilon_2 $-bubbling list**]{} to be a sequence of lists $\big\{ z_1^{(i)}, \ldots, z_\alpha^{(i)} \big\}$, satisfying
- $z_l^{(i)} = \big( s^{(i)}_l, t^{(i)}_l \big) \in U_j\simeq \Theta_=$ and $\lim_{i \to \infty} s^{(i)}_l \to +\infty$;
- for $l_1 \neq l_2$, $ d \big( z_{l_1}^{(i)}, z_{l_2}^{(i)} \big) \to +\infty$;
- for each $l$ we have $\displaystyle \liminf_{i \to \infty} e \big( {\bm u}^{(i)} \big)\big( z_l^{(i)} \big) \geq \epsilon_2$.
We call $\alpha\geq 1$ the length of an $\epsilon_2$-bubbling list. Then if we have an $\epsilon_2$-bubbling list, we can extract a subsequence (still indexed by $i$) for which locally near $z_l^{(i)}$ the sequence converges to a nontrivial soliton. It is easy to see, because there are only finitely many critical points of ${\widetilde}{W}_{\upgamma_j}|_{{\widetilde}{X}_{\upgamma_j}}$, the length of any $\epsilon$-bubbling list is uniformly bounded from above. So we take an $\epsilon_2$-bubbling list $\big\{ z_1^{(i)}, \ldots, z_{\alpha_j}^{(i)} \big\}$ of maximal length $\alpha_j$. By taking a subsequence, we may assume that $$\begin{aligned}
l_1 < l_2 \Longrightarrow s_{l_1}^{(i)} < s_{l_2}^{(i)}.\end{aligned}$$ Then we define $$\begin{aligned}
\big( u^{(i)}_l (s, t) , h^{(i)}_l (s, t) \big) = \big( u^{(i)} \big( s + s^{(i)}_l, t \big), h^{(i)} \big( s + s^{(i)}_l, t \big) \big),\ l=1, \ldots, \alpha.\end{aligned}$$ By choosing a further subsequence, we may assume that each $u^{(i)}_l$ converges to a nontrivial soliton $u_l: \Theta \to {\widetilde}{X}_{\upgamma_j}$. Denote ${\bm u}_j =(u_1, \ldots, u_{\alpha_j})$. Then for any sequence of points $z^{(i)} = \big( s^{(i)}, t^{(i)} \big)$, we have $$\begin{aligned}
\lim_{i \to \infty} \min_l d \big( z^{(i)}, z_l^{(i)} \big) = \infty \Longrightarrow \limsup_{i \to \infty} e \big( u^{(i)}, h^{(i)} \big) \big( z^{(i)} \big) < \epsilon_2.\end{aligned}$$ Otherwise it will contradicts with the fact that the $\epsilon_2$-bubbling list is of maximal length. Then it is easy to see that by Theorem \[thm43\], $(u_l)_+ = (u_{l+1})_-$ for $l = 1, \ldots, \alpha_j-1$ and $ev_j(A, u) = (u_1)_-$.
Therefore, the collection $\big( A, u, \{ {\bm u}_j\}_{z_j\ {\rm broad}}\big)$ forms a stable solution to the perturbed gauged Witten equation over $\vec{{\mathcal}C}$. Moreover, the subsequence constructed above and $\big(A, u , \{ {\bm u}_j\}_{z_j\ {\rm broad}} \big)$ satisfy Definition \[defn64\]. Hence the proof of Theorem \[thm65\] is complete.
Epsilon-regularity, etc. {#appendixa}
========================
Epsilon-regularity for Cauchy-Riemann equations
-----------------------------------------------
The Witten equation is an inhomogeneous Cauchy-Riemann equation. In this appendix we recall some basic estimates about Cauchy-Riemann equations.
We first recall the $\epsilon$-regularity result of [@IS_compactness] in the case of $J$-holomorphic curves with a continuous $J$. Let $Y$ be a manifold of dimension $2N$ and $Y' \subset Y$ be a subset. Let $h_0$ be a smooth Riemannian metric on $Y$ which we used as a reference to define the norms on function spaces on $Y$. For any $x\in Y$ and $\delta>0$, we use $B_\delta(x)$ to denote the open geodesic ball centered at $x$ with radius $\delta$.
\[defna1\]([@IS_compactness Definition 1.1]) A continuous almost complex structure $J$ on $Y$ is said to be [**uniformly continuous**]{} on $Y'$ (with respect to $h_0$), if the following is true. 1) $\left\| J \right\|_{L^\infty(Y')} < \infty$; 2) For any $\epsilon>0$, there is a number $\delta>0$ such that for any $x \in Y'$, there exists a $C^1$-diffeomorphism $\phi: B(x, \delta) \to B(0, \delta) \subset {{\mathbb}C}^N$ such that $$\begin{aligned}
\label{equationa1}
\big\| J - \phi^* J_{st} \big\|_{C^0( B_\delta(x)) \cap Y')} + \big\| h_0 - \phi^* h_{st} \big\|_{C^0(B_\delta(x) \cap Y'} < \epsilon,\end{aligned}$$ where $J_{st}$ is the standard complex structure and $h_{st}$ is the standard metric on ${{\mathbb}C}^N$.
For each $\epsilon>0$, the largest $\delta$ for which (\[equationa1\]) is true is called the modulus of uniform continuity, and is denoted by a function $\mu_J(\epsilon)$.
\[lemmaa2\][@IS_compactness Lemma 1.1] Let $J^*$ be a continuous almost complex structure on $Y$ which is uniformly continuous on $A\subset Y$. For every $p \in (2, +\infty)$, there exist constants ${\bm \epsilon_1} = {\bm \epsilon_1} (\mu_{J*}, A, h_0 )>0$, ${\bm \epsilon_p}>0$, ${\bm C_p} = C(p, \mu_{J^*}, A, h_0 )<\infty$ with the following property.
For any continuous almost complex structure $J$ on $Y$ with $\left\| J- J^* \right\|_{L^\infty(A)}< {\bm \epsilon_p}$ and for any $J$-holomorphic map $u \in C^0 \cap L_1^2(B_1, Y)$ such that $u(B_1) \subset A$ and $\left\| du \right\|_{L^2(B_1)} < {\bm \epsilon_1}$, we have $$\begin{aligned}
\label{equationa2}
\big\| du \big\|_{L^p(B_{1\over 2})} \leq {\bm C_p} \big\| du \big\|_{L^2(B_1)}.\end{aligned}$$
By Sobolev embedding $L_1^p \to C^{0, {2\over p} - 1}$, (\[equationa2\]) implies (using the same constant ${\bm C_p}$) $$\begin{aligned}
\label{equationa3}
{\rm diam} \big( u(B_{1\over 2}) \big) \leq {\bm C_p} \big\| du \big\|_{L^2(B_1)}.\end{aligned}$$
A consequence of Lemma \[lemmaa2\] is the following.
\[lemmaa4\] Let $(X, h_0)$ be a Riemannian manifold of dimension $2n$ and $J$ be a continuous almost complex structure on $X$ which is uniformly continuous on the whole (noncompact) manifold $X$ with respect to $h_0$. Then there exists ${\bm \epsilon_2} = {\bm \epsilon_2}( \mu_J, X, h_0 )>0$ satisfying the following condition.
Suppose $\rho \in (0, 1]$, $\nu\in C^0 ( B_\rho \times X, TX)$ and $u: B_\rho \to X$ satisfies the inhomogeneous equation $$\begin{aligned}
\label{equationa4}
{\partial u \over \partial {\overline}{z}} + \nu(u) = 0.\end{aligned}$$ If $$\begin{aligned}
\big\| du \big\|_{L^2(B_\rho)} \leq {\bm \epsilon_2},\ \rho \big\| \nu(u) \big\|_{L^\infty(B_\rho)} \leq {\bm \epsilon_p} {\bm \epsilon_2},\end{aligned}$$ then $$\begin{aligned}
{\rm diam} \big( u ( B_{\rho \over 2}) \big) \leq {\bm C_p} \Big( \big\| du \big\|_{L^2(B_\rho)} + \rho \big\| \nu(u) \big\|_{L^\infty(B_\rho)} \Big).\end{aligned}$$
Denote $Y = {{\mathbb}C} \times X$. Let ${\widetilde}{J}_0 = (J_0, J_{st})$ be the product almost complex structure on $Y$. Then ${\widetilde}{J}_0$ is uniformly continuous on $Y$ with respect to the product metric ${\widetilde}{h}_0 = (h_0, h_{st})$. We take $$\begin{aligned}
{\bm \epsilon_2} = {1\over 1 + \sqrt{\pi}} {\bm \epsilon_1} \Big( \mu_{{\widetilde}{J}_0}, {{\mathbb}C}\times X, {\widetilde}{h}_0 \Big).\end{aligned}$$
Indeed, denote $\kappa = \left\| \nu(u) \right\|_{L^\infty(B_\rho)}$, we define $$\begin{aligned}
\begin{array}{cccc}
{\widetilde}{v}: & B_1 & \to & B_{\rho \kappa \over {\bm \epsilon}_p} \times X\\
& w & \mapsto & \displaystyle \big( { \rho \kappa \over {\bm \epsilon_p}} w, u(\rho w) \big);
\end{array} \hspace{0.3cm}
\begin{array}{cccc}
{\widetilde}{\nu}: & \displaystyle B_{{\rho \kappa \over {\bm \epsilon}_p}} \times X & \to & TX \\
& (w, x) & \mapsto & \displaystyle {{\bm \epsilon}_p \over \kappa } \nu \big( {{\bm \epsilon}_p \over \kappa} w, x \big).
\end{array}\end{aligned}$$ Then define an almost complex structure ${\widetilde}{J}_{{\widetilde}{\nu}}$ on $B_{\rho\kappa \over {\bm \epsilon}_p} \times X$ by $$\begin{aligned}
{\widetilde}{J}_{{\widetilde}{\nu}} (\partial_s, X) = (\partial_t, J_0 X + {\widetilde}\nu ),\ {\widetilde}{J}_{{\widetilde}\nu}(\partial_t, X) = ( - \partial_s, J_0 X - J_0 {\widetilde}\nu).\end{aligned}$$ Then (\[equationa4\]) implies that ${\widetilde}{u}$ is holomorphic with respect to ${\widetilde}{J}_{{\widetilde}\nu}$. On the other hand, we have $$\begin{aligned}
\big\| {\widetilde}{J}_{{\widetilde}\nu} - {\widetilde}{J}_0 \big\|_{L^\infty({\widetilde}{v}(B_1))} \leq {\bm \epsilon_p},\ \big\| d {\widetilde}{v} \big\|_{L^2(B_1)} \leq \big\| du \big\|_{L^2(B_\rho)} + \sqrt{\pi} {\rho\kappa \over {\bm \epsilon}_p} \leq {\bm \epsilon_1}.\end{aligned}$$ Then Lemma \[lemmaa2\] and (\[equationa3\]) imply that $$\begin{aligned}
\big\| dv \big\|_{L^p(B_{1\over 2})} \leq {\bm C_p} \Big( \big\| d u \big\|_{L^2(B_\rho)} + \rho \big\| \nu(u) \big\|_{L^\infty(B_\rho)}\Big).\end{aligned}$$ $$\begin{aligned}
{\rm diam} \big( u ( B_{\rho\over 2} ) \big) = {\rm diam} \big( v ( B_{1\over 2}) \big) \leq {\bm C_p} \Big( \big\| d u \big\|_{L^2(B_\rho)} + \rho \big\| \nu(u) \big\|_{L^\infty(B_\rho)}\Big).\end{aligned}$$ The rescaling relation of $L^p$-norms implies that $$\begin{aligned}
\big\| d u \big\|_{L^p(B_{\rho \over 2})} \leq {\bm C_p} \rho^{{2\over p} - 1} \Big( \big\| d u \big\|_{L^2(B_\rho)} + \rho \big\| \nu(u) \big\|_{L^\infty(B_\rho)}\Big).\end{aligned}$$
Mean value estimates
--------------------
We quote several important mean value estimates for differential inequalities of the Laplace operator on the plane. Let $B_r$ be the radius $r$ open disk in ${{\mathbb}C}$ centered at the origin, with the standard coordinates $(s, t)$. Let $\Delta = \partial_s^2 + \partial_t^2$.
\[lemmaa5\]([@Salamon_lecture Page 156]) Suppose $f: B_r \to {{\mathbb}R}$ with $f(z) \geq 0$ be a smooth function, satisfying $$\begin{aligned}
\Delta f\geq -A - B f^2\end{aligned}$$ where $A\geq 0$, $B >0$. Then $$\begin{aligned}
\int_{B_r} f \leq {\pi \over 16 B} \Longrightarrow f(0) \leq {8 \over \pi r^2} \int_{B_r} f + {A r^2 \over 4}.\end{aligned}$$
Hofer’s lemma
-------------
In proving compactness we used the following lemma, which is due to Hofer.
[@McDuff_Salamon_2004 Lemma 4.6.4]\[lemmaa7\] Let $(X, d)$ be a metric space, $f: X \to {{\mathbb}R}$ be a non-negative continuous function. Suppose $x \in X$, $\delta>0$ and the closed ball ${\overline}{B}_{2\delta}(x) \subset X$ is complete. Then there exists $\xi \in X$, $\epsilon \in (0, \delta]$ such that $$\begin{aligned}
d(x, \xi) < 2\delta,\ \sup_{B_\epsilon(\xi)} \leq 2 f(\xi),\ \epsilon f(\xi) \geq \delta f(x).\end{aligned}$$
Equivariant topology {#appenxib}
====================
Suppose $G$ is a compact Lie group, $N$ is a $G$-manifold and $P\to M$ is a principal $G$-bundle over a closed oriented manifold $M$, then any continuous section $s$ of the associated bundle $P\times_K N$ defines a cycle in the Borel construction $N_G$, which represents an equivariant homology class $$\begin{aligned}
s_*[M] \in H_{{\rm dim}M}^G (N; {{\mathbb}Z}).\end{aligned}$$ In this current paper, we would like to define such an equivariant fundamental class for any solution $(A, u)$ to the perturbed gauged Witten equation by using the section $u$. However, since the monodromy of the $r$-spin structure at the punctures could be nontrivial, the image of the section $u$ is an equivariant cycle in $X$ only in the orbifold sense. So the contribution from the cylindrical ends $U_j$ should be weighted by a rational weight, and the fundamental class of a solution $(A, u)$ should be a class $$\begin{aligned}
\big[ A, u \big] \in H_2^G \big( {\widetilde}{X}; {{\mathbb}Z}[r^{-1} ] \big).\end{aligned}$$ We will carry this out explicitly in this subsection.
We first recall a general way of defining a rational fundamental class of an orbifold section of an associated bundle over an orbicurve. We assume that the reader is familiar with the notion of orbicurves (orbifold Riemann surfaces) and orbifold bundles over an orbicurve, so we will be sketchy when referring to such structures.
We assume that we have a compact Riemann surface $\Sigma$ with several distinct punctures $z_1, \ldots, z_k$. An orbifold chart near $z_j$ with local group $\Gamma_j \simeq {{\mathbb}Z}_{r_j}$ is a holomorphic map $$\begin{aligned}
\pi_j: {{\mathbb}D} \to \Sigma\end{aligned}$$ which maps $0$ to $z_j$ and can be expressed as $\zeta \mapsto \zeta^{r_j} $ in local coordinates. A collection of orbifold charts $\{\pi_j\}_{j=1}^k$ define an orbicurve structure. An equivalence relation can be defined among orbifold charts, and an equivalence class is called an orbicurve ${{\mathcal}C}$.
Now suppose for each $j$, we have an injective homomorphism $\chi_j: {{\mathbb}Z}_{r_j} \to G$. An orbifold $G$-bundle over ${{\mathcal}C}$ is a usual $G$-bundle over $\Sigma^*:= \Sigma \setminus \{z_1, \ldots, z_k\}$, together with a collection of “bundle charts” $$\begin{aligned}
\left( {\widetilde}\pi_j, \pi_j \right) : \left( {{\mathbb}D}^*\times G, {{\mathbb}D}^* \right) \to \left( P|_{\Sigma^*}, \Sigma^* \right),\ j=1, \ldots, k,\end{aligned}$$ where $\pi_j: {{\mathbb}D}^* \to \Sigma^*$ extends to an orbifold chart near $p_j$ and ${\widetilde}\pi_j$ covers $\pi_j$; moreover, ${\widetilde}\pi_j$ is invariant under the $\Gamma_j$-action on the left by $\upgamma \cdot( \zeta, k) = \left( \upgamma \zeta, \chi_j (\upgamma) k \right)$. An equivalence class of orbifold bundle charts defines an orbifold $G$-bundle ${{\mathcal}P}\to {{\mathcal}C}$. As a topological space, ${{\mathcal}P}$ is $$\begin{aligned}
{{\mathcal}P}:= P^* \cup \big( \bigcup_{j=1}^k {{\mathbb}D} \times G \big) /\sim\end{aligned}$$ with the equivalence relation generated by $p \sim (\zeta, k)$ if ${\widetilde}\pi_j(\zeta, k) = p$.
Now if $N$ is a $G$-manifold, we can have an “orbifold associated bundle” ${{\mathcal}Y}:= {{\mathcal}P}\times_G N$, which contains the usual associated bundle $Y^*:= P^* \times_G N$ as a proper subset. Each bundle chart ${\widetilde}\pi_j$ induces a chart ${\widetilde}\pi_j^N: {{\mathbb}D}^* \times N \to Y^*$ by $$\begin{aligned}
{\widetilde}\pi_j^N( \zeta, x ) = \left[ {\widetilde}\pi_j(\zeta, 1), x \right],\end{aligned}$$ which is invariant under the $\Gamma_j$-action $\upgamma(\zeta, x) = ( \upgamma \zeta, \upgamma x)$.
Suppose we have a continuous section $u: \Sigma^* \to Y^*$, identified with an equivariant map $U: P^* \to N$. Then the composition $$\begin{aligned}
U \circ {\widetilde}\pi_j: {{\mathbb}D}^* \times G \to N\end{aligned}$$ is again a $G$-equivariant map and invariant under the $\Gamma_j$-action. It can be viewed as a continuous section over the chart ${{\mathbb}D}^* \times N$. If it extends continuous to the origin $0\in {{\mathbb}D}_j$ for all $j$, then we have an orbifold section of ${{\mathcal}Y}\to {{\mathcal}C}$.
Now we can define the rational fundamental class of a continuous orbifold section of ${{\mathcal}Y}$. First, we construct a CW complex out of the orbicurve. The complement $\Sigma \setminus U$ is a surface with boundary, hence we can regard it as a CW complex in such a way that $\partial U$ is a subset of the 1-skeleton of $\Sigma \setminus U$. Then we take $k$ copies of 2-cells ${{\mathbb}D}_j$ and attach it to $\partial U$ by the $r_j$-to-1 map $\zeta_j \mapsto \zeta_j^{r_j}$. This CW complex is denoted by $|{{\mathcal}C}|$. Then, it is easy to see that the singular chain $$\begin{aligned}
\big[ {{\mathcal}C} \big]:= \big[ \Sigma \setminus U \big] + \sum_{j=1}^k {1\over r_j} \big| {{\mathbb}D}_j \big|\end{aligned}$$ defines a rational homology class in $H_2 \big( |{{\mathcal}C}|; {{\mathbb}Z}[ r^{-1}] \big)$, if $r$ is divisible by all $r_j$.
Moreover, the orbibundle charts defines a continuous $G$-bundle $|{{\mathcal}P}| \to |{{\mathcal}C}|$ (in the usual sense); the orbifold section $s$ defines a continuous section $|s|: |{{\mathcal}P}| \to N$. Hence we obtained a continuous map (up to homotopy) $|{{\mathcal}C}| \to N_G$. The pushforward of the rational class $[{{\mathcal}C}]$ is then a class $$\begin{aligned}
s_* \big[ {{\mathcal}C} \big] \in H_2 \big( N_G; {{\mathbb}Z}[r^{-1}] \big) = H_2^G \big( N; {{\mathbb}Z}[r^{-1}] \big).\end{aligned}$$
|
---
abstract: 'We study strain localization in slow shear flow focusing on layered granular materials. A heretofore unknown effect is presented here. We show that shear zones are refracted at material interfaces in analogy with refraction of light beams in optics. This phenomenon can be obtained as a consequence of a recent variational model of shear zones. The predictions of the model are tested and confirmed by 3D discrete element simulations. We found that shear zones follow Snell’s law of light refraction.'
author:
- Tamás Unger
bibliography:
- 'granu.bib'
title: Refraction of shear zones in granular materials
---
The motion of granular materials, such as sand, is difficult to predict, as they behave neither like elastic solids nor like normal fluids. When they yield under stress and start flowing, the material consists of large, almost solid parts, and the relative motion is confined to narrow regions between them, called shear zones or shear bands [@Mueth00; @Fenistein03; @GDRMiDi04]. Shear zones represent material failure and therefore play a crucial role in geophysics (geological faults), in engineering (building foundations) and in industrial processes. It is an important problem of these fields to predict where the failure takes place. This is a difficult task for those shear zones which arise in the bulk of the material far from the confining walls [@Fenistein03; @Fenistein04; @Luding04; @Cheng06; @Unger04a; @Torok06].
Nature often stratifies granular media, i.e. different materials are deposited in distinct layers. We study here the question what effect such inhomogeneities have on shear zones. Based on a recent theory and computer simulations we show that shear zones are refracted when they pass through the interface between different granular layers. Moreover, the angle of refraction obeys a law analogous to Snell’s law of light refraction which reveals an unexpected analogy between granular media and geometric optics. The effect of refraction presented here influences the position of shear zones in various kinds of materials including powders, sand, soil and rock layers providing implications on the behavior of geological faults [@Schultz05; @Scott96].
Recently the variational principle of minimum dissipation was successfully applied [@Unger04a; @Torok06] to describe the non-trivial shape of the shear zone in a homogeneous granular material which is sheared in a modified Couette cell [@Fenistein04; @Luding04; @Cheng06]. Minimum rate of energy dissipation is a widely applied selection principle [@Jaynes80] to determine which steady states are realized in nature. When applying this principle to the cylindrical geometry of the modified Couette cell [@Unger04a], not only the non-trivial position of the shear zone could be calculated without any fitting parameter, but new, closed shear zones were predicted which were discovered also in experiments [@Fenistein06] and simulations [@Cheng06]. In the following, a so far unknown but measurable effect will be derived from the same principle. It concerns the shape of shear zones in layered granular materials.
This phenomenon can be understood best, if the side effects of gravity and curved shearing are absent. Therefore a long cylindrical container is considered, which is cut along its axis into two halves (Fig. \[fig:refraction\]a). (In computer simulations periodic boundary conditions in the axial direction are applied.) The container is completely filled with two granular materials having different frictional properties. They are separated by a planar interface parallel to the axis, but at an angle to the cut of the container. The system is sheared quasi-statically [@GDRMiDi04] by moving the two halfs of the cylinder wall slowly parallel to the axis in opposite directions. This creates a shear zone starting and ending where the container wall is cut. In order to find the shape of the zone in between we apply the principle of minimum dissipation. This, in zero width approximation [@Unger04a], leads to the following variational problem (explained below): $$\int v p {{\mu^\text{eff}}}\text{d}S = \text{min.}$$ Here the local rate of energy dissipation is integrated over the whole surface of the shear zone. The shear zone is regarded as being infinitely thin, outside the shear zone no deformation and no dissipation takes place. The local dissipation rate per unit area $ v p {{\mu^\text{eff}}}$ is obtained by the sliding velocity $v$ between the two sides of the shear zone times the shear stress $p {{\mu^\text{eff}}}$. The parameters $p$ and ${{\mu^\text{eff}}}$ denote the overall pressure in the system and the coarse grained effective friction coefficient [@GDRMiDi04]. The effective friction has different values for the two materials while $v$ and $p$ are taken constant for our setup [^1]. The surface that minimizes the total rate of dissipation provides the shape of the shear zone. This means that the system yields along the surface where it has the least resistance against the external shear.
![\[fig:refraction\] (color online) a, Schematic view showing the shear cell and the refraction of the shear zone. Dense granular material is sheared between two half-cylinders. The shear direction is indicated by the arrows. b, A snapshot of the simulation. The upper light-gray (yellow) material and lower dark-gray (blue) material have different frictional properties (blue beads have stronger friction). If the velocity of any bead drops below half of the external shear velocity, it is overpainted by dark-gray (red). Despite the fluctuations of single bead-velocities, a dark (red) zone appears in the bulk that separates the two moving parts of the system. c, The spatial distribution of the shear rate which shows the structure of the shear zone. Lighter colors represent stronger deformations (orange means the largest shear rate, followed by red and blue, while black means almost no deformation). The straight line indicates the material interface.](Refr4-kpov-B.eps "fig:") ![\[fig:refraction\] (color online) a, Schematic view showing the shear cell and the refraction of the shear zone. Dense granular material is sheared between two half-cylinders. The shear direction is indicated by the arrows. b, A snapshot of the simulation. The upper light-gray (yellow) material and lower dark-gray (blue) material have different frictional properties (blue beads have stronger friction). If the velocity of any bead drops below half of the external shear velocity, it is overpainted by dark-gray (red). Despite the fluctuations of single bead-velocities, a dark (red) zone appears in the bulk that separates the two moving parts of the system. c, The spatial distribution of the shear rate which shows the structure of the shear zone. Lighter colors represent stronger deformations (orange means the largest shear rate, followed by red and blue, while black means almost no deformation). The straight line indicates the material interface.](Diffr.eps "fig:") ![\[fig:refraction\] (color online) a, Schematic view showing the shear cell and the refraction of the shear zone. Dense granular material is sheared between two half-cylinders. The shear direction is indicated by the arrows. b, A snapshot of the simulation. The upper light-gray (yellow) material and lower dark-gray (blue) material have different frictional properties (blue beads have stronger friction). If the velocity of any bead drops below half of the external shear velocity, it is overpainted by dark-gray (red). Despite the fluctuations of single bead-velocities, a dark (red) zone appears in the bulk that separates the two moving parts of the system. c, The spatial distribution of the shear rate which shows the structure of the shear zone. Lighter colors represent stronger deformations (orange means the largest shear rate, followed by red and blue, while black means almost no deformation). The straight line indicates the material interface.](KepernyorolShearzoneShearrateE.eps "fig:")
The system is translation invariant along the axis, therefore the above surface integral is reduced to a line integral. We have to find the path (cross section of the shear zone) which connects two fixed points and along which the integral of ${{\mu^\text{eff}}}$ is minimal. This problem has exactly the form of Fermat’s principle of geometric optics where the effective friction plays the role of the index of refraction. It is known that Fermat’s principle of the shortest travel time leads to refraction of light beams therefore one can expect a similar phenomenon of refraction also for shear zones.
In order to check this prediction we carried out computer simulations of $10^5$ frictional and hard spherical beads (Fig. \[fig:refraction\]b). Based on a standard discrete element method (DEM) [@Jean99; @Brendel04] the motion of each bead is determined by Newton’s equation. This grain-level dynamics leads to a collective steady state flow where we measure the coarse-grained velocity field. The coarse-grained velocities are parallel to the cylinder axis. The local strain rate, which is calculated from the velocity field, corresponds to pure shear deformation (without volumetric strain). The spatial distribution of the magnitude of the shear rate is shown in Fig. \[fig:refraction\]c. As predicted, the arising shear zone departs from the cut plane and its direction is changed significantly at the material interface (Fig. \[fig:refraction\]b, c). Note, that the shear zone we obtained is quite wide relative to the system size. The relative width becomes smaller for larger systems [@Fenistein04] which, however, would demand also a computational effort out of reach. Still, the effect of refraction is clearly shown.
The analogy with light refraction provides quantitative predictions that can be measured in numerical and real experiments. Fermat’s principle indicates that shear zones have to follow Snell’s law of light refraction which will be tested here by DEM simulations. In order to do so we characterize the behavior of the shear zone by two angles $\phi_1$ and $\phi_2$ which are analogous to the incident and refractive angles used in geometric optics. These are defined with the help of the shear deformation (Fig. \[fig:refraction\]c) as follows: first, the refraction point is identified as the point along the interface where the local shear rate has its maximum. Then the refraction point is connected to the starting and ending points of the shear zone by straight lines. Thus the direction of the shear zone on each side of the interface is determined. These directions provide the angles $\phi_1$ and $\phi_2$ with respect to the normal of the interface. Another angle, which is an important input parameter of the shear test, is the tilt angle of the interface $\alpha$ between the normal of the cut plane and the interface (Fig. \[fig:refraction\]c). For various values of $\alpha$ we measured $\phi_1$ and $\phi_2$. If the interface is perpendicular to the cut plane ($\alpha=0$) no refraction was observed, however, as $\alpha$ is increased the refraction of the shear zone becomes more and more pronounced.
If we apply Snell¢s law to our case it indicates that, no matter how the angles $\phi_1$ and $\phi_2$ are changed, the ratio of $\sin{\phi_1}$ and $\sin{\phi_2}$ has to be independent of the tilt angle $\alpha$. Furthermore it is expected that this ratio can be expressed by the effective frictions: $$\frac{\sin{\phi_1}}{\sin{\phi_2}} = \frac{{{\mu^\text{eff}}}_2}{{{\mu^\text{eff}}}_1}$$
First we report the influence of $\alpha$ observed in simulations. We used materials that contained hard spheres of radii distributed uniformly between ${{R_\text{min}}}$ and $1.3 \, {{R_\text{min}}}$. This polydispersity was needed to avoid shear induced crystallization [@Tsai03]. In order to control the frictional properties of the materials we varied the value of the microscopic friction $\mu$ which is the friction coefficient at the particle-particle contacts. It is important to note that the microscopic friction $\mu$ and the effective friction ${{\mu^\text{eff}}}$ are not the same although they are closely related as we will see later. We simulated three systems called $A$, $B$ and $C$. The total number of particles was $100000$ for system $A$ and $50000$ for systems $B$ and $C$. For each case the shear cell was filled with two materials with microscopic frictions $\mu_1$ and $\mu_2$. For systems $A$ and $B$ $\mu_1=0$ and $\mu_2=0.5$ while for system $C$ $\mu_1=0.1$ and $\mu_2=0.5$. For all systems the radius of the container was approximately $65 \, {{R_\text{min}}}$.
The influence of the tilt angle of the interface is presented in Fig. \[fig:Snellslaw\]. The computer simulations show that $\phi_1$ and $\phi_2$ vary strongly but $\sin{\phi_1} / \sin{\phi_2}$ remains constant for a wide range of the angle $\alpha$. This ratio seems to depend only on the materials in which the shear zone was created, in full agreement with the theoretical considerations.
![\[fig:Snellslaw\] (color online) $\phi_1$ (a), $\phi_2$ (b) and the relative index of refraction (c) measured for shear zones in computer simulations. The horizontal axis show the tilt angle of the interface between the two materials. Data points denoted by triangle, square and circle stand for systems A, B and C, respectively. The pairs of microscopic friction that are used in these systems are indicated in the figure. The straight lines show the theoretically predicted values (the upper line for system A and B, the lower one for system C).](CollectedRefrData-R5_C.eps "fig:") ![\[fig:Snellslaw\] (color online) $\phi_1$ (a), $\phi_2$ (b) and the relative index of refraction (c) measured for shear zones in computer simulations. The horizontal axis show the tilt angle of the interface between the two materials. Data points denoted by triangle, square and circle stand for systems A, B and C, respectively. The pairs of microscopic friction that are used in these systems are indicated in the figure. The straight lines show the theoretically predicted values (the upper line for system A and B, the lower one for system C).](CollectedRefrData-R5_A.eps "fig:")
Next we deal with the role of the effective friction. It is known that if a dense noncohesive granular material is sheared then after a short transient [@Craig04] it reaches a well defined resistance against shear deformation provided the flow is slow enough (quasistatic shear). Thus a material parameter, the effective friction coefficient, can be defined which turns out to be independent of the preparation history, the actual shear rate and the confining pressure [@GDRMiDi04; @Radjai04]. The effective friction ${{\mu^\text{eff}}}$ is given by the shear stress divided by the normal stress, both measured in the plane along which the shear deformation takes place.
The quantity ${{\mu^\text{eff}}}$ is not a microscopic input parameter of the computer simulation, thus the question arises what values ${{\mu^\text{eff}}}$ has for the materials that appeared in systems $A$, $B$ and $C$. This can be deduced from simulations which are independent of the previous refraction tests: we put the same materials into a 3D rectangular box under plane shear [@GDRMiDi04] measure the components of the stress tensor [@Christoffersen81] and calculate the value of ${{\mu^\text{eff}}}$. In this way we achieve a calibration curve that provides a one to one correspondence between microscopic and effective friction, see Fig. \[fig:mu-mueff\].
![\[fig:mu-mueff\] Effective friction versus microscopic friction. The open circles correspond to the materials used in systems $A$, $B$ and $C$.](mu-effmu-gray.eps)
With that we arrived to a point where we can make quantitative statements about the extent of the refraction based on Snell’s law. E.g. for system $A$, with help of the prescribed values of $\mu_1$ and $\mu_2$, the ratio ${{\mu^\text{eff}}}_2 / {{\mu^\text{eff}}}_1$ can be calculated and compared to the data $\sin{\phi_1} / \sin{\phi_2}$ recorded in the refraction test. This is done in Fig. \[fig:Snellslaw\] for systems $A$, $B$ and $C$. A surprisingly good agreement is found between the predicted and measured values which holds for all systems and for various tilt angles of the interface. This is our main result.
Dealing with shear localization in layered granular materials we investigated the predictions of a recent variational model and compared them to computer simulations. The results of the present work convey two messages. First, we gave verification of the variational approach to granular shear flows. We tested the model in a new situation and, according to the numerical data, it gave an excellent description of the behavior of the shear zones. Second, an interesting analogy between granular flow and geometric optics is revealed. We showed that shear zones are refracted at material interfaces similarly to light beams. The phenomenon presented here should be accessible by experiments.
We are grateful to J. Kertész and D.E. Wolf for their support and help. Support by grant OTKA T049403 and by the G.I.F. research grant I-795-166.10/2003 is acknowledged.
[^1]: By contrast, $p$ and $v$ depend strongly on the position in the modified Couette cell. This is caused by gravity (for $p$) and by curved shearing (for $v$) [@Unger04a]. For clarity, these effects are avoided in our case.
|
---
abstract: 'Implicit authentication (IA) is gaining popularity over recent years due to its use of user behavior as the main input, relieving users from explicit actions such as remembering and entering passwords. However, such convenience comes at a cost of authentication accuracy and delay which we propose to improve in this paper. Authentication accuracy deteriorates as users’ behaviors change as a result of mood, age, a change of routine, etc. Current authentication systems handle failed authentication attempts by locking the user out of her mobile device. It is unsuitable for IA whose accuracy deterioration induces high false reject rate, rendering the IA system unusable. Furthermore, existing IA systems leverage computationally expensive machine learning which can introduce large authentication delay. It is challenging to improve the authentication accuracy of these systems without sacrificing authentication delay. In this paper, we propose a multi-level privilege control (MPC) scheme that dynamically adjusts users’ access privilege based on their behavior change. MPC increases the system’s confidence on users’ legitimacy even when their behaviors deviate from historical data, thus improving authentication accuracy. It is a lightweight feature added to the existing IA schemes that helps avoid frequent and expensive retraining of machine learning models, thus improving authentication delay. We demonstrate that MPC increases authentication accuracy by 18.63% and reduces authentication delay by 7.02 minutes on average, using a public dataset that contains comprehensive user behavior data.'
author:
-
bibliography:
- 'reference.bib'
title: 'Dynamic Multi-level Privilege Control in Behavior-based Implicit Authentication Systems Leveraging Mobile Devices'
---
Introduction
============
Rich behavioral data gathered by various sensors embedded in smart devices facilitates the implicit authentication of users based on their behaviors [@yang2016personaia; @p9; @p62]. In general, IA systems authenticate a user by matching her real-time behavior to her historical behavior. Real-time behavior is obtained from one or more sensors whose data can uniquely characterize the user and distinguish her from other users, at the time of authentication. Similarly, historical behavior is obtained from the same sensors in the past and updated after new data is collected. IA schemes typically run at the background and stream data at an appropriate frequency to ensure that data is sufficiently collected and the battery consumption is reasonable. As with any other practical security system, IA systems need to strike a good balance between security and usability. However, it is highly challenging to achieve such balance due to the dynamically changing behaviors of users. On the one hand, we need the system to cope with a user’s behavior deviation [@yang2016personaia], e.g., a change of routine, and not falsely reject the user (usability). On the other hand, the system needs to differentiate between a legitimate user’s changed behavior and other users’ behaviors to prevent falsely accepting adversaries (security). Balancing between such false rejects and false accepts improves the authentication accuracy and is the main focus of this paper.
In addition to false reject rate, another important measure of system usability is authentication delay. Authentication delay mainly consists of training delay to obtain historical behaviors and behavior matching delay, which varies across different authentication schemes [@khan2014comparative] and is closely relevant to authentication accuracy. The amount and quality of sensor data collected by the system directly affects the authentication accuracy. Insufficient data collection can result in an inferior historical behavior model that is not representative of a user’s behavior. Low quality data can be caused by noisy behavior data (due to either a legitimate user’s behavior deviation or adversaries) or noisy sensor readings. Authentication delay is typically increased when the system attempts to improve upon the amount and quality of the collected data since retraining of the machine learning [@p67] model and additional data collection are needed. Balancing between authentication accuracy and delay is hence another problem this paper is trying to solve to further enhance usability.
Existing research on IA systems focuses on the effectiveness of IA schemes, i.e., finding suitable behavioral features such as touch, typing, and other motions that uniquely identify users [@p62; @p7; @bo2013silentsense; @feng2014tips; @shi2011senguard; @gurary2016implicit; @shahzad2013secure; @castelluccia2017towards; @p23]. Although authentication accuracy and delay were measured as performance indicators, none of these papers addressed methods to improve them to make the system more user-friendly. We argue that this is a rather important issue to consider since practicality is the key for IA systems to be widely deployed, and provide our solutions in this paper. Specifically, we propose a multi-level privilege control scheme, or MPC, that divides the single privilege level in current systems into multiple fine-grained privilege levels. The privilege levels are used to separate apps based on their level of security so that users can still access the less sensitive apps on their smart devices even if their behaviors change. The levels are dynamically adjusted to reflect the user’s dynamically changing behaviors, and therefore enhancing the system’s authentication accuracy by balancing between false rejects and false accepts. It is challenging to find such a balance because of the difficulty in distinguishing a user’s deviated behaviors from other users’ behaviors. In other words, decreased false reject rate may cause increased false accept rate, and vice versa. A fine line needs to be drawn to lower both rates and boost the system’s confidence on a user’s legitimacy, which requires in-depth analysis of the existing IA schemes and suitable mathematical modeling. Main contributions of this paper are summarized as follows:\
\
$\bullet$ We propose a multi-level privilege control scheme to address the usability of IA systems, a key issue the existing IA schemes are faced with, by improving the authentication accuracy and delay at the same time. The scheme solves the core problem of how to set and adjust the user’s privilege level such that both false reject rate and false accept rate are decreased, where the problem is modeled by applying physical laws that describe the motion of bodies under the influence of a system of forces. To further correct the privilege level adjustment and improve authentication accuracy, we employ a two-factor authentication mechanism in which the secondary factor provides feedback to identify the user’s behavior deviation and filter out noisy sensor readings using Kalman filter. The scheme does not rely on additional data collection and adds no authentication delay. The delay is in fact reduced due to the improve authentication accuracy.
$\bullet$ The proposed MPC can be generally applied to most of the current IA systems as long as the output (behavior scores) can be converted to probabilistic values.
$\bullet$ We demonstrate that MPC increases authentication accuracy by 18.63% and reduces authentication delay by 7.02 minutes on average. The experiments were conducted using a public dataset that contains comprehensive user behavior data, as opposed to proprietary datasets, and thus the repeatability is guaranteed.\
\
Preliminary
===========
In this section, we cover background information on support vector machine (SVM), kernel density estimator, and Kalman filter. The output of SVM needs to be converted to probabilistic values for MPC to handle, which will be discussed in Section \[ilm\]. Kernel density estimator will be used in Section \[da\] to estimate the occurrence frequency of a particular behavior score. Kalman filter filters out behavior data and sensor reading noises and will be elaborated in Section \[da\].
SVM Classifier \[psvm\]
-----------------------
SVM is the most widely adopted technique in IA systems [@frank2013touchalytics; @p65; @bo2013silentsense; @p62; @p66; @p67; @p35; @gascon2014continuous; @alzubaidi2016authentication]. Given a training dataset sampled from a group of people, SVM outputs a hyperplane located in a high dimensional space to cluster the data into two classes, the legitimate class and the illegitimate class. During authentication, new data sampled from the current user is verified according to its position in the hyperplane. The user is deemed legitimate if the new data falls in the legitimate class. In the proposed MPC, we need to calculate the distance between the hyperplane and the testing result in a high dimensional space which renders it difficult with SVM’s traditional output. Instead, we utilize the probability output calculated by fitting a sigmoid function, $\frac{1}{1+exp(Af_i+B)}$, to the margins of the SVM [@platt1999probabilistic], where $A$ and $B$ are the parameters required to estimate and $f_i$ indicates the margins of the SVM output. The probability output of SVM is called behavior score in this paper. Behavior scores represent users’ behaviors in numeric form and are used by the system to deduce users’ legitimacy.
Kernel Density Estimator
------------------------
Kernel density estimator [@scott2008kernel; @measurekernel; @bishop2006pattern] serves as a tool to analyze the usage pattern of the IA system, e.g., legitimate and illegitimate usages in a given time interval, by estimating how often a given behavior score occurs. This is necessary in distinguishing between the legitimate user’s deviated behaviors and other (illegitimate) users’ behaviors. Kernel density estimator divides the interval into small bins with length $h$, in each of which it calculates the number of behavior scores that fall into the bin. A distribution of the behavior scores is obtained by placing a Gaussian over each score and then adding up the contributions over the whole dataset. The kernel density model is $p(x)=\frac{1}{N}\sum_{n=1}^1\frac{1}{(2\pi h^2)^{D/2}}exp{-\frac{||x-x_n||^2}{2h^2}}$, where $D$ indicates $D-dimensional space$, $N$ is the total number of behavior scores, $x_n$ is the behavior score and $x$ indicates the center of each bin.
Kalman Filter
-------------
Kalman filter [@kalman] is employed in MPC to filter out sensor noise and help correct behavior deviation. The two types of noises it assumes, process noise and observation noise, can be used to model behavior deviation (or behavior noise) and sensor noise, respectively, making it an excellent tool for noise filtering in IA systems. In addition, Kalman filter is loop carried which means it automatically filters out noises at the time of authentication, instead of the need for more data to perform the filtering as in the existing literature [@bo2013silentsense; @frank2013touchalytics; @p65; @abramson2013user; @draffin2013keysens; @lee2015implicit; @wang2015understanding]. This property greatly reduces authentication delay.\
\
The Proposed MPC Scheme
=======================
We first provide a high-level overview of our MPC scheme before diving into technical details.
System Overview
---------------
Existing IA schemes such as [@bo2013silentsense; @p62] authenticate users by deriving a behavior score using data samples gathered in a period of time, called time window (or authentication cycle) which is a design-specific parameter. This score, $\epsilon$, is then compared with a threshold, e.g., $0.5$. If the threshold is passed, illegitimate usage is indicated and the system will lock the device. When legitimate and illegitimate users have vastly different behaviors, existing IA schemes can achieve high authentication accuracy. However, based on our preliminary simulation using the Friends and Family Dataset [@p21; @aharony2011social], more than 70% of users’ behavior data samples overlap and cannot be separated by simply setting a threshold . As an example, we randomly selected two participants from the dataset, one as the legitimate user and the other as the illegitimate user, and converted their SVM output to probabilistic behavior scores. The time window is set to 15 seconds. As shown in Fig. \[fig:eva\] (a) and (b), the legitimate and illegitimate users both have a large proportion of behavior scores located around the threshold 0.5 which are inseparable. The behavior overlapping problem can be exacerbated by mimicry attacks where the adversary imitates the legitimate user’s behaviors [@khan2016targeted]. The MPC scheme attempts to improve authentication accuracy even in the presence of this problem, by using the proposed initial mapping, privilege movement and domain expansion mechanisms which will be discussed in this section..
The first step of MPC is to obtain multiple privilege levels using initial mapping. It divides the single privilege level in the existing IA schemes into multiple privilege levels, where each level contains a subset of user installed apps. Users rank their apps based on the apps’ security requirements and map these apps to the privilege levels. For instance, in a system with four privilege levels $R_1$ through $R_4$, apps can be mapped to the levels as shown in Fig. \[fig:Sys\]. Apps with the highest security requirements such as banking, e-commerce, health and fitness, credit score, and password manager are mapped to the highest privilege level $R_1$. Apps with lower security requirements such as social media, contacts, games, and utility apps are mapped to lower levels $R_2$ and $R_3$. $R_4$ is the lowest privilege level which corresponds to locking the device and thus contains no app. Note that the number of privilege levels and the security requirements for the apps are system and user dependent. Although relevant, it is not the focus of initial mapping which is a generic method and will not be further discussed. After obtaining the privilege levels, the system needs to map the user to a specific level $R_c$ based on the user’s current behavior at the time of authentication. The level $R_c$ is called the user’s current level as shown in Fig. \[fig:Sys\]. This is performed in the second step of MPC, privilege movement. A user has access to all the apps contained in $R_c$ and the levels below $R_c$, but not the apps in the levels above $R_c$. Moreover, overlapping behaviors are effectively separated in this step. Finally, domain expansion is introduced to dynamically adjust the domain boundaries as more behavior data become available and filter out behavior and sensor noises.
An ideal IA scheme should always map the legitimate user to $R_1$, and illegitimate users to $R_4$. However, as mentioned before, when the legitimate user’s behavior deviates, it becomes harder to differentiate it from illegitimate users’ behaviors, which is why we need more intermediate levels for such differentiation to avoid locking the legitimate user out immediately. When behavior deviation happens, the legitimate user may be mapped to lower levels and not able to access high-privilege apps. To resume her full access privilege, the user can choose to proactively pass a second-factor authentication, or passively wait for the system to adjust her privilege in the next authentication cycle as more data becomes available. The two methods are captured in the privilege movement and domain expansion steps. Two-factor authentication has gained increasing popularity and deployment since it enhances security. We use it in our IA scheme with a twist, i.e., instead of having to pass the two factors at the same time, the user will be mapped back to $R_1$ if she passes the second factor authentication. The reason for such design is that since IA systems are still in their infancy, understanding their performance limitations is the most important first step before we can mature their design. The second factor serves as a feedback mechanism in MPC to help separate behavior deviation from illegitimate behaviors, and fundamentally improve the system’s false accept and false reject rates. Despite that we use password input as the second factor in this paper, any authentication scheme other than behavior-based IA can be used. Note that password input happens only when there is authentication failure in our system, much less frequently than using password as the main authentication scheme. After gaining enough insight, we will be able to enhance our IA scheme without the second factor in the future.
The majority of current IA research tends to gather their own data from a small number of volunteers [@khan2016targeted; @p65; @bo2013silentsense; @p62], rendering it difficult to repeat their tests. We use the public Friends and Family Dataset that contains comprehensive user behavior data for the presentation and evaluation of our MPC scheme. Specifically, the dataset contains 130 participants’ 8GB data collected in a 5-month period. Data consists of 9 main features: GPS, accelerometer, SMS, app installation, battery usage, call logs, app usage, blue-tooth devices log, and Wi-Fi access points. Some of them have many sub-features, e.g., battery usage includes battery level, plug status, health and brand information. In our tests, we randomly select one user as the legitimate user and use other users’ data as illegitimate behavior data.
![The system architecture.[]{data-label="fig:Sys"}](Architecture.pdf){width="3.6in" height="1.9in"}
Initial Mapping \[ilm\]
-----------------------
We mainly discuss applying MPC to SVM-based IA schemes. For the other IA schemes [@p14; @bo2013silentsense; @yang2016personaia], since their output is already a probabilistic behavior score, MPC can be directly applied.
DEFINITION 1. *Let behavior score $\epsilon \in$ \[0, 1\] denote the probabilistic output of SVM approximated by a two-parameter sigmoid function $\frac{1}{1+exp(Af_i+B)}$. In a specific training set , we further divide the interval \[0, 1\] into $n$ sub-intervals, called domains, denoted by $D_n\subset$ \[0, 1\]. The legitimate domain is the largest sub-interval that contains only true accept (TA) behavior scores. The illegitimate domain is the largest sub-interval that contains only true reject (TR) behavior scores. The slack domain is the sub-interval in between the legitimate domain and illegitimate domain.* The initial mapping mechanism is illustrated in Fig. \[fig:arc\] (a). The system first initializes the value of parameters $\alpha$ and $\beta$ by fitting the sigmoid function to the SVM output trained by data sampled from legitimate and illegitimate users. The legitimate and illegitimate domains are predefined based on these two parameters. Assuming the system has $n$ privilege levels, in each authentication cycle as new data is collected, SVM takes the data as input and outputs a new behavior score indicating the system’s authentication decision. If the new score falls in the legitimate domain, the system will move the user’s current privilege level $R_c$ to $R_1$ (if $R_c\neq R_1$) which grants the user full access. If the new score falls in the illegitimate domain, the system will lock the device. If the new score falls in the slack domain, the system will map $R_c$ to one of the observation levels $R_2$, $R_3$, ..., $R_{n-1}$, where the user has only limited access.
[.5]{} ![Initial mapping and Privilege movement. (a) Initial mapping. (b) Privilege movement.[]{data-label="fig:arc"}](basic.pdf "fig:"){width="2.0in" height="1.65in"}
[.5]{} ![Initial mapping and Privilege movement. (a) Initial mapping. (b) Privilege movement.[]{data-label="fig:arc"}](Movement.pdf "fig:"){width="2.0in" height="1.5in"}
As shown in Fig. \[fig:eva\] (a) and (b), the legitimate and illegitimate domains are \[0, $\alpha$\] and \[$\beta$, 1\], respectively. The slack domain is located in \[$\alpha$, $\beta$\], which contains ambiguous behavior scores that could come from either the legitimate user or illegitimate users and need separation. In a given dataset, we can easily find $\alpha$ and $\beta$ by searching for the largest and smallest behavior score $\epsilon$ derived from the legitimate user’s and illegitimate users’ training data, respectively. In this subsection and the next, we first assume that $\alpha$ and $\beta$ are fixed and focus on the mapping of the current privilege level $R_c$ to one of the observation levels in the slack domain. We then release this assumption in Section \[da\] when we complete our discussion with the possible movement of the domain boundaries. Compared to the existing IA schemes, the initial mapping in MPC balances between security and usability. Since the system only grants full access to the user who is most likely to be legitimate, security is enhanced. When the likelihood declines, instead of completely locking the user out, the system maps the user to an observation level that grants lower access rights. It enhances usability if the user is legitimate while limiting the security breach if the user is illegitimate. Nevertheless, initial mapping only handles failed authentications in a more gradual way by adding the slack domain and observation levels. It does not fundamentally ameliorate the false reject (FR) and false accept (FA) performance which will be the focus of privilege movement and domain expansion.
Privilege Movement \[dj\]
-------------------------
In initial mapping, the current privilege level $R_c$ is mapped to one of the defined privilege levels $[R_1, R_2, ..., R_n]$ when a new behavior score becomes available at the time of authentication and remains in that level until more data comes in. Such a mapping mechanism does not fundamentally improve the FR and FA performance since the system still needs a way to confirm the user’s legitimacy once her behavior score is mapped to the uncertain observation level. Recall that the system’s goal is to eventually grant the user full access if she is legitimate and lock the user out if otherwise. The slack domain is just a buffer for a more smooth transition. We introduce privilege movement in the mapping of $R_c$, where $R_c$ is moved up (towards $R_1$) or down (towards $R_n$) gradually out of the slack domain. We assume it takes the illegitimate users several tries before being able to impersonate (i.e., imitate the behavior or guess the password of) the legitimate user. We also assume that the IA scheme gives high authentication accuracy, i.e., the legitimate and illegitimate users’ behavior scores fall into their corresponding domains rather than the slack domain, when the scheme is newly trained. Authentication accuracy will gradually decline as more behavior data becomes available from either the legitimate user or illegitimate users after training. Retraining of the IA scheme may be needed which is covered in detail in [@p67].
{width="109.90000%" height="0.07\textheight"}
{width="109.90000%" height="0.07\textheight"}
{width="109.90000%" height="0.172\textheight"}
{width="109.90000%" height="0.172\textheight"}
{width="109.90000%" height="0.172\textheight"}
We summarize the privilege movement mechanism in Fig. \[fig:arc\] (b). The system keeps track of the user’s behaviors and once it observes a behavior score that falls into the slack domain, it searches through the previous scores to find a more definitive answer. If there were scores in the legitimate domain, the system leans towards regarding the user as legitimate and moves $R_c$ upward with distance $-\mu_l$ at the end of the current authentication cycle. This process is repeated until $R_c$ reaches $R_1$. However, if the system observes a behavior score that falls into the illegitimate domain instead, $R_c$ is moved downward with distance $+\mu_a$ at the end of the current authentication cycle. This process is repeated until $R_c$ reaches $R_n$. Similarly, if there were scores in the illegitimate domain, the system leans towards regarding the user as illegitimate and moves $R_c$ downward with distance $+\mu_a$ at the end of the current authentication cycle. This process is repeated until $R_c$ reaches $R_n$. However, if the system observes a behavior score that falls into the legitimate domain instead, $R_c$ is moved upward with distance $-\mu_l$ at the end of the current authentication cycle. This process is repeated until $R_c$ reaches $R_1$. If $R_c$ falls in between privilege levels, the user is assumed access privilege of the lower level. The movement distances $-\mu_l$ and $+\mu_a$ are design parameters that can be constants or variables. For the discussion in this subsection, we let $\mu_l=l/2$ and $\mu_a=l$ where $l$ is the fixed distance between privilege levels. The system is thus less tolerable and more restrictive when there is evidence that the current user is illegitimate. It is also more conservative in giving the user higher access privilege when the user’s legitimacy was confirmed in the past but is currently in doubt. Such design is to enhance security while not sacrificing usability. Moreover, the FR and FA performance is improved since the system always tries to move $R_c$ out of the slack domain based on evidence. The privilege movement mechanism has $O(1)$ time complexity, which renders the authentication delay the same as the IA schemes without MPC. In the next subsection, we discuss making $\mu_l$ and $\mu_a$ variables to improve authentication accuracy.
As an example, the behavior score distribution for the legitimate user and illegitimate user is shown in Fig. \[fig:eva\] (c) and (d), respectively, using the aforementioned simulation with two participants. The scores are grouped into five one-hour time slots, where each time slot contains multiple time windows. In each time slot, there are behavior scores belonging to the legitimate/illegitimate domain that co-occur with scores belonging to the slack domain. The scores that belong to the legitimate/illegitimate domain are used as evidence and guidance to move the scores in the slack domain. When behavior deviation happens, initial mapping may map the legitimate user to the observation level and still cause false rejects which are corrected with privilege movement. The same is true for false accepts. In addition, we randomly selected a time slot from Fig. \[fig:eva\] (c) and (d), and magnified it in Fig. \[fig:eva\] (e) where the threshold $\Omega$ is predefined to best separate the two users. For the ease of presentation, we assume that there is only one observation level and three privilege levels in total. In the first time window, the legitimate user’s behavior score falls in the legitimate domain (shown in the figure) but her $R_c$ has not reached $R_1$ (not shown in the figure). The system therefore moves $R_c$ upward for $l/2$. In the second through fourth time window, the score falls in the legitimate domain again but $R_c$ has reached $R_1$. So $R_c$ remains in $R_1$. In the fifth through tenth time window, $R_c$ falls in the slack domain. Since the system observed four behavior scores in the legitimate domain, $R_c$ remains in $R_1$. If the system observed scores in the illegitimate domain instead, $R_c$ would have been moved towards $R_n$. The illegitimate user in Fig. \[fig:eva\] (e) follows a similar privilege movement process. Using the Friend and Family Dataset, we were able to observe the co-occurrence of legitimate/illegitimate-domain behavior scores and slack-domain behavior scores for the same user in a reasonably short period of time (2-3 minutes), in all the two-participant simulations we conducted. There may be cases in reality where a user’s behavior score stays in the slack domain for a very long time without further evidence. We do not investigate this problem in this paper due to the lack of data and will leave it as our future work.
The effectiveness of privilege movement is highly dependent on the size of the legitimate and illegitimate domains. If $\alpha$ and $\beta$ are fixed, they may become less indicative as more behavior data from either the legitimate user or illegitimate users become available. This problem will be coped with in the domain expansion mechanism where the size of the domains is dynamically adjusted to reflect the behavior change and improve the authentication accuracy.
Domain Expansion \[da\]
-----------------------
We introduce domain expansion in which the domain boundaries $\alpha$ and $\beta$ are updated. In practice, due to behavior deviation and sensor noise, the initial setting of $\alpha$ and $\beta$ may become inaccurate. If behavior scores from the legitimate user keep falling in the slack domain, it may indicate that the legitimate domain is too small and needs to be expanded to reduce false rejects. Similarly, the illegitimate domain may need to be expanded to reduce false accepts. Authentication accuracy is improved as a result. As shown in Fig. \[fig:dj\], the original legitimate and illegitimate domains are \[0, $\alpha$\] and \[$\beta$, 1\], respectively. The new domains become \[0, $\alpha'$\] and \[$\beta'$, 1\] after expansion. In addition, authentication delay is reduced since less privilege movement is needed and the system can make decisions more quickly. In a given dataset, it is straightforward to find out whether the behavior scores that keep falling in the slack domain belong to the legitimate user. In reality however, it is difficult for the system to know in which case the second-factor authentication (password input for our discussion) is needed to provide feedback, as previously mentioned. We assume that the legitimate user will input the correct password and illegitimate users will input wrong passwords.
We model domain expansion by applying physical laws that describe the motion of bodies under the influence of a system of forces. Specifically, the expansion $S$ in time $t$ is defined as: $$\label{fun3}
\begin{split}
%\begin{aligned}
S=\frac{1}{2}(a-\hat{a})t^2+v_0t,
\end{split}$$ where $a$ denotes the acceleration of the expansion, $t$ denotes the number of time windows or authentication cycles, $v_0$ denotes the initial velocity of the expansion, and $\hat{a}$ is the resistance that slows down or stops the expansion. Every time the user inputs the correct password and the behavior score is outside of the legitimate domain, the system will expand the legitimate domain to contain the behavior score where the expansion is proportional to the distance between the behavior score and legitimate domain ($\epsilon-\alpha$). However, if the system observes frequent password input, it is an indication that the current machine learning model in the IA scheme is no longer suitable and needs to be retrained [@p67].
![Domain expansion.[]{data-label="fig:dj"}](DomainAdjustment.pdf){width="3.1in" height="1.4in"}
The acceleration of the expansion $a$ is defined as: $$\label{fun1}
\begin{split}
%\begin{aligned}
a=\frac{R_d*\varepsilon}{W_1}+W_2+\delta,
\end{split}$$ where $W_1=\frac{\sum_i n_l^{(i)}+n_a^{(i)}}{\sum_i N^{(i)}}$ is a balancing parameter that controls the expansion, $W_2$ is a constant representing the initial acceleration, $\sum_i n_l^{(i)}$ is the number of times the user inputs the correct password when her score is in the slack domain, $\sum_i n_a^{(i)}$ is the number of times the user inputs a wrong password when her score is in the slack domain, $\sum_i N^{(i)}$ is the total number of authentication cycles, $R_d$ is the distance between $R_c$ and $R_1$, $\varepsilon=\epsilon-\alpha$ is the distance between the behavior score and legitimate domain, and $\delta$ is the mixture of behavior noise and sensor noise.
The expansion of the legitimate domain may result in the inclusion of illegitimate users’ behavior scores that originally fall in the slack domain. To reduce such false accepts, we introduce the resistance $\hat{a}$ that constrains the expansion: $$\label{fun2}
\begin{split}
%\begin{aligned}
\hat{a}=a(\int_0^\alpha p(\varepsilon_a)d\varepsilon_a +\theta),
\end{split}$$ where $\theta$ is a constant that prevents $\alpha$ from surpassing $\beta$, $\int_0^\alpha p(\varepsilon_a)d\varepsilon_a$ denotes the probability that the legitimate domain contains behavior scores derived from illegitimate users in the training set, and $\varepsilon_a$ denotes the behavior score derived from illegitimate users’ data in the training set. $\int_0^\alpha p(\varepsilon_a)d\varepsilon_a$ be estimated using kernel density estimator.
Substituting (\[fun2\]) into (\[fun3\]) and assuming $t=1$, we have $$\label{fun4}
\begin{split}
%\begin{aligned}
S=\frac{1}{2}a(1-\int_0^\alpha p(\varepsilon_a)d\varepsilon_a-\theta)+v_0,
\end{split}$$ where $V=1-\int_0^\alpha p(\varepsilon_a)d\varepsilon_a-\theta$ controls when the expansion stops.
Substituting \[fun1\] into \[fun4\], we have $$\label{fun5}
\begin{split}
%\begin{aligned}
S=\frac{1}{2}(\frac{R_d*\varepsilon}{W_1}+W_2)V+v_0+\Delta,
\end{split}$$ where $\Delta=\frac{V*\delta}{2}$ is estimated and eliminated using Kalman filter.
In each authentication cycle, if the user inputs the correct password, the predicted state estimate $x_{k|k-1}$ which controls the expansion of the legitimate domain is defined as: $x_{k|k-1}=F_kx_{k-1|k-1}+B_ku_k$, where $F_k=\begin{bmatrix}
1 &t \\
0 &1 \\
\end{bmatrix}
$, $B_k=\begin{bmatrix}
\frac{t^2}{2}\\
t \\
\end{bmatrix}$ and $u_k=(\frac{R_d*\varepsilon_a}{W_1}+W_2)V$. The predicted estimate covariance $P_{k|k-1}$ is defined as: $P_{k|k-1}=F_kP_{k-1|k-1}F_k^T+Q_k$, where the process noise covariance is $Q_k=\begin{bmatrix}
\frac{t^4}{4} &\frac{t^3}{2} \\
\frac{t^3}{2} &t^2 \\
\end{bmatrix}*\sigma_a^2$ with $\sigma_a$ being the magnitude of the process noise (behavior noise). The innovation covariance is $S_k=H_kP_{k|k-1}H_k^T+R_k$, where $H_k=\begin{bmatrix}
1 \\
0 \\
\end{bmatrix}$ and$R_k$ is the covariance of the observation noise (sensor noise). Kalman gain is calculate as: $K_k=P_{k|k-1}H_k^TS_k^{-1}$. Since Kalman filter is loop carried, we update the state estimate and associate covariance at the end of each authentication cycle as: $x_{k|k}=x_{k|k-1}+K_k(z_k-H_kx_{k|k-1})$, and $P_{k|k}=(I-K_kH_k)P_{k|k-1}$. We calculate the expansion as $P_{k|k}H_k$ and need to rescale it before applying it to real systems.
If the user inputs a wrong password, we let $u_k=\frac{\varepsilon_l}{R_d*W_1}+W_2$ and a similar process happens for the expansion of the illegitimate domain. The illegitimate domain expansion is covered in detail in the full version of this paper [@appendix].
In addition to causing false accepts, the expansion of the legitimate domain also affects privilege movement, or more specifically, the distance of the movement $-\mu_l$ and $+\mu_a$. Now that the domain boundaries $\alpha$ and $\beta$ are dynamically adjustable, the distance of the movement needs to be adjusted accordingly. We let $-\mu_l=-\mu_l\frac{\int_0^\alpha p(\varepsilon_l)d\varepsilon_l}{\int_0^\alpha p(\varepsilon_a)d\varepsilon_a}$ and $+\mu_a=+\mu_a\frac{\int_\beta^1 p(\varepsilon_a)d\varepsilon_a}{\int_\beta^1 p(\varepsilon_l)d\varepsilon_l}$, where $\varepsilon_l$ and $\varepsilon_a$ denote the behavior scores derived from the legitimate user’s and illegitimate users’ data in the training set, respectively, $\int_0^\alpha p(\varepsilon_l)d\varepsilon_l$ and $\int_0^\alpha p(\varepsilon_a)d\varepsilon_a$ denote the probabilities that the legitimate domain contains behavior scores derived from the legitimate user’s and illegitimate users’ data in the training set, respectively, and $\int_\beta^1 p(\varepsilon_l)d\varepsilon_l$ and $\int_\beta^1 p(\varepsilon_a)d\varepsilon_a$ denote the probabilities that the illegitimate domain contains behavior scores derived from the legitimate user’s and illegitimate users’ data in the training set, respectively. If the ratio $\frac{\int_0^\alpha p(\varepsilon_l)d\varepsilon_l}{\int_0^\alpha p(\varepsilon_a)d\varepsilon_a}$ is large, it indicates that the legitimate user’s behavior scores still dominate the legitimate domain, and the distance of the privilege movement is appropriate. Otherwise, the distance needs to be adjusted.
Performance Evaluation \[pe\]
=============================
We have conducted comprehensive performance evaluation on a four-level MPC scheme that contains a top level, two observation levels and a bottom level. Most of the simulations use data from all 130 participants in 5 months, where we randomly select one participant as the legitimate user and mix her data with the data sampled from all other participants. The simulations are performed 130 times for each participant against all the other participants and averaged results are derived for each test. We keep the illegitimate users’ data portion in the range of 50% to 80% to simulate a more hostile environment. Features used in the evaluation include GPS, app installation, Bluetooth and battery usage. GPS contains longitude and latitude. App installation contains app package, installed apps, uninstalled apps, running apps and their corresponding class names. Bluetooth usage contains the participants’ IDs, address and duration. Battery usage contains health percentage, battery level, voltage, plug information and brand information.
Authentication Accuracy
-----------------------
The time window is set to 15 seconds which contains 1 KB user data. The data is sent to SVM for training and the output is converted to probabilities. To simulate real usage, we divide the whole dataset into 100 distinct subsets sorted based on time, and perform tests by gradually sending the subsets to the system. We use 5-fold cross validation in the training and testing. The parameters for SVM, including the separation threshold, are chosen to minimize false rejects and false accepts. Finally, we evaluate the average authentication accuracy in each time window for both MPC (applied to IA) and the traditional IA among all users. We use SVM with separation threshold $\Omega$ for the traditional IA. The results are shown in Fig. \[fig:acc\]. In Fig. \[fig:acc\] (a), the accuracies for both MPC and the traditional IA have some fluctuations in the first five subsets and become stable in the remaining subsets. The accuracy fluctuations of the traditional IA is larger than MPC due to behavior deviation and sensor noise that are left untreated. The accuracy improvement in MPC is obvious and stable across the all the subsets. In addition, we calculate MPC’s accuracy across all the subsets, which on average achieves 18.63% improvement compared with the traditional IA.
Fig. \[fig:acc\] (b) provides a more detailed view of accuracy improvement in the four features. The improvement in GPS is the highest due to the traditional IA’s low accuracy using the GPS feature. Similarly, the improvement in app installation is the lowest. The accuracy improvement for MPC is not stable during the first few authentications because of domain expansion. It becomes stable after the 70th subset for all four features. Generally, the accuracy improvement after applying MPC is between 0.04% to 0.35%.
[.5]{} ![Authentication accuracy. (a) Average accuracy for both MPC and the traditional IA. (b) Accuracy improvement for all four features with MPC.[]{data-label="fig:acc"}](AccuracyF.eps "fig:"){width="109.90000%" height="0.13\textheight"}
[.5]{} ![Authentication accuracy. (a) Average accuracy for both MPC and the traditional IA. (b) Accuracy improvement for all four features with MPC.[]{data-label="fig:acc"}](AccuracyIncrease.eps "fig:"){width="110.00000%" height="0.13\textheight"}
Authentication Delay
--------------------
[.5]{} ![Authentication delay. (a) Average delay reduction for all users. (b) Delay reduction for 18 randomly selected users.[]{data-label="fig:tr"}](TimeReduction.eps "fig:"){width="109.90000%" height="0.13\textheight"}
[.5]{} ![Authentication delay. (a) Average delay reduction for all users. (b) Delay reduction for 18 randomly selected users.[]{data-label="fig:tr"}](TimeReductionPeople.eps "fig:"){width="110.00000%" height="0.13\textheight"}
We calculate the authentication delay for both MPC and the traditional IA when they reach their corresponding highest accuracy, i.e., 99% and 82% on average, shown in Fig. \[fig:tr\]. The average authentication delay reductions for the four features with MPC are shown in Fig. \[fig:tr\] (a). For the app installation feature, MPC gives the lowest delay reduction since the traditional IA trained by the app data already achieves a good delay performance and there is little room for improvement. Similarly, for the Bluetooth feature, MPC provides the highest delay reduction. In addition, we calculate the overall delay among all features with MPC, which on average achieves a reduction of 7.02 minutes.
Moreover, we calculate the amount of time reduction for each user in MPC when it reaches the highest accuracy, by randomly selecting 18 users, as shown in Fig. \[fig:tr\] (b). We cluster the results based on the features. Users are marked by different colors and the corresponding delay reduction is shown in the y-axis. It is observed from the results that the delay reduction varies greatly among users and even for the same user, which reflects the complexity of human behaviors.
Performance under Long-term Usage
---------------------------------
We evaluate the performance of MPC under long-term usage using data from 19 randomly selected users in three time slots containing 200, 300, and 500 time windows. We calculate the accuracy (ACC), precision (PREC), true accept rate (TAR), true reject rate (TRR), false accept rate (FAR) and false reject rate (FRR) in Table \[tbperformance\] for both MPC and the traditional IA. As shown in Table \[tbperformance\], the performance improvement with MPC is significant compared with the traditional IA. Another important observation is that the performance of the traditional IA does not monotonically increase with time. In the other words, the authentication accuracy of the traditional IA does not always improve as we gather more behavior data. This is due to behavior deviation and sensor noise. MPC, on the other hand, is much more predictable in terms of improving the authentication accuracy since it automatically corrects behavior deviation and filters out noise in each authentication cycle.
Furthermore, as shown in Table \[tbperformance\], MPC’s accuracy improvement becomes smaller between the 300 and 500 time windows, compared with between the 200 and 300 time windows. As discussed previously, MPC reduces the impact of overlapping behavior scores in the slack domain using initial mapping, privilege movement and domain expansion. Since it is loop carried, the accuracy improvement is reflected gradually in each time window and the expansion becomes slower and more stable with time.
[|c|c|c|c|c|c|c|]{}\
Time\*&ACC %&PREC %&TAR %&TRR %&FAR %&FRR %\
200 &87.58&92.04&89.18&69.38&30.62&10.82\
300 &84.40&87.77&87.08&67.84&32.16&12.92\
500 &83.16&86.90&84.62&66.77&33.23&15.38\
\
200 &97.26&97.40&98.80&93.52&6.48&1.20\
300 &98.64&98.87&98.93&98.18&1.82&1.07\
500 &98.97&99.08&99.15&98.72&1.28&0.85\
Time stands for time window. $ACC=\frac{TA+TR}{TA+TR+FA+FR}$, $PREC=\frac{TA}{TA+FA}$, $TAR=\frac{TA}{TA+FR}$, $TRR=\frac{TR}{TR+FA}$, $FAR=\frac{FA}{FA+TR}$ and $FRR=\frac{FR}{FR+TA}$.
\[tbperformance\]
Other Performance Measures
--------------------------
We calculate the percentage of behavior scores that are mapped to each privilege level in MPC, for the legitimate user and illegitimate users. As shown in Fig. \[fig:levelbin\] (a), less than 3% of the behavior scores are mapped to the observation levels, which indicates that MPC is fast and highly effective in making the final decision. The number of behavior scores which fall in the observation levels, level 2 and level 3, is almost identical.
We also calculate the behavior score distributions in each privilege level for time windows 10 through 70 as shown in Fig. \[fig:levelbin\] (b). The z-axis indicates the number of behavior scores. The y-axis indicates the time windows. The x-axis indicates the privilege levels, where the left four levels are plotted from the legitimate user’s behavior scores and the right four levels are plotted from illegitimate users. For both users, the number of scores which fall in the observation levels is small, less than 10, which is similar to the result in Fig. \[fig:levelbin\] (a). The proportion of the behavior scores in the top and bottom privilege levels is different from Fig. \[fig:levelbin\] (a), since Fig. \[fig:levelbin\] (b) only considers limited time windows while Fig. \[fig:levelbin\] (a) considers data spanning 5 months.
[.5]{} ![The proportion of behavior scores in each level. (a) Average proportion. (b) Proportion in time windows 10 through 70.[]{data-label="fig:levelbin"}](levelbin.eps "fig:"){width="109.90000%" height="0.13\textheight"}
[.5]{} ![The proportion of behavior scores in each level. (a) Average proportion. (b) Proportion in time windows 10 through 70.[]{data-label="fig:levelbin"}](levelbin_detail.eps "fig:"){width="110.00000%" height="0.14\textheight"}
Related Work
============
The majority of the existing work [@p62; @p7; @bo2013silentsense; @feng2014tips; @shi2011senguard; @shahzad2013secure; @castelluccia2017towards] focuses on finding suitable behavioral features such as touch, typing, and other motions that uniquely identify users, ignoring the usability of the IA systems. The amount of data gathered by various sensors directly affects the accuracy of IA systems [@p62; @yang2016personaia; @p9]. By increasing the time spent in collecting users’ behavior data, the accuracy of IA can be improved [@p62; @p9]. However, this approach will also increase the authentication delay and undermine usability. In this paper, we proposed a multi-level privilege control scheme to address the usability of IA systems by improving authentication accuracy and delay.
To deal with the behavior and sensor noises, most of the existing IA schemes use simple approaches such as resampling [@p62], averaging the results [@bo2013silentsense], or no approach at all [@p65; @p35; @kate2017authentication]. Such noises will degrade system performance in terms of authentication accuracy and delay. The problem will be exacerbated as the size of the behavior data grows. We applied Kalman filter [@welch1995introduction] to correct behavior deviation and filter out sensor noise during the authentication. We showed that Kalman filter is naturally suitable for IA and can be implemented in practice to further improve authentication accuracy while reducing authentication delay.
The existing IA systems are evaluated using private datasets collected from their volunteers [@p62; @khan2016targeted; @frank2013touchalytics; @bo2013silentsense]. Such datasets may not be sharable due to the sensitivity of human behavior data. It is hence difficult to recreate their experiments, compare with their schemes, or use their datasets for future research. Our proposed MPC is evaluated using a public and comprehensive dataset [@p21; @aharony2011social]. The repeatability of our experiments and fair comparisons are guaranteed.
Conclusion and Future Work
==========================
In this paper, we proposed a multi-level privilege control scheme, MPC, to enhance the usability of IA systems. We evaluated MPC using a public dataset, which on average achieves 18.63% accuracy improvement and 7.02-minute authentication delay reduction. MPC is a lightweight solution that can be generally applied to IA schemes whose output can be converted to probabilistic values. As our future work, we plan to incorporate MPC into other non-SVM-based IA schemes and deploy our MPC-based IA system for use by recruiting volunteers.\
\
|
---
abstract: |
Cosmological hydrodynamic simulations predict the physical state of baryons in the circumgalactic medium (CGM), which can be directly tested via quasar absorption line observations. We use high resolution “zoom-in” simulations of 21 galaxies to characterize the distribution of neutral hydrogen around halos in the mass range $M_{\rm vir} \sim 2\times
10^{11} - 4\times 10^{12}~\rm M_\odot$ at $z\sim2$. We find that both the mass fraction of cool ($T \le 3\times 10^4~\rm K$) gas and the covering fraction of optically-thick Lyman limit systems (LLSs) depend only weakly on halo mass, even around the critical value for the formation of stable virial shocks. The covering fraction of LLSs interior to the virial radius varies between $f_{\rm c} \sim 0.05 - 0.2$, with significant scatter among halos. Our simulations of massive halos ($M_{\rm vir} \ge 10^{12}~\rm M_\odot$) underpredict the covering fraction of optically-thick gas observed in the quasar CGM by a large factor. The reason for this discrepancy is unclear, but several possibilities are discussed. In the lower mass halos ($M_{\rm vir} \ge 5 \times 10^{11}~\rm M_\odot$) hosting star-forming galaxies, the predicted covering factor agrees with observations, however current samples of quasar-galaxy pairs are too small for a conclusive comparison. To overcome this limitation, we propose a new observable: the small-scale auto-correlation function of optically-thick absorbers detected in the foreground of close quasar pairs. We show that this new observable can constrain the underlying dark halos hosting LLSs at $z\sim 2-3$, as well as the characteristic size and covering factor of the CGM.
author:
- 'Michele Fumagalli, Joseph F. Hennawi, J. Xavier Prochaska, Daniel Kasen, Avishai Dekel, Daniel Ceverino, Joel Primack'
title: 'Confronting Simulations of Optically Thick Gas in Massive Halos with Observations at $\lowercase{z}=2-3$'
---
Introduction {#intro}
============
Over the past several years, numerical simulations of galaxy formation have converged upon a paradigm for the accretion of gas into dark matter halos. One tenant of the model is that the majority of gas which travels to the central regions and contributes fuel for star-formation is cool, i.e. at a temperature $T \sim 10^4-10^5$K [@bir03; @ker05; @dek06; @ocv08; @ker09b; @dek09; @van11]. Importantly, the simulations reveal that this cool gas travels along relatively narrow (i.e. filamentary) structures often termed “cold streams”.
The existence of gas accretion to fuel star formation resembles in spirit early prescriptions for gas accretion from a hot halo in analytic calculations [e.g. @bin77; @ree77; @sil77; @whi78]. However, the origin, the morphology, and kinematics of the cold stream model are distinct, making this accretion mode the core element of a new paradigm for galaxy formation.
Despite a general acceptance of the cold accretion paradigm from a theoretical perspective, this model has been difficult to test empirically. A large body of literature explored the possibility of detecting Lyman-$\alpha$ emission from the accreting gas [e.g. @hai00; @far01; @fur05; @dij09; @goe10; @fau10; @ros12], powered by the potential energy of gravitational infall. Unfortunately, predictions for the surface brightness are exponentially sensitive to the conditions of the gas (e.g. temperature) and the signal may be confused by other sources of Lyman-$\alpha$ photons (e.g. ionization by stars or AGNs, that is active galactic nuclei, and scattered radiation). Furthermore, accurate modeling requires the solution of coupled hydrodynamic and radiative transfer equations [see e.g. @ros12], which is at present computationally expensive. To date, no compelling detection of the streams in emission has been achieved, although some tantalizing [Lyman-$\alpha$]{} observations of filamentary structures around high-redshift galaxies have been reported [e.g. @can12; @rau13; @hen13].
An alternate approach toward direct detection is to observe the cool gas via absorption arising from gas that is confined inside or in proximity to dark matter halos, within the so-called circumgalactic medium (CGM). High-resolution hydrodynamic simulations of galaxy formation predict that cold streams should be manifest as strong absorption systems with column densities ${N_{\rm HI}}\ge 10^{17.2} {\rm~cm^{-2}}$, such that they are optically thick to Lyman continuum radiation [e.g. @fau11; @fum11; @van12; @goe12; @she13]. Blind surveys along quasar sightlines for these so-called Lyman limit systems (LLSs) thus provide, in principle, a test for this scenario.
One approach is to compare the incidence of optically thick gas [e.g. @pow10; @ome13; @fum13], against global estimates for cold streams in the population of $z>2$ galaxies that are predicted to contain them [@alt11; @rah13a]. For instance, simulated massive galaxies with virial masses $M_{\rm vir}\gtrsim 10^{11}~\rm M_\odot$ at $z\sim3$ do not account for the entire population of LLSs alone, but consistency between models and observations could be achieved with an extrapolation to lower masses [@fum11; @van12; @fum13]. However, a detailed comparison to theoretical predictions is limited by the fact that these blind surveys, by construction, do not directly relate these absorbers to the galaxies and dark matter halos that they arise from.
The much more direct approach is to search for signatures of cold accretion in the vicinity of the $z\sim 2-3$ galaxies that are expected to host them. Analysis of the stacked spectra constructed from galaxies lying background to $z\sim 2.5$ star-forming galaxies (the Lyman break galaxies or LBGs) provide one such test [@ste10], and models of star-forming galaxies being fed by cold streams appear to match the average absorption to impact parameters of at least $\sim 100$kpc [@fum11; @she13]. However, such stacking analyses can only measure the average equivalent width of absorption and the flatness of the curve of growth unfortunately dictates that this method is mostly sensitive to kinematics and only weakly dependent on the total amount of absorbing material.
Ideally, one should probe LBGs with individual sightlines, at sufficiently high signal-to-noise ratio and resolution to characterize the column densities of absorbers that give rise to cold streams. @rud12 have reported on the [$N_{\rm HI}$]{} values measured in 10 quasar sightlines passing within 100kpc of a foreground LBG. They found evidence for optically-thick gas from the CGM in 3 cases. The implied covering fraction (here defined as the area subtended by optically-thick gas divided by a reference area) is $f_{\rm c} = (30 \pm 14)\%$ within the virial radius ([$R_{\rm vir}$]{}). And while future efforts will undoubtedly increase the samples of LBGs [e.g. @cri11], building up the data sets of $\sim 100$ sightlines required to make robust statistical measurements within [$R_{\rm vir}$]{} will be extremely telescope time intensive.
Recently, [@pro13] have expanded on previous efforts [@hen06b; @hen07; @pro09] to measure the incidence of optically-thick gas in the CGM of massive galaxies, specifically those hosting $z\sim 2$ quasars. Using pairs of quasars, they probed the halo gas that is physically associated to a foreground quasar host galaxy using a background sightline. Remarkably, this experiment reveals a high $f_{\rm c}$, in excess of $60\%$, for sightlines passing within the estimated virial radii of these massive galaxies ($\sim 150$kpc). Furthermore, the gas is enriched in heavy elements, showing large equivalent widths of low-ion absorption (e.g. 1334).
The strong clustering of $z\sim 2$ quasars indicates that they are hosted by massive dark matter halos $M_{\rm vir} \sim 10^{12.5}{~\rm M_\odot}$ [@whi12 e.g.], more than three times larger than the typical dark halos hosting LBGs with $M_{\rm vir}\sim 10^{12}~\rm M_\odot$ [@ade05] at $z\sim 2$. In the current picture of cold accretion, it is believed that at these high masses, virial shocks become stable and the CGM of such halos will become increasingly dominated by gas that is heated to about the virial temperature [@dek06]. Qualitatively, one would therefore expect that more massive dark matter halos have lower covering fraction of cold gas. For this reason, the results of @pro13 are very surprising, being indeed opposite to this naive expectation.
Motivated by this development, we expand our previous study of absorption line systems in the CGM of simulated galaxies [@fum11] by focusing on the properties of optically-thick gas in a larger suite of AMR simulations that have been presented in @cev10, @cev12, and @dek13. This new library increases by a factor of three the sample presented in @fum11 and includes for the first time galaxies hosted within massive dark matter halos ($M_{\rm vir} > 10^{12} {~\rm M_\odot}$) at $z\sim 2$. Following our previous work, we include in these simulations recipes for star-formation and its feedback and we post-process the outputs with radiative transfer calculations to estimate the ionization state of the hydrogen in the halos (Section \[sims\]).
The aim of this paper is to characterize from the theoretical point of view the distribution of the neutral hydrogen in cold-stream fed galaxies over more than a decade of halo mass (Section \[mass\]) and to perform direct comparisons to the new observational results derived in quasar-galaxy and quasar-quasar pairs, focusing on the incidence of optically-thick gas surrounding massive galaxies at $z \sim 2$ (Section \[qso\]). Further, since we will show that the current sample of quasar-galaxy pairs is too small for conclusive comparisons with simulations, we propose an additional direct test of the cold-stream paradigm by introducing the formalism to compute the auto-correlation function of LLSs, a quantity that can be used for statistical investigation of the spatial distribution of optically-thick gas in the CGM (Section \[future\]). The summary and conclusions follow in Section \[concl\]. Throughout this work, for consistency with the parameters used in the numerical simulations, we adopt a standard $\Lambda$CDM cosmology as described by $\Omega_{\rm m} = 0.27$, $\Omega_\Lambda = 0.73$, $\Omega_{\rm m} =0.045$, $h=0.7$ and $\sigma_8=0.82$ [@kom09].
Simulations and radiative transfer post-processing {#sims}
==================================================
We present the analysis of 21 galaxy halos at redshift $z \sim 3$ and $z \sim 2$, the properties of which are summarized in Table \[tab:galprop\]. In this section, we only briefly summarize the numerical techniques used to produce the final models. Additional information on these simulations can be found in @cev10, @cev12, and @dek13. The procedures adopted for the radiative transfer post-processing have been presented in @fum11 and are further discussed in Appendix \[ratraapp\].
Hydrodynamic simulations
------------------------
Each halo has been selected from a larger cosmological box and re-simulated with the adaptive mesh refinement (AMR) hydro-gravitational code ART [@kra97; @kra03]. The dark matter particle mass is $5.5 \times 10^5 {~\rm M_\odot}$ and the cell size on the finest level of refinement ranges between $35-70$ pc. At this resolution, densities of $n_{\rm H}\sim 10^3~\rm cm^{-3}$ can be reached. In these simulations, refinement occurs when the mass in stars and dark matter inside a cell is higher than $2\times10^6~\rm M_\odot$ (i.e. three times the dark matter particle mass), or the gas mass is higher than $1.5\times 10^6~\rm M_\odot$. The ART code incorporates the principal physical processes that are relevant for galaxy formation, including gas cooling and photoionization heating, star formation, metal enrichment and stellar feedback [@cev09; @cev10]. Both photo-heating and radiative cooling are modeled as a function of the gas density, temperature, metallicity, and UV background (UVB). During this calculation, self-shielding of gas is crudely modeled by suppressing the UVB intensity to $5.9 \times 10^{26}~\rm erg~s^{-1}~cm^{-2}~Hz^{-1}$ above hydrogen densities $n_{\rm H}=0.1$ cm$^{-3}$. Stochastic star formation occurs at a rate that is consistent with the Kennicutt-Schmidt law [@ken98] in cells with gas temperature $T \le 10^4~\rm K$ and densities $n_{\rm H} \ge 1~\rm cm^{-3}$, but more than half of the stars form at $T \lesssim 300~\rm K$ and $n_{\rm H} \ge 1~\rm cm^{-3}$.
To model feedback processes related to star formation, both the energy from stellar winds and supernova type II explosions are injected in the gas at a constant heating rate over 40 Myr, while the energy injection from supernovae type Ia is modeled with an exponentially declining heating rate with a maximum at 1 Gyr. Cooling is never prevented in these simulations and powerful outflows originate in regions where the thermal heating due to supernovae and stellar winds overcomes radiative cooling. In some cases, galactic outflows in these simulations reach high velocities, from few hundreds $\rm km~s^{-1}$ to a thousand $\rm km~s^{-1}$ [@cev09], but the mass loading factor is on average low ($\eta \sim 0.3$ at 0.5[$R_{\rm vir}$]{}). Star formation also enriches the interstellar medium (ISM) following the yields of @woo95 and the @mil79 initial mass function (IMF).
These simulations are able to reproduce the basic scaling relations observed in high redshift galaxies [see @cev10]. Nevertheless, because of the limited ability of the adopted subgrid prescriptions to model the complex baryonic processes that are associated to star formation and feedback, these simulations produce a factor of $\sim 2$ higher stellar mass and lower gas fractions by $z\sim 2$ [see a detailed discussion in @dek13]. Further, these simulations do not model feedback from an AGN. However, it has been shown by @van12 and @she13 that most of the optically-thick gas resides in filamentary structures associated to cold gas infall rather than in outflowing gas. Clearly, comparisons with other simulations that include different recipes for star formation and stellar winds are needed to verify the extent to which the results presented in this paper can be generalized, although at present there are only very few zoom-in simulations of halos with $M_{\rm vir} \ge 10^{12}{~\rm M_\odot}$ in the literature. Given the high resolution and the fact that AMR codes should capture most of the large-scale hydrodynamic processes that are relevant for galaxy formation, these simulations are among the best models currently available to investigate the distribution of hydrogen that originates from the cold streams that feed galaxies at high redshifts. We refrain instead from the analysis of the metal distribution and gas kinematics, two quantities that are most likely sensitive to the adopted feedback prescriptions [e.g. @she13].
Hydrogen neutral fraction
-------------------------
The ionization state of the gas in these simulations is computed in post-processing, under the simplistic assumption that the relevant time scales in the radiative transfer problem are shorter than the relevant time scales that govern the hydrodynamic equations. This approach however neglects the effects of radiative transfer on the hydrodynamics. Changes in the temperature and ionization fraction of the gas could in fact alter, for instance, the properties of the cooling function and the gas pressure [cf. @fau10; @ros12].
For each AMR cell, we compute the neutral fraction $x_{\rm HI}$ for atomic hydrogen with a Monte Carlo radiative transfer code, as detailed in Appendix \[ratraapp\]. Both ionization due to electron collisions and photons are included at equilibrium, but we neglect the ionization of helium. Because local sources of radiation are important contributors to the ionization of optically-thick hydrogen [@fum11; @rah13], in addition to the extragalactic UVB from @haa12, we include in our models the radiation from local stellar particles following a Kroupa IMF [@kro01] and we account for the presence of dust as described in @fum11. Our radiative transfer technique has been validated through one of the tests presented in @ili06 [see Appendix \[ratraapp\]]. Further, the escape fraction from the galaxy disks at the virial radius in these simulations is found to be below 10% [@fum11], consistent with current estimates [e.g. @nes13]. Finally, independent calculations by @rah13 have shown consistency with the results presented in our previous work [@fum11].
![The covering fraction of optically-thick neutral hydrogen as measured within the virial radius (top) and twice the virial radius (bottom). For each simulation, the data points and the error bars represent values measured along three orthogonal directions. Simulations at $z\sim 3$ are shown with blue squares, while models at $z\sim 2$ are shown with red circles. In each panel, we also display the covering fractions from simulations in the literature (green symbols). Open symbols are used for models without detailed radiative transfer post-processing. Simulated galaxies exhibit a wide range of covering fractions, mildly decreasing at fix halo mass from $z\sim3$ to $z\sim2$.[]{data-label="fig:cfall"}](cfplot_mass.eps)
The mass dependence of the neutral hydrogen covering fraction {#mass}
=============================================================
In this section, we investigate the mass dependence of the covering fraction of neutral hydrogen in these simulations. To this purpose, we generate maps of the neutral hydrogen column density in cylinders of radius 2[$R_{\rm vir}$]{} and height 4[$R_{\rm vir}$]{}. For each simulated galaxy, we generate three projections along the three orthogonal axes that are naturally defined by the AMR grid. The resolution of the projected maps is comparable to the resolution of the smallest cell in each simulation. For visualization purposes, we also generate temperature maps, which we construct similarly to the [$N_{\rm HI}$]{} maps by averaging the temperature of each cell along the line of sight with weights that are proportional to the total column density of hydrogen. Figure \[fig:allgal\] presents a gallery of these maps for the $z\sim 2$ galaxies.
Cold gas and the critical halo mass
-----------------------------------
Simply by inspecting Figure \[fig:allgal\], one can already infer the basic CGM properties of simulated $z\sim 2$ halos. Across one decade in virial mass ($M_{\rm vir} \sim 2\times 10^{11} - 4 \times 10^{12}~\rm M_\odot$), the average temperature of the lower column density gas ($N_{\rm HI} \lesssim 10^{16}~\rm cm^{-2}$) is increasing from a few $10^5~\rm K$ to a few $10^6~\rm K$. However, at all masses, pockets and narrow filaments of cooler ($T\lesssim 10^5~\rm K$) and higher column density ($N_{\rm HI} \gtrsim 10^{17}~\rm cm^{-2}$) gas persist within and beyond the virial radius.
More quantitatively, the volume averaged temperature within the virial radius at $z\sim 2$ is found to increase from $\langle T \rangle \sim 4 \times 10^5~\rm K$ at $M_{\rm vir} \sim 3 \times 10^{11}~\rm M_\odot$ to $\langle T \rangle \sim 3 \times 10^6~\rm K$ at $M_{\rm vir} \sim 4 \times 10^{12}~\rm M_\odot$. We exclude galactic gas in this calculation by ignoring regions inside 0.15[$R_{\rm vir}$]{}. For halos with virial masses $M_{\rm vir}\sim 5\times 10^{11} - 4 \times 10^{12}~\rm M_\odot$, which bracket the critical halo mass for the formation of stable virial shocks, $\langle T \rangle$ is consistent with the predicted post-shock temperature $T \sim \frac{3}{8} T_{\rm vir}$, where $T_{\rm vir}$ is the virial temperature [@bir03; @dek06]. Virial shocks are also visible in some of the temperature maps presented in Figure \[fig:allgal\]. A similar trend is found in simulations at $z\sim 3$, with $\langle T \rangle (z=3) \sim 1.3~ \langle T \rangle (z=2)$ at fixed halo mass, as expected from the redshift dependence of the virial scaling relations.
As already noted in Figure \[fig:allgal\], despite the increasing $\langle T \rangle$ as a function of halo mass, filaments of cooler and denser material are evident in the CGM of even the most massive halos. For gas to exhibit an appreciable fraction of neutral hydrogen in absorption, typical temperatures have to be $T\lesssim 3\times 10^4~\rm K$, while the volume density needs to be $n_{\rm H}\gtrsim 0.003~\rm cm^{-3}$ [e.g. @fum11]. Since gas slabs with these properties become optically-thick to the incident Lyman continuum radiation, LLSs that are relatively straightforward to identify in quasar spectra conveniently trace hydrogen with these physical conditions. Therefore, we restrict our analysis of the cool halo gas to column densities of $N_{\rm HI}\ge 10^{17.2}~\rm cm^{-2}$, which we can also compare to existing observations.
![The mass fraction of gas at the [*instantaneous*]{} temperature of $T<2.5\times 10^5~\rm K$ (top; circles) and $T<3\times 10^4~\rm K$ (bottom; squares) that is enclosed within $0.15 R_{\rm vir} < r < R_{\rm vir}$. Simulations at $z\sim3$ and $z\sim2$ are shown with open and filled circles, respectively. The distribution from the top panel is overlaid to the distribution in the bottom panel (and vice-versa) to facilitate comparisons. The fraction of gas with $T<2.5\times 10^5~\rm K$ decreases with increasing halo mass, while the mass fraction of the colder gas ($T<3\times 10^4~\rm K$) at any given redshift is only weakly dependent on mass.[]{data-label="fig:fcold"}](fcold.eps)
Figure \[fig:cfall\] summarizes the covering fractions of optically-thick gas in the CGM of the 21 simulations under examination, both at $z\sim 2$ and $z\sim 3$. In this paper, we focus on an empirical definition for the covering fraction because we aim to extract from simulations an observable quantity that can be directly compared to observations. Our covering fraction encompasses all gas that is optically-thick to a background source in projection, regardless to its kinematic state [cf. @van12], including gas that is associated to the central galaxies. In fact, in observations, one cannot trivially disentangle the contribution of halo gas from the contribution of the outskirts of galaxy disks. A subtlety arises, however, from the fact that galaxy-quasar pairs or quasar-quasar pairs are intrinsically rare at very small projected separations and, whenever possible, in the following we compare observations and simulations using the observed distribution of impact parameters. Furthermore, since observations cannot separate halo gas from gas associated to satellites, we include gas within satellite galaxies in our definition of $f_{\rm c}$ [see figure 7 in @fum11 for estimates of $f_{\rm c}$ with and without the contribution of satellites]. We emphasize that, since $f_{\rm c}$ includes also gas that is not infalling, this is an upper limit to the theoretical covering fraction of accreting gas within the CGM. Furthermore, given our (arbitrary) definition for gas inside galaxies ($R<r$[$R_{\rm vir}$]{} with $r=0.15$), one can trivially derive a lower limit to the covering fraction of halo gas without the galaxy contribution: $f'_{\rm c} \ge (f_{\rm c}/r^2-1)/(r^{-2}-1)$. As expected, the correction $f'_{\rm c}/f_{\rm c}$ is large for the few galaxies with small $f_{\rm c}$ (like SFG8 or MW11), but minor ($<20\%$) for most of the galaxies with $f_{\rm c} \ge 0.1$.
Figure \[fig:cfall\] shows that the range of $f_{\rm c}$ within the virial radius is between $f_{\rm c}\sim 0.05 - 0.2$ at $z\sim 2$. Variations resulting from projection effects, albeit quite large in some galaxies, are typically smaller than this scatter, which reflects instead an intrinsic variation in the gas accretion and merger history of halos. This large scatter should discourage one from generalizing results obtained from a single simulation, which has often been done in the literature. Due to the geometry of the filaments that extend radially outward, the covering fraction at 2[$R_{\rm vir}$]{} drops between $f_{\rm c} \sim 0.01 -
0.13$, implying that an approximately equal area is subtended by optically-thick gas within [$R_{\rm vir}$]{} and $R_{\rm vir} < R < 2R_{\rm
vir}$. Comparing the redshift evolution of individual galaxies, we find only a modest decrease in the covering fraction that at $z\sim2$ drops to $\sim 70\%$ of the value measured at $z\sim 3$ within 2[$R_{\rm vir}$]{} ($\sim 80\%$ at [$R_{\rm vir}$]{}).
Figure \[fig:cfall\] also shows a lack of any appreciable mass dependence of the covering fraction over one decade in virial mass, despite the fact that our sample brackets the critical mass of $\sim 5\times 10^{11}~\rm M_{\rm vir}$ above which virial shocks become stable [@dek06; @ocv08]. A general prediction of cosmological hydrodynamic simulations is that the fraction of cold gas decreases as a function of virial mass [e.g. @ocv08; @ker09b; @van11; @fau11b; @nel13]. This fact would naively suggest a lower covering fraction of neutral hydrogen in more massive halos, but Figure \[fig:cfall\] illustrate that is indeed not the case for our simulations.
Gas has been defined “cold” differently by various authors, and the accretion rates or the ultimate fate of the cold material falling onto galaxies are extensively discussed – and highly debated – in the literature. The goal of our analysis is not to determine the detailed evolution of cold gas in galaxies at $z\sim 2-3$, but instead we focus on predicting the covering fraction of optically-thick gas around galactic halos at any given time, and on understanding its relationship to the mass fraction of cold gas $\phi_{\rm cold}$. Predictions for $f_{\rm c}$ are of obvious interest for understanding the origin of LLSs, and furthermore this covering fraction is an observable quantity for which recent measurements exist. Thus, here we define cold gas using the instantaneous temperature at a given redshift, i.e. without considering the past or future thermal history of this gas.
To gain further insight into the weak mass dependence of the covering fraction in our simulations (Figure \[fig:cfall\]), we compute the fraction of cold gas $\phi_{\rm cold}$ within the virial radius for our simulated galaxies at $z\sim 2$ and $z\sim3$, which is shown in Figure \[fig:fcold\]. Here, $\phi_{\rm cold}$ is defined as the ratio of the cold gas mass to the total gas mass within a given radius. In agreement with previous work, gas within 0.15[$R_{\rm vir}$]{} has been excluded from the analysis and from the values listed in Table \[tab:galprop\] to avoid material residing in the galaxy disk. If we define gas as “cold” when the temperature is less than $2.5\times 10^5~\rm K$, we find a trend of decreasing $\phi_{\rm cold}$ with increasing virial mass (top panel of Figure \[fig:fcold\]), in qualitative agreement with previous simulations [e.g. @ocv08; @ker09b; @van11; @nel13]. However, when we refine the definition of cold gas to include only hydrogen that is likely to remain neutral when self-shielded from ionizing radiation (i.e. $T\sim 3 \times 10^{4}~\rm K$; Table \[tab:galprop\]), we find a shallower dependence of $\phi_{\rm cold}$ on halo mass (bottom panel of Figure \[fig:fcold\]).
Thus, in our simulations we observe both an increase in the “hot” gas fraction with virial mass and a mass-independent $f_{\rm c}$ for optically-thick gas. This result is in apparent contradiction with the naive expectations based on previous work which, however, did not directly characterize the mass dependence of the covering fraction of optically-thick gas at any given redshift, which is the observable quantity. In other words, the onset of stable virial shocks affects the temperature and the mass fraction of gas at $\gtrsim 10^5~\rm K$, without preventing the existence of colder and neutral gas pockets in galaxy halos, even for masses above the critical halo mass for shock formation. Qualitatively, this is consistent with the idea that filaments of cold gas survive above the transition mass at $z \ge 2$ [@ker05; @dek06; @ocv08; @ker09b].
Finally, a mass-independent covering fraction may appear in conflict with recent reports by @ste11a of a decreasing $f_{\rm c}$ once a galaxy crosses the critical mass for the formation of hot halos. However, it should be noted that these authors follow the redshift evolution of two halos, finding a drop in the covering fraction only for $z<1.5$. Therefore, in light of the previous discussion, we interpret the sudden decrease in $f_{\rm c}$ reported by @ste11a as not being simply due to the halo growing beyond the critical mass and the concomitant presence of shock heated gas. But rather other factors, including redshift evolution, have to play a role in shaping the covering fraction seen in these simulations. Furthermore, it should be noted that the critical mass does not coincide with exactly the same halo mass for all galaxies, but instead depends on when the virial shock is triggered. Values of critical mass can spread over more than a decade in mass [e.g. @ker05].
Comparisons with other simulations
----------------------------------
In Figure \[fig:cfall\], we compare the covering fractions measured in our simulated galaxies to values from other simulations published in the literature. The covering fraction of the [*Eris*]{} halo, simulated at $z\sim 2.8$ by @she13 with an SPH code, is consistent with the upper limit of our distribution at $z\sim 3$, although their analysis relies on simple approximations for the ionization state of the gas. Similar consistency is found for the SPH simulation of the Milky Way progenitor B1 by @fau11 at $z\sim 2$ and for the SPH models with virial masses between $\sim 3\times 10^{11} -6 \times 10^{11}~\rm M_\odot$ at $z\sim 2$ by @ste11.
There seems to be agreement in the covering fractions of halos simulated with different numerical techniques (AMR and SPH), but this comparison is at the moment rather crude, since it is based on a very basic metric. For instance, the covering fraction may not properly reflect the difficulties of classical SPH formulations in capturing contact discontinuities and instabilities [e.g. @age07; @sij12] or sub-sonic turbulence dissipation [@bau12] that can affect both the properties of hot halos and of cold filaments inside massive halos. We now await comparisons with simulations performed with new SPH implementations that mitigate these problems [@rea12; @hop13]. Moreover, as noted, some of these simulations do not incorporate a detailed radiative transfer post-processing which is crucial to correctly describe the neutral fraction at the column densities relevant to LLSs, nor do they implement the same prescriptions for sub-grid physics. Finally, as previously highlighted, the large scatter in $f_{\rm c}$ within our ensemble of simulated galaxies hampers a precise comparison simulations of individual halos. Future analysis, e.g. from the ongoing AGORA code comparison project, will provide a better characterization of the level of agreement between various simulations.
@bir13 compared halos from a cosmological box simulated with an SPH code and with the new moving-mesh code Arepo, without radiative transfer post-processing and at lower resolution. These authors concluded that SPH codes produce an excess of optically-thick gas around halos of $M_{\rm vir} > 10^{11}~\rm h^{-1}~M_\odot$ compared to Arepo simulations. Thus, one may conclude that galaxies simulated with Arepo have lower covering fractions than what is found in SPH simulations. Distressingly, this would worsen the current tension between numerical calculations and observations (Section \[qso\]). However, before drawing similar conclusions, we prefer to await additional comparisons between Arepo and SPH or AMR codes at the high resolutions that are comparable to the ones achieved by the simulations presented in Figure \[fig:cfall\], once radiative transfer post-processing has been included.
Finally, we acknowledge that other simulated halos with comparable redshifts and masses to those included in this study have been presented in the literature [e.g. @ros12; @hum13], but because these authors do not provide direct information on the covering fraction of optically-thick gas, these simulations do not appear in Figure \[fig:cfall\]. Nevertheless, the column density maps presented by @ros12 appear in qualitative agreement with the maps shown in Figure \[fig:allgal\]. Also, @hum13 comment on the agreement between their model and the simulations of @fau11.
![Comparison of the covering fraction of optically-thick gas in simulated and observed galaxies. Simulations at $z\sim3$ and $z\sim2$ are represented by empty and filled red circles, respectively. The observed covering fraction around $z\sim2$ quasar host galaxies is shown with a blue upward triangle, while observations for Lyman break galaxies at $z\sim 2.5$ are summarized by a green square. The horizontal error bars reflect the large uncertainty in the inferred halo mass for these objects. Simulations appear to systematically underpredict the observed covering fractions at the highest masses, while current samples are too small for conclusive comparisons with LBGs.[]{data-label="fig:cfobs"}](cfplot_data.eps)
Simulations versus observations {#qso}
===============================
Having characterized the covering fractions in galaxies at $z \sim
2-3$ from a theoretical point of view, in this section we directly compare the predictions from our simulations with observations.
Covering fractions within [$R_{\rm vir}$]{}
-------------------------------------------
In Figure \[fig:cfobs\], we show again the simulated covering fractions within [$R_{\rm vir}$]{} both at $z\sim 2$ and $z\sim 3$, but we now superimpose measurements of the covering fractions of optically-thick gas for LBGs [@rud12 N. Crighton et al., in prep.] and in quasar host galaxies [@pro13].
@rud12 have measured the covering fraction of optically-thick gas in a sample of 10 LBGs within 100 kpc from a bright background quasar. Typical halo masses for LBGs are inferred by comparing the observed clustering of galaxies to the clustering of dark matter halos in numerical simulations. In the following, we assume the mass interval of $10^{11.8} < M/{\rm M_\odot}< 10^{12.2}$ at $z\sim 2$ from @ade05 [see also @con08], where the uncertainty in the halo mass reflects the errors on the measured correlation function. However, different determinations may suffer from larger systematic uncertainties [see e.g. @bie13]. Assuming $R_{\rm vir}\sim 90~\rm kpc$ for galaxies at this mass, @rud12 find $f_{\rm c} = 0.30 \pm 0.14$ within 68% confidence interval inside the virial radius. A similar analysis by N. Crighton et al. (in prep.) yields a comparable covering fraction, with slightly larger error bars.
A subset of our simulated galaxies or the [*Eris*]{} simulation by @she13 approach the observed value. However, we emphasize that this comparison is subject to the uncertainties of the subgrid physics included in these simulations (see Section \[secwind\]). As a population, the covering fraction in simulations ($f_{\rm c} = 0.15 \pm 0.06$) is a factor of 2 lower than what suggested by observations [cf. @rud12], but is nevertheless consistent given the large error bars. The mean covering fraction in simulations is in fact in formal agreement with observations, lying within the 68% confidence interval. This comparison therefore highlights how current samples of LBGs at $z\sim 2-3$ in proximity to background quasars are too small to conclusively establish whether there is inconsistency between simulations and observations, limiting our ability to robustly test current theories for gas accretion onto galaxies.
The situation is instead different at larger masses, as shown in Figure \[fig:cfobs\]. Using a sample of 74 quasar pairs, @pro13 have measured the covering fractions of optically-thick gas in the surroundings of $z\sim 2$ quasar host galaxies at projected separations ranging from 30 to 300 kpc. Assuming a typical halo mass of $(2.85\pm 0.71)\times 10^{12}~M_\odot$ for the quasar host halos deduced from the clustering measurements of [@whi12], and a corresponding virial radius of $\sim 150~\rm kpc$, 27 optically-thick systems are found along the 41 sightlines that sample the halos within [$R_{\rm vir}$]{}. The inferred covering fraction is therefore $f_{\rm c} = 0.67
\pm 0.07$ (68% confidence interval). As evident from Figure \[fig:cfobs\], this covering fraction significantly exceeds the values measured in these simulations.
![Radial covering fractions for the five most massive halos simulated at $z \sim 2$ (red squares) and in observations of quasar host galaxies (blue circles). Error bars in the covering fractions around quasars represent the 68% confidence Wilson score interval. For simulations, instead, we show the mean covering fractions computed in 1000 trials, with the standard deviation shown by the red shaded area. The bottom panel shows the number of sightlines in each radial bin in simulations (red dashed histogram) and observations (shaded histogram). Due to the finite box size of simulations, the outermost radial bin is undersampled compared to observations. Within $\sim 200~\rm kpc$, simulations show a significantly lower covering fractions than what is found in quasar host galaxies.[]{data-label="fig:qsocf"}](cfrqso.eps)
The radial dependence of $f_{\rm c}$
------------------------------------
The @pro13 quasar pair sample is sufficiently large to enable measurements of the covering fraction as a function of projected separation from the foreground quasar. A comparison between observations and simulations for the 5 most massive halos above $M_{\rm vir}=1.6\times 10^{12}~\rm M_\odot$ in our sample (MW12, SFG1, SFG7, SFG9, and VL11) is presented in Figure \[fig:qsocf\]. The mean virial mass in this subset is $M_{\rm
vir}=2.2\times 10^{12}~\rm M_\odot$ (median $M_{\rm vir}=1.7\times 10^{12}~\rm M_\odot$), comparable to the typical halo mass of quasar host galaxies. In this figure, the covering fractions and the corresponding 68% Wilson confidence intervals deduced from the observations are shown in bins of projected separation between the foreground quasar and the background quasar sightline. We have assumed the quasar resides at the center of its host dark matter halo. For a consistent comparison, we generated 1000 realizations of the same experiment conducted by @pro13, but using our simulations. For each trial, we randomly sample the five simulated halos along three orthogonal axis using 74 sightlines at the exact same set of impact parameters as the observed quasar pair sample.
The mean of the covering fractions computed for each bin are compared to the measurements in Figure \[fig:qsocf\], where we also show the standard deviation of the distribution of covering fraction in each bin measured from our ensemble of realizations. Because of the limited size of our simulation cube (4[$R_{\rm vir}$]{} on a side), the bin at the largest impact parameters is slightly undersampled in our mock observations (see bottom panel of Figure \[fig:qsocf\]). The inconsistency between observations and simulations is readily apparent for separations $\lesssim 200~\rm kpc$. Given the limited number of sightlines within 50 kpc, the large difference between the observed and simulated covering fractions in the innermost bin is not statistically significant. However, in the interval between $50-200~\rm kpc$, all the simulated covering fractions are significantly below the observations, lying outside the 95% confidence interval measured for the quasar pair data. This striking discrepancy is also evident from the fact that there were 37 optically-thick systems found in the 74 observed sightlines, whereas we never found 37 or more such absorbers in our 1000 trials sampled at the same 74 impact parameters.
The picture that clearly emerges from this comparison is that the basic cosmological processes responsible for the assembly of massive galaxies, and particularly gas inflows, do not produce a sufficiently high covering fraction of optically-thick gas to explain the high value observed around quasar host galaxies. This is especially true given that our post processing radiative transfer does not include the effect of the additional ionizing photons from the quasar itself, which would even further reduce the covering fractions deduced from the simulations [@hen06b; @hen07; @pro13].
Impact of feedback mechanisms on comparisons between simulations and observations {#secwind}
---------------------------------------------------------------------------------
The foregoing analysis reveals that our understanding of the gas distribution in the massive galaxies that host quasars is incomplete. In this section, we briefly speculate on possible causes for the discrepancy highlighted by our study, focusing first on feedback mechanisms.
The simulations included in this study (as other simulations discussed in the recent literature) are imperfect models of our Universe, particularly because of the weak or ad-hoc implementation of feedback. As discussed in Section \[sims\], the average mass loading factors of the winds in these simulations is low, $\eta\sim 0.3$ at 0.5[$R_{\rm vir}$]{}. Therefore, these simulations overestimate the amount of stars formed by $z\sim2$ by a factor of $\sim 2$, and consequently underpredict the gas fractions within the galaxy disks. This fact may impact the simulated properties of the CGM in several ways.
For instance, strong outflows would prevent gas to be locked into stars at high redshift, and additional material may then available for later accretion [cf. @opp10]. At the same time, stronger outflows may interact with the accreting material shaping its structure [see a discussion in @fau11b; @pow11]. Further, a stronger implementation of stellar feedback [see e.g. @sti12; @she13; @cev13], or an additional form of feedback from the central AGN, may be the astrophysical process that is needed to boost the covering fractions of optically-thick gas in these simulations. Besides alleviating or even resolving the tension between observations and simulations, feedback processes may also be required to reproduce the large equivalent widths of metal lines (e.g. for ) that have been found within the virial radius of quasar host galaxies [@pro13].
However, detailed absorption line modeling and analysis of the physical properties of a single quasar absorption system in @pro09 indicated that the enriched gas detected in the quasar CGM was unlikely to represent material ejected from the AGN. Furthermore, as shown by @van12 and @she13, the majority of the cross section of optically-thick hydrogen lies in cold filaments, with only a small contribution originating in cold gas entrained within outflows. For these reasons, stronger feedback implementations that generate mostly hot winds may not significantly boost the cross section of optically-thick gas. Different implementations in which a larger fraction of cold material is entrained in the outflowing gas may be required to increase the cross section of optically-thick gas. Unfortunately, most of the relevant astrophysical and hydrodynamic processes that occur in winds are currently not fully resolved by cosmological hydrodynamic simulations [see @pow11; @jou12; @hop12; @cre13].
We also emphasize that resolution may play a significant role in shaping the structure of optically-thick gas in simulations, regardless to the adopted feedback models. At progressively lower resolution, high density peaks are smoothed out, and thus structures of size comparable to the grid cells are not properly capture in these simulations. Thus resolution may affect the resulting covering fraction directly, e.g. we could be missing small clumps of optically-thick gas, or indirectly, e.g. by altering the structure of the medium through which ionizing photons propagate during our radiative transfer post-processing.
Additional causes for the discrepancies between simulations and observations
----------------------------------------------------------------------------
Besides incomplete physics in our simulations, other reasons can be invoked to account for the current inconsistency between simulations and observations at the high-mass end. Because of our limited simulation volume, the optically-thick gas modeled in these simulations resides within $2R_{\rm vir}$ from the center of the halo. Conversely, the @pro13 analysis considered a velocity interval of $\pm 1500 \rm km~s^{-1}$, which was required because of significant uncertainties in the quasar redshifts [@ric02]. This velocity interval corresponds to $\pm 45~\rm Mpc$ along the line-of-sight, and thus optically-thick systems detected at small projected distances (e.g. within [$R_{\rm vir}$]{}) from a foreground quasar could in fact lie at larger line-of-sight separations, and hence larger physical separations than we have considered.
Given the observed number of LLS per unit redshift at $z\sim 2$ [@ome13], the probability to intercept a random LLS from the cosmic background over such a small redshift path is however negligible compared to the large covering factors observed. However, if quasars reside at the center of larger scale structures, such as group of galaxies, which are each surrounded by optically-thick halo gas, then the observed covering factor may include a contribution from optically-thick absorbers at distances larger than the $2R_{\rm vir}$ that we have considered. This effect needs to be investigated with simulations of larger cosmological volumes. However, we speculate that absorbers distance larger than $2R_{\rm vir}$ can ease but not resolve the discrepancy between observations and simulations. Indeed, @pro13 measure a dropoff of the $f_{\rm c}$ with impact parameter for $r > 200\,{\rm
kpc}$ (Figure \[fig:qsocf\]), which suggests that optically-thick gas is mostly contained in proximity to the central galaxy and argues against a large contribution to the covering fraction from Mpc scales.
Finally, if quasars mark a particular phase in the life of a galaxy in which the AGN activity is triggered by mergers [e.g. @san88; @dim05; @hop05], the observations of quasar pairs may provide only a biased view of the halo gas in massive galaxies. However, processes other than major mergers may be responsible for feeding AGNs, in particular at high redshifts [e.g. @dav09; @cio10; @bou11; @cis11; @dim12]. Furthermore, at the typical bolometric luminosity of the quasar pairs ($L_{\rm bol} = 10^{45.5}-10^{47}~\rm erg~s^{-1}$), observations implies a star formation rate of $\sim 10-100~\rm M_\odot~yr^{-1}$ [e.g. @tra10] which is comparable to the star formation rates observed in matched populations of non-active star forming galaxies [e.g. @sha10; @san12; @mul12; @har12; @ros13]. Thus, at present, there is no clear indication that quasar pairs reside in a population of halos that are systematically different than those described by our simulations.
A statistical view of the circumgalactic medium {#future}
===============================================
As shown in Section \[qso\], samples of LBG-quasar pairs are currently too limited in size for conclusive comparisons with simulations. In the second part of this paper, we therefore introduce the formalism for measuring the auto-correlation function of LLSs (Section \[llslls\]), which is based on an extension of the formalism used to measure the galaxy-LLS cross-correlation function (reviewed in Section \[gallls\]). The advantage of this experiment is to exploit larger samples of quasar pairs to statistically map the distribution of optically-thick hydrogen around galaxies at $z\sim 2-3$, avoiding the telescope-intensive task of finding many galaxy-quasar pairs.
The galaxy-LLS correlation function {#gallls}
-----------------------------------
In Figure \[fig:qsocf\], we have shown the radial dependence of the covering fraction in quasar host galaxies. This quantity, which is particularly useful to investigate the spatial extent of the CGM around galaxies of a given halo mass, can be recast in terms of the galaxy-LLS cross-correlation function $\xi_{gl}(r)$ [see @hen07; @pro13b]. The cross-correlation function contains the same information as the covering fraction, but it has the advantage of directly comparing gas around galaxies to the cosmic background abundance of optically-thick hydrogen absorbers that are intercepted randomly as intervening LLSs. Thus, it directly quantifies the spatial scales for which a statistically significant excess of optically-thick absorption is detected around galaxies.
This cross-correlation function can also be compared to, e.g. the auto-correlation function of the galaxies themselves as well as the underlying dark-matter distribution, to help further constrain the distribution of CGM gas relative to the large-scale structure [e.g. @sel00; @wei04; @coo02]. Indeed, cross-correlation functions between galaxies and absorbers have already been studied in the literature. For instance, @bou04 and @coo06 measured the correlation between LBGs and damped Lyman-$\alpha$ systems, while @hen07, @fon13, and @pro13b measured the clustering of either LLSs or the Lyman-$\alpha$ forest around quasars. Also, @tin08, @wil08, and @ade05b studied the correlation between galaxies or quasars and metal absorption lines.
With the exception of @hen07 and @pro13b, all previous work has measured clustering on scales larger than $\sim 1~\rm Mpc$, and these larger scale clustering measurements constrain the dark matter halos hosting absorbers [@tin08]. However, as we will argue below, the small-scale clustering (i.e. scales comparable to the virial radius) or “one-halo” term has the potential to provide a very sensitive test for simulations of the CGM around galaxies. In what follows, we briefly review the formalism to compute the galaxy-LLS correlation function, closely following the discussion in @hen07. We then show predictions of $\xi_{gl}(r)$ computed from numerical models which we will then compared to measurement for the LLS auto-correlation function.
### Formalism
For a given a population of galaxies with redshifts $z_0$ that are probed by background quasars at projected separations $r_\bot$, we describe the distribution of optically-thick gas around halos as an excess probability of finding a LLS in comparison to random expectation inside a velocity interval $\pm \Delta v$ that is centered at the galaxy systemic redshift.
The probability of finding a LLS at random in the corresponding redshift interval $\Delta z_{\rm 0} = 2 \Delta v (1+z_0)/c$ is $P(\Delta z_0,r_\bot)=\ell(z_0)\Delta z_0$, where $\ell(z)$ is the number of LLSs per unit redshift evaluated at $z_0$. This probability, which is independent of the projected separation, expresses the covering fraction of absorbers from the cosmic background population of random intervening LLSs. At a distance $r_\bot$ from a foreground galaxy (or quasar host galaxy), the probability of intercepting optically-thick gas is enhanced by clustering around the galaxy according to $$\label{plx}
F_{\rm c} (\Delta z_0,r_\bot) = \ell (z_0) [1+\chi_{\rm gl, \bot}
(\Delta z_{\rm 0}, r_\bot)] \Delta z_0\:.$$ Here, $\chi_{\rm gl, \bot} (\Delta z_{\rm 0}, r_\bot)$ is the projected galaxy-LLS cross-correlation function, which quantifies the excess probability above the cosmic mean of detecting LLSs near the galaxy in the corresponding redshift interval. As we will show below, this probability $F_{\rm c}$ is directly related to the covering fraction $f_{\rm c}$ of optically-thick gas around galaxies.
The projected correlation function $\chi_{\rm gl, \bot} (\Delta z_{\rm 0}, r_\bot)$ can be related to the real-space galaxy-LLS correlation function $\xi_{\rm gl}(r)$ with an average over the volume $V=\sigma_{\rm a,c}r_{||}$. Here, $$\label{rpar}
r_{||} = \frac{c}{H_0} \int_{\Delta z_0} \frac{dz}{\sqrt{(\Omega_{\rm m}(1+z)^3
+ \Omega_{\rm \Lambda})}} \approx
\frac{c \Delta z_0}{H(z)}$$ in a flat cosmology, and $\sigma_{\rm a,c}$ is the cross section of the absorbing clouds. Under the assumption that $r_\bot >> \sigma_{\rm a,c} ^{1/2}$, $$\label{intchi}
\chi_{\rm gl,\bot} (\Delta z_{\rm 0}, r_\bot) \approx \frac{1}{r_{||}} \int ^{+r_{||}/2}_{-r_{||}/2} \xi_{\rm gl} (r'_{||}, r_\bot) dr'_{||}\:.$$
For an ensemble of galaxy/quasar pairs, the projected galaxy-LLS correlation function $\chi_{\rm gl,\bot}$ can be evaluated in bins[^1] of $r_\bot$ as $$\label{binnedchi}
\chi_{\rm gl,\bot}(\Delta z_{\rm 0}, r_\bot) = \frac{N_{\rm lls}}{N_{\rm ran}}-1\:.$$ Here, $N_{\rm lls}$ is the number of LLSs detected around the galaxies in bins centered on $r_\bot$ and $N_{\rm ran}$ is the number of LLSs expected at random for a given $\ell(z)$. Given measurements of $\chi_{\rm gl,\bot}(\Delta z_{\rm 0}, r_\bot)$, one can determine the functional form for $\xi_{\rm gl}(r)$ that best describes the observations using Equation (\[intchi\]).
![The projected galaxy-LLS cross-correlation functions computed for different covering fractions ($f_{\rm c}=0.25, 0.10, 0.05$) of optically-thick gas (blue lines) around dark matter halos with masses $5\times 10^{11} - 5 \times 10^{12}~\rm M_\odot$ within a 250 cMpc/h cosmological box. The projected LLS auto-correlation function is shown with a red dashed line for $f_{\rm c}=0.25$, with a red dotted line for $f_{\rm c}=0.10$, and with a red dashed triple-dotted line for $f_{\rm c}=0.05$. The projected two-point correlation function of dark matter halos is shown by grey crosses. For these calculations, we assume a velocity window $\Delta v = \pm 400~\rm km~s^{-1}$. If LLSs statistically trace galaxies, the LLS auto-correlation function encodes the same information that is contained in the galaxy-LLS correlation function, only smoothed on scales of the gaseous halo.[]{data-label="fig:llslls"}](dm_chi.eps)
### Numerical models
Our goal is to show with simple numerical models how the LLS auto-correlation function can be used to gain insight into the properties of the CGM in comparison to the galaxy-LLS cross-correlation function, and not to produce detailed predictions for these two quantities. Therefore, we generate simple realizations of a universe in which LLSs are distributed around galaxies adopting the following prescriptions.
The spatial distribution of dark matter halos is given by the [rockstar]{} halo catalogue [@beh13] extracted form the Bolshoi simulation [@kly11], a dark matter only cosmological simulation in a box of 250 cMpc/h (comoving Mpc) on a side. We then model the spatial distribution of LLSs by populating dark matter halos at $z\sim2$ in the mass interval $5\times 10^{11} - 5 \times 10^{12}~\rm M_\odot$ (consistent with the range explored in the previous sections) with a varying covering fraction of $f_{\rm c}=0.05, 0.10$, and $0.25$ within 2[$R_{\rm vir}$]{}. These values can be compared to the results of the hydrodynamic zoom-in simulations presented in the first part of this paper or to the observed values around LBGs [@rud12].
In this way, we obtain realizations of a universe in which, by construction, a fraction $\ell(z)_{\rm halo}
\propto 4 \pi R_{\rm vir}^2 f_{\rm c} n_{\rm halo}$ of LLSs arise from halos in the specified mass interval, where $n_{\rm halo}$ is the volume density of dark matter halos in the selected mass range. To account for the remaining systems required to give the correct cosmic average line density of LLSs, i.e. $\ell(z)_{\rm obs}-\ell(z)_{\rm halo}$, we simply add a random population of absorbers that are not clustered to dark matter halos and hence to galaxies. In all that follows, we take the incidence to be $\ell(z)_{\rm obs} \equiv 1.5$, which is consistent with the observed value for $N_{\rm HI} \ge 10^{17.2}~\rm cm^{-2}$ LLSs at $z\sim2$ from @ome13.
In other words, this model assumes a random, i.e. non-clustered, background of LLSs and of a second populations of LLSs that are clustered to galaxies in a selected mass range. Clearly, this is a rather simplistic approach as, for instance, simulations suggest that LLSs are typically clustered to galaxies of different masses [e.g. @koh07; @alt11; @rah13]. However, this approximation is meant to describe the limit in which a subset of LLSs arise from either low mass galaxies that have a small bias compared to the halos here considered, or to a case in which a fraction of LLS absorption arises along filaments in the intergalactic medium (IGM) and instead traces the Lyman-$\alpha$ forest which has a very weak clustering compared to massive halos [@mcd03]. Albeit crude in its treatment of the baryon distribution around galaxies, this model accurately reproduces the spatial clustering on large scales that is imposed by structure formation.
For the analysis, we sample these mock universes with random sightlines and we compute the projected galaxy-LLS cross-correlation function as described in Equation (\[binnedchi\]) within a velocity window of $\Delta v = \pm 400~\rm km~s^{-1}$, corresponding to a redshift interval of $\Delta z_0 = 0.008$ or a depth of $12~\rm cMpc$ along the line of sight. This velocity window is suitable for comparisons with observations as it is large enough to encompass the majority of the denser gas ($n_{\rm H} \ge 0.1~\rm cm^{-2}$) within 2[$R_{\rm vir}$]{} from the galaxy center, after accounting for peculiar velocities along the line of sight. Note that this velocity window is also larger than typical redshift errors for LBGs ($\sim 150~\rm km~s^{-1}$). The resulting $\chi_{\rm gl,\bot}$ from the four different models with varying $f_{\rm c}$ are shown in Figure \[fig:llslls\] (blue lines) as a function of the projected separation between galaxies and LLSs. For comparison, we also show the projected two-point correlation function of dark matter halos (grey crosses), which we compute by comparing the number of galaxy pairs at projected distance $r_\bot$ within the Bolshoi simulation to the number of random pairs[^2].
Figure \[fig:llslls\] provides a schematic view of the CGM properties that can be extracted from the galaxy-LLS correlation function. First, one can see that at projected separations that are typical for the one-halo term ($\sim 0.3-0.4~\rm cMpc/h \sim 2$[$R_{\rm vir}$]{}), the projected correlation function is proportional to the covering fraction of optically-thick gas inside the dark matter halos. By construction, our models do not incorporate any radial dependence for $f_{\rm c}$ within $r_{\rm \bot} < 2$[$R_{\rm vir}$]{}. However, our zoom-in simulations exhibit only a shallow radial profile for the covering fraction (see e.g. Figure \[fig:qsocf\]), and a modest radial dependence for the projected correlation function up to $\sim 2R_{\rm vir}$ becomes a general prediction. If we adopted a power law form for the correlation function $\xi_{\rm gl}(r) \sim (r/r_0)^{-\gamma}$, and fitted only data interior to $r_{\rm \bot} < 2$[$R_{\rm vir}$]{} that are dominated by this flat one-halo term, we would infer a large correlation length $r_0$ or, equivalently, a shallow exponent $\gamma$. A quantitative comparison between the observed and predicted galaxy-LLS cross-correlation function offers therefore an additional test for theories of gas accretion around galaxies.
The second feature that is visible in Figure \[fig:llslls\] is that, around $\sim 0.5~\rm cMpc/h$, the projected cross-correlation function exhibits a break at the transition between the one-halo term and the two-halo term. This feature offers a natural way to define the typical extent of the CGM in the galaxy population under examination. Finally, at larger projected separations ($r_\bot \gtrsim 1~\rm cMpc/h$), the two-halo term of the cross-correlation function traces the (halo mass dependent) two-point correlation function of the dark matter halos that host LLSs. For models with large covering fractions such that $\ell(z)_{\rm obs} \sim \ell(z)_{\rm halo}$, the galaxy-LLS and halo correlation functions overlap, while for models with lower $f_{\rm c}$, the amplitude of the galaxy-LLS correlation function is suppressed compared to the halo correlation function due to the increasingly higher contribution from the background, which in this particular modelization is randomly distributed, and hence dilutes the clustering signal. Note however that the shape of the cross-correlation function is preserved for the case of a large random background.
Finally, Figure \[fig:llslls\] reveals that even models with modest covering fraction of optically-thick gas as predicted by our zoom-in simulations exhibit a high amplitude for the projected correlation function. This is a direct consequence of the limited number of LLSs that are expected at random within a velocity window of $\Delta v= \pm
400~\rm km~s^{-1}$ from a galaxy. Given the amplitude of the correlation function, for models with $f_{\rm c}=0.25$, samples of $\sim 30$ galaxies-quasar pairs are needed to detect the one-halo term of the galaxy-LLS correlation function at $\sim 3\sigma$. To place interesting constraints on models, samples with at least 80 galaxy-LLS sightlines are needed. Twice as much pairs are instead required for this measurement for the $f_{\rm c}=0.10$ case. The galaxy-LLS cross-correlation function has already been measured on small scales for the quasar host galaxies [@hen07; @pro13b]. However, building up the required statistics to make a measurement of comparable precision of the LBG-LLS cross-correlation is a more challenging task. While one can attempt to detect a signal with current data, samples that are 5 to 10 times larger than what currently available are needed to precisely characterize the distribution of optically-thick gas around galaxies.
The LLS auto-correlation function {#llslls}
---------------------------------
To circumvent the observational challenges of building up large foreground galaxy-background quasar samples, we propose that one measures the *auto-correlation* function of LLSs, using the large existing samples of close quasar pairs, with $\sim 300$ pairs currently known at $r_\perp < 200~\rm kpc$ [@hen04; @hen06a; @hen10]. Further, one can exploit samples of lensed quasars to extend this measurement to even smaller scales of $\lesssim 10~\rm kpc$ [e.g. @ina12]. The key advantage of this technique is that LLSs are easy to identify even in modest signal-to-noise spectra, and hence large samples of LLS pairs can be assembled at $z\sim 2-3$.
The idea of the LLS auto-correlation function builds on previous work that has shown the power of correlating absorption systems along multiple quasar sightlines to reveal the spatial distribution of hydrogen or metals in the IGM [e.g. @mcd03; @mar10; @slo11; @fon12]. We now generalize the formalism presented for the galaxy-LLS correlation function to the case of two intervening LLSs (i.e. systems that are not physically associated to the background quasars) in the foreground of quasar pairs with projected separation $r_{12}$. Next, we will show using numerical models that the LLS auto-correlation function contains the same information about the CGM of galaxies as is encoded in the galaxy-LLS cross-correlation function. Thus, if LLSs are associated to galaxies, searches for LLSs in quasar pairs provide a powerful statistical way to characterize the CGM in high redshift galaxies, without the need to identify individual galaxy-LLS associations.
### Formalism
The formalism to compute the LLS auto-correlation function closely follows the approach used to compute the galaxy-LLS cross-correlation function. For a random quasar sightline, the probability to find a LLS is $P_1=\ell(z)\Delta z_1$, where $\Delta z_1$ is the useful redshift path that can be searched for absorption lines. Once a LLS is found at redshift $z_{\rm lls,1}$ the probability of finding a second LLS within $\pm \Delta v$ from the redshift of the first LLS along a second sightline at distance $r_{\rm 12}$ is $$F_{\rm C} (\Delta z'_0,r_{\rm 12}) = \ell (z_{\rm lls,1}) [1+\chi_{\rm LL,\bot} (\Delta z'_{\rm 0}, r_{\rm 12})] \Delta z'_0\:,$$ where $\Delta z'_{\rm 0} = 2 \Delta v(1+z_{\rm lls, 1})/c$ and $\chi_{\rm LL,\bot}(\Delta z'_{\rm 0}, r_{\rm 12})$ expresses the projected LLS auto-correlation function. As we will show in the following, if LLSs mostly arise from galaxy halos, $F_{\rm C}$ is directly related to the covering fraction $f_{\rm c}$ of optically-thick gas in the CGM in the galaxy population from which LLSs arise.
As previously done for the galaxy-LLS correlation function, we can relate $\chi_{\rm LL,\bot}$ to the LLS auto-correlation function $\xi_{\rm LL}(r)$ in real space following Equation (\[intchi\]). Altogether, assuming $\ell (z_{\rm lls,2}) \approx \ell (z_{\rm lls,1})$, the probability to find a pair of LLSs in the foreground of a quasar pair becomes $$P_2 (\Delta z_1,\Delta z'_0,r_{\rm 12}) \approx \ell^2 (z_{\rm lls,1})
[1+\chi_{\rm LL,\bot} (\Delta z'_{\rm 0}, r_{\rm 12})] \Delta z'_0\Delta z_1\:.$$
For an ensemble of quasar pairs, one can measure the projected LLS auto-correlation function $\chi_{\rm LL,\bot}$ in bins of $r_{12}$ following Equation \[binnedchi\].
### Numerical models
Provided that the population of LLSs can be identified with the CGM of galaxies [@koh07; @fum11; @van12; @fum13], a LLS detected at redshift $z_{\rm lls,1}$ along one sightline signals the presence of a galaxy, which lie at an unknown projected distance $r_{\rm lg}$. Therefore, even without identifying the galaxies that are responsible for the absorption, one can use a second sightline at projected separation $r_{12}$ from the first quasar to probe the distribution of optically-thick gas in the galaxy halo. The LLS auto-correlation function is thus analogous to the galaxy-LLS correlation function, providing a statistical way of mapping the CGM of distant halos without explicitly identifying galaxy-LLS associations.
To illustrate this point with numerical models, we generate a new realization from the Bolshoi simulation assuming $f_{\rm c}=0.25$, such that the majority of LLSs arise from halos with masses $5\times 10^{11}-5\times 10^{12}~\rm M_\odot$. We then sample the simulated box with pairs of sightlines with separations $r_{12}$ and compute the projected LLS auto-correlation function as described in Equation (\[binnedchi\]), that is by comparing the pairs of LLSs with a given $r_{12}$ and within a velocity window of $\Delta v = \pm 400~\rm km~s^{-1}$ to the random expectation. The resulting LLS auto-correlation function is shown with a red dashed line in Figure \[fig:llslls\]. Note that in this figure the projected separation on the x-axis corresponds to the distance between quasar pairs for the LLS auto-correlation function, while it corresponds to the separation between a quasar sightline and a galaxy (assumed to be at the center of the dark matter halo in our models) for the galaxy-LLS correlation function.
As is evident from Figure \[fig:llslls\], the projected LLS auto-correlation function closely resembles the projected galaxy-LLS correlation function for $f_{\rm c}=0.25$. The only difference is that for the galaxy-LLS pairs, the galaxy is always at the center of the dark matter halo, whereas for the LLS-LLS pairs, the halo centers are offset by a random amount relative to the two quasars probing the LLSs, and thus $\chi_{\rm LL,\bot}(\Delta z'_{\rm 0}, r_{12})$ reflects the properties of the halo gas smoothed on scales that are comparable to the size of the CGM, or $\sim 2R_{\rm vir}$ in our numerical models. Nevertheless, the LLS auto-correlation function encodes all the information we previously discussed for the galaxy-LLS correlation function. This includes a flat one-halo term with an amplitude that varies with the covering fraction in the host halos, a one-halo to two-halo term transition that can be used to define the characteristic size of the CGM in the galaxies where LLSs arise, and a two-halo term which traces the large-scale clustering of the underlying dark matter halos hosting LLSs. The fact that the large scale LLS correlation traces the clustering of dark matter halos is a trivial consequence of how we constructed our models by associating LLSs only to dark matter halos in a selected mass range. However, this exercise shows how measurements of the large-scale LLS auto-correlation function can be used to determine the typical halo masses which host LLSs, a key unknown quantity that currently hampers the interpretation of the observed LLS properties [e.g. @fum13].
In Figure \[fig:llslls\], we also show the LLS auto-correlation function in a realization with $f_{\rm c}=0.10$ (red dotted line) and $f_{\rm c}=0.05$ (red dashed triple-dotted line). In the latter case, the majority of LLSs ($\sim 80\%$) are not clustered to galaxies, but they reside in a random background. As expected, one can see how the projected auto-correlation function approaches zero. A comparison between the three models with $f_{\rm c}=0.05,0.10,0.25$ is useful to highlight the two extreme behaviors that the LLS auto-correlation function may reflect. For high covering fractions, or more generally when the product of the covering fraction and the size of the CGM is large (as suggested by current observations), the number of LLSs that are associated to galaxies exceeds the number of LLSs in a random (non clustered) background. In this case, $\chi_{\rm LL,\bot}\gg 0$ and thus the LLS auto-correlation function yields information on the CGM properties. Conversely, if either $f_{\rm c}$ is small or the radial profile of optically-thick gas in the CGM is very steep, then the number of LLSs associated to galaxies is much smaller than the number of LLSs in a random background and $\chi_{\rm LL,\bot}\sim 0$. In this case, a measurement of the LLS auto-correlation function can be used to conclude that LLSs are not associated to massive galaxies, but rather they originate from a more weakly clustered population, e.g. the Lyman-$\alpha$ forest. It should also be noted that if the fraction of LLSs that are associated to galaxies evolves with redshift [cf. @fum13], then the auto-correlation function of LLSs will evolve accordingly.
From the above discussion, it follows the LLS auto-correlation function encodes information on the cross section of optically thick gas around galaxies, similarly to the measurement of the cross-correlation function of damped Lyman-$\alpha$ systems with the Lyman-$\alpha$ forest [see @fon12]. For instance, in constructing these simple models, we have assume a mass-independent covering fraction $f_{\rm c}$ within 2[$R_{\rm vir}$]{}, which implies a mass dependent cross section $\sigma_{\rm lls} (M) \propto M_{\rm vir}^{2/3}$. In computing the LLS auto-correlation function, $\sigma_{\rm lls}$ determines the weight with which each halo of a given mass contributes to the observed value of $\chi_{\rm LL}$. For this reason, a precise measurement of the auto-correlation of optically-thick systems around quasar pairs provides a way to constrain the mass-dependent cross section $\sigma_{\rm lls} (M) \propto M_{\rm vir}^\alpha$, a quantity for which theoretical predictions exist from hydrodynamic simulations [e.g. @bir13].
We conclude by noting that, thanks to the large samples of quasar spectra that currently are or will be soon available from surveys like the Sloan Digital Sky Survey [SDSS; e.g. @par12], a measurement of the LLS auto-correlation function can be obtained at large scales ($\gtrsim 1 h^{-1}\,{\rm Mpc}$) at redshifts $z\ge 3$. The minimum angular separation is set by the fiber collision limit in the spectroscopic survey, which severely limits the number of close quasar pairs with available spectroscopy, while the redshift constraint is currently set by the throughput of the survey spectrograph. However, because of the SDSS color-selection bias that preferentially selects quasars with LLS absorption at $z<3.6$ [@wor11; @fum13], additional investigation is needed to establish whether the redshift limit has to be restricted to $z\ge 3.6$.
To measure the LLS auto-correlation function on small scales, hundreds of spectroscopically confirmed quasar pairs with projected separations between $30\,h^{-1}\,{\rm kpc}-1 h^{-1}\,{\rm Mpc}$ have now been discovered via follow-up spectroscopy of the SDSS imaging [@hen04; @hen06a; @hen10]. This sample allows a precise measurement of the small-scale clustering at high confidence level ($>5\sigma$ for $f_{\rm c}=0.25$). Although the majority of useful pairs are currently between $z\sim 2-3$ requiring the use of space-based facilities to identify LLSs in the foreground of these quasar pairs, a precise measurement of the LLS auto-correlation function on all scales can potentially be achieved in the near future.
Summary and conclusions {#concl}
=======================
We have analyzed the hydrogen distribution in the surroundings of 21 galaxies at $z\sim2-3$ that have been simulated at high resolution with virial masses $M_{\rm vir} \sim 2\times 10^{11} - 4\times 10^{12}~\rm M_\odot$. After post-processing these simulations with a Monte Carlo radiative transfer code to identify regions that retain enough neutral hydrogen to remain optically thick to Lyman continuum radiation, we have directly compared the covering fraction of optically-thick gas in simulations and observations of $z\sim 2$ LBGs and quasar host galaxies. We have also presented a formalism to compute the galaxy-LLS cross-correlation function and the LLS auto-correlation function, and we have provided simple estimates for these quantities using numerical simulations. Our main findings can be summarized as follows.
The covering fractions of optically-thick gas within the virial radius of the simulated galaxies range between $f_{\rm c} \sim 0.05-0.2$, where the large scatter is driven by intrinsic variation in the gas distributions around individual halos. Within 2[$R_{\rm vir}$]{}, we have found instead $f_{\rm c} \sim 0.01 - 0.13$, implying that the area subtended by optically-thick gas within [$R_{\rm vir}$]{} and between $R_{\rm vir} < R <
2R_{\rm vir}$ is approximately the same. While our simulations exhibit the expected increase in the average gas temperature and an increase in the mass fraction of hot gas above virial masses for which stable virial shocks form, we have found that the mass fraction of cold gas with $T< 3\times 10^4~\rm K$ is only weakly dependent on halo mass. Further, at $z\ge 2$, we have not found any strong dependence of the covering fraction on the halo mass, even beyond the critical mass for the formation of virial shocks.
Once compared to observations of 10 galaxy-quasar pairs at $z\sim 2-3$, these simulations are statistically consistent with the observed covering fraction of optically-thick gas inside the virial radius. However, current samples are too small to make a conclusive comparison, preventing stringent tests for current theories of cold gas accretion. Conversely, simulated halos at $M_{\rm vir} \ge 10^{12}~\rm M_\odot$ exhibit covering fractions at all radii that significantly underestimate the values observed in the surroundings of quasar host galaxies. This discrepancy reveals that our numerical models do not fully capture all of the physical processes necessary to describe the gas distribution around massive halos. At present, we do not know the explanation for this disagreement, but issues that should be investigrated in future work are i) modeling the effects of stronger (AGN) feedback and/or small-scales hydrodynamic instabilities than what is currently implemented in our simulations, ii) or a better understanding on how the properties of quasar host galaxies, in particular their star-formation rates or gas masses, compare to other populations of star-forming galaxies such as the LBGs.
Further, we have showed how mesurements of the galaxy-LLS correlation function can be used to measure the ccovering fraction of LLSs around galaxies. The flat radial depence of the covering fraction interior to the $R_{\rm vir}$ predicted by our simulations, implies that the projected galaxy-LLS correlation function will exhibit a shallow radial dependence on small-scales that probe the one-halo term. We have also showed that the transition between the one-halo term and two-halo term imprints a feature in the projected cross-correlation function that can be used to define the spatial extent of the CGM.
Finally, under the assumption that LLSs are statistically associated to galaxy halos of a given mass range, we have proposed a measurement of the LLS auto-correlation function using quasar pair sightlines, to map the spatial distribution of optically-thick gas around galaxies, without the need to identify individual galaxy-LLS associations. Our numerical models show that the LLS auto-correlation function encodes the same information contained in the galaxy-LLS correlation function (both covering fraction of optically-thick gas, the characteristic size for the CGM), but smoothed on scales comparable to the typical size of the CGM. Furthermore, we have highlighted that at large separations the two-halo term of the LLS auto-correlation function traces the two-point correlation function of the dark matter halos hosting LLSs, providing long-sought information about the typical mass of the halos that host LLSs.
While our analysis underscores a still incomplete view of the gas distribution around massive galaxies, in this paper we have outlined a possible path towards an improved knowledge of the properties of the halo gas in the distant Universe. In the long term, the increasing availability of samples of quasar-galaxy pairs will offer a direct way to map the radial distribution of optically-thick gas at high redshift. Measurements of the galaxy-LLS correlation can be compared to different sets of simulations, providing additional insights into the processes that regulate the structure of the CGM and ultimately the formation and evolution of galaxies. Given the current availability of large spectroscopic samples of quasars and hundreds of quasar pairs with small projected separations, it is also possible to compute the LLS auto-correlation function to obtain the first view of the spatial distribution of optically-thick gas in the high-redshift Universe.
As discussed, this measurement would provide an important test for the cold-stream paradigm, as well as a solid empirical assessment of whether LLSs arise primarily in the CGM of galaxies at $z \sim 2-3$. Provided that the connection between LLSs and halo gas can be robustly established, analysis of the physical properties of these absorbers would then offer a powerful way to map the metal distribution in proximity to galaxies in the distant Universe. Further, better knowledge on the clustering of LLSs would affect estimates for the extragalactic UV background, that depend strongly on the distribution of optically-thick gas. It is therefore clear that an improved understanding of how LLSs cluster around galaxies and around themselves would constitute an important step forward that will impact several areas of study.
The simulations were performed at NASA Advanced Supercomputing (NAS) at NASA Ames Research Center, at the National Energy Research Scientific Computing Center (NERSC) at Lawrence Berkeley Laboratory, and in the astro cluster at The Hebrew University. We acknowledge useful conversations with Andrew Benson and we thank Claude-André Faucher-Giguère for his comments on this manuscript. We also thank the referee for suggestions that have improved this paper. Support for this work was provided by NASA to MF through Hubble Fellowship grant HF-51305.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. MF thanks the members of CCAPP and the astronomy department at Ohio State University for their hospitality during a visit made possible by the Price Prize and for interesting discussions on the LLS correlation function. JXP is supported by NSF grant AST-1010004. AD acknowledges support by ISF grant 24/12, by GIF grant G-1052-104.7/2009, by a DIP-DFG grant. AD and JP acknowledge support by NSF grant AST-1010033. DC is supported by the JdC subprogramme JCI-2010-07122.
Adelberger, K. L., Steidel, C. C., Pettini, M., et al. 2005, , 619, 697 Adelberger, K. L., Shapley, A. E., Steidel, C. C., et al. 2005, , 629, 636 Agertz, O., Moore, B., Stadel, J., et al. 2007, , 380, 963 Altay, G., Theuns, T., Schaye, J., Crighton, N. H. M., & Dalla Vecchia, C. 2011, , 737, L37 Bauer, A., & Springel, V. 2012, , 423, 2558 Behroozi, P. S., Wechsler, R. H., Wu, H.-Y., et al. 2013, , 763, 18 Bielby, R., Hill, M. D., Shanks, T., et al. 2013, , 430, 425 Binney, J. 1977, , 215, 483 Bird, S., Vogelsberger, M., Sijacki, D., et al. 2013, , 429, 3341 Birnboim, Y., & Dekel, A. 2003, , 345, 349 Bouch[é]{}, N., & Lowenthal, J. D. 2004, , 609, 513 Bournaud, F., Dekel, A., Teyssier, R., et al. 2011, , 741, L33 Cantalupo, S., Lilly, S. J., & Haehnelt, M. G. 2012, , 425, 1992 Cantalupo, S., & Porciani, C. 2011, , 411, 1678 Ceverino, D., Klypin, A., Klimek, E., et al. 2013, arXiv:1307.0943 Ceverino, D., & Klypin, A. 2009, , 695, 292 Ceverino, D., Dekel, A., Mandelker, N., et al. 2012, , 420, 3490 Ceverino, D., Dekel, A., & Bournaud, F. 2010, , 404, 2151 Ciotti, L., Ostriker, J. P., & Proga, D. 2010, , 717, 708 Cisternas, M., Jahnke, K., Bongiorno, A., et al. 2011, , 741, L11 Conroy, C., Shapley, A. E., Tinker, J. L., Santos, M. R., & Lemson, G. 2008, , 679, 1192 Cooke, J., Wolfe, A. M., Gawiser, E., & Prochaska, J. X. 2006, , 652, 994 Cooray, A., & Sheth, R. 2002, , 372, 1 Creasey, P., Theuns, T., & Bower, R. G. 2013, , 429, 1922 Crighton, N. H. M., Bielby, R., Shanks, T., et al. 2011, , 414, 28 Davies, R. I., Maciejewski, W., Hicks, E. K. S., et al. 2009, , 702, 114 Dekel, A., Zolotov, A., Tweed, D., et al. 2013, arXiv:1303.3009 Dekel, A., et al. 2009, , 457, 451 Dekel, A., & Birnboim, Y. 2006, , 368, 2 Di Matteo, T., Springel, V., & Hernquist, L. 2005, , 433, 604 Di Matteo, T., Khandai, N., DeGraf, C., et al. 2012, , 745, L29 Dijkstra, M., & Loeb, A. 2009, , 400, 1109 Fardal, M. A., Katz, N., Gardner, J. P., Hernquist, L., Weinberg, D. H., & Dav[é]{}, R. 2001, , 562, 605 Faucher-Gigu[è]{}re, C.-A., Kere[š]{}, D., & Ma, C.-P. 2011, , 417, 2982 Faucher-Gigu[è]{}re, C.-A., & Kere[š]{}, D. 2011, , 412, L118 Faucher-Gigu[è]{}re, C.-A., Kere[š]{}, D., Dijkstra, M., Hernquist, L., & Zaldarriaga, M. 2010, , 725, 633 Font-Ribera, A., Arnau, E., Miralda-Escud[é]{}, J., et al. 2013, JCAP, 5, 18 Font-Ribera, A., Miralda-Escud[é]{}, J., Arnau, E., et al. 2012, JCAP, 11, 59 Fumagalli, M., Prochaska, J. X., Kasen, D., et al. 2011, , 418, 1796 Fumagalli, M., O’Meara, J. M., Prochaska, J. X., & Worseck, G. 2013, , 775, 78 Furlanetto, S. R., Schaye, J., Springel, V., & Hernquist, L. 2005, , 622, 7 Genzel, R., Burkert, A., Bouch[é]{}, N., et al. 2008, , 687, 59 Goerdt, T., Dekel, A., Sternberg, A., Gnat, O., & Ceverino, D. 2012, , 424, 2292 Goerdt, T., Dekel, A., Sternberg, A., Ceverino, D., Teyssier, R., & Primack, J. R. 2010, , 407, 613 Haardt, F., & Madau, P. 2012, , 746, 125 Haiman, Z., Spaans, M., & Quataert, E. 2000, , 537, L5 Harrison, C. M., Alexander, D. M., Mullaney, J. R., et al. 2012, , 760, L15 Hennawi, J. F. 2004, Ph.D. Thesis Hennawi, J. F., & Prochaska, J. X. 2013, , 766, 58 Hennawi, J. F., Prochaska, J. X., Burles, S., et al. 2006, , 651, 61 Hennawi, J. F., Strauss, M. A., Oguri, M., et al. 2006, , 131, 1 Hennawi, J. F., & Prochaska, J. X. 2007, , 655, 735 Hennawi, J. F., Myers, A. D., Shen, Y., et al. 2010, , 719, 1672 Hopkins, P. F. 2013, , 428, 2840 Hopkins, P. F., Quataert, E., & Murray, N. 2012, , 421, 3522 Hopkins, P. F., Hernquist, L., Cox, T. J., et al. 2005, , 630, 705 Hummels, C. B., Bryan, G. L., Smith, B. D., & Turk, M. J. 2013, , 430, 1548 Iliev, I. T., Ciardi, B., Alvarez, M. A., et al. 2006, , 371, 1057 Inada, N., Oguri, M., Shin, M.-S., et al. 2012, , 143, 119 Kasen, D., Thomas, R. C., & Nugent, P. 2006, , 651, 366 Jefferies, J. T. 1968, A Blaisdell Book in the Pure and Applied Sciences, Waltham, Mass.: Blaisdell, 1968 Joung, M. R., Bryan, G. L., & Putman, M. E. 2012, , 745, 148 Kennicutt, R. C., Jr. 1998, , 498, 541 Kere[š]{}, D., Katz, N., Weinberg, D. H., & Dav[é]{}, R. 2005, , 363, 2 Kere[š]{}, D., Katz, N., Fardal, M., Dav[é]{}, R., & Weinberg, D. H. 2009, , 395, 160 Klypin, A. A., Trujillo-Gomez, S., & Primack, J. 2011, , 740, 102 Kohler, K., & Gnedin, N. Y. 2007, , 655, 685 Komatsu, E., et al. 2009, , 180, 330 Kravtsov, A. V., Klypin, A. A., & Khokhlov, A. M. 1997, , 111, 73 Kravtsov, A. V. 2003, , 590, L1 Kroupa, P. 2001, , 322, 231 Lucy, L. B. 2002, , 384, 725 Martin, C. L., Scannapieco, E., Ellison, S. L., et al. 2010, , 721, 174 McDonald, P. 2003, , 585, 34 Miller, G. E., & Scalo, J. M. 1979, , 41, 513 Mullaney, J. R., Pannella, M., Daddi, E., et al. 2012, , 419, 95 Nelson, D., Vogelsberger, M., Genel, S., et al. 2013, , 429, 3353 Nestor, D. B., Shapley, A. E., Kornei, K. A., Steidel, C. C., & Siana, B. 2013, , 765, 47 Ocvirk, P., Pichon, C., & Teyssier, R. 2008, , 390, 1326 O’Meara, J. M., Prochaska, J. X., Worseck, G., Chen, H.-W., & Madau, P. 2013, , 765, 137 Oppenheimer, B. D., & Schaye, J. 2013, arXiv:1302.5710 Oppenheimer, B. D., Dav[é]{}, R., Kere[š]{}, D., et al. 2010, , 406, 2325 P[â]{}ris, I., Petitjean, P., Aubourg, [É]{}., et al. 2012, , 548, A66 Powell, L. C., Slyz, A., & Devriendt, J. 2011, , 414, 3671 Prochaska, J. X., Hennawi, J. F., Simcoe, R. 2013, ApJ, 762, 19 Prochaska, J. X., Hennawi, J. F., Lee, K.-G., et al. 2013, arXiv:1308.6222 Prochaska, J. X., O’Meara, J. M., & Worseck, G. 2010, , 718, 392 Prochaska, J. X., & Hennawi, J. F. 2009, , 690, 1558 Rahmati, A., Pawlik, A. H., Rai[č]{}evic, M., & Schaye, J. 2013, , 430, 2427 Rahmati, A., Schaye, J., Pawlik, A. H., & Rai[č]{}evic, M. 2013, , 989 Rauch, M., Becker, G. D., Haehnelt, M. G., Gauthier, J.-R., & Sargent, W. L. W. 2013, , 429, 429 Read, J. I., & Hayfield, T. 2012, , 422, 3037 Rees, M. J., & Ostriker, J. P. 1977, , 179, 541 Richards, G. T., Vanden Berk, D. E., Reichard, T. A., et al. 2002, , 124, 1 Rosario, D. J., Mozena, M., Wuyts, S., et al. 2013, , 763, 59 Rosdahl, J., & Blaizot, J. 2012, , 423, 344 Rudie, G. C., Steidel, C. C., Trainor, R. F., et al. 2012, , 750, 67 Sanders, D. B., Soifer, B. T., Elias, J. H., et al. 1988, , 325, 74 Santini, P., Rosario, D. J., Shao, L., et al. 2012, , 540, A109 Seljak, U. 2000, , 318, 203 Shen, S., Madau, P., Guedes, J., et al. 2013, , 765, 89 Shao, L., Lutz, D., Nordon, R., et al. 2010, , 518, L26 Silk, J. 1977, , 211, 638 Sijacki, D., Vogelsberger, M., Kere[š]{}, D., Springel, V., & Hernquist, L. 2012, , 424, 2999 Slosar, A., Font-Ribera, A., Pieri, M. M., et al. 2011, Journal of Cosmology and Astroparticle Physics, 9, 1 Steidel, C. C., Erb, D. K., Shapley, A. E., Pettini, M., Reddy, N., Bogosavljevi[ć]{}, M., Rudie, G. C., & Rakic, O. 2010, , 717, 289 Stewart, K. R., Kaufmann, T., Bullock, J. S., et al. 2011b, , 738, 39 Stewart, K. R., Kaufmann, T., Bullock, J. S., et al. 2011a, , 735, L1 Stinson, G. S., Brook, C., Prochaska, J. X., et al. 2012, , 425, 1270 Tinker, J. L., & Chen, H.-W. 2008, , 679, 1218 Trainor, R. F., & Steidel, C. C. 2012, , 752, 39 Trakhtenbrot, B., & Netzer, H. 2010, , 406, L35 van de Voort, F., Schaye, J., Altay, G., & Theuns, T. 2012, , 421, 2809 van de Voort, F., Schaye, J., Booth, C. M., Haas, M. R., & Dalla Vecchia, C. 2011, , 414, 2458 Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, , 465, 487 Verner, D. A., & Ferland, G. J. 1996, , 103, 467 Weinberg, D. H., Dav[é]{}, R., Katz, N., & Hernquist, L. 2004, , 601, 1 White, S. D. M., & Rees, M. J. 1978, , 183, 341 White, M., Myers, A. D., Ross, N. P., et al. 2012, arXiv:1203.5306 Wild, V., Kauffmann, G., White, S., et al. 2008, , 388, 227 Worseck, G., & Prochaska, J. X. 2011, , 728, 23 Wood, K., & Loeb, A. 2000, , 545, 86 Woosley, S. E., & Weaver, T. A. 1995, , 101, 181
Method for Calculating Ionizing Radiation Transport {#ratraapp}
===================================================
To calculate the transport of ionizing radiation, we used a 3D Monte Carlo (MC) code, derived from the SEDONA code framework [@kas06] and using an approach similar to that described in [@woo00]. The radiation field is represented by a large number ($N_{\rm p} \sim 10^8$) of discrete photon packets, which are propagated throughout absorption or scattering events until they escape the simulation domain. This approach permits an arbitrary number of individual sources and conserves energy by construction. In addition, we are able to properly model the diffuse ionizing radiation field arising from recombinations to the ground state – i.e., we do not make the “on-the-spot” approximation. The main disadvantage of MC methods is that a large number of photon packets must be used to overcome statistical noise, however the method scales well and can be run on massively parallel machines.
The calculation in this paper ignores time-dependent effects and makes the assumption of ionization equilibrium [cf. @can11; @opp13]. Each photon packet carries a “luminosity” of $L_{\rm p} = (L_{\rm ubv} + L_{\rm loc})/N_{\rm p}$, where $L_{\rm ubv}$ and $L_{\rm loc}$ are the total ionizing luminosities from the UVB and from local sources, respectively. A fraction $f = L_{\rm ubv}/(L_{\rm ubv} + L_{\rm loc})$ of the photon packets are selected to represent the UVB. These packets are initially distributed randomly over the surface of a sphere of radius $R_{\rm ubv}$, chosen to be larger than the simulation box. The effective luminosity of the UVB from this outer surface is given by $$L_{\rm ubv} = 4 \pi R_{\rm ubv}^2 \int d\nu~ {{\ensuremath{J_{\rm bk}}}}(\nu),$$ where we choose a UVB mean intensity at $z \sim 2$ and $z\sim 3$ (according to the redshift of each snapshot) from @haa12. For simplicity, the spectrum of the UVB is assumed to be flat over the energy range $1 - 4$ Rydberg ($238-912$ Å) corresponding to the wavelength region from the HeII ionization threshold to the hydrogen threshold.
To produce a homogenous UVB radiation field within the simulation domain, the directionality of the packets is sampled from a $\cos \theta$ angular distribution, where $\theta$ is the angle between the packet direction vector and the inward local radial vector. To test the validity of this approach, we ran a calculation using $N_p = 10^8$ packets and assuming that the opacity throughout the domain was zero everywhere. A nearly uniform UVB of the desired mean intensity was achieved in the simulation box, with random errors of the order of $1\%$.
The remaining $1-f$ of the packets are selected to represent the ionizing radiation from local sources, and are emitted isotropically from locations given by the star particles from the hydrodynamic simulation. The probability of a packet being emitted from a given star particle is proportional to the UV luminosity of that star particle [see @fum11]. As with the UVB, the spectrum of local sources is assumed to be flat between 1 and 4 Rydberg.
Packets are propagated through the AMR grid until they are absorbed or escape the domain. The mean free path to a photoionization interaction with neutral hydrogen is given by $1/n_{\rm HI} \sigma_p(\nu)$ where $\sigma_p(\nu)$ is the photoionization cross-section. Whenever a photon ionizes a hydrogen atom, the atom is assumed to recombine either to the ground state or to an excited state followed by a cascade to the ground state. The former case corresponds to an “effective scattering” in which an ionizing photon is immediately re-emitted in a new direction. The probability of this occurring is $P_s = \alpha_1/\alpha_A$ where $\alpha_1$ is the recombination coefficient to the ground state while $\alpha_A$ is the coefficient for recombination to all levels including the ground state. We assume a constant value $P_s = 0.38$ appropriate for gas at $T = 10^4$ K, as this quantity is only weakly dependent on temperature. At each ionization interaction event, a random number is chosen to determine whether recombination to the ground state occurs; if so, the photon packet is redirected isotropically, assigned a new wavelength from the local emissivity function, and its propagation continues until a true absorption occurs.
To calculate the state of the gas, ionization equilibrium is assumed, $$R_{\rm pi} + R_{\rm ci} = R_{\rm rr},
\label{Eq:ion_eq}$$ where $R_{\rm pi}$ and $R_{\rm ci}$ are, respectively, the hydrogen photoionization and collisional ionization rates, and $R_{\rm rr}$ is the hydrogen radiative recombination rate (all per cm$^3$). At these low densities collisional recombination can be ignored. The photoionization rate is given by $$R_{\rm pi} = 4 \pi n_{\rm H} x_{\rm HI} \int d\nu~ \frac{J_\nu(\nu) }{h \nu} \sigma_p(\nu)
\label{Eq:photoion}$$ where $n_{\rm H}$ is the total hydrogen density, $x_{\rm HI}$ is the fraction of hydrogen in the neutral state, and $J_\nu$ is the mean intensity of the ionizing radiation field. The radiative recombination rate is $$R_{\rm rr} = n_e n_p \alpha_{A}$$ where $n_p$ is the proton density, and $n_e$ the electron density. For these calculations, we assume helium is neutral and does not contribute to the free electron density. In that case, we take $n_e = n_p = (1 - x_{\rm HI}) n_{\rm H}$. Expressions for $\sigma_p$ and $\alpha_{A}(T)$ are taken from @ver96 and @ver96b, respectively, and the collisional ionization rate from @jef68.
During the MC procedure, an estimate of the photoionization rate (Equation (\[Eq:photoion\])) is constructed in each cell by tallying all traversing packets [e.g., @luc02; @woo00] $$R_{\rm pi}/x_{\rm HI} = \frac{n_{\rm H}}{V} \sum_i \frac{L_p \sigma_p(\nu)}{h \nu} l_i
\label{Eq:estimate}$$ where $V$ is the cell volume, and the sum runs over all steps of length $l_i$ that occur for packets passing through the cell.
An iterative approach is used to converge the model to ionization equilibrium. Initially, a guess is made as to the neutral fraction $x_{\rm HI}$ in all cells. We then follow the MC transport and construct the estimator Equation (\[Eq:estimate\]). By solving Equation (\[Eq:ion\_eq\]), we obtain a new value for the neutral fraction in each cell. The MC transport routine is then rerun, and a new estimate of $x_{\rm HI}$ is derived. This procedure is iterated until the ionization state no longer changes significantly from one iteration to the next. To speed convergence, we adopt as an initial guess that hydrogen is completely ionized everywhere, as in this case photons packets can propagate information across the entire domain. We find that 12 iterations are sufficient for convergence.
To validate the photoionization code, we perform test 1 of @ili06 which consists of a box of dimension $R_{\rm box}=6.6$ kpc with uniform gas number density $n_H = 10^{-3}~{\ensuremath{{\rm cm^{-3}}}}$. ú A source of ionizing photons with a production rate $Q = 5 \times 10^{48}~{\rm s^{-1}}$ is placed at the center of the box. Figure \[Fig:uv\_test\] shows the resulting equilibrium ionization structure, which is in agreement with the converged structure presented in figure 8 of @ili06.
![Results for test 1 of @ili06 used to validate the photoionization calculation. The problem consists of a homogenous distribution of gas of number density $n_H = 10^{-3}~{\ensuremath{{\rm cm^{-3}}}}$ with a central source of ionizing luminosity $Q = 5 \times 10^{48}~{\rm s^{-1}}$. The solid line shows the neutral fraction and the dashed line the ionized fraction of hydrogen, plotted as a function of distance from the center. The results can be compared the calculations shown in the right hand panel of figure 8 of @ili06. \[Fig:uv\_test\]](ilievtest.eps)
[^1]: For alternative method for computing the projected correlation function without binning data see, e.g., @hen07.
[^2]: The two-point correlation function for dark matter halos flattens at scales of $\lesssim 0.8~\rm cMpc/h$ because of halo exclusion effects for which two halos cannot occupy the same volume.
|
---
abstract: 'In this work we introduce volume constraint problems involving the nonlocal operator $(-\Delta)_{\delta}^{s}$, closely related to the fractional Laplacian $(-\Delta)^{s}$, and depending upon a parameter $\delta>0$ called horizon. We study the associated linear and spectral problems and the behavior of these volume constraint problems when $\delta\to0^+$ and $\delta\to+\infty$. Through these limit processes on $(-\Delta)_{\delta}^{s}$ we derive spectral convergence to the local Laplacian and to the fractional Laplacian as $\delta\to 0^+$ and $\delta \to +\infty$ respectively, as well as we prove the convergence of solutions of these problems to solutions of a local Dirichlet problem involving $(-\Delta)$ as $\delta\to0^+$ or to solutions of a nonlocal fractional Dirichlet problem involving $(-\Delta)^s$ as $\delta\to+\infty$.'
address:
- 'José C. BellidoE.T.S.I. Industriales & INEI, Department of Mathematics University of Castilla-La Mancha Ciudad Real, E-13071 Spain '
- 'Alejandro OrtegaE.T.S.I. Industriales, Department of Mathematics University of Castilla-La Mancha Ciudad Real, E-13071 Spain '
author:
- 'José C. Bellido'
- Alejandro Ortega
title: A restricted nonlocal operator bridging together the Laplacian and the fractional Laplacian
---
[^1] [^2]
Introduction
============
Nonlocal and fractional elliptic problems have attracted a great attention in the mathematical community in the last two decades, coming from fields, among others, as nonlocal diffusion [@BuVal; @Mazon], statistical mechanics [@AlBe], continuum mechanics, including peridynamics, [@EvBe; @KassMenScott; @MenDu; @Si], and imaging processing [@KinOJo; @GilStan; @BoElPonScher]. For a good account on nonlocal modeling we refer to [@Du]. Nonlocal variational problems are also of importance in the characterization Sobolev spaces [@BourBrezMiro; @Ponce; @GioSpec; @Spec; @MenSpec; @Men]. Interesting surveys, which include an exhaustive list of references, on the fractional Laplacian and nonlocal elliptic problems are [@RosOton; @MR4043885].
In this work we study volume constraint elliptic problems driven by a nonlocal operator closely related to the well-known fractional Laplace operator. In particular, given an open bounded domain $\Omega\subset\mathbb{R}^N$ with Lipschitz boundary and $\delta>0$, a parameter called *horizon*, let us define the problem, $$\label{problema}
\left\{\begin{array}{rl}
(-\Delta)_{\delta}^su=f &\quad\mbox{in}\quad \Omega,\\
u=0 &\quad\mbox{on}\quad \partial_{\delta}\Omega,\\
\end{array}\right.
\tag{$P_{\delta}^{s}$}$$ where, $$\label{defLaplacian}
(-\Delta)_{\delta}^su(x)=c_{N,s}P.V.\int_{B(x,\delta)}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,$$ with $c_{N,s}=\frac{2^{2s}s\Gamma(\frac{N}{2}+s)}{\pi^{\frac{N}{2}}\Gamma(1-s)}$ a normalization constant and $\partial_\delta\Omega$ the nonlocal boundary given by $$\partial_{\delta}\Omega=\{y\in\mathbb{R}^N\backslash\Omega: |x-y|<\delta\ \text{for}\ x\in\Omega\}.$$ The operator $(-\Delta)_\delta^s$ is not new, and it has been addressed in different studies in the literature before. In view of the definition of $(-\Delta)_{\delta}^s$, it is clear that long-range interactions are neglected and only those exerted at distance smaller than $\delta>0$ are taken into account, i.e., the horizon $\delta>0$ represents the range of interactions. In this sense, the operator $(-\Delta_\delta^s)$, pertaining to the class of nonlocal elliptic operators, it is clearly inspired by peridynamics, and it could actually be seen as a *peridynamic fractional Laplacian*. Peridynamics is a nonlocal continuum model for Solid Mechanics proposed by Silling in [@Si]. The main difference with classical theory relies on the nonlocality, reflected in the fact that points separated by a positive distance exert a force upon each other. Since the use of gradients is avoided, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fracture, dislocation, or, in general, multiscale materials. Operator $(-\Delta)_\delta^s$ fits into bond-based peridynamics (see [@Si]), where the elastic energy is computed through a double integral of a pairwise potential function. In [@Siwei] a numerical study comparing $(-\Delta)_\delta^s$ with the fractional Laplacian, the spectral fractional Laplacian and the regional Laplacian is performed. In [@Alali], the Fourier multiplier associated to $(-\Delta)_\delta^s$ is computed and, as a consequence, convergence of $(-\Delta)_\delta^su(x)$ to $(-\Delta)u(x)$, for sufficiently smooth $u$, is obtained as $\delta\to 0^+$ or $s\to 1^-$. Also, $(-\Delta)_\delta^s$ was studied in [@DeliaGun] in connection with the fractional Laplacian, $(-\Delta)^s=(-\Delta)_{\infty}^s$, and with the motivation of computing numerical approximations of its solutions. Notice that taking the limit as $\delta\to+\infty$ one recovers, at least formally, the usual nonlocal elliptic problem driven by the fractional Laplace operator with boundary condition on the complementary of the domain $\Omega$. Following the probabilistic interpretation for the fractional Laplacian $(-\Delta)_{\infty}^s$, the operator $(-\Delta)_\delta^s$ can be seen as the infinitesimal generator of a symmetric $2s$-stable Lévy process restricted to $B(x,\delta)$.
At this point, it is worth mentioning that problem and the underlying boundary geometry $\partial_{\delta}\Omega$ play and intermediate role between the classical local problem driven by the Laplace operator $(-\Delta)$, where the boundary condition is imposed on $\partial\Omega$, and the nonlocal problem driven by the standard fractional Laplacian $(-\Delta)_{\infty}^s$, where the boundary condition is imposed on the whole $\mathbb{R}^N\backslash\Omega$.
By means of a change of variables, we can write the singular integral as a weighted second order differential quotient so that the operator $(-\Delta)_{\delta}^s$ admits the representation $$\label{seconddiff}
(-\Delta)_{\delta}^su(x)=-\frac12c_{N,s}\int_{B(0,\delta)}\frac{u(x+y)-2u(x)+u(x-y)}{|y|^{N+2s}}dy.$$ As it happens with the standard fractional Laplacian $(-\Delta)_{\infty}^s$, because of , the operator $(-\Delta)_{\delta}^s$ has the following monotonicity property: if $u$ has a global maximum at $x$, then $(-\Delta)_{\delta}^s u(x) \geq 0$ for any horizon $\delta>0$. In addition, if $u$ has a local maximum, then there exists an horizon $\delta=\delta(u)>0$ such that $(-\Delta)_{\delta}^s u(x) \geq 0$.
In this paper the limit properties of $(-\Delta)_\delta^s$, both as $\delta \to 0^+$ and as $\delta \to +\infty$, are addressed. We show convergence of solutions and spectral stability, i.e., convergence of eigenvalues and eigenfunctions, to the local Laplacian and to the fractional Laplacian as $\delta \to 0^+$ and as $\delta \to +\infty$ respectively. Therefore, the operator $(-\Delta)_\delta^s$ is an intermediate operator in between the local Laplacian and the fractional Laplacian.
This investigation fits into the framework of $\Gamma$-convergence (cf. [@Braides]). The results for the case $\delta \to 0^+$ rely on a general $\Gamma$-convergence result from [@BellCorPed], whereas the results for the case $\delta \to+\infty$ are consequence of the $\Gamma$-convergence of the energy associated to $(-\Delta)_\delta^s$ to the one corresponding to $(-\Delta)_\infty^s$ induced by the monotone convergence of the sequence of energies.
References related to this work are the following. Also in the framework of $\Gamma$-convergence is [@BrascoPariniSquassina], where spectral convergence of the fractional $p\,$-Laplacian to the classical $p\,$-Laplacian as $s\to 1^-$ is shown. Closely related to this work is [@AndresMunoz], where spectral stability for certain nonlocal problems in the case $\delta\to 0^+$ is shown without explicitly appealing to $\Gamma$-convergence. As mentioned before, in [@DeliaGun], the convergence phenomena as $\delta\to+\infty$ is addressed for a class of nonlocal linear problems, including the operator $(-\Delta)_\delta^s$ as a particular case, with a direct approach relying on the linearity of those problems.
**Organization of the paper**: In Section \[functionalsetting\] we introduce the appropriate functional setting to deal with problems involving the operator $(-\Delta)_{\delta}^s$ and we present the main results proved in this paper. In Section \[preliminaryresults\] some results about $\Gamma$-convergence that will be essential to prove the main results of this work are included. In Section \[horizon0\] we prove the results concerning the behaviour of and the eigenvalue problem associated to $(-\Delta)_{\delta}^s$ when one takes the horizon $\delta\to0^+$. Finally, in Section \[horizoninfty\] we prove the results concerning the behaviour of and the eigenvalue problem associated to $(-\Delta)_{\delta}^s$ when one takes the horizon $\delta\to+\infty$.
Functional setting and Main Results {#functionalsetting}
===================================
Let us start recalling the fractional-order Sobolev space $H^s(\Omega)$. Given a regular bounded domain $\Omega\subset\mathbb{R}^N$, let us set $$H^s(\Omega)=\{v\in L^2(\Omega):\|v\|_{H^s(\Omega)}<\infty\},$$ where $$\|v\|_{H^s(\Omega)}^2=\|v\|_{L^2(\Omega)}^2+|v|_{H^s(\Omega)}^2$$ with $|\cdot|_{H^s(\Omega)}$ denoting the Gagliardo semi-norm, $$\label{Gagliseminorm}
|v|_{H^s(\Omega)}^2=\int_{\Omega}\int_{\Omega}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx.$$ Also, let $H_0^s(\Omega)$ be the completion of $C_0^{\infty}(\Omega)$ under the norm $\|\cdot\|_{H^s(\Omega)}^2$, i.e. $$H_0^s(\Omega)=\overline{\mathcal{C}^{\infty}_0 (\Omega)}^{\| \cdot \|_{H^s(\Omega)}}.$$ Due to [@LionMag Theorem 11.1], if $0<s\leq\frac{1}{2}$ then $H_0^s(\Omega)=H^s(\Omega)$ while for $\frac{1}{2}<s<1$ we have the strict inclusion $H_0^s(\Omega)\subsetneq H^s(\Omega)$. Next, denoting by $\Omega^c=\mathbb{R}^N\backslash\Omega$, let us set the energy space $$\mathcal{H}_0^s(\Omega)=\{v\in H^s(\mathbb{R}^N): v=0 \text{ on }\Omega^c\}$$ endowed with the norm inherited from $H^s(\mathbb{R}^N)$. Let us note that, given $v\in \mathcal{H}_0^s(\Omega)$, although $v=0$ on $\Omega^c$, the norms $\|v\|_{H^s(\Omega)}$ and $\|v\|_{\mathcal{H}_0^s(\Omega)}$ are not the same. Indeed, denoting by $\mathcal{D}=\big(\mathbb{R}^N\times\mathbb{R}^N\big)\big\backslash\big(\Omega^c\times\Omega^c\big)$, we have the strict inclusion $\Omega\times\Omega\subsetneq\mathcal{D}$. In other words, the norm $\|\cdot\|_{\mathcal{H}_0^s(\Omega)}$ takes into account the interaction between $\Omega$ and $\Omega^c$, i.e., $$\begin{split}
\|v\|_{\mathcal{H}_0^s(\Omega)}^2=&\|v\|_{H^s(\mathbb{R}^N)}^2\\
=&\|v\|_{L^2(\mathbb{R}^N)}^2+\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx\\
=&\|v\|_{L^2(\Omega)}^2+\iint\limits_{\mathcal{D}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx.
\end{split}$$ Therefore, the space $\mathcal{H}_0^s(\Omega)$ is the appropriate space to deal with homogeneous elliptic boundary value problems involving the fractional Laplace operator, $$\label{fracclaplacian}
(-\Delta)_{\infty}^s u(x)=c_{N,s}P.V.\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy.$$ On the other hand, by [@DiNezzaPalaValdi Theorem 6.5], there exists a constant $S(N,s)>0$ such that $$\|v\|_{L^{2_{s}^{*}}(\mathbb{R}^N)}^2\leq S(N,s)\iint\limits_{\mathcal{D}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx, \quad \forall\,v\in\mathcal{H}_0^s(\Omega),$$ with $2_s^*=\frac{2N}{N-2s}$, the critical fractional Sobolev exponent. Hence, for all $v\in\mathcal{H}_0^s(\Omega)$, $$\iint\limits_{\mathcal{D}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx\leq \|v\|_{\mathcal{H}_0^s(\Omega)}^2\leq C_1(\Omega,N,s)\iint\limits_{\mathcal{D}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx,$$ with $C_1(\Omega,N,s)=S(N,s)|\Omega|^{\frac{2_s^*-2}{2_s^*}}$. Therefore, we can renormize the space $\mathcal{H}_0^s(\Omega)$ and consider it endowed with the norm $$\label{normHfrac}
{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert v
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathcal{H}_0^s}^2=\iint\limits_{\mathcal{D}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx.$$ Next, given an horizon $\delta>0$, let us define the ([*nonlocally*]{}) completed domain $$\Omega_{\delta}=\Omega\cup\partial_{\delta}\Omega=\{y\in\mathbb{R}^N:\ |x-y|<\delta,\ \text{for}\ x\in\Omega\},$$ and the energy space associated to $(-\Delta)_{\delta}^s$ as $$\mathbb{H}^s(\Omega_{\delta})=\{v\in L^2(\Omega_{\delta}): \|v\|_{\mathbb{H}^s(\Omega_{\delta})}<\infty\},$$ where $$\|v\|_{\mathbb{H}^s(\Omega_{\delta})}^2=\|v\|_{L^2(\Omega_{\delta})}^2+| v |_{\mathbb{H}^s(\Omega_{\delta})}^2,$$ with $$| v |_{\mathbb{H}^s(\Omega_{\delta})}^2=\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx.$$ The next result links the semi-norm $|\cdot|_{\mathbb{H}^s(\Omega_{\delta})}$ with the Gagliardo semi-norm .
\[belcor\][@BellCor Proposition 6.1] Let $s\in(0,1)$, $1\leq p<\infty$, $\delta>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded Lipschitz domain. Then, there exists $C=C(\delta)>0$ such that for all $u\in W^{s,p}(\Omega)$, $$\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dydx\leq C\int_{\Omega}\int_{\Omega\cap B(x,\delta)}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dydx.$$
Because of Proposition \[belcor\], the space $\mathbb{H}^s(\Omega_{\delta})$ is isomorphic to the (classical) fractional-order Sobolev space $H^s(\Omega_\delta)$. In order to deal with the boundary value problem , we define the energy space $$\mathbb{H}_0^{\delta,s}(\Omega)=\{v\in \mathbb{H}^s(\Omega_{\delta}):v\equiv 0\ \text{on}\ \partial_{\delta}\Omega\},$$ endowed with the norm inherited from $\mathbb{H}^s(\Omega_{\delta})$. Let us notice that, given a function $v\in \mathbb{H}_0^{\delta,s}(\Omega)$, although we have $v=0$ on $\partial_{\delta}\Omega=\Omega_{\delta}\backslash\Omega$, the norms $\|v\|_{\mathbb{H}^s(\Omega)}$ and $\|v\|_{\mathbb{H}_0^{\delta,s}(\Omega)}$ are not the same. Indeed, if $v=0$ on $\Omega^c$, since $$\mathbb{H}^s(\Omega)=\{v\in L^2(\Omega): \|v\|_{\mathbb{H}^s(\Omega)}<\infty\},$$ where $$\|v\|_{\mathbb{H}^s(\Omega)}^2=\|v\|_{L^2(\Omega)}^2+| v |_{\mathbb{H}^s(\Omega)}^2,$$ with $$| v |_{\mathbb{H}^s(\Omega)}^2=\int_{\Omega}\int_{\Omega\cap B(x,\delta)}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx,$$ denoting by $$\mathcal{D}_{\delta}=\Big(\Omega_{\delta}\times\big(\Omega_{\delta}\cap B(x,\delta)\big)\Big)\Big\backslash\Big(\partial_{\delta}\Omega\times\big(\partial_{\delta}\Omega\cap B(x,\delta)\big)\Big),$$ we have the strict inclusion $\big(\Omega\times(\Omega\cap B(x,\delta))\big)\subsetneq\mathcal{D}_{\delta}$. Hence, the norm $\|\cdot\|_{\mathbb{H}_0^{\delta,s}(\Omega)}$ takes into account the interaction between $\Omega$ and $\partial_{\delta}\Omega$ in the sense that $$\begin{split}
\|v\|_{\mathbb{H}_0^{\delta,s}(\Omega)}^2=&\|v\|_{\mathbb{H}^s(\Omega_{\delta})}^2\\
=&\|v\|_{L^2(\Omega_{\delta})}^2+\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx\\
=&\|v\|_{L^2(\Omega)}^2+\iint\limits_{\mathcal{D}_{\delta}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx.
\end{split}$$ Therefore, the space $\mathbb{H}_0^{\delta,s}(\Omega)$ is the appropriate space to deal with homogeneous elliptic boundary value problems involving the operator $(-\Delta)_{\delta}^s$.Some comments are in order. Comparing the norms $\|\cdot\|_{\mathcal{H}_0^s(\Omega)}$ and $\|\cdot\|_{\mathbb{H}_0^{\delta,s}(\Omega)}$ we observe that $\partial_{\delta}\Omega$ plays the role of $\Omega^c$. Indeed, the sets $\Omega_{\delta}$ and $\Omega_{\delta}\cap B(x,\delta)$ will lead to the complete space $\mathbb{R}^N$ in the limit $\delta\to+\infty$, the set $\Omega\cap B(x,\delta)$ will eventually reach the set $\Omega$ for $\delta>0$ big enough and the sets $\partial_{\delta}\Omega$ and $\partial_{\delta}\Omega\cap B(x,\delta)$ will reach $\Omega^c$ in the limit $\delta\to+\infty$. In fact, $$\label{contenido}
\mathcal{D}_{\delta_1}\subset\mathcal{D}_{\delta_2}\quad\text{ for }\delta_1<\delta_2,$$ and[^3] $$\mathcal{D}_{\delta}\to\mathcal{D}\quad\text{ as }\delta\to+\infty.$$ To continue, we recall now a Poincaré-type inequality.
[@BellCor Lemma 6.2]\[poincare\] Let $\Omega\subset\mathbb{R}^N$ a bounded Lipschitz domain and let $\Omega_{D}$ a measurable set of $\Omega$ of positive measure. Let $s\in(0,1)$ and $1\leq p<\infty$. Then, there exists $C>0$ such that for all $u\in W^{s,p}(\Omega)$ with $u=0$ a.e. on $\Omega_{D}$ we have $$\|u\|_{L^p(\Omega)}\leq C\ |u|_{W^{s,p}(\Omega)}=C\left(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dydx\right)^{\frac{1}{p}}.$$
Because of Proposition \[belcor\] and Lemma \[poincare\], for a positive constant $c\in\mathbb{R}$, $$\label{normequiv2}
|v|_{\mathbb{H}^s(\Omega_{\delta})}\leq\| v \|_{\mathbb{H}^s(\Omega_{\delta})}\leq c |v |_{\mathbb{H}^s(\Omega_{\delta})},$$ then, we can renormize the space $\mathbb{H}_0^{\delta,s}(\Omega)$ and consider it endowed with the norm $$\label{normHHfrac}
{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert v
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathbb{H}_0^{\delta,s}}^2=\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx.$$ As a consequence, we have the following.
\[lemmaHilbert\] The space $\mathbb{H}_0^{\delta,s}(\Omega)$ is a Hilbert space endowed with norm ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \cdot
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathbb{H}_0^{\delta,s}}$ induced by the scalar product $$\langle u,v\rangle_{\mathbb{H}_0^{\delta,s}}=\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}dydx.$$
\[lemmaconvergence\] Let $\{v_j\}_{j\in\mathbb{N}}$ be a bounded sequence in $\mathbb{H}_0^{\delta,s}(\Omega)$. Then, there exists $v\in L^p(\Omega)$ such that, up to a subsequence, $$v_j\to v\quad \text{in}\ L^p(\Omega),\ \text{as}\ j\to+\infty,$$ for any $p\in[2,2_s^*)$, where $2_s^*=\frac{2N}{N-2s}$ is the critical (fractional) Sobolev exponent.
Thanks to Proposition \[belcor\] and Lemma \[poincare\], the proofs of Lemma \[lemmaHilbert\] and Lemma \[lemmaconvergence\] follow similarly as to the case involving the standard fractional Laplacian $(-\Delta)_{\infty}^s$ and the space $\mathcal{H}_0^s(\Omega)$, (cf. [@SerVal2 Lemma 7] and [@SerVal2 Lemma 8] respectively).Next, to study convergence phenomena when one takes $\delta\to+\infty$, it will be essential to study the relation between the spaces $\mathcal{H}_0^s(\Omega)$ and $\mathbb{H}_0^{\delta,s}(\Omega)$. To that end, we show the following, whose proof is deferred to Section \[horizoninfty\].
\[isomorfia\] For any $\delta>0$, the spaces $\mathbb{H}_0^{\delta,s}(\Omega)$ and $\mathcal{H}_0^s(\Omega)$ are isomorphic. In particular, there exists a constant $C=C(\delta)>1$ such that $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \cdot
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathbb{H}_0^{\delta,s}}^2\leq{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \cdot
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathcal{H}_0^s}^2\leq C(\delta){{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \cdot
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathbb{H}_0^{\delta,s}}^2\quad\text{for all }\delta>0.$$ Moreover, $C(\delta)\to 1$ as $\delta \to +\infty$.
Let us notice that Lemma \[isomorfia\] implies the following convergence of spaces, $$\mathbb{H}_0^{\infty,s}(\Omega)\vcentcolon=\lim\limits_{\delta\to\infty}\mathbb{H}_0^{\delta,s}(\Omega)\equiv \mathcal{H}_0^s(\Omega).$$ Now we make precise the definition of weak solution of problem $(P_\delta^s)$.
We say that $u\in\mathbb{H}_0^{\delta,s}(\Omega)$ is a weak solution to problem if $$\frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}dydx=\int_{\Omega}f(x)v(x)dx,$$ for all $v\in \mathbb{H}_0^{\delta,s}(\Omega)$, or written in a more compact way, $$\frac{c_{N,s}}{2}\langle u,v\rangle_{\mathbb{H}_0^{\delta,s}}=\langle f,v\rangle_{L^2(\Omega)},\quad\text{ for all } v \in \mathbb{H}_0^{\delta,s}(\Omega).$$
Once we have introduced the functional setting we continue with some existence results dealing with the operator $(-\Delta)_{\delta}^s$.
\[thlineal\] Given an horizon $\delta>0$ and a reaction term $f\in L^2(\Omega)$ there exists a unique solution $u^{\delta,s}$ to problem . Moreover, such a solution is the unique minimizer of the energy functional associated to , i.e., $$\mathcal{J}_{\delta,s}(u^{\delta,s})=\min\limits_{v\in \mathbb{H}_0^{\delta,s}(\Omega)}\mathcal{J}_{\delta,s}(v)=\min\limits_{v\in \mathbb{H}_0^{\delta,s}(\Omega)}\left\{\frac{c_{N,s}}{4}{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert v
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathbb{H}_0^{\delta,s}}^2-\int_{\Omega}fv \, dx\right\}.$$
The proof of Theorem \[thlineal\] is, thanks to the renormization of the space $\mathbb{H}_0^{\delta,s}(\Omega)$ in terms of ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \cdot
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathbb{H}_0^{\delta,s}}$ provided by Proposition \[belcor\] and Lemma \[poincare\], an immediate consequence of Lax-Milgram Theorem, so we omit this proof.Next we study existence and stability issues for the eigenvalue problem $$\label{eigenproblem}
\left\{\begin{array}{rl}
(-\Delta)_{\delta}^s\varphi=\lambda\varphi&\quad\mbox{in}\quad \Omega,\\
\varphi\!=0\ \ &\quad\mbox{on}\quad \partial_{\delta}\Omega.\\
\end{array}\right.
\tag{$EP_{\delta}^{s}$}$$
\[propeigen\] Let $\delta>0$, $s\in(0,1)$, $N>2s$ and $\Omega\subset\mathbb{R}^N$ an open bounded set with Lipschitz boundary. Then, the following hold:
1. Problem has a first positive eigenvalue that can be characterized as $$\label{minlambda1}
\lambda_{1}^{\delta,s}=\min\limits_{\substack{ u\in \mathbb{H}_0^{\delta,s}(\Omega)\\ \|u\|_{L^2(\Omega)}=1}}\frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx,$$ or, equivalenty, $$\lambda_{1}^{\delta,s}=\min\limits_{\substack{ u\in \mathbb{H}_0^{\delta,s}(\Omega)}}\frac{c_{N,s}}{2}\frac{\displaystyle\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx}{\displaystyle\int_{\Omega}|u(x)|^2dx}.$$ Moreover, there exists a nonnegative function $\varphi_1^{\delta,s}\in\mathbb{H}_0^{\delta,s}(\Omega)$, which is an eigenfunction corresponding to $\lambda_1^{\delta,s}$, attaining the minimum in , i.e., $\|\varphi_1^{\delta,s}\|_{L^2(\Omega)}=1$ and $$\label{minfirsteigenfunction}
\lambda_{1}^{\delta,s}=\frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|\varphi_1^{\delta,s}(x)-\varphi_1^{\delta,s}(y)|^2}{|x-y|^{N+2s}}dydx.$$ In addition, the first eigenvalue $\lambda_1^{\delta,s}$ is simple, i.e., if $\psi\in\mathbb{H}_0^{\delta,s}(\Omega)$ is such that, for all $\phi\in\mathbb{H}_0^{\delta,s}(\Omega)$, $$\label{simple}
\frac{c_{N,s}}{2}\langle \psi,\phi\rangle_{\mathbb{H}_0^{\delta,s}}=\lambda_1^{\delta,s}\int_{\Omega}\psi(x)\phi(x)dx,$$ then, $\psi=c\,\varphi_1^{\delta,s}$ with $c\in\mathbb{R}$.
2. The eigenvalues of are a countable set $\{\lambda_{k}^{\delta,s}\}_{k\in\mathbb{N}}$ satisfying $$\label{increasingeigenvalues}
0<\lambda_1^{\delta,s}<\lambda_2^{\delta,s}\leq\ldots\leq\lambda_k^{\delta,s}\leq\ldots$$ and $$\label{eigenvaluetoinfty}
\lambda_k^{\delta,s}\to +\infty\qquad\text{as } k\to +\infty.$$ Furthermore, for any $k\in\mathbb{N},\ k\geq2$ the eigenvalues can be characterized as $$\label{minlambdak}
\lambda_{k}^{\delta,s}=\min\limits_{\substack{ u\in \mathbb{P}_{k}^{\delta}\\ \|u\|_{L^2(\Omega)}=1}}\frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx,$$ or, equivalently, $$\lambda_{k}^{\delta,s}=\min\limits_{\substack{ u\in \mathbb{P}_{k}^{\delta}}}\frac{c_{N,s}}{2}\frac{\displaystyle\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx}{\displaystyle\int_{\Omega}|u(x)|^2dx},$$ where $$\label{projectspace}
\mathbb{P}_{k}^{\delta}=\{u\in\mathbb{H}_0^{\delta,s}(\Omega):\langle u,\varphi_j^{\delta,s}\rangle_{\mathbb{H}_0^{\delta,s}}=0,\ j=1,\ldots,k-1\}.$$ In addition, for any $k\in\mathbb{N},\ k\geq2$, there exists a function $\varphi_k^{\delta,s}\in \mathbb{P}_{k}^{\delta}$, which is an eigenfunction corresponding to $\lambda_k^{\delta,s}$, attaining the minimum in , i.e., $\|\varphi_k^{\delta,s}\|_{L^2(\Omega)}=1$ and $$\label{minksteigenfunction}
\lambda_k^{\delta,s}=\frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|\varphi_k^{\delta,s}(x)-\varphi_k^{\delta,s}(y)|^2}{|x-y|^{N+2s}}dydx.$$
3. The set of eigenfunctions $\{\varphi_k^{\delta,s}\}_{k\in\mathbb{N}}$ is an orthogonal basis of $\mathbb{H}_0^{\delta,s}(\Omega)$ and an orthonormal basis of $L^2(\Omega)$.
4. For any $k\in\mathbb{N}$, the eigenvalue $\lambda_k^{\delta,s}$ has finite multiplicity, i.e., if $\lambda_k^{\delta,s}$ satisfies $$\label{multiplicity}
\lambda_{k-1}^{\delta,s}<\lambda_k^{\delta,s}=\ldots=\lambda_{k+n}^{\delta,s}<\lambda_{k+n+1}^{\delta,s},$$ for some $n\in\mathbb{N},\ n\geq1$, then the set of all the eigenfunctions corresponding to $\lambda_k^{\delta,s}$ belongs to $$span\{\varphi_k^{\delta,s},\ldots,\varphi_{k+n}^{\delta,s}\}.$$ In other words, denoting by $m_k^{\delta,s}$ the multiplicity of the eigenvalue $\lambda_k^{\delta,s}$, then $$\label{finitemultiplicity}
1\leq m_k^{\delta,s}<\infty,\quad\text{for all } k\in\mathbb{N}.$$
\[boundinftyeigen\] Let $\varphi_k^{\delta,s}\in\mathbb{H}_0^{\delta,s}(\Omega)$ be an eigenfunction of , then $\varphi_k^{\delta,s}\in L^{\infty}(\Omega)$ for any $k\in\mathbb{N}$.
The proofs of Proposition \[propeigen\] and Lemma \[boundinftyeigen\] can be also done similarly as to the case dealing with the standard fractional Laplacian $(-\Delta)_{\infty}^s$ and the space $\mathcal{H}_0^s(\Omega)$. Indeed, up to minor modifications involving the joint use of Proposition \[belcor\] and the fractional Sobolev inequality (cf.[@DiNezzaPalaValdi Theorem 6.5]), the proofs can be done following step by step the proofs of [@SerVal Proposition 9] and [@Servadei2013 Proposition 4] respectively, so we omit the details for the sake of brevity.
We present now the main results of the work, dealing with the behavior of and when the horizon $\delta\to0^+$ and $\delta\to+\infty$. To that end, let us consider the following problems. $$\label{problemareescaled}
\left\{\begin{array}{cl}
(-\Delta)_{\delta}^{s}u=\frac{\delta^{2(1-s)}}{\kappa(N,s)}f &\quad\mbox{in}\quad \Omega,\\
u=0 &\quad\mbox{on}\quad \partial_{\delta}\Omega,\\
\end{array}\right.
\tag{$RP_{\delta}^{s}$}$$ where $$\label{constantkappa}
\kappa(N,s)=\frac{4N(1-s)}{\sigma_{N-1}c_{N,s}},$$ with $\sigma_{N-1}$ the surface of the unitary sphere $\mathbb{S}^{N-1}$. As the limit $\delta\to0^+$ encodes a concentration phenomena, the rescaling on the reaction term $f$ of , namely $\frac{\delta^{2(1-s)}}{\kappa(N,s)}f$ in , is needed in order to avoid the degeneracy in the limit process.
The constant $\kappa(N,s)$ is well defined when $s\to 1^-$. Indeed, cf. [@Bucur; @Stinga], we have $$\label{limitecns}
c_{N,s}=\frac{2^{2s}s\Gamma(\frac{N}{2}+s)}{\pi^{\frac{N}{2}}\Gamma(1-s)}\to0\quad\text{as }s\to1^-,$$ and, (cf. [@DiNezzaPalaValdi Corollary 4.2]), $$\label{limitecns2}
\lim\limits_{s\to 1^-}\frac{c_{N,s}}{(1-s)}=\lim\limits_{s\to 1^-}\frac{2^{2s}s\Gamma(\frac{N}{2}+s)}{\pi^{\frac{N}{2}}\Gamma(2-s)}=\frac{4\Gamma(\frac{N}{2}+1)}{\pi^{\frac{N}{2}}}=\frac{4N}{\frac{2\pi^{\frac{N}{2}}}{\Gamma(\frac{N}{2})}}=\frac{4N}{\sigma_{N-1}}.$$ As a consequence, $$\label{limitkapa}
\lim\limits_{s\to 1^-}\kappa(N,s)=1.$$
Next, let us consider the classical local linear problem, $$\label{problema0}
\left\{\begin{array}{rl}
(-\Delta)u=f &\quad\mbox{in}\quad \Omega,\\
u=0 &\quad\mbox{on}\quad \partial\Omega,\\
\end{array}\right.
\tag{$P_{0}^{1}$}$$ and the linear problem driven by the standard fractional Laplacian, $$\label{problemainf}
\left\{\begin{array}{rl}
(-\Delta)_{\infty}^su=f &\quad\mbox{in}\quad \Omega,\\
u=0 &\quad\mbox{on}\quad \partial_{\infty}\Omega\equiv\mathbb{R}^{N}\backslash\Omega,\\
\end{array}\right.
\tag{$P_{\infty}^{s}$}$$ with $(-\Delta)_{\infty}^s$ defined in . Our main results as regarding the relationship between the linear problems , , and are the following.
\[linear0\] Let $u^{\delta,s}$ and $u^{0,1}$ be the solutions of and respectively. Then, up to a subsequence, $$u^{\delta,s}\to u^{0,1}\ \text{in }L^2(\Omega)\quad\text{as }\delta\to0^+.$$
\[linearinf\] Let $u^{\delta,s}$ and $u^{\infty,s}$ be the solutions of and respectively. Then, up to a subsequence, $$u^{\delta,s}\to u^{\infty,s}\ \text{in }L^2(\Omega)\quad\text{as }\delta\to+\infty.$$
On the other hand, let us consider the classical eigenvalue problem, $$\label{eigenproblem01}
\left\{\begin{array}{rl}
(-\Delta)\varphi=\lambda\varphi&\quad\mbox{in}\quad \Omega,\\
\varphi=0\ \ &\quad\mbox{on}\quad \partial\Omega.
\end{array}\right.
\tag{$EP_{0}^{1}$}$$ It is well known (cf. [@Brezis]) that the problem has a countable set of eigenvalues that we denote by $\{\lambda_{k}^{0,1}\}_{k\in\mathbb{N}}$ and such that $$0<\lambda_{1}^{0,1}<\lambda_{2}^{0,1} \leq\ldots\leq\lambda_{k}^{0,1}\leq\ldots ,\\$$ with $$\lambda_{k}^{0,1}\to+\infty\ \text{as}\ k\to+\infty,$$ and, denoting by $m_k^{0,1}$ the multiplicity of the eigenvalue $\lambda_k^{0,1}$, $$1\leq m_k^{0,1}<\infty,\quad\text{for all }k\in\mathbb{N}.$$ Moreover, there exists a countable set of eigenfunctions $\{\varphi_k^{0,1}\}_{k\in\mathbb{N}}$ that is an orthogonal basis of $H_0^1(\Omega)$ and an orthonormal basis of $L^2(\Omega)$. The first eigenvalue is simple ($m_1^{0,1}=1$) and, by the Maximum Principle, $\varphi_1^{0,1}>0$ in $\Omega$. Finally, let us also consider the nonlocal eigenvalue problem $$\label{eigenprobleminfs}
\left\{\begin{array}{rl}
(-\Delta)_{\infty}^s\varphi=\lambda\varphi&\quad\mbox{in}\quad \Omega,\\
\varphi=0\ \ &\quad\mbox{on}\quad \mathbb{R}^N\backslash\Omega.
\end{array}\right.
\tag{$EP_{\infty}^{s}$}$$ with $(-\Delta)_{\infty}^s$ defined in . Concerning this problem, Servadei and Valdinoci proved, cf. [@SerVal], that has a countable set of eigenvalues that we denote by $\{\lambda_{k}^{\infty,s}\}_{k\in\mathbb{N}}$ and such that $$0<\lambda_{1}^{\infty,s}<\lambda_{2}^{\infty,s}\leq\ldots\leq\lambda_{k}^{\infty,s}\leq\ldots,\\$$ with $$\lambda_{k}^{\infty,s}\to+\infty\ \text{as}\ k\to+\infty,$$ and, denoting by $m_k^{\infty,s}$ the multiplicity of the eigenvalue $\lambda_k^{\infty,s}$, $$1\leq m_k^{\infty,s}<\infty,\quad\text{for all }k\in\mathbb{N}.$$ Moreover, there exists a countable set of eigenfunctions $\{\varphi_k^{\infty,s}\}_{k\in\mathbb{N}}$ that is an orthogonal basis of $$\mathbb{H}_0^{\infty,s}(\Omega)=\left\{v\in L^2(\Omega): \iint\limits_{\mathcal{D}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx<\infty,\ v=0 \text{ a.e. } \text{on } \mathbb{R}^N\backslash\Omega\right\},$$ and an orthonormal basis of $L^2(\Omega)$. Again, the first eigenvalue is simple ($m_1^{\infty,s}=1$) and $\varphi_1^{\infty,s}$ can chosen so that $\varphi_1^{\infty,s}\geq0$ in $\Omega$. Finally, we relate the eigenvalues and eigenfunctions of to those of the eigenvalue problems and through the following results.
\[theigen\] Let $\{(\lambda_{k}^{\delta,s},\varphi_k^{\delta,s})\}_{k\in\mathbb{N}}$ be the set of eigenvalues and eigenfunctions of $(-\Delta)_{\delta}^s$ with homogeneous Dirichlet boundary condition on $\partial_{\delta}\Omega$ and let$\{(\lambda_{k}^{0,1},\varphi_k^{0,1})\}_{k\in\mathbb{N}}$ be the set of eigenvalues of $(-\Delta)$ with homogeneous Dirichlet boundary condition on $\partial\Omega$. Then, $$\kappa(N,s)\frac{\lambda_{k}^{\delta,s}}{\delta^{2(1-s)}}\to\lambda_{k}^{0,1}\quad \text{as}\ \delta\to0^+,$$ for the constant $\kappa(N,s)$ appearing in , and there exists a subsequence (that we do not relabel) such that $$\varphi_{k}^{\delta,s}\to\varphi_{k}^{0,1}\ \text{in}\ L^2(\Omega)\quad \text{as}\ \delta\to0^+,$$ for every $k\in\mathbb{N}$. As a consequence, $m_k^{\delta,s}\to m_k^{0,1}$ as $\delta \to 0^+$, for any $k\ge 1$.
We would like to emphasize that, as Theorems \[linear0\] and \[theigen\] shows, even though the fractionality parameter $s$ keeps fixed, the local problem driven by $(-\Delta)$ is recovered, under the appropriate rescaling, as $\delta\to 0^+$. A different issue is the limit as $s\to 1^-$. This has been addressed in the case of the fractional Laplacian in several references, both operator convergence (cf. [@DiNezzaPalaValdi]) and spectral stability (cf. [@BrascoPariniSquassina]). Let us see now that the same behavior holds for $(-\Delta)_\delta^s$ as $s\to 1^-$.
First of all, because of [@DiNezzaPalaValdi Proposition 4.4], for any $u\in C_0^{\infty}(\mathbb{R}^N)$, we have the pointwise convergence $$\lim\limits_{s\to 1^-}(-\Delta)_{\infty}^su(x)=(-\Delta)u(x).$$ Indeed, the hypothesis $u\in C_0^{\infty}(\mathbb{R}^N)$ can be relaxed to $u\in\mathcal{C}^2(B(x,R))\cap L^{\infty}(\mathbb{R}^N)$ for some $R>0$, cf. [@Stinga Proposition 5.3]. Following [@DiNezzaPalaValdi Proposition 4.4], by means of a change of variable in , we have $$(-\Delta)_{\infty}^su(x)=-\frac{c_{N,s}}{2}\int_{\mathbb{R}^N}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}dy,$$ so that, for $u\in L^{\infty}(\mathbb{R}^N)$, $$\begin{split}
\left|\int_{\mathbb{R}^N\backslash B(0,\delta)}\mkern-10mu\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}dy\right|&\leq 4\|u\|_{L^{\infty}(\mathbb{R}^N)}\int_{\mathbb{R}^N\backslash B(0,\delta)}\frac{1}{|y|^{N+2s}}dy\\
&\leq4\sigma_{N-1}\|u\|_{L^{\infty}(\mathbb{R}^N)}\int_{\delta}^{\infty}\frac{1}{\rho^{2s+1}}d\rho\\
&=\frac{2\sigma_{N-1}}{s\delta^{2s}}\|u\|_{L^{\infty}(\mathbb{R}^N)}.
\end{split}$$ Thus, by , we find that, for any $\delta>0$ fixed, $$\lim\limits_{s\to 1^-}\left|\frac{c_{N,s}}{2}\int_{\mathbb{R}^N\backslash B(0,\delta)}\mkern-10mu\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}dy\right|=0.$$ Therefore, recalling , $$\label{limitesto1}
\begin{split}
\lim\limits_{s\to 1^-}(-\Delta)_{\infty}^su(x)&=\lim\limits_{s\to 1^-}-\frac{c_{N,s}}{2}\int_{B(0,\delta)}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}dy\\
&=\lim\limits_{s\to 1^-}(-\Delta)_{\delta}^su(x).
\end{split}$$ We conclude that, for any $\delta>0$ fixed and a given $u\in L^{\infty}(\mathbb{R}^N)$, the behavior of $(-\Delta)_\infty^su(x)$ and $(-\Delta)_\delta^su(x)$ as $s\to 1^-$ coincides. Hence, under sufficient smothness asumptions, $$\lim\limits_{s\to 1^-}(-\Delta)_{\delta}^su(x)=(-\Delta)u(x)\quad\text{for all }\delta>0\text{ fixed}.$$ On the other hand, because of [@Servadei2013 Proposition 4], the eigenfunctions of the fractional Laplacian $(-\Delta)_{\infty}^s$ are bounded, i.e., $\varphi_k^{\infty,s}\in L^{\infty}(\mathbb{R}^N)$ for all $k\in\mathbb{N}$. Moreover, these eigenfunctions are Hölder continuous, cf. [@Servadei2014; @RosOton2012]. Also, because of Lemma \[boundinftyeigen\], we have $\varphi_k^{\delta,s}\in L^{\infty}(\mathbb{R}^N)$ for all $k\in\mathbb{N}$. Therefore, using it is not difficult to show that, for any $\delta>0$ fixed, $$\label{limitequal}
\lim\limits_{s\to 1^-}\lambda_k^{\delta,s}=\lim\limits_{s\to 1^-}\lambda_k^{\infty,s}\quad\text{for all }k\in\mathbb{N}.$$ Next, by [@BrascoPariniSquassina Theorem 1.2] with $p=2$, $$\label{rem}
\lim\limits_{s\to 1^-}\lambda_{k}^{\infty,s}=\lambda_{k}^{0,1}.$$ Observe that no normalization constant is used in [@BrascoPariniSquassina] to define the fractional operator, just a factor $2$ instead of $c_{N,s}$. On the other hand, in [@BrascoPariniSquassina Theorem 1.2] the limit process is normalized by a factor $(1-s)$, so that adjusting constants and taking in mind and below, equality follows accordingly to our normalization setting. Finally, because of , and , $$\lim\limits_{s\to 1^-}\kappa(N,s)\frac{\lambda_{k}^{\delta,s}}{\delta^{2(1-s)}}=\lim\limits_{s\to 1^-}\lambda_{k}^{\delta,s}=\lim\limits_{s\to1^-}\lambda_{k}^{\infty,s}=\lambda_k^{0,1}.$$
\[theigen2\] Let $\{(\lambda_{k}^{\delta,s},\varphi_{k}^{\delta,s})\}_{k\in\mathbb{N}}$ be the set of eigenvalues and eigenfunctions of $(-\Delta)_{\delta}^s$ with homogeneous Dirichlet boundary condition on $\partial_{\delta}\Omega$ and let$\{(\lambda_{k}^{\infty,s},\varphi_{k}^{\infty,s})\}_{k\in\mathbb{N}}$ be the set of eigenvalues of $(-\Delta)_{\infty}^{s}$ with homogeneous Dirichlet boundary condition on $\mathbb{R}^N\backslash\Omega$. Then, $$\lambda_{k}^{\delta,s}\to\lambda_{k}^{\infty,s}\quad \text{as}\ \delta\to+\infty,$$ and there exists a subsequence (that we do not relabeled) such that $$\varphi_{k}^{\delta,s}\to\varphi_{k}^{\infty,s}\ \text{in}\ L^2(\Omega)\quad \text{as}\ \delta\to+\infty,$$ for every $k\in\mathbb{N}$. As a consequence, $m_k^{\delta,s}\to m_k^{\infty,s}$ as $\delta \to \infty$, for any $k\ge 1$.
An interesting question, out of the scope of this paper, is whether the spectral structure of $(-\Delta)$ and $(-\Delta)_{\infty}^s$ coincides. In other words, whether the multiplicity $m_k^{\delta,s}$ remains constant as a function of the horizon $\delta>0$. It seems a very hard task to solve, since not so much is known about the multiplicity of eigenvalues, even for the classical Laplace operator $(-\Delta)$. Nevertheless, because of the above results, $m_k^{0,1}$ and $m_k^{\delta,s}$ coincide, at least for $\delta>0$ small enough, and $m_k^{\infty,s}$ and $m_k^{\delta,s}$ coincide, at least, for $\delta>0$ big enough. Roughly speaking, for $\delta<<1$ the operator $(-\Delta)_{\delta}^s$ is almost a local operator, in the sense that it takes into account what happens in a neighborhood of size $\delta$ while for $\delta>>1$ the operator $(-\Delta)_{\delta}^s$ takes into account almost the same as $(-\Delta)_{\infty}^s$.
Preliminary results: $\Gamma$-convergence and localization {#preliminaryresults}
==========================================================
This section includes some results that play a crucial role to study problems and when we take the limit $\delta\to0^+$. We start with the classical [*localization*]{} result of Bourgain, Brezis and Mironescu, cf. [@BourBrezMiro], that provides us with a first hint towards Theorem \[theigen\]. Let $\{\rho_{n}(x)\}_{n\in\mathbb{N}}$ be a sequence of radial mollifiers, i.e. $$\rho_n(x)=\rho_n(|x|),\ \rho_n(x)\geq0,\ \int\rho_n(x)dx=1$$ satisfying $$\lim\limits_{n\to\infty}\int_{\varepsilon}^{\infty}\rho_n(r)r^{N-1}=0,\ \text{for every}\ \varepsilon>0$$
\[BoBrMi\][@BourBrezMiro Theorem 2] Assume $u\in L^p(\Omega)$, $1<p<\infty$. Then, for a constant $C=C(N,p)>0$, we have $$\lim\limits_{n\to\infty}\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^p}\rho_n(x-y)dydx=C\int_{\Omega}|\nabla u|^pdx.$$ with the convention that $\int_{\Omega}|\nabla u|^pdx=\infty$ if $u\notin W^{1,p}(\Omega)$.
$\Gamma$-Convergence
--------------------
The limit process in the sense of $\Gamma$-convergence, denoted by $\overset{\Gamma}{\to}$, is the right concept of limit for variational problems since it, together with equicoercivity or compactness, implies that minimizers of $I_{\delta}$ converge to minimizers of $I$ as well as their energies. A nice account on $\Gamma$-convergence is provided in [@Braides]. We present now a result about $\Gamma$-convergence of functionals proved in [@BellCorPed] that is the core of the proof of Theorem \[linear0\] and Theorem \[theigen\]. Let us consider a functional of the form $$\label{funcional}
\int_{\Omega}\int_{\Omega\cap B(x,\delta)}\omega(x-y,u(x)-u(y))dydx,$$ with $u:\Omega\mapsto\mathbb{R}$. In our case the *potential function* $\displaystyle\omega(\overline{x},\overline{y})=\frac{|\overline{y}|^2}{|\overline{x}|^{N+2s}}$ fits into the cases studied in [@BellCorPed], where the type of functional considered orbits around the important case $$\omega(\overline{x},\overline{y})=\frac{|\overline{y}|^p}{|\overline{x}|^{\alpha}},\quad \text{where}\ 1<p<+\infty,\,0\leq\alpha<N+p,$$ related to the fractional $p\,$-Laplacian. In particular, the result establishes that, under some hypotheses to be stated below and for some $\beta>0$ to be specified, the sequence of rescaled functionals $$\label{scaledfunctional}
I_{\delta}(u)=\frac{N+\beta}{\delta^{N+\beta}}\int_{\Omega}\int_{\Omega\cap B(x,\delta)}\omega(x-y,u(x)-u(y))dydx.$$ has a $\Gamma$-limit (under the strong-$L^p(\Omega)$ topology) given by the local functional $$I_{\delta}(u)\overset{\Gamma}{\to}I_0(u)=\int_{\Omega}W(\nabla u)dx,\quad \text{as}\ \delta\to0^+,$$ for some function $W(\cdot)$ that we construct next. Specifically, the $\Gamma$-limit is recovered in several steps:
1. *Scaling*: we scale the functional to obtain the functional $I_{\delta}(u)$ given in with $\beta$ given in the next step.
2. *Blow-up at zero*: we assume that there exists some $\beta\in\mathbb{R}$ such that the following limit exists, $$\label{blowup}
\omega^{\circ}(\overline{x},\overline{y})=\lim\limits_{t\to0^+}\frac{1}{t^{\beta}}\omega(t\overline{x},t\overline{y}).$$
3. *Limit density*: the limit density $\overline{\omega}:\Omega\times\mathbb{R}^{1\times N}\mapsto\mathbb{R}$ is given by $$\label{limitdensity}
\overline{\omega}(F)\vcentcolon=\int_{\mathbb{S}^{N-1}}\omega^{\circ}(z,Fz)d\sigma(z),$$ where $\mathbb{S}^{N-1}$ is the $(N-1)$-dimensional unitary sphere.
4. *Convexification*: The $\Gamma$-limit of $I_{\delta}$ as $\delta\to0^+$ is given by $$\label{quasiconvex}
I_0(u)=\int_{\Omega}\overline{\omega}^{c}(\nabla u)dx,$$ where, $$\overline{\omega}^{c}(F)\vcentcolon=\sup\{v(F):v(\cdot)\leq \overline{\omega}(\cdot)\ \text{and}\ v(\cdot)\ \text{convex}\}.$$
The right scaling required in for the *potential function* $$\omega(\tilde{x},\tilde{y})=\frac{|\tilde{y}|^p}{|\tilde{x}|^\alpha}$$ is then given by $\beta=p-\alpha$. The key result we use to prove the main results of the paper is the following, which is a simplified version of [@BellCorPed Theorem 1] but enough for our aim here. Let us denote $\tilde{\Omega}\vcentcolon=\{ z=x-y\;:\; x,\,y\in \Omega\}$.
[@BellCorPed Theorem 1]\[belcorped\] Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with Lipschitz boundary and $\omega:\Omega\times\tilde{\Omega}\times\mathbb{R}\mapsto\mathbb{R}$ satisfying the hypotheses (H1)-(H5) below.
- [*Compactness:*]{} For each $\delta>0$ let $$u_{\delta}\in \mathcal{A}_{\delta}\vcentcolon=\{v\in L^p(\Omega): v=0\ \text{on}\ \partial_{\delta}\Omega\}\quad \text{with}\ \sup\limits_{\delta>0} I_{\delta}(u_{\delta})<+\infty.$$ Then, there exist $$u\in\mathcal{B}\vcentcolon=\{v\in W^{1,p}(\Omega): v=0\ \text{on}\ \partial\Omega\},$$ such that, $$u_{\delta}\to u\ \text{in}\ L^p(\Omega)\quad \text{as}\ \delta\to0^+.$$
- [*Liminf inequality:*]{} For each $\delta>0$ let $u_{\delta}\in\mathcal{A}_{\delta}$ and $u\in\mathcal{B}$ such that $u_{\delta}\to u$ in $L^p(\Omega)$ as $\delta\to0^+$. Then, $$I_0(u)\leq \liminf\limits_{\delta\to0^+}I_{\delta}(u_{\delta}).$$
- [*Limsup inequality:*]{} For each $\delta>0$ and $u\in\mathcal{B}$ there exist $u_{\delta}\in\mathcal{A}_{\delta}$, called *recovery sequence*, such that $u_{\delta}\to u$ in $L^p(\Omega)$ as $\delta\to0^+$ and $$\limsup\limits_{\delta\to0^+}I_{\delta}(u_{\delta})\leq I_0(u).$$
For a general potential function $\omega(\overline{x},\overline{y})$, the hypotheses required in Theorem \[belcorped\] are quite involved but, as it is noted in [@BellCorPed], for a potential function of the form $$\omega(\tilde{x},\tilde{y})=f(\tilde{x})g(\tilde{y}),$$ the necessary hypotheses are the following:
- $f$ is Lebesgue measurable and $g$ is Borel measurable.
- $g$ is convex.
- There exists $c_0>0$ such that $$\frac{c_0}{|\tilde{x}|^{\alpha}}\leq f(\tilde{x})\quad \text{and}\quad c_0|\tilde{y}|^p\leq g(\overline{y})\qquad\text{for}\ \tilde{x}\in\widetilde{\Omega},\ \tilde{y}\in\mathbb{R}.$$
- There exists $c_1>0$ and $h\in L^1(\mathbb{S}^{N-1})$ with $h\geq0$ such that $$f(\tilde{x})\leq h\left(\frac{\tilde{x}}{|\tilde{x}|}\right)\frac{1}{|\tilde{x}|^{\alpha}}\ \ \text{and}\ \ g(\tilde{y})\leq c_1 |\tilde{y}|^p\qquad\text{for}\ \tilde{x}\in\widetilde{\Omega},\ \tilde{y}\in\mathbb{R}.$$
- The functions $f^{\circ}:\mathbb{R}\backslash\{0\}\mapsto\mathbb{R}$ and $g^{\circ}:\mathbb{R}\mapsto\mathbb{R}$ defined as $$f^{\circ}(\tilde{x})\vcentcolon=\lim\limits_{t\to0^+}t^{\alpha}f(t\tilde{x})\quad\text{and}\quad g^{\circ}(\tilde{y})\vcentcolon=\lim\limits_{t\to0^+}\frac{1}{t^p}g(t\tilde{y}),$$ are continuous and, for each compact $K\subset\mathbb{R}$, $$\lim\limits_{t\to0^+}\sup\limits_{\tilde{x}\in\mathbb{S}^{N-1}}|t^{\alpha}f(t\tilde{x})-f^{\circ}(\tilde{x})|=0\quad\text{and}\quad \lim\limits_{t\to0^+}\sup\limits_{K\subset\mathbb{R}}|\frac{1}{t^p}g(t\tilde{y})-g^{\circ}(\tilde{y})|=0.$$
A straightforward consequence, cf. [@Braides], of $\Gamma$-convergence and the compactness property (Theorem \[belcorped\]-b) and c) and Theorem \[belcorped\]-a) respectively), is the following corollary. Notice that under previous hypothesis existence of minimizers for $I_\delta$ is guaranteed, cf. [@BellCor].
\[corGammaconv\] In the conditions of Theorem \[belcorped\], let $u_\delta\in\mathbb{H}_0^{\delta,s}(\Omega)$ be a minimizer of $I_\delta$, for any $\delta>0$. Then, there exists $u_0\in H_0^1(\Omega)$ a minimizer of $I_0$ such that, up to a subsequence, $$u_\delta\to u_0\mbox{ strong in }L^2(\Omega)\quad\text{as }\delta\to 0^+,$$ and $$I_\delta(u_\delta)\to I_0(u_0)\quad\text{as }\delta\to 0^+.$$
Taking the horizon $\delta\to0^+$ {#horizon0}
=================================
In this section we prove Theorem \[linear0\] and Theorem \[theigen\]. To that end, we prove first a preliminary result concerning the $\Gamma$-convergence.
\[Glimit\] Let us consider the scaled functional $$\label{funcionaldeltas}
I_{\delta,s}(u)=\frac{2(1-s)}{\delta^{2(1-s)}}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx.$$ defined on $\mathbb{H}_0^{\delta,s}(\Omega)$. Then, the $\Gamma$-limit of $I_{\delta,s}(u)$ as $\delta\to 0^+$ is given by $$\label{GlimitFunc}
I_0(u)=\gamma\int_{\Omega}|\nabla u(x)|^2dx,$$ for a constant $\gamma=\gamma(N)=\frac{\sigma_{N-1}}{N}>0$, being $\sigma_{N-1}$ the measure of the $(N-1)$-dimensional unit sphere.
Let us follow the construction process of the limit density as specified in -. First, let us observe that, since the potential function $$\omega(\tilde{x},\tilde{y})=\frac{|\tilde{y}|^2}{|\tilde{x}|^{N+2s}},$$ then $\omega(\tilde{x},\tilde{y})=f(\tilde{x})g(\tilde{y})$ with $f(\tilde{x})=\frac{1}{|\tilde{x}|^{N+2s}}$ and $g(\tilde{y})=|\tilde{y}|^2$. Hence, in our case, $p=2$, $\alpha=N+2s<N+2$ and hypotheses (H1)-(H5) are clearly satisfied. Next we construct the function given in . Since $\omega(\tilde{x},\tilde{y})=\frac{|\tilde{y}|^2}{|\tilde{x}|^{N+2s}}$ is homogeneous, it is immediate that, taking $\beta=2-(N+2s)$ in , $$\begin{split}
\omega^{\circ}(\tilde{x},\tilde{y})&=\lim\limits_{t\to0^+}\frac{1}{t^\beta}\omega(t\tilde{x},t\tilde{y})=\lim\limits_{t\to0^+}\frac{1}{t^{2-(N-2s)}}\frac{t^2|\tilde{y}|^2}{t^{N+2s}|\tilde{x}|^{N+2s}}=\frac{|\tilde{y}|^2}{|\tilde{x}|^{N+2s}}\\
&=\omega(\tilde{x},\tilde{y}).
\end{split}$$ We continue by constructing the function $\overline{\omega}(F)$ given in . Because of $$\omega^{\circ}(\tilde{x},\tilde{y})=\omega(\tilde{x},\tilde{y})=\omega(|\tilde{x}|,|\tilde{y}|),$$ we find that, for a given vector $F\in\mathbb{R}^{1\times N}$, the function $\overline{\omega}(F):\Omega\times\mathbb{R}^{1\times N}\mapsto\mathbb{R}$ is given by $$\begin{split}
\overline{\omega}(F)&=\int_{\mathbb{S}^{N-1}}\omega^{\circ}(z,F\cdot z)d\sigma=\int_{\mathbb{S}^{N-1}}\omega(|z|,|F\cdot z|)d\sigma=\int_{\mathbb{S}^{N-1}}\omega(1,|F\cdot z|)d\sigma\\
&=\int_{\mathbb{S}^{N-1}}|F\cdot z|^2d\sigma=|F|^2\int_{\mathbb{S}^{N-1}}\frac{|F\cdot z|^2}{|F|^2}d\sigma=|F|^2\int_{\mathbb{S}^{N-1}}\left|\frac{F}{|F|}\cdot z\right|^2d\sigma\\
&=|F|^2\int_{\mathbb{S}^{N-1}}|e\cdot z|^2d\sigma,
\end{split}$$ with $e=\frac{F}{|F|}\in\mathbb{S}^{N-1}$ an unitary vector. Moreover, it is easy to see that, given a unitary vector $e\in\mathbb{S}^{N-1}$, $$\label{contbr}
\int_{\mathbb{S}^{N-1}}|e\cdot z|^2d\sigma=\frac{\sigma_{N-1}}{N},$$ where $\sigma_{N-1}$ is the area of the unitary sphere $\mathbb{S}^{N-1}$. Hence, we conclude $$\overline{\omega}(F)=\frac{\sigma_{N-1}}{N}|F|^2=\gamma(N)|F|^2.$$ At last, since the function $\overline{\omega}(F)=\gamma(N) |F|^2$ is convex, the convexification $\overline{\omega}^{c}(F)$ coincides with the function $\overline{\omega}(F)$ itself. Therefore, we conclude that the $\Gamma$-limit of the functional $I_{\delta,s}(\cdot)$ as $\delta\to 0^+$ is given by $$I_0(u)=\int_{\Omega}\overline{\omega}^{c}(\nabla u(x))dx=\int_{\Omega}\overline{\omega}(\nabla u(x))dx=\gamma(N) \int_{\Omega}|\nabla u(x)|^2dx.$$
At this point it is worth mentioning that, up to the correct scaling $\displaystyle\frac{2(1-s)}{\delta^{2(1-s)}}$, all the functionals $I_{\delta,s}$ will $\Gamma$-converge to the same limit independently of $s$, because, in the previous proof, the denominator of $\omega^{\circ}(\overline{x},\overline{y})$ plays no role in the integration over the unit sphere $\mathbb{S}^{N-1}$.
Let $\mathcal{F}(u)$ be the energy functional associated to , i.e., $$\mathcal{F}(u)=\frac{c_{N,s}}{4}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx-\frac{\delta^{2(1-s)}}{\kappa(N,s)}\int_{\Omega}fudx,$$ so that the solution $u^{\delta,s}$ to is a minimum of $\mathcal{F}(u)$. Next, let us define $\kappa(N,s)\vcentcolon=\frac{4(1-s)}{c_{N,s}\gamma(N)}$, with $\gamma(N)$ the constant given in Lemma \[Glimit\] and consider the rescaled functional $$\begin{split}
\mathcal{F}_{\delta,s}(u)&\vcentcolon=\frac{1}{\delta^{2(1-s)}}\mathcal{F}(u)\\
&=\frac{c_{N,s}}{4(1-s)}\frac{(1-s)}{\delta^{2(1-s)}}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx-\frac{1}{\kappa(N,s)}\int_{\Omega}fudx\\
&=\frac{1}{\kappa(N,s)}\left(\frac{1}{2\gamma(N)}I_{\delta,s}(u)-\int_{\Omega}fudx\right).
\end{split}$$ Clearly, if $u^{\delta,s}$ is a solution to , or equivalently, a minimizer of $\mathcal{F}(u)$, then it is also a minimizer of the rescaled functional $\mathcal{F}_{\delta,s}$. As we see in the proof of Lemma \[Glimit\], the hypotheses (H1)-(H5) of Theorem \[belcorped\] are satisfied and the $\Gamma$-limit of $I_{\delta,s}(u)$ as $\delta\to 0^+$ is given by . Hence, the $\Gamma$-limit of $\mathcal{F}_{\delta,s}$ as $\delta\to 0^+$ is $$\begin{split}
\mathcal{F}_0(u)&=\frac{1}{\kappa(N,s)}\left(\frac{1}{2} I_0(u)-\int_{\Omega}fudx\right)\\
&=\frac{1}{\kappa(N,s)}\left(\frac{1}{2} \int_{\Omega}|\nabla u|^2dx-\int_{\Omega}fudx\right).
\end{split}$$ We conclude that the $\Gamma$-limit functional is, up to a constant, the energy functional associated to . On the other hand, as $\Gamma$-convergence implies convergence of optimal energies, if $u^{\delta,s}\in \mathbb{H}_0^{\delta,s}(\Omega)$ is the minimizer of $\mathcal{F}_{\delta,s}$ and $u^{0,1}$ the minimizer of $\mathcal{F}_0$, then $$\lim_{\delta\to 0^+} \mathcal{F}_{\delta,s}(u^{\delta,s})=\mathcal{F}_{0}(u^{0,1}),$$ Indeed, if $u^{\delta,s}\in \mathbb{H}_0^{\delta,s}(\Omega)$ is the minimizer of $\mathcal{F}_{\delta,s}$, so that $u^{\delta,s}$ is a solution to problem , we have $$\mathcal{F}_{\delta,s}(u_\delta)=-\frac{1}{2\kappa(N,s)} I_{\delta,s}(u^{\delta,s}).$$ Therefore, the sequence $\{I_{\delta,s}(u^{\delta,s})\}_{\delta>0}$ is bounded and, by Theorem \[belcorped\], there exists $u^{0,1}\in H_0^1(\Omega)$ such that, up to a subsequence $u^{\delta,s}\to u^{0,1}$ strong in $L^2(\Omega)$. Moreover, $\Gamma$-convergence of $\mathcal{F}_{\delta,s}$ to $\mathcal{F}_0$ implies that $u^{0,1}$ is the unique minimizer of $\mathcal{F}_0$, and therefore the solution of $(P_0^1)$.
First we prove the result for the sequence of first eigenvalues and corresponding eigenfunctions. Recall that first eigenvalues, $\lambda_1^{\delta,s}$ and $\lambda_1^{0,1}$, are simple for both $(EP_\delta^s)$, for any $\delta>0$, and for the limit problem $(EP_0^1)$ respectively. Let $I_\delta^{(1)}$ and $I^{(1)}$ be the restricted functionals $$I_{\delta,s}^{(1)}(u)=\left\{
\begin{array}{ll}
I_{\delta,s}(u) &\mbox{if }\|u\|_{L^2(\Omega)}=1,\\
+\infty &\mbox{otherwise},
\end{array}\right.$$ and $$I^{(1)}(u)=\left\{
\begin{array}{ll}
I(u) &\mbox{if }\|u\|_{L^2(\Omega)}=1,\\
+\infty &\mbox{otherwise},
\end{array}\right.$$ respectively, with $I_{\delta,s}(u)$ defined in and $I(u)=\int_\Omega |\nabla u|^2\,dx$. Let us show that $I_{\delta,s}^{(1)}\overset{\Gamma}{\to} I^{(1)}$. Actually, this is derived easily from Lemma \[Glimit\] and Theorem \[belcorped\]:
1. [*Liminf inequality:*]{} Given $u_\delta\to u$ strong in $L^2(\Omega)$, then $$I^{(1)}(u)\le \liminf_{\delta\to 0^+} I_{\delta,s}^{(1)}(u_\delta).$$ this is consequence of Theorem \[belcorped\]-b), and the fact that strong convergence in $L^2(\Omega)$ implies convergence of the norms.
2. [*Limsup inequality:*]{} Given $u\in H_0^1(\Omega)$, with $\|u\|_{L^2(\Omega)}=1$, by Theorem \[belcorped\]-c), there exists $u_\delta\in \mathbb{H}_0^{\delta,s}(\Omega)$ such that $u_\delta\to u$ strong in $L^2(\Omega)$ as $\delta \to 0^+$, and $$\limsup_{\delta\to 0^+}I_{\delta,s} (u_\delta) \le I(u).$$ Calling $v_\delta =\frac{u_\delta}{\|u_\delta\|_{L^2(\Omega)}}$, it is elementary to check that $v_\delta\to u$ strong in $L^2(\Omega)$ as $\delta \to 0^+$ and $I_\delta^{(1)}(v_\delta)=\frac{I_\delta(u_\delta)}{\|u_\delta\|^2}$, so that $$\limsup_{\delta\to 0^+} I_{\delta,s}^{(1)}(v_\delta)=\limsup_{\delta\to 0^+} I_{\delta,s}(u_\delta)\le I^{(1)}(u).$$
Since $I_{\delta,s}^{(1)}$ and $I_0^{(1)}$ are restricted functionals of $I_{{\delta},s}$ and $I_0$, compactness hold by Theorem \[belcorped\]-a). Therefore, by Corollary \[corGammaconv\], we have the convergence of optimal energies, $$\kappa(N,s)\frac{\lambda_1^{\delta,s}}{\delta^{2(1-s)}}\to \lambda_1^{0,1},\quad\text{as }\delta\to 0^+,$$ and there exists a subsequence $\delta_n\to 0$ as $n\to\infty$, such that $$\varphi_1^{\delta_n,s}\to \varphi_1^{0,1}\quad \text{as }n\to+\infty.$$ Now we prove the result for the second eigenvalue and the corresponding eigenfunction. As above, we show $\Gamma$-convergence of the restricted functionals $$I_{\delta,s}^{(2)}(u)=\left\{\begin{array}{rl}
I_{\delta,s}(u) &\quad\mbox{if}\quad u\in \mathbb{P}_{2}^{\delta}\text{ and }\|u\|_{L^2(\Omega)}=1,\\
+\infty &\quad\mbox{otherwise,}
\end{array}\right.$$ and $$I^{(2)}(u)=\left\{\begin{array}{rl}
I(u) &\quad\mbox{if}\quad u\in \mathbb{P}_{2}^{0}\text{ and }\|u\|_{L^2(\Omega)}=1,\\
+\infty &\quad\mbox{otherwise,}
\end{array}\right.$$ where $\mathbb{P}_{2}^{\delta}$ is defined in , i.e., $$\mathbb{P}_{2}^{\delta}=\left\{u\in\mathbb{H}_0^{\delta,s}(\Omega):\langle u,\varphi_1^{\delta,s}\rangle_{L^2(\Omega)}=0\right\},$$ and $$\mathbb{P}_{2}^{0}=\left\{u\in H_0^1(\Omega):\langle u,\varphi_1^{0,1}\rangle_{L^2(\Omega)}=0\right\}.$$ $\Gamma$-convergence of $I_{\delta,s}^{(2)}$ to $I^{(2)}$ is again consequence of Lemma \[Glimit\] and Theorem \[belcorped\]:
1. [*Liminf inequality:*]{} Given $u_\delta\to u$ strong in $L^2(\Omega)$, with $\|u_\delta\|_{L^2(\Omega)}=1$, then $\|u\|_{L^2(\Omega)}=1$ and, hence, up to a subsequence, $$\langle u_{\delta},\varphi_1^{\delta,s}\rangle_{L^2(\Omega)}\to \langle u,\varphi_1^{0,1}\rangle_{L^2(\Omega)} \quad\text{as }\delta\to0^+,$$ since $\varphi_1^{\delta,s}\to \varphi_1^{0,1}$ strong in $L^2(\Omega)$, up to a subsequence. Finally, because of Theorem \[belcorped\]-b), we find $$\label{liminfi}
I^{(2)}(u)\leq\liminf\limits_{\delta}I_{\delta,s}^{(2)}(u_{\delta}).$$
2. [*Limsup inequality:*]{} Given $u\in \mathbb{P}_2^0$, with $\|u\|_{L^2(\Omega)}=1$, we construct a recovery sequence $u_\delta \in \mathbb{P}_2^\delta$, with $\|u_\delta\|_{L^2(\Omega)}=1$, such that $u_\delta \to u$ strong $L^2(\Omega)$ and $$\limsup_{\delta\to 0^+} I_{\delta,s}^{(2)}(u_\delta) \le I^{(2)}(u).$$ Define $$u_{\delta}=\eta_{\delta}\varphi_{1}^{\delta,s}+\mu_{\delta}u,$$ where the constants $\eta_\delta$, $\mu_\delta$ are determined by imposing $u_{\delta}\in\mathbb{P}_{2}^{\delta}$ together with $\|u_{\delta}\|_{L^2(\Omega)}=1$. These two conditions give us $$\left\{\begin{array}{l}
\eta_{\delta}+\mu_\delta\langle u,\varphi_1^{\delta,s}\rangle_{L^2(\Omega)}=0,\\
\eta_{\delta}^2+2\eta_\delta\mu_\delta\langle u,\varphi_1^{\delta,s}\rangle_{L^2(\Omega)}+\mu_\delta^2=1.
\end{array}\right.$$ Since $\langle u,\varphi_1^{\delta,s}\rangle_{L^2(\Omega)}\to 0$, then $\eta_\delta\to0$ and $\mu_\delta\to1$ as $\delta\to0^+$. Then, noticing that $I_{\delta,s}$ is quadratic, $$I_{\delta,s}^{(2)}(u_\delta) =\eta_{\delta}^2 I_{\delta,s}(\varphi_{1}^{\delta,s})+2\eta_{\delta}\mu_{\delta}\langle u,\varphi_1^{\delta,s}\rangle_{L^2(\Omega)}+\mu_{\delta}^2I_{\delta,s}(u).$$ and taking $\delta \to 0^+$, we arrive at $$\begin{split}
\lim\limits_{\delta\to0^+} I_{\delta,s}^{(2)}(u_\delta)&= \lim\limits_{\delta\to0^+}\left(\eta_{\delta}^2 I_{\delta,s}(\varphi_{1}^{\delta,s})+2\eta_{\delta}\mu_{\delta}\langle u,\varphi_1^{\delta,s}\rangle_{L^2(\Omega)}+\mu_{\delta}^2I_{\delta,s}(u)\right)\\
&=\lim_{\delta \to 0^+} I_{\delta,s}(u)=I_0(u),
\end{split}$$ where the last equality is due to Theorem \[BoBrMi\] (see Remark \[remarkBBM\] below).
Consequently, by Corollary \[corGammaconv\], $$\kappa(N,s)\frac{\lambda_2^{\delta,s}}{\delta^{2(1-s)}}\to \lambda_2^{0,1},\quad\text{as }\delta\to0^+,$$ and, for any sequence $\{\varphi_2^{\delta,s}\}_{\delta>0}$ such that $\varphi_2^{\delta,s}$ is an eigenfunction of $(EP_\delta^s)$ associated to $\lambda_2^{\delta,s}$, there exists a subsequence $\{\varphi_2^{\delta_n,s}\}_{\delta_n>0}$ with $\delta_n\to 0$ as $n\to+\infty$ and $\varphi_2^{0,1}\in H_0^1(\Omega)$, an eigenfunction of $(EP_0^1)$ associated to $\lambda_2^{0,1}$, such that $$\varphi_2^{\delta_n,s}\to \varphi_2^{0,1},\quad\text{as }n\to+\infty.$$ Once we have proved the convergence for the second eigenvalue and the second eigenfunction, the rest of the argument follows by induction interatively. Therefore, we have the convergences $$\kappa(N,s)\frac{\lambda_k^{\delta,s}}{\delta^{2(1-s)}}\to\lambda_k^{0,1}\quad \text{as }\delta\to 0^+,\quad \text{for all }k\in\mathbb{N}$$ and, up to a subsequence, $$\label{cnveigenf}
\varphi_k^{\delta,s}\to\varphi_k^{0,1}\ \text{in }L^2(\Omega)\quad \text{as }\delta\to 0^+,\quad \text{for all }k\in\mathbb{N}.$$ Moreover, because of , we also have the convergence of the orthogonal spaces $$\label{cnvproject}
\mathbb{P}_k^{\delta}\to\mathbb{P}_k^{0}\quad \text{as }\delta\to 0^+,\quad \text{for all }k\in\mathbb{N}.$$ Next, given an eigenvalue $\lambda_k^{\delta,s}$, $k\in\mathbb{N}$, that, by has finite multiplicity, i.e., $1\leq m_k^{\delta,s}<\infty$, let us denote by $\{\varphi_{k,i}^{\delta,s}\}_{i=1}^{m_k^{\delta,s}}$ the basis of the subspace of eigenfunctions associated to the eigenvalue $\lambda_k^{\delta,s}=\ldots=\lambda_{j+m_k^{\delta,s}-1}^{\delta,s}$. Observe that, by , $$\label{chain}
0<\lambda_1^{\delta,s}<\lambda_2^{\delta,s}\leq\ldots\leq\lambda_{k-1}^{\delta,s}<\lambda_k^{\delta,s}=\ldots=\lambda_{k+m_k^{\delta,s}-1}^{\delta,s}<\lambda_{k+m_k^{\delta,s}}^{\delta,s}\leq\ldots$$ Let us rewrite without repeat the eigenvalues according its multiplicity, i.e., we have a strictly increasing sequence of eigenvalues $$\label{chain2}
0<\lambda_1^{\delta,s}<\lambda_2^{\delta,s}<\ldots<\lambda_{k-1}^{\delta,s}<\lambda_k^{\delta,s}<\lambda_{k+1}^{\delta,s}<\ldots$$ with finite multiplicities $\{m_k^{\delta,s}\}_{k\in\mathbb{N}}$. Because of and we conclude $$m_k^{\delta,s}\to m_k^{0,1}\ \text{as }\delta\to 0^+,\quad \text{for all }k\in\mathbb{N}.$$
\[remarkBBM\] In order to clarify why the scaling in $\Gamma$-convergence result is natural in our context, it is of interest to obtain the upper bound $$\lim_{\delta \to 0^+} \frac{\kappa(N,s)}{\delta^{2(1-s)}}\lambda_1^{\delta,s}\le \lambda_1^{0,1}$$ as a consequence of Theorem \[BoBrMi\]. Since $H_0^1(\Omega)\subset\mathbb{H}_0^{\delta,s}(\Omega)$ for all $\delta>0$ (it is enough to extend $\varphi\in H_0^1(\Omega)$ as $\varphi\equiv0$ on $\partial_{\delta}\Omega$), then $$\begin{split}
\lambda_1^{\delta,s}&=\min\limits_{\substack{ u\in \mathbb{H}_0^{\delta,s}(\Omega)\\ \|u\|_{L^2(\Omega)}=1}}\frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx\\
&\leq \frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|\varphi_1^{0,1}(x)-\varphi_1^{0,1}(y)|^2}{|x-y|^{N+2s}}dydx,
\end{split}$$ where $\varphi_1^{0,1}$ is the first eigenfunction of the Laplace operator with $\|\varphi_1^{0,1}\|_{L^2(\Omega)}=1$. In order to apply Theorem \[BoBrMi\], let us rewrite the above inequality as $$\begin{split}
\lambda_1^{\delta,s}&\leq \frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|\varphi_1^{0,1}(x)-\varphi_1^{0,1}(y)|^2}{|x-y|^{N+2s}}dydx\\
&=\frac{c_{N,s}}{2}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}}\frac{|\varphi_1^{0,1}(x)-\varphi_1^{0,1}(y)|^2}{|x-y|^{2}}\frac{\chi_{B(0,\delta)}(|x-y|)}{|x-y|^{N+2(s-1)}}dydx\\
&=\int_{\Omega_{\delta}}\int_{\Omega_{\delta}}\frac{|\varphi_1^{0,1}(x)-\varphi_1^{0,1}(y)|^2}{|x-y|^{2}}\rho_{\delta}(|x-y|)dydx,
\end{split}$$ with $\displaystyle\rho_{\delta}(z)=\frac{c_{N,s}}{2}\frac{\chi_{B(0,\delta)}(|z|)}{|z|^{N+2(s-1)}}$ and $\chi_A$ the characteristic function of the set $A$. In order to fulfill the hypotheses of Theorem \[BoBrMi\] we normalize $\rho_{\delta}(z)$. Since $$\int \rho_{\delta}(z)dz=\frac{\sigma_{N-1}c_{N,s}}{4(1-s)}\delta^{2(1-s)},$$ with $\sigma_{N-1}$ the surface of the unitary sphere $\mathbb{S}^{N-1}$, the sequence of radial mollifiers $$\overline{\rho}_{\delta}(z)=\frac{4(1-s)}{\sigma_{N-1}}\frac{1}{\delta^{2(1-s)}}\frac{\chi_{B(0,\delta)}(|z|)}{|z|^{N+2(s-1)}},$$ satisfy the hypotheses of Theorem \[BoBrMi\]. Observe that the scaling in $\delta$ coincides with the one of Theorem \[belcorped\]. Therefore, we get $$\frac{4(1-s)}{\sigma_{N-1}c_{N,s}}\frac{\lambda_1^{\delta,s}}{\delta^{2(1-s)}}\leq \int_{\Omega_{\delta}}\int_{\Omega_{\delta}}\frac{|\varphi_1^{0,1}(x)-\varphi_1^{0,1}(y)|^2}{|x-y|^{2}}\overline{\rho}_{\delta}(|x-y|)dydx,$$ and because of Theorem \[BoBrMi\], we conclude $$\label{eigenpre}
\begin{split}
\lim\limits_{\delta\to0^+}\frac{4(1-s)}{\sigma_{N-1}c_{N,s}}\frac{\lambda_1^{\delta,s}}{\delta^{2(1-s)}}&\leq \lim\limits_{\delta\to0^+}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}}\frac{|\varphi_1^{0,1}(x)-\varphi_1^{0,1}(y)|^2}{|x-y|^{2}}\overline{\rho}_{\delta}(|x-y|)dydx\\
&=\gamma\int_{\Omega}|\nabla \varphi_1^{0,1}|^2dx
\end{split}$$ The constant $\gamma=\gamma(N,p)$ appearing in Theorem \[BoBrMi\] takes the form (see [@BourBrezMiro]) $$\gamma(N,p)=\frac{1}{\sigma_{N-1}}\int_{\mathbb{S}^{N-1}}|z\cdot e|^pd\sigma,$$ for any unitary vector $e\in\mathbb{S}^{N-1}$. Then, for $p=2$, we have $\gamma(N,2)=\frac{1}{N}$. Simplifying and taking in mind that $\|\varphi_1^{0,1}\|_{L^2(\Omega)}=1$, we conclude $$\lim\limits_{\delta\to0^+}\kappa(N,s)\frac{\lambda_1^{\delta,s}}{\delta^{2(1-s)}}\leq\int_{\Omega}|\nabla \varphi_1^{0,1}|^2dx=\lambda_1^{0,1},$$ where $\kappa(N,s)=\frac{4N(1-s)}{\sigma_{N-1}c_{N,s}}$ is the constant appearing in .
Taking the horizon $\delta\to+\infty$ {#horizoninfty}
=====================================
Because of the definition of the operator $(-\Delta)_{\delta}^s$, as a restriction of the fractional Laplacian, it is plausible that if we take $\delta\to+\infty$ one recovers the definition of the standard fractional Laplacian, namely $$\lim\limits_{\delta\to +\infty}(-\Delta)_{\delta}^s u(x)=c_{N,s}P.V.\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy.$$ Our interest is to explore this convergence by proving Theorems \[linearinf\] and \[theigen2\]. Prior to that, we would like to mention that a result in this line was given in [@DeliaGun Theorem 3.1], where it is showed the explicit convergence rate $$\|u^{\delta,s}-u^{\infty,s}\|_{\mathbb{H}_0^{\delta,s}}\le \frac{c}{(\delta-I)^{2s}}\|u^{\infty,s}\|_{L^2(\Omega)},$$ being $u^{\delta,s}$ and $u^{\infty,s}$ the solutions of $(P_\delta^s)$ and $(P_\infty^s)$ respectively, $c>0$ is a constant independent of $\delta$ and $I=\min\{R>0:\Omega\subset B(x,R)\ \forall x\in\Omega\}$. This is an important result from the point of view of the numerical approximation of problems involving the fractional Laplacian. Nevertheless, this reference does not address spectral problems, and the proof of [@DeliaGun Theorem 3.1] strongly relies on the linearity of the problem . Instead, the proof of Theorem \[linearinf\] and Theorem \[theigen2\] are based on a general result about $\Gamma$-convergence that works for both the linear and nonlinear setting (we address the $p\,$-fractional Laplacian case in a forthcoming paper).
Using and , we have $$\label{normcomparison3}
{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert v
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathcal{H}_0^s}^2-{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert v
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathbb{H}_0^{\delta,s}}^2=\iint\limits_{\mathcal{D}\backslash\mathcal{D}_{\delta}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx\geq0,$$ because $\mathcal{D}_{\delta}\subset\mathcal{D}$ for all $\delta>0$. Thus, given $v\in\mathcal{H}_0^s(\Omega)$, since $v=0$ on $\Omega^c$ we have $v=0$ on $\partial_{\delta}\Omega$ and, then, the restriction operator, $$\begin{array}{l}
R:\mathcal{H}_0^s(\Omega)\mapsto\mathbb{H}_0^{\delta,s}(\Omega)\\
\mkern+69.7mu v\mapsto R[v]=v\big|_{\Omega_{\delta}}
\end{array}$$ is a continuous linear mapping. Hence, $\mathcal{H}_0^s(\Omega)$ can be continuously embedded into $\mathbb{H}_0^{\delta,s}(\Omega)$. On the other hand, for a given $v\in\mathbb{H}_0^{\delta,s}(\Omega)$, extending $v$ by 0 on $\mathbb{R}^N\backslash \Omega_{\delta}$, we have $$\begin{split}
\iint\limits_{\mathcal{D}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx=&\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx\\
&+\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\backslash B(x,\delta)}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx.
\end{split}$$ Using the variational characterization of the eigenvalue $\lambda_{1}^{\delta,s}$, we find $$\begin{split}
\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\backslash B(x,\delta)}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx
\leq&\frac{1}{\delta^{N+2s}}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\backslash B(x,\delta)}|v(x)-v(y)|^2dydx\\
\leq&\frac{2}{\delta^{N+2s}}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\backslash B(x,\delta)}|v(x)|^2+|v(y)|^2dydx\\
\leq&\frac{2}{\delta^{N+2s}}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}}|v(x)|^2+|v(y)|^2dydx\\
=&\frac{4}{\delta^{N+2s}}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}}|v(x)|^2dydx\\
=&\frac{4|\Omega_{\delta}|}{\delta^{N+2s}}\|v\|_{L^2(\Omega)}^{2}\\
\leq&\frac{4|\Omega_{\delta}|}{\delta^{N+2s}}\frac{1}{\lambda_1^{\delta,s}}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dydx.
\end{split}$$ Thus, ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert v
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathcal{H}_0^s}^2\leq C(\delta){{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert v
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mathbb{H}_0^{\delta,s}}^2$, where $C(\delta)=\left(1+\frac{4|\Omega_{\delta}|}{\delta^{N+2s}}\frac{1}{\lambda_1^{\delta,s}}\right)$. As a consequence, given $v\in\mathbb{H}_0^{\delta,s}(\Omega)$, the extension operator $$\begin{array}{l}
E:\mathbb{H}_0^{\delta,s}(\Omega)\mapsto\mathcal{H}_0^s(\Omega)\\
\mkern+79.2mu v\mapsto E[v]= \left\{\begin{array}{rl}
v&\mbox{in}\quad \Omega_{\delta},\\
0&\mbox{in}\quad \mathbb{R}^N\backslash\Omega_{\delta},\\
\end{array}\right. \end{array}$$ is a linear continuous mapping so that $\mathbb{H}_0^{\delta,s}(\Omega)$ can be continuously embedded into $\mathcal{H}_0^s(\Omega)$. Now, by and , for any positive $\delta_1,\,\delta_2$ with $\delta_1<\delta_2$, $$\mathbb{H}_0^{\delta_1,s}(\Omega)\subset\mathbb{H}_0^{\delta_2,s}(\Omega)\subset \mathcal{H}_0^s(\Omega).$$ In particular, the sequence of eigenvalues $\{\lambda_1^{\delta,s}\}_{\delta>0}$ is increasing in $\delta$ and uniformly bounded from above by the first eigenvalue $\lambda_{1}^{\infty,s}$ of the fractional Laplacian $(-\Delta)_{\infty}^{s}$. Therefore, the constant $C(\delta)$ satisfies $$\begin{split}
C(\delta)&=1+\frac{4|\Omega_{\delta}|}{\delta^{N+2s}}\frac{1}{\lambda_1^{\delta,s}}\leq1+c\frac{(diam(\Omega)+2\delta)^N}{\delta^{N+2s}}\frac{1}{\lambda_1^{\delta,s}}\to 1,\quad\text{as }\delta\to+\infty.
\end{split}$$
The following $\Gamma$-convergence result is in the core of the proofs of Theorems \[linearinf\] and Theorem \[theigen2\].
\[Gconvergenceinf\]Let us consider the functional $$\mathcal{E}_{\delta,s}(u)=\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx.$$ defined on $\mathbb{H}_0^{\delta,s}(\Omega)$. Then, the $\Gamma$-limit of $\mathcal{E}_{\delta,s}(u)$ is given by $$\label{GlimitFuncinf}
\mathcal{E}_{\infty,s}(u)=\int_{\mathbb{\mathbb{R}}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx\quad\text{as }\delta\to+\infty.$$
The sequence of functionals $\mathcal{E}_{\delta,s}(u)$ with $\delta\to+\infty$ is a monotone increasing sequence and functionals $\mathcal{E}_{\delta,s}$ are lower semicontinuous, cf. [@BellCor]. Therefore, because of [@Braides Remark 1.40], $$\mathcal{E}_{\delta,s}(u)\overset{\Gamma}{\to}\mathcal{E}_{\infty,s}(u)=\int_{\mathbb{\mathbb{R}}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx.$$
Let us consider the energy functional associated to problem , $$\begin{split}
\mathcal{J}_{\delta,s}(u)&=\frac{c_{N,s}}{4}\int_{\Omega_{\delta}}\int_{\Omega_{\delta}\cap B(x,\delta)}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx-\int_{\Omega}fudx\\
&=\frac{c_{N,s}}{4}\mathcal{E}_{\delta,s}(u)-\int_{\Omega}fudx.
\end{split}$$ Then, by Lemma \[Gconvergenceinf\], we conclude $$\mathcal{J}_{\delta,s}(u)\overset{\Gamma}{\to}\mathcal{J}_{\infty,s}(u)=\frac{c_{N,s}}{4}\int_{\mathbb{\mathbb{R}}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dydx-\int_{\Omega}fudx,$$ that is the energy functional associated to . Now, if $u^{\delta,s}$ is the minimizer of $\mathcal{J}_{\delta,s}$, the sequence $\{\mathcal{J}_{\delta,s}(u^{\delta,s})\}_{\delta>0}$ is monotone increasing in $\delta$ and bounded from above by $\mathcal{J}_{\infty,s}(u^{\infty,s})$. Indeed, given $\delta_1<\delta_2$, then $$\mathcal{J}_{\delta_1,s}(u^{\delta_1,s})\le \mathcal{J}_{\delta_1,s}(u^{\delta_2,s}) \le \mathcal{J}_{\delta_2,s}(u^{\delta_2,s}).$$ On the other hand, since $$\mathcal{J}_{\delta,s}(u^{\delta,s})=-\frac{c_{N,s}}{2} {{\vert\kern-0.25ex\vert\kern-0.25ex\vert u^{\delta,s}
\vert\kern-0.25ex\vert\kern-0.25ex\vert}}_{\mathbb{H}_0^{\delta,s}}^2,$$ the sequence ${{\vert\kern-0.25ex\vert\kern-0.25ex\vert u^{\delta,s}
\vert\kern-0.25ex\vert\kern-0.25ex\vert}}_{\mathbb{H}_0^{\delta,s}}$ is decreasing and bounded. Consequently, by Lemma \[isomorfia\], ${{\vert\kern-0.25ex\vert\kern-0.25ex\vert u^{\delta,s}
\vert\kern-0.25ex\vert\kern-0.25ex\vert}}_{\mathcal{H}_0^{s}}$ is bounded and, by Lemma \[isomorfia\] and the compact embedding of $\mathcal{H}_0^s(\Omega)$ into $L^2(\Omega)$, cf. [@DiNezzaPalaValdi Corollary 7.2], there exists a subsequence (that we do not relabel) and $u^{\infty,s}\in \mathcal{H}_0^s(\Omega)$ such that $$u^{\delta,s}\to u^{\infty,s}\quad\text{as }\delta\to+\infty.$$ The $\Gamma$-convergence of $\mathcal{J}_{\delta,s}$ to $\mathcal{J}_{\infty,s}$ implies that $u^{\infty,s}$ is the unique minimizer of $\mathcal{J}_{\infty,s}$.
The proof follows by combining Lemma \[Gconvergenceinf\] and Lemma $\ref{isomorfia}$ with the arguments used in the proof of Theorem \[theigen\].
[10]{}
B. Alali and N. Albin, *Fourier multipliers for nonlocal Laplace operators*, Applicable Analysis, [**0**]{} (2019), pp. 1–21.
G. Alberti and G. Bellettini, *A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies*, European J. Appl. Math., [**9**]{} (1998), pp. 261–284.
F. Andrés and J. Muñoz, *A type of nonlocal elliptic problem: existence and approximation through a Galerkin-Fourier method*, SIAM J. Math. Anal., [**47**]{} (2015), pp. 498–525.
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, *Nonlocal diffusion problems*, vol. [**165**]{} of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010.
J. C. Bellido and C. Mora-Corral, *Existence for nonlocal variational problems in peridynamics*, SIAM J. Math. Anal., [**46**]{} (2014), pp. 890–916.
J. C. Bellido, C. Mora-Corral and P. Pedregal, *Hyperelasticity as a [$\Gamma$]{}-limit of peridynamics when the horizon goes to zero*, Calc. Var. Partial Differential Equations, [**54**]{} (2015), pp. 1643–1670.
J. Boulanger, P. Elbau, C. Pontow and O. Scherzer, *Non-local functionals for imaging*, in Fixed-point algorithms for inverse problems in science and engineering, vol. [**49**]{} of Springer Optim. Appl., Springer, New York, 2011, pp. 131–154.
J. Bourgain, H. Brezis and P. Mironescu, *Another look at Sobolev spaces*, in Optimal control and partial differential equations, IOS, Amsterdam, 2001, pp. 439–455.
A. Braides, *[$\Gamma$]{}-convergence for beginners*, vol. [**22**]{} of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002.
L. Brasco, E. Parini and M. Squassina, *Stability of variational eigenvalues for the fractional [$p$]{}-Laplacian*, Discrete Contin. Dyn. Syst., [**36**]{} (2016), pp. 1813–1845.
H. Brezis, *Functional analysis, Sobolev spaces and partial differential equations*, Universitext, Springer, New York, 2011.
C. Bucur, *Some observations on the Green function for the ball in the fractional Laplace framework*, Commun. Pure Appl. Anal., [**15**]{} (2016), pp. 657–699.
C. Bucur and E. Valdinoci, *Nonlocal diffusion and applications*, vol. [**20**]{} of Lecture Notes of the Unione Matematica Italiana, Springer, \[Cham\]; Unione Matematica Italiana, Bologna, 2016.
M. D’Elia and M. Gunzburger, *The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator*, Comput. Math. Appl., [**66**]{} (2013), pp. 1245–1260.
E. Di Nezza, G. Palatucci and E. Valdinoci, *Hitchhiker’s guide to the fractional Sobolev spaces*, Bull. Sci. Math., [**136**]{} (2012), pp. 521–573.
Q. Du, *Nonlocal modeling, analysis, and computation*, vol. [**94**]{} of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019.
S. Duo, H. Wang and Y. Zhang, *A comparative study on nonlocal diffusion operators related to the fractional Laplacian*, Discrete Contin. Dyn. Syst. Ser. B, [**24**]{} (2019), pp. 231–256.
A. Evgrafov and J. C. Bellido, *From non-local Eringen’s model to fractional elasticity*, Math. Mech. Solids, [**24**]{} (2019), pp. 1935–1953.
G. Gilboa and S. Osher, *Nonlocal operators with applications to image processing*, Multiscale Model. Simul., [**7**]{} (2008), pp. 1005–1028.
M. Kassmann, T. Mengesha and J. Scott, *Solvability of nonlocal systems related to peridynamics*, Commun. Pure Appl. Anal., [**18**]{} (2019), pp. 1303–1332.
S. Kindermann, S. Osher and P. W. Jones, *Deblurring and denoising of images by nonlocal functionals*, Multiscale Model. Simul., [**4**]{} (2005), pp. 1091–1115.
G. Leoni and D. Spector, *Characterization of Sobolev and [$BV$]{} spaces*, J. Funct. Anal., [**261**]{} (2011), pp. 2926–2958.
J.-L. Lions and E. Magenes, *Non-homogeneous boundary value problems and applications. Vol. I*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.
A. Lischke, G. Pang, M. Gulian and et al., *What is the fractional Laplacian? A comparative review with new results*, J. Comput. Phys., [ **404**]{} (2020), 109009, 62 pp.
T. Mengesha, *Nonlocal Korn-type characterization of Sobolev vector fields*, Commun. Contemp. Math., [**14**]{} (2012), 1250028, 28 pp.
T. Mengesha and Q. Du, *Characterization of function spaces of vector fields and an application in nonlinear peridynamics*, Nonlinear Anal., [ **140**]{} (2016), pp. 82–111.
T. Mengesha and D. Spector, *Localization of nonlocal gradients in various topologies*, Calc. Var. Partial Differential Equations, [**52**]{} (2015), pp. 253–279.
A. C. Ponce, *A new approach to Sobolev spaces and connections to [$\Gamma$]{}-convergence*, Calc. Var. Partial Differential Equations, [**19**]{} (2004), pp. 229–255.
X. Ros-Oton, *Nonlocal elliptic equations in bounded domains: a survey*, Publ. Mat., [**60**]{} (2016), pp. 3–26.
X. Ros-Oton and J. Serra, *Fractional Laplacian: Pohozaev identity and nonexistence results*, Comptes Rendus Mathematique, [**350**]{} (2012), pp. 505–508.
R. Servadei and E. Valdinoci, *A Brezis-Nirenberg result for non-local critical equations in low dimension*, Communications on Pure [&]{} Applied Analysis, [**12**]{} (2013), pp. 2445–2464.
R. Servadei and E. Valdinoci, *Mountain pass solutions for non-local elliptic operators*, J. Math. Anal. Appl., [**389**]{} (2012), pp. 887–898.
R. Servadei and E. Valdinoci, *Variational methods for non-local operators of elliptic type*, Discrete Contin. Dyn. Syst., [**33**]{} (2013), pp. 2105–2137.
R. Servadei and E. Valdinoci, *On the spectrum of two different fractional operators*, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, [**144**]{} (2014), pp. 831–855.
S. A. Silling, *Reformulation of elasticity theory for discontinuities and long-range forces*, J. Mech. Phys. Solids, [**48**]{} (2000), pp. 175–209.
D. Spector, *On a generalization of [$L^p$]{}-differentiability*, Calc. Var. Partial Differential Equations, [**55**]{} (2016), Art. 62, 21 pp.
P. R. Stinga and J. L. Torrea, *Extension problem and Harnack’s inequality for some fractional operators*, Comm. Partial Differential Equations, [**35**]{} (2010), pp. 2092–2122.
[^1]: This work has been supported by the [*Agencia Estatal de Investigación, Ministerio de Ciencia e Innovación*]{} (Spain) through project MTM2017-83740-P
[^2]: This paper is in final form and no version of it will be submitted for publication elsewhere.
[^3]: This convergence can be understood in the sense of $cap(\mathcal{D}\backslash\mathcal{D}_{\delta})\to0$ as $\delta\to+\infty$, being $cap(E)=\inf \{ \|v\|_{H^1(\mathbb{R}^N)}^2 : v \in C_0^\infty(\mathbb{R}^N),\ v \geq 1 \text{ on }E\}$, the capacity of the set $E$.
|
harvmac epsf amssym
§
\#1[\#1]{} \#1 \#1 \#1 \#1\#2[[\#1\#2]{}]{} \#1[[*\#1*]{}]{}
\#1
[\#1]{}
\#1\#2 \#1\#2 \#1\#2
\#1[\#1]{}
Daniel Green$^1$ and David Shih$^2$
$^1$[*School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 USA*]{}
$^2$[ *NHETC, Dept. of Physics, Rutgers University, Piscataway, NJ 08854 USA*]{}
The operator product expansion (OPE) in 4d (super)conformal field theory is of broad interest, for both formal and phenomenological applications. In this paper, we use conformal perturbation theory to study the OPE of nearly-free fields coupled to SCFTs. Under fairly general assumptions, we show that the OPE of a chiral operator of dimension $\Delta = 1+\epsilon$ with its complex conjugate always contains an operator of dimension less than $2 \Delta$. Our bounds apply to Banks-Zaks fixed points and their generalizations, as we illustrate using several examples.
Conformal field theories (CFTs) and superconformal field theories (SCFTs) play a valuable role in our understanding of quantum field theories. Many interacting theories are believed to flow to (S)CFTs, while many more can be understood as small deformations away from conformality. Furthermore, through the AdS/CFT correspondence, CFTs are a useful tool for understanding quantum gravity.
Combined with unitarity, the enhanced symmetry of these theories has long been known to impose significant constraints on the spectrum of operator dimensions . These “unitarity bounds" have played a valuable role in our understanding of the phases of supersymmetric gauge theories . In addition, $a$-maximization has made it possible to compute the dimensions of chiral operators in proposed IR SCFTs, and unitarity bounds can be used to test the consistency of such proposals.
In recent years, there has been significant progress in using conformal invariance to constrain the spectrum of primaries that can appear in the operator product expansion (OPE) of scalar operators. Specifically, given a CFT with a scalar primary operator $X$ of dimension $\Delta_X$, the OPE takes the form where the $c_i$ are the OPE coefficients, $\Delta_i$ are the dimensions of the scalar primary operators ${\cal S}_i$ and $\ldots$ signifies higher spin operators and conformal descendants. Applying the OPE to the four point function of $X$, were able to place an upper bound on where $\Delta_{min}$ denotes the smallest $\Delta_i$ that appears in the OPE. Subsequent work has extended these results, including to supersymmetric theories .
In the context of supersymmetry, there has been much interest in theories with large anomalous dimensions, in part because of their many applications to model building, see e.g. (for non-SUSY motivation, see e.g. and references therein). For example, one can try to resolve the $\mu$/$B \mu$ problem by using an SCFT with $\delta_{min}>0$ . We often imagine generating $\mu$ and $B \mu$ at the SUSY breaking scale through the F-term of a chiral operator $X$ via the effective lagrangian Electroweak symmetry breaking requires $c_{B\mu}\sim c_\mu^2$ at the scale SUSY breaking. Unfortunately, models almost invariably give rise to $c_{B\mu} \gg c_{\mu}^2$ at the scale where they are generated – this is known as the $\mu/B\mu$ problem (see for a review and original references). However, if the theory flows near a SCFT in between these scales, the “$X^{\dagger} X$" is replaced by the operators in the OPE of $X^{\dagger}$ and $X$. If all such operators have dimension $\Delta_i > 2 \Delta_X$, then RG flow will suppress $c_{B\mu}$ down to acceptable levels in the IR.
Although the bounds in allow for positive $\delta_{min}$, to date there is no existence proof showing that this is possible. In , SCFTs with weakly coupled AdS duals were studied, and it was shown that $\delta_{min}$ could have either sign as far as the 5d effective gravity theory was concerned. If these models could be UV completed into full string compactifications, then they would furnish existence proofs of positive $\delta_{min}$. Whether this is possible remains an interesting open question.
In this paper, we will take a complementary approach, and investigate the quantity $\delta_{min}$ in the context of SCFTs in which $X$ is a chiral primary operator with dimension $\Delta_X=1+\epsilon$, with $\epsilon\ll 1$. When $\Delta_X =1$, the unitarity bounds require that $X$ is a free field and its OPE is trivially determined. One might suspect that for $\Delta_X=1+\epsilon$, the bounds on operators in the OPE are accessible perturbatively in $\epsilon$. We will develop a general formalism for computing $\delta_{min}$ based on conformal perturbation theory, with the assumption that $X$ acquires its anomalous dimension solely through a coupling in the superpotential to a chiral operator $\CO$ of dimension $\Delta_\CO=2-\epsilon$. Under such circumstances, the theory will flow to a fixed point where $X$ has the required dimension, and we will prove that the OPE of $X$ and $X^{\dagger}$ contains an operator of dimension at most $\Delta = 2\Delta_X + |\cO(\epsilon^2)|$. We will further show that $\delta_{\rm min} \sim - \epsilon^2/ \nu_i < 0$ when the original SCFT contains real operators $L_i$ of dimension $\Delta_i = 2 + \nu_i$ with $\nu_i \ll 1$. As an application of our formalism, we will prove that in fully perturbative examples, such as generalized Banks-Zaks fixed points, one always finds $\delta_{min}<0$.
While these assumptions may seem restrictive, it is plausible that this setup actually applies to all examples of SCFTs with a chiral operator with dimension close to one. In the limit $\epsilon \to 0$, the operator $X$ must become free and will decouple from any other fields. Furthermore, $X$ must be gauge invariant. Provided there is some Lagrangian description of $X$, this suggests the above description should hold in some duality frame.
The organization of the paper is as follows: In Section 2, we will explain how the bounds on scalar operators in the OPE of chiral primaries in an SCFT can be understood from conformal perturbation theory. The presentation in this section will be mostly general, making only a few simplifying assumptions about the form of the SCFT. In Section 3, we will compare these results with concrete examples derived from Banks-Zaks-like SCFTs. The results in this section agree with the results of conformal perturbation theory but are derived in a more conventional language. We will summarize our results and mention some interesting open problems in Section 4. Appendix A contains the details of the conformal perturbation theory calculations. Appendix B addresses complications arising from the presence of global symmetries. Appendix C contains further details regarding the matching between the results of Section 2 and 3.
Consider a 4d interacting SCFT $\cP_1$ which contains a chiral operator $\CO$ of dimension $\Delta_\CO = 2-\epsilon$. The OPE of $\CO$ and $\CO^\dagger$ is given by: where $L_i$ are real scalar multiplets with dimension $\Delta_i = 2+\nu_i$, $\nu_i\ge 0$, and $\dots$ denotes descendants and operators with higher spin. All primary scaling operators in this paper will be “CFT-canonically normalized" as in , i.e. with the coefficient of the leading singularity set to one.
Now consider adding to $\cP_1$ a free chiral superfield $X$, and the deformation where we have introduced the Wilsonian cutoff scale $\Lambda$ in order to make $\lambda$ dimensionless. Note that $X$ is not a canonically normalized field; this is to obey the more natural CFT normalization described in the previous paragraph. When $0< \epsilon \ll 1$, this theory will flow to a new fixed point, $\cP_2$, with $\lambda \sim \cO(\sqrt{\epsilon})$. At the new fixed point, we would like to know what is the lowest dimension scalar operator that appears in the OPE of $X$ and $X^{\dagger}$. Candidate operators include $X^\dagger X$ and the $L_i$. In general, these are not scaling operators at the new fixed point $\cP_2$, but will instead mix with one another. As we will see in the following subsections, when the dimensions of $L_i$ are approximately 2, this mixing is the dominant contribution to $\delta_{\rm min}$.
We will focus on the case where conserved currents are absent in the $X^\dagger(x)X(0)$ OPE. In the free theory, $X^{\dagger} X$ is the conserved current associated with the symmetry $U(1)_X$. This current is broken by but, in general, new conserved currents could arise from a combination of $U(1)$ currents broken by the superpotential interaction, i.e. a combination of $U(1)_X$ and $U(1)$’s from $\CP_1$ under which $\CO$ is charged. The unitarity bounds ensure that any such current would have dimension two, and therefore would trivially satisfy $\Delta_{\min} < 2 \Delta_X$ for $\Delta_X>1$.
To simplify the discussion in this section, we will assume that $\CO$ is a singlet under all the global symmetries of $\cP_1$, i.e. the $\nu_i$ are strictly positive in . This ensures that the deformation breaks the $U(1)_X$ symmetry completely, and then absence of conserved currents is trivially guaranteed. In appendix B, we will generalize to the (considerably more complicated) case of charged $\CO$. With some mild assumptions, all the results of this section carry over to charged $\CO$, provided that conserved currents remain absent in the $X^\dagger X$ OPE.
Under the assumption that $\epsilon \ll 1$, we can understand the RG flow induced by using conformal perturbation theory. Without loss of generality, $\lambda$ will be real, as its phase can be removed by a field redefinition of $X$. As above, $\Lambda$ will denote the sliding Wilsonian cutoff scale.
Once we introduce the deformation by a single operator, as in , other operators will also contribute to the RG flow at higher orders in $\lambda$. Therefore, the RG is actually described by the flow in a larger space of couplings. We will include these additional operators in the Lagrangian as Here $Z_X$ is the wavefunction renormalization of $X$ and $y_i$ are coupling constants. The $\ldots$ denote operators that that do not appear in the $\CO^\dagger\CO$ OPE. The counterterms for these can only arise at higher orders in $\lambda$, and so they will not contribute to our analysis in this paper. We have also dropped operators of the form $X^{\dagger} X L_i$ as they contribute at the same order as $L_i$ of dimension $\Delta_i > 4$ and will be negligible in the regimes of interest. We have included explicit $\Lambda$ dependence to make all the couplings dimensionless. By a non-renormalization theorem , we have chosen a holomorphic scheme such that no additional operators appear in the superpotential.
If we work in a holomorphic scheme, there is no renormalization of $\lambda$ and its beta function $\beta_\lambda\equiv {d \lambda\over d\log\Lambda}$ is given purely by the classical contribution, $\beta_\lambda = - \epsilon \lambda$. In addition, there is wavefunction normalization, namely $Z_X = \Big( \frac{\Lambda_0}{\Lambda} \Big)^{2 \gamma_X (\lambda, y_i)} = 1 + 2 \gamma_X (\lambda, y_i) \log(\Lambda_0 /\Lambda) + \ldots$ (where $\Lambda_0$ is some arbitrary reference scale). In this scheme, as we flow to the IR ($\Lambda \to 0$), $\lambda$ and $Z_X$ will run to infinity. To see that we reach a fixed point, we redefine $X$ and $\lambda$ to make the former (nearly) canonical After the redefinition, the beta function of $\lambda$ (henceforth dropping the “phys" superscripts) becomes Regardless of the form of $\gamma_X (\lambda, y_i)$, we see that the fixed point will arise at $\gamma_X (\lambda_\star, y_{i \, \star}) = \epsilon$. As expected, the dimension of $X$ at $\cP_2$ is $\Delta_X = 1 + \gamma_X (\lambda_\star, y_{i \, \star}) = 1 + \epsilon$.
In order to the determine the dimensions of operators at $\cP_2$, we will need to know the functional form of $\gamma_X$. Expanding $\gamma_X$ in powers of $\lambda$ and $y_i$, we find We have relegated the detailed computations to appendix A. Here $c_i$ is the OPE coefficient of $\cO \cO^{\dagger} \to L_i$ in . $\CI$ is a function of $\nu_i$ and $\epsilon$ which is smooth everywhere, and which we have normalized so that: The $\dots$ in includes important corrections at $\CO(\lambda^4)$. Ideally we would also compute these corrections for completeness. However, the calculation is third order in conformal perturbation theory and beyond the scope of this work.
Finally, we also need the leading-order beta functions for the couplings $y_i$. There is a classical contribution to the beta functions from the explicit scale dependence, $\beta_{y_i} = \nu_i y_i + \ldots$. The leading order correction in $\lambda$ was already determined using conformal perturbation theory in ; we have redone it in appendix A with the notation and conventions of this paper for completeness sake. As a result of this calculation, one finds where $c_i$ is again the same OPE coefficient as appears in and . We see that $y_i$ will flow to a fixed point $y_{i \star} \approx \frac{1}{2} \frac{c_i}{\nu_i} \lambda_\star^2$.
For the theory to be under control and conformal perturbation theory to be valid, we obviously need $y_{i\star} \ll 1$. This places a restriction on the range of $\nu_i$ relative to the other parameters. In general, we need If $c_i\sim\CO(1)$, then $\nu_i$ must be parametrically larger than $\epsilon$. If, on the other hand, $c_i\ll 1$ (as in the perturbative examples we will study in section 3), then $\nu_i$ can be much smaller. Of course, theories where $\nu_i$ violates are perfectly valid, they just cannot be described using conformal perturbation theory around $y_i=0$. Instead, it makes more sense to treat the corresponding operators $L_i$ as nearly-conserved currents. In appendix B, we outline how to treat exactly conserved currents; we expect that nearly-conserved currents behave in a similar way. We will leave a complete analysis to future work.
In summary, the RG flow of is described by the system of beta functions The fixed point $\cP_2$ is attractive and occurs when the couplings are This is sufficient information to determine the dimensions of the non-chiral, scalar operators at $\cP_2$, which we will turn to in the next subsection.
Given the beta functions that describe an RG flow, it is straightforward to determine the set of scaling operators and their dimensions that describe the infinitesimal deformations away from a fixed point. Specifically, if we deform any fixed point by $\int d^4 x\, \kappa\, \Lambda^{4 - \Delta} \Sigma$ where $\Sigma$ is some operator of dimension $\Delta$, the beta function for the coupling $\kappa$ must take the form $\beta_{\kappa} = (\Delta- 4)\kappa+ \cO(\kappa^2)$. Reversing this logic, if we are given the beta functions for some couplings $\kappa_i$ near a fixed point, we can determine the anomalous dimensions of operators at that fixed point by Taylor expanding and diagonalizing the beta functions around it, The eigenvectors and eigenvalues of $\partial_{\kappa_j} \beta_{\kappa_i}$ determine the scaling operators and anomalous dimensions, respectively.
Applying this to the beta functions at the fixed point , we find the matrix of anomalous dimensions at $\CP_2$: As discussed in the introduction, we are interested in the anomalous dimensions relative to $2\gamma_X=2\epsilon$. It is convenient to introduce the shifted matrix The eigenvalues $\delta$ of this matrix satisfy the characteristic equation where we have used to replace $\lambda_\star$ with $\sqrt{\epsilon}/\pi$ (again, dropping higher order corrections for now). The minimum eigenvalue $\delta_{min} \equiv \Delta_{\rm min} - 2 \Delta_X$ is the quantity of interest, introduced in . In order to understand the behavior of $\delta_{min}$, it will now be useful to distinguish different scenarios for the $\nu_i$:
$\nu_i \ll 1$ for some $i$. This scenario applies to all fully perturbative SCFTs (generalized Banks-Zaks models), because in these theories the kinetic terms are always approximately dimension 2. In the $\nu_i\ll 1$ regime, two things happen. First, $\CI(\nu_i,\epsilon)\to 1$ as shown in appendix A. Then $\Delta\Gamma$ is related by a similarity transform to a symmetric matrix The minimum eigenvalue of any symmetric matrix is always smaller than all the diagonal elements, so we conclude that in this regime. The second thing that happens in this regime is that the mixing with $X^\dagger X$ is enhanced, making its contribution to the anomalous dimension reliable compared to the $\CO(\epsilon^2)$ from higher order in conformal perturbation theory. The situation is clearest if $\epsilon\ll\nu_i\ll 1$, where we find from This is indeed enhanced relative to the $\CO(\epsilon^2)$ corrections to the beta functions which we have not computed. If on the other hand $\epsilon\sim\nu_i\ll 1$, then no longer applies and we have to do degenerate perturbation theory instead. But, in this case, it is easy to see that the enhancement is strictly larger – instead of , $\delta_{min}<0$ scales as $\CO(\epsilon)$. Finally, if $\nu_i\ll\epsilon$, then there is already an operator $L_i$ in the OPE of $X^\dagger X$ with much smaller anomalous dimension than $2\epsilon$, so $\delta_{min}<0$ trivially.
$\nu_i \sim 1$ for all $i$. Theories in this class necessarily contain some strong dynamics. When all the operators $L_i$ have $\CO(1)$ anomalous dimensions, the mixing with $X^\dagger X$ is not enhanced, and we find from . In this case, we cannot determine the sign of $\delta_{min}$. Even if we knew the sign of $\CI(\nu_j,\epsilon)$, it turns out higher order corrections in conformal perturbation theory are equally important to the contribution . In particular, $\delta_{min}$ is sensitive to $\CO(\lambda^4)$ terms in $\beta_\lambda$ which we have not computed (the $\dots$ in ). Nevertheless, we see that the lowest dimension operator in the $X^{\dagger}X$ OPE has dimension $\Delta_{min} = 2\Delta_X + \delta_{min} = 2 + 2\epsilon+\CO(\epsilon^2)$ to leading order. This is consistent with the bounds of , who found that $\Delta_{min} \leq 2 + 2\epsilon + 2.683 \epsilon^2$. It would be very interesting to see whether this bound is saturated after including the $\CO(\lambda^4)$ contributions. If this bound is the strongest possible one, then it may be possible to compute the numerical coefficient at $\CO(\epsilon^2)$ directly from conformal perturbation theory.
To summarize: in this section, we found that the lowest dimension operator that can appear in the OPE of $X$ and $X^{\dagger}$ has dimension that is at most $\Delta_{min} = 2 \Delta_X + |\cO(\epsilon^2)|$. When $\nu_i \ll 1$ for some $i$, as is the case for all fully perturbative SCFTs, we found that $\Delta_{min} - 2 \Delta_X \sim - \epsilon^2 / \nu_i < 0$. The bounds when $\nu_i \ll 1$ are stronger than those found in . It would be interesting to determine all $\cO(\epsilon^2)$ corrections to see if this is true in general.
In the previous subsections, we determined the fixed point and the matrix of anomalous dimensions by neglecting higher order corrections in conformal perturbation theory. Now let us discuss the conditions under which this is valid.
One basic assumption which we have been making implicitly so far is that there are no large hierarchies in the OPE coefficients. Without this assumption, leading-order conformal perturbation theory is not necessarily valid. For instance, if the coefficient of $y\lambda^2$ in is anomalously small, then higher orders in conformal perturbation theory involving unsuppressed OPE coefficients could dominate.
In other words, we are assuming that parametrically, and are really Here $\tilde c_i$ and all the coefficients in $\dots$ are assumed to be $\CO(1)$. $\kappa$ can either be (at most) $\CO(1)$ or parametrically small. In the BZ models of Section 3, $\kappa\sim 1/N$.
Having clarified this point, let us now systematically consider higher-order corrections to $\gamma_X$ and $\beta_{y_i}$, the $\dots$ in and . Corrections to $\gamma_X$ must be proportional to $\lambda^2$; combining this with , we conclude that they are always subleading for determining the fixed point couplings. As discussed in the previous subsection, they are also subleading for determining the anomalous dimensions, provided that there are $L_i$ operators with dimensions close to two. All higher-order corrections to $\beta_{y_i}$ proportional to $\lambda^2$ are also subleading for determining the fixed point couplings and the anomalous conditions, in general.
This leaves corrections to $\beta_{y_i}$ which are $\CO(\lambda^0)$, i.e. which involve the $y$’s only. Since these should be compared with the classical term $\nu_i y_i$, they can conceivably be important when $\nu_i\ll 1$. For instance, consider an $\CO(y^2)$ correction to $\beta_{y_i}$: These will affect the matrix of anomalous dimensions as $\delta\Gamma_{ij} = c_{ijk}y_k\sim {c_{ijk} c_k \epsilon\over \nu_k}$. So if $\nu_i$ is too small, even if it satisfies , this perturbation to the matrix of anomalous dimensions could be larger than the classical dimension. For this paper, we will simply [*assume*]{} that these higher order terms are subleading. Nevertheless, it is plausible that this assumption could be promoted to a consequence of $y_{i\star}\ll 1$. Because the two-point function $\langle (Q^2 L_i) (\bar Q^2 L_i) \rangle \propto \nu_i$, we must have $\langle (Q^2 L_i) (Q^2 L_j) \tilde \CO \rangle \propto \nu_i \nu_j$ in order for the $ ( Q^2 L_{i,j} )\tilde \CO$ OPE coefficient to be finite as $\nu_i \to 0$. The resulting $\nu_{i} \nu_{j}$ suppression of the $(Q^2 L_i) (\bar Q^2 L_j)$ OPE ensures that $\delta \Gamma_{ij}$ is negligible when $y_{i,j} \ll 1$. However, a careful derivation valid to all orders in perturbation theory is beyond the scope of this work.
More generally, as mentioned already in the previous subsection, in order for our analysis in this paper to be valid, we must avoid extremely small values of $\nu_i$. The classical approximation of $\beta_{y_i}=\nu_i y_i+\dots$ should be a reliable starting point for conformal perturbation theory. For this to be true, $\nu_i$ must be parametrically larger than any of the corrections to the dimension of $L_i$ coming from conformal perturbation theory. In the BZ models with $\kappa\sim 1/N$ being the smallest parameter, this is automatically guaranteed.
Overall, the complication presented in this general analysis is that, a priori, all the OPE coefficients and dimensions appear to be independent parameters. If one of these numbers is anomalously small or large, it could invalidate our leading order calculation. In practice, a given theory (or class of theories) is controlled by a much smaller set of parameters. Using these parameters, it is easier to establish the regime of validity of conformal perturbation theory.
In the previous section we presented general results on the spectrum of operators in the OPEs of a nearly free, chiral superfield $X$. Now, we will illustrate these general results with broad class of simple, calculable examples, namely supersymmetric Banks-Zaks (BZ) fixed points and their generalizations. We will examine the dimensions of operators at BZ fixed points from a more conventional perspective and show that the results agree with the analysis using conformal perturbation theory.
Before we start with a detailed analysis, let us make a few general remarks about perturbative SCFTs. In four dimensions, gauge fields are required in order to produce a non-trivial, perturbative fixed point. This requirement has several important consequences: (1) Charged matter cannot produce chiral operators of dimension near one because the matter fields are not gauge invariant. (2) To produce an operator of approximate dimension one, we must introduce a singlet, $X$, under the gauge group; therefore, the only marginal or relevant interactions are through operators in the superpotential. (3) The kinetic terms for the matter fields are approximately dimension two and will appear in the OPE of $X$ and $X^{\dagger}$. (4) Therefore, all the models of this type are described by “scenario 1" of the previous section and we must have $\delta_{min}<0$ on general grounds. The two examples in this section will serve as illustrations of these basic observations.
The simplest supersymmetric Banks-Zaks fixed points arise in ${\cal N}=1$, $SU(N_c)$ gauge theories with $N_f = {3N_c/ (1+\epsilon)}$ flavors (we assume $N_f$, $N_c\to\infty$ throughout). The only coupling in these theories is the gauge coupling; since we are at large $N$, we will use instead the ’t Hooft coupling Its RG flow is described (in a suitable scheme) by the NSVZ beta function : where we have defined $f(\hat g)$ so that $f(\hat g)=1+\dots$, and is the anomalous dimension of the fundamental flavors. We have written the anomalous dimension in the form to emphasize the nature of the double expansion in $1/N_c$ and $\hat g$. The Banks-Zaks fixed point occurs at $\beta_{\hat g}=0$, which according to is equivalent to or in terms of the ’t Hooft coupling, Here the 0 subscript on $\hat g_*$ is to emphasize that this is the fixed point in the undeformed BZ theory. Shortly we will deform the theory by coupling it to $X$, and that will change the value of $\hat g_*$.
The BZ fixed point will play the role of $\cP_1$ in Section 2. We would now like to identify some linear combination of the gauge invariant operators $Q_f\widetilde Q_{\tilde g}$ with the operator $\CO$ in section 2. According to , these have dimension close to 2: Note that since $Q$ and $\tilde Q$ are not gauge invariant operators, there are no chiral primaries with dimension near 1. This is a general feature of all BZ-type theories.
As discussed in Section 2, the principal requirement on the operator $\CO$ (other than having dimension close to 2) is that after deforming by $\int d^2\theta\,\lambda X \CO$, the $U(1)_X$ symmetry is completely broken, so that no conserved currents appear in the $X^\dagger X$ OPE. Modulo $SU(N_f)_L\times SU(N_f)_R$ rotations, this condition is uniquely satisfied with the choice The normalization $a = {4\pi^2\over \sqrt{N_f N_c}}(1 + \cO(g^2))$ is chosen so that $\CO$ is CFT-canonically normalized (with $Q$, $\tilde Q$ canonically normalized). The deformation breaks the flavor group $U(1)_X \times SU(N_f) \times SU(N_f) \to SU(N_f)_{\rm diag}$, so there is no unbroken $U(1)$ which can appear in the OPE of $X$ and $X^\dagger$.
To make contact with Section 2, we must also determine the OPE coefficients and dimensions at $\cP_1$. Given , the $\CO^\dagger\CO$ OPE is Here $c$ is the OPE coefficient, and $L$ is given by To leading order in perturbation theory (i.e. with free-field contractions), the coefficients $b$ and $c$ are given by The additional scalar operators in can be safely ignored. Any $L_i$ must be a $SU(N_f)_{\rm diag}\times U(1)_R$ singlet and be invariant under $Q\leftrightarrow \tilde Q$, and is the unique such operator with approximate dimension 2. Note that the OPE coefficient is $1/N$ suppressed. This is a general feature of BZ-type theories and is important for the decoupling of global symmetry currents explicitly broken by the deformation of $\CP_1$. We also note that $L$ is a singlet under the full $SU(N_f) \times SU(N_f)$ which leads to a further decoupling of these currents.
We see from that $L$ is nothing but the anomalous Konishi current of the SQCD theory, which satisfies the anomaly equation $\bar D^2 L \propto {\rm Tr} W_{\alpha} W^{\alpha}$ . Therefore the anomalous dimensions of $L$ and $\Tr\,W_\alpha^2$ are the same. The latter can be derived by expanding the beta function for the gauge coupling around the fixed point (see the related discussion at the beginning of Section 2.3). We conclude that where $f(\hat g)$ was defined in , and we have used and in the last equality. The anomalous dimension of $L$ is $\CO(\epsilon^2)$.
Finally, let us deform the theory by and work out the matrix of anomalous dimensions in this simple example. Here $a$ was defined below . We will first compute the anomalous dimensions using a more conventional perturbative approach. Then we will compare this with the general conformal perturbation theory of Section 2.
In the deformed theory, the anomalous dimensions of $X$ and $Q$ are given by the following: where we have introduced a large $N$ coupling for $\lambda$, $ \hat\lambda\equiv {N_c^2\,a^2\,\lambda^2\over8\pi^2}$. The beta functions consist of , together with While these equations are exactly solved by the couplings at the fixed point are determined perturbatively using , At the new fixed point, we note that $\hat g_*$ differs from $\hat g_{*,0}$ at $\CO(\epsilon/N_c^2)$.
Given the couplings, we find the matrix of anomalous dimensions is given by: The eigenvectors of this matrix correspond to scaling operators formed from $L$ and $X^{\dagger}X$, with anomalous dimensions given by the eigenvalues: In and , we have explicitly included all the terms that are reliably computed with one-loop anomalous dimensions. At the same time, we should emphasize that the $\nu_L$ appearing in and is the [*exact*]{} anomalous dimension of $L$ in the undeformed BZ theory, i.e. including all higher loop corrections. It is important to capture these corrections even though they cannot be computed explicitly, so that the deformation only shifts $\nu_L$ by $\CO(1/N_c^2)$.
We can arrive at the same results using the general formalism from section 2. According to , we should have so the eigenvalues are: If we plug in $c_L = \sqrt{2\over N_fN_c} = \sqrt{2(1+\epsilon)\over 3N_c^2}$, then we find perfect agreement with the correction to the $2\epsilon$ eigenvalue in . On the other hand, our $1/N_c^2$ correction to the $\nu_L$ eigenvalue does not match. This is because we are keeping enough terms in conformal perturbation theory to determine the correction to $\Delta_{X^\dagger X}$, but not enough terms for the correction to $\Delta_L$. For instance, a term $\sim c_i y_i\lambda^2$ in $\beta_{y_i}$ would correct $\nu_L$ at the same order as what one gets from mixing (but it doesn’t affect the correction to $2\epsilon$ to leading order). While it would be interesting to also get these terms right, it is not necessary for our purposes and is beyond the scope of this paper.
Of course, given , we know that this example does not come close to testing whether the dimension of $X^\dagger X$ can be larger than $2 \Delta_X$; in terms of the classification of section 2.3, we are in the $\nu\ll\epsilon$ regime of “scenario 1." This occurs in most BZ-like fixed points – the $X^\dagger X$ OPE always contains an operator associated with an anomalous $U(1)$, like $L$, whose anomalous dimension $\nu_L$ typically obeys $\nu_L \ll \epsilon$. As we will show in the next subsection, it is possible to construct a model with all $\nu_L>2\epsilon$, but at the cost of significant tuning of parameters.
=0.8
[*:[*$\;$ Quiver diagram for a viable generalized BZ model, where it is possible to achieve $\nu_L\gg \epsilon$ for all $L$. Circles denote $SU(N)$ gauge groups; squares denote $SU(N)_L\times SU(N)_R$ global symmetries. Links between nodes denote vector-like bifundamental fields. Finally, every triangle corresponds to a gauge and global-symmetry invariant Yukawa coupling.* ]{}*]{}
Clearly, to get a meaningful example with all $\nu_L> 2\epsilon$, we need to consider generalized BZ fixed points which have multiple small parameters and the freedom to dial $\epsilon\to 0$ without simultaneously dialing any of the $\nu_L\to 0$.
Shown in fig. 1 is a quiver diagram for a viable generalized BZ model. Although the model is quite complicated, it is the simplest example we have been able to find. The nodes in fig. 1 are labelled by $N_i$, $i=1,\dots,6$; circles indicate $SU(N_i)$ gauge groups, and boxes indicate $SU(N_i)_L\times SU(N_i)_R$ flavor symmetries. Links between nodes (there are 10 altogether) denote vector-like bifundamental fields; we will label these fields by the pair of numbers corresponding to the nodes that they connect, e.g. the bifundamentals between nodes $N_1$ and $N_2$ will be called $Q_{12}$ and $ Q_{21}$. Our convention will be that the field is always a fundamental under the first node and an anti-fundamental under the second node. Finally, every triangle corresponds to two gauge and global-symmetry invariant Yukawa couplings with different orientations. For instance, the 1-2-3 triangle corresponds to $\delta W = \lambda_{123} {\rm Tr}\,Q_{12}Q_{23}Q_{31} +\lambda_{123}' {\rm Tr}\, Q_{21} Q_{32} Q_{13}$.
The non-anomalous, unbroken flavor symmetry of this model is $\prod_{i=4}^6 SU(N_i)_L\times SU(N_i)_R$, together with 5 baryonic symmetries left unbroken by the Yukawa couplings. In addition, at the fixed point there is an unbroken ${\Bbb Z}_2$ symmetry under which $Q_{ij}\leftrightarrow Q_{ji}$. We will choose to preserve this ${\Bbb Z}_2$ symmetry along the entire flow, to avoid unnecessary clutter. As a result, $\gamma_{ij}=\gamma_{ji}$ and $\lambda_{ijk}=\lambda_{kji}$. One can also include the ${\Bbb Z}_2$ odd couplings and verify this explicitly.
As we will show, in this model $\epsilon\equiv -2 \gamma_{12}$ can be dialed to zero while keeping all the gauge and Yukawa couplings (and hence all the $\nu_L$) nonzero. Therefore, we will couple $X$ to Before we launch into a more detailed analysis, let us discuss some general points:
[1.]{} $\CO$ appears to be the unique operator in this model whose anomalous dimension can be dialed to zero independently of the couplings.
[2.]{} Unlike in the previous example, $\CO$ is a singlet under the gauge and global symmetry of the quiver theory. The analysis of section 2 will apply directly, without having to deal with any of the complications discussed in appendix B.
[3.]{} In general, there are a number of approximately dimension two scalar operators $L_i$ that could appear in the OPE of $X^\dagger$ and $X$. Clearly, any such operator must be a gauge and global singlet, and they must be invariant under $Q_{ij}\leftrightarrow Q_{ji}$. The complete list of compatible dimension two operators is There are 10 such operators, and they are all explicitly broken by Konishi anomalies and the Yukawa couplings. There are an additional ten, ${\Bbb Z}_2$-odd operators of dimension near two, of which five are the conserved baryonic currents. These twenty operators can be combined to form the twenty $U(1)$ currents of the free theory that commute with the non-abelian flavor symmetry ($Q_{ij} \to e^{i \theta_{ij}} Q_{ij}$).
[4.]{} For the choice of , only one operator, $L_{12}$, appears in the $X^\dagger X$ OPE to leading order. Its leading order OPE coefficient is:
[5.]{} This model has both Yukawa couplings and multiple gauge groups. Both are necessary conditions – otherwise sending $\gamma_{ij}\to 0$ necessarily forces $\nu_{L}\to 0$ faster for some $L$.
Now let us embark on a more detailed analysis of this model. We will follow the notation and conventions established in the previous subsection, with some minor additions. First, because of the complexity of this model, we will not be as careful about the double expansion in $N$ and $\epsilon$ as we were previously; we will stick to leading-order one-loop anomalous dimensions at large $N$. Second, since here we have many $N_i\to \infty$, we will introduce an auxiliary parameter $N$, and define where $b_i=3N_i-\sum_j N_j$ are the coefficients of the one-loop beta functions. We will take $N\to \infty$ holding fixed $x_i$ and $\epsilon_i$.
For a general quiver of this type, the gauge beta functions are (we neglect the denominator in the NSVZ formula, because it will not matter for our discussion): Here $\hat g_i=0$ if $i$ is not a gauge node, and $\hat \lambda_{ijk}=0$ if $(ijk)$ is not a triangle. One can check that all of the anomalous dimensions $\gamma_{ij}$ (and hence all of the couplings $\hat g_i$, $\hat\lambda_{ijk}$) are fully determined by setting the beta functions to zero.
In , the gauge and Yukawa interactions contribute with opposite signs to $\gamma_{ij}$. This is ultimately what allows us to tune $\gamma_{12}\to 0$ while keeping all the couplings nonzero (and real and positive). To demonstrate this explicitly, consider taking with $\epsilon,\,\xi\ll 1$. The origin of coupling-space corresponds to $\epsilon=\xi=0$. Solving , one finds independent of $\xi$. Thus we can dial $\gamma_{12}\to 0$ while maintaining $\xi$, and hence the couplings, nonzero (and real and positive). Correspondingly, at $\epsilon=0$, the dimensions of the $L_{ij}$ operators are all nonzero: with $\dots$ denoting higher powers of $\xi$.
To complete the construction of the model, we couple $X$ to $a\Tr\,Q_{12}Q_{21}$ via a Yukawa coupling $\lambda$. This introduces the beta function for $\hat\lambda\equiv {N^2a^2\lambda^2\over 8\pi^2}$ and modifies the anomalous dimension of $Q_{12}$, as in the previous subsection. Along with , the RG flow is now described by At the new fixed point, we again have $\gamma_X=-2\gamma_{12}=\epsilon$.
So as we dial $\epsilon\to 0$ while keeping the couplings nonzero, we will reach a regime where $\gamma_X=\epsilon \ll \nu_{L_{ij}} \ll 1$. This in turn provides a concrete example of the nontrivial part of “scenario 1" from section 2.3, where mixing with the $L$ operators gives the dominant contribution to $\delta_{min}$ and forces it to be negative. We can see this explicitly by using the formulas to calculate the correction to the $X^\dagger X$ anomalous dimension in the limit . For $\epsilon \ll \xi^2$, we find: This is the parametric behavior of $\delta_{min}\sim -c_i^2\epsilon^2/\nu_i$ expected from Section 2, with the smallest $\nu_i\sim \xi^2$ dominating from , and $c_i\sim 1/N$ according to . We will leave a more detailed matching with conformal perturbation theory to Appendix C.
In summary, we have constructed an example that realizes all the regimes of “scenario 1" of our conformal perturbation theory calculation. This example serves two purposes. First, it demonstrates that there is no fundamental obstacle to varying $\epsilon$ and $\nu_i$ independently. Second, it illustrates how the bound derived in Section 2 arises in an actual example.
In our treatment of the generalized BZ theories, we have neglected an important subtlety: the presence of double-trace operators. In large $N$ theories, the OPE of single trace operators will contain double trace operators with $\CO(1)$ OPE coefficients.
When all the OPE coefficients are comparable in size, we are justified in focusing on the operators of the lowest dimension, as they will have the largest mixing with $X^{\dagger} X$. In the weakly-coupled BZ theories, we determined the dimension of $X^{\dagger} X$ by isolating the mixing with single trace operators of approximate dimension 2, such as the Konishi currents. However, the OPE coefficients that control the mixing with these operators are suppressed by $\CO(1/N)$. Therefore, when $N \to \infty$, single trace operators do not mix with $X^{\dagger} X$ and do not alter its scaling dimension. One may worry that the dominant effect arises from the double-trace operator $L_4= \CO^\dagger\CO$ which has approximate dimension 4 but an [*unsuppressed*]{} OPE coefficient with $\CO$. Since $1/N$ is the smallest parameter in the model, if $L_4$ has any unsuppressed mixing with $X^\dagger X$, then it will dominate the anomalous dimension calculation, despite being of much higher dimension
Fortunately, the double-trace operator is also decoupled from $X^\dagger X$ to leading-order in $1/N$. We see this from the calculation in appendix A: the $y\lambda^2$ term in $\gamma_X$ vanishes when $\Delta_L=2\Delta_\CO$. Deviations from this are proportional to $\Delta_L-2\Delta_\CO\sim 1/N^2$ for the double-trace operator in question. The double-trace operator then contributes at the same order as any other real scalar operator with $\nu\sim\CO(1)$, and its mixing with $X^\dagger X$ is subleading relative the Konishi currents which have $\nu\ll 1$.
What about at higher orders in conformal perturbation theory? These correspond to higher order (four points and higher) correlation functions involving $\CO$, $\CO^\dagger$, and $L_4$. In the $N \to \infty$ limit, we may use $L_4= \CO^\dagger\CO$ to find that non-vanishing correlation functions are disconnected and do not require additional counterterms. New counter terms arise from connected correlation functions of $\CO$ and $\CO^{\dagger}$ and are necessarily $1/N$ suppressed, as standard $1/N$ power-counting shows. We conclude that the decoupling of the double-trace operators is robust to all orders in perturbation theory.
A very interesting question, with numerous potential phenomenological and formal applications, is: what are the bounds on the lowest dimension operator that can appear in the OPE of scalar primaries? To date, most work on this question has focused on the constraints of unitarity and crossing symmetry , or on theories with gravity duals . In this paper, we address how such bounds arise in perturbative SCFTs. Using conformal perturbation theory, we found that a bound $\Delta_{X^{\dagger}X} < 2 \Delta_X$ can be seen directly from the beta functions.
The calculations in this paper were performed as a perturbative expansion in $\epsilon = \Delta_X - 1$. In perturbative theories, the $\CO(\epsilon^2)$ contribution to $\Delta_{X^{\dagger} X}$ was computable because it is dominated by mixing with other operators with dimensions near two. These results are sufficient to cover a very broad class of perturbative SCFTs, including Banks-Zaks theories and their generalizations.
Extending our results to all SCFTs would require a more comprehensive treatment of conformal perturbation theory in several regimes:
[1.]{} We did not include all $\cO(\epsilon^2)$ contributions to the beta functions, including an $\CO(|\lambda|^4)$ contribution to the anomalous dimension of $X$. These corrections will be relevant to strongly coupled theories.
[2.]{} When the deformation breaks a global symmetry, the RG flow is more complicated and a complete analysis might be insightful. In the appendix, we simply showed that the contributions to $\Delta_{X^{\dagger}X}$ from the broken currents are small in the cases of interest.
[3.]{} We did not completely elucidate the limit where the dimension of a non-chiral operator approaches two. As these operators become conserved currents in this limit, a description using [*approximately*]{} conserved currents would likely be more useful.
[4.]{} Finally, as with all perturbative arguments, our results do not apply if contributions at a given order in perturbation theory are anomalously small. A more systematic understanding of conformal perturbation theory would be helpful to address these and other questions.
The calculations in this paper could be useful for understanding RG flows in other applications. One advantage of our approach is that we have made all properties of the RG flow manifest. The results in the paper are organized explicitly as a perturbative expansion in $\epsilon$ and are non-perturbative in any other small parameters present in explicit examples. For this reason, our analysis could be applied to strongly coupled SCFTs that are weakly coupled to a free field. For example, many theories contain a regime of parameters where the dimension of a gauge invariant chiral operator violates the unitarity bound and is believed to become a free field . It is plausible that our results could describe these theories as the dimension of this operator approaches one.
**Acknowledgements**
We would like to thank N. Craig, T. Dumitrescu, G. Festuccia, T. Hartman, S. Knapen, Z. Komargodski, D. Poland and N. Seiberg for helpful discussions. The research of D.G is supported by the DOE under grant number DE-FG02-90ER40542 and the Martin A. and Helen Chooljian Membership at the Institute for Advanced Study. The research of D.S. partly supported by a DOE Early Career Research Award.
In this appendix, we will explicitly calculate the contributions to the beta functions quoted and . We will largely follow the approach taken in . However, we do offer an improved understanding of total derivative counterterms, and to our knowledge we perform the first “two-loop" calculation in 4d conformal perturbation theory.
We would like to understand the renormalization group flow of the Langrangian . For convenience we will write the action as $S= S_{\cP_1} + \delta S$ with As in the main text, we have taken $\Delta_{\cO} = 2 - \epsilon$ and $\Delta_{L_i} = 2 + \nu_i$. Here we are being careful to show explicitly the dependence on the superspace coordinates $(z^\mu)^\pm=x^\mu\pm i\theta\sigma^\mu\bar\theta$. In order to achieve finite results, we will impose UV and IR cutoffs $\Lambda$ and $\ell^{-1}$ respectively. We are free to choose a holomorphic renormalization scheme in which $\lambda$ only depends on the cutoff through its classical scaling, $\lambda = \lambda_0 (\Lambda/\Lambda_0)^{-\epsilon}$, where $\Lambda_0$ is some fixed scale. Also, by definition, $y_i$ refers to the part of the $L_i$ coupling which has the classical scaling, $y_i =y_{i0}(\Lambda/\Lambda_0)^{\nu_i}$.
We will determine the beta functions by requiring that the correlation functions are independent of the cutoff scale $\Lambda$. The counterterms $\delta Z_X$ and $\delta y_i$ in are fixed by this requirement. Let us write $\delta Z_X$ and $\delta y_i$ as an expansion in the couplings: By expanding a general correlation function in powers of $\lambda$ and $y_i$ and performing the OPE repeatedly, we can fix the counterterms order by order.
For our calculations, we will need the $\CO\CO^\dagger$, $X X^\dagger$ two-point functions and the $L\CO\CO^\dagger$ three point function in superspace. Superconformal invariance determine these up to overall normalization . We fix the two-point function normalization to one, and then the three-point function normalization is the OPE coefficient: where is a supertranslation invariant interval. From , we can read off some superspace OPEs that we will need for the calculation: and To be precise, the OPE limit we are taking in is $x_2\to x_1$, $\theta_2\to\theta_1$, $\bar\theta_1\to\bar\theta_2$. Here $\dots$ denote superconformal descendants, i.e. derivatives in position and in superspace acting on the superfield primary. These will not matter for any of our beta function calculations in the next subsection.
Finally, one comment about our regularization procedure. As mentioned above, we will impose UV and IR cutoffs of $1/\Lambda$ and $\ell$ respectively. These will be imposed as a hard point-splitting cutoff between any two operators. It is crucial for our calculations that this point-splitting cutoff be implemented in a supersymmetric way. We have found that the easiest way to do this is to work in superspace instead of components, and to be careful to always impose the cutoff on supertranslation invariant intervals $X_{ij}^+$ defined in . Other ways of imposing the cutoff, for instance on position differences $x_{ij}$, appear to break supersymmetry, leading to non-supersymmetric counterterms.
Now we are finally ready to calculate the counterterms. Let us imagine expanding a general correlation function in powers of $y_i$ and $\lambda$.
At $\CO(y)$, we find: The $y_i$ term is cutoff independent, and the other terms must be absent, since they are cutoff-dependent. So we conclude that Note that we are not allowed to use the equations of motion of the free theory to set $X^\dagger X$ to zero, since $X^\dagger X$ appears in the correlation function as an integrated operator, and the equations of motion can fail due to contact terms when operators coincide.
[**2.**]{} At $\CO(|\lambda|^2)$, we find (now and henceforth dropping the $\langle \dots \rangle$): The $a_{1}$ and $b_{1i}$ counterterms are required in order to cancel off the cutoff dependence in the third term when the integrated operators fuse to give $ X^\dagger X$ and $L_i$. To see this, we use the OPE on the third term: Then we are free to change integration variables $z_2^-\to X_{21}^+$ and $z_1^+\to x_1$ with trivial Jacobian, so that becomes Performing the $X_{21}^+$ integrals with a short-distance cutoff $1/\Lambda$, plugging this back into , and requiring that the counterterms in cancel off the result, we finally obtain The appearance of the $\epsilon^{-1}$ in the counter-terms here is no different than the $(4-d)^{-1}$ that appears in counter-terms in dimensional regularization. Although the counter-term will diverge in the $\epsilon \to 0$ limit, the $\epsilon^{-1}$ does not appear in the beta functions and RG flow is well behaved.
[**3.**]{} At $\CO(y\lambda)$, we find (taking into account ): These terms must be cutoff independent, because in our holomorphic scheme, there are no counterterms which can cancel them. Indeed, when $L_i$ approaches $\CO X$ or $\CO^\dagger X^\dagger$, we can use the OPEs in . Substituting these into yields We are free to change integration variables from $x_3\to X_{31}^+$ and $\theta_3\to \theta_{31}$; then is annihilated by the $d^4\theta_{31}$ integral. This change of variables also ensures that all the superconformal descendants in the $L\CO$ OPE do not contribute to renormalization of $\CO X$ in the superpotential.
[**4.**]{} At $ \CO(y|\lambda|^2)$, we find: The first term gives rise to cutoff dependence that must be cancelled by the counterterms. We already fixed $a_{1}$ at a lower order in perturbation theory. Clearly, $a_{1}$ does not play an important role – $L$ and $X^\dagger X$ have a trivial OPE in the unperturbed theory and $a_1$ will cancel the contribution to this operator from the first term. The $b_{1i}$ and $b_{2ij}$ terms are relevant for the renomalization of $y_k L_k$ and other non-chiral operators of $\cP_1$, but these contributions are negligible for our present purposes. What we are interested in is the $X^\dagger X$ counterterm represented by $a_{2i}$ that is required to cancel off the cutoff dependence when $L_i\CO\CO^\dagger\to1$ in the first term of . In more detail, using the OPEs , we have from the first term: Comparing with , we see that the required counterterm is where the integrals should be performed at fixed $z_1^+$, $\theta_1$, and $\bar\theta_2$. Using to evaluate the three-point function, and changing integration variables from $(z_2^-,\,x_3,\,\theta_3,\bar\theta_3)\to (X_{23}^+,X_{31}^+,\theta_{31},\bar\theta_{31})$ (with trivial Jacobian), this becomes where we have defined the dimensionless integral function In general, this is a complicated function of $\nu_i$ and $\epsilon$, because we must take care to regularize the integral so that $|X_{23}^+|>1/\Lambda$, $|X_{31}^+|>1/\Lambda$, and $|X_{23}^++X_{31}^+|>1/\Lambda$. In two special cases, things simplify:
[a.]{} If $\nu_i \ll 1$, then the singularities at $X_{23}^+=0$, $X_{31}^+=0$, and $X_{23}^++X_{31}^+=0$ are separately integrable, so the integral is dominated by the common singularity at $X_{23}^+,\,X_{31}^+\to 0$. We should be able to calculate this dominant part by doing the $X_{23}^+$ integral without regard to the $X_{23}^+=0$ and $X_{23}^+=-X_{31}^+$ singularities, and then doing the $X_{31}^+$ integral with a $1/\Lambda$ cutoff. Doing so, we obtain $\CI=1$ plus higher orders in $\nu_i$ and $\epsilon$.
[b.]{} If $\Delta_{L_i}=2\Delta_\CO$ (i.e. $\nu_i=2-2\epsilon$), as is the case for the double-trace operators discussed in section 3.3, then simply vanishes. Therefore, in this case, the wavefunction renormalization of $X$ is independent of $y_i$ to this order.
With the counterterms in hand, it is straightforward to compute the beta functions for the couplings, keeping in mind we are interested in the [*physical*]{} couplings, Using the counter terms and classical running of the bare couplings, we find: where $\gamma_X \equiv -{1\over2}{d \log Z_X \over d\log\Lambda}$. Substituting , , , and , we obtain for the anomalous dimension of $X$: At this order in perturbation theory, we are free to substitute $\lambda^{phys}$, $y_i^{phys}$ for $\lambda$, $y_i$ in and . This completes our derivation of the beta functions used in the body of the paper.
In this appendix, we will generalize the discussion in Section 2 to situations where $\cO$ transforms in some nontrivial representation ${\bf r}$ of the global symmetry group $G$ of $\cP_1$.
By definition, when $\cO$ is charged, the appropriate conserved current multiplets $J^a$ will appear in the $\cO$-$\cO^{\dagger}$ OPE: Here $m$, $n$ and $p$ are global symmetry indices. $T^a_{m\bar n}$ are the generators of $G$ in the representation ${\bf r}$; their normalization is fixed by the requirement that the $J^a$’s are CFT-canonically normalized. We are free to choose $g_{m\bar n} = \delta_{m\bar n}$ at $\cP_1$. In writing , we have allowed for the possibility that $L_i$ transforms in some representation ${\bf h}$ of $G$. Allowing for charged $\CO$ and $L_i$ complicates the situation considerably.
As before, we will deform the theory by the superpotential $W \supset \frac{1}{2\pi} \lambda \cO X$, where now $\lambda$ transforms in the ${\bf \bar r}$ representation of $G$ (the global symmetry indices on $\lambda$ and $\CO$ are implicit). This deformation breaks the global symmetry, allowing the currents to acquire anomalous dimensions and mix with $J^X \equiv X^\dagger X$ and $L_i$.
In order to arrive at a non-trivial bound, we should forbid the appearance of conserved currents in the OPE of $X$ and $X^{\dagger}$ at $\cP_2$. That is, we will demand that our theory does not contain an unbroken $U(1)$ current formed from a linear combination of $J^X$ and $J^a$. These satisfy Therefore $\lambda T^a$ and $\lambda$ should be linearly independent vectors. Stated more formally, we require that the only real solution to $D^2( \sum_a C_a J^a + C_X J^X ) = 0$ is $C_a = C_X = 0$. Using , this requirement can rewritten (to leading order) as
After deforming the theory by $\lambda \cO X$, the theory will undergo an RG flow to the new fixed point $\cP_2$. As before, we will work in a holomorphic basis where the only renormalization is of operators in the Kahler potential. We should therefore consider the following action At one loop, $\delta Z_a = - 2 \pi^2 \lambda T^a \lambda^\dagger \, {\rm log}(\Lambda) $ .
The beta function for $y_i$ is calculated in the same way as before: The beta function for $\lambda$ is not useful in the holomorphic basis, as the fixed point appears at $\lambda \to \infty$. To study the fixed point, we will move to a non-holomorphic basis by absorbing $Z_a$ and $Z_X$ into the definition of $\lambda$. This can be accomplished using to rewrite The RG flow for $\lambda$ is governed by where $\gamma_a = - \frac{1}{2} \frac{\partial \log Z_a}{\partial \log \Lambda}$. We compute the contributions to $Z_X$ as before to find (at small $\nu_i$) We will not compute all the contributions to $\gamma_a$, but we will simply note that it takes the form: From , we see that $\beta_\lambda = 0$ requires that To leading order, this implies $ \lambda T^a \lambda^\dagger = 0$ for every $a$.
We now wish to compute the scaling operators around the fixed point as before. In general, $\lambda$ is an $r$-dimensional vector of couplings that admits many independent deformations. We are only interested in those “directions" that correspond irrelevant operators, namely the broken currents $J^X$, $J^a$. Other deformations either correspond to exactly marginal operators or total derivatives. In order to focus on just the broken currents, let us specialize to the deformations around the fixed point which correspond to them via : with $\eta_0$, $ \eta_a$ real. In other words, we are trading the vector of couplings $\lambda$ for a subset parametrized by real $(\eta_0,\eta_a)$ which correspond to the broken currents. Note that the unbroken currents automatically drop out of this since they satisfy $\lambda_* T^a =0$.
We would like to know the beta functions for $(\eta_0,\eta_a)$ to linear order. We can read them off from , using the linear independence of $\lambda_*$ and $\lambda_* T^a $ and the fact that $-\epsilon+\gamma_X,\,\gamma_a\sim \CO(\eta)$: Combining this and , we finally obtain the matrix of beta function derivatives: We can diagonalize the matrix $ 2 \pi^2 \lambda \{T^b,\, T^a\} \lambda^\dagger \to \tilde \nu^a \delta^{a b}$, and shift by $2\epsilon$, to write this as In a general model, the mixing of $J_X$ with $J_a$ is potentially important. To get a feeling for this, let us simplify things somewhat by ignoring the distinction between $\lambda$ and $y$ and just taking $\lambda^2\sim y \sim \epsilon$. Also, let us drop the higher orders in , as these are irrelevant for our discussion. Then the characteristic equation for the eigenvalues of is The last term in can be traced back to the $\CO(\lambda^2y)$ and $\CO(y^2)$ corrections to $\gamma_a$ which we have not computed. If $\tilde\nu^a$ is not close to $2\epsilon$, this term is higher order in $\epsilon$ and can be safely ignored, meaning the leading order for $\delta$ is as before. However, if $\tilde\nu^a=2\epsilon + \CO(\epsilon^2)$, then the last term in cannot be ignored for determining the leading order $\delta$.
To summarize so far: we have seen that it is valid to neglect the mixing with the $J_a$ operators, as long as their anomalous dimensions are not extremely degenerate with $2\epsilon$. If $\tilde\nu_a-2\epsilon\sim\CO(\epsilon^2)$, then the mixing with $J_a$ cannot be ignored, and more study (beyond the scope of this work) is required.
We will now conclude our discussion by describing two important classes of models where the mixing can always be neglected, regardless of the $J_a$ anomalous dimensions.
The correlation functions of $L_i$ with $J^a$ are restricted by the representation of $L_i$ under $G$. When all the $L_i$ are singlets of $G$, Ward identities require that $ \langle L_i L_j J^a \rangle = 0$ (or equivalently that $J^a$ is not in the OPE of $L_i$ and $L_j$). As a result, the higher order correction to $\gamma_a$ take the form Furthermore, $y_i$ is a flavor singlet and therefore with the $c_i$’s now just numbers, not matrices. As a result, Here we have used $\lambda T^a \lambda^{\dagger} \sim \epsilon^2$. Up to order $\epsilon^2$ corrections, $\tilde \nu_a$ are eigenvalues and do not mix with $J_X$ and $L_i$. So after diagonalizing the $\tilde\nu^a$, $\nu_i$ block as above, one finds that the contribution from mixing with $\tilde\nu^a$ is subleading in $\lambda$, $y$, even if they are completely degenerate with $2\epsilon$. So the results of Section 2 are unchanged.
In the Banks-Zaks examples we studied in Section 3, the OPE coefficients are controlled by a separate small parameter, $1/N$. More generally, if the theory has a large $N$ parameter which controls the sizes of the $\CO^\dagger\CO$ OPE coefficients, and such that other OPE coefficients are also suppressed by powers of $N$, specifically then the conclusions of section 2 are robust. (This should include all generalized BZ theories and also theories with weakly-coupled gravity duals.) Using and , the large $N$ counting of is The $\CO(1/N^{1+\alpha})$ came from two contributions to $\gamma_a$ – the contribution at $\CO(y \lambda^2)$ involves $\CO\CO^\dagger L_i\to J^a$, which involves two OPE coefficients; and the contribution at $\CO(y^2)$ arises from $L L\to J$, plus the fact that $y\sim \CO(1/N)$ from . So we conclude that the correction from mixing with $\tilde \nu^a$ is suppressed by $1/N^{2+\alpha}$ and vanishes relative to the $\CO(1/N^2)$ contributions computed in Section 2. As a result, the conclusions are unchanged.
In this appendix, we would like to make contact between the analysis of section 2 and our generalized Banks-Zaks example, by rederiving using the general formula .
In order to use the results of section 2, we must determine the operators in the OPE of $\CO \propto Q_{12}Q_{21}$ and $\CO^{\dagger}$ along with their OPE coefficients and dimensions. This example is controlled by the large $N$ expansion and we will only be concerned with results at leading order in $1/N$. As discussed in section 3, the only dimension approximate dimension two operator in the OPE is $\cO(x) \cO^{\dagger} (0)\to L_{12}$ where with OPE coefficient The problem with directly applying is that $L_{12}$ is not a scaling operator in the original undeformed model. So instead of , we should have where $\Gamma$ is the matrix of anomalous dimensions in the $L_{ij}$ basis. The last remaining step is to find the transformation into this basis starting from the matrix of anomalous dimensions obtained from the beta functions . This change of basis is implemented by the Konishi anomaly and $U(1)$ equations of motion. Adding $y_{ij} L_{ij}$ to the Kähler potential is equivalent to rotating $Q_{ij}$ and $Q_{ji}$ by the axial phase where the charge $q_{ij} \propto 1/\sqrt{x_ix_j}$ can be read off of the $\langle Q_{ij}^\dagger Q_{ij} L_{ij}\rangle$ three point function. This in turn induces a change in the superpotential and gauge couplings, via the current non-conservation equation (for a nice discussion of this, see ) The third and fourth terms correspond to the Konishi anomaly terms, and they are present only if the $i$th or $j$th node is gauged, respectively. Finally, from , we read off the shift in the couplings (our conventions for the gauge kinetic term are the standard ones, in which $\CL\supset {1\over 4g^2}\int d^2\theta\, {\rm Tr}\,W_\alpha^2$) So this is a linear map $S$ which takes the couplings $y_{ij}$ to the couplings $(\hat g,\,\hat\lambda)$. If we then compute $\Gamma$ using the beta functions $\betarew$, we obtain it in the $L_{ij}$ basis by acting with $\Gamma\to S^{-1}\Gamma S$. Substituting this into , we find perfect agreement with .
|
---
abstract: 'Automatic identification of animal species by their vocalization is an important and challenging task. Although many kinds of audio monitoring system have been proposed in the literature, they suffer from several disadvantages such as non-trivial feature selection, accuracy degradation because of environmental noise or intensive local computation. In this paper, we propose a deep learning based acoustic classification framework for Wireless Acoustic Sensor Network (WASN). The proposed framework is based on cloud architecture which relaxes the computational burden on the wireless sensor node. To improve the recognition accuracy, we design a multi-view Convolution Neural Network (CNN) to extract the short-, middle-, and long-term dependencies in parallel. The evaluation on two real datasets shows that the proposed architecture can achieve high accuracy and outperforms traditional classification systems significantly when the environmental noise dominate the audio signal (low SNR). Moreover, we implement and deploy the proposed system on a testbed and analyse the system performance in real-world environments. Both simulation and real-world evaluation demonstrate the accuracy and robustness of the proposed acoustic classification system in distinguishing species of animals.'
address:
- 'Department of Computer Science, City University of Hong Kong, Hong Kong'
- 'School of Computer Science and Engineering, University of New South Wales, Australia'
- 'Department of Computer and Information Sciences, Northumbria University, UK'
author:
- Weitao Xu
- Xiang Zhang
- Lina Yao
- Wanli Xue
- Bo Wei
bibliography:
- 'main.bib'
title: 'A Multi-view CNN-based Acoustic Classification System for Automatic Animal Species Identification'
---
Wireless acoustic sensor network ,Animal identification ,Deep learning ,CNN
Introduction {#sec:intro}
============
Wireless Acoustic Sensor Network (WASN) based animal monitoring is of great importance for biologists to monitor real-time wildlife behavior for long periods and under variable weather/climate conditions. The acquired animal voice can provide valuable information for researchers, such as the density and diversity of different species of animals [@hao2013monitoring; @luque2016evaluation; @akyildiz2005underwater]. For example, Hu et al. proposed a WASN application to census the populations of native frogs and the invasive introduced species (Cane Toad) in Australia [@hu2009design]. There are also several important commercial applications of acoustic animal detection. For instance, America imports billions of dollars of timber from Aisa every year. However, the inadvertent introduction of the Asian Longhorn Beetle has cost USA government millions of dollars to eradicate the Beetle population [@nowak2001potential]. Therefore, a wireless monitoring system is imperative to detect the distribution of these insects.
There are a large volume of audio monitoring systems in the literature [@hu2009design; @anderson1996template; @kogan1998automated; @fagerlund2007bird; @guo2003content; @huang2009frog; @acevedo2009automated; @banerjee2014partial; @dutta2013energy; @diaz2012use]. In the early stage, biologists have traditionally deployed audio recording systems over the natural environment where their research projects were developed [@anderson1996template; @kogan1998automated]. However this procedure requires human presence in the area of interest at certain moments. In recent years, with the development of WSN, some researchers have proposed remotely accessible systems in order to minimize the impact of the presence of human beings in the habitat of interest [@hu2009design; @banerjee2014partial; @dutta2013energy; @diaz2012use].
Despite much effort in this area, previous studies suffer from several disadvantages. First, traditional methods usually first extract a number of appropriate features and then employ classic machine learning methods such as Support Vector Machine (SVM) or K-Nearest Neighbours (KNN) to detect the species of the animals. Features, such as statistical features through statistical anlaysis (e.g., variance, mean, median), Fast Fourier Transmission (FFT) spectrum, spectrograms, Wigner-Ville distribution (WVD), Mel-frequecy cepstrum coefficient (MFCC) and wavelets have been broadly used. However, extracting robust features to recognizing noisy field recordings is non-trivial. While these features may work well for one , it is not clear whether they generalize to other species. The specific features for one application do not necessarily generalize to others. Moreover, a significant number of calibrations are required for both manually feature extraction and the classification algorithms. This is because the performance of the traditional classifiers such as SVM and KNN [@fagerlund2007bird; @huang2009frog; @acevedo2009automated] highly depends on the quality of the extracted features. However, handcrafting features relies on a significant amount of domain and engineering knowledge to translate insights into algorithmic methods. Additionally, manual selection of good features is slow and costly in effort. Therefore, these approaches lack scalability for new applications. Deep learning technologies can solve these problems by using deep architectures to learn feature hierarchies. The features that are higher up in deep hierarchies are formed by the composition of features on lower levels. These multi-level representations allow a deep architecture to learn the complex functions that map the input (such as digital audio) to output (e.g. classes), without the need of dependence on manual handcrafted features.
Secondly, these approaches suffer from accuracy degradation in real-world applications because of the impact of environmental noise. The voice recorded from field usually contains much noise which poses a big challenge to real deployment of such system. To address this problem, Wei et al. [@wei2013real] proposed an *in-situ* animal classification system by applying sparse representation-based classification (SRC). SRC uses $\ell_1$-optimization to make animal voice recognition robust to environmental noise. However, it is known that $\ell_1$-optimization is computationally expensive [@shen2014face; @xu2016sensor], which limits the application of their system in resource-limited sensor nodes. Additionally, in order to make SRC achieve high accuracy, a large amount of training data is required. This means a wireless sensor node can only store a limited number of training classes because of the limited storage.
Recently, deep learning has emerged as a powerful tool to solve various recognition tasks such as face recognition [@sun2014deep], human speech recognition [@hinton2012deep; @graves2013speech] and natural language processing [@collobert2008unified]. The application of deep learning in audio signal is not new; however, most previous studies focus on human speech analysis to obtain context information [@hinton2012deep; @graves2013speech; @wang2016attention]. Limited efforts have been devoted to applying deep learning in WASN to classify different species of animals. To bridge this gap, we aim to design and implement a acoustic classification framework for WASN by employing deep learning techniques. Convolutional Neural Network (CNN), as a typical deep learning algorithm, has been widely used in high-level representative feature learning. In detail, CNN is enabled to capture the local spatial coherence from the input data. In our case, the spatial information refers to the spectral amplitude of the audio signal. However, one drawback of the standard CNN structure is that the filter length of the convolution operation is fixed. As a result, the convolutional filter can only discover the spatial features with the fixed filter range. For example, CNN may explore the short-term feature but fail to capture the middle- and long-term features. In this paper, we propose a multi-view CNN framework which contains three convolution operation with three different filter length in parallel in order to extract the short-, middle-, and long-term information at the same time. We conduct extensive experiments to evaluate the system on real datasets. More importantly, we implement the proposed framework on a testbed and conduct a case study to analyse the system performance in real environment. To the best of our knowledge, this is the first work that designs and implements a deep learning based acoustic classification system for WASN.
The main contributions of this paper are threefold:
- We design a deep learning-based acoustic classification framework for WASN, which adopts a multi-view convolution neural network in order to automatically learn the latent and high-level features from short-, middle- and long-term audio signals in parallel.
- We conduct extensive evaluation on two real dataset (Forg dataset and Cricket dataset) to demonstrate the classification accuracy and robustness of the proposed framework to environmental noise. Evaluation results show that the proposed system can achieve high recognition accuracy and outperform traditional methods significantly especially in low SNR scenarios.
- We implement the proposed system on a testbed and conduct a case study to evaluate the performance in real world environments. The case study demonstrate that the proposed framework can achieve high accuracy in real applications.
The rest of this paper is organized as follows. Section \[sec:relatedwork\] introduces related work. Then, we describe system architecture in Section \[sec:system\] and evaluate the system performance in Section \[sec:evaluation\]. We implement the system on a testbed and conduct user study to evaluate the system in Section \[sec:testbed\]. Finally, Section \[sec:conclusion\] concludes the paper.
{width="6.5in"}
Related Work {#sec:relatedwork}
============
Animal voice classification has been extensively studied in the literature. At the highest level, most work extract sets of features from the data, and use these features as inputs for standard classification algorithms such as SVM, KNN, decision tree, or Bayesian classifier. Previous studies have involved a wide range of species which include farm animals [@jahns1997sound], bats [@preatoni2005identifying], birds [@fagerlund2007bird; @acevedo2009automated; @hodon2015monitoring], pests [@eliopoulos2016estimation], insects [@ntalampiras2019automatic] and anurans [@vaca2010using]. The works of Anderson et al. [@anderson1996template] and Kogan et al. [@kogan1998automated] were among the first attempts to recognize bird species automatically by their sounds. They applied dynamic time warping and hidden Markov models for automatic song recognition of Zebra Finche and Indigo Punting. In [@luque2016evaluation], the authors focus on classifying two anuran species: Alytes obstetricans and Epidalea calamita using generic descriptors based on an MPEG-7 standard. Their evaluation demonstrate that MPEG-7 descriptors are suitable to be used in the recognition of different patterns, allowing a high scalability. In [@hao2013monitoring], the authors propose to classify animal sounds in the visual space, by treating the texture of animal sonograms as an acoustic fingerprint. Their method can obviate the complex feature selection process. They also show that by searching for the most representative acoustic fingerprint, they can significantly outperform other techniques in terms of speed and accuracy.
The WSN has been massively applied in sensing the environment and transferring collected samples to the server. However, it is challenging to realize in-network classification system because of the limited computational ability of wireless sensor node. Recently, several research works regarding in-network classification have been proposed. Sun et al. [@sun2008dynamic] dynamically select the feature space in order to accelerate the classification process. A hybrid sensor networks is designed by Hu et al. [@hu2009design] for in-network and energy-efficient classification in order to monitor amphibian population. Wei et al. [@wei2013real] proposed a sparse representation classification method for acoustic classification on WSN. A dictionary reduction method was designed in order to improve the classification efficiency. The sparse representation classification method was also used by face recognition on resource-constrained smart phones to improve the classification performance [@shen2014face; @xu2016sensor].
Deep learning has achieved great success over the past several years for the excellent ability on high-level feature learning and representative information discovering. Specifically, deep learning has been widely used in a number of areas, such as computer version [@wen2016discriminative], activity recognition [@chen2018interpretable; @luo2019brush], sensory signal classification [@zhang2018multi; @lan2019entrans; @xu2019energy], and brain computer interface [@zhang2018mindid]. Wen et al. [@wen2016discriminative] propose a new supervision signal, called center loss, for face recognition task. The proposed center loss function is demonstrated to enhance the discriminative power of the deeply learned features. Chen et al. [@chen2018interpretable] propose an interpretable parallel recurrent neural network with convolutional attentions to improve the activity recognition performance based on Inertial Measurement Unit signals. Zhang et al. [@zhang2018multi] combine deep learning and reinforcement learning to deal with multi-modal sensory data (e.g., RFID, acceleration) and extract the latent information for better classification. Recently, deep learning involves in the brain signal mining in brain computer interface (BCI). Zhang et al. [@zhang2018mindid] propose an attention-based Encoder-Decoder RNNs (Recurrent Neural Networks) structure in order to improve the robustness and adaptability of the brainwave based identification system.
There are also several works that apply deep learning techniques in embedded devices. Lane et al. [@lane2015can] propose low-power Deep Neural Network (DNN) model for mobile sensing. CPU and DSP in one mobile device are exploited for activity recognition. Lane et al. [@lane2015can] also design a DNN model for audio sensing in mobile phone by using dataset from 168 places for the training purpose. A framework DeepX is further proposed for software accelerating on mobile devices [@lane2016deepx].
In terms of animal voice classification, Zhang et al. [@zhang2018automatic], Oikarinen et al. [@oikarinen2018deep] study animal voice classification using deep learning techniques. Our method is different from these two works. Our studies focus on voice classification in noisy environment while the voice data in [@zhang2018automatic] are collected from controlled room without environmental noise. Instead of classifying different animals, [@oikarinen2018deep] analyses different call types of marmoset monkeys such as Trill, Twitter, Phee and Chatter. Moreover, we implement the proposed system on a testbed and evaluate its performance in real world environment. In another work [@ntalampiras2018bird], Stavros Ntalampiras used transfer learning to improve the accuracy of bird classification by exploiting music genres. Their result show that the transfer learning scheme can improve classification accuracy by $11.2\%$. Although the goal of this work and our study is to improve the recognition accuracy with deep learning technology, the methodologies are different. Our approach analyses the inherent features of audio signal and propose a multi-view CNN model to improve the accuracy. Instead of looking at the bird audio signals alone, Stavros Ntalampiras proposed to statistically analyse audio signals using their similarities with music genres. Their method, however, is only effective for a limited number of bird species because they need to perform feature transformation again when a new bird species comes in. In comparison, our approach is applicable for a large number of bird species. A number of studies also apply deep learning technologies in bird voice classification [@potamitis2016deep; @koops2014deep; @goeau2016lifeclef], however, they only use conventional deep learning approaches such as CNN and do not make any novel improvement. In this paper, we propose a multi-view CNN model and evaluation results show that the proposed model outperforms the conventional CNN.
System Design {#sec:system}
=============
System Overview
---------------
As shown in Figure \[fig:systemoverview\], our proposed framework consists of two parts: WASN and server. In the WASN, the wireless nodes will detect and record animal voices and then perform local processing which include silence removal, segmentation and FFT. We process signal *in-situ* before uploading because of the high sampling frequency of audio signal and energy inefficiency of wireless communication [@barr2006energy; @wei2013real]. The spectrum signal obtained from FFT can save half spaces since FFT is symmetric. On the server side, the spectrum signal will be fed into a deep neural network to obtain the species of the animal. The classification results can be used by biologists to analyze the density, distribution and behavior of animals.
Wireless sensors are usually resource-poor relative to server, and not able to run computationally expensive algorithms such as deep learning models. Therefore, we assume all the wireless sensors can connect to a server via wireless communication technologies, such as ZigBee, Wi-Fi, and LoRa [@xu2019measurement]. However, there may be network failure, server failure, power failure, or other disruption makes offload impossible. While such failures will hopefully be rare, they cannot be ignored in a cloud-based system. In this case, the node can transmit the data to the gateway or a nearby server which are usually resource-rich and capable of running deep learning models. Alternatively, the classification can be performed in the node to recognize only a few species, pre-defined by the user. When offloading becomes possible again, the system can revert to recognizing its full range of species.
In the following parts, we will describe the design details of each component.
Local Processing
----------------
**Silence Removal.** The collected audio signal usually contains a large amount of silent signal when this is no animal present. Therefore, we apply a simple silence removal method on the raw signal to delete not-of-interest area. The procedure is explained in Algorithm \[al:silence\]. We first calculate the root mean square (RMS) of each window which contains 1s samples and then compare it with a pre-defined threshold learned from the environment. The windows of samples whose RMS above the threshold will be kept. The threshold is determined by exhaustive search. To be specific, we increase the threshold from 0 to 0.5 with an increment of 0.01, then choose the one that can achieve the best performance (0.03 in this paper).
, where $N$ is the total number of segments and $\rho$ is the threshold Remove $(S_i)$
Figure \[fig:silenceremoval\] shows an example of silence removal on an animal voice recording. We can see that it can effectively remove the silent periods and detect the present of animals.
![Silence removal.[]{data-label="fig:silenceremoval"}](silenceremoval.pdf){width="3in"}
**Segmentation and FFT.** After silence removal, we obtain audio signals containing animal vocalization only. The audio signal is segmented into consecutive sliding windows with $50\%$ overlap. Hamming window is used in this paper to avoid spectral leakage. Each window contains $2^{14}$ samples which is chosen to balance the trade-off between classification accuracy and latency as discussed in Section \[sec:evaluation\]. The overlap in sliding window is used to capture changes or transitions around the window limits. Then we perform FFT on each segment to calculate spectrum energy (i.e. the magnitude of the FFT coefficients). As an example, Figure \[fig:fft\] shows the sound in time and frequency domain of two different frog species: Cultripes and Litoria Caerulea. It is conspicuous that they have different spectrum distributions. The graphs are plotted by audio signal analysis software Audacity.
Multi-view Convolutional Neural Networks
----------------------------------------
We propose a deep learning framework in order to automatically learn the latent and high-level features from the processed audio signals for better classification performance. Among deep learning algorithms, CNN is widely used to discover the latent spatial information in applications such as image recognition [@ciresan2011convolutional], ubiquitous [@ning2018deepmag], and object searching [@ren2017faster], due to their salient features such as regularized structure, good spatial locality and translation invariance. CNN applies a convolution operation to the input, passing the result to the next layer. Specifically, CNN captures the distinctive dependencies among the patterns associated to different audio categories. However, one drawback of the standard CNN structure is that the filter length of the convolution operation is fixed. As a result, the convolutional filter can only discover the spatial features with the fixed filter range. For example, CNN may explore the short-term feature but fail to capture the middle- and long-term features.
{width="5in"}
To address the mentioned challenge, we propose a multi-view CNN framework which applies three different filter length to extract the short-, middle-, and long-term features in parallel. As shown in Figure \[fig:CNN\_workflow\], the proposed framework regards the processed audio signals as input and feed into three views at the same time. Each view contains three convolutional layers. $Conv_h^k$ denotes the $k$-th convolutional operation in the $h$-th view. The convolutional layer contains a set of filters to convolve the audio data followed by the nonlinear transformation to extract the geographical features. The filter length keeps invariant in the same view while varies in different views. The extracted features from the multi-view pipe are stacked together and then through the max pooling operation for dimension reduction. Afterward, a fully-connected layer, a softmax layer and the output layer work as a classifier to predict the audio label. The proposed multi-view CNN has several key differences from the inception module [@szegedy2015going] although the ideas are similar. First, [@szegedy2015going] has a $1 \times 1$ convolutional filter in the module in order to prevent the information corruption brought by inter-channel convolutions. The proposed multi-view CNN does not has this component. This is because in our case the input data are naturally formed as a vector which represents the spectral information of the acoustic signals. Moreover, [@szegedy2015going] adds an alternative parallel pooling path in the middle layer to acquire additional beneficial effect. However, we believe this may cause information loss and only perform the pooling operation after the concentration of the results of various views.
Suppose the input audio data $\bm{E}$ has shape $[M, L]$ with depth as $1$. The chosen three convolutional filters with size in short-, middle-, and long-term views are $[M, 10]$, $[M, 15]$, $[M, 20]$, respectively. The stride sizes keep $[1, 1]$ for all the convolutional layers. The stride denotes the x-movements and y-movements distance of the filters. Since the audio signals are arranged as 1-dimension data, we set $M = 1$. Same shape zero padding is used, which keeps the sample shape constant during the the convolution calculation. In the convolutional operation, the feature maps from the input layer are convolved with the learnable filters and fed to the activation function to generate the output feature map. For a specific convolutional area (also called perceptive area) $\bm{x}$ which has the same shape as the filter, the convolutional operation can be described as $$\bm{x}' = tanh(\sum_{i}\sum_{j}\bm{f}_{ij}*\bm{x}_{ij})$$ where $\bm{x}'$ denotes the filtered results while $\bm{f}_{ij}$ denotes the $i$-th row and the $j$-th column element in the trainable filter. We adopt the widely used *tanh* activation function for nonlinearity. The depth of input sample transfers to $D$ through the convolutional layer and the sample shape is changed to $[M, L, D]$. In particular, the corresponding depth $D_h= 2, 4, 8$ for three convolutional layers. The features learned from the filters are concatenated and flattened to $[1, M*L*\sum_{h=1}^{3}D_h]$. The max pooling has $[1, 3]$ as both pooling length and strides. Therefore, the features with shape $[1, M*L*\sum_{h=1}^{3}D_h/3]$ after the pooling operation, which are forwarded to the fully-connected layer. The operation between the fully-connected layer and the output layer can be represented by $$\bm{y} = softmax(\bar{\bm{w}}\bm{E^{FC}} + \bar{\bm{b}})$$ where $FC$ denotes the fully-connected layer while the $\bar{\bm{w}}$ and $\bar{\bm{b}}$ denote the corresponding weights matrix and biases. The softmax function is used for activation. For each sample, the corresponding label information is presented by one-hot label $\bm{y} \in \mathbb{R}^H$ where $H$ denotes the category number of acoustic signals. The error between the predicted results and the ground truth is evaluated by cross-entropy $$loss = - \sum_{h=1}^{H}\bm{y}_h log(p_h)$$ where $p_h$ denotes the predicted probability of observation of an object belonging to category $h$. The calculated error is optimized by the AdamOptimizer algorithm [@kingma2014adam]. To minimize the possibility of overfitting, we adopt the dropout strategy and set the drop rate to 80%.
Evaluation {#sec:evaluation}
==========
Goals, Metrics, and Methodology {#subsec:goals}
-------------------------------
In this section, we evaluate the performance of the proposed system based on two real datasets. The goals of the evaluation are twofold: 1) evaluate the performance of the proposed system under different settings; 2) compare the proposed system with previous animal vocalization system.
![Confusion matrix of frog dataset.[]{data-label="fig:confusionmatrix_frog"}](confusionmatrix.png){width="3.2in"}
We use two datasets collected from real-world for evaluation. The first dataset contains audio signals recorded from fourteen different species of frogs. The sampling frequency for this dataset is 24Khz. More details about this dataset can be found in [@wei2013real]. The second dataset [^1] contains audio signals recorded from different species of crickets. The data consists of twenty species of crickets, eight of which are Gryllidae and twelve of which are Tettigoniidae. The sampling frequency is also 24Khz. More details about this dataset can be found in [@hao2013monitoring]. For completeness, Table. \[tab:species\] lists all the species we used in the experiments.
Our method CNN SRC SVM-MFCC SVM-Spectrum KNN-MFCC KNN-Spectrum
----------- ------------ ------- ------- ---------- -------------- ---------- --------------
Accuracy **94.7%** 82.7% 53.4% 24.4% 40.5% 20.1% 26.4%
Precision **93.1%** 81.6% 54.2% 25.9% 43.5% 19.9% 25.1%
Recall **94.3%** 82.4% 53.7% 24.7% 41.2% 21.5% 27.1%
F1-score **92.9%** 81.2 52.1% 25.1% 39.6% 20.7% 25.7%
In this paper, we use SVM and KNN to benchmark ASN classification because they have been widely used in WASN classification systems [@fagerlund2007bird; @huang2009frog; @acevedo2009automated]. We evaluate the performance of SVM and KNN by using frequency domain and Mel-frequency cepstral coefficients (MFCCs), respectively. The parameters in SVM and KNN are well tuned to give highest accuracy. In addition, we compare the accuracy of our system with a recent work which is based on SRC [@wei2013real] and conventional CNN. In total, we compare our method with six classifiers: CNN, SRC, SVM-MFCC, SVM-spectrum, KNN-MFCC and KNN-spectrum. For each classifier, we perform 10-fold cross-validation on the collected dataset. In the original dataset, the data only contain little environment noise. Therefore, to demonstrate the robustness of the proposed framework, we add different scales of environmental noise to create different SNRs. This is used to simulate the real environment because the recorded animal voices are usually deteriorated by environmental noise in real WASN. In the evaluation, it is done by adding different scales of random Gaussian noise to the original audio data.
In this paper, we focus on the following four metrics: *accuracy*, *precision*, *recall* and *F1-score*. We plot the results of the average values and stand deviation obtained from 10 folds cross-validation.
Performance of Frog Dataset
---------------------------
### Impact of parameters
We first evaluate the impact of important parameters in our system. On the node’s side, the important parameters include window size of segment. On the server’s side, the important parameters include the number of iterations in training, the dropout rate and learning rate in CNN, the size of training dataset. Dropout is a technique where randomly selected neurons are ignored during training. For example, the dropout rate of $80\%$ means that we randomly select $20\%$ of the neurons and drop them (force the values of them as 0). The dropout strategy is widely used to enhance the generalization of a machine learning model and prevent overfitting. The learning rate is a hyper-parameter that controls how much we are adjusting the weights of the neuron network with respect to the loss gradient.
To evaluate the impact of window size, we vary the window size from $2^{11}$ to $2^{15}$ samples and calculate the accuracy of our scheme. From the results in Figure \[fig:impactofwindow\_Frog\], we can see that there is a performance gain when we increase the window size and the improvement reduces after $2^{14}$ samples. Although we can achieve higher accuracy with more samples, the resource consumption of FFT operation which runs on the wireless sensor node also increases. Therefore, we choose to use $2^{14}$ window size to balance the trade-off between accuracy and resource consumption.
Figure \[fig:impactofiteration\_Frog\] shows the accuracy along with different training iterations. We can see that the proposed method converges to its highest accuracy in less than 200 iterations. The results show that the proposed framework can finish training quickly. Figure \[fig:impactofdropout\_Frog\] plots the accuracy of various dropout rates. We can observe that the accuracy fluctuates first and then becomes stable after the dropout rate is greater than 0.8. Therefore, we set the default dropout rate to be 0.8. Moreover, we can infer from Figure \[fig:impactofdropout\_Frog\] that our model is not very sensitive to the dropout rate. This is because the Frog dataset matches well with the proposed multi-view CNN, as a result, the convergence suffers less from overfitting which can be demonstrated by the good convergence property as shown in Figure \[fig:impactofiteration\_Frog\]. Figure \[fig:impactoflearningrate\_Frog\] shows the accuracy under different learning rates. We can see that it achieves the highest accuracy when the learning rate is $0.5\times 10^{-3}$ and $1 \times 10^{-3}$. Correspondingly, we choose 0.001 to reduce the training time because the smaller the learning rate is, the slower the training process is. From Figure \[fig:impactoflearningrate\_Frog\], we can observe that the performance varies dramatically with the increasing of learning rate. One possible reason for this is that the gradient surface of our loss function is not smooth and very sensitive to the learning rate. The optimiser is easy to step over the local optima while the learning rate is larger than a threshold.
Next, we evaluate the accuracy of the proposed system under different sizes of training dataset. In this experiment, we use different proportions of the whole dataset for training, and use the left dataset for testing. The proportion increases from $10\%$ to $90\%$ with an increment of $10\%$. For example, the proportion of $10\%$ means we use $10\%$ of the dataset for training, and use the left dataset for testing. For comparison purpose, we also calculate the accuracy of CNN, SRC, SVM and KNN. From the results in Figure \[fig:impactoftrainingsize\_Frog\], we can see that our method continuously achieves the highest accuracy, and the accuracy becomes relatively stable after $60\%$ of the dataset is used for training. We also notice that the improvement of our method from $10\%$ to $90\%$ is remarkable. More specifically, when the proportion of the training dataset increases from $10\%$ to $90\%$, the accuracy improvement of our method is $40.1\%$ while the improvement of CNN, SRC, SVM and KNN are $34.7\%$, $29.3\%$, $26.7\%$ and $22.4\%$, respectively. In this experiment, we do not test SVM-MFCC and KNN-MFCC because their accuracy is poor as will be shown later.
Our method CNN SRC SVM-MFCC SVM-Spectrum KNN-MFCC KNN-Spectrum
----------- ------------ ------- ------- ---------- -------------- ---------- --------------
Accuracy **86.4%** 76.6% 42.4% 22.1% 36.8% 19.4% 28.5%
Precision **86.9%** 76.2% 42.5% 23.6% 36.3% 18.2% 28.6%
Recall **85.1%** 75.3% 41.2% 22.7% 37.3% 19.9% 29.8%
F1-score **86.1%** 74.6% 41.7% 21.8% 38.1% 19.5% 29.5%
### Comparison With Other Methods
We now compare the performance of proposed scheme with previous approaches. As mentioned above, we compare the accuracy of the proposed system with conventional CNN, SRC, SVM-MFCC, SVM-spectrum, KNN-MFCC and KNN-spectrum. The MFCC of each window is calculated by transforming the power spectrum of each window into the logarithmic mel-frequency spectrum. We calculate the accuracy of different methods under different SNRs by adding different scales of environmental noise.
As we can see from Figure \[fig:comparisonwithothers\_Frog\], SVM-MFCC and KNN-MFCC performs the worst which suggests that different frog species are not distinguishable in MFCC feature space. The results also explains why MFCC-based methods usually requires other carefully selected features [@vaca2010using]. We find that when the animal voice is overwhelmed by environmental noise (low SNR), the accuracy of our system is significantly higher than the other methods. For example, when $SNR=-6dB$, the accuracy of our method is $12\%$ higher than CNN, $41\%$ higher than SRC, $70\%$ higher than SVM-MFCC, $53.9\%$ higher than SVM-spectrum, $74.3\%$ higher than KNN-MFCC, and $68\%$ higher than KNN-spectrum. The robustness to noise makes the proposed system suitable for real deployment in noisy environments. Moreover, the results also indicate that our system needs less sensors to cover a certain area because our system can classify low SNR signals which are usually collected from longer distance.
To take a closer look at the result, we summarize the results of different methods in Table \[tab:comparison\_frog\] and plot confusion matrix in Figure \[fig:confusionmatrix\_frog\] when SNR is -6dB. We can see that each class can achieve high accuracy and the overall average accuracy is $94.7\%$.
Performance of Cricket Dataset
------------------------------
Similar to Frog dataset, we also evaluate the impact of window size, the number of iterations, the dropout rate and learning rate in CNN, and the size of training dataset using Cricket dataset. The procedures are the same as above and the results are shown in Figure \[fig:cricketdataset\]. We can see that it shows similar patterns as Frog dataset which suggests that the proposed framework is robust to different species. In terms of the dropout rate and learning rate, the optimal values for dropout rate and learning rate are 0.7 and 0.0005 which is slightly different from that of Frog dataset.
As mentioned in [@hao2013monitoring], the cricket dataset consists of twenty species of insects, eight of which are Gryllidae and twelve of which are Tettigoniidae. Thus, we can treat the problem as either a two-class genus level problem, or twenty-class species level problem. We first treat the classification as a two-class level problem and calcualte the accuracy of different methods under different SNRs. From the results in Figure \[fig:comparison\_cricket2\], we can see that our method, SRC, SVM-spectrum and KNN-spectrum can achieve high accuracy. However, our method still outperforms all the other classifiers. Thereafter, we treat the classification as a twenty-class species level problem and plot the accuracy of different methods in Figure \[fig:comparison\_cricket2\]. We can see that the proposed method significantly outperforms the other methods when SNR is low. Table \[tab:comparison\_cricket\] summarizes the results of each method in detail. The results above demonstrate the advantage of our method in classifying more species in noisy environment.
Case Study on Testbed {#sec:testbed}
=====================
To validate the feasibility of the proposed framework in real environment, we implement the system on an outdoor ASN testbed which is located in Brisbane, Australia. As shown in Figure \[fig:local\], the testbed is composed of five nodes which are configured as *Ad-hoc* mode with a star network topology. Its task is to evaluate the system’s capability of recognizing bird vocalization in real world environment.
**Module** **Consumption (W)**
------------------- ---------------------
CPU 2.05
CPU + microphone 2.1
CPU + Wifi (idle) 2.45
CPU + Wifi (Rx) 2.67
CPU + Wifi (Tx) 2.78
: Power Consumption.[]{data-label="tab:powerconsumption"}
\[tab:energy\]
The hardware platform used in the testbed is based on a Pandaboard ES with an 1.2Ghz OMAP 4460, 1GB Ram and 4GB SD-card. Additionally, Pandaboard includes an 802.11 interface for wireless connection. Microphones are connected to the Pandaboard via USB port to record bird voice with 24Khz sampling rate. All the nodes are connected via the local Wi-Fi network. The data collected from Node 2, 3, 4 and 5 will be first transferred to Node 1. Then, all the data will be uploaded from Node 1 to the local server. The acoustic date from different nodes are classified separately in the system.
In the testbed, each node is powered by a rechargeable battery (12V, 7.2Ah), and an optional solar panel (5W, 12V). The power consumption of each module is given in Table \[tab:powerconsumption\]. Compared to SolarStore testbed [@solarstore] which consumes 10W (low load) and 15W (high load) energy, our testbed is approximately 3.5 to 5.4 times more energy efficient. Without solar panel, a node in our ASN testbed will run continuously for more than 31 hours, which is significantly longer than the previous platforms such as ENSBox [@Girod2006:AENSBox]. We find that if a solar panel is exposed to direct sunlight for 8 hours per day, the node can maintain a 50% duty cycle at 85% solar charge efficiency.
The nodes use Network Time Protocol (NTP) for time synchronization. We use one node as the NTP server, and the other nodes as the NTP clients. The NTP clients send request for time synchronization every 10 seconds. The accuracy of time synchronization is about 25 ms, which is good enough for our distributed real-time system because the length of each testing signal segment is 400ms.
During deployment, we found that the recorded voice is deteriorated by wind. To solve this problem, we take two measures. First, we install foam and fur windscreen around each microphone. Second, we apply a Butterworth high pass filter with 200Hz cut-off frequency to filter out unwanted noise. This is because most of the wind audio energy lies in the frequency band below 200Hz, while most of the vocalization energy of the birds is in the frequency band higher than 200Hz.
After implementing the proposed framework on the testbed, we calculate the computation time on the node’s side and classification accuracy on the server’s side. On the node’s side, we find that the node in our testbed can process all the captured acoustic data in real time. From Table \[tab:performancetestbed\], we can see the silence removal and FFT take 20.38 ms and 15.33 ms, respectively.
Silence Removal FFT
----------- ------------------ ------------------
Time (ms) 20.38 $\pm$ 2.04 15.33 $\pm$ 0.63
: Computation time of local processing.[]{data-label="tab:performancetestbed"}
Our system CNN SRC SVM-Spectrum KNN-Spectrum
----------- ------------ ------- ------- -------------- --------------
Accuracy **90.3%** 84.4% 72.3% 65.7% 68.8%
Precision **91.2%** 82.1% 72.6% 66.4% 69.2%
Recall **89.4%** 84.6% 70.9% 65.6% 67.1%
F1-score **91.1%** 83.7% 71.8% 66.4% 70.5%
In this study, we choose two common bird species in the area of interest: Anseriformes and Galliformes (Figure \[fig:bird\]). Our goal is to classify the voice into three classes: Anseriformes, Galliformes and others. The testbed runs for 30 days and the data is labeled manually. Table \[tab:performancetestbed\] lists the results of different methods for classification in the server. We find that the proposed system achieve $90.3\%$ classification accuracy which outperforms other methods significantly. The results in turn suggest that the proposed framework is robust to environmental noise and can achieve high classification accuracy in real-world WASN. We also notice that the results of the case study is slightly lower than the simulation results in Section \[sec:evaluation\]. This is because the public dataset are collected in a controlled manner and the signals are well trimmed and processed. However, the data we used in our case study are collected in a totally automatic manner.
Conclusion {#sec:conclusion}
==========
In this paper, we design and implement a CNN-based acoustic classification system for WASN. To improve the accuracy in noisy environment, we propose a multi-view CNN framework which contains three convolution operation with three different filter length in parallel in order to extract the short-, middle-, and long-term information at the same time. Extensive evaluations on two real datasets show that the proposed system significantly outperforms previous methods. To demonstrate the performance of the proposed system in real world environment, we conduct a case study by implementing our system in a public testbed. The results show that our system works well and can achieve high accuracy in real deployments. In our future work, we will deploy the proposed framework in wider area and evaluate its performance in different environments.
Acknowledgement {#acknowledgement .unnumbered}
===============
The work described in this paper was fully supported by a grant from City University of Hong Kong (Project No.7200642)
[^1]: http://alumni.cs.ucr.edu/ yhao/animalsoundfingerprint.html
|
---
abstract: 'Magnetar flares excite strong [Alfvén ]{}waves in the magnetosphere of the neutron star. The wave energy can (1) dissipate in the magnetosphere, (2) convert to “fast modes” and possibly escape, and (3) penetrate the neutron star crust and dissipate there. We examine and compare the three options. Particularly challenging are nonlinear interactions between strong waves, which develop a cascade to small dissipative scales. This process can be studied in the framework of force-free electrodynamics (FFE). We perform three-dimensional FFE simulations to investigate [Alfvén ]{}wave dissipation, how long it takes, and how it depends on the initial wave amplitude on the driving scale. In the simulations, we launch two large [Alfvén ]{}wave packets that keep bouncing on closed magnetic field lines and collide repeatedly until the full turbulence spectrum develops. Besides dissipation due to the turbulent cascade, we find that in some simulations spurious energy losses occur immediately in the first collisions. This effect occurs in special cases where the FFE description breaks. It is explained with a simple one-dimensional model, which we examine in both FFE and full magnetohydrodynamic settings. We find that magnetospheric dissipation through nonlinear wave interactions is relatively slow and more energy is drained into the neutron star. The wave energy deposited into the star is promptly dissipated through plastic crustal flows induced at the bottom of the liquid ocean, and a fraction of the generated heat is radiated from the stellar surface.'
author:
- Xinyu Li
- Jonathan Zrake
- 'Andrei M. Beloborodov'
bibliography:
- 'ms.bib'
title: Dissipation of Alfvén Waves in Relativistic Magnetospheres of Magnetars
---
= 1
Introduction
============
Magnetars are neutron stars hosting ultra-strong magnetic fields of $10^{14}-10^{16}$ G (see for a recent review). They exhibit a broad range of X-ray activity, including giant flares with luminosities up to $10^{47}$ erg/s. The flares are likely powered by a sudden magnetospheric rearrangement that dissipates magnetic energy [@1995MNRAS.275..255T; @1996ApJ...473..322T]. A slower mode of dissipation is invoked to explain persistent hard X-ray emission [@2013ApJ...762...13B]. Magnetic energy dissipation in the magnetosphere generally plays a key role in magnetar activity.
One proposed dissipation mechanism is the turbulent cascade of magnetospheric [Alfvén ]{}waves excited by a starquake [@1996ApJ...473..322T]. [Alfvén ]{}waves are also excited when the magnetosphere is slowly “overtwisted” and loses equilibrium, as observed in simulations by [@2013ApJ...774...92P]. The excited waves can have large amplitudes and carry a significant fraction of the magnetospheric energy.
@1996ApJ...473..322T proposed that the waves will cascade to small dissipative scales and convert to heat, creating an energetic “fireball” of thermalized $e^\pm$ plasma. However, there are two competing processes that can remove the wave energy. First, [Alfvén ]{}waves can convert to so-called “fast modes” capable of escaping the magnetosphere. Unlike [Alfvén ]{}waves, which are ducted along the magnetic field lines and trapped in the closed magnetosphere, fast modes can propagate across the field lines. Secondly, @2015ApJ...815...25L showed that [Alfvén ]{}waves bouncing in the magnetosphere are gradually drained into the stellar crust, where they initiate plastic flows and dissipate. About 10 bouncing cycles are typically sufficient to damp the waves by this mechanism, and @2015ApJ...815...25L suggested that this may occur faster than dissipation of waves through a turbulent cascade in the magnetosphere. Evaluating the efficiency of the latter process requires a detailed calculation of nonlinear processes, which can be done numerically. We attempt this calculation in the present paper.
The theory of turbulent cascades has a long history. The MHD cascade is different from the hydrodynamic cascade where energy transfer is mediated by interacting vortices. The difference is seen already in the simplest, incompressible, non-relativistic MHD, where only [Alfvén ]{}waves are present. In the case of weak turbulence (meaning that the time for energy transfer across spatial scales is longer than the wave period), the three-wave interaction is prohibited by kinetic constraints [@1994ApJ...432..612S]. Then nonlinear interactions are dominated by the four-wave interactions among [Alfvén ]{}waves and give rise to the anisotropic energy cascade in the direction perpendicular to the background field. For strong turbulence, a $k_\perp^{-5/3}$ spectrum was predicted from detailed balance [@1995ApJ...438..763G].
More work was done later to include compressive modes – the fast and slow magnetosonic waves. Using the random phase approximation @2001JETP...93.1052K found a weak-turbulence spectrum $k_{\perp}^{-2}$. Relativistic nonlinear Alfvénic turbulence was studied using numerical simulations [@2005ApJ...621..324C; @2012ApJ...744...32Z; @2016ApJ...817...89Z; @2016ApJ...831L..11T; @2017MNRAS.472.4542T], but far less than in the non-relativistic setting. Simulations by @2016ApJ...831L..11T [@2017MNRAS.472.4542T] suggested that compressible modes are strongly coupled with [Alfvén ]{}waves and participate in the energy cascade.
Similar to the previous works, we are interested in low-frequency waves, which are described by relativistic magneto-hydrodynamics (RMHD). In the magnetically-dominated limit (negligible plasma inertia), a simpler approximation of “force-free electrodynamics” (FFE) becomes useful. In this approximation, the plasma energy and momentum are neglected and the stress-energy tensor of the electromagnetic field $T^{\mu\nu}$ satisfies $\nabla_\mu T^{\mu\nu}=0$. Both RMHD and FFE support [Alfvén ]{}waves, which transport energy along the direction of the background field, and also support the fast modes.[^1]
An analytical study of wave interaction in FFE by @1998PhRvD..57.3219T found that an [Alfvén ]{}wave pair can convert to a fast wave via three-wave interactions. In contrast to [Alfvén ]{}waves, the group velocity of the fast modes can be in any direction and they can possibly escape the magnetosphere, carrying energy away.
In this paper we use FFE and RMHD simulations to investigate the efficiency of nonlinear processes in removing wave energy through dissipation and escape. Our goals are to determine the fate of wave energy in the context of giant magnetar flares, and to study the nonlinear dynamics of interacting [Alfvén ]{}waves from a physics perspective. We present numerical simulations of relativistic [Alfvén ]{}wave turbulence operating in a toy magnetosphere replaced by a rectangular box. We utilize a high-order conservative finite differencing scheme to evolve the FFE equations, and devote particular attention to the code’s modeling of energy dissipation. We discuss the various modes by which energy is removed numerically, and point out (via direct comparison with a relativistic MHD code) circumstances when FFE wrongly models the energy dissipation rate.
Our paper is organized as follows. In Section 2, we give a brief description of wave modes and nonlinear interactions in FFE. In Section 3, we outline our numerical scheme and discuss energy dissipation channels it admits. 2D and 3D numerical results for [Alfvén ]{}wave turbulence driven by colliding wave packets are presented in Section 4. In Section 5, we show that FFE simulations can badly over-predict the energy dissipation rate, as the result of a commonly employed technique for maintaining magnetic dominance ($E < B$). The final section is devoted to a discussion of our results in the context of the fireball model for magnetar giant flares. Throughout this paper we utilize units in which speeds are measured in units of the speed of light $c$, and electric ($\boldsymbol{E}$) and magnetic ($\boldsymbol{B}$) field values are normalized by $\sqrt{4 \pi}$.
Waves and Nonlinear Interactions in FFE
=======================================
Equations of FFE
----------------
FFE describes relativistic magnetically dominated plasma, where the plasma inertia can be neglected, i.e. $\rho \ll B^2/2$ where $\rho$ is the mass density of plasma. The dynamical equations are given by Maxwell’s equations, $$\begin{aligned}
\label{Maxwell}
\frac{\partial\boldsymbol{B}}{\partial t}+\nabla\times \boldsymbol{E},
\qquad
\frac{\partial\boldsymbol{E}}{\partial t}-\nabla\times \boldsymbol{B} &=& -\boldsymbol{J} \, ,\end{aligned}$$ together with the vanishing force condition $\nabla_\mu T^{\mu\nu}=0$ or $$\label{ffe_condition}
\rho_e \boldsymbol{E}+\boldsymbol{J}\times \boldsymbol{B} = 0 \, ,$$ where $\rho_e=\nabla\cdot{\boldsymbol{E}}$ is the charge density. The force-free condition, Equation \[ffe\_condition\], requires ${\boldsymbol{E}}\cdot {\boldsymbol{B}}= 0$ and $E < B$. Equation \[ffe\_condition\] and $\partial_t({\boldsymbol{E}}\cdot {\boldsymbol{B}}) = 0$ together yield the the following expression for the electric current density (e.g. @2002MNRAS.336..759K), $$\label{current}
{\boldsymbol{J}}=\boldsymbol{J}_{\rm FFE} \equiv \rho_e \frac{\boldsymbol{E}\times\boldsymbol{B}}{B^2}+\frac{\boldsymbol{B}\cdot\nabla\times\boldsymbol{B}-\boldsymbol{E}\cdot\nabla\times\boldsymbol{E}}{B^2}\boldsymbol{B} \, .$$ ${\boldsymbol{J}}_{\rm FFE}$ introduces nonlinearity into the Maxwell equations.
Since FFE neglects the plasma energy, the total energy of the system is given by $$U_{\rm tot} = \int{\mathrm{d}}V\;\frac{1}{2}(B^2+E^2) \, .$$ This energy is formally conserved because Equation \[ffe\_condition\] guarantees ${\boldsymbol{E}}\cdot {\boldsymbol{J}}= 0$.
Wave solutions in FFE {#waves}
---------------------
We will use the temporal gauge where the electric scalar potential $\varphi$ is set to zero, and the vector potential ${\boldsymbol{A}}$ fully specifies the electromagnetic field, $$\begin{aligned}
\label{potential}
\boldsymbol{B} = \nabla\times {\boldsymbol{A}}, \qquad \boldsymbol{E} = - \frac{\partial {\boldsymbol{A}}}{\partial t} \, .\end{aligned}$$ The Maxwell equations then reduce to $$\label{reducedMax}
\frac{\partial^2{\boldsymbol{A}}}{\partial t^2} + \nabla\times \nabla\times{\boldsymbol{A}}= {\boldsymbol{J}}\, .$$ We approximate the steady background magnetic field ${\boldsymbol{B}}^{(0)}$ as uniform (i.e. limit our consideration to waves much shorter than the variation scale of the background field), and choose the $z$-axis along ${\boldsymbol{B}}^{(0)}$ and the $y$-axis along ${\boldsymbol{A}}^{(0)}$, $${\boldsymbol{A}}^{(0)}=B_{0}x\,\hat{\boldsymbol{y}}, \qquad
{\boldsymbol{B}}^{(0)} = B_{0}\, \hat{\boldsymbol{z}}, \qquad \boldsymbol{E}^{(0)}=0.$$ $A^{(0)}$ has no time dependence, and so there is no background electric field.
Approximate solutions for waves and their interactions may be obtained by use of a perturbative expansion, $$\begin{aligned}
\label{pert}
{\boldsymbol{A}}= {\boldsymbol{A}}^{(0)}
+\epsilon {\boldsymbol{A}}^{(1)}+\epsilon^2{\boldsymbol{A}}^{(2)}+\cdots,\end{aligned}$$ where $\epsilon\ll 1$. We seek solutions for the perturbed quantities of the form $$\label{Fourier}
{\boldsymbol{A}}^{(n)}(t,\boldsymbol{r}) \propto \exp[i(\boldsymbol{k}^{(n)} \cdot \boldsymbol{r} - \omega^{(n)} t)]\, ,
\qquad n\geq 1,$$ where $\boldsymbol{r}=(x,y,z)$ is the position vector.
Inserting Equation (\[potential\]) into the expression for ${\boldsymbol{J}}$ (Equation \[current\]), substituting the result into Equation \[reducedMax\], and keeping only terms up to the first order in $\epsilon$ yields the linear equation for ${\boldsymbol{A}}^{(1)}$ of the form $$\label{Maxwell1}
\mathcal{L}[{\boldsymbol{A}}^{(1)}]=0,$$ where $$\mathcal{L}\equiv \frac{\partial^2}{\partial t^2} + (\nabla\times \nabla\times)_\perp$$ is a linear differential operator. The operator becomes algebraical when it is applied to the Fourier modes (Equation \[Fourier\]), and the wave equation becomes $L{\boldsymbol{A}}^{(1)}=0$, where $L(\omega,\boldsymbol{k})$ is a matrix. The condition $\det L=0$ for the existence of solutions ${\boldsymbol{A}}^{(1)}\neq 0$ gives two pairs of roots $\omega(\boldsymbol{k})$, which describe the dispersion relations of the propagating eigen modes. The corresponding eigen vectors $\boldsymbol{e}_m$ represent the wave polarization, and each eigen mode may be written in the form $${\boldsymbol{A}}^{(1)}_m=\Lambda_m\,\boldsymbol{e}_m,$$ where $\Lambda_m$ represents the wave amplitude.
For any Fourier mode the induction equation $\partial {\boldsymbol{B}}/\partial t=-\nabla\times{\boldsymbol{E}}$ implies $\omega{\boldsymbol{B}}=\boldsymbol{k}\times{\boldsymbol{E}}$, and hence the condition ${\boldsymbol{E}}\cdot{\boldsymbol{B}}=0$ is automatically satisfied. In our setting, the first-order expansion of ${\boldsymbol{E}}\cdot{\boldsymbol{B}}={\boldsymbol{E}}^{(0)}\cdot{\boldsymbol{B}}^{(1)}+{\boldsymbol{E}}^{(1)}\cdot{\boldsymbol{B}}^{(0)}\propto {\boldsymbol{A}}^{(1)}\cdot{\boldsymbol{B}}^{(0)}$ implies $$\label{eq:Az}
A^{(1)}_z=0,$$ i.e. the polarization vectors $\boldsymbol{e}_m$ must be perpendicular to the background magnetic field.
A straightforward calculation shows that two distinct modes are supported by FFE:
1. [Alfvén ]{}wave — This mode has the dispersion relation $\omega(\boldsymbol{k})=\pm k_z$ and the polarization vector $$\label{polA}
\boldsymbol{e}_\mathcal{A} = \frac{\boldsymbol{k}_\perp}{\sqrt{\omega}|\boldsymbol{k}_\perp|}\,,$$ where $\boldsymbol{k}_\perp$ is the component of wave vector perpendicular to the background field ${\boldsymbol{B}}^{(0)}$. The electric field in the wave ${\boldsymbol{E}}^{(1)}=-i\omega{\boldsymbol{A}}^{(1)}$ is along $\boldsymbol{k}_\perp$, and the magnetic field ${\boldsymbol{B}}^{(1)}=i\boldsymbol{k}\times{\boldsymbol{A}}^{(1)}$ is along $\hat{\boldsymbol{z}}\times\boldsymbol{k}_\perp$. [Alfvén ]{}waves have group velocity along $\pm \hat{\boldsymbol{z}}$, and therefore can only transport energy parallel (or anti-parallel) to the background field. The sign in the dispersion relation indicates the direction of the wave. The current associated with [Alfvén ]{}waves is ${\boldsymbol{J}}_{\mathcal{A}} \propto
k_\perp \sqrt{\omega} \hat{\boldsymbol{z}}$, which is non-zero for $k_\perp \ne 0$.
2. Fast wave — The dispersion relation is $\omega(\boldsymbol{k})=\pm |\boldsymbol{k}|$ with the polarization vector $$\label{polF}
\boldsymbol{e}_\mathcal{F} = \frac{\boldsymbol{k}_\perp\times\hat{\boldsymbol{z}}}{\sqrt{\omega}|\boldsymbol{k}_\perp|} \, .$$ Then ${\boldsymbol{E}}^{(1)}$ is along $\boldsymbol{k}_\perp\times\hat{\boldsymbol{z}}$, and ${\boldsymbol{B}}^{(1)}=(\boldsymbol{k}\times{\boldsymbol{E}}^{(1)})/\omega$ is in the $\boldsymbol{k}_\perp$-$\hat{\boldsymbol{z}}$ plane and perpendicular to $\boldsymbol{k}$. Fast waves in FFE create no charge density $\rho_e=\nabla\cdot {\boldsymbol{E}}^{(1)}=i\boldsymbol{k}\cdot {\boldsymbol{E}}^{(1)}=0$, and also no current density, ${\boldsymbol{J}}_{\mathcal{F}}=0$. Therefore, the fast waves propagate as vacuum electromagnetic waves.
When $k_\perp=0$, the two wave modes become degenerate. Notably, while these wave solutions have been derived from the linearized equations, they are in fact exact nonlinear solutions to the FFE equations.
The polarization vectors $\boldsymbol{e}_{\mathcal{A},\mathcal{F}}$ in Equations \[polA\] and \[polF\] are normalized so that the energy of an ensemble of fast and [Alfvén ]{}waves takes the form $$U =\sum\limits_{m=\mathcal{A},\mathcal{F}} \sum\limits_{\boldsymbol{k}}\omega_m
\Lambda_m^\star(\boldsymbol{k})
\Lambda_m(\boldsymbol{k}).$$
Wave-wave interactions
----------------------
Nonlinear interactions between waves arise from the current density ${\boldsymbol{J}}$. The lowest order interaction involves three waves, where two waves generate a third. These three-wave interactions are identified by inserting the expansion for two modes, ${\boldsymbol{A}}^{(1)} = {\boldsymbol{A}}^{(1)}_1 + {\boldsymbol{A}}^{(1)}_2$, into Maxwell’s equations, and equating the second order terms, $$\begin{aligned}
\label{Maxwell2}
\mathcal{L}[{\boldsymbol{A}}^{(2)}] = {\boldsymbol{J}}^{(2)}_{\rm nl} \, ,\end{aligned}$$ The second order term ${\boldsymbol{J}}^{(2)}_{\rm nl}$ in the-force free current is cumbersome and presented in Appendix. It is instructive to consider the following variants of the incoming waves ${\boldsymbol{A}}^{(1)}_1+{\boldsymbol{A}}^{(1)}_2$.
For two incoming fast modes one finds that $\boldsymbol{J}_{\rm nl}^{(2)}\neq 0$ is possible (in contrast to their $\boldsymbol{J}^{(1)}=0$). However in this case, $\boldsymbol{J}_{\rm nl}^{(2)}$ is parallel to the guide field ${\boldsymbol{B}}^{(0)}$, and sources ${\boldsymbol{A}}^{(2)}$ along $\hat {\boldsymbol{z}}$. There are no propagating modes with $A_z\neq 0$ (see Equation (\[eq:Az\])), and so the three-wave interaction with two incoming fast modes is suppressed.
For two incoming [Alfvén ]{}waves propagating in the same direction along the guide field ($k^{(1)}_{1,z}$ has the same sign as $k^{(1)}_{2,z}$), one finds that ${\boldsymbol{J}}^{(2)}_{\rm nl}$ vanishes. Therefore, only counter-propagating [Alfvén ]{}waves can generate new waves through 3-wave interaction. The generated wave has wavevector $\boldsymbol{k}^{(2)} = \boldsymbol{k}^{(1)}_1+ \boldsymbol{k}^{(1)}_2$ and frequency $\omega^{(2)} = \omega^{(1)}_1+\omega^{(1)}_2$. The excitation of the second-order wave is enhanced for the resonant three-wave interaction, meaning that ${\boldsymbol{A}}^{(2)}$ is also a linear eigen mode. One can show that $\boldsymbol{k}^{(2)}$ and $\omega^{(2)}$ may satisfy the dispersion relation of [Alfvén ]{}waves only if one of the incoming waves has $k_z = 0$, and such modes do not propagate, as they have $\omega=0$ according to the dispersion relation $\omega=\pm k_z$. Therefore, two counter-propagating [Alfvén ]{}waves can only participate in resonant interactions where the outgoing wave is a fast mode ($\mathcal{A}+\mathcal{A}\rightarrow \mathcal{F}$).
Resonant three-wave interactions are also possible between an incoming [Alfvén ]{}wave and an incoming fast wave, and the outgoing wave can be either a fast wave or an [Alfvén ]{}wave ($\mathcal{A}+\mathcal{F}\rightarrow \mathcal{A/F}$).
Numerical setup
===============
Computational setting
---------------------
We perform numerical simulations in a fully periodic domain with uniform guide magnetic field aligned with the $z$-axis, $B_z=1$. The box extends from 0 to 1 along each axis, and the wave crossing time is also unity. We utilize initial conditions comprising a pair of counter-propagating [Alfvén ]{}wave packets. Due to the use of periodic boundary conditions, the wave packets collide repeatedly, twice each time they traverse the computational domain; the time interval between successive collisions is $\tau = 0.5$. This setting simulates [Alfvén ]{}wave packets propagating on closed field lines anchored on the magnetar surface, neglecting geometric effects due to the field-line curvature. The periodic boundary condition corresponds to an idealized situation where [Alfvén ]{}waves are perfectly reflected when hitting the magnetar surface.
Solution scheme {#scheme}
---------------
We numerically evolve the FFE equations using a third-order in time, fifth-order in space, flux-conservative scheme based on the WENO method [@doi:10.1137/070679065] adapted to FFE [@2011MNRAS.411.2461Y]. We define a vector of primitive variables, $${\boldsymbol{P}}= \left(B_x,B_y,B_z, E_x,E_y,E_z\right)^\top \, ,$$ and rewrite the Maxwell equations in the flux-conservative form $$\label{maxwell-fv}
\partial_t {\boldsymbol{P}}+ \partial_x{\boldsymbol{F}}^x + \partial_y{\boldsymbol{F}}^y + \partial_z{\boldsymbol{F}}^z = \boldsymbol{T} \, ,$$ where the source term $\boldsymbol{T}$, and the flux functions are given by $$\begin{aligned}
\boldsymbol{T} &=& \left(0,0,0, -J_x, -J_y, -J_z\right)^\top \nonumber\\
{\boldsymbol{F}}^x &=& \left(0,-E_z,E_y,0,B_z,-B_y\right)^\top \nonumber\\
{\boldsymbol{F}}^y &=& \left(E_z,0,-E_x,-B_z,0,B_x\right)^\top \nonumber\\
{\boldsymbol{F}}^z &=& \left(-E_y,E_x,0, B_y,-B_x,0 \right)^\top \, .\end{aligned}$$ The components of electric current appearing in $\boldsymbol{T}$ are computed using standard forth order finite differencing on the volume-centered values of ${\boldsymbol{E}}$ and ${\boldsymbol{B}}$, according to Equation \[current\].
Time stepping is accomplished using the third-order TVD Runge-Kutta (RK) method [@Gottlieb:1998:TVD:279724.279737]. Each RK sub-step updates the primitive variables by adding the source term and face-centered fluxes in a finite-volume form of Equation \[maxwell-fv\] $${\boldsymbol{P}}^{n + 1} = {\boldsymbol{P}}^n + \boldsymbol{T}^n \Delta t - \frac{\Delta t}{\Delta V} \sum_{\rm{faces}} \Delta S_i \hat{\boldsymbol{F}}^i\, .$$ Here $\hat{\boldsymbol{F}}^i$ are the face-centered fluxes, evaluated from ${\boldsymbol{P}}^{n}$, and Roe’s Riemann solver, $$\begin{aligned}
\hat{{\boldsymbol{F}}}^i_{j+1/2} &=& \frac{1}{2}\Big[{\boldsymbol{F}}^i({\boldsymbol{P}}^+_{j+1/2}) + {\boldsymbol{F}}^i({\boldsymbol{P}}^-_{j+1/2}) \nonumber\\
&& -\sum\limits_{m=1}^6 \left|\lambda^i_{m, j+1/2}\right| \alpha^i_{m, j+1/2}\boldsymbol{v}^i_{m, j+1/2} \Big]\end{aligned}$$ where $\lambda^i_{m, j+1/2}$ and $\boldsymbol{v}^i_{m, j+1/2}$ are eigenvalues and eigenvectors of the Jacobian matrix $\partial {\boldsymbol{F}}^i / \partial {\boldsymbol{P}}$, and $$\alpha^i_{m, j+1/2,j,k} = \left({\boldsymbol{P}}^+_{j+1/2}-{\boldsymbol{P}}^-_{j+1/2}\right) \cdot \boldsymbol{v}^i_{m, j+1/2}$$ is the projection of the difference between left and right states to the eigenvectors. The left and right states ${\boldsymbol{P}}^{\pm}$ are reconstructed from $\boldsymbol{P}^n$ using the fifth order WENO method.
Constraint preservation {#constraint}
-----------------------
In order to keep the magnetic field divergence-free, we utilize the hyperbolic divergence cleaning approach outlined in @2002JCoPh.175..645D. In practice, we find this approach maintains $\nabla \cdot {\boldsymbol{B}}= 0$ to high precision. Note that violations in $\nabla \cdot {\boldsymbol{B}}= 0$ only arise from the truncation error of the numerical scheme, and so the modifications to the solution introduced by the hyperbolic cleaning step converge away with increasing numerical resolution. Hyperbolic divergence cleaning involves the addition of an auxiliary scalar field $\Psi$ and a corresponding evolution equation. For brevity, we have excluded this equation from the description of our numerical scheme in Section \[scheme\]. For details we refer the reader to [@2002JCoPh.175..645D].
Small violations in the ${\boldsymbol{E}}\cdot{\boldsymbol{B}}= 0$ constraint also arise at the level of truncation error. Instead of removing the parallel component of ${\boldsymbol{E}}$ at each time step, we introduce a correction term to the force-free Ohm’s law that allows for time-resolved damping of ${\boldsymbol{E}}_\parallel$ [@2017MNRAS.469.3656P], $$\begin{aligned}
\label{modified-ffe-ohm}
{\boldsymbol{J}}_m &=& \rho_e \frac{{\boldsymbol{E}}\times{\boldsymbol{B}}}{B^2 +\tilde{E}^2} \nonumber\\
&& + \frac{{\boldsymbol{B}}\cdot\nabla\times {\boldsymbol{B}}- {\boldsymbol{E}}\cdot\nabla\times{\boldsymbol{E}}+ \gamma {\boldsymbol{E}}\cdot{\boldsymbol{B}}}{B^2}{\boldsymbol{B}}\, .\end{aligned}$$ Here, $1 / \gamma$ is a time scale for the damping of ${\boldsymbol{E}}_\parallel$ (typically chosen to be several times $\Delta t$), and the modified electric field magnitude $\tilde E^2$ appearing in the denominator of the first term in Equation \[modified-ffe-ohm\] is defined as [@2012ApJ...746...60L] $$\tilde{E}^2 = \frac{1}{2} \left(\sqrt{\chi^4 + 4 {\boldsymbol{E}}\cdot {\boldsymbol{B}}} + \chi^2\right) - \chi^2 \, ,$$ where $\chi^2 \equiv B^2 - E^2$. When ${\boldsymbol{E}}\cdot {\boldsymbol{B}}= 0$, the modified current density Equation \[modified-ffe-ohm\] reduces to Equation \[current\].
Maintaining magnetic dominance {#magnetic-dominance}
------------------------------
Self-consistent evolution of the FFE equations requires magnetic dominance ($E < B$) to be maintained. However, non-linear FFE solutions in which $E$ remains everywhere smaller than $B$ generally exist only for finite time. This reflects that realistic plasma systems, having small but finite rest-mass energy, inevitably develop regions where the thermal pressure gradient or the MHD inertial term becomes important. Such conditions arise either where $B^2 / 2$ drops below $\rho$ (e.g. near magnetic null points), or where plasma is accelerated to high Lorentz factor. In such regions the electric current deviates significantly from ${\boldsymbol{J}}_{\rm FFE}$, enabling momentum transfer between the plasma and the electromagnetic field.
A standard procedure to continue numerical evolution of the FFE equations is to artificially reduce the magnitude of $E$ wherever $E > B$, $$\label{shrinkE}
{\boldsymbol{E}}\rightarrow\sqrt{\frac{B^2}{E^2}} {\boldsymbol{E}}\, .$$ This procedure is commonly interpreted as modeling a dissipative process [@2006MNRAS.367.1797M; @2006ApJ...648L..51S], such as the rapid acceleration of charged particles enabled by $E > B$. However, violation of the force-free condition in real plasma systems does not necessarily lead to energy dissipation. We will demonstrate this explicitly in Section \[sec:breakffe\] by comparing our FFE solutions with those of strongly magnetized relativistic MHD. The MHD solutions reveal that breaking of the force-free condition leads to time-reversible momentum exchange between the field and the plasma. We will thus conclude that electromagnetic dissipation is not properly modeled by FFE when significant energy is lost as a consequence of the procedure in Equation \[shrinkE\].
Dissipation channels in FFE simulations {#dissipation-channels}
---------------------------------------
Although FFE formally conserves energy, numerical evolution schemes require some dissipation in order to maintain stability, satisfy constraints, and keep the solution magnetically dominated. There are four channels for energy dissipation in our numerical simulations:
(i) The hyperbolic divergence cleaning step, which leads to $$\partial_t U = -{\boldsymbol{B}}\cdot \nabla \Psi \, ,$$ where $\Psi$ is the auxiliary scalar function discussed in Section \[constraint\].
(ii) Dissipation introduced by the modified force-free current ${\boldsymbol{J}}_m$ in Equation \[modified-ffe-ohm\]. $$\begin{aligned}
\partial_t U &=& - {\boldsymbol{J}}_m \cdot{\boldsymbol{E}}\nonumber \\
&=& -({\boldsymbol{E}}\cdot{\boldsymbol{B}})\frac{{\boldsymbol{B}}\cdot\nabla\times {\boldsymbol{B}}-{\boldsymbol{E}}\cdot\nabla\times{\boldsymbol{E}}}{B^2} \nonumber \\
&&-\frac{\gamma ({\boldsymbol{E}}\cdot{\boldsymbol{B}})^2}{B^2}.
\end{aligned}$$
(iii) Reduction of electrical field when $E>B$ (Equation \[shrinkE\]).
(iv) Subtraction of short-wavelength field oscillations at the grid scale, referred to as “grid heating.”
Channels (i) and (ii) become less significant with increasing grid resolution, because the numerical values of $\Psi$ and ${\boldsymbol{E}}\cdot{\boldsymbol{B}}$ are proportional to the truncation error of the numerical scheme. In the results presented in Sections \[sec:spec\] and \[sec:fate\], these channels do not contribute significantly to the measured energy dissipation rate.
Channel (iii) does not in general become small as the grid resolution increases. As mentioned in Section \[magnetic-dominance\], this dissipation may be artificially strong. Therefore, we consider our measurements of the energy dissipation rate to be reliable only when dissipation is not dominated by this effect.
Channel (iv), energy removal by grid heating, may or may not “converge away” with increasing resolution. For example, the FFE wave solutions discussed in Section \[waves\] evolve without any significant grid heating, provided their wavelength is well resolved. Generally, the rate of grid heating of isolated waves will depend on the numerical resolution. In contrast, non-linear numerical solutions can exhibit significant energy loss over time, at a rate that becomes independent of grid resolution. Such behavior usually reflects the presence of a forward energy cascade, in which the rate of high frequency wave damping is determined by the rate of energy transfer into high frequency modes by numerically resolved non-linear interactions. In such cases, grid heating is expected to capture the true dissipation rate.
Numerical diagnostics
---------------------
A useful diagnostic in our analysis will be the free energy $U$, which we define to be the total electromagnetic energy of the system, but with the contribution from the background magnetic field $B_z$ removed, $$U \equiv U_{\rm tot} - \frac{1}{2} \int {\mathrm{d}}V\; B_z^2 \, .$$ We will also utilize the power spectra $P(k_\parallel)$ and $P(k_\perp)$, representing the distribution of electromagnetic energy in wavenumber parallel and perpendicular to the background field. The power spectra are obtained by binning the square of the discrete Fourier modes $\tilde{{\boldsymbol{B}}}(\boldsymbol{k})$ and $\tilde{{\boldsymbol{E}}}(\boldsymbol{k})$ by the wavenumber components $k_\parallel$ and $k_\perp$, $$\label{eqn:power-spectrum}
P(k_{\parallel,\perp}) \Delta k_{\parallel,\perp}\equiv \frac{1}{2 U_0} \sum\limits_{\boldsymbol k \in \Delta k_{\parallel, \perp}} |\tilde{{\boldsymbol{B}}}(\boldsymbol{k})|^2 + |\tilde{{\boldsymbol{E}}}(\boldsymbol{k})|^2 \, .$$ Note that the spectra are normalized to the free energy $U_0$ in the system at $t=0$, such that $$U = U_0 \sum
P(k_\parallel)\Delta k_\parallel =
U_0 \sum
P(k_\perp) \Delta k_\perp \, .$$ In 2D and 3D runs, the spectral bins $\Delta k_\parallel$ are planar slabs orthogonal to the background field. The spectral bins $\Delta k_\perp$ are planar slabs in 2D and cylindrical annuli in 3D.
Spectral evolution of wave turbulence {#sec:spec}
=====================================
3D simulations {#sec:3d}
--------------
In order to study the interaction between [Alfvén ]{}modes in a three-dimensional setting, we initialize two counter-propagating wave packets, each perturbing the background field within a spherical volume. We utilize initial conditions ${\boldsymbol{B}}= B_0\hat{\boldsymbol{z}}+ \nabla \times(\phi \hat{\boldsymbol{z}})$ with scalar field $$\label{3dpacket}
\phi = \xi\ell\sum\limits_{i=1,2} \exp\left(- \frac{|\boldsymbol{r}-\boldsymbol{r}_i|^2}{\ell^2} \right),$$ where $\boldsymbol{r}_1 = (0.5,0.5,0.25)$, $\boldsymbol{r}_2 = (0.5,0.5,0.75)$ are the wave packet center positions and $\ell = 0.1$ is the width of packets. The electric field is set by ${\boldsymbol{E}}= \pm \hat{\boldsymbol{z}} \times {\boldsymbol{B}}$ with opposite sign for each wave packet. The amplitude $\xi$ characterizes the size of largest perturbation imposed on the background field, $$\begin{aligned}
\xi \simeq \max \left|\frac{{\boldsymbol{B}}- B_0\hat{\boldsymbol{z}}}{B_0}\right|.\end{aligned}$$
![Free energy evolution for different resolutions in 3D simulations of colliding [Alfvén ]{}wave packets with amplitude $\xi=0.5$.[]{data-label="energy3D"}](convergence3D){width="50.00000%"}
We observe that in 3D simulations, the collision of counter-propagating [Alfvén ]{}waves results in a forward energy cascade, in which energy is dissipated primarily by grid-heating (Channel (iv) in Section \[dissipation-channels\]). This interpretation is supported by (1) consistency of the overall energy dissipation rate with increasing grid resolution, (2) observation of a definite time $t_{\rm onset}$ at which energy dissipation commences, and (3) formation of a Kolmogorov-type energy spectrum.
Figure \[energy3D\] shows the time series of electromagnetic free energy $U(t)$ for the same model, $\xi = 0.5$, at different grid resolutions. All of the simulations exhibit an initial phase with slow dissipation lasting $t_{\rm onset}\sim 24 \tau$, a fast dissipation phase between $\sim 24\tau$ and $\sim 40\tau$, and a subsequent gradual relaxation phase. The difference between the initial slow and fast dissipation phases becomes more pronounced as the grid resolution increases; the rate of energy dissipation prior to $t_{\rm onset}$ diminishes with increasing numerical resolution. Meanwhile, the energy lost by the system at late times $>40\tau$ is independent of the grid resolution to within roughly $5\%$.
![Development of turbulent spectrum (snapshots at $t/\tau=0$, 10, 30, 80) in the simulation with the initial packet amplitude $\xi=0.5$ and grid resolution $512^3$. Upper panel: spectrum in $k_\parallel$ (parallel to the background field). Lower panel: spectrum in $k_\perp$(perpendicular to the background field). The dashed line indicates the slope $P(k_\perp)\propto k_\perp^{-2}$. []{data-label="spec3Dev"}](spec3Dev_parallel "fig:"){width="50.00000%"} ![Development of turbulent spectrum (snapshots at $t/\tau=0$, 10, 30, 80) in the simulation with the initial packet amplitude $\xi=0.5$ and grid resolution $512^3$. Upper panel: spectrum in $k_\parallel$ (parallel to the background field). Lower panel: spectrum in $k_\perp$(perpendicular to the background field). The dashed line indicates the slope $P(k_\perp)\propto k_\perp^{-2}$. []{data-label="spec3Dev"}](spec3Dev "fig:"){width="50.00000%"}
Figure \[spec3Dev\] shows the power spectrum evolution for a simulation with resolution $512^3$. Over time, the system develops waves at progressively increasing wavenumber, indicating a forward energy cascade. The spectrum extends to a maximum wavenumber $k_{\rm max}(t)$, which is seen to increase between $t=0$ and $t_{\rm onset}$. At $t_{\rm onset}$, $k_{\rm max}$ reaches the nominal dissipation wavenumber $k_{\rm diss} \sim N / 10$, where $N$ is the number of grid points in each $(x,y,z)$ direction. Figure \[spec3Dev\] also reveals that the spectrum is significantly anisotropic, with $P(k_\perp) > P(k_\parallel)$ at all but the largest scales, indicating that energy cascades primarily in the direction perpendicular to the background field. As energy moves from large to small (perpendicular) scales, the energy around large $k_\perp$ increases monotonically up until $t_{\rm onset}$, at which time modes around $k_{\rm diss}$ become significantly populated. Subsequently, some energy is reflected back toward low wave numbers, causing the power at large scales to grow between $t_{\rm onset}$ and $80\tau$. The perpendicular spectrum eventually relaxes to a power-law consistent with $k_\perp^{-2}$ at $\sim 80\tau$. Such spectral slope is consistent with the so-called weak MHD wave turbulence spectrum, as reported by [@2001JETP...93.1052K].
![ Turbulence spectrum in $k_\perp$ at $t=80\tau$ in four simulations with different initial packet amplitudes $\xi$. []{data-label="spec3Dst"}](spec3Dst){width="50.00000%"}
The perpendicular power spectrum exhibits oscillations in $k_\perp$ at times earlier than $80\tau$. This is due to a known feature of wave turbulence [@2011LNP...825.....N], that energy is transferred mainly through resonant interactions; only a discrete set of secondary modes are excited by the primary modes. The resonant secondary modes then couple with the primaries and further drive the same secondaries. This leads to disproportionate energy transfer between particular sectors of the $k$-space, enhancing the energy concentration around preferred wavenumbers. Figure \[spec3Dst\] shows the perpendicular power spectrum after 80 collisions for various amplitudes $\xi$ of the initial wave packets in the range $0.5 - 3$. The slopes of the perpendicular spectra are all close to $k_\perp^{-2}$. We observe that the spectral oscillations are weaker for larger values of $\xi$. This is because the strength of nonlinear interactions increases with $\xi$ and energy is distributed across a larger number of modes within a given time.
The existence of a universal time $t_{\rm onset}$ at which dissipation commences is consistent with a forward energy cascade, and a spectral energy distribution having finite *energy capacity*, meaning that $$\lim \limits_{k_{\rm max} \rightarrow \infty} \int \limits_{k_0}^{k_{\rm max}} P(k) dk < \infty \, .$$ This is the case for any power-law spectra $P(k)$ steeper than $k^{-1}$. Such spectra have the property that the energy stored at wavenumbers higher than $k$ asymptotes toward zero with increasing $k$. As energy cascades toward smaller scales, $k_{\rm max}$ must either increase without bound (thus exciting modes at the dissipation scale, however small), or some of the energy must be reflected toward larger scales. We do see evidence in Figure \[spec3Dev\] for such energy reflection, as the power around $k_\perp \sim 4$ first drops, but then rises again at $t \sim t_{\rm onset}$. However, the energy distribution subsequently equilibrates to a Kolomogorov spectrum, with energy transferring continuously into the dissipation range, and leading to the divergence of $k_{\rm max}$. The rapid increase of $k_{\rm max}$ implies that $t_{\rm onset}$ becomes insensitive to $k_{\rm diss}$, and thus to the grid resolution. Therefore the energy spectrum $P(k_\perp) \propto k_\perp^{-2}$ seen in 3D simulations is compatible with the time series in Figure \[energy3D\] which suggests a universal value of $t_{\rm onset}$. In the next section we will show that 2D settings exemplify the opposite behavior, where the spectrum is very shallow, having infinite energy capacity. Those 2D systems will not display numerical consistency of the dissipation onset time.
2D simulations {#d-simulations}
--------------
In our 2D simulations, the field is independent of the $y$ coordinate. We utilize initial conditions ${\boldsymbol{B}}= B_0\hat{\boldsymbol{z}}+ B_y\hat{\boldsymbol{y}}$ in the $x-z$ plane, where two [Alfvén ]{}wave packets have Gaussian profiles, $$\label{eqn:2d-wave-packets}
B_y = \xi \sum\limits_{i=1,2} \exp\left(- \frac{|\boldsymbol{r}-\boldsymbol{r}_i|^2}{\ell^2} \right) \, .$$ Here, $\boldsymbol{r}_1 = (0.5,0.25)$ are $\boldsymbol{r}_2 = (0.5,0.75)$ are the center positions of the two wave packets in the $x-z$ plane. The width of the wave packet is the same ($\ell=0.1$) as in our 3D simulations. The electric field is again ${\boldsymbol{E}}= \pm \hat{\boldsymbol{z}}\times {\boldsymbol{B}}$, with opposite sign for each wave packet. The wave packets travel toward one another along the guide field (in the $\pm \hat{\boldsymbol{z}}$ directions), and have a cylindrical envelope in which the magnetic field is perturbed along the cylinder axis $\hat {\boldsymbol{y}}$.
![ Free energy evolution for different resolutions in the 2D model with packet amplitude $\xi=0.4$.[]{data-label="energy2D"}](convergence2D){width="50.00000%"}
In 2D simulations, we observe that collisions between counter-propagating wave packets also result in a forward energy cascade. However, unlike in the 3D case, 2D systems do not exhibit consistency of the overall dissipation rate for different grid resolutions. Figure \[energy2D\] shows the time series of electromagnetic free energy for amplitude $\xi = 0.4$, with different numerical resolution. The amount of energy $\Delta U$ dissipated before a given time $t_0$ is a decreasing function of the grid resolution, showing no trend toward a universal value. Moreover, the onset of dissipation occurs at later times; there is no asymptotic $t_{\rm onset}$ that was observed in Section \[sec:3d\] for the 3D case.
![Spectrum evolution for the 2D simulation with $2048^2$ resolution and amplitude $\xi =0.4$. The dashed line indicates the slope $P(k_\perp)\propto k_\perp^{-1}$.[]{data-label="spec2Dev"}](spec2Dev){width="50.00000%"}
The energy spectrum in 2D simulations is also different from the 3D case. Figure \[spec2Dev\] shows the spectral evolution for a model with $\xi = 0.4$ and grid resolution of $2048^2$. Over the course of tens of collisions, energy is gradually redistributed toward smaller scales, with a perpendicular spectrum $P(k_\perp) \propto k_\perp^{-1}$, significantly shallower than the 3D case.
The different energy dissipation rates seen in 2D versus 3D settings can be explained by the difference in their spectral slopes. In particular, the energy spectrum in 2D has an unbounded energy capacity, because $$\label{eqn:energy-cap}
U_0 = \alpha \int\limits_{k_{0}} ^{k_{\rm max}} {\mathrm{d}}k_{\perp} \; k_\perp^{-1} = \alpha
\,\ln
\frac{k_{\rm max}}{k_0}$$ would diverge if $k_{\rm max}\rightarrow \infty$. Here $\alpha$ is a normalization factor which may evolve with time. As energy cascades toward smaller scales, $k_{\rm max}$ increases, but remains finite. This fact is consistent with the increasing delay of the dissipation onset with increasing resolution, as it takes longer for $k_{\max}$ to reach $k_{\rm diss}$.
The evolution of the $k_\perp^{-1}$ turbulence spectrum is determined by the evolution of its normalization $\alpha(t)$. Suppose that $k_{\rm max}$ increases as a power law with time, $$k_{\rm max} \propto t^q, \qquad q>0.$$ The normalization $\alpha$ must decrease as $k_{\rm max}$ increases, $$\label{eqn:alpha-of-t}
\alpha(t) = \frac{U_0}{\ln(k_{\rm max}/k_0)} = \frac{U_0}{q\ln t+\beta},$$ where $\beta$ is a constant. When $k_{\rm max}$ reaches $k_{\rm diss}$, grid heating begins to remove energy on scales smaller than $k_{\rm diss}^{-1}$, and the turbulence energy $U$ decreases below $U_0$, $$\label{eqn:U-of-t}
U(t) = \alpha(t) \int\limits_{k_{0}} ^{k_{\rm diss}} {\mathrm{d}}k_{\perp}\; k_\perp^{-1} = \alpha(t) \ln \frac{k_{\rm diss}}{k_0}\, .$$ Equations \[eqn:alpha-of-t\] and \[eqn:U-of-t\] together yield the relation $$\label{eqn:U-model}
\frac{U_0}{U(t)} = \frac{q\ln t + \beta}{\ln(k_{\rm diss} / k_0)}.$$ This description assumes that $\alpha(t)$ (or the value of $q$) is independent of grid dissipation at high $k_\perp$. The value of $k_{\rm diss}$ is proportional to the grid resolution $N$ and the evolution of $U$ depends on $N$.
The predicted relation (\[eqn:U-model\]) can be tested by measuring $U(t)$ in the simulations with different resolutions and checking (1) whether $U_0/U(t)$ is indeed a linear function of $\ln t$, and (2) whether $q$ indeed has a universal value. This test is shown in Figure 6 for five different resolutions $N$ that span a factor of $16$. In each case, after $k_{\max}$ reaches $k_{\rm diss}$ we observe a linear growth of $U_0/U(t)$ with $\ln t$. We have measured its slope $s$ as a function of $\ln k_{\rm diss}$ and then calculated $q$ from the linear realtion inferred from Equation (\[eqn:U-model\]), $$\frac{1}{s}=\frac{1}{q}\ln\frac{k_{\rm diss}}{k_0}.
$$ For all five resolutions, the values of $(\ln k_{\rm diss},\,1/s)$ are found to follow the same line with $q\approx 1.75$, confirming the above analytical picture of the turbulence spectrum evolution. In contrast to the 3D simulations, dissipation slows down with increasing $N$.
![ The evolution of $U_0/U(t)$ in 2D simulations. Dashed lines show the best-fit slopes of the linear relation between $U_0/U$ and $\ln t$. The slope value $s$ is indicated next to each curve.[]{data-label="fit"}](model2D){width="50.00000%"}
Local dissipation and escape of waves {#sec:fate}
=====================================
As discussed in Section \[waves\], nonlinear interactions between [Alfvén ]{}waves can excite fast modes. These modes are not ducted along the magnetic field lines; they have group velocity in any direction and may escape the magnetosphere. In this section we examine the competition between the two energy sinks: local dissipation of the turbulent cascade and the escape of generated fast waves. We use the 3D setup of colliding [Alfvén ]{}wave packets as described in Section 3, and compare two sets of simulations, with and without wave escape, as explained below.
Turbulent dissipation rate without wave escape
----------------------------------------------
{width="50.00000%"} {width="50.00000%"}
The periodic boundary conditions for all ($x,y,z$) directions imply that waves cannot escape the computational box; they can only dissipate. The dissipation efficiency depends on the amplitude of the colliding [Alfvén ]{}packets $\xi$ and their sizes $\ell$. We have studied this dependence by calculating models with seven different values of $\xi$ between 0.5 and $3$, at fixed resolution of $512^3$. The results are presented in Figure 7, which shows evolution of the dissipated energy fraction, $$f(t)=\frac{U_0-U(t)}{U_0}.$$ One can see that $f(t)$ is small in the first collisions, and its time dependence is step-like, because dissipation occurs only during the collisions, when the two packets overlap. (The duration of overlap is significantly shorter than the time between the collisions, because the packet width $\ell$ is smaller than the computational box size.) At later times, the field line bundle carrying the two packets becomes increasingly filled with strong [Alfvén ]{}turbulence, capable of dissipating energy outside the packets; then the dissipation curve $f(t)$ becomes smoother.
As expected, $f(t)$ is higher for the simulations with larger packet amplitudes $\xi$, because of the higher effectiveness of the nonlinear interaction. The dissipated fraction after the first collision, $f_1=f(\tau)$, is plotted in the right panel of Figure \[casraterate3D\] for different values of $\xi$. We find that $f_1$ is a very sensitive function of $\xi$, rising sharply from $10^{-5}$ at $\xi = 1$ to $10^{-2}$ at $\xi = 2$.
Turbulent cascade with escaping fast modes
------------------------------------------
To evaluate the energy radiated away by fast modes, we have introduced a “sponge” layer along the transverse domain boundaries, intended to absorb fast waves reaching the boundaries. The elimination of energy transported by fast waves into the sponge layer simulates their escape.
The sponge layer is implemented by adding an Ohmic-like dissipation term $-\sigma_s {\boldsymbol{E}}$ to the force-free current in Equation \[current\]. This term leads to exponential damping of the electric field on the timescale $\sigma_s^{-1}$. We adopt a spatial profile of $\sigma_s(x,y)$ that leads to faster damping of the electric field near the boundary of the computational domain in $x$-$y$ plane ($0<x<1$; $0<y<1$), $$\sigma_s = \frac{1}{2 \tau} \left( 1 - e^{-8 \delta^4} \right), \qquad \delta = \max \left(\frac{r_\perp - r_0}{d - r_0}, 0 \right),$$ where $r_\perp = \sqrt{(x - d)^2+(y - d)^2}$, $d = 0.5$ is half of the transverse domain scale, and $r_0 = 0.3$ is the distance from the $z$-axis within which absorption is switched off completely, $\sigma_s=0$. Near the boundary, the damping time scale $\sigma_s^{-1}$ drops to $2 \tau$ (equal to the light crossing time of the computational box). The energy dissipated in the sponge layer is a proxy for energy escaping the system in the form of fast waves.
The loss of fast modes at the boundaries leads to a faster decline of the free energy in the box $U(t)$ compared with the simulations without wave escape. The magnitude of this effect is a measure of the effectiveness of the energy loss through the boundaries compared with local dissipation through the turbulent cascade. Figure \[3Dfastwave\] shows the comparison of four pairs of simulations with and without the sponge layer (identical otherwise). All eight simulations have resolution $512^3$. The four different amplitudes $\xi=0.5, 1, 2, 3$ are chosen to investigate how the competition between wave escape and turbulent damping depends on the initial amplitudes of the packets.
For instance, in the simulation with $\xi = 0.5$, during the first $10$ collisions (prior to $t_{\rm onset}$), fast waves carry away $\sim 2\%$ of the initially available free energy $U_0$, compared to $\sim 1\%$ taken by turbulent dissipation. Radiation of fast waves is thus the primary energy loss channel prior to the onset of developed turbulence. After turbulence is fully developed, at times $\gtrsim t/\tau=100$, the energy radiated by fast waves accounts for only $6\%$ of $U_0$ while turbulent dissipation accounts for nearly $60\%$.
A smaller energy fraction is carried away by fast waves in the simulations with larger $\xi$. In the run with $\xi=3$, the sponge layer accounts for only $3\%$ of $U_0$. This trend is the result of a stronger coupling of fast waves in nonlinear interactions. As the fast waves participate in the turbulence to a greater degree, they are scattered and damped more efficiently, dissipating more energy locally in the magnetosphere.
![Comparison between free energy evolution $U(t)$ in the simulations with (dashed curves) and without (solid curves) damping of fast modes at the boundary. The difference between solid and dashed curves shows the effect of fast mode escape compared with local dissipation through the turbulent cascade. []{data-label="3Dfastwave"}](casrate3Dohm){width="50.00000%"}
Enhanced immediate dissipation and FFE failure {#sec:breakffe}
================================================
The results of the preceding sections show that wave damping takes many crossing times $\tau$, even at very high amplitudes $\xi>1$. Since this conclusion is based on numerical simulations with a concrete setup, one would like to know whether the conclusion is robust. To address this question we have tried to vary the initial setup of the [Alfvén ]{}wave packets in search for more efficient damping, and found that in some cases our FFE simulations predict much quicker dissipation, which occurs immediately in the first collisions of the wave packets, even before the development of turbulence.
Immediate dissipation observed in FFE simulations
-------------------------------------------------
The immediate dissipation effect is sensitive to the initial polarization of the wave packets. It is maximized (and most convenient to study) in the simplest 1D “slab” setup. Then the initial perturbations of the magnetic field in the two colliding packets, ${\boldsymbol{B}}_1$ and ${\boldsymbol{B}}_2$, can point in any two chosen directions in the $x$-$y$ plane perpendicular to the guide field ${\boldsymbol{B}}_0$. The angle between ${\boldsymbol{B}}_1$ and ${\boldsymbol{B}}_2$ will be denoted by $\theta$. The relative polarization $\theta$ is an important parameter of the packet collision problem, in addition to the packet amplitude $\xi$.
The simple 1D setup is better suited for the study of polarization effects than the 3D and 2D setups of the preceding sections. Recall that in the 3D setting the spherical packets could not have a single direction for ${\boldsymbol{B}}_1$ or ${\boldsymbol{B}}_2$, and thus the polarization angle was not well defined. In the 2D setting explored in Section 4.2, we chose ${\boldsymbol{B}}_1$ and ${\boldsymbol{B}}_2$ along the $y$-axis perpendicular to the simulation plane $x$-$z$, which allowed us to confine the packets in a circular region in the $x$-$z$ plane. However, the requirement of ${\boldsymbol{B}}_{1,2}$ being parallel (or anti-parallel) to the $y$-axis leaves only two possibilities for the relative polarization, $\theta=0$ or $180^\circ$, and in Section 4.2, we stuck to the case of $\theta = 0^\circ$. Therefore, in both 3D and 2D simulations presented in Section 4 we observed dissipation only through turbulence cascade to the grid scale, which takes a significant time.
The reason for immediate damping discussed in the present section is the activation of the dissipation channel (iii) listed in Section 3.5. Our FFE simulations show, for some values of $\theta$ and $\xi$, field evolution that violates the condition $E<B$, and then the procedure of enforcing this condition (Section 3.4) creates strong dissipation.
One can see the role of relative polarization $\theta$ for this effect from a simplified consideration that neglects the nonlinear character of packet collisions and merely looks at the linear superposition of the colliding packets. When ${\boldsymbol{B}}_1$ and ${\boldsymbol{B}}_2$ are parallel ($\theta=0$), the magnetic fields of the packets add constructively while their electric fields add destructively — the opposite Poynting fluxes of the two packets ${\boldsymbol{E}}_1\times {\boldsymbol{B}}_1$ and ${\boldsymbol{E}}_2\times{\boldsymbol{B}}_2$ require them to have antiparallel electric fields ${\boldsymbol{E}}_1$ and ${\boldsymbol{E}}_2$. By contrast, when the packets have nearly anti-aligned ${\boldsymbol{B}}_1$ and ${\boldsymbol{B}}_2$, the electric fields ${\boldsymbol{E}}_1$ and ${\boldsymbol{E}}_2$ become parallel and add constructively making it possible for $E$ to exceed $B$ for sufficiently large amplitudes $\xi$.
A simple estimate gives the range of $\theta$ and $\xi$ where this effect may be expected. Let us consider two counter-propagating packets of amplitude $\xi$ centered at $z_1(t)$ and $z_2(t)$. The packets can have, for example, a Gaussian shape, $B_{1,2}=\xi B_0\exp[-(z-z_{1,2})^2/\ell^2]$. Let us choose the $y$-axis along ${\boldsymbol{B}}_1$; then $$\begin{aligned}
{\boldsymbol{B}}_1&=&\xi \exp\left[-\frac{(z-z_1)^2}{\ell^2}\right]\,\boldsymbol{b}_1, \quad
\boldsymbol{b}_1=(0,1,0) \\
{\boldsymbol{B}}_2&=&\xi \exp\left[-\frac{(z-z_2)^2}{\ell^2}\right]\,\boldsymbol{b}_2, \quad
\boldsymbol{b}_2=(\sin\theta,\cos\theta,0).\end{aligned}$$ where we use the units $B_0=1$, . The corresponding electric fields are ${\boldsymbol{E}}_{1,2}=\mp \hat{\boldsymbol{z}} \times {\boldsymbol{B}}$, so the angle between the electric fields is $180^\circ - \theta$. If the non-linear interaction of the packets is neglected, then at the point of maximum overlap (at $z=z_1=z_2$) the superposed field magnitudes would be $$\begin{aligned}
({\boldsymbol{B}}_1+{\boldsymbol{B}}_2)^2 &=& 2\xi^2(1+\cos\theta) + 1 \\ \nonumber
({\boldsymbol{E}}_1+{\boldsymbol{E}}_2)^2 &=& 2\xi^2(1-\cos\theta) \, .\end{aligned}$$ Magnetic dominance would thus be lost when $$\label{cr1}
-4\xi^2\cos\theta >1 \, .$$ This condition can be satisfied if $\theta>90^\circ$, and is easiest to satisfy if $\theta=180^{\circ}$. In the latter case, a moderately strong amplitude $\xi>0.5$ is required.
![Dissipated energy fraction in the 1D wave packet collision as a function of the relative polarization angle $\theta$, for various amplitudes $\xi$ of the colliding packets.[]{data-label="dis1d"}](dis1D){width="50.00000%"}
![ Comparison of the FFE simulation (dashed curve) with the full MHD simulation that tracks both plasma energy and electromagnetic energy (solid curves). The colliding [Alfvén ]{}wave packets have amplitude $\xi=0.8$ and relative polarization angle $\theta = 180^\circ$. Numerical convergence of the MHD simulation is shown by plotting the results obtained with resolutions $N=256,512,1024,2048$ (the increasing line thickness corresponds to the increasing resolution). []{data-label="rmhd"}](alfven180_deltaB08_single_collision){width="50.00000%"}
In general, such large amplitude waves interact non-linearly and their magnitude cannot be determined by linear superposition. Remarkably however, we observe from Equation \[current\] that in FFE the non-linear terms vanish for 1D anti-polarized plane waves counter-propagating along ${\boldsymbol{B}}$. Thus, when $\theta = 180^\circ$, the wave packets pass through one another unchanged, as long as $E<B$, and the linear superposition estimate for the loss of $E<B$ condition should be accurate.
{width="\textwidth"}
These expectations are tested in Figure 9, which presents the simulation results for the 1D setup described above. It shows the fraction of energy dissipated after a single collision of the two packets, $f_1=|\Delta U|/U_0$, for different values of the relative polarization angle $\theta$ and packet amplitude $\xi$. We find that strong dissipation, caused by the imposed shortening of the electric field to sustain $E<B$, becomes active for nearly anti-polarized packets if $\xi \gtrsim 0.5$. For example, we see 20% energy loss after a single collision when $\xi = 0.6$ and $\theta = 180^\circ$. As $\xi$ increases to 1.4, we observe $f_1$ growing to $\sim 90$% for $\theta$ approaching $180^\circ$, and exceeding 10% for $\theta>135^\circ$.
Figure \[dis1d\] also reveals a second peak in $f_1(\theta)$ near $\theta=90^\circ$. This peak is not predicted by the above linear superposition estimate, and thus has a nonlinear origin. However, it is also caused by the violation of $E<B$ condition. It can be understood by looking at the evolution of ${\boldsymbol{E}}\cdot {\boldsymbol{B}}$ in the linear superposition approximation, ${\boldsymbol{E}}\cdot {\boldsymbol{B}}= 2\xi^2\sin\theta$. The linearly superposed fields would violate the FFE condition ${\boldsymbol{E}}\cdot {\boldsymbol{B}}=0$, and nonlinear effects are responsible for sustaining this condition: the system is forced to generate a longitudinal electric field $E_z \propto \xi^2$, and for large enough $\xi$ this electric field component leads to the loss of magnetic dominance. This effect is proportional to $\sin\theta$ and thus strongest at $\theta\approx 90^\circ$.
MHD simulations and the spurious character of immediate dissipation in FFE
--------------------------------------------------------------------------
One may conclude from the FFE simulations that the collisions of large-amplitude [Alfvén ]{}waves with favorable polarization gives strong immediate dissipation. The dissipation mechanism in this case is the result of the customary procedure of shortening $E$ to sustain $E<B$. However, we point out that there is no guarantee that this procedure correctly captures the true field evolution. The true behavior of the system when $E$ reaches $B$ is outside the realm of FFE and can be understood only with a more complete physical model. The model must explicitly include a component of the system that takes the energy (and momentum) lost by the field.
Therefore, we have performed similar simulations of the 1D packet collisions in the full relativistic MHD, which does not neglect the plasma stress-energy tensor, and conserves the total energy and momentum of field and plasma. Plasma moves with a subliminal velocity $\boldsymbol{v}$, and MHD simulations never break the condition $E<B$, since the electric field ${\boldsymbol{E}}= -{\boldsymbol{v}}\times {\boldsymbol{B}}$ is obtained from the primitive variables, rather than evolved independently as it is in FFE. We use the relativistic MHD code `Mara` [@2012ApJ...744...32Z].
MHD is expected to approach the FFE regime in the limit of high magnetization $\sigma\gg 1$. Therefore, in our simulations we choose a high $\sigma=25$, where $\sigma=B_0^2/\rho_0$ is defined for the background magnetic field $B_0$ and $\rho_0$ is the initial (uniform) plasma density in the computational box. Otherwise, the simulation setup is the same as described in Section 6.1 for the 1D FFE simulations.
Figure \[rmhd\] compares the MHD and FFE results. For this test we chose the case where the colliding packets have amplitude $\xi=0.8$ and are anti-polarized ($\theta=180^\circ$). We see that MHD and FFE predict similar evolution of the free electromagnetic energy $U(t)$ up through the peak of the collision. Then the MHD evolution strongly deviates from the prediction of the FFE simulation. Importantly, the electromagnetic energy lost in the MHD simulation is compensated by a gain in the plasma kinetic energy, while the FFE code removes that same energy $\Delta U$ from the simulation irreversibly by the $E < B$ fix.
Note that the electromagnetic fields of the two packets initially carry a significant $y$-momentum of the same sign. The MHD simulation shows that during the collision a large fraction of this momentum is taken by the plasma. The plasma momentum density is enhanced at the interface between the colliding packets by a factor $\sim 20$, comparable to $\sigma$. This enhancement is caused by two factors: the plasma is compressed by a factor of $\sim 6$ and the Lorentz factor of its transverse drift ($\boldsymbol{v}_\perp={\boldsymbol{E}}\times{\boldsymbol{B}}_0\parallel -\hat{\boldsymbol{y}}$) exceeds 3. As a result, a large fraction of the packet electromagnetic energy temporarily converts into bulk kinetic energy of the plasma accelerated along the $y$-axis. Once the collision is over, the accelerated plasma is stopped by magnetic stresses, restoring the electromagnetic field energy nearly to its initial value.
Figure \[snapshots\] shows in more detail the evolution of the field and plasma in the FFE and MHD simulations. Unlike the FFE, the MHD system does not reach the “floor” $B^2-E^2=0$, because it would correspond to the drift speed equal to the speed of light and hence infinite kinetic energy of the plasma. The plasma is strongly accelerated when $B^2-E^2$ is reduced, and the subsequent dynamics are completely different in the two simulations. We conclude that the strong dissipation effect observed in FFE simulations is spurious. It is caused by the failure of FFE simulations to keep track of energy that is temporarily removed from the electromagnetic field when $E$ approaches $B$.
Discussion {#sec:discussion}
==========
The fate of [Alfvén ]{}waves excited in a magnetar magnetosphere is interesting from observational point of view if the wave energy eventually converts to radiation. In particular, the hot plasma fireball formed in giant flares could be powered by dissipation of waves [@1995MNRAS.275..255T], or the waves may be absorbed by the neutron star and feed surface afterglow emission [@2015ApJ...815...25L]. In the relativistic magnetospheres, where the magnetic field dominates over the plasma rest mass and all waves propagate with the speed of light, the nonlinear behavior of wave turbulence is poorly known. In this paper we employed numerical simulations to systematically study turbulence excited by colliding packets of [Alfvén ]{}waves and the resulting dissipation. The packets are assumed to be launched by an unspecified triggering event (e.g. a fast displacement of the crust or a global magnetospheric instability), which determines the initial packet amplitude $\xi = \delta B / B_0$ and size $\ell$.
Summary of results
------------------
Most of our results are obtained from high-resolution FFE simulations in a Cartesian box, using the 5th order conservative finite differencing scheme described in [@2011MNRAS.411.2461Y]. We have also explored situations where the FFE approximation becomes insufficient; then we employed relativistic MHD simulations. Our results are as follows.
\(1) Our 3D simulations of packet collisions show that significant dissipation begins when a turbulence cascade develops down to the grid scale. The cascade is dominated by modes with wavevectors orthogonal to the background magnetic field, $k_\perp\gg k_\parallel$, and its spectrum is steep, with a slope close to $-2$. We observed consistency of the cascade and the resulting dissipation rate with increasing grid resolution, and concluded that dissipation of the 3D turbulence is well modeled by “grid heating” (the removal of high-$k$ modes on the grid scale). The simulations reveal that even for wave packets of enormous amplitudes $\xi=1-3$ dissipation develops slowly, over many (10-100) collisions of the packets bouncing in the magnetosphere. The main reason for the dissipation delay is the relatively slow development of the broad spectrum of high-frequency modes and the onset of a persistent energy flow in the cascade down to the dissipation scale.
\(2) We have found that it is essential to calculate the wave turbulence in three dimensions. Similar simulations restricted to two dimensions (where the fields are assumed to be independent of one coordinate running transverse to the guide field ${\boldsymbol{B}}_0$) are deficient. They produce qualitatively different results and show no convergence with increasing resolution, because the 2D cascade has a flat spectrum with an infinite energy capacity.
\(3) [Alfvén ]{}waves are trapped, because they are ducted along the magnetic field lines, however their collisions generate fast modes that can carry energy away across the field lines. We have measured energy loss due to fast mode escape and found this effect to be weaker than energy dissipation on the field lines carrying the [Alfvén ]{}waves.
\(4) When two strong [Alfvén ]{}waves collide, the electromagnetic field can experience immediate significant energy loss. This effect is qualitatively different from the cascade dissipation on the grid scale. It occurs when the Lorentz invariant $B^2-E^2$ is pushed to zero during the field evolution in some parts of the colliding packets, threatening to violate the condition $E<B$. This effect is possible only for certain relative polarizations of the colliding waves and is maximum when the waves have anti-aligned magnetic fields, as demonstrated by a simple 1D model. Our simulations revealed that $B^2-E^2$ can be pushed to zero also in a collision of waves with orthogonal polarizations; this occurs due to a non-linear effect responsible for sustaining ${\boldsymbol{E}}\cdot{\boldsymbol{B}}=0$.
\(5) We have shown that the permanent energy loss in FFE simulations caused by $B^2-E^2\rightarrow 0$ is spurious. FFE has no component of the system other than the electromagnetic field and thus has no choice but to permanently remove the energy lost by the field. Our relativistic MHD simulations revealed that in fact this energy is temporarily stored in the plasma that is compressed and accelerated to a high Lorentz factor perpendicular to the background magnetic field ${\boldsymbol{B}}_0$. As the two colliding wave packets finish their interaction, the relativistic plasma motion is eventually decelerated and most of its energy is returned to the electromagnetic field. Our results indicate that damping of [Alfvén ]{}waves in the magnetosphere is surprisingly slow even at extremely high amplitudes, and so the waves can bounce in the magnetosphere for many crossing times. We have discussed the physical reasons for this behavior, and conclude that the slow damping is likely a true feature, not an artifact of our approximations. However, one should bear in mind the following simplifications adopted in our simulations.\
(1) Our computational box was rectangular and filled with a uniform background magnetic field ${\boldsymbol{B}}_0$. Magnetic field lines in a real magnetosphere are curved and can extend far from the star, where the field is much weaker. [Alfvén ]{}waves bouncing on such extended field lines will significantly increase their amplitudes as they propagate in the outer weak-field region.\
(2) We focused on magnetospheres with energy density $B^2/8\pi$ much greater than the plasma rest mass. This regime almost always holds for the magnetosphere of a neutron star. However, during a giant flare, a significant fraction of the magnetic energy may be dissipated and stored in the electron-positron fireball trapped in the magnetosphere. Then the plasma inertia can become a significant factor in the evolution of [Alfvén ]{}wave turbulence.\
(3) Our simulations assumed perfect reflection at the boundaries that represent the stellar surface in the computational box. Since the two colliding packets are symmetric in our simulation setup, their perfect reflection at the surface is equivalent to periodic boundary conditions. In reality, the reflection coefficient is slightly below unity, and $\sim 10$% of the packet energy is transmitted into the star [@2015ApJ...815...25L].\
(4) We described the spurious immediate dissipation in packet collisions using only 1D (FFE and MHD) simulations. As one can see from Figure \[snapshots\], a short-lived current sheet forms at the packet collision interface, in the $x$-$y$ plane perpendicular to ${\boldsymbol{B}}_0$. If the current sheet becomes tearing unstable, magnetic reconnection will occur and dissipate some energy. The tearing is not allowed in 1D models, and so 2D or 3D simulations are required to investigate the possible reconnection in the current sheet. As a first step, we have ran several test 2D simulations using kinetic code TRISTAN-MP. We found that magnetic reconnection is important only when the packet amplitude $\xi$ is much larger than unity. We leave the detailed study of magnetic reconnection in packet collisions to a future paper.
Fate of wave energy in magnetar flares
--------------------------------------
One implication of our results is that dissipation of [Alfvén ]{}waves in the magnetosphere is less efficient than their absorption by the neutron star. @2015ApJ...815...25L showed that $\sim 10$ interactions of the wave packet with the stellar crust is sufficient to absorb a large fraction of its energy. That work simulated [Alfvén ]{}wave packets hitting the neutron star crust with realistic density profile $\rho(z)$ and obtained the reflection and transmission coefficients for this interaction, $\mathcal{R}$ and $\mathcal{T}$. The numerical results were also found consistent with an analytical estimate for wave tunneling into the crust using WKB approximation. For typical magnetar fields $B_0>10^{14}$ G and sizes of the wave packet $\ell\sim 10$ km (comparable to the star radius), the transmission coefficient is $\mathcal{T}=10-20\%$. It increases for stronger $B_0$, because it implies a higher [Alfvén ]{}speed inside the magnetar crust.
The shear wave transmitted into the heavy crust is much slower than the magnetospheric [Alfvén ]{}wave. It continues to propagate into the deeper and denser crustal layers with a decreasing speed and a diminishing amplitude. However the strain in the wave [*grows*]{} as $\propto \rho^{1/4}$ and eventually induces a plastic flow. As a result, the wave energy converts to heat, melting the solid material at the bottom of the liquid ocean, which is $\sim 100$ m deep in magnetars. Thus, most of the magnetospheric wave energy is expected to convert to heat at $\sim 100$ m below the stellar surface. @2015ApJ...815...25L also calculated how the heat diffuses from this depth and is mostly lost to neutrino emission; a fraction $\sim 0.1$ of the heat will reach the surface and feed the surface afterglow weeks to months after the event that triggered the magnetospheric waves.
Only a fraction ${f_{\rm diss}}$ of the magnetospheric wave energy will be dissipated locally in the magnetosphere (and an even smaller fraction ${f_{\rm esc}}$ will convert to fast modes that escape the field lines carrying the [Alfvén ]{}waves). In particular, in our simulations, the wave energy fraction dissipated per collision of packets is $ \simlt 1\%$, which is $\simgt 10$ times lower than $\mathcal{T}$. Therefore, we roughly estimate ${f_{\rm diss}}\simlt 0.1$. It may still be interesting for powering fireball radiation. However, a more promising source for fireball energy appears to be magnetic reconnection in a global instability of the over-twisted magnetosphere, as observed in the simulations of Parfrey et al. (2013). The reconnection event immediately dissipates significant energy. It also launches strong waves, which dissipate with efficiency ${f_{\rm diss}}$ in the magnetosphere, but mostly disappear into the star and feed its invisible neutrino emission and a delayed afterglow from the stellar surface.
X.L. and J.Z. appreciate useful input from Maxim Lyutikov, in particular the suggestion to simplify the description of FFE eigen modes by using the temporal gauge. A.M.B. is supported by NASA grant NNX17AK37G and a grant from the Simons Foundation \#446228.
Resonant Three Wave Interactions
================================
The second order nonlinear current ${\boldsymbol{J}}^{(2)}_{\rm nl}$ in Equation \[Maxwell2\] reads $$\begin{aligned}
\label{J2}
{\boldsymbol{J}}^{(2)}_{\rm nl} &=& \nabla\cdot{\boldsymbol{E}}^{(1)}\frac{{\boldsymbol{E}}^{(1)}\times\hat{{\boldsymbol{z}}}}{B_0} + \frac{\hat{{\boldsymbol{z}}}\cdot\nabla\times{\boldsymbol{B}}^{(1)}}{B_0}{\boldsymbol{B}}^{(1)} \nonumber\\
&&+\frac{{\boldsymbol{B}}^{(1)}\cdot\nabla\times{\boldsymbol{B}}^{(1)} - {\boldsymbol{E}}^{(1)}\cdot\nabla\times{\boldsymbol{E}}^{(1)}}{B_0}\hat{{\boldsymbol{z}}}-2\hat{{\boldsymbol{z}}}\cdot(\nabla\times{\boldsymbol{B}}^{(1)})\frac{\hat{{\boldsymbol{z}}}\cdot{\boldsymbol{B}}^{(1)}}{B_0}\hat{{\boldsymbol{z}}} \, .\end{aligned}$$ Note that the terms in the second line of the above equation (those proportional to $\hat{{\boldsymbol{z}}}$) only source $A_z$,and thus do not excite any propagating waves.
Let us consider the interaction of a pair of linear waves ${\boldsymbol{A}}^{(1)}_{1}$ and ${\boldsymbol{A}}^{(1)}_{2}$. Then we substitute into Equation (\[J2\]) ${\boldsymbol{E}}^{(1)}$ and ${\boldsymbol{B}}^{(1)}$ obtained from ${\boldsymbol{A}}^{(1)}={\boldsymbol{A}}^{(1)}_{1}+{\boldsymbol{A}}^{(1)}_{2}$. This yields the second order current, $$\label{J2b}
{\boldsymbol{J}}^{(2)}_{\rm nl} = \nabla\cdot{\boldsymbol{E}}_1^{(1)}\frac{{\boldsymbol{E}}_2^{(1)}\times\hat{{\boldsymbol{z}}}}{B_0} + \frac{\hat{{\boldsymbol{z}}}\cdot\nabla\times{\boldsymbol{B}}_1^{(1)}}{B_0}{\boldsymbol{B}}_2^{(1)} + (1\leftrightarrow 2) + (\mathrm{terms \ proportional \ to \ } \hat{{\boldsymbol{z}}}) \, ,$$ where $(1\leftrightarrow 2)$ means the repetition of previous terms but with subscript $1$ and $2$ exchanged. We seek a solution ${\boldsymbol{A}}^{(2)}$ of Equation (\[Maxwell2\]) which is sourced by ${\boldsymbol{J}}_{\rm nl}^{(2)}$, is itself an eigen mode, and whose amplitude grows in time. Our ansatz is thus ${\boldsymbol{A}}^{(2)}(\boldsymbol{r}, t) = \Lambda_m(t)\boldsymbol{e}_m\exp[i(\boldsymbol{k}^{(2)}\cdot\boldsymbol{r}-\omega^{(2)}t)]$ where $\omega^{(2)}$ and $\boldsymbol{k}^{(2)}$ satisfy either the fast or [Alfvén ]{}wave dispersion relations. Inserting ${\boldsymbol{A}}^{(2)}(\boldsymbol{r}, t)$ into Equation (\[Maxwell2\]), we obtain the evolution equation for the wave amplitude $\Lambda_m(t)$, $$\label{ampevol}
\partial^2_t \Lambda_m(t) - 2i\omega^{(2)} \partial_t \Lambda_m(t) =\omega^{(2)} {\boldsymbol{J}}^{(2)}_{\rm nl}\cdot\boldsymbol{e}_m e^{i(\omega^{(2)}t-\boldsymbol{k}^{(2)}\cdot\boldsymbol{r})} \, .$$ Inspection of Equation (\[J2b\]) reveals that ${\boldsymbol{J}}^{(2)}_{\rm nl}$ is proportional to $\exp[i(\boldsymbol{k}_{12} \cdot\boldsymbol{r}-\omega_{12} t)]$ where $\boldsymbol{k}_{12}=\boldsymbol{k}_1^{(1)}+\boldsymbol{k}_2^{(1)}$ and $\omega_{12}=\omega_1^{(1)}+\omega_2^{(1)}$. The right hand side of Equation (\[ampevol\]) may thus be written as $$\label{eqn:lambda-ode1}
\partial^2_t \Lambda_m(t) - 2i\omega^{(2)} \partial_t \Lambda_m(t) =C_{12m} e^{i (\boldsymbol{k}_{12}-\boldsymbol{k}^{(2)})\cdot\boldsymbol{r}} e^{-i (\omega_{12}-\omega^{(2)})t}\, ,$$ where $C_{12m}$ has no space or time dependence (these coefficients describe the strength of wave-wave interactions and are evaluated below for each of the allowed channels). The absence of spatial dependence on the left hand side of Equation (\[eqn:lambda-ode1\]) implies that its right hand side is independent of $\boldsymbol{r}$, which requires $\boldsymbol{k}^{(2)}=\boldsymbol{k}_{12}$. The temporal evolution of $\Lambda_m(t)$ satisfies the equation, $$\label{eqn:lambda-ode2}
\partial^2_t \Lambda_m(t) - 2i\omega^{(2)} \partial_t \Lambda_m(t) = C_{12m} e^{-i (\omega_{12}-\omega^{(2)})t} \, .$$ The general solution of Equation (\[eqn:lambda-ode2\]) subject to the initial condition $\Lambda_m(0)=0$ (and neglecting the constant of integration) is given by $$\Lambda_m(t) = C_{12m} \frac{1 - e^{-i(\omega_{12} - \omega^{(2)})t}}{\omega_{12}^2 - (\omega^{(2)})^2} \, .$$ For arbitrary values of $\omega^{(2)}$, the amplitude oscillates in time. However, as $\omega^{(2)} \rightarrow \omega_{12}$, the oscillation period grows longer, and when the resonance condition is met precisely, $\Lambda_m(t) \rightarrow i C_{12m} t / 2 \omega_{12}$. Energy transfer from the primary waves is only possible for such resonant interactions.
Below we list the expressions for $C_{12m}$ for each allowed resonant channel.
1. $\mathcal{A}+\mathcal{A}\rightarrow\mathcal{F}'$ $$C_{\mathcal{A}\mathcal{A}\mathcal{F}'}=\frac{-i \omega}{B_0\sqrt{\omega \omega_1\omega_2}} \frac{\Lambda_1\Lambda_2}{k_{\perp}k_{1\perp}k_{2\perp}}\left[\left( \omega_1\omega_2 - k_{1z}k_{2z} \right) \left(2k_{1\perp}^2k_{2\perp}^2 + (k_{1\perp}^2+k_{2\perp}^2)\boldsymbol{k}_{1\perp}\cdot \boldsymbol{
k}_{2\perp} \right)\right]\, .$$ Here two interacting [Alfvén ]{}waves with frequencies $\omega_1(\boldsymbol{k}_1)$ and $\omega_2(\boldsymbol{k}_2)$, and amplitudes $\Lambda_1$ and $\Lambda_2$, generate a fast mode $\omega(k)$ that satisfies the resonance conditions $\boldsymbol{k}=\boldsymbol{k}_1+\boldsymbol{k}_2$ and $\omega=\omega_1+\omega_2$.
2. $\mathcal{A}+\mathcal{F}\rightarrow\mathcal{F}'$ $$C_{\mathcal{A}\mathcal{F}\mathcal{F}'}=\frac{ i \omega}{B_0\sqrt{\omega \omega_\mathcal{F}\omega_\mathcal{A}}} \frac{\Lambda_{\mathcal{A}}\Lambda_{\mathcal{F}}}{k_{\perp}k_{\mathcal{A}\perp}k_{\mathcal{F}\perp}}\left[\left( \omega_\mathcal{A}\omega_\mathcal{F} - k_{\mathcal{A}z}k_{\mathcal{F}z} \right) k_{\mathcal{A}\perp}^2\left( \boldsymbol{k}_{\mathcal{F}\perp}\times \boldsymbol{k}_{\mathcal{A}\perp} \right)\cdot\hat{{\boldsymbol{z}}} \right]\, .$$ Here an [Alfvén ]{}wave with frequency $\omega_{\mathcal{A}}(\boldsymbol{k}_{\mathcal{A}})$ and amplitude $\Lambda_{\mathcal{A}}$ interacts with a fast mode with frequency $\omega_{\mathcal{F}}(\boldsymbol{k}_{\mathcal{F}})$ and amplitude $\Lambda_{\mathcal{F}}$. The interaction generates a new fast mode $\omega(\boldsymbol{k})$ that satisfies $\omega=\omega_{\mathcal{A}}+\omega_{\mathcal{F}}$ and $\boldsymbol{k}=\boldsymbol{k}_{\mathcal{A}}+\boldsymbol{k}_{\mathcal{F}}$.
3. $\mathcal{A}+\mathcal{F}\rightarrow\mathcal{A}'$ $$C_{\mathcal{A}\mathcal{F}\mathcal{A}'}=\frac{ i \omega}{B_0\sqrt{\omega \omega_\mathcal{F}\omega_\mathcal{A}}} \frac{\Lambda_{\mathcal{A}}\Lambda_{\mathcal{F}}}{k_{\perp}k_{\mathcal{A}\perp}k_{\mathcal{F}\perp}}\left[\left( \omega_\mathcal{A}\omega_\mathcal{F} - k_{\mathcal{A}z}k_{\mathcal{F}z} \right) k_{\mathcal{A}\perp}^2\left( \boldsymbol{k}_{\mathcal{F}\perp}\cdot \boldsymbol{k}_{\mathcal{A}\perp} + k^2_{\mathcal{F}\perp} \right) \right]\, .$$ Here the interaction is similar to the previous one, except that the third (generated) wave $\omega(\boldsymbol{k})$ is an [Alfvén ]{}wave rather than a fast mode.
[^1]: In FFE, name “fast” is somewhat misleading, because all waves propagate with the speed of light.
|
---
abstract: 'High-field antiferromagnetic-resonance (AFMR) spectra were obtained in the frequency range 60 GHz $<\nu <$ 700 GHz and for magnetic fields up to 8 T in twin-free single crystals of La$_{0.95}$Sr$_{0.05}$MnO$_{3}$. At low temperatures two antiferromagnetic modes were detected, which reveal different excitation conditions and magnetic field dependencies. No splitting of these modes was observed for any orientation of the static magnetic field excluding the phase-separation scenario for this composition. Instead, the full data set including the anisotropic magnetization can be well described using a two-sublattice model of a canted antiferromagnetic structure.'
address: |
$^{1}$Experimentalphysik V, EKM, Universität Augsburg, 86135 Augsburg, Germany\
$^{2}$General Physics Institute of the Russian Acad. Sci., 117942 Moscow, Russia\
$^{3}$Moscow Power Engineering Institute, 105835 Moscow, Russia
author:
- 'A. Pimenov$^{1}$, M. Biberacher$^{1}$, D. Ivannikov$^{1}$, A. Loidl$^{1}$, V. Yu. Ivanov $^{2}$, A. A. Mukhin $^{2}$, and A. M. Balbashov$^{3}$'
title: 'High-field AFMR in single-crystalline La$_{0.95}$Sr$_{0.05}$MnO$_{3}$: Experimental evidence for the existence of a canted magnetic structure'
---
[2]{}
The idea of phase separation in the manganite perovskites (R$_{1-x}$M$_{x}$MnO$_{3}$, R=La, Pr ..., M=Ca, Sr ...) is one of the most controversially discussed topics concerning the electronic properties of these compounds. After the pioneering works of Jonker and van Santen[@jonker] and of Wollan and Koehler[@wollan], de Gennes[@degennes] developed a model in which the purely antiferromagnetic and insulating LaMnO$_{3}$ on increasing doping passes through a canted (CAF) ground state and arrives at a purely ferromagnetic and metallic state at high doping level ($x\gtrsim 0.2
$). This phase diagram was calculated using competing superexchange (SE) and double exchange (DE)[@Zener] interactions.
However, in recent years a number of theoretical models predicted that the CAF structure becomes unstable against electronic phase separation into ferromagnetic (FM) and antiferromagnetic (AFM) regions[@nagaev; @yunoki]. As discussed by Yunoki et al.[@yunoki], the tendency to the phase separation seems to be an intrinsic property of the double-exchange model. A number of experimental data including neutron scattering[@hennion] and NMR[@allodi] pointed toward the existence of the phase separation in different types of manganites. For discussion of the recent results see Refs.[@yunoki; @goodenough].
It has to be pointed out, that the experimental observation of electronic phase separation is rather difficult. Already more than forty years ago, Wollan and Koehler[@wollan] stated that, on the basis of neutron diffraction experiments, it is impossible to decide whether the structure of the doped manganites is homogeneously canted or inhomogeneously mixed FM and AFM. Only a few experimental methods can distinguish between inhomogeneous and homogeneous magnetic phases because the technique has to be sensitive to the local magnetic structure of the sample. In addition, a sample quality appears to be of major importance for these experiments. Antiferromagnetic resonance (AFMR) seems to be an excellent tool for the solution of the phase-separation problem. The main parameters of the resonance lines, like position, excitation conditions, behavior in magnetic field etc., sensitively depend on the local environment of the magnetic moments. Recently, using this method, we have investigated the concentration dependence of AFMR-lines in low-doped La$_{1-x}$Sr$_{x}$MnO$_{3}$ without external magnetic field[@europhys]. The results were explained within the frame of a two-sublattice model, which strongly supported the existence of a canted magnetic structure. Most La$_{1-x}$Sr$%
_{x}$MnO$_{3}$ crystals of this series were twinned. However, the samples with 5% Sr concentration were identified as untwinned single crystals. This fact allowed the unambiguous determination of the excitation conditions of AFMR lines and to carry out detailed investigations in static magnetic fields. In this paper we present, in addition to the results of the magnetic-field experiments, anisotropic magnetization curves and compare the observed data with the predictions of a two-sublattice model. The possible explanations within phase separation models are also discussed.
La$_{0.95}$Sr$_{0.05}$MnO$_{3}$ single crystals were grown by a floating zone method with radiation heating[@preparation]. X-ray powder diffraction measurements showed that the crystals were single-phase. Four-circle X-ray analysis showed the twin-free structure of the crystal. The temperature dependence of the dc-resistivity of these samples has been published previously[@jetp] and agrees well with literature data[@urushibara]. Plane-parallel plates of size approximately 8$\times $8$\times $1 mm$^{3}$ were used for optical measurements. The magnetic measurements were carried out on small pieces of the same crystals.
The magnetization curves of La$_{0.95}$Sr$_{0.05}$MnO$_{3}$ were measured using a SQUID magnetometer in fields up to 6.5 T. The transmission spectra in the frequency range 40 GHz $\leq \nu \leq $ 700 GHz were recorded using a quasioptical technique utilizing backward-wave oscillators as coherent light sources[@volkov]. Combining this method with a superconducting split-coil magnet equipped with optical windows allows to carry out transmission experiments in fields up to 8T. The data were obtained in the frequency-sweep mode at constant magnetic field. However, in some cases field-sweep measurements were performed because this procedure enhances the accuracy of determination of the resonance frequency. The frequency-dependent transmission spectra were analyzed using the Fresnel optical formulas for a transmission coefficient of a plane-parallel plate[@born]. The relative transparency of the sample in the frequency range investigated resulted in the observation of interference patterns in the spectra. The observation of these interferences allowed the calculation of the optical parameters of the sample without measuring the phase shift of the transmitted signal. The dispersion of the magnetic permeability was taken into account assuming a harmonic oscillator model for the complex magnetic permeability: $$\mu ^{*}(\nu )=\mu _{1}+i\mu _{2}=1+\Delta \mu \nu _{0}^{2}/(\nu
_{0}^{2}-\nu ^{2}+i\nu g) \label{eqreson}$$ where $\nu _{0}$, $\Delta \mu $ and $g$ are eigenfrequency, mode strength and width of the resonance respectively. The dielectric parameters of the sample $(n^{*}=n+ik)$ were assumed to behave regular in the vicinity of the resonance frequency. Hence, the frequency-sweep measurements allowed to obtain absolute values of the parameters of AFMR lines.
Fig.\[figmagn\] shows the low-temperature magnetization curves of single-crystalline La$_{0.95}$Sr$_{0.05}$MnO$_{3}$ for different orientations of the magnetic field. The magnetization along the c-axis shows a spontaneous magnetization and therefore is identified as the direction of the weak ferromagnetic moment. The magnetization along the crystallographic b-axis reveals the weakest field dependence and thus resembles the data of a simple antiferromagnet along the easy (antiferromagnetic) axis[@kittel].
In order to understand the magnetization measurement and the submillimeter spectra quantitatively, a two-sublattice model has been adopted. The two-sublattice model is a widely used approximation to describe the properties of magnetically ordered materials [@moria]. It was originally applied to the spin-wave spectrum of manganites by de Gennes [@degennes]. For a realistic description, additional contributions to the free energy, i.e. the single ion anisotropy $D_{x}\Sigma _{i}S_{xi}^{2}+D_{z}\Sigma
_{i}S_{zi}^{2}$ and the Dzyaloshinsky-Moria (D-M) antisymmetric exchange interactions $\Sigma _{i,j}{\bf d}_{ij}[S_{i}S_{j}]$ have to be taken into account. In the classical approximation the free energy at T=0 is given by:
$$\begin{array}{l}
F({\bf m,l})=\frac{1}{2}A{\bf m}^{2}-B|{\bf m}|+\frac{1}{2}%
K_{x}(m_{x}^{2}+l_{x}^{2})+ \\
+\frac{1}{2}K_{z}(m_{z}^{2}+l_{z}^{2})-d(m_{z}l_{y}-m_{y}l_{z})-M_{0}{\bf mH}
\end{array}
\label{eqfree}$$
In Eq.(\[eqfree\]) the (x,y,z) axes of the coordinate system are directed along the crystallographic axes (a,b,c) of the sample (a=5.547Å,b=5.666Å,c=7.725Å). The first and the second terms of Eq.(\[eqfree\]) describe antiferromagnetic and ferromagnetic (double) exchange, the third and fourth terms give the single ion anisotropy, the fifth term describes the D-M exchange while the last term takes into account effects of an external magnetic field. In Eq.(\[eqfree\]), ${\bf m}$ and ${\bf l}$ are dimensionless ferro- and antiferromagnetic vectors, which are defined as $%
{\bf m}=({\bf M}_{1}+{\bf M}_{2})/2M_{0}$, ${\bf l}=({\bf M}_{1}-{\bf M}%
_{2})/2M_{0}$ and satisfy the conditions ${\bf ml}=0$, ${\bf m}^{2}+{\bf l}%
^{2}=1$ since the sublattices ${\bf M}_{1}$ and ${\bf M}_{2}$ are assumed to be saturated at $T=0$. The parameter $B$ describes the DE interaction. $K_{x,z}>0$ are anisotropy constants stabilizing the $A_{y}F_{z}
$ configuration in pure LaMnO$_{3}$. $d$ is the interlayer antisymmetric exchange constant. $M_{0}=0.95M_{0}(Mn^{3+})+0.05M_{0}(Mn^{4+})=3.95\mu _{%
\text{B}}$, is the saturation magnetization of the sublattices. The equilibrium arrangement of the sublattices has been obtained minimizing the free energy given by Eq.(\[eqfree\]). The frequencies of the resonance modes were calculated in the limit of small perturbations from the equations of motion $d{\bf M}_{i}/dt=\gamma [{\bf M}_{i}\times \partial F/\partial
{\bf M}_{i}],$ $(i=1,2)$, where $\gamma $ is the gyromagnetic ratio.
The solutions of Eq.(\[eqfree\]) for the Hc were published previously [@europhys]. The full set of solution for the field orientation along the a and b-axes is too lengthy and will be published in full form elsewhere[@formeln]. However, in order to understand the experimental data qualitatively some approximations can easily be made. Assuming $(B,d,K_{x},K_{z},M_{0}H)\ll A\,$, the approximate solution for the magnetization can be written as:
$$M_{x}\equiv M_{0}m_{x}=\chi _{\perp }H_{x}(B+d)/d\text{ , }{\bf H}%
=(H_{x},0,0) \label{eqmx}$$
$$M_{y}\equiv M_{0}m_{y}=\chi _{rot}H_{y}\text{ , }{\bf H}=(0,H_{y},0)
\label{eqmy}$$
$$M_{z}\equiv M_{0}m_{z}=M_{z}^{0}+\chi _{\perp }H_{z}\text{ , }{\bf H}%
=(0,0,H_{z}) \label{eqmz}$$
where $M_{z}^{0}\equiv M_{s}=M_{0}(B+d)/(A+K_{z})$ is the spontaneous magnetic moment along the c-axis, $\chi _{\perp }=M_{0}^{2}/(A+K_{z})$ and $%
\chi _{rot}=M_{s}^{2}/K_{z}$ are the transverse and rotational susceptibilities respectively.
The analysis of Eq.(\[eqmz\]) shows that the z-axis exhibits weak ferromagnetism as the magnetization is nonzero in the absence of an external magnetic field. The magnetization along the y-axis (Eq. \[eqmy\]) is determined by the small rotational susceptibility and disappears in the pure antiferromagnetic case $(B=d=0)$. The low-field susceptibility along the x-direction (Eq. \[eqmx\]) is enhanced compared to the z-axis by the factor $(B+d)/d$. Qualitatively similar behavior of the magnetization is observed in Fig. \[figmagn\]. The solid lines in Fig. \[figmagn\] were calculated using the exact expressions based on Eq.(\[eqfree\]) and describe the experimental data reasonably well. A small static moment along the y-axis appears to be strongly angle dependent and possibly is due to some residual influence of the spontaneous moment along the z-axis. The absolute values of the parameters of the model were obtained by simultaneously fitting the magnetization curves and the values of the resonance frequencies in the absence of magnetic field. Despite the relatively large number of parameters in Eq.(\[eqfree\]) $%
(A,B,d,K_{x},K_{z})$, the requirement of a simultaneous fit allows the unambiguous determination of the parameters: $A=4.67\cdot 10^{7}$ erg/g, $%
B=7.4\cdot 10^{6}$ erg/g, $K_{z}=3.33\cdot 10^{6}$ erg/g, $K_{x}=3.42\cdot
10^{6}$ erg/g, $d=2.1\cdot 10^{6}$ erg/g, and $M_{0}=92.14$ emu/g. From the values of $A$ and $K_{x}$ the interlayer exchange ($%
J_{2}=-0.37meV$) and the single-ion anisotropy ($C=0.11meV$) constants can be calculated[@europhys] which are in good agreement with neutron scattering data for La$_{0.95}$Ca$_{0.05}$MnO$_{3}$[@laca05] and for La$%
_{0.95}$Sr$_{0.06}$MnO$_{3}$[@hennion].
Fig. \[figspectra\] shows the transmission spectra of the La$_{0.95}$Sr$%
_{0.05}$MnO$_{3}$ single crystal at low temperatures. The solid lines were calculated using the Fresnel equations and Eq.(\[eqreson\]) as described above. The parameters of the magnetic mode were obtained by fitting the transmission spectra. In addition, the frequency position and the appearance of the modes were also examined using field sweeps at fixed frequencies. From the data shown in Fig.\[figspectra\] two peculiarities of the observed AFMR modes immediately become clear: i) the observed lines have unique excitation conditions: $\widetilde{h}\parallel $ c -axis for the high-frequency mode and $\widetilde{h}\parallel $ b -axis for the low-frequency mode, and ii) no splitting of the AFMR lines is observed in finite magnetic fields for any geometry of the experiment. Both conclusions are characteristic properties of a canted antiferromagnetic structure[@herrmann] and follow naturally from the solution of Eq.(\[eqfree\]).
The magnetic field dependencies of the resonance frequencies of both AFMR lines are shown in Fig.\[figmode\]. The solid lines in Fig.\[figmode\] were calculated on the basis of the two-lattice model, discussed above. However, the parameters of the model [*were already fixed*]{} by fitting the magnetization curves and absolute values of the AFMR frequencies in the absence of magnetic field. Having this in mind, the theoretical curves describe the experimental data reasonably well. The most important feature of Fig. \[figmode\] is the softening of the FM-mode for B$\parallel $b. This softening represents a common property of magnetic resonance in antiferromagnets and is followed by the field-induced rearrangement of the magnetic structure (spin-flop) at a critical value of magnetic field. The softening of the FM-mode at low fields is in good agreement with the model calculations. However, the behavior for higher fields (B$\sim $7T) significantly deviates from the model predictions. These deviations are most probably due to the extreme sensitivity of the data with respect to the exact orientation of the static magnetic field and the neglect of the higher-order terms in Eq.(\[eqfree\]). The angular dependence of a critical behavior in a canted antiferromagnet has been calculated in details by Hagedorn and Gyorgy [@angledep]. These calculations show that already a misalignment of the magnetic field as low as one degree strongly suppress the softening of the FM-line in the vicinity of the critical field. Most probably similar effects explain the deviations observed in Fig. \[figmode\].
Within the presented model it is also possible to calculate the absolute intensities of the AFMR modes. In the absence of static field the solution of Eq. (\[eqfree\]) gives $\Delta \mu _{xx}=0.0080$ and $\Delta \mu
_{zz}=0.0136$, using the parameters obtained above. These values are in good agreement with the experimental values $\Delta \mu _{xx}=0.012\pm 0.003$ and $\Delta \mu _{zz}=0.0120\pm 0.0010$.
Finally, we discuss the possible explanation of the above-presented data within the concept of phase separation in the ferromagnetic droplets in an antiferromagnetic matrix. Already the magnetization data (Fig. \[figmagn\]) impose a set of constraints on the possible configuration of the phases. E.g. ferromagnetic moments have to be parallel to the c-axis and the b-axis has to be the antiferromagnetic easy axis. However, the most important consequences of a possible electronic phase separation follow for the properties of the magnetic resonances:
- the antiferromagnetic phase would reveal an AFMR mode with a resonance frequency similar to the frequencies in pure LaMnO$_{3}$ ($\nu \sim $18 cm$%
^{-1}$).
- this AFMR mode should split into two modes in the presence of magnetic field, as was observed by Mitsudo et al.[@mitsido] in pure LaMnO$_{3}$.
- a ferromagnetic line arising from the ferromagnetic droplets has to be observed. The frequency of this mode is expected at substantially lower frequencies ($\nu \sim 10GHz$) as observed by Lofland et al.[@lofland] in La$_{0.9}$Sr$_{0.1}$MnO$_{3}$. The resonance frequency of this mode should increase roughly linear with external magnetic field up to frequencies 150-250 GHz for B=7T.
None of these properties could be detected in the present experiment. Instead, the observed picture can be well described using the canted magnetic structure.
In conclusion, twin-free single crystals of La$_{0.95}$Sr$_{0.05}$MnO$_{3}$ were grown by the floating-zone method. The low-temperature magnetization was measured along the principal crystallographic directions. High field AFMR spectra of this sample were investigated in the frequency range 60-700 GHz and for magnetic fields up to B=8T. Two AFMR lines having different excitation conditions were detected at low temperatures. The softening of the low-frequency mode was observed for the orientation of the static field Bb and is explained by approaching to the spin-flop transition. The full data set, obtained for the La$_{0.95}$Sr$_{0.05}$MnO$_{3}$ single crystal, can be easily explained as arising from a canted magnetic structure and clearly contradicts the concept of phase separation into ferro- and antiferromagnetic regions.
This work was supported in part by BMBF (13N6917/0 - EKM), by DFG (Pi 372/1-1), by RFBR (99-02-16849), and by INTAS (97-30850).
G. H. Jonker and J. H. van Santen, Physica [**16**]{}, 337 (1950); [**19**]{}, 120 (1953)
E. O. Wollan and W. C. Koehler, Phys. Rev. [**100**]{}, 545 (1955).
P.-G. de Gennes, Phys. Rev. [**118**]{}, 141 (1960).
C. Zener, Phys. Rev. [**82**]{}, 403 (1951).
E. L. Nagaev, Sov. Phys.- Uspekhi [**39**]{}, 781 (1996); Phys. Rev. B [**58**]{}, 2415 (1998); L. P. Gor’kov, Sov. Phys.- Uspekhi [**41**]{}, 589 (1998)
S. Yunoki et al., Phys. Rev. Lett. [**80**]{}, 845 (1998); S. Yunoki, A. Moreo, and E. Dagotto, Phys. Rev. Lett. [**81**]{}, 5612 (1998); A. Moreo et al., cond-mat/9901057.
J. B. Goodenough and J.-S. Zhou, Nature [**386**]{}, 229 (1997); P. Littlewood, Nature [**399**]{}, 529 (1999).
M. Hennion et al., Phys. Rev. Lett. [**81**]{}, 1957 (1998); cond-mat/9910361 (1999).
G. Allodi et al., Phys. Rev. B [**56**]{}, 6036 (1997); Phys. Rev. B [**57**]{}, 1024 (1998).
A. A. Mukhin et al., Europhys. Lett., in press.
A. M. Balbashov et al., J. of Cryst. Growth, [**167**]{}, 365 (1996).
A. A. Mukhin et al., JETP Lett., [**68**]{}, 356 (1998).
A. Urushibara et al., Phys. Rev. B [**51**]{}, 14103 (1995).
G. V. Kozlov and A. A. Volkov in [*Millimeter and Submillimeter Wave Spectroscopy of Solids*]{}, Ed. by G. Grüner (Springer, Berlin, 1998), p. 51.
M. Born and E. Wolf, [*Principles of optics*]{} (Pergamon, Oxford, 1986).
Ch. Kittel, [*Einführung in die Festkörperphysik*]{}, (Oldenburg, München, 1991) p. 528.
T. Morya, in [*Magnetism*]{}, edited by G. T. Rado and H. Suhl, Vol. I (Academic Press, NY, 1984) p. 85.
A. A. Mukhin et al., to be published.
F. B. Hagedorn and E. M. Gyorgy, Phys. Rev. [**174**]{}, 540 (1968).
M. Hennion et al., Phys. Rev. B [**56**]{}, 497 (1997); F. Moussa et al., [*ibid*]{}. [**60**]{}, 12299 (1999).
G. F. Herrmann, J. Phys. Chem. Sol. [**24**]{}, 597 (1963); Phys. Rev. [**133**]{}, A1334 (1964).
S. Mitsudo et al., J. Magn. Magn. Mater. [**177-181**]{}, 877 (1998).
S. E. Lofland et al., J. Phys.: Cond. Matt. [**9**]{}, L633 (1997).
|
---
abstract: 'Malicious software are categorized into families based on their static and dynamic characteristics, infection methods, and nature of threat. Visual exploration of malware instances and families in a low dimensional space helps in giving a first overview about dependencies and relationships among these instances, detecting their groups and isolating outliers. Furthermore, visual exploration of different sets of features is useful in assessing the quality of these sets to carry a valid abstract representation, which can be later used in classification and clustering algorithms to achieve a high accuracy. In this paper, we investigate one of the best dimensionality reduction techniques known as t-SNE to reduce the malware representation from a high dimensional space consisting of thousands of features to a low dimensional space. We experiment with different feature sets and depict malware clusters in 2-D. Surprisingly, t-SNE does not only provide nice 2-D drawings, but also dramatically increases the generalization power of SVM classifiers. Moreover, obtained results showed that cross-validation accuracy is much better using the 2-D embedded representation of samples than using the original high-dimensional representation.'
author:
- Mohamed Nassar
- Haidar Safa
bibliography:
- 'malwareViz.bib'
title: Throttling Malware Families in 2D
---
Introduction
============
Security breaches are executed through malwares and are a major threat to the Internet today. There are several forms of malware ranging from viruses and spam bots to trojan horses and rootkits [@stallings2016cryptography]. Recently, the Petya ransomware [@akkas2017malware] crashed shipping companies, ports, and law agencies. This malware targets the master boot record of a machine and prohibits the operating system from normal execution. It then spreads and encrypts all the system files. A message appears on the screen stating the amount of ransom to decrypt the files. The payment is through crypto-currencies. The prominent expansion of malwares is due to their metamorphic and polymorphic techniques that give the ability to change their code as they propagate. In addition, malwares adopt new ways to detect the environments where they are running, hence hindering their detection and making dynamic analysis difficult if not impossible.
Visual analytics provides approaches to obtain an understanding from complex data. It aims at developing methods that allow analysts to examine the processes underlying the data [@ellis2010mastering]. Visual exploration of malware families is a pre-processing step of a more in-depth malware family analysis, as it allows for the development of intuitions and hypotheses about the discriminative power of a set of contextual or behavioral features. However, visualizing malware families in low dimensional space (2-D, 3-D) is a topic that received little attention in the literature. Malware data are fundamentally different than text and images[^1] which motivates investigating ways to adapt existing approaches or inventing new ones. For instance a byte in malware has different meanings in different contexts, in contrast to a byte representing pixel intensity in an image.
In this paper, we experiment with the best low-dimensional embedding technique known as t-SNE (Student-t distribution – Stochastic Neighborhood Embedding) for depicting malware clusters and features. We propose a pipeline for feature extraction and selection, followed by visualization. We compare the raw classification accuracy at the high-dimensional and low-dimensional spaces for n-grams features. Finally We propose a new first-insight classifier based on t-SNE and SVM. Note that our goal is not to propose a very high accuracy classifier or to compete with extensive feature selection approaches. Instead, we aim at exploring the visualization space of malware families and to which extent such a pre-processing procedure might be useful for analyzing a typical malware dataset. The remaining of the paper is organized as follows. Section 2 surveys relevant related work. Section 3 presents our proposed methodology. In Section 4, we discuss implementation and results. We finally conclude in Section 5 and present future research directions.
Background and Related Work
===========================
Classifying and Clustering malware families were addressed in many recent work in the literature. In [@li2010challenges], the current automated approaches for malware clustering were summarized. The paper considered the high accuracy obtained by six commercial anti-viruses as biased since unbalanced datasets were used where most malware instances are easy to classify. A plagiarism detector algorithm was applied on the same dataset and yielded the same accuracy results compared to those of the anti-viruses, though the plagiarism detector does not have any expert knowledge about malwares. In [@bayer2009scalable] a scalable, behavior-based malware clustering approach was proposed. This approach aims to isolate outliers that exhibit a novel behavior to be further analyzed. It used a recent technique called taint tracking to build behavioral profiles and locality sensitive hashing for clustering these profiles. Malware instance visualization was proposed in [@nataraj2011malware]. This approach suggested transforming the binary into a vector of 8-bit integers, which can be reshaped into a matrix and therefore viewed as a gray-scale image. This technique proves useful in increasing the accuracy of malware classifiers. The work presented in this paper is different such that we focus on visualizing families of malwares as scatter plots using t-SNE [@maaten2008visualizing], an embedding technique that allows visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. The technique is a variation of Stochastic Neighbor Embedding (SNE).
SNE starts by converting the high-dimensional Euclidean distances between datapoints into conditional probabilities that represent similarities. The similarity of datapoint $x_j$ to datapoint $x_i$ is the conditional probability, $p_{j|i}$ , that $x_i$ would pick $x_j$ as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at $x_i$. For the low-dimensional counterparts $y_i$ and $y_j$ of $x_i$ and $x_j$, it is possible to compute a similar conditional probability $q_{j|i}$. SNE aims to find a low-dimensional data representation that minimizes the mismatch between $p_{j|i}$ and $q_{j|i}$. To do so, SNE minimizes the sum of Kullback-Leibler divergences over all datapoints using a gradient descent method. t-SNE differs from the old SNE in two ways: (1) it uses a symmetrized version of the SNE cost function with simpler gradients and (2) it uses a heavy-tailed Student-t distribution rather than a Gaussian distribution to compute the similarity between two points in the low-dimensional space. This allows t-SNE to alleviate both the so-called crowding problem and the optimization problems of SNE [@maaten2008visualizing].
The dataset we use is the training set from Microsoft Malware Classification Challenge [@ronen2018microsoft] released in 2015, which since has been studied by researchers in more than 50 publications targeting feature engineering, deep learning, clustering and classification approaches. For instance, [@gibert2016convolutional] proposes using convolutional neural networks for feature extraction and classification based on the binary and reconstructed assembly files. Our work takes another direction and focuses on the visualization question.
Proposed Methodology
====================
In this section we describe our proposed methodology which copes with relatively large data sets. We have used the training dataset from Microsoft malware classification challenge (Big 2015) having 10868 labeled instances [@ronen2018microsoft]. The standard version of t-SNE having quadratic complexity in terms of the number of instances $O(N^2)$ might be applied. For larger data-sets, other versions of t-SNE are proposed such as random-walk based sampling of landmark points or using specialized data structures leading to $O(N \log N)$ complexity [@van2014accelerating]. We particularly used the version of scikit-learn with the Barnes-Hut approximation running in $O(N \log N)$ time.
Fig. \[fig1\] shows the pipeline of our proposed methodology. Starting from a labeled corpus of files containing the malwares payload in hexadecimal format, we extract n-byte grams (3, 4 and 5). Note that the number of features grow exponentially such that for 3-byte grams we have $2^{8*3}$ possible words, for 4-byte grams it is $2^{8*4}$ and so on. Most current machine learning libraries cannot handle this number of features even in sparse format. For example, Scikit-learn accepts feature indices less than a positive 4-bytes signed integer ($2^{31}-1$). Therefore, in the proposed methodology, we hash the feature indices to 22-bits integers. We also proceed by early removal of rare words that appear less than k times (we assumed k = 3). These words probably represent addresses in memory or literals and have little differentiation power. This technique is very efficient in reducing the storage amount of the features set. We store the resulting features along with the instance label in sparse format (LibSVM/SVMLight format) as one line per instance in the output text file. Each feature is represented by an index (the hash of the n-bytes gram) and a value which is the number of occurrences of this n-bytes gram in the malware instance. This stage is implemented by using the Sally tool [@rieck2012sally] which is an efficient feature extraction tool that generates n-grams besides other features such as TF-IDF [@chowdhury2010introduction].
We proceed with a first feature selection stage using the $\chi^2$ statistical test-based selector where our target is to reduce the number of features to 1,000. This limit reduces the complexity of computing the pair-wise distances in t-SNE. However, we are starting with a much larger number ($2^{22} = 4,194,304$ feature-space). The time complexity of the $\chi^2$ selector is $O(n_\text{classes} * n_\text{features}$). The features that are the most likely to be independent of class and therefore irrelevant for classification are removed.
We also can reduce the space complexity of this stage (mainly because of memory limitations) by sampling the instances in equal proportions to their family sizes. If sampling is used than we must use the generated $\chi^2$ selector model to transform the complete dataset and keep the top $1,000$ features of each instance. For the dataset in question we did not use sampling.
![Pipeline for malware family visualization.[]{data-label="fig1"}](pipeline2){width="\textwidth"}
Optionally, we apply a PCA (Principal Component Analysis) transformation to reduce further the number of features to the range of 30-50 features. This speeds up the computation of pairwise distances between the data-points in the next stage and suppresses some noise without severely distorting the inter-point distances [@maaten2008visualizing]. t-SNE then embeds these features in 2 dimensions. The malware instances are depicted as scatter plots. t-SNE outperforms other data embedding techniques such as PCA, Sammon mapping, Isomap and LLE.
Implementation, Preliminary Results, and Interpretation
========================================================
In this section, we describe the implementation environment and setup, the dataset used, the hyper parameters then we discuss the obtained results.
Setup, implementation, and tools
--------------------------------
All our experiments were performed using a commercial off-the-shelf laptop with a 64-bit Ubuntu 16.04 LTS operating system, an Intel core i5-5200U CPU (4 cores, 2.20GHz) 8 GB RAM and 1 TB Hard disk. Our implementation was based on Python v3.6, numpy v1.13, scipy v0.19 and the Scikit-learn v0.19 library. The plots are generated using the BokehJS v0.12 library. We have used Sally [@rieck2012sally] for feature extraction. Our code is available at <https://github.com/mnassar/malware-viz> under form of Python Jupyter notebooks for further exploration and result reproducibility.
Dataset
-------
The dataset is the training set from Microsoft Malware Classification Challenge [@ronen2018microsoft], which includes 10868 labeled samples. For each sample, the raw data and meta data are provided. The raw data contains the hexadecimal representation of the file’s binary content, without the Portable Executable (PE) header to ensure sterility. The metadata manifest is a log containing various metadata information extracted from the binary, such as function calls, strings, etc. This was generated using the IDA disassembler tool. In our implementation, we focused solely on the raw data. However, we consider augmenting our visualization with the meta data for future work. The raw training data volume is about 41 GBytes. However, our feature selection method reduces this to about 6.4 GBytes for the sum of three feature sets (3, 4 and 5-grams). The dataset contains malwares belonging to the following 9 families: Ramnit, Lollipop, Kelihos Ver. 3, Vundo, Simda, Tracur, Kelihos Ver. 1, Obfuscator.ACY and Gatak. One challenge of this data set is the unbalanced sizes of different families. The distribution of instances for the training dataset is shown in Table \[tab1\]. It is shown that classes 4, 5, 6 and 7 are underrepresented as compared to the other families.
Class Family Type Nb. Of Instances Percentage (%)
------- ---------------- -------------------- ------------------ ----------------
1 Ramnit Worm 1541 14.20
2 Lollipop Adware 2478 22.80
3 Kelihos\_ver 3 Backdoor 2942 27.07
4 Vundo Trojan 475 4.37
5 Simda Backdoor 42 0.39
6 Tracur Trojan Downloader 751 6.91
7 Kelihos\_ver 1 Backdoor 398 3.66
8 Obfuscator.ACY Obfuscated malware 1228 11.30
9 Gatak Backdoor 1013 9.32
: Distribution of samples among the families.[]{data-label="tab1"}
t-SNE parameters
----------------
t-SNE implementation in Sklearn has many tunable parameters. We show the most important ones with a short description in Table \[tab2\]. Note that the t-SNE method is known to be little sensitive to these parameters. Nevertheless, a good tuning enhances the quality of the obtained clusters sometimes.
**Parameter** **Description** **Typical/default value**
---------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------
*Random\_state* This is the seed of random initialization in the embedded space. It is an important parameter to accomplish comparisons under same initial conditions (used with init = ’random’) No typical value. We have chosen 42 as our default.
*N\_iter* Maximum number of iterations for the optimization. Default: 1000
*Perplexity* The perplexity can be interpreted as a smooth measure of the effective number of neighbors. 5 – 50 Our default is 40
*Early exaggeration* Larger values of this parameter tend to start with distant clusters in the embedded space Default: 12
*metric* The distance between instances where each instance is a feature array. Euclidean
*Learning rate (lr)* It must not be too low or too high. If the cost function gets stuck in a bad local minimum increasing the learning rate may help. 10 – 1000 We have chosen 200 as our default.
: Tunable Parameters in t-SNE.[]{data-label="tab2"}
Sample of results
-----------------
Due to lack of space we do not show all obtained visual plots. However, we depict and interpret the most important ones. Fig. \[fig2\] shows the scatter plot of the 9 classes for 4-byte grams (no PCA). In this figure, it is directly noticed that each family is composed of one or multiple clusters. No one clear cluster per family exists as in the case of the MNIST dataset [@lecun1998gradient]. This is expected due to the polymorphic nature of malware data which has much more noise than what can be carried by handwritten digits or letters. Common evasion and obfuscation techniques may also explain why some families have intersection regions in between their clusters. Still we can identify big clusters for each of the big families (1, 2, 3, 8, 9) and even some smaller clusters for the families 4 (Vundo) and 7 (Kelihos\_ver 1). It is also important to notice the existence of some outliers.
To give a chance to the underrepresented families, we started over with only their instances in the pipeline. Obtained results are shown in Fig. \[fig3\]. These results are much better, and we can observe clear groups for each of these small families. Next, we take only the largest two classes (2 and 3) to compare plots between different sets of features (3-bytes grams, 4-bytes grams and 5-bytes grams). Obtained results are illustrated in Fig. \[fig4\]. They show that to some extent, 5-grams (Fig. \[fig4\](c)) have less overlap between clusters than 3-grams (Fig. \[fig4\](a)) or 4-grams (Fig. \[fig4\](b)). We wanted to correlate this with the classification accuracy on the original dataset (the one after feature selection which is 5420 instances \* 1000 features) and the transformed dataset (5420 instances \* 2 features). Results in Table \[tab3\] shows that training accuracy is almost perfect for all three feature sets. However, training accuracy is often not a good metric since the classifier might overfit the training set.
![4-grams, no-PCA, 9 classes, perplexity=40, lr=200.[]{data-label="fig2"}](4grams-no-pca-9classes){width="70.00000%"}
![4-grams, no-PCA, classes 4,5,6,7, perplexity=40, lr=200.[]{data-label="fig3"}](4grams-no-pca-4567classes){width="70.00000%"}
![(a) Classes 2 and 3 (3-grams, perplexity=40, lr=200); (b) Classes 2 and 3 (4-grams, perplexity=40, lr=200); (c) Classes 2 and 3 (5-grams, perplexity=40, lr=200) []{data-label="fig4"}](3grams "fig:"){width="48.00000%"} ![(a) Classes 2 and 3 (3-grams, perplexity=40, lr=200); (b) Classes 2 and 3 (4-grams, perplexity=40, lr=200); (c) Classes 2 and 3 (5-grams, perplexity=40, lr=200) []{data-label="fig4"}](4grams "fig:"){width="48.00000%"}\
(a) (b)\
![(a) Classes 2 and 3 (3-grams, perplexity=40, lr=200); (b) Classes 2 and 3 (4-grams, perplexity=40, lr=200); (c) Classes 2 and 3 (5-grams, perplexity=40, lr=200) []{data-label="fig4"}](5grams "fig:"){width="48.00000%"}\
(c)
We compute the two-fold cross validation accuracy which is much more indicative of the generalization power of the classifier. Astonishingly, the two-fold cross validation accuracy is very poor on the original dataset and dramatically much better on the embedded dataset. This can be explained by the fact that t-SNE groups the datapoints into separate clusters in a low dimensional space, which is at the origin of the design of support vector classifiers with the Radial Basis Function (RBF) kernel. These classifiers shine under this kind of settings. The SVC accuracy (%) for the three feature sets with default Sklearn parameters (RBF kernel, $C=1.0$, $\gamma=1/n_\text{features}$) are shown in Table \[tab3\]. Better SVC accuracy results are usually obtained after a grid search for the best hyperparameters $C$ and $\gamma$.
-------------- ---------- ---------- --------------------- ---------------------
Feature Set Training Training Two-fold cross- Two-fold cross-
Accuracy Accuracy Validation Accuracy Validation Accuracy
(1000-d) (2-d) (1000-d) (2-d)
3-byte grams 99.98 99.98 67.91 99.57
4-byte grams 100.00 99.96 64.06 99.88
5-byte grams 99.98 99.98 68.13 99.92
-------------- ---------- ---------- --------------------- ---------------------
: SVC Accuracy on different feature-sets (classes 2 and 3).[]{data-label="tab3"}
Next, we examine the performance of t-SNE on unbalanced families. We take the extreme case by choosing the largest family (Kelihos\_ver 3 – class 3) and the smallest family (Simda – class 5). Results that are presented in Fig. \[fig5\] show that t-SNE can isolate the class 5 in a small cluster. They also show that 4-grams and 5-grams perform better than 3-grams in isolating clusters of the two classes. Fig. \[fig5\](d) shows that choosing a bad perplexity value might degrade the clustering quality.
![(a) Classes 3 and 5 (3-grams, perplexity=40, lr=20); (b) Classes 3 and 5 (4-grams, perplexity=40, lr=200); (c) Classes 3 and 5 (5-grams, perplexity=40, lr=200); (d) Classes 3 and 5 (5-grams, perplexity=5, lr=200) []{data-label="fig5"}](largest_vs_smallest_3grams "fig:"){width="45.00000%"} ![(a) Classes 3 and 5 (3-grams, perplexity=40, lr=20); (b) Classes 3 and 5 (4-grams, perplexity=40, lr=200); (c) Classes 3 and 5 (5-grams, perplexity=40, lr=200); (d) Classes 3 and 5 (5-grams, perplexity=5, lr=200) []{data-label="fig5"}](largest_vs_smallest_4grams "fig:"){width="45.00000%"}\
(a) (b)\
![(a) Classes 3 and 5 (3-grams, perplexity=40, lr=20); (b) Classes 3 and 5 (4-grams, perplexity=40, lr=200); (c) Classes 3 and 5 (5-grams, perplexity=40, lr=200); (d) Classes 3 and 5 (5-grams, perplexity=5, lr=200) []{data-label="fig5"}](largest_vs_smallest_5grams "fig:"){width="45.00000%"} ![(a) Classes 3 and 5 (3-grams, perplexity=40, lr=20); (b) Classes 3 and 5 (4-grams, perplexity=40, lr=200); (c) Classes 3 and 5 (5-grams, perplexity=40, lr=200); (d) Classes 3 and 5 (5-grams, perplexity=5, lr=200) []{data-label="fig5"}](largest_vs_smallest_perp_5 "fig:"){width="45.00000%"}\
(c) (d)\
t-SNE shows similar performance in enhancing the classification accuracy as shown in Table \[tab4\]. Note that since the classes are severely unbalanced, an accuracy of 98.60 would be simply obtained if the classifier considers all the data points a belonging to the majority class 5. The two-fold cross validation accuracies on the original dataset are bad in this sense. However, we notice that the two-fold cross validation accuracy is much better in the embedded space. 3-grams features perform the worst as it can be expected by examining the corresponding scatter plot.
-------------- ---------- ---------- --------------------- ---------------------
Feature Set Training Training Two-fold cross- Two-fold cross-
Accuracy Accuracy Validation Accuracy Validation Accuracy
(1000-d) (2-d) (1000-d) (2-d)
3-byte grams 100.00 99.93 98.52 (bad) 99.83
4-byte grams 100.00 99.93 98.62 (bad) 99.90
5-byte grams 100.00 99.97 98.62 (bad) 99.90
-------------- ---------- ---------- --------------------- ---------------------
: SVC Accuracy on different feature sets (classes 3 and 5)[]{data-label="tab4"}
Finally, we want to validate this hypothesis on the complete dataset (9-classes). Results are shown in Table \[tab5\]. The training accuracy in 2-d is a bit smaller than in 1000-d but allows much better generalization of the classification model as clearly inferred from the cross-validation accuracy results.
-------------- ---------- ---------- --------------------- ---------------------
Feature Set Training Training Two-fold cross- Two-fold cross-
Accuracy Accuracy Validation Accuracy Validation Accuracy
(1000-d) (2-d) (1000-d) (2-d)
3-byte grams 99.66 96.84 56.76 94.26
4-byte grams 99.58 96.22 55.26 93.13
5-byte grams 98.25 95.69 60.24 92.58
-------------- ---------- ---------- --------------------- ---------------------
: SVC Accuracy on different feature sets (all classes)[]{data-label="tab5"}
Testing Accuracy
----------------
In this section we further assess the idea of squeezing the dimensions into a small hyperspace using t-SNE than expanding it back to infinite dimensional space using the RBF kernel with SVM. We wanted to estimate the testing accuracy using the raw unlabeled test dataset (10873 instances). Note that t-SNE is a non-parametric mapping, therefore we cannot use the learnt model to map the test datapoints to the embedded space which is formed by the train datapoints. As an alternative we run t-SNE on the full dataset composed of both train and test datapoints (Another approach is to train a multivariate regressor to predict the map location from the input data [@van2009learning]). We then fit an SVC model solely based on the train embedded datapoints. This SVC model is used to estimate probabilistic predictions of the membership of each embedded test point to each of the 9 possible classes. The pipeline of this approach is depicted in Fig. \[fig6\].
![Testing Accuracy Pipeline[]{data-label="fig6"}](testing_){width="100.00000%"}
Note that the labels of these instances are not available, and the only way of evaluation is to obtain the multi-class logarithmic loss by submitting our predictions in probabilistic form to the dataset hosting platform (Kaggle) online. The equation for the logarithmic loss is: $$\text{logloss}= -\frac{1}{10873}\sum_{i=1}^{10873}\sum_{j=1}^{9}y_{ij}\log p_{ij}$$ where $y_{ij}=1$ if $i$ belongs to class $j$ and $0$ otherwise, $\log$ is the natural logarithm and $p_{ij}$ is the probability that $i$ belongs to class $j$ as given by the classifier.
Our classifier achieves a testing logloss of 0.1719. This is fairly acceptable given the simplicity of the employed feature set (1000 best features among 1,2,3,4 and 5 n-bytes grams) and without recurring to any involved feature engineering. A clueless classifier scores 2.1972. For this experiment we have used an m5.4xlarge AWS EC2 instance (64 GB RAM) and a 500 GB volume.
Conclusion
==========
In this paper, we have successfully applied feature extraction, selection, embedding and visualization over a recent malware dataset. We have proposed a pipeline that can cope with dataset of similar or larger size. We use the t-SNE algorithm to embed the malware datapoints in 2D and visualize them as scatter plots. A very interesting result is that compressing the data using t-SNE dramatically enhances the cross-validation accuracy of Support Vector Machines classifiers. t-SNE shapes the clusterability of datapoints in the embedded space, which is very appealing to SVM classifiers with the RBF kernel.
In future work, we aim to experiment with other feature sets, for instance by analyzing the assembly data files. We also want to assess the viability of the SVM–t-SNE classifier over other data sets. Another direction is to work on implementing t-SNE in a 3D WebGL framework and integrate it in Jupyter notebooks. Available GPUs can also be used to bear some of the tedious computations.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported in part by a grant from the University Research Board of the American University of Beirut, Lebanon.
[^1]: <http://www.jsylvest.com/blog/2017/12/malconv/>
|
---
abstract: 'Metal halide perovskites are a remarkable class of materials with particular promise for application in optoelectronics. Like in any semiconductor, defects in perovskites play a critical role in determining their properties and performance in optoelectronic devices, however many open questions regarding the nature of ionic defects remain unanswered. In this work we apply impedance spectroscopy and deep-level transient spectroscopy to characterize the ionic defect landscape in methylammonium lead triiodide (MAPbI~3~) perovskites in which defects were purposely introduced by fractionally changing the precursor stoichiometry. Our results indicate that variation of the ionic defect landscape has a profound influence on the electronic landscape, exemplified by its impact on the device built-in potential, and consequently, the open-circuit voltage. Moreover, we find that all measured ionic defects fulfill the Meyer–-Neldel rule with a characteristic energy, which corresponds to the underlying ionic hopping process in perovskite materials. These findings allow a defect categorization of our data and literature values.'
author:
- Sebastian Reichert
- Qingzhi An
- 'Young-Won Woo'
- Aron Walsh
- Yana Vaynzof
- Carsten Deibel
title: Probing the ionic defect landscape in halide perovskite solar cells
---
[^1]
Introduction
============
Triggered by the first demonstration of a perovskite solar cell in 2009,[@Kojima2009] significant research efforts have been devoted to the field of perovskite photovoltaics leading to a record power conversion efficiency of $25.2~\%$.[@Nrel] This remarkable performance is made possible by a combination of advantageous properties of perovskite materials, among which most noteworthy are the low exciton binding energies, high absorption coefficients, high charge carrier diffusion lengths and correspondingly long lifetimes of free charge carriers.[@Miyata2015; @Stranks2013; @Chouhan2017; @Leguy2016] Additionally, significant progress has been made over the last decade in the development of novel fabrication methods and device architectures as well as optimization by interfacial engineering.[@Saliba2018; @Li2018; @Bing2019; @Song2016; @Shi2018; @Tai2019]
Despite these advancements, several aspects of perovskite solar cells remain a challenge. For example, in many different fabrication approaches, mobile ions have proven to be a major limitation.[@Jacobs2017; @Yuan2016; @Rivkin2018; @Lee2019; @Kim2018] Mobile ions or ionic defects were shown to be the source of current density–voltage hysteresis and were linked to a reduced stability of devices.[@Snaith2014; @Tress2015; @Chen2015; @Jacobs2017; @Miyano2016; @Rivkin2018] Moreover, ionic defects that form states within the bandgap which act as recombination centers, can reduce the photovoltaic performance of the device.[@Meggiolaro2019; @Du2014] Despite their importance, characterization of ionic defects and their properties in perovskite materials is incomplete. According to calculations and experimental reports, the most likely native point defects in methylammonium lead triiodide (MAPbI~3~) perovskites are charged vacancies such as $V_\mathrm{I}^+$ and $V_\mathrm{MA}^-$ and interstitials such as $\mathrm{I}_i^-$ and $\mathrm{MA}_i^+$.[@Yin2014; @Buin2014; @Buin2015; @Senocrate2018; @Senocrate2019] Experimentally, ionic defects and their migration has been observed by a range of methods.[@Li2016_2; @Yuan2015_2; @Yang2017_2]
Noteworthy is the work by Futscher et al.[@Futscher2019], who employed transient capacitance measurements on MAPbI~3~ solar cells to reveal both a fast ($t<\mathrm{ms}$) and relatively slow ($t\sim\mathrm{s}$) species which varied by several orders of magnitude in both their concentration and diffusion coefficient. While in all their measurements the authors assigned the fast species to $\mathrm{I}_i^-$ and the slow species to $\mathrm{MA}_i^+$, they also observed variations in activation energies, diffusion coefficients and ion concentrations when measuring different samples fabricated either in their laboratory or that of others. This observation is not uncommon, especially in light of the wide range of reported defect parameters presented in literature for the same perovskite material.[@Futscher2019; @Eames2015; @Azpiroz2015; @Yin2014; @Yang2016; @Yang2017; @Rosenberg2017; @Samiee2014; @Duan2015; @Xu2019] One contributing factor to this observation is related to the method of evaluation of the transient ion-drift measurements. Recently, we developed an extended regularization algorithm for inverse Laplace transform for deep-level transient spectroscopy (DLTS) that reveals distributions of migration rates for ionic species instead of single migration rates.[@Reichert2020] This finding suggests that in part, the differences and inconsistencies reported in literature can originate from the fact that various experimental methods may probe different parts of the same ionic defect distribution.
Another significant contributing factor, is the high sensitivity of perovskite materials to their fabrication conditions. Subtle changes in the atmospheric environment[@Sheikh2015], annealing process[@Li2016], or perovskite precursor stoichiometry [@Fassl2018; @Falk2020] have all been shown to affect the properties of the perovskite layers. These changes will also influence the properties of the ionic defects. For example, the model reported by Meggiolaro et al.[@Meggiolaro2019] describes the dependence of defect formation energies on the microstructure of the perovskite layer and is in good agreement with the experimental results of Xing et al.[@Xing2016] Taken together, these observations highlight the need to investigate more deeply the ionic defect landscape in perovskite materials and identify fundamental processes that govern their formation and physical properties.
In this work, we purposefully tune the ionic defect landscape of MAPbI~3~ perovskite samples by fractionally modifying the stoichiometry of the perovskite precursor solution. This results in a gradual change in the densities of the various types of defects as suggested by both X-ray photoemission spectroscopy [@Fassl2018] and photoluminescence microscopy measurements.[@Fassl2019] Herein, we directly probe the variations to the ionic defect landscape by impedance spectroscopy (IS) and DLTS, and reveal the interplay between this defect landscape and the electronic landscape of the device. We compared ionic migration rates with literature values, and found that the systematic variation in our study allows to categorize the results from literature, leading to a remarkably good agreement. Moreover, we show that the temperature dependent diffusion parameters of all the ionic defects fulfill the Meyer–-Neldel rule, which we link to the fundamental hopping process of mobile ion transport in halide perovskite solar cells.
Results {#sec:results}
=======
To controllably tune the defect landscape in MAPbI~3~ perovskite solar cells, we exploited the method developed by Fassl et al.[@Fassl2018] to fabricate a series of samples from precursor solutions with gradually changing stoichiometry. In short, we start by intentionally preparing an understoichiometric solution, in which a slight deficiency of methylammonium iodide (MAI) is expected to result in films rich in vacancies such as $V_\mathrm{I}^+$ and $V_\mathrm{MA}^-$. By gradually increasing the MAI content in the solution, a stoichiometric ratio is reached, followed by a transition to an overstoichimetric regime, in which access of MAI increases the densities of $\mathrm{I}_i^-$ and $\mathrm{MA}_i^+$ interstitials. To eliminate the influence of different extraction layers, all devices share a common architecture, in which poly(3,4-ethylenedioxythiophene)-poly(styrenesulfonate) (PEDOT:PSS) is used for hole extraction, while \[6,6\]-phenyl-C~61~-butyric acid methyl ester (PC~61~BM) is used for electron extraction. A thin layer of bathocuproine (BCP) is introduced between the PC~61~BM layer and the Ag contact in order to achieve efficient hole-blocking.[@Wang2014_2; @An2019] The current density–voltage ($JV$) characteristics of the resulting photovoltaic devices are shown in Fig. S1. A very small hysteresis, often associated with the presence of mobile ions,[@Lee2019; @Kim2018; @Contreras2016; @Weber2018; @Calado2016] appears when sweeping in both voltage directions. The solar cell parameters averaged over both scan directions are shown in Fig. S2 and are in agreement with the previous report by Fassl et al.[@Fassl2018] In short, while the fill factor ($FF$) and short-circuit current ($J_\mathrm{sc}$) are only very slightly influenced by the changes in stoichiometry, the open-circuit voltage ($V_\mathrm{oc}$) and consequently the power conversion efficiency ($PCE$) strongly increase for increasing stoichiometric ratios.
Capacitance–voltage profiling {#sec:CV}
-----------------------------
![Dependence of power conversion efficiency $PCE$, built-in potential $V_\mathrm{bi}$ and effective doping density $N_\mathrm{eff}$ on stoichiometry (MAI:PbAc~2~) of the precursor solution. $PCE$ values are taken from Fig. S2 for comparison. $V_\mathrm{bi}$ can be corrected by estimating the potential drop caused by mobile ions at the interfaces as described in Sec. \[sec:discussion\].[]{data-label="fig:CV_parameter"}](CV_parameters_1.pdf)
Capacitance–voltage (CV) measurements may offer first insights into the ionic defect landscape of the devices. We performed these measurements at an ac frequency of 80 kHz using a fast sweep rate of 30 V/s in the reverse scan direction. Following the methodology of Fischer et al.,[@Fischer2018] the devices were pre-biased for 60 s at 1 V, in order to minimize the influence of mobile ions present at the interfaces of the active layer. The results of the CV measurements (Fig. S3) were evaluated using the Mott–Schottky approach,[@Sze2006; @Grundmann2010] $$\label{eq:MS}
1/C^2=\frac{2(V_\mathrm{bi}-V)}{e\epsilon_0\epsilon_\mathrm{R}N_\mathrm{eff}}.$$ where $V$ is the applied external voltage, $e$ is the elementary charge, $\epsilon_0$ is the absolute permittivity and $\epsilon_\mathrm{R}$ is the relative permittivity. By applying Eqn. (\[eq:MS\]) to the range dominated by the depletion capacitance, the built-in potential ($V_\mathrm{bi}$) and effective doping density ($N_\mathrm{eff}$) can be extracted. In Fig. \[fig:CV\_parameter\], these values are compared to the $PCE$ values from Fig. S2. The results indicate that both $V_\mathrm{bi}$ and $N_\mathrm{eff}$ increase with increasing stoichiometry. The increase in $V_\mathrm{bi}$ is in agreement with the findings of Fassl et al.,[@Fassl2018] where a shift in the exponential diode characteristics revealed a similar trend in $V_\mathrm{bi}$. Interestingly, the increase in $N_\mathrm{eff}$ suggests an overall higher defect density for overstoichiometric samples since ions introduce additional charges and affect the net doping concentration.[@Yuan2015; @Kim2014; @Wang2014] This is in agreement with the experimental observation of a lower photoluminescence quantum efficiency for samples with higher stoichiometry.[@Fassl2019]
Determining the defect landscape by IS and DLTS {#sec:IS_DLTS}
-----------------------------------------------
Advanced spectroscopic techniques such as impedance spectroscopy (IS) and DLTS offer further insights into the defect landscape of the devices. In an IS experiment, the current response to an externally applied alternating voltage at a certain frequency $\omega$ is measured and considered as a capacitance signal by taking into account the imaginary part of the impedance $Z$,[@Schroder2005; @Rau2016] $$\label{eq:Z_to_C}
C=\frac{\mathrm{Im}(1/Z)}{\omega},$$ by modeling the solar cell as a capacitor in parallel to a shunt resistance. To obtain a complete picture of the defects and to quantify their physical properties, we performed IS measurements over a wide frequency range ($0.6~\mathrm{Hz}<\omega<3.2~\mathrm{MHz}$) and at different temperatures (200 K to 350 K in 5 K increments).
![DLTS spectra (top) and IS spectra (bottom) for the samples with precursor stoichiometry: 2.96, 3.00 and 3.04. Areas are shaded differently in order to highlight where each defect is dominant. Defect $\gamma$ is not completely visible for the chosen rate window $t_2/t_1=10$. For a more detailed overview see Fig. S5 and S8.[]{data-label="fig:DLTS_IS"}](DLTS_IS.pdf)
There are two responses in the representative IS spectra as shown in Figs. \[fig:DLTS\_IS\] and S4: a low frequency response ($<10^2~\mathrm{Hz}$) at high temperatures ($>315~\mathrm{K}$) and a step at higher frequencies ($>10^2~\mathrm{Hz}$) and lower temperatures ($<285~\mathrm{K}$) for each of the investigated samples. These responses can be assigned to two different defects. Particularly noteworthy is the increase of the low frequency section of the spectra with increasing stoichiometry, which indicates its impact on the properties of the corresponding ionic defect. From the capacitance spectra, we are able to extract the ion (defect) diffusion coefficient $D$ based on the equation: $$\label{eq:D}
D=D_0\exp{\left(-\frac{E_\mathrm{A}}{k_\mathrm{B}T}\right)}.$$ where the $k_\mathrm{B}$ is the Boltzmann constant, $T$ is the temperature, $D_0$ is the diffusion coefficient at infinite temperature and $E_\mathrm{A}$ is the activation energy for ion migration. This is done by extracting the ion migration rates (emission rates is the corresponding term from DLTS when applied to study electronic defects in semiconductors) [@Heiser1993; @Zamouche1995; @Yang2016; @Futscher2019] as defined by $$\label{eq:en}
e_\mathrm{t}=\frac{e^2N_\mathrm{eff}D_\mathrm{0}}{k_\mathrm{B}T\epsilon_\mathrm{0}\epsilon_\mathrm{R}}\exp{\left(-\frac{E_\mathrm{A}}{k_\mathrm{B}T}\right)},$$ from the maxima of the derivative $-\omega \mathrm{d}C/\mathrm{d}\omega$, shown in Fig. S5. The presence of two maxima in these spectra reveal two distinct ionic defects, $\beta$ and $\gamma$. We summarized the migration rates associated with these two defects in an Arrhenius diagram (Fig. S6) and calculated the activation energies $E_\mathrm{A}$ and the diffusion coefficients $D_\mathrm{300K}$ at 300 K based on Eqn. (\[eq:en\]).
Interestingly, the defects $\beta$ and $\gamma$ show opposing trends in terms of $E_\mathrm{A}$, $D_\mathrm{300K}$, and $N_\mathrm{ion}$ with varying stoichiometric ratios (Fig. \[fig:defect\_parameters\]). For $\beta$, the activation energy decreases whereas the diffusion coefficient at 300 K increases for increasing stoichiometry, whereas $\gamma$ shows the inverse behavior. This suggests that by increasing the sample stoichiometry, ion migration of defect $\gamma$ is suppressed, while the defect $\beta$ becomes more mobile.
The accumulation of mobile ions at the interfaces of the active layer, driven by the internal electric field of the photovoltaic devices, was reported in several studies.[@Ebadi2019; @Calado2016; @Lee2019; @Zhu2019] The resultant inhomogeneity of the ionic distribution in the perovskite active layer leads to the formation of a Debye layer of cations at the hole transport layer and a Debye layer of anions at the electron transport layer.[@Richardson2016; @Bertoluzzi2019] As a result of the inhomogeneity, the defect density from IS measurements cannot be determined by using the approach by Walter et al.[@Walter1996] for semiconductor defects. A more feasible approach can be found by taking into account the capacitance of the ionic Debye layer,[@Almora2015] $$\label{eq:Nt}
N_\mathrm{ion}=\frac{k_\mathrm{B}T\Delta C^2}{e^2\epsilon_\mathrm{0}\epsilon_\mathrm{R}}.$$ In this case, $\Delta C$ is proportional to the capacitance step observed in Fig. S4. Following this approach, the ionic defect concentration of $\beta$ slightly decreases with increasing stoichiometry, while $N_\mathrm{ion}$ of $\gamma$ shows a notable increase as shown in Fig. \[fig:defect\_parameters\]. We conclude that defect $\gamma$ dominates the behavior of overstoichiometric samples, while the more mobile defect $\beta$ dominates the understoichiometric ones.
To expand the insights gained by IS, we performed DLTS measurements on the same set of solar cells. For DLTS, a voltage filling pulse (from 0 V to 1 V for a duration of 100 ms) is applied to the devices, while measuring the capacitance response at 80 kHz until the solar cell returns to equilibrium conditions.[@Lang1974; @Rau2016] During the filling pulse, mobile ions are pushed from both interfaces of the perovskite layer into the perovskite bulk until they reach a new steady state condition. After the filling pulse, the mobile ions move back to the interfaces caused by the internal field, which introduces a change of the solar cell capacitance.[@Futscher2019] The resulting transients, shown in Fig. S7, were averaged over 35 single measurements to yield a high signal-to-noise ratio, and were measured within the same temperature range as the IS measurements. For the evaluation, we performed the commonly utilized boxcar method[@Lang1974] as shown in Fig. \[fig:DLTS\_IS\] and S8.
The analysis of the DLTS data reveals three different temperature-dependent peaks associated with three distinct defect states. Two of these defects exhibit high migration rates at low to medium temperature range, while the third shows low migration rates at higher temperatures. Following the good agreement in the peak position shown in Fig. \[fig:DLTS\_IS\] and that of the migration rates plotted in the Arrhenius diagram (Fig. S6), we conclude that one of the two defect states with high migration rates corresponds to defect $\beta$ previously identified by IS. The defect exhibiting low migration rates is attributed to $\gamma$ in agreement with IS. Similarly to the IS data, defect $\gamma$ dominates the boxcar spectrum for high stoichiometry samples. The remaining defect with comparably high migration rates was not observed in IS measurements and was labeled $\delta$.
Unlike IS, DLTS data allows to distinguish between positive and negative ionic defects, i.e. anions and cations. As shown in Fig. \[fig:DLTS\_IS\] and S8, defects $\beta$ and $\delta$ have a positive sign and correspond therefore to anions, whereas $\gamma$ corresponds to a cation. As mentioned above, the migration rates ($e_\mathrm{t}=1/\tau$) of these ionic defects can be extracted from the position of the peaks shown in Fig. S8 and complement the results of IS measurements when plotted in the same Arrhenius diagram (Fig. S6). We note that since the slow response of $\gamma$ dominates for overstoichiometric devices, the transients for these devices at very high temperatures did not return to equilibrium within the recorded transient time length of 30 s (Fig. S7). We excluded these non-equilibrium transients from the determination of defect parameters, as they lead to overestimated migration rates for a given temperature.
The overall trend of $E_\mathrm{A}$ and $D_\mathrm{300K}$ for defects $\beta$ and $\gamma$, shown in Fig. \[fig:defect\_parameters\], is comparable with the results obtained by IS. We note that while defect parameters extracted using IS and DLTS exhibit the same general trend, they do show some variance in the absolute values of the extracted defect parameters. These differences, which are visible as an offset in the defect parameters, might arise from the broad distributions of ionic defects, reported in our recent work.[@Reichert2020] Different parts of the same defect distribution are probed by each of the two methods. For defect $\delta$, identified solely via DLTS, we observe a significant increase in $E_\mathrm{A}$ for overstoichiometric samples, accompanied by a strong decrease in $D_\mathrm{300K}$. The ionic defect concentration, $N_\mathrm{ion}$, can be extracted from DLTS measurements by using the ratio between the capacitance change $\Delta C$ caused by the ionic movement and the steady state capacitance $C_\infty$, given by: $$\label{eq:Nion}
N_\mathrm{ion}\propto\frac{\Delta C}{C_\infty}N_\mathrm{eff}~\textrm{ if}~N_\mathrm{ion}\ll N_\mathrm{eff}.$$ As shown in Fig. \[fig:defect\_parameters\], the trend of $N_\mathrm{ion}$ for defects $\beta$ and $\gamma$ with changing stoichiometry is also in agreement with the results obtained with IS. For defect $\delta$, the ionic defect concentration is found to increase with stoichiometry, similar to the behavior of defect $\gamma$.
As part of our scenario in our recent work,[@Reichert2020] we assign the anion $\beta$ to $V_\text{MA}^-$ and $\delta$ to $\text{I}_i^-$. The cation $\gamma$ is attributed to $\text{MA}_i^+$. This assignment is in good agreement with the results of Fassl et al.,[@Fassl2018] where XPS measurements showed an increase in the I/Pb and N/Pb ratios with increasing stoichiometry. We note that the assignment of cation $\gamma$ to $\text{MA}_i^+$ may appear in contrast to the reports by Maier and coworkers that claim that methylammonium cations are only mobile in terms of reorientation, ruling out the migration of this species.[@Senocrate2018_2; @Senocrate2018_3] However, it was shown that rotational dynamics of methylammonium cations occurs with relaxation times in the ps timescale at room temperature,[@Mosconi2014; @Chen2015_2; @Kanno2017] which would be too fast to explain hysteresis. Other groups propose that methylammonium can slowly migrate,[@Yuan2015; @Lee2019; @Eames2015] since other possible cations, such as iodine vacancies, are expected to have far higher diffusion coefficient.[@Azpiroz2015; @Barboni2018; @Senocrate2017] Nevertheless, we stress that DLTS provides information solely on the charge of the ionic defects and cannot directly determine the specific ionic species.
Discussion {#sec:discussion}
==========
Interplay between the ionic and electronic landscapes {#sec:Vdrop}
-----------------------------------------------------
The mixed ionic–electronic conducting nature of perovskites dictates that the ionic and electronic landscapes of these materials cannot easily be decoupled.[@Kerner2017; @Tessler2020] One aspect linking the two is related to the effect of ion accumulation at the interfaces of the perovskite layer and the extraction layers that sandwich it.[@Courtier2019] Such ionically charged interfacial layers influence the internal electric field and the built-in potential of the device, suggesting that the estimation of $V_\mathrm{bi}$ from CV measurements as discussed in Sec. \[sec:CV\] needs to be re-evaluated.[@Almora2016] The validity of the Mott–Schottky relation (Eqn. (\[eq:MS\])) is based on the assumption that the charge carrier density within the perovskite layer is homogeneously distributed, which may not be the case for perovskite solar cells. While we pre-biased the devices before measuring CV in an attempt to eliminate the accumulation of ions at the interfaces, the resultant trend in $V_\mathrm{bi}$ is consistent with what has been observed by diode J-V characterization, for which no pre-biasing was applied.[@Fassl2018] This might indicate that ions still accumulate at the interfaces, resulting in a voltage drop that changes the $V_\mathrm{bi}$. A simple model that accounts for this voltage drop can be constructed by considering these interfacial ion densities as Debye layers.[@Almora2015; @Richardson2016; @Courtier2019] The overall charge for one ionic species can be expressed by $\Delta Q=eN_\mathrm{ion}L_\mathrm{D}$, where $L_\mathrm{D}$ is the Debye length according to $$\label{eq:debyeL}
L_\mathrm{D}=\sqrt{\frac{\epsilon_\mathrm{R} \epsilon_0 k_\mathrm{B}T}{e^2N_\mathrm{ion}}}.$$ In order to account for the potential drop $\Delta V$ caused by the mobile ion density of cations $N_\mathrm{C}$ and anions $N_\mathrm{A}$, we assumed a series connection of the capacitance caused by cations $C_\mathrm{A}$ and anions $C_\mathrm{A}$, $$\label{eq:Vdrop}
\Delta V=\sqrt{\epsilon_\mathrm{R} \epsilon_0 k_\mathrm{B}T}\left(\frac{1}{C_\mathrm{C}}+\frac{1}{C_\mathrm{A}}\right)(\sqrt{N_\mathrm{C}}-\sqrt{N_\mathrm{A}}).$$ With Eqn. (\[eq:Vdrop\]) the corrected $V_\mathrm{bi,corr}$ can be obtained by correcting the determined built-in potential by CV measurements with the voltage drop caused by mobile ions at the interfaces, $$\label{eq:Vbi_corr}
V_\mathrm{bi,corr}=V_\mathrm{bi,CV}-\Delta V.$$ The capacitance responses $C_\mathrm{A}$ and $C_\mathrm{C}$ by anions and cations, respectively, can be estimated by the capacitance steps in the IS spectra from Fig. S4, as they correspond to the ion density $N_\mathrm{ion}$ according to Eqn. (\[eq:Nt\]). Taking into consideration the ionic interfacial layers to suppress the trend observed in $V_\mathrm{bi}$, a more consistent value of around 1.1 V can be obtained as shown in Fig. \[fig:CV\_parameter\]. This result is more expected, since all the devices share the same extraction layers and contacts. As shown in a recent study,[@Tessler2020] a shift in $V_\mathrm{bi}$ can also be caused by electronic charge carrier accumulation at the hole transport layer interface. While our calculation correct the influence of ions on $V_\mathrm{bi}$, we cannot rule out an additional electronic influence. However, the result of our correction suggests that, here, the consideration of ions is sufficient.
![Activation energy $E_\mathrm{A}$ of the ionic defects plotted in dependence of the built-in potential $V_\mathrm{bi,CV}$ extracted from CV measurements for all investigated perovskite solar cells. The dashed lines with slopes of $\pm 1$ are guides for the eye.[]{data-label="fig:Vbi_relation"}](Vbi_relation.pdf)
One interesting, and seemingly contradicting, observation is related to the observed increase in $V_\mathrm{oc}$, which coincides with an increase in the overall ionic defect density with increasing stoichiometry. Recent studies suggest that mobile ions may act as non-radiative recombination centers,[@Meggiolaro2019; @Du2014; @Yang2019] evidenced, for example, by a decrease in the photoluminescence quantum efficiency (PLQE). Indeed, overstoichiometric samples exhibit a markedly lower PLQE than understoichiometric ones.[@Fassl2019] Based on these results, one might expect for overstoichiometric devices lower open-circuit voltages than for understoichiometric ones,[@Goetz2020] in contrast to the experimental observation shown in Fig. S2. However, this apparent discrepancy can be reconciled when taking into account the substantial increase in $V_\mathrm{bi}$ with higher stoichiometric ratio. Consequently, while a high ionic defect concentration has a negative effect on $V_\mathrm{oc}$ due to increase of non-radiative recombination, this effect is weaker than the considerable increase introduced by changes to the energetic alignment between MAPbI~3~ and the transport layers and the resultant change in $V_\mathrm{bi}$.[@Fassl2018] Moreover, the impact of ions on the energy landscape is in agreement with a recent study by the group of Maier,[@Kim2019] where the authors report an increase of band bending in MAPbI~3~ toward the electron transport layer originating from an ionically dominated space charge. The interplay of ions and the space charge potential enable device improvements by interfacial engineering.
The intricacy of the interplay between the ionic and electronic landscapes is exemplified by plotting the $E_\mathrm{A}$ versus the $V_\mathrm{bi,CV}$ as shown in Fig. \[fig:Vbi\_relation\]. The $E_\mathrm{A}$ of defects $\beta$ and $\gamma$ are of similar magnitude and show a broadly linear dependence on the built-in potential, albeit with slopes of opposing signs. Straight dashed lines with slopes of $\pm 1$ were added to Fig. \[fig:Vbi\_relation\] as a guide to the eye. These similar, but opposing trends in slope supports our earlier assignment of these defects to be related to the same type of MA ion. We can associate defect $\beta$ with an MA vacancy with negative charge, and $\gamma$ with a positively charged MA interstitial. Our interpretation of defect $\delta$ cannot be confirmed in this manner, since we do not observe the corresponding defect species with an opposing charge.
The dependence of $E_\mathrm{A}$ on $V_\mathrm{bi}$ might be a consequence of the band bending introduced by the interfacial ion accumulation. As the ion concentration increases, stronger band bending at the interfaces leads to higher fields that impedes the ionic hopping process at the interfaces lowering their overall mobility. This explanation is supported by plotting $E_\mathrm{A}$ and $D_\mathrm{300K}$ versus $N_\mathrm{ion}$ (see Fig. S9). Although the trends are less clear, we generally observe an increase of $E_\mathrm{A}$ and a decrease in $D_\mathrm{300K}$ for higher defect concentrations. The dependence is in agreement with our finding that mobile ions are pushed stronger towards the interfaces caused by the relation between the internal electric field and the ion density. As a result, the diffusion coefficient decreases with higher ion density. This interaction between the ionic and electronic landscapes highlights the need to construct a clearer picture of the underlying defect physics in perovskite devices.
Unraveling the defect landscape across the literature
-----------------------------------------------------
![Migration rates reported in literature[@Samiee2014; @Yang2017; @Rosenberg2017; @Xu2019; @Futscher2019] (often called emission rates in the papers) plotted in an Arrhenius diagram for comparison to our findings (species $\beta$, $\gamma$, $\delta$). All reported emission rates can be associated with two regimes at low emission rates and high temperatures and at high emission rates at high and middle temperatures.[]{data-label="fig:Masterarrhenius"}](Masterarrhenius.pdf)
To evaluate our results in a broader context, we compared the migration rates measured herein, with data available from literature. We chose several studies with similar measurement methods such as DLTS and IS, but with a selection of different perovskite materials and transport layers (as summarized in Tab. 1 in SI). Included in Fig. \[fig:Masterarrhenius\] are the results of Samiee et al.[@Samiee2014] who observed two different defects in a mixed halide perovskite using IS, and three defects (attributed to cations) probed by Yang et al.[@Yang2017] using DLTS on FAPbI~3~. Additionally, included are the emission rates of two defects measured using current DLTS by Rosenberg et al.[@Rosenberg2017] in MAPbBr~3~ single crystals and those probed by Xu et al.[@Xu2019] on FAPbI~3~ light-emitting diodes. Finally, the results of Futscher et al.[@Futscher2019] using transient ion-drift measurements (which is DLTS under a different name) were added, which exhibit two ionic species assigned as $\text{I}_i^-$ and $\text{MA}_i^+$ interstitials in MAPbI~3~ solar cells.
This comparison reveals a remarkable agreement between reports despite the use of different perovskite compositions and device structures. The reported emission rates broadly fall into two categories: those with low emission rates at high temperatures or those with high emission rates at high or medium temperatures. This assessment indicates that there are most likely two dominant underlying ionic defects which can be universally observed in all perovskite materials investigated thus far.
Meyer–Neldel Rule
-----------------
To gain an understanding of the underlying mechanism for ion transport, we examine the relationship between the diffusion coefficient $D_0$ (at infinite temperature) and the activation energy $E_A$ according to Eqn. (\[eq:D\]) and Fig \[fig:defect\_parameters\]. Fig. \[fig:MN\]a reveals a clear linear dependence between these two values for each of the ionic defects. Such a linear relation is known as the Meyer–Neldel rule, which is often used to describe thermally activated processes.[@Meyer1937] According to the Arrhenius Eqn. (\[eq:D\]), the Meyer–Neldel rule states that the pre-factor $D_0$ itself depends on the activation energy $E_\mathrm{A}$ via: $$\label{eq:MN}
D_0=D_\mathrm{00}\exp{\left( \frac{E_\mathrm{A}}{E_\mathrm{MN}} \right)} \textrm{ with } E_\mathrm{MN}=k_\mathrm{B}T_\mathrm{MN},$$ where $D_\mathrm{00}$ refers to the critical diffusion coefficient, $E_\mathrm{MN}$ is the characteristic energy and $T_\mathrm{MN}$ is the corresponding characteristic temperature. Eqn. (\[eq:MN\]) yields very similar values for $E_\mathrm{MN}$: 28 meV, 30 meV and 35 meV for $\beta$, $\delta$ and $\gamma$, respectively. The critical diffusion coefficient $D_\mathrm{00}$ of $\beta$ with $3\cdot 10^{-7}~\mathrm{cm^2/s}$ is one order of magnitude higher than for $\delta$ ($1\cdot 10^{-8}~\mathrm{cm^2/s}$). $\gamma$ has the lowest $D_\mathrm{00}$ which is equal to $4\cdot 10^{-10}~\mathrm{cm^2/s}$. As a consequence of the Meyer–Neldel rule, the migration rates shown in Fig. S6 and the diffusion coefficients (presented in Fig. \[fig:MN\]b) that are associated with the same defect, intersect at $1000/T_\mathrm{MN}$. At this intersection point, which is different for each defect species, the migration rates become independent of stoichiometry. In other words, the ionic defect landscape is no longer affected by stoichiometry at $T_\mathrm{MN}$. As guide to the eye, we added dashed lines to Fig. \[fig:MN\]b, which were extracted from the values of the fits presented in Fig. \[fig:MN\]a.
We propose two possible origins for the Meyer–Neldel behavior. The first is based on disordered organic semiconductors, where the legitimacy of the Meyer–Neldel rule was linked to a Gaussian distribution of defect levels or hopping energies.[@Metselaar1984] Despite being crystalline materials, it is well established that halide perovskites contain a significant amount of disorder due to spatial and temporal variations of octahedral tilts and molecular rotations.[@beecher2016direct] A range of defect environments and transition pathways are therefore expected. Indeed, in our recent work, we demonstrated a distribution of migration rates for each of the reported defects.[@Reichert2020] This is also supported by a combined experimental–theoretical work where modeling of ion migration induced PL quenching was only possible by applying a Gaussian distribution of ion migration rates.[@Fassl2018_2]
A second explanation arises if a multi-excitation entropy model is considered. A single hopping event is usually the result of a multi-phonon excitation, since the activation energy for ion migration is large compared to the phonon energy (e.g. 16.5 meV for optical phonons).[@Sendner2016; @Kirchartz2018] Consequently, a large number of activation pathways are available for each hopping event. A higher activation energy results in a larger number of distinct pathways expressed by the entropy, which is proportional to the exponential pre-factor $D_0$.[@Fishchuk2014; @Przybytek2018] Based on this model, the Meyer–Neldel rule originates from the absorption of $N_\textsubscript{A}$ phonons with $N_\textsubscript{A}\cdot E_\mathrm{ph}=E_\mathrm{A}$, which transfers the ion first to an activated state, and then—accompanied by the emission of multiple phonons—to the target site of the perovskite lattice.[@Shimakawa2012]
![ (a) Calculated 2D energy surface on a 20$\times$20 real-space grid for V$_{\mathrm{I}}^+$ migration in a (001) plane of tetragonal MAPbI$_3$ from initial state A to final state B with along the configurational coordinate Q. (b) The projected 1D pathway showing the associated activation energy and the effective frequencies for the initial and final states. []{data-label="fig:DFT"}](DFT.png)
To probe the atomistic nature of a typical diffusion process, we performed first-principles calculations of charged vacancy migration in the room temperature phase of MAPbI$_3$ using the technical setup reported elsewhere.[@Eames2015] We consider a low energy transition in the (001) plane as illustrated in Fig \[fig:DFT\]. The associated migration barrier of 0.55 eV and it follows a curved diffusion pathway. Even in a single plane, due to the presence of MA, the initial and final states differ in energy by 60 meV, which supports the first disorder explanation. We further determine the vibrational frequency at T = 300 K around using `CarrierCapture`.[@Kim2020] Effective frequencies of 0.4–0.7 THz represent the curvature of the potential energy surface along the directions of ion diffusion. These are unusually soft owing to a combination of the heavy elements and the flexible perovskite structure.
A simple estimation of $N_A$ suggests that hundreds of phonon modes are involved in a single hopping process, which supports the second explanation.
While we cannot yet assign the validity of the Meyer–Neldel rule to a single origin, it offers interesting insights into the physical mechanisms of ion migration in halide perovskites.
Summary
=======
In summary, we investigated the ionic defect landscape of MAPbI$_3$ samples with gradually varying defect densities, introduced by fractionally varying the stoichiometry of the perovskite precursor solution (MAI:PbAc~2~). By combining the results of IS and DLTS measurements, we identify three ionic defect, which we attribute to $V_\text{MA}^-$, $\text{I}_i^-$ and $\text{MA}_i^+$. We explore the tight link between the ionic defect and electronic landscapes in perovskite devices and reveal that the accumulation of defects at the interfaces of the perovskite layer results in an increase of the built-in potential for increasing stoichiometry, which we show to be the dominant factor influencing the open-circuit voltage of the devices. The presence of ionic interfacial layers is also shown to affect the $E_\mathrm{A}$ of the various defects, by impeding their transport due to high electric fields they introduce. We compared the temperature dependent ion migration rates to the literature, and were able to categorize defect parameters of different perovskite materials and device architectures. Importantly, we find that the ionic defects we observed fulfill the Meyer–Neldel rule. We propose that the origin of the Meyer–Neldel rule lies either in the distribution of migration pathways or the multi-phonon emission process that characterizes the hopping of ions. Our results offer significant insights into the defect physics of perovskite materials and progress the current understanding of the underlying processes that govern the properties of this phenomenal class of materials.
Methods
=======
**Device fabrication:** Pre-patterned indium tin oxide (ITO) coated glass substrates (PsiOTech Ltd., $15~\Omega/\square$) were ultrasonically cleaned with $2~\%$ Hellmanex detergent, deionized water, acetone, and isopropanol, followed by 10 min oxygen plasma treatment. Modified poly(3,4-ethylene-dioxythiophene):poly(styrenesulfonate) (m-PEDOT:PSS) was spin cast on the clean substrates at 4000 rpm for 30 s and annealed at $150~^\circ\mathrm{C}$ for 15 min to act as hole transport layer.[@Zuo2016] The MAPbI~3~ active layer was formed using the lead acetate trihydrate route following previous works.[@An2019; @Fassl2018] In short, the perovskite solution (at different stoichiometry of 2.96:1 to 3.06:1 in 0.02 steps PbAc~2~:MAI) was spin cast at 2000 rpm for 60 s in a dry air filled glovebox (relative humidity $<0.5~\%$). After blowing 25 s and drying 5 min, the as-spun films were annealed at $100~^\circ\mathrm{C}$ for 5 min forming a uniform perovskite layer. The prepared samples were transferred to a nitrogen filled glove box, where an electron transport layer \[6,6\]-phenyl-C61-butyric acid methylester (PC~61~BM), 20 mg/ml dissolved in chlorobenzene, was dynamically spin cast at 2000 rpm for 30 s on the perovskite layer followed by a 10 min annealing at $100~^\circ\mathrm{C}$. Sequentially, a bathocuproine (BCP), 0.5 mg/ml dissolved in isopropanol, hole blocking layer was spin cast on top of the PC~60~BM. The device was completed with a thermally evaporated 80 nm thick silver layer.
**jV characterization:** The current density–voltage (jV) characteristics were measured by a computer controlled Keithley 2450 Source Measure Unit under simulated AM 1.5 sunlight with $100~\mathrm{mW/cm}^2$ irradiation (Abet Sun 3000 Class AAA solar simulator). The light intensity was calibrated with a Si reference cell (NIST traceable, VLSI) and corrected by measuring the spectral mismatch between the solar spectrum, the spectral response of the perovskite solar cell and the reference cell.
**Defect spectroscopy measurements:** All defects were measured using a setup consisting of a Zurich Instruments MFLI lock-in amplifier with MF-IA and MF-MD options, a Keysight Technologies 33600A function generator and a cryo probe station Janis ST500 with a Lakeshore 336 temperature controller. We performed the defect spectroscopy in the temperature range of 200 K to 350 K in 5 K steps, controlled accurately within 0.01 K, using liquid nitrogen for cooling. DLTS, IS and CV measurements were done applying an AC frequency of 80 kHz with amplitude of $V_\mathrm{ac}=20~\mathrm{mV}$. For DLTS, the perovskite solar cells were biased from 0 V to 1 V for 100 ms. The transients were measured over 30 s and averaged over 35 single measurements. For CV profiling, the solar cells were pre-biased at 1 V for 60 s and rapidly swept with 30 V/s in reverse direction.
C.D. and S.R. acknowledge financial support by the Bundesministerium für Bildung und Forschung (BMBF Hyper project, contract no. 03SF0514C) and thank their project partners from the University of Würzburg and ZAE Bayern for interesting discussions. Y.W.W. thanks Sunghyun Kim for assistance. Via our membership of the UK’s HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202), this work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk). This work was also supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (no. 2018R1C1B6008728). Y.V. and C.D. thank the DFG for generous support within the framework of SPP 2196 project (PERFECT PVs). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC Grant Agreement no. 714067, ENERGYMAPS).
[94]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](https://doi.org/10.1021/ja809598r) @noop , (), [****, ()](https://doi.org/10.1038/nphys3357) [****, ()](https://doi.org/10.1126/science.1243982) [****, ()](https://doi.org/10.1016/j.cap.2017.07.005) [****, ()](https://doi.org/10.1039/c5nr05435d) [****, ()](https://doi.org/10.1021/acs.chemmater.8b00136) [****, ()](https://doi.org/10.1039/c8ta04120b) [ ()](https://doi.org/10.1002/ente.201901114) [****, ()](https://doi.org/10.1117/1.jpe.6.022001) [****, ()](https://doi.org/10.3390/ma11050729) [****, ()](https://doi.org/10.1039/c9ee01479a) [****, ()](https://doi.org/10.1039/c6cp06989d) [****, ()](https://doi.org/10.1021/acs.accounts.5b00420) [****, ()](https://doi.org/10.1021/acsomega.8b01626) [****, ()](https://doi.org/10.1063/1.5085643) [****, ()](https://doi.org/10.1038/s41563-018-0038-0) [****, ()](https://doi.org/10.1021/jz500113x) [****, ()](https://doi.org/10.1039/c4ee03664f) [****, ()](https://doi.org/10.1021/acs.jpclett.5b02229) [****, ()](https://doi.org/10.1021/acs.jpclett.6b00579) [****, ()](https://doi.org/10.1021/acsenergylett.9b00247) [****, ()](https://doi.org/10.1039/c4ta01198h) [****, ()](https://doi.org/10.1063/1.4864778) [****, ()](https://doi.org/10.1021/nl502612m) [****, ()](https://doi.org/10.1021/acs.chemmater.5b01909) [****, ()](https://doi.org/10.1016/j.ssi.2018.03.029) [****, ()](https://doi.org/10.1021/jacs.8b13594) [****, ()](https://doi.org/10.1002/adma.201503832) [****, ()](https://doi.org/10.1002/aenm.201501803) [**** ()](https://doi.org/10.1002/adfm.201770158) [****, ()](https://doi.org/10.1039/c9mh00445a) [**** ()](https://doi.org/10.1038/ncomms8497) [****, ()](https://doi.org/10.1039/c5ee01265a) [****, ()](https://doi.org/10.1021/acs.chemmater.6b01348) [****, ()](https://doi.org/10.1126/science.aan2301) [****, ()](https://doi.org/10.1063/1.4995970) [****, ()](https://doi.org/10.1063/1.4897329) [****, ()](https://doi.org/10.1039/c4cp04479g) [****, ()](https://doi.org/10.1038/s41566-019-0390-x) [**** ()](https://doi.org/10.1103/physrevapplied.13.034018) [****, ()](https://doi.org/10.1016/j.solmat.2015.01.023) [**** ()](https://doi.org/10.1038/ncomms10214) [****, ()](https://doi.org/10.1039/c8ee01136b) [****, ()](https://doi.org/10.1002/ente.201900737) [****, ()](https://doi.org/10.1039/c6cp06496e) [****, ()](https://doi.org/10.1039/c8tc05998e) [****, ()](https://doi.org/10.1063/1.4899051) [****, ()](https://doi.org/10.1002/aenm.201970126) [****, ()](https://doi.org/10.1039/c6cp05851e) [****, ()](https://doi.org/10.1039/c8ee01447g) [**** ()](https://doi.org/10.1038/ncomms13831) [****, ()](https://doi.org/10.1021/acsaem.8b01119) [**](https://doi.org/10.1002/0470068329) (, ) [**](https://doi.org/10.1007/978-3-642-13884-3) (, ) [****, ()](https://doi.org/10.1002/aenm.201500615) [****, ()](https://doi.org/10.1021/jz500370k) [****, ()](https://doi.org/10.1039/c4ee00233d) [**](https://doi.org/10.1002/0471749095) (, ) in [**](https://doi.org/10.1002/9783527699025.ch4), (, ) Chap. , pp. [****, ()](https://doi.org/10.1007/bf00332285) [****, ()](https://doi.org/10.1063/1.114142) [**** ()](https://doi.org/10.1038/s41598-019-48505-6) [****, ()](https://doi.org/10.1021/acs.jpcc.9b01059) [****, ()](https://doi.org/10.1039/c5ee02740c) [ ()](https://doi.org/10.1016/j.joule.2019.10.003) [****, ()](https://doi.org/10.1063/1.363401) [****, ()](https://doi.org/10.1021/acs.jpclett.5b00480) [****, ()](https://doi.org/10.1063/1.1663719) [****, ()](https://doi.org/10.1021/acs.jpcc.8b06814) [****, ()](https://doi.org/10.1039/c8cp01535j) [****, ()](https://doi.org/10.1039/c4cp00569d) [****, ()](https://doi.org/10.1039/c5cp05348j) [****, ()](https://doi.org/10.1021/acs.jpcc.7b04589) [****, ()](https://doi.org/10.1039/c8ee01697f) [****, ()](https://doi.org/10.1002/anie.201701724) [****, ()](https://doi.org/10.1021/acs.jpclett.7b02401) [ ()](https://doi.org/10.1021/acsenergylett.0c00172) [****, ()](https://doi.org/10.1039/c8ee01576g) [****, ()](https://doi.org/10.1063/1.4966127) [ ()](https://doi.org/10.1021/acsenergylett.9b01956) [ ()](https://doi.org/10.1002/adfm.201910004) @noop (), @noop [****, ()]{} [****, ()](https://doi.org/10.1016/0022-4596(84)90284-6) @noop [****, ()]{} [****, ()](https://doi.org/10.1021/acsami.8b16460) [****, ()](https://doi.org/10.1039/c6mh00275g) [****, ()](https://doi.org/10.1021/acs.jpclett.7b03414) [**** ()](https://doi.org/10.1103/physrevb.90.245201) [**** ()](https://doi.org/10.1103/physrevb.98.115159) [****, ()](https://doi.org/10.1007/s00706-012-0835-0) [****, ()](https://doi.org/10.21105/joss.02102) [****, ()](https://doi.org/10.1002/aenm.201601193)
[^1]: Corresponding author: deibel@physik.tu-chemnitz.de
|
---
abstract: 'We investigate magnetic properties of strongly interacting four component spin-3/2 ultracold fermionic atoms in the Mott insulator limit with one particle per site in an optical lattice with honeycomb symmetry. In this limit, atomic tunneling is virtual, and only the atomic spins can exchange. We find a competition between symmetry breaking and liquid like disordered phases. Particularly interesting are valence bond states with bond centered magnetizations, situated between the ferromagnetic and conventional valence bond phases. In the framework of a mean-field theory, we calculate the phase diagram and identify an experimentally relevant parameter region where a homogeneous SU(4) symmetric Affleck-Kennedy-Lieb-Tasaki-like valence bond state is present.'
author:
- 'D. Jakab'
- 'E. Szirmai'
- 'M. Lewenstein'
- 'G. Szirmai'
bibliography:
- 'magnetism.bib'
title: 'Competing valence bond and symmetry breaking Mott states of spin-3/2 fermions on a honeycomb lattice'
---
Introduction {#sec:intro}
============
Mott insulators with antiferromagnetic spin correlations are in the center of interest in condensed matter physics due to their relation to high-$T_c$ superconductivity [@lee2006doping; @anderson1973resonating; @fazekas1974on]. They are also intensively investigated in quantum information, because of their potential applicability for quantum computing [@nielsen2000quantum]. In particular, these systems can be used to realize states required for measurement-based quantum computation (MBQC) [@raussendorf2001one; @verstraete2004valence; @van2006universal; @gross2007novel; @briegel2009measurement]. In addition they may exhibit nontrivial topology [@yang93a; @bauer2014chiral; @nasu15a; @hickey2015haldane]; accordingly, they play an important role in the studies of the topological states of matter. For experimental studies of these phenomena ultracold atomic systems provide one of the most promising and efficient playgrounds. There is a still ongoing progress in ground breaking experiments with ultracold atoms in order to realize quantum emulators of magnetic systems [@lewenstein2012ultracold]. The main advantage is that in these systems there is an unprecedented control over the parameters describing almost every feature of their physics [@jordens2010quantitative; @trotzky2010controlling; @greif2011probing; @nascimbene2012experimental; @fukuhara2013quantum; @fukuhara2013microscopic; @imriska2014thermodynamics; @*imriska2014erratum]. The atoms are trapped optically; both the potential height and the lattice periodicity can be adjusted by tuning the amplitude, phase and wavelength of the lasers. Even the geometry of the lattice can be changed in situ [@tarruell2012creating]. Further advantage of such systems is that interaction between the atoms can be controlled in a wide range through the access of various scattering resonances [@inouye1998observation; @fedichev1996influence; @petrov2001interatomic; @chin2010feshbach]. As a result of this versatility, many antiferromagnets, either encountered in real materials, or proposed by theorists for academic interest, can now be potentially realized [@hermele2009mott; @gorshkov2010two; @szirmai2011exotic; @szirmai2011gauge; @sinkovicz2013spin; @hazzard2012high; @pinheiro2013xyz; @cai2013pomeranchuk; @cai2013quantum; @cazalilla2014ultracold; @grass2014quantum]. In the first experiments the Mott insulator state was realized with a dilute gas sample of alkaline atoms loaded to an optical lattice formed by counterpropagating laser beams [@greiner2002quantum; @jordens2008mott; @greif2014short]. Later, trapping of higher spin alkalies [@krauser2012coherent; @Pagano14] and cooling of alkaline-earth-metal atoms to quantum degeneracy [@taie2010realization; @desalvo2010degenerate] has opened the way to Mott insulators with higher spin atoms [@taie2012su; @scazza2014observation]. Most recently, the first steps were made towards the study of the direct effects of spin-exchange interactions [@Zhang14a; @Cappellini14a; @Scazza14a]
The word ’antiferromagnet’ refers to a state of matter, where the total magnetization of the sample is zero, but magnetic correlations differ from that of the trivial paramagnetic phase. The simplest possible antiferromagnetic state has Néel order: in a square lattice in 2D, for instance, opposing spins are arranged in a checkerboard configuration. Such a state is symmetry breaking, and an order parameter can be introduced, which is the magnetization of the sublattice formed by every second site. According to the Mermin-Wagner-Hohenberg theorem [@mermin1966absence; @hohenberg1967existence] symmetry breaking cannot take place in two dimensions at finite temperature, while in one dimension fluctuations destroy long-range order even at $T=0$ [@sachdev2011quantum]. Therefore, another completely different antiferromagnetic state can be expected in low dimensions: the singlet covering of the lattice, where pairs of sites form a two-particle spin-singlet state [@fazekas1999lecture]. Similar state was discovered by Majumdar and Ghosh in a 1D model [@majumdar1969on], where the ground state is a periodic lattice of independent singlet pairs. A translational invariant generalization of such a valence-bond-solid (VBS) state was introduced by Anderson [@anderson1973resonating] to describe frustrated antiferromagnetism on a triangular lattice; it is called the resonating valence bond (RVB) state.
The direct consequences of an antiferromagnetic spin exchange can be studied most clearly in the Mott insulator state of matter [@lee2006doping; @imada1998metal; @fisher1999mott; @auerbach1994interacting; @fazekas1999lecture]. In this case the charge degree of freedom is frozen out — e.g. as a consequence of a strong repulsive interaction — and the low energy physics of the system is governed by the spin degrees of freedom, which in turn are governed by an effective Heisenberg-like exchange interactions. However, as mentioned above, it is not obvious, particularly in 2D, that even in the case of strong antiferromagnetic exchange interactions, a Néel-type spin ordered state will emerge. Instead, valence bonds can form and the SU(2) spin rotational invariance can remain preserved. Nowadays the concept of the valence bond picture is the most commonly used to describe the underlying order of the various spin liquid states. With the help of the valence bond picture a series of phenomena with nontrivial magnetic origin can be described like various exotic VBS states [@wu06a; @hermele2009mott; @szirmai2011exotic; @hermele11a; @szirmai2011gauge; @ganesh2011quantum], or even chiral spin liquid (CSL) states with non-trivial topology [@hermele2009mott; @szirmai2011gauge; @hermele11a; @sinkovicz2013spin; @hickey2015haldane]. In Ref. [@szirmai14a] it was shown that a topological charge-Haldane state can emerge on a spin-3/2 fermionic ladder. The SU(4) symmetric spin-3/2 system has been studied intensively in the last few years [@wu06a; @szirmai2011exotic; @corboz11a; @corboz12a; @xu08a; @hung11a], mostly on the square lattice. Although some recent numerical results suggest that a weak SU(4) symmetry breaking order can emerge [@corboz11a], it is mostly accepted that the ground state is a VBS state consisting disconnected RVB plaquettes. In Ref. [@corboz12a] a pure SU(4) Heisenberg model was studied on a honeycomb lattice; the authors found that a spin-orbital liquid phase can emerge in the system which collapse into a tetramerized VBS-like state in the presence of next nearest neighbor exchange [@lajko2013tetramerization].
The states, which can be suitable as universal resource for MBQC, could be searched among these spin liquid-like states, which preserve the usual SU(2) spin rotational invariance. The classification of these states, just as the quest for physical systems realizing them and harvesting their potential use as universal resource states, is a big challenge of quantum information theory. A fundamental requirement is that they have to be the unique ground states of a Hamiltonian with gapped spectrum and short-range interactions in order to assure their robustness [@Wei15]. One of the most promising states is the AKLT-state introduced by Affleck, Kennedy, Lieb and Tasaki who proposed several models in ’dimension one and more’ where the ground state is a unique VBS [@affleck1987rigorous; @affleck1988valence]. In the contemporary technical “slang” these are states that are exactly described by the Matrix Product States (MPS) or Pair Entangled Projected States (PEPs) with the lowest possible (nontrivial) [*local dimension*]{} (cf. [@lewenstein2012ultracold]).
However, the ’parent’ Hamiltonians of these idealized states are not easy to realize as low energy effective Hamiltonian of a fermionic or bosonic Mott insulator with $SU(2)$ invariant interactions. Let us remind the reader that when the internal states of particles correspond to the degenerate manifold of the hyperfine atomic level $F$, i.e. $|F,m_F\rangle$ of an ultracold spinor gas, the resulting Hamiltonians, in the absence of symmetry breaking fields, must be $SU(2)$ symmetric. The corresponding spin Hamiltonians are given then by powers of the nearest neighbor Heisenberg interactions $H=\sum_{<i,j>}\sum_k a_k({\bf S}_i\cdot
{\bf S}_j)^k$, where $a_k$ are numbers that depend on scattering lengths (for a review see for instance [@Zawitkowski] for $F=1,3/2,2,5/2$, and for general Fermi systems c.f. [@Hofstetter; @congjunwureview; @hermele2009mott]).
The situation for the Mott insulators with one particle per site might be summarized as described below; the case of Mott insulators with two or more particles per site is much more complex (cf. [@Zawitkowski]).
- The original AKLT state was proposed for a spin $F=1$ in 1D. $F=1$ can $s$-wave collide in the $F_{tot}=0, 2$ channels, and thus are characterized in general by the two distinct scattering lengths. The effective Hamiltonian in the super-exchange limit reads then $H=\sum_{\left<i,j\right>}\sum^{k=2}_{k=0} a_k({\bf
S}_i\cdot {\bf S}_j)^k$ or equivalently $H=\sum_{\left<i,j\right>}\sum_{S=0, 2}[ q_S\mathcal{P}_S({\bf
S}_i + {\bf S}_j)]$, where $q_S$ are numbers, and $P_S$ are projections on the total spin $S$. If one could control the two scattering lengths independently and arbitrarily, one could realize the case when $H=\sum_{\left<i,j\right>}\mathcal{P}_2({\bf
S}_i + {\bf S}_j)]$, which in 1D corresponds exactly to the AKLT case. Unfortunately, such control of scattering lengths nowadays is hardly possible – see ref. [@Zawitkowski] for details.
- The Hamiltonian $H=\sum_{\left<i,j\right>}\sum^{k=2}_{k=0}({\bf S}_i\cdot {\bf
S}_j)^k$ for $F=1$ particles is known as [*biquadratic-bilinear*]{} Hamiltonian and at least in 1D has been studied very intensively (cf. [@Demler2002; @Yip2003; @Imambekov; @oriol2007; @gabriele2011] and references therein). It is known that the antiferromagnetic regime exhibits a robust gapped phase that does not break the $SU(2)$, the celebrated Haldane phase [@Haldane1; @Haldane2].
- For $F=2$ bosons there are three possible $s$-wave scattering channels and three scattering length, respectively. The effective Hamiltonian has the form $H=\sum_{\left<i,j\right>}\sum_{S=0, 2, 4}[ q_S\mathcal{P}_S({\bf
S}_i + {\bf S}_j)]$, and can in principle be reduced to $H=\sum_{\left<i,j\right>}\mathcal{P}_4({\bf S}_i + {\bf S}_j)]$ by adjusting the scattering lengths. That would correspond to AKLT model on the square lattice or the 3D lattice with coordination number 4. Again, in practice the necessary control of scattering length is not possible.
- For $F=3/2$ fermions there are two possible $s$-wave scattering channels and, consequently, two scattering length, respectively. The effective Hamiltonian has the form $H=\sum_{\left<i,j\right>}\sum_{S=0, 2}[ q_S\mathcal{P}_S({\bf
S}_i + {\bf S}_j)]$, and cannot be by any means reduced to $H=\sum_{\left<i,j\right>}\mathcal{P}_3({\bf S}_i + {\bf S}_j)]$, which on the honeycomb lattice in 2D corresponds to the AKLT model. Similar situation holds for higher half-integer spin.
- The effective model with $F=3/2$ was studied extensively, and already few years ago there has been a lot of progress in understanding the special properties of $F=3/2$ and $F=5/2$ Fermi gases. In spin-3/2 systems with contact interaction, Wu [*et al.*]{} realized that a generic $SO(5)$ symmetry exists [@Wu38]. These authors also found novel competing orders [@Wu39; @shu2005exact], suggesting a [*quartetting*]{} phase and the $s$ -[*wave quintet*]{} Cooper pairing phase.
Let us recapitulate: the parent Hamiltonian of the two-dimensional generalization of the AKLT-state is a spin-Hamiltonian of spin-3/2 operators $\mathbf{S}_i$ on a two-dimensional honeycomb lattice: $$H=g_3 \sum_{\left<i,j\right>} \mathcal{P}_3 (\mathbf{S}_i + \mathbf{S}_j) ,$$ where $\mathcal{P}_3 (\mathbf{S}_i + \mathbf{S}_j)$ projects to the total spin-3 subspace: $ (\mathbf{S}_i +
\mathbf{S}_j)^2=3(3+1)$. In the next section we will demonstrate in detail that the effective spin Hamiltonian derived from a usual Hubbard-like model does not contain the $\mathcal{P}_3$ projector. Although, in a more general model this term would appear, its role remains always secondary, unless we assume “all mighty” control over the scattering length – typically this term is generated by weaker interactions: as a perturbation it can change the ground state even drastically, but it will never become dominant.
In this paper we show that a spin-3/2 ultracold fermionic system loaded into a two-dimensional honeycomb lattice has a ground state similar to the two-dimensional generalization of the AKLT-state for an extended parameter range of the coupling constants describing the on-site fermion-fermion interaction. Our analysis is based on a mean-field study of the system with a suitable ansatz to describe the coexistence of or even the competition between site- and bond-centered spin orders. We find that the AKLT-like homogeneous state is the lowest energy solution in an extended experimentally reachable parameter region. This state competes on one hand with the usual site ordered homogeneous ferromagnetic, and Néel-type antiferromagnetic phases, at appropriate coupling constant values, while on the other hand we find competition also with exotic spin-Peierls-like dimerized orders. Our results, for the SU(4) symmetric point, are in agreement with the algebraic color liquid state found in Ref. [@corboz12a].
The paper is organized as follows. In Sec. \[sec:model\] we start with the generalized Hubbard model describing the four-component Fermi gas on an optical lattice. Then, we briefly summarize the steps needed to derive a superexchange model for the system in the Mott insulator limit with one particle per site. In Sec. \[sec:meanfield\] a mean-field approximation is applied to the magnetic superexchange model. By analyzing the mean-field solutions we characterize and discuss the possible ground state phases of the model in Sec. \[sec:phdiag\]. Finally, we conclude and summarize our results in Sec. \[sec:sum\].
Model {#sec:model}
=====
We consider a system of ultracold spin-3/2 atoms on an optical lattice with honeycomb structure. The atom can be any of the alkali- or alkali-earth-metal-atomic species with total hyperfine spin-3/2. At low temperatures and for sufficiently deep optical lattices the atoms occupy the lowest band formed by the lowest states of the individual sites, and the system is described by the generalized Hubbard model. $$\label{eq:hubham}
H=-t\sum_{\langle i,j\rangle,\alpha}{c^\dagger_{i,\alpha}}{c^{}_{j,\alpha}}+\frac{1}{2}\sum_{i,\alpha,\beta}V^{\alpha,\beta}_{\gamma,\delta}{c^\dagger_{i,\alpha}}{c^\dagger_{i,\beta}}{c^{}_{i,\delta}} {c^{}_{i,\gamma}},$$ where the first sum runs over nearest-neighbor pairs and spin components $\alpha=\lbrace -3/2,-1/2,1/2,3/2\rbrace$, while the second sum runs over the single sites of the lattice and spin components $\alpha$ and $\beta$. We use the convention that Greek letters denote the $z$-component of the atomic hyperfine spin. From now on, an implicit summation over repeated Greek indices is also assumed. The operators ${c^\dagger_{i,\alpha}}$ and ${c^{}_{i,\alpha}}$ respectively create and annihilate a single-particle state at site $i$ with spin projection $\alpha$. The spin-dependent on-site interaction is described by the tensor $V^{\alpha,\beta}_{\gamma,\delta}$. The total spin and its $z$-component of the colliding particles are conserved. The most general form of such spin dependence can be expressed with the help of projection matrices that project to the two particle tensor product space with a given total hyperfine spin of the two colliding particles. $$\label{eq:spinmat}
V^{\alpha,\beta}_{\gamma,\delta}=g_0\,(P_0)^{\alpha,\beta}_{\gamma,\delta}+g_2\, (P_2)^{\alpha,\beta}_{\gamma,\delta}.$$ $P_0$ and $P_2$ are the projectors projecting to total spin 0 and 2, respectively. These operators are antisymmetric under the exchange of their upper or their lower indices, i.e. $(P_{e})^{\alpha,\beta}_{\gamma,\delta}=-(P_{e})^{\beta,\alpha}_{\gamma,\delta}=-(P_{e})^{\alpha,\beta}_{\delta,\gamma}$, where $e=0,2$. We note, that $P_1$ and $P_3$ is missing from the sum, since they are symmetric in the respective spin indices and due to the Pauli principle their contribution cancels in Eq. . The coupling constants $g_0$ and $g_2$ are expressed in the usual way together with the hopping amplitude $t$ with the help of the Wannier-function overlap integrals [@altland2010condensed].
When the number of atoms is equal to the number of lattice sites, furthermore, the on-site interaction is much larger than the tunneling amplitude, then multiple occupancy becomes energetically costly. In this limit, all of the low energy states have exactly one particle per site. The dynamics restricted to this low energy sector contains only spin fluctuations. The superexchange Hamiltonian, $\tilde H$, governing such a Mott insulator dynamics can be obtained with the help of perturbation theory. Here we follow the procedure and notations of Ref. [@szirmai2011exotic]. $$\label{eq:effham}
\tilde H=-\sum_{\langle i,j\rangle} \Bigg[ \frac{2t^2}{g_0}(P_0)^{\alpha,\beta}_{\gamma,\delta}+\frac{2t^2}{g_2}(P_2)^{\alpha,\beta}_{\gamma,\delta}\Bigg]{c^\dagger_{i,\alpha}}{c^{}_{i,\gamma}}{c^\dagger_{j,\beta}}{c^{}_{j,\delta}}.$$ In the following we express the projectors with the help of the SU(2) spin operators. For a spin-3/2 system, the single spin Hilbert space is 4 dimensional, and the spin matrices are $4\times4$ Hermitian matrices. We use the representation, where the single site spin basis vectors are eigenvectors of $F_z$. The spin matrices are $$\begin{aligned}
F_z&=\left[\begin{array}{c c c c}
\frac{3}{2}&0&0&0\\
0&\frac{1}{2}&0&0\\
0&0&-\frac{1}{2}&0\\
0&0&0&-\frac{3}{2}
\end{array}\right],\quad
F_x=\left[\begin{array}{c c c c}
0&\frac{\sqrt{3}}{2}&0&0\\
\frac{\sqrt{3}}{2}&0&1&0\\
0&1&0&\frac{\sqrt{3}}{2}\\
0&0&\frac{\sqrt{3}}{2}&0
\end{array}\right],\\
F_y&=\left[\begin{array}{c c c c}
0&-i\frac{\sqrt{3}}{2}&0&0\\
i\frac{\sqrt{3}}{2}&0&-i&0\\
0&i&0&-i\frac{\sqrt{3}}{2}\\
0&0&i\frac{\sqrt{3}}{2}&0
\end{array}\right].\end{aligned}$$ The two spin tensor product space is 16 dimensional. $P_0$ projects to the $S=0$ (spin singlet) subspace, which is 1 dimensional, while $P_2$ projects to the 5 dimensional $S=2$ (quintet) subspace. Therefore, the total antisymmetric sector is 6 dimensional. In the antisymmetric sector these two orthogonal projectors span the whole space, i.e.
$$(P_0)^{\alpha,\beta}_{\gamma,\delta}+(P_2)^{\alpha,\beta}_{\gamma,\delta}=\frac{1}{2}(\delta_{\alpha,\gamma}\delta_{\beta,\delta}-\delta_{\alpha,\delta}\delta_{\beta,\gamma})\equiv(\mathbf{E}^{\text{(as)}})^{\alpha,\beta}_{\gamma,\delta}.$$
Here, the notation with the superscript ’(as)’ is introduced as a shorthand for antisymmetrization. Furthermore [@szirmai2011exotic], $$\begin{gathered}
-\frac{15}{4}(P_0)^{\alpha,\beta}_{\gamma,\delta}-\frac{3}{4}(P_2)^{\alpha,\beta}_{\gamma,\delta}=
\frac{1}{2}(\mathbf{F}_{\alpha,\gamma}\cdot\mathbf{F}_{\beta,\delta}-\mathbf{F}_{\alpha,\delta}\cdot\mathbf{F}_{\beta,\gamma})\\
\equiv(\mathbf{F}_1\cdot\mathbf{F}_2)^{\text{(as)}},\end{gathered}$$
where $\mathbf{F}=(F_x,F_y,F_z)$ is the 3 component vector of the spin-3/2 matrices. With a straightforward calculation one obtains the Hamiltonian $$\begin{gathered}
\label{Hsajt}
\tilde H =\sum_{\langle i,j\rangle}\bigg[a_n (n_i\,n_j+\chi_{i,j}\,\chi^\dagger_{i,j}-n_i)\\
+a_s\bigg(\mathbf{S}_i\mathbf{S}_j+\mathbf{B}_{i,j}\mathbf{B}_{i,j}^\dagger-\frac{15}{4}n_i\bigg)\bigg],\end{gathered}$$ where we have introduced the following two-fermion operators:
\[eqs:twoferops\] $$\begin{aligned}
n_i&={c^\dagger_{i,\alpha}}{c^{}_{i,\alpha}},\\
\chi_{i,j}&={c^\dagger_{i,\alpha}}{c^{}_{j,\alpha}}, \label{eq:chi}\\
\mathbf{S}_i&=\mathbf{F}_{\alpha,\beta}\,{c^\dagger_{i,\alpha}}{c^{}_{i,\beta}},\\
\mathbf{B}_{i,j}&=\mathbf{F}_{\alpha,\beta}\,{c^\dagger_{i,\alpha}}{c^{}_{j,\beta}}.\label{eq:B}\end{aligned}$$
The quantities $n_i$ and $\mathbf{S}_i$ describe the density and spin on site i, respectively. In the Mott insulator state with one particle per site, $n_i\equiv1$ on the whole lattice. The operators $\chi_{i,j}$ and $\mathbf{B}_{i,j}$ are bond operators describing nearest-neighbor correlations; $\chi_{i,j}$ is the SU(4) symmetric part, while $\mathbf{B}_{i,j}$ is for correlations violating the spin rotation symmetry. The coupling constants in the SU(4) symmetric and symmetry-breaking channels are $a_n=-t^2(5g_0-g_2)/(4g_0g_2)$ and $a_s=-t^2(g_0-g_2)/(3g_0g_2)$, respectively. We assume that the quintet coupling constant ($g_2$) never dominates so strongly over the singlet one ($g_0$) that the SU(4) invariant interaction would become attractive, otherwise formation of four-particle composite particles would be also expected which is beyond the applicability of our treatment. Accordingly, the coupling constant $a_n$ is always negative. Contrary, the coupling $a_s$ can both be positive or negative. Accordingly, the spin anisotropic interaction can be tuned from a predominantly ferromagnetic exchange to an antiferromagnetic one.
Mean-field theory {#sec:meanfield}
=================
In order to describe the system with the help of a mean-field theory, we assume that both the site and bond operators can (but not necessarily do) have a nonzero classical value. These classical values of the operators will be denoted by a horizontal bar above the operator. Furthermore, we assume that the fluctuations around these classical values are small: $0\approx(\bar\chi_{i,j}-\chi_{i,j})(\bar\chi_{i,j}^*-\chi_{i,j}^\dagger)$. This way, $\chi_{i,j}\chi_{i,j}^\dagger \approx \bar\chi_{i,j}^*\chi_{i,j} + \bar\chi_{i,j} \chi_{i,j}^\dagger-|\chi_{i,j}|^2$. We assume similar relations for $\mathbf{S}_i$ and $\mathbf{B}_{i,j}$.
By dropping the constant terms from the Hamiltonian and using the above approximation together with the definitions in Eqs. we arrive to the mean-field Hamiltonian, $$\begin{gathered}
\label{parmaggio}
H_{\text{mf}} = \sum_{\langle i,j\rangle}\bigg[a_n \Big(\bar\chi_{i,j}^*\,{c^\dagger_{i,\alpha}}{c^{}_{j,\alpha}} + \bar\chi_{i,j}\, {c^\dagger_{j,\alpha}}{c^{}_{i,\alpha}}-|\chi_{i,j}|^2\Big)\\
+a_s\Big( \bar{\mathbf{S}}_i\cdot\mathbf{F}_{\alpha\beta}\,{c^\dagger_{j,\alpha}}{c^{}_{j,\beta}} +\bar{\mathbf{S}}_j\cdot\mathbf{F}_{\alpha\beta}\,{c^\dagger_{i,\alpha}}{c^{}_{i,\beta}}-\bar{\mathbf{S}}_i\cdot\bar{\mathbf{S}}_j\\
+\bar{\mathbf{B}}_{i,j}^*\cdot\mathbf{F}_{\alpha,\beta}\,{c^\dagger_{i,\alpha}}{c^{}_{j,\beta}}+
\bar{\mathbf{B}}_{i,j}\cdot\mathbf{F}_{\alpha,\beta}\,{c^\dagger_{j,\alpha}}{c^{}_{i,\beta}}-|\bar{\mathbf{B}}_{i,j}|^2\Big)\bigg]\\
-\sum_i \varphi_i ({c^\dagger_{i,\alpha}}{c^{}_{i,\alpha}}-1).\end{gathered}$$ The last term can be regarded as a Lagrange multiplier enforcing the single particle per site constraint. The Hamiltonian is now quadratic in the fermion operators, and therefore, it can be diagonalized directly. Once the spectrum and eigenvectors of the Hamiltonian are known, every physical quantity can be calculated. To this end, quasi-particles are introduced in terms of which the Hamiltonian is diagonal. Their occupation number is given by the Fermi-Dirac distribution function. Most importantly, there is a set of self-consistency equations
\[gorgonzola\] $$\begin{aligned}
1=\bar n_i&=\langle {c^\dagger_{i,\alpha}}{c^{}_{i,\alpha}} \rangle,\\
\bar \chi_{i,j}&=\langle {c^\dagger_{i,\alpha}}{c^{}_{j,\alpha}} \rangle,\label{eq:chiexp}\\
\bar{\mathbf{S} }_i&=\mathbf{F}_{\alpha\beta}\langle {c^\dagger_{i,\alpha}}{c^{}_{i,\beta}} \rangle,\label{eq:Sexp}\\
\bar{\mathbf{B} }_{i,j}&=\mathbf{F}_{\alpha\beta}\langle {c^\dagger_{i,\alpha}}{c^{}_{j,\beta}} \rangle,\label{eq:Bexp}\end{aligned}$$
which has to be solved to find the mean fields.
![(Color online) The choice of the unit cell. There are 8 nonequivalent lattice points inside the unit cell, which are illustrated as numbers inside the circles. There are 8 different bonds running inside the unit cell and 4 bonds connecting sites between neighboring cells. The bonds are directed and are enumerated with numbers. Their orientation is represented by an arrow. The coordinates in parenthesis refer to the position of the unit cell. The shading in the figure is only a guide to the eye, for locating the possible tetramerized clusters.[]{data-label="fig:unitcell"}](unit_cell){width="0.95\columnwidth"}
In order to solve the self-consistency equations, one needs to introduce a unit cell repeating periodically on the lattice. The choice of the unit cell is a key assumption in the applied mean-field calculation. In this system, we expect the competition between symmetry-breaking states with non-vanishing, on-site, classically ordered spins and states with bond-centered mean fields with symmetry breaking or spin-disordered nature. The unit cell has to be compatible with all these assumptions.
States with classically ordered, site-centered spins does not require any special unit cell. In a ferromagnetic state all of the mean-field spins point to the same direction. In a Néel-like state, the site spins are alternating in a checkerboard manner. Thus on the honeycomb lattice, even the smallest unit cell, containing two sites, can describe both states. The non-zero mean-field value of bond-centered operators signals the presence of spin correlations on clusters of sites. In the spin-3/2 case a completely antisymmetric SU(4) singlet can be formed with 4 sites. In the simplest case, these 4 sites occupy the smallest tetramer, 3 sites surrounding a central one [@lajko2013tetramerization]. In order to describe these states too, the unit cell needs to contain all these four sites. If we want to include also competition with states forming larger clusters, we need to introduce a further enlarged unit cell. To this end, we introduce a unit cell containing 8 sites, as depicted in Fig. \[fig:unitcell\].
We have $12$ $\bar\chi$ variables and $12\times3$ $\bar{\mathbf{B}}$ ones corresponding to the $12$ bonds, from which 8 is connecting sites inside the unit cell and 4 is linking together neighboring unit cells. These, together with the $8\times3$ spin variables and the $8$ Lagrange multipliers add up to a total of $48$ complex and $32$ real, that is overall $128$ real variables. The solution strategy is the following. The $128$ mean fields and Lagrange multipliers are assumed to be the variables we are looking for. The set of $128$ equations are those in Eqs. . The right hand side of the equations depend on the variables implicitly through the expectation values of the quasiparticles and the eigenvectors of the mean-field Hamiltonian. In the ground state ($T=0$), the quasiparticle occupation is 1 (0) for single-particle states below (above) the Fermi energy. Because of the jump in the occupation numbers at the Fermi energy, the nonlinear solver of the self-consistency equations can get stuck. To find the solutions we first go to finite but low temperatures, where the Fermi-Dirac distribution is more smooth. The occupation numbers are calculated with the help of the thermal average defined by the grand canonical density matrix $\rho=\exp(-\beta H_{\text{mf}})/Z$, with the Hamiltonian , $\beta=1/(kT)$ the inverse temperature and $Z=\mathrm{Tr}[\exp(-\beta H_{\text{mf}})]$ the grand canonical partition function. Then, we follow the solution by lowering the temperature to a much lower value. In our calculation the characteristic energy of the problem is $|a_n|$. By choosing a final inverse temperature $\beta=100\times|a_n|^{-1}$ we get results corresponding to the zero temperature limit up to our numeric accuracy.
Even though Eq. is derived from the Hubbard Hamiltonian , it has a much higher symmetry. Namely, Eq. is invariant under local $\mathrm{U}(1)$ transformations, ${c^{}_{j,\alpha}}\rightarrow e^{i\theta_j}\,{c^{}_{j,\alpha}}$, which is not true for Eq. . This emergent gauge symmetry is the consequence of the one particle per site local constraint of the Mott insulating state. The bond operators are not gauge invariant, they transform as, $\chi_{i,j}\rightarrow \chi_{i,j}\,e^{-i(\theta_i-\theta_j)}$, and similarly, ${\mathbf{B}}_{i,j} \rightarrow {\mathbf{B}}_{i,j}\,e^{-i(\theta_i-\theta_j)}$. Consequently, a state of the system is characterized by the full set of the mean fields, $\{\bar \chi_{i,j},\bar{\mathbf{S} }_i,\bar{\mathbf{B} }_{i,j}\}\rightarrow|\Psi_{\text{mf}}^{\bar\chi,\bar{\mathbf{S}},\bar{\mathbf{B}}}\rangle$, but in such a way, that those states are equivalent, whose mean fields are related to each other by a gauge transformation. Therefore, the states can only be characterized through gauge invariant quantities. The magnitude of the $\bar\chi_{i,j}$ bonds is gauge invariant. Instead of their phase, which is not gauge invariant, we can use the phase $\Phi$ of the Wilson loops $\Pi=|\Pi|e^{i\Phi}$, which are the products of the nonzero $\bar \chi_{i,j}$ bonds. Note, that when a Wilson loop is zero, due to the vanishing of a bond expectation value, one can always choose the bonds to be real along that loop. Inside the periodically repeating unit cell of our choice, there are four distinct elementary plaquettes, whose Wilson loops are:
\[eqs:WL\]
\_1 & =|\_1 |\_2 |\_3 |\_4 |\_5 |\_6 ,\[eq:WL1\]\
\_2 & =|\_1\^\* |\_9\^\* |\_[12]{}\^\* |\_4\^\* |\_7 |\_[11]{} , \[eq:WL2\]\
\_3 & =|\_6\^\* |\_8 |\_7\^\* |\_3\^\* |\_[10]{}\^\* |\_9 , \[eq:WL3\]\
\_4 & =|\_5\^\* |\_[12]{} |\_[10]{} |\_2\^\* |\_[11]{}\^\* |\_8\^\* . \[eq:WL4\]
Here the bond indices follow the convention used in Fig. \[fig:unitcell\]. However, the big Wilson loop encircling all 4 plaquettes is inevitably real. It can be easily checked by taking the product of all 4 Wilson loops Eqs. -. Thus, the phase of the fourth loop is just the opposite of the sum of the previous three.
The site spins, $\mathbf{S}_i$, are gauge invariant, and therefore their expectation values are physical quantities. The bond spins, $\mathbf{B}_{i,j}$, on the other hand, are not gauge invariant, and loop operators can not be introduced so straightforwardly. However, we can introduce a bond spin with the following definition: $$\label{eq:bondspin}
\bar{\mathbf{b}}_{i,j}\equiv\bar{\mathbf{B}}_{i,j}\frac{\bar\chi_{i,j}^*}{|\bar\chi_{i,j}|}.$$ The bond spin is gauge invariant by construction, and is the last element needed to characterize the state. In summary, the mean-field solutions are uniquely (up to a gauge transformation) characterized by the absolute values of the 12 $\chi$ bonds, the 8 site spins Eq. , the 12 bond spins Eqs. and 3 of the 4 Wilson loops Eqs. .
The equivalence of mean-field states related to each other by gauge transformation is obvious on the physical state vector of the spin system, i.e. on the spin wavefunction [@lee2006doping]. $$\label{eq:Gutzwiller}
\Psi(\alpha_1,\alpha_2,\ldots,\alpha_N)=\Big\langle 0\Big| {c^{}_{i_1,\alpha_1}} {c^{}_{i_2,\alpha_2}} \ldots {c^{}_{i_N,\alpha_N}}\Big|\Psi_{\text{mf}}^{\bar\chi,\bar{\mathbf{S}},\bar{\mathbf{B}}}\Big\rangle.$$ The construction of the physical state vector from the mean fields is the so-called Gutzwiller projection. It is easy to see that all of the gauge equivalent states lead to the same physical spin wave function, apart from an unimportant global phase, after Gutzwiller projection [@lee2006doping].
Phase diagram {#sec:phdiag}
=============
![(Color online) Energy per site vs coupling strength. On the bottom of the graph, the various phases (SB stands for spin bond) and approximate phase boundaries are indicated. The lines are guides to the eye for the coupling strength dependence of the energy per particle.[]{data-label="fig:energy"}](energies){width="0.97\columnwidth"}
In order to obtain the zero temperature phase diagram, we performed a massive search for the solutions of the self-consistency equations numerically, starting from random initial configurations. The phase diagram of the system is determined by the properties of the lowest energy solutions. The energy of various solutions are plotted in Fig. \[fig:energy\] for different coupling constant ratios. We measure the energy in units of $|a_n|$ (remember that $a_n<0$), and therefore only the ratio $a_s/|a_n|$ is relevant when studying the properties of the ground state.
The $\pi$-flux state
--------------------
![(Color online) Illustration of the $\pi$-flux state. The sites are represented by red ellipses, while the homogeneous $\chi_{i,j}$ bonds are drawn with blue lines.[]{data-label="fig:pifluxstate"}](MF0043.pdf)
![(Color online) The fermion energy spectrum of the $\pi$-flux color liquid phase.[]{data-label="fig:vbsenerg"}](spectrum3D0043.pdf){width=".97\columnwidth"}
Let us start with the center of the phase diagram, where the couplings in the singlet and quintet channels are equal ($g_0=g_2$), and thus the spin flipping interaction vanishes ($a_s=0$). In this case, the effective Hamiltonian is SU(4) symmetric. In this high symmetry point, all of the low energy mean-field solutions preserve the SU(4) symmetry: there is neither site- nor bond spin order, $\bar{\mathbf{S}}_i=0$, $\bar{\mathbf{B}}_{i,j}=0$. Such a state is exclusively characterized by the plaquette Wilson loops. For the lowest energy solution, we found that all of the SU(4) symmetric bond averages have the same magnitude, $|\bar{\chi}_{i,j}|\approx0.771$, as it is illustrated in Fig. \[fig:pifluxstate\]. The Wilson loops are $\Pi_1=\Pi_2=\Pi_3=\Pi_4\approx-0.21$. As the Wilson loops on the elementary plaquettes have a uniform negative value, this state is a homogeneous $\pi$-flux state. The energy of this state is found to be $E/N\approx-0.892|a_n|$ per site. In a $\pi$-flux state, time reversal symmetry is preserved, therefore — despite the nonzero flux passing through the elementary plaquettes — this state is nondegenerate, as it is expected from a potential universal resource state. The single fermion excitation spectrum of this homogeneous SU(4) state is shown in Fig. \[fig:vbsenerg\]. Due to the complete rotational invariance in the 4 dimensional spin space, and also to the choice of our unit cell, the spectrum consists of 4 bands, each of them are 8-fold degenerate. In our case of 1/4 filling, the lowest band is completely filled. There is a Dirac cone touching between the fully filled lowest band and the empty second band. Therefore, massless Dirac-fermion excitations determine the low energy properties of the homogeneous $\pi$-flux state. We found that this state remains stable at higher temperatures, too, up to a critical temperature around $k_B T_c \approx 0.75|a_n|$. The magnitude of the homogeneous order parameter is plotted versus the temperature in Fig. \[fig:chivstemp\]. Close to the critical temperature, the order parameter $|\chi_{i,j}|$ disappears at $T_c$ like a square root of the reduced temperature, according to the mean-field exponent $\beta=1/2$, as thermal fluctuations destroy the VBS order.
![(Color online) The temperature dependence of the dimensionless order parameter $|\chi_{i,j}|$ of the $\pi$-flux color-liquid phase at $a_s=0$.[]{data-label="fig:chivstemp"}](EvsB.pdf){width=".97\columnwidth"}
We also found that this SU(4) symmetric $\pi$-flux state is robust against the presence of weak SU(4) symmetry violating perturbations, like a nonzero $a_s$. It is particularly remarkable, that despite the Hamiltonian has only SU(2) global symmetry for nonzero $a_s$, the ground state is still invariant under the higher SU(4) symmetry. This symmetry enlargement is unusual, although not unique in high spin systems[@assaraf2004dynamical]. Furthermore, we found that in the homogeneous $\pi$-flux state the order parameter $\bar{\chi}_{i,j}$ does not depend on the SU(4) symmetry-breaking coupling $a_s$ at all. As a consequence, the energy of the solution does not depend on $a_s$, and from Eq. its explicit form is given by $E_{\pi\mathrm{flux}}/N= a_n \sum_{\left<i,j\right>} |\bar{\chi}_{i,j}|^2\approx - 0.771^2 |a_n|\cdot 12/8\approx -0.892|a_n|$, where 12 is the number of the independent bonds, 8 is the number of the sites in a unit cell. The energy per particle of the solution is indicated in Fig. \[fig:energy\] by a solid line.
The $\pi$-flux state is identical to the one reported as the ground state (algebraic color liquid) of the pure SU(4) Heisenberg model ($a_s=0$) in Ref. [@corboz12a]. Interestingly, in our mean-field calculation the $\chi_{i,j}$ parameters are obtained in a self-consistent way, and even without performing the Gutzwiller projection, the calculated energy of our state ($E/N\approx-0.892|a_n|$) compares remarkably well to the one after the Gutzwiller projection performed with Monte-Carlo calculation: $E/N\approx-0.894|a_n|$ of Ref. [@corboz12a]. Note, that the fermion spectrum, Fig. \[fig:vbsenerg\] also agrees to the one found in Ref. [@corboz12a].
As the strength of the spin flipping interaction $|a_s|$ is increasing, the homogeneous $\pi$-flux phase remains the lowest energy mean-field solution up to the two distinct critical values in the ferromagnetic and Néel sides. Even beyond the critical value of $a_s$, the homogeneous, SU(4) symmetric $\pi$-flux state remains a higher energy solution above the symmetry-breaking states, although, above $a_s^{c,1}\approx 0.26|a_n|$, the antiferromagnetic order becomes energetically more favorable. On the other side, for negative values of $a_s$, aready above the critical $a_s^{c,2}\approx -0.2|a_n|$ coupling, the ferromagnetic spin exchange starts to favor a symmetry-breaking state. However, before the fully polarized ferromagnetic order wins, in an extended region we found an intermediate state where spin and bond orders coexist, as it will be discussed in Section \[sec:spin-bond\].
Conventional symmetry-breaking states
-------------------------------------
![(Color online) Illustration of the Néel (a) and ferromagnetic (b) states. The site spins are represented with blue arrows.[]{data-label="fig:SBstates"}](SBstates.pdf)
[*Néel state:*]{} — In the Néel state both the SU(4) symmetric bond and spin-bond expectation values are zero, $\bar\chi_{i,j}=\bar{\mathbf{B}}_{i,j}=0$. In contrary, the site spins $\bar{\mathbf{S}}_i$ are nonzero. They have a homogeneous magnitude with maximal spin projection: $\pm$3/2, but spins on neighboring sites are pointing to opposite directions. A nice example of the emerging antiferromagnetic order is shown in Fig. \[fig:SBstates\] a). We have seen above, that when the coupling $a_s$ becomes larger than the critical value $a_s^{c,1}\approx 0.26 |a_n|$, the energy of the Néel state goes below that of the $\pi$-flux state. Note, that below $a_s^{c,1}$ iterations of the self-consistency equations starting from random initial mean-field values usually converge to non-symmetry-breaking states. This also indicates the extreme robustness of the SU(4) preserving ground state even in presence of weak $a_s$ couplings. The energy of this classical Néel order shows the usual linear dependence on the exchange coupling $a_s$: $E_{\mathrm{Neel}}/N=-a_s (3/2)^3$, as it can be observed in Fig. \[fig:energy\], too.
[*Ferromagnetic state:*]{} — For large and negative $a_s/|a_n|$, the spins prefer parallel alignment, due to the strong ferromagnetic coupling dominating the exchange processes. Accordingly, this homogeneous spin ordered state is a usual ferromagnetic state, in which only the spin expectation values $\bar{\mathbf{S} }_i$ are nonzero, and their values are independent of $i$. A typical ferromagnetic state is illustrated in Fig. \[fig:SBstates\] b). Note, that we found the ground state to be fully polarized ($|\bar{\mathbf{S} }_i|=3/2$) as soon as the ferromagnetic state becomes the lowest energy state. This happens for $a_s<a_s^{c,3}\approx-0.36|a_n|$. Similarly to the classical Néel state, in the fully polarized system the energy dependence on the spin exchange is $E_{\mathrm{FM}}/N= a_s (3/2)^3$. The line corresponding to this energy scaling is also plotted in Fig. \[fig:energy\].
The spin-bond ferromagnetic state {#sec:spin-bond}
---------------------------------
![(Color online) Illustration of the columnar-bond (a) and zip-bond (b) ferromagnetic state. Note the alignment of the dimers on the two subplots. The site spins are represented with blue arrows, while the bond spins are shown with green arrows.[]{data-label="fig:spinbondstates"}](spinbondstates.pdf)
On the ferromagnetic side, between the homogeneous $\pi$-flux and fully polarized ferromagnetic phases, the spin anisotropic and the SU(4) symmetric exchange become competitive. In their delicate balance, a mixed intermediate state between $a_s^{c,3}$ and $a_s^{c,2}$ wins, where both spin order and valence bond expectation values on nearest neighbor dimers coexist. Thus, in addition to the homogeneous nonzero spin averages, both of the two link operators $\chi_{i,j}$ and $\mathbf{B}_{i,j}$ have nonzero expectation values along the dimers.
The above state, in which a spin anistropic dimer order becomes superimposed upon the ferromagnetic background, has the lowest energy between $a_s^{c,3}$ and $a_s^{c,2}$, i.e. in the vicinity of the $\pi$-flux state. There are two gauge nonequivalent states, which are not related to each other by lattice symmetries either: in one case the dimers form a staggered columnar pattern \[Fig. \[fig:spinbondstates\] a)\], and in the other case, they lay in alternating directions in the neighboring columns forming a zip-like pattern \[see Fig. \[fig:spinbondstates\] b)\]. We found that this state is also robust, the magnitudes of the nonzero order parameters are not sensitive to the tuning of the spin flipping interactions: $|\bar{\mathbf{S} }_i|\approx |\bar{\mathbf{B}}_{i,j}|\approx |\bar{\chi}_{i,j}|\approx 1$. The mean-field energy of the solution from Eq. is $E_{\mathrm{SB}}/N = \sum_{\left<i,j\right>} ( a_n |\bar{\chi}_{i,j}|^2 + a_s |\bar{\mathbf{B}}_{i,j}|^2 + a_s| \bar{\mathbf{S}}_i| |\bar{\mathbf{S}}_j| ) \approx (a_n+4 a_s)/2$, where we used that the number of dimers per unit cell is 4. In Fig. \[fig:energy\] the $a_s$ dependence of the energy of the dimer spin-bond solution is also shown with a line. The energy of this dimer spin-bond state goes above that of the ferromagnetic state for $a_s<a_s^{c,3}$, but it still remains a solution, as the magnitude of the ferromagnetic exchange is increasing. Contrary, above the upper critical coupling ($a_s>a_s^{c,2}\approx -0.2 |a_n|$) the SU(4) invariant exchange destroys spin-bond order.
This intermediate spin-bond phase can be regarded as a polarized version of the SU(4)-symmetric dimer phase. To our knowledge, such a state has not yet been found in numerically exact calculations; however, in Ref. [@savary2012coulombic], in the framework of a slave-particle theory, a similar, polarized quantum spin-liquid phase was found.
In the uniform ferromagnetic part of the phase diagram, but not so far from the dimer spin-bond order ($a_s^{c,4}\approx-0.66|a_n|<a_s<a_s^{c,3}$) we found that the competition between the two exchange channels is even stronger leading to various spin-bond states in which the nonzero site spin and bond averages form different complex patterns. The energy of these spin-bond configurations lies lower than that of the dimer spin-bond state. These solutions are not robust, and show a large diversity.
Other competing states
----------------------
![(Color online) Illustration of the disconnected-chain state. The $\chi_{i,j}$ bonds are drawn with blue lines. Other mean fields are zero.[]{data-label="fig:chainstate"}](MF0038.pdf)
Above the lowest energy states, we found several other states, some competing with the ground state and some which are situated energetically well above. The competing states located energetically close to the homogeneous $\pi$-flux state in the $a_s^{c,2}<a_s<a_s^{c,1}$ region are especially interesting. These states are all similar to the ground state in many aspects: they are also SU(4) invariant VBS without any spin order. That is, for these states only the link order parameter $\chi_{i,j}$ has nonzero expectation value, and they are robust against the presence of weak SU(4) symmetry breaking spin flipping processes. However, these states are not homogeneous anymore, usually the translational invariance is preserved in one direction. Therefore these states are three-fold degenerate, corresponding to the lattice symmetry of rotation by $60$ degrees. We mostly found various weakly coupled ladder patterns with stronger bonds along the legs and rungs. The states with different patterns lay energetically close to each other. In Fig. \[fig:chainstate\] we present an extreme of such states, where complete dimension reduction can be observed: the bonds along the rungs, and between the ladders are zero, i.e. the nonzero bonds form disconnected chains. The value of the order parameter along the chains is $|\bar{\chi}_{i,j}|\approx 0.897$. As a consequence of the dimension reduction, the fermion spectrum is flat in the reciprocal space along the direction orthogonal to the orientation of the chain.
For $a_s<a_s^{c,3}$, with energies well above the ferromagnetic, and lowest energy spin-bond states, we found several higher energy solutions of the self-consistency equations where the site centered spin order coexists with valence bond order. In these states, the bond spins are disordered and only site spins exhibit the ferromagnetic order.
Summary {#sec:sum}
=======
In this paper we have studied the magnetic properties of Mott insulators realized with ultracold fermions on an optical lattice with honeycomb symmetry. In the framework of a mean-field theory, incorporating both site and bond orderings on equal footing, we calculated the phase diagram and identified the low energy competing magnetic states. We found, that even though the AKLT parent Hamiltonian can not be realized with ultracold atoms, a color liquid state with a $\pi$-flux per plaquette emerges as the ground state for an extended parameter region surrounding the SU(4) symmetric point. This state is characterized by a completely homogeneous set of valence bonds, similar to the AKLT state.
When the spin changing coupling constant is sufficiently large compared to the SU(4) symmetric coupling constant, the color liquid state goes to a symmetry breaking state. In the antiferromagnetic side there is a direct transition to the classical Néel order from the homogeneous $\pi$-flux state. Contrary, in the ferromagnetic side of the interaction, the transition to the fully polarized ferromagnetically ordered state goes through an intermediate phase, which is characterized both by site and bond spin orders. In this intermediate phase we found a narrow region where the emerging state can also be regarded as a dimerized VBS state, but with anisotropic nearest-neighbor correlations. For stronger ferromagnetic exchange we found a spin-bond “disordered” state, in which the nonzero links form various disordered patterns as a consequence of the competition between the SU(4) symmetry breaking($a_s$) and preserving ($a_n$) couplings. When the ferromagnetic coupling is further increased, the system ultimately goes to a ferromagnetic state with only site spin order. In the SU(4) symmetric point, our mean-field results are supported by the numerically exact methods of Ref. [@corboz12a], therefore we believe that at the SU(4) point, and also in a vicinity of it, the mean-field result can be trusted. Also, for high values of $|a_s|$, where one anticipates symmetry-breaking solutions, the conclusions of the mean-field theory look solid. Between these different symmetry limiting cases there has to be at least one phase transition. As the mean-field theory is not suitable to describe criticality, it is hard to tell the precise location of the transition point(s). The intermediate spin-bond phase, between the ferromagnetic and color-liquid phases can not be justified, to our knowledge, by earlier numerically exact calculations. However, a similar phase was found in Ref. [@savary2012coulombic] for a different model in a slave-particle calculation.
These phases can be distinguished in experiments by measuring quantities sensitive to the type of ordering. The most striking difference between the various phases is exhibited in nearest-neighbor spin correlations, which can be measured e.g. with the superlattice technique [@trotzky2010controlling]. Another way of testing the magnetic properties is by measuring the spin structure factor by spin-sensitive Bragg scattering [@hart2015observation]. An important question is how low the temperature should be in order to access these correlated phases. The color-liquid phase was found to be robust up to a critical temperature in the order of $|a_n|$, that is a few nanoKelvin in ultracold atom experiments on optical lattices. As for the symmetry breaking phases, we don’t expect them at finite temperatures in an infinite system due to the Mermin-Wagner theorem. However, in experiments in a finite system a tendency towards ordering can happen even in finite temperatures. The zero temperature results can be valid up to the gap of the excitations, which goes approximately with $|a_s|$.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge support from the Hungarian Scientific Research Fund (OTKA) (Grant Nos. PD104652, K105149 and K100908), Spanish MINECO Project FOQUS (FIS2013-46768), ERC AdG OSYRIS, EU IP SIQS, EU STREP EQuaM, and EU FETPROACT QUIC. GSz. and ESz. also acknowledge support from the János Bolyai Scholarship.
|
---
abstract: 'We review observations on the chemical enrichment of the intracluster medium (ICM) performed using BeppoSAX MECS data. The picture emerging is that non-cooling flow clusters have flat metallicity profiles, whereas a strong enhancement in the abundance is found in the central regions of the cooling flow clusters. All the non-cooling flow clusters present evidence of recent merger activity suggesting that the merger events redistributes efficiently the metal content of the ICM. The observed abundance excess in the central regions of cooling flow clusters is probably due to metals ejected from the cD galaxy located in the cluster core. Cooling flow cluster have also enhanced Nickel abundances in their cores with respect to the non cooling flow clusters.'
author:
- Sabrina De Grandi
- Silvano Molendi
title: The chemical enrichment of the ICM with BeppoSAX
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
The X-ray emission in clusters of galaxies originates from the hot gas permeating the cluster potential well. The continuum emission is dominated by thermal bremsstrahlung, which is proportional to the square of the gas density times a function of the gas temperature (i.e. the cooling function). From the shape and the normalization of the spectrum we derive the gas temperature and density. At the high temperatures measured in clusters the ICM is highly ionized (H and He are completely ionized). A measure of the equivalent width of a spectral line is a direct measure of the relative abundance of a given element. This comes from the fact that both the continuum and line emissions are two-body processes with the continuum emissivity proportional to the electron density times the proton density, and the line emissivity proportional to the electron density times the density of a given element. From the definition of equivalent width it is easily derived that this quantity is proportional to the ratio between the ion and proton densities.
In the 2-10 keV band Iron is the most easily measurable element in the X-ray spectrum. This is true for several reasons: the atomic physics of K-shell lines is well known; Fe is an abundant element in nature; the Fe-K$\alpha$ line is well isolated in the spectrum and it is located in a region of the spectrum where the spectral resolution of the X-ray detectors is usually good. Other elements measurable in the X-rays are K-shell lines of C, N, O, Ne, Mg, Si, S, Ar, Ca and Ni, and L-shell lines of Fe and Ni.
It is well known that the global Fe abundance in clusters is about a third of the solar value. The evolution of metal abundances of the ICM in clusters has been investigated by various authors and a common result is the lack of evolution within redshift about 0.3-0.4 (e.g. Allen & Fabian 1998). This is also confirmed by BeppoSAX measures on distant clusters (Della Ceca et al. 2000; Ettori, Allen & Fabian 2001).
While the origin of the metals observed in the ICM is clear (SNs make metals), less clear is the transfer mechanism of these metals from stars to the ICM. The main mechanisms that have been proposed for the metal enrichment in clusters are: enrichment of gas during the formation of the proto-cluster (e.g. Gnedin 1998, Kauffmann & Charlot 1998) ; ram pressure stripping of metal enriched gas from cluster galaxies (e.g. Gunn & Gott 1972, Toniazzo & Schindler 2001) ; stellar winds AGN- or SN-induced in Early-type galaxies (e.g. Matteucci & Vettolani 1988, Renzini 1997). As we shall see below clues to metal enrichment mechanisms can be derived from spatially resolved analysis of metal abundances.
An interesting result is that found by Allen & Fabian (1998) for a sample of cooling flow (CF) and non-cooling-flow (non-CF) clusters, where these authors found that CF clusters have global metallicities about 1.8 times higher that that of non-CF clusters. This difference could be explained by assuming that CF clusters have abundances excess in their cores. Understanding if the metal production in the ICM is segregated or if it is constant with the cluster radius is important because it leads to a more precise estimate of the metal amount in the ICM (e.g. the total iron mass) and gives clues to the transport mechanism(s) from galaxies to the ICM. Moreover, the comparison between the spatial distribution of metals and the optical light gives information on the galaxies which have contributed to the enrichment of the ICM.
Spatially resolved abundance measures in clusters observed with ASCA and BeppoSAX
=================================================================================
ASCA and BeppoSAX have been decisive missions to explore spatially resolved abundance measurements in clusters. Various works have identified abundance gradients in a few cooling flow clusters, e.g. Ikebe et al. (1997) on Hydra A, Molendi et al. (1998) on Perseus cluster, Dupke and White (2000) on A496.
Metal abundance derived from ASCA data for samples of rich clusters are reported in Kikuchi et al. (1999), White (2000), Fukazawa et al. (1998), Fukazawa et al. (2000), Dupke and White (2000), Finoguenov, David and Ponman (2000). A general result from these works is the evidence of abundance gradients in several clusters, with an indication that abundance gradients are common in CF clusters, however the shape of these gradients is poorly determined. In fact the quality of the ASCA data does not allow a detailed investigation of the abundance gradients. A typical example is the case of A496 where abundance profile has been derived from ASCA data by three different working groups. In the first case Dupke & White (2000) derived an abundance gradient (see Figure 1 top panel) dividing the cluster emission into two bins only, one from 0 to 2 arcmin and the second one from 2 to 12 arcmin. In the second case Finoguenov et al. (2000) by increasing the number of bins (at the price of increasing the errors in the abundance measurements) up to four within 13 arcmin (Figure 1 middle panel) find evidences of for a smoothly declining gradient. In the last case, similarly to Finoguenov et al, White (2000) increased the numbers of bins up to 5 within 12 arcmin finding again an evidence for a smoothly declining gradient with large errors (see Figure 1 bottom panel). A comparison of all these three cases shows a poor determination of the real shape of the abundance gradient in A496. In Figure 2 we show the abundance profile as observed with BeppoSAX, the data are able to constrain the shape of the gradient revealing that the abundance excess is concentrated in the cluster core.
The difference between the ASCA and BeppoSAX metallicity profiles is due to the differences in the PSFs. The ASCA PSF is broad (HPR $\sim
2$ arcmin), strongly energy dependent and non-radially symmetric, therefore the analysis of extended sources required complicated correction procedures (e.g. Markevitch et al. 1998, White & Buote 2000). On the other hand, BeppoSAX PSF is sharper (HPR $\sim 1$ arcmin), is almost energy independent (D’Acri et al. 1997) and radially symmetric, therefore the analysis of extended sources is relatively straightforward (e.g. De Grandi & Molendi 2001). Moreover BeppoSAX exposure times are typically about three times larger than ASCA exposure times. From all these considerations it follows that BeppoSAX MECS data are better suited than ASCA to investigate abundance profiles for galaxy clusters.
Up to date there are two works based on BeppoSAX MECS samples which are systematically searching for abundance gradients in clusters. The first is an analysis of 12 clusters performed by Irwin & Bregman (2001). This analysis is limited to the 9 innermost arcmin of the cluster emission (i.e. radii $\simlt 20\%$ of the virial radius) and do not explore systematically the difference between CF and non-CF clusters. Irwin & Bregman (2001) find a general evidence for a negative abundance gradient in most of the clusters.
The other work is that of De Grandi & Molendi (2001) on a sample of 17 clusters analyzed considering the whole field of view of the MECS (which corresponds to radii $\simlt 50\%$ of the virial radius). The projected abundance profiles of the non-CF systems (Figure 3 top panel), are consistent with being constant with the radius. On the contrary the metallicity profiles of the CF clusters (Figure 3 bottom panel) is completely different showing a clear evidence of an abundance gradient declining with the radius in most of the systems.
In Figure 4 we compare the mean error-weighted abundance profile for CF and non-CF clusters. The metal abundances of the CF clusters are larger than 0.4 of the solar value in the central regions and decrease rapidly to values similar to those of the non-CF clusters at radii $\simgt 0.25~ \rm {r_{180}}$. The profile for non-CF clusters is much flatter, a fit with a constant to all non-CF abundance measurements is statistically acceptable. However, a small gradient is present in the data (significant at more than $99.5\%$ level on the basis of an F-test). The comparison of the abundance profile for CF and non-CF clusters supports the scenario where major merger events disrupt the central regions of clusters thereby re-mixing the gas within these regions and therefore destroying pre-existing abundance gradients. The modest gradient observed in non-CF clusters is quite likely the relic of a much stronger gradient which has not been completely wiped out by merger events.
Implication for the enrichment mechanism of the ICM in CF cluster cores
-----------------------------------------------------------------------
If each merger events redistributes efficiently the metals within the ICM then the metal excess we see in the core of CF clusters should be directly related to the enrichment processes which have occurred in the cluster core since the last major merger. Thus, just as the global metallicity of clusters is an indicator of the global star formation history within the whole cluster, the abundance excess we see in the core of CF clusters is an indicator of the star formation history in the core of the cluster since the last major merger. In the light of the above statement, we have tried to test whether the metal abundance excess we see is due to metals expelled from early type galaxies located in the core of the cluster. More specifically we have computed the metal abundance excess profile expected when the metal excess distribution traces the light distribution of early type galaxies, included the cD galaxy (details are given in De Grandi & Molendi 2001). We have performed this computation for the 4 cooling flow objects where the metal abundance profile is best measured and optical data is available, namely: A85, A496, A2029 and Perseus (see Figure 5).
For A496 and A2029 the predicted projected metal abundance excess can be reconciled with the observed one, while for A85 the predicted projected metal abundance excess appears to be slightly more centrally concentrated than the observed one. The more interesting case is that of Perseus cluster, where the two profiles are substantially different.
We find that the abundance excess in the expected profiles is completely due to the cD galaxy. Fukazawa et al. (2000) computed that the amount of metal excess at the cluster center can be provided by the cD alone. Therefore we conclude that we are probably just observing the accumulation of metal ejection from the cD galaxy into the ICM, and suggest that a possible way of reconciling the observed and predicted abundance excess profile of Perseus is to assume that metals ejected from the central cD have drifted away by about 50 kpc.
BeppoSAX Nickel measurements
============================
For an updated sample of 22 clusters (11 with CF and 10 without CF) we have measured Nickel abundances in the innermost regions of the clusters, i.e. for radii smaller than 200-400 kpc. The results are reported in Figure 6. This figure shows the Fe segregation between CF and non-CF clusters already discussed in the previous sections, as well as a difference for the Ni abundances.
The mean Fe abundance values for CF and non-CF clusters are, in solar units, $0.45\pm0.01$ and $0.30\pm0.01$, whereas the mean Ni abundances are $1.10\pm0.12$ and $0.16\pm0.17$, respectively. We find a $\sim
5\sigma$ evidence for a Ni excess in the central regions of the CF clusters with respect to the central regions of non-CF clusters. This Nickel excess is probably associated to the cD galaxy as the Iron excess, indeed both elements are mostly produced by SN type Ia which are dominating the enrichment of the cluster central regions. The Nickel-to-Iron abundance ratio produced by the cD galaxy, defined as $[Ni/Fe]_{cD} = ([Ni]_{CF} - [Ni]_{non-CF}) / ([Fe]_{CF} -
[Fe]_{non-CF})$, is $6.3\pm1.4$, normalized to the solar ratio.
References
==========
Allen, S. W., & Fabian, A., C. 1998, MNRAS, 297, L63 (AF98)\
Anders, E., & Grevesse, N. 1989, Geochim. Cosmoshim. Acta, 53, 197\
D’Acri, F., De Grandi, S., & Molendi S. 1998, Nuclear Physics, 69/1-3, 581 (astro-ph/9802070)\
De Grandi, S. & Molendi, S. 2001, ApJ, 551, 153\
Della Ceca, R., Scaramella, R., Gioia, I. M., Rosati, P., Fiore, F., Squires, G. 2000, A&A, 353, 498\
Dupke, R. A. & White R. 2000, ApJ, 528, 139\
Ettori, S., Allen, S. W., Fabian, A. C. 2001, MNRAS, 322, 187\
Finoguenov, A., David, L. P. & Ponman, T. J. 2000, ApJ, 544, 188\
Fukazawa, Y., Makishima, K., Tamura, T., Ezawa, H., Xu, H., Ikebe, Y., Kikuchi, K. & Ohashi, T. 1998, PASJ, 50, 187\
Fukazawa, Y., Makishima, K., Tamura, T., Nakazawa, K., Ezawa, H., Gnedin, N. Y. 1998, MNRAS, 294, 407\
Gunn, J. E. & Gott J. R. 1972, ApJ, 176, 1\
Ikebe, Y., Kikuchi, K., Ohashi, T. 2000, MNRAS, 313, 21\
Ikebe, Y. et al. 1997, ApJ, 481, 660\
Irwin, J. A. & Bregman J. N. 2001, ApJ, 546, 150\
Kauffmann, G. & Charlot, S. 1998, MNRAS, 294, 705\
Kikuchi, K., Furusho, T., Ezawa, H., Yamasaki, N. Y., Ohashi, T., Fukazawa, Y.; Ikebe, Y. 1999, PASJ, 51, 301\
Markevitch, M., Forman, W. R., Sarazin, C. L. & Vikhlinin, A. 1998, ApJ, 503, 77\
Matteucci, F. & Vettolani, G. 1988, A&A, 202, 21\
Molendi, S., Matt, G., Antonelli, L. A., Fiore, F., Fusco-Femiano, R., Kaastra, J., Maccarone, C. & Perola, C. 1998, 499, 608\
Renzini, A. 1997, ApJ, 488, 35\
Toniazzo, T. & Schindler, S. 2001, MNRAS in press (astro-ph/0102204)\
White, D. A. 2000, MNRAS, 312, 663\
White, D. A. & Buote, D. A., 2000, MNRAS, 312, 649\
|
---
abstract: 'A new technique was developed to measure the lifetimes of neutron unbound nuclei in the picosecond range. The decay of $^{26}$O$\rightarrow$$^{24}\mathrm{O}+n+n$ was examined as it had been predicted to have an appreciable lifetime due to the unique structure of the neutron-rich oxygen isotopes. The half-life of $^{26}$O was extracted as 4.5$^{+1.1}_{-1.5}~(stat.) \pm 3 (sys.)$ ps. This corresponds to $^{26}$O having a finite lifetime at an 82$\%$ confidence level and, thus, suggests the possibility of two-neutron radioactivity.'
author:
- 'Z. Kohley'
- 'T. Baumann'
- 'D. Bazin'
- 'G. Christian'
- 'P.A. DeYoung'
- 'J.E. Finck'
- 'N. Frank'
- 'M. Jones'
- 'E. Lunderberg'
- 'B. Luther'
- 'S. Mosby'
- 'T. Nagi'
- 'J. K. Smith'
- 'J. Snyder'
- 'A. Spyrou'
- 'M. Thoennessen'
title: 'Study of two-neutron radioactivity in the decay of $^{26}$O'
---
Drastic changes in the structure, properties, and available decay-modes of isotopes with extreme neutron-to-proton ratios have been observed in comparison to their stable counterparts [@BAUMANN12; @Pfu12; @Tho04; @Han01; @BROWN01; @Ots01]. In 1960 Goldansky predicted that the unique properties of very proton-rich nuclei would produce scenarios in which one- and two-proton radioactivity could be observed [@Gold60]. These exotic modes of radioactivity were later verified through measurements of the one-proton decay of $^{151}$Lu [@Hof82] and $^{147}$Tm [@Klep82] and the two-proton decay of $^{45}$Fe [@Pfu02; @Gio02]. In particular, a measurement by Miernik *et al.* allowed for the exotic two-proton radioactivity of $^{45}$Fe to be fully characterized for the first time providing insight into the three-body structure of this proton dripline nucleus [@Mie07]. New modes of radioactivity have also been observed for neutron-rich nuclei including $\beta$-delayed two- [@Asu79], three- [@Azu80], and four-neutron decays [@Duf88].
Recently, Grigorenko *et al.* calculated the lifetimes of one-, two-, and four-neutron decays from unbound nuclei [@Gri11]. While long lifetimes for proton decay can arise from the presence of both the Coulomb and angular momentum barriers, the neutron decay must only overcome the angular momentum barrier and will therefore have a much shorter lifetime. However, Grigorenko *et al.* showed that it would be possible for some neutron-rich unbound nuclei to have long lifetimes, reaching the limit of radioactivity, depending on the nuclear structure [@Gri11]. While an exact limit for a lifetime to be considered radioactivity does not exist, different arguments have been presented suggesting a lower limit between 10$^{-14}$ s and 10$^{-12}$ s (see discussion in Refs. [@Pfu12] and [@Tho04]). For the two-neutron unbound $^{26}$O a lifetime on the order of 10$^{-12}$ s was predicted for a ground-state resonance energy of about 150 keV and a pure $\nu$($d_{3/2}$)$^{2}$ configuration for the valence neutrons [@Gri11].
The recent measurement of the $^{26}$O ground state resonance at $E_{decay}$ = 150$^{+50}_{-150}$ keV by Lunderberg *et al.* opened up the exciting possibility for a new type of radioactivity to be discovered [@LUN12]. This limit was confirmed by the GSI-LAND group which determined the $^{26}$O ground state to be unbound by less than 120 keV [@Cae12]. Upper-limits have already been placed on the lifetime of $^{26}$O from previous experiments. Searches for $^{26}$O using fragment separators placed a lifetime limit of $<$ 200 ns (roughly the flight time through the separator) based on the non-observation of the bound nucleus [@Gui90; @Tar97]. The GSI-LAND group improved that limit by an order of magnitude to $<$ 5.7 ns based on the reconstruction of an $^{26}$O fragment which decayed in flight outside the target [@Cae12]. The $^{26}$O lifetime is therefore too short to be observed by traditional implantation/decay methods, where lifetimes down to several hundred nanoseconds have been achieved using digital electronics [@Lid06], and must be measured in-flight. For two-proton emission, reconstruction of the decay vertex by Mukha *et al.* [@Muk07] and an adaption of the Recoil Distance Doppler Shift (RDDS) technique by Voss *et al.* [@Voss12] were used to measure the lifetime in-flight of $^{19}$Mg.
In this letter, the lifetime of $^{26}$O is extracted using a novel technique based on the velocity difference of the emitted neutrons and residual charged fragment traveling through a thick target. This approach is a variance on the Doppler-Shift Attenuation Method (DSAM) in which an excited nucleus is slowed in a solid material resulting in a distribution of Doppler shifted $\gamma$-ray energies that can be related to the lifetime [@Sch68; @Nol79; @Dew12]. This concept is extended in the present work for nuclei which decay by neutron emission. The analysis is completed using the experimental data from the work of Lunderberg *et al.* [@LUN12], in which the $^{26}$O ground state resonance was originally measured.
Since the experimental details have been provided previously in Ref. [@LUN12], only a brief overview is presented. The $^{26}$O was produced using a one-proton knockout reaction from a 82 MeV/u $^{27}$F beam produced at the National Superconducting Cyclotron Laboratory at Michigan State University. To produce the $^{27}$F beam, a 140 MeV/u primary beam of $^{48}$Ca bombarded a 1316 mg/cm$^{2}$ $^{9}$Be production target. The A1900 fragment separator [@Mor03] was used to select the desired $^{27}$F fragments which were then impinged on a 705 mg/cm$^{2}$ $^{9}$Be reaction target in the experimental vault. The $^{26}$O$\rightarrow^{24}\mathrm{O}+n+n$ decay was measured using the Modular Neutron Array (MoNA) and the 4 Tm superconducting dipole magnet [@SWEEPER]. The dipole magnet bent the charged fragments about 43$^{\circ}$ into a suite of charged particle detectors, which allowed for the mass, charge, kinetic energy, and angle of the charged particle to be reconstructed from its track through the magnet [@Chr12]. MoNA was placed 6.05 m from the reaction target and provided the measurement of the velocity and angle of the neutrons.
The three-body decay energy of the $^{24}\mathrm{O}+n+n$ system was calculated as $E_{\mathrm{decay}} = M_{^{26}\mathrm{O}} - M_{^{24}\mathrm{O}} - 2M_{n}$, where $M_{^{26}\mathrm{O}}$ ($M_{^{24}\mathrm{O}}$) is the mass of $^{26}$O ($^{24}$O) and $M_{\mathrm{n}}$ is the neutron mass. The invariant mass, $M_{^{26}\mathrm{O}}$, was calculated from the experimentally measured four-momenta of the $^{24}$O and two neutrons. The three-body decay spectra requires a triple coincidence of two interactions in MoNA that pass the causality cuts and a $^{24}$O fragment. The causality cuts are used to select true $2n$ events from multiple scattering of a single neutron and are discussed in detail in Ref. [@LUN12].
![\[f:edecay\] (Color online) Experimental $^{24}\mathrm{O}+n+n$ decay energy spectrum (solid black points) with causality cuts applied is compared with the Monte Carlo simulation (solid black line) with two components: (red long-dashed line) the $^{26}$O ground state resonance and (green short-dashed line) the first excited state. The vertical dotted line represents the selection of $^{26}$O events used in the analysis.](Fig1.eps){width="40.00000%"}
A detailed Monte Carlo simulation was used to fit the experimental spectrum as described in Ref. [@LUN12]. The simulation included all relevant components of the experimental setup. In particular, special care was taken in reproducing the neutron interaction observables in MoNA using the `Geant4` framework with the custom neutron interaction model <span style="font-variant:small-caps;">menate\_r</span> [@Koh12]. As shown in Fig. \[f:edecay\], the experimental three-body decay spectrum was well reproduced by the Monte Carlo simulation including the decay from both the ground state (red long-dashed line) and first excited state (green dashed line). It is important to note that the ground state resonance was determined from a fit of the data [@LUN12] while the placement of the first excited state was taken from predictions from the continuum shell-model [@Volya06].
Two scenarios for the decay of $^{26}$O with different lifetimes are illustrated in Fig. \[f:cartoon\]. The $^{27}$F beam enters the 3815 $\mu$m (705 mg/cm$^{2}$) $^{9}$Be target followed by the one-proton knockout reaction producing the $^{26}$O. If the reaction was to occur at the beginning of the target where the $^{27}$F beam is traveling at about 11.8 cm/ns and the $^{26}$O had a very short lifetime (top of Fig. \[f:cartoon\]) then the neutrons would be emitted with an average velocity of 11.8 cm/ns. In the other case if the $^{26}$O had a lifetime of 30 ps (bottom of Fig. \[f:cartoon\]), which is roughly the time of flight through the target, then the neutrons would be emitted with an average velocity of 10.9 cm/ns due to the energy loss of the $^{26}$O fragment traveling through the target. Thus, the observation of a shift in the expected neutron velocity can provide a measure of the lifetime of $^{26}$O.
![\[f:cartoon\] The decay of $^{26}$O within the thick $^{9}$Be target is illustrated for two cases: (top) very short lifetime corresponding to an immediate decay and (bottom) a lifetime around 30 ps which allows the $^{26}$O to exit the target before decaying.](Fig2.eps){width="40.00000%"}
The relative velocity between the neutrons and fragment is defined as $V_{rel} = V_{n} - V_{frag}$, where $V_{n}$ ($V_{frag}$) is the velocity of the neutron (fragment) in the laboratory frame. The relative velocity was examined to remove the effect of the momentum dispersion of the $^{27}$F beam ($\Delta p/p$ = 2$\%$). Thus, the variation in the incoming velocity of the $^{27}$F is removed event by event. Since the reaction point in the target is unknown, the fragment velocity ($V_{frag}$) is calculated assuming the reaction occurs at the center of the target. The width of the $V_{rel}$ distribution will be dependent on the target thickness and magnitude of the decay energy (both of which will increase the width). If the reaction point was known on an event-by-event basis and the decay energy was very small then the $V_{rel}$ distribution should be narrow and centered around zero. While the width of the experimental $V_{rel}$ distribution will be increased, the mean velocity will still be centered around zero if the $^{26}$O lifetime is short. Thus, a shift in the relative velocity away from zero would indicate a long-lived component of the decay.
The $^{26}$O events were selected from events passing the causality cut criteria and having a $E_{decay}<1.0$ MeV, as indicated from the dotted grey line in Fig. \[f:edecay\]. This selection should maximize the statistics and minimize the contamination of other decay channels. Based on the fit of Fig. \[f:edecay\] the $^{26}$O ground state resonance accounts for 96$\%$ of the events with $E_{decay}<1.0$ MeV. The relative velocity between the $^{24}$O and each of the emitted neutrons is shown in Fig. \[f:vrel\](a). The experimental $V_{rel}$ (solid black points) is shifted away from zero with an average $V_{rel}<0$. This implies that the neutron velocity is smaller than the fragment velocity (at half target thickness). This would be the case if the neutrons were emitted after the $^{26}$O traveled through a portion of the target decreasing its velocity.
It is important to understand the calibration, resolution, and accuracy of the neutron and fragment velocities. The neutron time-of-flight (and therefore velocity) was calibrated based on the time-of-flight of the gamma-rays produced at the target traveling to MoNA. From the width of the gamma peak the relative resolution (FWHM/centroid) of the neutron velocity is about 3$\%$. In comparison to the resolution, the accuracy of the neutron velocity is determined from the accuracy of the time-of-flight measurement and location of the MoNA detector. The neutron velocity accuracy was determined to be 0.03 cm/ns at beam velocity (11.8 cm/ns). The fragment velocity is determined from the track of the fragment through the dipole magnet which is measured using two Cathode Readout Drift Chambers (CRDCs). The accuracy of the fragment velocity is related to the accuracy of the magnetic field map and measured position of the CRDCs. Reasonable variation of these parameters showed the fragment velocity to be 0.02 cm/ns at beam velocity. While the resolutions will determine the width of the $V_{rel}$ distribution, the accuracy of the centroid is related to the accuracy of the neutron and fragment velocity measurements. The detector resolutions were included in the Monte Carlo simulation.
![\[f:vrel\] (Color online) (a) Experimental $V_{rel}$ distribution from the decay of $^{26}$O compared to the Monte Carlo simulation where the $^{26}$O half-life is set as 0 ps, 4 ps, and 10 ps. (b) Negative log-likelihood ($-$ln$[L]$) as a function of the half-life from the unbinned fit of the experimental $V_{rel}$ data set. The 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence interval are indicated.](Fig3.eps){width="38.00000%"}
In order to extract a half-life limit ($T_{1/2}$) of $^{26}$O from the $V_{rel}$ distribution, the Monte Carlo simulation was modified such that the probability distribution for the $^{26}$O decay based on $T_{1/2}$ was included. Thus, after the one-proton reaction occurred within the target (at a random position) the $^{26}$O was propagated for a time $t$ determined from the probability distribution before decaying into $^{24}\mathrm{O}+n+n$. The resulting $V_{rel}$ distributions from the simulation with $T_{1/2}$ = 0, 4, and 10 ps is compared to the experiment in Fig. \[f:vrel\](a). The $V_{rel}$ distribution with $T_{1/2}$ = 0 is unable to reproduce the experimental data. A much better fit is achieved with $T_{1/2}$ = 4 ps, which shows a similar shift in the simulation as the experiment. This suggests that $^{26}$O did not decay instantaneously but had an appreciable lifetime. The shape of the decay energy spectrum (Fig. \[f:edecay\]) would also be affected by the finite lifetime of $^{26}$O. The Monte Carlo simulations showed that significant changes in the $E_{decay}$ spectrum would be observed for $T_{1/2} \gtrsim$ 10 ps.
Due to the low statistics of the experiment, the $\chi^{2}$ analysis was observed to be dependent on the binning of the data. Therefore, a unbinned maximum likelihood technique was employed to determine the statistical significance of the results. This procedure is described in Ref. [@Sch93] and was recently used in the analysis of $^{27,28}$F measured with MoNA [@Chr12; @Chr12PRL]. The negative log-likelihood ($-$ln$[L]$) is plotted as function of $T_{1/2}$ in Fig. \[f:vrel\](b). A minimum in $-$ln$[L]$ is found at 4.5 ps. The $n\sigma$ confidence intervals are calculated as ln$[L_{max}] - $ln$[L]\leq n^{2}/2$. The 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence intervals are shown in Fig. \[f:vrel\](b). The statistical significance of the results indicate that $^{26}$O has a half-life of about 4.5 ps.
In addition to the statistical significance, it is important to account for possible systematic uncertainties. As previously discussed, the accuracy of the neutron and fragment velocities was 0.03 and 0.02 cm/ns, respectively. This represents a total systematic uncertainty of 0.05 cm/ns in the $V_{rel}$ distribution, which corresponds to a 1.7 ps systematic uncertainty, and indicates a finite half-life of $^{26}$O at 95$\%$ confidence level. The systematic uncertainty was also estimated through examining the neutron decay of the first excited state of $^{23}$O$^{*} \rightarrow ^{22}$O$ + n$, which was also populated during the experiment from the $^{27}$F beam. Thus, the $^{9}$Be target, $B\rho$ of the dipole, MoNA configuration, Sweeper detector settings, and calibrations were identical to the $^{26}$O measurement and can provide an estimate of any unknown systematic errors. Since the decay of $^{23}$O should not have a long-lived component, the relative velocity spectrum should not be shifted away from $V_{rel}$ = 0 (see Fig. \[f:o23\]). Following the same half-life analysis discussed above, the upper limit on $1\sigma$ $T_{1/2}$ was 3 ps for the $^{23}$O distribution. Therefore, the systematic uncertainty was estimated as 3 ps in comparison to 1.7 ps determined above. The half-life of $^{26}$O is then taken as 4.5$^{+1.1}_{-1.5}~(stat.) \pm 3 (sys.)$ ps, which gives $T_{1/2} > 0$ at 82$\%$ confidence level. A new measurement with improved statistics would allow for both the statistical and systematic uncertainties to be reduced.
![\[f:o23\] Experimental $V_{rel}$ distribution from the $^{23}$O$^{*} \rightarrow ^{22}$O$ + n$ decay. The Gaussian fit (dashed line) is shown to guide the eye. ](Fig4.eps){width="30.00000%"}
![\[f:grig\] (Color online) Predications of Grigorenko *et al.* for the true two-neutron emission width (half-life) as a function of the decay energy. The grey region represents the $3\sigma$ statistical limit on the $T_{1/2}$ from the current work and the ground state resonance energy limit from Ref. [@LUN12]. This figure was adapted from Ref. [@Gri11]. ](Fig5.eps){width="41.00000%"}
Possible systematic effects related to the selection of the $^{24}$O events and the application of the $2n$ causality cuts were also investigated. The selection of the $^{24}$O events (shown in Fig. 2 of Ref. [@LUN12]) based on the time-of-flight was varied to examine the dependence of the $V_{rel}$ distribution (or half-life). The results showed minor variations and the half-life remained within the 1$\sigma$ limit shown in Fig. \[f:vrel\]. While the causality cuts allow for the removal of the large majority of all false $2n$ events, it is important to verify that shift in the $V_{rel}$ distribution is not created by the causality cuts. The causality cuts were removed from both the experimental and simulated data and the shift in the $V_{rel}$ distribution was maintained.
The results of the lifetime analysis are compared with the prediction of Grigorenko *et al.* [@Gri11] in Fig. \[f:grig\]. The solid grey region is defined from the $3\sigma$ statistical limit on the $T_{1/2}$ and the ground state resonance energy limit of $E<200$ keV [@LUN12] for $^{26}$O. As shown the agreement with the predictions will depend greatly on improving the constraints on the energy and the configuration of the ground state. For example, a small $\nu(s)^{2}$ component in the $^{26}$O ground state will require the resonance energy to be very small for $T_{1/2}$ = 4.5 ps.
In summary, a new technique for measuring the lifetimes of neutron unbound nuclei has been presented and applied to the case of $^{26}$O. A shift in the relative velocity between the $^{24}$O and emitted neutrons was observed and shown to be related to the lifetime of $^{26}$O. Detailed Monte Carlo simulations, in which the half-life of $^{26}$O could be varied, were compared to the experimental data. The extracted $^{26}$O half-life was 4.5$^{+1.1}_{-1.5}~(stat.) \pm 3 (sys.)$ ps. This corresponds to $^{26}$O having a finite lifetime with an 82$\%$ confidence level and suggests the possibility of two-neutron radioactivity. It appears that, much like the proton dripline, the unique structure of the neutron dripline nuclei opens the door for observations of new modes of radioactivity. Future experimental work is needed to confirm this observation and provide stringent constraints on the properties of the $^{26}$O ground state.
The authors would like to thank H. Attanayake, D. Divaratne, S. M. Grimes, A. Haagsma, and A. Schiller from Ohio University for help during the experiment. The authors gratefully acknowledge the support of the NSCL operations staff for providing a high quality beam. This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-NA0000979 and DOE Award number DE-FG02-92ER40750. This work was also supported by the National Science Foundation under Grant Nos. PHY06-06007, PHY08-55456, PHY09-22335, PHY09-69058, and PHY11-02511.
[33]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [“,” ]{} (), @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{}
|
---
abstract: 'This contribution describes fast time-stamping cameras sensitive to optical photons and their applications.'
address: 'Brookhaven National Laboratory, Upton NY 11973, USA'
author:
- Andrei Nomerotski
bibliography:
- 'ulitima\_full.bib'
title: Imaging and time stamping of photons with nanosecond resolution in Timepix based optical cameras
---
time-stamping camera ,imaging mass spectrometry ,fluorescent lifetime imaging ,single photon imaging ,Tpx3Cam ,TimepixCam
Imaging with photon counting
============================
Imaging of fast processes with nanosecond-scale timing resolution is necessary in many applications. Detection of individual photons is ultimately the best route to capture all available information for the process in which they were created. This information then can be used to do basic imaging by counting all detected photons or, if desired, to perform more complex operations with the data, which could involve timing of individual hits or some correlation analysis. Photon counting is already a widely used modality in x-ray imaging, where the signal is large enough to detect individual photons directly without external amplification, and to enable measurement of the time and energy for each of them. Here, we describe the concept of time stamping for optical photons, including single optical photons, and review recent results on TimepixCam and Tpx3Cam cameras, which can be used for this purpose.
[**Framing versus time-stamping**]{}: In conventional imagers the signal is integrated in a slice of time and stored in the pixel for consequent readout, frame by frame. Currently, this approach makes it impossible to achieve nanosecond resolution for continuous readout because the data rates are becoming prohibitively high. Silicon based imagers for the 400-1000nm wavelength range are commercially available for frame rates of up to 10kHz, with their pixel output rate of Gpix/s. This corresponds to the time resolution of 100 $\mu$s, which is orders of magnitude inferior to the nanosecond-scale requirement. More specialized cameras based on CCD or CMOS technology are becoming available, which can achieve faster rates by buffering multiple frames in the sensor [@etoh2013; @kirana]. However, this approach is not scalable and cannot be used for continuous readout. When the buffer is full one must stop and transfer the data to the outside world so the overall duty cycle is small.
An alternative approach is based on the so-called data-driven readout when only the data of interest is captured. To reduce the rate and support continuous readout, only those pixels in which the signal exceeds a certain predefined level are measured and read out. Of course, this approach is favored over the full frame readout when the occupancy of the sensor is not too high, typically less than 10-20%. In the context of light detection, this technique was implemented in the PImMS camera for mass-spectrometry applications [@pimms1; @pimms2], and, as described in the following, in the optical cameras based on readout chips of the Timepix family [@timepix; @timepix3].
In the data-driven approach the signal shaping in the front-end electronics is fast, with peaking time of $\sim100$ ns in order to be compatible with nanosecond timing resolution. With enough statistics, images can be formed by counting the photons, and also more complex analyses can be performed. For example, one could use the measured information to determine time coincidences between the photons, to calculate correlations, invariant masses etc. This has close parallels with the registration of x-rays at the synchrotron light sources and of ionizing particles in the high energy physics experiments, where they are detected as standalone objects.
The high rate capability of the readout electronics is essential for fast accumulation of statistics. Assuming a similar back-end bandwidth as for the framing approach, Gpix/sec, and a 0.1-1 Mpix array, it is easy to calculate that this would correspond to an average pixel and, therefore, an average frame readout rate of 1-10 kHz. Thus, the average “frame” rate for a data-driven system is similar to the discussed above framing approach but in addition to the “normal” imaging one has a precise time-stamp for each photon. The price for this, of course, is complexity of the pixel as the data-driven approach needs to accommodate considerably more than just a few transistors as in simple framing architectures. It is typical to have hundreds of transistors per pixel. This also leads to a larger pixel size: 55x55 and 70x70 square micron respectively for Timepix and PImMS sensors. An overview comparison of these time-stamping optical cameras is given in Table \[tab:comparison\], with more detail on the Timepix-based cameras provided in the further sections.
\[tab:comparison\]
PImMS-1 PImMS-2 TimepixCam Tpx3Cam
------------------------ ---------------- ----------- ------------- -------------
year 2009 2012 2015 2017
array 72 x 72 324 x 324 256 x 256 256 x 256
pixel size, $\mu$m$^2$ 70 x 70 70 x 70 55 x 55 55 x 55
time resolution 12.5 ns 12.5 ns 10 ns 1.6 ns
pixel information TOA TOA TOA TOA & TOT
max rate 1200 Hz 60 Hz 2000 Hz 80 Mpix/s
max QE, % 8 8 90 90
technology MAPS [@pimms2] MAPS hybrid CMOS hybrid CMOS
: Comparison of the existing time-stamping optical cameras.[]{data-label="tab:milestones"}
[**Optical versus direct detection**]{}: The pixel noise is too high to be sensitive to single particles. Their detection for the applications discussed in this paper currently require external amplification in the form of a micro-channel plate (MCP), which produces an avalanche of electrons in the MCP pores. In principle, these electrons can be collected to sense electrodes on the readout chip directly. Indeed, the Timepix2 chip was used before to directly detect electrons from the MCP both for optical photons and for ion imaging applications [@Jungmann2010; @Jungmann2013; @valerga2014]. Another approach advocated here would be to send the MCP electrons to a thin layer of fast scintillator so they produce a flash of light, which can be registered with an optical camera. This approach is very common in the ion imaging and in the intensified cameras for night vision, though it is normally used for much slower scintillators and cameras. The two approaches are illustrated in Figure \[fig:optical\].
{width="0.8\linewidth"}
Both PImMS and Timepix cameras adopted the optical approach, which has three important advantages over the direct detection approach. Firstly, the direct collection of MCP electrons requires close ($\sim$ mm) proximity of the readout chip, which must be placed inside the experimental volume, to the kV-scale voltages at the MCP. This is a considerable complication because of possible sparking. It also requires a solution for cooling of the chip in a vacuum. The optical approach completely avoids these issues. Secondly, the camera is placed outside of the vacuum, so it is fully decoupled from the rest of the setup. In many cases it just replaces a slower camera used in these experiments before, and furthermore, the approach allows for a straightforward upgrade path as technology improves, without interfering with the rest of the experiment. Lastly, the optical approach allows flexible mapping between the scintillator screen and sensor by introducing demagnification, so a larger scintillator can be fully imaged in a small sensor. Other optical schemes with magnification, relay lenses and mirrors are also possible.
Fast optical cameras based on Timepix2 and Timepix3
===================================================
The fast cameras described below are based on the so-called hybrid pixel detectors: a pixelated optical sensor with a high quantum efficiency (QE) is bump-bonded to a Timepix ASIC (application specific integrated circuit). The design of the back-side illuminated silicon sensor, in particular its thin entrance window, was inspired by the fully depleted astronomical CCDs, such as used, for example, in LSST [@Radeka2009], while the readout chip is a product of the Medipix Collaborations[^1] [@medipix] led by CERN employing technologies developed for the LHC experiments. The data acquisition system for the cameras used commercial readouts of x-ray detectors. The first fast camera based on the Timepix2 readout chip [@timepix], TimepixCam [@timepixcam], was built in 2015, followed in 2017 by the next generation camera, Tpx3Cam [@tpx3cam], based on Timepix3 [@timepix3]. Both cameras employ the same sensor. Figure \[fig:camera\] shows the optical sensor inside Tpx3Cam and intensified version of the camera with attached image intensifier and 50 mm f/0.95 Navitar lens. Below we provide more details about Tpx3Cam as the most advanced camera.
{width="0.50\linewidth"}
In Timepix3 each pixel has a predefined threshold so only the pixels above the threshold perform the time measurements and are read out, making the chip completely data-driven. It uses a free running 40MHz clock and does not require any external triggering. The fired pixels provide the time-of-arrival (TOA) information with 1.56 ns granularity, and time-over-threshold (TOT) information with 25 ns granularity. The individual pixel deadtime is equal to the pixel TOT + 475 ns.
Fast scintillator P47, used in combination with MCP has the rise and decay time of 7ns and 100ns, respectively, and maximum emission in blue at 430 nm [@P47]. The absorption depth for the blue photons is only 250 nm, so it is very important that the passive layer on the sensor surface is thin enough to let the photons through. At the same time it should be conductive to ensure uniform electric field and full depletion of the 300 micron thick silicon sensor because the created charge carriers need to drift from the sensor window on the back side to the collection pads on the front-side, which are bump-bonded to Timepix. In this case holes are collected as the sensor is of a p-on-n type. Anti-reflective coating on the sensor was optimized for the P47 emission spectrum. Figure \[fig:QE\] shows the measured quantum efficiency as a function of the wavelength for several types of sensors with varying thickness of passivation layer and with/without anti-reflective coating (ARC). The measurements are described in detail in [@characterizationTpxCam] together with other test results for the optical sensors.
{width="0.8\linewidth"}
The Tpx3Cam camera readout is based on the SPIDR data acquisition system [@spidr1; @spidr2], which is available commercially [@ASI]. The SPIDR maximum output rate is 80 Mpix/s. It also implements a time-digital-converter (TDC), which provides a time-stamp with 0.26 ns granularity for an input signal and is synchronized to the Timepix3 data. This input can provide a precise time reference for the Timepix3 hits using, for example, a pulse synchronous to the laser employed in the experiments. It also can be used to synchronize multiple cameras to each other and to external devices.
Detection of single photons
---------------------------
Detection of single photons requires external amplification, so the intensified version of the camera employs an image intensifier, an off-the-shelf vacuum device with photocathode followed by a MCP and scintillator. The MCP can operate at gains up to few times 10$^5$ making the device sensitive to single photons. The back-end of the intensifier used for the measurements discussed below consists of the MCP/P47 assembly, so it is very similar to the assembly routinely used for the ion imaging, as illustrated in Figure \[fig:single\]. This similarity in detection of various particles: ions, electrons, single photons and, as proposed below, x-rays and neutrons, makes this approach very versatile, because the same camera can be used for all of them without any modifications.
{width="0.8\linewidth"}
The photocathode in the intensifier can be selected to match the QE requirements for a particular application. There is a wide choice of available photocathodes with different spectral sensitivities. An example is given in Figure \[fig:photonisQE\], which provides QE as function of wavelength for a variety of Photonis photocathodes [@Photonis]. The 18mm Photonis intensifier tested with the camera had High-QE Red photocathode with QE equal to $18\%$ at 800 nm, the wavelength relevant for the quantum information science applications [@qis2018], and dark count rate of $\sim80$ kHz over the full photocathode area at room temperature.
{width="0.9\linewidth"}
The intensifier in a cricket [@cricket] is shown together with the camera in the right part of Figure \[fig:camera\]. The cricket integrates into a single unit the intensifier, power supply and collimating relay optics between the intensifier output and camera sensor. The cricket is fetched with C-mounts on both ends for attachment to the camera and to a lens.
{width="0.49\linewidth"} {width="0.49\linewidth"}
Figure \[fig:photons1\] shows examples of single photon hits recorded by the camera in a time slice of $5~\mathrm{ms}$. The hits are shown as heatmaps in TOT representation (left) and TOA representation (right). One can see that each hit typically consists of several pixels, which have signals above the threshold. Since all fired pixels measure TOA and TOT independently and have position information, it can be used for centroiding to determine the photon coordinates. The centroiding considerably improves the spatial resolution, easily to sub-pixel values. The timing resolution can be improved at the pixel level correcting for the time-walk, an effect caused by the dependence of the front-end pixel electronics time response on the amplitude of the input signal [@Turecek_2016; @tpx3cam]. Since the latter is measured as TOT, the TOA can be corrected achieving 2 ns timing resolution (rms) [@tpx3cam; @qis2018]. The resolution per photon can be further improved by combining timing information in multiple pixels, which belong to the same photon hit.
Applications
------------
During the last three years the Timepix-based time-stamping cameras have been used in a variety of different applications, briefly reviewed below.
[**Ion and electron imaging**]{}: In this case the ions impinge a micro-channel plate producing light flashes in the fast scintillator behind the MCP, which are imaged by the Timepix camera. This approach works particularly well for the coincidence velocity map imaging (VMI), an essential tool in the study of reaction dynamics and strong field laser-matter interactions. VMI projects the transverse momenta of charged particles to positions on a 2D detector such that for a given particle species, its distance to the center of the detector is proportional to the initial transverse velocity. It also requires simultaneous measurement of all reaction fragments, which is not possible with a slow camera. The measurements with TimepixCam and Tpx3Cam were performed at FLASH in DESY [@flash2018] and in the ultrafast spectroscopy lab at Stony Brook University [@tpx3cam]. Several other groups interested in the ion and electron imaging used the cameras with the data analysis currently in progress.
[**Phosphorecent lifetime imaging**]{}: The camera performance for single photons was first validated in experiments on the photon counting phosphorescence lifetime imaging. In these types of measurements the camera is time-stamping photons, which are emitted with characteristic lifetime after excitation of a material under study with a short laser pulse. The results are described in [@FLIM1] and [@FLIMCFN].
[**Imaging of entangled photons**]{}: in the single photon mode the camera was also used for characterization of sources of entangled photons and for measurements of their temporal and spatial correlations [@qis2018].
[**Optical readout of TPC**]{}: recently Tpx3Cam was employed for the first demonstration of three-dimensional optical readout of a dual phase liquid Ar TPC (time projection chamber) [@tpc2018]. The approach is described in detail in [@tpc2009; @tpc2015].
[**Other possible applications**]{}: the camera can be used to register light flashes in thin scintillators produced by x-rays with energy 10keV and higher, where the direct detection with silicon sensors becomes inefficient. This approach, currently under testing, requires an intensified version of the camera since only a handful of photons will be collected per x-ray. However, it should allow a simple technique of time-stamping for individual x-rays with nanosecond resolution. Also it will avoid placing the detector in the direct beam of x-rays. A similar approach of registration in a thin scintillator can be employed for neutrons.
Future R&D directions
=====================
It is not possible to detect a single optical photon with good time resolution in a silicon sensor without amplification because of the noise. The amplification can be achieved outside of the sensor using an image intensifier, as described above or, alternatively, inside the sensor. Sensors with internal amplification are based on technologies such as SPAD (single photon avalanche devices) [@Perenzoni2016] and LGAD (low gain avalanche devices) [@Pellegrini2016; @lgadbnl], which have a multitude of applications, including high-energy physics. Advances in the CMOS technology are enabling integration of these devices in to pixelated silicon sensors. This lead to production of the first SPAD arrays capable to count and time stamp single photons with resolution below 100 ps [@Lee2018]. Infrared HgCdTe imagers with internal amplification, which are compatible with single photon detection, also received a lot of attention in astronomy in the context of extremely low light level applications [@saphira]. In terms of charge collection those sensors should be able to support fast timing. All the above sensors should be also compatible with the hybrid approach as presented here, which will benefit from future improvements in readout ASICs as, for example, in Timepix4, which is aiming at the 200 ps timing resolution [@timepix4].
Acknowledgements
================
The author is grateful to all his collaborators who helped in the implementation and testing of the first cameras, in particular, to Merlin Fisher-Levine, Peter Svihra, Martin van Beuzekom, Irakli Chakaberia, Peter Takacs, Thomas Tsang, Bram Bouwens, Erik Maddox, Jan Visser, Arthur Zhao, Thomas Weinacht, Vaclav Vrba, Zdenko Janoska, Mael Flament, Eden Figueroa, Daniel Rolles, Rebecca Boll, Heinz Graafsma, David Pennicard, Milija Sarajlic, Jochen Kupper, Igor Rubinski, Sebastian Trippel, Ruth Livingstone, Liisa Hirvonen, Klaus Suhling, Adam Roberts, Kostas Mavrokoridis, Mark Brouard, Michael Burt and Ilia Vardishvili. This work was supported by the BNL LDRD grants 13-006 and 18-030.
References {#references .unnumbered}
==========
[^1]: The Timepix2 chip used in TimepixCam was developed by the Medipix2 collaboration, while Timepix3 used in Tpx3Cam was developed by the Medipix3 collaboration.
|
[**PERFORMANCE OF A SPECKLE INTERFEROMETER** ]{}
S K Saha$^{1}$, A P Jayarajan$^{1}$, G Sudheendra$^{2}$ and A Umesh Chandra$^{2}$\
$^{1}$Indian Institute of Astrophysics, Bangalore 560034, India\
$^{2}$Central Manufacturing Technology Institute, Bangalore 560022, India\
[**Abstract**]{} An interferometer system for use at the 2.34 meter Vainu Bappu Telescope (VBT), situated at Vainu Bappu Observatory (VBO), Kavalur, to obtain speckle-grams of astronomical objects in the visible wavelength, has been developed. Laboratory tests of resultant intensity distributions due to a point source, with phase modulation screens, as well as the images obtained using this interferometer at the cassegrain end of the said telescope are discussed.
Key words: Interferometer, Speckle Imaging, Image Reconstruction.
1\. [**Introduction**]{}
Atmospherically induced phase fluctuations distort otherwise flat wavefronts from distant stars which reach the entrance pupil of a telescope with patches of random excursions in phase. Such phase distortions restrict the effective angular resolution of most telescopes to 1 second of arc or worse. Random phase excursions are attributed to the refractive index fluctuations created by pockets of inhomogeneities in the atmosphere characterized by the value of quasi-coherent areas of diameter r$_{o}$, (r$_{o}$ $\sim$ 10 cm under general conditions), known as Fried parameter (Fried, 1966). If the exposure time is shorter ($<$ 20 millisecond) than the evolution time of the phase inhomogeneities, then each patch of the wavefront would act independently of the rest of the wavefront resulting in multiple images of the source, called, ’Speckle’ (the term ’Speckle’ refers to a grainy structure observed when an uneven surface of an object is illuminated by a coherent source). Its structure in astronomical images is the result of constructive and destructive bi-dimensional interferences between rays coming from different zones of incident wave. These speckles can occur randomly along any direction within an angular patch of diameter $\lambda$/r$_{o}$. The sum of several statistically uncorrelated speckle patterns from a point source can result in an uniform patch of light a few seconds of arc wide (conventional image).
The technique of speckle interferometry (Labeyrie, 1970) has become an invaluable tool for astronomical research in obtaining diffraction limited spatial Fourier spectrum and image features of the object intensity distribution from a series of short exposure images through a narrow band filter. The intensity distribution in the focal plane in case of quasi-monochromatic incoherent source can be described by the following equations.
[c(x,y) = [o(x,y) p(x,y)]{}]{}
where $c(x,y)$ is the image, $o(x,y)$ is object intensity distribution, $p(x,y)$ is the telescope-atmosphere point spread function (PSF) and $\ast$ denotes convolution.
The Fourier space relationships between objects and their images are
[C(u,v) = [O(u,v) P(u,v)]{}]{}
Here, $O(u,v)$ is the object spectrum and $P(u,v)$ is the transfer function. In the conventional speckle interferometry, the ensemble averaged power spectrum is obtained for a large set of short exposure images.
[<C(u,v) \^[2]{}> = [O(u,v) \^[2]{} ]{}]{}
where $< >$ indicates the ensemble average and $\mid \mid$ the modulus.
The form of transfer function $<\mid P(u,v) \mid^{2}>$ can be obtained by calculating Weiner spectrum of the instantaneous intensity from the unresolved star (reference).
The 2.34 meter Vainu Bappu Telescope (VBT) at Vainu Bappu Observatory (VBO), Kavalur can be used extensively to study high resolution features of many types of celestial objects. These may be in the form of separation and orientation of close binary stars (separation $<$ 1 arc second), shapes of asteroids, sizes of certain types of circumstellar envelopes, the structure of active galactic nuclei etc. The latitude of this observatory gives us access to almost 70$^{o}$ south of the celestial equator. Most of the observational results beyond 30$^{o}$ south of zenith at high latitude stations obtained earlier require to be confirmed utilizing positional advantage of VBT. This paper presents the design and performance of the new speckle camera system suitable for operation at the prime focus of this telescope.
2\. [**Speckle Interferometer**]{}
A speckle interferometer is a high quality diffraction limited camera where magnified ($\sim$ f/100) short exposure images can be recorded. Additional element for atmospheric dispersion corrections is necessary to be incorporated. At increasing zenith distance speckles get elongated owing to this effect. Either a pair of Risley prism must be provided for the corrections or the observation may be carried out using a narrow bandwidth filter. In this set up, we have used a narrow band filter to minimize the effect.
To arrive at the design of this equipment, preliminary investigations were made by us with a modest equipment (Saha [*et al.*]{}, 1987) consisting of a Barlow lens, a broad band filter in the blue region and a movie camera. Speckles and interference fringes of various bright stars were recorded with the f/13 beam of the 1 meter telescope at VBO. The power of this technique was demonstrated by the detection of some telescope aberrations (Saha [*et al.*]{}, 1987). Owing to the low quantum efficiency of photographic emulsion, use of an image intensifier becomes essential (Breckinridge [*et al.*]{}, 1979) to record speckle-grams of faint astronomical objects. New developments in instrumentation technology enable us to detect the photon events per frame (Blazit [*et al.*]{}, 1977, Blazit, 1986) up to a frame rate of $\sim$ 50 Hz. To detect individual photon event, recording time resolution $\sim$ 1 $\mu$s has been successfully employed (Papaliolios [*et al.*]{}, 1985, Durand [*et al.*]{}, 1987). We have modified the afore-mentioned set up by replacing movie camera with an EEV uncooled intensified CCD (ICCD) camera (385 $\times$ 288) which provides a standard CCIR video output (Chinnappan [*et al.*]{}, 1991) and were able to obtain speckle-grams of several close binaries through a 5 nm filter centered on H$\alpha$ with the cassegrain end (f/13) beam of the 2.34 meter VBT at VBO. Two of these binaries were processed using Blind Iterative De-convolution (BID) technique and reconstructed the Fourier phases (Saha and Venkatakrishnan, 1996).
Figure 1 depicts the optical design of the speckle camera system. Provisions have been made to observe at both the foci (Prime f/3.25 as well as cassegrain f/13) of the aforementioned telescope. Before arriving at the final concept of the design of this interferometer, the optical alignment was optimized at the laboratory. An artificial star image and the telescope with f/3.25 beam were generated. Atmospheric seeing was simulated by introducing various static dielectric cells (SDC) of different sizes etched in glass plate with hydrofluoric acid. Several glass plates with both regular and random distribution of SDCs of known sizes have been made and were used to produce speckles at the laboratory. The image was magnified to discern the individual speckles with a microscope objective. A 10 nm filter centered on OI\[5577\] was used to reduce the chromatic blurring. For an accurate evaluation of the performance of the design of this interferometer, we have also obtained fringes by placing a mask with multiple aperture in front of the telescope in the laboratory and compared those with the computer simulation of the intensity distribution (Saha [*et al.*]{},1988). The similarity of the observed image shape obtained at the laboratory, as well as the computer simulated image pattern proved the perfection of the design of this interferometer.
One of the most difficult problems was solved by us by making a focal plane optical flat of a low expansion glass with high precision hole of aperture $\sim$350$\mu$, at an angle of 15 degree on its surface. A novel technique was developed to make this aperture in the laboratory. This aperture is equivalent to a field of $\sim$9 arc sec at the Prime focus and to a field of $\sim$2.25 arc sec at Cassegrain focus of the VBT. The rear side of the flat was shaped suitably to enable the microscope objective, to be brought very close to the focal plane ($\sim$1mm). The field covered by this aperture of the flat at Prime focus of the VBT allows us to observe both the object and the reference star simultaneously, if the latter is located within the iso-planatic domain around the object. The image of the object is passed on to the microscope objective through this aperture. The surrounding star field of $\sim$4.5 arc min. at the Prime focus and $\sim$1.25 arc min. at the Cassegrain focus is re-imaged on an Intensified CCD for guiding.
Before finalizing the mechanical design for the mounts and the housing of this interferometer, Finite Element Method (FEM) analysis was performed to have prior informations about the deflections, deformation, stress, flexure etc. of the material. The computer simulation test had shown that the instrument can hold any detector of 10 kgs. weight kept at a distance of 200 mm away from the rear end of the interferometer with a flexure of $\sim$1.3$\mu$. The model was analyzed for strength and deflection for a load of 20 kgs. over the span of the instrument. The analysis shows a deflection of $\sim$0.7$\mu$. Figure 2 shows the FEM model of the structure of speckle interferometer.
The mechanical design requirements which have to be met while designing the interferometer are: (i) high accuracy, (ii) light weight, (iii) minimum deflection at various orientations of the telescope, (iv) provision for fine adjustment of the microscope without allowing rotation of the image, (v) provision for fine focusing of the camera without disturbing / rotating the image, (vi) provision for adjusting the inclined beam, (vii) rigid locking of the lens positions during use of the instrument.
To assure high accuracy even at varying ambient temperatures, a Martensitic variety of stainless steel material ( SS 410 ) with low coefficient of linear expansion has been used. The instrument has been conceived as a box made of two end plates of thickness 8 mm and joined by means of struts of section 22 mm square. The struts have been machined from a cylinder of 25 mm diameter of required length ground at the ends. The struts are provided with spigots of 18 mm diameter at ends for locating the end plates with corresponding holes. The concentricity between spigots can be ensured by grinding. The end plates are machined together and the four holes which receive the spigots as well as the central hole are machined in one setting on wire cut Electro-discharge Machining (EDM) to assure required center distance accuracies. The end plates when locked with the struts, form a box structure of light weight with required strength to house the desired mounts with minimum deflection. A bottom plate is clamped on to two of the struts and this forms a platform on which the various mounts for the microscope, flat and the lens can be mounted. The mounts themselves are designed and machined in such a way that lens mounting holes are at an accurate distance from their bases. In case of any error, the base can be ground to get the required center height accuracy. Thus, the plane in which the light travels through the microscope and mirrors is maintained accurately. The individual mechanisms in each of the mounts provide for fine adjustment which help in fine focusing of the image. Plate I shows the photograph of the interferometer. In order to avoid the reflection from other surfaces, black chrome plating was done to achieve blackening of the stainless steel.
3\. [**Observations**]{}
We have successfully made an attempt to observe a few close binary systems using this newly built interferometer at the Cassegrain focus of the VBT. The image scale at the Cassegrain focus (6.7 arc second per mm.) of this telescope was magnified to 0.67 arc second per mm, using a microscope objective. This enlarged image was recorded through a 5 nm filter centered on H$\alpha$ using the ICCD camera (Chinnappan [*et al.*]{}, 1991). The images were acquired with exposure times of 20 ms using a Data Translation[$^T$$^M$]{} frame-grabber card DT-2861 and subsequently stored on to the hard disk of a PC486 computer. This computer allowed us to record 64 images continuously in a few seconds time. Figure 3 shows the speckle-gram of the $\alpha$-Andromeda. The observing conditions were fair with an average seeing of $\sim$2 arcseconds during the nights of 29/30 November 1996.
4\. [**Discussion and Conclusions**]{}
Studies involving the optimization of speckle imaging, are part of the important groundwork for addressing the basic astrophysical problems. The quality of the image degrades due to the following reasons: (i) variations of air mass X ($\sim$ 1/cosZ) or of its time average between the object and the reference, (ii) seeing differences between the modulation transfer function (MTF) for the object and its estimation from the reference, (iii) deformation of mirrors or to a slight misalignment while changing its pointing direction, (iv) bad focussing, (v) thermal effect from the telescope etc. These may lead to a dangerous artifact, yielding a wrong identification of the companion star. A group of observers found a spurious secondary peak in the mean auto-correlation at a distance equal to the first Airy ring (Foy, 1988). Labeyrie (Foy, 1988) opined that spider-induced temperature gradient might be the cause of this phenomenon since the upper faces of spider cool fast. Here, we have fabricated a mechanical spacer of 450 mm to keep the instrument away from the rear end of the Cassegrain end of the VBT. Care has been taken while designing the spacer (see Plate I) that no hot air would be trapped in and around the interferometer, which may help to get rid of such phenomena.
To estimate the MTF, it is neccessary to calibrate it on unresolved star, for which all observing conditions are required to be identical to those for the object. In this respact, we have taken a precautionary measure by optimising the size of the aperture in the focal point flat of this interferometer. In the prime focus of the VBT, this aperture will allow us to observe both the object, as well as reference star simultaneously, if the latter is located in the isoplanatic domain.
Depending on the variability in temperature, air currents etc., the atmospheric time constant is generally assumed to range between 1 and 100 msec. If the integration time is too long, there is a degradation of signal to noise ratio. The need for fast exposure times implies the use of detectors facilitating desired integration time. The recently acquired Peltier-cooled Intensified CCD (386 $\times$ 578) camera, which has the option of various exposures, viz., 5msec, 10msec, 20msec etc. will be used to record the speckle-grams of faint objects. But photon-counting detectors with frame integration are subject to limitations in detecting fast photon-event pairs. A pair of photons closer than a minimum separation cannot be detected as a pair by the afore-mentioned sensor. This yields a loss in HF information which, in turn, produces a hole in the centre of the auto-correlation, known as Centreur hole (Foy, 1988). Therefore, it is desirable to have a detector which facilitates the oversample image of the order of $\sim$ f/500 in place of the former typical $\sim$ f/100, with a high degree of accuracy to determine the time coordinate of each event. The present interferometer has the capability of oversampling the image of the order of f/520 at the Cassegrain focus of the VBT. If the modern detector, viz., PAPA (Precision Analogue Photon Address), or IPD (Image Photon Detector) is made available, this camera will be able to record speckle-grams of the objects of the faintest limiting magnitudes.
To reiterate, optimised instruments and meticulous observing procedures are essential for obtaining error-free information. It is required to have a simulation bench consisting of an artificial star, a telescope with various focal ratios, a reducing optical system for calibration of the focal point instruments. Calibration is needed both before and after the on-site observations. Systematic use of simulated image to validate the image processing algorithms could be beneficial in retrieving the diffraction limited information.
[**Acknowledgement**]{} The authors are grateful to Prof. J C Bhattacharyya, for his constant encouragement during the progress of this piece of work. The personnel of the mechanical division of IIA, in particular Messrs B R Madhava Rao, R M Paulraj, K Sagayanathan and A Selvaraj, provided excellent support during execution of the work. The help rendered by Mr. J R K Murthy of C M T I, for computer analysis of the design and by Dr. Indira Rajagopal of National Aerospace Laboratory, Bangalore for the black chrome plating are also gratefully acknowledged.
[**References**]{}
Blazit, A., 1986, Proc. ’Image detection and quality’ - SFO ed SPIE, [**702**]{}, 259.\
Blazit, A., Bonneau, D., Koechlin, L. and Labeyrie, A., 1977, Ap. J., [**214**]{}, L79.\
Breckinridge, J. B., McAlister, H. A. and Robinson, W. A., 1979, App. Opt., [**18**]{}, 1034.\
Chinnappan, V., Saha, S. K. and Faseehana, 1991, Kod. Obs. Bull. [**11**]{}, 87.\
Durand, D., Hardy, E. and Couture, J., 1987, Astron. Soc. Pacific. [**99**]{}, 686\
Foy, R., 1988, Proc. ’Instrumentation for Ground-Based Optical Astronomy - Present and Future’, ed. L. Robinson, Springer-Verlag, New York, 345.\
Fried, D. L., 1966, J. Opt. Soc. Amm., [**56**]{}, 1372.\
Labeyrie, A., 1970, Astron. Astrophys., [**6**]{}, 85.\
Papaliolios, C., Nisenson, P. and Ebstein, S., 1985, App. Opt., [**24**]{}, 287.\
Saha, S. K., Jayarajan, A. P., Rangarajan, K. E. and Chatterjee, S., 1988, Proc. ESO-NOAO conf. ’High Resolution Imaging Interferometry’, ed. F. Merkle, Garching bei Munchen, FRG, 661.\
Saha, S. K. and Venkatakrishnan, P., 1996, Submitted to Bull. Astron. Soc. Ind.\
Saha, S. K., Venkatakrishnan, P., Jayarajan, A. P. and Jayavel, N., 1987, Curr. Sci., [**57**]{}, 985.\
[**Figure captions**]{}
1\. Fig. 1: depicts the optical design of the speckle interferometer.
2\. Fig. 2: 2a, 2b show the Finite element model of the structure, and the same with the load of 20 kgs. over the span of the instrument respactively.
3\. Fig. 3: shows the contours of the speckle image of $\alpha$-Andromeda.
Plate I. shows the photograph of the speckle interferometer.
|
---
abstract: 'The rise and fall of the superconducting transition temperature $T_c$ upon tuning carrier density or external parameters, such as pressure or magnetic field, is ubiquitously observed in a wide range of quantum materials. In order to investigate such domes of $T_c$, we go beyond the prototypical Hubbard model, and consider a lattice model of electrons coupled via instantaneous, spatially extended, attractive interactions. Using mean-field theory and functional renormalization group (FRG) methods, we find that for a characteristic interaction range $\ell$, there exists a dome in $T_c$ around $k_F \ell \! \sim \! {\cal O}(1)$. This result can intuitively be understood from the geometric relation between the Fermi surface and the interaction range. We show that our results generally hold for single-band as well as multi-band superconductors, in both two and three dimensions. Our model may be relevant for domes of $T_c$ in dilute weakly coupled superconductors or in engineered cold atom systems.'
author:
- Nazim Boudjada
- Finn Lasse Buessen
- Arun Paramekanti
bibliography:
- 'dome.bib'
title: 'Domes of $T_c$ in superconductors with finite-range attractive interactions'
---
[*Introduction.–*]{} Domes in the superconducting (SC) transition temperature $T_c$, observed across a broad range of quantum materials, typically reflect some form of underlying dynamical competition in the electronic fluid. In heavy fermion compounds [@Stewart_RMP1984; @QSi_Science2010; @WirthSteglich_Nature2016] and iron pnictide materials [@Wen_ARCMP2011; @Si_Nature2016; @Fernandes_2016], for instance, the SC dome emerges around magnetic or nematic quantum critical points (QCPs). In SrTiO$_3$, SC domes may possibly be driven by proximity to a ferroelectric QCP [@GASTIASORO2020168107; @PhysRevMaterials.3.091401; @gastiasoro2020anisotropic; @PhysRevB.100.094504; @PhysRevB.98.024521; @PhysRevB.100.094504; @PhysRevLett.115.247002; @Ahadieaaw0120; @PhysRevB.98.104505; @Neaton2019; @atkinson2017influence]. In the cuprates, even aside from the physics of QCPs, a decrease in the hole concentration can enhance spin-fluctuation mediated pair formation while simultaneously suppressing the superfluid density as a consequence of Mott physics or competing orders. This interplay yields the highest $T_c$ at an optimal doping [@Anderson_RVB2004; @Keimer_Nature2015]. SC near QCPs has also been found in numerical simulations and field theory studies [@Berg_Science2012; @Chowdhury_PRB2015; @Raghu_PRB2015; @Wang_PRL2016; @Torroba_PRB2017; @Berg_ARCMP2019; @Chowdhury2019]. Finally, for ultracold atomic fermions, the highest $T_c$ appears near unitary scattering which marks the BCS-BEC crossover from weak to strong coupling SC [@Randeria_ARCMP2014; @STRINATI_BCSBEC2018].
In this Letter, we discuss a geometric picture of superconducting domes in systems with (non-retarded) finite-range attractive interactions. Our proposal is motivated by the following observation. In a system where electrons attract each other over a fixed characteristic range $\ell$ in real space, the typical momentum transfer in electron-electron scattering processes is $\Delta k \!\sim\! 1/\ell$. Thus, in dilute systems with $k_F \!\ll\! \Delta k$, such interactions can efficiently scatter electrons across any two points on the Fermi surface (FS). In the opposite limit, however, when $k_F \!\gg \! \Delta k$ the aforementioned interactions lead to small-angle scattering, making it more challenging for electrons to explore the full FS. Therefore, the phase space which is accessible in a single electron-scattering event initially increases with the size of the FS, before dropping at high densities when the locality of the interactions in momentum space suppresses global SC. Consequently, a dome-like dependence of $T_c$ on the electron density emerges around some intermediate Fermi momentum $k_F^\star$ which satisfies $k_F^\star \ell \!\sim \! {\cal O}(1)$. The dome thus marks the crossover from predominantly global interactions to local interactions in momentum space. From a real-space perspective, the highest $T_c$ occurs when the interaction range $\ell$ becomes comparable to the interparticle spacing. This geometric picture may be most clearly appreciated in dilute systems, when the FS is far from van Hove singularities. Our work does not address the microscopic origin of such a pairing interaction or the length scale $\ell$, which are important issues in their own right [@PhysRevB.99.094524; @PhysRevB.94.224515; @PhysRevB.98.104505; @Gorkov201604145; @Gorkov], but it is reminiscent of the geometric Mott-Ioffe-Regel criterion which marks the crossover from coherent to incoherent electronic transport [@Hussey_PhilMag2004] without reference to an underlying mechanism.
Expanding on the above physical arguments, we study both single band and multiband models in two and three dimensions (2D and 3D). Our work may be useful as a toy model for systems with critical modes or soft bosons which may induce such long-range attractive interactions, such as for fermions experiencing fluctuating zero-momentum orders. We thus make some qualitative comparisons with results on dilute electron gases in bulk SrTiO$_3$. Models similar in spirit to our study have previously been explored in the context of cuprates [@KunYang_PRB2000], FeSe on SrTiO$_3$ [@Rademaker_2016; @dhlee_ARCMP2018], and ultracold atomic fermions [@Parish_PRB2005]. Previous work has also discussed how density-dependent screening might lead to domes of $T_c$ in SrTiO$_3$ [@PhysRevB.94.224515; @Gorkov201604145; @Gorkov]. However, the universal geometric picture for $T_c$ domes we discuss does not appear to have been highlighted. Our work may also be relevant to ultracold Bose-Fermi mixtures, where $\ell$ could be set by the correlation length associated with the superfluid to Mott insulator transition [@KunYang_PRB2008]. We emphasize, however, that the $T_c$ dome we uncover is not inherently a strong-coupling phenomenon. We therefore employ mean-field theory and FRG methods below to study this problem.
[*Model Hamiltonian. –*]{} We consider a tight-binding Hamiltonian parametrized as $$\mathcal{H}_0=\sum_{{{{\bf{k}}}}\mu\nu} c^\dagger_\mu({{{\bf{k}}}})H^{\mu\nu}_0({{{\bf{k}}}})c_\nu({{{\bf{k}}}}) \,,$$ where $\mu,\nu$ stand for generic orbital and spin indices which give a matrix structure to the Hamiltonian $H_0$. The electrons are assumed to interact via an instantaneous attractive interaction $$\label{eqn:interaction}
\mathcal{H}_{\mathrm{int}}=\frac{1}{2} \int\!\! {{\mathrm{d}}}^d{{{\bf{r}}}}\!\! \int \!\! {{\mathrm{d}}}^d{{{\bf{r}}}}' ~{\cal V}({{{\bf{r}}}}-{{{\bf{r}}}}') \hat{n}({{{\bf{r}}}}) \hat{n}({{{\bf{r}}}}') \,,$$ with $\hat{n}({{{\bf{r}}}}) = \sum_\mu c^\dagger_\mu({{{\bf{r}}}})c_\mu({{{\bf{r}}}})$ being the density operator at position ${{{\bf{r}}}}$, and ${\cal V}({{{{\bf{r}}}}}-{{{{\bf{r}}}}}') \! < \!0$ being the interaction potential.
Anticipating a singlet superconducting instability, we Fourier transform the interaction to momentum space and focus on the zero center of mass pairing channel, which leads to the effective Hamiltonian $$\label{eqn:MFHamiltonian}
\!\!\!\mathcal{H}^{\rm BCS}_{\mathrm{int}}\!=\! \frac{1}{2N} \!\sum_{{{{\bf{k}}}}{{{\bf{k}}}}'}c^\dagger_\mu({{{\bf{k}}}})c^\dagger_\nu(-{{{\bf{k}}}}) V({{{\bf{k}}}}\!-\!{{{\bf{k}}}}') c_\nu(-{{{\bf{k}}}}')c_\mu({{{\bf{k}}}}') \,,$$
where $N$ is the total number of lattice sites and summation over repeated indices is implied. The interaction $V({{{\bf{k}}}}-{{{\bf{k}}}}')$ is the Fourier transform of ${{\cal V}({{{\bf{r}}}}-{{{\bf{r}}}}')}$. We decouple the interaction, using a Hubbard-Stratonovich transformation, via complex bosonic fields $\Delta_{\mu\nu}({{{\bf{k}}}})$ and integrate out the fermions (see the Supplemental Material (SM) for details). The resulting self-consistent matrix gap equation is given by $$\small
\!\!\!\Delta({{{\bf{k}}}})\!=\! - \frac{1}{N}\! \sum_{{{{\bf{k}}}}'} V({{{\bf{k}}}}\!-\!{{{\bf{k}}}}') U({{{\bf{k}}}}')\frac{\tanh\!\left[\!\frac{E({{{\bf{k}}}}')}{2T}\!\right]}{2 E({{{\bf{k}}}}')} U^\dagger({{{\bf{k}}}}')\Delta({{{\bf{k}}}}') \,,
\label{eq:gapeqn}$$ where $T$ is the temperature, $E({{{\bf{k}}}})$ is a diagonal matrix comprising the square roots of the eigenvalues of ${H_0({{{\bf{k}}}})H^\dagger_0({{{\bf{k}}}})+\Delta({{{\bf{k}}}})\Delta^\dagger({{{\bf{k}}}})}$, and $U({{{\bf{k}}}})$ is the corresponding eigenvector matrix. For a one-band model this expression reduces to the familiar single-gap equation. We assume a Gaussian interaction ${{\cal V}({{{\bf{r}}}}) \!=\! -g_0 {{\mathrm{e}}}^{-|{{{\bf{r}}}}|^2/2\ell^2}}$, so that ${V({{\bf{q}}})\!=\!- g_0(2\pi\ell^2)^{d/2}{{\mathrm{e}}}^{-|{{\bf{q}}}|^2\ell^2/2}}$ [^1] in $d$ spatial dimensions. Here $g_0\!>\! 0$ is the pairing strength, and $\ell$ sets the range of the potential in real space (in units of the lattice constant). For $\ell \!\to\! 0$, the interaction reduces to a Hubbard model, while a large value of $\ell$ favors small momentum scattering. Our results are qualitatively unchanged if we use alternative potentials, such as a Lorentzian or a hard-sphere, with a similar characteristic range $\ell$. To explore the full density dependence and multiband examples, we numerically solve the gap equation for $T_c$ and the momentum dependence of the gap $\Delta_{\mu\nu}({{{\bf{k}}}})$. For a fixed density, we also simultaneously solve for the chemical potential.
[1Band\_2D.pdf]{} (80,33)[(0,1)[9]{}]{} (85,15)[(0,-1)[7]{}]{}
We begin by discussing the geometric origin of domes of $T_c$ for interactions with finite range $\ell$. As illustrated in Fig. \[fig:geometry\](a), for a given momentum point on the 2D FS, the arc length of the FS which lies within the (momentum-space) interaction range $\sim\!\! 1/\ell$ depends on the electron density. Up to $k_F \ell \!=\! 1/2$, the full FS circumference $2\pi k_F$ is accessible in a single electron-scattering event. Beyond this value, the accessible part of the FS shrinks to $2/\ell \! \ll \! 2\pi k_F$ at large $k_F$. The functional form of the geometric arc length $\mathcal{R}(\bar{n})$ versus the electron density $\bar{n}$, displayed in Fig. \[fig:geometry\](b) for a 2D FS with $\ell=1$, exhibits a sharp peak at a density corresponding to $k_F \ell \!=\! 1/2$. Since $\mathcal{R}$ is a geometric measure of the phase space available for Cooper pairs, we expect the peak in ${\mathcal R}$ to be reflected as a dome in $T_c$.
Although $T_c$ is not a purely geometric quantity, depending also on the form of the potential and the energy-dependent density of states, we shall demonstrate that a smoothed version of this geometric maximum generically persists. To illustrate this in a simple one-band example, we compute the mean-field singlet pairing $T_c$ for a 2D square lattice with dispersion $\xi({{{\bf{k}}}}) \!=\!-2t_1 (\cos k_x \!+\! \cos k_y) \!-\! \mu$. Using $\ell\!=\! 5$ and $g_0\!=\! 1$ (in units of $t_1$), we observe a peak in $T_c$ at an electron density $n^\star \!\approx\!0.04$ in the dilute limit as shown in Fig. \[fig:geometry\](c). At higher densities $T_c$ exhibits additional peaks: near half-filling $\bar{n}\!=\!1$, and at $\bar{n}=2-n^\star$. The former peak stems from the combination of an enhanced density of states near the van Hove singularity and an increased geometric overlap with states in the second Brillouin zone (BZ) (see the SM). The latter is a geometric peak arising from small hole pockets near $(\pi,\pi)$, when the hole density becomes dilute.
Our numerical solution of the gap equation shows that domes of $T_c$ also appear in 3D. To investigate the role of the coupling strength $g_0$ in the occurrence of the dome, we compute $k_F^\star \ell$ as a function of $g_0$, where $k_F^\star$ is the angle-averaged Fermi wave vector associated with the density $n^\star$ [^2]. The results for the 2D square lattice as well as for the 3D cubic lattice show that the dome shifts towards smaller densities as $g_0$ is reduced, see Fig. \[fig:geometry\](d). However, we emphasize that the dome persists in the weak coupling limit. Indeed, extrapolating our results to $g_0 \!\to \! 0$, we find a finite value $k_F^\star \ell \! \approx \! 0.2$ in 2D and $k_F^\star \ell \! \approx \! 0.86$ in 3D. At the same time, the ratio of the critical temperature and the Fermi energy $T^\star_c/\epsilon^\star_F$ at the geometric peak remains moderate (with $T^\star_c/\epsilon^\star_F \!<\! 1$ for $g_0=1$, and decreasing for smaller $g_0$), implying that the dome is not a strong-coupling phenomenon.
[*Multiband case. –*]{} Next, we study the generalization of our mean-field results to multiband cases and demonstrate that the geometric interpretation of domes still holds. Furthermore, we shall see that in multiband systems, it is possible to obtain multiple domes of $T_c$ as new FSs appear with increasing density. To this end, we consider the two-orbital model $$\!\!\!\mathcal{H}_0= \sum_{{{{\bf{k}}}}} \! \begin{pmatrix}
X^\dagger_{{{{\bf{k}}}}{\uparrow}} & \! Y^\dagger_{{{{\bf{k}}}}{\uparrow}}
\end{pmatrix}\!\!
\begin{pmatrix}
\xi_{X}({{{\bf{k}}}}) & \! \delta \\
\delta & \! \xi_{Y}({{{\bf{k}}}})
\end{pmatrix}\!\!
\begin{pmatrix}
X_{{{{\bf{k}}}}{\uparrow}} \\
Y_{{{{\bf{k}}}}{\uparrow}}
\end{pmatrix},
\label{eqn:twoorbitals}$$ where ${\xi_X({{{\bf{k}}}}) \!=\! -2t_1\cos k_x\!-\! 2t_2\cos k_y\!-\! \mu}$ and ${\xi_Y({{{\bf{k}}}})\!=\!-2t_2\cos k_x\!-\!2t_1\cos k_y\!-\!\mu}$, with $\delta$ being the momentum-independent interorbital hybridization. As before, we set $t_1\!=\!1$, $\ell\!=\!5$, and $g_0\!=\!1$, and choose, for illustrative purposes, $t_2 \! = \! \delta \!=\! 0.2$; additional examples are discussed in the SM. In the low-density regime ($\bar{n}\ll 0.1$), only one band crosses the Fermi level and the physics is analogous to the single orbital model, i.e. a geometric dome forms at a density corresponding to the optimal value of $k_F\ell$ (orange star in Fig. \[fig:Tc2Bands\]). The parameters are chosen such that the Lifshitz transition, i.e. the appearance of the second band at the Fermi level, occurs near the maximum of the dome. The second band then gives rise to a second geometric peak at a slightly higher density (blue star), yielding an overall double peak structure. We note, however, that for different parameter choices, the Lifshitz transition does not necessarily coincide with the first peak in $T_c$: for example, increasing $\ell$ pushes the geometric peak to lower densities (thus keeping $k_F^\star\ell\sim\mathcal{O}(1)$), but has no impact on the Lifshitz transition point.
![Superconducting transition temperature $T_c$ for a two-orbital model showing a double geometric peak in the low-density regime. Colored stars correspond to the FSs in the insets. Vertical dashed line marks the Lifshitz transition when the second band appears at the Fermi level.[]{data-label="fig:Tc2Bands"}](Tc_2Band_2D.pdf){width="\linewidth"}
[*Functional RG. –*]{} In deriving the gap equation Eqn. , we have explicitly assumed Cooper pair formation in the singlet channel. While yielding a structurally simple, self-consistent mean-field theory, the decoupling comes at the price of being inherently biased to favor the specific type of superconductivity encoded in the ansatz, potentially neglecting any competing phases. When $k_F \ell \!\gg\! 1$, however, different patches on the FS could effectively decouple and many angular momentum pairing channels become quasi-degenerate as seen from the eigenfunctions of the subleading instabilities in the linearized mean field gap equation (see SM). Additionally, attractive interactions could make the systems unstable towards phase separation. Such effects can lead to a breakdown of coherent superconductivity.
![Effective interaction in the patching approximation. Normalized color code shows the value of the flowing interaction vertex $u_T(n_1,n_2,1)$, where $(n_1,n_2)$ enumerate momentum patches around the FS. (a) Density $\bar{n}\!=\!0.17$ at $T\!=\!T_\mathrm{max}$, (b) $\bar{n}\!=\!0.17$ at $T\!=\!T_\mathrm{min}$, (c) $\bar{n}\!=\!0.94$ at $T\!=\!T_\mathrm{min}$.[]{data-label="fig:vertex"}](vertex.pdf){width="\linewidth"}
To investigate this breakdown – or, conversely, justify the mean-field ansatz – we employ the FRG approach which treats all competing interaction channels on equal footing [@Metzner2012; @Gersch2008; @Eberlein2014]. The resulting RG flow equations, which relate the bare interaction as defined in Eqn. to an effective low-energy theory by continuously tracing its evolution under infinitesimal reductions of the temperature [@Honerkamp2001], naturally have a more complex structure than the self-consistent mean-field equation, and in general cannot be solved exactly. For weak coupling, however, it is sufficient to include only the one-loop contributions to the flow equations for the two-particle interaction, neglecting higher-order processes [@Halboth2000], and to treat the interaction vertex in a momentum patching approximation which resolves its angular dependence around the FS. In this way, a finite set of differential equations is obtained which can be solved numerically to determine the effective interaction vertex $u_T(n_1,n_2,n_3)$, where the $n_i$ enumerate the momentum patches around the FS. The specific choice of momentum patches, as well as the detailed FRG flow equations, are outlined in the SM.
We perform calculations on the single band model at fixed $\ell\!=\!1$ and $g_0\!=\!\frac{3}{2\pi}$, while varying the density $\bar{n}$ to assess the role of competing interaction channels. The RG flow is initialized at an upper temperature $T_\mathrm{max}=4t_1$, which is comparable to the bandwidth, and stopped at a temperature scale $T_\mathrm{min}$ when the maximum component of the vertex exceeds $18t_1$, which is large compared to the bandwidth. The onset of strong interactions at $T_\mathrm{min}$ can then be related to a putative phase transition [^3]. In the dilute limit, $k_F \ell \ll 1$, the bare interaction at $T=T_\mathrm{max}$ has negligible momentum dependence on the FS. The Gaussian profile becomes visible only at slightly larger $\bar{n}$ as seen in Fig. \[fig:vertex\](a). The effective low-temperature vertex, however, for a wide range of $\bar{n}$ is dominated by a distinct attractive interaction between momentum patches which lie on opposite sides of the FS, indicating impending zero-momentum Cooper pair formation, see Fig. \[fig:vertex\](b). However, at large densities $\bar{n}>0.88$, the initial Gaussian profile of the bare interaction sharpens throughout the RG flow as shown in Fig. \[fig:vertex\](c), so that the renormalized $\ell \!\to\! \infty$, and forward scattering gets enhanced. This indicates the breakdown of Cooper pairing and coherent superconductivity.
![Characteristic temperature $T_\mathrm{min}$ for the Gaussian interaction potential with $\ell = 1$, as determined from FRG calculations which include only particle-particle scattering (blue curve) or all interaction channels (orange curve). Curves plotted in opaque colors are computed using $N_p=96$ momentum patches, curves in lighter colors are for $N_p=72$ and $N_p=48$. In regimes I and II, the effective vertex at $T_\mathrm{min}$ captures superconducting pairing, while in regime III the interaction becomes increasingly localized in momentum space leading to a breakdown of SC. Inset shows the geometric dome of $T_c$ at low density.[]{data-label="fig:phasediagramFRG"}](phasediagramFRG.pdf){width="\linewidth"}
To better understand this breakdown, we resolve the role of additional interaction channels by comparing the full FRG calculations with reduced flow equations that only include the particle-particle forward scattering as also captured by the mean-field ansatz. We find that we can divide the FRG phase diagram shown in Fig. \[fig:phasediagramFRG\] into three regimes. In regime I ($\bar{n}<0.58$) the superconducting $T_c$ is suppressed by fluctuations in additional interaction channels. Nevertheless, as shown in the inset to Fig. \[fig:phasediagramFRG\], the full FRG calculation yields a dome of $T_c$, in qualitative agreement with the simplified mean-field approach. In regime II ($0.58<\bar{n}<0.88$) on the other hand, unlike what is seen for the attractive Hubbard model, the finite-range character of the interactions leads to an enhancement of $T_\mathrm{min}$ by the additional interaction channels. Finally, in regime III ($\bar{n}>0.88$), mean-field theory formally yields a finite $T_c$, while the full FRG approach reveals the breakdown of superconductivity. We tentatively identify this regime, where the renormalized $\ell \! \to \! \infty$, with phase separation induced by extended attractive interactions.
[*Conclusion. –*]{} In this Letter, we have provided a geometric phase space argument for the formation of $T_c$ domes in systems with spatially extended interactions. We have shown that for multiband systems a scenario with two or more domes can arise naturally. In order to apply this picture to 3D bulk SrTiO$_3$, we note that the first dome with maximum transition temperature $T_c \!\approx\!0.2$K is centered at a density $\bar{n} \!\approx\! 1.2 \times 10^{18}\;{\rm cm}^{-3}$ with a Fermi energy $\epsilon_F\!\approx\! 2$meV. Demanding $k_F \ell \!\sim\! 1$ at the center of the dome yields $\ell \! \sim\! 30$Å, while requiring $T_c/\epsilon_F \! \sim \! 10^{-2}$ at this point fixes $g_0 \!\sim\! 4.5$meV. The inferred length scale $\ell$ may reflect the range of attractive interactions between polaron quasiparticles which have been reported in bulk SrTiO$_3$ [@Swartz_PNAS2018] and its interfaces [@Baumberger_NMat2016; @Strocov_NComms2016]. The microscopic theory of SC of such dilute polarons remains an open issue. It would be interesting to explore such $T_c$ domes in a wider range of experimental systems including atomic Bose-Fermi mixtures, and to extend the FRG results by incorporating the frequency dependence of the interaction vertex. Such studies may shed light on the interplay of spatially extended interactions with retardation effects in driving SC near QCPs.
We thank M. M. Scherer for discussions. This work was funded by NSERC of Canada and FRQNT of Quebec. The numerical simulations were performed on the JURECA cluster at the Forschungszentrum Juelich, and on the Cedar and Niagara clusters enabled by support provided by Compute Ontario, SciNet, Westgrid, and Compute Canada. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.
Supplemental Material {#supplemental-material .unnumbered}
=====================
Derivation of multiband gap equation {#derivationGE}
------------------------------------
We start by writing the imaginary-time action for free fermions: $$S_0=\frac{1}{V^2}\int{{\mathrm{d}}}^d{{{\bf{r}}}}\;\int{{\mathrm{d}}}^d{{{\bf{r}}}}'\int_0^\beta{{\mathrm{d}}}\tau\;\bar{\psi}_\mu({{{\bf{r}}}},\tau)[\partial_\tau\delta_{\mu\nu}-H_0^{\mu\nu}({{{\bf{r}}}},{{{\bf{r}}}}')]\psi_\nu({{{\bf{r}}}}',\tau),$$ where $V=Na^d$ is the volume of the $d$-dimensional cubic system with $N$ sites of lattice constant $a$. Working in units where $a=1$, we can Fourier transform $S_0$ to momentum and Matsubara frequency space for a translationally invariant system: $$\begin{aligned}
S_0&=&\frac{1}{N}\sum_{{{{\bf{k}}}}\omega_n}\bar{\psi}_\mu({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)[{{\mathrm{i}}}\omega_n\delta_{\mu\nu}-H_0^{\mu\nu}({{{\bf{k}}}})]\psi_\nu({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)\nonumber\\
&=&\frac{1}{2N}\sum_{{{{\bf{k}}}}\omega_n}\left(\bar{\psi}_\mu({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)\underbrace{[{{\mathrm{i}}}\omega_n\delta_{\mu\nu}-H_0^{\mu\nu}({{{\bf{k}}}})]}_{G_{0p}^{-1}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)}\psi_\nu({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)+\psi_\mu(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\underbrace{[{{\mathrm{i}}}\omega_n\delta_{\mu\nu}+H_0^{\nu\mu}(-{{{\bf{k}}}})]}_{G_{0h}^{-1}(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)}\bar{\psi}_\nu(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\right)\nonumber\\
&=&\frac{1}{2N}\sum_{{{{\bf{k}}}}\omega_n}
\begin{pmatrix}\bar{\psi}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n) & \psi(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\end{pmatrix}
\begin{pmatrix}G_{0p}^{-1}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n) & 0 \\
0 & G_{0h}^{-1}(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\end{pmatrix}
\begin{pmatrix}\psi({{{\bf{k}}}},{{\mathrm{i}}}\omega_n) \\ \bar{\psi}(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\end{pmatrix} \,,\end{aligned}$$ where $G^{-1}_{0p}$ and $G^{-1}_{0h}$ are the matrix non-interacting Green’s functions, neglecting self-energy corrections. The instantaneous interaction is, in real-space: $$S_{\mathrm{int}}=\frac{1}{2V^2}\int{{\mathrm{d}}}^d{{{\bf{r}}}}\;\int{{\mathrm{d}}}^d{{{\bf{r}}}}'\int_0^\beta{{\mathrm{d}}}\tau\;\bar{\psi}_\mu({{{\bf{r}}}},\tau)\bar{\psi}_\nu({{{\bf{r}}}}',\tau)\mathcal{V}({{{\bf{r}}}}-{{{\bf{r}}}}')\psi_\nu({{{\bf{r}}}}',\tau)\psi_\mu({{{\bf{r}}}},\tau) \,,$$ and in ${{{\bf{k}}}}$-space, if we only keep zero centre of mass momentum terms: $$S_{\mathrm{int}}=\frac{1}{2\beta N^2}\sum_{{{{\bf{k}}}}{{{\bf{k}}}}'}\sum_{\omega_n\omega_m}\bar{\psi}_\mu({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)\bar{\psi}_\nu(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)V({{{\bf{k}}}}-{{{\bf{k}}}}')\psi_\nu(-{{{\bf{k}}}}',-{{\mathrm{i}}}\omega_m)\psi_\mu({{{\bf{k}}}}',{{\mathrm{i}}}\omega_m) \,,$$ with $V({{{\bf{k}}}}-{{{\bf{k}}}}')<0$ the Fourier transform of $\mathcal{V}({{{\bf{r}}}})$. We now introduce the complex fields $\Delta_{\mu\nu}({{{\bf{k}}}})$: $$\begin{aligned}
{{\mathrm{e}}}^{-S_{\mathrm{int}}}\propto\int\mathcal{D}[\bar{\Delta},\Delta]\;\exp\bigg(&-\frac{1}{2N}\sum_{{{{\bf{k}}}}\omega_n}\Big(\frac{\beta}{N}\sum_{{{{\bf{k}}}}'}\Delta_{\mu\nu}({{{\bf{k}}}}')\Delta_{\nu\mu}^*({{{\bf{k}}}})F({{{\bf{k}}}}-{{{\bf{k}}}}') \nonumber\\
&+\psi_\nu(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\Delta_{\nu\mu}^*({{{\bf{k}}}})\psi_\mu({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)+\bar{\psi}_\mu({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)\Delta_{\mu\nu}({{{\bf{k}}}})\bar{\psi}_\nu(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\Big)\bigg)\end{aligned}$$ with $F({{{\bf{k}}}}-{{{\bf{k}}}}')=\frac{1}{V}\int{{\mathrm{d}}}^d{{{\bf{r}}}}\;\frac{{{\mathrm{e}}}^{{{\mathrm{i}}}({{{\bf{k}}}}-{{{\bf{k}}}}')\cdot {{{\bf{r}}}}}}{\mathcal{V}({{{\bf{r}}}})}$. The total partition function, $\mathcal{Z}=\int\mathcal{D}[\bar{\psi},\psi]{{\mathrm{e}}}^{-(S_0+S_{\mathrm{int}})}$ up to normalization constants, becomes quadratic in the fermion fields: $$\begin{aligned}
\hspace{-5mm}
\mathcal{Z}&=&\int\mathcal{D}[\bar{\Delta},\Delta]\exp\left(-\frac{\beta}{2N^2}\sum_{{{{\bf{k}}}}{{{\bf{k}}}}'}\Delta_{\mu\nu}({{{\bf{k}}}}')\Delta_{\nu\mu}^*({{{\bf{k}}}})F({{{\bf{k}}}}-{{{\bf{k}}}}')\right)\nonumber\\
&\times&\int\mathcal{D}[\bar{\psi},\psi]\exp\left(-\frac{1}{2N}\sum_{{{{\bf{k}}}}\omega_n}
\underbrace{\begin{pmatrix}\bar{\psi}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n) & \psi(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\end{pmatrix}}_{\bar{\Psi}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)}
\underbrace{\begin{pmatrix}G_{0p}^{-1}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n) & \Delta({{{\bf{k}}}}) \\
\Delta^\dagger({{{\bf{k}}}}) & G_{0h}^{-1}(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\end{pmatrix}}_{\mathcal{G}^{-1}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)}
\underbrace{\begin{pmatrix}\psi({{{\bf{k}}}},{{\mathrm{i}}}\omega_n) \\ \bar{\psi}(-{{{\bf{k}}}},-{{\mathrm{i}}}\omega_n)\end{pmatrix}}_{\Psi({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)}\right) \,.\end{aligned}$$ And we can proceed by integrating out the fermions and obtain the effective action $\mathcal{Z}=\int\mathcal{D}[\bar{\Delta},\Delta]{{\mathrm{e}}}^{-S_{\mathrm{eff}}}$: $$S_{\mathrm{eff}}=\underbrace{\frac{\beta}{2N^2}\sum_{{{{\bf{k}}}}{{{\bf{k}}}}'}\Delta_{\mu\nu}({{{\bf{k}}}}')\Delta_{\nu\mu}^*({{{\bf{k}}}})F({{{\bf{k}}}}-{{{\bf{k}}}}')}_{S_1}+\underbrace{\frac{1}{2N}\sum_{{{{\bf{k}}}}\omega_n}\mathrm{tr}\log\begin{pmatrix}{{\mathrm{i}}}\omega_n\mathbb{I}-H_0({{{\bf{k}}}}) & \Delta({{{\bf{k}}}}) \\
\Delta^\dagger({{{\bf{k}}}}) & {{\mathrm{i}}}\omega_n\mathbb{I}+H_0^\mathrm{T}(-{{{\bf{k}}}})\end{pmatrix}}_{S_2},$$ To obtain the equation of motion, we need to set $\frac{\delta S_{\mathrm{eff}}}{\delta \Delta^*_{\sigma\lambda}({{{\bf{p}}}})}=0$. Varying $S_1$ is straighforward: $$\frac{1}{\beta}\frac{\delta S_1}{\delta \Delta^*_{\sigma\lambda}({{{\bf{p}}}})}=\frac{1}{2N^2}\sum_{{{{\bf{k}}}}{{{\bf{k}}}}'}\delta_{\nu\sigma}\delta_{\mu\lambda}\delta^{(d)}({{{\bf{p}}}}-{{{\bf{k}}}})\Delta_{\mu\nu}({{{\bf{k}}}}')F({{{\bf{k}}}}-{{{\bf{k}}}}')=\frac{1}{2N}\sum_{{{{\bf{k}}}}'}\Delta_{\lambda\sigma}({{{\bf{k}}}}')F({{{\bf{p}}}}-{{{\bf{k}}}}') \,.$$ Varying $S_2$ leads to: $$\begin{aligned}
\frac{1}{\beta}\frac{\delta S_2}{\delta \Delta^*_{\sigma\lambda}({{{\bf{p}}}})}&=\frac{1}{2\beta N}\sum_{{{{\bf{k}}}}\omega_n}\mathrm{tr}\left(\frac{\delta\log(\mathcal{G}^{-1}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n))}{\delta\mathcal{G}^{-1}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)}\frac{\delta\mathcal{G}^{-1}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)}{\delta \Delta^*_{\sigma\lambda}({{{\bf{p}}}})}\right) \nonumber\\
&=\frac{1}{2\beta N}\sum_{{{{\bf{k}}}}\omega_n}\left[\mathcal{G}({{{\bf{k}}}},{{\mathrm{i}}}\omega_n)\delta^{(d)}({{{\bf{p}}}}-{{{\bf{k}}}})\begin{pmatrix} 0 & 0 \\ \delta_{\sigma\lambda} & 0 \end{pmatrix}\right] \nonumber\\
&=\frac{1}{2\beta}\sum_{\omega_n}\left[\mathcal{G}({{{\bf{p}}}},{{\mathrm{i}}}\omega_n)\right]_{\lambda\sigma}^{12} \,.
\label{eqn:dS2}\end{aligned}$$ $\left[\mathcal{G}({{{\bf{p}}}},{{\mathrm{i}}}\omega_n)\right]_{\lambda\sigma}^{12}$ refers to the $(\lambda,\sigma)$ matrix element of the $(1,2)$ block (i.e. top right block) of the $\mathcal{G}$ matrix. In order to invert a matrix containing square block matrices, we make use of the following identity:
$$\begin{pmatrix} A & B \\
C & D\end{pmatrix}^{-1}=\begin{pmatrix} A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1} & -A^{-1}B(D-CA^{-1}B)^{-1} \\
-(D-CA^{-1}B)^{-1}CA^{-1} & (D-CA^{-1}B)^{-1}\end{pmatrix} \,.$$
Focusing on the top right corner, we obtain after some algebra and using the particle-hole symmetry of the Hamiltonian: $$\begin{aligned}
\left[\mathcal{G}({{{\bf{p}}}},{{\mathrm{i}}}\omega_n)\right]^{12}&=&-[\omega_n^2+{{\mathrm{i}}}\omega_n(H_0({{{\bf{p}}}})-\Delta({{{\bf{p}}}}) H_0^\mathrm{T}(-{{{\bf{p}}}})\Delta^{-1}({{{\bf{p}}}}))+\Delta({{{\bf{p}}}})H_0^\mathrm{T}(-{{{\bf{p}}}})\Delta^{-1}({{{\bf{p}}}})H_0({{{\bf{p}}}})+\Delta({{{\bf{p}}}})\Delta^\dagger({{{\bf{p}}}})]^{-1}\Delta({{{\bf{p}}}})\nonumber\\
&=&-({{\mathrm{i}}}\omega_n\mathbb{I}-\mathcal{E}({{{\bf{p}}}}))^{-1}({{\mathrm{i}}}\omega_n\mathbb{I}+\mathcal{E}({{{\bf{p}}}}))^{-1}\Delta({{{\bf{p}}}})\end{aligned}$$ and $\mathcal{E}^2({{{\bf{p}}}})\equiv H_0({{{\bf{p}}}})H^\dagger_0({{{\bf{p}}}})+\Delta({{{\bf{p}}}})\Delta^\dagger({{{\bf{p}}}})$. This matrix can be diagonalized via $\mathcal{E}^2({{{\bf{p}}}})=U({{{\bf{p}}}})E^2({{{\bf{p}}}})U^\dagger({{{\bf{p}}}})$ and the Matsubara frequency summation performed: $$\begin{aligned}
\frac{1}{\beta}\frac{\delta S_2}{\delta \Delta^*_{\sigma\lambda}({{{\bf{p}}}})}&=\frac{1}{2\beta}\sum_{{{\mathrm{i}}}\omega_n}-({{\mathrm{i}}}\omega_n\mathbb{I}-\mathcal{E}({{{\bf{p}}}}))_{\sigma m}^{-1}({{\mathrm{i}}}\omega_n\mathbb{I}+\mathcal{E}({{{\bf{p}}}}))_{m\alpha}^{-1}\Delta_{\alpha\lambda}({{{\bf{p}}}}) \nonumber\\
&=-\frac{1}{2\beta}\sum_{{{\mathrm{i}}}\omega_n}U_{\sigma m}({{{\bf{p}}}})({{\mathrm{i}}}\omega_n-E({{{\bf{p}}}}))_{m}^{-1}({{\mathrm{i}}}\omega_n+E({{{\bf{p}}}}))_m^{-1}U_{m\alpha}^*({{{\bf{p}}}})\Delta_{\alpha\lambda}({{{\bf{p}}}}) \nonumber\\
&=\frac{1}{2}U_{\sigma m}({{{\bf{p}}}})\frac{1}{2E_m({{{\bf{p}}}})}\tanh\left(\frac{\beta E_m({{{\bf{p}}}})}{2}\right)U_{m\alpha}^*({{{\bf{p}}}})\Delta_{\alpha\lambda}({{{\bf{p}}}})\end{aligned}$$ with implied sums over repeated indices. In matrix notation: $$\frac{1}{\beta}\frac{\delta S_\mathrm{eff}}{\delta \Delta^\dagger({{{\bf{p}}}})}=\frac{1}{2N}\sum_{{{{\bf{k}}}}'} F({{{\bf{p}}}}-{{{\bf{k}}}}')\Delta({{{\bf{k}}}}')+\frac{1}{2}U({{{\bf{p}}}})\frac{1}{2E({{{\bf{p}}}})}\tanh\left(\frac{\beta E({{{\bf{p}}}})}{2}\right)U^\dagger({{{\bf{p}}}})\Delta({{{\bf{p}}}})=0 \,.$$ For an inversion symmetric scattering potential $\mathcal{V}({{{\bf{r}}}})=\mathcal{V}(-{{{\bf{r}}}})$, this can be rewritten in terms of the Fourier transform $V({{{\bf{k}}}})$: $$\begin{aligned}
\Delta_{\mu\nu}({{{\bf{k}}}})&=-\frac{1}{N}\sum_{{{{\bf{k}}}}'}\sum_{m\lambda} V({{{\bf{k}}}}-{{{\bf{k}}}}') U_{\mu m}({{{\bf{k}}}}')\frac{1}{2E_m({{{\bf{k}}}}')}\tanh\left(\frac{E_m({{{\bf{k}}}}')}{2T}\right)U^*_{m\lambda}({{{\bf{k}}}}')\Delta_{\lambda\nu}({{{\bf{k}}}}') \nonumber\\
&=\sum_{{{{\bf{k}}}}'}V({{{\bf{k}}}}-{{{\bf{k}}}}')M_{\mu\lambda}({{{\bf{k}}}}')\Delta_{\lambda\nu}({{{\bf{k}}}}') \,,\end{aligned}$$ with $M_{\mu\lambda}({{{\bf{k}}}}')\equiv-\frac{1}{N}\sum_m U_{\mu m}({{{\bf{k}}}}')\frac{1}{2E_m({{{\bf{k}}}}')}\tanh\left(\frac{E_m({{{\bf{k}}}}')}{2T}\right)U^*_{m\lambda}({{{\bf{k}}}}')$. Numerically, it is useful to write this as a matrix equation at temperature $T$: $$\vec{\Delta}(T)=\mathcal{M}(T)\;\vec{\Delta}(T)\iff\Delta_i(T)=\mathcal{M}_{ij}(T)\Delta_j(T) \,.$$ For a Hamiltonian comprising $b$ bands and discretized on a momentum mesh with $N$ points, the indices ${i,j\in[1,...,b^2N]}$ which makes the matrix $\mathcal{M}$ of dimensions $b^2N\times b^2N$. More explicitly: $$\begin{pmatrix}
\Delta_{11}({{{\bf{k}}}}_1) \\ \Delta_{12}({{{\bf{k}}}}_1) \\ \vdots \\ \Delta_{1b}({{{\bf{k}}}}_1) \\ \vdots \\ \Delta_{bb}({{{\bf{k}}}}_1) \\ \vdots \\ \Delta_{bb}({{{\bf{k}}}}_N)
\end{pmatrix}
\!\!=\!\!
\begin{pmatrix}
V({{{\bf{k}}}}_1-{{{\bf{k}}}}_1)[M({{{\bf{k}}}}_1)]\otimes\mathbb{I}_b & V({{{\bf{k}}}}_1-{{{\bf{k}}}}_2)[M({{{\bf{k}}}}_2)]\otimes\mathbb{I}_b & \hdots & V({{{\bf{k}}}}_1-{{{\bf{k}}}}_N)[M({{{\bf{k}}}}_N)]\otimes\mathbb{I}_b\\
V({{{\bf{k}}}}_2-{{{\bf{k}}}}_1)[M({{{\bf{k}}}}_1)]\otimes\mathbb{I}_b & V({{{\bf{k}}}}_2-{{{\bf{k}}}}_2)[M({{{\bf{k}}}}_2)]\otimes\mathbb{I}_b & \hdots & V({{{\bf{k}}}}_2-{{{\bf{k}}}}_N)[M({{{\bf{k}}}}_N)]\otimes\mathbb{I}_b\\
\vdots & \vdots & \ddots & \vdots \\
V({{{\bf{k}}}}_N-{{{\bf{k}}}}_1)[M({{{\bf{k}}}}_1)]\otimes\mathbb{I}_b & V({{{\bf{k}}}}_N-{{{\bf{k}}}}_2)[M({{{\bf{k}}}}_2)]\otimes\mathbb{I}_b & \hdots & V({{{\bf{k}}}}_N-{{{\bf{k}}}}_N)[M({{{\bf{k}}}}_N)]\otimes\mathbb{I}_b
\end{pmatrix}
\!\!
\begin{pmatrix}
\Delta_{11}({{{\bf{k}}}}_1) \\ \Delta_{12}({{{\bf{k}}}}_1) \\ \vdots \\ \Delta_{1b}({{{\bf{k}}}}_1) \\ \vdots \\ \Delta_{bb}({{{\bf{k}}}}_1) \\ \vdots \\ \Delta_{bb}({{{\bf{k}}}}_N)
\end{pmatrix} \,,
\label{eqn:matrixDelta}$$ where $\mathbb{I}_b$ is the $b\times b$ identity matrix.
[GapTc1B.pdf]{}
At $T=T_c$, $E_m({{{\bf{k}}}})\rightarrow\xi_m({{{\bf{k}}}})$ and $U_{\mu m}({{{\bf{k}}}})\rightarrow W_{\mu m}({{{\bf{k}}}})$ where $\xi_m({{{\bf{k}}}})=W^*_{m\lambda}({{{\bf{k}}}})H_0^{\lambda\sigma}({{{\bf{k}}}})W_{\sigma m}({{{\bf{k}}}})$ such that $M$ no longer depends on $\Delta$. Eqn. reduces to an eigenvalue equation and $T_c$ is obtained when the largest eigenvalue of $\mathcal{M}$ reaches 1 (for $T\ll T_c$ all the eigenvalues are larger than 1 while for $T\gg T_c$ all the eigenvalues vanish.)
At $T=0$, the equation is non-linear as the matrix $M$ depends on $\Delta$ and we must solve by (i) guessing an initial $\Delta^{(0)}({{{\bf{k}}}})$, (ii) diagonalizing $H_0({{{\bf{k}}}})H_0^\dagger({{{\bf{k}}}})+\Delta^{(0)}({{{\bf{k}}}})[\Delta^{(0)}({{{\bf{k}}}})]^\dagger=U({{{\bf{k}}}})[E^{(0)}({{{\bf{k}}}})]^2U^{-1}({{{\bf{k}}}}')$, (iii) constructing the $\mathcal{M}$ matrix and (iv) multiplying by the ‘vectorized’ $\vec{\Delta}^{(0)}$ to obtain a new vector $\vec{\Delta}^{(1)}$ which can then be used to repeat the procedure. The gap function $\Delta(T=0)$ is the ‘fixed point’ of this equation.
Finally, since the instability is expected near the Fermi momenta (i.e. where the denominator $E_m({{{\bf{k}}}})\rightarrow 0$) it is useful to only store momenta within a given range of ${{{\bf{k}}}}_F$. The lengthscale $\ell^{-1}$ provides a natural cutoff and we found that only keeping momenta within $\pm 3\ell^{-1}$ of ${{{\bf{k}}}}_F$ is sufficient to reach convergent results.
Momentum structure of the gap functions
---------------------------------------
At $T=T_c$, the spectrum of $\mathcal{M}$ tells us about the modes of the gap. The largest eigenvalue is the lowest energy state into which the system will condense first and the corresponding eigenvector shows the momentum dependence of the gap. When $\ell\rightarrow\infty$, all eigenvalues converge to 1 and the corresponding eigenvectors become localized to single momentum points on the Fermi surface. In that case, any linear combination of the eigenvectors would be a solution to the gap equation and all momenta condense simultaneously. However, for finite values of $\ell$, this doesn’t happen and we generally have one eigenvalue reaching unity before the others. The closest eigenvalues correspond to eigenvectors with higher energy and in general get closer to each other for high densities (as this is equivalent to increasing $\ell$). For a generic density and pairing lengthscale shown in Fig. \[fig:gapTc1B\], we show that the largest eigenvalue is nondegenerate and has $s$-wave symmetry, while the next eigenvalues are doubly degenerate with $p_x$ and $p_y$ symmetry and the next two have $d$-wave symmetry. In Fig. \[fig:gaps\], we show the gaps at various densities at $T=0$, which are obtained by solving the non-linear gap equation.
[Gaps.pdf]{}(2,94)[(a)]{}(53,94)[(b)]{}(2,45)[(c)]{}(53,45)[(d)]{}
The van Hove peak
-----------------
[MFGeometric.pdf]{}(16,95)[(a)]{} (62,95)[(b)]{}(16,44)[(c)]{}
The critical temperature in 2D over all densities is shown in Fig. \[fig:Tc2D\] (a) for the same choice of parameters as Fig. \[fig:geometry\] of the main text. For a one band model with isotropic dispersion there are three peaks. The first peak (dashed line) is the electron geometric peak and was discussed in the main text. We also have the hole version of this geometric peak marked with a dot-dashed line. The corresponding gaps at $T=0$ are shown in Figs. \[fig:gaps\](a) and (d), respectively. Since $1/\ell$ is much larger than the size of the hole pocket, the gap doesn’t peak on the Fermi surface but at the center of the pockets, i.e. the M points. This is similar to the electron pockets where the gap peaks at the $\Gamma$ point in the radial direction. The middle peak (dotted line) is exactly at half-filling when the van Hove singularity occurs in two-dimensions: the density of states (DOS) is largest and we expect an enhancement of $T_c$ from a BCS-like picture. The amplitude of the gap $|\Delta({{{\bf{k}}}},T=0)|$ is shown in Fig. \[fig:gaps\] (c) with the FS overlaid on top. Tangentially to the Fermi surface, it peaks at the van Hove points $(\pi,0)$ and $(0,\pi)$ while in the perpendicular direction it peaks at ${{{\bf{k}}}}_F$ and decays a distance $\sim 1/\ell$ away from the FS. Although the existence of the van Hove peak is expected independently of the lengthscale $\ell$, its appearance overlaps with a different, $\ell$-dependent effect: As the van Hove point is approached, Umklapp processes become allowed and the scattering phase space is enhanced for $|{{{\bf{k}}}}_F-({{{\bf{k}}}}'_F\pm{{{\bf{G}}}})|\ell\sim 1$ where ${{{\bf{G}}}}$ is a reciprocal lattice vector. This is also seen in the geometric weight calculation where $\mathcal{R}$ goes up again as parts of the FS from the neighboring BZ become accessible (Fig. \[fig:Tc2D\] (b)). This situation is shown pictorially in Fig. \[fig:Tc2D\] (c) where a point on the FS of the left Brillouin zone starts to ‘sense’ its neighbor on the other side of the BZ because of the finite potential range $\sim1/\ell$.
Anisotropic dispersions and $\ell$ dependence
---------------------------------------------
[Tc\_1Band\_2D\_L\_Anisotropic.pdf]{}(16,47)[(a)]{} (62,47)[(b)]{}
As we tune the value of $\ell$ we observe that as $\ell$ is increased the location of the geometric peak moves to lower densities as expected (thus keeping $k_F^\star\ell\sim1$). On the other hand, although the van Hove peak remains at half filling (Fig. \[fig:Tc1B2DLAni\] (a)), its tail gets sharper for small $\ell$ because Umklapp scattering only kicks in at increasingly large fillings. For an anisotropic FS with dispersion ${\xi({{{\bf{k}}}})=-2t_1\cos(kx)-2t_2\cos(k_y)-\mu}$ and $t_1\neq t_2$, the location of the van Hove peak shifts to lower densities since the elongated part of the elliptical FS hits the BZ boundary at densities $\bar{n}<1$. For a very elliptical FS ($t_2/t_1\ll 1$ or $t_2/t_1\gg1$), the van Hove point is at smaller densities and $k_F$ acquires a strong angular modulation ${k_F^{\mathrm{long}}\gg k_F^{\mathrm{short}}}$. In turn, this introduces a new condition that as soon as $k_F^{\mathrm{long}}\ell\gg1$, the available phase space for scattering starts to decrease even if ${k_F^{\mathrm{long}}\ell\lesssim1}$. In this scenario, the van Hove peak shifts to lower densities while the geometric peak is pushed to higher densities and the two eventually merge into a single peak (Fig. \[fig:Tc1B2DLAni\] (b)). The peak value of $T_c$ is enhanced when $\ell$ is large and it can be shown analytically that the mean-field $T_c$ reaches an upper limit of $g_0/4$ when $\ell\rightarrow\infty$ ($V({{{\bf{k}}}})\rightarrow-g_0\delta^{(d)}({{{\bf{k}}}})$); this value is however suppressed when vertex corrections are included as discussed in the main text. In all cases, $T_c$ hardly changes by a factor of 2-3 over many orders of magnitude of $\bar{n}$.
Finite potential difference
---------------------------
![Critical temperature for a two-orbital model (black curve) with a finite potential difference $2V_0=2$ and a single orbital model (red curve) showing a perfect agreement at low densities when only one band is populated. The insets show the Fermi surfaces at the correspondingly colored stars. []{data-label="fig:V02D"}](Tc_2Band_2D_V0.pdf){width="\linewidth"}
In this section, we consider a modified version of the Hamiltonian presented in Eqn. of the main text $$\!\!\!\mathcal{H}_0=\sum_{{{{\bf{k}}}}} \! \begin{pmatrix}
X^\dagger_{{{{\bf{k}}}}{\uparrow}} & \! Y^\dagger_{{{{\bf{k}}}}{\uparrow}}
\end{pmatrix}\!\!
\begin{pmatrix}
\xi_{X}({{{\bf{k}}}})+V_0 & \! 0 \\
0 & \! \xi_{Y}({{{\bf{k}}}})-V_0
\end{pmatrix}\!\!
\begin{pmatrix}
X_{{{{\bf{k}}}}{\uparrow}} \\
Y_{{{{\bf{k}}}}{\uparrow}}
\end{pmatrix},$$
and we study the impact of a finite potential difference between the two orbitals, keeping $t_1=t_2=1$. This corresponds to two independent bands separated in energy by $2V_0$. The resulting $T_c$ is illustrated in Fig. \[fig:V02D\] with the Fermi surfaces at the three peak densities in electron-doped regime shown in the insets and the dashed red curve showing the one-orbital model with the same parameters.
In the low-density regime, the two curves agree exactly, showing that only the lower band dictates $T_c$: a first geometric peak (marked by a green star) is reached at $\bar{n}=n^\star\approx0.046$ which corresponds to $k_F^\star\ell\approx2.85$. Near the Lifshitz transition (vertical dashed curve), the one-orbital and two-orbital models start to diverge since the higher band crosses the Fermi level and starts to contribute to $T_c$. A second geometric peak (orange star) is reached precisely when the new band’s Fermi wavevector is such that $k_F^\star\ell\approx2.85$. At higher densities, the lower band reaches the van Hove point and we see the corresponding peak (blue star). As for the one-orbital case, particle-hole symmetry dictates $T_c(\bar{n})=T_c(4-\bar{n})$ and an exact copy of the three peaks is obtained in the hole-doped regime $\bar{n}>2$.
FRG flow equations {#sec:appendix:frg}
------------------
![Partitioning of the Brillouin zone into a set of patches $\mathcal{P}_1,\dots,\mathcal{P}_N$, illustrated for $N=24$. All momentum points within a single patch $\mathcal{P}_m$ are projected onto the representative momentum ${{{\bf{p}}}}_m$ which lies on the Fermi surface. []{data-label="fig:appendix:patching"}](patchingScheme.pdf){width="0.9\linewidth"}
![Diagrammatic representation of contributions to the flow of the effective interaction. Singling out particle-particle scattering (first term) is equivalent to a Bethe-Salpeter like resummation and generates results on the level of self-consistent mean-field theory. The inclusion of additional direct particle-hole (terms two to four) and crossed particle-particle (last term) scattering allows to model the interplay of competing interaction channels.[]{data-label="fig:appendix:FRGchannels"}](flowequation.pdf){width="\linewidth"}
The general form of the effective two-particle interaction vertex is given by $$\label{eq:appendix:frg:vertex}
V(K_1',K_2';K_1,K_2) \,,$$ where the parameters $K_n$ are composite indices denoting tuples $({{{\bf{k}}}}_n,\omega_n,\alpha_n)$ of momentum, Matsubara frequency, and spin, respectively. In this most general form, the fermionic interaction vertex must be antisymmetric under the pairwise exchange of its arguments. For the SU(2)-symmetric model at hand, however, it is more convenient to constrain the effective interaction to a form which is inherently encoded to be SU(2) symmetric; To this end, we parameterize the effective interaction by two terms – spin-conserving and spin-exchange terms – which span a full basis for SU(2) invariant interactions: $$\begin{aligned}
V(K_1',K_2';K_1,K_2)&= U(k_1',k_2';k_1,k_2)\delta_{\alpha_1' \alpha_1}\delta_{\alpha_2'\alpha_2} \nonumber\\
&- U(k_1',k_2';k_2,k_1) \delta_{\alpha_1' \alpha_2}\delta_{\alpha_2'\alpha_1} \,.\end{aligned}$$ Here, the composite indices $k_n$ denote pairs $({{{\bf{k}}}}_n,\omega_n)$ of momentum and Matsubara frequency, while the spin index $\alpha$ is written out explicitly. The basis function $U(k_1',k_2';k_1,k_2)$ is symmetric under simultaneous exchange of ingoing and outgoing indices.
For further simplification of the vertex parametrization we resort to the momentum space patching approximation outlined in Ref. [@Halboth2000], which is suitable in the weak coupling limit. In this approximation, the frequency dependence of the vertex is neglected, while the momentum dependence is parametrized such that it resolves the angular dependence around the Fermi surface, but it neglects any dependence in the radial direction. This is achieved by partitioning the Brillouin zone into a set of *patches* $\{\mathcal{P}_1 \dots \mathcal{P}_N\}$ as shown in Fig. \[fig:appendix:patching\], and projecting all momentum points within a patch $\mathcal{P}_m$ onto a single representative point ${{{\bf{p}}}}_m$ on the Fermi surface, i.e. the vertex function is assumed to be constant within the entire patch. The parametrization of the vertex function can thus be written as $$\begin{aligned}
&U(k_1',k_2';k_1,k_2) = \sum\limits_{i_1, i_2, i_3} u(n_{i_1}, n_{i_2}, n_{i_3}) \nonumber\\
&\quad\times \delta({{{\bf{k}}}}_1'+{{{\bf{k}}}}_2'-{{{\bf{k}}}}_1-{{{\bf{k}}}}_2) \delta_{{{{\bf{k}}}}_1' \in \mathcal{P}_{i_1}} \delta_{{{{\bf{k}}}}_2' \in \mathcal{P}_{i_2}} \delta_{{{{\bf{k}}}}_1 \in \mathcal{P}_{i_3}} \,,\end{aligned}$$ where the indices $n_m$ enumerate momentum patches and the symbol $\delta_{q\in \mathcal{P}_n}=1$ if momentum $q$ lies within patch $\mathcal{P}_n$ and zero otherwise.
The FRG flow equations are obtained by introducing an additional dependence of the interaction vertex on some RG cutoff. We follow the temperature flow RG scheme outlined in Ref. [@Honerkamp2001], where the temperature itself assumes to role of the RG cutoff and the flow equations take the form $$\frac{\mathrm{d}}{\mathrm{d}T}u_T(n_1, n_2, n_3) = \mathcal{T}_{PP,T} + \mathcal{T}_{PH,T}^d + \mathcal{T}_{PH,T}^c \,,$$ where the three interaction channels (particle-particle, direct particle-hole, and crossed particle-hole interaction, respectively) are given by (terms in the same order as shown in Fig. \[fig:appendix:FRGchannels\])
$$\begin{aligned}
\mathcal{T}_{PP,T}(n_1, n_2, n_3) &= -\sum\limits_{n} u_T(n_1, n_2, n) u_T(n, -n+n_1+n_2, n_3) L^+_T(n, n_1+n_2) \nonumber\\
\mathcal{T}^d_{PH,T}(n_1, n_2, n_3) &= -\sum\limits_{n} \Big(
-2u_T(n_1,n,n_3) u_T(n+n_1-n_3,n_2,n) \nonumber\\
&\quad\quad\quad\quad\quad +u_T(n_1,n,n+n_1-n_3) u_T(n+n_1-n_3,n_2,n) \nonumber\\
&\quad\quad\quad\quad\quad +u_T(n_1,n,n_3) u_T(n_2,n+n_1-n_3,n)
\Big) L^-_T(n, n_1-n_3) \nonumber\\
\mathcal{T}^c_{PH,T}(n_1, n_2, n_3) &= -\sum\limits_{n} u_T(n_1,n+n_2-n_3,n) u_T(n,n_2,n_3) L^-_T(n, n_2-n_3) \,.
\end{aligned}$$
The internal propagator bubble is defined as $$L^\pm_T(n,m) = \int_{{{{\bf{k}}}}\in\mathcal{P}_n} \mp \frac{\lambda\left(\xi({{{\bf{k}}}})\right)\pm \lambda\left(\xi(\mp {{{\bf{k}}}}+ {{{\bf{p}}}}_m)\right)}{\xi({{{\bf{k}}}})\pm\xi(\mp {{{\bf{k}}}}+ {{{\bf{p}}}}_m)} \,,$$ where $\xi({{{\bf{k}}}})$ is the dispersion of the noninteracting system and $\lambda(\xi)$ is the temperature derivative of the Fermi distribution function $\lambda(\xi) = \xi {{\mathrm{e}}}^{\xi/T}[T^2 \left( {{\mathrm{e}}}^{\xi/T}+1 \right)^2]^{-1}$.
In this form, the flow equations can be solved numerically to connect the high-temperature limit, in which the effective interaction vertex Eqn. equals the bare interaction as defined by the Hamiltonian $\mathcal{H}_\mathrm{int}$, to the effective low-energy theory.
Hubbard model
-------------
![Benchmark of the characteristic temperature $T_\mathrm{min}$, as determined in FRG calculations (including only particle-particle scattering channel), with the mean-field critical temperature of the Hubbard model. The Hubbard limit $\ell=0$ shows excellent agreement for fillings $n\gtrsim 10^{-3}$. The generalized Gaussian potential ($\ell=1$) approaches the Hubbard limit in the very dilute regime. []{data-label="fig:appendix:hubbard:hubbardFRG"}](hubbardFRG.pdf){width="\linewidth"}
The FRG flow equations derived in Sec. \[sec:appendix:frg\] are suited for the weak-coupling limit [@Honerkamp2001]. In the dilute limit, however, when the Fermi energy scale becomes small compared to the interaction potential, the weak-coupling scenario may be violated. In order to convince ourselves that the approach produces meaningful results nevertheless, we benchmark the implementation against the mean-field solution of the Hubbard model at small densities.
To this end, we consider again the general Gaussian potential introduced in the main article and set $\ell = 0$, while fixing the prefactor to $g_0=\frac{3}{2\pi}$. As displayed in Fig. \[fig:appendix:hubbard:hubbardFRG\], the characteristic temperature scale $T_\mathrm{min}$ obtained from the FRG solution is in excellent agreement with the critical temperature as determined by the mean-field approach. Only at extremely low densities, below fillings relevant for our studies of geometric domes of $T_c$, deviations manifest. The FRG results remain consistent when a finite $\ell=1$ is considered in the sense that the result smoothly connects to the Hubbard limit in the dilute limit where $k_F \ell \ll 1$. This is to be expected since the width of the Gaussian profile becomes large compared to the size of the Fermi surface and the interaction potential effectively appears almost constant.
[^1]: Properly speaking, we set $V({{\bf{q}}})$ to be a periodic Gaussian, given by $V({{\bf{q}}})= -g_0(2\pi\ell^2)^{d/2} \sum_{{{\bf{G}}}}{{\mathrm{e}}}^{-|{{\bf{q}}}+{{{\bf{G}}}}|^2\ell^2/2}$, where ${{{\bf{G}}}}$ are reciprocal lattice vectors.
[^2]: We define $k_F^\star = (2\pi n^\star)^{1/2}$ in 2D, and $k_F^\star = (3\pi^2 n^\star)^{1/3}$ in 3D
[^3]: For the single band Hubbard model, this choice reproduces the superconducting $T_c$ as shown in the SM
|
---
abstract: 'Given a continuous monadic functor $T:{\mathbf{Comp}}\to{\mathbf{Comp}}$ in the category of compacta and a discrete topological semigroup $X$ we extend the semigroup operation $\varphi:X\times X\to X$ to a right-topological semigroup operation $\Phi:T\beta X\times T\beta X\to T\beta X$ whose topological center $\Lambda_\Phi$ contains the dense subsemigroup $T_f X$ consisting of elements $a\in T\beta X$ that have finite support in $X$.'
address:
- 'Ivan Franko National University of Lviv, Ukraine'
- 'Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine'
author:
- Taras Banakh and Volodymyr Gavrylkiv
title: 'Extending binary operations to funtor-spaces'
---
One of powerful tools in the modern Combinatorics of Numbers is the method of ultrafilters based on the fact that each binary operation $\varphi:X\times X\to X$ defined on a discrete topological space $X$ can be extended to a right-topological operation $\Phi:\beta X\times \beta X\to\beta X$ on the Stone-Čech compactification $\beta X$ of $X$, see [@HS], [@P]. The extension of $\varphi$ is constructed in two step. First, for every $x\in X$ extend the left shift $\varphi_x:X\to X$, $\varphi_x:y\mapsto \varphi(x,y)$, to a continuous map $\overline{\varphi}_x:\beta X\to\beta X$. Next, for every $b\in\beta X$, extend the right shift $\bar \varphi^b:X\to\beta X$, $\bar\varphi^b:x\mapsto\bar\varphi_x(b)$, to a continuous map $\Phi^b:\beta X\to\beta X$ and put $\Phi(a,b)=\Phi^b(a)$ for every $a\in\beta X$. The Stone-Čech extension $\beta X$ is the space of ultrafilters on $X$. In [@G2] it was observed that the binary operation $\varphi$ extends not only to $\beta X$ but also to the superextension $\lambda X$ of $X$ and to the space $GX$ of all inclusion hyperspaces on $X$. If $X$ is a semigroup, then $GX$ is a compact Hausdorff right-topological semigroup containing $\lambda X$ and $\beta X$ as closed subsemigroups.
In this note we show that an (associative) binary operation $\varphi:X\times X\to X$ on a discrete topological space $X$ can be extended to an (associative) right-topological operation $\Phi:T\beta X\times T\beta X\to T\beta X$ for any monadic functor $T$ in the category ${\mathbf{Comp}}$ of compact Hausdorff spaces. So, for the functors $\beta,\lambda$ or $G$, we get the extensions of the operation $\varphi$ discussed above.
Monadic functors and their algebras
===================================
Let us recall [@McL VI], [@TZ §1.2] that a functor $T:{\mathcal C}\to{\mathcal C}$ in a category ${\mathcal C}$ is called [*monadic*]{} if there are natural transformations $\eta:{\mathrm{Id}}\to T$ and $\mu:T^2\to T$ making the following diagrams commutative: $$\xymatrix{T \ar[r]^{\eta T} \ar[d]_{T\eta} \ar[rd]^{1_{T}}&
T^2 \ar[d]^{\mu} & T^3 \ar[r]^{\mu T} \ar[d]_{T\mu} & T^2
\ar[d]^{\mu}\\
T^2 \ar[r]_{\mu} & T & T^2 \ar[r]_{\mu} & T}$$ In this case the triple ${\mathbb T}=(T,\eta,\mu)$ is called a [*monad*]{}, the natural transformations $\eta:{\mathrm{Id}}\to T$ and $\mu:T^2\to T$ are called the [*unit*]{} and [*multiplication*]{} of the monad ${\mathbb T}$, and the functor $T$ is the [*functorial part*]{} of the monad ${\mathbb T}$.
A pair $(X,\xi)$ consisting of an object $X$ and a morphism $\xi:TX\to X$ of the category ${\mathcal C}$ is called a [*${\mathbb T}$-algebra*]{} if $\xi\circ\eta_X={\mathrm{id}}_X$ and the square $$\xymatrix{T^2X\ar[d]_\mu\ar[r]^{T\xi}&TX\ar[d]^\xi\\TX\ar[r]_\xi&X}$$ is commutative. For every object $X$ of the category ${\mathcal C}$ the pair $(TX,\mu)$ is a ${\mathbb T}$-algebra called the [*free ${\mathbb T}$-algebra over $X$*]{}.
For two ${\mathbb T}$-algebras $(X,\xi_X)$ and $(Y,\xi_Y)$ a morphism $h:X\to Y$ is called a [*morphism of ${\mathbb T}$-algebras*]{} if the following diagram is commutative: $$\xymatrix{TX\ar[d]_{\xi_X}\ar[r]^{Th}&TY\ar[d]^{\xi_Y}\\ X\ar[r]_h&Y}$$ The naturality of the multiplication $\mu:T^2\to T$ of the monad ${\mathbb T}$ implies that for any morphism $f:X\to Y$ in ${\mathcal C}$ the morphism $Tf:TX\to TY$ is a morphism of free ${\mathbb T}$-algebras.
Each morphism $h:TX\to Y$ from the free ${\mathbb T}$-algebra into a ${\mathbb T}$-algebra $(Y,\xi)$ is uniquely determined by the composition $h\circ\eta$.
\[free\] If $h:TX\to Y$ is a morphism of a free ${\mathbb T}$-algebra $TX$ into a ${\mathbb T}$-algebra $(Y,\xi)$, then $h=\mu\circ T(h\circ \eta)=\mu\circ Th\circ T\eta$.
Consider the commutative diagram $$\xymatrix{
X\ar[d]_{\eta}\ar[r]^{\eta}&TX\ar@<2pt>[d]^{\eta_{T}}\ar[r]^h&Y\\
TX\ar@/_2pc/[rr]_{T(h\circ\eta)}\ar@<2pt>[r]^{T\eta}&T^2X\ar@<2pt>[l]^{\mu}
\ar@<2pt>[u]^{\mu}\ar[r]^{Th}&TY\ar[u]_{\xi}
}$$and observe that $$h=h\circ \mu\circ\eta_{T}=\xi\circ Th\circ\eta_T=\xi\circ Th\circ T\eta\circ\mu\circ\eta_T=\xi\circ T(h\circ\eta).$$
By a [*topological category*]{} we shall understand a subcategory of the category ${\mathbf{Top}}$ of topological spaces and their continuous maps such that
- for any objects $X,Y$ of the category ${\mathcal C}$ each constant map $f:X\to Y$ is a morphism of ${\mathcal C}$;
- for any objects $X,Y$ of the category ${\mathcal C}$ the product $X\times Y$ is an object of ${\mathcal C}$ and for any object $Z$ of ${\mathcal C}$ and morphisms $f_X:Z\to X$ and $f_Y:Z\to Y$ the map $(f_X,f_Y):Z\to X\times Y$ is a morphism of the category ${\mathcal C}$.
A discrete topological space $X$ is called [*discrete in ${\mathcal C}$*]{} if $X$ is an object of ${\mathcal C}$ and each function $f:X\to Y$ into an object $Y$ of the category ${\mathcal C}$ is a morphism of ${\mathcal C}$. It is clear that any bijection $f:X\to Y$ between discrete objects of the category ${\mathcal C}$ is an isomorphism in ${\mathcal C}$.
From now on we shall assume $({\mathbb T},\eta,\mu)$ is a monad in a topological category ${\mathcal C}$ such that for any discrete objects $X,Y$ in ${\mathcal C}$ the product $X\times Y$ is discrete in ${\mathcal C}$.
Binary operations and their ${\mathbb T}$-extensions
====================================================
By a [*binary operation in the category ${\mathcal C}$*]{} we understand any function $\varphi:X\times Y\to Z$ where $X,Y,Z$ are objects of the category ${\mathcal C}$. For any $a\in X$ and $b\in Y$ the functions $$\varphi_a:Y\to Z,\;\;\varphi_a:y\mapsto\varphi(a,y)$$and $$\varphi^b:X\to Z,\;\;\varphi^b:x\mapsto\varphi(x,b),$$are called the [*left*]{} and [*right*]{} [*shifts*]{}, respectively.
A binary operation $\varphi:X\times Y\to Z$ is called [*right-topological*]{} if for every $y\in Y$ the right shift $\varphi^y:X\to Z$, $\varphi^y:x\mapsto\varphi(x,y)$, is continuous. The [*topological center*]{} of a right-topological binary operation $\varphi:X\times Y\to Z$ is the set $\Lambda_\varphi$ of all elements $x\in X$ such that the left shift $\varphi_x:Y\to Z$ is continuous.
\[Text\] Let $\varphi:X\times Y\to Z$ be a binary operation in the category ${\mathcal C}$. A binary operation $\Phi:TX\times TY\to TZ$ is defined to be a [*${\mathbb T}$-extension*]{} of $\varphi$ if
1. $\Phi(\eta_X(x),\eta_Y(y))=\eta_Z(\varphi(x,y))$ for any $x\in X$ and $y\in Y$;
2. for every $b\in TY$ the right shift $\Phi^b:TX\to TZ$, $\Phi^b:x\mapsto\Phi(x,b)$, is a morphism of the free ${\mathbb T}$-algebras $TY$, $TZ$;
3. for every $x\in X$ the left shift $\Phi_{\eta(x)}:TY\to TZ$, $\Phi_{\eta(x)}:y\mapsto \Phi(\eta(x),y)$, is a morphism of the free ${\mathbb T}$-algebras $TX$, $TZ$.
This definition implies that for any binary operation $\varphi:X\times Y\to Z$ its ${\mathbb T}$-extension $\Phi:TX\times TY\to TZ$ is a right-topological binary operation whose topological center $\Lambda_\Phi$ contains the set $\eta(X)\subset TX$.
\[unique\] Let $\varphi:X\times Y\to Z$ be a binary operation in the category ${\mathcal C}$.
1. The binary operation $\varphi$ has at most one ${\mathbb T}$-extension $\Phi:TX\times TY\to TZ$.
2. If $X,Y$ are discrete in ${\mathcal C}$, then $\varphi$ has a unique ${\mathbb T}$-extension $\Phi:TX\times TY\to TZ$.
1\. Let $\Phi,\Psi:TX\times TY\to TZ$ be two ${\mathbb T}$-extensions of the operation $\varphi$. By the condition (3) of Definition \[Text\], for every $x\in X$ and $a=\eta_X(x)\in TX$, the left shifts $\Phi_a,\Psi_a:TY\to TZ$ are morphisms of the free ${\mathbb T}$-algebras.
By the condition (1) of Definition \[Text\], $$\Phi_a\circ \eta_Y=\eta_Z\circ\varphi_x=\Psi_a\circ\eta_Y.$$ Then Lemma \[free\] implies that $$\Phi_a=\mu\circ T(\Phi_a\circ\eta_X)=\mu\circ T(\eta_Z\circ\varphi_x)=\mu\circ T(\Psi_a\circ\eta_X)=\Psi_a.$$
The equality $\Phi=\Psi$ will follow as soon as we check that $\Phi^b=\Psi^b$ for every $b\in TY$. Since $\Phi^b,\Psi^b:TX\to TZ$ are morphisms of the free ${\mathbb T}$-algebras $TX$ and $TZ$, the equality $\Phi^b=\Psi^b$ follows from the equality $$\Phi^b\circ\eta(x)=\Phi_{\eta(x)}(b)=\Psi_{\eta(x)}(b)=\Psi^b\circ\eta(x),\;\;x\in X$$according to Lemma \[free\].
2\. Now assuming that the spaces $X,Y$ are discrete in ${\mathcal C}$, we show that the binary operation $\varphi:X\times Y\to Z$ has a ${\mathbb T}$-extension. For every $x\in X$ consider the left shift $\varphi_x:Y\to Z$. Since $Y$ is discrete in ${\mathcal C}$, the function $\varphi_x$ is a morphism of the category ${\mathcal C}$. Applying the functor $T$ to this morphism, we get a morphism $T\varphi_x:TY\to TZ$. Now for every $b\in TY$ consider the function $\varphi^b:X\to TZ$, $\varphi^b:x\mapsto T\varphi_x(b)$. Since the object $X$ is discrete, the function $\varphi^b$ is a morphism of the category ${\mathcal C}$. Applying to this morphism the functor $T$, we get a morphism $T\varphi^b:TX\to T^2Z$. Composing this morphism with the multiplication $\mu:T^2Z\to TZ$ of the monad ${\mathbb T}$, we get the function $\Phi^b=\mu\circ T\varphi^b:TZ\to TZ$. Define a binary operation $\Phi:TX\times TY\to TZ$ letting $\Phi(a,b)=\Phi^b(a)$ for $a\in TX$.
\[cl1\]$\Phi(\eta(x),b)=T\varphi_x(b)$ for every $x\in X$ and $b\in TY$.
The commutativity of the diagram $$\xymatrix{
X\ar[d]_{\eta}\ar[r]^{\varphi^b}&TZ\ar[d]_{\eta}\\
TX\ar[r]_{T\varphi^b}\ar[ur]^{\Phi^b\!\!\!}&T^2Z\ar@/_1pc/[u]_{\mu}
}$$implies the desired equality $$\Phi(\eta(x),b)=\mu\circ T\varphi^b(\eta(x))=\varphi^b(x)=T\varphi_x(b).$$
Now we shall prove that $\Phi$ is a ${\mathbb T}$-extension of $\varphi$.
i\) For every $x\in X$ and $y\in Y$ we need to prove the equality $$\Phi(\eta_X(x),\eta_Y(y))=\eta_Z\circ\varphi(x,y).$$ By Claim \[cl1\], $$\Phi(\eta_X(x),\eta_Y(y))=T\varphi_x\circ \eta_Y(y)=\eta_Z\circ\varphi_x(y)=\eta_Z\circ \varphi(x,y).$$ The latter equality follows from the naturality of the transformation $\eta:{\mathrm{Id}}\to T$.
ii\) The definition of $\Phi$ implies that for every $b\in TY$ the right shift $\Phi^b=\mu_Z\circ T\varphi^b$ is a morphism of free ${\mathbb T}$-algebras, being the compositions of two morphisms $T\varphi^b:TX\to T^2Z$ and $\mu_Z:T^2Z\to TZ$ of free ${\mathbb T}$-algebras.
iii\) Claim \[cl1\] guarantees that for every $x\in X$ the left shift $\Phi_{\eta(x)}=T\varphi_x:TY\to TZ$ is a morphism of the free ${\mathbb T}$-algebras.
Let $\varphi:X\times Y\to Z$, $\psi:X'\times Y'\to Z'$ be two binary operations in ${\mathcal C}$, $\Phi:TX\times TY\to TZ$, $\Psi:TX'\times TY'\to TZ'$ be their ${\mathbb T}$-extensions, and $h_X:X\to X'$, $h_Y:Y\to Y'$, $h_Z:Z\to Z'$ be morphisms in ${\mathcal C}$. If $\psi(h_X\times h_Y)=h_Z\circ \phi$, then $T\Psi(Th_X\times Th_Y)=Th_Z\circ \Phi$.
Observe that for any $x\in X$ and $x'=h_X(x)$, the commutativity of the diagrams $$\xymatrix{
Y\ar[r]^{\varphi_x}\ar[d]_{h_Y}&Z\ar[d]^{h_Z}\\
Y'\ar[r]_{\psi_{x'}}&Z'
}\quad\quad\quad
\xymatrix{
TY\ar[r]^{T\varphi_x}\ar[d]_{Th_Y}&TZ\ar[d]^{Th_Z}\\
TY'\ar[r]_{T\psi_{x'}}&TZ'
}$$ imply that $Th_Z\circ T\varphi_x(b)=T\psi_{x'}(b')$ for every $b\in TY$ and $b'=Th_Y(b)\in TY'$.
It follows from Lemma \[cl1\] that $\Phi_{\eta(x)}=T\varphi_x:TY\to TZ$ and $\Psi_{\eta(x')}=T\psi_{x'}:TY'\to TZ'$. Consequently, $$Th_Z\circ \Phi^b(\eta(x))=Th_Z\circ \Phi_{\eta(x)}(b)=Th_Z\circ T\varphi_x(b)=T\psi_{x'}(b')=\Psi_{\eta(x')}(b')=\Psi^{b'}(\eta(x'))$$and hence $$Th_Z\circ\Phi^b\circ\eta=\Psi^{b'}\circ \eta\circ h_X.$$ Applying the functor $T$ to this equality, we get $$T^2h_Z\circ T(\Phi^b\circ\eta)=T(\Psi^{b'}\circ\eta)\circ Th_X.$$ Since $\Phi^b:TX\to TZ$ and $\Psi^{b'}:TX'\to TZ'$ are homomorphisms of the free ${\mathbb T}$-algebras, we can apply Lemma \[free\] and conclude that $\Phi^b=\mu\circ T(\Phi^b\circ\eta)$ and hence $$Th_Z\circ\Phi^b=Th_Z\circ\mu_Z\circ T(\Phi^b\circ\eta)=\mu_{Z'}\circ T^2h_Z\circ T(\Phi^b\circ\eta)=\mu_{Z'}\circ T(\Psi^{b'}\circ\eta)\circ Th_X=\Psi^{b'}\circ Th_X.$$ Then for every $a\in TX$ we get $$Th_Z\circ\Phi(a,b)=Th_Z\circ\Phi^b(a)=\Psi^{b'}\circ Th_X(a)=\Psi(Th_X(a),Th_Y(b)).$$
Binary operations and tensor products
=====================================
In this section we shall discuss the relation of ${\mathbb T}$-extensions to tensor products. The tensor product is a function $\otimes:TX\times TY\to T(X\times Y)$ defined for any objects $X,Y\in{\mathcal C}$ such that $X$ is discrete in ${\mathcal C}$.
For every $x\in X$ consider the embedding $i_x:Y\to X\times Y$, $i_x:y\mapsto (x,y)$. The embedding $i_x$ is a morphism of the category ${\mathcal C}$ because the constant map $c_x:Y\to\{x\}\subset X$ and the identity map ${\mathrm{id}}:Y\to Y$ are morphisms of the category and ${\mathcal C}$ contains products of its objects. Applying the functor $T$ to the morphism $i_x$, we get a morphism $Ti_x:TY\to T(X\times Y)$ of the category ${\mathcal C}$. Next, for every $b\in TY$ consider the function $Ti^b:X\to T(X\times Y)$, $Ti^b:x\mapsto Ti_x(b)$. Since $X$ is discrete in ${\mathcal C}$, the function $Ti^b$ is a morphism of the category ${\mathcal C}$. Applying the functor $T$ to this morphism, we get a morphism $TTi^b:TX\to T^2(X\times Y)$. Composing this morphism with the multiplication $\mu:T^2(X\times Y)\to T(X\times Y)$ of the monad ${\mathbb T}$, we get the morphism $\otimes^b=\mu\circ TTi^b:TX\to T(X\times Y)$. Finally define the tensor product $\otimes:TX\times TY\to T(X\times Y)$ letting $a\otimes b=\otimes^b(a)$ for $a\in TX$.
The following proposition describes some basic properties of the tensor product. For monadic functors in the category $\mathbf{Comp}$ of compact Hausdorff spaces those properties were established in [@TZ 3.4.2].
\[tensor\]
1. The diagram $\xymatrix@1{X\times Y\ar@/^1pc/[rr]^{\eta}\ar[r]_-{\eta\times\eta}&TX\times TY\ar[r]_-\otimes &T(X\times Y)}$ is commutative for any discrete object $X$ and any object $Y$ of ${\mathcal C}$;
2. the tensor product is natural in the sense that for any morphisms $h_X:X\to X'$, $h_Y:Y\to Y'$ of ${\mathcal C}$ with discrete $X,Y$, the following diagram is commutative: $$\xymatrix{
TX\times TY\ar[d]_{Th_X\times Th_Y}\ar[r]^{\otimes}&T(X\times Y)\ar[d]^{T(h_X\times h_Y)}\\
TX'\times TY'\ar[r]^{\otimes}& T(X'\times Y')}$$
3. the tensor product is associative in the sense that for any discrete objects $X,Y,Z$ of ${\mathcal C}$ the diagram $$\xymatrix{TX\times TY\times TZ\ar[d]_{{\mathrm{id}}\times\otimes}\ar[r]^{\otimes\times {\mathrm{id}}}&T(X\times Y)\times TZ\ar[d]^{\otimes}\\
TX\times T(Y\times Z)\ar[r]_{\otimes}&T(X\times Y\times Z)}$$is commutative, which means that $(a\otimes b)\otimes c=a\otimes(b\otimes c)$ for any $a\in TX$, $b\in TY$, $c\in TZ$.
1\. Fix any $y\in Y$ and consider the element $b=\eta_Y(y)\in TY$. The definition of the right shift $\otimes^b$ implies that the following diagram is commutative: $$\xymatrix{
X\ar[d]_{\eta}\ar[r]^-{Ti^b}&T(X\times Y)\\
TX\ar[r]_-{TTi^b}\ar[ru]^{\otimes^b}&T^2(X\times Y)\ar[u]_{\mu}}$$ Consequently, for every $x\in X$ we get $$\eta(x)\otimes \eta(y)=\otimes^b\circ\eta(x)=Ti^b\circ\eta(x)=Ti_x(\eta(y))=\eta(i_x(y))=\eta(x,y).$$ The latter equality follows from the diagram $$\xymatrix{Y\ar[d]_\eta\ar[r]^-{i_x}&X\times Y\ar[d]^{\eta}\\
TY\ar[r]_-{Ti_x}&T(X\times Y)}$$whose commutativity follows from the naturality of the transformation $\eta:{\mathrm{Id}}\to T$.
2\. Let $h_X:X\to X'$ and $h_Y:Y\to Y'$ be any functions between discrete objects of the category ${\mathcal C}$. Let $Z=X\times Y$, $Z'=X'\times Y'$ and $h_Z=h_X\times h_Y:Z\to Z'$. Given any point $b\in TY$, consider the element $b'=Th_Y(b)\in TY'$. The statement (2) will follow as soon as we check that $Th_Z\circ\otimes^b=\otimes^{b'}\circ Th_X$. By Lemma \[free\], this equality will follow as soon as we check that $Th_Z\circ\otimes^b\circ\eta_X=\otimes^{b'}\circ Th_X\circ \eta_X=\otimes^{b'}\circ \eta_{X'}\circ h_X$. The last equality follows from the naturality of the transformation $\eta:{\mathrm{Id}}\to T$. As we know from the proof of the preceding item, $\otimes^{b'}\circ \eta_{X'}(x')=Ti_{x'}(b')$ for any $x'\in X'$. For every $x\in X$ and $x'=h_X(x)$ we can apply the functor $T$ to the commutative diagram $$\xymatrix{
Y\ar[d]_{h_Y}\ar[r]^{i_x}&Z\ar[d]^{h_Z}\\
Y'\ar[r]_{i_{x'}}&Z'}$$and obtain the equality $Th_Z\circ Ti_x=Ti_{x'}\circ Th_Y$ which implies the desired equality: $$\otimes^{b'}\circ\eta_{X'}\circ h_X(x)=\otimes^{b'}\circ\eta_{X'}(x')=Ti_{x'}(b')=Th_Z\circ Ti_x(b)=Th_Z\circ\otimes^b\circ \eta(x).$$
3\. The proof of the associativity of the tensor product can be obtained by literal rewriting the proof of Proposition 3.4.2(4) of [@TZ].
\[ext-tensor\] Let $\varphi:X\times Y\to Z$ be a binary operation in the category ${\mathcal C}$ and $\Phi:TX\times TY\to TZ$ be its ${\mathbb T}$-extension. If $X$ is a discrete object in ${\mathcal C}$, then $\Phi(a,b)=T\varphi(a\otimes b)$ for any elements $a\in TX$ and $b\in TY$.
Our assumptions on the category ${\mathcal C}$ guarantee that the product $X\times Y$ is a discrete object of ${\mathcal C}$ and hence $\varphi:X\times Y\to Z$ is a morphism of the category ${\mathcal C}$. So, it is legal to consider the morphism $T\varphi:T(X\times Y)\to TZ$. We claim that the binary operation $$\Psi:TX\times TY\to TZ,\;\;\Psi:(a,b)=T\varphi(a\otimes b),$$ is a ${\mathbb T}$-extension of $\varphi$.
1\. The first item of Definition \[Text\] follows Proposition \[tensor\](1) and the naturality of the transformation $\eta:{\mathrm{Id}}\to T$: $$\Psi(\eta_X(x),\eta_Y(y))=T\varphi(\eta_X(x)\otimes \eta_Y(y))=T\varphi\circ\eta_{X\times Y}(x,y)=\eta_Z\circ\varphi(x,y).$$
2\. For every $b\in TY$ the morphism $$\Psi^b=T\varphi\circ\otimes^b=T\varphi\circ \mu\circ TTi^b$$ is a morphism of the free ${\mathbb T}$-algebras $TX$ and $TZ$.
3\. For every $x\in X$ we see that $$\Psi_{\eta(x)}(b)=T\varphi(\otimes^b(\eta(x)))=T\varphi\circ\mu\circ TTi^b\circ\eta(x)=T\varphi\circ\mu\circ \eta\circ Ti^b(x)=T\varphi\circ Ti^b(x)$$ is a morphism of the free ${\mathbb T}$-algebras $TY$ and $TZ$.
Thus $\Psi$ is a ${\mathbb T}$-extension of the binary operation $\varphi$. By the Uniqueness Theorem \[unique\](1), $\Psi$ coincides with $\Phi$ and hence $\Phi(a,b)=\Psi(a,b)=T\varphi(a\otimes b)$.
The topological center of ${\mathbb T}$-extended operation
==========================================================
Definition \[Text\] guarantees that for a binary operation $\varphi:X\times Y\to Z$ in ${\mathcal C}$ any ${\mathbb T}$-extension $\Phi:TX\times TY\to TZ$ of $\varphi$ is a right-topological operation whose topological center $\Lambda_\varphi$ contains the subset $\eta_X(X)$. In this section we shall find conditions on the functor $T$ and the space $X$ guaranteeing that the topological center $\Lambda_\Phi$ is dense in $TX$.
We shall say that the functor $T$ is [*continuous*]{} if for each compact Hausdorff space $K$ that belongs to the category ${\mathcal C}$ and any object $Z$ of ${\mathcal C}$ the map $T:{\mathrm{Mor}}(K,Z)\to{\mathrm{Mor}}(TK,TZ)$, $T:f\mapsto Tf$, is continuous with respect to the compact-open topology on the spaces of morphisms (which are continuous maps).
\[cont\] Let $\varphi:X\times Y\to Z$ be a binary operation in ${\mathcal C}$ and $\Phi:X\times Y\to Z$ be its ${\mathbb T}$-extension. If the object $X$ is finite and discrete in ${\mathcal C}$, $TX$ is locally compact and Hausdorff, and the functor $T$ is continuous, then the operation $\Phi$ is continuous.
Since the space $X$ is discrete, the condition (2) of Definition \[Text\] implies that the map $\Phi_\eta:X\times TY\to TZ$, $\Phi_\eta:(x,b)\mapsto \Phi(\eta(x),b)$ is continuous. Since $X$ is finite, the induced map $$\Phi_\eta^{(\cdot)}:TY\to {\mathrm{Mor}}(X,TZ),\;\;\Phi_\eta^{(\cdot)}:b\mapsto\Phi_\eta^b\mbox{ \ where \ } \Phi_\eta^b:x\mapsto \Phi(\eta(x),b),$$ is continuous. By the continuity of the functor $T$, the map $T:{\mathrm{Mor}}(X,TZ)\to {\mathrm{Mor}}(TX,T^2Z)$, $T:f\mapsto Tf$, is continuous and so is the composition $T\circ\Phi_\eta^{(\cdot)}:TY\to {\mathrm{Mor}}(TX,T^2Z)$. Since $TX$ is locally compact and Hausdorff, we can apply [@En 3.4.8] and conclude that the map $$T\Phi^{(\cdot)}_\eta:TX\times TY\to T^2Z,\;\;T\Phi_{\eta}^{(\cdot)}:(a,b)\mapsto T\Phi^b_\eta(a)$$ is continuous and so is the composition $\Psi=\mu\circ T\Phi_\eta^{(\cdot)}:TX\times TY\to TZ$. Using the Uniqueness Theorem \[unique\](1), we can prove that $\Psi=\Phi$ and hence the binary operation $\Phi$ is continuous.
Let $X$ be an object of the category ${\mathcal C}$. We say that an element $a\in FX$ has [*discrete (finite) support*]{} if there is a morphism $f:D\to X$ from a discrete (and finite) object $D$ of the category ${\mathcal C}$ such that $a\in Ff(FD)$. By $T_d X$ (resp. $T_f X$) we denote the set of all elements $a\in TX$ that have discrete (finite) support. It is clear that $T_f X\subset T_d X\subset TX$.
\[top-cent\] Let $\varphi:X\times Y\to Z$ be a binary operation and $\Phi:TX\times TY\to TZ$ be a ${\mathbb T}$-extension of $\varphi$. If the functor $T$ is continuous, and for every finite discrete object $D$ of ${\mathcal C}$ the space $TD$ is locally compact and Hausdorff, then the topological center $\Lambda_\Phi$ of the binary operation $\Phi$ contains the subspace $T_f X$ of $TX$. If $T_f X$ is dense in $TX$, then the topological center $\Lambda_\Phi$ of $\Phi$ is dense in $TX$.
We need to prove that for every $a\in T_f X$ the left shift $\Phi_a:TY\to TZ$, $\Phi_a:b\mapsto \Phi(a,b)$, is continuous. Since $a\in T_f X$, there is a finite discrete object $D$ of the category ${\mathcal C}$ and a morphism $f:D\to X$ such that $a\in Ff(FD)$. Fix an element $d\in FD$ such that $a=Ff(d)$.
Consider the binary operations $$\psi:D\times Y\to Z,\;\psi:(x,y)\mapsto \phi(f(x),y),$$ and $$\Psi:TD\times TY\to TZ,\;\Psi:(a,b)\mapsto \Phi(Ff(a),b).$$ It can be shown that $\Psi$ is a ${\mathbb T}$-extension of $\psi$.
By Theorem \[cont\], the binary operation $\Psi$ is continuous. Consequently, the left shift $\Psi_d:TY\to TZ$, $\Psi_d:b\mapsto\Psi(d,b)$, is continuous. Since $\Psi_d=\Phi_a$, the left shift $\Phi_a$ is continuous too and hence $a\in \Lambda_\Phi$.
The associativity of ${\mathbb T}$-extensions
=============================================
In this section we investigate the associativity of the ${\mathbb T}$-extensions. We recall that a binary operation $\varphi:X\times X\to X$ is [*associative*]{} if $\varphi(\varphi(x,y),z)=\varphi(x,\varphi(y,z))$ for any $x,y,z\in X$. In this case we say that $X$ is a [*semigroup*]{}.
A subset $A$ of a set $X$ endowed with a binary operation $\varphi:X\times X\to X$ is called a [*subsemigroup*]{} of $X$ if $\varphi(A\times A)\subset A$ and $\varphi(\varphi(x,y),z)=\varphi(x,\varphi(y,z))$ for all $x,y,z\in A$.
\[asl\] Let $\varphi:X\times X\to X$ be an associative operation in ${\mathcal C}$ and $\Phi:TX\times TX\to TX$ be its ${\mathbb T}$-extension.
1. for any morphisms $f_A:A\to X$, $f_B:B\to X$ from discrete objects $A,B$ in ${\mathcal C}$, the map $\varphi_{AB}=\varphi(f_A\times f_B):A\times B\to X$ is a morphism of ${\mathcal C}$ such that $\Phi(Tf_A(a),Tf_B(b))=T\varphi_{AB}(a\otimes b)$ for all $a\in TA$ and $b\in TB$;
2. $\Phi(T_d X\times T_d X)\subset T_d X$ and $\Phi(T_f X\times T_f X)\subset T_f X$;
3. $\Phi((a,b),c)=\Phi(a,\Phi(b,c))$ for any $a,b,c\in T_d X$.
1\. Let $f_A:A\to X$, $f_B:B\to X$ be morphisms from discrete objects $A,B$ of ${\mathcal C}$ and $\varphi_{AB}=\varphi(f_A\times f_B):A\times B\to X$. By our assumption on the category ${\mathcal C}$, the product $A\times B$ is a discrete object in ${\mathcal C}$ and hence $\phi_{AB}$ is a morphism in ${\mathcal C}$. Consider the binary operation $\Phi_{AB}:TA\times TB\to TX$ defined by $\Phi_{AB}(a,b)=\Phi(Tf_A(a),Tf_B(b))$. The following diagram implies that $\Phi_{AB}$ is a ${\mathbb T}$-extension of $\varphi_{AB}$: $$\xymatrix{
TX\times TX\ar[rrr]^{\Phi}& & &TX\\
& X\times X\ar[ul]_{\eta\times\eta}\ar[r]^-{\varphi}&X\ar[ur]^{\eta}&\\
&A\times B\ar[dl]^{\eta\times\eta}\ar[u]^{f_A\times f_B}\ar[r]^-{\varphi_{AB}}&X\ar@{<->}[u]_{{\mathrm{id}}}\ar[rd]_{\eta}&\\
TA\times TB\ar[uuu]^{Tf_A\times Tf_B}\ar[rrr]_{\Phi_{AB}}&&&TX\ar@{<->}[uuu]_{{\mathrm{id}}}
}$$ By Theorem \[ext-tensor\], $$\Phi(Tf_A(a),Tf_B(b))=\Phi_{AB}(a,b)=T\varphi_{AB}(a\otimes b)$$for all $a\in TA$ and $b\in TB$.
2\. Given elements $a,b\in T_d X$, we need to show that the element $\Phi(a,b)\in TX$ has discrete support. Find discrete objects $A,B$ in ${\mathcal C}$ and morphisms $f_A:A\to X$, $f_B:B\to X$ such that $a\in Ff_A(FA)$ and $b\in f_B(FB)$. Fix elements $\tilde a\in FA$, $\tilde b\in FB$ such that $a=Ff_A(\tilde a)$ and $b=Ff_B(\tilde b)$. Our assumption on the category ${\mathcal C}$ guarantees that $A\times B$ is a discrete object in ${\mathcal C}$.
Consider the binary operations $\psi:A\times B\to X$ and $\Psi:FA\times FB\to FZ$ defined by the formulae $\psi=\phi\circ(f_A\times f_B)$ and $\Psi=\Phi\circ (Tf_A\times Tf_B)$. Let $\tilde c=\tilde a\otimes\tilde b\in T(A\times B)$. By the first statement, $\Phi(a,b)=T\psi(\tilde a\otimes \tilde b)=T\psi(\tilde c)\in T\psi(A\times B)$ witnessing that the element $\Phi(a,b)$ has discrete support and hence belongs to $T_d X$.
By analogy, we can prove that $\Phi(T_f X\times T_f X)\subset T_f X$.
3\. Given any points $a,b,c\in T_d X$, we need to check the equality $$\Phi(\Phi(a,b),c)=\Phi(a,\Phi(b,c)).$$
Find discrete objects $A,B,C$ in ${\mathcal C}$ and morphisms $f_A:A\to X$, $f_B:B\to X$, $f_C:C\to X$ such that $a\in Tf_A(TA)$, $b\in Tf_B(TB)$, and $c\in Tf_C(TC)$. Fix elements $\tilde a\in TA$, $\tilde b\in TB$, and $\tilde c\in TC$ such that $a=Tf_A(\tilde a)$, $b=Tf_B(\tilde b)$, and $c=Tf_C(\tilde c)$.
Consider the morphisms $\varphi_{AB}=\varphi(f_A\times f_B):A\times B\to X$, $\varphi_{BC}=\varphi(f_B\times f_C):B\times C\to X$ and $\varphi_{ABC}=\varphi(\varphi_{AB}\times f_C)=\varphi(f_A\times\varphi_{BC}):A\times B\times C\to X$. Consider the following diagram: $$\xymatrix{
TX\times TX\times TX\ar[ddd]_{{\mathrm{id}}\times\Phi}\ar[rrr]^{\Phi\times{\mathrm{id}}}& & & TX\times TX\ar[ddd]^{\Phi}\\
& TA\times TB\times TC\ar[lu]_{Tf_A\times Tf_B\times Tf_C}\ar[d]_{{\mathrm{id}}\times\otimes}\ar[r]^{\otimes\times {\mathrm{id}}}&T(A\times B)\times TC\ar[d]^{\otimes}\ar[ru]^{T\varphi_{AB}\times Tf_C}&\\
&TA\times T(B\times C)\ar[ld]^{Tf_A\times T\varphi_{BC}}\ar[r]_\otimes &T(A\times B\times C)\ar[rd]_{T\varphi_{ABC}}&\\
TX\times TX\ar[rrr]_{\Phi}& & & TX
}$$ In this diagram the central square is commutative because of the associativity of the tensor product $\otimes$. By the item (1) all four margin squares also are commutative. Now we see that $$\begin{aligned}
&\Phi(\Phi(a,b),c))=\Phi(\Phi(Tf_A(\tilde a),Tf_B(\tilde b)),Tf_C(\tilde c))=\\ &\Phi(T\varphi_{AB}(\tilde a\otimes \tilde b),Tf_C(\tilde c))=T\varphi_{ABC}((\tilde a\otimes \tilde b)\otimes \tilde c)=T\varphi_{ABC}(\tilde a\otimes(\tilde b\otimes \tilde c))=\\
&\Phi(Tf_A(\tilde a),T\varphi_{BC}(\tilde a\otimes \tilde b))=\Phi(Tf_A(\tilde a),\Phi(Tf_B(\tilde b),Tf_C(\tilde c)))=\Phi(a,\Phi(b,c)).
\end{aligned}$$
Combining Lemma \[asl\] with Theorem \[top-cent\], we get the main result of this paper:
\[ast\] Assume that the monadic functor $T$ is continuous and for each finite discrete space $F$ in ${\mathcal C}$ the space $TF$ is Hausdorff and locally compact. Let $\varphi:X\times X\to X$ be an associative binary operation in ${\mathcal C}$ and $\Phi:X\times X\to X$ be its ${\mathbb T}$-extension. If the set $T_f X$ of elements with finite support is dense in $TX$, then the operation $\Phi$ is associative.
By Theorem \[top-cent\], the set $T_f X$ lies in the topological center $\Lambda_\Phi$ of the operation $\Phi$ and by Lemma \[asl\], $T_f X$ is a subsemigroup of $(TX,\Phi)$. Now the associativity of $\Phi$ follows from the following general fact.
\[asp\] A right topological operation $\cdot:X\times X\to X$ on a Hausdorff space $X$ is associative if its topological center contains a dense subsemigroup $S$ of $X$.
Assume conversely that $(xy)z\ne x(yz)$ for some points $x,y,z\in X$. Since $X$ is Hausdorff, the points $(xy)z$ and $x(yz)$ have disjoint open neighborhoods $O((xy)z)$ and $O(x(yz))$ in $X$. Since the right shifts in $X$ are continuous, there are open neighborhoods $O(xy)$ and $O(x)$ of the points $xy$ and $x$ such that $O(xy)\cdot z\subset O((xy)z)$ and $O(x)\cdot(yz)\subset O(x(yz))$. We can assume that $O(x)$ is so small that $O(x)\cdot y\subset O(xy)$. Take any point $a\in O(x)\cap S$. It follows that $a(yz)\in O(x(yz))$ and $ay\in O(xy)$. Since the left shift $l_a:\beta S\to\beta S$, $l_a:y\mapsto ay$, is continuous, the points $yz$ and $y$ have open neighborhoods $O(yz)$ and $O(y)$ such that $a\cdot O(yz)\subset O(x(yz))$ and $a\cdot O(y)\subset O(xy)$. We can assume that the neighborhood $O(y)$ is so small that $O(y)\cdot z\subset O(yz)$. Choose a point $b\in O(y)\cap S$ and observe that $bz\in O(y)\cdot z\subset O(yz)$, $ab\in a\cdot O(y)\subset O(xy)$, and thus $(ab)z\in O(xy)\cdot z\subset O((xy)z)$. The continuity of the left shifts $l_b$ and $l_{ab}$ allows us to find an open neighborhood $O(z)\subset\beta S$ of $z$ such that $b\cdot O(z)\subset O(yz)$ and $ab\cdot O(z)\subset O((xy)z)$. Finally take any point $c\in S\cap O(z)$. Then $(ab)c\in ab\cdot O(z)\subset O((xy)z)$ and $a(bc)\subset a\cdot O(yz)\subset O(x(yz))$ belong to disjoint sets, which is not possible as $(ab)c=a(bc)$.
${\mathbb T}$-extension for some concrete monadic functors
==========================================================
In this section we consider some examples of monadic functors in topological categories. Let ${\mathbf{Tych}}$ denote the category of Tychonov spaces and their continuous maps and ${\mathbf{Comp}}$ be the full subcategory of the category ${\mathbf{Tych}}$, consisting of compact Hausdorff spaces.
Discrete objects in the category ${\mathbf{Tych}}$ are discrete topological spaces while discrete objects in the category ${\mathbf{Comp}}$ are finite discrete spaces.
Consider the functor $\beta:{\mathbf{Tych}}\to {\mathbf{Comp}}$ assigning to each Tychonov space $X$ its Stone-Čech compactification and to a continuous map $f:X\to Y$ between Tychonov spaces its continuous extension $\beta f:\beta X\to \beta Y$. The functor $\beta$ can be completed to a monad ${\mathbb T}_\beta=(\beta,\eta,\mu)$ where $\eta:X\to\beta X$ is the canonical embedding and $\mu:\beta(\beta X)\to\beta X$ is the identity map. A pair $(X,\xi)$ is a ${\mathbb T}_\beta$-algebra if and only if $X$ is a compact space and $\xi:\beta X\to X$ is the identity map.
Combining Theorems \[unique\], \[ast\] we get the following well-known
Each binary right-topological operation $\varphi:X\times Y\to Z$ in ${\mathbf{Tych}}$ with discrete $X$ can be extended to a right-topological operation $\Phi:\beta X\times\beta Y\to\beta Z$ containing $X$ in its topological center $\Lambda_\Phi$. If $X=Y=Z$ and the operation $\varphi$ is associative, then so is the operation $\Phi$.
Now let ${\mathbb T}=(T,\eta,\mu)$ be a monad in the category ${\mathbf{Comp}}$. Taking the composition of the functors $\beta:{\mathbf{Tych}}\to{\mathbf{Comp}}$ and $T:{\mathbf{Comp}}\to{\mathbf{Comp}}$, we obtain a monadic functor $T\beta:{\mathbf{Tych}}\to{\mathbf{Comp}}$.
Each binary right-topological operation $\varphi:X\times Y\to Z$ in the category ${\mathbf{Tych}}$ with discrete $X$ can be extended to a right-topological operation $\Phi:T\beta X\times T\beta Y\to T\beta Z$ that contain the set $\eta(X)\subset T\beta X$ in its topological center $\Lambda_\Phi$. If the functor $T$ is continuous, then the set $T_f X$ of elements $a\in T\beta X$ with finite support is dense in $T\beta X$ and lies in the topological center $\Lambda_\Phi$ of the operation $\Phi$. Moreover, if $X=Y=Z$ and the operation $\varphi$ is associative, the so is the operation $\Phi$.
By Theorem \[unique\], the binary operation $\varphi$ has a unique ${\mathbb T}$-extension $\Phi:TX\times TY\to TZ$. By Definition \[Text\], the set $\eta(X)\subset T\beta X$ lies in the topological center $\Lambda_\varphi$ of $\varphi$.
Now assume that the functor $T$ is continuous. First we show that the set $T_f X$ is dense in $T\beta X$. Fix any point $a\in F\beta X$ and an open neighborhood $U\subset T\beta X$ of $a$. Then $[a,U]=\{f\in {\mathrm{Mor}}(F\beta X,F\beta X):f(a)\in U\}$ is an open neighborhood of the identity map ${\mathrm{id}}:F\beta X\to F\beta X$ in the function space ${\mathrm{Mor}}(F\beta X,F\beta X)$ endowed with the compact-open topology. The continuity of the functor $T$ yields a neighborhood ${\mathcal U}({\mathrm{id}}_{\beta X})$ of the identity map ${\mathrm{id}}_{\beta X}\in{\mathrm{Mor}}(\beta X,\beta X)$ such that $Tf\in[a,U]$ for any $f\in{\mathcal U}({\mathrm{id}}_{\beta X})$. It follows from the definition of the compact-open topology, that there is an open cover ${\mathcal U}$ of $\beta X$ such that a map $f:\beta X\to\beta X$ belongs to ${\mathcal U}({\mathrm{id}}_{\beta X})$ if $f$ is ${\mathcal U}$-near to ${\mathrm{id}}_{\beta X}$ in the sense that for every $x\in\beta X$ there is a set $U\in{\mathcal U}$ with $\{x,f(x)\}\subset U$. Since $\beta X$ is compact, we can assume that the cover ${\mathcal U}$ is finite. Since $X$ is discrete, the space $\beta X$ has covering dimension zero [@En 7.1.17]. So, we can assume that the finite cover ${\mathcal U}$ is disjoint. For every $U\in{\mathcal U}$ choose an element $x_U\in U\cap X$. Those elements compose a finite discrete subspace $A=\{x_U:U\in {\mathcal U}\}$ of $X$. let $i:A\to X$ be the identity embedding and $f:X\to A$ be the map defined by $f^{-1}(x_U)=U$ for $U\in{\mathcal U}$. It follows that $i\circ f\in{\mathcal U}({\mathrm{id}}_{\beta X})$ and thus $T(i\circ f)\in [a,U]$ and $Ti\circ Tf(a)\in U$. Now we see that $b=Tf(a)\in TA$ and $c=Ti(b)\in T_f X\cap U$, so $T_f X$ is dense in $\beta X$.
By Theorem \[top-cent\], the set $T_f X$ lies in the topological center $\Lambda_\Phi$ of $\Phi$.
Now assume that the operation $\varphi$ is associative. By Lemma \[asl\], $T_f X$ is a subsemigroup of $(X,\Phi)$. Since $T_f X$ is dense and lies in the topological center $\Lambda_\Phi$, we may derive the associativity of $\Phi$ from Proposition \[asp\].
Given a discrete semigroup $X$ investigate the algebraic and topological properties of the compact right-topological semigroup $T\beta X$ for some concrete continuous monadic functors $T:{\mathbf{Comp}}\to{\mathbf{Comp}}$.
This problem was addressed in [@G1], [@G2] for the monadic functor $G$ of inclusion hyperspaces, in [@BG1]–[@BG4] for the functor of superextension $\lambda$, in [@BCHR], [@Heyer], [@Par] for the functor $P$ of probability measures and in [@BHr], [@Ber], [@BL], [@Trn] for the hyperspace functor $\exp$.
In [@Zar] it was shown that for each continuous monadic functor $T:{\mathbf{Comp}}\to{\mathbf{Comp}}$ any continuous (associative) operation $\varphi:X\times Y\to Z$ in ${\mathbf{Comp}}$ extends to a continuous (associative) operation $\Phi:TX\times TY\to TZ$.
For which monads ${\mathbb T}=(T,\eta,\mu)$ in the category ${\mathbf{Comp}}$ each right-topological (associative) binary operation $\varphi:X\times Y\to Z$ in ${\mathbf{Comp}}$ extends to a right-topological (associative) binary operation $\Phi:TX\times TY\to TZ$? Are all such monads power monads?
[9]{}
T.Banakh, M.Cencelj, O.Hryniv, D.Repovs, [*Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups*]{} // preprint (http://arxiv.org/abs/0811.1026).
T. Banakh, V. Gavrylkiv, O. Nykyforchyn, [*Algebra in superextensions of groups, I: zeros and commutativity*]{} // Algebra Discrete Math. no.3 (2008), 1-29.
T. Banakh, V. Gavrylkiv, [*Algebra in superextension of groups, II: cancelativity and centers*]{} // Algebra Discrete Math. no.4 (2008) 1-14.
T. Banakh, V. Gavrylkiv, [*Algebra in the superextensions of groups, III: minimal left ideals*]{} // Mat. Stud. 31:2 (2009) 142-148.
T. Banakh, V. Gavrylkiv, [*Algebra in the superextensions of groups, IV: representation theory*]{} // preprint (http://arxiv.org/abs/0811.0796).
T. Banakh, O. Hryniv, [*Embedding topological semigroups into the hyperspaces over topological groups*]{} // Acta Univ. Carolinae, Math. et Phys.[**48**]{}:2 (2007), 3–18.
S.G. Bershadskii, [*Imbeddability of semigroups in a global supersemigroup over a group*]{} // in: Semigroup varieties and semigroups of endomorphisms, Leningrad. Gos. Ped. Inst., Leningrad, 1979, 47–49.
R.G. Bilyeu, A. Lau, [*Representations into the hyperspace of a compact group*]{} // Semigroup Forum [**13**]{} (1977), 267–270.
R.Engelking, [*General Topology*]{}, PWN, Warsaw, 1977.
V.Gavrylkiv, [*The spaces of inclusion hyperspaces over noncompact spaces*]{} // Mat. Stud. [**28:1**]{} (2007), 92–110.
V. Gavrylkiv, [*Right-topological semigroup operations on inclusion hyperspaces*]{} // Mat. Stud. [**29**]{}:1 (2008), 18–34.
H. Heyer, [*Probability Measures on Locally Compact Groups*]{}, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 94. Springer-Verlag, Berlin-New York, 1977.
N.Hindman, D.Strauss, [Algebra in the Stone-Čech compactification]{}, de Gruyter, Berlin, New York, 1998.
S. MacLane, Categories for working mathematicican, Springer, 1971.
K.R. Parthasarathy, [*Probability Measures on Metric Spaces*]{}, AMS Bookstore, 2005.
I. Protasov, Combinatorics of Numbers, VNTL, Lviv, 1997.
A. Teleiko, M. Zarichnyi, Categorical Topology of Compact Hausdofff Spaces, VNTL, Lviv, 1999.
V. Trnkova, [*On a representation of commutative semigroups*]{} // Semigroup Forum [**10:3**]{} (1975), 203–214.
M. Zarichnyi, A. Teleiko, [*Semigroups and monads*]{} // in: Algebra and Topology, Lviv Univ. Press, 1996, P.84–93 (in Ukrainian).
|
---
address: |
DESY - Notkestra[ß]{}e 85 D-22607 Hamburg Germany\
On behalf of the CDF and DO Collaborations.
author:
- 'Stefano Camarda[^1]'
title: 'W/Z+Jets and W/Z+HF Production at the Tevatron'
---
Introduction
============
The study of the production of electroweak bosons in association with jets of hadrons constitutes a fundamental item in the high-$\rm p_T$ physics program at the Tevatron. Vector bosons plus jets final states are a major background to many interesting physics processes like single and pair top quarks production, Higgs, and super-symmetry. Precise measurements of W/Z + jets production provide a stringent test of perturbative QCD predictions [@QCD; @MCFM; @Blackhat; @Loopsim; @QCDEW] at high $Q^2$, and offer the possibility to validate Monte Carlo simulation tools [@Alpgen; @Powheg; @Pythia; @Herwig; @Sherpa; @HEJ]. The latest vector boson plus jets results at the Tevatron are reviewed and discussed.
W/Z + jets measurements
=======================
The CDF experiment recently measured Z + jets production cross sections with the full Tevatron run II dataset, corresponding to $9.64~\textrm{fb}^{-1}$ of integrated luminosity [@Camarda:2012yha]. Differential cross sections as a function of several variables have been measured, including jet , jet rapidity and jet multiplicity, angular variables like di-jet $\Delta \phi$ and $\Delta y$, and $H_T^{\textrm{jet}} = \sum p_T^{\textrm{jet}}$. Events are required to have two electrons or muons with a reconstructed invariant mass in the range $66 \leqslant M_{Z}
\leqslant 116 \rm ~GeV/c^{2}$ around the Z boson mass. Jets are clustered with the Midpoint algorithm [@midpoint] in a cone radius of $0.7$, and are required to have $p_T \geqslant
30~ \rm GeV/c$ and $|y| \leqslant 2.1$. The background estimation is performed both with data-driven and Monte Carlo techniques, the QCD and W+jet backgrounds are estimated from data, other backgrounds contributions like , diboson and $Z/\gamma^* \rightarrow \tau^+ \tau^-$ are estimated from Monte Carlo simulation. The cross sections are unfolded back to the particle level accounting for acceptance and smearing effects employing <span style="font-variant:small-caps;">alpgen+pythia</span> Monte Carlo. The measured cross sections are compared to predictions from the Monte Carlo event generators <span style="font-variant:small-caps;">alpgen+pythia</span>[@Alpgen], <span style="font-variant:small-caps;">powheg+pythia</span>[@Powheg], and to the fixed order perturbative QCD predictions <span style="font-variant:small-caps;">mcfm</span>[@MCFM], <span style="font-variant:small-caps;">blackhat+sherpa</span>[@Blackhat], <span style="font-variant:small-caps;">loopsim+mcfm</span>[@Loopsim] and to a fixed order prediction including NLO EW corrections[@QCDEW]. Fixed order predictions include parton-to-particle correction factors that account for the non-perturbative underlying event and fragmentation effects, estimated with <span style="font-variant:small-caps;">alpgen+pythia</span> Monte Carlo simulation. Figure \[fig:zjets\] shows the measured cross section as a function of $H_T^{\textrm{jet}}$ in $\rm+ \geqslant 1~jet$ events.
![Measured differential cross sections as a function of $H_T^{\textrm{jet}}$ in $\rm+ \geqslant 1~jet$ events. Data (black dots) are compared to <span style="font-variant:small-caps;">loopsim</span> prediction (open circles). The shaded bands show the total systematic uncertainty, except for the 5.8$\%$ luminosity uncertainty, the blue area represents simultaneous variation of renormalization and factorization scales.[]{data-label="fig:zjets"}](CDF_htjet){width="90.00000%"}
A new measurement of $W \rightarrow e \nu$ + jets production cross section has been performed with the D0 experiment with $3.7~\rm fb^{-1}$ of integrated luminosity [@D0Wjets], including a comprehensive study of several kinematics variables. Events are selected with a reconstructed electron of $p_T
\geqslant 15$ GeV/c and $|\eta| \leqslant 1.1$, the transverse mass of the W, reconstructed with the electron and $\MET$, is required to be $M_{T}^{W} \geqslant 40~ \rm GeV/c^2$, jets are reconstructed with the Midpoint algorithm in a radius $R=0.5$ and required to have $p_T \geqslant 20$ GeV/c and $|y| \leqslant 3.2$. Data are compared to several Monte Carlo generators, <span style="font-variant:small-caps;">alpgen+pythia</span>, <span style="font-variant:small-caps;">alpgen+herwig</span>[@Alpgen], <span style="font-variant:small-caps;">pythia</span>[@Pythia], <span style="font-variant:small-caps;">herwig</span>[@Herwig] and <span style="font-variant:small-caps;">sherpa</span>[@Sherpa], to perturbative NLO QCD predictions from <span style="font-variant:small-caps;">blackhat+sherpa</span>[@Blackhat], and to the all order resummed prediction <span style="font-variant:small-caps;">hej</span>[@HEJ]. Figure \[fig:D0wjets\] shows the average number of jets and the probability of third jet emission as a function of the $\Delta y$ between the most rapidity-separated jets, in events with $W \rightarrow e \nu \rm+ \geqslant 2~jets$.
![Measured average number of jets (left) and probability of third jet emission (right) as a function of the $\Delta y$ between the most rapidity-separated jets, in events with $W \rightarrow e \nu \rm+ \geqslant 2~jets$. Data (open black dots) are compared to several Monte Carlo generators and predictions. The lower pane shows the theory/data ratio.[]{data-label="fig:D0wjets"}](D0_Wnjdy "fig:"){width="45.00000%"} ![Measured average number of jets (left) and probability of third jet emission (right) as a function of the $\Delta y$ between the most rapidity-separated jets, in events with $W \rightarrow e \nu \rm+ \geqslant 2~jets$. Data (open black dots) are compared to several Monte Carlo generators and predictions. The lower pane shows the theory/data ratio.[]{data-label="fig:D0wjets"}](D0_Wpdy "fig:"){width="45.00000%"}
W/Z + heavy flavor jets production
==================================
The measurement of vector boson production with associated heavy flavor jets provides an important test of perturbative QCD predictions, and can be used to improve the determination of PDF. Understanding these processes is also critical for the measurement of Higgs boson production in association with a W or Z and in the search for SUSY.
The W + charm production cross section has been measured by CDF with $4.3~\rm fb^{-1}$ of integrated luminosity[@WcharmCDF]. Charm jets are identified with an algorithm which identifies soft leptons coming from the semileptonic decay of the charm. The measurement exploits the charge correlation between the soft lepton and the lepton coming from the leptonic decay of the W to reduce the background contamination. The measured cross section of $\rm 13.6 \pm 2.2(stat) ^{+2.3}_{-1.9} (syst) \pm 1.1 (lum)$ pb is in good agreement with the NLO prediction of $11.4 \pm 1.3$ pb from MCFM.
The W + b-jet cross section has been measured by D0 with $6~ \rm fb^{-1}$ of integrated luminosity[@D0Wbjet]. $W \rightarrow e \nu$ and $W \rightarrow \mu \nu$ decay channels are combined, and a multivariate technique is used to identify b-jets. The measured cross section of $1.05 \pm 0.12$ pb is in good agreement with the perturbative QCD NLO prediction from MCFM of $1.34 ^{+0.41}_{-0.34}$ pb, and consistent with predictions from the Monte Carlo generators <span style="font-variant:small-caps;">sherpa</span> ($1.08$ pb) and <span style="font-variant:small-caps;">madgraph</span> ($1.44$ pb).
The CDF and D0 collaborations recently measured differential cross sections of Z + b-jet production with the full Tevatron run II dataset[@CDFZbjet; @D0Zbjet], with integrated luminosities corresponding to $9.7~\rm fb^{-1}$ and $9.13~\rm fb^{-1}$ respectively.
![Measured Z + b-jet differential cross sections as a function of b-jet $p_T$ with the CDF (left) and D0 (right) detectors. Data (black dots) are compared to MCFM NLO prediction and Monte Carlo generators.[]{data-label="fig:Zbjet"}](CDF_Zbptjet "fig:"){width="40.00000%"} ![Measured Z + b-jet differential cross sections as a function of b-jet $p_T$ with the CDF (left) and D0 (right) detectors. Data (black dots) are compared to MCFM NLO prediction and Monte Carlo generators.[]{data-label="fig:Zbjet"}](D0_Zbptjet "fig:"){width="35.00000%"}
Figure \[fig:Zbjet\] shows the measured cross sections unfolded to particle level and compared to NLO perturbative QCD predictions from MCFM and Monte Carlo generators <span style="font-variant:small-caps;">sherpa</span> and <span style="font-variant:small-caps;">alpgen+pythia</span>. The measured cross sections are in reasonable agreement with theory, within large experimental and theoretical uncertainties.
Summary {#summary .unnumbered}
=======
W/Z + jets and W/Z + heavy flavor measurements belong to the Tevatron legacy. All the measurements are in general good agreement with the perturbative NLO QCD predictions, in the tail of some distributions like $H_T^{\rm jet}$ and di-jet $\Delta y$ the inclusion of higher order corrections improves the agreement between data and theory. Detailed studies of differential distributions in W/Z + jets and W/Z + heavy flavour production provide an important test of the different Monte Carlo generators and theoretical predictions, and a fundamental validation of the background modeling of such processes in the search for new physics.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to the ATLAS-DESY group for supporting my participation.
References {#references .unnumbered}
==========
[99]{} D.J. Gross and F. Wilczek, **8**, 3633 (1973).
J.Campbell and R.K. Ellis, **65**, 113007 (2002).
C. F. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, H. Ita, D. A. Kosower and D. Maitre, , 036003 (2008). M. Rubin, G. P. Salam and S. Sapeta, JHEP [**1009**]{} (2010) 084 A. Denner, S. Dittmaier, T. Kasprzik and A. Muck, JHEP [**1106**]{} (2011) 069 M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A. D. Polosa, JHEP [**0307**]{} (2003) 001 S. Frixione, P. Nason and C. Oleari, JHEP [**0711**]{} (2007) 070 T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP [**0605**]{} (2006) 026 M. Bahr, S. Gieseke, M. A. Gigg, D. Grellscheid, K. Hamilton, O. Latunde-Dada, S. Platzer and P. Richardson [*et al.*]{}, Eur. Phys. J. C [**58**]{} (2008) 639 T. Gleisberg, S. .Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert and J. Winter, JHEP [**0902**]{} (2009) 007 J. R. Andersen and J. M. Smillie, JHEP [**1106**]{} (2011) 010 S. Camarda, FERMILAB-THESIS-2012-12. A. Abulencia et al. (CDF Collaboration), , 071103(R) (2006).
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], arXiv:1302.6508 \[hep-ex\]. T. Aaltonen [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett. [**110**]{} (2013) 071801 V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Lett. B [**718**]{} (2013) 1314 T. Aaltonen [*et al.*]{} \[CDF Collaboration\], CDF public note 10594
V. M. Abazov [*et al.*]{} \[D0 Collaboration\], arXiv:1301.2233 \[hep-ex\].
[^1]: formerly IFAE - Barcelona
|
---
abstract: 'Evolution strategies (ES) are a family of black-box optimization algorithms able to train deep neural networks roughly as well as Q-learning and policy gradient methods on challenging deep reinforcement learning (RL) problems, but are much faster (e.g. hours vs. days) because they parallelize better. However, many RL problems require directed exploration because they have reward functions that are sparse or deceptive (i.e. contain local optima), and it is unknown how to encourage such exploration with ES. Here we show that algorithms that have been invented to promote directed exploration in small-scale evolved neural networks via populations of exploring agents, specifically novelty search (NS) and quality diversity (QD) algorithms, can be hybridized with ES to improve its performance on sparse or deceptive deep RL tasks, while retaining scalability. Our experiments confirm that the resultant new algorithms, NS-ES and two QD algorithms, NSR-ES and NSRA-ES, avoid local optima encountered by ES to achieve higher performance on Atari and simulated robots learning to walk around a deceptive trap. This paper thus introduces a family of fast, scalable algorithms for reinforcement learning that are capable of directed exploration. It also adds this new family of exploration algorithms to the RL toolbox and raises the interesting possibility that analogous algorithms with multiple simultaneous paths of exploration might also combine well with existing RL algorithms outside ES.'
author:
- |
Edoardo Conti Vashisht Madhavan[^1] Felipe Petroski Such\
**Joel Lehman** **Kenneth O. Stanley** **Jeff Clune**\
Uber AI Labs\
title: 'Improving Exploration in Evolution Strategies for Deep Reinforcement Learning via a Population of Novelty-Seeking Agents'
---
Introduction {#intro}
============
In RL, an agent tries to learn to perform a sequence of actions in an environment that maximizes some notion of cumulative reward [@sutton1998reinforcement]. However, reward functions are often *deceptive*, and solely optimizing for reward without some mechanism to encourage intelligent exploration can lead to getting stuck in local optima and the agent failing to properly learn [@liepins:deceptiveness; @lehman; @sutton1998reinforcement]. Unlike in supervised learning with deep neural networks (DNNs), wherein local optima are not thought to be a problem [@kawaguchi2016deep; @dauphin:arxiv14], the training data in RL is determined by the actions an agent takes. If the agent greedily takes actions that maximize reward, the training data for the algorithm will be limited and it may not discover alternate strategies with larger payoffs (i.e. it can get stuck in local optima) [@liepins:deceptiveness; @lehman; @sutton1998reinforcement]. Sparse reward signals can also be a problem for algorithms that only maximize reward, because at times there may be no reward gradient to follow. The possibility of deceptiveness and/or sparsity in the reward signal motivates the need for efficient and *directed* exploration, in which an agent is motivated to visit unexplored states in order to learn to accumulate higher rewards. Although deep RL algorithms have performed amazing feats in recent years [@mnih2015; @a3c; @schulman2015trust], they have mostly done so despite relying on simple, *undirected* (aka dithering) exploration strategies, in which an agent hopes to explore new areas of its environment by taking random actions (e.g. epsilon-greedy exploration) [@sutton1998reinforcement].
A number of methods have been proposed to promote directed exploration in RL [@schmidhuber2010formal; @oudeyer2009intrinsic], including recent methods that handle high-dimensional state spaces with DNNs. A common idea is to encourage an agent to visit states it has rarely or never visited (or take novel actions in those states). Methods proposed to track state (or state-action pair) visitation frequency include (1) approximating state visitation counts based on either auto-encoded latent codes of states [@tang2017exploration] or pseudo-counts from state-space density models [@bellemare2016unifying; @ostrovski2017count], (2) learning a dynamics model that predicts future states (assuming predictions will be worse for rarely visited states/state-action pairs) [@stadie2015incentivizing; @houthooft2016vime; @pathak2017curiosity], and (3) methods based on compression (novel states should be harder to compress) [@schmidhuber2010formal].
Those methods all count each state separately. A different approach to is to hand-design (or learn) an abstract, holistic description of an agent’s lifetime behavior, and then encourage the agent to exhibit different behaviors from those previously performed. That is the approach of novelty search (NS) [@lehman] and quality diversity (QD) algorithms [@cully:nature15; @mouret2015illuminating; @pugh:frontiers16], which are described in detail below. Such algorithms are also interestingly different, and have different capabilities, because they perform exploration with a population of agents rather than a single agent (discussed in SI Sec. \[popVsSingleAgentExploration\]). NS and QD have shown promise with smaller neural networks on problems with low-dimensional input and output spaces [@lehman:gecco11; @cully:nature15; @mouret2015illuminating; @pugh:frontiers16; @velez:gecco14; @huizinga2016does]. Evolution strategies (ES) [@rechenberg1978evolutionsstrategien] was recently shown to perform well on high-dimensional deep RL tasks in a short amount of wall clock time by scaling well to many distributed computers. In this paper, for the first time, we study how these two types of algorithms can be hybridized with ES to scale them to deep neural networks and thus tackle hard, high-dimensional deep reinforcement learning problems, without sacrificing the speed/scalability benefits of ES. We first study NS, which performs exploration *only* (ignoring the reward function) to find a set of novel solutions [@lehman]. We then investigate algorithms that balance exploration and exploitation, specifically novel instances of QD algorithms, which seek to produce a set of solutions that are both novel and high-performing [@lehman:gecco11; @cully:nature15; @mouret2015illuminating; @pugh:frontiers16]. Both NS and QD are explained in detail in Sec. \[algos\].
ES directly searches in the parameter space of a neural network to find an effective policy. A team from OpenAI recently showed that ES can achieve competitive performance on many reinforcement learning (RL) tasks while offering some unique benefits over traditional gradient-based RL methods [@es]. Most notably, ES is highly parallelizable, which enables near linear speedups in runtime as a function of CPU/GPU workers. For example, with hundreds of parallel CPUs, ES was able to achieve roughly the same performance on Atari games with the same DNN architecture in 1 hour as A3C did in 24 hours [@es]. In this paper, we investigate adding NS and QD to ES only; in future work, we will investigate how they might be hybridized with Q-learning and policy gradient methods. We start with ES because (1) its fast wall-clock time allows rapid experimental iteration, and (2) NS and QD were originally developed as neuroevolution methods, making it natural to try them first with ES, which is also an evolutionary algorithm.
Here we test whether encouraging novelty via NS and QD improves the performance of ES on sparse and/or deceptive control tasks. Our experiments confirm that NS-ES and two simple versions of QD-ES (NSR-ES and NSRA-ES) avoid local optima encountered by ES and achieve higher performance on tasks ranging from simulated robots learning to walk around a deceptive trap to the high-dimensional pixel-to-action task of playing Atari games. Our results add these new families of exploration algorithms to the RL toolbox, opening up avenues for studying how they can best be combined with RL algorithms, whether ES or others.
Background {#prelim}
==========
Evolution Strategies {#evolution strategies details}
--------------------
Evolution strategies (ES) are a class of black box optimization algorithms inspired by natural evolution [@rechenberg1978evolutionsstrategien]: At every iteration (generation), a population of parameter vectors (genomes) is perturbed (mutated) and, optionally, recombined (merged) via crossover. The fitness of each resultant offspring is then evaluated according to some objective function (reward) and some form of selection then ensures that individuals with higher reward tend to produce offspring for the next generation. Many algorithms in the ES class differ in their representation of the population and methods of recombination; the algorithms subsequently referred to in this work belong to the class of Natural Evolution Strategies (NES) [@wierstra2008natural; @sehnke2010parameter]. NES represents the population as a distribution of parameter vectors $\theta$ characterized by parameters $\phi$: $p_{\phi}(\theta)$. Under a fitness function, $f(\theta)$, NES seeks to maximize the average fitness of the population, $\mathbb{E}_{\theta \sim p_{\phi}}[f(\theta)]$, by optimizing $\phi$ with stochastic gradient ascent.
Recent work from OpenAI outlines a version of NES applied to standard RL benchmark problems [@es]. We will refer to this variant simply as ES going forward. In their work, a fitness function $f(\theta)$ represents the stochastic reward experienced over a full episode of agent interaction, where $\theta$ parameterizes the policy $\pi_{\theta}$. From the population distribution $p_{\phi_{t}}$ , parameters $\theta^{i}_{t} \sim \mathcal{N}(\theta_{t},\,\sigma^{2}I)$ are sampled and evaluated to obtain $f(\theta^{i}_{t})$. In a manner similar to REINFORCE [@williams1992simple], $\theta_{t}$ is updated using an estimate of approximate gradient of expected reward: $$\begin{gathered}
\label{eq:1}
\scalebox{.9}{$
\nabla_{\phi}\mathbb{E}_{\theta \sim \phi}[f(\theta)] \approx \frac{1}{n}\sum_{i=1}^{n}f(\theta^{i}_{t})\nabla_{\phi} \log p_{\phi}(\theta^{i}_{t})
$}\end{gathered}$$ where $n$ is the number of samples evaluated per generation. Intuitively, NES samples parameters in the neighborhood of $\theta_{t}$ and determines the direction in which $\theta_{t}$ should move to improve expected reward. Since this gradient estimate has high variance, NES relies on a large $n$ for variance reduction. Generally, NES also evolves the covariance of the population distribution, but for the sake of fair comparison with @es we consider only static covariance distributions, meaning $\sigma$ is fixed throughout training.
To sample from the population distribution, @es apply additive Gaussian noise to the current parameter vector : $\theta^{i}_{t} = \theta_{t} + \sigma\epsilon_{i}$ where $\epsilon_{i} \sim \mathcal{N}(0,\,I)$. Although $\theta$ is high-dimensional, previous work has shown Gaussian parameter noise to have beneficial exploration properties when applied to deep networks [@sehnke2010parameter; @plappert2017parameter; @fortunato2017noisy]. The gradient is then estimated by taking a sum of sampled parameter perturbations weighted by their reward: $$\begin{gathered}
\label{eq:2}
\scalebox{.9}{$
\nabla_{\theta_{t}}\mathbb{E}_{\epsilon \sim \mathcal{N}(0,\,I)}[f(\theta_{t} + \sigma\epsilon)] \approx \frac{1}{n\sigma}\sum_{i=1}^{n}f(\theta^{i}_{t})\epsilon_{i}
$}\end{gathered}$$ To ensure that the scale of reward between domains does not bias the optimization process, we follow the approach of @es and rank-normalize $f(\theta^{i}_{t})$ before taking the weighted sum. Overall, this NES variant exhibits performance on par with contemporary, gradient-based algorithms on difficult RL domains, including simulated robot locomotion and Atari environments [@bellemare2013arcade].
Novelty Search (NS) {#NS overview}
-------------------
Inspired by nature’s drive towards diversity, NS encourages policies to engage in notably different behaviors than those previously seen. The algorithm encourages different behaviors by computing the *novelty* of the current policy with respect to previously generated policies and then encourages the population distribution to move towards areas of parameter space with high novelty. NS outperforms reward-based methods in maze and biped walking domains, which possess deceptive reward signals that attract agents to local optima [@lehman]. In this work, we investigate the efficacy of NS at the scale of DNNs by combining it with ES. In NS, a policy $\pi$ is assigned a domain-dependent *behavior characterization* $b(\pi)$ that describes its behavior. For example, in the case of a humanoid locomotion problem, $b(\pi)$ may be as simple as a two-dimensional vector containing the humanoid’s final $\{x,y\}$ location. Throughout training, every $\pi_{\theta}$ evaluated adds $b(\pi_{\theta})$ to an archive set $A$ with some probability. A particular policy’s novelty $N(b(\pi_{\theta}), A)$ is then computed by selecting the k-nearest neighbors of $b(\pi_{\theta})$ from $A$ and computing the average distance between them: $$\begin{gathered}
\scalebox{.9}{$N(\theta, A) = N(b(\pi_{\theta}), A) = \frac{1}{\vert{S}\vert}\sum_{j\in{S}}\vert\vert{b(\pi_{\theta}) - b(\pi_{j})}\vert\vert_{2}$}\\
\scalebox{.9}{$S = kNN(b(\pi_{\theta}), A)$}\\
\scalebox{.9}{$
= \{b(\pi_{1}), b(\pi_{2}), ..., b(\pi_{k})\}$}\end{gathered}$$ Above, the distance between behavior characterizations is calculated with an $L2$-norm, but any distance function can be substituted. Previously, NS has been implemented with a genetic algorithm [@lehman]. We next explain how NS can now be combined with ES, to leverage the advantages of both.
Methods {#algos}
=======
NS-ES {#NSES Overview}
-----
We use the ES optimization framework, described in Sec. \[evolution strategies details\], to compute and follow the gradient of expected novelty with respect to $\theta_{t}$. Given an archive $A$ and sampled parameters $\theta^{i}_{t} = \theta_{t} + \sigma\epsilon_{i}$, the gradient estimate can be computed: $$\begin{gathered}
\scalebox{.9}{$
\nabla_{\theta_{t}}\mathbb{E}_{\epsilon \sim \mathcal{N}(0,\,I)}[N(\theta_{t} + \sigma\epsilon, A) | A] \approx \frac{1}{{n}\sigma}\sum_{i=1}^{n}N(\theta^{i}_{t}, A)\epsilon_{i}
$}\end{gathered}$$ The gradient estimate obtained tells us how to change the current policy’s parameters $\theta_{t}$ to increase the average novelty of our parameter distribution. We condition the gradient estimate on A, as the archive is fixed at the beginning of a given iteration and updated only at the end. We add only the behavior characterization corresponding to each $\theta_{t}$, as adding those for each sample $\theta_{t}^{i}$ would inflate the archive and slow the nearest-neighbors computation. As more behavior characterizations are added to $A$, the novelty landscape changes, resulting in commonly occurring behaviors becoming “boring." Optimizing for expected novelty leads to policies that move towards unexplored areas of behavior space.
NS-ES could operate with a single agent that is rewarded for acting differently than its ancestors. However, to encourage additional diversity and get the benefits of population-based exploration described in SI Sec. \[popVsSingleAgentExploration\], we can instead create a population of $M$ agents, which we will refer to as the *meta-population*. Each agent, characterized by a unique $\theta^{m}$, is rewarded for being different from all prior agents in the archive (ancestors, other agents, and the ancestors of other agents), an idea related to that of @liu2017stein, which optimizes for a distribution of M diverse, high-performing policies. We hypothesize that the selection of $M$ is domain dependent and that identifying which domains favor which regime is a fruitful area for future research.
We initialize $M$ random parameter vectors and at every iteration select one to update. For our experiments, we probabilistically select which $\theta^{m}$ to advance from a discrete probability distribution as a function of $\theta^{m}$’s novelty. Specifically, at every iteration, for a set of agent parameter vectors $\Pi = \{\theta^{1}, \theta^{2}, ..., \theta^{M}\}$, we calculate each $\theta^{m}$’s probability of being selected $P(\theta^{m})$ as its novelty normalized by the sum of novelty across all policies: $$\begin{gathered}
\label{eq:novelty}
\scalebox{.9}{$
P(\theta^{m}) = \frac{N(\theta^{m}, A)}{\sum_{j=1}^{M}N(\theta^{j}, A)}
$}\end{gathered}$$ Having multiple, separate agents represented as independent Gaussians is a simple choice for the *meta-population* distribution. In future work, more complex sampling distributions that represent the multi-modal nature of parameter vectors could be tried.
After selecting an individual $m$ from the meta-population, we compute the gradient of expected novelty with respect to $m$’s current parameter vector, $\theta^{m}_{t}$, and perform an update step accordingly: $$\begin{gathered}
\label{eq:5}
\scalebox{.9}{$
\theta^{m}_{t+1} \leftarrow \theta^{m}_{t} + \alpha \frac{1}{n\sigma} \sum_{i=1}^{n} N(\theta_{t}^{i,m}, A) \epsilon_{i}
$}\end{gathered}$$ Where $n$ is the number of sampled perturbations to $\theta^{m}_{t}$, $\alpha$ is the stepsize, and $\theta^{i,m}_{i} = \theta^{m}_{t} + \sigma\epsilon_{i}$, where $\epsilon_{i} \sim \mathcal{N}(0,\,I)$. Once the current parameter vector is updated, $b(\pi_{\theta^{m}_{t+1}})$ is computed and added to the shared archive $A$. The whole process is repeated for a pre-specified number of iterations, as there is no true convergence point of NS. During training, the algorithm preserves the policy with the highest average episodic reward and returns this policy once training is complete. Although @es return only the final policy after training with ES, the ES experiments in this work return the best-performing policy to facilitate fair comparison with NS-ES. Algorithm \[alg:nses\] in SI Sec. \[ns-es-algo\] outlines a simple, parallel implementation of NS-ES. It is important to note that the addition of the archive and the replacement of the fitness function with novelty does not damage the scalability of the ES optimization procedure (SI Sec. \[scalable\_si\]).
QD-ES Algorithms: NSR-ES and NSRA-ES {#qd_section}
------------------------------------
NS-ES alone can enable agents to avoid deceptive local optima in the reward function. Reward signals, however, are still very informative and discarding them completely may cause performance to suffer. Consequently, we train a variant of NS-ES, which we call NSR-ES, that combines the reward (“fitness") and novelty calculated for a given set of policy parameters $\theta$. Similar to NS-ES and ES, NSR-ES operates on entire episodes and can thus evaluate reward and novelty simultaneously for any sampled parameter vector: $\theta^{i,m}_{t} = \theta^{m}_{t} + \epsilon_{i}$. Specifically, we compute $f(\theta_{t}^{i,m})$ and $N(\theta_{t}^{i,m}, A)$, average the two values, and set the average as the weight for the corresponding $\epsilon_{i}$. The averaging process is integrated into the parameter update rule as: $$\begin{gathered}
\scalebox{.9}{$
\theta^{m}_{t+1} \leftarrow \theta^{m}_{t} + \alpha \frac{1}{n\sigma} \sum_{i=1}^{n} \frac{f(\theta_{t}^{i,m}) + N(\theta_{t}^{i,m}, A)}{2}\epsilon_{i}
$}\end{gathered}$$ Intuitively, the algorithm follows the approximated gradient in parameter-space towards policies that both exhibit novel behaviors and achieve high rewards. Often, however, the scales of $f(\theta)$ and $N(\theta, A)$ differ. To combine the two signals effectively, we rank-normalize $f(\theta_{t}^{i,m})$ and $ N(\theta_{t}^{i,m}, A)$ independently before computing the average. Optimizing a linear combination of novelty and reward was previously explored in @cuccu2011novelty and @cuccu2011restart, but not with large neural networks on high-dimensional problems. The result of NSR-ES is a set of $M$ agents being optimized to be both high-performing, yet different from each other.
NSR-ES has an equal weighting of the performance and novelty gradients that is static across training. We explore a further extension of NSR-ES called NSRAdapt-ES (NSRA-ES), which takes advantage of the opportunity to dynamically weight the priority given to the performance gradient $f(\theta_{t}^{i,m})$ vs. the novelty gradient $ N(\theta_{t}^{i,m}, A)$ by intelligently adapting a weighting parameter $w$ during training. By doing so, the algorithm can follow the performance gradient when it is making progress, increasingly try different things if stuck in a local optimum, and switch back to following the performance gradient once unstuck. For a specific $w$ at a given generation, the parameter update rule for NSRA-ES is expressed as follows: $$\begin{gathered}
\scalebox{.9}{$
\theta^{m}_{t+1} \leftarrow \theta^{m}_{t} + \alpha \frac{1}{n\sigma} \sum_{i=1}^{n} w f(\theta_{t}^{i,m})\epsilon_{i} + (1 - w)N(\theta_{t}^{i,m}, A)\epsilon_{i}
$}\end{gathered}$$ We set $w=1.0$ initially and decrease it if performance stagnates across a fixed number of generations. We continue decreasing $w$ until performance increases, at which point we increase $w$. While many previous works have adapted exploration pressure online by learning the amount of noise to add to the parameters [@plappert2017parameter; @wierstra2008natural; @sehnke2010parameter; @hansen2003reducing], such approaches rest on the assumption that an increased amount of parameter noise leads to increased *behavioral diversity*, which is often not the case (e.g. too much noise may lead to degenerate policies) [@lehman:gecco11]. Here we directly adapt the weighting between behavioral diversity and performance, which more directly controls the trade-off of interest. SI Sec. \[ns-es-algo\] provides a more detailed description of how we adapt $w$ as well as pseudocode for NSR-ES and NSRA-ES. Source code and hyperparameter settings for our experiments can be found here: <https://github.com/uber-research/deep-neuroevolution>
Experiments
===========
Simulated Humanoid Locomotion problem
-------------------------------------
We first tested our implementation of NS-ES, NSR-ES, and NSRA-ES on the problem of having a simulated humanoid learn to walk. We chose this problem because it is a challenging continuous control benchmark where most would presume a reward function is necessary to solve the problem. With NS-ES, we test whether searching through novelty alone can find solutions to the problem. A similar result has been shown for much smaller neural networks ($\mathtt{\sim}$50-100 parameters) on a more simple simulated biped [@lehman:gecco11], but here we test whether NS-ES can enable locomotion at the scale of deep neural networks on a much more sophisticated environment. NSR-ES and NSRA-ES experiments then test the effectiveness of combining exploration and reward pressures on this difficult continuous control problem. SI Sec. \[mujoco\_train\] outlines complete experimental details.
The first environment is in a slightly modified version of OpenAI Gym’s Humanoid-v1 environment. Because the heart of this challenge is to learn to walk efficiently, not to walk in a particular direction, we modified the environment reward to be isotropic (i.e. indifferent to the direction the humanoid traveled) by setting the velocity component of reward to distance traveled from the origin as opposed to distance traveled in the positive $x$ direction.
As described in section \[NS overview\], novelty search requires a domain-specific behavior characterization (BC) for each policy, which we denote as $b(\pi_{\theta_{i}})$. For the Humanoid Locomotion problem the BC is the agent’s final $\{x,y\}$ location, as it was in @lehman:gecco11. NS also requires a distance function between two BCs. Following @lehman:gecco11, the distance function is the square of the Euclidean distance: $$\begin{gathered}
\scalebox{.9}{$
dist(b(\pi_{\theta_{i}}), b(\pi_{\theta_{j}})) = \vert\vert b(\pi_{\theta_{i}}) - b(\pi_{\theta_{j}})\vert\vert_{2}^2
$}\end{gathered}$$ The first result is that ES obtains a higher final reward than NS-ES ($p < 0.05$) and NSR-ES ($p < 0.05$); these and all future $p$ values are calculated via a Mann-Whitney U test. The performance gap is more pronounced for smaller amounts of computation (Fig. \[fig:es\_vs\_nses\_muj\] (c)). However, many will be surprised that NS-ES is still able to consistently solve the problem despite ignoring the environment’s reward function. While the BC is *aligned* [@pugh2015confronting] with the problem in that reaching new $\{x,y\}$ positions tends to also encourage walking, there are many parts of the reward function that the BC ignores (e.g. energy-efficiency, impact costs).
We hypothesize that with a sophisticated BC that encourages diversity in all of the behaviors the multi-part reward function cares about, there would be no performance gap. However, such a BC may be difficult to construct and would likely further exaggerate the amount of computation required for NS to match ES. NSR-ES demonstrates faster learning than NS-ES due to the addition of reward pressure, but ultimately results in similar final performance after 600 generations ($p > 0.05$, Fig. \[fig:es\_vs\_nses\_muj\] (c)). Promisingly, on this non-deceptive problem, NSRA-ES does not pay a cost for its latent exploration capabilities and performs similarly to ES ($p > 0.05$).
The Humanoid Locomotion problem does not appear to be a deceptive problem, at least for ES. To test whether NS-ES, NSR-ES, and NSR-ES specifically help with deception, we also compare ES to these algorithms on a variant of this environment we created that adds a deceptive trap (a local optimum) that must be avoided for maximum performance (Fig. \[fig:es\_vs\_nses\_muj\] (b)). In this new environment, a small three-sided enclosure is placed at a short distance in front of the starting position of the humanoid and the reward function is simply the distance traveled in the positive $x$ direction.
Fig. \[fig:es\_vs\_nses\_muj\] (d) and SI Sec. \[mujoco tabular results\] show the reward received by each algorithm and Fig. \[fig:trap\_comp\] shows how the algorithms differ qualitatively during search on this problem. In every run, ES gets stuck in the local optimum due to following reward into the deceptive trap. NS-ES is able to avoid the local optimum as it ignores reward completely and instead seeks to thoroughly explore the environment, but doing so also means it makes slow progress according to the reward function. NSR-ES demonstrates superior performance to NS-ES ($p < 0.01$) and ES ($p < 0.01$) as it benefits from both optimizing for reward and escaping the trap via the pressure for novelty. Like ES, NSRA-ES learns to walk into the deceptive trap initially, as it initially is optimizing for reward only. Once stuck in the local optimum, the algorithm continually increases its pressure for novelty, allowing it to escape the deceptive trap and ultimately achieve much higher rewards than NS-ES ($p < 0.01$) and NSR-ES ($p < 0.01$). Based just on these two domains, NSRA-ES seems to be the best algorithm across the board because it can exploit well when there is no deception, add exploration dynamically when there is, and return to exploiting once unstuck. The latter is likely why NSRA-ES outperforms even NSR-ES on the deceptive humanoid locomotion problem.
![**Humanoid Locomotion Experiment.** The humanoid locomotion task is shown without a deceptive trap (a) and with one (b), and results on them in (c) and (d), respectively. Here and in similar figures below, the median reward (of the best seen policy so far) per generation across 10 runs is plotted as the bold line with 95% bootstrapped confidence intervals of the median (shaded). Following @es, policy performance is measured as the average performance over $\mathtt{\sim}$30 stochastic evaluations.[]{data-label="fig:es_vs_nses_muj"}](muj_vert_figure "fig:"){width="0.85\columnwidth"} -0.1cm
-0.01in
Fig. \[fig:trap\_comp\] also shows the benefit of maintaining a meta-population ($M=5$) in the NS-ES, NSR-ES, and NSRA-ES algorithms. Some lineages get stuck in the deceptive trap, incentivizing other policies to explore around the trap. At that point, all three algorithms begin to allocate more computational resources to this newly discovered, more promising strategy via the probabilistic selection method outlined in Sec. \[NSES Overview\]. Both the novelty pressure and having a meta-population thus appear to be useful, but in future work we look to disambiguate the relative contribution made by each.\
![**ES gets stuck in the deceptive local optimum while NS-ES, NSR-ES & NSRA-ES explore to find better solutions.** An overhead view of a representative run is shown for each algorithm on the Humanoid Locomotion with Deceptive Trap problem. The black star represents the humanoid’s starting point. Each diamond represents the final location of a generation’s policy, i.e. $\pi(\theta_{t})$, with darker shading for later generations. For NS-ES, NSR-ES, & NSRA-ES plots, each of the $M=5$ agents in the meta-population and its descendants are represented by different colors. Similar plots for all 10 runs of each algorithm are provided in SI Sec. \[overhead\_plots\].[]{data-label="fig:trap_comp"}](trap_comp_vert){width="0.75\columnwidth"}
Atari {#atari_results}
-----
We also tested NS-ES, NSR-ES, and NSRA-ES on numerous games from the Atari 2600 environment in OpenAI Gym [@brockman]. Atari games serve as an informative benchmark due to their high-dimensional pixel input and complex control dynamics; each game also requires different levels of exploration to solve. To demonstrate the effectiveness of NS-ES, NSR-ES, and NSRA-ES for local optima avoidance and directed exploration, we tested on 12 different games with varying levels of complexity, as defined by the taxonomy in @bellemare2016unifying. Primarily, we focused on games in which, during preliminary experiments, we observed ES prematurely converging to local optima (Seaquest, Q\*Bert, Freeway, Frostbite, and Beam Rider). However, we also included a few other games where ES did not converge to local optima to understand the performance of our algorithm in less-deceptive domains (Alien, Amidar, Bank Heist, Gravitar, Zaxxon, and Montezuma’s Revenge). SI Sec. \[atari\_train\] describes additional experimental details. We report the median reward across 5 independent runs of the best policy found in each run (see Table \[atari-table\]).
For the behavior characterization, we follow an idea from @naddaf2010game and concatenate Atari game RAM states for each timestep in an episode. RAM states in Atari 2600 games are integer-valued vectors of length 128 in the range \[0, 255\] that describe all the state variables in a game (e.g. the location of the agent and enemies). Ultimately, we want to automatically learn behavior characterizations directly from pixels. A plethora of recent research suggests that this is a viable approach [@lange2010deep; @kingma2013auto; @bellemare2016unifying]. For example, low-dimensional, latent representations of the state space could be extracted from auto-encoders [@tang2017exploration; @van2016conditional] or networks trained to predict future states [@pathak2017curiosity; @stadie2015incentivizing]. In this work, however, we focus on learning with a pre-defined, informative behavior characterization and leave the task of jointly learning a policy and latent representation of states for future work. In effect, basing novelty on RAM states provides a confirmation of what is possible in principle with a sufficiently informed behavior characterization. We also emphasize that, while during training NS-ES, NSR-ES, and NSRA-ES use RAM states to guide novelty search, the policy itself, $\pi_{\theta_{t}}$, operates only on image input and can be evaluated without any RAM state information. The distance between behavior characterizations is the sum of L2-distances at each timestep $t$: $$\begin{gathered}
\scalebox{.9}{$
dist(b(\pi_{\theta_{i}}), b(\pi_{\theta_{j}})) = \sum_{t=1}^{T} \vert\vert(b_{t}(\pi_{\theta_{i}})) - b_{t}(\pi_{\theta_{j}}))\vert\vert_{2}
$}\end{gathered}$$ For trajectories of different lengths, the last state of the shorter trajectory is repeated until the lengths of both match. Because the BC distance is not normalized by trajectory length, novelty is biased to be higher for longer trajectories. In some Atari games, this bias can lead to higher performing policies, but in other games longer trajectories tend to have a neutral or even negative relationship with performance. In this work we found it beneficial to keep novelty unnormalized, but further investigation into different BC designs could yield additional improvements.
Table \[atari-table\] compares the performance of each algorithm discussed above to each other and with those from two popular methods for exploration in RL, namely Noisy DQN [@fortunato2017noisy] and A3C+ [@bellemare2016unifying]. Noisy DQN and A3C+ only outperform all the ES variants considered in this paper on 3/12 games and 2/12 games respectively. NSRA-ES, however, outperforms the other algorithms on 5/12 games, suggesting that NS and QD are viable alternatives to contemporary exploration methods.
While the novelty pressure in NS-ES does help it avoid local optima in some cases (discussed below), optimizing for novelty alone does not result in higher reward in most games (although it does in some). However, it is surprising how well NS-ES does in many tasks given that it is not explicitly attempting to increase reward. Because NSR-ES combines exploration with reward maximization, it is able to avoid local optima encountered by ES while also learning to play the game well. In each of the 5 games in which we observed ES converging to premature local optima (i.e. Seaquest, Q\*Bert, Freeway, Beam Rider, Frostbite), NSR-ES achieves a higher median reward. In the other games, ES does not benefit from adding an exploration pressure and NSR-ES performs worse. It is expected that if there are no local optima and reward maximization is sufficient to perform well, the extra cost of encouraging exploration will hurt performance. Mitigating such costs, NSRA-ES optimizes solely for reward until a performance plateau is reached. After that, the algorithm will assign more weight to novelty and thus encourage exploration. We found this to be beneficial, as NSRA-ES achieves higher median rewards than ES on 8/12 games and NSR-ES on 9/12 games. It’s superior performance validates NSRA-ES as the best among the evolutionary algorithms considered and suggests that using an *adaptive* weighting between novelty and reward is a promising direction for future research.
In the game Seaquest, the avoidance of local optima is particularly important (Fig. \[fig:squest\_main\_plot\]). ES performance flatlines early at a median reward of 960, which corresponds to a behavior of the agent descending to the bottom, shooting fish, and never coming up for air. This strategy represents a classic local optima, as coming up for air requires temporarily foregoing reward, but enables far higher rewards to be earned in the long run (@es did not encounter this particular local optimum with their hyperparameters, but the point is that ES without exploration can get stuck indefinitely on whichever major local optima it encounters). NS-ES learns to come up for air in all 5 runs and achieves a slightly higher median reward of 1044.5 ($p < 0.05$). NSR-ES also avoids this local optimum, but its additional reward signal helps it play the game better (e.g. it is better at shooting enemies), resulting in a significantly higher median reward of 2329.7 ($p < 0.01$). Because NSRA-ES takes reward steps initially, it falls into the same local optimum as ES. Because we chose (without performing a hyperparameter search) to change the weighting $w$ between performance and novelty infrequently (only every 50 generations), and to change it by a small amount (only 0.05), 200 generations was not long enough to emphasize novelty enough to escape this local optimum. We found that by changing $w$ every 10 generations, this problem is remedied and the performance of NSRA-ES equals that of NSR-ES ($p > 0.05$, Fig. \[fig:squest\_main\_plot\]). These results motivate future research into better hyperparameters for changing $w$, and into more complex, intelligent methods of dynamically adjusting $w$, including with a population of agents with different dynamic $w$ strategies.
The Atari results illustrate that NS is an effective mechanism for encouraging directed exploration, given an appropriate behavior characterization, for complex, high-dimensional control tasks. A novelty pressure alone produces impressive performance on many games, sometimes even beating ES. Combining novelty and reward performs far better, and improves ES performance on tasks where it appears to get stuck on local optima.
-0.1in
![**Seaquest Case Study.**[ By switching the weighting between novelty and reward, $w$, every 10 generations instead of every 50, NSRA-ES is able to overcome the local optimum ES finds and achieve high scores on Seaquest.]{}[]{data-label="fig:squest_main_plot"}](squest_main_plot "fig:"){width="0.5\columnwidth"} -0.1cm
-0.1in
Discussion and Conclusion
=========================
NS and QD are classes of evolutionary algorithms designed to avoid local optima and promote exploration in RL environments, but have only been previously shown to work with small neural networks (on the order of hundreds of connections). ES was recently shown to be capable of training deep neural networks that can solve challenging, high-dimensional RL tasks [@es]. It also is much faster when many parallel computers are available. Here we demonstrate that, when hybridized with ES, NS and QD not only preserve the attractive scalability properties of ES, but also help ES explore and avoid local optima in domains with deceptive reward functions. To the best of our knowledge, this paper reports the first attempt at augmenting ES to perform directed exploration in high-dimensional environments. We thus provide an option for those interested in taking advantage of the scalability of ES, but who also want higher performance on domains that have reward functions that are sparse or have local optima. The latter scenario will likely hold for most challenging, real-world domains that machine learning practitioners will wish to tackle in the future.
Additionally, this work highlights alternate options for exploration in RL domains. The first is to holistically describe the behavior of an agent instead of defining a per-state exploration bonus. The second is to encourage a population of agents to simultaneously explore different aspects of an environment. These new options thereby open new research areas into (1) comparing holistic vs. state-based exploration, and population-based vs. single-agent exploration, more systematically and on more domains, (2) investigating the best way to combine the merits of all of these options, and (3) hybridizing holistic and/or population-based exploration with other algorithms that work well on deep RL problems, such as policy gradients and DQN. It should be relatively straightforward to combine NS with policy gradients (NS-PG). It is less obvious how to combine it with Q-learning (NS-Q), but may be possible.
As with any exploration method, encouraging novelty can come at a cost if such an exploration pressure is not necessary. In Atari games such as Alien and Gravitar, and in the Humanoid Locomotion problem without a deceptive trap, both NS-ES and NSR-ES perform worse than ES. To avoid this cost, we introduce the NSRA-ES algorithm, which attempts to invest in exploration only when necessary. NSRA-ES tends to produce better results than ES, NS-ES, and NSR-ES across many different domains, making it an attractive new algorithm for deep RL tasks. Similar strategies for adapting the amount of exploration online may also be advantageous for other deep RL algorithms. How best to dynamically balance between exploitation and exploration in deep RL remains an open, critical research challenge, and our work underscores the importance of, and motivates further, such work. Overall, our work shows that ES is a rich and unexploited parallel path for deep RL research. It is worthy of exploring not only because it is an alternative algorithm for RL problems, but also because innovations created in the ES family of algorithms could be ported to improve other deep RL algorithm families like policy gradients and Q learning, or through hybrids thereof.
### Acknowledgments {#acknowledgments .unnumbered}
We thank all of the members of Uber AI Labs, in particular Thomas Miconi, Rui Wang, Peter Dayan, John Sears, Joost Huizinga, and Theofanis Karaletsos, for helpful discussions. We also thank Justin Pinkul, Mike Deats, Cody Yancey, Joel Snow, Leon Rosenshein and the entire OpusStack Team inside Uber for providing our computing platform and for technical support.
Supplementary Information {#supplemental}
=========================
Videos of agent behavior {#videos}
------------------------
Videos of example agent behavior in all the environments can be viewed here: <https://goo.gl/cVUG2U>.
Population-based exploration vs. single-agent exploration {#popVsSingleAgentExploration}
---------------------------------------------------------
As mentioned in the introduction, aside from being holistic vs. state-based, there is another interesting difference between most exploration methods for RL [@bellemare2016unifying; @ostrovski2017count; @stadie2015incentivizing; @houthooft2016vime; @pathak2017curiosity] and the NS/QD family of algorithms. We do not investigate the benefits of this difference experimentally in this paper, but they are one reason we are interested in NS/QD as an exploration method for RL. One commonality among the previous methods is that the exploration is performed by a *single* agent, a choice that has interesting consequences for learning. To illustrate these consequences, we borrow an example from @stanton2016curiosity. Imagine a cross-shaped maze (Fig. \[fig:stanton\_maze\]) where to go in each cardinal direction an agent must master a different skill (e.g. going north requires learning to swim, west requires climbing mountains, east requires walking on sand, and south requires walking on ice). Assume rewards may or may not exist at the end of each corridor, so all corridors need to be explored. A single agent has two extreme options, either a depth-first search that serially learns to go to the end of each corridor, or a breadth-first search that goes a bit further in one direction, then comes back to the center and goes a bit further in another direction, etc. Either way, to get to the end of each hallway, the agent will have to at least have traversed each hallway once and thus will have to learn all four sets of skills. With the breadth-first option, all four skillsets must be mastered, but a much longer total distance is traveled.
![**Hypothetical Hard Exploration Maze.** In this maze, the agent needs to traverse 4 different terrains to obtain rewards associated with the “?” boxes. Traversing each terrain requires learning a certain skill (i.e. climbing, swimming, etc.). The sprites are from a Super Mario Bros. environment introduced by @paquette2016.[]{data-label="fig:stanton_maze"}](stanton_maze){width="0.35\columnwidth"}
In both cases, another problem arises because, despite recent progress [@kirkpatrick2017overcoming; @velez2017diffusion], neural networks still suffer from catastrophic forgetting, meaning that as they learn new skills they rapidly lose the ability to perform previously learned ones [@french1999catastrophic]. Due to catastrophic forgetting, at the end of learning there will be an agent specialized in one of the skills (e.g. swimming), but all of the other skills will have been lost. Furthermore, if any amount of the breadth-first search strategy is employed, exploring each branch a bit further will require relearning that skill mostly from scratch each iteration, significantly slowing exploration. Even if catastrophic forgetting could be solved, there may be limits on the cognitive capacity of single agents (as occurs in humans), preventing one agent from mastering all possible skills.
A different approach is to explore with a *population* of agents. In that case, separate agents could become experts in the separate tasks required to explore in each direction. That may speed learning because each agent can, in parallel, learn only the skills required for its corridor. Additionally, at the end of exploration a specialist will exist with each distinct skill (versus only one skill remaining in the single-agent case). The resulting population of specialists, each with a different skill or way of solving a problem, can then be harnessed by other machine learning algorithms that efficiently search through the repertoire of specialists to find the skill or behavior needed in a particular situation [@cully:nature15; @cully2013behavioral]. The skills of each specialist (in any combination or number) could also then be combined into a single generalist via policy distillation [@rusu2015policy]. A further benefit of the population-based approach is, when combining exploration with some notion of quality (e.g. maximizing reward), a population can try out many different strategies/directions and, once one or a few promising strategies are found, the algorithm can reallocate resources to pursue them. The point is not that population-based exploration methods are better or worse than single-agent exploration methods (when holding computation constant), but instead that they are a different option with different capabilities, pros, and cons, and are thus worth investigating [@stanton2016curiosity]. Supporting this view, recent work has demonstrated the benefits of populations for deep learning [@jaderberg2017population; @miikkulainen:arxiv17].
Choosing an appropriate behavior characterization
-------------------------------------------------
Although optimizing for novelty can be a useful exploration signal for RL agents, the efficacy of optimizing for novelty is determined by the choice of behavior characterization (BC). Often, a BC can be difficult to specify for complex environments, for which the principal axes of effective exploration cannot be enumerated. However, choosing an informative BC, like choosing an informative reward function, is a useful way to inject domain knowledge to induce specific agent behavior. In cases where designing the BC is cumbersome, methods do exist to systematically derive or learn them [@gomes2014systematic; @meyerson:gecco16]. The Atari results for NSR-ES and NSRA-ES also suggest that even if the BC is not selected carefully for the domain (the RAM state was not designed to be a BC, and thus being diverse in some RAM state dimensions does not lead to fruitful exploration), novelty with respect to the BC is still a useful exploration signal when combined with reward. Although determining appropriate BCs is out of the scope of this work, we believe it is fruitful to investigate the effect of BC choice on exploration in future work and that doing so may further improve performance.
Preserving scalability {#scalable_si}
----------------------
As shown in @es, ES scales well with the amount of computation available. Specifically, as more CPUs are used, training times reduce almost linearly, whereas DQN and A3C are not amenable to massive parallelization. NS-ES, NSR-ES and NSRA-ES, however, enjoy the same parallelization benefits as ES because they use an almost identical optimization process. The addition of an archive between agents in the meta-population does not hurt scalability because $A$ is only updated after $\theta^{m}_{t}$ has been updated. Since $A$ is kept fixed during the calculation of $N(\theta_{t}^{i,m}, A)$ and $f(\theta^{i,m}_{t})$ for all $i=1...n$ perturbations, the coordinator only needs to broadcast $A$ once at the beginning of each generation. In all algorithms, the parameter vector $\theta^{i}_{t}$ must be broadcast at the beginning of each generation and since $A$ generally takes up much less memory than the parameter vector, broadcasting both would incur effectively zero extra network overhead. NS-ES, NSR-ES, and NSRA-ES do however introduce an additional computation conducted on the coordinator node. At the start of every generation we must compute the novelty of each candidate $\theta^{m}_{t}; m\in
\{1,...,M\} $. For an archive of length $n$ this operation is $O(Mn)$, but since $M$ is small and fixed throughout training this cost is not significant in practice. Additionally, there are methods for keeping the archive small if this computation becomes an issue [@lehman:ecj11].
NS-ES, NSR-ES, and NSRA-ES Algorithms {#ns-es-algo}
-------------------------------------
learning rate $\alpha$, noise standard deviation $\sigma$, number of policies to maintain $M$, iterations $T$, behavior characterization $b(\pi_{\theta})$ $M$ randomly initialized policy parameter vectors $\{\theta^{1}_{0}, \theta^{2}_{0}, ..., \theta^{M}_{0}\}$, archive $A$, number of workers $n$ Compute $b(\pi_{\theta^{j}_{0}})$ Add $b(\pi_{\theta^{j}_{0}})$ to $A$ Sample $\theta^{m}_{t}$ from $\{\theta^{1}_{t}, \theta^{2}_{t}, \ldots, \theta^{M}_{t}\}$ via eq.\[eq:novelty\] Sample $\epsilon_{i} \sim \mathcal{N}(0, \sigma^{2}I)$ Compute $\theta_{t}^{i,m} = \theta^{m}_{t} + \epsilon_{i}$ Compute $b(\pi_{\theta_{t}^{i,m}})$ Compute $N_{i} = N(\theta_{t}^{i,m}, A)$ Send $N_{i}$ from each worker to coordinator Set $\theta^{m}_{t+1}$ = $\theta^{m}_{t} + \alpha \frac{1}{n\sigma} \sum_{i=1}^{n} N_{i} \epsilon_{i}$ Compute $b(\pi_{\theta^{m}_{t+1}})$ Add $b(\pi_{\theta^{m}_{t+1}})$ to $A$
learning rate $\alpha$, noise standard deviation $\sigma$, number of policies to maintain $M$, iterations $T$, behavior characterization $b(\pi_{\theta})$ $M$ sets of randomly initialized policy parameters $\{\theta^{1}_{0}, \theta^{2}_{0}, ..., \theta^{M}_{0}\}$, archive $A$, number of workers $n$ Compute $b(\pi_{\theta^{j}_{0}})$ Add $b(\pi_{\theta^{j}_{0}})$ to $A$ Sample $\theta^{m}_{t}$ from $\{\theta^{0}_{t}, \theta^{1}_{t}, \ldots, \theta^{M}_{t}\}$ via eq. \[eq:novelty\] Sample $\epsilon_{i} \sim \mathcal{N}(0, \sigma^{2}I)$ Compute $\theta_{t}^{i,m} = \theta^{m}_{t} + \epsilon_{i}$ Compute $b(\pi_{\theta_{t}^{i,m}})$ Compute $N_{i} = N(\theta_{t}^{i,m}, A)$ Compute $F_{i} = f(\theta_{t}^{i,m})$ Send $N_{i}$ and $F_{i}$ from each worker to coordinator Set $\theta^{m}_{t+1}$ = $\theta^{m}_{t} + \alpha \frac{1}{n\sigma} \sum_{i=1}^{n} \frac{N_{i} + F_{i}}{2} \epsilon_{i}$ Compute $b(\pi_{\theta^{m}_{t+1}})$ Add $b(\pi_{\theta^{m}_{t+1}})$ to $A$
learning rate $\alpha$, noise standard deviation $\sigma$, number of policies to maintain $M$, iterations $T$, behavior characterization $b(\pi_{\theta})$ $M$ sets of randomly initialized policy parameters $\{\theta^{1}_{0}, \theta^{2}_{0}, ..., \theta^{M}_{0}\}$, archive $A$, number of workers $n$, initial weight $w$, weight tune frequency $t_w$, weight delta $\delta_{w}$ Compute $b(\pi_{\theta^{j}_{0}})$ Add $b(\pi_{\theta^{j}_{0}})$ to $A$ $f_{best} = -\infty$ $t_{best} = 0$ Sample $\theta^{m}_{t}$ from $\{\theta^{0}_{t}, \theta^{1}_{t}, \ldots, \theta^{M}_{t}\}$ via eq. \[eq:novelty\] Sample $\epsilon_{i} \sim \mathcal{N}(0, \sigma^{2}I)$ Compute $\theta_{t}^{i,m} = \theta^{m}_{t} + \epsilon_{i}$ Compute $b(\pi_{\theta_{t}^{i,m}})$ Compute $N_{i} = N(\theta_{t}^{i,m}, A)$ Compute $F_{i} = f(\theta_{t}^{i,m})$ Send $N_{i}$ and $F_{i}$ from each worker to coordinator Set $\theta^{m}_{t+1}$ = $\theta^{m}_{t} + \alpha \frac{1}{n\sigma} \sum_{i=1}^{n} w * N_{i}\epsilon_{i} + (1 - w) * F_{i} \epsilon_{i}$ Compute $b(\pi_{\theta^{m}_{t+1}})$ Compute $f(\theta^{m}_{t+1})$ $ w = min(1, w + \delta_{w})$ $t_{best} = 0$ $f_{best} = f(\theta^{m}_{t+1})$ $t_{best} = t_{best} + 1$ $w = max(0, w - \delta_{w})$ $t_{best} = 0$ Add $b(\pi_{\theta^{m}_{t+1}})$ to $A$
For the NSRA-ES algorithm, we introduce 3 new hyperparameters: $w$, $t_w$, $\delta_w$. These quantities determine the agent’s preference for novelty or reward at various phases in training. For all of our experiments, $w=1.0$ initially, meaning that NSRA-ES initially follows the gradient of reward only. If the best episodic reward seen, $f_{best}$, does not increase in $t_w=50$ generations, $w$ is decreased by $\delta_w=0.05$ and gradients with respect to the new weighted average of novelty and reward are followed. The process is repeated until a new, higher $f_{best}$ is found, at which point $w$ is increased by $\delta_w$. Intuitively the algorithm follows reward gradients until the increases in episodic reward plateau, at which point the agent is increasingly encouraged to explore. The agent will continue to explore until a promising behavior (i.e. one with higher reward than seen so far) is found. NSRA-ES then increases $w$ to pivot back towards exploitation instead of exploration.
Atari training details {#atari_train}
----------------------
Following @es, the network architecture for the Atari experiments consists of 2 convolutional layers (16 filters of size 8x8 with stride 4 and 32 filters of size 4x4 with stride 2) followed by 1 fully-connected layer with 256 hidden units, followed by a linear output layer with one neuron per action. The action space dimensionality can range from 3 to 18 for different games. ReLU activations are placed between all layers, right after virtual batch normalization units [@salimans2016]. Virtual batch normalization is equivalent to batch normalization [@ioffe2015batch], except that the layer normalization statistics are computed from a reference batch chosen at the start of training. In our experiments, we collected a reference batch of size 128 at the start of training, generated by random agent gameplay. Without virtual batch normalization, Gaussian perturbations to the network parameters tend to lead to single-action policies. The lack of action diversity in perturbed policies cripples learning and leads to poor results [@es].
The preprocessing is identical to that in @a3c. Each frame is downsampled to 84x84 pixels, after which it is converted to grayscale. The actual observation to the network is a concatenation of 4 subsequent frames and actions are executed with a *frameskip* of 4. Each episode of training starts with up to 30 random, no-operation actions (no-ops). Policies are also evaluated using a random number (sampled uniformly from 1-30) of no-op starts, whereas @a3c evaluates policies using starts randomly sampled from the initial portion of human expert trajectories (a dataset we do not have access to).
For all experiments, we fixed the training hyperparameters for fair comparison. Each network is trained with the Adam optimizer [@kingma2014adam] with a learning rate of $\eta=10^{-2}$ and a noise standard deviation of $\sigma=0.02$. The number of samples drawn from the population distribution each generation was $n=5000$. For NS-ES, NSR-ES, and NSRA-ES we set $M=3$ as the meta-population size and $k=10$ for the nearest-neighbor computation, values that were both chosen through an informal hyperparameter search. We lowered $M$ because the Atari network is much larger and thus each generation is more computationally expensive. A lower $M$ enables more generations to occur in training. We trained ES, NS-ES, NSR-ES, and NSRA-ES for a the same number of generations $T$ for each game. The value of $T$ varies between 150 and 300 depending on the number of timesteps per episode of gameplay (i.e. games with longer episodes are trained for 150 generations and vice versa). The figures in SI Sec. \[atari\_learn\_plots\] show how many generations of training occurred for each game.
Humanoid Locomotion problem training details {#mujoco_train}
--------------------------------------------
The domain is the MuJoCo Humanoid-v1 environment in OpenAI Gym [@brockman]. In it, a humanoid robot receives a scalar reward composed of four components per timestep. The robot gets positive reward for standing and velocity in the positive $x$ direction, and negative reward for ground impact energy and energy expended. These four components are summed across every timestep in an episode to get the total reward. Following the neural network architecture outlined by @es, the neural network is a multilayer perceptron with two hidden layers containing 256 neurons each, resulting in a network with 166.7K parameters. While small (especially in the number of layers) compared to many deep RL architectures, this network is still orders of magnitude larger than what NS has been tried with before. The input to the network is the observation space from the environment, which is a vector $\in \mathbb{R}^{376}$ representing the state of the humanoid (e.g. joint angles, velocities) and the output of the network is a vector of motor commands $\in \mathbb{R}^{17}$ [@brockman].
For all experiments, we fixed the training hyperparameters for fair comparison. Each network was trained with the Adam optimizer [@kingma2014adam] with a learning rate of $\eta=10^{-2}$ and a noise standard deviation of $\sigma=0.02$. The number of samples drawn from the population distribution each generation was $n=10000$. For NS-ES, NSR-ES, and NSRA-ES we set $M=5$ as the meta-population size and $k=10$ for the nearest-neighbor computation, values that were both chosen through an informal hyperparameter search. We trained ES, NS-ES, NSR-ES, and NSRA-ES for the same number of generations $T$ for each game. The value of $T$ is 600 for the Humanoid Locomotion problem and 800 for the Humanoid Locomotion with Deceptive Trap problem.
Humanoid Locomotion problem tabular results {#mujoco tabular results}
-------------------------------------------
Environment ES NS-ES NSR-ES NSRA-ES
------------- ------------ -------- -------- ---------- --
Isotropic **8098.5** 6010.5 6756.9 7923.1
Deceptive 5.3 16.5 31.2 **37.1**
: **Final results for the Humanoid Locomotion problem.** The reported scores are computed by taking the median over 10 independent runs of the rewards of the highest scoring policy per run (each of which is the mean over $\mathtt{\sim}$30 evaluations). []{data-label="mujoco-table"}
-0.1in
Plots of Atari learning across training (generations) {#atari_learn_plots}
-----------------------------------------------------
![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](alien_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](amidar_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](bankheist_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](beamrider_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](freeway_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](frostbite_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](grav_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](montezuma_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](pacman_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](qbert_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](squest_si_plot "fig:"){width="0.3\columnwidth"} ![**Comparison of ES, NS-ES, NSR-ES, and NSRA-ES learning on 12 Atari games.**[]{data-label="fig:alien_plot"}](zaxxon_plot "fig:"){width="0.3\columnwidth"}
Overhead plots of agent behavior on the Humanoid Locomotion with Deceptive Trap Problem. {#overhead_plots}
----------------------------------------------------------------------------------------
![**Overhead plot of ES (left) and NS-ES (right) across 10 independent runs on the Humanoid Locomotion with Deceptive Trap problem.**[]{data-label="fig:es_overhead"}](es_comp "fig:"){width="0.4\columnwidth"} ![**Overhead plot of ES (left) and NS-ES (right) across 10 independent runs on the Humanoid Locomotion with Deceptive Trap problem.**[]{data-label="fig:es_overhead"}](nses_comp "fig:"){width="0.4\columnwidth"}
![**Overhead plot of NSR-ES (left) and NSRA-ES (right) across 10 independent runs on the Humanoid Locomotion with Deceptive Trap problem.**[]{data-label="fig:nsres_overhead"}](nsres_comp "fig:"){width="0.5\columnwidth"} ![**Overhead plot of NSR-ES (left) and NSRA-ES (right) across 10 independent runs on the Humanoid Locomotion with Deceptive Trap problem.**[]{data-label="fig:nsres_overhead"}](nsraes_comp "fig:"){width="0.4\columnwidth"}
[^1]: Equal contribution, corresponding authors: `vashisht@uber.com, edoardo.conti@gmail.com`.
|
---
author:
- Skylar Sible
- 'Rodrigo Iza-Teran'
- Jochen Garcke
- Nikola Aulig
- Patricia Wollstadt
bibliography:
- 'deformation\_modes.bib'
title: A Compact Spectral Descriptor for Shape Deformations
---
INTRODUCTION
============
Modern engineering product design relies heavily on computer-aided engineering (CAE) methods such as finite-element based simulation or computational optimization. With increasing computational capabilities and shorter design cycles in many industries, CAE methods are applied more and more extensively. As a result, a tremendous amount of data has become available, especially due to many components being repeatedly re-designed or optimized. This data provides potential to apply state-of-the-art machine learning techniques [@Burrows2011; @Garcke2017; @Zhao2010] including deep learning approaches [@Georgiou2018; @Guo2018; @Umetani2017] in order to increase the efficiency and quality of the design process, but also to handle the amount of generated data itself [@Burrows2011; @Spruegel2018]. One obstacle are the unstructured and finely detailed mesh representations typically used for design parts resulting in high-dimensional data vectors. Hence, to enable the application of computational tools, designers have to find suitable representations and parameterizations for both, components under development as well as related design criteria [@Graening2014; @Spruegel2018].
A critical design criterion in many application domains is the plastic deformation of a component under stress. For example, in automotive design, components are developed to exhibit a specific plastic deformation behavior during a crash in order to protect vehicle occupants [@DuBois2004]. A component’s crash performance is improved if its plastic deformation avoids intrusion into the passenger cabin or allows for high energy absorption [@DuBois2004; @Fang2017; @Liu2014]. Hence, the ability to computationally analyze or optimize for specific deformation behaviors is a powerful design tool.
Current CAE-approaches typically study deformation behavior of a design part by representing the part as a polygon surface mesh and quantifying the displacement of a subset of selected nodes, for instance, during a crash simulation [@Fang2017] (see also [@Redhe2004] for an example). Such a parameterization is computationally efficient because it requires to only monitor a small set of quantities of interest during the simulation—yet, it is limited in terms of the geometric complexity it is able to represent. For example, a bending of a part may be easily quantified by the displacement of a single node in a specified direction (Figure \[fig:deformation\_modes\]A), whereas more complex deformations, such as an axial crushing of the part, is not as easily quantified by the displacement of pre-selected mesh nodes (Figure \[fig:deformation\_modes\]B). Hence, the evaluation of deformation behavior, e.g., as a result of a crash simulation, requires an expert to manually check the resulting deformation and to alter the design if necessary. As a result, the development of a component requires many iterations of simulation and visual inspection (e.g., [@Tang2016]), which makes the design process costly.
![Modes of deformation typically encountered in automotive engineering [@DuBois2004]: (A) Upward and downward bending deformation mode, (B) axial crushing or folding deformation mode.[]{data-label="fig:deformation_modes"}](deformation_modes.png){width="47.50000%"}
In this paper we present a novel representation for geometric deformations using a compact descriptor that allows for the representation of complex deformation modes using methods from spectral mesh processing (e.g., [@Sorkine2005; @Zhang2010]). Spectral mesh processing represents geometric shapes as a set of coefficients with respect to an eigenbasis obtained from spectral decomposition of a discrete mesh operator (see next section). We propose to represent deformations by identifying the subset of spectral coefficients, sufficient to describes a shape’s plastic deformation with respect to its original form, and present a workflow to select these relevant spectral coefficients. In the following, we will first introduce spectral mesh processing and its application in the engineering domain, before we formally introduce our proposed method and demonstrate its application in a filtering task, relevant to engineering contexts. We conclude with a discussion of the results and directions for further research.
SPECTRAL MESH PROCESSING IN THE ENGINEERING DOMAIN
==================================================
Spectral mesh processing is an approach to geometry processing that, similar to Fourier analysis in the 1D-domain, is based on the eigendecomposition of a suitable, discrete operator defined on the mesh (e.g., a discrete Laplace-Beltrami operator [@Zhang2010]). The decomposition returns eigenvectors that form an orthonormal basis into which any discrete function defined on the mesh can be projected, yielding a set of spectral coefficients. A common approach is to interpret the mesh vertices’ Euclidean coordinates as mesh functions and project them into the eigenbasis to obtain a spectral representation of the mesh geometry. This transformation is invertible such that the geometry’s spatial representation can be reconstructed from its spectral representation.
Representing mesh geometries in the spectral domain has interesting properties and allows to address various geometric processing tasks more easily. For example, obtained eigenvectors represent different geometric contributions to a geometry, where eigenvectors associated with smaller eigenvalues describe low-frequency functions on the mesh, while larger eigenvalues describe high-frequency functions [@Sorkine2005] (where the ordering depends on the operator used). This property is used in spectral mesh compression [@Karni2000], which efficiently stores geometries in terms of their first $M$ low-frequency eigenvectors and corresponding spectral coefficients. This approach assumes that most geometries can be approximated sufficiently by their first $M$ low-frequency, spectral components. From this reduced set of spectral components, the geometry’s spatial representation can be reconstructed with acceptable loss. Similar spectral approximations have been used to realize shape matching and retrieval based on a subset of eigenvectors [@Reuter2005; @Reuter2006]. Selecting a subset of spectral components for reconstruction using an orthogonal basis also relates to nonlinear approximation, see e.g. DeVore [@devore], and has also been used in image processing [@Peyre].
In the engineering domain, it has been shown that spectral mesh representations using the Laplace-Beltrami operator are an adequate choice for representing several deformations in a compact way [@Garcke2017; @IzaTeran2016Thesis; @IzaTeran2018]. Here, an approach has been proposed that uses the decomposition of one Laplace-Beltrami operator for a series of deformations, which are assumed to be isometric to a base shape. Spectral coefficients obtained by projecting several mesh deformations in $x$-, $y$- and $z$-directions into a common eigenbasis are used as a dimensionality reduction and visualization technique for large sets of deformed shapes resulting from crash simulations. It has also been proposed that certain eigenvectors can be interpreted as specific geometric operations on the shape, e.g., a translation in the Euclidean space [@IzaTeran2018]. Furthermore, the presented approach allows to handle arbitrary discrete functions defined on the mesh, so that additional functional properties of a part, such as plastic strain or stress, may be represented in the spectral domain.
Finally, note that in geometry processing the spectral representation of a shape is often used to find pose invariant representations, i.e., to distinguish a shape independent of pose changes by using the eigenfunctions as shape features. On the other hand, in the studied engineering applications part of the objective is to distinguish them in a pose dependent way, which is achieved by projecting the deformations as functions into the (joint) eigenbasis.
PROPOSED METHOD
===============
Building on previous work in spectral mesh processing, the present paper proposes to solve the problem of efficiently representing geometric deformations by constructing a compact, spectral descriptor through the targeted and adaptive selection of spectral components. In particular, we propose to select the subset of only those components that are relevant for representing application-specific deformations in the spatial domain, and not just use the eigenvalue order as a fixed global criteria for the subset selection. Relevant components can be identified by the relative magnitude of their spectral coefficients, which indicates the importance of an eigenvector for representing the geometry in the spatial domain. In other words, opposed to previous applications such as mesh compression, we propose to select a subset of spectral components not based on their corresponding geometric frequency, but based on their contribution to the spatial representation irrespective of frequency, as indicated by the magnitude of their spectral coefficients.
In particular, we propose a workflow to find a compact spectral descriptor for deformations found in a set of geometries in three steps: a) compute a common spectral basis that can be used for all simulations, b) identify deformation modes in the set of simulations (e.g., through clustering in the spectral domain), and c) obtain spectral descriptors by identifying relevant spectral components for each deformation mode.
The obtained spectral descriptor is significantly smaller than the full mesh representation in the spatial domain, while preserving a high amount of relevant geometric information, yielding an efficient representation of geometric deformations for further computational analysis. Furthermore, we demonstrate empirically that the resulting descriptor provides an abstract representation of the deformation behavior by applying it in a nearest-neighbor search to identify similar simulation results in a filtering task.
Spectral mesh representation
----------------------------
In order to describe the generation of the descriptor, we first provide a formal definition of spectral mesh processing. Let $K=(G, P)$ be a triangle mesh embedded in $\mathbb{R}^3$ with a graph $G=(V\!,\,E)$ describing the connectivity of the mesh, where $V$ are mesh vertices with $|V|=N$ and $E
\subseteq V \times V$ the set of edges. The matrix $P \in \mathbb{R}^{N \times
3}$ describes the coordinates of mesh vertices in Euclidean space such that each vertex has coordinates $\mathbf{p}_i = (x_i, y_i, z_i)$. We view $K$ as an approximation of the Riemannian manifold $\mathcal{M}$, isometrically embedded into $\mathbb{R}^3$. Furthermore, let $f: \mathcal{M} \rightarrow \mathbb{R}$ be a continuous function on $\mathcal{M}$. Evaluating this function at vertices $V$ yields the discrete mesh function $f_K : K \rightarrow \mathbb{R}$.
We may now define a discrete, linear operator on $\mathcal{M}$ (see [@Zhang2010] for a discussion of possible operators). We here consider the Laplace-Beltrami operator, which is defined as the divergence of the gradient in the intrinsic geometry of the shape and is a generalization of the Laplace operator to Riemannian manifolds. The operator is invariant under isometric transformations, i.e., transformations that preserve geodesic distances on the shape. In the following, we thus assume that we operate on sets of shapes that are isometries of each other in order to assume constant eigenbases between deformed shapes. In practice, numerically $\epsilon$-isometries will be present, which will result in only approximately the same eigenbasis. Spectral mesh decomposition still can be used accordingly since under suitable conditions the Laplacians only differ by a scaling factor in such a case [@IzaTeran2018].
The eigendecomposition of the operator returns eigenvectors $E=\left[\psi_1,
\psi_2, \ldots, \psi_N \right]$, ordered by the magnitude of the corresponding eigenvalues, $\lambda_1<\lambda_2<\ldots<\lambda_N$, where each eigenvector corresponds to a frequency component of the mesh function in increasing order [@Sorkine2005]. Note that opposed to classical Fourier transform where basis functions are fixed, the orthonormal basis obtained from the spectral decomposition of a mesh operator depends on the mesh geometry and operator used. Often, only the set of the first $M$ eigenvectors ordered by the magnitude of eigenvalues, $E_M$, is used for further processing such that high-frequency components are discarded.
The normalized eigenvectors $E$ of a symmetric operator form an orthonormal basis. We therefore may map any discrete mesh function $f_K$, given by a vector $\mathbf{f}$, into this basis to obtain a representation, or encoding, in the spectral domain,
$$\hat{\mathbf{f}} = E^\top\mathbf{f},
\label{eq:spectrum_mesh_function_mat}$$
where the columns of $E$ are eigenvectors $\{\psi_i\}^N_{i=1}$ and $\hat{\mathbf{f}}$ contains corresponding spectral coefficients $\{\alpha_i\}_{i=1}^N$. The spectral coefficients are thus obtained by calculating $\{\alpha_i\}_{i=1}^N=\langle \mathbf{f}, \psi_i\rangle$. The inverse transform reconstructs, or decodes, the mesh function in the spatial domain,
$$\mathbf{f} = E\,\hat{\mathbf{f}}.
\label{eq:spectrum_mesh_function_mat_inverse}$$
By considering Euclidean coordinates, $P = \left[ \mathbf{f}_x, \mathbf{f}_y,
\mathbf{f}_z \right] $, as mesh functions we project the mesh geometry into the spectral domain,
$$\hat{P} = E^\top P,
\label{eq:spectrum_mesh_coords_mat}$$
such that each row of $\hat{P}$, $\hat{\mathbf{p}}_i = \left[
\alpha^x_i, \alpha^y_i, \alpha^z_i \right], i=1,\ldots,N$, contains spectral coefficients that can be used to express $x$-, $y$-, and $z$-coordinates as
$$\mathbf{f}_x = \sum_{i=1}^N \alpha_i^x \psi_i ,\,\,
\mathbf{f}_y = \sum_{i=1}^N \alpha_i^y \psi_i ,\,\,
\mathbf{f}_z = \sum_{i=1}^N \alpha_i^z \psi_i.
\label{eq:spectrum_cart_coords_reconstruction}$$
An approximation of the mesh geometry is obtained by using the coefficients corresponding to the first $M \ll N$ eigenfunctions, or suitably selected ones.
In a CAE-application, the spectral representation may now be used to project a set of geometries, each represented as three-valued mesh functions, into the same spectral basis, allowing for a joint handling of the data in a common space. This approach assumes that geometries are available in a regular mesh format and that the mesh is the same, or is suitably interpolated, for all geometries in the set. A single Laplace-Beltrami operator is then computed for the set of deformations [@Garcke2017; @IzaTeran2018]. Observe that the operator is computed using geodesic distances along the surface of a shape, where one assumes that the deformations do not modify this distance. As a result, the approach yields a common representation for deformations using the spectral coefficients obtained by projecting three functions, each for mesh deformations in $x$-, $y$- and $z$-directions, to the eigenvectors of only one shape.
We would like to explain some of the properties of the data representation by comparing to principal component analysis (PCA). In that data-driven approach, a data matrix (e.g., comprising of the deformations as vectors) gets compressed into a small number components based on the variance of the data, where the largest variance will be contained in the first principal component. As an example let us consider a series of deformations that puts two areas of a part nearby. Here, the PCA will provide principal components that concentrate on those sections of the part and will reproduce behavior similar to this one. Now, let us assume one did not include data for a deformation that affects only one area of the part. The principal components will not be able to suitably represent such an unseen and strongly different deformation behavior since it was not trained for that.
On the other hand, take the basis obtained from the Laplace-Beltraim operator, where only the shape geometry is taken into account. In this basis, the (new) deformation of the shape can be reproduced in the same fashion as the earlier ones. In other words, whereas in the PCA higher variance, which can be interpreted as higher frequencies of the input data, is discarded, in the Laplace-Beltrami basis higher (geometric) frequencies of the underlying shape are discarded. Note here also, it was shown using suitable assumptions that for functions bounded in the $H^1$-Sobolev norm the $L_2$-approximation using the orthonormal basis obtained from the Laplace operator is optimal in a certain best basis sense [@Brezis2017]; this result can be extended to the Laplace-Beltrami operator and functions in the Sobolev space $H^{2,2}$ on the underlying manifold [@Tesch2018].
Finding an efficient spectral descriptor for plastic deformations
-----------------------------------------------------------------
In the spectral domain, individual eigenvectors, $\psi_i$, can be interpreted as geometric contributions to the spatial representation of the shape (geometric frequencies), where in general eigenvectors with low eigenvalues represent more low-frequency contributions and those with high eigenvalues high-frequency contributions (depending on the operator used). Furthermore, eigenvectors may be associated with specific geometric transformations of the shape, such that changes in the corresponding spectral coefficients can even have a mathematical (e.g., rotation of a shape in the underlying space) or physical interpretation (e.g., deformation in parts of the shape).
When projecting Euclidean coordinates, $P = \left[\mathbf{f}_x, \mathbf{f}_y,
\mathbf{f}_z \right] $, into the eigenbasis, the magnitude of resulting spectral coefficients associated with each eigenvector, $\{\alpha_i^x,\alpha_i^y,\alpha_i^z\}^N_{i=1}$, represents the relevance of that eigenvector’s geometric contribution to the deformed shape’s spatial representation [@IzaTeran2018]. Based on this property, we propose to find a compact spectral descriptor, $\mathbf{S}$, for a deformation, by selecting only those coefficients, $\alpha^{(\cdot)}_j$, that have a high relative magnitude compared to a suitable baseline. This is a further filtering in comparison to [@IzaTeran2018] or mesh compression, where a fixed number of the spectral components is used, based on the order of the corresponding eigenvalues. We assume that coefficients with high values indicate that corresponding eigenvectors, $\psi_j$, represent geometric information relevant for the description of the deformation in the spatial domain. Figure \[fig:workflow\] shows an exemplary workflow for finding $\mathbf{S}$, which is described in detail in the following.
= \[rectangle, draw, text width=18em, text centered, minimum height=3em\] = \[rectangle, draw, text width=7em, text centered, minimum height=4em\] = \[rectangle, draw, text width=18em, text centered, minimum height=4em\] = \[ text width=8em\]
(start) [start]{}; (selectgeom) [provide mesh representing desired deformation behavior]{};
(operator) [calculate mesh operator]{}; (eigproblem) [calculate eigendecomposition of mesh operator]{}; (spectralrep) [obtain spectral coefficients for mesh geometry]{}; at (0.1,-3.1) ;
(setthresh) [set initial $t$]{}; (identify) [obtain $\mathbf{S}$ by selecting coefficients $> t$]{}; (evaluate) [reconstruct spatial representation from $\mathbf{S}$]{}; (decide) [spatial reconstruction sufficient?]{}; (update) [adjust $t$]{}; at (0.1,-8.5) ;
(return) [return $\mathbf{S}$]{}; (apply) [apply $\mathbf{S}$: calculate coefficients in $\mathbf{S}$ for new geometry]{}; at (0.1,-19.5) ; (end) [end]{}; (start) – (selectgeom); (selectgeom) – (operator); (operator) – (eigproblem); (eigproblem) – (spectralrep); (spectralrep) – (setthresh); (setthresh) – (identify); (identify) – (evaluate); (evaluate) – (decide); (decide) -| node \[near start\] [no]{} (update); (update) |- (identify); (decide) – node [yes]{}(return); (return) – (apply); (apply) – (end); ($(operator.north west)+(-4.1,{0.4})$) rectangle ($(spectralrep.south east)+(0.44,-{0.4})$); ($(setthresh.north west)+(-4.1,{0.4})$) rectangle ($(return.south east)+(0.44,-{0.4})$); ($(apply.north west)+(-4.1,{0.4})$) rectangle ($(apply.south east)+(0.44,-{0.4})$);
To find $\mathbf{S}$, we first have to provide a mesh representing the geometry of a desired deformation behavior. For example, in the automotive context we may wish to describe an axial folding of a beam during a frontal crash, because this deformation mode leads to high energy absorption. Such a geometry may be selected from a set of $k$ existing, different simulations. The desired deformation may be identified either through manual selection by an expert motivated by functional requirements, or through data-driven approaches in either the spatial or spectral domain. For example, clustering of spectral coefficients may be used to reveal main modes of deformations present in the set of simulations (Figure \[fig:spectral\_basis\_cluster\_modes\]), e.g., bending or folding behavior (Figure \[fig:deformation\_modes\]). We may either cluster the geometries based on their deformation in the final time step of a simulation run, the transient data from the full simulation run over time, or some other physical quantities such as plastic strain or stress on the surface of the geometry. In particular, a suitable visual representation of the coarse behavior of the deformation can often be obtained by using the $x$-, $y$-, $z$-spectral coefficients of the first eigenvector in eq. (\[eq:spectrum\_cart\_coords\_reconstruction\]) [@Garcke2017; @IzaTeran2018].
Note that the application of the descriptor requires that the deformation, or other quantities of interest, is sufficiently represented by the initial surface mesh. In particular, the mesh resolution has to be fine enough to represent the deformation. For application domains concerned with more high-frequency deformations, this results in fine meshes with a high number of nodes. Here, the applicability of the method may be limited by the practical run time of the proposed method, which depends on the number of nodes in the mesh (see also the asymptotic run time of the proposed approach, discussed in section *Computational complexity of the proposed workflow*).
{width="80.00000%"}
Once a mesh representing the target deformation is identified, we calculate its spectral representation according to eq. (\[eq:spectrum\_mesh\_coords\_mat\]) (Figure \[fig:workflow\], red box) to obtain coefficients $\hat{\mathbf{p}}_i$. Now, several approaches to defining $\mathbf{S}$ by selecting relevant coefficients are conceivable. In many applications, e.g., computer graphics, typically the first $M$ eigenfunctions are used. In contrast to this approach, we propose to identify relevant eigenfunctions based on the magnitude of their *coefficients*. For a single shape exhibiting the targeted deformation, we may thus select all coefficients exceeding a threshold $t$,
$$\mathbf{S} = \left\{\alpha_j^x,\alpha_j^y,\alpha_j^z | \alpha_j^x > t \vee \alpha^y_j > t \vee \alpha^z_j > t\right\},
\label{eq:descriptor_single}$$
where $t$ may be set based on a statistical criterion, e.g., relative to the mean of coefficients, to identify those coefficients that indicate a high relevance for representing the shape.
Alternatively, the shape describing the desired deformation may be contrasted against the undeformed baseline shape, such that relevant coefficients can be identified by the largest differences in coefficients between baseline and deformed shape (Figure \[fig:data\_generation\]B,C). Here again the setting of a threshold $t$ for selecting the highest differences is required.
Both procedures identify components that contribute to the spatial representation of the targeted deformation while ignoring less relevant geometric components. Hence, we obtain a sparse description of the geometric information relevant to characterize a specific deformation in the spatial domain.
For both approaches, $t$ may be either set based on a statistical criterion, but may also be found through an iterative process that alternates between lowering or increasing $t$ and evaluating the reconstruction quality in the spatial domain (Figure \[fig:workflow\], blue box). Reconstruction quality may either be judged through visual inspection by an expert or through calculation of an error metric between the original and reconstructed shape. Adjusting $t$ controls for the size of the descriptor, $|\mathbf{S}| = M$, and thus allows for a trade-off between compactness and the level of geometric detail captured by the descriptor. Note that recovering the original mesh representation requires *all* spectral coefficients (according to eq. \[eq:spectrum\_mesh\_function\_mat\_inverse\]), hence—even though $\mathbf{S}$ is sufficient to represent the geometric deformation of a part—it is not possible to recover the original mesh representation from $\mathbf{S}$ alone. Nevertheless, an approximate reconstruction is still possible.
In our experiment, we demonstrate that, for a part typically encountered in engineering contexts, setting $t$ based on a statistical criterion resulted in a small descriptor size, $M \ll N$, where $N$ is the number of nodes in a triangular surface mesh. This descriptor was able to robustly filter simulation results based on their deformation mode despite the considerable reduction in description length, indicating that a high amount of application-relevant geometric detail was retained.
Application scenarios
---------------------
Once the descriptor $\mathbf{S}$ is constructed, it can be applied in further computational tasks (Figure \[fig:workflow\], green box). For example, the descriptor may be used as a feature in a machine learning task, encoding the plastic deformation behavior of a part. A possible application is meta-modelling, which aims at replacing costly simulations or optimization runs by a cheaper evaluation of statistical models. The descriptor can be used in case the design process investigates the relationship between a geometry and an objective function of interest. Learning such a meta-model requires the efficient parametrizations of the properties of interest, such as deformation behavior. Further tasks may include optimization of the part with respect to some property, e.g., material thickness, while the shape descriptor is used as a constraint or part of the objective function to ensure the desired deformation behavior. Here, the descriptor can be used to describe the targeted deformation behavior and to describe the deformation behavior of parts in intermediate steps of the optimization. As soon as the desired and actual deformation behavior diverge, measurable by an increasing distance between the descriptors, the optimization can be stopped automatically. Alternatively, the distance may be used as part of the objective function, e.g., when using evolutionary optimization techniques.
A further task, common in the engineering design process, is the filtering and verification of simulation results based on geometric properties. Often, large numbers of simulation runs are performed in order to investigate the impact of variations in design parameters on the component under development. This results in large amounts of simulation data that typically require time-intensive visual inspection by an expert in order to verify the success of individual simulations [@Zhao2010; @Burrows2011]. Our descriptor can be used both, to automatically verify the outcome of a simulation with respect to the desired deformation behavior, and to filter simulation results based on the deformation behavior (see section *Experiments*). In particular, the proposed workflow may again be used to find the spectral descriptor for the targeted deformation behavior and the actual deformation behavior of the part being optimized. To verify a result, a threshold on the acceptable distance between both descriptors can be set, allowing for the verification of large sets of simulation results without the need for manual inspection by an expert.
Note that since the Laplace-Beltrami operator is only invariant under isometric transformations, the presented approach is limited to scenarios where all shapes involved in finding and using the descriptor represent the same baseline geometry in different states of (isometric) deformation. This is however the case for many engineering applications, where properties of the structure are determined by variables that can be varied independently of the mesh geometry. Two examples are thickness of shell finite elements or material properties of finite elements such as failure criteria. Another application scenario is the variation of loading or boundary conditions whose variation can lead to different deformation behaviors while the base geometry of the component of interest stays the same. In these scenarios, our method has relevance for robustness or optimization studies, where certain deformation modes are desired as a main objective. Additionally, small changes to the geometry can be handled by interpolating the mesh to a joint reference mesh.
Computational complexity of the proposed workflow
-------------------------------------------------
In terms of computational efficiency, the most costly operations in constructing the descriptor are the preprocessing phase (Figure \[fig:workflow\], red box) comprising the calculation of the mesh operator, its eigendecomposition, and the projection of Euclidean coordinates into the eigenbases, eq. (\[eq:spectrum\_mesh\_coords\_mat\]). The asymptotic time complexity of the preprocessing is dominated by the matrix multiplications performed as part of the operator definition and its eigendecomposition, which is cubic in the number of mesh vertices, $\mathcal{O}(N^3)$. This operation is done only once and can be performed offline with respect to the application of the descriptor in a subsequent task (Figure \[fig:workflow\], green box). The time complexity is cubic if a naive algorithm is used, but also algorithms with subcubic runtimes are available (e.g., [@Cormen2009]). Furthermore, since the discrete Laplace-Beltrami matrix is inherently data sparse, fast computations of the first, say, eigenvectors using numerical approaches exploiting this data sparseness seem possible, which is part of future work. In any case, this one-time pre-processing step is small in comparison to the runtime of a single numerical simulation performed to generate one datum.
The application of the descriptor to new geometries in the online phase (Figure \[fig:workflow\], green box) requires the projection of the new geometry’s Euclidean coordinates, $P$, into the bases indicated by the descriptor, $E_{\mathbf{S}}$. The projection consists of a matrix multiplication, $E_{\mathbf{S}}^TP$, eq. (\[eq:spectrum\_mesh\_coords\_mat\]), which has asymptotic time complexity $\mathcal{O}(3MN)$, with $M \ll N$.
EXPERIMENTS
===========
Data generation
---------------
To demonstrate the effectiveness of our method, we used the proposed descriptor to represent the deformation behavior of a hat section beam in a crash simulation and filter simulation results based on deformation mode.
A hat section beam is a structural element common in engineering domains such as civil or automotive engineering (Figure \[fig:data\_generation\]A). It consists of a top-section with a hat-shaped cross-section that is joined together with a base plate.
We simulated an axial crush of the beam using LS-DYNA mpp R7.1.1 (Figure \[fig:data\_generation\]B,C), double precision, while inducing various deformation modes. On the resulting data, we used the proposed descriptor to filter results based on a desired deformation behavior.
Variation in the deformation behavior was introduced by adding “notches” in the flange of the hat section at defined locations along the length of the part (Figure \[fig:data\_generation\]A) and varying the material thickness of the part between and . A notch was simulated by setting the material thickness to in order to simulate a removal of the material at that point. The setup resulted in a simulation bundle consisting of $k=100$ simulations with a large variety of deformation modes, each with several saved time steps.
![ (A) Hat section beam used for data generation. Colored markers indicate notches introduced at various locations along the length of the shape, which lead to various different deformation behaviors shown to the left (notches denote areas of material thickness of ). (B) Simulation setup with undeformed baseline shape. (C) Exemplary final time step of simulation run showing deformed shape.[]{data-label="fig:data_generation"}](data_generation.png)
Calculation of mesh spectral coefficients and selection of shape descriptor
---------------------------------------------------------------------------
For further analysis, we considered the set of deformed shapes from the final time step of all simulation runs. The spectral representation according to eq. (\[eq:spectrum\_mesh\_coords\_mat\]) was computed, where we used the first eigenvectors with smallest eigenvalues. We selected geometries representative of one of three deformation modes, namely *upward bend*, *downward bend*, and *axial crush* (Figure \[fig:deformation\_modes\]A, B), identified through clustering of the first coefficients in the spectral domain. Alternatively, Figure \[fig:spectral\_basis\_cluster\_modes\] shows the spectral coefficients of $x$-, $y$-, $z$-coordinates corresponding to the first eigenvector for all simulation runs and time steps, from this visualization the different modes could also be selected.
For each deformation mode we obtained the spectral descriptor according to eq. (\[eq:descriptor\_single\]) by setting $t$ to one standard deviation above the mean over all coefficients for a representative shape. This approach resulted in descriptor sizes of $M=14$ for axial crush, $M=17$ for upward bend, $M=16$ for downward bend, compared to an original mesh size, $N$, of around nodes. The quality of the descriptor was validated through visual inspection of the spatial reconstruction (Figure \[fig:experiments\]A). Note that to allow for proper reconstruction, for each selected coefficient, $\alpha^{(\cdot)}_j
\in \mathbf{S}$, also the spectral coefficients corresponding to the remaining two 3D coordinates were added if not already contained in the descriptor. Furthermore, eigenvectors $\psi_1$ and $\psi_2$ were added to the reconstruction.
![(A) Mesh reconstruction for three deformation modes (color scale indicates displacement in $z$-direction): top row shows mesh reconstructions from the first 500 eigenvectors ordered by magnitude of eigenvalues; middle row shows reconstruction from the proposed descriptor with sizes $M=14$ for axial crush, $M=17$ for upward bend, $M=16$ for downward bend, bottom row shows reconstruction from the first $M=14$, $M=17$, and $M=16$ eigenvectors ordered by magnitude of their eigenvalues. (B) Cosine similarity between axial crush descriptor and all simulation runs. (C) First nine most similar simulation results for axial crush spectral descriptor (color scale indicates displacement in $z$-direction). []{data-label="fig:experiments"}](axial_crush_similarity_col.png){width="35.00000%"}
We compared the spatial reconstruction of the meshes from our descriptor of length $M$ to the reconstruction from the first $M$ low-frequency components (eigenvectors with smallest eigenvalues), i.e. $N=M$ in eq. (\[eq:spectrum\_cart\_coords\_reconstruction\]) (Figure \[fig:experiments\]A, bottom row). The latter is a common approach in dimensionality reduction or compression applications [@Karni2000; @Garcke2017]. The comparison shows that our descriptor obtained a better reconstruction quality than the reconstruction from low-frequency components alone. In particular, the high-frequency components included in the descriptor captured also finer detail on the mesh, for example, the edge of the hat section in the center of the bent parts. An approximate decoding from the sparse encoding of size $M$ is therefore possible, where the quality is good enough to reconstruct the main behavior, but not the details, e.g., the location of the axial crush is not preserved.
Application of spectral descriptor for filtering and shape retrieval
--------------------------------------------------------------------
We used the spectral descriptors to filter the set of simulation results for geometries exhibiting a specific crash behavior. As described in section *Application scenario*, filtering large-scale simulation runs based on geometric properties is a common application scenario in the engineering design process.
To filter simulation runs using the proposed descriptor, $\mathbf{S}$, we first obtained spectral coefficients for all shapes through projection into the spectral domain; we then calculated the similarity between these coefficients and coefficients in $\mathbf{S}$ using the cosine similarity. Figure \[fig:experiments\]B shows the similarity between the descriptor of the axial crush deformation mode and all simulation results. Our approach correctly identified all simulation results exhibiting an axial crushing as most similar to the spectral descriptor, with the exception of one geometry showing an upward bend. Figure \[fig:experiments\]C shows the nine simulation results most similar to the descriptor, which all show an axial crushing of the beam. Note that the approach was able to identify simulation results exhibiting an axial crushing irrespective of the exact location of the axial folding along the part.
In summary, the descriptor provided an abstract description of the deformation behavior that did not require the specification of an exact deformation, e.g., in terms of the displacement of individual nodes. The descriptor successfully represented application-relevant geometric information through the targeted selection of coefficients while being of much smaller size than the full mesh representation in the spatial domain.
CONCLUSION AND FUTURE WORK
==========================
We proposed a novel approach for the efficient representation of geometric deformations using a spectral descriptor comprising components selected in a targeted and adaptive fashion. The selection procedure ensures that only geometric components relevant for specifying the targeted deformation are included, which makes the descriptor much smaller in size than a full geometric representation of the deformation, e.g., by a surface mesh, and also smaller and more focused than using the first few hundred spectral components as in [@IzaTeran2018]. Despite its compactness, the descriptor was able to capture a high amount of application-relevant geometric information and provided the necessary descriptive power to distinguish between deformation modes in a filtering task. The trade-off between size and represented geometric detail makes the descriptor a promising tool for the parametrization of also complex geometric deformations in various computational tasks.
Future work may explore the applicability of the descriptor, for example, in tasks such as structural optimization with plastic deformation as design criterion. Here, the proposed descriptor offers a powerful design tool and may improve on current approaches that require extensive manual intervention and evaluation by an expert. Furthermore, the descriptor may be used in post-processing of simulation data by identifying or filtering results based on geometric deformation. Especially in large-scale data sets, such an approach allows to automatically verify the success of a simulation instead of requiring the visual inspection of large sets of simulation results by an expert. Lastly, the descriptor may be used as a feature in machine learning tasks on large-scale engineering data sets, representing a central design property in many application domains.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors thank Emily Nutwell of the OSU SIMCenter for support in generating the hat section model.
|
[**An efficient Gehan-type estimation for the accelerated failure time model with clustered and censored data**]{}
Liya Fu$^{a}$, Zhuoran Yang$^{a}$, Yan Zhou$^{b,*}$ and You-Gan Wang$^{c}$
[*$^a$School of Mathematics and Statistics, Xi’an Jiaotong University, China\
$^b$ College of Mathematics and Statistics, Institute of Statistical Sciences, Shenzhen University, China\
$^c$ School of Mathematical Sciences, Queensland University of Technology, Australia*]{}\
$^*$[*Email:zhouy1016@szu.edu.cn*]{}
**ABSTRACT**
In medical studies, the collected covariates usually contain underlying outliers. For clustered/longitudinal data with censored observations, the traditional Gehan-type estimator is robust to outliers existing in response but sensitive to outliers in the covariate domain, and it also ignores the within-cluster correlations. To take account of within-cluster correlations, varying cluster sizes, and outliers in covariates, we propose weighted Gehan-type estimating functions for parameter estimation in the accelerated failure time model for clustered data. We provide the asymptotic properties of the resulting estimators and carry out simulation studies to evaluate the performance of the proposed method under a variety of realistic settings. The simulation results demonstrate that the proposed method is robust to the outliers existing in the covariate domain and lead to much more efficient estimators when a strong within-cluster correlation exists. Finally, the proposed method is applied to a medical dataset and more reliable and convincing results are hence obtained.
[**Keywords**]{}: Censored data; Induced smoothing; Robust.
Introduction
=============
Censored data are very common in biomedical studies. A popular method for analyzing censored data is the Cox proportional hazards model (Cox 1972). However, when the proportional hazards assumption is violated, the Cox model may derive inconsistent parameter estimators. A semiparametric accelerated failure time model is an alternative to the Cox proportional hazards model, which is a linear model for the logarithm of the failure time and covariates with error distribution unspecified (Kalbfleish and Prentice 2002). Rank-based estimation for the accelerated failure time model with clustered/longitudinal data has been studied by some researchers in recent years (Lee et al. 1993; Lin et al. 1998; Jin et al. 2006; Wang and Fu 2011; Chiou et al. 2015; Chiou et al. 2014). The analysis of the clustered failure time data is much more complicated due to the potential within-cluster correlations and the nature of censoring. Lee et al. (1993) studied the weighted log-rank statistic and the Buckley-James method for the correlated and censored data, and provided the covariance matrix estimation of the estimating functions, which does not require specifying the error distribution. Jin et al. (2006) proposed rank-based estimating functions for multivariate failure time data and developed a novel resampling method for the covariance matrix estimation of regression parameters. Chiou et al. (2015) presented weighted rank-based estimating equations for fitting the AFT model with clustered failure times from stratified random sampling, and used the induced smoothing approach proposed by Brown and Wang (2005) to reduce the computational burden. This approach has been adapted to clustered failure time data by Johnson and Strawderman (2009) and Wang and Fu (2011).
The aforementioned methods are based on the independence working model assumption and ignore the underlying within-cluster correlations. To take account of the within-cluster correlations and improve the efficiency of estimators with similar computational complexity for clustered survival data analysis, Wang and Fu (2011) proposed splitting the Gehan weight estimating function to the between- and within-cluster estimating functions and recombining the two resultant estimators. Chiou et al. (2014) extended the generalized estimating equations approach to the clustered and censored data. In longitudinal studies, some potential outliers exist in response and/or covariates, which often result in serious problems for parameter estimation in the AFT model. The rank-based method is robust against outliers in response. However, as far as we know, the literature on parameter estimators against outliers in covariates for clustered and censored data is quite limited. Luo et al. (2014) proposed robust approaches based on the smoothed Gehan rank estimation methods, but their method was based on an independence model. This leads us to seek an efficient and doubly robust method for clustered and censored data with outliers in covariates and/or response.
In this paper, we propose weighted Gehan-type estimating functions with the induced smoothing approach, which take account of the within-cluster correlations, varying cluster sizes, and outliers in covariates and/or response. Therefore, the proposed method is robust against outliers existing in covariates and/or response. Furthermore, the induced smoothing method is utilized to eliminate computational issues resulting from the unsmoothness of the estimating functions and multiple solutions. The induced estimating functions are continuous and differentiable, which make the statistical inference convenient and provide both regression parameter estimation and their covariance matrices. The asymptotic properties of the estimators from the nonsmoothed weighted rank-based estimating functions are established. The estimators from the smoothed estimating functions are shown to be consistent and have the same asymptotic distribution as those from the nonsmooth version. The covariance of the estimators is estimated by a sandwich formula.
In Section 2, we briefly review the accelerated failure time model, and present the weighted rank-based estimating equations for the AFT models with clustered data. In Section 3, we provide computational procedures for computing the parameter estimates and their covariance and carry out simulation studies to evaluate the performance of the proposed method. In Section 4, we analyze two real medical datasets for illustration. Some conclusions are summarized in Section 5.
Weighted estimating functions
===============================
The AFT model
--------------
Suppose that there are $N$ independent clusters, and their respective cluster sizes are $n_1,\cdots,n_N$. Let $T_{ik}$ and $C_{ik}$ denote the failure time and censoring time for the $k$th member of the $i$th cluster, and let $X_{ik}$ be the corresponding $p \times 1$ vector of covariates. We assume that $(T_{i1}, \cdots, T_{i n_i} )'$ and $(C_{i1},\cdots, C_{i n_i} )'$ are independent conditional on the covariates $(X_{i1}, \cdots, X_{i
n_i} )'$. The accelerated failure time model is $$\begin{aligned}
\label{eq:model}
\log T_{ik}=X_{ik}^{\rm T}\beta+\epsilon_{ik}, \ \ k=1,\cdots,
n_i;\ \ i=1,\cdots ,N,\end{aligned}$$ where $\beta$ is the unknown regression parameter vector corresponding to the covariate vector $X_{ik}$ of dimension $p$, and $(\epsilon_{i1}, \cdots, \epsilon_{ik_i} )' $ are independent random error vectors for $i = 1, \cdots, N$. However, for each cluster, the error terms $\epsilon_{i1}, \cdots, \epsilon_{in_i}$ may be correlated. If $\tilde{T}_{ik}=T_{ik}\wedge C_{ik}$, and $\Delta_{ik}=I(T_{ik}\leq C_{ik})$, where $I(\cdot)$ is the indicator function, then the observations consist of $(\tilde{T}_{ik},\Delta_{ik},X_{ik})$.
Weighted estimating functions for AFT models
---------------------------------------------
Let $e_{ik}=\log\tilde T_{ik}-X_{ik}^{\rm T}\beta$. The rank-based estimating functions of dimension $p$ using the Gehan-type weight take the following form, $$\begin{aligned}
\label{gehan}
S_{G}(\beta)=N^{-2}\sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^{n_i}
\sum_{l=1}^{n_j}\Delta_{ik}(X_{ik}-X_{jl})I(e_{ik}-e_{jl}\leq 0),\end{aligned}$$ which is monotonic with respect to $\beta$ (Fygenson and Ritov 1994). Let $\hat\beta_G$ be the resultant estimator from (\[gehan\]), which can be also obtained by minimizing the following scalar objective function, $$\begin{aligned}
L_{G}(\beta) &=&N^{-2}\sum_{i=1}^N\sum_{j=1}^N \sum_{k=1}^{n_i}
\sum_{l=1}^{n_j}\Delta_{ik}(e_{ik}-e_{jl})^{-},\end{aligned}$$ where $e^{-}=|e|I(e<0)$.
Because $S_G(\beta)$ is based on the independent working correlation assumption, the efficiency of $\hat\beta_G$ can be enhanced by accounting for the within-cluster correlations and the impacts of varying cluster sizes. Furthermore, $\hat\beta_G$ is robust against outliers in response and is sensitive to outliers in covariates. To seek doubly robust and efficient parameter estimates, we propose the following weighted estimating functions $$\begin{aligned}
S_{\omega
h}(\beta)=N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}
\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}\Delta_{ik}(X_{ik}-X_{jl})I(e_{jl}<
e_{ik}),\end{aligned}$$ where $\omega_i$ and $h_{ik}$ are weights to be specified. Let $\hat\beta_{\omega h}$ be the estimator from $S_{\omega
h}(\beta)=0$, which can be also derived by minimizing the following objective function $$\begin{aligned}
L_{\omega h}(\beta) &=&N^{-2}\sum_{i=1}^N\sum_{j=1}^N
\sum_{k=1}^{n_i}
\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}\Delta_{ik}(e_{ik}-e_{jl})^{-}.\end{aligned}$$
For $\omega_i$, in a general way, we can select weights including $\omega_i= 1$, $\omega_i = 1/n_i$, and $\omega_i = \{1 + (n_i -
1)\bar{\rho}\}^{-1}$, where $\bar{\rho}$ is the average within-cluster correlation and is obtained using the moment estimator from Wang and Carey (2003) given a consistent estimation for $\beta$, $$\hat{\bar{\rho}}=\frac{\sum_{i=1}^N\sum_{j\neq l}^{n_i} (r_{ij}-\bar{r})(r_{il}-\bar{r})}{\sum_{i=1}^N(n_i-1)\sum_{j=1}^{n_i}(r_{ij}-\bar{r})^2},$$ where $r_{ij}$ is the rank of the corresponding residual term $e_{ij}$, and $\bar{r} = (M+1)/2$ is the average of the rank sum of all $\{e_{ik}\}$. In this paper, we use the third weight to incorporate the within-cluster correlations. For weight $h_{ik}$, we use the generalized rank (GR) estimation (Naranjo and Hettmansperger 1994) defined by $$h_{ik}=\min \left\{1,\left[\frac{c}{d^2_i(X_{ik})}\right]^{\alpha/2}\right\}.$$ Here, $c$ and $\alpha$ correspond to tuning constants and $d^2_i(X_{ik})$ denotes the squared Mahalanobis distance of $X_{ik}$ based on the robust estimates of location and dispersion for the design set $\{X_{ik}\}$ (Rousseeuw and van Zomeren 1990). For the tuning parameters $\alpha$ and $c$, we use $\alpha=2$ and $c=\chi_{0.95}^2(p)$ which is the $95$th percentile of a $\chi^2(p)$-distribution. Specifically, when $\omega_i=\omega_j= 1$ and $h_{ik}=h_{jl}=1$, $S_{\omega h}(\beta)$ corresponds to the classical Gehan-type estimating function $S_G(\beta)$.
Denote $\beta_0$ as the true value of $\beta$. According to Lee et al. (1993) and Jin et al. (2006), under some regularity conditions, the limiting distribution of $\sqrt{N}(\hat\beta_{\omega
h}-\beta_0)$ follows a zero-mean multivariate normal distribution, and the asymptotic covariance matrix of $\sqrt{N}\hat\beta_{\omega
h}$ is $$\label{eq:sigmaW}
\Sigma_{\omega h}=\{D_{\omega h}(\beta_0)\}^{-1}V_{\omega h}\{D_{\omega h}(\beta_0)\}^{-1},$$ where $D_{\omega h}(\beta)=\partial{E(S_{\omega
h}(\beta))}/\partial{\beta^{\rm T}}$ and $V_{\omega
h}=\lim_{N\rightarrow \infty}\Cov\{\sqrt{N}S_{\omega h}(\beta_0)\}$. According to Lee et al. (1993), we deduce the limiting variance matrix of $\sqrt{N}S_{\omega h}(\beta_0)$ given as follows: $$\hat V_{\omega h}=\frac{1}{N}\sum_{i=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_i}\omega_i^2h_{ik}h_{il} \hat\xi_{ik}(\beta)\hat\xi_{il}^{\rm
T}(\beta),$$ where $$\begin{aligned}
\tiny\hat{\xi}_{ik}(\beta)&=&\sum_{j=1}^N\sum_{f=1}^{n_j}\omega_j\left\{\frac{\Delta_{ik}}{N}h_{jf}(X_{ik}-X_{jf})I(e_{ik}<
e_{jf})
- \frac{\Delta_{jf}}{N}z_{ikrs}I(e_{ik}\geq
e_{jf})\right\},\end{aligned}$$ $$\begin{aligned}
z_{ikrs}=\frac{\sum_{r=1}^N\sum_{s=1}^{n_r}\omega_rh_{rs}(X_{ik}-X_{rs})I(e_{rs}\geq e_{jf})}{\sum_{m=1}^N\sum_{t=1}^{n_m}I(e_{mt}\geq e_{jf})}.\end{aligned}$$
Matrix $D_{\omega h}(\beta)$ depends on the error distributions, which are unknown and usually difficult to estimate. If $S_{\omega
h}(\beta)$ is a smooth function of $\beta$, $D_{\omega h}(\beta)$ can be estimated by $\partial{D_{\omega
h}(\beta)}/\partial{\beta^{\rm T}}$ evaluated at an estimate of $\beta$. However, $S_{\omega h}(\beta)$ is a step function, Hence, its derivative does not exist, which makes parameter estimates and computation of $\Sigma_{\omega h}$ difficult. Moreover, calculational issues often arise when minimizing $L_{\omega
h}(\beta)$ or solving $S_{\omega h}(\beta)=0$, and the solution in general is not unique and consists of a single interval or even multiple intervals, although the length of these intervals converges asymptotically to zero (Jin et al. 2006).
Smoothed weighted estimating function
--------------------------------------
To overcome difficulties with the lack of smoothness of the estimating functions, we now introduce the induced smoothing method given by Brown and Wang (2005). Assume that $Z\sim N(0, I_p)$ and is independent of the data, where $I_p$ denotes the $p \times p$ identity matrix. Let $\Gamma$ be a $p \times p$ positive definite matrix and satisfy $||\Gamma||=O(1)$. Then, the induced smoothing version of $S_{\omega h}(\beta)$ is $\tilde S(\beta) = {\rm E}_Z \{S_{\omega h}(\beta+ N^{-1/2}\Gamma Z)\}$, where the expectation is taken with respect to $Z$. By some simple calculation, we have $$\tilde{S}_{\omega
h}(\beta)=N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}\Delta_{ik}(X_{ik}-
X_{jl})\Phi\left(\frac{N^{1/2}(e_{jl}- e_{ik})}{r_{ikjl}}\right),
\label{eq:jy3}$$ where $r^2_{ikjl}=(X_{ik}-X_{jl} )^T\Gamma^2(X_{ik}-X_{jl})$. Let $\phi(\cdot)$ be the standard normal density function. Similarly, we can obtain the induced smoothing version of $L_{\omega h}(\beta)$, $$\tilde{L}_{\omega
h}(\beta)=N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}\Delta_{ik}\left[(e_{jl}-e_{ik})
\Phi\left(\frac{N^{1/2}(e_{jl}-
e_{ik})}{r_{ikjl}}\right)+\frac{r_{ikjl}}{N^{1/2}}\phi\left(\frac{N^{1/2}(e_{jl}-
e_{ik})}{r_{ikjl}}\right) \right].$$ Then we can obtain $\tilde{\beta}$ by minimizing $\tilde{L}_{\omega
h}(\beta)$. Alternatively, $\tilde{\beta}_{\omega h}$ can be derived as the multivariate root of $\tilde S_{\omega h}(\beta)=0$. The derivative of $\tilde S_{\omega h}(\beta)$ can be easily derived, $$\begin{aligned}
\tilde D_{\omega h}(\beta) & = & N^{-2}\sum_{i=1}^N \sum_{j=1}^N
\sum_{k=1}^{n_i}
\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}\Delta_{ik}
\frac{(X_{ik}-X_{jl})^T(X_{ik}-X_{jl})}{r_{ikjl}}\phi\left(\frac{N^{1/2}(e_{jl}-e_{ik})}{r_{ikjl}}\right) .\end{aligned}$$ Before giving the asymptotic properties of the smoothed versions and the resultant estimators, the following regularity conditions are required.\
C1. The parameter vector $\beta$ lies in a compact subset $\mathbb{B}$ of $\mathbb{R}^{p}$.\
C2. For $i=1,\cdots, N$, $n_i$ are bounded and $\max_{1\leq k \leq n_i, 1\leq i \leq N}||X_{ik}||^2=o(\sqrt{N})$, where $ ||\cdot||$ is the Euclidean norm.\
C3. ${\rm E}(\epsilon_{ik}^2)\leq M<\infty$, where $M$ is a constant.\
C4. The common marginal density function of the errors $\epsilon_{ik}$, $f_{0}(\cdot)$, and its derivative $f_0'(\cdot)$ are bounded and satisfy $\int[f'_0(t)/f_0(t)]^2f_0(t)\mathrm{d}t<\infty.$\
C5. The marginal distribution of $C_{ik}$ is absolutely continuous and the corresponding density function is bounded on $\mathbb{R}$, for any $i$ and $k$.\
C6. The matrices $D_{\omega h}(\beta)$ and $\tilde D_{\omega
h}(\beta)$ are non-singular.
[**Theorem 1.**]{}[*Under conditions C1-C5, $\sqrt{N}\{\tilde S_{\omega h}(\beta)-S_{\omega h}(\beta)\}=o_p(1)$ uniformly in $\beta$.*]{}
[**Theorem 2.**]{}[*Let $\Gamma^2$ be any symmetric and positive definite matrix with $\|\Gamma\|<\infty$. Under conditions C1-C6, $\tilde{\beta}_{\omega h}$ is a strongly consistent estimator of $\beta_0$.*]{}
[**Theorem 3.**]{}[*Let $\Gamma^2$ be any symmetric and positive definite matrix with $\|\Gamma\|<\infty$. Under conditions C1-C6, $N^{1/2}(\tilde{\beta}_{\omega h}-\beta_0)$ converges in distribution to $N(0, \Sigma_{\omega h})$, where $\Sigma_{\omega h}$ is given by (\[eq:sigmaW\]).*]{}
Theorem 1 indicates that the difference between the smoothed estimating functions and unsmoothed version is negligible. Theorems 1 and 2 indicate that $\tilde D_{\omega h}$ is a consistent estimator of $D_{\omega h}$. Therefore, $\Sigma_{\omega h}$ can be consistently estimated using $\hat\Sigma_{\omega
h}=\tilde{D}_{\omega h}^{-1}(\beta)\hat{V}_{\omega
h}\tilde{D}_{\omega h}^{-1}(\beta).$ The proofs of theorems 1 and 2 are given in the Appendices A and B. The proofs of Theorem 3 can be established by following similar lines as in established similar results for the independence estimator (Johnson and Strawderman 2009). According to Brown and Wang (2005), an iteration procedure to simultaneously obtain the smoothed estimate $\tilde{\beta}_{\omega h}$ and its covariance matrix estimate can be described by the following steps:\
Step 1. Choose an initial value (e.g. $I_p$) for the working covariance matrix ${\Gamma}^{(0)}$ and a consistent estimator for $\beta$ to evaluate $w_i$ and $w_j$.\
Step 2. In the $k$-th iteration, update $\tilde\beta_{\omega
h}^{(k)}$ by minimizing $\tilde L_{\omega h}(\beta)$ or solving $\tilde S_{\omega h}(\beta)=0$.\
Step 3. Update $\Gamma^{(k)}$ based on $ \Gamma^2=\{\tilde D_{\omega
h}(\beta)\}^{-1}\hat{V}_{\omega h}\{\tilde D_{\omega
h}(\beta)\}^{-1}$ using the current values $\tilde\beta_{\omega
h}^{(k)}$ and $\Gamma^{(k-1)}$.\
Step 4. Repeat Steps 2-3 until a convergence criterion is satisfied.
In our experience, in general, the algorithm converges after only a few iterations. The final values of $\tilde\beta_{\omega h}$ and $\Gamma^2$ can be as estimates of $\beta$ and $\Sigma_{\omega h}$.
Simulation studies
==================
In this section, we carry out simulation studies to evaluate the performance of the proposed estimator $\hat\beta_{\omega h}$ by comparing the biases and mean squared errors (MSE) with the Gehan-estimator $\hat\beta_G$, $\hat\beta_{\omega}$ derived from $S_{\omega h}(\beta)=0$ with $h_{ik}=h_{jl}=1$, and the smoothed estimator $\tilde\beta_{\omega h}$ from $\tilde S_{\omega h}(\beta)=0$.
In the simulation studies, we generate the data from model (\[eq:model\]) with $p=2$, and $\beta=(1.2,1.5)^{\rm T}$. Cluster sizes $n_i$ are sampled from $3$ to $10$ with equal probability. The censoring times $C_{ik}$ are generated from a uniform distribution $U(0,\tau)$, where $\tau$ controls the rate of the censoring. The rates of the censoring are taken as $15\%$ and $30\%$. The error terms are generated from a multivariate normal distribution $\mbox{N}(0,\Sigma(\rho))$ and a multivariate t-distribution $\mbox{T}(0,\Sigma(\rho))$ with three degrees of freedom, where $\Sigma(\rho)$ is an exchangeable matrix with parameter $\rho=0.5$ and $0.8$. Note that the correlation coefficient between $T_{ik}$ and $T_{il}$ $(k \neq l)$ is $(e^{\rho}-1)/(e-1)$. The covariate $X_{ik1}$ is a cluster-level covariate in which $X_{ik1}$ does not change within each subject or cluster, and independently generated from the standard normal distribution. The covariate $X_{ik2}$ are with-cluster covariate in which the covariate varies within each subject or cluster, and independently generated from the standard normal distribution. The covariate $X_{ik2}$ is contaminated by adding an outlier equal to $5$ with a probability of $0$ or $5\%$. For the case, $1000$ simulations are carried out. The simulation results are given in Tables \[case1\]-\[case4\] present the biases and mean squared errors for each case. Tables \[casev1\]-\[casev2\] show the empirical variances (Evar) and the variances (Ivar) using the iterative method of §2 for simultaneously estimating the regression parameters and covariance matrix.
From Tables \[case1\]-\[case2\], we can see that both biases and MSEs of the estimates increase as censoring rate increases, and all the estimates are unbiased when there are no outliers. The mean square errors decrease as the number of cluster increases. When there exist outliers (Tables \[case3\]-\[case4\]), the proposed estimate $\hat\beta_{\omega h}$ has much smaller biases and MSEs than $\hat\beta_G$ and $\hat\beta_{\omega}$. The mean squared errors of the smoothed estimate $\tilde \beta_{\omega h}$ are similar to those of the nonsmoothed estimate $\hat\beta_{\omega}$ across all cases. When the covariate is a within-cluster covariate, the estimates $\hat\beta_G$ corresponding to $\beta_1$ performs better than $\hat\beta_{\omega}$ and $\hat\beta_{\omega h}$. However, when the covariate is cluster-level covariate, $\hat\beta_{\omega}$ and $\hat\beta_{\omega h}$ corresponding to $\beta_1$ perform better than $\hat\beta_G$. From Tables \[casev1\]-\[casev2\], we can see that the variance estimates obtained via the iterate method (for simultaneously estimating the regression parameters and covariance matrix) for $\beta_1$ and $\beta_2$ are accurate and similar to the empirical variance estimates across all simulation studies. Overall, the results presented in Tables \[casev1\]-\[casev2\] suggest that the smoothing parameter has a minimal impact on the bias or actual variance of the regression parameter estimates, and the proposed estimate $\hat\beta_{\omega h}$ are robust and efficient.
Analysis of real medical data
=============================
In this section, we illustrate the proposed method by analyzing two real longitudinal data sets.
The first one is a longitudinal and survival dataset collected in a recent clinical trial, which was described by Guo and Carlin (2004). In this trial, a total of $467$ HIV-infected patients were enrolled and randomly assigned to receive either didanosine (ddI) or zalcitabine (ddC). CD4 counts were recorded at study entry and again at $2$, $6$, $12$, and $18$-month visits, and the times to death were also recorded. Due to death or censoring of the patients, the data is unbalanced. For full details regarding the conduct of the trial the reader is referred to Abrams et al. (1994) and Goldman et al. (1996). The dataset is available in the JM package in statistical software R.
In this paper, we are interested in whether the time to death or censoring of the patients is different for the ddI and ddC groups. Let $T_{i}$ be the time to death or censoring of the $i$th patient. We include five covariates as main effects in our analysis: CD4 counts, observation time at which the CD4 cells count was recorded (obstime), drug ($\mbox{ddI}=1$, $\mbox{ddC}=0$), gender ($\mbox{male}=1$, $\mbox{female}=-1$), PrevOI (previous opportunistic infection (AIDS diagnosis) at study entry $= 1$, no AIDS diagnosis $=-1$), and AZT (AZT failure $= 1$, AZT intolerance $=-1$). Note that covariates are cluster-level covariates except CD4 and $\mbox{obstime}$. Figure \[real1fig\] indicates that the CD4 may include some underlying outliers which are larger than $281$. We analyze the data by the following AFT model, $$\log(T_{i})=\beta_0+\beta_1\mbox{CD}_{ik}+\beta_2\mbox{obstime}_{ik}+\beta_3
\mbox{drug}_i+\beta_4\mbox{gender}_i+\beta_5 \mbox{prevOI}_i+\beta_6
\mbox{AZT}_i+\epsilon_{ik}.$$ We estimate the parameters by the same method in simulation studies. Parameter estimates and their standard errors are given in Table \[real1\]. We can see that the estimates obtained from different methods are similar. Furthermore, $\tilde\beta_{\omega h}$ has smaller standard errors than $\hat\beta_G$ and $\hat\beta_{\omega}$ for the cluster-level covariates. However, for within-cluster covariates, CD4 and $\mbox{obstime}$, the standard errors of $\tilde\beta_{\omega h}$ are larger than those of $\hat\beta_G$ and $\hat\beta_{\omega}$, which are consistent with the findings in our simulation studies.
The second example is from the standard and new anti-epileptic drugs (SANAD) study [@mar07a; @mar07b] with the aim to know whether the new drug lamotrigine (LTG) is superior to the standard drug carbamazepine (CBZ) for patients with epilepsy. There were $605$ patients in the trial treated with LTG or CBZ randomly. We consider the effects of six covariates on the time of drug withdrawal: dose, treatment (LTG=1, CBZ=0), age, gender (male=1, female=0), and two indicator variables, with.use (1=withdrawal due to unacceptable adverse effects, 0=otherwisw) and with.isc(1= withdrawal because of inadequate seizure control, 0=otherwise). It is noticeable that these covariates are cluster-level except dose, and there may be some underlying outliers in dose according to Figure \[real2fig\]. We use the following AFT model to analyze the data. $$\log(T_{i})=\beta_0+\beta_1\mbox{dose}_{ik}+\beta_2\mbox{treatment}_{i}+\beta_3
\mbox{age}_i+\beta_4\mbox{gender}_i+\beta_5
\mbox{with.uae}_i+\beta_6 \mbox{with.isc}_i+\epsilon_{ik}.$$ The parameter estimates and their standard errors using different methods are shown in Table \[real2\].
In Table \[real2\], it is shown that using different methods obtains similar estimates. The effect of treatment is positive. In other words, the conclusion is that LTG is superior to CBZ, which has been found in [@mar07a; @mar07b; @wil08]. Moreover, the standard error of $\tilde{\beta}_{wh}$, the coefficient of the within-cluster covariate dose, is comparable with those obtained by other methods. However, the standard errors of $\tilde{\beta}_{wh}$ for cluster-level covariates are the smallest one among the five methods.
Discussion
==========
The Gehan weight estimating function is monotonic with respect to regression parameters, is robust against outliers in response, and has a unique solution. Therefore, many researchers have utilized it to estimate parameters in the AFT model (Fygenson and Ritov 1994; Jin et al. 2006). However, they did not consider the possible outliers existing in covariates. Furthermore, the within-cluster correlation was often ignored for the clustered and censored data. The proposed method is as simple as the independence model of Jin et al. (2006), but takes account of within-cluster correlations and varying cluster sizes. Moreover, it is robust against outliers in covariates and/or response. When there exist outliers in covariates, the proposed method leads to substantial improvement over the commonly used Gehan estimator. Furthermore, the calculation burden can be greatly reduced by the induced smoothing method (Brown and Wang 2005; Johnson and Strawderman 2009).
In this paper, we only considered the linear regression model. In fact, the idea can be easily extended to the partial linear model (Cheng and Wei 2000). The simulation results indicate that the proposed method depends on the covariate design and is not fully efficient; that is because the correlations are still not well considered and quantified in this paper. Further work is therefore needed to incorporate within-cluster correlations into the optimal parameter estimation.
**Acknowledgements** {#acknowledgements .unnumbered}
====================
Yan Zhou’s research was supported by the National Natural Science Foundation of China (Grant No. 11701385), the National Statistical Research Project (Grant No. 2017LY56), the Doctor Start Fund of Guangdong Province (Grant No. 2016A030310062). Liya Fu’s research was supported by the National Natural Science Foundation of China (11871390), the Fundamental Research Funds for the Central Universities (No. xjj2017180), the Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ1006) and the Doctoral Programs Foundation of the Ministry of Education of China (20120201120053). You-Gan Wang wishes to thank the support from the Australian Research Council grant (DP160104292).
Abrams, D. I., Goldman, A. L, Launer, C, Korvick, J. A., Neaton, J. D., Crane, L. R., Grodesky, M., Wakefield, S., Muth, K., Kornegay, S., Cohn, D. J., Haris, A., Luskin-Hawk, R., Markowitz, N., Sampson, J. H., Thompson, M., Deyton, L., and the Terry Beim Community Programs for Clinical Research on AIDS (1994), Comparative Trial of Didanosine and Zalcitabine in Patients with Human Immunodeficiency Virus Infection Who are Intolerant of or Have Failed Zidovudine Therapy, [ New England Journal of Medicine]{} [ 330]{}, 657–662.
Brown, B. M. and Wang, Y-G. (2005). Standard errors and covariance matrices for smoothed rank estimators. [ Biometrika]{} [ 92,]{} 149–58.
Cox, D. R. (1972). Regression models and life tables (with discussion). [ Journal of the Royal Statistical Society, Ser. B]{} [ 34,]{} 187–220.
Cheng, S. C. and Wei, L. J. (2000). Inference for a seiparametric model with panel data. Biometrika 87, 89–97.
Chiou S. H., Kang, S. W. and Jun Y. (2015) Semiparametric accelerated failure time modeling for clustered failure times from stratified sampling. [ Journal of the American Statistical Association]{}, [110]{}, 621–629
Chiou S. H., Kang S. W., Kim J. and Jun Y. (2014) Marginal semiparametric multivariate accelerated failure time model with generalized estimating equations. [ Lifetime Data Analysis.]{} [ 20,]{} 599–618.
Fygenson, M. & Ritov, Y. (1994). Monotone estimating equations for censored data. [ The Annals of Statistics]{} [ 22,]{} 732–746.
Guo, X. and Carlin, B. P. (2004). Separate and joint modeling of longitudinal and event time data using standard computer packages. [ The American Statistician]{} [ 58,]{} 16–24.
Goldman, A. L, Carlin, B. P., Crane, L. R., Launer, C, Korvick, J. A., Deyton, L., and Abrams, D. I. (1996), Response of CD4+ and Clinical Consequences to Treatment Using ddl or ddC in Patients with Advanced HIV Infection, [ Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology]{} [ 11]{}, 161–169.
(2006). Rank regression analysis of multivariate failure time data based on marginal linear models. [ Scandinavian Journal of Statistics]{} [ 33,]{} 1–23.
Johnson, L. M. and Strawderman, R. L. (2009). Induced smoothing for the semiparametric accelerated failure time model: Asymptotics and extensions to clustered data. [ Biometrika]{} [ 96,]{} 577–590.
Kalbfleisch, J. D. & Prentice, R. L. (2002). [*The Statistical Analysis of Failure Time Data*]{}, 2nd ed. Hoboken: Wiley.
Lee, E. W., Wei, L. J. & Ying, Z. (1993). Linear regression analysis for highly stratified failure time data. [ Journal of the American Statistical Association]{} 88, 557–565.
Lin, D. Y., Wei, L. J., Ying, Z., (1998). Accelerated failure time models for counting processes.[ Biometrika]{} [ 85]{}, 605–618.
Luo J., Li H. F. Zhang J. J. (2014). Robust smoothed rank estimation methods for accelerated failure time model allowing clusters. Communications in Statistics-Simulation and Computation. DOI:10.1080/03610918.2014.882944.
Marson A.G., Appleton R., Baker G.A., Chadwick, D.W., Doughty, J., Eaton, B., Gamble, C., Jacoby, A., Shackley, P., Smith, D.F., Tudur-Smith, C., Vanoli, A.,and Williamson P.R. 2007. A randomised controlled trial examining longer-term outcomes of standard versus new antiepileptic drugs, Health Technology Assessment [ 11(37)]{}, 1-154.
Marson, A.G., Al-Kharusi, A.M., Alwaidh, M., Appleton, R., Baker, G.A., Chadwick, D.W., Cramp, C., Cockerell, O.C., Cooper, P.N., Doughty, J., Eaton, B., Gamble, C., Goulding, P.J., Howell, S.J.L., Hughes, A., Jackson, M., Jacoby, A., Kellett, M., Lawson, G.R., Leach, J.P., Nicolaides, P., Roberts, R., Shackley, P., Shen, J., Smith, D.F., Smith, P.E.M., Smith, T.C., Vanoli, A. and Williamson, P.R. on behalf of the SANAD Study group. 2007. The SANAD study of effectiveness of carbamazepine, gabapentin, lamotrigine, oxcarbazepine, or topiramate for treatment of partial epilepsy: an unblinded randomised controlled trial, Lancet [ 365]{}, 1000-1015.
Naranjo, J. D., and Hettmansperger, T. P. (1994). Bounded-Influence rank regression. Journal of the Royal Statistical Society, Ser. B [ 56]{}, 209–220.
Neuhaus, J. M. and Kalb eish, J. D. (1998). Between- and within-cluster covariate effects in the analysis of clustered data. [ Biometrics]{} [ 54]{} 638–645.
Rousseeuw, P. J. and van Zomeren, B. C. (1990). Unmasking multivariate outliers and leverage points. Journal of the American Statistical Association [ 85]{}, 633–639.
Wang, Y-G. and Carey, V. J. (2003). Working correlation structure misspecification, estimation and covariate design: Implications for generalized estimating equations performance. [ Biometrika]{} [ 90]{}, 29–41.
Wang, Y-G. and Fu, L. Y. (2011). Rank regression for accelerated failure time model with clustered and censored data. [ Computational Statistics and Data Analysis]{} [ 55]{}, 2334–2343.
Williamson, P.R., Kolamunnage-Dona, R., Philipson, P., Marson, A.G. 2008. Joint modelling of longitudinal and competing risks data, Statistics in Medicine [ 27(30)]{}, 6426-6438.
[**Appendix A: Proof of Theorem 1**]{}
[*Proof:*]{} Suppose $G_{ik}(\cdot)$ and $g_{ik}(\cdot)$ are distribution and marginal density functions of $\tilde
C_{ik}=\log{C_{ik}}-X_{ik}^{\rm T}\beta_0$ conditional on covariates $(X_i,X_j)$, respectively. A bar above a distribution denotes a survival function, such as $\bar F_0(\cdot)=1-F_0(\cdot)$.
We first prove $\sqrt{N}{\rm E}\|\tilde{U}_{\omega
h}(\beta)-U_{\omega h}(\beta)\|=o(1)$. Because $$\begin{aligned}
\Delta_{ik} &=& I(T_{ik}\leq C_{ik}) = I(\log{T_{ik}}-X_{ik}^{\rm T}\beta_0)\leq \log{C_{ik}}-X_{ik}^{\rm T}\beta_0)\\
% &=& I(\epsilon_{ik}\leq \log{C_{ik}}-X_{ik}^{\rm T}\beta_0)\\
&=& I(\epsilon_{ik}\leq \tilde{C_{ik}}),\\
I(e_{ik}-e_{jl}\leq 0) &=& I\left\{\log({T_{ik}\wedge C_{ik}})-X_{ik}^{\rm T}\beta_0-[\log({T_{jl}\wedge C_{jl}})-X_{jl}^{\rm T}\beta_0]+a(\beta)\right\}\\
% &=& I\left\{\log({T_{ik}\wedge C_{ik}})-X_{ik}^{\rm T}\beta_0-X_{jl}^{\rm T}\beta_0+a_{ikjl} \leq \log({T_{jl}\wedge C_{jl}}) \right\}\\
&=& I\left\{\log({T_{ik}\wedge C_{ik}})-X_{ik}^{\rm T}\beta_0-X_{jl}^{\rm T}\beta_0+a(\beta)\leq \log{T_{jl}} \right\}\\
&\times& I\left\{\log({T_{ik}\wedge C_{ik}})-X_{ik}^{\rm T}\beta_0-X_{jl}^{\rm T}\beta_0+a(\beta)\leq C_{jl}
\right\},\end{aligned}$$ here $a(\beta)=d_{ikjl}^{\rm T}(\beta_0-\beta)$, where $d_{ikjl}=X_{ik}-X_{jl}$. Let $f_{ikjl}(\cdot)$ be the joint density function of $(\epsilon_{ik},\epsilon_{jl},\tilde C_{ik},\tilde
C_{jl})$ conditional on the covariates $(X_i,X_j)$. For any $i\neq
j$, $f_{ikjl}(\Theta)=f_{ik}(\epsilon_{ik})f_{jl}(\epsilon_{jl})g_{ik}(\tilde
C_{ik})g_{jl}(\tilde C_{jl})$, where $\Theta=(\epsilon_{ik},\epsilon_{jl},\tilde C_{ik},\tilde C_{jl})$. Therefore, $$\begin{aligned}
{\rm E}\{U_{\omega
h}(\beta)\}&=& N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\E\left\{ \omega_i\omega_jh_{ik}h_{jl}\Delta_{ik}(X_{ik}-X_{jl})I(e_{ik}-e_{jl}\leq 0)\right\}\\
&=& N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}d_{ikjl}\int_{-\infty}^{+\infty}\bar G_{ik}(u)\bar G_{jl}(u+a(\beta))\bar
F_{jl}(u+a(\beta))f_{ik}(u)\mathrm{d}u.\end{aligned}$$ Define $m_{jl}(s)=\bar G_{jl}(s)f_{jl}(s)+g_{jl}(s)\bar F_{jl}(s)$. Some tedious algebra leads to $$\begin{aligned}
{\rm E}\{\tilde U_{\omega h}(\beta)\}&=& N^{-2}\sum_{i,j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\E\left\{\omega_i\omega_jh_{ik}h_{jl}(X_{ik}-X_{jl})\Delta_{ik}\Phi\left(\frac{e_{jl}-e_{ik}}{N^{-1/2}r_{ikjl}}\right)\right\}\\
&=& N^{-2}\sum_{i,j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}d_{ikjl}\iint\limits_{\mathbb{R}^2}m_{jl}(s)\Phi\left(\frac{s-u-a(\beta)}{N^{-1/2}r_{ikjl}}\right)\bar G_{ik}(u)f_{ik}(u)\mathrm{d}u\mathrm{d}s\\
&=& N^{-2}\sum_{i,j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}d_{ikjl}\iint\limits_{\mathbb{R}^2}\bar G_{jl}(v_{ikjl})\bar F_{jl}(v_{ikjl})\bar G_{ik}(u)f_{ik}(u)\phi(t)\mathrm{d}t\mathrm{d}u,\end{aligned}$$ where $v_{ikjl}=u+a(\beta)+\frac{r_{ikjl}t}{\sqrt{N}}$.\
For $\bar G_{jl}(v_{ikjl})$ and $\bar
F_{jl}(v_{ikjl})$, taking a second-order Taylor series expansion and simple calculation leads to, $${\rm E}\{\tilde U_{\omega h}(\beta)\}={\rm E}\{U_{\omega
h}(\beta)\}+N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}d_{ikjl}(Q_1-Q_2+Q_3),$$ where $$Q_1=N^{-1}r^2_{ikjl}\iint\limits_{\mathbb{R}^2}\{\bar
G_{ji}(u+a(\beta))f'_{jl}(u^{**})+g'_{jl}(u^{*})\bar
F_{jl}(u+a(\beta))\}\bar
G_{ik}(u)f_{ik}(u)\phi(t)t^2\mathrm{d}u\mathrm{d}t,$$ $$Q_2=N^{-3/2}r_{ikjl}^3\iint\limits_{\mathbb{R}^2}\{g_{jl}(u+a(\beta))f_{jl}(u^{**})+g'_{jl}(u^{*})f_{jl}(u+a(\beta))\}\bar
G_{ik}(u)f_{ik}(u)\phi(t)t^3\mathrm{d}t\mathrm{d}u,$$ and $Q_3=N^{-2}r_{ikjl}^4\iint\limits_{\mathbb{R}^2}g'_{jl}(u^*)f'_{jl}(u^{**})\bar
G_{ik}(u)f_{ik}(u)\phi(t)t^4\mathrm{d}t\mathrm{d}u$, where $u^*$ and $u^{**}$ lie between $u+a(\beta)$ and $u+a(\beta)+N^{-1/2}r_{ikjl}t$. Therefore, $$\begin{aligned}
\sqrt{N}\|{\rm E}\{\tilde U_{\omega h}(\beta)-U_{\omega h}(\beta)\}\|& = & N^{-3/2}\|\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}d_{ikjl}(Q_1-Q_2+Q_3)\|\\
&\leq & N^{-3/2} \sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}\|d_{ikjl}\|(Q_1+|Q_2|+Q_3).\end{aligned}$$ Under condition C4 and C5, there exist constants $M_1, M_2, M_3$, which satisfy $Q_1 \leq N^{-1}r^2_{ikjl}M_1$, $Q_2 \leq
N^{-3/2}r_{ikjl}^3\iint\limits_{\mathbb{R}^2}M_2f_{ik}(u)\phi(t)|t^3|\mathrm{d}t\mathrm{d}u=\sqrt{2/\pi}N^{-3/2}r_{ikjl}^3M_2$, and $Q_3\leq
N^{-2}r_{ikjl}^4\iint\limits_{\mathbb{R}^2}M_3f_{ik}(u)\phi(t)t^4$\
$\mathrm{d}t\mathrm{d}u=6N^{-2}r_{ikjl}^4M_3$. Moreover, because $\Gamma=O(1)$, $r_{ikjl}=\sqrt{d_{ikjl}^{\rm T}
\Gamma^2 d_{ikjl}} \leq \sqrt{2}\max_{1\leq i \leq
N}\|X_{ik}\|O(1)$. Therefore, $Q_1 \leq 2M_1\max_{1\leq i \leq
N}\|X_{ik}\|^2O(N^{-1})$, $Q_2 \leq 4M_2\max_{1\leq i \leq
N}\|X_{ik}\|^3O(N^{-3/2})$, and $Q_3 \leq 12M_3\max_{1\leq i \leq
N}\|X_{ik}\|^4O(N^{-2})$. Under condition C2, we obtain $\sqrt{N}\|{\rm E}\{\tilde U_{\omega h}(\beta)-U_{\omega
h}(\beta)\}\|=o(1)$. By Chebyshev inequality, $\sqrt{N}\{\tilde
U_{\omega h}(\beta)-U_{\omega h}(\beta)\}=o_p(1)$. $\Box$
[**Appendix B: proof of Theorem 2**]{}
The following lemma is required in order to prove Theorem 2.
[*Lemma 1.*]{} Under conditions C1-C5, $$\sup_{\beta\in\mathbb{B}}|\tilde L_{\omega h}(\beta)-L_{\omega
h}(\beta)|\xrightarrow {a.s.}0.$$ [*Proof.*]{} $$\begin{aligned}
|\tilde L_{\omega h}(\beta)-L_{\omega h}(\beta)|& = &
N^{-2}\left|\sum_{i,j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}
\omega_i\omega_jh_{ik}h_{jl}\Delta_{ik}Z_{ikjl}\right|\\
&\leq & N^{-2}\sum_{i,j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}
\omega_i\omega_jh_{ik}h_{jl}\left|\Delta_{ik}Z_{ikjl}\right|\\
% &\leq& N^{-2}\sum_{i \neq j}\sum_k\sum_l\left|
% e_{jlik}\left[\Phi(\frac{e_{jlik}}{\sigma_{ikjl}})-I(e_{ikjl}\leq0)\right]\right|
% +N^{-2}\sum_{i \neq j}\sum_k\sum_l\left|\sigma_{ikjl}\phi(\frac{e_{jlik}}{\sigma_{ikjl}})\right|\\
&=&H_1+H_2,
% &=&N^{-2}\sum_{i \neq j}\sum_k\sum_l\left|
% \sigma_{ikjl}\{\frac{e_{jlik}}{\sigma_{ikjl}}\}\left[\Phi(\frac{e_{jlik}}{\sigma_{ikjl}})-I\{\frac{e_{ikjl}}{\sigma_{ikjl}}\leq0\}\right]+\sigma_{ikjl}\phi(\frac{e_{jlik}}{\sigma_{ikjl}})\right|\\\end{aligned}$$ where $$\begin{aligned}
Z_{ikjl}=\left\{e_{jlik}\left[\Phi\left(\frac{\sqrt{N}e_{jlik}}{r_{ikjl}}\right)-I(e_{ikjl}\leq0)\right]
+ \frac{1}{\sqrt{N}}r_{ikjl}\phi\left(\frac{{\sqrt{N}}e_{jlik}}{r_{ikjl}}\right)\right\},\end{aligned}$$
$$\begin{aligned}
H_1 &=& N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}\left| e_{jlik}\left[\Phi\left(\frac{{\sqrt{N}}e_{jlik}}{r_{ikjl}}\right)-I(e_{ikjl}\leq0)\right]\right|,\\
%&=& N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}\left|
% N^{-1/2}r_{ikjl}\{\frac{e_{jlik}}{N^{-1/2}r_{ikjl}}\}\left[\Phi(\frac{e_{jlik}}{N^{-1/2}r_{ikjl}})-I\{\frac{e_{ikjl}}{N^{-1/2}r_{ikjl}}\leq0\}\right]\right|,\end{aligned}$$
and $$H_2=N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}
\left|N^{-1/2}r_{ikjl}\phi\left(\frac{{\sqrt{N}}e_{jlik}}{r_{ikjl}}\right)\right|.$$ Let $t_{ikjl}=e_{ikjl}/(N^{-1/2}r_{ikjl})$, we have $$\begin{aligned}
H_1 &=&
N^{-2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}|N^{-1/2}r_{ikjl}
t_{ikjl}\{\Phi(-t_{ikjl})-I(t_{ikjl}\leq
0)\}|\\
&=& N^{-5/2}\sum_{i=1}^N\sum_{j=1}^N\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}r_{ikjl}
t_{ikjl}\Phi(-|t_{ikjl}|)\sgn(t_{ikjl}).\end{aligned}$$ Let $t \in \mathbb{R}$, because $\lim_{t\rightarrow \infty}
t\Phi(-|t|)\sgn(t)=0$, hence $t\Phi(-|t|)\sgn(t)$ is bounded. By condition C5, it follows that $\sup_{\beta\in \mathbb{B}}H_1
\xrightarrow {a.s.}0$, as $N\rightarrow \infty.$ Furthermore, $|\phi(N^{1/2}e_{jlik}/r_{ikjl})| \leq 1/\sqrt{2\pi}$, thus $\sup_{\beta\in \mathbb{B}}H_2 \xrightarrow {a.s.}0$, and $\sup_{\beta\in\mathbb{R}}|L_{\omega h}(\beta)-\tilde L_{\omega
h}(\beta)|\xrightarrow {a.s.}0.$
[***Proof of Theorem 2.***]{} According to Lemma 1 in Johnson and Strawderman (2009), we can get the similar result under condition C1-C3 as follows: $$\sup_{\beta\in\mathbb{B}}| L_{\omega
h}(\beta)-L_0(\beta)|\xrightarrow {a.s.}0.$$
where $$\begin{aligned}
L_0(\beta)&=&\frac{H_{12}+H_{21}}{2}+
\sum_{k=1}^{n_1}\sum_{l=1}^{n_1}\omega^2_1h_{1k}h_{1l}{\rm E}\left\{\Delta_{1k}(e_{1l}-e_{1k})I(e_{1k}-e_{1l}\leq0)\right\},\\
H_{ij}&=&\sum_{k=1}^{n_i}\sum_{l=1}^{n_j}\omega_i\omega_jh_{ik}h_{jl}{\rm
E}\left\{\Delta_{ik}(e_{jl}-e_{ik})I(e_{ik}-e_{jl}\leq0)\right\}.\end{aligned}$$ Then, combining Lemma 1 and the triangle inequality $$|\tilde L_{\omega h}(\beta)-L_0(\beta)|\leq |\tilde L_{\omega
h}(\beta)-L_{\omega h}(\beta)|+|L_{\omega}(\beta)-L_0(\beta)|,$$ we obtain $\sup_{\beta\in\mathbb{B}}|\tilde L_{\omega
h}(\beta)-L_{0}(\beta)|\xrightarrow {a.s.}0$. In other words, $\tilde L_{\omega h}(\beta)$ converges almost surely and uniformly to the convex function $L_{0}(\beta)$ for $\beta \in \mathbb{B}$. By condition C6, $L_{0}(\beta)$ is strictly convex at $\beta_0$, and $\beta_0$ is a unique minimizer of $L_{0}(\beta)$. Therefore, $\tilde\beta_{\omega h}\xrightarrow {a.s.} \beta_0$.
[lrrrrrrrrrr]{}\
$\rho$=0.5 & & &&\
\
$\beta$ & C &$\hat\beta_G$&$\hat\beta_{\omega}$& $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & &$\hat\beta_G$& $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& 0.0074 & 0.0075 & 0.0023 & 0.0065 && 0.0155 & 0.0149 & 0.0154 & 0.0156\
&$30\%$& 0.0109 & 0.0113 & 0.0027 & 0.0102 && 0.0171 & 0.0164 & 0.0168 & 0.0171\
$\beta_2=1.5 $&$15\%$& 0.0062 & 0.0060 & 0.0005 & 0.0059 && 0.0038 & 0.0041 & 0.0041 & 0.0042\
&$30\%$& 0.0103 & 0.0100 & 0.0007 & 0.0101 && 0.0048 & 0.0051 & 0.0051 & 0.0052\
$\rho$=0.8 & & &&\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$ & 0.0076 & 0.0076 & 0.0024 & 0.0067 && 0.0224 & 0.0209 & 0.0218 & 0.0220\
&$30\%$ & 0.0122 & 0.0126 & 0.0039 & 0.0115 && 0.0246 & 0.0230 & 0.0236 & 0.0240\
$\beta_2=1.5$ &$15\%$ & 0.0061 & 0.0063 & 0.0009 & 0.0062 && 0.0039 & 0.0042 & 0.0043 & 0.0043\
&$30\%$ & 0.0108 & 0.0108 & 0.0014 & 0.0109 && 0.0048 & 0.0052 & 0.0051 & 0.0053\
\
$\rho$=0.5 & & &&\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & &$\hat\beta_G$& $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$ & -0.0004 &-0.0001 & -0.0021 &-0.0000 &&0.0067 & 0.0061 & 0.0063 & 0.0063\
&$30\%$ & 0.0013 & 0.0018 & -0.0018 & 0.0019 &&0.0075 & 0.0069 & 0.0070 & 0.0071\
$\beta_2=1.5 $&$15\%$ & 0.0034 & 0.0036 & 0.0008 & 0.0035 &&0.0018 & 0.0020 & 0.0020 & 0.0020\
&$30\%$ & 0.0049 & 0.0049 & 0.0000 & 0.0048 &&0.0024 & 0.0026 & 0.0026 & 0.0026\
$\rho$=0.8 & & &&\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$ & -0.0000 & 0.0004 & -0.0016 & 0.0005 && 0.0098 & 0.0087 & 0.0090 & 0.0090\
&$30\%$ & 0.0020 & 0.0025 & -0.0010 & 0.0027 && 0.0108 & 0.0097 & 0.0099 & 0.0100\
$\beta_2=1.5 $&$15\%$ & 0.0038 & 0.0041 & 0.0014 & 0.0041 && 0.0020 & 0.0021 & 0.0022 & 0.0022\
&$30\%$ & 0.0053 & 0.0055 & 0.0008 & 0.0056 && 0.0026 & 0.0028 & 0.0029 & 0.0029\
[lrrrrrrrrrr]{}\
\
$\rho=0.5$ & & & &\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& 0.0042 & 0.0053 & 0.0014 & 0.0050 & & 0.0222 & 0.0201 & 0.0199 & 0.0202\
&$30\%$& 0.0103 & 0.0091 & 0.0020 & 0.0089 & & 0.0272 & 0.0247 & 0.0244 & 0.0248\
$\beta_2=1.5 $&$15\%$& 0.0039 & 0.0036 & -0.0001 & 0.0040 & & 0.0056 & 0.0062 & 0.0064 & 0.0065\
&$30\%$& 0.0131 & 0.0133 & 0.0040 & 0.0122 & & 0.0076 & 0.0084 & 0.0084 & 0.0086\
$\rho=0.8$ & & & &\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& 0.0112 & 0.0092 & 0.0054 & 0.0091 & & 0.0340 & 0.0307 & 0.0303 & 0.0307\
&$30\%$& 0.0167 & 0.0143 & 0.0077 & 0.0144 & & 0.0385 & 0.0344 & 0.0339 & 0.0345\
$\beta_2=1.5 $&$15\%$& 0.0062 & 0.0074 & 0.0028 & 0.0070 & & 0.0055 & 0.0064 & 0.0065 & 0.0066\
&$30\%$& 0.0134 & 0.0138 & 0.0045 & 0.0124 & & 0.0074 & 0.0082 & 0.0081 & 0.0084\
\
$\rho=0.5$ & & & &\
\
$\beta$ & C &$\hat\beta_G$& $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& 0.0072 & 0.0067 & 0.0048 & 0.0066 & & 0.0108 & 0.0101 & 0.0100 & 0.0101\
&$30\%$& 0.0081 & 0.0077 & 0.0045 & 0.0079 & & 0.0126 & 0.0114 & 0.0113 & 0.0114\
$\beta_2=1.5 $&$15\%$& 0.0028 & 0.0027 & -0.0001 & 0.0021 & & 0.0030 & 0.0033 & 0.0033 & 0.0034\
&$30\%$& 0.0028 & 0.0030 & -0.0012 & 0.0029 & & 0.0038 & 0.0042 & 0.0043 & 0.0044\
$\rho=0.8$ & & & &\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& 0.0033 & 0.0025 & 0.0005 & 0.0024 & & 0.0150 & 0.0138 & 0.0137 & 0.0138\
&$30\%$& 0.0035 & 0.0049 & 0.0017 & 0.0053 & & 0.0191 & 0.0175 & 0.0175 & 0.0176\
$\beta_2=1.5 $&$15\%$& 0.0032 & 0.0029 & 0.0001 & 0.0023 & & 0.0031 & 0.0035 & 0.0036 & 0.0036\
&$30\%$& 0.0072 & 0.0070 & 0.0031 & 0.0072 & & 0.0039 & 0.0043 & 0.0045 & 0.0045\
[lrrrrrrrrrr]{}\
\
$\rho=0.5$ & & & &\
\
$\beta$ & C &$\hat\beta_G$& $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& -0.0085 & -0.0089 & -0.0087 & -0.0043 & & 0.0149 & 0.0141 & 0.0129 & 0.0129\
&$30\%$& -0.0160 & -0.0167 & -0.0142 & -0.0065 & & 0.0164 & 0.0156 & 0.0143 & 0.0142\
$\beta_2=1.5 $&$15\%$& -0.5184 & -0.5182 & -0.0837 & -0.0814 & & 0.2958 & 0.2972 & 0.0115 & 0.0112\
&$30\%$& -0.5114 & -0.5107 & -0.0949 & -0.0885 & & 0.2898 & 0.2908 & 0.0145 & 0.0134\
$\rho=0.8$ & & & &\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& -0.0091 & -0.0099 & -0.0105 & -0.0061 & & 0.0209 & 0.0192 & 0.0183 & 0.0183\
&$30\%$& -0.0158 & -0.0168 & -0.0147 & -0.0068 & & 0.0226 & 0.0210 & 0.0200 & 0.0200\
$\beta_2=1.5 $&$15\%$& -0.5150 & -0.5147 & -0.0844 & -0.0820 & & 0.2927 & 0.2946 & 0.0116 & 0.0112\
&$30\%$& -0.5089 & -0.5082 & -0.0952 & -0.0887 & & 0.2877 & 0.2890 & 0.0146 & 0.0135\
\
$\rho=0.5$ & & & &\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ &$\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& -0.0115 & -0.0119 & -0.0079 & -0.0056 & & 0.0082 & 0.0075 & 0.0070 & 0.0070\
&$30\%$& -0.0218 & -0.0221 & -0.0128 & -0.0088 & & 0.0091 & 0.0086 & 0.0078 & 0.0078\
$\beta_2=1.5 $&$15\%$& -0.5008 & -0.5006 & -0.0823 & -0.0810 & & 0.2634 & 0.2641 & 0.0091 & 0.0089\
&$30\%$& -0.5013 & -0.5007 & -0.0942 & -0.0909 & & 0.2646 & 0.2649 & 0.0118 & 0.0112\
$\rho=0.8$ & & & &\
\
$\beta$ & C &$\hat\beta_G$&$\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ &$\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& -0.0116 & -0.0122 & -0.0081 & -0.0058 & & 0.0115 & 0.0104 & 0.0100 & 0.0100\
&$30\%$& -0.0231 & -0.0222 & -0.0123 & -0.0082 & & 0.0119 & 0.0111 & 0.0103 & 0.0102\
$\beta_2=1.5 $&$15\%$& -0.4991 & -0.4993 & -0.0817 & -0.0806 & & 0.2620 & 0.2638 & 0.0091 & 0.0089\
&$30\%$& -0.5023 & -0.5030 & -0.0911 & -0.0879 & & 0.2664 & 0.2685 & 0.0114 & 0.0108\
[lrrrrrrrrrr]{}\
\
$\rho=0.5$ & & & &\
\
$\beta$ & C &$\hat\beta_G$& $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & &$\hat\beta_G$& $\hat\beta_{\omega}$ &$\hat\beta_{\omega h}$ &$\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& -0.0076 & -0.0082 & -0.0084 & -0.0051 & & 0.0249 & 0.0240 & 0.0203 & 0.0205\
&$30\%$& -0.0106 & -0.0080 & -0.0070 & 0.0024 & & 0.0272 & 0.0256 & 0.0234 & 0.0238\
$\beta_2=1.5 $&$15\%$& -0.5704 & -0.5721 & -0.0889 & -0.0878 & & 0.3533 & 0.3569 & 0.0153 & 0.0152\
&$30\%$& -0.5647 & -0.5656 & -0.1083 & -0.1025 & & 0.3493 & 0.3523 & 0.0208 & 0.0199\
$\rho=0.8$ & & & &\
\
$\beta$ & C &$\hat\beta_G$& $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ &$\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& 0.0011 & -0.0004 & 0.0023 & 0.0059 & & 0.0393 & 0.0363 & 0.0325 & 0.0330\
&$30\%$& -0.0061 & -0.0061 & -0.0041 & 0.0026 & & 0.0422 & 0.0384 & 0.0350 & 0.0354\
$\beta_2=1.5 $&$15\%$& -0.5589 & -0.5577 & -0.0893 & -0.0880 & & 0.3421 & 0.3431 & 0.0153 & 0.0151\
&$30\%$& -0.5543 & -0.5556 & -0.1005 & -0.0952 & & 0.3396 & 0.3446 & 0.0196 & 0.0188\
\
\
$\rho=0.5$ & & & &\
\
$\beta$ & C &$\hat\beta_G$ & $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & & $\hat\beta_G$& $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& -0.0016 & -0.0013 & 0.0013 & 0.0032 & & 0.0129 & 0.0121 & 0.0105 & 0.0106\
&$30\%$& -0.0173 & -0.0166 & -0.0098 & -0.0062 & & 0.0132 & 0.0123 & 0.0111 & 0.0111\
$\beta_2=1.5 $&$15\%$& -0.5785 & -0.5792 & -0.0938 & -0.0932 & & 0.3516 & 0.3531 & 0.0126 & 0.0125\
&$30\%$& -0.5615 & -0.5618 & -0.1039 & -0.1011 & & 0.3314 & 0.3331 & 0.0153 & 0.0148\
$\rho=0.8$ & & & &\
\
$\beta$ & C &$\hat\beta_G$& $\hat\beta_{\omega}$ & $\hat\beta_{\omega h}$ & $\tilde{\beta}_{\omega h}$ & &$\hat\beta_G$& $\hat\beta_{\omega}$& $\hat\beta_{\omega h}$ &$\tilde{\beta}_{\omega h}$\
$\beta_1=1.2$ &$15\%$& -0.0014 & -0.0030 & -0.0011 & 0.0007 & & 0.0186 & 0.0166 & 0.0151 & 0.0152\
&$30\%$& -0.0168 & -0.0164 & -0.0108 & -0.0072 & & 0.0186 & 0.0167 & 0.0151 & 0.0152\
$\beta_2=1.5 $&$15\%$& -0.5738 & -0.5747 & -0.0947 & -0.0940 & & 0.3469 & 0.3495 & 0.0130 & 0.0129\
&$30\%$& -0.5650 & -0.5674 & -0.1045 & -0.1019 & & 0.3382 & 0.3422 & 0.0159 & 0.0154\
[lrrrrrrrrrrrr]{}\
& & & &\
\
& & & && & & &\
& C & Ivar & Evar && Ivar &Evar & & Ivar & Evar &&Ivar &Evar\
$\rho$=0.5 &$15\%$ & 0.0152 &0.0155 && 0.0036 & 0.0041 & &0.0073 & 0.0063 && 0.0018 & 0.0020\
&$30\%$ & 0.0159 &0.0170 && 0.0044 & 0.0051 & &0.0077 & 0.0071 && 0.0023 & 0.0026\
$\rho=0.8$ &$15\%$ & 0.0196 &0.0220 && 0.0038 & 0.0043 & &0.0093 & 0.0090 && 0.0019 & 0.0022\
&$30\%$ & 0.0208 &0.0238 && 0.0047 & 0.0052 & &0.0101 & 0.0100 && 0.0024 & 0.0029\
\
& & & &\
\
& & & & & & & &\
& C & Ivar & Evar && Ivar &Evar & & Ivar & Evar &&Ivar &Evar\
$\rho=0.5$ &$15\%$ &0.0218 &0.0202 & &0.0053 &0.0065 & &0.0104 &0.0100 & &0.0027 &0.0034\
&$30\%$ &0.0255 &0.0248 & &0.0071 &0.0085 & &0.0118 &0.0114 & &0.0038 &0.0044\
$\rho=0.8$ &$15\%$ &0.0292 &0.0306 & &0.0057 &0.0066 & &0.0136 &0.0138 & &0.0030 &0.0036\
&$30\%$ &0.0331 &0.0343 & &0.0075 &0.0083 & &0.0155 &0.0176 & &0.0039 &0.0045\
[lrrrrrrrrrrrr]{}\
\
& & & &\
\
& & & & & & & &\
& C & Ivar & Evar & & Ivar & Evar & & Ivar & Evar & & Ivar & Evar\
$\rho=0.5$ &$15\%$ &0.0136 &0.0129 & &0.0040 &0.0046 & &0.0065 &0.0070 & &0.0020 &0.0023\
&$30\%$ &0.0143 &0.0142 & &0.0050 &0.0055 & &0.0069 &0.0077 & &0.0026 &0.0030\
$\rho=0.8$ &$15\%$ &0.0181 &0.0182 & &0.0042 &0.0045 & &0.0085 &0.0100 & &0.0022 &0.0025\
&$30\%$ &0.0191 &0.0199 & &0.0052 &0.0056 & &0.0093 &0.0101 & &0.0027 &0.0031\
\
& & & &\
\
& & & & & & & &\
& C & Ivar & Evar & & Ivar & Evar & & Ivar & Evar & & Ivar & Evar\
$\rho=0.5$ &$15\%$ &0.0213 &0.0205 & &0.0063 &0.0074 & &0.0102 &0.0106 & &0.0031 &0.0038\
&$30\%$ &0.0236 &0.0238 & &0.0081 &0.0094 & &0.0112 &0.0111 & &0.0039 &0.0046\
$\rho=0.8$ &$15\%$ &0.0289 &0.0329 & &0.0063 &0.0074 & &0.0132 &0.0152 & &0.0036 &0.0041\
&$30\%$ &0.0311 &0.0354 & &0.0086 &0.0097 & &0.0150 &0.0152 & &0.0042 &0.0050\
Method CD4 obstime drug gender prevOI AZT
------------------------ ---------- ---------- ---------- ---------- ---------- ---------- -- --
$\hat\beta_G$ 0.0050 0.0981 -0.1330 0.1051 -0.1977 -0.0053
(SE) (0.0068) (0.0215) (0.1862) (0.2196) (0.1641) (0.0893)
$\hat\beta_{\omega}$ 0.0055 0.1215 -0.1600 0.1432 -0.2271 -0.0129
(SE) (0.0066) (0.0219) (0.1617) (0.1682) (0.1548) (0.0774)
$\hat\beta_{\omega h}$ 0.0090 0.1285 -0.1436 0.1596 -0.2579 -0.0302
(SE) (0.0050) (0.0180) (0.1246) (0.1255) (0.1211) (0.0666)
: The estimates and their standard errors (SE) of the coefficients in the AFT model for the HIV data. []{data-label="real1"}
Method dose treatment age gender with.uae with.isc
------------------ ---------- ----------- ---------- ---------- ---------- ---------- -- --
$\hat\beta_I$ 0.1914 0.0055 0.0101 0.1307 -4.9838 -4.4585
(SE) (0.0540) (0.1777) (0.0050) (0.1723) (0.3901) (0.4035)
$\hat\beta_w$ 0.3034 0.1575 0.0059 0.1058 -4.6099 -4.1111
(SE) (0.0600) (0.1727) (0.0049) (0.1637) (0.3402) (0.3590)
$\hat\beta_{wh}$ 0.3245 0.2217 0.0051 0.1461 -3.3281 -2.8242
(SE) (0.0710) (0.1553) (0.0048) (0.1513) (0.3307) (0.3498)
: The estimates and their standard errors (SE) of the coefficients in the AFT model for the SANAD data. []{data-label="real2"}
![The boxplot of the CD4 values. []{data-label="real1fig"}](boxplot.eps){width="80mm" height="80mm"}
![The boxplot of the dose in the SANAD study. []{data-label="real2fig"}](dose.eps){width="80mm" height="80mm"}
|
---
abstract: |
Surrogate-modelling techniques including Polynomial Chaos Expansion (PCE) is commonly used for statistical estimation (aka. Uncertainty Quantification) of quantities of interests obtained from expensive computational models. PCE is a data-driven regression-based technique that relies on spectral polynomials as basis-functions. In this technique, the outputs of few numerical simulations are used to estimate the PCE coefficients within a regression framework combined with regularization techniques where the regularization parameters are estimated using standard cross-validation as applied in supervised machine learning methods.
In the present work, we introduce an efficient method for estimating the PCE coefficients combining Elastic Net regularization with a data-driven feature ranking approach. Our goal is to increase the probability of identifying the most significant PCE components by assigning each of the PCE coefficients a numerical value reflecting the magnitude of the coefficient and its stability with respect to perturbations in the input data. In our evaluations, the proposed approach has shown high convergence rate for high-dimensional problems, where standard feature ranking might be challenging due to the curse of dimensionality.
The presented method is implemented within a standard machine learning library (scikit-learn [@Scikit_Learn]) allowing for easy experimentation with various solvers and regularization techniques (e.g. Tikhonov, LASSO, LARS, Elastic Net) and enabling automatic cross-validation techniques using a widely used and well tested implementation. We present a set of numerical tests on standard analytical functions, a two-phase subsurface flow model and a simulation dataset for CO2 sequestration in a saline aquifer. For all test cases, the proposed approach resulted in a significant increase in PCE convergence rates.
author:
- 'Alexander Tarakanov [^1]'
- 'Ahmed H. Elsheikh'
bibliography:
- 'sample.bib'
title: 'Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models[^2] '
---
Introduction
============
Uncertainty Quantification (UQ) and Uncertainty Propagation (UP) in the subsurface flow related problems have been the subject of intensive research activities over the last decades [@TUGAN2018107; @WANG2019105; @CHEN2017328; @MA20114696; @UQ_CO2_1]. For instance, UQ of oil production forecasts from a given reservoir has far-reaching economical consequences [@ELGSAETER20084540]. Also, the accurate risk assessment of CO2 trapping in an underground reservoir [@JIA2018104] is of high importance from ecological and social perspectives [@Lackner1677].
The main challenge for UQ in subsurface flow related tasks is the complexity of the modeled physical systems [@CO2_factors] and the lack of information about the rock properties that determine underground flow [@Book_on_Geostatistics]. Therefore, UQ for a Quantity of Interest (QoI) is usually performed numerically through multiple evaluations of expensive reservoir simulations [@DAI2019519]. This corresponds to significant computational resources especially when dealing with a high number of uncertain parameters [@UQ2] and for the cases where model high resolution is requirement [@Optimization]. Several lines of research have been pursued to address this challenge. For instance, models of multiple continuum media [@ZHANG2018484; @LI2018127; @AKKUTLU201765], dual mesh approaches [@Dual_Mesh1; @Dual_Mesh2; @KHOOZAN2011195], upscaling [@VASILYEVA2019; @Upscaling1; @Upscaling2; @c98b95f9037d457990cec445e43f1fdd] and model reduction [@Model_Reduction1; @Model_Reduction2; @Model_Reduction3; @2ae730559d3d4dfa92f1da3e3188cc79] techniques have been developed to decrease the run-time of a single simulation. Also, various surrogate modeling techniques [@Proxy1; @Proxy2; @Proxy3] emerged in order to reduce the cost of evaluating a large number of expensive numerical simulations.
In the current manuscript, we focus on surrogate modeling approaches using PCE-based response surfaces. There are several advantages of using PCE as a proxy model. First of all, surrogate models based on sparse PCE do not require significant computational resources to compute a value at any given point within the interpolation domain as it is simply a direct polynomial function evaluation. Secondly, important statistical properties such as moments and sensitivities can be computed directly from the PCE coefficients without the need for a Monte-Carlo simulations [@PCE_Stat1]. This is attributed to a special design properties of PCE that links the probability distribution of random variables with the orthogonality of polynomial basis functions [@PCE_Stat1].
Generally, two techniques could be utilized to estimate the PCE coefficients: collocation-based and regression-based methods. For collocation approaches, the QoI values are evaluated at pre-specified set of points called collocation nodes [@PCE_quadrature]. These specific points are designed in such a way that the PCE coefficients can be expressed as linear combination of the QoI values, allowing for direct computation of the PCE coefficients. The optimal choice of the collocation points especially for high-dimensional problems is a subject of extensive research activities [@WU2018997; @XU201824; @PALAR2018175]. In regression-based approaches, the PCE coefficients correspond to the solution of an error-minimization problem [@PCE_Regression]. It is simple to show that the mean-square error minimization can be reduced to a linear regression problem to estimate the PCE expansion coefficients. Designing fast and accurate solution techniques to this minimization problem including various preconditioning methods is also a subject of intensive research activities [@PCE_Preconditioning; @ABOLGHASEMI2012999; @6484193]. @PCE_Hybrid developed a hybrid collocation and regression technique, where the training points for the surrogate model are generated with collocation techniques while the PCE coefficients are estimated by solving an error-minimization problem. One of the advantages of this approach is the better conditioning of the regression problem when compared to training using random samples [@PCE_Hybrid].
In generic cases, sparse collocation techniques and hybrid approaches provide accurate response surfaces using reasonable computational resources [@Hosder_2010; @Yan_2012; @good_collocation]. However, these methods rely on evaluating the QoI at specific set of points. This strategy can be successfully adopted for UQ of oil production and CO2 storage capacity [@Babaei_Paper1; @Babaei_Paper2; @Petvipusit2014]. However, computation of QoI values in the case of subsurface flow problems can be challenging if the collocation points correspond to extreme values of parameters that significantly affect convergence properties of the numerical scheme. Therefore, such collocation nodes can either increase computational costs of the response surface construction or reduce the overall accuracy of the surrogate model if significant numerical error is introduced to the QoI values at collocation nodes. Additionally, for many practical problems sampling of data points can not be controlled. For instance, samples could be generated randomly (e.g. Latin hypercube sampling), or in accordance with a prescribed probability distribution [@ELSHEIKH2014515], or based on another meta-modeling technique used in combination with PCE for model stacking [@Model_Selection]. Under these conditions, collocation techniques cannot be directly applied. For this reason, regression based PCE (utilized in this manuscript) have wider applicability for any set of training samples where optimal response surfaces could be built.
In regression methods, PCE coefficients are computed through the minimization of mean-square error over the training data. Therefore, for low-dimensional problems, a direct approach could be adopted. In generic setting, the number of PCE coefficients for a problem with $n$ variables can be expressed as follows: $$\label{eq:dimension_of_the_problem}
D = D(n, d) = \binom{n+d}{n}$$ where $D$ is the number of PCE coefficients and $d$ is the degree of polynomials used. It is simple to observe the fast growth of $D$ with both $d$ and $n$. This exponential growth of PCE coefficients imposes significant constraints on building PCE-based response surfaces. First of all, solving the error-minimization regression problem in high dimensions is a challenging task, because of the high number of numerical operations needed till convergence. Secondly, the number of QoI values (i.e. training samples) needed for accurate estimation of the PCE coefficients increases with $D$, which corresponds to additional runs of an expensive numerical simulator. In other words, the curse of dimensionality makes it impractical to solve for PCE coefficients directly. However, for a large class of problems it was observed that PCE coefficients are sparse [@HAMPTON201820; @BAZARGAN2015385]. Therefore, various techniques for sparse regression can be adopted. For example, $\ell^1$ regularization techniques [@Sparse_PCE_L1] can be considered as a first step towards enforcing sparsity on the PCE regression coefficients. This approach is widely adopted and will be referred to as standard PCE [@classic_sparse_PCE] in the rest of this manuscript. Further dimension reduction could be achieved through fitting both the data and the QoI derivatives at the training points [@PCE_Derivatives2]. The additional information from the gradients increases the quality of PCE response surface [@PCE_Derivatives1]. Unfortunately, for many problems it is not possible to obtain the gradient information at the training points. Another line of research focuses on reducing the problem dimension by using advanced methods for solving nonlinear regression problems. For instance, sparse PCE coefficients can be computed efficiently through the application of support vector regression [@PCE_SVR] or preconditioned conjugate gradient [@PCE_PCG] techniques. Another direction of development relies on coupling the iterative solvers with algorithms for ranking the importance of the basis polynomials (e.g. orthogonal matching pursuit [@PCE_Regression]) or ranking based on the impact on the residual [@PCE_Residual_Based_Ranking]. Further reduction of dimension could also be achieved by adaptive truncation of the spectrum of the expansion. For instance, it has been observed empirically for a broad class of problems, that higher order interactions between the polynomial basis from different dimension have less impact on the quality of the response surface when compared to the one dimensional low order polynomials. This empirical observation is the foundation for hyperbolic truncation techniques [@Hyperbolic_Truncation_Scheme]. Moreover, the performance of all regression-based approaches could be improved by transformation of the input variables (e.g. scaling, normalization). For example, variable rotations [@PCE_Rotations] or generic linear transformations [@PCE_Linear_Combination_of_Variables] could significantly reduce the complexity of the error minimization problem corresponding to finding the PCE coefficients.
In the current paper, we focus on further improvement of dimension reduction techniques for regression based PCE. We present a novel iterative approach for solving the error minimization problem. We introduce a new data-driven ranking procedure for sequential identification of the most significant PCE basis functions with the closest relation to the interpolated QoI values. The ranking procedure is based on the correlation between the basis functions and the QoI values penalized by factors that measure the sensitivity of the corresponding coefficient to the noise in the input data. The aim of the introduction of correction/penalty factors is to avoid overestimation of the significance of a given polynomial basis function due to occasional location of data-points. The introduced ranking approach enables us to determine the most significant PCE terms and subsequently solving a reduced regression problem at each iteration. The new method could be easily combined with various regularization techniques.
The proposed approach has been integrated in scikit-learn [@Scikit_Learn], a widely used machine learning library. This integration enables uniform testing of a huge variety of techniques such as Lasso, Lars and Elastic Net [@Regularized_Linear_Regression] in order to formulate and solve the regularized regression problem. We implement PCE as an input feature transformation using machine learning terminology. Therefore, PCE can be naturally included in any machine-learning pipeline allowing one to combine different methods for variable transformation with advanced cross-validation techniques. Moreover, this implementation allows for an easy comparisons to alternative machine-learning techniques (e.g. Random Forests, Support Vector Machines). In the numerical evaluation section, we compare the proposed approach to classical methods for sparse PCE namely, the Orthogonal Matching Pursuit (OMP) and Least Angular Regression (LARS). We consider four data-sets for evaluation. The first two data-sets are generated using analytical functions and the last two data-sets are based on subsurface simulations of fluid flow in porous media. In all the test cases, extensive comparisons are performed in terms of Mean-Square Error (MSE) using a hold-out (aka. validation) set of points following the best practices in the machine learning literature.
The rest of this manuscript is organized as follows: In the following section, a general introduction to PC is presented followed by the proposed ranking procedure. In section \[sec:numerical\_examples\] we present a set of numerical examples. Finally, the conclusion of our work is presented in section \[sec:conclusions\].
Methodology {#sec:Methodology}
===========
Polynomial chaos expansion PCE is a meta-modeling technique that relies on orthogonal polynomials. One of the main advantages of PCE when compared to alternative surrogate modeling techniques is the ability to estimate the QoI sensitivity to given combination of variables through simple analytical formulae. This is only possible due to the close relation between the orthogonality of basis polynomials and the probability distribution of the input variables. This relation is explained in subsection \[subsec:overview\_of\_polynomial\_chaos\], along with an overview of basic ideas of PCE. The proposed reordering of PCE basis is then introduced in subsection \[subsec:regression\_based\_pce\].
Basics of Polynomial Chaos {#subsec:overview_of_polynomial_chaos}
--------------------------
The essence of PCE is the relation between the statistics of input data and orthogonality of the utilized basis polynomials. The relation concerned gives a powerful tool for calculating the PCE coefficients and for further statistical analysis of the data. We first explain this relation for the single-variate case and then extend this formulation to multi-variate cases. Additionally, examples of applying this concept to study the statistical properties of PCE are presented.
For single-variate function $f(x)$, a PCE is defined as series of orthogonal polynomials: $$\label{eq:generic_1d_PC_expansion}
f(x) = \sum_{\alpha} \textit{c}_{\alpha}p_{\alpha}(x)$$ where $p_{\alpha}(x)$ is an orthogonal single-variate polynomial with the index $\alpha$ and $c_{\alpha}$ is the corresponding PCE coefficient. The specific type of utilized orthogonal polynomials is not of a principal importance in the definition introduced in Eq. . Therefore, PCE can be naturally formulated for all well-known families of orthogonal polynomials. For example Hermite, Legendre and Chebyshev polynomials [@GUO2019129].
The analysis of the PCE relies on the orthogonality of the basis polynomials, which is introduced through the notion of an inner product defined as following: $$\label{eq:generic_inner_product}
\langle g_1, g_2 \rangle = \int_{-\infty}^{+\infty} \operatorname{\mathcal{K}}(x)g_1(x)g_2(x) \mathrm{d}x$$ where $g_1(x)$ and $g_2(x)$ are certain square-integrable functions and $\operatorname{\mathcal{K}}(x)$ is a non-negative function referred to as the kernel function or simply the kernel. Classical families of orthogonal polynomials are related to a specific form of the kernel function. For instance, Hermite polynomials correspond to a kernel function identical to Gaussian distribution function with zero mean and unit variance [@PC_Hermite]: $$\label{eq:Hermite_kernel}
\operatorname{\mathcal{K}}(x) = \frac{1}{\sqrt{2\pi}} e^{- {{x^2}/{2}}}$$ For a generic case, the PCE basis functions are constructed by applying Gram-Schmidt orthogonalization to the set of monomial functions (e.g. $1,~x,~x^2,~\dots$) [@Book_on_Orthogonal_Polynomials]. Therefore, PCE techniques could be naturally extended to any arbitrary kernel functions $\operatorname{\mathcal{K}}(x)$.
A central idea of PCE is the statistical interpretation of $\operatorname{\mathcal{K}}(x)$ as probability density function for a given random variable [@CORTES20171]. This interpretation allows one to reformulate the inner product defined in Eq. in terms of expectations: $$\label{eq:generic_inner_product_and_probability}
\langle g_1, g_2 \rangle = \int_{-\infty}^{+\infty} \operatorname{\mathcal{K}}(x)g_1(x) g_2(x) \mathrm{d}x = \operatorname{\mathbb{E}}[g_1, g_2]$$ In the setting, the orthogonality of polynomials $p_{\alpha}(x)$ with respect to the inner product can be reformulated as: $$\label{eq:orthogonality_def}
\langle p_{\alpha}, p_{\beta} \rangle = \operatorname{\mathbb{E}}[p_{\alpha}, p_{\beta}] = \|p_{\alpha}\|^2 \delta_{\alpha\beta}$$ where $\delta_{\alpha\beta}$ is a Kronecker symbol. In the present work, we consider orthonormal polynomials with $\|p_{\alpha}\|^2 = 1$ in order to simplify the numerical analysis of PCE. Therefore, Eq. can be transformed as follows: $$\label{eq:orthonormality_def}
\langle p_{\alpha}, p_{\beta} \rangle = \operatorname{\mathbb{E}}[p_{\alpha}, p_{\beta}] = \delta_{\alpha\beta}$$ Moreover, the basis polynomials orthogonality can be used to estimate the PCE coefficients: $$\label{eq:PCE_coef_generic}
c_{\alpha} = \langle f, p_{\alpha} \rangle = \operatorname{\mathbb{E}}[f, p_{\alpha}]$$
For multi-variate functions, similar analysis could be performed through the introduction of the tensor-product concept where the set of multivariate basis functions is formed as products of single-variate polynomials: $$\label{eq:tensor_product}
p_A(\textbf{x}) = p_{\alpha_1}^{(1)}(x_1) p_{\alpha_2}^{(2)}(x_2) \dots p_{\alpha_n}^{(n)}(x_n)$$ where $\alpha_k$ is the degree of single-variate polynomial, $p_{\alpha_k}^{(k)}(x_k)$ is a uni-variate polynomial that depends only on the $k$-th coordinate of the vector $\mathbf{x}$. The degree of polynomial $p_A(\textbf{x})$ is defined as: $$\label{eq:degree_of_polynomials}
\text{deg}(p_A(\textbf{x})) = \sum_k \text{deg}(p_{\alpha_k}^{(k)}(x_k)) = \sum_k \alpha_k$$ Similar to Eq. , the PCE of multivariate function $f(\textbf{x})$ is defined as: $$\label{eq:generic_PC_md_expansion}
f(\textbf{x}) = f(x_1,\dots,x_n) = \sum_A c_A p_A(\textbf{x})$$ where $c_A$ is the PCE coefficient corresponding to polynomial basis function with multi-index $A$. The inner product in multi-dimensional case is defined as: $$\label{eq:md_inner_product}
\langle g_1, g_2 \rangle = \int \operatorname{\mathcal{K}}(\mathbf{x}) g_1(\mathbf{x}) g_2(\mathbf{x}) \mathrm{d}\mathbf{x}$$ where $\operatorname{\mathcal{K}}(\mathbf{x})$ is a multi-variate kernel function and $g_1(\mathbf{x})$, $g_2(\mathbf{x})$ are certain square-integrable functions. It is important to note that the polynomial basis functions obtained by tensor multiplications inherit the orthogonality and orthonormality from single-variate PCE if the multi-variate kernel function $\operatorname{\mathcal{K}}(\mathbf{•}{x})$ equals the product of single-variate kernel functions: $$\label{eq:kernel_product}
\operatorname{\mathcal{K}}(\textbf{x}) = \operatorname{\mathcal{K}}_1(x_1) \cdots \operatorname{\mathcal{K}}_n(x_n)$$ where $\operatorname{\mathcal{K}}_1(x_1), \cdots , \operatorname{\mathcal{K}}_n(x_n)$ are single-variate kernel functions. From a probabilistic point of view, this is equivalent to the mutual independence of the coordinates of the vector $\textbf{x}$.
The inner product defined in Eq. can be utilized to derive an expression for the PCE coefficients similar to Eq. : $$\label{eq:PCE_coef_generic_multi_variate}
c_{A} = \langle f, p_{A} \rangle = \operatorname{\mathbb{E}}[p_A, f]$$ The relation between the input data statistics and the polynomial basis orthogonality can be used to derive analytical expressions for the mean, variance and Sobol’ indices of the function $f(\textbf{x})$. For example, the mean can be estimated by: $$\label{eq:mean_calculation}
\operatorname{\mathbb{E}}[f] = \langle 1, f \rangle = \sum_A c_A \langle 1, p_A \rangle = c_{0, \dots, 0} = c_0$$ Where $c_{0, \dots, 0}$ is the constant polynomial coefficient. In the present work, we simplify the notations and use $c_{0}$ instead of $c_{0, \dots, 0}$. The mean-square deviation can be calculated in the similar fashion: $$\label{eq:variance_calculation}
\sigma^2 = \operatorname{\mathbb{E}}[(f-c_0), (f-c_0)] = \sum_{A_1, A_2 \neq 0} c_{A_1} c_{A_2} \delta_{A_1 A_2} = \sum_{A>0} c_A^2$$ Calculation of other quantities for sensitivity analysis and UQ such as partial variances and Sobol’ indices could be performed naturally with PCE. A partial standard deviation represents the sensitivity to a given combination of variables. It is defined as the standard deviation of the function $f(\mathbf{x})$ averaged with respect to certain collection of variables [@PALAR2018175]: $$\label{eq:partial_variance}
\sigma^2_{r_1, \dots, r_k}(f) = \sigma^2(\operatorname{\mathbb{E}}_{t_1, \dots, t_{n-k}}[f])$$ where $\sigma_{r_1, \dots, r_k}$ is the standard deviations with respect to the components of the vector $\mathbf{x}$ with indices $r_1, \cdots, r_k$ and $\operatorname{\mathbb{E}}_{t_1, \dots,t_{n-k}}[f]$ is the average with respect to components of the vector $\mathbf{x}$ with indices $t_1, \cdots, t_{n-k}$ that form a complement to $r_1, \cdots, r_k$ [@PCE_Stat1], $n$ is the dimension of $\mathbf{x}$ and $k$ is a certain integer number from $1$ to $n$. Sobol’ indices are commonly used as a measure for sensitivity and are defined as the normalized partial standard deviations: $$\label{eq:sobol_index}
S_{r_1, \dots, r_k}(f) = \frac{ \sigma^2_{r_1, \dots, r_k}(f)} {\sigma^2(f)}$$ For the response function $f$ with a PCE representation, the partial standard deviations can be calculated in a similar fashion as the normal standard deviation defined in Eq. following [@PALAR2018175]: $$\label{eq:partial_variance_pc}
\sigma^2_{r_1, \dots, r_k}(f) = \sum_{\alpha_{r_l} > 0, \alpha_{t_j} = 0 } c_A^2$$
The relation between orthogonality of polynomial basis functions and the probability distribution of input data has an important consequence on the numerical calculation of the PCE coefficients. In practice, for regression based response surfaces, the PCE coefficients for a given function $f(x)$ can be computed for given input data through the minimization of the mean-square error (MSE) functional: $$\label{eq:mean_square_functional}
\operatorname{\mathcal{F}}(\mathbf{c}) = \sum_i \frac{(y_i - \sum_A \textit{c}_Ap_A(\textbf{x}_i))^2} {N}$$ where $\textbf{x}_i$ is the $i^{\text{th}}$ vector of input variables, $N$ is the number of data points and $y_i$ is the value of the function $f$ at the point $\textbf{x}_i$. In the present work the spectrum of PCE is truncated to a certain polynomial degree $d$. Therefore, the dimension of $\mathbf{c}$ is given by Eq. .
It is simple to see that minimizing the functional defined in Eq. is equivalent to solving a system of linear equations: $$\label{eq:linear_problem}
\textbf{M}_{AB}c_B = \textbf{V}_A$$ where the square matrix $\mathbf{M}$ and vector $\mathbf{V}$ are defined as: $$\label{eq:matrix_and_vector_notations}
\textbf{M}_{AB} = \sum_i \frac{p_A(\textbf{x}_i) p_B(\textbf{x}_i)}{N}, \qquad
\textbf{V}_A = \sum_i \frac{y_i p_A(\textbf{x}_i)}{N}$$ The relation between the basis orthogonality and the statistical distribution of the input data imposes several constraints on the value of the matrix $\textbf{M}$ and the vector $\textbf{V}$. If the data is sampled in agreement with the probability distribution determined by the kernel function defined in Eq. , then the matrix $\textbf{M}$ should converge to a unit matrix: $$\label{eq:estimate_for_the_matrix_element}
\textbf{M}_{AB} = \sum_i \frac{p_A(\textbf{x}_i) p_B(\textbf{x}_i)}{N} = \operatorname{\mathbb{E}}[p_Ap_B] + \operatorname{\mathcal{O}}\bigg ( \frac{1}{\sqrt{N}}\bigg ) = \delta_{AB} + \operatorname{\mathcal{O}}\bigg ( \frac{1}{\sqrt{N}}\bigg )$$ where the term $\operatorname{\mathcal{O}}\big( \frac{1}{\sqrt{N}}\big)$ represents the convergence in accordance with the law of large numbers [@Probability_Textbook]. Similar reasoning could be applied to the vector $\textbf{V}$ showing the close correlation between the data and the basis functions: $$\label{eq:estimate_for_the_vector}
\textbf{V}_A = \sum_i \frac{p_A(\textbf{x}_i) y_i}{N} = \operatorname{\mathbb{E}}[y(x)p_A(x)] + \operatorname{\mathcal{O}}\bigg ( \frac{1}{\sqrt{N}}\bigg ) = \textit{c}_A + \operatorname{\mathcal{O}}\bigg ( \frac{1}{\sqrt{N}}\bigg )$$ Eq. and Eq. , simply means that the coefficients $\textbf{c}$ minimizing the MSE functional defined in Eq. is close to the correlation vector $\textbf{V}$ if a sufficient number $N$ of training data points is available. Moreover, the difference between $\textbf{V}$ and $\textbf{c}$ can be estimated as follows: $$\label{eq:inequality_for_coefficients}
|V_A - c_A| \leq \frac{k_A}{\sqrt{N}}$$ Where $k_A$ is a positive number. In other words, $\textbf{V}$ provides a reasonable approximation for $\textbf{c}$ if a sufficient number of data-points is available. We utilize this observation to introduce a novel ranking-based approach to approximate the PCE coefficients as described in the next subsection.
Ranking based sparse PCE {#subsec:regression_based_pce}
------------------------
In the present work, we estimate the PCE coefficients by minimizing the mean-square error functional defined in Eq. . It is well-known that a straight-forward minimization of mean square errors could provide an unstable solution or a response surface that is not quite accurate at points that are not included in the training data-set. Therefore, we utilize a mixed $\ell_1$ and $\ell_2$ regularization technique known as Elastic Net model [@Elastic_Net1] (i.e., combined Tikhonov and Lasso regularization). This results in a regularized functional for error minimization of the following form: $$\label{eq:minimization_problem}
\textbf{c} = \underset{\textbf{c}}{\arg\min} \operatorname{\mathcal{L}}(\textbf{c}) = \underset{\textbf{c}}{\arg\min} \bigg( \operatorname{\mathcal{F}}(\textbf{c}) + \lambda_1 \sum_{A} |c_A| + \lambda_2 \sum_{A} c_A^2\bigg)$$ where $\operatorname{\mathcal{L}}(\textbf{c})$ is a functional for minimization and $\lambda_1, \lambda_2$ are hyperparameters that could be tuned in order to maximize the quality of the surrogate model. In the present work, $\lambda_1$ and $\lambda_2$ are determined through cross-validation.
We utilize a coordinate descent algorithm [@Regularized_Linear_Regression] in order to find the solution for the minimization problem defined in Eq. . This is an iterative algorithm that sequentially updates the solution vector $\textbf{c}$ by minimizing the functional $\operatorname{\mathcal{L}}(\textbf{c})$ with respect to one of the coordinates at each step as summarized in Algorithm \[alg:coordinate\_descent\].
$\textbf{c} = 0 $ Select a value $k$ from $1$ to $\text{dim}(\textbf{c})$ $c_{k} = \underset{c_k}{\arg\min} \operatorname{\mathcal{L}}(\mathbf{c})$ Update $\Delta \operatorname{\mathcal{L}}$ **return** $\mathbf{c}$
One of the essential parts in Algorithm \[alg:coordinate\_descent\] is the selection of the next component for update. Classical approaches include: random selection or selection based on the cyclic order on the set of components [@Regularized_Linear_Regression]. In the present work, we introduce a novel scheme for reordering the polynomial basis functions that increases the algorithm convergence rate and increases the response surface quality when utilizing small number of training samples. The aim of the reordering procedure is to identify the polynomial basis functions with the highest PCE coefficients in order to determine its values first. It should be noted that the assumption about the agreement between sampling of training data and orthogonality of basis polynomial functions Eq. is of principal importance for the proposed reordering procedure. For the cases where this assumption is violated, data transformation techniques should be applied before using the proposed reordering approach. For instance, the desired distribution of input variables can be achieved through quantile transformation [@Quantile] or Rosenblatt transformation [@rosenblatt1952].
The reordering technique utilizes a ranking of PCE coefficients inspired by Eq. , which states that the vector of moments is close to the actual PCE coefficients given a sufficient number of training points. However, for certain polynomial basis functions the difference between $c_A$ and $V_A$ can be significant leading to an overestimation of the importance of those components due to the lack of the available data, which can be considered as a noise. In order to address this issue, we introduce a ranking of polynomial basis functions in a form of the signal-to-noise ratio which is a correlation coefficient divided by a correction factor that quantifies the sensitivity of a given PCE coefficient to the data noise.
Two sources of noise are considered in the current work: noise in the values of QoI and noise in the deviation of matrix $\mathbf{M}$ due to the random sampling of the training data. In order to quantify both sources of noise, we perform two series of Monte-Carlo simulations. In the first series of Monte-Carlo simulations, the sensitivity of the correlation vector $\mathbf{V}$ to the QoI values is estimated. For that purpose, we introduce random perturbations $\theta_i$ to each of the training data-points. In the present study, the noise part is sampled from a normal distribution with zero mean and unit variance. The correlation of the basis polynomial $p_A(\mathbf{x})$ with perturbed data to the QoI is estimated using: $$\label{eq:U_definition}
{\bf U}_A = \sum_i \frac{(y_i+\theta_i) p_A(\textbf{x}_i)}{N}$$ The mean-square deviation $\sigma_{Y,A}$ of ${\bf U}_A$ from $\bf{V}_A$ is considered as a measure for stability: $$\label{eq::sensitivity_measure_for_Y}
\sigma_{Y,A} = \sqrt{\operatorname{\mathbb{E}}_{\theta}[({\bf U}_A-{\bf V}_A)^2]}$$ where the mean $\operatorname{\mathbb{E}}_{\theta}$ is taken over several realization of $\theta$.
The second series of Monte-Carlo simulations is performed to quantify the stability with respect to the location of training points. As long as the location of training data points $\tilde{\textbf{x}}$ is considered as a random parameter, a set of $N$ points is generated at each Monte-Carlo simulation. Then the mean-square deviation $\sigma_{X, A}$ of ${\mathbf M}_{AA}$ from the unit matrix can be computed numerically as follows: $$\label{eq:sensitivity_measure_for_X}
\sigma_{X,A} = \sqrt{E_{\tilde{\textbf{x}}}[({\mathbf M}_{AA} - \mathbf{I})^2]}$$ where the mean $\operatorname{\mathbb{E}}_{\tilde{\textbf{x}}}$ is taken over a number of realizations of $\tilde{\mathbf{x}}$. Finally, the ranking coefficient for the basis polynomial $p_A(\mathbf{x})$ is defined as: $$\label{eq:ranking_coefficient_final}
r_A = \frac{1} {\sqrt{\sigma_{Y,A}^2 + \sigma_{X,A}^2}} \frac{|\textbf{V}_A|} {H}$$ where parameter $H$ is introduced for normalization purposes. The value of $H$ is given by the expression: $$\label{eq:D_coef}
H = \frac{1}{2}\min_A|\textbf{V}_A| + \frac{1}{2}\max_A|\textbf{V}_A|$$ where minimum and maximum are taken over all values of multi-index $A$. In this work we consider high values of $r_A$ as an indicator of a high value of the corresponding PCE coefficient.
In the present work, the ranking parameter $r_A$ is used within the coordinate descent Algorithm \[alg:coordinate\_descent\] to select the next PCE coefficient for updates. Therefore, we solve iteratively for PCE coefficients by performing the following steps sequentially: ranking of basis functions based on the residual $\eta^{(k)}$ at the step $k$, select the first $N_B$ basis functions and solve for the corresponding PCE coefficients using coordinate descent method. These steps are combined in Algorithm \[alg:PCE\_solver\]. In our numerical testing, we set $N_B=5$ based on some initial testing. However, a more rigorous approach could utilize cross validation to select the optimal number of $N_B$.
k=0 $\eta^{(0)}$ = y $\textbf{c} = 0 $ $\textbf{r} = \textbf{r}(\eta^{(k)})$ Select $A_1, \cdots , A_{N_B}$ Solve $\Delta \mathbf{c} = \underset{\Delta \mathbf{c}}\arg\min(\operatorname{\mathcal{L}}(\Delta c_{A_1}, \cdots , \Delta c_{A_{N_B}}))$ with respect to selected components Update coefficients: $\mathbf{c} = \mathbf{c} + \mathbf{\Delta c}$ Update residual: $\eta^{(k+1)} = y - \sum_A c_A p_A(\mathbf{x})$ $k = k+1$ **return** $c$
Numerical Examples {#sec:numerical_examples}
==================
In this section, the proposed ranking based sparse PCE is evaluated on four test cases. The first test case is the Ishigami function [@Modified_Ishigami], the second test case is a ten-dimensional Ackley function, the third case is a waterflooding problem with uncertain permeability field and the forth test case utilizes a data-set from simulations of CO2 injection [@UQ_CO2_2]. In all test cases, the proposed PCE approach is compared to two standard techniques for sparse regression-based PCE: Least Angular Regression [@efron2004] and the Orthogonal Matching Pursuit (OMP) algorithm [@singaravelu:06:eurosys]. The numerical implementations are all based on scikit-learn, a machine library including with standard implementation of the LARS, OMP and coordinate descent algorithm. Moreover, cross-validation tools within this library are used to select the optimal regularization parameters for the Elastic Net functional defined in Eq. .
Test case 1: Ishigami Function {#test-case-1-ishigami-function .unnumbered}
------------------------------
Ishigami function is one of the standard benchmarks [@Modified_Ishigami]: $$\label{eq:modified_Ishigami_Function}
y = 1 + \frac{1 + \pi^4/10 + \sin(\pi x_1) + 7\sin^2(\pi x_2) + 0.1 (\pi x_3)^4 \sin(\pi x_1)}{9 + \pi^4/5}$$ This three dimensional function shows a strong nonlinear behavior and is commonly used as a test function for evaluating different response surface techniques. Typically, the evaluation domain is $[-\pi, \pi]^3$. In the present work, PCE with Legendre polynomials is used because of the finite length of the interval concerned. We have rescaled the input parameters linearly to the interval $[-1,1]$, given that the Legendre polynomials are defined over the interval $[-1,1]$. For each of the rescaled variables, a uniform distribution over the interval $[-1,1]$ is assumed. We consider two training sets of $200$ samples and $2000$ samples. Another set of $2000$ points uniformly sampled over the cube $[-1,1]^3$ is utilized for out-of-sample MSE calculations (aka. test set). We construct a PCE of polynomial functions up to degree $d=10$. The aim of the example is to compare the convergence rates of the proposed ranking based PCE approach against the standard sparse regularization techniques (i.e. LARS and OMP), while increasing the number of free coefficients $N_D$ available for fitting by these iterative techniques. In the case of LARS and OMP, the value of $N_D$ is well-defined. For the proposed ranking based approach, $N_D$ is defined as: $$\label{eq:number_of_degrees_of_freedom}
N_D = N_I N_B$$ where $N_I$ is the number of iterations and $N_B$ is the number of PCE coefficients that can be modified by coordinate descent solver after each ranking update in Algorithm \[alg:PCE\_solver\]. It should be emphasized that PCE coefficients are selected solely based on ranking. In other words, the overlapping with previously selected PCE coefficients can occur. Therefore, the value of $N_D$ given by Eq. is a conservative upper bound for the total number of polynomial basis functions involved in the response surface construction (i.e. with non-zero coefficient).
The numerical level of tolerance has been set to $10^{-6}$ in all numerical schemes. Fig. \[fig:Ish\_Convergence\_n\_200\] and Fig. \[fig:Ish\_Convergence\_n\_2000\] shows the MSE for each method versus the number of free coefficients $N_D$ for $200$ and $2000$ training points, respectively.
[0.45]{} ![Mean-square error on the test data set versus the number of free coefficients for the Ishigami function.[]{data-label="fig:Ish_Convergence"}](af1_ndata_200.pdf "fig:"){width="1.0\linewidth"}
[0.45]{} ![Mean-square error on the test data set versus the number of free coefficients for the Ishigami function.[]{data-label="fig:Ish_Convergence"}](af1_ndata_2000.pdf "fig:"){width="1.0\linewidth"}
The results presented in Fig. \[fig:Ish\_Convergence\], shows that response surface built using the ranking based sparse PCE is of higher quality when compared to those obtained by the standard LARS or OMP algorithm, especially when the size of training data is limited. However, all three techniques perform similarly in the case with higher number of training data-points as shown in Fig. \[fig:Ish\_Convergence\_n\_2000\]. This is a major advantage when collecting training samples corresponds to running computationally expensive simulations.
Test case 2: Ackley Function {#test-case-2-ackley-function .unnumbered}
----------------------------
In this example, we build a response surface for a $10$-dimensional Ackley function [@Ackley_Function] of the form: $$\label{eq:Ackley_function}
y = -20 \exp\Bigg ( -0.2 \bigg ( {\frac{1}{n} \sum_{k=1}^n x_k^2} \bigg )^{1/2} \Bigg ) - \text{exp}\Bigg ( {\frac{1}{n} \sum_{k=1}^n \cos{2 \pi x_k}} \Bigg ) + 20 + \exp \big( 1 \big)$$ where $n$ is the dimension of the input vector. This function shows a strong nonlinear behavior with plenty of local minimums and is frequently used as a benchmark for optimization algorithm. The setup of the current numerical example is similar to the first test case. However, we assume that each of the input variables is uniformly distributed in the interval $[-5, 5]$. Legendre polynomials are utilized as basis function for the PCE. Therefore, input rescaling is applied to map all input variables to the interval $[-1, 1]$. In other words, we consider data to be uniformly distributed in the cube $[-1,1]^{10}$. Similar to the first test case, two training sets sizes are considered ($200, 2000$ samples) and $2000$ samples points uniformly distributed in the cube $[-1,1]^{10}$ are set aside as a test set for calculating the out-of-sample MSE. We truncate the PCE spectrum to polynomial functions up to degree $d=8$.
[0.45]{} ![Mean-square error on the test data set versus the number of free coefficients for a ten dimensional Ackley function.[]{data-label="fig:Ackley_Convergence"}](af2_ndata_200.pdf "fig:"){width="1.0\linewidth"}
[0.45]{} ![Mean-square error on the test data set versus the number of free coefficients for a ten dimensional Ackley function.[]{data-label="fig:Ackley_Convergence"}](af2_ndata_2000.pdf "fig:"){width="1.0\linewidth"}
Figure \[fig:Ackley\_Convergence\] shows the MSE convergence for the ranking based sparse PCE versus LARS and OMP algorithms. Similar to the first test case, the proposed approach produces a response surface of higher quality than LARS or OMP if the size of training data is limited as shown in Fig. \[fig:Ackley\_Convergence\_n\_200\], while all techniques perform similarly a higher number of training data-points is used and a high number of polynomial basis functions is utilized as shown in Fig. \[fig:Ackley\_Convergence\_n\_2000\].
Test case 3: Waterflooding problem {#test-case-3-waterflooding-problem .unnumbered}
----------------------------------
In the present test case, we evaluate the developed PCE approach on an uncertainty propagation for a waterflooding problem with uncertain permeability field. Dimension reduction using PCA technique is applied to the spatial field as an effective parametrization techniques [@MA20117311]. Waterflooding is a commonly used process within the petroleum industry for achieving higher hydrocarbon recovery rates. The essence of this approach is to inject water through a number of wells in a given reservoir in order to displace the existing oil and increase the oil production from another set of wells, which are commonly spatially scattered to surround the injection wells. The increased productivity is observed until the injected water starts to appear at the production wells. Thus estimating when water will appear at the production wells (commonly known as the water breakthrough time [@AHMED2019901]) is of significant practical importance. We note, that this time is commonly measured in terms of volume of water injected relative to the total reservoir pore volume (PVI). Prediction of the water breakthrough time $t_b$ is of high importance for hydrocarbon field development because of the economical effects associated with it. In addition, $t_b$ is highly sensitive to the spatial distribution of reservoir properties [@HENDERSON2017178] (e.g. porosity, permeability) which are highly uncertain because of lack of observations. Moreover, reliable forecast for hydrocarbon production rate after the water breakthrough is significant for economical decisions. Therefore, in the present test case we develop a surrogate model for the production rate at late stages of the well life. In particular, we focus on the oil production rate $q_{\text{oil}}$ when $60\%$ of PVI has been injected.
The waterflooding system is modeled via mass and momentum conservation laws coupled with Darcy’s law. A simplified model for waterflooding is utilized where flow of two incompressible fluids (water and oil) is considered. In this setting, we are interested in predicting the spatial distribution of volumetric fractions $s_a$ (saturation) of each of the fluids. The index $a$ could be replaced by either $w$ or $o$ for water and oil, respectively. The evolution of saturations is governed by mass and momentum conservation laws expressed through the following partial differential equation (PDE): $$\label{eq:mass_conservation}
\frac{\partial \phi s_a \rho_a}{\partial t} - \sum_{\gamma = 1}^{3} \frac{\partial}{\partial x^{\gamma}} \bigg ( \frac{\rho_a k k_a}{\mu_a} \frac{\partial P} {\partial x^{\gamma}} \bigg ) = Q_{a}$$ where $\gamma = 1, 2, 3$ is a spatial index of the coordinate vector $\mathbf{x}$, $\rho_a = \rho_a(\mathbf{x})$ and $\mu_a = \mu_a(\mathbf{x})$ are the density and viscosity of fluid $a$ at the point $\mathbf{x}$ respectively, $k = k(\mathbf{x})$ is the permeability, $\phi = \phi(\mathbf{x})$ is the porosity at a given point, $P(\textbf{x})$ is a pressure at point $\mathbf{x}$, $k_a(s)$ is a relative phase permeability of fluid $a$, $s = s(\mathbf{x})$ saturations of fluids at the point $\mathbf{x}$, $Q_a = Q_a(\mathbf{x})$ is a source term for fluid $a$ at the point $\mathbf{x}$. Generally, the permeability $k$ is a tensor. In the present example, we assume $k$ to be a spherical tensor which can vary in space. Therefore, it is fully described by a single spatial function $k = k(\mathbf{x})$. In the present work we neglect capillary pressure effects. Therefore, both fluids are subjected to the same pressure at any given point. The source terms $Q_a$ are considered to be non-zero only for cells with injection and production wells. Finally, incompressible fluids and rocks (solid matrix) are considered. Therefore, Eq. could be simplified: $$\label{eq:mass_conservation_volumetric}
\phi \frac{\partial s_a}{\partial t} - \sum_{\gamma = 1}^{3} \frac{\partial}{\partial x^{\gamma}} \bigg ( \frac{k k_a}{\mu_a} \frac{\partial P} {\partial x^{\gamma}} \bigg ) = q_{a}$$ where $q_a = Q_a/\rho_a$ is the source term for fluid $a$ normalized to the density of corresponding fluid. For calculation of relative phase permeabilities, Brooks-Corey model [@Relative_Phase_Permeability] is used: $$\label{eq:Corey_model}
\begin{gathered}
k_{w}(S_{\text{wn}}) = k_w^{(0)} S_{\text{wn}} ^ {p_w} \\
k_{w}(S_{\text{wn}}) = k_o^{(0)} (1-S_{\text{wn}}) ^ {p_o}
\end{gathered}$$ where $k_w$ and $k_o$ are the values of relative phase permeability for water and oil, respectively and $k_w^{(0)}$ and $k_o^{(0)}$ are maximum the values of relative phase permeability for water and oil, respectively. The values $p_w$ and $p_o$ are dimensionless parameters of the model and $S_{\text{wn}}$ is the normalized water saturation defined as: $$\label{eq:normalized_water_saturation}
S_{\text{wn}} = \frac{S-S_{\text{wir}}} {1 - S_{\text{wir}} - S_{\text{owr}}}$$ where $S_\text{wir}$ and $S_\text{owr}$ are irreducible water and oil saturations, respectively.
In this test case, we consider a five-spot injection pattern where an injection well is located in the center of a square surrounded by four production wells. Given the symmetry of this pattern, only one quarter of the domain is modeled with one producer and one injector located at the opposite corners of a square domain. The length of the edge of that square is set to $L=640 \text{m}$. The thickness of the reservoir is $h=10 \text{m}$. We do not consider discretization along the vertical direction and we only consider a two-dimensional flow problem. For the purposes of simplicity, incompressible immiscible fluids is considered while neglecting gravity effects. A uniform square grid is used for simulations and the dimensions of each grid-block is $10 \text{m}$ by $10 \text{m}$ by $10 \text{m}$. in other words, a $64$ by $64$ by $1$ mesh is used for discretization. The porosity of the reservoir is assumed to be constant and equal to $0.2$. Both injection and production rates are considered to be constant and equal to $10\ \text{m}^3/\text{day}$. The fluid properties and parameters of Corey model are presented in the table \[tab:parameters\].
$\mu_o$, cP $\mu_w$ $p_o$ $p_w$ $k_o^{(0)}$ $k_w^{(0)}$
------------- --------- ------- ------- ------------- -------------
$10.0$ $1.0$ $2.0$ $2.0$ $1.0$ $1.0$
: Fluid properties and parameters of the model for relative-phase permeability.[]{data-label="tab:parameters"}
In the present work, the reservoir permeability $k(\textbf{x})$ is assumed to be a random field with a predefined distribution given the correlation between values at different points within the domain. In reservoir modeling, it is natural to assume that the values of logarithm of permeability $\log(k(\mathbf{r}))$ at different points $\mathbf{r}_1$ and $\mathbf{r}_2$ are exponentially correlated [@MA20117311]: $$\label{eq:exponential_correlation}
\langle \log(k(\mathbf{r}_1)) , \log(k(\mathbf{r}_2)) \rangle = \exp\bigg( -\frac{|\mathbf{r}_1-\mathbf{r}_2|}{L_c} \bigg )$$ where $L_c$ is a correlation length. In the present example, the correlation length is set to $L_c = 1/4 L = 160 \textit{m}$. The utilized distribution of log-permeability allows one to implement Karhunen-Loeve expansion and express $\log(k(\mathbf{r}))$ as a linear combination of mutually independent random variables: $$\label{eq:KL_expansion}
\log(k(\mathbf{r})) = \sum_{\alpha} \theta_{\alpha} \lambda_{\alpha} \xi_{\alpha}(\mathbf{r})$$ where $\lambda_{\alpha}, \xi_{\alpha}(\mathbf{r})$ are the eigen-values and eigen-functions of the KL expansion, respectively. The $\theta_{\alpha}$ are random mutually independent coefficients. In the present example $\theta_{\alpha}$ are considered as input random variables for the PCE response surface. The permeability field is normalized such that a zero value of $\log(k(\mathbf{r}))$ corresponds to a permeability of $1\ \text{mD}$.
We truncate the KL expansion spectrum by taking the first $5$, $15$ or $45$ KL components. Because of the long correlation length with respect to the size of the domain, a significant part of the energy of the spectrum is captured in all truncation scenarios. In this work, the fraction of the energy of the spectrum is defined as following: $$\label{eq:energy_fraction}
H(n) = \frac{\sum_{\alpha=1}^{n}\lambda_{\alpha}^2}{\sum_{\alpha=1}^{\infty}\lambda_{\alpha}^2}$$ where $n$ is the number of components in the truncated KL expansion. In the present example, $H(5) = 0.9898$, $H(15) = 0.9948$ and $H(45) = 0.9972$. It is important to notice that despite the fact that KL expansion captures significant portion of the energy spectrum, it provides smooth reconstruction of the permeability field as shown in Figure \[fig:log\_perm\].
[.22]{} ![Permeability realizations projected to a different number of KL-terms []{data-label="fig:log_perm"}](WF_logperm_nc_5.png "fig:"){width="0.99\linewidth"}
[.22]{} ![Permeability realizations projected to a different number of KL-terms []{data-label="fig:log_perm"}](WF_logperm_nc_15.png "fig:"){width="0.99\linewidth"}
[.22]{} ![Permeability realizations projected to a different number of KL-terms []{data-label="fig:log_perm"}](WF_logperm_nc_45.png "fig:"){width="0.99\linewidth"}
[.22]{} ![Permeability realizations projected to a different number of KL-terms []{data-label="fig:log_perm"}](WF_logperm_nc_4096.png "fig:"){width="0.99\linewidth"}
Three different PCE response surfaces are built corresponding to the $5$, $15$, $45$-KL terms where $1000$ samples (i.e. reconstructed permeability realizations) are evaluated. Each of these samples has been generated from a normal distribution of the coordinates $\theta_{\alpha}$ corresponding to the truncated eigen-vectors of the KL expansion. Water breakthrough times are estimated through numerical simulations for each of the permeability realization using a forward simulation run. A training set of $750$ samples is used for building the PCE response surface and the remaining $250$ samples are used for testing. Legendre polynomials with degree $d \leq 5$ are considered as basis functions. The tolerance in all numerical schemes used to estimate the PCE coefficients has been set to $10^{-6}$.
[0.30]{} ![Mean square error (MSE) over the test data for breakthrough time (a, b, c) and for oil production rate (d, c, e) for different PCE algorithms versus the response surface free parameters.[]{data-label="fig:mse_test3"}](btime_5.pdf "fig:"){width="1.0\linewidth"}
[0.30]{} ![Mean square error (MSE) over the test data for breakthrough time (a, b, c) and for oil production rate (d, c, e) for different PCE algorithms versus the response surface free parameters.[]{data-label="fig:mse_test3"}](btime_15.pdf "fig:"){width="1.0\linewidth"}
[0.30]{} ![Mean square error (MSE) over the test data for breakthrough time (a, b, c) and for oil production rate (d, c, e) for different PCE algorithms versus the response surface free parameters.[]{data-label="fig:mse_test3"}](btime_45.pdf "fig:"){width="1.0\linewidth"}
[0.30]{} ![Mean square error (MSE) over the test data for breakthrough time (a, b, c) and for oil production rate (d, c, e) for different PCE algorithms versus the response surface free parameters.[]{data-label="fig:mse_test3"}](flow_rate_5.pdf "fig:"){width="1.0\linewidth"}
[0.30]{} ![Mean square error (MSE) over the test data for breakthrough time (a, b, c) and for oil production rate (d, c, e) for different PCE algorithms versus the response surface free parameters.[]{data-label="fig:mse_test3"}](flow_rate_15.pdf "fig:"){width="1.0\linewidth"}
[0.30]{} ![Mean square error (MSE) over the test data for breakthrough time (a, b, c) and for oil production rate (d, c, e) for different PCE algorithms versus the response surface free parameters.[]{data-label="fig:mse_test3"}](flow_rate_45.pdf "fig:"){width="1.0\linewidth"}
[0.32]{} ![Cross-plots of breakthrough time and oil production rate in PVI units for Rank PCE algorithm. Top row (a, b, c) shows the results for test samples for breakthrough time and the bottom row (d, e, f) shows the results for oil production rate[]{data-label="fig:cross_plot_test3"}](btime_test_leg_npca_5.pdf "fig:"){width="0.99\linewidth"}
[0.32]{} ![Cross-plots of breakthrough time and oil production rate in PVI units for Rank PCE algorithm. Top row (a, b, c) shows the results for test samples for breakthrough time and the bottom row (d, e, f) shows the results for oil production rate[]{data-label="fig:cross_plot_test3"}](btime_test_leg_npca_15.pdf "fig:"){width="0.99\linewidth"}
[0.32]{} ![Cross-plots of breakthrough time and oil production rate in PVI units for Rank PCE algorithm. Top row (a, b, c) shows the results for test samples for breakthrough time and the bottom row (d, e, f) shows the results for oil production rate[]{data-label="fig:cross_plot_test3"}](btime_test_leg_npca_45.pdf "fig:"){width="0.99\linewidth"}
[0.32]{} ![Cross-plots of breakthrough time and oil production rate in PVI units for Rank PCE algorithm. Top row (a, b, c) shows the results for test samples for breakthrough time and the bottom row (d, e, f) shows the results for oil production rate[]{data-label="fig:cross_plot_test3"}](wf_test_leg_npca_5.pdf "fig:"){width="0.99\linewidth"}
[0.32]{} ![Cross-plots of breakthrough time and oil production rate in PVI units for Rank PCE algorithm. Top row (a, b, c) shows the results for test samples for breakthrough time and the bottom row (d, e, f) shows the results for oil production rate[]{data-label="fig:cross_plot_test3"}](wf_test_leg_npca_15.pdf "fig:"){width="0.99\linewidth"}
[0.32]{} ![Cross-plots of breakthrough time and oil production rate in PVI units for Rank PCE algorithm. Top row (a, b, c) shows the results for test samples for breakthrough time and the bottom row (d, e, f) shows the results for oil production rate[]{data-label="fig:cross_plot_test3"}](wf_test_leg_npca_45.pdf "fig:"){width="0.99\linewidth"}
Figure \[fig:mse\_test3\] shows the MSE for various KL truncation levels. The results presented in this figure, demonstrate that the proposed rank based PCE technique has similar accuracy when compared to the OMP algorithm for low-dimensional problems. However, the rank based PCE has clear advantages in higher dimensions. Also, both the Rank-PCE and OMP methods, perform slightly better than LARS. However, all three techniques do not perform perfectly, because the MSE is around $5\%$ of the mean-value of the QoI. The cross-plot shown in Figure \[fig:cross\_plot\_test3\] demonstrates how the quality of prediction is affected by the dimension of the problem or truncation scheme for KL expansion. The best accuracy of the response surface has been achieved for the problem with the lowest dimension corresponding to $5$-KL truncation level (left). The numerical error is the highest for $45$-KL truncation level (right). The reason for such behavior is that the training set of the same cardinality has been used for all truncation schemes. It should be noted that the accuracy of the permeability representation increases with the increase of parameter space dimension. However, capabilities of the response surface to reproduce the simulation results for a fixed number of direct simulations drop with dimension. In other words, more training data is required to build a high quality response surface for the high-dimensional case when compared with problems of lower dimensionality.
Test case 4: Data from CO2 injection simulations {#test-case-4-data-from-co2-injection-simulations .unnumbered}
------------------------------------------------
In this test case, PCE-based response surface is used as a fast emulator for CO2 injection process. The QoI is the mass of CO2 in a gas phase after given time period from the end of CO2 injection [@UQ_CO2_2]. The data is based on the simulations results developed by @UQ_CO2_2. The key uncertain parameters in this simulations are: average field porosity, average field permeability, regional hydraulic gradient relative phase permeability, capillary pressure and the permeability anisotropy $k_v/k_h$ ratio. More detailed description of this problem can be found in [@UQ_CO2_2]. The average field porosity $\phi$ and permeability $k$ are considered as independent continuous variables with a uniform probability distribution via density-function variable transformation [@rosenblatt1952]: $$\label{eq:variable_transformation}
\begin{cases}
\phi = f_{\phi}(x_1) \\
k = f_{k}(x_2) \\
\end{cases}$$ where $x_1$ and $x_2$ are independent random variables uniformly distributed in the interval $[-1;1]$, $f_{\phi}$ and $f_{k}$ are functions for transformation of variables. All other variables are considered as discrete variables with equal probabilities over all discretized values. Table \[tab:variables\] summarizes the variable names and types used in this test case.
Variable Notation Type
----------------------------- ---------- ------------------------------------------------
Porosity $x_1$ Continuous, $\operatorname{\mathcal{U}}[-1;1]$
Permeability $x_2$ Continuous, $\operatorname{\mathcal{U}}[-1;1]$
Relative phase permeability $x_3$ Discrete, $10$ different models
Regional hydraulic gradient $x_4$ Discrete, $2$ different values
Capillary pressure $x_5$ Discrete, $2$ different models
Permeability anisotropy $x_6$ Discrete, $3$ different values
: Summary of variables notations and types.[]{data-label="tab:variables"}
We note that this test case includes categorical variables in the input space. In order to handle this type of data, we utilize Chebyshev polynomials for categorical data. Additionally, we establish a one-to-one correspondence between the values of a given categorical variable and Chebyshev nodes: $$\label{eq:Chebyshev_nodes}
t_m \rightarrow \cos \bigg ( \frac{2m-1}{2M}\pi\bigg)$$ where $M$ is the total number of possible values for a given categorical variable. The mapping given by this equation is illustrated in Figure \[fig:Chebyshev nodes\].
[0.45]{} ![Location of Chebyshev nodes corresponding to roots of the polynomials with the same degrees as the number of distinct values present in the categorical variable.[]{data-label="fig:Chebyshev nodes"}](Chebyshev_Nodes_5.pdf "fig:"){width="\textwidth"}
[0.45]{} ![Location of Chebyshev nodes corresponding to roots of the polynomials with the same degrees as the number of distinct values present in the categorical variable.[]{data-label="fig:Chebyshev nodes"}](Chebyshev_Nodes_10.pdf "fig:"){width="\textwidth"}
We note that Gauss-quadrature rules for Chebyshev polynomials have the same weight [@Book_on_Orthogonal_Polynomials] for each of the nodes. This justifies using Chebyshev polynomials for categorical data and the corresponding mapping to the Chebyshev nodes presented in Eq. especially when training samples are uniformly distributed over the distinct categories. Therefore, the polynomials orthogonality and the distribution of categorical variables are consistent with each other. $$\label{eq:orthogonality_of_Chebyshev_polynomilas}
\int_{-1}^{+1} \frac{ p_{\alpha}(t) p_{\beta}(t)}{ \sqrt{1-t^2}} \mathrm{d}t = \sum_m \frac{\pi }{M} p_{\alpha}(t_m) p_{\beta}(t_m) = \pi \operatorname{\mathbb{E}}[p_{\alpha}p_{\beta}]$$ In the present work we utilize normalized polynomials $p_{\alpha}(t)$ to $q_{\alpha}(t)$: $$\operatorname{\mathbb{E}}[q_{\alpha}q_{\beta}] = \delta_{\alpha\beta}$$ Using Chebyshev polynomials provides a natural extension of standard PCE to problems with categorical variables while preserving the fundamental relation between the orthogonality of basis functions and probability distribution as defined in Eq. .
In the current example, sampling of the data is performed using uniform distributions over the parameter ranges. A total of $998$ data points are generated in accordance with the proposed probability distributions of variables and we used $250$ data-points for training (i.e. constructing the PCE) and the remaining data points are used for testing. The mass of CO2 injection is computed via detailed numerical simulations (see [@UQ_CO2_2] for more details). We normalized the QoI such that following equality holds for the training data: $$\label{eq:normalization_of_training_data}
\sum_i \frac{y_i^2}{N} = 1$$ We observed empirically that the QoI is highly sensitive to the permeability and relative phase permeability. Therefore, we constructed two evaluation cases with the same data set. For the first case which we refer to as the reduced case, we built a two-dimensional response surface using the permeability and relative phase permeability only as an input. The second case, which we denote as the full case, we utilize all the six uncertain variables in the response surface. In both cases, we evaluate the proposed ranking based sparse PCE against standard sparse regression PCE algorithms (i.e. LARS and OMP methods) for different numbers of expansion coefficients $N_D$. For both the reduced and full problems, PCE is performed with polynomials of degree $d \leq 10$. The number of terms in PCE varies from $5$ to $50$ and the tolerance has been set similar to all other test cases to $10^{-6}$. Legendre polynomials were used for continuous variables $x_1, x_2$ and Chebyshev polynomials were used for the discrete/categorical variables.
[0.45]{} ![Mean square error (test data) for the different PCE algorithm versus the response surface free parameters.[]{data-label="fig:mass_CO2"}](mse_CO2_2d.pdf "fig:"){width="1.0\linewidth"}
[0.45]{} ![Mean square error (test data) for the different PCE algorithm versus the response surface free parameters.[]{data-label="fig:mass_CO2"}](mse_CO2_6d.pdf "fig:"){width="1.0\linewidth"}
Figure \[fig:mass\_CO2\], shows the mean square error over the test data set for both the reduced and full cases in Fig. \[fig:mass\_CO2\_2d\_Convergence\] and Fig. \[fig:mass\_CO2\_6d\_Convergence\], respectively. The introduced ranking based approach shows better convergence rates for both problems. Moreover, the results in Fig. \[fig:mass\_CO2\_6d\_Convergence\] demonstrate that advantages of the proposed Rank-PCE are more pronounced for higher dimensional problems, where the search space inside the iterative solver is large. For this case, the introduced ranking step allows for an efficient identification of the most significant components of PCE resulting in a higher quality response surfaces.
Concluding remarks {#sec:conclusions}
==================
In the current manuscript, we introduced a ranking based sparse PCE technique (Rank-PCE). The core idea of the proposed approach is to rank the PCE features in accordance with the magnitude of a given PCE coefficient based on the correlation with data while estimating for the accuracy of computed correlations. We demonstrated, via a set of numerical examples, the superior performance of Rank-PCE when compared to standard sparse regularization techniques. Rank-PCE resulted in an increase in convergence rates for generative function with sparse spectrum. We also noticed that the improvements in convergence is more pronounced for high-dimensional problems, enabling the application of PCE to problems with significant number of independent variables. Moreover, the advantages of Rank-PCE are also evident for problems with limited number of training samples as demonstrated in the analytical test cases.
In addition to novel ranking procedure, we presented an extension of PCE response surfaces to problems with both continuous and categorical data through the utilization of Chebyshev polynomials to represent the discrete variables. The proposed technique might be not optimal for general cases, however under the uniform sampling conditions, it provides a simple approach to handle categorical data in PCE that is consistent with the statistical properties of PCE for sensitivity analysis and UQ. In other words, the proposed approach maintains the relation between basis orthogonality and statistics of the input variables, which is fundamental for UQ with PCE. This technique is also easy to implement given the availability of Chebyshev polynomials in most scientific computing libraries.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 653718. We thank J.C. Manceau and J. Rohmer (BRGM, France) for providing the CO2 injection dataset used in test cases. We also thank Dr. Shing Chan (formally at HWU and currently at the Oxford Big Data Institute) for developing the reservoir simulation codes used in the third example.
[^1]: Corresponding author.\
E-mail addresses: `a.tarakanov@hw.ac.uk` (Alexander Tarakanov), `a.elsheikh@hw.ac.uk` (Ahmed H. Elsheikh).
[^2]: Published in Journal of Computational Physics (2019). DOI: [10.1016/j.jcp.2019.108909](https://doi.org/10.1016/j.jcp.2019.108909)
|
---
address: |
INFN, sezione di Ferrara, Via Paradiso, 12 - 44100 Ferrara, Italy\
and\
ITEP, B. Cheremushkinskaya 25, Moscow, 117259, Russia
author:
- 'A. D. DOLGOV'
title: PARTICLE PRODUCTION IN COSMOLOGY AND IMAGINARY TIME METHOD
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Personal Recollections. \[s:psrn\]
==================================
It is difficult to write about a close friend who died so unexpectedly and so early. Time runs fast and it seems that it was just yesterday that we saw each other and talked a lot on every possible subject. I really miss these discussions now. Last time that I saw Misha Marinov was spring 1999. I was visiting Weizmann Institute of Science in frameworks of Landau-Weizmann Program and used this opportunity to come to Haifa where Misha lived and worked in Technion. It was a nice sunny and fresh morning when we arrived with my wife Inna to Haifa railway station, where Misha met us and drove along a beautiful road up the Carmel Mountain to his home where Lilia, his wife, waited us with a delicious lunch. Misha was in high spirits, the four of us being old friends, were very glad to see each other, but slightly complained about, as he said, small pain in his spine. None of us knew at that moment that it was a first sign of fast and terrible disease.
Our friendship with Misha began, I think, in 1965 when we both were graduate students. We happened to be in the same plane on the way to Yerevan to the First International Nor-Ambert School on Particle Physics. Together with another young physicist from ITEP, Misha Terentev, we shared a small hut and enjoyed first in our lives international conference and charming and hospitable town of Yerevan.
Misha was two years older than me but I had a feeling that his knowledge of physics, especially of mathematical physics, was at a professor level and benefited a lot from our communications. Later we both worked in ITEP theory group and it was always instructive and interesting to talk with him not only about physics but practically any subject, especially history where Misha had unusually deep and wide knowledge would it be ancient or modern.
Our friendship turned into friendship between families when in 1970 we started to live in the same apartment building near ITEP and the distance between our apartments was only 1-2 minute walk up or down the stairs. In 1979 Misha quit his position in ITEP and applied for permission to emigrate with the aim to live in Israel. Immediately life became much harder for him. It was difficult to find a job that could give enough money to support his family of four. Special rules existed at that time in the Soviet Union to prevent people from working without strict state control. In summer seasons (plus a part of spring and autumn) Misha worked as a construction worker, building small private houses (dachas) in the country. In winter he did some work for the official “Center of Translations” translating scientific papers or books from English into Russian or vice versa. However this kind of job was allowed only if one has another permanent place of work at one or other state enterprise, which Misha had not. At some stage they requested from him a certificate that he had such a job and since nothing can be presented, he was fired. So my wife, Inna, formally took this job and fetch for him papers to translate from the Center. Misha translated them, Inna presented the translated papers to the Center, received the money and brought it to Misha or his wife, Lilia. That’s how it worked.
I have to confess that I also participated in a similar deception activity, a few papers and books translated into Russian under my name were in fact translated by Misha. In particular, the review paper by S. Coleman “The magnetic monopole fifty years later”[@coleman82] was translated for Uspekhi Fiz. Nauk, vol. 144, by Marinov but under my name. Moreover, the editors wanted to have a short review on the activity related to magnetic monopoles to the moment when the paper was translated, i.e. two years after the original one had been written. Again Misha wrote the paper and I only signed it (honestly I also read it and liked it very much). So he got the money and I got the fame. Now I have to restore justice and to change the reference[@dolgov84] into[@marinov84].
Only in 1987 Marinovs received permission to emigrate and left for Israel. As we all thought that time, emigration meant leaving for good with practically zero chances to see or contact each other again. However things were changing fast and freedom to travel abroad, unbelievable at Soviet times, came to our country. In 1990 Inna was able to go to Israel and for a whole month enjoyed friendly atmosphere of Marinov’s home. She and Lilia even now recall with mutual pleasure how nice was that time. After a couple of years Misha’s life in Israel was successfully arranged. He got a professor position in Technion and was happy to be there. I remember how proudly he showed me the campus, labs, students during my first visit to Haifa. He enthusiastically returned to research that was interrupted for 6-7 years. As I can judge by what and how I learned from him, he was a very good teacher and did teaching with vigor and love. On the other hand, he kept warm feelings toward ITEP, and often going in the morning to his office in Technion he used to say, addressing Inna and Lilia, “Buy girls, I am going to ITEP”.
There are several fields where Misha made very important contributions, despite a long break in his scientific activity. But I am not going to describe them all, since, I think, this will be described in the Introduction to this volume. I will mention only two which have some relation to me. Misha’s results on application of path integral methods to complicated quantum systems are internationally renowned and I am proud that I recommended Maurice Jacob to publish Marinov’s review on the subject in Physics Reports[@marinov79]. This was the last paper written by Misha, while he was still in ITEP. Another subject where M. Marinov made a very essential contribution together with V. Popov was electron-positron production by an external electromagnetic field[@marinov72]. The method developed in these works was applied to the non-perturbative calculations of cosmological particle production by scalar (inflaton) field in our paper with D. Kirilova[@dolgov90], which is discussed below.
Particle Production in Cosmology; Brief Historical Review \[s:hstr\]
====================================================================
There are two different cases of quantum particle production by external classical fields that are cosmologically interesting. The first is the production by time-dependent background metric or, in other words, by gravitational field and the second is the transformation of classical (oscillating) inflaton field into elementary particles and the corresponding universe (re)heating. Particle production by gravity might be essential in the very early universe near cosmological singularity when the strength of gravitational field was close to the Planck value. Creation of particles by isotropic Friedman-Robertson-Walker (FRW) metric was pioneered by Parker[@parker68] and further developed in a series of papers[@parker69; @grib69; @chernikov72]. Particle production by gravity in anisotropic cosmologies was considered in refs.[@zeldovich70]. As argued in these papers, particle production in anisotropic case creates anisotropic distribution of matter and back reaction of the created matter on the metric could lead to isotropization of the latter. Thus, in principle, the observed FRW cosmology might originate from a rather general initial state. More references to the subsequent works and detailed discussion can be found in the books[@books].
There is an important difference between particle production in isotropic and anisotropic cosmologies. Isotropic FRW metric is known to be conformally flat, i.e. after a suitable coordinate transformation it can be reduced to the form: ds\^2 = g\_ dx\^dx\^= a\^2 (r,) ( d\^2 - dr\^2 ) \[ds2\] From this expression follows, in particular, that FRW metric cannot create massless particles if the latter are described by conformally invariant theory[@parker68]. If the particle mass, $m$, is non-vanishing but the interactions are conformally invariant, their production rate is suppressed as a power of the ratio $(m/m_{Pl})$. (Of course, non-vanishing masses break conformal invariance.) These statements can be easily checked in perturbation theory. The coupling of gravity to matter fields is given by $\left(g^{\mu\nu}-\eta_{\mu\nu}\right) T^{\mu\nu}$, where $\eta^{\mu\nu}$ is the Minkowsky metric tensor and $ T^{\mu\nu}$ is the energy-momentum tensor of matter. If the metric tensor is given by expression (\[ds2\]), the coupling to matter is proportional to the trace of the energy-momentum tensor that vanishes in conformally invariant theory.
A well known example of the theory which is conformally invariant at classical level (i.e. without quantum corrections) is electrodynamics with massless charged fermions, or any other (possibly non-abelian) gauge theory describing interacting massless gauge bosons and fermions. However quantum trace anomaly[@chanowitz73] breaks conformal invariance and gives rise to a non-zero trace of $T_{\mu\nu}$. In $SU(N)$ gauge theory with $N_f$ number of fermions the trace of the energy-momentum tensor of matter is equal to: T\^\_= ( [11 N 3]{} - [2N\_f 3]{} ) G\_ G\^ \[tmumu\] where $G_{\mu\nu}$ is the gauge field strength tensor. This anomaly could strongly enhance generation of electromagnetic field (or any other gauge fields) in the early universe[@dolgov80].
Another simple and important theory of a free massless scalar field $\phi$ is not conformally invariant even at the classical level if $\phi$ is minimally coupled to gravity (that is through covariant derivatives only). The energy-momentum tensor of such field is given by the expression: T\_ () = (1/2)\_ \_- (1/4) g\_ \_ \^\[tmunuphi\] and its trace $T^\mu_\mu = -(1/2)\partial_\alpha \phi\,\partial^\alpha \phi$ is generally non-vanishing. Conformal invariance can be restored if one adds to the free Lagrangian the nonminimal coupling to gravity in the form $R \phi^2 /12$ (see e.g. refs. [@books]). However it would be better not to restore it because generation of primordial density perturbations at inflation[@guth82], which serve as seeds for large scale structure formation, is possible only for non-conformal fields.
Another realistic example of conformally non-invariant theory with massless fields is gravity itself. It was shown that gravitational waves are not conformally invariant in the standard General Relativity[@grishchuk75]. This explains efficient production of gravitational waves during inflationary stage[@starobinsky79].
A renewed interest to gravitational particle production arose in connection with a possible explanation of the observed ultra-high energy cosmic rays by heavy particle decays[@berezinsky97]. There are two competing mechanisms of creation of such particles in cosmology: by background metric and by inflaton field. The former was considered in refs. [@kuzmin98] (for a review see[@kuzmin99]), while particle production by inflaton will be discussed below.
In the earlier papers[@dolgov82] the universe (re)heating at the final stage of inflation through particle production by the oscillating inflaton field was treated in a simplified perturbation theory approximation. First non-perturbative treatment was performed in two papers[@dolgov90; @traschen90]. In what follows we concentrate on the approach of ref.[@dolgov90] where the imaginary time method was used. In both papers[@dolgov90; @traschen90] a possibility of parametric resonance enhancement of particle production rate, noticed long ago[@narozhnyi73], was mentioned. However, it was argued in the first of them that the resonance was not effective because the produced particles were quickly removed from the resonance band by the cosmological expansion and elastic scattering on the background. A more careful analysis of the subsequent paper[@traschen90] showed that under certain condition expansion might be irrelevant and did not destroy the resonance. In this case a strong amplification of the production probability and much faster process of post-inflationary (re)heating could be expected. The issue of the parametric resonance (re)heating attracted great attention after the paper[@kofman94], and now the number of published papers on the subject is measured by a few hundreds. However, a review of this activity is outside the scope of the present paper and below we will confine ourselves to the problem of fermion production by a time dependent scalar field where parametric resonance is not effective.
Concerning production of fermions, there is a contradiction in the literature between the paper[@dolgov90], where non-perturbative production of fermions was pioneered, and the subsequent ones. While in the paper[@dolgov90] was stated that fermion production is always the strongest in perturbation theory regime, and in the opposite, quasiclassical limit the production is noticeably weaker, in subsequent works was argued that in non-perturbative regime fermion production was strongly enhanced so that it could even compete with resonant boson production. Calculations in ref. [@dolgov90] have been performed by imaginary time method, while other works either used numerical calculations or some approximate analytical estimates. I will argue in what follows that there is practically no difference between the results of all calculations, earlier and later ones, but the difference is in the interpretation of the results and fermion production by the inflaton is always weak, weaker than that found in perturbation theory.
Particle Production in Perturbation Theory \[s:pert\]
=====================================================
Let us start from consideration of production in the case when perturbation theory is applicable and calculations are straightforward and simple. In this section we will neglect the universe expansion and assume that the external scalar field periodically changes with time according to: (t) = \_0 t \[phioft\] Here $\phi_0$ is the amplitude of the field, it can be slowly varying function of time, and the frequency of oscillations $\omega$ coincides with the mass of $\phi$ if the latter lives in the harmonic potential $U(\phi) = m^2_\phi \phi^2/2$.
We assume that $\phi$ is coupled to fermions through the Yukawa interaction: \_ = |( i[ /]{}+m\_0 ) + g |\[lpsi\] Perturbation theory would be valid if the coupling constant is small, $g\ll 1$, which is well fulfilled for the inflaton field, and if the fermion mass is smaller then the mass of the inflaton, $m_\phi = \omega$. The last condition may not be true even if $m_0 < m_\phi$ because the interaction with $\phi$ introduces effective time-dependent mass m\_1 (t) = g \_0 t \[m1oft\] and for a large amplitude $\phi_0$ the latter may be large in comparison with $\omega$ for most of the oscillation period, except for a small part, when $\cos \omega t$ is close to zero. In this case perturbation theory is invalid.
It is practically evident, even without calculations, that in perturbative case the rate of particle production is equal to the width of the decay of the scalar boson $\phi$ into a pair of fermions: n\_/n\_= \_= g\^2 /8\[dotn/n\] where $n_{\psi ,\phi}$ are the number densities of $\psi$ and $\phi$ particles per unit volume respectively and we assumed for simplicity that the fermion mass $m_0 =0$ (it is straightforward to lift this restriction).
Still to make comparison with subsequent non-perturbative calculations we will sketch below the derivation of this result. According to general rules of quantum field theory the amplitude of production of a pair of particles with momenta $\vec p_1$ and $\vec p_2$ by an external time-dependent field $\phi(t)$ in first order in perturbation theory is given by A(p\_1,p\_2) = gd\^4 x (t) p\_1,p\_2 | |(x) (x) | vac \[apt\] where the state $\langle \vec p_1,\vec p_2 |$ is produced by action on vacuum of the creation operators in the standard second-quantized decomposition of Dirac operators $\psi $ and $\bar \psi$: (x) = \_s \[u\_k\^s b\_k\^s e\^[-ik x]{} + v\_k\^s d\_k\^[s ]{} e\^[ik x]{}\] \[psiofx\] where $b_k^s$ and $d_k^{s{\dagger}}$ are respectively annihilation and creation operators for particles and antiparticles with momentum $k$ and spin $s$.
After the usual anti-commutation algebra we will arrive to the integral d\^3 k d\^3 k’ (k - p\_1) (k’- p\_2) e\^[i(E+E’)t - i(k+ k’) x ]{} \[intdp\] The integral can be trivially taken and substituted into the integral over $d^3x dt$ (\[apt\]). Integration over $d^3 x$ gives $\delta (\vec p_1 +\vec p_2)$ and we are left with the Fourier transform: A(p\_1,p\_2) \~g\^2 \^[(3)]{} (p\_1 +p\_2) dt (t) e\^[i(E\_1+E\_2)t]{} \[adelta\] (for details and more rigorous consideration in terms of Bogolyubov coefficients see e.g. appendix A in ref. [@dolgov96]).
The probability of particle production is proportional to $ |A(\vec p_1,\vec p_2) |^2$ and contains the square of momentum delta-function. The latter is treated in the standard way, \^2 = 2V (p\_1 +p\_2) \[delta2\] where $V$ is the total space volume. The origin of the volume factor is evident: since the external field is space-point independent, so is the probability of production per unit volume and the total probability is proportional to the total volume.
Similar situation is realized for the time dependence in the case of periodic external fields, if one neglects back reaction of the produced particles on the field evolution and on the probability of production. The former can be taken into account by a (slow) decrease of the field amplitude $\phi_0(t)$, while the latter is determined by the statistics of the produced particles: the probability of boson production is proportional to the phase space density of already produced bosons, $(1+f_k)$, while the probability of fermion production is inhibited by the factor $(1-f_k)$. This back reaction effect is absent for Boltzmann statistics, which we will mostly assume in what follows. Thus, for a periodic external field one would expect that the probability of production is proportional to the total time interval, during which the external field was operating. In the idealistic case of $\phi \sim \exp (i\omega t)$, its Fourier transform gives $\delta (2E -\omega)$ and the square of the latter is, as above, $t_{tot} \delta (2E -\omega)$. The second factor ensures energy conservation and is infinitely large for $E=\omega/2$. It means that the phase space density of the produced particles becomes very large after period of time when the energy conservation is approximately established. One can check that this time is much shorter than $1/\Gamma$ (where $\Gamma$ is the perturbative decay rate) but still the time of transition of energy from the inflaton field to the produced fermions is given by $1/\Gamma$. This fact is commonly agreed upon in the case of perturbative production. The statements in the literature that in non-perturbative regime fermion production could be very strong is possibly related to this trivial rise of the occupation numbers and does not mean that fermion production can compete with production of bosons (see below).
In the case when external field operates during a finite period of time, starting e.g. from $t=0$, or if one is interested in the number of produced particles at the running moment $t$, the integral in expression (\[adelta\]) should be taken in the limits $(0,t)$ and for the particular case of $\phi = \phi_0 \cos \omega t$ one obtains: I(t; E, ) && \_0\^t dt e\^[2iEt]{} t\
&=&e\^[i(E-/2) t]{}\[int0t\]
For $E$ close to $\omega/2$ the first term dominates and the number of produced fermions rises as $t^2$ till $t \sim 1/|2E-\omega|$. At larger times it oscillates. The same phenomena was found in non-perturbative calculations. Indeed, the phase space number density of the produced particles (we use this term interchangeably with the “occupation number”) is given by f\_p = g\^2 \_0\^2 I (t; E,) \^2 \[fpptbl\] As we have argued above, usually one has $ \mid I (t; E,\omega) \mid^2 = 2\pi t\,\delta (2E-\omega)$. In this case the number density of the produced particles as a function of time is given by: n(t) = f\_p = [g\^2 8]{}\_0\^2t = n\_t \[noftbl\] where $n_\phi = \phi_0^2 \omega$ is the number density of $\phi$-bosons and $\Gamma$, given by eq. (\[dotn/n\]), is their decay width.
A detailed explanation of the discussed phenomena can be found in textbooks on quantum mechanics in the section where perturbation theory for time dependent potential is presented, see e.g.[@landau].
Returning to the occupation number (\[fpptbl\]) we see that for $(\omega- 2E)t <1$ it evolves as $f_p \approx g^2\phi_0^2 t^2$ and reaches unity for $t=t_1 = 1/g\phi_0$. This is much earlier than $t_d = 1/\Gamma$ which is the characteristic decay time of $\phi (t)$: t\_d /t\_1 = (8/g)(\_0 /) \[tdt1\] Formally taken this ratio may reach the value $10^8-10^9$. This is an explanation of statement that fermions could be very quickly produced by inflaton. On the other hand, though some fermionic bands (approximately satisfying energy conservation) might be quickly populated, the total transfer of energy from the inflaton to the produced particles is determined by the total decay rate and is much slower. Roughly speaking $f_p = 1$ corresponds to production of only one pair of fermions and, of course, the energy of this pair is negligible in comparison with the total energy accumulated in the classical field $\phi (t)$.
Perturbation theory is not applicable if the effective mass of fermions $m_{eff}= (m_0 + g\phi_0)$ is larger than the frequency of the oscillations of the scalar field. For example, the probability of pair production by two-quanta process, when the energy of each produced fermion would be equal to $\omega$, is related to one-quantum process, when $E=\omega/2$, as $W_2/W_1 \sim (g\phi_0 /\omega)^2$. It is still possible that $\phi_0/g\omega \gg 1$, while $g\phi_0/\omega <1$, so that perturbation theory is reliable and the relation $t_d/t_1 \gg 1$ still holds. However in many practically interesting cases $g\phi_0/\omega >1$ and in this range of parameters the result obtained above can serve only for the purpose of illustration and for more precise statements we have to go beyond perturbation theory. This will be done in the following section by the imaginary time method[@nikishov69; @popov71; @marinov72]. (For recent applications of this method and a more complete list of references see[@ringwald01].) Qualitatively clear that non-perturbative effects could only diminish the rate of particle production because the non-perturbative calculations take into account non-vanishing and large value of the effective mass of the produced particles and this leads to a smaller rate of the production in comparison with the case when the interaction is taken in the form $g\phi\bar\psi\psi$ but its contribution into fermion effective mass is neglected. As we see below, the suppression of the production rate in nonperturbative regime[@dolgov90] in comparison with perturbation theory is given by the factor $(\omega/g\phi)^{1/2}$ in qualitative agreement with these simple arguments.
Effects of quantum statistics were neglected above, and thus the results obtained are valid only if $f_p <1$. The corresponding corrections can be approximately introduced by multiplication of the r.h.s. of eq. (\[fpptbl\]) by the factor $(1\pm f_p)$ and correspondingly $f_p^{(f,b)} = g^2 \phi_0^2|I|^2 / (1 \pm g^2 \phi_0^2 |I|^2)$, where the signs $''\pm''$ refer for fermions and bosons respectively. One sees that the production of fermions effectively stops (as one should expect) when $f_p^{(f)} \sim 1$, while production of bosons tends to infinity. Presumably a more accurate treatment would not allow bosons to reach infinitely large density in a finite time but the message is clear, the production of bosons becomes explosive in perturbation theory with characteristic time of the order of $t_1 = 1/(g\phi_0)$ and all the energy of the inflaton would go into that of the produced bosons during approximately this time. There are several effects that can weaken this conclusion. One is a possible inapplicability of perturbation theory for a large $g\phi_0/\omega$. This effect qualitatively acts in the same way as in fermionic case discussed above. Still, even if the $g\phi_0/\omega>>1$ the effect of explosive production of bosons would survive due to parametric resonance in equation of motions for the produced modes[@dolgov90; @traschen90; @kofman94]. Another two effects that could diminish the production are the cosmological red-shift of momenta of the produced particles and their scattering on other particles in the background. Both would push the produced particles away from the resonance band and could significantly slow down the production in the case of narrow resonance[@dolgov90], while in the case of wide resonance the effect survives[@traschen90; @kofman94].
On the other hand, both red-shift and scattering of the produced fermions back react on their production in exactly opposite (to bosons) way. These phenomena “cleans” the occupied zone and allows for production of more fermions.
Quasiclassical Limit; Imaginary Time Method. \[s:imt\]
======================================================
Small Mass Case. \[ss:small\]
-----------------------------
Usually non-perturbative calculations are not simple but in the case that we are considering there is a fortunate circumstance that in the anti-perturbative limit quasiclassical approximation works pretty well. The latter can be efficiently treated by the imaginary time method[@nikishov69; @popov71; @marinov72]. Below we will essentially repeat the paper[@dolgov90] correcting some typos and algebraic errors, though the basic results of the paper remain intact.
The coupling of $\phi(t)$ to the produced particles is equivalent to prescription of the time dependent mass to the latter, $m(t) = m_0 + g\phi(t)$. The classical Lagrange function for a relativistic particle with such a mass has the form L\_[cl]{} = - m(t) ( 1 - V\^2)\^[1/2]{} \[lcl\] where $\vec V$ is the particle velocity. The corresponding Hamiltonian is = \^[1/2]{} (t) \[ham\] The quantization of this system can be achieved by the path integral method. The Green’s function of the quantum particle has the form (see e.g. [@marinov79]): G(x\_f,t\_f; x\_i, t\_i) = Dp Dx . \[gxtint\] The functional integral in this case can be easily taken, giving: G(x\_f,t\_f; x\_i, t\_i) = \[gxt\]
According to the general rules of quantum mechanics the amplitude of the transition from the state given by the initial wave function $\Psi_i$ into that given by $\Psi_f$ is equal to A(p\_1,p\_2) = d\^3 x\_i d\^3 x\_f \^\*\_f (x\_f) G(x\_f,t\_f; x\_i, t\_i) \_i (x\_i). \[ap1p2\] where for $\Psi_{i,f}$ plane waves are usually substituted.
If we want to obtain the amplitude of creation of a pair of particles the contour of integration over time should be shifted into complex $t$-plane in such a way that it goes around the branching point of the energy $\Omega$ in the direction of changing the sign of energy from negative to positive one. This corresponds to transition from the lower continuum of the Dirac sea to the upper one, i.e. to pair creation. Thus we find: A(p\_1,p\_2) = ( 2)\^3 ( p\_1 + p\_2 ) , \[apair\] where the contour $C(t_i,t_f)$ starts at $t=t_i$ and ends at $t=t_f$ and turns around the branching point of $\Omega$ in the way specified above.
The position of the branching points $t_b = t' +it'' $ can be found from the equation: p\^2 + ( m\_0 + g \_0 t )\^2 =0. \[brpoint\] Correspondingly m\_0 + g\_0 ( ’” -i ’ ” ) = i p, \[tt\] where $\tau = \omega t$. In what follows we assume that $m_0 = 0$ and it will grossly simplify technical details. In this limit $\tau' = \pi/2 + n\pi$ and $\sinh \tau'' = \pm (p/g\phi_0)$.
The integral along the cut $\tau = \tau' +i\eta$ is real and, according to our prescription, negative. It gives exponential suppression factor for the production probability, $W \sim \exp (-2Q)$, with Q =(2/) \_0\^[”]{} d( p\^2 - g\^2\_0\^2\^2 )\^[1/2]{}. \[Q\] This integral can be expressed through complete elliptic functions[@grad]: Q = [2 ]{} \[qell\] where m\_1 = g\_0,[and]{} = p/. \[beta\] For small $\beta$ these functions can be expanded as $K(\beta) \approx (\pi/2) (1+\beta^2 /4)$ and $E(\beta) \approx (\pi/2) (1-\beta^2 /4)$, so that $Q\approx (\pi/2) (p^2/ \omega \, m_1 ) $.
The total production amplitude is equal to the sum of expressions (\[apair\]) with all the contours encircling the proper branch points between $t_i$ and $t_f$. Since the integrals along imaginary direction $id\eta$ are all real and have the same value for all branch points, their contribution to the amplitude gives the common factor $\exp (-Q)$. The integrals over real time axis corresponding to different contours $C$ around neighboring branch points differ by the phase factor $A_{n+2}/A_n =\exp (2i\alpha)$, because the energy changes sign after the integration contour turns around branch points. The absence of the contribution from the nearest cut is related to the particle statistics and is discussed e.g. in ref.[@popov71; @marinov72]. The phase $\alpha$ is given by: = \_0\^[2]{} dt = [4 ]{} E() \[alpha\] All this is true if the free fermion mass is vanishing, $m_0 =0$, otherwise equations become significantly more complicated. In the limit of small $\beta$ we find[@grad]: E()\[E\] while for $\beta$ close to 1 the necessary expressions are presented after eq. (\[beta\]) with the interchange $\beta^2 \leftrightarrow (1-\beta^2)$.
Summing over all branch points we obtain: A(p\_1,p\_2) = ( 2)\^3 ( p\_1 + p\_2 ) [(N ) -1 ( ) -1]{} \[atot\] where $N$ is the total number of branch points included in the amplitude; it is approximately equal to the total time in units $1/\omega$ during which the particles are produced, $N= {\rm Integer}[ (t_f-t_i)/\omega]$. The last factor reminds that coming from the integration over time in perturbation theory discussed in sec. \[s:pert\] and in fact its physical nature is the same. For very large $N$, formally for $N\rightarrow\infty$ it tends to \_j ( -j ) \[delofal\] These delta-functions impose energy conservation for the production of pair of particles by $j$ quanta of the field $\phi$. Note that in contrast to the lowest order perturbation theory, when only a single quanta production is taken into account, the expression (\[atot\]) includes production of a pair by many quanta of the field $\phi$. For example, in the limit of high momenta of the produced particles these delta-functions are reduced to $\delta (2p -j\omega)$, the same as in perturbation theory for j-quanta production.
Treating again, as in sec \[s:pert\], the square of delta-function as a product of the single delta-function and $\delta (0) = \pi N$ with $N$ expressed through the total time $t$, during which the particles have been produced, as the integer part of $t\omega $, we find the following expression for the rate of production per unit time and unit volume[@dolgov90]: n =\_j (-2Q) ( - j) \[dotn\] In the limit of $m_1 \gg \omega$ one obtains: Q && [2]{}[p\^2 m\_1]{}\
&& [4m\_1 ]{}, \[Qalpha\] and hence n = [12]{}\_[j\_m]{} \[dotnpj\] Here summation starts from the minimum integer value $j_m \geq (4m_1/\pi\omega)$ and $p_j$ is determined from the equation $\alpha = j\pi$, i.e. p\_j\^2 (m\_1 /2)(j-4m\_1/) /\[(4m\_1/p\_j) +1\] \[pj2\] A rough estimate gives $\dot n \sim \omega^{5/2} m_1^{3/2}$. Correspondingly the characteristic rate of the inflaton decay in the quasiclassical approximation is given by \_[q]{} = n/n\_=n /(\^2\_0) \~( /m\_1 )\^[1/2]{} \[gammaq\] where $\Gamma$ is the decay rate in perturbation theory (\[dotn/n\]). One sees that in the quasiclassical limit the decay rate is suppressed in comparison with the formal result of perturbation theory as a square root of the ratio of the oscillation frequency to the amplitude of the scalar field. This suppression can be understood as follows[@dolgov90]. Most of the time the instant value of the field $\phi(t)$ and the effective mass of the fermions, $m_{eff}=g\phi_0\,\cos \omega t$ are large in comparison with the oscillation frequency. As is well known (see also sec. \[ss:large\] below) the probability of particle production in this case is exponentially suppressed. However, when $\cos \omega t$ is very close to zero the effective mass of the produced particles would be smaller than $\omega$ and they are essentially produced at this short time moments. This results in a much milder suppression of the production, not exponential but only as $(\omega/g\phi_0)^{1/2}$.
For the case of finite and not too big $N$ we will see that, according to the calculations of reference[@dolgov90] presented above, the occupation number $f_p$ would reach unity in a much shorter time than $1/\Gamma_q$. This result was rediscovered later in the papers[@baacke98; @green99] by numerical calculations and reconfirmed by analytical methods in ref. [@peloso00]. However, as it has been already argued, this does not mean that non-perturbative production of fermions is strong, it is always weaker than the perturbative one.
The calculations presented above do not include the effects of quantum statistics, so strictly speaking, they are valid for “boltzons”. Thus, they present an upper bound for the production of fermions. In the fermionic case, the production would stop when the occupation number, $f_p$, approaches unity, while production of “boltzons” would go unabated. However, if the particles from the occupied Fermi band are quickly removed by scattering or red-shift (as we discussed above) the production of fermions would go essentially with the same rate as production of “boltzons”.
For a finite number of oscillations $N$ the occupation number of the produced particles is equal to (see eq. (\[atot\])): f\_p (N) = (-2Q)( [(N ) -1 ( ) -1]{} )\^2 \[fpnonpt\] The last factor is rather similar to that in eq. (\[int0t\]). This is an oscillating function of $N$. For $\alpha= \pi (1-\epsilon)$ with a small $\epsilon$ it rises roughly as $N^2$ during $N= 1/(2\epsilon)$ oscillations. The occupation number increases with time discontinuously as a series of discrete jumps as time $t/\omega$ reaches integer values. During this stage $f_p$ may quickly rise with the speed much faster then the rate $\Gamma_q$ (\[gammaq\]) in complete analogy with the perturbative case considered in sec. \[s:pert\]. However, as we have already stressed, this does not mean that the production of fermions goes faster than in perturbation theory.
After this period of increase, $f_p$ starts to go down and approaches zero at $N_0\approx 1/\epsilon$. This oscillating behavior of the number of produced particles was noticed long ago in the problem of $e^+e^-$-pair creation by periodic electric field (for the list of references see e.g. the book by Grib et al in ref. [@books]). Thus it looks as though particles are produced by the field and after a while they all are absorbed back. This behavior is difficult to digest. Note that it is absent if time is very large, tending to infinity, as is discussed above. In this case the energy conservation is strictly imposed by the delta-function, $\alpha = \pi n$ (where $n$ is an integer), or in other words $\epsilon =0$ and $N_0 \rightarrow \infty$.
Possibly this mysterious phenomenon of re-absorption of the produced particles is related to the fact that during finite time the external field $\phi(t)$ does not disappear and the particle vacuum is not well defined over this time dependent background. To resolve the ambiguity one may calculate the transition of energy from the time-varying field $\phi (t)$ into other quantum states which are not necessarily determined in terms of particles. Energy density, in contrast to the particle number density, can be unambiguously defined in terms of local fields operators and does not suffer from any ambiguity related to the non-local character of the latter. The energy density of the quantum field $\psi$, defined as the expectation value of the time-time component of its energy-momentum operator, may also exhibit the oscillating behavior described above but the correct interpretation is possibly not production of $\psi$-particles but some excitation (“classical”?) of the (fermion) field $\psi$ coupled to $\phi(t)$.
Large Mass Case. \[ss:large\]
-----------------------------
Let us now consider the case when the fermion mass $m_0$ is large in comparison with the oscillation frequency $\omega$ and with the amplitude of the oscillations, $m_0 \gg g\phi_0$, so that the total effective fermion mass, $m_{tot} = m_0 + g\phi_0 \cos \omega t$ never vanishes and always large. The calculations for this case have been only done in ref. [@dolgov90] and we will reproduce them here. To be more precise we will reproduce only imaginary time part, while in ref. [@dolgov90] the method of Bogolyubov coefficients was used as well.
Following this paper we will consider production of bosons. It will be technically simpler allowing to make all calculations analytically, but qualitatively the same results should be valid also for fermions, because for a large $m_0$ the production is weak and the occupation numbers remain small. We assume that the effective mass has the form m\^2(t) = m\_0\^2 + g\^2 \_0\^2 \^2 t \[m2oft\] This case is realized if the interaction of the inflaton field with the produced particles ($\chi$-bosons) has the form $g^2|\chi^2| \phi^2$. The probability of production can be found from the expressions of the previous subsection by the substitution $p^2 \rightarrow p^2 +m_0^2$. In particular, the exponential damping factor is given, instead of (\[qell\]), by: Q’ = [2 ]{} \[qell’\] where (’)\^2 1- u\^2 = 1 - [m\_1\^2 m\_0\^2 +m\_1\^2 + p\^2]{} \[beta’2\] and the complete elliptic integrals in the case of small $k$ are expanded as[@grad]: K(’) && + [u\^24]{} ( -1)\
E(’) && 1 + [u\^22]{} ( -[12]{} ) \[KE\] The phase difference over the period of oscillations is now given by: ’ &=& [4 ]{} E()\
&& [2 ]{} \[alpha’\]
We can repeat the same calculations as in the previous subsection to find the occupation number and the number density of the produced particles. The production probability is now exponentially suppressed, as $\exp \{{-2 \sqrt{m_0^2+m_1^2}\ln [16(m_0^2+m_1^2)/m_1^2] /\omega}\}$. For a sufficiently large ratio $m_0/\omega$ the production would be very weak, all occupation numbers would be small in comparison with unity and bosons and fermions would be equally poorly produced.
Back Reaction and Cosmological Expansion Effects. \[s:evol\]
============================================================
.
Now we briefly comment on applicability of the results discussed above to realistic case of universe (re)heating after inflation. We have neglected universe expansion and damping of the field $\phi$ due to energy transfer to the produced particles. The effect of expansion can be easily taken into account in conformal coordinates where the metric takes the form (\[ds2\]) with space point independent cosmological scale factor $a(\tau)$. Under transformation to conformal coordinates and simultaneous redefinition of the gravitational, scalar, and fermionic fields respectively as $g_{\mu\nu} \rightarrow a^2 g_{\mu\nu}$, $\phi \rightarrow \phi/a$, and $\psi \rightarrow \psi /a^{3/2}$, the mode equation for the scalar field takes the form: \_k” +(k\^2 + m\^2 a\^2 - a”/a ) \_k = 0, \[phi”\] where the derivatives are taken with respect to conformal time and $k$ is comoving momentum. The presence of the term $a''/a$ demonstrates breaking of conformal invariance even for massless scalar field, as has been already mentioned in sec. \[s:hstr\]. All masses enter equation of motion in the combination $ma$, so mass terms explicitly break conformal invariance. The interactions of the types $g\phi \bar \psi \psi$, $\lambda \phi^4$ and $f\phi^2 \chi^* \chi$ are invariant with respect to the transformation of the fields specified above (note that the presence of the $\sqrt{det[g_\mu\nu]}$ in the action integral gives the necessary factor $a^4$ to ensure this invariance).
The expressions for the scale factors through conformal time in three most interesting cosmologies are the following: a&\~& e\^[Ht]{} = - [1/ H]{}, [De Sitteruniverse, inflation]{},\
a&\~& t\^[1/2]{} \~, [radiationdominance]{},\
a&\~& t\^[2/3]{} \~\^2, [matterdominance]{}. \[expregm\] In particular, in the radiation dominated universe with conformally invariant interactions, scalar field is conformally invariant but this is not true for other expansion regimes. Correspondingly, particles production by massless scalar field with the self-potential $\lambda \phi^4$ can be reduced to the flat space case discussed in the previous section. The difference between the potentials of $\phi$ in these two cases, $\omega^2 \phi^2$ and $\lambda \phi^4$, is not essential and the obtained above results can be easily translated to the $\lambda \phi^4/4$ potential. Indeed, the equation of motion of spatially homogeneous field $\phi$ in flat space-time (in conformal coordinates) has the form: ” +\^3 = 0 \[ddotphi\] This equation is solved in terms of Jacobi elliptic functions[@grad]: () &=& \_0 [cn]{} ( \_0; 2 )\
&=& [2 ]{} \_[n=1]{} [1 + ]{} \[cnoft\] where $\kappa = \Gamma^2 (1/4) /4\sqrt{\pi}$. The expansion is well approximated by the first term and particle production rate can be estimated using results of the previous section. Significant deviations from those results can be expected only in the case of heavy particle production when higher frequency terms in expansion (\[cnoft\]) may compete with the exponentially suppressed contribution coming from lower terms (see eq. (\[qell’\])).
It should be repeated, however, that these results are true only for radiation dominated regime of expansion. For other cosmologies the term $a''/a$ in eq. (\[phi”\]) is non-vanishing and must be taken into account.
Another effect, in addition to expansion, that results in a decrease of the amplitude of the field $\phi(t)$, is back reaction of the particle production. Energy that is transferred to the produced particles is taken from the field $\phi$ so the energy density of the latter should become smaller. For harmonic oscillations (in the case of the potential $\omega^2 \phi^2$) only the amplitude of the field diminishes, while frequency remains the same. For quartic potential both the frequency and the amplitude of oscillations go down, as one can see from eq. (\[cnoft\]) with $\phi_0 (t)$.
In the case of quickly oscillating field the effect can be easily estimated in adiabatic approximation. One has to solve the equation for energy balance in expanding background: = -3H ( + P ) \[dotrho\] where $\rho$ and $P$ are respectively energy and pressure densities of the field $\phi$ and the produced particles. For the former the solution of the standard equation of motion without interactions should be substituted with the effect of production included in a slow decrease of the amplitude $\phi_0$.
More accurate consideration demands using equation of motion modified by the production process. Usually this is described by the introduction into equation of motion, in addition to Hubble friction, the “production friction term”: + 3H + U’() = -. \[dotphi\] where $U(\phi)$ is the potential of $\phi$ and derivative is taken with respect to $\phi$. This anzats gives reasonable results only for harmonic potential but in all other cases this approximation is not satisfactory. A better approximation has been derived in refs. [@dolgov95; @dolgov99]. One starts with exact quantum operator equation of motion for the field $\phi$ and some other fields $\chi$ that are coupled to $\phi$. The production of the latter by oscillations of $\phi$ results in a damping term in the equation of motion for $\phi$. As an example let us consider a simple case of scalar $\chi$ with trilinear coupling $f\phi \chi^2$. The corresponding equations of motion have the form (expansion neglected for simplicity): -+ V\^[’]{}() &=& f \^2 , \[eqvp\]\
\^2 +m\^2\_ &=& 2 f . \[eqchi\] The next step is to make quantum averaging of these equations in the presence of classical field $\phi_c (t)$ (in what follows we omit sub-c and neglect the mass of $\chi$). This can be easily done in one-loop approximation (some subtleties related to renormalization of mass and coupling constants are discussed in ref. [@dolgov99]) and one comes to the equation that contains only the field $\phi$ and accounts for the backreaction from the production of the quanta of $\chi$: + V’() = \_0\^[t-t\_[in]{}]{} (t-) , \[scalres\] where $t_{in}$ is an initial time, when the particle production was switched on (it is assumed that $t>t_{in}$). The term in the r.h.s. that describes the influence of the particle production is non-local in time as one should have expected because the impact of the produced particle on the evolution of $\phi$ depends upon all the previous history. To use this equation for realistic calculations one has to make proper renormalization procedure. It is described in detail in ref. [@dolgov99]. The coupling to fermions as well as quartic coupling $\lambda' \phi^2 \chi^2$ are also considered in that paper. Similar one-loop approach was used in ref. [@baacke98] but no self-contained equation for $\phi$ was derived there.
Both effects of cosmological expansion and of damping of $\phi$ due to particle production can be easily incorporated into imaginary time method. This is especially simple in the case of fast oscillations and slow decrease of the amplitude of $\phi$. In this case the results obtained above practically do not change. One should only substitute there $\phi_0 (t)$ and to determine the law of the evolution of the latter from the energy balance equation (\[dotrho\]) or, more accurately, from eq. (\[scalres\]).
One more phenomenon deserves a comment here. As we have already mentioned, production of bosons may be strongly amplified due to the presence of the earlier produced bosons in the same final state. In classical language this effect is described by the parametric resonance in the equation of motion of the produced particles, while in quantum language it is the so called stimulated emission well known in laser physics. When the amplitude of the driving field $\phi$ drops below a certain value, the resonance would not be excited and the rest of $\phi$ would decay slowly. If the mass of $\phi$ is non-zero, this field would behave as non-relativistic matter and its cosmological energy density would drop as $1/a^3$. On the other hand, the produced particles are mostly relativistic with energy density decreasing as $1/a^4$. Thus for a sufficiently slow decay rate of $\phi$ the latter may dominate the cosmological energy density once again, when previously produced particles are red-shifted away. This would result in a low second reheating temperature, much lower than in parametric resonance scenario. On the other hand, the phenomenon of stimulated emission persists in perturbation theory even with a very small amplitude of $\phi$. Possibly even in this limit the production is not very fast as well, because the width of the band is quite narrow and the produced bosons are quickly pushed away from the band due to cosmological red-shift and collisions. More detailed consideration is desirable here.
Conclusion. \[s:concl\]
=======================
It is demonstrated that imaginary time method very well describes particle production by scalar field. It is very simple technically and permits to obtain physically transparent results. The calculations here were done for a particular case of periodic or quasiperiodic oscillations of the field but, as shows the experience with production of $e^+e^-$-pairs by electric field (for a review see e.g. third paper in ref.[@marinov72]), the method also works well in the opposite case of short pulse fields. The method is applicable in the quasiclassical limit. In the opposite case perturbation theory is applicable and hence one can obtain simple and accurate (semi)analytical estimates practically in all parameter range.
The results of calculations in the quasiclassical limit are in a good agreement with subsequent numerical ones[@baacke98; @green99]. An important difference between the latter papers and the initial one[@dolgov90] lays in the interpretation of the results. According to all these papers the occupation numbers of the produced particles quickly approaches unity but, in contrast to refs.[@baacke98; @green99], it is argued in the paper[@dolgov90] that the total production rate is nevertheless suppressed in comparison to perturbation theory and the production of fermions by the inflaton with Yukawa coupling to fermions is always weak. This conclusion is verified above. As is shown in this paper, the occupation numbers may quickly reach unity both in perturbation theory and in non-perturbative case. Still even the production rate of particles obeying Boltzmann statistics is very weak to ensure fast (pre,re)heating. In the case of fermion production the rate is evidently much weaker because the production must stop when the occupation number reaches unity and to continue the process the produced fermions should be eliminated from the band. As is argued in sec. \[ss:small\], the non-perturbative effects can only diminish the production rate.
The bosonic case is opposite: more bosons are in the final state, the faster is production. Thus even in perturbation regime the boson production can be strongly amplified because their occupation number may reach unity in much shorter time than $1/\Gamma$ and the energy may be transferred from the inflaton to the produced bosons much faster than is given by the original perturbative estimates[@dolgov82], where the effect of stimulated emission was not taken into account. Of course to realize this regime the band should not be destroyed by expansion and scattering, as argued in ref. [@dolgov90].
To summarize, we have shown that perturbation theory gives a good estimate of production of light fermions and bosons if Fermi exclusion principle or stimulated emission respectively are taken into account. The formally calculated production rate in perturbation theory is always larger than the non-perturbative one, at least in the simple cases that we have considered. So the results of perturbation theory may be used as upper bounds for production rates. Moreover, perturbation theory helps to understand physical meaning of the obtained results and to interpret them correctly.
In many realistic cases (e.g. for large $g\phi_0$ or $m_0$) perturbation theory is not applicable and to calculate the real production rate (not just an upper bound) one has to make more involved non-perturbative calculations. In quasiclassical (anti-perturbative) limit imaginary time method permits to obtain accurate and simple results and to avoid complicated numerical procedure
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to S. Hansen and A. Vainshtein for critical comments on the manuscript.
References {#references .unnumbered}
==========
[99]{}
S.R. Coleman, HUTP-82/A032, Jun 1982; in [*The Unity of the Fundamental Interactions*]{}, proceedings of the 19th International School of Subnuclear Physics, Erice, Italy, Jul 31 - Aug 11, 1981. Edited by Antonino Zichichi. Plenum Press, 1983. 817p. (Subnuclear Series, v. 19).
A.D. Dolgov, [*Usp. Fiz. Nauk*]{}, [**144**]{}, 341 (1984), English translation: [*Sov. Phys. Usp.*]{}, [**27**]{}, 786 (1984).
M.S. Marinov, [*Usp. Fiz. Nauk*]{}, [**144**]{}, 341 (1984), English translation: [*Sov. Phys. Usp.*]{}, [**27**]{}, 786 (1984).
M.S. Marinov, [*Phys. Repts.*]{}, [**60**]{}, 1 (1980).
M.S. Marinov, V.S. Popov, [*Yad. Fiz.*]{}, [**15**]{}, 1271 (1972);\
M.S. Marinov, V.S. Popov, [*Yad. Fiz.*]{}, [**16**]{}, 809 (1972);\
M.S. Marinov, V.S. Popov, [*Fortsch. Phys.*]{}, [**25**]{}, 373 (1977);\
V.S. Popov, V.L. Eletsky, M.S. Marinov, [*Zh. Eksp. Teor. Fiz.*]{}, [**73**]{} 1241 (1977).
A.D. Dolgov, D.P. Kirilova, [*Yad. Fiz.*]{}, [**51**]{}, 273 (1990); English translation: [*Sov. J. Nucl. Phys.*]{}, [**51**]{}, 172 (1990).
L. Parker, , [**21**]{}, [562]{} [(1968)]{}.
L. Parker, [*Phys. Rev.*]{}, [**183**]{}, 1057 (1969);\
L. Parker, [*Phys. Rev.*]{}, [**D3**]{}, 346 (1971);\
L. Parker, , [**28**]{}, 705 [(1972)]{};\
L. Parker, [*Phys. Rev.*]{}, [**D7**]{}, 976 (1973).
A.A. Grib, S.G. Mamaev, [*Yad. Fiz.*]{}, [**10**]{}, 1276 (1969);\
A.A. Grib, S.G. Mamaev, [*Yad. Fiz.*]{}, [**14**]{}, 800 (1971).
N.A. Chernikov, N.S. Savostina, [*Teor. Mat. Fiz.*]{}, [**16**]{}, 77 (1973).
Ya.B. Zeldovich, [*Pis’ma ZhETF*]{}, [**12**]{}, 443 (1970);\
Ya.B. Zeldovich, A.A. Starobinsky, [*ZhETF*]{}, [**61**]{}, 2161 (1971);\
B.L. Hu, C.A. Fulling, L. Parker, [*Phys. Rev.*]{}, [**D8**]{}, 2377 (1973);\
B.L. Hu, [*Phys. Rev.*]{}, [**D9**]{}, 2377 (1974);\
B.K. Berger, [*Ann. Phys.*]{}, [**83**]{}, 458 (1974);\
V.N. Lukash, A.A. Starobinsky, [*ZhETF*]{}, [**66**]{}, 1515 (1974).
N.D. Birrel, P.C.W. Davies, [*Quantum Fields in Curved Space*]{} (Cambridge University Press, Cambridge, 1982);\
Ya.B. Zel’dovich, I.D. Novikov, [*Structura i Evolyutsiya Vselennoi*]{} (in Russian), (Nauka, Moscow, 1975); [*Structure and Evolution of the Universe*]{} (University of Chicago Press, Chicago, 1983);\
A.A. Grib, S.G. Mamayev, V.M. Mostepanenko, [*Vacuum Quantum Effects in Strong Fields*]{} (Friedman Laboratory Publishing, St. Petersburg, 1995).
M. Chanowitz, J. Ellis, [*Phys. Rev.*]{}, [**D7**]{}, 2490 (1973).
A.D. Dolgov, [*Pis’ma ZhETF*]{}, [**32**]{}, 673 (1980);\
A.D. Dolgov, [*ZhETF*]{}, [**81**]{}, 417 (1981), English translation: [*Sov. Phys. JETP*]{}, [**54**]{}, 23 (1981);\
A.D. Dolgov, [*Phys. Rev.*]{}, [**D48**]{}, 2499 (1993).
A.H. Guth, S.Y. Pi, [*Phys. Rev. Lett.*]{}, [**49**]{}, 1110 (1982);\
S.W. Hawking, [*Phys. Lett.*]{}, [**115B**]{}, 295 (1982);\
A.A. Starobinsky, [*Phys. Lett.*]{}, [**117B**]{}, 175 (1982);\
J.M. Bardeen, P.J. Steinhardt, M.S. Turner, [*Phys. Rev.*]{}, [**D28**]{}, 679 (1983).
L.P. Grishchuk, [*ZhETF*]{}, [**67**]{}, 825 (1975), English translation: [*Sov. Phys. JETP*]{}, [**40**]{}, 409 (1975).
A.A. Starobinsky, [*Pis’ma ZhETF*]{}, [**30**]{}, 719 (1979);\
V.A. Rubakov, M.V. Sazhin, A.V. Veryaskin, [*Phys. Lett.*]{}, [**115B**]{}, 189 (1982).
V. Berezinsky, M. Kachelreiss, A. Vilenkin, [*Phys. Rev. Lett.*]{}, [**79**]{}, 4302 (1997);\
V.A. Kuzmin, V.A. Rubakov, [*Yad. Fiz.*]{}, [**61**]{}, 1122 (1998).
V.A. Kuzmin, I.I. Tkachev, [*JETP Lett.*]{}, [**68**]{} 271 (1998);\
V.A. Kuzmin, I.I. Tkachev, [*Phys. Rev.*]{}, [**D59**]{}, 123006 (1999);\
D.J.H. Chung, E.W. Kolb, A. Riotto, [*Phys. Rev.*]{}, [**D60**]{}, 063504 (1999).
V.A. Kuzmin, I.I. Tkachev, [*Phys. Rept.*]{}, [**320**]{}, 199 (1999).
A.D. Dolgov, A.D. Linde, [*Phys. Lett.*]{}, [**116B**]{}, 329 (1982);\
L.F. Abbott, E. Farhi, M.B. Wise, [*Phys. Lett.*]{}, [**117B**]{}, 29 (1982);\
A. Albrecht, P. Steinhardt, M.S. Turner, F. Wilczek, [*Phys. Rev. Lett.*]{}, [**48**]{}, 1437 (1982).
J. Traschen, R. Brandenberger, [*Phys. Rev.*]{}, [**D42**]{}, 2490 (1990).
N.B. Narozhnyi, A.I. Nikishov, [*ZhETF*]{}, [**65**]{}, 862 (1973);\
V.M. Mostepanenko, V.M. Frolov, [*Yad. Fiz.*]{}, [**19**]{}, 885 (1974).
L. Kofman, A. Linde, A. Starobinsky, [*Phys. Rev. Lett.*]{}, [**73**]{}, 3195 (1994).
A. Dolgov, K. Freese, R. Rangarajan, M. Srednicki, [*Phys. Rev.*]{}, [**D56**]{}, 6155 (1997).
L. Landau, E. Lifshitz, [*Quantim Mechanics. Nonrelativistic Theory.*]{} (Moscow, Mir).
A.I. Nikishov, [*ZhETF*]{}, [**57**]{}, 1210 (1969).
V.S. Popov, [*ZhETF*]{}, [**61**]{}, 1334 (1971);\
V.S. Popov, [*Yad. Fiz.*]{}, [**19**]{}, 1140 (1974).
A. Ringwald, [*Phys. Lett.*]{}, [**B510**]{} 107 (2001);\
A. Ringwald, hep-ph/0112254;\
V.S. Popov, [*JETP Lett.*]{}, [**74**]{}, 133 (2001);\
V.S. Popov, [*Pisma Zh. Eksp. Teor. Fiz.*]{}, [**74**]{}, 151 (2001).
I.S. Gradshteyn, I.M. Ryzhik, [*Tables of Integrals, Series, and Products*]{}, Academic Press, 1980.
J. Baacke, K. Heitmann, C. Patzold, [*Phys.Rev.*]{}, [**D58**]{}, 125013 (1998);\
[green99]{} P.B. Greene, L. Kofman, [*Phys.Lett.*]{}, [**B448**]{}, 6 (1999).
M. Peloso, L. Sorbo, [*JHEP*]{}, [**0005**]{}, 016 (2000).
[dolgov95]{} A. Dolgov, K. Freese, [*Phys. Rev.*]{}, [**D51**]{}, 2693 (1995).
[dolgov99]{} A.D. Dolgov, S.H. Hansen, [*Nucl. Phys.*]{}, [**B548**]{}, 408 (1999).
|
---
abstract: 'We study steady-state spherically symmetric accretion of a galileon field onto a Schwarzschild black hole in the test fluid approximation. The galileon is assumed to undergo a stage of cosmological evolution, thus setting a non-trivial boundary condition at spatial infinity. The critical flow is found for some parameters of the theory. There is a range of parameters when the critical flow exists, but the solution is unstable. It is also shown that for a certain range of parameters the critical flow solution does not exist. Depending on the model the sound horizon of the flow can be either outside or inside of the Schwarzschild horizon. The latter property may make it problematic to embed the galileon theory in the standard black hole thermodynamics.'
author:
- 'E. Babichev'
bibliography:
- 'bibliography.bib'
title: Galileon accretion
---
Ł
Introduction
============
Galileon is a scalar field theory, introduced in [@Nicolis:2008in]. This theory has a remarkable property: its Lagrangian contains higher-order derivative terms, while the equations of motion are of only second derivatives of the field. The galileon theory is invariant under the galilean transformation of the field. It was originally introduced as a generalization of the scalar field theory, left over in the decoupling limit [@Nicolis:2004qq] of Dvali-Gabadadze-Porrati (DGP) model [@Dvali:2000hr] and which allows the implementation of the Vainshtein mechanism [@Vainshtein:1972sx]. The covariant version of galileon (covariant galileon) was found in [@Deffayet:2009wt], where a model yielding only up to second-order derivatives in both field and metric was constructed. Other aspects of the galileon and related ideas were developed later [@Deffayet:2009mn; @*deRham:2010eu; @*Deffayet:2010zh; @*Padilla:2010de; @*Hinterbichler:2010uq; @*Padilla:2010tj; @*Andrews:2010kx; @*Goon:2010fk].
The galileon and related models were also applied for construction of cosmological models [@Chow:2009fm] (a more generic scalar-tensor model was suggested in [@Babichev:2009ee] where the gravity is modified in infra-red, while the nonlinear kinetic coupling provides the restoration of general relativity). More recently, several galileon-related models were studied in the cosmological context, see e.g. [@Deffayet:2010qz; @Creminelli:2010ba; @*Felice:2010fk; @*Kobayashi:2010cm; @*DeFelice:2010nf; @Silva:2009km; @*Kobayashi:2009wr; @*Kobayashi:2010wa; @*Gannouji:2010au; @*Ali:2010gr].
The galileon model is interesting in several aspects. First of all, this is a theory with non-quadratic kinetic coupling. This leads to propagation of perturbations in an effective metric, which is generically different from the gravitational one. In particular, the propagations may have sub- or superluminal velocities on non-trivial backgrounds. And indeed, as it was shown in [@Nicolis:2008in], galileon generically contains superluminal excitations (the same happens in for a scalar left over in the decoupling limit of DGP [@Adams:2006sv]). The existence of faster-than-light propagation on non-trivial backgrounds does not necessary imply inconsistencies [@Babichev:2007dw; @Bruneton:2006gf; @*Bruneton:2007si; @*Geroch:2010fk]. In particular, “superluminal” DBI inflation [@Mukhanov:2005bu] or accreting k-essence [@Babichev:2006vx; @Babichev:2007wg] are free of any time paradoxes (Note, that in [@Adams:2006sv], however, it was argued that superluminality of certain theories might indicate the problem for embedding them into a UV complete theory). Another interesting feature of galileon was recently discussed in [@Deffayet:2010qz; @Creminelli:2010ba; @*Felice:2010fk; @*Kobayashi:2010cm; @*DeFelice:2010nf]: cosmological models with the phantom behavior were constructed, and it was argued that perturbations are ghost-free for some choice of parameters and initial conditions.
In the original non-covariant version of the galileon it was found that the coefficient of the Lagrangian must be constrained, depending on the background [@Nicolis:2008in]. The check was performed by constructing static spherically symmetric solutions with a source in the de Sitter universe. It is thus seems important to continue this route and to test the galileon model further against pathologies. In particular, the covariant version of the galileon must be checked for existence of solutions and their stability. One of the useful area to test theories for unusual/pathological behavior is black hole physics. Indeed, previously the study of models in the black hole background has revealed interesting (and sometimes undesired) features of underlying models, e.g. it was shown that the phantom field accretion decreases the black hole mass [@Babichev:2004yx; @*Babichev:2005py; @*Babichev:2008jb]; the ghost condensate plus a black hole forms a perpetuum mobile of the 2nd kind [@Dubovsky:2006vk; @*Eling:2007qd] (however in [@Mukohyama:2009rk] it was argued that ghost condensate does not violate the second law of thermodynamics); and the superluminal k-essence allows one to send signals from inside of the event horizon of a black hole [@Babichev:2006vx; @*Babichev:2007wg].
In this paper we consider the galileon accretion onto a Schwarzschild black hole. The paper is organized as follows. In Sec. 2 we specify the action of the theory, give equations of motion and the energy-momentum tensor for different types of galileon. Section 3 is devoted to the galileon of the certain type, namely, containing only the terms left over in the decoupling limit of DGP. We construct the family of solutions describing a steady-state inflow of the scalar field. The unique physical solution is fixed by the requirement that it is regular both at the Schwarzschild and sound horizons. In Sec. 4 we widen our study to include another form of the galileon term. In Sec. 5 we briefly summarize our results, discuss applicability of our study and physical consequences. In particular, for the galileon of the DGP type we find no constrains on the Lagrangian. When another galileon term is included, we obtain an additional constraint as compared to [@Nicolis:2008in]. For a certain choice of parameters, the galileon forms the sound horizon inside the Schwarzschild one, thus giving an opportunity to send signals from the inside of the Schwarzschild horizon. We also discuss accretion of the “phantom” type of galileon.
Model
=====
The general scalar field covariant galileon action reads [@Deffayet:2009wt] \[action0\] S\_= d\^4 xŁ\_, where Lagrangian density for the can be written as a linear combination of terms, \[L\] Ł\_= \_[i=1]{}\^[i=5]{} c\_iŁ\_i, where, \[L1234\]
Ł\_1 &= , Ł\_2 = \_[;]{} \^[;]{}, Ł\_3 = \_[;]{} \^[;]{} ,\
Ł\_4 &= (\_[;]{}\^[;]{}),
and the term $\L_5$ has a more complicated structure and contains higher order derivatives in $\pi$, we are not going to study it in this paper[^1]. We will also set $c_1$ to zero in the following study, i.e. the “potential” term is not included. Otherwise, if $c_1\neq 0$, there is no steady-state solution for the accretion (this is similar to the case of the k-esence field see, e.g. [@Akhoury:2008nn]). The constant $c_2$ in (\[L\]) is dimensionless, $c_3$ has dimension $-3$, in the context of the DGP model this coefficient is usually written as $r_c/M_{\rm Pl}^2$ where $r_c$ is the so called cross-over scale and $M_{\rm Pl}$ is the Planck mass. The coefficient $c_4$ has mass dimension $-6$.
The energy-momentum tensor derived from (\[L\]), (\[L1234\]) as $T^{(i)}_{\m\n} \equiv 2/\sqrt{-g} \left( \delta S_{(i)}/\delta g^{\m\n}\right)$ is \[emt\] $$\begin{aligned}
%
T_{\m\n}^{(2)}&= 2\pi_{,\m} \pi_{,\n} - g_{\m\n}\left(\pd\pi\right)^2, \label{emt2}\\
%
T_{\m\n}^{(3)}&= 2\pi_{,\mu} \pi_{,\nu}\Box\pi - 2\pi_{,(\mu}\nabla_{\n)} \left(\pd\pi\right)^2
+ g_{\m\n}\pi^{,\a} \nabla_\a \left(\pd\pi\right)^2, \\
%
T_{\mu \nu}^{(4)} &=
-8
\left(\Box \pi\right)\pi^{;\rho}\bigl[\pi_{;\mu}\,\pi_{;\rho\nu}+\pi_{;\nu}\,\pi_{;\rho\mu}\bigr]
+4 \left(\Box \pi\right)^2 \left(\pi_{;\mu}\,\pi_{;\nu}\right)
-4 \left(\Box \pi\right)\left(\pi_{;\lambda}\,\pi^{;\lambda}\right) \left(\pi_{;\mu\nu}\right)
\nonumber \\
& -8
\left(\pi_{;\lambda}\,\pi^{;\lambda\rho}\,\pi_{;\rho}\right) \left(\pi_{;\mu\nu}\right)
+8 \left(\pi^{;\lambda}\,\pi_{;\lambda\mu}\right)\left(\pi^{;\rho}\,\pi_{;\rho\nu}\right)
-4 \left(\pi_{;\lambda\rho}\,\pi^{;\lambda\rho}\right)
\left(\pi_{;\mu}\,\pi_{;\nu}\right)
\nonumber \\
&+4
\left(\pi_{;\lambda}\,\pi^{;\lambda}\right)
\left(\pi_{;\mu\rho} \, \pi^{;\rho}_{\hphantom{;\nu}\nu}\right) + 8 \,\pi_{;\lambda}\,\pi^{;\lambda\rho} \bigl[\pi_{;\rho\mu}\,\pi_{;\nu}
+\pi_{;\rho\nu}\,\pi_{;\mu}\bigr] +2 \left(\Box \pi\right)^2 \left(\pi_{;\lambda}\,\pi^{;\lambda}\right)g_{\mu \nu}
\nonumber \\
&+ 8 \left(\Box \pi\right)
\left(\pi_{;\lambda}\,\pi^{;\lambda\rho}\,\pi_{;\rho}\right) g_{\mu \nu}
-8 \left(\pi_{;\lambda}\,\pi^{;\lambda\rho}\,\pi_{;\rho\sigma}\,\pi^{;\sigma}\right) g_{\mu \nu}
\nonumber \\
&-2 \left(\pi_{;\lambda}\,\pi^{;\lambda}\right)
\left(\pi_{;\rho\sigma}\,\pi^{;\rho\sigma}\right) g_{\mu \nu} -2 \left(\pi_{;\lambda}\,\pi^{;\lambda}\right)
\left(\pi_{;\mu}\,\pi_{;\nu}\right) R + \frac{1}{2} \left(\pi_{;\lambda}\,\pi^{;\lambda}\right) \left(\pi_{;\rho}\,\pi^{;\rho}\right)R g_{\mu \nu}
\nonumber \\
& +4 \left(\pi_{;\lambda}\,\pi^{;\lambda}\right) \pi^{;\rho} \bigl[R_{\rho\mu}\,\pi_{;\nu}
+R_{\rho\nu}\,\pi_{;\mu}\bigr] - \left(\pi_{;\lambda}\,\pi^{;\lambda}\right) \left(\pi_{;\rho}\,\pi^{;\rho}\right) R_{\mu \nu}
\nonumber \\
& - 4 \left(\pi_{;\lambda}\,\pi^{;\lambda}\right)
\left(\pi_{;\rho}\,R^{\rho \sigma}\,\pi_{;\sigma}\right) g_{\mu \nu}
+4 \left(\pi_{;\lambda}\,\pi^{;\lambda}\right) \left(\pi^{;\rho}\,\pi^{;\sigma}\,R_{\mu\rho\nu\sigma}\right). \label{emt4}\end{aligned}$$ Note that the expression for ${ T}^{(4)}_{\mu \nu}$ differs from the one derived in [@Deffayet:2009wt] by factor $-2$, because of the difference in the definition for the energy-momentum tensor[^2]. Equations of motion are obtained by the variation of (\[action0\]) with respect to $\pi$, $\E_{(i)} \equiv (-g)^{-1/2}\delta S_{(i)}/\delta\pi =0$, with \[eomDGP\]
\_2 &= - 2 ,\
\_3 &= -2 ( ()\^2 - ()\^2 - R\^\_[,]{} ),\
\_4 &= - 4 ()\^3 -8 (\_[;]{}\^\_[;]{}\^\_[;]{}\^) +12 () (\_[;]{}\^[;]{} ) + 2 () (\_[;]{}\^[;]{}) R\
&+ 4 (\_[;]{}\^[;]{}\_[;]{}) R +8 () (\_[;]{} R\^\_[;]{}) - 4(\_[;]{}\^[;]{}) (\_[;]{}R\^)\
& -16 (\_[;]{}\^[;]{}R\_\^[;]{}) - 8 (\_[;]{}\_[;]{}\_[;]{}R\^).
where $R_{\m\n}$ and $R_{\m\n\alpha\beta}$ are the Ricci and Riemann tensors correspondingly.
In what follows it will be useful to introduce dimensionless quantities in order to simplify formulae. Having in mind that we will apply our formalism to the problem of accretion, a convenient rescaling is \[rescale\] x\^r\_g x\^,Cr\_g , where $r_g=2M$ is the gravitational radius of a black hole, and the constant $C$ (with mass dimension 2) will be associated with the cosmological value of $\pd_t\pi$, see below.
Case $\L_2 + \L_3$
==================
In this section we consider the accretion of a galileon, having nonzero $\L_2$ and $\L_3$, while other terms set to zero. This type of action (up to coefficients in front of $\L_2$ and $\L_3$) appears as the effective scalar field action, left over after taking the decoupling limit in DGP model of gravity. In the rescaled units the action reads, \[L2L3action\] S\_= r\_g\^4 C\^2d\^4 x, where $\e = 0,\, \pm1$, $\kappa = C c_3/r_g$ and we let $\kappa$ to be positive or negative. Positive $\e$ corresponds to the canonical kinetic term, while positive both $\e$ and $\kappa$ correspond to the decoupling limit of DGP[^3]. Note that the the constraints from [@Nicolis:2008in] imply $\e>0$ and $\kappa\geq 0$ in the asymptotically flat space-time. In this case, for the static solution far from the source, $\pi' \propto r^{-2}$ (such that $\pi'>0$), and the perturbations $\delta\pi$ on the background $\pi$ propagate superluminally. In this paper, we, however, do not restrict ourselves to only positive $\e$ and $\kappa$.
The equations of motion derived from (\[L2L3action\]) are \[eom\] \_j\^= 0,j\_ 2 \_[,]{} + (2 \_[,]{}- \_()\^2), or, \[eom1\] +( ()\^2 - ()\^2 - R\^\_[,]{} )= 0.
We will also need the equation for perturbations $\delta\pi$ on a non-trivial background $\pi(t,x)$, in the limit of high frequencies[^4]. From (\[eom1\]), we find the equation, \[eomG\] G\^\_\_= 0, where \[G\] G\^ = (+2)g\^ - 2\^\^. Note that in deriving (\[eomG\]), (\[G\]) we neglected the term $\sim \pd\pi\pd\pi \delta R$, although there is a contribution of this term in the effective metric (\[G\]), see [@Deffayet:2010qz]. However, this term is suppressed by $M_{\rm Pl}^2$ [@Deffayet:2010qz], so we can safely neglect it in our calculations (see also the discussion in Sec. 5).
The propagation vectors for small perturbations can be found from the relation (see, e.g. [@Babichev:2007dw]), \[vectors\] \_\^\^= 0, where $\tilde{G}_{\mu\nu}$ is the inverse matrix to $G^{\mu\nu}$, $\tilde{G}_{\mu\nu} G^{\mu\nu} =1$.
Since we are interested in solutions of a scalar field including some region inside the Schwarzschild horizon, we will be needing a regular (on the Schwarzschild horizon) coordinate system. E.g. we can take ingoing Eddington-Finkelstein (EF) coordinates. The EF coordinates $(v,\, r)$ are connected to the usual Schwarzschild coordinates $(t,\, r)$ in the following way, \[EFcoord\] v= t+ ,r=r,where \[f\] f=1-, in the rescaled units. The metric of a Schwarzschild black hole metric in EF coordinates reads, \[EF\] d s\^2 = -f d v\^2 + 2d vdr + r\^2 d , To describe a steady-state accretion we take the ansatz, \[ansatz\] (v,r) = v - + (r). Note that by taking the ansatz (\[ansatz\]) we used the freedom in choosing the rescaling in (\[rescale\]), thus we took $C$ to be equal $\pd_t\pi$ at the spatial infinity, \[C\] C = \_t|\_[r=]{} = \_v|\_[r=]{},thus setting the coefficient in front of $v$ in (\[ansatz\]) to unity. The term $\int f^{-1}dr$ in (\[ansatz\]) was introduced for convenience, such that for the homogeneous solution (with no black hole in the Universe), $\psi(r) = {\rm const}$. Since the current depends only on $r$, Eq. (\[eom\]) can be integrated once to give \[1int\] r\^2 j\^r = A, where $A$ is a constant determining the total flux (to be fixed below). For the ansatz (\[ansatz\]) the $r$-components of the current reads, \[jr\] j\^r = 2 f ’+ (-+f f’ ’\^2+). Eqs. (\[1int\]), (\[jr\]) can be obtained in a different way, namely, by integrating the equation for the energy-momentum conservation, $T^{\mu\nu}_{\hphantom{\mu;};\nu}=0$, which gives $r^2T_v^r = {\rm const}$, where for the ansatz (\[ansatz\]) one can find $T_v^r = j^r$. The Eqs.(\[1int\]) and (\[jr\]) form an algebraic equation on $\psi'$. The solution contains constant $A$ as a free parameter, $
\psi' = \psi'(A,r).
$ The physical solution will be chosen by the requirement that it is neither singular at the Schwarzschild horizon nor at the sound horizon. This seemingly “weak” requirement choses the solution uniquely, which in hydrodynamics is called transonic branch. To fix the transonic solution one needs to find a position of the sound horizon, i.e. one needs to study perturbations on the background solution $\pi_A(v,r)$. There are generically two solutions of (\[1int\]) and (\[jr\]), \[sol12\] ’\_[(2,3)]{}= - , where the subscript $(2,3)$ means that the solution was obtained for the theory with $\L_2$ and $\L_3$ terms in the Lagrangian. Below we consider solutions different cases corresponding to the choice of coefficients in the action (\[L2L3action\]).
Canonical kinetic term, $\e= 1$, $\kappa = 0$.
----------------------------------------------
The standard case is recovered from (\[sol12\]) by setting $\e=1$ and taking the limit $\kappa\to 0$ for the solution with the minus sign (alternatively, one can solve (\[1int\]) and (\[jr\]) setting $\kappa=0$), \[solst\] ’\_[can]{} (A) = . It is not hard to see that the physical solution corresponds to the choice $A=2$ in (\[solst\]). The sound horizon $r_*$ coincides with the Schwarzschild horizon, $r_*=1$, which is not surprising, since the speed of excitations for the the canonical term is the speed of light. The solution for $\pi'(r)$ is shown in Fig \[fig DGP\].
No canonical term, $\e = 0$, $\kappa \neq 0$.
---------------------------------------------
It is interesting to look at the case when the quadratic kinetic term is absent. This case is simpler than the generic case (which we consider below), and in meantime it gives an insight to the full problem. From Eq. (\[sol12\]), setting $\e=0$ and substituting $f(r)$ from (\[f\]), we obtain, \[solNC\] ’\_[(3)]{}(A)= . For the solution with the plus sign the singularity coming from the denominator of (\[solNC\]) at $r=1$ is cancelled by the term $\left(-\int f^{-1}dr\right)$ in (\[ansatz\]) for any $A$; while the solution with the minus sign leads to a divergency of physical quantities. Another potentially dangerous point is at $r=3/4$. The only way to avoid the singular behavior at this point is to to choose $A=3\kappa$. Thus, \[solNCph\] ’\_[[(no can)]{}]{}= is the physical solution, in Fig. \[fig DGP\] the corresponding solution for $\pi'$ is shown. Let us now analyze propagation of perturbations. From (\[G\]) setting $\e=0$ one can find, \[GNC\] G\^[00]{}=-2 (+”), G\^[01]{} = (f’ ’-+), G\^[11]{}= . Then solving the equation (\[vectors\]) with the covariant metric $\tilde{G}_{\m\n}$, being the inverse of (\[GNC\]), we find two characteristics of the differential equation, $\eta\equiv dv/dr$, \[etaNC\] \_[(3)]{} = .The divergence of one the characteristic $\eta_{(3)}$ (with the minus in the denominator) at $r=3/4$, indicates the position of the sound horizon, $r_* = 3/4$. Note that in this case the sound horizon is inside the Schwarzschild one. The other propagation vector, which corresponds to the perturbations being sent into the black hole, is finite everywhere at $0<r<\infty$.
Generic case, $\e \neq 0$, $\kappa \neq 0$.
-------------------------------------------
Let us now turn to the generic case, i.e. when both terms in (\[L2L3action\]) are nonzero. First we consider the quadratic term in action (\[L2L3action\]) to be canonical, $\e=1$. The solution with the plus sign (\[sol12\]), for any $A$, at infinity is approximated by \[sol1inf\] ’\_[(2,3)+]{} = - r + [O]{}(1),thus violating our assumption on the homogeneity of the solution at the spatial infinity. Meantime it is regular at $r=3/4$. The other solution, on the contrary, is well-behaved at infinity, \[sol2inf\] ’\_[(2,3)-]{} = (A+) + ( + ) + (r\^[-4]{}). However, the solution (\[sol2inf\]) diverges at $r=3/4$ along with the characteristics $\eta_{(2,3)}$ for the generic choice of $A$. Thus, similar to the standard problem of fluid accretion, one needs to find such a value of $A$ that at some point $r=r_*$ the solutions $\psi_{(2,3)\pm}$ and their derivatives match. The surface $r=r_*$ (called the critical point), is analogous to the acoustic horizon, from inside of which perturbations of $\pi$ cannot escape to the asymptotically flat regions (see below). At the critical point one of the propagation vectors $\eta_{(2,3)}$ diverges, i.e. in our case $\tilde{G}_{vv}=0$, implying $G^{rr}=0$. However, up to the non-relevant conformal factor, $G^{rr}$ is also being the coefficient in front of $\psi''$ in the equation of motion for the background solution. In order for the solution to be smooth at the sound horizon, the rest of equation of motion for $\psi$ (containing $\psi$, $\psi'$ and terms) should also be zero at the sound horizon. Thus the procedure is as follows. Differentiating (\[1int\]) once, we obtain the second derivative equation on $\psi$. Then one can express $\psi''$ in terms of $\psi'$ (and there is no $A$ in the expression, because it disappears during the differentiation). Now, the critical point is where both denominator and numerator are equal to zero. Solving simultaneously these two equations, one obtains an algebraical equation $P(r)=0$, where $P(r)$ is a polynomial 6th order in our case. The position of the sound horizon, $r_*$, can be then found numerically from this equation. After finding $r_*$, the constant of integration $A$ is fixed. There are generically two possibilities, one of them corresponds to (physical) inflow and the other to (unphysical) outflow.
![In the case $\L\sim \L_2 + \L_3$, solutions for $\pi' = -1/f+\psi'(r)$ are shown for different parameters of the model. The positions of sound horizon are shown by the dots. In case $\kappa <0$ the sound horizon is outside the Schwarzschild horizon, while for $\kappa>0$ the sound horizon is inside the Schwarzschild horizon.[]{data-label="fig DGP"}](dgp_pi_cases.eps){width="50.00000%"}
For a positive $\kappa$, corresponding to the decoupling limit of the DGP model, the sound horizon is inside the Schwarzschild horizon. The latter could be anticipated, since in this case the perturbations of the scalar field on nontrivial background of $\pi$ are superluminal. There is a regular solution in the whole range of $r$, $0<r<\infty$, see Fig. \[fig DGP\]. For $\kappa<0$, on the contrary, the sound horizon is outside the Schwarzschild horizon. The solution is smooth outside of the Schwarzschild horizon, and it becomes singular at some point $r<3/4$, which can be seen by analyzing the square root in (\[sol12\]). This solution is nevertheless physical, since the pathological behavior happens inside of both acoustic and Schwarzschild horizons.
After fixing the critical inflow solution, one can analyze the characteristics of the equations for perturbations, $\eta_{(2,3)}$ and to check that $r_*$ is indeed the sound horizon. Fig. \[fig DGP\] shows behavior of the solutions for $\pi'$ for various cases.
Note that one can also relax the requirement of positivity of $\e$ (being the requirement that the Lagrangian is having healthy canonical term) and to consider the case when the canonical kinetic term has a negative sign (ghost-like). Such a Lagrangian was, e.g. considered in the cosmological context in [@Deffayet:2010qz]. In our study, this situation can be easily reduced to one of the studied cases. Indeed, if both $c_2$ and $c_3$ are negative, then action is equivalent to the case $c_2 > 1$, $\kappa >0$, up to the total sign in front of the action, so that the equations of motion do not change. The action with $c_2<0$, $c_3<0$, is equivalent to the studied above case $c_2 > 0$, $c_3 <0$, up to the total sign in front of the action. Note that we do not address the problem of ghosts here, we are only interested in the existence and stability of the classical solutions.
Case of non-zero $\L_4$ {#L4}
=======================
In this section we consider the case of nonzero $\L_4$, \[L2L4\] S\_= r\_g\^4 C\^2d\^4 x, where $\bar{\kappa} \equiv c_4 C^2/r_g^2$. In this case the problem of accretion becomes quite cumbersome. To start with, let us first consider the case when only $\L_4$ term is present in the action (\[action0\]).
No canonical term, $\e=0$.
--------------------------
One can use the fact that the energy-momentum tensor for $\L_4$ is conserved, $T^{\mu\nu}_{\hphantom{\mu;};\nu}=0$ [@Deffayet:2009wt]. Since $T^\mu_\nu$ does not depend on $v$ explicitly for the ansatz (\[ansatz\]), we obtain, \[consT\] T\_v\^r = , where from (\[emt4\]), applying the ansatz (\[ansatz\]), after lengthy but straightforward calculations we find, \[emt4ans\] T\^[r]{}\_v = - (-r\^2 f\^2 f” ’\^2+r\^2 f”-6 r f’ (f\^2 ’\^2-1)-4 f\^3 ’\^2), where without loss of generality we have set the coefficient in front of $\L_4$ to unity. Now, the problem of accretion for non-zero $\L_4$ (and $\L_i=0$ for $i\neq 4$) is reduced to the solving (\[consT\]) and (\[emt4\]) and choosing the physical solution, if any. Combining (\[consT\]) and (\[emt4ans\]) we obtain three branches of solutions $\psi'_{(4)}$. For $A<0$ or $A>8$ there is no well-behaved solution. It is interesting that for any $A$ in the range $0 \leq A <8$ a smooth solution exists for $\pi'$, covering the whole space, $0<r<\infty$, with corresponding characteristics finite everywhere, see Fig. \[fig L4 A\]. The critical point is absent and the solution at infinity is, \[L4 sol inf\] ’\_[(4)]{} = + + [O]{}(), 0 A <8. The critical point exists for $A=8$ at $r_* = 1/3$; one can check that the sound horizon is indeed at $r=1/3$. The solution critical solution is then, \[L4 crit sol inf\] ’\_[(4),\*]{} = , which implies $\pi'_{(4),*} = 0 $, see Fig. \[fig L4 A\].
![Case, $\L\sim \L_4$. Solutions for $\pi' = -1/f+\psi'(r)$ are shown for various values of $A$. The critical flow is for $A=8$ (solid thick line), the critical point is indicated by the black dot, $r_* = 1/3$. For the cases $A= 4$, $1$, $0$ the solution is shown by dotted lines, from top to bottom. []{data-label="fig L4 A"}](L4_A.eps){width="50.00000%"}
![Case, $\L\sim \L_2+ \L_4$. Solutions for $\pi' = -1/f+\psi'(r)$ are shown for different $\bar{\kappa}$: $\bar{\kappa}=-100$, $-1$, $-0.04$ (thick lines, from the top to bottom) and $\bar\kappa=0$, i.e. only canonical term (thin line). The positions of sound horizon are shown by the dots, for $\bar\kappa <0$ they are outside the Schwarzschild horizon. For $\bar\kappa <0$ solutions are singular.[]{data-label="fig L4"}](ctL4.eps){width="50.00000%"}
Generic case, $\e\neq 0$, $\bar{\kappa}\neq 0$.
-----------------------------------------------
Let us now consider the case when besides the term $\L_4$ in the action, there is also the canonical kinetic term, $\L\sim \L_2 + \L_4$. Depending on the signs in front of $\L_2$ and $\L_4$, one can distinguish, in principle, four different cases. Let us first take $\L_2$ to be canonical, i.e. $c_2=1$ and also $c_4>0$. In this case a family of smooth solutions exists for a certain range of positive $A$, $0<A\leq A_*(\bar\kappa)$. For negative $A$ the solutions diverge at the Schwarzschild horizon. The critical solution, realized for $A=A_*(\bar\kappa)$, is regular for $0<r<\infty$ with the critical surface inside the Schwarzschild horizon. Interestingly, however, that even though smooth background solutions exist for $0<A\leq A_*(\bar\kappa)$, the propagation vectors are nonexistent for some radii outside the Schwarzschild horizon. In fact, the equation of motion for perturbations become elliptic, indicating that solutions are catastrophically unstable.
On the other hand, for $\e=1$ and $\bar{\kappa}<0$, a critical solution exists, with well-behaved propagation vectors, at least up to the sound horizon. The asymptotic form of the physical solution is, \[can L4 sol inf\] ’\_[(2,4)\*]{} = + [O]{}(). Note that the leading asymptotic behavior at infinity is the same as in the case of canonical term, up to the difference in constant $A$, which is simply the reflection of the fact that at infinity the term $\L_4$ becomes unimportant compared to $\L_2$.
Results and discussion
======================
The results of the paper can be summarized as follows. For the Lagrangian (\[L2L3action\]), $\L \sim \L_2 + \L_3$, we have found,
- case $c_2=0$ (canonical kinetic term is absent). Critical solution exists, Eq. (\[solNCph\]), with the critical point at $r_*=3/4$, see Fig. \[fig DGP\], upper (red) line.
- case $c_3/c_2>0$. Critical solution exists, with asymptotic form (\[sol2inf\]) at infinity, the critical radius $r_*$ is inside the Schwarzschild horizon.
- case $c_3/c_2<0$. Critical solution exists as well, with the asymptotic form (\[sol2inf\]) at infinity, however the critical radius $r_*$ is ouside the Schwarzschild horizon.
For the Lagrangian $\L \sim \L_2 + \L_4$,
- case $c_2= 0$ (no kinetic term). There is a family of solutions parametrized by the value total influx, $A$. These solutions and the corresponding propagation vectors are smooth for $0<r<\infty$. Critical solution also exists for a particular value of the influx (for $A=8$), with the critical point at $r_* = 1/3$, see Fig. \[fig L4 A\]
- case $c_4/c_2>0$. A family of smooth solution exist for some range of positive $A$. The critical solution also exists with a critical point $r_*<1$. However, for the found solutions the propagation vectors do not exist in some region of $r$, $r>1$.
- case $c_4/c_2<0$. There is the well-behaved critical solution and propagation vectors, corresponding to the solution. The sound horizon is outside the Schwarzschild horizon, see Fig. \[fig L4\].
It is interesting to see if we get any new constraints on the Lagrangian compared to those obtained for the non-covariant version of galileon. Since in our study the space-time is asymptotically flat, it would be natural to apply constraints on $c_i$ obtained in [@Nicolis:2008in] for flat space-time[^5]. For the Lagrangian containing only $\L_2$ and $\L_3$ we did not find any constraints on the coefficients $c_2$, $c_3$. Note that $c_2>0$ was found in [@Nicolis:2008in] by requiring the model to be ghost-free. Since we did not address the issue of ghosts in our study[^6], it is not surprising that we did not find any constraints on $c_2$. It is more interesting, that if we impose positivity of $c_2$ we still do not get any constraints on $c_3$ (meantime the study in [@Nicolis:2008in] implies that $c_3>0$ if applied to our case). There is a subtle point here: for our solution the case $c_3/c_2<0$ corresponds to the critical point outside the Schwarzschild horizon, and formally the solution diverges at some point inside both horizons. This behavior, however, cannot be used to invalidate the solution, because the region inside of both horizons is causally disconnected from the asymptotically free region, so, in principle, inside the Schwarzschild horizon the solution can change to another one (the one which does not satisfy the steady-state ansatz).
For the Lagrangian $\L\sim \L_2 + \L_4$ we also obtained an interesting result. In particular, for both $c_2$ and $c_4$ positive (and $c_3=0$), the solution for the steady-state accretion is unstable, because the propagation vectors do not exist in some interval of $r$ outside the Schwarzschild horizon. On the other hand, for $c_4/c_2 <0$, we obtained a well-behaved critical solution with corresponding propagating vectors. These results are in contrast to those obtained in [@Nicolis:2008in] for a static configuration of non-covariant galileon, where for $c_3 =0 $ and asymptotically flat space-time the coefficient $c_4$ has to be positive or zero.
The propagation of small perturbations for galileon is superluminal for a range of parameters. Thus, it is not surprising that in this range of parameters the sound horizon is inside the Schwarzschild horizon, implying that one can in principle communicate with the interior part of a black hole, $r>r_*$. The same effect was found for “superluminal” k-essence [@Babichev:2006vx; @Babichev:2007wg]. Similarly to k-essence accretion, the acoustic space-time for accreting galileon is globally hyperbolic, in spite of the superluminal perturbations. This can be explicitly demonstrated by constructing a Schwarzschild-like metric of the perturbations of the field. We take the case $\L \propto \L_3$ for simplicity[^7]. Substituting (\[EF\]) and (\[solNCph\]) into (\[G\]) one can find the effective contravariant metric. By inverting the obtained metric, one finds the covariant effective metric $\tilde{G}_{\m\n}$. Making successive change of coordinates, $dv = dT - G_{tr}/G_{tt} dr$, $r = R^{3/4}$ and $T\to \sqrt{12} T$, we finally get, \[eff S\] dS\^2 = - (4R\^[4/7]{} - 3) R\^[2/7]{} dT\^2 + dR\^2 + R\^2d. The above expression for the effective metric is similar to the Schwarzschild metric, with the important difference that the position of the event horizon is now at $R^{4/7}= 3/4$.
Although the global hyperbolicity is not affected by the presence of the superluminal propagation in this case of steady-state accretion of galileon, it is still unclear, however, how to embed the existence of the new “event” horizon into the standard thermodynamics of black holes. In particular, since the sound horizon is inside the Schwarzschild one, the latter loses its meaning of the universal event horizon, therefore one may need to modify the entropy assigned to a black hole.
It is interesting to note, that for a range of parameters, our solutions for accretion possesses the analogue of the Vainshtein behavior, similar to the case of static sources.[^8] Indeed, let us take the case $\L\sim \L_2 + \L_3$ for definiteness and $\e>0$. In the limit $r\to \infty$ the solution is given by (\[sol2inf\]), which has the same form as the solution for the standard kinetic term (\[solst\]), apart from the overall constant $A\to A+\kappa$. In this regime the kinetic term is dominant, and assuming that non-minimal coupling is given by a term $\sim \pi T/M_P$ (in the Einstein frame) a test particle would experience an additional force, $F_\pi$, which is to be compared to the gravitational force $F_{\rm grav}$, \[5th\] \~ \~, where we restored the physical units. On the other hand, if also $\kappa>0$, then the solution changes its behavior at the radius, \[Vainshtein\] r\_r\_g\^[1/3]{}.where $r_\star$ is a analogue of the Vainshtein radius. The “accretion” Vainshtein radius is related to the “usual” one, $r_V$, defined for a static source as follows, \[VV\] \~()\^[1/3]{}.Thus, for the range of distances $r_g < r < r_\star$, the ratio of the galileon and gravitational forces is, \[5thscreened\] \~ ()\^[3/2]{},which has an additional small factor $(r/r_\star)^{3/2}$ compared to (\[5th\]), being a manifestation of the Vainshtein effect. This can also be seen in Fig. (\[fig DGP\]): the behavior of $\pi'$ for $\kappa>0$ is smoother closer to the horizon, compared to the canonical kinetic term. Note, that if $\e$ and $\kappa$ have different signs (e.g. $\e>0$ and $\kappa<0$), then solutions do not exhibit the Vainshtein behavior, due to the fact that the term in (\[sol12\]), which responsible for this (the second term inside the square root) is negative and therefore it can never be dominant. A similar analysis can be made for the Lagrangians of the form $\L\sim \L_2 + \L_4$, and one can find that the solutions have the Vainshtein behavior in this case as well.
It is worth to mention the change of the black hole mass during the accretion of the galileon. Since by definition the problem we studied was steady-state in the test-fluid approximation, the rate of the change can be found via the total flux at the infinity, $r\to \infty$. In the Schwarzschild coordinate system the total flux is $\propto (r^2 T_t^r)$. Expressing $T_t^r$ through the components in the EF coordinates, we obtain[^9], \[BHmass\] = A r\_g\^2 \^2\_, where the physical units have been restored. In particular, for the case when only the canonical term is present, $A=2$, one gets the known result (compare, i.e. with [@Babichev:2004yx; @Babichev:2005py], the additional factor 2 in the above expression is due to the choice of the canonical term $(\pd\pi)^2$, instead of $1/2(\pd\pi)^2$). The flux can be easily made negative by changing of the overall sign of the Lagrangian for $\pi$, so that the black hole will loose mass instead of gaining it. Usually, such a change of sign is accompanied by ghosts. However, e.g. in [@Deffayet:2010qz] it was shown that even though the $\L_2$ term is ghost-like, nevertheless the full Lagrangian, $\L\sim\L_2 + \L_3$, is ghost-free in the vicinity of the cosmological attractor. One can think of exploiting this particular feature to construct negative flow onto a black hole. In this case, however, one immediately stumbles the following problem. In our study the space-time is taken to be asymptotically flat, implying that the term $\L_3$ is subdominant in comparison to $\L_2$, implying the appearance of ghosts in the model studied in [@Deffayet:2010qz]. (Note that the expansion of the universe in the attractor is the key to to avoid ghosts in [@Deffayet:2010qz]). Thus, in the region of space-time where the gravitation of a black hole is stronger than the cosmological expansion, one may naively expect the appearance of ghost. This question, however, deserves a separate and more careful study.
Let us in conclusion discuss the assumptions we made to find the above results. First of all we assumed that the accretion is steady-state, which is physically reasonable when the Lagrangian is shift-symmetric and it is enough time for the system to evolve before the steady-state is established. We also neglected the backreaction of the accreting fluid on the metric, since the backreaction can be made parametrically small. Indeed, the total flux onto the black hole is proportional to $r_g^2 \dot{\pi}_\infty^2$ (\[BHmass\]). In the limit $c_3 = {\rm const}$, $r_g\to 0$, $\dot{\pi}_\infty\to 0$ (and keeping $\dot{\pi}_\infty/r_g = {\rm const}$, so that the solution, expressed in dimensionless units is not affected[^10]), the total flux goes as $\sim r_g^4$, while the mass of the black hole $\sim r_g$, thus validating our the assumption. Another source of possible inaccuracy in our calculations is that we also neglected the kinetic mixing term of perturbations and gravity. E.g. the term $\sim\pd\pi\pd\pi\delta R$ was neglected in (\[G\]). The contribution of this term is to compared to the contributions of the terms left over in (\[G\]), one can check that in order to neglect the backreaction of the perturbations, one needs $(\pd\pd\pi) \gg c_3 \left(\pd\pi\right)^4/M_P^2$, which again can be maid parametrically small by taking the limit we used before.
To summarize, in this paper we studied the steady-state accretion of the covariant galileon onto a Schwarzschild black hole. Subject to the assumptions we made, our study of existence and classical stability of solutions did not show any constraints on the parameters of the Lagrangian of the form $\L\sim \L_2 + \L_3$. In the event when $ \L_4$ is present, the constraints on the parameters of the theory we obtained are in a sense orthogonal to the constraints found in other studies. We also notice that for some range of parameters of the theory signals can be sent from the inside of the Schwarzschild horizon, being a consequence of superluminal propagation of perturbations. The accretion of the energy flow can be made negative by appropriate choice of the coefficients in the Lagrangian.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Slava Mukhanov for discussions and Alex Vikman for discussions, interesting suggestions and a critical reading of the manuscript. I would also like to thank the referee for a suggestion to check the Vainshtein behavior of the solutions. The work was supported by the TRR 33 “The Dark Universe”.
[^1]: Throughout this paper we use the signature $(-,+,+,+)$. We also follow here the notations of Ref. [@Deffayet:2009wt], which are different to ones in [@Nicolis:2008in]. Namely, neglecting gravity, and up to the integration by parts, $\L_2 = -2 \L_2^{NRT}$, $\L_3 = -2 \L_3^{NRT}$ and $\L_4 = -4 \L_4^{NRT}$, where $\L_i^{NRT}$ are the definitions of [@Nicolis:2008in]. To avoid confusions with signs we also assume that our full action, including gravity, has the overall minus sign compared to [@Nicolis:2008in], so that, e.g. the constraint $d_2>0$ in [@Nicolis:2008in], for asymptotically flat space-time implies $c_2>0$ in our notations.
[^2]: In [@Deffayet:2009wt] the energy-momentum tensor was defined as $T^{\m\n} \equiv 1/\sqrt{-g} \left( \delta S/\delta g_{\m\n}\right)$.
[^3]: Note that the overall sign in the action is opposite to [@Nicolis:2008in], so that in our notations the canonical kinetic (no-ghost) term is $+(\pd \pi)^2$.
[^4]: Not to be confused with “perturbation” which sometimes used in a different context: as a deviation of the solution from the homogeneous configuration due do the presence of a static source.
[^5]: Then in [@Nicolis:2008in] one should make substitution $d_i\to c_i$.
[^6]: In fact, the presence of ghost does not affect the solutions we found, since the overall sign in front of the action is unimportant.
[^7]: A similar arguments can be repeated for other models of galileon, considered above.
[^8]: Note, however, that in order to observe the Vainshtein effect, the matter must be coupled to the galileon non-minimally, so that the fifth force (due to the direct coupling) leads to the deviation from General relativity on large scales, while on small scales the galileon is screened. Meanwhile, our results for the accretion do not depend on coupling to matter, since the solution is matter-free.
[^9]: The same expression can be obtained by calculating the flux at of the Killing energy across the Schwarzschild horizon, $dM/dt = Area\times T_{vv}$.
[^10]: Here we restored the physical units, so that $\dot{\pi}$ has dimension 2.
|
---
abstract: 'The optimal reconstruction of cosmic metric perturbations and other signals requires knowledge of their power spectra and other parameters. If these are not known a priori, they have to be measured simultaneously from the same data used for the signal reconstruction. We formulate the general problem of signal inference in the presence of unknown parameters within the framework of information field theory. To solve this, we develop a generic parameter uncertainty renormalized estimation (PURE) technique. As a concrete application, we address the problem of reconstructing Gaussian signals with unknown power-spectrum with five different approaches: (i) separate maximum-a-posteriori power spectrum measurement and subsequent reconstruction, (ii) maximum-a-posteriori reconstruction with marginalized power-spectrum, (iii) maximizing the joint posterior of signal and spectrum, (iv) guessing the spectrum from the variance in the Wiener filter map, and (v) renormalization flow analysis of the field theoretical problem providing the PURE filter. In all cases, the reconstruction can be described or approximated as Wiener filter operations with assumed signal spectra derived from the data according to the same recipe, but with differing coefficients. All of these filters, except the renormalized one, exhibit a perception threshold in case of a Jeffreys prior for the unknown spectrum. Data modes with variance below this threshold do not affect the signal reconstruction at all. Filter (iv) seems to be similar to the so called Karhune-Loève and Feldman-Kaiser-Peacock estimators for galaxy power spectra used in cosmology, which therefore should also exhibit a marginal perception threshold if correctly implemented. We present statistical performance tests and show that the PURE filter is superior to the others, especially if the post-Wiener filter corrections are included or in case an additional scale-independent spectral smoothness prior can be adopted.'
author:
- 'Torsten A. En[ß]{}lin'
- Mona Frommert
bibliography:
- 'bibtex/ift.bib'
title: |
Reconstruction of signals with unknown spectra\
in information field theory with parameter uncertainty
---
Introduction
============
The generic sensing problem
---------------------------
Reception of a signal is strongly aided by prior knowledge of the signals properties. This is especially true in low signal to noise (S/N) situations, in which proper knowledge can make the difference between recognition of a signal and blindness. Our human senses like vision and hearing are strongly enhanced by our knowledge on the possible signals present in the data-stream entering the human brain. The very same is true for signal reception by artificial sensor systems, since signal knowledge permits us to construct optimal filters, suppressing the noise as far as possible while focusing on the data modes with stronger S/N. If sufficient training data are available, or theoretical reasoning permits us to predict signal properties, optimal filter design is possible and relatively straightforward.
However, there are situations, where such knowledge is not available, or is to be excluded on purpose from the analysis, in order to have a prejudice-free signal reconstruction. In such a situation the required parameters have to be measured simultaneously from the same data which is used for the signal reconstruction. Due to the interdependence of reconstructed signal and parameters, the problem becomes non-trivial and in general non-linear, even if the original inference problem was linear for fixed parameter values.
Let us provide a concrete example in cosmology. The cosmic matter distribution and its imprinted metric fluctuations on large scales can be well approximated to be a Gaussian random field obeying statistical isotropy and homogeneity. Knowledge of the power spectrum of these fields permits us to construct optimal and linear reconstruction filters for data of any linear tracers like the cosmic microwave background, the galaxy distribution (approximatively), or the gravitational lensing signature. For a set of cosmological parameters (e.g. Hubble constant, cosmic matter content, ...) these power spectra are known and can be used. However, the cosmological parameters themselves are not precisely known, and our best knowledge might come from the data-set we are analyzing. Furthermore, if we want to be open for non-standard cosmological scenarios, we might not want to put any prior assumption on the functional form of the power spectrum into our signal reconstruction problem.
Therefore, we need signal reconstruction methods, which are capable of dealing with uncertainties in the parameters of the problem. Such methods would be very useful in many situations, where prior knowledge on signal properties are absent or should be avoided. Some loss in fidelity compared to the case where these parameters are known can be expected, however, such methods can be expected to be flexible and robust due to their generic nature and self-tuning abilities.
For the problem of the reconstruction of the cosmic large-scale structure, the key parameter is the cosmic matter power-spectrum. It is known in the field of signal detection, that a statistical verification of the presence of a signal due to an increase in the data variance is possible well before the signal can be reconstructed itself. Thus, a measurement of the signal power-spectrum is already possible while the S/N-ratio is too low for map-making, and is therefore immediately available for filter optimization as soon as the critical S/N-ratio is achieved.
Derived filters {#sec:derivedfilters}
---------------
The signal reconstruction filters derived in this work can all be regarded as or approximated by an application of a data-dependent Wiener filter operator onto the data, which results in a non-linear transformation of the data. The Wiener filter construction requires the knowledge of the signal covariance, or spectrum, the instrument response and the noise covariance. The signal covariance has to be extracted from the data itself, and therefore introduces a data dependence into the filter. The five filters presented in this work differ in the way the assumed covariance is constructed, due to the different philosophies:
1. **MAP spectrum filter:** The maximum a posteriori (MAP) of the spectrum given the data should be a reasonable guess for the signal spectrum assumed in the Wiener reconstruction.
2. **Classical map:** The inference problem should be marginalized over all possible power spectra. In doing so, and deriving the classical filter equation by extremizing the resulting effective posterior, a data-dependent Wiener filter is derived, in which an effective spectrum emerges. This spectrum differs in general from the MAP-spectrum.
3. **Joint MAP filter:** Instead of marginalizing the joint posterior of signal and spectrum and then extremizing it with respect to one of those, we can maximize it with respect to both, leading to the joint MAP filter.
4. **Critical filter:** This filter results if one requires the covariance of the Wiener filter map to exhibit exactly its expected variance, while taking the power loss due to the filter operation into account. The critical filter implements accurately the idea behind frequently used power spectrum estimation schemes used in cosmology, like the Karhunen-Loève (KL, [@1947KarhunenK; @1978LoeveM; @2002MNRAS.335..887T]) and Feldman-Kaiser-Peacock (FKP, [@1994ApJ...426...23F]) estimators. In case of a Jeffreys prior on the spectral normalisation, it exhibits a marginal perception threshold and marks the demarcation line between filter with, as the three above, and filter without such a threshold, as the next one.
5. **PURE filter:** Our ultimate filter would implement the Baysian mean of the signal posterior marginalized over the unknown spectral parameter. Only this provides the optimal reconstruction algorithm in the sense of minimizing the reconstruction error variance. This can only be done by a full field theoretical treatment which incorporates spectrum-uncertainty effects correcting for imbalances of the induced errors due to over- and underestimations of the signal spectrum. Here, we incorporate such a correction by virtue of an uncertainty-renormalization calculation. The resulting parameter uncertainty renormalized estimation (PURE) filter appears only to be a Wiener filter in case only an infinitesimal amount of uncertainty is added. The renormalized-optimal spectrum as a fixed point of this uncertainty adding operation is different from the spectra of the other filter. In case a finite amount of uncertainty is added, the PURE filter contains corrections terms which can not be described exactly as Wiener filtering.
Previous works
--------------
The PURE approach is derived within information field theory (IFT). This deals with the information of data on spatially distributed quantities, and is a statistical field theory. The connection of inference problems and statistical field theories was discovered independently by several authors in cosmology [@1985ApJ...289...10F; @1987ApJ...323L.103B; @2009PhRvD..80j5005E], statistical field theory [@1987PhRvL..58..741B; @1988PhRvL..61.1512B; @1996PhRvL..77.4693B], and quantum mechanics [@2000FBS....29...25L; @2000PhRvL..84.2068L; @2000PhRvL..84.4517L; @2000PhLA..276...19L; @2001EPJB...20..349L; @2005EPJB...46...41L]. A pedagogical introduction into IFT can be found in [@2009PhRvD..80j5005E].
The uncomfortable dependence of information theoretical methods on signal prior information have lead several authors to think about methods to extract this information at least partly from the data. For example a smoothness prior for the signal can be used, where an “optimal” value for the smoothness controlling parameter derives from the data themself [@1996PhRvL..77.4693B]. The optimal smoothness constraint for a Gaussian signal is provided by its covariance, as known from Wiener filter theory [@1949wiener]. A natural proposal is therefore to measure the power spectrum (or any characteristics of the signal covariance) from the data and to use this for Wiener filtering or other signal reconstruction methods [@1987AJ.....93..968R; @1992ApJ...398..169R; @1996ITSP...44.1469L; @1999ISPL....6..205S]. Data gaps complicate the power spectrum measurement step, but extensions of such methods to this case exist [@2000AJ....120.2163S]. However, a more theoretical understanding of the inference problem and the assumptions implicitly made by these methods would be beneficial to answer several questions. How should the spectrum be measured optimally? How can spectral prior information be incorporated into the filter? And is the best spectral estimator really the best choice for the spectrum assumed in the Wiener filter?
Only Bayesian approaches, which are explicitely dealing with all relevant prior information, can answer these questions accurately. For example, it is possible to use the MAP approach to the problem of Wiener filtering if the overall amplitude of the signal covariance is unknown, even on a logarithmic scale [@1986WRR....22..499K]. For a white signal, where all pixels are statistically independent, this can be generalized to the case that all pixels amplitudes are drawn from a scale-free distribution function [@2008ApJ...675.1304R].
In precision cosmology, the problem of inferring the image and its power spectrum simultaneously is very prominent in cosmic microwave studies and cosmography of the large scale structure. It has been addressed rigorously via the Gibbs sampling scheme [@2004PhRvD..70h3511W; @2004ApJS..155..227E; @2004ApJ...609....1J; @2010MNRAS.406...60J]. Since this approach samples the full joint posterior of maps and spectra, it provides the full solution to the problem. However, the computational costs of Gibbs sampling are high. Also obtaining analytical insights into the general behavior of the scheme is not trivial. Computationally cheaper and analytically simpler, or even just alternative methods are therefore interesting and and some of the algorithms provided by this work are good candidates for being this.
Structure of the work
---------------------
We introduce IFT with parameter uncertainties in Sec. \[sec:IFTPU\]. In Sec. \[sec:ssu\] the problem of signal spectrum uncertainty is introduced, and the four of the mentioned filters are derived from MAP principles. To go beyond the MAP approximation the generic PURE approach is developed in Sec. \[sec:uncertainty renormalization flow\], where for any case with fourth order interaction terms the generic uncertainty renormalization flow equation is provided. The specific application of this approach is given in Sec. \[sec:applyingPURE\], where the PURE filter for the problem of reconstruction without spectral knowledge is derived. The perception thresholds of all these filters are investigated in Sec. \[sec:perception threshold\], and their fidelity in Sec. \[sec:compare\], where also a PURE filter with spectral smoothness prior is presented. Finally, we conclude in Sec. \[sec:conclusion\].
Information field theory with parameter uncertainty {#sec:IFTPU}
===================================================
Information field theory {#sec:ift}
------------------------
We briefly introduce the concepts of IFT and extend them to the case of parameter uncertainties. A more pedagogical introductions, as well as more details on terminology and notation of the framework can be found in [@2009PhRvD..80j5005E]. An information field is simply a spatially extended signal, where a signal $s$ is any quantity a scientist might be interested in measuring. We treat the signal $s(x) = s_x$, a function of a spatial coordinate $x$, as an abstract vector in Hilbert space with the scalar product $j^\dagger s = \int dx \, \overline{j(x)}\, s(x).$
The goal of IFT is to make statements on the signal field, which is constrained by prior knowledge and observational data. Since we are usually dealing with a finite number of noisy data points, a precise reconstruction of a signal field with its infinite number of degrees of freedom is rarely possible. Our aim is therefore to investigate the probability function of $s$ given the data $d$, the so called posterior $P(s|d)$. The posterior is usually constructed from the signal prior $P(s)$ and the likelihood of the data $P(d|s)$ using Bayes theorem $$P(s|d) = \frac{P(d|s)\, P(s)}{P(d)}.$$ The normalisation constant here, the so called evidence $P(d)$, is given by a marginalization of the signal field $$P(d) = \int \! \mathcal{D}s\, P(d,s),$$ where $P(d,s)=P(d|s)\, P(s)$ is the joint probablity density function of data and signal. The phase space or path integral $\int \! \mathcal{D}s$ goes over all possible signal field configurations, weighted with $P(d, s)$.
In IFT, we rewrite Bayes theorem in the language of a statistical field theory, namely as $$P(s|d) = \frac{e^{-H[s]}}{Z},$$ where the information Hamiltonian $H[s] = - \log P(d,s)$ and the partition function $Z = P(d)$ are actually only a renaming of (the negative logarithm of) the joint probability and evidence. This change in language, however, permits to transfer many results from statistical field theory to tackle IFT problems.
The goal of an IFT analysis could be to calculate moments of the signal field averaged in a similar path integral over the posterior $P(s|d)$, e.g. in order to know the mean signal $$m = \langle s \rangle_{(s|d)} = \int \! \mathcal{D}s \, s \, P(s|d) .$$ This mean is of special interest, since it is optimal in an $\mathcal{L}^2$-error norm sense. It minimizes the expected error variance $\langle (s-m)^\dagger (s-m)\rangle_{(s|d)} $ among all possible $m$.
In practical applications, we often discretize the signal field in $N_\mathrm{pix}$ pixels at locations $x_i$. Then the discretized path integral for any signal function $f(s)$ is $$\int \! \mathcal{D}s\, f(s) = \left( \prod_{i=1}^{N_\mathrm{pix}} \, \int \! ds(x_i) \right) f(s({x_1}), \ldots, s({x_{N_\mathrm{pix}}})).$$ If possible, we try to avoid to evaluate such very high dimensional integrals nummerically. We use the fact that a multimodal Gaussian probability density function as given by $$\label{eq:GaussPrior}
{\mathcal{G}}(s,S) \equiv \frac{1}{|2\pi\, S|^\frac{1}{2}}\, \exp\left( -\frac{1}{2} s^\dagger S^{-1} s \right)$$ (with $|S|$ denoting the determined of the matrix $S$) can be integrated analytically: $ \int \! \mathcal{D}s\,\mathcal{G}(s,S) = 1$. Many functional integrals can be derived from this, like the moments of a Gaussian, and path-integrals of any quadratic functional of the integrated field. Non-quadratic exponents can be expanded around the multivariate Gauss integral in terms of diagramatic pertubation series. For further details, the reader is refered to [@2009PhRvD..80j5005E] and any standard book on field theory.
In the simplest case of the theory, signal and noise are independent Gaussian random variables, and the data depend linearly on them. This so-called free theory can be treated analytically and is our starting point. It has been analyzed in depth before and leads to the so called Wiener-filter theory [@1949wiener]. However, usually the assumption that all parameters $p$ of the problem like instrument calibration, or signal covariance, are known is used. This assumption will be dropped in the following, and we will see, that the otherwise trivial case gets interesting complications and the corresponding free IFT is enriched by interaction terms.
Free theory from a Gaussian data model
--------------------------------------
We assume that the signal we want to reconstruct is a Gaussian random field, with a probability distribution prior to any measurement described as $P(s|p) = {\mathcal{G}}(s,S_p)$, where $S_p=\langle s\,s^\dagger \rangle_{(s|p)}$ is the signal covariance given the parameter $p$, which itself might be a vector or even a field over some space. The subscript $(s|p)$ on the brackets of the expectation value indicate that the average should be done over the probability distribution $P(s|p)$. Thus, the individual elements of the signal covariance matrix read $$(S_p)_{xy} =\langle s(x)\,\overline{s(y)} \rangle_{(s|p)} = \int \! \mathcal{D}s\, s(x) \overline{s(y)}\, P(s|p).$$
We further assume that the signal is processed by a linear measurement device with response matrix $R$ and additive noise $n$ according to: $$d = R\, s + n.$$ In general, response and noise can also depend on unknown parameters and the general theory developed here can also be applied to that case. To focus the discussion, we only consider here the concrete example of a parameter dependent signal covariance, and assume the response and noise statistics to be known. We assume the noise to be signal-independent and Gaussian, and thus $$P(n|s,p) = {\mathcal{G}}(n,N),$$ where $N= \langle n\, n^{\dagger} \rangle_{(n)}$ is the noise covariance matrix. Since the noise is just the difference of the data to the signal-response, $n=d-R\,s$, the likelihood of the data is $$\label{eq:PsPosterior}
P(d|s,p) = P(n= d-R\,s|s,p) = {\mathcal{G}}(d-R\,s,N).$$
The information Hamiltonian as defined in [@2009PhRvD..80j5005E] is the negative logarithm of the joint probability function of signal and data for given and fixed parameters: $$H_{p}[s] = - \log P(d,s|p) = - \log\left[ P(d|s,p) \,P(s|p)\right].$$ Thus the Hamiltonian of the Gaussian theory, $$\begin{aligned}
\label{eq:freeHamiltonian}
H_{p}^{\mathcal{G}}[s] =
\frac{1}{2} s^{\dagger} D_p^{-1} s - j^{\dagger} s + H_{0,p}^{{\mathcal{G}}},\end{aligned}$$ is only quadratic in the signal, and therefore corresponds to a free field theory. Here $$\label{eq:D}
D_p= \left[ S^{-1}_p + M \right]^{-1},\; \mbox{with}\; M= R^{\dagger} N^{-1} R,$$ is the information propagator, which depends on the unknown spectral parameters. The information source, $$j = R^{\dagger} N^{-1} d,$$ depends linearly on the data in a response-over-noise weighted fashion. Finally, $$H_{0,p}^{{\mathcal{G}}} = \frac{1}{2}\, d^\dagger\,N^{-1}\,d + \frac{1}{2}\,\log\left( |2\,\pi\,S_p|\,|2\,\pi\,N| \right)$$ absorbs all $s$-independent normalization constants. It can not be ignored here, since it depends on $p$.
The key quantity, from which all relevant moments of the signal can be estimated, is the partition function, $$Z_p[J] =
\int \!\mathcal{D}s \, e^{-H_p[s] + J^{\dagger}s}.$$
For the free field theory the partition function is $$\label{eq:ZdfreeTheory}
Z_p^{{\mathcal{G}}}[J] = \sqrt{|2\pi\, D_p|}\,
\exp\left\lbrace +\frac{1}{2} (J+j)^{\dagger} D_p (J+j) -H_{0,p}^{{\mathcal{G}}} \right\rbrace\!.$$ This explicit formula permits us to calculate the expectation of the signal given the data (and the parameters), in the following called the map $m_{p}$: $$\begin{aligned}
\label{eq:WFmap}
m_{p} &=& \langle s \rangle_{(s|d,p)} = \left. \frac{\delta \log Z_p^{{\mathcal{G}}}}{\delta J}\right|_{J=0}
=D_p\,j \nonumber\\
&=& \underbrace{\left[ S_p^{-1} + R^{\dagger} N^{-1} R\right]^{-1} R^{\dagger} N^{-1}}_{F_{p}} d.\end{aligned}$$ The last expression shows that the map is given by the data after applying a generalized Wiener filter, $m_{p} = F_{p} \,d$, which depends on the parameter $p$ of the signal covariance.
Similarly, the quadratic uncertainty of the signal map can be worked out. It turns out that for a free theory it is the propagator itself $$\label{eq:mapuncertainty}
\langle (s-m_p) (s-m_p)^{\dagger}\rangle_{(s|d,p)} =
\langle s \, s^{\dagger}\rangle_{(s|d,p)} - m_p\,m_p^\dagger =
D_p .$$ The first identity follows from $\langle s\,m_p^\dagger\rangle_{(s|d,p)} = \langle s\rangle_{(s|d,p)} \,m_p^\dagger = m_p m_p^\dagger$ due to the fact, that the reconstructed map $m_p$ is solely determined by the data, and therefore given in this average. The second identity holds due to the identity of the connected correlation function and the propagator, $\langle s\, s^\dagger \rangle^\mathrm{c}_{(s|d,p)} = \delta^2 \log Z_p^{\mathcal{G}}/\delta Z^2|_{J=0} = D_p$.
Classical field theory
----------------------
In case of the free theory, the map, Eq. \[eq:WFmap\], would also be obtained from a classical treatment of the Hamiltonian by extremizing it: $$\frac{\delta H_p[s]}{\delta s} = 0.$$ For a Hamiltonian with interaction terms the classical field (in field-theoretical language) or MAP estimator (in signal processing language) is a useful approximation to the correct expectation value. The inverse Hessian in the signal Hilbert space around this map, $$\left( \frac{\delta^2 H_p[s]}{\delta s \, \delta s^\dagger}\right)^{-1},$$ characterizes the uncertainty. For the free theory, this is the propagator, as given by Eq. \[eq:mapuncertainty\].
The identity of fully field theoretical and classical results holds only for the case of a free theory. However, the latter is often an acceptable approximation to the former, while much easier to derive. Therefore, we will make also use of the classical approximation in the following.
Parameter uncertainty and posterior
-----------------------------------
In many applications, there are parameters specifying the likelihood and prior, and thereby the coefficients of the Hamiltonian, which are not precisely known. These parameters, in the following denoted by the abstract vector $p$, are either to be determined from the data, to be marginalized over, or to be simultaneously determined with the signal.
In such a case we have to construct the joint posterior of the signal and the parameter given the data. This is given according to Bayes’ theorem as $$\begin{aligned}
P(s,p|d) &=&\frac{P(d,s,p)}{P(d)} = P(s|d,p) \, \frac{P(d|p)}{P(d)}\, P(p)
\,,\end{aligned}$$ where we had to introduce the parameter prior $P(p)$. The last expression contains a Bayes factor $P(d|p)/P(d)$, the ratio of the evidence of data for a specific parameter set to that of the model at all. Thus, the joint posterior is weighted towards model-parameters for which the data provide larger evidence in addition to any prior-weighting.
The definition of the Hamiltonian for fixed-parameters as $H_p[s] = - \log P(d,s|p)$ permits us to construct the joint partition function $$\begin{aligned}
Z[J,K] \!\!&\equiv &\!\! \!\int\! dp \!\int \!\mathcal{D}s \, P(s,p|d)\,P(d)\, e^{J^\dagger s + K^\dagger p}\\
\!\!&=&\!\! \!\int\! dp\,P(p)\, e^{K^\dagger p} \!\underbrace{\int \!\mathcal{D}s \,\overbrace{P(d|s,p)\,P(s|p)}^{P(d,s|p)=e^{-H_p[s]}}\,e^{J^\dagger s}}_{Z_p[J]},\nonumber\end{aligned}$$ which is built upon $Z_p[J]$, the partition function of the theory for given parameters $p$. The information field estimators, marginalized for the unknown parameters, is then simply given by $$\begin{aligned}
\label{eq:m_eff}
\langle s \rangle_{(s|d)} &=& \left. \frac{\delta\,\log Z[J,K] }{\delta J} \right|_{J,K=0} \nonumber\\
&=& \left. \frac{1}{Z} \, \int dp\,P(p) \, \frac{\delta Z_p[J]}{\delta J}\right|_{J=0}\\
&=& \int dp\,\underbrace{P(p)\, \frac{Z_p}{Z}}_{P(p|d)}\, \langle s \rangle_{(s|d,p)}. \nonumber\end{aligned}$$ The aim of this work is to provide schemes to calculate this parameter marginalized signal mean. It is not just the signal estimator multiplied by the parameter prior $P(p)$, but is additionally weighted by a parameter likelihood factor $P(d|p) = Z_p$, so that the parameter-dependent signal means are averaged over the parameter posterior $P(p|d)$. Therefore, parameter values which are especially compatible with the data get automatically a larger weight, as recognized before.
Effective marginalized Hamiltonian {#sec:effectiveH}
----------------------------------
If a parameter-dependent Hamiltonian $H_p[s] = - \log P(d,s|p)$ describes the conditional probability of the signal and data given the parameters, an effective, parameter-marginalized Hamiltonian $H[s]$ is defined by $$\begin{aligned}
\label{eq:effectiveHamiltonianGeneral}
e^{-H[s]} &\equiv& \int \!\! dp\, P(d,s,p)
= \int \!\! dp\, P(d,s|p)\, P(p) \nonumber\\
&=&
\int \!\! dp \, e^{-H_p[s]- E_p} \,,\end{aligned}$$ with $E_p = -\log P(p)$ the parameter-prior-energy. It is crucial, that $H_p[s]$ obeys the correct normalization condition, $\int \!\! \mathcal{D}d \,\int \!\! \mathcal{D}s \, \exp(-H_p[s])=1$, otherwise a hidden prior on $p$ may enter the calculation.
In many cases, an analytical calculation of the effective Hamiltonian will be out of reach. Since the perturbative field theoretical treatment requires a polynomial representation anyway, it is often easier to obtain the coefficients of the effective Hamiltonian separately by Taylor-Frechét expansion around a reference field configuration $t$, so that $s = t + \phi$. The Hamiltonian for $\phi$ is then $$\begin{aligned}
\label{eq:HeffTaylor}
H[\phi] &= & H_0 - j^\dagger \phi +
\frac{1}{2}\,\phi^\dagger D^{-1}\, \phi + \nonumber\\
&&
\sum_{n=3}^{\infty} \frac{1}{n!} \,\Lambda^{(n)}_{x_1\ldots x_n}\, \phi_{x_1}\cdots \phi_{x_n}, \;\mbox{with}\nonumber\\
H_0 &=& H[t] = -\log \int \!\!dp\, e^{-H_p[t]-E_p},\nonumber\\
j_x &=& - \left. \frac{\delta H[s]}{\delta s_x} \right|_{s=t} =
- \left\langle \frac{\delta H_p[s]}{\delta s_x}\right\rangle_{(p|d, s=t)},\\
D^{-1}_{x\, y} &=& \left. \frac{\delta^2 H[s]}{\delta s_x\,\delta s_y} \right|_{s=t} \nonumber\\
&=& \left\langle \frac{\delta^2 H_p[s]}{\delta s_x\,\delta s_y}
- \frac{\delta H_p[s]}{\delta s_x} \, \frac{\delta H_p[s]}{\delta s_y}
\right\rangle_{(p|d, s=t)} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+j_x\,j_y,\; \mbox{and} \nonumber\\
\Lambda^{(n)}_{x_1\ldots x_n} &= & \frac{1}{n!} \sum_{\pi \epsilon \mathcal{P}} \left. \frac{\delta^n \,H[s] }{\delta s(x_{\pi(1)})\cdots \delta s(x_{\pi(n)})} \right|_{s=t}. \nonumber\end{aligned}$$
Here, $\langle \ldots \rangle_{(p|d, s)} \equiv \int \! dp\, \ldots\,P(p|d,s)$ provides expectation values with respect to the parameter $p$ given the data $d$ and the signal $s$. Repeated coordinate indices are thought to be integrated over. The interaction coefficients $\Lambda^{(n)}_{x_1\ldots x_n}$ are symmetrized by averaging over all possible permutations $\pi$ from the space of permutations $\mathcal{P}$. In general, $D^{-1}_{x\, y}$ needs to be symmetrized, too, but we have left out the symmetrization in the above equation for convenience, since in the cases we consider $D^{-1}_{x\, y}$ is already symmetric.
In case the expansion was around $t=0$, then $$\begin{aligned}
\label{eq:HOeffs=0}
H_0 &=& -\log \int \!\! dp \, e^{-H_{0,p} - E_p},\nonumber\\
\label{eq:jeffs=0}
j &=& \langle j_{p} \rangle_{(p|d, s=0)}, \nonumber\\
\label{eq:Deffs=0}
D^{-1} &=& \langle D_{p}^{-1} - j_{p} \, j_{p}^\dagger \rangle_{(p|d, s=0)} +j\,j^\dagger, \;\mbox{and}\\
\label{eq:L3effs=0}
\Lambda^{(3)} &=& \langle\Lambda^{(3)}_p + 3\, D_{p}^{-1} \otimes j_p - j_p j_p j_p\rangle_{(p|d, s=0)}\nonumber\\
&& - 3\, D^{-1} \otimes j + jjj\nonumber\\
\label{eq:L4effs=0}
\Lambda^{(4)} &=& \langle\Lambda^{(4)}_p + 4\, \Lambda^{(3)}_p \otimes j_p -3\, D_{p}^{-1} \otimes D_{p}^{-1} +6\, D_{p}^{-1} \otimes j_p j_p^\dagger \nonumber\\
&& - j_p j_p j_p j_p\rangle_{(p|d, s=0)}- 4\, \Lambda^{(3)} \otimes j +3\, D^{-1} \otimes D^{-1}\nonumber\\
&& -6\, D^{-1} \otimes j j^\dagger + j j j j, \;\ldots \nonumber\end{aligned}$$
Here, an implicit tensor notation was used, with e.g. $(j\,j\,j)_{xyz} \equiv
j_x\, j_y\, j_z$ and we defined the symmetrized tensor product $(A\otimes j)_{x_1 x_2 x_3} \equiv \frac{1}{3!} \sum_{\pi \in
\mathcal{P}} A(x_{\pi(1)}, x_{\pi(2)}) \, j(x_{\pi(3)})$. For higher rank tensors, the symmetrized tensor product is defined in an analogous way.
Signal spectrum uncertainty {#sec:ssu}
===========================
Spectrum parameterization
-------------------------
Our example application of IFT with parameter uncertainties is the reconstruction of a Gaussian signal with unknown variance, which we introduce now.
The signal covariance $(S_p)_{x\,y} = \langle s_x \overline{s_y}\rangle_{(s|p)}$ may exhibit any dependence on the spatial coordinates as long as the matrix is symmetric and positive definite. In the cosmological relevant case of translationally and rotationally invariant signal statistics, the signal covariance is fully characterized by its power spectrum. This means, there is an orthonormal basis $O$ of the signal Hilbert space which diagonalizes $S_p$: $$(O\,S_p O^\dagger)_{k\,q} \equiv O_{k\, x} (S_p)_{x\, y} \overline{O_{q\,y}} = 1_{k\,q}\, P_{S_p}(k),$$ with $1_{k\,q}$ the identity in the transformed basis, $P_{S_p}(k)$ the power-spectrum, and using Einstein sum convention. In case we are dealing with a signal over a $d$-dimensional Cartesian space, $O_{k\, x} = \exp(i\,k\,x)$ is simply a Fourier transformation and the Fourier space identity is $1_{k\,q} = (2\,\pi)^d \, \delta(k-q)$, provided the scalar product in Fourier space is adopted as $a^\dagger b = (2\,\pi)^{-d}\, \int dk \,\overline{a(k)}\, b(k) $. However, since the theory should also be applicable in curved spaces like the sphere, or even in spaces without translational invariance, we formulate it in an abstract way and just assume that the basis $O$ diagonalizes the signal covariance, which is always possible.
In general, the signal covariance $S_p$ may also exhibit any dependence on the unknown parameter $p$ of the problem, as the power spectrum in cosmology is a complicated function of the cosmological parameters. However, in order not to depend on a specific model, we model the power spectrum as being a linear combination of a number of positive basis functions $f_i(k)$ with disjunct supports (the spectral bands) with respect to the basis $O_{k\,x}$, so that $$P_{S_p}(k) = \sum_i p_i f_i(k)$$ is positive for all $k$ (all coefficients of $p=(p_i)_i$ are positive and the spectral bands cover the full $k$-space domain). We define $$(S_i)_{xy} = \overline{O_{k\,x}} \, f_i(k) \, O_{k\, y}$$ and therefore have $$S_p = \sum_i p_i S_i.$$ Since we also need the inverse of the covariance matrix we further define $$g_i(k) = \left\{ \begin{array}{ll}
1/f_i(k) & f_i(k)>0 \\
0 & f_i(k) =0
\end{array}
\right.$$ and the pseudo-inverse of the band-variances, $$S_i^{-1} = \overline{O_{k\,x}}\, g_i(k)\, O_{k\, y},$$ so that $$S_p^{-1} = \sum_i p_i^{-1} S_i^{-1},$$ is the inverse of $S_p$, as one can easily verify.
Spectral prior and joint Hamiltonian
------------------------------------
For definiteness, we assume that the individual signal-band amplitudes $p_i$ have independent prior distributions, $$\label{eq:pall-prior}
P(p) = \prod_i P(p_i),$$ with the individual priors being given by inverse Gamma distributions, which are power-laws with exponential low-amplitude cutoff at $q_{i}$ : $$\label{eq:p-prior}
P(p_i) = \frac{1}{q_i\,\Gamma(\alpha_i -1)} \,\left(\frac{p_i}{q_i}\right)^{-\alpha_i} \,
\exp \left(-\frac{q_i}{p_i}\right).$$ For $\alpha_i \gg 1$ this is an informative prior, where $q_i/\alpha_i$ determines the preferred value. A non-informative prior would be given by Jeffreys prior with $\alpha_i = 1$ and $q_i=0$.[^1]
The joint Hamiltonian is therefore $$\label{eq:joint Hamiltonian of our problem}
H[s,p] = H^{\mathcal{G}}_p[s] + E(p)$$ with the parameter prior energy $$E(p) = \sum_i \left[\frac{q_i}{p_i} + \alpha_i \log\left(\frac{p_i}{q_i}\right) + \log(q_i\,\Gamma(\alpha_i -1))\right].$$
Generic filter formula
----------------------
In the following, we derive five approximate filters for this problem. It will turn out that they can all be cast into a single set of determining equations, with different coefficients. This generic filter formula should be presented first, before we discuss the individual approaches.
All of the derived filters can be expressed as Wiener filters for some specific spectrum $S_{p^*} = \sum_i p_i^* S_i$, with different spectral parameters $p^*$. The signal map and the spectrum assumed for its construction have to be calculated self-consistently from $$\begin{aligned}
\label{eq:characteristic equation}\label{eq:general-spectrum}
m_{p^*} &=& D_{p^*}\, j,\;\mbox{and}\\
p_i^* &=& \frac{1}{\gamma_i+\varepsilon_i} \left( q_i + \frac{1}{2} \mathrm{Tr}[(m_{p^*} m_{p^*}^\dagger + \delta_i\, D_{p^*})\, S_i^{-1}]\right),\nonumber\end{aligned}$$ for example by simply iterating these two equations.
Here, the filter-specific parameters are $\varepsilon_i$, $\delta_i$, and $\gamma_i = \alpha_i - 1 + \varrho_i/2$, where $\varrho_i = \mathrm{Tr}[S_i^{-1}S_i]$ is the number of degrees of freedom of the $i$th spectral band. In order to simplify notation, we drop in the following the $*$ from $p^*$, assuming that the context makes it clear wether we talk about the unknown parameter $p$ or a parameter choice $p^*$ for a specific filter.
In order to develop a filter for our signal, we have to decide according to which principle the signal or the power spectrum used in the Wiener filtering is determined. In the space of all possibilities for the signal and its power spectrum the joined probability function $P(s,p|d)$ has to be asked. There are different hyperplanes in this space along which this function can be cut, marginalized, and maximized. The ultimate answer of the PURE approach will come from marginalizing $p$ and calculating the signal mean. However, first we want to establish more traditional signal estimators, using largely the MAP principle along different cuts through the joint signal and spectral parameter space.
In case a Jeffreys prior is adopted ($q_i=0$ and $\alpha_i=1$) it will turn out that the trivial filter $m(d)=0$ would be the preferred solution in all cases. However, since Jeffreys prior is an improper prior which is convenient to represents the class of very broad, but proper priors, we should not hesitate to remove the trivial filter solution by hand. Otherwise we would need to enter the discussion about an appropriate informative prior, which we like to avoid for simplicity. This can not be decided generically, but only for any concrete inference problems individually.
The parameters of the filters described in Sec. \[sec:derivedfilters\] and derived in the next few subsections are summarized in Fig. \[fig:diagram\].
The critical filter {#sec:critical filter}
-------------------
Our first filter can be understood without any reference to statistical inference and is along the lines of the well known Karhunen-Loève (KL, [@1947KarhunenK; @1978LoeveM; @2002MNRAS.335..887T]) and Feldman-Kaiser-Peacock (FKP, [@1994ApJ...426...23F]) estimators for power spectra. The Wiener filter map $m_p = D_p\, j$ (with $D_p = (S_p^{-1} + R^\dagger N^{-1} R)^{-1}$ and $j = R^\dagger N^{-1} d$) of a data realization of a Gaussian random signal with a known covariance $S_p$ will have on average the covariance $$\langle m_p \, m_p^\dagger \rangle_{(d,s|p)} = S_p - D_p,$$ as one can verify with a short calculation.[^2] The propagator on the rhs just accounts for the power lost in measurement and filtering. Now we assume that our data and our Wiener filter map are so rich or typical that this equation also holds for our individual data realization. Thus we drop the expectation angles, apply $\mathrm{Tr}[\times\,S_i^{-1}]$, and get the critical filter recipe in the form of Eq. \[eq:characteristic equation\] with parameters $\delta_i = 1$, $\varepsilon_i =0$, $\alpha_i = 1$, and $q_i =0$. The last two parameters are characteristic for Jeffreys prior, which we obviously have assumed implicitely, since no prior information on the spectrum, or even its magnitude on a logarithmic scale, has entered the critical filter scheme.
The name *critical filter* should become clear in Sec. \[sec:perception threshold\]. There, we show that at least in cases where the different spectral parameters are independent of each other, the different filters can be cast into two classes, such with and such without perception threshold. The critical filter marks the demarcation line between these phases.
The critical filter has recently been applied successfully by [@2010arXiv1008.1246O] to reconstruct an all sky map of the galactic Faraday depth from sparse and noisy measurements.
Joint MAP filter {#sec:jMAP}
----------------
Extremizing the joint Hamiltonian, Eq. \[eq:joint Hamiltonian of our problem\], with respect to $p$ and $s$ yields the joint MAP filter parameters $(\delta_i, \varepsilon_i)= (0,1)$. We note, that if we extremize with respect to the log-spectral amplitudes $\tau_i= \log p_i$, the parameters $(\delta_i, \varepsilon_i)= (0,0)$ would have resulted due to the effect of the Jacobian of the prior transformation. This latter filter is identical to the classical one derived below in Sec. \[sec:classical filter\].
![The effective signal Hamiltonian, Eq. \[eq:MAPmapHamilton\], without the normalization constant $H_0$ in case of Jeffreys prior and for a single, independent signal $s=s_i$ and data point $d=d_i$. The parameters are $R_{ij}=N_{ij}=\delta_{ij}$ and $S_{ij} = p_i\, \delta_{ij}$. The different curves show the Hamiltonian for representative data values. The triangle symbols mark the results of the inverse response estimator $m_\mathrm{ir} = R^{-1} d$ on the corresponding curves. The large open and small filled circles mark the renormalized and classical map estimator results, respectively. The existence of a classical perception threshold can be seen: for $-2<d<2$, the classical map is exact zero since no non-trivial stationary point of the Hamiltonian exists. The thin dotted line shows the renormalized Hamiltonian for the case $d=3$, as provided by $\frac{1}{2}(s-m_p)^\dagger D_p^{-1} (s-m_p)$.[]{data-label="fig:H(s)"}](hamiltons.pdf){width="\columnwidth"}
MAP spectrum filter {#sec:MAPspecFilter}
-------------------
Marginalizing the joint Hamiltonian Eq. \[eq:joint Hamiltonian of our problem\] over the signal space provides the spectrum Hamiltonian $$\begin{aligned}
\label{eq:MAPspecHamilton}
H(p) &=& - \log(P(d,p)) = - \log(P(d|p)\, P(p))\nonumber\\
&=& \frac{1}{2}\log| 1+ Q_p| - \frac{1}{2} j^\dagger D_p\, j + H_0'\nonumber\\
&+& \sum_i \left(\frac{q_i}{p_i} + \alpha_i \log \left( \frac{p_i}{q_i}\right) \right),\; \mathrm{with}\\
D_p &=& (S_p^{-1} + M)^{-1},\; Q_p = S_p \, M,\; \mathrm{and}\nonumber\\
H_0' &=& \frac{1}{2}\log|N| + \frac{1}{2} d^\dagger N\, d + \sum_i \, \log( q_i\, \Gamma(\alpha_i -1)).\nonumber\end{aligned}$$ Here we used Eq. \[eq:ZdfreeTheory\] for $P(d|p)$. A data-space view on this likelihood is given in Appendix \[sec:signal covariance like\]. Extremizing $H(p)$ with respect to $p_i$ and sorting for terms linear in it provides the MAP-spectrum parameter $(\delta_i,\varepsilon_i ) = (1,1)$.
If we extremize with respect to the parameters $\tau_i = \log p_i$, we get $(\delta_i, \varepsilon_i)= (1,0)$, the parameters of the critical filter. Thus, the critical filter can be regarded as the one resulting from a MAP spectrum estimation on a logarithmic scale. Note that MAP estimators are sensitive to the coordinate system in which parameters are expressed.
Classical map estimator {#sec:classical filter}
-----------------------
The effective, parameter marginalized signal Hamiltonian (Eq. \[eq:effectiveHamiltonianGeneral\]) can be calculated analytically:[^3] $$\begin{aligned}
\label{eq:MAPmapHamilton}
H[s] &=& \frac{1}{2} \, s^\dagger M\, s - j^\dagger s + \sum_i \gamma_i\, \log\left(q_i+ \frac{1}{2} \, s^\dagger S_i^{-1} s\right)\nonumber\\
&+& H_0
,\;\mbox{with}
\nonumber\\
H_{0} &=& \frac{1}{2} d^\dagger N^{-1} d + \frac{1}{2} \log\left(|2\,\pi\,N|\right)\nonumber\\
&-& \log\left( \prod_i \frac{\Gamma(\gamma_i)\, q_i^{\alpha_i -1}}{\Gamma(\alpha_i -1)\,|2\,\pi\,S_i|^{\frac{1}{2}}}\right).\end{aligned}$$
The classical mapping equation results from extremizing this Hamiltonian and is provided by Eq. \[eq:general-spectrum\] for $(\delta_i, \varepsilon_i) = (0,0)$. This can be regarded as a poor man’s critical filter, since only the power in the map is used to determine the signal covariance, and no correction for the power lost in the filtering is applied. In case of a single independent data and signal point, the effective Hamiltonian is an one dimensional function in signal space and is shown in Fig. \[fig:H(s)\].
Uncertainty renormalization flow {#sec:uncertainty renormalization flow}
================================
General remarks
---------------
Although the MAP methods often provide acceptable signal estimators, they are not optimal in an $\mathcal{L}^2$-error norm sense. In case of a skewed posterior, such reconstructions are suboptimal. Our goal is to calculate moments of the signal field averaged over the effective posterior, as e.g. $\langle s \rangle_{(s|d)}$ given by Eq. \[eq:m\_eff\], since those optimize the $\mathcal{L}^2$-error. For this we might construct the effective Hamiltonian exactly or in terms of a Taylor expansion as in Eqs. \[eq:HeffTaylor\] and \[eq:Deffs=0\].
Such an expansion of the effective Hamiltonian around a reference field is expected to work best when the parameter prior is well localized around a specific value. The effective Hamiltonian will then be close to the original, parameter-dependent one for this parameter value. In case the original theory was free, the effective theory will have only small interaction terms. Diagrammatic expansions can then be conducted and truncated at low order.
Unfortunately, in many practical applications, the uncertainties of the parameters are substantial, and not described by a well localized prior. In this case it might be possible to construct the effective Hamiltonian by repeatedly adding smaller portions of parameter-uncertainty, with each uncertainty dose so small that the resulting Hamiltonian has only weak interactions, which can be re-absorbed into renormalized, effective propagator and data source terms. The accumulated uncertainty can thereby become large and equal to the required amount of entropy for the unknown parameter of the theory. In the following we will explain the basics of this uncertainty renormalization flow.
Parameter uncertainty renormalization
-------------------------------------
A broad prior for a parameter $p$ may be decomposed into a number $N$ of narrow and mutually independent priors for some auxiliary variables $\tau_j$ (with $j \in \{1,...,N\}$): $$P(p) = \left( \prod_{j=1}^{N}\, \int \! d\tau_j \, P(\tau_j)\right) \delta(p - \sum_{j=0}^{N} \tau_j).$$ We have chosen here the parameter to be the sum of the auxiliary variables for definiteness and simplicity, but other relations can be worked out in a similar way or be mapped onto this case. Also the mutual independence of the auxiliary variable is mostly a technical convenience and not a strict requirement. Note, that we have included a starting parameter value of $\tau_0$ into the sum. Since it would be convenient to identify this with the prior expectation value $ \langle p \rangle_{(p)} $ throughout the full renormalization procedure we require $$\label{eq:unbiased auxiliary priors}
\langle \tau_j \rangle_{(\tau_j)} = \delta_{j \, 0} \, p_0,$$ with $p_0 = \langle p \rangle_{(p)}$. We further introduce the $l$-th parameter residual as $r_l = \tau_0 + \sum_{j=l+1}^{N} \tau_j$, so that $r_0 = \sum_{j=0}^{N} \tau_j = p$ and $r_N = \tau_0 = p_0$. The effective Hamiltonian can now be expressed as $$\begin{aligned}
e^{-H[s]} &=& \int \!dp \, P(p) \, e^{-H_p[s]} \nonumber\\
&=& \int \!dp \, \left( \prod_{j=1}^{N}\, \int \! d\tau_j \, P(\tau_j)\right) \delta(p - \sum_{j=0}^{N} \tau_j)\, e^{-H_p[s]} \nonumber\\
&=& \left( \prod_{j=1}^{N}\, \int \! d\tau_j \, P(\tau_j)\right) \, e^{-H_{r_0·}[s]} \nonumber\\
&=& \left( \prod_{j=2}^{N}\, \int \! d\tau_j \, P(\tau_j)\right) \, \underbrace{ \int \! d\tau_1 \, P(\tau_1)\, e^{-H_{\tau_1+r_1·}[s]}}_{\equiv e^{-H_{r_1·}^{(1)}[s]} } \nonumber\\
&=& \left( \prod_{j=3}^{N}\, \int \! d\tau_j \, P(\tau_j)\right) \, \underbrace{\int \! d\tau_2 \, P(\tau_2)\, e^{-H_{\tau_2+r_2·}^{(1)}[s]} }_{\equiv e^{-H_{r_2·}^{(2)}[s]} } \nonumber\\
&=& ... \nonumber\\
&=& \int \! d\tau_N \, P(\tau_N) \, e^{-H_{\tau_N·+p_0}^{(N-1)}[s]} = e^{-H_{r_N=p_0}^{(N)}[s]},\end{aligned}$$ where $H_{p_0}^{(N)}[s] = H[s]$. This means that a series of effective Hamiltonians with increasing accumulated parameter uncertainty is defined, and an uncertainty adding operator: $$\label{eq:Hrecast}
H_{r_n}^{(n)} \mapsto H_{r_{n+1}}^{(n+1)} \equiv - \log \int \! d\tau_{n+1} \, P(\tau_{n+1})\, e^{-H_{r_{n+1}+\tau_{n+1}}^{(n)}}.$$ Note that $H_{r_0}^{(0)}= H_{p=r_0}$ and $H_{r_N}^{(N)} = H$. This uncertainty renormalization can be done using Eq. \[eq:HeffTaylor\] if it is not possible to do it analytically. To each Hamiltonian a time-like variable $t$ can be assigned, which measures the amount of uncertainty accumulated so far. A suitable variable is the accumulated uncertainty dispersion, $$t_n = \sum_{j=1}^{n} \sigma_{\tau_j}^2 = \sum_{j=1}^{n} \left( \langle \tau_j^2 \rangle_{(\tau_j)} - \langle \tau_j \rangle_{(\tau_j)}^2\right)
= \sum_{j=1}^{n} \langle \tau_j^2 \rangle_{(\tau_j)} ,$$ where we used Eq. \[eq:unbiased auxiliary priors\]. In case all auxiliary variables have the same prior, we find $t_n = n \, t_1 = \frac{n}{N}\, \langle (p-p_0)^2 \rangle_{(p)}$.
At each time-step a renormalization of the Hamiltonian can be done, in which it is cast back into the structure it had before, e.g. in our example of reconstruction with unknown power spectrum the free Hamiltonian of Eq. \[eq:freeHamiltonian\], just with modified coefficients (propagator, source and interaction terms).
In our example the recast Hamiltonian is free, which implies that we are constructing a Gaussian approximation of the parameter marginalized signal posterior to be used for inference. It is shown in [@2010PhRvE..82e1112E] that the chosen Gaussian seems to be optimal in an information theoretical and thermodynamical sense. It maximizes the cross information with the correct effective posterior.
A renormalization flow can further be established by letting the individual time-steps of size $t_1$ become infinitesimally small, however, their number $N$ infinitely large, while keeping the total added uncertainty constant, $t= N\, t_1$. The result are the renormalization flow equations for the coefficients of the Hamiltonian. The actual form of these equations depends on the Hamiltonian and is much simpler if the Hamiltonian has less interactions.
Therefore, even a free Hamiltonian as in Eq. \[eq:freeHamiltonian\] should be further simplified by suppressing the linear term $j^{\dagger}_p s $ as far as possible. This is done for the value of $p=p_0$, which is our starting point in parameter space, by changing to a new field variable $\phi = s -m_0$ with $ m_0 \equiv D_{0} \, j_{0}$, $D_{0} \equiv D_{p_0} $, and $j_{0}\equiv j_{p_0}$. The Hamiltonian reads now $$\begin{aligned}
\label{eq:defHamiltonianwithPara1}
H'_p[\phi] &=& \frac{1}{2} \phi^\dagger D_p^{-1} \phi - {j'_p}^\dagger \phi + H'_{0,p},\;\mbox{with}\nonumber\\
\phi &=& s - m_0 , \nonumber\\
j'_p &=& j_p - D_p^{-1}\,m_0 = j_p-D_p^{-1}\,D_0\, j_0,\;\mbox{and}\\
H'_{0,p} &=& H_{0,p} + \frac{1}{2} m_0^{\dagger} D_p^{-1} m_0 - j_p^\dagger m_0\nonumber\end{aligned}$$ and is especially simple for $p=p_0$, since then $j'_0 = 0$. Now, the effective Hamiltonian is calculated and expanded according to the recipe in Sec. \[sec:effectiveH\] for a parameter prior well localized on $p=p_0$. The localization of the prior is typically characterized by a small parameter $\delta t= \sigma_\tau^2 $, which also appears as a pre-factor of the various coefficients of the effective Hamiltonian.
4$^\mathrm{th}$ order interactions
----------------------------------
In order to perform the renormalization step, the recasting of the uncertainty marginalized Hamiltonian in Eq. \[eq:Hrecast\] into its original form, let us be a bit more specific about the effective Hamiltonian for definiteness. By virtue of our foresight on the calculations in Sec. \[sec:lognormal prior\] we assume that up to linear order in $\delta t$ the effective Hamiltonian is given by $$\begin{aligned}
H'[\phi] &= & H_0' + \Lambda^{(1)\dagger} \phi +\frac{1}{2}\,\phi^\dagger \left( {D}_0^{-1} +\Lambda^{(2)} \right) \, \phi \\
&& + \frac{1}{3!} \,\Lambda^{(3)}\, [\phi, \phi, \phi] + \frac{1}{4!} \,\Lambda^{(4)}\, [\phi, \phi, \phi, \phi]+ \mathcal{O}(\delta t^2),\nonumber\end{aligned}$$ with $ \Lambda^{(1)}$, $\Lambda^{(2)} $, $\Lambda^{(3)} $, and $\Lambda^{(4)} $ being of order $\mathcal{O}(\delta t)$. Here, Eq. \[eq:HeffTaylor\] or \[eq:Deffs=0\] might have been used.
The corrections can be expected to be small of $\mathcal{O}(\delta t)$, since our originally free Hamiltonian, Eq. \[eq:defHamiltonianwithPara1\], should be recovered in the limit of vanishing parameter uncertainty, $\delta t \rightarrow 0$. All higher order interaction terms are of higher order in $\delta t$ and therefore ignored in the following. For our later convenience we introduce $$\lambda^{(n)} =\lim_{\delta t \rightarrow 0} \frac{\Lambda^{(n)}}{\delta t}.$$
Now, we can renormalize by absorbing all diagrams of order $ \mathcal{O}(\delta t)$ into renormalized propagator and source terms, in order to obtain a free Hamiltonian. Since $j'$ in Eq. \[eq:defHamiltonianwithPara1\] is already of order $\delta t$ and only the three- and four-leg vertices have contributions of order $\mathcal{O}(\delta t)$, only uncertainty loop corrections have to be taken into account. We can therefore define the renormalized data-source vertex of the effective $\phi$-theory, $$\begin{aligned}
j_\# &=&
\includegraphics[width=0.23\fgwidth]{fg_c1.pdf}+
\includegraphics[width=0.8\fgwidth]{fg_c2.pdf}+\mathcal{O}(\delta t^2)\nonumber\\
&=& -\Lambda^{(1)} - \frac{1}{2}\, \Lambda^{(3)} [\cdot, D]\\
&=& -\Lambda^{(1)}_x - \frac{1}{2}\, \Lambda^{(3)}_{x y z} D_{y z},\nonumber\end{aligned}$$ which takes the dominant uncertainty-loop correction into account. We dropped the subscript at $D_0$ and use the Feynman rules provided in [@2009PhRvD..80j5005E]. The renormalized propagator up to linear order in $\delta t$ is $$\begin{aligned}
D_\# &=& \includegraphics[width=\fgwidth]{fg_a6.pdf}+\includegraphics[width=\fgwidth]{fg_b6.pdf}+\includegraphics[width=\fgwidth]{fg_a5.pdf}+\mathcal{O}(\delta t^2)\nonumber\\
&=& D - D\, \Lambda^{(2)} \, D - \frac{1}{2}\,D\, \Lambda^{(4)}[\cdot, D, \cdot]\, D\\
&=& D_{x y} - D_{x z} \, \Lambda^{(2)}_{z z'} \, D_{z' y} - \frac{1}{2}\, D_{x x'} \, \Lambda^{(4)}_{x' z z' y'}\,D_{z z'}\,D_{y' y}.\nonumber\end{aligned}$$ The inverse renormalized propagator up to first order is $$\begin{aligned}
D_\#^{-1} &=& D^{-1} + \Lambda^{(2)} + \frac{1}{2} \, \Lambda^{(4)} [\cdot, D, \cdot]\,.\end{aligned}$$
These coefficients now define a renormalized effective Hamiltonian, $H'_\#[\phi] = \frac{1}{2} \phi^\dagger D_\#^{-1} \phi - j_\#^\dagger \phi + H_{\#0}$, which belongs to a free theory, and is similar to $H'[\phi]$, in that it has the same mean and uncertainty dispersion by construction. Higher order uncertainty correlations differ certainly, due to the approximation of the renormalization step. In contrast to the original Hamiltonian $H_{p_0}[s]$ in Eq. \[eq:freeHamiltonian\], which was also free, the renormalized Hamiltonian has some amount of parameter uncertainty corrections included.
Now, the original field $s= m_0+\phi $ can be restored, leading to a free Hamiltonian with $$\begin{aligned}
D_{t+\delta t} &=& D_\#,\;\mbox{and} \\
j_{t+\delta t} &=& j_\# +D_\#^{-1} m_t,\nonumber\end{aligned}$$ where the subscript $t+\delta t$ indicates that the parameter uncertainty is increased by $\delta t$ from its original value of $t$. Since $\delta t$ can be made arbitrarily small, a system of differential equations can be derived, $$\begin{aligned}
\label{eq:Ddot}
\frac{dD_t}{dt} &=& \lim_{\delta t \rightarrow 0} \frac{D_{t+\delta t}-D_{t}}{\delta t}\nonumber\\
&=& - D_t\, \lambda^{(2)} \, D_t - \frac{1}{2}\,D_t\, \lambda^{(4)}[\cdot, D_t, \cdot]\, D_t
,\;\mbox{and} \nonumber\\
\frac{dj_t}{dt} &=& \lim_{\delta t \rightarrow 0} \frac{j_{t+\delta t}-j_{t}}{\delta t}\\
&=& -\lambda^{(1)} + \lambda^{(2)} D_t\, j_t - \frac{1}{2}\, \lambda^{(3)} [\cdot, D_t]\nonumber\\
&+& \frac{1}{2} \, \lambda^{(4)} [\cdot, D_t, D_t\, j_t ],\nonumber\end{aligned}$$ which form the uncertainty renormalization flow equations. The pseudo-time $t$ measures the accumulated dispersion of the resulting prior probability. These equations can be transformed into the more compact form $$\begin{aligned}
\frac{dD_t^{-1}}{dt} &=& \lambda^{(2)} + \frac{1}{2}\, \lambda^{(4)}[\cdot, D_t, \cdot]
,\;\mbox{and} \\
\frac{dm_t}{dt} &=& -D_t\,\lambda^{(1)} - \frac{1}{2}\, D_t\, \lambda^{(3)} [\cdot, D_t],\nonumber
\label{eq:mdot}\end{aligned}$$ where $m_t = D_t \, j_t$.
The renormalization equations so far are evolution equations for operators. If they should become ordinary partial differential equations, e.g. in our case in terms of spectral parameters, some sort of closure is required. This should ensure that the renormalized Hamiltonian gets its original structure, so that it is clear which terms are affected by the parameter uncertainty adding operation. Ideally, the change in the Hamiltonian can be mapped onto changes of effective parameter values.
After the repeated adding of small amounts of parameter uncertainty, the resulting effective parameter prior distribution can be expected to be a Gaussian, due to the central limit theorem of statistics, $$P(p) = {\mathcal{G}}(p-p_0, t).$$
Signal reconstruction with PURE {#sec:applyingPURE}
===============================
Lognormal spectral prior {#sec:lognormal prior}
------------------------
Now we want to apply the PURE scheme to our example problem from Sect \[sec:ssu\] of how to reconstruct a Gaussian signal with unknown covariance.
First, we have to express our spectral prior in a way that we can apply the PURE method developed in the previous section. For this we need some additive auxiliary random variables into which we can decompose our (unknown) spectral amplitudes. These variables should each have an unbiased distribution with zero mean according to Eq. \[eq:unbiased auxiliary priors\]. For the moment, we concentrate on a single spectral parameter $p_i$ and change to the parameter variable $\tau_i = \log p_i$, which can be split up into additive auxiliary variables: $\tau_i = \sum_j {\tau_{ij}}$. For convenience we assume $p_{ij} = e^{\tau_{ij}}$ to be distributed according to Eq. \[eq:p-prior\], with properly chosen parameters $\alpha_{ij}$, and $q_{ij}$, as detailed in the Appendix \[sec:central limit theorem\]. There, it is shown that $$P(\tau_i) \longrightarrow {\mathcal{G}}(\tau_i,t_i)$$ for the limit of an infinite number of auxiliary parameters, with a finite total uncertainty dispersion of $t_i = \langle \tau_{i}^2 \rangle_{(\tau_{i})} - \langle \tau_{i} \rangle_{(\tau_{i})}^2$, as expected from the central limit theorem of statistics. The resulting statistics for $p_i = e^{\tau_{i}}$ is therefore log-normal. If we take the limit $t_i\rightarrow \infty $ we obtain Jeffreys prior, which is flat on a logarithmic scale, and which conveniently permits us to compare the PURE filter to the others.
Uncertainty renormalization
---------------------------
In the following we assume that all spectral coefficients receive uncertainty with the same infinitesimal rate, so that the prior distributions in Eq. \[eq:p-prior\] are all the same and narrowly centered on $p_i=1$, which implies $\delta t_i = 1/(\alpha_i -1)= \delta t$ and $q_i=\alpha_i-3/2 = \delta t^{-1} -1/2$ (see Appendix \[sec:central limit theorem\]).
Expanding the Hamiltonian in Eq. \[eq:MAPmapHamilton\] around the reference map $m=D \, j$ recovers the original free Hamiltonian, shifted to $\phi = s -m$, and perturbed by some additional interaction terms $\Lambda^{(n)} = \delta t \,\lambda^{(n)}+ \mathcal{O}(\delta t^2)$ with $$\begin{aligned}
\lambda^{(1)} &=& \sum_i \frac{1}{2} \left(\varrho_i +1 - p^{-1} m^\dagger S_i^{-1} m\right) S_i^{-1} p_i^{-1} m , \nonumber \\
\lambda^{(2)} &=& \sum_i \frac{1}{2} \left( \varrho_i +1 - p_i^{-1} m^\dagger S_i^{-1} m\right) \,
S_i^{-1} p_i^{-1} \nonumber \\
&-& S_i^{-1} m m^\dagger S_i^{-1} p_i^{-2} , \nonumber\\
\lambda^{(3)} &=& \lambda^{(4)}[\cdot, \cdot,\cdot, m] , \\
\lambda^{(4)} &=& -3 \sum_i \,p_i^{-2}\, S_i^{-1} \otimes S_i^{-1} , \; \mbox{and}\nonumber\\
\lambda^{(n)} &=& 0 \; \mbox{for}\; n>4. \nonumber\end{aligned}$$ Here we have reinserted $p_i$ in order to have variables which capture the evolution of the renormalization flow dynamics. The renormalization flow equations are given by inserting the latter terms into two independent equations out of Eqs. \[eq:Ddot\] - \[eq:mdot\]: $$\begin{aligned}
\!\!\!\!\!\!\frac{dD^{-1}}{dt}\!\!\! &=& \!\!\!\sum_i \!\!
\left[ \frac{1}{2}\left((1+\varrho_i) p_i -\mathrm{Tr}[B_i]\right) S_i^{-1} \!-\! S_i^{-1} B_i \right]\!\! p_i^{-2}\!\!\!\!\!,\,\,\,\,\, \label{eq:D-1dot}\nonumber\\
\frac{dj}{dt} &=& - \sum_i p_i^{-2} (m^{\dagger} S_i^{-1} m) \, S_i^{-1} m
, \;\; \mathrm{with} \label{eq:djdt}\\
B_i &=& (m\, m^\dagger + D)\, S_i^{-1} \;\; \mathrm{and} \;\; m = D\, j .\nonumber\end{aligned}$$ This system of integro-differential equation represents the most accurate form of the PURE filter for this application. It is, however, in general quite expensive to implement numerically, since it requires to follow the evolution of matrices.
Projection onto spectral parameterization
-----------------------------------------
To simplify the PURE filter equations, we want to recast the system into the original from, which assumes $D^{-1} = (S_p^{-1}+ M)$ with $S^{-1}=\sum_i p_i^{-1}\, S_i^{-1}$. Thus evolution equations for the $p_i$s are needed. Since $\frac{d}{dt}D^{-1} = \sum_i S_i^{-1} \, \frac{d}{dt} p_i^{-1}+\frac{d}{dt}M$ contains the parameter evolution one has to specify how to split the evolution equation of the inverse propagator.
A natural way is to require all terms of the rhs of Eq. \[eq:D-1dot\], which are parallel to the inverse signal covariance bands, to contribute to their evolution, and the ones which are orthogonal, to contribute to the evolution of $M$. The part of a matrix $A$ parallel to $S_i$ is obtained by the projector $$\mathcal{P}_i \, A \equiv \frac{1}{\varrho_i}\mathrm{Tr}\left[A\, S_i \right] \, S_i^{-1}$$ and the orthogonal part by $(1-\mathcal{P}_i)\,A$. Splitting the evolution equation this way yields $$\begin{aligned}
\label{eq:dpidt}
\frac{dp_i}{dt} &=& \beta_i\, p_i, \;\;\mathrm{or}\;\; \frac{d\tau_i}{dt} = \beta_i, \;\;\mathrm{and} \\
\frac{dM}{dt} &=& \sum_i \, p_i^{-2} S_i^{-1} \left(
\frac{1}{\varrho_i}\mathrm{Tr}\left[B_i \right] - B_i \right), \;\;\mathrm{with} \nonumber\\
\beta_i &=& \left(\frac{1}{2}+\frac{1}{\varrho_i}\right) \mathrm{Tr} \left[ B_i \right] p_i^{-1} - \frac{1+\varrho_i}{2}.\nonumber\end{aligned}$$ With this, the fastest evolution is assigned to the signal strength, whereas the inverse noise term evolves on much longer time-scales for $\varrho_i \gg 1$. Actually, $M$ evolves only in directions orthogonal to all $S_i$, since $$\frac{d}{dt} (\mathcal{P}_i \,M) = 0,$$ meaning that the power within the spectral bands of $M$ gets only reshuffled, but is conserved. This implies that the evolution of $M$ interferes very little with the spectral evolution, since all changes to $M$ happen in directions which are projected out for $S_p$. The reverse is not true, since $M$ couples to the value of $p$. For an accurate reconstruction the evolution of $M$ needs to be followed, since it determines $D$ and thereby $m=D\, j$. However, we focus now only on the signal spectrum evolution and ignore the slow and perpendicular $M$ evolution.
The evolution equation for $p$ and $j$ have to be solved simultaneously as a function of $t$ up to the spectral uncertainty $t_\mathrm{max} = \langle (\log p - \log p_0)^2 \rangle_{(p)}$ of the original problem. This version of the PURE filter for spectrally uncertain Gaussian random signals with a lognormal spectral prior is projected onto our spectral parametrization, but not yet onto our generic filter formula.
Jeffreys prior
--------------
Let us see if there is a stationary asymptotic for the limit of infinite spectral uncertainty. The resulting filter for $t \rightarrow \infty$ (which implies a Jeffreys prior) seems to be trivial, since $j\rightarrow 0$ and therefore $m_p\rightarrow 0$ in this limit.
This can actually be understood intuitively. On the logarithmic scale $\tau_i = \log p_i$ Jeffreys prior becomes flat in $\tau_i$. Thus an arbitrary negative $\tau_i$ (and therefore infinitesimally small $p_i$) is as probable a priori as an arbitrary large $\tau_i$ (and therefore basically infinite large $p_i$). However, the likelihood $P(d|p) = \int \!\mathcal{D}s\, P(d,s|p)$ discriminates clearly between those cases.
For $p\approx 0$ we expect $s\approx 0$, which means that the data must be purely noise, which has a low, but finite likelihood. This likelihood does not decrease significantly if $\tau \rightarrow -\infty$ and $p$ and $s$ become exactly zero, since the amount of noise stays constant. It has to be identical to the data in this case.
However, for $p_i\rightarrow + \infty$, while the data stays finite, either the more and more unlikely case of a low signal realization for an increasing variance must have happend, or the more and more unlikely case of a noise canceling the large amplitude signal must have happend.
Thus, the a priori as probable case $\tau_i \rightarrow +\infty$ is heavily penalized by the likelihood with respect to the case $\tau_i \rightarrow -\infty$. Since the PURE filter aims to estimate the mean signal averaged over all $\tau_i $, this imbalance of the likelihood factor lets the regime $\tau_i \rightarrow -\infty$ dominate this average leading to $\langle s \rangle_{(s|d)} =0$.
Projection onto generic filter formula {#sec:PUREprojected}
--------------------------------------
We can artificially remove the trivial solution of the PURE filter in case of Jeffreys prior by imposing $dj/dt = 0$ instead of Eq. \[eq:djdt\]. This should be understood as looking for a stationary point of the $p$-evolution alone. Thus, we are asking for the unique spectrum, which taken as a sharp prior would remain unchanged if a small amount of spectral uncertainty is added. This fix point is given by $\beta_i=0$ and therefore $$\label{eq:renormalisedspec}
p_i = \frac{1+\frac{2}{\varrho_i}}{\varrho_i+1} \mathrm{Tr}\left[ B_i \right].$$
Although we have derived this filter only for Jeffreys prior, it is quite plausible to assume that the general spectrum formula, Eq. \[eq:characteristic equation\], with $(\delta_i,\varepsilon_i)=(1,-0.5/(1+2/\varrho_i))$ also holds for $(\alpha_i, q_i)\neq (1,0)$. We leave a formal proof of this for future work. In this form the PURE filter for a Jeffreys prior is projected into the $\delta \epsilon$-plane of the representation Eq. \[eq:characteristic equation\] for the MAP filters, which is displayed in Fig. \[fig:diagram\].
Perception threshold {#sec:perception threshold}
====================
Critical perception
-------------------
{width="\columnwidth"} {width="\columnwidth"}
In case of Jeffreys prior ($q_i =0$, $\alpha_i=1$, and $t = \infty$), the spectral coefficients $p_i$ used by some of our filters are only non-zero for spectral bands with a data variance above some threshold. Bands with lower band power are fully suppressed in the reconstructed map, since the Wiener filter removes completely any fluctuations in bands for which the assumed signal covariance is zero. Thus, a perception threshold appears for filters within a certain critical line in the $\delta\varepsilon$-plane, which we calculate in the following.
Filter without perception threshold have to exhibit $p_i>0$, even when the data has no power at all. Thus we investigate the extreme case $d=0$ by inserting $m_p=0$ into Eq. \[eq:characteristic equation\] and find after some algebra $$1 + \frac{2\,\varepsilon_i}{\varrho_i} = \delta_i \, \underbrace{\frac{1}{\varrho_i} \mathrm{Tr}\left( (1+Q_p)^{-1}I_i \right)}_{\le 1},$$ with $I_i = S^{-1}_i S_i$ the unit matrix restricted to the $i$-th band. Since the marked expression on the rhs is one only for vanishing $p$, we find the critical line to be given by $$\label{eq:critical line}
\delta_i^\mathrm{crit} = 1 + \frac{2\,\varepsilon_i}{\varrho_i} .$$ Filters with $\delta_i> \delta_i^\mathrm{crit}$ do not exhibit a perception threshold, since even for $d=0$ all $p_i >0$. Filters with $\delta_i < \delta_i^\mathrm{crit}$ exhibit a perception threshold. We note that a non-Jeffreys prior with $\alpha_i>1$ but still $q_i=0$ can also be included into this classification scheme, by just adding $\alpha_i -1$ to $\varepsilon_i$. Filters with $q_i>0$ obviously do not exhibit a perception threshold, since even in the limit of vanishing data and vanishing propagator Eq. \[eq:characteristic equation\] has the positive solution $p_i = q_i/(\gamma_i+{\varepsilon}_i)$.
The point $(\delta_i, \varepsilon_i)=(1,0)$ lies on top of the critical line, as can be seen in Fig. \[fig:diagram\], and therefore the term *critical filter* seems to be appropriate for it.
Translation invariant data model
--------------------------------
Here, we calculate the perception thresholds of our filters in the case of a translationally invariant data model. Although a general criterion for the position of the threshold in data space can easily be worked out, it is more instructive to investigate a simplified case. We assume the signal and noise to live in the same spatial space, and their covariances to be fully characterized by power spectra in Fourier space, $$\begin{aligned}
S(k,q) &=& (2\pi)^n\, \delta(k-q) \,P_{S}(k),\nonumber\\
N(k,q) &=& (2\pi)^n\, \delta(k-q) \,P_{N}(k),\end{aligned}$$ with $P_s(k) = \langle |s(k)|^2 \rangle /V$, and $P_N(k) = \langle |n(k)|^2
\rangle /V$, where $V$ is the observed volume. We define spectral bands with band spectra $P_{S_i}(k)$, so that $P_S(k) = \sum_i p_i \, P_{S_i}(k)$. We assume further that the signal processing can be completely described by a convolution with an instrumental beam, $$d(x) = \int
dy\, R(x-y) \, s(y) + n(x),$$ where the response-convolution kernel has a Fourier power spectrum $P_R(k) = |R(k)|^2$ (no factor $1/V$).
In this case $D$ can be fully described by a power spectrum, $$\begin{aligned}
D(k,q) &=& (2\pi)^n\, \delta(k-q) \,P_{D}(k),\end{aligned}$$ with $P_D(k) = (P_S^{-1}(k) + P_R(k)\, P_N^{-1}(k))^{-1}$ and all spectral bands decouple.
Approximative treatment
-----------------------
The generic filter equations, Eqs. \[eq:characteristic equation\], now separate into independent equations for the individual $p_i$. Let us look first at the trace-terms in this equation, which now read $$\begin{aligned}
\mathrm{Tr}\left[ m_p\,m_p^\dagger S_i^{-1}\right] &=& V \, \int _i \frac{dk}{(2\,\pi)^n} \, \frac{P_d(k) \, { p_i}^2\,P_{Q_i}(k)}{P_N(k) \,(1 + p_i\, P_{Q_i}(k))^2},\nonumber\\
\mathrm{Tr}\left[ D_p S_i^{-1}\right] &=& V \, \int _i \frac{dk}{(2\,\pi)^n} \, \frac{p_i}{1 + p_i\, P_{Q_i}(k)}.\end{aligned}$$ We define the data power $P_d(k) = |d(k)|^2/V$ and the $i$-band fidelity power $P_{Q_i}(k) = (P_{S_i} \, P_R/P_N)(k)$. We further use the approximation $ V\, \int _i {dk}/{(2\,\pi)^n} f(k) \approx \varrho_i \, f(k_i)$, which assumes that $f(k)$, a combination of spectra, does not vary significantly over the narrow spectral band $i$. This permits us to write the generic filter formula, Eq. \[eq:characteristic equation\], which determines the filter band coefficients $p_i$ as an algebraic and dimensionless expression: $$\label{eq:characteristic}
x = \frac{1+y}{y^2}\, \left[(t\,y -u)\,(1+y) - \delta\, y \right].$$ Here, we have dropped the index $i$ and defined the noise-normalized data power $x = P_d(k_i)/P_N(k_i)$ and the measurement fidelity $y = p_i \, P_{Q_i}(k_i)$. The numerical coefficients are $$\begin{aligned}
t &=& \frac{2}{\varrho_i}\,(\gamma_i + \varepsilon_i) = 1 + \frac{2}{\varrho_i}\,(\alpha_i -1 + \varepsilon_i),\nonumber\\
u &=& \frac{2}{\varrho_i}\,q_i\,P_{Q_i}(k_i),\;\mbox{and}\; \delta = \delta_i.\end{aligned}$$ In case of Jeffreys prior, these simplify to $u=0$ and $t=1 + 2\,\varepsilon_i/\varrho_i$ and the recast generic filter formula Eq. \[eq:characteristic\] has the following solutions $$\begin{aligned}
y &=& 0,\;\mbox{and}\\
y &=& \frac{x-x_0}{2\,t} \pm \sqrt{\left( \frac{x-x_0}{2\,t}\right)^2 - 1 + \frac{\delta}{t}},
\;\mbox{with}
\nonumber\\
x_0 &=& 2\,t-\delta = 2+4\,\varepsilon_i/\varrho_i -\delta_i.\nonumber\end{aligned}$$ Although there might be up to three simultaneous real solutions for a given $x$, always the largest value should be taken. This is in line with our decision to ignore the trivial solution and the expectation that the assumed spectral amplitude $y$ should increase with increasing data power $x$, an not decrease as the lower branch of the square root does. The largest solution is non-zero only if $$\begin{aligned}
x &\ge& x_\mathrm{th} =
\left\{
\begin{array}{ll}
0 & x_0 < 1, \\
x_0 + 2 \, \sqrt{t\,(t-\delta)} & x_0 \ge 1.
\end{array}
\right.\end{aligned}$$ The assumed dimensionless signal power $y$ is shown in Fig. \[fig:perceptionTh\] as a function of the dimensionless data variance $x$. Asymptotically, for $x\gg x_0$, we have a linear increase of assumed signal strength and data variance $y(x) = x-x_0$. The critical filter is special in that this relation holds exactly for the full region $x\ge x_\mathrm{th} = x_0$. All of the MAP estimators in this work have $x_\mathrm{th}>x_0$ and exhibit a jump from $y=0$ to $y=\sqrt{1-\delta/t}$ at $x=x_\mathrm{th}$, followed by an approach to the linear asymptotic. The threshold approaches $x_\mathrm{th} \rightarrow 1$ from above for $\varrho_i \rightarrow \infty $ for the MAP spectrum filter, however, it is always $x_\mathrm{th}=4$ for the classical filter, independent of the spectral bin size $\varrho_i$.
The PURE filter as given in Sec. \[sec:PUREprojected\] is the only one of our sample, which has no perception threshold since $y(x)$ is positive for all $x$. Even in case the data exhibits negligible variance $x\ll 1$, the filter still uses a non-negligible spectral amplitude, since $y(0) \approx 1/(\varrho_i+1)$. This might surprise, since the implied assumption of a significant signal variance is obviously not supported by the data. However, the renormalized filter aims for an optimal reconstruction, and not for an accurate power spectrum measurement, and letting some fraction of some data band with apparently low noise realization pass (remember $x\ll 1$) does not spoil this.
The combination of signal measurement and filtering can be regarded as a single response operator $R'$, with $R'\, s= \langle m \rangle_{(d|s)} = F_{p}\, R\, s = D\, M \, s$, which decomposes into separate pass-through factors for the individual bands, $R_i' = P_D(k_i) \, P_R(k_i)\,P_N^{-1}(k_i) = y/(1+y)$. This is also shown in Fig. \[fig:perceptionTh\].
Consequences for cosmological practice
--------------------------------------
The critical filter estimates the power spectrum of a Wiener map, which is (iteratively) filtered with this very same spectrum (until convergence), while correcting the spectra for an estimate of the filtered-out power during each iteration. Similar procedures are widely used in cosmology under the names Karhunen-Loève (KL, [@1947KarhunenK; @1978LoeveM; @2002MNRAS.335..887T]) and Feldman-Kaiser-Peacock (FKP, [@1994ApJ...426...23F]) estimators to measure power spectra of galaxy cataloges. As the critical filter, these should therefore also exhibit a perception threshold for spectral modes with a data variance not significantly exceeding the noise variance. Therefore, one would expect that cosmological spectra obtained by these estimator should exhibit modes with zero power. However, in applications of these scheme in the cosmological literature, the iterations of filtering and spectral measurements are usually not repeated until convergence.[^4] Thus the knowledge system keeps some memory of the initial power spectrum choice, which can be regarded as a hidden prior regularizing the spectrum and preventing the perception threshold that a correctly implemented KL or FKP estimator would exhibit (see also [@2002MNRAS.335..887T] for a discussion on this).
{width="\textwidth"}
{width="100.00000%"}
Comparison of the map making algorithms {#sec:compare}
=======================================
The test case
-------------
We want to examine the filter performances with an instructive test case. In case the spectral uncertainty is small, all filters in this work can be expected to provide comparable results since they Wiener filter the data with basically the prior spectrum with small differences. Thus, in order to see the differences in performance more clearly, we again adopt Jeffreys prior for our spectral parameters ($\alpha_i=1$ and $q_i=0$, well, for numerical reasons $q_i = 0.01$). A spectrum, which naturally implements this distribution is the famous $1/f$-spectrum, which has equal power per decade in frequency space. To have a finite zero mode and signal variance, we adopt $$P_S(k)= P_0 \, (1+(k/k_0)^2)^{-\frac{1}{2}},$$ with $P_0= 5$ and $k_0= 2$. We further assume some white noise with $P_N(k)=\sigma_n^2 = 0.1$.
In case the response would be constant or a convolution, the spectral inference problems would be separable in Fourier space, as we have shown in the last section. In order to have a more complex problem, with coupling between the different unknown spectral parameter, we introduce a non-homogeneous observational signal response $R$ over the 257 pixel of our signal space, as displayed in Fig. \[fig:data1\] together with a test data set. We split the Fourier space in 64 disjunct spectral bands, with $\rho_i=4$ for all but the lowest band, which has $\rho_0=5$, since it also contains the zero mode. Since we are dealing with a real-number signal in a discrete space, we have to take care of the negative frequency spectrum being identical to the positive ones, and therefore our bands are split into identical positive and negative parts, except the zero-band, which is continuous.
The signal reconstructions of the five filters are also shown in Fig. \[fig:data1\], and the used spectra in Fig. \[fig:spec1\]. The spectra are roughly ordered the way we expect them to be following our perception threshold analysis in Sec. \[sec:perception threshold\]. However, there is the suprising modification that even the renormalized filter seems to suffer from a slight perception threshold, since many of the higher $k$-vector bands with lower signal to noise ratio are nearly free of power. A more informative prior for the power distribution would cure this, but this would limit the generality of our filter. So we should look for other yet unexploited prior information.
Spectral smoothness regularization
----------------------------------
The $1/f$ signal spectrum adopted in our example is a member of the large class of smooth spectra, which do not exhibit spectral lines, jumps and edges. Spectral smoothness information can easily be incorporated into the framework. Since we do not want to specify a specific smoothness length scale, we require the double logarithmic derivative of the spectrum to be of limited variance. This can be done by introducing an additional prior energy for non-smoothness $$\begin{aligned}
E_\mathrm{reg} &=&
\frac{1}{2\, \sigma_P^2} \, \int d\log k \,\left( \frac{d\log P_S(k)}{d\log k}\right)^2\\
&\approx&
\frac{1}{4\, \sigma_P^2} \, \sum_i \frac{k_i+k_{i-1}}{k_{i}-k_{i-1}}
(\tau_i - \tau_{i-1})^2 \equiv \frac{1}{2} \,\tau^\dagger T\, \tau. \nonumber\end{aligned}$$ Here we have (re-)introduced the logarithm of the power spectrum parameters $\tau_i = \log p_i$, have discretized the integral and derivatives, and collected all coefficients in a matrix $T$. The quadratic form in $\tau$ in the last line shows that this is actually a log-normal prior contribution, which can be combined with the log-normal prior appearing in the renormalization calculation. Instead of repeating that calculation with now interdependent parameters, we just use our physical intuition to obtain the regularized filter equation for the filter spectrum, and leave any proof or improvement for future work.
The unregularized evolution equation for $\tau$, Eq. \[eq:dpidt\], can just be equipped with a regularizing force $-d E_\mathrm{reg}/d\tau$: $$\frac{d\tau}{dt} = \beta(\tau) - T\,\tau.$$ The regularized Jeffreys prior case is then given by the fix point specified by $\beta(\tau) = T\,\tau$ and reached asymptotically for $t \rightarrow\infty$. The matrix $T$ couples the neighboring bands together and thereby produces much smoother filter spectra without the gaps the other filter spectra exhibit, as can be seen in Fig. \[fig:spec1\], where the regularized filter spectrum for $\sigma_P = 2$ is shown.
Full PURE filter
----------------
Spectral smoothness can not always be assumed, and therefore we should also think of other ways to improve the filter fidelity. One way is to be more precise in the PURE filter derivation. The largest approximation made was probably the neglection of the $dj/dt$ term, which for infinite spectral uncertainty, $t\rightarrow \infty$, leads to a trivial solution of $m=0$. If we want to include this term, we can therefore only apply it for a finite amount of uncertainty, say up to $t=1$. This implies that the initial starting point of the spectral renormalization flow would influence our result. In case of a concrete application, this might be very desirable, since there a good initial guess for the spectrum might be available.
In our more abstract discussion here, we want to avoid such choices, also in order to be sure not to have included too much spectral prior knowledge into the filter preventing a fair comparison to the others. Therefore we start the renormalisation flow including the $dj/dt$ term with the fix point spectrum of the approximated PURE filter (without this term) and stop it at $t=1$. This way we have both, independence of any prior spectrum and inclusion of non-Wiener corrections. The resulting filter seems to be partly cured from too generous predictions in regions without data while the results in better determined regions are practically unchanged, as can be seen in Fig. \[fig:data1\].
This can be understood in the following way. We have roughly $dj/dt \propto -S^{-1}m$, since $m^\dagger S_i^{-1}m \approx \varrho_i$ for most modes. If there is power at a poorly observed location in the map $m$ on a level comparable to the well observed ones, $j$ evolves in both regions with similar speed. However, the effect of this evolution to the map $m = D\,j$ is larger in regions with larger uncertainties, since $D$ is larger there. Thus, any power spilled into observational gaps is removed faster than power in well observed regions. The full PURE filter seems to be aware of the lower certainty of the former.
Statistical comparison
----------------------
A statistical assessment of the different filters is also shown in Fig. \[fig:data1\]. There it is apparent that the filters derived from MAP principles are worse than the PURE filter, with only the critical filter being comparable in performance. The underestimation of the power spectra due to the perception threshold obviously reduces the fidelity of those filters.
The spectral smoothness regularized, renormalized filter clearly outperforms the unregularized ones, probably due to the lack of spectral gaps. Its performance is comparable to that of the Wiener filter using the correct signal power spectrum $P_S(k)$. The error variance for the latter filter is also displayed in Fig. \[fig:data1\] in comparison to its theoretical value given by the Wiener variance $D_{xx}$ (see Eq. \[eq:mapuncertainty\]). Finally, also the full PURE filter as described in the last section is shown. Its fidelity is comparable to the spectrally regularized one, without that any spectral smoothness assumptions had to be made. Of course, such assumptions could also be included into this filter.
Conclusions {#sec:conclusion}
===========
We showed how to deal with parameter uncertainties in information field theory by introducing an effective Hamiltonian over the joint space of the signal field and the parameters. In order to go beyond a classical, or Maximum a Posteriori treatment of the problem we presented an uncertainty renormalization scheme, in which the parameter uncertainty is successively fed into the knowledge system. The resulting parameter uncertainty renormalized estimation, PURE, can be used to tackle many signal inference problems including calibration uncertainties.
It seems that the PURE provides a Gaussian approximation to the full posterior probability function, which has maximal cross-information with it, as thermodynamic considerations in [@2010PhRvE..82e1112E] have shown.
To demonstrate the advantage of PURE with a concrete example, we investigated the general problem of inferring a Gaussian signal with unknown spectrum from noisy data, which follows from a linear, but inhomogeneous data model. Following the parameter uncertainty renormalization and various classical approaches, four classical and one renormalized filter were derived. All filters can be regarded as Wiener filter operations with assumed signal spectra to be calculated from the data by a single recipe, Eq. \[eq:characteristic equation\], with just differences in two of its numerical coefficients.
The computational complexity of all those filters is therefore very similar and should not be a reason to prefer one over the other. Their signal fidelity, however, differs significantly. In case a non-informative Jeffreys prior is adopted for the spectral amplitudes, all classical filters suffer from a perception threshold. Spectral bands, which do not show more data power than the threshold, are completely filtered out. Three out of the four classical filters investigated have a perception threshold which requires data variance significantly above the noise level. The fourth one, the critical filter, lives on the critical line between filters with and without perception threshold in our space of filter parameters. The critical filter tries to match the correct spectrum on a logarithmic power scale. Its perception starts therefore for modes with a variance just above the noise level, as soon the data indicates some potential signal power. It has recently been applied successfully to the reconstruction of an all sky map of the galactic Faraday depth [@2010arXiv1008.1246O].
The critical filter coresponds in general to the Karhunen-Loève method [@1947KarhunenK; @1978LoeveM; @2002MNRAS.335..887T], and for an infinite window function to the FKP method [@1994ApJ...426...23F] frequently used in cosmology to estimate power spectra of galaxy catalogs. It seems that the perception threshold of this method is often ‘cured’ in applications by a truncation of the full iterative scheme. This implies the presence of a hidden spectrum prior in such estimates.
The PURE filter precepts also for spectral bands, which by chance exhibit less power than expected for the noise alone. This might appear as being too generous – the signal spectrum adopted by this filter is typically larger than the correct and therefore optimal, but unknown one. However, the PURE filter exhibits the largest fidelity of our filter sample, even slightly better than that of the critical filter. The reason lies in the asymmetric fidelity loss for under- and overestimating the true signal spectrum. Spectrum underestimation is much worse than overestimation in terms of signal reconstruction accuracy. The renormalized filter knows about this and adds a safety margin to any spectral band. This margin is inversely proportional to the number of data degrees of freedom informing about the signal spectrum in this band. Thus, in the limit of a large number of data points determining the band spectrum the renormalized filter approaches the critical one, but always from the perception threshold free side.
Although the classical filter resulted from maximizing the exact effective, parameter marginalized Hamiltonian (Eq. \[eq:MAPmapHamilton\]), it performs much worse than the critical and PURE filters. Thus, this is an example where the MAP principle, or equivalently a tree-level IFT calculation, provides a poorly performing algorithm, and uncertainty loop corrections as explicitely included in the PURE filter or even the critical filter are essential.
The PURE filter, as well as the others, can be further improved by adding any additional spectral information. One way is to use informative priors on the spectral behavior, which instantaneously cure the perception threshold problem. However, even in case no information on the location of the spectrum is available, information about its smoothness as a function of the Fourier space coordinate may be exploited. We show that the performance of the PURE filter with spectral smoothness prior approaches that of the optimal Wiener filter for known signal power spectrum.
Since the computational complexity of the renormalized filter is identical to the critical one already used in cosmology, there exists no reason not to use it for Wiener filtering of signals with unknown spectra. One only has to keep in mind that the internally used spectrum of the filter is not the best estimate of the signal spectrum, but an overestimate. The critical spectrum provides such an estimate, using the posterior maximum for the logarithm of the spectral amplitudes.
The full PURE filter, which contains non-Wiener filter corrections and requires the more expensive evaluation of the renormalization flow equation, performs best among all spectrally unregularized filters. Spectral smoothness information can also be incorporated into it if available.
To conclude, the PURE scheme to construct optimized filters presented in this work is very general and should also be applicable to the problems of inference with uncertainties in the instrument response, the typical calibration problem, and for measurements without known noise level. A better understanding of the implications and assumptions of the commonly used process of self-calibration should be feasible, and possibly also improvements thereof. The pseudo-time parameter appearing in the renormalization flow, the amount of uncertainty or parameter dispersion fed into the knowledge system, may be connected under certain circumstances to real physical time. For measurement devices with drifting calibration or noise parameters, and also for signals with a slow, but unknown time evolution of their signal spectra, the parameter uncertainty renormalization equation offers a natural possibility to model this. Once the amount of uncertainty dispersion per physical time is fixed, the equation permits to continuously update the unknown parameters by combining past and novel information in an optimal, and controlled way. The PURE approach may thereby make contributions to the technologically important field of optimal control and time dependent instrument calibration.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge helpful discussions with and comments on the manuscript by Jens Jasche, Henrik Junklewitz, Cornelius Weig, Marco Selig, Niels Oppermann, Gerhard B[ö]{}rner, Benjamin Wandelt, and three anonymous referees.
Signal covariance likelihood {#sec:signal covariance like}
============================
In order to find the posterior of the signal covariance, we have to calculate $P(p)\,Z_p/Z$. We show below that the evidence $Z_p[0]$ for any parameter $p$ of the free Hamiltonian, given by Eq. \[eq:ZdfreeTheory\], is $$\label{eq:dataLikelihooparam}
Z_p = P(d|p) = {\mathcal{G}}(d,R\,S_p\,R^\dagger + N ).$$ This formula can also intuitively be read as the data likelihood given $p$, since it compares the power in the data to their expected fluctuations level $\langle d\,d^\dagger \rangle_{(d,s|p)} = R\,S_p\,R^\dagger + N$. It can therefore be used for a Bayesian estimate of any model-parameter of the free theory, not only for spectral parameters as in this work.
Proofs for Eq. \[eq:dataLikelihooparam\] can be found in [@2008MNRAS.389..497K; @2008MNRAS.391.1315F]. However, these proofs rely on either on the very special assumption fo $R$ being invertible [@2008MNRAS.389..497K] or on a Taylor expansion of the logarithm of a marix [@2008MNRAS.391.1315F], which has actually a limited convergence radius and therefore is not sufficient for a general proof. A proof without such limitations goes as follows: First, we concentrate only on the dependence of $P(d|p)$ on the data $d$, $$\begin{aligned}
Z_p &=& P(d|p) = \int \!\mathcal{D}s\, P(d,s|p)\nonumber\\
&\propto & \!\!\! \int \!\mathcal{D}s \exp\!\left(\! -\frac{1}{2}\! \left( s^\dagger S_p^{-1} s + (d-R\,s)^{\dagger} N^{-1} \, (d-R\, s)\right)\! \right) \nonumber\\
&\propto & \exp \left(- \frac{1}{2} \left(d^\dagger (N^{-1} - N^{-1}R\, D_p R^\dagger N^{-1}) \, d \right) \right)
\nonumber\\
& \propto&
\exp \left(- \frac{1}{2} d^\dagger (R\,S_p\,R^\dagger + N )^{-1} d\right)\nonumber\\
& \propto& {\mathcal{G}}(d,R\,S_p\,R^\dagger + N ).\nonumber\end{aligned}$$ Here, we used $D_p = (S_p^{-1} + M)^{-1}$ with $M = R^\dagger N^{-1} R$. The second last step relied on $R\,S_p\,R^\dagger + N$ being the inverse of $N^{-1} - N^{-1}R\, D_p R^\dagger N^{-1}$: $$\begin{aligned}
(R\,S_p\,R^\dagger + N)\, (N^{-1} - N^{-1}R\, D_p R^\dagger N^{-1}) &=& \nonumber\\
R\,(S_p - S_p\,M\, D_p - D_p) R^\dagger N^{-1} + 1 &=&\nonumber\\
R\,(S_p - S_p\,(D_p^{-1} -S_p^{-1})\, D_p - D_p) R^\dagger N^{-1} + 1 \nonumber
&=&1.\end{aligned}$$ Second, we have to show that $Z_p$ has the same normalization the Gaussian in Eq. \[eq:dataLikelihooparam\] has. This is most easily seen by $$\begin{aligned}
\int \! \mathcal{D}d\, Z_p &=& \int \!\mathcal{D}d \int \!\mathcal{D}s\, P(d,s|p)\nonumber\\
&=&\int \!\mathcal{D}s\, P(s|p) \int \!\mathcal{D}d \, P(d|s) \nonumber\\
&=& \int \!\mathcal{D}s\, {\mathcal{G}}(s, S_p) \int \!\mathcal{D}n \, {\mathcal{G}}(n, N)= 1,\nonumber\end{aligned}$$ where in the last line we replaced the data space integration variable $d $ by a linear shift with the noise variable $n = d- R\, s$ and used the fact that Gaussians are normalized to unity. Thus, Eq. \[eq:dataLikelihooparam\] is proven.
Derivation of the Gaussian prior {#sec:central limit theorem}
================================
Here we show how the different auxiliary variables $\tau_{ij}$ combine into a normal distribution for $\tau_i = \sum_j \tau_{ij}$, as was assumed in Sec. \[sec:lognormal prior\]. We drop in the following the index $i$, which labels the signal bands. Since we assume $e^{\tau_{j}}$ to be distributed according to Eq. \[eq:p-prior\], we have $$P(\tau_j) = \frac{\exp \left[ {-(\alpha -1)\, (\tau_j -\log q) - q\, e^{-\tau_j}} \right] }{\Gamma[\alpha -1]}$$ The non-bias condition, Eq. \[eq:unbiased auxiliary priors\], translates into $$\langle \tau_j \rangle_{(\tau_j)} = \log q - \psi_0(\alpha-1) = 0,$$ with $ \psi_n(z) $ being the Polygamma function. This condition fixes $q(\alpha) = e^{ \psi_0(\alpha-1) }$, which for large values of $\alpha$, and thereby for well localized auxiliary parameters, is asymptotically $q = \alpha - \frac{3}{2}$. The dispersion of the auxiliary variables is $$\delta t = \langle \tau_j^2 \rangle_{(\tau_j)} = \psi_1(\alpha-1),$$ which asymptotically is $\delta t=1/(\alpha -1)$ for large $\alpha$.
Now, we can work out the total prior resulting from the combination of $N =t/\delta t$ auxiliary variables, where $t$ is the uncertainty level of the prior, and $\delta t$ that of the individual variables: $$\begin{aligned}
P(\tau) \!\!\!\! \!\!&=& \!\! \!\! \!\!\left( \prod_{j=1}^N \int\! d\tau_j \, P(\tau_j)\right) \, \delta(\tau - \sum_{j=1}^N \tau_j)\nonumber\\
\!\! \!\! \!\!&=& \!\! \!\!\!\! \int\! \frac{dk}{2\pi}\,\left( \prod_{j=1}^N \int\! d\tau_j \, P(\tau_j)\right) \, e^{-i\,k\,(\tau- \sum_{j=1}^N \tau_j)}\nonumber\\
\!\! \!\! \!\!&=& \!\!\!\! \!\! \int\! \frac{dk}{2\pi} \left( \int\! \frac{d\tau_j \,q^{\delta t^{-1}}\,\exp \left[ {-(\delta t^{-1} -i\, k)\, \tau_j - q\, e^{-\tau_j}} \right]
}{\Gamma[\delta t^{-1}]} \right)^N \!\! \!\! \!\!\nonumber\\
&\times & e^{-i\,k\,\tau}\nonumber\\
\!\! \!\!\!\!&=& \!\! \!\! \!\! \int\! \frac{dk}{2\pi}\,\left( \frac{\Gamma[\delta t^{-1} - i\, k]}{\Gamma[\delta t^{-1}]}\, q^{i\,k}\right)^N e^{-i\,k\,\tau}\nonumber\\
\!\! \!\!\!\!&=& \!\! \!\! \!\!\int\! \frac{dk}{2\pi}\,\exp\left[ {-i\,k\,\tau} + N\,\log \left(\frac{\Gamma[\delta t^{-1} - i\, k]}{\Gamma[\delta t^{-1}]}\right) \right. \nonumber \\
&&\,\, \,\, \,\,\,\, \,\,\,\, \,\, \,\,\,\, \,\,\,\,\,\,\left.
-i\,k\,N\,\psi_0(\alpha-1)
\begin{array}{c}
\\ \\
\end{array}\!\!\!\!
\right] \nonumber\\
\!\! \!\! \!\!&=& \!\! \!\! \!\!\int\! \frac{dk}{2\pi}\,\exp\left[ {-i\,k\,\tau} - \frac{t}{2}\,k^2 + \mathcal{O}(\delta t\,k)\,k^2 \right] \, \nonumber\\
\!\! \!\!&\longrightarrow & {\mathcal{G}}(\tau,t)\;\mbox{for}\; \delta t\rightarrow 0,
$$ as also expected from the central limit theorem of statistics. Thus, the resulting distribution for the $p_i$ parameter is log-normal.
[^1]: Since this would result in an improperly normalized prior, we understand this as $\alpha_i= 1+ \epsilon $, $q_i = \epsilon$, and $\lim_{\epsilon \rightarrow 0}$ at the end of the calculation. We note, that this limit might not exist, or that it provides trivial results. I.e. we will find in Sec. \[sec:lognormal prior\] that in this limit the signal reconstructed with the full field theory turns out to be zero and the data is assumed to be purely made of noise. Thus the improper Jeffreys prior is actually inappropriate for the full problem, although interesting.
[^2]: Using the abbreviation $M = R^\dagger N^{-1} R$ we write $\langle m_p \, m_p^\dagger \rangle_{(d,s|p)} =
D_p\, \langle j \, j^\dagger \rangle_{(d,s|p)} \, D_p =
D_p R^\dagger N^{-1} ( R\, S_p R + N) \, N^{-1} R \, D_p =
D_p (M\, S_p M + M) \, D_p =
D_p M \, (1+ S_p M) \, (1+ S_p M)^{-1} S_p =
D_p M \,S_p =
S_p (1 + M\, S_p)^{-1} (1+ M\, S_p -1) =
S_p - D_p$.
[^3]: The term $|S_i|$ has to be read as the determinant within the non-zero subspace of $S_i$.
[^4]: Some random examples: @2002MNRAS.335..887T, @2007ApJ...657..645P as well as @1994ApJ...426...23F use a fixed and constant spectrum in the optimal data weighting step of the KL and FKP schemes, and do not iterate at all.
|
---
abstract: 'In this note we formulate recent stability results for Hardy inequalities in the language of Folland and Stein’s homogeneous groups. Consequently, we obtain remainder estimates for Rellich type inequalities on homogeneous groups. Main differences from the Euclidean results are that the obtained stability estimates hold for any homogeneous quasi-norm.'
address:
- '$^{1}$ Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom'
- '$^{2}$ Institute of Mathematics and Mathematical Modelling, 125 Pushkin str., 050010 Almaty, Kazakhstan.'
author:
- 'Michael Ruzhansky,$^1$ Durvudkhan Suragan$^2$$^{*}$'
date: 'Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. $^{*}$Corresponding author'
title: A note on stability of Hardy inequalities
---
[^1]
Introduction {#SEC:intro}
============
Recall the $L^{p}$-Hardy inequality $$\label{Lp_Hardy}
\int_{\mathbb{R}^{n}}|\nabla f|^{p}dx\geq\left(\frac{n-p }{p}\right)^{p}\int_{\mathbb{R}^{n}}\frac{|f|^{p}}{|x|^{p}}dx$$ for every function $f\in C_{0}^{\infty}(\mathbb{R}^{n})$, where $2\leq p<n$.
Cianchi and Ferone [@CF08] showed that for all $1<p<n$ there exists a constant $C=C(p,n)$ such that $$\int_{\mathbb{R}^{n}}|\nabla f|^{p}dx\geq\left(\frac{n-p}{p}\right)^{p}\int_{\mathbb{R}^{n}}\frac{|f|^{p}}{|x|^{p}}dx\,(1+Cd_{p}(f)^{2p^{*}})$$ holds for all real-valued weakly differentiable functions $f$ in $\mathbb{R}^{n}$ such that $f$ and $|\nabla f|\in L^{p}(\mathbb{R}^{n})$ go to zero at infinity. Here $$d_{p}f=\underset{c\in \mathbb{R}}{\rm inf}\frac{\|f-c|x|^{-\frac{n-p}{p}}\|_{L^{p^{*},\infty}(\mathbb{R}^{n})}}{\|f\|_{L^{p^{*},p}(\mathbb{R}^{n})}}$$ with $p^{*}=\frac{np}{n-p}$, and $L^{\tau, \sigma}(\mathbb{R}^{n})$ is the Lorentz space for $0<\tau\leq \infty$ and $1\leq\sigma\leq\infty$. Sometimes the improved versions of different inequalities, or remainder estimates, are called stability of the inequality if the estimates depend on certain distances: see, e.g. [@BJOS16] for stability of trace theorems, [@CFW13] for stability of Sobolev inequalities, etc. For more general Lie group discussions of above inequalities we refer to recent papers [@Ruzhansky-Suragan:Layers], [@RS-identities] and [@Ruzhansky-Suragan:squares] as well as references therein.
Recently Sano and Takahashi obtained the improved versions of Hardy inequalities in their works [@S17], [@ST15a], [@ST15b] and [@ST15]. The aim of this note is to formulate their results one of the largest classes of nilpotent Lie groups on $\mathbb{R}^{n}$, namely, homogeneous Lie groups since obtained results give new insights even for the Abelian groups in term of arbitrariness of homogeneous quasi-norm.
Preliminaries {#SEC:prelim}
=============
First let us shortly review some main concepts of homogeneous groups following Folland and Stein [@FS-Hardy] (see also recent books [@BLU07] and [@FR] on this topic). We also recall a few other facts that will be used in the proofs. A connected simply connected Lie group $\mathbb G$ is called a [*homogeneous group*]{} if its Lie algebra $\mathfrak{g}$ is equipped with a family of the following dilations: $$D_{\lambda}={\rm Exp}(A \,{\rm ln}\lambda)=\sum_{k=0}^{\infty}
\frac{1}{k!}({\rm ln}(\lambda) A)^{k},$$ where $A$ is a diagonalisable positive linear operator on $\mathfrak{g}$, and every $D_{\lambda}$ is a morphism of $\mathfrak{g}$, that is, $$\forall X,Y\in \mathfrak{g},\, \lambda>0,\;
[D_{\lambda}X, D_{\lambda}Y]=D_{\lambda}[X,Y],$$ holds. We recall that $Q := {\rm Tr}\,A$ is called the homogeneous dimension of $\mathbb G$. The Haar measure on a homogeneous group $\mathbb{G}$ is the standard Lebesgue measure for $\mathbb{R}^{n}$ (see, for example [@FR Proposition 1.6.6]).
Let $|\cdot|$ be a homogeneous quasi-norm on $\mathbb G$. Then the quasi-ball centred at $x\in\mathbb{G}$ with radius $R > 0$ is defined by $$B(x,R):=\{y\in \mathbb{G}: |x^{-1}y|<R\}.$$ We refer to [@FS-Hardy] for the proof of the following important polar decomposition on homogeneous Lie groups, which can be also found in [@FR Section 3.1.7]: there is a (unique) positive Borel measure $\sigma$ on the unit quasi-sphere $$\label{EQ:sphere}
\wp:=\{x\in \mathbb{G}:\,|x|=1\},$$ so that for every $f\in L^{1}(\mathbb{G})$ we have $$\label{EQ:polar}
\int_{\mathbb{G}}f(x)dx=\int_{0}^{\infty}
\int_{\wp}f(ry)r^{Q-1}d\sigma(y)dr.$$
We use the notation $$\label{dfdr}
\mathcal{R} f(x):= \mathcal{R}_{|x|} f(x) = \frac{d}{d|x|}f(x)=\mathcal{R}f(x), \;\forall x\in \mathbb G,$$ for any homogeneous quasi-norm $|x|$ on $\mathbb G$.
We will also use the following result:
\[CKN\] Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q$. Let $|\cdot|$ be any homogeneous norm on $\mathbb{G}$. Then for $u\in C_{0}^{\infty}(\mathbb{G}\backslash\{0\})$ and $u_{R}=u\left(R\frac{x}{|x|}\right)$ we have $$\label{ScalHardy4}
\left\|\frac{u-u_{R}}{|x|^{\frac{Q}{p}}\log\frac{R}{|x|}}\right\|_{L^{p}(\mathbb{G})}
\leq\frac{p}{p-1}\left\||x|^{\frac{p-Q}{p}}
\mathcal{R}u \right\|_{L^{p}(\mathbb{G})}, \;\;1<p<\infty,$$ for all $R>0$, and the constant $\frac{p}{p-1}$ is sharp.
In the abelian isotropic case, the following result was obtained in [@MOW15]. In the case $\gamma=p$ this result on the homogeneous group was proved in [@Ruzhansky-Suragan:critical].
We will also use the following known relations
\[ab\_relation\] Let $a,b\in \mathbb{R}$. Then
- $$|a-b|^{p}-|a|^{p}\geq-p|a|^{p-2}ab,\quad p\geq 1.$$
- There exists a constant $C=C(p)>0$ such that $$|a-b|^{p}-|a|^{p}\geq-p|a|^{p-2}ab+C|b|^{p},\quad p\geq 2.$$
- If $a\geq 0$ and $a-b\geq 0$. Then $$(a-b)^{p}+pa^{p-1}b-a^{p}\geq |b|^{p},\quad p\geq 2.$$
Stability of $L^{p}$-Hardy inequalities
=======================================
Let us set $$d_H (u;R) := \left(\int_{\mathbb{G}} \frac{\left|u(x)-R^{\frac{Q-p}{p}}u\left(R\frac{x}{|x|}\right)|x|^{-\frac{Q-p}{p}}\right|^p}{|x|^p|\log\frac{R}{|x|}|^p} dx \right)^{\frac{1}{p}},\; x\in\mathbb{G},\; R>0.$$
\[stab\_H\] Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q$. Let $|\cdot|$ be any homogeneous quasi-norm on $\mathbb{G}$. Then there exists a constant $C>0$ for all real-valued functions $u\in C_{0}^{\infty}(\mathbb{G})$ we have $$\label{stab_H_eq}
\int_{\mathbb{G}} \left|\mathcal{R} u \right|^p dx - \left(\frac{Q-p}{p}\right)^p \int_{\mathbb{G}} \frac{|u|^p}{|x|^p} dx \geq C \sup_{R>0} d^p_H(u;R),\; 2 \leq p< Q,$$ where $\mathcal{R}:=\frac{d}{d|x|}$ is the radial derivative.
Let us introduce polar coordinates $x=(r,y)=(|x|, \frac{x}{\mid x\mid})\in (0,\infty)\times\wp$ on $\mathbb{G}$, where $\wp$ is the unit quasi-sphere $$\wp:=\{x\in \mathbb{G}:\,|x|=1\},$$ and
$$\label{trasn}
v(ry):= r^{\frac{Q-p}{p}}u(ry),$$
where $u\in C_{0}^{\infty}(\mathbb{G})$. This follows that $v(0)=0$ and $\underset{r\rightarrow \infty }{\lim}v(ry)=0$ for $y \in \wp$ since $u$ is compactly supported. Using the polar decomposition on homogeneous groups (see ) and integrating by parts, we get $$\begin{aligned}
D:&= \int_{\mathbb{G}} \left|\mathcal{R} u \right|^p dx - \left(\frac{Q-p}{p}\right)^p \int_{\mathbb{G}} \frac{|u|^p}{|x|^p} dx \\
& = \int_{ \wp}\int_{0}^{\infty} \left| -\frac{\partial }{\partial r} u(ry)\right|^p r^{Q-1} - \left(\frac{Q-p}{p}\right)^p |u(ry)|^p r^{Q-p-1} drdy \\
& = \int_{ \wp}\int_{0}^{\infty} \left| \frac{Q-p}{p} r^{-\frac{Q}{p}} v(ry) - r^{-\frac{Q-p}{p}} \frac{\partial}{\partial r} v(ry)\right|^p r^{Q-1} \\ & - \left(\frac{Q-p}{p}\right)^p |v(ry)|^p r^{-1} drdy.
\end{aligned}$$ Now using the second relation in Lemma \[ab\_relation\] with the choice $a = \frac{Q-p}{p}r^{-\frac{Q}{p}}v(ry)$ and $b = r^{-\frac{Q-p}{p}}\frac{\partial }{\partial r}v(ry)$, and using the fact $\int_{0}^{\infty}|v|^{p-2}v\left(\frac{\partial}{\partial r}v\right)dr =0$, we obtain $$\begin{aligned}
\label{1_3.2}
D &\geq \int_{ \wp}\int_{0}^{\infty} -p \left(\frac{Q-p}{p} \right)^{p-1} |v(ry)|^{p-2} v(ry) \frac{\partial }{\partial r} v(ry)
\\ & +C\left|\frac{\partial}{\partial r}v(ry)\right|^p r^{p-1} drdy \\
& = C \int_{\mathbb{G}} |x|^{p-Q} \left| \mathcal{R} v \right|^p dx. \nonumber
\end{aligned}$$ Finally, combining and Lemma \[CKN\], we arrive at $$\begin{aligned}
D &\geq C \int_{\mathbb{G}} \frac{|v(x) -v(R\frac{x}{|x|})|^p}{|x|^Q |\log \frac{R}{|x|}|^p} dx = C \int_{ \wp}\int_{0}^{\infty} \frac{|v(ry) -v(Ry)|^p}{r\left|\log \frac{R}{r}\right|^p} drdy \\
& =C\int_{ \wp}\int_{0}^{\infty} \frac{|u(ry)-R^{\frac{Q-p}{p}}u(Ry)r^{-\frac{Q-p}{p}}|^p}{r^{1+p-Q}|\log \frac{R}{r}|^p} drdy \nonumber
\end{aligned}$$ for any $R >0$. This proves the desired result.
Stability of critical Hardy inequalities
========================================
In this section we establish a stability estimate for the critical Hardy inequality involving the distance to the set of extremisers: Let us denote $$\label{aremterm7}
f_{T,R}(x)=T^{\frac{Q-1}{Q}}u\left(Re^{-\frac{1}{T}}\frac{x}{|x|}\right)\left(\log \frac{R}{|x|}\right)^{\frac{Q-1}{Q}}$$ and the following ’distance’ $$\label{aremterm8}
d_{cH}(u;T, R):= \left( \int_{B(0,R)} \frac{|u(x)-f_{T,R}(x)|^Q}{|x|^Q\left|\log \frac{R}{|x|}\right|^Q\left|T\log \frac{R}{|x|}\right|^Q} dx\right)^{\frac{1}{Q}},$$ for some parameter $T>0$, functions $u$ and $f_{T,R}$ for which the integral in is finite.
\[stab\_CH\] Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q\geq2$. Let $|\cdot|$ be any homogeneous quasi-norm on $\mathbb{G}$. Then there exists a constant $C>0$ for all real-valued functions $u\in C_{0}^{\infty}(B(0,R))$ we have $$\label{aremterm9}
\int_{B(0,R)} \left| \mathcal{R} u(x) \right|^Q dx - \left(\frac{Q-1}{Q}\right)^{Q} \int_{B(0,R)} \frac{|u(x)|^Q}{|x|^Q (\log \frac{R}{|x|})^Q}dx \geq C \sup_{T>0} d^Q_{cH} (u;T, R)$$ where $\mathcal{R}:=\frac{d}{d|x|}$ is the radial derivative.
Introducing polar coordinates $(r,y)=(|x|, \frac{x}{\mid x\mid})\in (0,\infty)\times\wp$ on $\mathbb{G}$, where $\wp$ is the sphere as in , we have $u(x)=u(ry)\in C_0^{\infty}(B(0,R))$. In addition, let us set $$\label{5.4}
v (sy): = \left( \log \frac{R}{r}\right)^{-\frac{Q-1}{Q}}u(ry),\; y \in \wp,$$ where $$s=s(r):=\left( \log \frac{R}{r}\right)^{-1}.$$ Since $u \in C_0^{\infty}(B(0,R))$ we have $v(0)=0$ and $v$ has a compact support. Moreover, it is straightforward that $$\frac{\partial }{\partial r} u(ry) = - \left(\frac{Q-1}{Q}\right)\left(\log \frac{R}{r}\right)^{-\frac{1}{Q}}\frac{v(sy)}{r} + \left(\log \frac{R}{r}\right)^{\frac{Q-1}{Q}}\frac{\partial}{\partial s} v(sy)s'(r).$$ A direct calculation gives $$\begin{aligned}
\label{5.5}
S: & = \int_{B(0,R)} \left| \mathcal{R} u \right|^Q dx - \left(\frac{Q-1}{Q}\right)^Q \int_{B(0,R)} \frac{|u|^Q}{|x|^Q \left(\log \frac{R}{|x|}\right)^Q} dx
\\ & = \int_{\wp} \int_{0}^{R} \left|\frac{\partial }{\partial r}u(ry) \right|^Q r^{Q-1} - \left(\frac{Q-1}{Q}\right)^Q \frac{|u(ry)|^Q}{r\left(\log \frac{R}{r}\right)^Q} dr dy \\
&= \int_{\wp} \int_{0}^{R} \left| \left(\frac{Q-1}{Q}\right)\left(r\log \frac{R}{r}\right)^{-\frac{1}{Q}}v(sy)
+ \left(r\log \frac{R}{r}\right)^{\frac{Q-1}{Q}}\frac{\partial}{\partial s} v(sy)s'(r) \right|^Q \\
&- \left(\frac{Q-1}{Q}\right)^Q \frac{|v(sy)|^Q}{r\log \frac{R}{r}} dr dy.
\end{aligned}$$ Now by applying the second relation in Lemma \[ab\_relation\] with the choice $$a = \frac{Q-1}{Q} \left(r \log \frac{R}{r}\right)^{-\frac{1}{Q}}v(sy) \quad \text{and} \quad b = \left(r \log \frac{R}{r}\right)^{\frac{Q-1}{Q}} \frac{\partial}{\partial s} v(sy)s'(r),$$ and by using the facts $v(0)=0$ and $\underset{r\rightarrow \infty}{\lim}v(ry)=0$, we obtain $$\begin{aligned}
S & \geq \int_{\wp} \int_{0}^{R} -Q \left(\frac{Q-1}{Q}\right)^{Q-1} |v(sy)|^{Q-2}v(sy) \frac{\partial }{\partial s}v(sy)s'(r) \nonumber \\
& + C \left|\frac{\partial }{\partial s}v(sy) \right|^Q (s'(r))^Q \left(r \log \frac{R}{r}\right)^{Q-1} drdy \nonumber \\
& =\int_{\wp} \int_{0}^{R} -Q \left(\frac{Q-1}{Q}\right)^{Q-1} |v(sy)|^{Q-2}v(sy) \frac{\partial }{\partial s}v(sy)s'(r) \nonumber \\
& + C \left|\frac{\partial }{\partial s}v(sy) \right|^Q \frac{1}{r^{Q}\left( \log \frac{R}{r}\right)^{2Q}} \left(r \log \frac{R}{r}\right)^{Q-1} drdy \nonumber \\
& =\int_{\wp} \int_{0}^{R} -Q \left(\frac{Q-1}{Q}\right)^{Q-1} |v(sy)|^{Q-2}v(sy) \frac{\partial }{\partial s}v(sy)s'(r) \nonumber \\
& + C \left|\frac{\partial }{\partial s}v(sy) \right|^Q \frac{1}{\left( \log \frac{R}{r}\right)^{Q-1}} s'(r) drdy \nonumber \\
& = \int_{\wp} \int_{0}^{R} -Q \left(\frac{Q-1}{Q}\right)^{Q-1} |v(sy)|^{Q-2}v(sy) \frac{\partial }{\partial s} v(s)
\\ & +C \left|\frac{\partial }{\partial s}v(sy) \right|^Q s^{Q-1}dsdy \nonumber \\
& = C \int_{\mathbb{G}} \left| \mathcal{R} v \right|^Q dx, \nonumber
\end{aligned}$$ that is, $$\label{5.6}
S \geq C \int_{\mathbb{G}} \left| \mathcal{R} v \right|^Q dx.$$
According to Lemma \[CKN\] with $v \in C^{\infty}_0(\mathbb{G}\backslash\{0\})$ with $p=Q$ and , it implies that $$\begin{aligned}
S &\geq C \int_{\mathbb{G}} \frac{|v(x) - v(T\frac{x}{|x|})|^Q}{|x|^Q|\log \frac{T}{|x|}|^Q} dx = C \int_{\wp} \int_{0}^{\infty} \frac{|v(sy)-v(Ty)|^Q}{s|\log \frac{T}{s}|^Q} ds dy \\
& = C \int_{\wp} \int_{0}^{R} \frac{\left|\left(\log \frac{R}{r}\right)^{-\frac{Q-1}{Q}}u(ry) - T^{\frac{Q-1}{Q}} u(Re^{-\frac{1}{T}}y) \right|^Q}{r(\log \frac{R}{r})|\log (T\log\frac{R}{r} )|^Q} dr dy \\
& = C \int_{\wp} \int_{0}^{R} \frac{\left|u(ry) - T^{\frac{Q-1}{Q}} u(Re^{-\frac{1}{T}}y)(\log \frac{R}{r})^{\frac{Q-1}{Q}} \right|^Q}{r(\log \frac{R}{r})^Q|\log (T\log\frac{R}{r} )|^Q} dr dy.
\end{aligned}$$ Thus, we arrive at $$S \geq C \int_{B(0,R)} \frac{\left|u(x)-T^{\frac{Q-1}{Q}} u\left(Re^{-\frac{1}{T}}\frac{x}{|x|}\right)\left(\log \frac{R}{|x|}\right)^{\frac{Q-1}{Q}}\right|^Q}{|x|^Q\left|\log \frac{R}{|x|}\right|^Q \left|\log \left(T \log \frac{R}{|x|}\right)\right|^Q} dx$$ for all $T>0$. The proof is complete.
Improved critical Hardy and Rellich inequalities for radial functions
=====================================================================
\[radial\_rem\] Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q\geq2$. Let $|\cdot|$ be a homogeneous quasi-norm on $\mathbb{G}$. Let $q>0$ be such that $$\label{1_1.5}
\alpha = \alpha(q,L): = \frac{Q-1}{Q}q +L +2 \leq Q,$$ for $-1 <L<Q-2$. Then for all real-valued positive non-increasing radial functions $u\in C_0^{\infty}(B(0,R))$ we have $$\begin{aligned}
\label{1.6}
\int_{B(0,R)} |\mathcal{R} u|^Q dx &- \left( \frac{Q-1}{Q}\right)^Q \int_{B(0,R)} \frac{|u(x)|^Q}{|x|^Q \left(\log \frac{R e}{|x|}\right)^Q} dx \\
& \geq |\wp|^{1-\frac{Q}{q}} C^{\frac{Q}{q}} \left(\int_{B(0,R)}\frac{|u(x)|^q}{|x|^Q \left(\log \frac{R e}{|x|}\right)^{\alpha}} dx \right)^{\frac{Q}{q}}, \nonumber
\end{aligned}$$ where $|\wp| $ is the measure of the unit quasi-sphere in $\mathbb{G}$ and $$\begin{aligned}
C^{-1}=C(L,Q,q)^{-1} & := \int_{0}^{1} s^L \left(\log \frac{1}{s}\right)^{\frac{Q-1}{Q}q} ds \\ &= (L+1)^{-\left(\frac{Q-1}{Q}q+1\right)}\Gamma \left(\frac{Q-1}{Q}q +1\right)
\end{aligned}$$ here $\Gamma (\cdot)$ is the Gamma function.
As in previous proofs we set $$\begin{aligned}
\label{2.2}
&v(s) = \left(\log \frac{R e}{r}\right)^{-\frac{Q-1}{Q}} u(r), \quad \text{where} \quad r =|x|,s=s(r)=\left(\log \frac{R e}{r}\right)^{-1}, \\
&s'(r) = \frac{s(r)}{r \log \frac{Re}{r}} \geq 0. \nonumber
\end{aligned}$$ Simply we have $v(0)=v(1)=0$ since $u(R)=0$, moreover, $$\begin{gathered}
\label{2.3}
u'(r) =-\left(\frac{Q-1}{Q}\right)\left(\log \frac{R e}{r}\right)^{-\frac{1}{Q}} \frac{v(s(r))}{r} \\+ \left(\log \frac{R e}{r}\right)^{\frac{Q-1}{Q}} v'(s(r))s'(r)\leq 0.
\end{gathered}$$ It is straightforward that $$\begin{aligned}
& I:= \int_{B(0,R)} |\mathcal{R} u|^Q dx - \left(\frac{Q-1}{Q}\right)^Q \int_{B(0,R)} \frac{|u|^Q}{|x|^Q \left(\log \frac{R e}{|x|}\right)^Q}dx \\
& = |\wp| \int_{0}^{R} |u'(r)|^Q r^{Q-1} dr - \left(\frac{Q-1}{Q}\right)^Q |\wp| \int_{0}^R \frac{|u(r)|^Q}{r \left(\log \frac{R e}{r}\right)^Q}dr
\\
& = |\wp| \int_{0}^R \left( \frac{Q-1}{Q} \left(\log \frac{R e}{r}\right)^{-\frac{1}{Q}} \frac{v(s(r))}{r} - \left(\log \frac{R e}{r}\right)^{\frac{Q-1}{Q}} v'(s(r))s'(r)\right)^Q r^{Q-1}dr
\\
& - \left(\frac{Q-1}{Q}\right)^Q |\wp| \int_{0}^R \frac{|u(r)|^Q}{r \left(\log \frac{R e}{r}\right)^Q}dr.
\end{aligned}$$ By applying the third relation in Lemma \[ab\_relation\] with $$a = \frac{Q-1}{Q} \left( \log \frac{Re}{r} \right)^{-\frac{1}{Q}} \frac{v(s(r))}{r} \quad \text{and} \quad b = \left( \log \frac{Re}{r} \right)^{\frac{Q-1}{Q}} v'(s(r))s'(r),$$ and dropping $a^Q \geq 0$ as well as using the boundary conditions $v(0)=v(1)=0$, we get $$\begin{aligned}
\label{2.5}
I &\geq -|\wp| Q\left(\frac{Q-1}{Q}\right)^{Q-1} \int_0^R v(s(r))^{Q-1}v'(s(r))s'(r)dr \\
& + |\wp| \int_{0}^{R} |v'(s(r))|^Q (s'(r))^Q \left(r \log \frac{Re}{r} \right)^{Q-1} dr\nonumber \\
& =-|\wp| Q\left(\frac{Q-1}{Q}\right)^{Q-1} \int_0^R v(s(r))^{Q-1}v'(s(r))s'(r)dr \\
& + |\wp| \int_{0}^{R} |v'(s(r))|^Q \frac{1}{r^{Q}\left(\log \frac{Re}{r} \right)^{2Q}} \left(r \log \frac{Re}{r} \right)^{Q-1} dr\nonumber \\
&=-|\wp| Q\left(\frac{Q-1}{Q}\right)^{Q-1} \int_0^R v(s(r))^{Q-1}v'(s(r))s'(r)dr \\
& + |\wp| \int_{0}^{R} |v'(s(r))|^Q s(r)^{Q-1}s'(r) dr\nonumber \\
&= - |\wp| Q \left(\frac{Q-1}{Q}\right)^{Q-1} \int_{0}^{1} v(s)^{Q-1} v'(s)ds
\\ & + |\wp| \int_{0}^{1} |v'(s)|^Qs^{Q-1}ds \nonumber\\
& =|\wp| \int_{0}^{1} |v'(s)|^Q s^{Q-1}ds.\nonumber
\end{aligned}$$ Moreover, by using the inequality $$\begin{aligned}
|v(s)|= \left| \int_{s}^{1} v'(t)dt \right| & = \left| \int_{s}^{1} v'(t) t^{\frac{Q-1}{Q} -\frac{Q-1}{Q}} dt\right| \\ & \leq \left(\int_{0}^{1} |v'(t)|^Qt^{Q-1}dt\right)^{\frac{1}{Q}} \left(\log \frac{1}{s}\right)^{\frac{Q-1}{Q}},
\end{aligned}$$ we obtain $$\int_{0}^{1} |v(s)|^q s^Lds \leq \left(\int_{0}^{1} |v'(s)|^Q s^{Q-1} ds \right)^{\frac{q}{Q}} \int_{0}^{1} s^L \left(\log \frac{1}{s}\right)^{\frac{Q-1}{Q}q} ds$$ for $-1 <L<Q-2$. Thus, we have $$\label{2.6}
\int_{0}^{1} |v'(s)|^Q s^{Q-1} ds \geq C^{\frac{q}{Q}} \left(\int_{0}^{1} |v(s)|^q s^L ds\right)^{\frac{Q}{q}}.$$ Now it follows from and that $$\begin{aligned}
I &\geq |\wp| C^{\frac{Q}{q}}\left(\int_{0}^{1} |v(s)|^q s^L ds\right)^{\frac{Q}{q}} = |\wp|C^{\frac{Q}{q}} \left( \int_{0}^{R} \frac{|u(r)|^q}{r\left(\log \frac{Re}{r}\right)^{\alpha}}dr\right)^{\frac{Q}{q}} \\
& = |\wp|^{1-\frac{Q}{q}} C^{\frac{Q}{q}} \left( \int_{0}^{R} \frac{|u(x)|^q}{|x|^Q\left(\log \frac{Re}{|x|}\right)^{\alpha}}dx\right)^{\frac{Q}{q}}.
\end{aligned}$$ where $\alpha = \alpha(q,L)=\frac{Q-1}{Q}q+L+2$. The proof is complete.
The method used in the previous section also allows one to obtain the following stability inequality for Rellich type inequalities:
\[Rellichtype\] Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q$. Let $|\cdot|$ be a homogeneous quasi-norm on $\mathbb{G}$ and $p\geq 1$. Let $k \geq 2, k \in \mathbb{N}$ be such that $kp<Q$. Then for all real-valued radial functions $u\in C_0^{\infty}(\mathbb{G})$ we have $$\begin{gathered}
\int_{\mathbb{G}} \frac{|\tilde{\mathcal{R}} u|^p}{|x|^{(k-2)p}} dx - K^p_{k,p} \int_{\mathbb{G}} \frac{|u|^p}{|x|^{kp}}dx \\ \geq C \sup_{R>0} \int_{\mathbb{G}}\frac{\left| |u(x)|^{\frac{p-2}{2}}u(x) -R^{\frac{Q-kp}{2}}|u(R)|^{\frac{p-2}{2}}u(R)|x|^{-\frac{Q-kp}{2}} \right|^2}{|x|^{kp}\left|\log \frac{R}{|x|}\right|^2} dx,
\end{gathered}$$ where $$\tilde{\mathcal{R}}f = \mathcal{R}^{2}f + \frac{Q -1}{|x|} \mathcal{R}f$$ is the Rellich type operator on $\mathbb{G}$ and $K_{k,p} = \frac{(Q-kp)[(k-2)p+(p-1)Q]}{p^2}$.
For $k \geq 2, k \in \mathbb{N}$ and $kp<Q$ let us set $$\label{4.2}
v(r): = r^{\frac{Q-kp}{p}}u(r), \quad \text{where} \quad r \in [0, \infty).$$ Thus, $v(0)=0$ and $v(\infty) = 0$.
We have $$\begin{aligned}
-\tilde{\mathcal{R}} u & =- \mathcal{R}^{2}\left(r^{\frac{kp-Q}{p}}v(r)\right) - \frac{Q-1}{r} \mathcal{R}\left(r^{\frac{kp-Q}{p}}v(r)\right)
\\ &=- \mathcal{R}\left( \frac{kp-Q}{p}r^{\frac{kp-Q}{p}-1}v(r)
+r^{\frac{kp-Q}{p}}\mathcal{R}v(r)\right)
\\ &- \frac{Q -1}{r}\frac{kp-Q}{p}r^{\frac{kp-Q}{p}-1}v(r)
-\frac{Q -1}{r}r^{\frac{kp-Q}{p}}\mathcal{R}v(r)
\\ & = -\frac{kp-Q}{p}\left(\frac{kp-Q}{p}-1\right) r^{\frac{kp-Q}{p}-2}v(r)
-\frac{kp-Q}{p}r^{\frac{kp-Q}{p}-1}\mathcal{R}v(r)
\\ &- \frac{kp-Q}{p}r^{\frac{kp-Q}{p}-1}\mathcal{R}v(r)
-r^{\frac{kp-Q}{p}}\mathcal{R}^{2}v(r)
\\ &- \frac{Q -1}{r}\frac{kp-Q}{p}r^{\frac{kp-Q}{p}-1}v(r)
-\frac{Q -1}{r}r^{\frac{kp-Q}{p}}\mathcal{R}v(r)
\\ &=-r^{\frac{kp-Q}{p}-2}\left(\frac{(kp-Q)(kp-Q-p)}{p^{2}}+\frac{(Q-1)(kp-Q)}{p} \right)v(r)
\\ &-r^{\frac{kp-Q}{p}-2}r^{2}\left(\mathcal{R}^{2}v(r)+\frac{1}{r}\left( \frac{2(kp-Q)}{p}+(Q-1)\right) \mathcal{R} v(r) \right)
\\ & = r^{k-2-\frac{Q}{p}}(K_{k,p}v(r)-r^2 \tilde{\mathcal{R}}_{k} v(r)),
\end{aligned}$$ where $$\tilde{\mathcal{R}}_{k}f = \mathcal{R}^{2}f + \frac{2k + \frac{Q(p-2)}{p} -1}{r} \mathcal{R}f$$ and $K_{k,p} = \frac{(Q-kp)[(k-2)p+(p-1)Q]}{p^2}$. By using the first inequality in Lemma \[ab\_relation\] with $a=K_{k,p}v(r)$ and $b = r^2 \tilde{\mathcal{R}}_{k} v(r)$, and the fact $\int_{0}^{\infty}|v|^{p-2}vv'dr = 0$ since $v(0)=0$ and $v(\infty) = 0$, we obtain $$\begin{aligned}
J &:= \int_{\mathbb{G}} \frac{|\tilde{\mathcal{R}} u|^p}{|x|^{(k-2)p}} dx - K^p_{k,p} \int_{\mathbb{G}} \frac{|u|^p}{|x|^{kp}}dx \\
&= |\wp|\int_{0}^{\infty} |-\tilde{\mathcal{R}} u(r)|^p r^{Q-1-(k-2)p} dr - K^p_{k,p} |\wp|\int_{0}^{\infty} |u(r)|^p r^{Q-kp-1}dr \\
&= |\wp|\int_{0}^{\infty} \left( |K_{k,p} v(r)-r^2\tilde{\mathcal{R}}_{k} v(r)|^p - (K_{k,p}v(r))^p \right)r^{-1}dr \\
& \geq - p |\wp|K^{p-1}_{k,p} \int_{0}^{\infty} |v|^{p-2} v \tilde{\mathcal{R}}_{k} v r dr\\
& = - p |\wp|K^{p-1}_{k,p} \int_{0}^{\infty} |v|^{p-2} v \left(v'' +\frac{2k + \frac{Q(p-2)}{p}-1}{r}v' \right)rdr \\
& =- p |\wp|K^{p-1}_{k,p} \int_{0}^{\infty} |v|^{p-2} vv''rdr.
\end{aligned}$$ On the other hand, we have $$\begin{aligned}
-\int_{0}^{\infty} |v|^{p-2} vv''rdr &= (p-1) \int_{0}^{\infty} |v|^{p-2} (v')^2 r dr + \int_{0}^{\infty} |v|^{p-2} vv'dr \\
& = (p-1) \int_{0}^{\infty} |v|^{p-2} (v')^2 r dr
\\ &= \frac{4(p-1)}{p^2} \int_{0}^{\infty} \left(
\frac{p-2}{2}\right)^{2} |v|^{p-2}(v')^2dr
\\ & +\frac{4(p-1)}{p^2} \int_{0}^{\infty}(p-2) |v|^{p-2}(v')^2+|v|^{p-2}(v')^2 r dr
\\&= \frac{4(p-1)}{p^2} \int_{0}^{\infty} \left(\left(|v|^{\frac{p-2}{2}}\right)'v+|v|^{\frac{p-2}{2}}v'\right)^2 r dr \\
&= \frac{4(p-1)}{p^2} \int_{0}^{\infty} |(|v|^{\frac{p-2}{2}}v)'|^2 r dr \\
&= \frac{4(p-1)}{|\wp_{2}| p^2} \int_{\mathbb{G}_{2}} \left| \mathcal{R}(|v|^{\frac{p-2}{2}}v) \right|^2 dx,
\end{aligned}$$ where $\mathbb{G}_{2}$ is a homogeneous group of homogeneous degree $2$ and $|\wp_{2}|$ is the measure of the corresponding unit $2$-quasi-ball. By using Lemma \[CKN\] for $|v|^{\frac{p-2}{2}}v \in C^{\infty}_{0} (\mathbb{G}_{2} \backslash \{0\})$ in $p=Q=2$ case, and combining above equalities, we obtain $$\begin{aligned}
J &\geq C_{1} \int_{\mathbb{G}_{2}} \frac{\left| |v(x)|^{\frac{p-2}{2}} v(x) - |v(R\frac{x}{|x|})|^{\frac{p-2}{2}} v(R\frac{x}{|x|} )\right|^2}{|x|^2 \left|\log \frac{R}{|x|}\right|^2} dx \\
& = C_{1} \int_{0}^{\infty} \frac{\left| |v(r)|^{\frac{p-2}{2}} v(r) - |v(R)|^{\frac{p-2}{2}} v(R) \right|^2}{r \left|\log \frac{R}{r}\right|^2} dr \\
& = C_{1} \int_{0}^{\infty} \frac{\left| |u(r)|^{\frac{p-2}{2}} u(r) - R^{\frac{Q-kp}{2}}|u(R)|^{\frac{p-2}{2}} u(R) r^{-\frac{Q-kp}{2}} \right|^2}{r^{1-Q+kp} \left|\log \frac{R}{r}\right|^2} dr
\end{aligned}$$ for any $R>0$. That is, $$\begin{aligned}
J \geq C \sup_{R>0} \int_{\mathbb{G}} \frac{\left| |u(x)|^{\frac{p-2}{2}}u(x) -R^{\frac{Q-kp}{2}}|u(R)|^{\frac{p-2}{2}}u(R)|x|^{-\frac{Q-kp}{2}} \right|^2}{|x|^{kp}\left|\log \frac{R}{|x|}\right|^2} dx.
\end{aligned}$$ The proof is complete.
[**Acknowledgments.**]{} The authors were supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02, as well as by the MESRK grant 5127/GF4. No new data was collected or generated during the course of research.
[99]{}
N. Bez, C. Jeavons, T. Ozawa and M. Sugimoto, *Stability of trace theorems on the sphere*, arXiv:1611.00928 (to appear).
A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, *Stratified Lie groups and potential theory for their sub-Laplacians*, Springer-Verlag, Berlin, Heidelberg, 2007.
A. Cianci and A. Ferone, *Hardy inequalities with non-standard remainder terms*, Ann. Inst. H. Poincaré. Anal. Nonlinéaire, [**25**]{} (2008), 889–906.
S. Chen, R. Frank and T. Weth, *Remainder terms in the fractional Sobolev inequality*, Indiana Univ. Math. J. [**62**]{} (2013), 1381–1397.
V. Fischer and M. Ruzhansky, *Quantization on nilpotent Lie groups*, volume 314 of Progress in Mathematics, Birkhäuser, 2016.
G. B. Folland and E. M. Stein, *Hardy spaces on homogeneous groups*, volume 28 of Mathematical Notes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.
S. Machihara, T. Ozawa, and H. Wadade, *Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space*, J. Inequal. Appl. **281** (2015), 1–13.
M. Ruzhansky and D. Suragan, *Critical [H]{}ardy inequalities*, arXiv: 1602.04809 (to appear).
M. Ruzhansky and D. Suragan, *Layer potentials, [K]{}ac’s problem, and refined [H]{}ardy inequality on homogeneous [C]{}arnot groups*, Adv. Math. **308** (2017), 483–528.
M. Ruzhansky and D. Suragan, *Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups*, Adv. Math. **317** (2017), 799–822.
M. Ruzhansky and D. Suragan, *Local [H]{}ardy and Rellich inequalities for sums of squares of vector fields*, Adv. Diff. Equations, **22** (2017), 505–540.
M. Sano, *Scaling invariant Hardy type inequalities with non-standard remainder terms*, preprint (2016), http://www.sci.osaka-cu.ac.jp/math/OCAMI/preprint/2016/$16_{-}$15.pdf.
M. Sano and F. Takahashi, *Scale invariance structures of the critical and the subcritical Hardy inequalities and their improvements*, Calc. Var. Partial Differential Equations, **56** (2017), no. 3, Paper No. 69.
M. Sano and F. Takahashi, *Some improvements for a class of the Caffarelli-Kohn-Nirenberg inequalities*, Differ. Integral Equ. (2017) (to appear).
M. Sano and F. Takahashi, *Improved Hardy inequality in a limiting case and their applications*, preprint (2015), http://www.sci.osaka-cu.ac.jp/math/OCAMI/preprint/2015/$15_{-}$03.pdf
[^1]: Copyright 2016 by the Tusi Mathematical Research Group.
|
---
abstract: 'The paper aims to investigate the classification problem of low dimensional complex none Lie filiform Leibniz algebras. There are two sources to get classification of filiform Leibniz algebras. The first of them is the naturally graded none Lie filiform Leibniz algebras and the another one is the naturally graded filiform Lie algebras [@GO]. Here we do consider Leibniz algebras appearing from the naturally graded none Lie filiform Leibniz algebras. According to the theorem presented in [@AO] this class can be splited into two subclasses. However, isomorphisms within each class there were not investigated. In [@BR] U.D.Bekbaev and I.S.Rakhimov suggested an approach to the isomorphism problem in terms of invariants. This paper presents an implementation of the results of [@BR] in low dimensional cases. Here we give the complete classification of complex none Lie filiform Leibniz algebras in dimensions at most 8 from the first class of the above mentioned result of [@AO] and give a hypothetic formula for the number of isomorphism classes in finite dimensional case.'
---
I.S. Rakhimov$^1$, S.K. Said Husain$^2$
$^1$isamiddin@science.upm.edu.my $\&$ risamiddin@mail.ru
$^2$skartini@science.upm.edu.my
Institute for Mathematical Research (INSPEM) $\&$ Department of Mathematics, FS,\
43400 UPM, Serdang, Selangor Darul Ehsan, (Malaysia)
[^1]
**2000 MSC:** *17A32, 17B30.*
**Key-Words:** *filiform Leibniz algebra, invariant, isomorphism.*
Preliminaries
=============
[**Definition 1.1**]{} An algebra $L$ over a field $K$ is called a [*Leibniz algebra*]{} if it satisfies the following Leibniz identity: $$x(yz)=(xy)z-(xz)y.$$
Let $Leib_{n}(K)$ be a subvariety of $Alg_{n}(K)$ consisting of all $n$-dimensional Leibniz algebras over $K$. It is invariant under the isomorphic action of $GL_n(K)$ (“transport of structure”). Two algebras are isomorphic if and only if they belong to the same orbit under this action. As a subset of $Alg_{n}(K)$ the set $Leib_{n}(K)$ is specified by the system of equations with respect to structural constants $\gamma_{ij}^{k}$: $$\sum\limits_{\emph{l}=1}^{\emph{n}}{(\gamma_{\emph{jk}}^{\emph{l}}
\gamma_{\emph{il}}^{\emph{m}}-\gamma_{\emph{ij}}^{\emph{l}}\gamma_{\emph{lk}}^{\emph{m}}+
\gamma_{\emph{ik}}^{\emph{l}}\gamma_{\emph{lj}}^{\emph{m}})}=0.$$
It is easy to see that if the multiplication in Leibniz algebra happens to be anticommutative then it is a Lie algebra. So Leibniz algebras are “noncommutative” generalization of Lie algebras. As for Lie algebras case they are well known and several classifications of low dimensional cases have been given. But unless simple Lie algebras the classification problem of all Lie algebras in common remains a big problem. Yu.I.Malcev [@Mal] reduced the classification of solvable Lie algebras to the classification of nilpotent Lie algebras. Apparently the first non-trivial classification of some classes of low-dimensional nilpotent Lie algebra are due to Umlauf. In his thesis [@Um] he presented the redundant list of nilpotent Lie algebras of dimension less seven. He gave also the list of nilpotent Lie algebras of dimension less than ten admitting a so-called adapted basis (now, the nilpotent Lie algebras with this property are called *filiform Lie algebras*). It was shown by M.Vergne [@Vr] the importance of filiform Lie algebras in the study of variety of nilpotent Lie algebras laws.
Further if it is not asserted additionally all algebras assumed to be over the field of complex numbers.
Let $L$ be a Leibniz algebra. We put: $ L^1=L,\quad
L^{k+1}=[L^k,L],\enskip k\in N.$
[**Definition 1.2**]{} A Leibniz algebra $L$ is said to be [*nilpotent*]{} if there exists an integer $s\in N,$ such that $L^{1}
\supset L^{2} \supset ... \supset L^{s}=\{0\}.$ The smallest integer $s$ for which $L^{s}=0$ is called [*the nilindex*]{} of $L$.
[**Definition 1.3**]{} An $n$-dimensional Leibniz algebra $L$ is said to be [*filiform*]{} if $dim L^i =n-i,$ where $2\le i\le n.$
[**Theorem 1.4 [@AO]**]{} Arbitrary complex non-Lie filiform Leibniz algebra of dimension $n+1$ obtained from naturally graded none Lie filiform Leibniz algebra is isomorphic to one of the following filiform Leibniz algebras with the table non zero multiplications for basis vectors $\{e_0,e_1, \ldots, e_n\}$:
$ \mbox{ a) (The first class):} \left\{
\begin{array}{lll}
e_{0}e_{0}=e_{2},\\
e_ie_{0}=e_i+1, \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \ \ 1\leq i\leq n-1 \\
e_{0}e_{1}]= \alpha_{3} e_{3}+ \alpha_{4} e_{4}+...+\alpha_{n-1}
e_{n-1}+
\theta e_{n}, \\
e_{j}e_{1}=\alpha_{3}e_{j+2}+
\alpha_{4}e_{j+3}+...+\alpha_{n+1-j}e_{n}, \qquad 1\leq j\leq n-2
\end{array}
\right.
$
$ \mbox{b) (The second class):} \left\{
\begin{array}{lll}
\emph{e}_{0}\emph{e}_{0}=\emph{e}_{2},\\
e_{i}e_{0}=e_{i+1}, \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \ \ 2\leq i\leq n-1 \\
e_{0}e_{1}= \beta_{3} e_{3}+ \beta_{4}
e_{4}+...+\beta_{n} e_{n}, \\
e_{1}e_{1}=\gamma e_{n},\\
e_{j}e_{1}=\beta_{3}e_{j+2}+
\beta_{4}e_{j+3}+...+\beta_{n+1-j}e_{n}, \qquad 2\leq j\leq n-2
\end{array}
\right.$
Note that the algebras from the first class and the second class never are isomorphic to each other.
In this paper we will deal with the first class of algebras of the above Theorem, they will be denoted $L(\alpha_{3},\alpha_{4},...,\alpha_{\emph{n}},\theta)$, meaning that they are defined by parameters $\alpha_{3},\alpha_{4},...,\alpha_{\emph{n}},\theta$. The class here will be denoted as $FLeib_{n+1}$. As for the second class it will be considered somewhere else.
Using the method of simplification of the basis transformations in [@GO] the following criterion on isomorphism of two $(n+1)$-dimensional filiform Leibniz algebras was given. Namely: let $n\geq 3$.
[**Theorem 1.5**]{} Two algebras $L(\alpha)$ and $L(\alpha')$ from $FLeib_{n+1}$, where $\alpha=(\alpha_{3},\alpha_{4},...,\alpha_{\emph{n}},\theta)$ and $\alpha'=(\alpha'_{3},\alpha'_{4},...,\alpha'_{\emph{n}},\theta')$, are isomorphic if and only if there exist complex numbers $\emph{A},\emph{B}$ such that $\emph{A}(\emph{A}+\emph{B})\neq$ 0 and the following conditions hold:
$ \left\{
\begin{array}{lll}
\alpha'_{3}=\frac{A+B}{A^{2}}\alpha_{3},\\
\alpha'_{t}=\frac{1}{A^{t-1}}((A+B)\alpha_{t}- \sum
\limits_{k=3}^{t-1}(C_{k-1}^{k-2}A^{k-2}B \alpha_{t+2-k}+
C_{k-1}^{k-3}A^{k-3}B^{2} \sum \limits_{i_{1}=k+2}^{t}
\alpha_{t+3-i_{1}}\cdot\alpha_{i_{1}+1-k}+\\
C_{k-1}^{k-4}A^{k-4}B^{3}\sum\limits_{i_{2}=k+3}^{t}
\sum\limits_{i_{1}=k+3}^{i_{2}}\alpha_{t+3-i_{2}}\cdot\alpha_{i_{2}+3-i_{1}}\cdot
\alpha_{i_{1}-k}+...+\\
C_{k-1}^{1}AB^{k-2}\sum\limits_{i_{k-3}=2k-2}^{t}
\sum\limits_{i_{k-4}=2k-2}^{i_{k-3}}...
\sum\limits_{i_{1}=2k-2}^{i_{2}}\alpha_{t+3-i_{k-3}}\cdot\alpha_{i_{k-3}+3-i_{k-4}}
\cdot...\cdot\alpha_{i_{2}+3-i_{1}}\cdot\alpha_{i_{1}+5-2k}\\+B^{k-1}
\sum\limits_{i_{k-2}=2k-1}^{t}
\sum\limits_{i_{k-3}=2k-1}^{i_{k-2}}...
\sum\limits_{i_{1}=2k-1}^{i_{2}}\alpha_{t+3-i_{k-2}}\cdot\alpha_{i_{k-2}+3-i_{k-3}}
\cdot...\cdot\alpha_{i_{2}+3-i_{1}}\alpha_{i_{1}+4-2k})\cdot\alpha'_{k}), \\
\mbox{where} \ \ 4 \leq t \leq n.\\
\theta'=\frac{1}{A^{n-1}}(A\theta+B\alpha_{n}- \sum
\limits_{k=3}^{n-1}(C_{k-1}^{k-2}A^{k-2}B \alpha_{n+2-k}+
C_{k-1}^{k-3}A^{k-3}B^{2} \sum \limits_{i_{1}=k+2}^{n}
\alpha_{n+3-i_{1}}\cdot\alpha_{i_{1}+1-k}+\\
C_{k-1}^{k-4}A^{k-4}B^{3}\sum\limits_{i_{2}=k+3}^{n}
\sum\limits_{i_{1}=k+3}^{i_{2}}\alpha_{n+3-i_{2}}\cdot\alpha_{i_{2}+3-i_{1}}
\cdot\alpha_{i_{1}-k}+...+\\
C_{k-1}^{1}AB^{k-2}\sum\limits_{i_{k-3}=2k-2}^{n}
\sum\limits_{i_{k-4}=2k-2}^{i_{k-3}}...
\sum\limits_{i_{1}=2k-2}^{i_{2}}\alpha_{n+3-i_{k-3}}\cdot\alpha_{i_{k-3}+3-i_{k-4}}
\cdot...\cdot\alpha_{i_{2}+3-i_{1}}\cdot\alpha_{i_{1}+5-2k}\\+B^{k-1}
\sum\limits_{i_{k-2}=2k-1}^{n}
\sum\limits_{i_{k-3}=2k-1}^{i_{k-2}}...
\sum\limits_{i_{1}=2k-1}^{i_{2}}\alpha_{n+3-i_{k-2}}\cdot\alpha_{i_{k-2}+3-i_{k-3}}
\cdot...\cdot\alpha_{i_{2}+3-i_{1}}\cdot\alpha_{i_{1}+4-2k})\cdot\alpha'_{k}),
\end{array}
\right.
$\
It is not difficult to notice that the expressions for $\alpha'_{t}$, $\theta'$ in Theorem 1.5 can be represented in the following form: $\alpha'_{t}=\frac{1}{A^{t-2}}\varphi_{t}(\frac{B}{A};\alpha),$ where $\alpha=(\alpha_{3},\alpha_{4},...,\alpha_{n}, \theta)$ and $\varphi_{t}(y;z)=\varphi_{t}(y;z_{3},z_{4},...,z_{n},z_{n+1})=$ $$\begin{array}{lll} ((1+y)z_{t}- \sum \limits_{k=3}^{t-1}(C_{k-1}^{k-2}y z_{t+2-k}+
C_{k-1}^{k-3}y^{2} \sum \limits_{i_{1}=k+2}^{t} z_{t+3-i_{1}}
\cdot z_{i_{1}+1-k}+\\
\\
C_{k-1}^{k-4}y^{3}\sum\limits_{i_{2}=k+3}^{t}
\sum\limits_{i_{1}=k+3}^{i_{2}} z_{t+3-i_{2}} \cdot
z_{i_{2}+3-i_{1}} \cdot z_{i_{1}-k}+...+\\
\\
C_{k-1}^{1}y^{k-2}\sum\limits_{i_{k-3}=2k-2}^{t}
\sum\limits_{i_{k-4}=2k-2}^{i_{k-3}}...
\sum\limits_{i_{1}=2k-2}^{i_{2}} z_{t+3-i_{k-3}} \cdot
z_{i_{k-3}+3-i_{k-4}} \cdot...\cdot z_{i_{2}+3-i_{1}} \cdot
z_{i_{1}+5-2k}+\\
\\y^{k-1} \sum\limits_{i_{k-2}=2k-1}^{t}
\sum\limits_{i_{k-3}=2k-1}^{i_{k-2}}...
\sum\limits_{i_{1}=2k-1}^{i_{2}} z_{t+3-i_{k-2}} \cdot
z_{i_{k-2}+3-i_{k-3}} \cdot...\cdot z_{i_{2}+3-i_{1}}
z_{i_{1}+4-2k}) \cdot \varphi_{k}(y;z)),\\
\\
\mbox{for} \ 3 \leq t \leq n.\end{array}$$ $\theta'=\frac{1}{A^{n-2}}\varphi_{n+1}(\frac{B}{A};\alpha),
\mbox{where}\
\varphi_{n+1}(y;z)=\varphi_{n+1}(y;z_{3},z_{4},...,z_{n},z_{n+1})=$ $$\begin{array}{lll}
(z_{n+1}+y z_{n}- (1+y)\sum \limits_{k=3}^{n-1}(C_{k-1}^{k-2}y
z_{n+2-k}+ C_{k-1}^{k-3}y^{2} \sum \limits_{i_{1}=k+2}^{n}
z_{n+3-i_{1}} \cdot z_{i_{1}+1-k}+\\
\\
C_{k-1}^{k-4}y^{3}\sum\limits_{i_{2}=k+3}^{n}
\sum\limits_{i_{1}=k+3}^{i_{2}} z_{n+3-i_{2}} \cdot
z_{i_{2}+3-i_{1}}
\cdot z_{i_{1}-k}+...+\\
\\
C_{k-1}^{1}y^{k-2}\sum\limits_{i_{k-3}=2k-2}^{n}
\sum\limits_{i_{k-4}=2k-2}^{i_{k-3}}...
\sum\limits_{i_{1}=2k-2}^{i_{2}}z_{n+3-i_{k-3}} \cdot
z_{i_{k-3}+3-i_{k-4}} \cdot...\cdot z_{i_{2}+3-i_{1}}\cdot
z_{i_{1}+5-2k}\\
\\
+y^{k-1} \sum\limits_{i_{k-2}=2k-1}^{n}
\sum\limits_{i_{k-3}=2k-1}^{i_{k-2}}...
\sum\limits_{i_{1}=2k-1}^{i_{2}}z_{n+3-i_{k-2}}\cdot
z_{i_{k-2}+3-i_{k-3}} \cdot...\cdot z_{i_{2}+3-i_{1}}\cdot
z_{i_{1}+4-2k})\cdot\varphi_{k}(y;z)).\end{array}$$
For transition from the $(n+1)$-dimensional filiform Leibniz algebra $L(\alpha)$ to the $(n+1)$-dimensional filiform Leibniz algebra $L(\alpha')$ we will write $\alpha'=\rho(\frac{1}{A},\frac{B}{A};\alpha)$, where $\alpha=(\alpha_{3},\alpha_{4},...,\alpha_{n},\theta)$,
$$\rho(\frac{1}{A},\frac{B}{A};\alpha)=(\rho_{1}(\frac{1}{A},\frac{B}{A};\alpha),
\rho_{2}(\frac{1}{A},\frac{B}{A};\alpha),...,\rho_{n-1}(\frac{1}{A},\frac{B}{A};
\alpha)),$$
$$\rho_{t}(x,y;z)=x^{t}\varphi_{t+2}(y;z)\ \mbox{for} \ 1 \leq t \leq
n-2$$ and $$\rho_{n-1}(x,y;z)=x^{n-2}\varphi_{n+1}(y,z)$$
Here are the main properties of the operator $\rho$, derived from the fact that $\rho(\frac{1}{A},\frac{B}{A};\cdot)$ is an action of a group.
$$\begin{array}{lll}{1^{0}. \ \ \rho(1,0;\cdot) \ \ \mbox{is the identity
operator}.}\\
\\ 2^{0}. \
\rho(\frac{1}{A_{2}},\frac{B_{2}}{A_{2}};\rho(\frac{1}{A_{1}},
\frac{B_{1}}{A_{1}};\alpha))=\rho(\frac{1}{A_{1}A_{2}},
\frac{A_{1}B_{2}+A_{2}B_{1}+B_{1}B_{2}}{A_{1}A_{2}};\alpha).\\
\\ 3^{0}. \ \ \mbox{If} \ \
\alpha'=\rho(\frac{1}{A},\frac{B}{A};\alpha) \ \ \mbox{then} \ \
\alpha=\rho(A,-\frac{B}{A+B};\alpha').\end{array}$$
From here on $n$ is a positive integer. We assume that $n\geq4$ since there are complete classifications of complex nilpotent Leibniz algebras of dimension at most four [@AOR].
We first present the result of [@BR] that underlies our classification result.
We consider the following presentation of $FLeib_{n+1}:$ $FLeib_{n+1}=U_{1} \cup F$, where $U_1= \{L(\alpha):
\alpha_{3}(\alpha_{4}+2\alpha_{3}^{2}) \neq 0\}$, $F= \{L(\alpha):
\alpha_{3}(\alpha_{4}+2\alpha_{3}^{2}) = 0\}.$
[**Theorem 1.6**]{} [@BR] $i)$ Two algebras $L(\alpha)$ and $L(\alpha')$ from $\emph{U}_1$ are isomorphic if and only if $$\rho_{i}(\frac{2\alpha_{3}}{\alpha_{4}+ 2
\alpha_{3}^{2}},\frac{\alpha_{4}}{2 \alpha_{3}^{2}};\alpha) =
\rho_{i}(\frac{2\alpha_{3}'}{\alpha_{4}'+ 2
\alpha_{3}'^{2}},\frac{\alpha_{4}'}{2 \alpha_{3}'^{2}};\alpha')$$ whenever $\emph{i}=\overline{3,n-1}.$
$ii)$ For any $(a_3,a_4,...,a_{n-1})\in C^{n-3}$ there is an algebra $L(\alpha)$ from $\emph{U}_1$ such that $$\rho_{i}(\frac{2\alpha_{3}}{\alpha_{4}+ 2
\alpha_{3}^{2}},\frac{\alpha_{4}}{2 \alpha_{3}^{2}};\alpha)= a_i \
\ \mbox{for all} \ \ \emph{i}=\overline{3,n-1}.$$
Note that the above theorem describes the field of invariant rational functions on $Leib_{n+1}$ under the action of the adapted subgroup of $GL_{n+1}(\mathbf{C})$ and the part $ii)$ means the algebraic independence of the generators.
The procedure that we are applying works by the following way: first we present $FLeib_{n+1}$ as a disjoint union of subsets then formulate for each subset the analogue of the above theorem.
The list of algebras
====================
In this section we will present the list of none Lie complex filiform Leibniz algebras from $FLeib_{n+1}$ for $n=4,5,6,7.$ Later on $\Delta_4=\alpha_4+2\alpha_3^2, \ \
\Delta_5=\alpha_5-5\alpha_3^3, \ \ \Delta_6=\alpha_6+14\alpha_3^4,
\ \ \Delta_7=\alpha_7-42\alpha_3^5, \ \
\ \ \Theta_i=\theta-\alpha_i, \ \
i=4,5,6,7$ and the same letters $\Delta$ and $\Theta$ with $\prime$ will denote the same expression depending on parameters $\alpha_3', \alpha_4', \alpha_5', \alpha_6',\alpha_7', \theta'.$ Notice that $\Delta_i=\alpha_i$ $(i=4,5,6,7)$ when $\alpha_3=0.$
Let $N_n$ denote the number of isomorphism classes in dimension $n$ (each parametric family here will be considered as a one class).
**2.1 Dimension 5**
The class $FLeib_{5}$ can be represented as a disjoint union of the following subsets:
$\qquad FLeib_{5}=U_{1}\bigcup U_{2}\bigcup U_{3}\bigcup
U_{4}\bigcup U_{5}\bigcup U_{6}\bigcup U_{7},$ where
$\qquad U_1=\{L(\alpha)\in FLeib_5:\alpha_3\neq 0, \Delta_4 \neq 0
\},$
$\qquad U_2=\{L(\alpha)\in FLeib_5:\alpha_3\neq 0, \Delta_4=
0,\Theta_4\neq0\} ,$
$\qquad U_{3}=\{L(\alpha )\in FLeib_{5}:\alpha _{3}\neq
0,\Delta_4=0,\Theta_4=0 \},$
$\qquad U_{4}=\{L(\alpha )\in FLeib_{5}:\alpha _{3}=0,\Delta
_{4}\neq 0,\Theta_4\neq0\},$
$\qquad U_{5}=\{L(\alpha )\in FLeib_{5}:\alpha _{3}=0,\Delta
_{4}\neq 0,\Theta_4=0\},$
$\qquad U_{6}=\{L(\alpha )\in FLeib_{5}:\alpha _{3}=0,\Delta
_{4}=0,\Theta_4\neq0\},$
$\qquad U_{7}=\{L(\alpha )\in FLeib_{5}:\alpha _{3}=0,\Delta
_{4}=0,\Theta_4=0\}.$\
Now we will investigate the isomorphism problem for each of these sets separately.
**Proposition 2.1.1** Two algebras $L(\alpha )$ and $L(\alpha
^{\prime })$ from $U_1$ are isomorphic if and only if
$$\left( \frac{\alpha _{3}}{\Delta_4}\right)
^{2}\Theta_4=\left(\frac{\alpha^{\prime}_3}{\Delta_4'}\right)^{2}\Theta_4'$$
This means that the expression $$\left( \frac{\alpha
_{3}}{\Delta_4}\right) ^{2}\Theta_4$$ can be taken as a parameter $\lambda$ and then algebras from the set $U_{1}$ can be parameterized as $L(1,0,\lambda ).$\
**Proposition 2.1.2.**
a\) All algebras from the set $U_2$ are isomorphic to $L(1,-2,0);$
b\) All algebras from the set $U_{3}$ are isomorphic to $L(1,-2,-2);$
c\) All algebras from the set $U_{4}$ are isomorphic to $L(0,1,0);$
d\) All algebras from the set $U_{5}$ are isomorphic to $L(0,1,1);$
e\) All algebras from the set $U_{6}$ are isomorphic to $L(0,0,1);$
f\) All algebras from the set $U_{7}$ are isomorphic to $L(0,0,0).$\
**Theorem 2.1.3.** Let $L$ be a none Lie complex filiform Leibniz algebra in $FLeib_{5}$. Then it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras:
$1)\ \ L(1,0,{\lambda}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq
i\leq3,\ \ e_{0}e_{1}=e_{3}+\lambda e_{4}, \ \ e_{1}e_{1}=e_{3},$
$\qquad \qquad e_{2}e_{1}=e_{4},\ \ \lambda \in
\mathbf{C}.$\
$2) \ \ L(1,-2,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq
i\leq3,\ \ e_{0}e_{1}=e_{3},\ \ e_{1}e_{1}=e_{3}-2e_{4},$
$\qquad \qquad e_{2}e_{1}=e_{4}.$\
$3) \ \ L(1,-2,-2):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq
i\leq3 ,\ \ e_{0}e_{1}=e_{3}-2e_{4}, $
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4},\ \ e_{2}e_{1}=e_{4}
.$\
$4) \ \ L(0,1,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,},\ \ 1\leq
i\leq3 ,\ \ e_{1}e_{1}=e_{4}. $\
$5) \ \ L(0,1,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,},\ \ 1\leq
i\leq3 ,\ \ e_{0}e_{1}=e_{4},\ \ e_{1}e_{1}=e_{4}.$\
$6) \ \ L(0,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,},\ \ 1\leq
i\leq3 ,\ \ e_{0}e_{1}=e_{4}.$\
$7) \ \ L(0,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i\leq3.$\
The number of isomorphism classes $N_5=7.$\
**2.2 Dimension 6**
Now we consider the six dimensional case. The set $FLeib_6$ can be represented as a disjoint union of the subsets:
$\qquad FLeib_{6}=U_{1}\bigcup U_{2}\bigcup U_{3}\bigcup
U_{4}\bigcup U_{5}\bigcup U_{6}\bigcup U_{7}\bigcup U_{8}\bigcup
U_{9}\bigcup U_{10}\bigcup U_{11},$ where
$\qquad U_1=\{L(\alpha)\in FLeib_6:\alpha_3\neq 0, \Delta_4 \neq 0
\},$
$\qquad U_{2}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}\neq
0,\Delta_4=0,\Delta_5 \neq 0,\Theta_5\neq0\},$
$\qquad U_{3}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}=0,\Delta
_{4}\neq 0,\Delta _{5}\neq 0\},$
$\qquad U_{4}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}\neq
0,\Delta_4=0,\Delta_5 \neq 0,\Theta_5=0\},$
$\qquad U_{5}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}\neq
0,\Delta_4=0,\Delta_5=0,\Theta_5\neq0\},$
$\qquad U_{6}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}=0,\Delta
_{4}\neq 0,\Delta _{5}=0,\Theta_5\neq 0\},$
$\qquad U_{7}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}=0,\Delta
_{4}\neq 0,\Delta_{5}=0,\Theta_5 =0\},$
$\qquad U_{8}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}=0,\Delta
_{4}=0,\Delta _{5}\neq 0,\Theta_5\neq0\},$
$\qquad U_{9}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}=0,\Delta
_{4}=0,\Delta _{5}\neq 0,\Theta_5=0\},$
$\qquad U_{10}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}=0,\Delta
_{4}=0,\Delta _{5}=0,\Theta_5 \neq 0\},$
$\qquad U_{11}=\{L(\alpha )\in FLeib_{6}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Theta_5 =0\}.$
**Proposition 2.2.1.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_1$ are isomorphic if and only if
$$\frac{\alpha _{3}(\Delta_{5}+5\alpha
_{3}\Delta_4)}{\Delta_4^{2}}=\frac{\alpha
_{3}'(\Delta_{5}'+5\alpha _{3}'\Delta_4')}{\Delta_4'^{2}}$$ $$\frac{\alpha _{3}^{3}\Theta_{5}}{\Delta_4^{3}}=\frac{\alpha
_{3}'^{3}\Theta_{5}'}{\Delta_4'^{3}}$$
Thus the following two expressions can be taken as parameters $\lambda_{1},\lambda_{2}:$ $$\frac{\alpha _{3}(\Delta_{5}+5\alpha
_{3}\Delta_4)}{\Delta_4^{2}},$$ $$\frac{\alpha _{3}^{3}\Theta_{5}}{\Delta_4^{3}}$$ and algebras from $U_{1}$ can be parameterized as $$L(1,0,{\lambda }_{1},{\lambda }_{2}).$$
**Proposition 2.2.2.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_{2}$ are isomorphic if and only if
$$\frac{\Delta_5^{3}}{\alpha _{3}^{3}\Theta_5^{2}}=\
\frac{\Delta_5'^{3}}{\alpha _{3}'^{3}\Theta_5'^{2}}.$$
Thus in the set $U_{2}$ the expression $$\frac{\Delta_5^{3}}{\alpha _{3}^{3}\Theta_5^{2}}$$ can be taken as a parameter and algebras from $U_{2}$ can be parameterized as $$L(1,-2,{\lambda },2{\lambda }-5).$$
**Proposition 2.2.3.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $ U_{3} $ are isomorphic if and only if
$$\frac{\alpha _{4}^{3}\Theta_5}{\alpha _{5}^{3}}=\frac{\alpha
_{4}'^{3}\Theta_5'}{\alpha _{5}'^{3}}.$$
So as a parameter $\lambda$ in $U_{3}$ we will take the expression $$\frac{\alpha _{4}^{3}\Theta_5}{\alpha _{5}^{3}}$$ and write algebras from the set $U_{3}$ as $$L(0,1,1,{\lambda }).$$
**Proposition 2.2.4.**
a\) All algebras from the set $U_{4}$ are isomorphic to the algebra $L(1,-2,0,0);$
b\) All algebras from the set $U_5$ are isomorphic to the algebra $L(1,-2,5,0);$
c\) All algebras from the set $U_{5}$ are isomorphic to the algebra $L(0,1,0,1);$
d\) All algebras from the set $U_{6}$ are isomorphic to the algebra $L(0,1,0,0);$
e\) All algebras from the set $U_{7}$ are isomorphic to the algebra $L(0,0,1,0);$
f\) All algebras from the set $U_{8}$ are isomorphic to the algebra $L(0,0,1,1);$
g\) All algebras from the set $U_{9}$ are isomorphic to the algebra $L(0,0,0,1);$
h\) All algebras from the set $U_{10}$ are isomorphic to the algebra $L(0,0,0,0).$\
**Theorem 2.2.5.** Let $L$ be a none Lie complex filiform Leibniz algebra in $FLeib_{6}$. Then it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras:
$1)\ \ L(1,0,{\lambda} _{1},{\lambda} _{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{3}+\lambda _{2}e_{5},$
$\qquad \qquad e_{1}e_{1}=e_{3}+{\lambda }_{1}e_{5},\ \
e_{2}e_{1}=e_{4},\ \ e_{3}e_{1}=e_{5},\ \ {\lambda }_{1},{\lambda
}_{2}\in \mathbf{C}.$\
$2)\ \ L(1,-2,{\lambda} ,2{\lambda} -5):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{3}-2e_{4}+(2{\lambda }-5)e_{5},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+{\lambda }e_{5},\ \
e_{2}e_{1}=e_{4}-2e_{5},\ \ e_{3}e_{1}=e_{5},\ \ {\lambda }\in \mathbf{C}.$\
$3)\ \ L(0,1,1,{\lambda} ):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{4}+{\lambda }e_{5},$
$\qquad \qquad e_{1}e_{1}=e_{4}+e_{5},\ \ e_{2}e_{1}=e_{5},\ \
{\lambda }\in \mathbf{C}.$\
$4)\ \ L(1,-2,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{3}-2e_{4},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}, \ \
e_{2}e_{1}=e_{4}-2e_{5},\ \ e_{3}e_{1}=e_{5}.$\
$5)\ \ L(1,-2,5,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{3}-2e_{4},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+5e_{5},\ \
e_{2}e_{1}=e_{4}-2e_{5},\ \ e_{3}e_{1}=e_{5}.$\
$6)\ \ L(0,1,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{4}+e_{5},\ \ e_{1}e_{1}=e_{4},$
$\qquad \qquad e_{2}e_{1}=e_{5}.$\
$7)\ \ L(0,1,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{4},\ \ e_{1}e_{1}=e_{4},\ \
e_{2}e_{1}=e_{5}.$\
$8)\ \ L(0,0,1,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{1}e_{1}=e_{5}.$\
$9)\ \ L(0,0,1,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{5},\ \ e_{1}e_{1}=e_{5}.$\
$10)\ \ L(0,0,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 4,\ \ e_{0}e_{1}=e_{5}.$\
$11)\ \ L(0,0,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1}, \ \ 1\leq i\leq 4.$\
The number of isomorphism classes $N_6=11.$\
**2.3 Dimension 7**
$FLeib_{7}=U_{1}\bigcup U_{2}\bigcup U_{3}\bigcup U_{4}\bigcup
U_{5}\bigcup U_{6}\bigcup U_{7}\bigcup U_{8}\bigcup U_{9}\bigcup
U_{10}\bigcup U_{11}\bigcup U_{12}\bigcup $
$\qquad \qquad \ U_{13}\bigcup U_{14}\bigcup U_{15}\bigcup
U_{16}\bigcup U_{17}, $
where
$U_{1}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}\neq 0,\Delta_4\neq
0\},$
$U_{2}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}\neq 0,\Delta_4=0,
\Delta_5\neq 0,\Delta_{6}+6\alpha _{3}\Delta_5\neq 0\},$
$U_{3}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}\neq 0,\Delta_4=0,
\Delta_5\neq 0, \Delta_{6}+6\alpha _{3}\Delta_5=0\},$
$U_{4}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}\neq
0,\Delta_4=0,\Delta_{5}=0,\Delta_6 \neq 0,\Theta _{6}\neq 0\},$
$U_{5}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}\neq 0\},$
$U_{6}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}\neq 0,\Theta_{6}\neq
0\},$
$ U_{7}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}\neq 0,\Delta_{6}\neq 0\},$
$U_{8}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}\neq
0,\Delta_4=0,\Delta_5=0, \Delta_6\neq 0,\Theta _{6}=0\},$
$U_{9}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}\neq
0,\Delta_4=0,\Delta_5=0, \Delta_6=0,\Theta_{6}\neq 0\},$
$ U_{10}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}\neq 0,\Theta_{6}=0\},$
$U_{11}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}=0,\Theta_{6}\neq 0\},$
$U_{12}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}\neq 0,\Delta_{6}= 0,\Theta_6 \neq 0\},$
$ U_{13}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta
_{4}=0,\Delta _{5}\neq 0,\Delta_{6}= 0,\Theta_6 =0\},$
$ U_{14}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}\neq 0,\Theta_{6}\neq 0\},$
$U_{15}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}\neq 0,\Theta_{6}=0\},$
$U_{16}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}=0,\Theta_6 \neq 0\},$
$ U_{17}=\{L(\alpha )\in FLeib_{7}:\alpha _{3}=0,\Delta
_{4}=0,\Delta _{5}=0,\Delta_{6}=0,\Theta_6 =0\}.$\
**Proposition 2.3.1.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_1$ are isomorphic if and only if
$$\frac{\alpha _{3}(\Delta_{5}+5\alpha
_{3}\Delta_4)}{\Delta_4^{2}}=\frac{\alpha
_{3}'(\Delta_{5}'+5\alpha _{3}'\Delta_4')}{\Delta_4'^{2}}$$ $$\frac{\alpha
_{3}(\alpha_{3}\Delta_{6}+6\alpha_{3}^{2}\Delta_{5}-3\Delta_{4}\Delta_{5}+9\alpha_{3}^{3}\Delta_{4}-12\alpha_{3}
\Delta_{4}^{2})}{\Delta_4^{3}}=$$ $$\frac{\alpha
_{3}'(\alpha_{3}'\Delta_{6}'+6\alpha_{3}'^{2}\Delta_{5}'-3\Delta_{4}'\Delta_{5}'+9\alpha_{3}'^{3}\Delta_{4}'-12\alpha_{3}'
\Delta_{4}'^{2})}{\Delta_{4}'^{3}}$$
$$\frac{\alpha _{3}^4\Theta_6}{\Delta_{4}^4}=\frac{\alpha
_{3}'^4\Theta_6'}{\Delta_{4}'^4}.$$
Thus, in this case algebras from the set $U_{1}$ can be parameterized as $L(1,0,\lambda _{1},\lambda _{2},\lambda
_{3}).$\
**Proposition 2.3.2** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{2}$ are isomorphic if and only if
$$\frac{\Delta_5^{3}}{\alpha_3(\Delta_6+6\alpha_3\Delta_5)^2}=\frac{\Delta_5'^{3}}{\alpha_3'(\Delta_6'+6\alpha_3'\Delta_5')^2}$$ $$\begin{aligned}
\frac{\Delta_5^{4}\Theta_{6}}{\left(\Delta_6+6\alpha_3\Delta_5\right)
^{4}}
=\frac{\Delta_5'^{4}\Theta_{6}'}{\left(\Delta_6'+6\alpha_3'\Delta_5'\right)
^{4}}\end{aligned}$$
The expressions the above can be taken as parameters in $U_{2}$ and the set $U_{2}$ can be represented as $L(1,-2,\lambda _{1},
-5\lambda _{1}-14,\lambda _{2}).$\
**Proposition 2.3.3.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{3}$ are isomorphic if and only if
$$\frac{\alpha _{3}^{4}\Theta_{6}^{2}}{\Delta_{5}^{4}}=\frac{\alpha
_{3}'^{4}\Theta_{6}'^{2}}{\Delta_{5}'^{4}}.$$
The parameter $\lambda$ for algebras from the set $U_{3}$ is $$\frac{\alpha _{3}^{4}\Theta_{6}}{\Delta_{5}^{4}}$$ and $U_{3}$ can be parameterized as $L(1,-2,0,16,\lambda ).$\
**Proposition 2.3.4.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{4}$ are isomorphic if and only if
$$\frac{\Delta_6^{4}}{\alpha
_{3}^{4}\Theta_{6}^{3}}=\frac{\Delta_6'^{4}}{\alpha
_{3}'^{4}\Theta_{6}'^{3}}.$$
$U_{4}$ can be parameterized as $L(1,-2,5,\lambda ,2\lambda -14).$\
**Proposition 2.3.5.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{5}$ are isomorphic if and only if
$$\frac{\Delta_{4}(\Delta_{6}+3\Delta_{4}^{2})}{\Delta
_{5}^{2}}=\frac{\Delta_{4}'(\Delta_{6}'+3\Delta_{4}'^{2})}{\Delta
_{5}'^{2}},$$
$$\left(\frac{\Delta
_{4}}{\Delta_{5}}\right)^4\Theta_{6}=\left(\frac{\Delta
_{4}'}{\Delta_{5}'}\right)^4\Theta_{6}'.$$
The set $U_{5}$ can be parameterized as $L(0,1,1,\lambda _{1},\lambda _{2}).$\
**Proposition 2.3.6.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{6}$ are isomorphic if and only if
$$\frac{(\Delta_{6}+3\Delta_{4}^{2})^2}{\Delta_{4}^{2}\Theta_{6}}=\frac{(\Delta_{6}'+3\Delta_{4}'^{2})^2}{\Delta_{4}'^{2}\Theta_{6}'}.$$
$U_{6}$ can be represented as a parameterized family of algebras $L(0,1,0,\lambda ,2\lambda -3).$\
**Proposition 2.3.7.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{7}$ are isomorphic if and only if
$$\left(\frac{\Delta_{5}}{\Delta_{6}}\right)^4\Theta_6=\left(\frac{\Delta_{5}'}{\Delta_{6}'}\right)^4\Theta_6'.$$
We will get one parametric family of algebras for the set $U_{7}:$ $L(0,0,1,1,\lambda ).$\
**Proposition 2.3.8**
a\) All algebras from the set $U_{8}$ are isomorphic to $L(1,-2,5,0,0);$
b\) All algebras from $U_{9}$ are isomorphic to $L(1,-2,5,14,0);$
c\) Algebras from $U_{10}$ are isomorphic to $L(0,1,0,0,0);$
d\) All algebras from $U_{11}$ are isomorphic to $L(0,1,0,-3,0);$
e\) All algebras from $U_{12}$ are isomorphic to $L(0,0,1,0,1);$
f\) All algebras from $U_{13}$ are isomorphic to $L(0,0,1,0,0);$
g\) All algebras from $U_{14}$ are isomorphic to $L(0,0,0,1,0);$
h\) Algebras from $U_{15}$ are isomorphic to $L(0,0,0,1,1);$
i\) Algebras from $U_{16}$ are isomorphic to $L(0,0,0,0,1);$
j\) Algebras from $U_{17}$ are isomorphic to $L(0,0,0,0,0).$\
**Theorem 2.3.9.** Let $L$ be a none Lie complex filiform Leibniz algebra in $FLeib_{7}$. Then it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras:
$1)\ \ L(1,0,{\lambda} _{1},{\lambda} _{2},{\lambda} _{3}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{3}+{\lambda }_{1}e_{5}+{\lambda
}_{3}e_{6},$
$\qquad \qquad e_{1}e_{1}=e_{3}+{\lambda} _{1}e_{5}+{\lambda}
_{2}e_{6},\ \ e_{2}e_{1}=e_{4}+{\lambda} _{1}e_{6},\ \
e_{3}e_{1}=e_{5},\ \ e_{4}e_{1}=e_{6},$
$\qquad \qquad {\lambda }_{1},{\lambda }_{2},{\lambda }_{3}\in \mathbf{C}.$\
$2)\ \ L(1,-2,{\lambda} _{1},-5{\lambda} _{1}-14,{\lambda} _{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{3}-2e_{4}+{\lambda }_{1}e_{5}+{\lambda
}_{2}e_{6},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+{\lambda}
_{1}e_{5}+(-5{\lambda} _{1}-14)e_{6},\ \
e_{2}e_{1}=e_{4}-2e_{5}+{\lambda} _{1}e_{6},$
$\qquad \qquad e_{3}e_{1}=e_{5}-2e_{6},\ \ e_{4}e_{1}=e_{6},\ \ {\lambda }_{1},{\lambda }_{2}\in \mathbf{C}.$\
$3)\ \ L(1,-2,0,16,{\lambda} ):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{3}-2e_{4}+{\lambda }e_{6},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+16e_{6},\ \
e_{2}e_{1}=e_{4}-2e_{5},\ \ e_{3}e_{1}=e_{5}-2e_{6},$
$\qquad \qquad e_{4}e_{1}=e_{6},\ \ {\lambda }\in
\mathbf{C}.$\
$4)\ \ L(1,-2,5,{\lambda},2{\lambda} -14):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{3}-2e_{4}+5e_{5}+(2{\lambda }-14)e_{6},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+5e_{5}+{\lambda} e_{6},\ \
e_{2}e_{1}=e_{4}-2e_{5}+5e_{6},\ \ e_{3}e_{1}=e_{5}-2e_{6},$
$\qquad \qquad e_{4}e_{1}=e_{6},\ \ {\lambda }\in
\mathbf{C}.$\
$5)\ \ L(0,1,1,{\lambda} _{1},{\lambda} _{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{4}+e_{5}+{\lambda }_{2}e_{6},$
$\qquad \qquad e_{1}e_{1}=e_{4}+e_{5}+{\lambda }_{1}e_{6},\ \
e_{2}e_{1}=e_{5}+e_{6},\ \ e_{3}e_{1}=e_{6},\ \ {\lambda }_{1},{\lambda }_{2}\in \mathbf{C}.$\
$6)\ \ L(0,1,0,{\lambda} ,2{\lambda}-3): $
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{4}+(2{\lambda }-3)e_{6},$
$\qquad \qquad e_{1}e_{1}=e_{4}+{\lambda }e_{6},\ \
e_{2}e_{1}=e_{4},\ \ e_{3}e_{1}=e_{6},\ \ {\lambda }\in
\mathbf{C}.$\
$7)\ \ L(0,0,1,1,{\lambda}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{5}+{\lambda }e_{6},\ \
e_{1}e_{1}=e_{5}+e_{6},$
$\qquad \qquad e_{2}e_{1}=e_{6},\ \ {\lambda }\in \mathbf{C}.\
$\
$8)\ \ L(1,-2,5,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{3}-2e_{4}+5e_{5},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+5e_{5},\ \
e_{2}e_{1}=e_{4}-2e_{5}+5e_{6},\ \ e_{3}e_{1}=e_{5}-2e_{6},\ \
e_{4}e_{1}=e_{6}.$\
$9)\ \ L(1,-2,5,14,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{3}-2e_{4}+5e_{5},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+5e_{5}-14e_{6},\ \
e_{2}e_{1}=e_{4}-2e_{5}+5e_{6},\ \ e_{3}e_{1}=e_{5}-2e_{6},$
$\qquad \qquad e_{4}e_{1}=e_{6}.$\
$10)\ \ L(0,1,0,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{4},\ \ e_{1}e_{1}=e_{4},\ \
e_{2}e_{1}=e_{5},$
$\qquad \qquad e_{3}e_{1}=e_{6}.$\
$11)\ \ L(0,1,0,-3,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{4},\ \ e_{1}e_{1}=e_{4}-3e_{6},$
$\qquad \qquad e_{2}e_{1}=e_{5},\ \ e_{3}e_{1}=e_{6}.$\
$12)\ \ L(0,0,1,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{5}+e_{6},\ \ e_{2}e_{1}=e_{6}.$\
$13)\ \ L(0,0,1,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{5},\ \ e_{2}e_{1}=e_{6}.$\
$14)\ \ L(0,0,0,1,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq
i\leq 5,\ \ e_{1}e_{1}=e_{6}.$\
$15)\ \ L(0,0,0,1,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{6,}\ \ e_{1}e_{1}=e_{6}.$\
$16)\ \ L(0,0,0,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq
i\leq 5,\ \ e_{0}e_{1}=e_{6}.$\
$17)\ \ L(0,0,0,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i\leq 5.$\
The number of isomorphism classes $N_7=17.$\
**2.4 Dimension 8**
$FLeib_{8}=U_{1}\bigcup U_{2}\bigcup U_{3}\bigcup
U_{4}\bigcup U_{5}\bigcup U_{6}\bigcup U_{7}\bigcup U_{8}\bigcup
U_{9}\bigcup U_{10}\bigcup U_{11}\bigcup U_{12}\bigcup U_{13}
\bigcup$
$\qquad \qquad \ \ U_{14}\bigcup U_{15}\bigcup U_{16}\bigcup
U_{17}\bigcup U_{18}\bigcup U_{19}\bigcup U_{20}\bigcup \bigcup
U_{21}\bigcup U_{22}\bigcup U_{23}\bigcup U_{24}\bigcup U_{25},$
where
$U_{1}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}\neq 0,\Delta _{4}\neq
0\},$
$U_{2}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}\neq 0,\Delta
_{4}=0,\Delta_{5}\neq 0,\Delta_{6}+6\alpha _{3}\Delta_{5}\neq
0\},$
$U_{3}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}\neq 0,\Delta
_{4}=0,\Delta_{5}\neq 0,\Delta_{6}+6\alpha _{3}\Delta_{5}\ =0,
\Theta_{7}\neq 0\},$
$U_{4}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}\neq 0,\Delta
_{4}=0,\Delta_{5}\neq 0, \Delta_{6}+6\alpha _{3}\Delta_{5}\
=0,\Theta_{7}= 0\},$
$U_{5}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}\neq 0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}\neq 0, \Delta_{7}+7\alpha
_{3}\Delta_{6}\neq 0\},$
$U_{6}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}\neq
0,\Delta_{4}=0,\Delta_{5}=0, \Delta _{6}\neq 0, \Delta_{7}+7\alpha
_{3}\Delta_{6}=0\},$
$U_{7}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}\neq 0,\Delta
_{4}=0,\Delta_{5}=0, \Delta_{6}=0, \Delta_{7}\neq 0\},$
$U_{8}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}\neq 0\},$
$ U_{9}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}\neq 0,\Delta _{7}\neq
0\},$
$ U_{10}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta _{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}\neq 0,\Theta _{7} \neq
0\},$
$ U_{11}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}=0,\Delta _{7}\neq
0,\Theta_7\neq 0\},$
$ U_{12}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}\neq 0,\Delta_{6}\neq 0\},$
$U_{13}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}\neq 0,\Delta_{6}= 0,\Delta_{7}\neq 0,\Theta
_{7}\neq 0\},$
$U_{14}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}\neq 0,\Delta_{7}\neq 0\},$
$ U_{15}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}\neq 0,\Delta
_{7}=0,\Theta_7=0\}, $
$U_{16}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}=0,\Delta_{7}\neq
0,\Theta_7= 0\},$
$U_{17}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta_{4}\neq
0,\Delta_{5}=0,\Delta_{6}+3\Delta_{4}^{2}=0,\Delta_{7}=0,\Theta_7
\neq 0\},$
$U_{18}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}\neq 0,\Delta_{6}= 0,\Delta_{7}\neq
0,\Theta_{7}=0\},$
$U_{19}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}\neq 0,\Delta_{6}= 0,\Delta_{7}=0,\Theta_7\neq
0\},$
$U_{20}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}\neq 0,\Delta_{7}=0,\Theta_7 \neq
0\},$
$U_{21}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}\neq 0,\Delta_{7}=0,\Theta_7 =0\},$
$U_{22}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}=0,\Delta_{7}\neq 0,\Theta_7\neq
0\},$
$U_{23}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta _{5}=0,\Delta_{6}=0,\Delta_{7}\neq 0,\Theta_7=0\},$
$U_{24}=\{L(\alpha )\in FLeib_{8}:\alpha
_{3}=0,\Delta_{4}=0,\Delta_{5}=0,\Delta_{6}=0,\Delta_{7}=0,\Theta_7
\neq 0\},$
$U_{25}=\{L(\alpha )\in FLeib_{8}:\alpha _{3}=0,\Delta
_{4}=0,\Delta_{5}=0,\Delta_{6}=0,\Delta_{7}=0,\Theta_7=0\}.$\
**Proposition 2.4.1.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_1$ are isomorphic if and only if
$$\frac{\alpha _{3}(\Delta_{5}+5\alpha
_{3}\Delta_4)}{\Delta_4^{2}}=\frac{\alpha
_{3}'(\Delta_{5}'+5\alpha _{3}'\Delta_4')}{\Delta_4'^{2}}$$ $$\frac{\alpha
_{3}(\alpha_{3}\Delta_{6}+6\alpha_{3}^{2}\Delta_{5}-3\Delta_{4}\Delta_{5}+9\alpha_{3}^{3}\Delta_{4}-12\alpha_{3}
\Delta_{4}^{2})}{\Delta_4^{3}}=$$ $$\frac{\alpha
_{3}'(\alpha_{3}'\Delta_{6}'+6\alpha_{3}'^{2}\Delta_{5}'-3\Delta_{4}'\Delta_{5}'+9\alpha_{3}'^{3}\Delta_{4}'-12\alpha_{3}'
\Delta_{4}'^{2})}{\Delta_{4}'^{3}}$$
$$\frac{\alpha _{3}^{3}\Delta_{7}+28\alpha _{3}^{4}\Delta
_{4}^{2}+7\alpha _{3}^{6}\Delta_4+14\alpha _{3}^{5}\Delta
_{5}+7\alpha _{3}^{4}\Delta_6+7\alpha
_{3}^{3}\Delta_{4}\Delta_{5}}{\Delta_{4}^{4}}=$$ $$\frac{\alpha _{3}'^{3}\Delta_{7}'+28\alpha
_{3}'^{4}\Delta _{4}'^{2}+7\alpha _{3}'^{6}\Delta_4'+14\alpha
_{3}'^{5}\Delta _{5}'+7\alpha _{3}'^{4}\Delta_6'+7\alpha
_{3}'^{3}\Delta_{4}'\Delta_{5}'}{\Delta_{4}'^{4}}$$
$$\frac{\alpha _{3}^{5}\Theta_{7}}{\Delta_{4}^{5}} =\frac{\alpha
_{3}'^{5}\Theta_{7}'}{\Delta_{4}'^{5}}$$
$U_{1}$ is parameterized as $L(1,0,\lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4}).$\
**Proposition 2.4.2.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{2}$ are isomorphic if and only if
$$\frac{\Delta_{5}^{3}}{\alpha _{3}(\Delta_{6}+6\alpha
_{3}\Delta_{5})^2}=\frac{\Delta_{5}'^{3}}{\alpha
_{3}'(\Delta_{6}'+6\alpha _{3}'\Delta_{5}')^2}$$ $$\frac{\Delta_{5}^{4}(\Delta_{7}+7\alpha _{3}\Delta_{6}+14\alpha
_{3}^2\Delta_{5})}{\alpha _{3}(\Delta_{6}+6\alpha _{3}\Delta
_{5})^{4}}=\frac{\Delta_{5}'^{4}(\Delta_{7}'+7\alpha
_{3}'\Delta_{6}'+14\alpha _{3}'^2\Delta_{5}')}{\alpha
_{3}'(\Delta_{6}'+6\alpha _{3}'\Delta _{5}')^{4}}$$ $$\frac{\Delta_{5}^{5}\Theta_{7}}{(\Delta_{6}+6\alpha _{3}\Delta
_{5})^{5}} =\frac{\Delta_{5}'^{5}\Theta_{7}'}{(\Delta_{6}'+6\alpha
_{3}'\Delta _{5}')^{5}}$$
The set $U_{2}$ can be parameterized as $L(1,-2,\lambda
_{1},-5\lambda _{1}-14,\lambda _{2},\lambda _{3}).
$\
**Proposition 2.4.3.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{3}$ are isomorphic if and only if
$$\left( \frac{\Delta_{5}}{\alpha
_{3}}\right)^{5}\frac{1}{\Theta_{7}^{2}}=\left(\frac{\Delta_{5}'}{\alpha
_{3}}'\right)^{5}\frac{1}{\Theta_{7}'^{2}}$$ $$\frac{\Delta_{5}^{8}(\Delta_{7}-28\alpha _{3}^{2}\Delta
_{5})}{\alpha _{3}^{9}\Theta_{7}^{4}}
=\frac{\Delta_{5}'^{8}(\Delta_{7}'-28\alpha _{3}'^{2}\Delta
_{5}')}{\alpha _{3}'^{9}\Theta_{7}'^{4}}$$
The algebras from the set $U_{3}$ can be parameterized as $L(1,-2,\lambda _{1},-6\lambda _{1}-14,\lambda _{2},\lambda
_{1}^{2}).$\
**Proposition 2.4.4.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{4}$ are isomorphic if and only if
$$\frac{\alpha _{3}(\Delta_{7}-28\alpha _{3}^{2}\Delta_{5})}{\Delta
_{5}^{2}}=\frac{\alpha _{3}'(\Delta_{7}'-28\alpha
_{3}'^{2}\Delta_{5}')}{\Delta _{5}'^{2}}.$$
The set $U_4$ can be parameterized as $L(1,-2,0,16,\lambda ,\lambda ).$\
**Proposition 2.4.5.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{5}$ are isomorphic if and only if
$$\frac{\Delta_{6}^{4}}{\alpha _{3}(\Delta_{7}+7\alpha _{3}\Delta
_{6})^{3}}\\
=\frac{\Delta_{6}'^{4}}{\alpha _{3}'(\Delta_{7}'+7\alpha
_{3}'\Delta _{6}')^{3}}.$$ $$\left(\frac{\Delta_{6}}{\Delta_{7}+7\alpha _{3}\Delta
_{6}}\right)^5\Theta_7
=\left(\frac{\Delta_{6}'}{\Delta_{7}'+7\alpha _{3}'\Delta
_{6}'}\right)^5\Theta_7'$$
$U_{5}$ is parameterized as $L(1,-2,5,\lambda _{1},-6\lambda _{1}+42,\lambda _{2}).$\
**Proposition 2.4.6.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{6}$ are isomorphic if and only if
$$\left(\frac{\alpha
_{3}}{\Delta_{6}}\right)^{5}\Theta_{7}^{3}=\left(\frac{\alpha
_{3}'}{\Delta_{6}'}\right)^{5}\Theta_{7}'^{3}$$
The set $U_{6}$ can be parameterized as $L(1,-2,5,\lambda ,-7\lambda +42,\lambda ^{2}).$\
**Proposition 2.4.7.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_{7}$ are isomorphic if and only if
$$\left(\frac{\Delta_{7}}{\alpha
_{3}}\right)^5\frac{1}{\Theta_{7}^{4}}=\left(\frac{\Delta_{7}'}{\alpha
_{3}'}\right)^5\frac{1}{\Theta_{7}'^{4}}$$ $U_{7}$ is parameterized as $L(1,-2,5,-14,\lambda,2({\lambda} +21)).$\
**Proposition 2.4.8.** Two algebras $L(\alpha )$ and $L(\alpha')$ from $U_{8} $ are isomorphic if and only if $$\frac{\Delta_{4}(\alpha _{6}+3\alpha _{4}^{2})}{\Delta
_{5}^{2}}=\frac{\Delta_{4}'(\alpha _{6}'+3\alpha
_{4}'^{2})}{\Delta _{5}'^{2}}$$
$$\frac{\Delta_{4}^{2}(\Delta_{7}+7\Delta_{4}\Delta_{5})}{\Delta
_{5}^{3}}=\frac{\Delta_{4}'^{2}(\Delta
_{7}'+7\Delta_{4}'\Delta_{5}')}{\Delta _{5}'^{3}}.$$ $$\left(\frac{\Delta_{4}}{\Delta_{5}}\right)^{5}\Theta_{7}=\left(\frac{\Delta_{4}'}{\Delta_{5}'}\right)^{5}\Theta_{7}'.$$
$U_8$ can be parameterized as $L(0,1,1,\lambda _{1},\lambda _{2},\lambda _{3}).$\
**Proposition 2.4.9.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_{9} $ are isomorphic if and only if
$$\frac{(\Delta_{6}+3\Delta_{4}^{2})^{3}}{\Delta_{4}\Delta
_{7}^{2}}=\frac{(\Delta_{6}'+3\Delta_{4}'^{2})^{3}}{\Delta_{4}'\Delta
_{7}'^{2}}$$$$\left(\frac{\Delta_{6}+3\Delta
_{4}^{2}}{\Delta_{7}}\right)^{5}\Theta_{7}=\left(\frac{\Delta_{6}'+3\Delta
_{4}'^{2}}{\Delta_{7}'}\right)^{5}\Theta_{7}'.$$
Thus the algebras from the set $U_{9}$ can be parameterized as $L(0,1,0,\lambda _{1},\lambda _{1}+3,\lambda _{2}).$\
**Proposition 2.4.10.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_{10}$ are isomorphic if and only if
$$\left(\frac{\Delta_{6}+3\Delta_{4}^{2}}{\Delta_{4}}\right)^{5}\Theta_7^2
=\left(\frac{\Delta_{6}'+3\Delta_{4}'^{2}}{\Delta_{4}'}\right)^{5}\Theta_7'^{2}.$$
$U_{10}$ can be parameterized as $L(0,1,0,\lambda ,0,\lambda ^{2}).$\
**Proposition 2.4.11.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_{11}$ are isomorphic if and only if
$$\left(\frac{\Delta_{4}}{\Delta_{7}}\right)^{5}\Theta
_{7}^{3}=\left(\frac{\Delta_{4}'}{\Delta_{7}'}\right)^{5}\Theta
_{7}'^{3}.$$
$L(0,1,0,-3,\lambda ,\lambda ^{2}+\lambda )$ are representatives of $U_{11}.$\
**Proposition 2.4.12.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_{12}$ are isomorphic if and only if
$$\frac{\Delta_{5}\Delta_{7}}{\Delta
_{6}^{2}}=\frac{\Delta_{5}'\Delta_{7}'}{\Delta _{6}'^{2}}$$ $$\left(\frac{\Delta_{5}}{\Delta_{6}}\right)^{5}\Theta_{7}=\left(\frac{\Delta_{5}'}{\Delta_{6}'}\right)^{5}\Theta_{7}'.$$
Thus, the algebras from the set $U_{12}$ can be parameterized as $L(0,0,1,1,\lambda _{1},\lambda _{2}).$\
**Proposition 2.4.13.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_{13}$ are isomorphic if and only if
$$\left(\frac{\Delta_{7}}{\Delta_{5}}\right)^{5}\frac{1}{\Theta_{7}^{2}}=\left(\frac{\Delta_{7}'}{\Delta_{5}'}\right)^{5}\frac{1}{\Theta_{7}'^{2}}.$$
Thus, the algebras from the set $U_{13}$ can be parameterized as $L(0,0,1,0,\lambda ,\lambda ^{2}+\lambda ).$\
**Proposition 2.4.14.** Two algebras $L(\alpha )$ and $L(\alpha ^{\prime })$ from $U_{14}$ are isomorphic if and only if
$$\left(\frac{\Delta_{6}}{\Delta_{7}}\right)^{5}\Theta_{7}=\left(\frac{\Delta_{6}'}{\Delta_{7}'}\right)^{5}\Theta_{7}'.$$
$L(0,0,0,1,1,\lambda )$ is a parametrization of $U_{14}.$\
**Proposition 2.4.15**
a\) All algebras from the set $U_{15}$ are isomorphic to $L(0,1,0,0,0,0);$
b\) All algebras from the set $U_{16}$ are isomorphic to $L(0,1,0,-3,1,1);$
c\) All algebras from the set $U_{17}$ are isomorphic to $L(0,1,0,-3,0,1);$
d)All algebras from the set $U_{18}$ are isomorphic to $L(0,0,1,0,1,1);$
e\) All algebras from the set $U_{19}$ are isomorphic to $L(0,0,1,0,0,1);$
f\) All algebras from the set $U_{20}$ are isomorphic to $L(0,0,0,1,0,1);$
g\) All algebras from the set $U_{21}$ are isomorphic to $L(0,0,0,1,0,0);$
h\) All algebras from the set $U_{22}$ are isomorphic to $L(0,0,0,0,1,0);$
i\) All algebras from the set $U_{23}$ are isomorphic to $L(0,0,0,0,1,1);$
j\) All algebras from the set $U_{24}$ are isomorphic to $L(0,0,0,0,0,1);$
k\) All algebras from the set $U_{25}$ are isomorphic to $L(0,0,0,0,0,0).$\
**Theorem 2.4.16.** Let $L$ be a none Lie complex filiform Leibniz algebra in $FLeib_{8}$. Then it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras:
$1)\ \ L(1,0,{\lambda} _{1},{\lambda} _{2},{\lambda}
_{3},{\lambda} _{4}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i
\leq 6,\ \ e_{0}e_{1}=e_{3}+{\lambda} _{1}e_{5}+{\lambda}
_{2}e_{6}+{\lambda} _{4}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{3}+{\lambda} _{1}e_{5}+{\lambda} _{2}e_{6}+{\lambda} _{3}e_{7},\ \ e_{2}e_{1}=e_{4}+{\lambda} _{1}e_{6}+{\lambda}_{3}e_{7},$
$\qquad \qquad e_{3}e_{1}=e_{5}+{\lambda }_{1}e_{7},\ \
e_{4}e_{1}=e_{6},\ \
e_{5}e_{1}=e_{7},\ \ {\lambda }_{1},{\lambda }_{2},{\lambda }_{3},{\lambda }_{4}\in \mathbf{C}.$\
$2)\ \ L(1,-2,{\lambda} _{1},-(5{\lambda} _{1}+14),{\lambda}
_{2},{\lambda} _{3}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i
\leq 6,\
\ e_{0}e_{1}=e_{3}-2e_{4}+{\lambda} _{1}e_{5}+{\lambda}_{2}e_{6}+{\lambda}_{3}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+{\lambda}
_{1}e_{5}-(5{\lambda} _{1}+14)e_{6}+{\lambda} _{2}e_{7},$
$\qquad \qquad e_{2}e_{1}=e_{4}-2e_{5}+{\lambda}
_{1}e_{6}-(5{\lambda} _{1}+14)e_{7},\ \
e_{3}e_{1}=e_{5}-2e_{6}+{\lambda}_{1}e_{7},$
$\qquad \qquad e_{4}e_{1}=e_{6}-2e_{7},\ \ e_{5}e_{1}=e_{7},\ \ {\lambda }_{1},{\lambda }_{2},{\lambda }_{3}\in \mathbf{C}.$\
$3)\ \ L(1,-2,{\lambda} _{1},-(6{\lambda} _{1}+14),{\lambda}
_{2},{\lambda} _{1}^{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i
\leq 6,\ \ e_{0}e_{1}=e_{3}-2e_{4}+{\lambda} _{1}e_{5}+{\lambda}
_{2}e_{6}+{\lambda} _{1}^{2}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+{\lambda}
_{1}e_{5}-(6{\lambda} _{1}+14)e_{6}+{\lambda} _{2}e_{7},$
$\qquad \qquad e_{2}e_{1}=e_{4}-2e_{5}+{\lambda}
_{1}e_{6}-(6{\lambda} _{1}+14)e_{7},\ \
e_{3}e_{1}=e_{5}-2e_{6}+{\lambda} _{1}e_{7},$
$\qquad \qquad e_{4}e_{1}=e_{6}-2e_{7},\ \ e_{5}e_{1}=e_{7},\ \ {\lambda }_{1},{\lambda }_{2}\in \mathbf{C}.$\
$4)\ \ L(1,-2,0,16,{\lambda} ,{\lambda} ):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq 6,\ \ e_{0}e_{1}=e_{3}-2e_{4}+16e_{6}+{\lambda} e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+16e_{6}+{\lambda} e_{7},\ \
e_{2}e_{1}=e_{4}-2e_{5}+16e_{7},$
$\qquad \qquad e_{3}e_{1}=e_{5}-2e_{6},\ \
e_{4}e_{1}=e_{6}-2e_{7},\ \ e_{5}e_{1}=e_{7},\ \ {\lambda }\in
\mathbf{C}.$\
$5)\ \ L(1,-2,5,{\lambda}_{1},-6({\lambda} _{1}-7),{\lambda}
_{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq 6,\ \ e_{0}e_{1}=e_{3}-2e_{4}+5e_{5}+{\lambda}
_{1}e_{6}+{\lambda} _{2}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+5e_{5}+{\lambda} _{1}e_{6}-6({\lambda}_{1}-7)e_{7},$
$\qquad \qquad e_{2}e_{1}=e_{4}-2e_{5}+5e_{6}+{\lambda}
_{1}e_{7},\ \ e_{3}e_{1}=e_{5}-2e_{6}+5e_{7},$
$\qquad \qquad e_{4}e_{1}=e_{6}-2e_{7},\ \ e_{5}e_{1}=e_{7},\ \ {\lambda }_{1},{\lambda }_{2}\in \mathbf{C}.$\
$6)\ \ L(1,-2,5,{\lambda} ,-7({\lambda}-6),{\lambda} ^{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq 6,\ \ e_{0}e_{1}=e_{3}-2e_{4}+5e_{5}+{\lambda}
e_{6}+{\lambda}^{2}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+5e_{5}+{\lambda}
e_{6}-7({\lambda}-6)e_{7},$
$\qquad \qquad e_{2}e_{1}=e_{4}-2e_{5}+5e_{6}+{\lambda} e_{7},\ \
e_{3}e_{1}=e_{5}-2e_{6}+5e_{7},$
$\qquad \qquad e_{4}e_{1}=e_{6}-2e_{7},\ \ e_{5}e_{1}=e_{7},\ \ {\lambda }\in \mathbf{C}.$\
$7)\ \ L(1,-2,5,-14,{\lambda},2({\lambda} +21)):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq 6,$
$\qquad \qquad e_{0}e_{1}=e_{3}-2e_{4}+5e_{5}-14e_{6}+2({\lambda}
+21)e_{7}, $
$\qquad \qquad e_{1}e_{1}=e_{3}-2e_{4}+5e_{5}-14e_{6}+{\lambda}
e_{7},\ \ e_{2}e_{1}=e_{4}-2e_{5}+5e_{6}-14e_{7},$
$\qquad \qquad e_{3}e_{1}=e_{5}-2e_{6}+5e_{7},\ \
e_{4}e_{1}=e_{6}-2e_{7},\ \ e_{5}e_{1}=e_{7},\ \ {\lambda}
\in \mathbf{C}.$\
$8)\ \ L(0,1,1,{\lambda}_{1},{\lambda} _{2},{\lambda} _{3}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq 6,\ \ e_{0}e_{1}=e_{4}+e_{5}+{\lambda} _{1}e_{6}+{\lambda}
_{3}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{4}+e_{5}+{\lambda}
_{1}e_{6}+{\lambda} _{2}e_{7},\ \
e_{2}e_{1}=e_{5}+e_{6}+{\lambda}_{1}e_{7},$
$\qquad \qquad e_{3}e_{1}=e_{6}+e_{7},\ \ e_{4}e_{1}=e_{7},\ \ {\lambda }_{1},{\lambda }_{2},{\lambda }_{3}\in \mathbf{C}.$\
$9)\ \ L(0,1,0,{\lambda}_{1},{\lambda}_{1}+3,{\lambda} _{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i1}e_{0}=e_{i+1,}\ \ 1\leq
i \leq 6,\ \ e_{0}e_{1}=e_{4}+{\lambda} _{1}e_{6}+{\lambda}
_{2}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{4}+{\lambda} _{1}e_{6}+({\lambda}
_{1}+3)e_{7},\ \ e_{2}e_{1}=e_{5}+{\lambda} _{1}e_{7},\ \
e_{3}e_{1}=e_{6},$
$\qquad \qquad e_{4}e_{1}=e_{7},\ \ {\lambda }_{1},{\lambda }_{2}\in \mathbf{C}.$\
$10)\ \ L(0,1,0,{\lambda} ,0,{\lambda} ^{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i
\leq 6,\ \ e_{0}e_{1}=e_{4}+{\lambda} e_{6}+{\lambda} ^{2}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{4}+{\lambda }e_{6},\ \ e_{2}e_{1}=e_{5}+{\lambda }e_{7},\ \ e_{3}e_{1}=e_{6},\ \ e_{4}e_{1}=e_{7},\ \
{\lambda }\in \mathbf{C}.$\
$11)\ \ L(0,1,0,-3,{\lambda} ,{\lambda} ^{2}+{\lambda} ):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i
\leq6,\ \ e_{0}e_{1}=e_{4}-3e_{6}+({\lambda} ^{2}+{\lambda}
)e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{4}-3e_{6}+{\lambda} e_{7},\ \
e_{2}e_{1}=e_{5}-3e_{7},\ \ e_{3}e_{1}=e_{6},\ \
e_{4}e_{1}=e_{7},$
$\qquad \qquad {\lambda }\in \mathbf{C}.$\
$12)\ \ L(0,0,1,1,{\lambda}_{1},{\lambda} _{2}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6, \ \ e_{0}e_{1}=e_{5}+e_{6}+{\lambda} _{2}e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{5}+e_{6}+{\lambda }_{1}e_{7},\ \
e_{2}e_{1}=e_{6}+e_{7},\ \ e_{3}e_{1}=e_{7},\ \ {\lambda }_{1},{\lambda }_{2}\in \mathbf{C}.$\
$13)\ \ L(0,0,1,0,{\lambda} ,{\lambda} ^{2}+{\lambda}):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6, \ \ e_{0}e_{1}=e_{5}+({\lambda}^{2}+{\lambda} )e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{5}+{\lambda }e_{7},\ \
e_{2}e_{1}=e_{6},e_{3}e_{1}=e_{7},\ \ {\lambda }\in
\mathbf{C}.$\
$14)\ \ L(0,0,0,1,1,{\lambda} ):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6,\ \ e_{0}e_{1}=e_{6}+{\lambda} e_{7},\ \
e_{1}e_{1}=e_{6}+e_{7},$
$\qquad \qquad e_{2}e_{1}=e_{7},\ \ {\lambda }\in
\mathbf{C}.$\
$15)\ \ L(0,1,0,0,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i
\leq6,\ \ e_{0}e_{1}=e_{4},\ \ e_{1}e_{1}=e_{4},$
$\qquad \qquad e_{2}e_{1}=e_{5},\ \ e_{3}e_{1}=e_{6},\ \ e_{4}e_{1}=e_{7}.$\
$16)\ \ L(0,1,0,-3,1,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i
\leq6,\ \ e_{0}e_{1}=e_{4}-3e_{6}+e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{4}-3e_{6}+e_{7},\ \
e_{2}e_{1}=e_{5}-3e_{7},\ \ e_{3}e_{1}=e_{6},\ \
e_{4}e_{1}=e_{7}.$\
$17)\ \ L(0,1,0,-3,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6, \ \ e_{0}e_{1}=e_{4}-3e_{6}+e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{4}-3e_{6},\ \
e_{2}e_{1}=e_{5}-3e_{7},e_{3}e_{1}=e_{6},\ \
e_{4}e_{1}=e_{7}.$\
$18)\ \ L(0,0,1,0,1,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6, \ \ e_{0}e_{1}=e_{5}+e_{7},$
$\qquad \qquad e_{1}e_{1}=e_{5}+e_{7},\ \ e_{2}e_{1}=e_{6},\ \
e_{3}e_{1}=e_{7}.$\
$19)\ \ L(0,0,1,0,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6, \ \ e_{0}e_{1}=e_{5}+e_{7},\ \ e_{1}e_{1}=e_{5},$
$\qquad \qquad e_{2}e_{1}=e_{6},\ \ e_{3}e_{1}=e_{7}.$\
$20)\ \ L(0,0,0,1,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1,}\ \ 1\leq i
\leq6,\ \ e_{0}e_{1}=e_{6}+e_{7},\ \ e_{1}e_{1}=e_{6},$
$\qquad \qquad e_{2}e_{1}=e_{7}.$\
$21)\ \ L(0,0,0,1,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1}, \ \ 1\leq
i \leq6 ,\ \ e_{0}e_{1}=e_{6},\ \ e_{1}e_{1}=e_{6},\ \
e_{2}e_{1}=e_{7} .$\
$22)\ \ L(0,0,0,0,1,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6,\ \ e_{1}e_{1}=e_{7}.$\
$23)\ \ L(0,0,0,0,1,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6, \ \ e_{0}e_{1}=e_{7},\ \ e_{1}e_{1}=e_{7}.$\
$24)\ \ L(0,0,0,0,0,1):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6, \ \ e_{0}e_{1}=e_{7}.$\
$25)\ \ L(0,0,0,0,0,0):$
$\qquad \qquad e_{0}e_{0}=e_{2},\ \ e_{i}e_{0}=e_{i+1},\ \ 1\leq i
\leq6. $\
The number of isomorphism classes $N_8=25.$\
**Conjecture.** The number of isomorphism classes $N_n$ of $n$-dimensional none Lie complex filiform Leibniz algebras in $FLeib_{n}$ can be found by the formula: $$N_n=n^2-7n+17.$$
Note that the validity of the above formula is confirmed in dimension $9$ as well.
We would like to thank U.D.Bekbaev and B.A.Omirov for their helpful discussions.
[7]{}
A.Malcev. On Solvable Lie algebras,Izv. Acad. Nauk SSSR, ser. Math., 9(1945), 329-356. K.A.Umlauf, $\ddot{U}ber$ die Zusammmensetzung der endlichen continuierlichen Transformationsgrouppen insbesondere der Gruppen vom Range null, Thesis, Leipzig, 1891. M.Vergne, Cohomologie des algebres de Lie nilpotentes. Application $\grave{a}$ l’$\acute{e}$tude de la vari$\acute{e}$t$\acute{e}$ des alg$\grave{e}$bres de Lie nilpotentes, Bull. Soc. Math. France 98(1970), 81-116. Sh.A.Ayupov, B.A.Omirov, On some classes of nilpotent Leibniz algebras. // Sib. Math. J. (2001). V. 42, 1. 18-29.(in Russian) Albeverio S., Omirov B.A., Rakhimov I.S. Classification of four-dimensional nilpotent complex Leibniz algebras. Extracta Math. 21(3) (2006), 197-210. U. D.Bekbaev, I.S.Rakhimov. On classification of finite dimensional complex filiform Leibniz algebras (part 1). http://front.math.ucdavis.edu/, ArXiv:math. RA/01612805.(2006). J.R.Gomez, B.A.Omirov. On classification of complex filiform Leibniz algebras. arXive:math/0612735 v1 \[math.R.A.\] 23 dec 2006.
[^1]: The research is supported by Grant 04-01-06 SF01-22 MOSTI (Malaysia)
|
---
address:
- 'Universität Essen, FB6 Mathematik, 45117 Essen, Germany'
- 'The Chinese University of Hong Kong, Department of Mathematics, Shatin, Hong Kong'
author:
- Eckart Viehweg
- Kang Zuo
title: Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks
---
\[section\] \[thm\][Theorem]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Addendum]{} \[thm\][Variant]{} \[thm\][Notations]{} \[thm\][Question]{} \[thm\][Problem]{} \[thm\][Remark]{} \[thm\][Remarks]{} \[thm\][Definition]{} \[thm\][Claim]{} \[thm\][Assumption]{} \[thm\][Assumptions]{} \[thm\][Properties]{} \[thm\][Example]{} ‘=11 \#1\#2[\#1]{} \#1[[**\#1**]{}]{} @\#1\#2 @\#1\#2 @rs@nd@@ @.2326ex @ @ @16.08739@ @ @2.5pc @ \#1[@\#1]{}
CD[\#\#1A\#\#2A[&&]{}\#\#1V\#\#2V[&&]{}&.5em.5em&\#\#1 @true]{} \#1>\#2>[@rs@nd@@ne@ @ @@@@ @>@@@ ne>@@ne @.5em @>@ @.5em@rs@nd@]{} \#1<\#2<[@rs@nd@@ne@ @ @@@@ @>@@@ ne>@@ne @.5em @>@ @.5em@rs@nd@]{}
CDS[\#\#1A\#\#2A[&]{}\#\#1V\#\#2V[&]{}&.5em.5em& \#\#1E\#\#2E[(0,18)(4,-3)[\#\#1]{}[\#\#2]{}&]{} \#\#1W\#\#2W[(24,18)(-4,-3)[\#\#1]{}[\#\#2]{}&]{} \#\#1E\#\#2E[(0,0)(4,3)[\#\#1]{}[\#\#2]{}&]{} \#\#1W\#\#2W[(24,0)(-4,3)[\#\#1]{}[\#\#2]{}&]{} (\#\#1)(\#\#2)\#\#3\#\#4[6.5pt]{} \#\#1 @true]{} @ @
TriCDV[@true@[6]{}TriCD]{} TriCDA[@false@[10]{}TriCD]{} TriCD\#1\#2\#3\#4\#5\#6[0 @=0 @ 24pt @ -1em \#\#1E\#\#2E[(0,18)(2,-3)[\#\#1]{}[\#\#2]{}&]{} \#\#1W\#\#2W[(12,18)(-2,-3)[\#\#1]{}[\#\#2]{}&]{} \#\#1E\#\#2E[(0,0)(2,3)[\#\#1]{}[\#\#2]{}&]{} \#\#1W\#\#2W[(12,0)(-2,3)[\#\#1]{}[\#\#2]{}&]{} (\#\#1)(\#\#2)\#\#3\#\#4[ 6.5pt]{} = \#\#1>>[@@@ @>@@@ .5em .5em]{} \#\#1<<[@@@ @>@@@ ]{} @true endTriCD]{}
[^1]
Introduction
============
Let $f:V\to U$ be a smooth projective morphism with connected fibres over a complex quasi-projective manifold $U$.
\[variation\]
1. ${\rm Var}(f)$ is the smallest integer $\eta$ for which there exists a finitely generated subfield $K$ of $\overline{{{\mathbb C}}(U)}$ of transcendence degree $\eta$ over ${{\mathbb C}}$, a variety $F'$ defined over $K$, and a birational equivalence $$V\times_U {\rm Spec}(\overline{{{\mathbb C}}(U)}) \sim F'\times_{{\rm Spec}(K)}
{\rm Spec}(\overline{{{\mathbb C}}(U)}).$$
2. $f:V\to Y$ is birationally isotrivial if ${\rm Var}(f)= 0$, hence if there exists some generically finite covering $U'\to U$, a projective manifold $F'$, and a birational map $$V\times_UU' \sim U'\times F'.$$
3. $f:V\to U$ is (biregulary) isotrivial if there exists a generically finite covering $U'\to U$, a projective manifold $F$ and an isomorphism $$V\times_UU' \simeq U'\times F.$$
So the variation of a morphisms counts the number of parameters controlling the birational structure of the fibres of $F$.
Maehara has shown in [@Mae] that under the assumption that $\omega_F$ is semi-ample and big for a general fibre $F$ of $f$, a family is birationally isotrivial, if and only if it is biregulary isotrivial. In different terms, for families of minimal models of complex manifolds of general type, ${\rm Var}(f)$ measures the number of directions where the structure of $F$ varies. As shortly discussed in \[sectcan\] his result today follow immediately from the existence of the moduli scheme $M_h$ of canonically polarized manifolds, and from the description of an ample invertible sheaf on $M_h$.
We will slightly extend the methods used to prove [@Vie], Theorem 6.24, to show that for families with $\omega^\delta_F={{\mathcal O}}_F$, for some $\delta >0$, the same holds true. We will show that for a given projective manifold $F'$ the set of minimal models is discrete, hence that there are no non-trivial families of minimal models.
Let us fix some polarization ${{\mathcal L}}$ of $f:V\to U$, with Hilbert polynomial $h$. If $\omega_{V/U}$ is $f$-ample we will choose ${{\mathcal L}}=\omega_{V/U}^\rho$, for some $\rho >0$. By [@Vie] there exists a quasi-projective moduli schemes $M_h$, parameterizing polarized manifolds $(F,{{\mathcal L}})$ with $\omega_F$ semiample and with $h(\nu)=\chi({{\mathcal L}}^\nu)$. The family $f:V\to U$ together with ${{\mathcal L}}$ induces a map $$\varphi:U \to M_h$$ Since we require $\varphi:U\to M_h$ to be induced by a family it factors through the moduli stack ${{\mathcal M}}_h$.
\[stthm\] Let $f:V\to U$ be a family of polarized manifolds. Assume that $\omega_F^\delta={{\mathcal O}}_F$, for some $\delta >0$ (or that all fibres $F$ of $f$ are canonically polarized). Let $\varphi:U\to M_h$ be the induced morphism to the moduli scheme. Then ${\rm Var}(f)=\dim(\varphi(U))$.
If in Theorem \[stthm\] the morphism $f:V\to U$ is birationally isotrivial, $\varphi(U)$ must be zero dimensional, hence $f$ is biregulary isotrivial.
As Y. Kawamata told us, Theorem \[stthm\] remains true for families of polarized manifolds with $\omega_{X/Y}$ $f$-semiample. Here however one has to replace the moduli scheme $M_h$ by the moduli scheme $P_h$ of polarized manifolds, up to numerical equivalence (see [@Vie]). So given a family $f:V\to U$ of polarized manifolds with $\omega_{V/U}$ $f$-semiample, let $\psi:U
\to P_h$ be the morphism to the moduli scheme of polarized manifolds up to numerical equivalence. Then ${\rm
Var}(f)=\dim(\psi(U))$.
For some applications to subvarieties of moduli stacks of polarized $n$-folds of Kodaira dimension $0<\kappa<n$, one still has to understand the structure of the ample sheaves on $P_h$, less transparent than the ones on $M_h$. Nevertheless, we hope that most of the results stated in the second half of this note remain true for all Kodaira dimensions, with $M_h$ replaced by $P_h$.
Theorem \[stthm\], or more generally the equivalence of biregular and birational isotriviality allows to extends some of the results obtained in [@VZ3] for canonically polarized manifolds to families of manifolds $F$ with $\omega_F^\delta={{\mathcal O}}_F$ (see Theorem \[stthm2\], i) and ii) and Section \[sectappl\]). This is done in the second half of this article, a continuation of [@VZ3]. What methods are concerned, the reader familiar with [@VZ3] will find nothing new. In fact we just sketch the changes needed to extend some of the results to this case.
In the final Section \[rigidity\] we will state a criterion for the rigidity of non-isotrivial families over curves, and its translation to curves in the moduli stack of minimal polarized manifolds of Kodaira dimension zero, or of canonically polarized manifolds. This criterion is implicitly used in [@VZ3], Proof of 6.4 and 6.5, but it was not explicitly stated there.
A slightly weaker statement (\[proprigidsa\]) extends to all families with $\omega_F$ semiample. A similar criterion has been shown by S. Kovács and, for families of Calabi-Yau manifolds by K. Liu, A. Todorov, S.-T. Yau and the second named author in [@LTYZ]. As a corollary one obtains (see \[corrigid2\]):
\[corrigid3\] Let $M_h$ be either the moduli scheme of canonically polarized manifolds or the moduli scheme of polarized manifolds $F$ with $\omega_F^\delta={{\mathcal O}}$ for some $\delta >0$. There are only finitely many morphisms $\varphi:U\to M_h$ which are induced by a smooth family $f:V\to U$ with:\
For a general fibre $F$ of $f$ the $n$-th wedge product $$0\neq \wedge^{n} \xi \in H^{n}(F,\omega^{-1}_F),$$ where $\xi\in H^1(F,T_F)$ denotes the Kodaira Spencer class corresponding to the deformation $f:V\to U$ of $F$.
The first half of this article presents a proof of Theorem \[stthm\], hopefully of interest independently of the applications to subvarieties of moduli stacks. In the first section, we will show, that the proof of Theorem \[stthm\] can be reduced to families over a curve. Next we recall and strengthen a Positivity Theorem from [@Vie]. It allows to reinterpret Maehara’s Result in section \[sectcan\]. The case of minimal models of Kodaira dimension zero is handled in Section \[sectzero\].
Reduction to families over curves {#sectred}
=================================
In Definition \[variation\], i), we may choose a finitely generated subfield $L$ of $\overline{{{\mathbb C}}(U)}$ which contains ${{\mathbb C}}(U)$ and $K$. Let $U'$ be the normalization of $U$ in $L$, and let $T$ be a smooth quasi-projective variety with function field ${{\mathbb C}}(T)=K$. Replacing $T$ by some open subscheme, we may assume that there exists a smooth projective morphism $g:Z\to T$ with general fibre $F'$, and replacing $U'$ by some open subscheme one finds morphisms $\tau:U'\to U$ and $\pi:U'\to T$ fitting into a diagram $$\@CD
V \<<< V' \>\sim >> Z' \>>> Z\\
{{\mathbb V}}f VV {{\mathbb V}}f' VV {{\mathbb V}}g' VV {{\mathbb V}}g VV\\
U \< \tau << U' \> = >> U' \> \pi >> T,
\@endCD$$ where $V'\to Z'$ is a birational equivalence, and where the right and left hand squares are fibre products. For a point $t \in T$ in general position, one has $$\dim(\pi^{-1}(t))= \dim(U')-\dim(T) = \dim(U) - {\rm Var}(f).$$ If under the assumptions made in Theorem \[stthm\] ${\rm
Var}(f) < \dim(\varphi(U))$, for a point $\eta \in \varphi(U)$ in general position, $$\dim(\tau^{-1}\varphi^{-1}(\eta))= \dim(U')-\dim(\varphi(U))
< \dim(\pi^{-1}(t)),$$ hence there exists a curve $C$ in $\pi^{-1}(t)$ with $\tau\circ\varphi|_C$ finite. In order to prove Theorem \[stthm\] one just has to show that such a curve can not exist. Theorem \[stthm\] follows from
\[mainprop\] Let $U$ be a non-singular irreducible curve and let $F'$ be a projective manifold. Let $f:V\to U$ be a family of polarized manifolds. Assume that there exists a birational equivalence $V \sim U\times F'$ over $U$. If either the fibres $F$ of $f$ are canonically polarized, or if $\omega_F^\delta={{\mathcal O}}_F$, for some $\delta >0$, then the induced morphism $\varphi:U\to M_h$ is constant.
In particular, $f:V\to U$ is biregulary isotrivial.
In order to prove Proposition \[mainprop\] we may replace $U$ by a finite covering. Doing so one can assume that $f:V\to U$ extends to a semistable morphism $f:X\to Y$ of projective manifolds, hence that $\Delta=f^{-1}(S)$ is a reduced normal crossing divisor, for $S=Y\setminus U$.
Moreover, we may replace the given polarization by some power. In fact, the corresponding map of the moduli schemes $M_h$ is a finite map. This allows to assume that for all fibres $F$ of $V\to U$ the polarization ${{\mathcal L}}$ is very ample, and without higher cohomology.
Positivity of direct image sheaves {#sectpos}
==================================
Recall that a locally free sheaf ${{\mathcal E}}$ on a projective non-singular curve $Y$ is numerically effective (nef), if for all finite morphisms $\tau:Z\to Y$ and for all invertible quotients ${{\mathcal L}}$ of $\tau^*({{\mathcal E}})$ the degree $\deg({{\mathcal L}})\geq 0$.
Fujita’s positivity theorem (today an easy corollary of Kollár’s vanishing theorem) says that $f_* \omega_{X/Y}$ is nef. By [@VZ1], 2.3, one obtains as a direct consequence.
\[fujita\] Let $f:X\to Y$ be a morphism from a normal projective variety $X$ to a curve $Y$, with connected fibres. Assume that $X$ has at most rational double points as singularities. Let ${{\mathcal N}}$ be an invertible sheaf on $X$ and $\Gamma$ an effective divisor. Assume that for some $N > 0$ there exists a nef locally free sheaf ${{\mathcal E}}$ on $Y$ and a surjection $$f^* {{\mathcal E}}\>>> {{\mathcal N}}^N (- \Gamma).$$ Then $$f_* \left({{\mathcal N}}\otimes \omega_{X/Y} \left\{ - \frac{\Gamma}{N}
\right\} \right)$$ is nef.
Here $\omega_{X/Y} \left\{ - \frac{\Gamma}{N} \right\}$ denotes the (algebraic) multiplier sheaf (see for example [@EV], 7.4, or [@Vie], section 5.3). If $\tau: X' \to X$ is any blowing up with $\Gamma' = \tau^* \Gamma$ a normal crossing divisor, then $$\omega_{X/Y} \left\{ - \frac{\Gamma}{N} \right\} = \tau_*
\left(\omega_{X'/Y} \left( - \left[ \frac{\Gamma'}{N} \right]
\right)\right).$$ As in [@EV], § 7 and [@Vie], section 5.3, we are mainly interested in the case where the multiplier sheaf on a general fibre $F$ is isomorphic to $\omega_F$. The corresponding threshold is defined for any effective divisor $\Pi$ or any invertible sheaf ${{\mathcal L}}$ on $F$ with $H^0 (F, {{\mathcal L}}) \neq 0$. $$\begin{gathered}
e(\Pi) = {\rm Min} \left\{ N \in {{\mathbb N}}- \{ 0 \} ; \ \omega_F \left\{-
\frac{\Pi}{N} \right\} = \omega_F \right\} \ \ \mbox{ \ \ \
\ \ and}\\
e ({{\mathcal L}}) = {\rm Max} \left\{ e (\Pi); \ \Pi \ \mbox{the zero set of} \
\sigma \in H^0 (F, {{\mathcal L}}) - \{ 0 \} \right\}.\end{gathered}$$
For smooth morphisms $f:X\to Y$ and for an $f$-ample sheaf ${{\mathcal L}}$ on $X$ we obtained in [@Vie], 6.24 and 7.20, strong positivity theorems. Their proof, in case $Y$ is a curve, can easily be extended to semistable morphisms $f:X\to Y$.
\[positiv\] Let $Y$ be a curve, let $f:X\to Y$ be a semistable morphism between projective manifolds with connected fibres, and let ${{\mathcal M}}$ be an invertible sheaf on $X$. Let $U\subset Y$ be an open dense subscheme with $V=f^{-1}(U) \to U$ smooth. Assume that for all fibres $F$ of $V\to U$ the canonical sheaf $\omega_F$ is semiample, that ${{\mathcal M}}|_F$ is very ample and without higher cohomology. Then for $$\begin{gathered}
e \geq c_1({{\mathcal M}}|_F)^{\dim(F)}+2, \ \ \ \ \
r={{\rm rank}}(f_*{{\mathcal M}})\\
\mbox{ and \ \ \ }r(\nu)={{\rm rank}}(f_*({{\mathcal M}}^\nu\otimes \omega^{e\cdot\nu}_{X/Y})):\end{gathered}$$
1. For all $\nu >0$ $$\big( \bigotimes^r f_*({{\mathcal M}}^\nu\otimes
\omega_{X/Y}^{e\cdot\nu})\big)
\otimes\det(f_*{{\mathcal M}})^{-\nu}$$ is nef.
2. If the invertible sheaf $\det(f_*({{\mathcal M}}^\nu\otimes\omega^{e\cdot\nu}_{X/Y}))\otimes\det(f_*{{\mathcal M}})^{-\nu
r(\nu)}$ is ample for some $\nu>0$, $$\big( \bigotimes^r f_*({{\mathcal M}}\otimes \omega_{X/Y}^{e})\big)
\otimes\det(f_*{{\mathcal M}})^{-1}$$ is ample.
3. If for all $\nu > 0$ the degree of $$\det(f_*({{\mathcal M}}^\nu\otimes\omega^{e\cdot\nu}_{X/Y}))\otimes\det(f_*{{\mathcal M}})^{-\nu
r(\nu)}$$ is zero, then $f:V\to U$ is biregulary isotrivial as a family of polarized manifolds, i.e. there exists some finite covering $U'\to U$, a projective manifold $F'$, invertible sheaves ${{\mathcal L}}'$ on $F'$ and ${{\mathcal B}}$ on $U'$, and an isomorphism $$\pi:V'=X\times_YU' \to F'\times U'$$ with $$pr_1^*{{\mathcal M}}= \pi^*( pr_1^*{{\mathcal L}}\otimes pr_2^*{{\mathcal B}}).$$
As indicated already, the proof of parts a) and b) will follow the arguments used in [@Vie], 194–196, to prove 6.20. We just have to take care, that for a semistable family over a curve the sheaves are at most getting larger. So we repeat the arguments.
For a) let us fix some $\nu >0$. For b) we assume that $$\det(f_*({{\mathcal M}}^\nu\otimes\omega^{e\cdot\nu}_{X/Y}))
\otimes\det(f_*{{\mathcal M}})^{-\nu \cdot r(\nu)}$$ is ample.
The semicontinuity of the threshold, shown in [@Vie], 5.17 for example, allows to find some $\gamma \geq e\cdot\nu$ with $$\label{gamma}
e ({{\mathcal M}}|_F ^{\nu \cdot e} \otimes
\omega^{e \cdot \nu \cdot (e-1)}_{F} ) \leq \gamma$$ for all fibres $F$ of $V \to U$.
For b) we will show that $$\begin{gathered}
S^{\gamma} \big(\big( \bigotimes^{r\cdot r(\nu)} f_*({{\mathcal M}}\otimes \omega_{X/Y}^{e})\big)
\otimes\det(f_*{{\mathcal M}})^{-{r(\nu)}}\big)\otimes\\
\det(f_*({{\mathcal M}}^\nu\otimes\omega^{e\cdot\nu}_{X/Y}))^{-r\cdot(e-1)}
\otimes\det(f_*{{\mathcal M}})^{\nu\cdot
r\cdot(e-1) \cdot r(\nu)}\end{gathered}$$ is nef. Hence for both, a) or b), it is sufficient to prove the corresponding statements for the pullback of the sheaves to any finite covering $Y'$ of $Y$. Since we assumed $X\to Y$ to be semistable, the fibre product $X'=X\times_YY'$ is a normal variety with at most rational double points. Flat base change allows to replace $Y$ by such a covering and $(f:X\to
Y,{{\mathcal M}})$ by a desingularization of the pullback family.
Doing so, we may assume that $\det(f_*{{\mathcal M}})$ is the $r$-th power of an invertible sheaf, and since all the sheaves occurring in a), b) or c) are compatible with changing the polarization by the pullback of an invertible sheaf on $Y$, we can as well assume that $\det(f_*{{\mathcal M}})={{\mathcal O}}_Y$. Under this additional assumption we have to verify in a) that $$f_*({{\mathcal M}}^\nu\otimes \omega_{X/Y}^{e\cdot\nu})$$ is nef. For part b) we may assume in addition that $$\det(f_* ({{\mathcal M}}^{\nu} \otimes \omega^{e \cdot \nu}_{X/Y} ))^{r\cdot (e-1)}
= {{\mathcal O}}_Y(\gamma \cdot H)$$ for some effective divisor $H$ supported in $U$. We have to prove that $$\big( \bigotimes^{r\cdot r(\nu)} f_*({{\mathcal M}}\otimes \omega_{X/Y}^{e})\big)
\otimes {{\mathcal O}}_Y(-H)$$ is nef.\
Let $f^s:X^s \to Y$ be the $s$-fold fibre product. $X^s$ is normal with at most rational double points (see [@Mor], page 291, for example). Consider $${{\mathcal P}}= \bigotimes^{s}_{i=1} pr^{*}_{i} {{\mathcal M}}.$$ By flat base change one obtains $$f^{s}_{*}( {{\mathcal P}}^{\alpha} \otimes \omega^{\beta}_{X^s /Y}) =
\bigotimes^s f_* ({{\mathcal M}}^{\alpha} \otimes \omega^{\beta}_{X/Y})$$ for all $\alpha , \beta$. The restriction of ${{\mathcal P}}^{\nu} \otimes
\omega^{e\cdot\nu - \iota}_{X^s /Y} $ to ${f^s}^{-1}(U)=V^s$ is $f^s$-ample for all $\iota \leq e\cdot\nu $. Let us write $\epsilon= e\cdot\nu$ or $\epsilon=e\cdot\nu -1$, where $\nu$ may be any positive integer.
If $\Gamma$ is the zero divisor of a section of ${{\mathcal P}}$, which does not contain any fibre $F^s$ of $V^s\to U$ the compatibility of the threshold with products and its semicontinuity imply (see [@Vie], 5.14 and 5.21) $$\label{threshold}
e (\Gamma |_{F^s} ) \leq e ({{\mathcal P}}|_{F^s} ) =
e ({{\mathcal M}}|_{F} ) < e \mbox{ \ \ and \ \ }
e (\Gamma|_{V^s} ) < e.$$ In fact, as shown in [@Vie], 5.11), one has $$e({{\mathcal M}}|_F) \leq c_1({{\mathcal M}}|_F)^{\dim(F)} + 1.$$ Moreover, by the choice of $\epsilon$ $$\label{threshold2}
e(\nu\cdot\Gamma|_{F^s}) \leq \nu\cdot e(\Gamma|_{F^s}) \leq
\nu \cdot (c_1({{\mathcal M}}|_{F})^{\dim(F)} + 1) \leq \epsilon.$$ Let ${{\mathcal H}}$ be an ample invertible sheaf on $Y$.
\[claim\] Assume that for some $\rho \geq 0$, $N >0$, $M_0 >0$ and for all multiples $M$ of $M_0$, the sheaf $$f_* (({{\mathcal M}}^{\nu} \otimes \omega^{\epsilon}_{X/Y} )^{M\cdot N} ) \otimes
{{\mathcal H}}^{\rho \cdot \epsilon \cdot N \cdot M}$$ is nef. Then $$f_* (({{\mathcal M}}^{\nu} \otimes \omega^{\epsilon}_{X/Y} )^N ) \otimes {{\mathcal H}}^{\rho
\cdot (\epsilon \cdot N -1)}$$ is nef.
Let us choose $s=r$. The determinant gives an inclusion $${\rm det} (f_* {{\mathcal M}}) = {{\mathcal O}}_Y \>>> f^{r}_{*} {{\mathcal P}}= \bigotimes^r f_* {{\mathcal M}},$$ which splits locally. Hence the zero divisor $\Gamma$ of the induced section of ${{\mathcal P}}$ does not contain any fibre of $V^r\to U$. For $${{\mathcal N}}= {{\mathcal P}}^{\nu \cdot N} \otimes \omega^{\epsilon\cdot N -1}_{X^r /Y}
\otimes f^{r *} {{\mathcal H}}^{\rho \cdot (\epsilon \cdot N -1) \cdot r}$$ one obtains that the restriction of $${{\mathcal N}}^{\epsilon} (-\nu \cdot \Gamma ) = ({{\mathcal P}}^{\nu} \otimes
\omega^{\epsilon}_{X^r /Y} \otimes f^{r*} {{\mathcal H}}^{\rho \cdot \epsilon \cdot r}
)^{(\epsilon \cdot N -1)}$$ to $V^s$ is $f^r$-ample. If $M'$ is a positive integer, divisible by $M_0 \cdot N$ the sheaf $$f^{r}_{*} ({{\mathcal N}}^{\epsilon} (-\nu \cdot \Gamma )^{M'}) =
\bigotimes^r (f_* ({{\mathcal M}}^{\nu} \otimes \omega^{\epsilon}_{X/Y} )^{(\epsilon
\cdot N -1) \cdot M'} \otimes {{\mathcal H}}^{\rho \cdot \epsilon
\cdot r (\epsilon \cdot N -1) \cdot M'} )$$ is nef. Choose $M'$ such that $$f^*f_*(({{\mathcal M}}^\nu\otimes\omega^\epsilon_{X/Y})^{(\epsilon
\cdot N -1) \cdot M'}) \>>> ({{\mathcal M}}^\nu\otimes\omega^\epsilon_{X/Y})^{(\epsilon
\cdot N -1) \cdot M'}$$ is surjective over $U$.
\[fujita\] implies that the subsheaf $f^{r}_{*} ({{\mathcal N}}\otimes
\omega_{X^r /Y} \left\{ -\frac{\nu\cdot\Gamma}{\epsilon}\right\})$ of $$f^{r}_{*} ({{\mathcal N}}\otimes \omega_{X^r /Y} ) = \bigotimes^r (f_* (
{{\mathcal M}}^{\nu \cdot N} \otimes \omega^{\epsilon \cdot N}_{X/Y} ) \otimes
{{\mathcal H}}^{\rho \cdot (\epsilon \cdot N -1)} )$$ is nef. On the other hand, (\[threshold\]) and (\[threshold2\]) imply that both sheaves coincide on $U$.
Choose some $N_0>0$ such that for all multiples $N$ of $N_0$ and for all $M>0$ the multiplication maps $$m: S^M (f_* ({{\mathcal M}}^{\nu \cdot N} \otimes \omega^{\epsilon \cdot N}_{X/Y}
)) \>>> f_* ({{\mathcal M}}^{\nu \cdot N \cdot M} \otimes \omega^{\epsilon \cdot N
\cdot M}_{X/Y} )$$ are surjective over $U$. Define $$\rho = {\rm Min } \{ \mu >0 ; \ f_* ({{\mathcal M}}^{\nu \cdot N} \otimes
\omega^{\epsilon \cdot N}_{X/Y} ) \otimes {{\mathcal H}}^{\mu \cdot \epsilon \cdot N} \
\mbox{is nef} \} .$$ The surjectivity of $m$ implies that $$f_* ({{\mathcal M}}^{\nu \cdot N \cdot M} \otimes \omega^{\epsilon \cdot N \cdot
M}_{X/Y} ) \otimes {{\mathcal H}}^{\rho \cdot \epsilon \cdot N \cdot M}$$ is nef for all $M >0$. By \[claim\] $$f_* ({{\mathcal M}}^{\nu \cdot N} \otimes \omega^{\epsilon \cdot N}_{X/Y} )
\otimes {{\mathcal H}}^{\rho \cdot (\epsilon \cdot N -1)} .$$ is nef, hence by the choice of $\rho$ $$(\rho -1)\cdot \epsilon \cdot N < \rho \cdot (\epsilon \cdot N -1)$$ or equivalently $\rho < \epsilon \cdot N$. Then $$f_* ({{\mathcal M}}^N \otimes \omega^{\epsilon \cdot N}_{X/Y} ) \otimes {{\mathcal H}}^{\epsilon^2
\cdot N^2}$$ is nef. This remains true if one replaces $Y$ by any finite covering, and by [@VZ1], 2.2, one obtains that $f_* ({{\mathcal M}}^N \otimes \omega^{\epsilon \cdot N}_{X/Y} )$ is nef. Applying \[claim\] a second time, for the numbers $(N',N_0)$ instead of $(N,M_0)$ and for $\rho =0$, one finds
\[nef2\] For $\nu >0$ and $\epsilon=e\cdot\nu$ or $\epsilon=e\cdot\nu -1$ and for all $N'>0$ the sheaf $$f_* (({{\mathcal M}}^{\nu} \otimes \omega^{\epsilon}_{X/Y} )^{N'})$$ is nef.
In particular, choosing $N'=1$ and $\epsilon=\nu e$ one obtains a).\
For b) we consider the $s$-fold product $f^s : X^s \to Y$ for $s = r \cdot r (\nu).$ One has natural inclusions, splitting locally, $${{\mathcal O}}_Y = {\rm det} (f_* {{\mathcal M}})^{r (\nu)} \>>> f^{s}_{*} {{\mathcal P}}= \bigotimes^s f_*{{\mathcal M}}$$ and $$\det (f_* ({{\mathcal M}}^{\nu} \otimes \omega^{e \cdot
\nu}_{X/Y} ))^{r} \>>> f^{s}_{*} ({{\mathcal P}}^{\nu} \otimes
\omega^{e \cdot \nu}_{X^s /Y} ) = \bigotimes^s f_*
({{\mathcal M}}^{\nu} \otimes \omega^{e \cdot \nu}_{X/Y} ).$$ If $\Delta_1$ and $\Delta_2$ denote the corresponding zero-divisors on $X^s$ then $\Delta_1 + \Delta_2$ does not contain any fibre of $V^s \to U$. Then $${{\mathcal P}}^{e\cdot\nu}\otimes\omega_{X^s/Y}^{e\cdot\nu\cdot(e-1)}
= {f^s}^*\det (f_* ({{\mathcal M}}^{\nu} \otimes \omega^{e \cdot
\nu}_{X/Y} ))^{r\cdot(e-1)}\otimes {{\mathcal O}}_{X^s}((e -1) \cdot \Delta_2 + \nu \cdot \Delta_1 ),$$ and $${{\mathcal P}}^{\gamma} \otimes \omega^{\gamma \cdot (e -1)}_{X^s/Y}
= ({{\mathcal P}}\otimes \omega_{X^s/Y}^{e-1})^{\gamma-\nu e}\otimes {{\mathcal O}}_X( \gamma \cdot
{f^s}^* H +(e -1) \cdot \Delta_2 + \nu \cdot \Delta_1)).$$ By \[nef2\] the sheaf $${f^s}_* (({{\mathcal P}}\otimes \omega_{X^s/Y}^{e-1})^{\gamma-\nu \cdot e})^M
= \bigotimes^s f_* (({{\mathcal P}}\otimes \omega_{X/Y}^{e-1})^{\gamma-\nu \cdot e})^M$$ is nef for all $M>0$. \[fujita\] implies that $${{\mathcal P}}\otimes \omega^{e-1}_{X^s/Y} \otimes \omega_{X^s/Y}\left\{-
\frac{\gamma \cdot{f^s}^*H +(e -1) \cdot \Delta_2 + \nu \cdot \Delta_1}{\gamma}
\right\}$$ is nef. By (\[threshold\]) and (\[gamma\]) $$\begin{gathered}
e(((e -1) \cdot \Delta_2 + \nu \cdot \Delta_1) |_{F^{s}} ) \leq e ({{\mathcal P}}|_{F^{s}}^{\nu
\cdot e } \otimes \omega^{e \cdot \nu \cdot
(e -1)}_{F^{s}} ) =\\ e ({{\mathcal M}}|_{F}^{\nu \cdot e} \otimes
\omega^{e \cdot \nu \cdot (e-1)}_{F} )\leq \gamma\end{gathered}$$ for all fibres $F$ of $V \to U$. Hence the cokernel of $$\omega_{X^s/Y}\left\{
\frac{-\gamma \cdot {f^s}^* H +(e -1) \cdot \Delta_2 + \nu \cdot \Delta_1}{\gamma}
\right\} \to \omega_{X^s/Y}(-{f^s}^* H)$$ lies in $X^s\setminus V^s$, and thereby $${f^s}_* ({{\mathcal P}}\otimes \omega^{e}_{X^s/Y}) \otimes {{\mathcal O}}_Y(-H)=
\big( \bigotimes^{r\cdot r(\nu)} f_*({{\mathcal M}}\otimes \omega_{X/Y}^{e})\big)
\otimes {{\mathcal O}}_Y(-H)$$ is nef.\
Part c) follows from part a) and Kollar’s ampleness criterion (see [@Vie], 4.34). Again we may assume that $\det(f_*{{\mathcal M}})={{\mathcal O}}_Y$. By part a) for all $\eta >0$ the sheaf ${{\mathcal E}}=f_*({{\mathcal M}}^\eta \otimes
\omega_{X/Y}^{e\cdot\eta})$ is nef. Choose $\nu>0$ such that the multiplication map $$\mu: S^\nu({{\mathcal E}}^\eta) \to f_*({{\mathcal M}}^{\nu\cdot\eta}\otimes \omega_{X/Y}^{e\cdot\nu \cdot \eta})$$ is surjective over $U$. By [@Vie], 4.34, $\det({\rm Im}(\mu))$ is ample, if the kernel ${{\mathcal K}}$ of the multiplication map is of maximal variation. Let us recall the definition. For a point $y\in U$ choose a local trivialization of ${{\mathcal E}}$. Then ${{\mathcal K}}_y={{\mathcal K}}\otimes {{\mathbb C}}(y)$ as a subvectorspace of $S^\nu({{\mathbb C}}^{r(\eta)})$, defines a point $[{{\mathcal K}}_y]$ in the Grassmann variety $${{\mathbb G}}r ={\rm Grass}(r(\nu\cdot\eta), S^\nu({{\mathbb C}}^{r(\eta)})).$$ The group $G={\rm Sl}(r(\eta),{{\mathbb C}})$ acts on $S^\nu({{\mathbb C}}^{r(\eta)})$, hence on ${{\mathbb G}}r$. Let $G_y$ denote the orbit of $[{{\mathcal K}}_y]$. The kernel has maximal variation, if $$\{ z \in Y; \ G_{z}=G_y\}$$ is finite, as well as the stabilizer of $[{{\mathcal K}}_y]$.
The second condition holds true for $\eta$ and $\nu$ sufficiently large. In fact, ${{\mathcal K}}_y$ determines the fibre $f^{-1}(y)$ as a subvariety of $${{\mathbb P}}(H^0(f^{-1}(y),({{\mathcal M}}^\eta
\otimes \omega_{X/Y}^{e\cdot\eta})|_{f^{-1}(y)})),$$ and $[{{\mathcal K}}_y]$ is nothing but the point of the Hilbert scheme $\rm Hilb$ parameterizing subvarieties of this projective space. By [@Vie], 7.2, the stabilizer of such a point is finite.
The assumption in c) implies that ${{\mathcal K}}$ is not of maximal variation. Hence for all points $z$ in a neighborhood $U_y$ of $y$, the orbits $G_z$ coincide. In different terms the images of $z\in U_y$ in $\rm Hilb$ all belong to the same $G$-orbit. Since $M_h$ is a quotient of a subscheme of $\rm Hilb$ by the $G$-action, the morphism $\varphi:U \to M_h$ is constant, as claimed in c).
Families of canonically polarized manifolds {#sectcan}
===========================================
For families with $\omega_F$ big and semi-ample the equivalence of birational and biregular isotriviality has been shown by Maehara in [@Mae]. For families of canonically polarized manifolds, one just has to use, that the fibres are their own canonical model.
Or, to formulate the proof parallel to the one given below in the Kodaira dimension zero case, one could argue in the following way. Assume that $U$ is a curve, and choose a semistable compactification $f:X\to Y$ of $V\to U$. By assumption $X$ is birational to the trivial family $F'\times Y$ over $Y$, hence $f_*\omega_{X/Y}^\nu$ is a direct sum of copies of ${{\mathcal O}}_Y$, for all $\nu$. Obviously, if ${{\mathcal M}}$ is some power of $\omega_{X/Y}$ this implies that $$\det(f_*({{\mathcal M}}^\nu\otimes\omega^{e\cdot\nu}_{X/Y}))\otimes\det(f_*{{\mathcal M}})^{-\nu
r(\nu)}={{\mathcal O}}_Y,$$ and by \[positiv\], c), one finds $V\to U$ to be biregulary isotrivial.
Families of manifolds of Kodaira dimension zero {#sectzero}
===============================================
Let $U$ be a curve and $f:V\to U$ be a family of polarized manifolds $F$ with $\omega_F^\delta={{\mathcal O}}_F$. In order to prove \[mainprop\] we may replace $U$ by some finite cover, and we may choose a compactification $f:X\to Y$ satisfying the assumptions made in Theorem \[positiv\].
By assumption, $\lambda=f_*\omega_{X/Y}^{\delta}$ is an invertible sheaf, and the natural map $f^*\lambda\to \omega_{X/Y}^\delta$ is an isomorphism over $V$. Let $E$ be the zero divisor of this map, i.e. $$\omega_{X/Y}^\delta = f^*\lambda\otimes{{\mathcal O}}_X(E).$$ $E$ is supported in $\Delta=X\setminus V$ and, since the fibres of $f$ are reduced divisors, $E$ can not contain a whole fibre. Hence for all $\mu >0$ $f_*{{\mathcal O}}_X(\mu E)={{\mathcal O}}_Y$ and $f_*\omega_{X/Y}^{\mu\cdot\delta}=\lambda^\mu$.
Let ${{\mathcal M}}'$ be any polarization which is very ample and without higher cohomology on the fibres of $V\to U$. The sheaf $$f_*({{\mathcal M}}'\otimes{{\mathcal O}}_X(*E))=f_*({{\mathcal M}}'\otimes(\lim_{\mu>0}{{\mathcal O}}(\mu E)))$$ is coherent, hence locally free.
In fact, locally étale or locally analytic we can choose over a neighborhood ${{\mathcal U}}$ of $s\in S=Y\setminus U$ a section $\sigma:{{\mathcal U}}\to X$ with image $C$, not meeting the support of $E$. If $I$ denotes the ideal sheaf of $C$, for some $\rho \gg 0$ $$f_*({{\mathcal M}}'|_{f^{-1}({{\mathcal U}})}\otimes I^\rho)=0.$$ Since direct images are torsion free, and since $E$ is supported in fibres, $$f_*(({{\mathcal M}}'\otimes{{\mathcal O}}_X(*E))|_{f^{-1}({{\mathcal U}})}\otimes I^\rho)=0.$$ Then $f_*({{\mathcal M}}'\otimes{{\mathcal O}}_X(*E))|_{{\mathcal U}}$ is a torsion free subsheaf of $$f_*(({{\mathcal M}}'\otimes{{\mathcal O}}_X(*E))|_{f^{-1}({{\mathcal U}})}\otimes {{\mathcal O}}_{f^{-1}({{\mathcal U}})}/I^\rho)=
f_*({{\mathcal M}}'|_{f^{-1}({{\mathcal U}})}\otimes {{\mathcal O}}_{f^{-1}({{\mathcal U}})}/I^\rho),$$ hence coherent.
Let ${{\mathcal M}}$ be the reflexive hull of the image of $$f^*f_*({{\mathcal M}}'\otimes{{\mathcal O}}_X(*E))\>>> {{\mathcal M}}'\otimes{{\mathcal O}}_X(*E).$$ ${{\mathcal M}}$ is again coherent, and it must be contained in ${{\mathcal M}}'\otimes{{\mathcal O}}_X(\alpha\cdot E)$ for some $\alpha$. Since it is reflexive, it is an invertible sheaf. By construction ${{\mathcal M}}|_V\simeq {{\mathcal M}}'|_V$ and $$\begin{gathered}
f_*({{\mathcal M}}'\otimes{{\mathcal O}}_X(*E))= f_*f^*f_*({{\mathcal M}}'\otimes{{\mathcal O}}_X(*E)) \subset\\
f_*{{\mathcal M}}\subset f_*({{\mathcal M}}\otimes{{\mathcal O}}_X(*E)) \subset f_*({{\mathcal M}}'\otimes{{\mathcal O}}_X(*E)),\end{gathered}$$ hence all those sheaves coincide. We found an invertible sheaf ${{\mathcal M}}$ satisfying the assumptions made in \[positiv\] with the additional condition $$f_*({{\mathcal M}}\otimes \omega_{X/Y}^e)=f_*({{\mathcal M}}\otimes {{\mathcal O}}_X(\frac{e}{\delta}\cdot E)
\otimes f^*\lambda^{\frac{e}{\delta}})=
f_*{{\mathcal M}}\otimes \lambda^{\frac{e}{\delta}},$$ for all multiples $e$ of $\delta$. For those $e$ $$\begin{gathered}
\label{direct}
\big( \bigotimes^r f_*({{\mathcal M}}\otimes \omega_{X/Y}^{e})\big)
\otimes\det(f_*{{\mathcal M}})^{-1} =\\
\big( \bigotimes^r (f_*{{\mathcal M}})\otimes \lambda^{\frac{e}{\delta}}
\big) \otimes\det(f_*{{\mathcal M}})^{-1}.\end{gathered}$$
\
By assumption $f:X\to Y$ is birational over $Y$ to the trivial family $pr_2:F'\times Y \to Y$, hence $$\lambda=f_*\omega^\delta_{X/Y}={{\mathcal O}}_Y.$$ So the sheaf in (\[direct\]) is $$\big( \bigotimes^r (f_*{{\mathcal M}}) \big) \otimes\det(f_*{{\mathcal M}})^{-1}.$$ Since its determinant is of degree zero, it can not be ample. (\[direct\]) and \[positiv\], b), imply that for no $\nu >0$ the sheaf $$\det(f_*({{\mathcal M}}^\nu\otimes\omega^{e\cdot\nu}_{X/Y}))\otimes\det(f_*{{\mathcal M}})^{-\nu
\cdot r(\nu)}$$ is ample. By \[positiv\], a), it is of non negative degree, and \[positiv\], c), implies that the family $V\to U$ is biregulary isotrivial.
Assume that $\delta=1$, hence that $\omega_{V/U}=f^*\lambda|_U$. The argument used in the proof of \[mainprop\] shows in this particular case that “$f$ non-isotrivial” implies that on the compactification $Y$ of $U$ $$\deg(f_*\omega_{X/Y}) >0.$$ Since the same holds true for all finite coverings of $U$, one obtains that the fibres of the period map from $M_h$ to the period domain classifying the corresponding variations of Hodge structures can not contain a quasi-projective curve. Of course this is a well known consequence of the local Torelli Theorem for manifolds with a trivial canonical bundle.
Kodaira-Spencer maps {#kernels}
====================
Recall first the following definition, replacing of nef and ample, on projective manifolds $Y$ of higher dimension.
\[positivity\] Let ${{\mathcal F}}$ be a torsion free coherent sheaf on a quasi-projective normal variety $Y$ and let ${{\mathcal H}}$ be an ample invertible sheaf.
1. ${{\mathcal F}}$ is generically generated if the natural morphism $$H^0 (Y, {{\mathcal F}}) \otimes {{\mathcal O}}_Y \longrightarrow {{\mathcal F}}$$ is surjective over some open dense subset $U_0$ of $Y$. If one wants to specify $U_0$ one says that ${{\mathcal F}}$ is globally generated over $U_0$.
2. ${{\mathcal F}}$ is weakly positive if there exists some dense open subset $U_0$ of $Y$ with ${{\mathcal F}}|_{U_0}$ locally free, and if for all $\alpha > 0$ there exists some $\beta > 0$ such that $$S^{\alpha \cdot \beta} ({{\mathcal F}}) \otimes {{\mathcal H}}^{\beta}$$ is globally generated over $U_0$. We will also say that ${{\mathcal F}}$ is weakly positive over $U_0$, in this case.
3. ${{\mathcal F}}$ is big if there exists some open dense subset $U_0$ in $Y$ and some $\mu > 0$ such that $$S^{\mu} ({{\mathcal F}}) \otimes {{\mathcal H}}^{-1}$$ is weakly positive over $U_0$. Underlining the role of $U_0$ we will also call ${{\mathcal F}}$ ample with respect to $U_0$.
Here, as in [@Vie] and [@VZ3], we use the following convention: If ${{\mathcal F}}$ is a coherent torsion free sheaf on a quasi-projective normal variety $Y$, we consider the largest open subscheme $i: Y_1 \to
Y$ with $i^* {{\mathcal F}}$ locally free. For $$\Phi = S^{\mu}, \ \ \
\Phi =\bigotimes^\mu \mbox{ \ \ \ or \ \ \ }\Phi = \det$$ we define $$\Phi ({{\mathcal F}}) = i_* \Phi (i^* {{\mathcal F}}).$$
Again, $f:V\to U$ denotes a smooth family of manifolds over a quasi-projective manifold $U$, which is allowed to be of dimension larger than one. We choose non-singular projective compactifications $Y$ of $U$ and $X$ of $V$, such that both $S=Y\setminus U$ and $\Delta=X\setminus V$ are normal crossing divisors and such that $f$ extends to $f:X\to Y$. As usual $\eta$ will denote a closed point in sufficient general position on $U$ and $X_\eta$ the fibre of $f$ over $\eta$. We will write $T^i_{X_\eta}$ (or $T^i_{X/Y}(-\log \Delta)$ $\ldots$) for the $i$-th wedge product of $T_{X_\eta}$ (or of $T_{X/Y}(-\log
\Delta)=\Omega^1_{X/Y}(\log \Delta)^\vee$ $\ldots$).
Let $T_\eta$ denote the restriction $T_U\otimes {{\mathbb C}}$ of the tangent sheaf of $U$ to $\eta$. The Kodaira-Spencer map $$T_\eta \>>> H^1(X_\eta,T_{X_\eta})$$ gives rise to $$\bigotimes^\nu T_\eta \>>> \bigotimes^\nu H^1(X_\eta,T_{X_\eta}) \>>>
H^\nu(X_\eta,T^\nu_{X_\eta}).$$ The composite map factors through $$S^\nu(T_\eta)\>>> H^\nu(X_\eta,T^\nu_{X_\eta}).$$ One defines $$\begin{gathered}
\mu(f)={\rm Max}\{\nu\in{{\mathbb N}}-\{0\}; \ S^\nu(T_\eta)\>>> H^\nu(X_\eta,T^\nu_{X_\eta})
\mbox{ is non zero}\}.\end{gathered}$$ Of course, $\mu(f) \leq n=\dim(X_\eta)$. We do not know any criterion, implying that for $f:V\to U$ one has $\mu(f)=\dim(V)-\dim(U)$.
For example, if $U$ is a curve and $f:V\to U$ a family of polarized manifolds, restricting the tautological sequence to ${X_\eta}=f^{-1}(\eta)$ one obtains an extension $$0 \>>> T_{X_\eta} \>>> T_X|_{X_\eta} \>>> \mathcal O_{X_\eta} \>>> 0$$ and the induced class $\xi_\eta\in H^1({X_\eta},T_{X_\eta})$. Then $\mu(f)=\mu$ if and only if for $\eta$ in general position, the wedge product $\wedge^\mu \xi_\eta \in H^\mu({X_\eta},T_{X_\eta}^{\mu})$ is non-zero, whereas $\wedge^{\mu+1} \xi_\eta \in H^{\mu+1}({X_\eta},T_{X_\eta}^{\mu+1})$ is zero.
\[problem2\] Are there properties of ${X_\eta}$ which imply that for all families $V\to U$ over a curve $U$, with general fibre ${X_\eta}=f^{-1}(\eta)$ the class $$\wedge^n \xi_\eta \in H^n({X_\eta},\omega_{X_\eta}^{-1})$$ is non-zero?
Being optimistic, one could try in \[problem2\] the condition “$\Omega_{X_\eta}^1$ ample”.\
A slight extension of the main result of [@VZ3] says:
\[stthm2\] Assume that for a general fibre $X_\eta$ of $f:X \to Y$ either $\omega_{X_\eta}$ is ample, or $\omega^\delta_{X_\eta}={{\mathcal O}}_{X_\eta}$, for some $\delta$.
1. Then for some $m >0$ the sheaf $S^m(\Omega_Y^1(\log S))$ contains an invertible subsheaf ${{\mathcal M}}$ of Kodaira dimension ${\rm Var}(f)$.
2. If ${\rm Var}(f)=\dim(U)$ the sheaf $S^{\mu(f)}(\Omega_Y^1(\log S))$ contains a big coherent subsheaf ${{\mathcal P}}$.
3. Let $Z$ be a submanifold of $Y$ such that $S_Z=S\cap Z$ remains a normal crossing divisor, and such that $W=X\times_YZ$ is non-singular. For the induced family $h:W \to Z$ assume that $\mu(f)=\mu(h)$. Then, if ${\rm Var}(f)=\dim(U)$, the restriction of the sheaf ${{\mathcal P}}$ from part ii) to $S^{\mu(f)}(\Omega^1_Z(\log S_Z))$ is non trivial.
4. Assume in iii) that $h:W\to Z$ is a desingularization of the pullback of a family $h':W'\to Z'$ under $\pi:Z\to Z'$, with $Z'$ non-singular and with $h'$ smooth over $Z'\setminus S_{Z'}$ for a normal crossing divisor $S_{Z'}$. Then then the restriction of the sheaf ${{\mathcal P}}$ to $S^{\mu(f)}(\Omega^1_Z(\log S_Z))$ lies in $S^{\mu(f)}(\pi^*(\Omega^1_{Z'}(\log S_{Z'})))$.
Parts i) and ii) have been shown in [@VZ3], 1.4, for canonically polarized manifolds with $\mu(f)$ replaced by the fibre dimension $n$. We will just sketch the changes which allow to extend the arguments used in [@VZ3] to cover \[stthm2\], ii), iii) and iv), for canonically polarized manifolds. Next we will try to convince the reader, that the same proof goes through for minimal models of Kodaira dimension zero.
As in [@VZ3] we drop the assumption that $Y$ is projective. Leaving out a codimension two subscheme, we may assume that $f$ is flat and that $\Delta$ is a relative normal crossing divisor. Then we have the tautological exact sequence $$0 \>>> f^*\Omega^1_Y(\log S) \>>> \Omega^1_X(\log \Delta)
\>>> \Omega^1_{X/Y}(\log \Delta) \>>> 0$$ and the wedge product sequences $$\begin{gathered}
\label{exact}
0\>>> {f}^*\Omega^1_Y(\log S)\otimes
\Omega^{p-1}_{X/Y}(\log \Delta)
\>>> \\ {\mathfrak g \mathfrak r}(\Omega_X^p(\log
\Delta)) \>>> \Omega_{X/Y}^p(\log \Delta)\>>> 0,\end{gathered}$$ where $$\begin{gathered}
{\mathfrak g \mathfrak r}(\Omega_X^p(\log \Delta))=
\Omega_X^p(\log \Delta)
/f^*\Omega^2_Y(\log S)\otimes \Omega^{p-2}_{X/Y}(\log
\Delta).\end{gathered}$$ For the invertible sheaf ${{\mathcal L}}=\Omega_{X/Y}^n(\log \Delta)$ we consider the sheaves $$F^{p,q}:=R^qf_*(\Omega^{p}_{X/Y}(\log
\Delta)\otimes{{\mathcal L}}^{-1})$$ together with the edge morphisms $$\tau_{p,q}:F^{p,q}\>>> F^{p-1,q+1}\otimes \Omega^1_{Y}(\log
S),$$ induced by the exact sequence (\[exact\]), tensored with ${{\mathcal L}}^{-1}$. As explained in [@VZ3], Proof of 4.4 iii), over $U$ the edge morphisms $\tau_{p,q}$ can also be obtained in the following way. Consider the exact sequence $$0 \>>> T_{V/U} \>>> T_V \>>> f^*T_U \>>> 0,$$ and the induced wedge product sequences $$0 \>>> T^{n-p+1}_{V/U} \>>> \tilde T^{n-p+1}_V \>>> T^{n-p}_{V/U}\otimes f^*T_U \>>> 0,$$ where $\tilde{T}^{n-p+1}_{V}$ is a subsheaf of $T^{n-p+1}_V$. One finds edge morphisms $$\tau^\vee_{p,q}:(R^{q}f_*T^{n-p}_{V/U}) \otimes T_U \>>> R^{q+1}f_*T^{n-p+1}_{V/U}.$$ Restricted to $\eta$ those are just the wedge product with the Kodaira-Spencer class. Moreover, tensoring with $\Omega_U^1$ one gets back $\tau_{p,q}|_U$. Hence $\mu(f)$ is the smallest number $m$ for which the composite $$\begin{gathered}
\tau^m:F^{n,0}={{\mathcal O}}_Y \> \tau_{n,0} >> F^{n-1,1}\otimes \Omega_{U}^1
\> \tau_{n-1,1} >>
F^{n-2,2}\otimes S^2(\Omega^1_U) \>>> \cdots \\
\> \tau_{n-m+1,m-1}>> F^{n-m,m}\otimes S^m(\Omega^1_U)\end{gathered}$$ is non-zero. Next we used that (replacing $Y$ by some covering) there is an ample invertible sheaf ${{\mathcal A}}$ on $Y$ such that the kernel ${{\mathcal K}}$ of $${\rm id}_{{{\mathcal A}}}\otimes\tau_{n-m,m}: {{\mathcal A}}\otimes F^{n-m,m} \>>>
{{\mathcal A}}\otimes F^{n-m-1,m+1} \otimes \Omega^1_{Y}(\log S)$$ is negative, or to be more precise, that its dual is weakly positive. This gives a non-trivial map $$\upsilon:{{\mathcal A}}\otimes {{\mathcal K}}^\vee \>>> S^m(\Omega^1_Y(\log S))$$ and we take for ${{\mathcal P}}$ its image.
The number $m$ was used in the proof of [@VZ3], 1.4 ii), hence there is no harm to replace the upper bound $n$, used there, by the more precise number $\mu(f)$ in \[stthm2\], ii).
The sheaves $F^{p,q}$ are compatible with restriction to the subvariety $Z$. The assumption $\mu(f)=\mu(h)$ implies that the restriction $$\upsilon|_Z:{{\mathcal A}}|_{Z} \otimes {{\mathcal K}}^\vee|_{Z} \>>> S^m(\Omega^1_{Z}(\log S_Z))$$ is non-trivial. In fact, the kernel ${{\mathcal K}}'$ of $${\rm id}_{{{\mathcal A}}_{Z}}\otimes\tau^{Z}_{n-m,m}: {{\mathcal A}}|_{Z}\otimes F^{n-m,m}|_{Z}
\>>> {{\mathcal A}}|_{Z} \otimes F^{n-m-1,m+1}|_{Z} \otimes \Omega^1_{Z}(\log S_Z)$$ contains ${{\mathcal K}}|_{Z}$, and the diagram $$\@CD
{{\mathcal A}}|_{Z} \otimes {{\mathcal K}}^\vee|_{Z} \>>> S^m(\Omega^1_{Y}(\log S))|_{Z}\\
{{\mathbb A}}A A {{\mathbb V}}V V \\
{{\mathcal A}}|_{Z} \otimes {{{\mathcal K}}'}^\vee \>>> S^m(\Omega^1_{Z}(\log S_Z))
\@endCD$$ is commutative. One obtains iii).\
Since the sheaves $F^{p,q}$ and the maps $\tau_{p,q}$ are compatible with pullbacks, under the additional assumptions made in iv), the image of $${{\mathcal A}}|_{Z} \otimes {{{\mathcal K}}'}^\vee \>>>
S^m(\Omega^1_{Z}(\log S_Z))$$ lies in $S^m(\pi^*(\Omega^1_{Z'}(\log S_{Z'})))$ and the same holds true for the restriction of ${{\mathcal P}}$.\
If one considers the proof of [@VZ3], 1.4, i) and ii), the assumption that the fibres are canonically polarized is used twice. First of all, since we apply in the proof of 4.4, iv), the Akizuki-Kodaira-Nakano vanishing theorem to the restriction of $\omega_F$ to a smooth multicanonical divisor $B$. If some power of $\omega_F$ is trivial the divisor $B$ is empty, and there is nothing to show.
The second time is in the proof of [@VZ3], 4.8. We use the diagram (2.8.1) and the fact that the morphism $Z^{\#} \to Y^{\#}$ considered there is of maximal variation. The construction of (2.8.1) just uses the existence of the moduli scheme $M_h$, and it provides a morphism $Z^{\#} \to Y^{\#}$ induced by a generically finite morphism $Y^{\#} \to M_h$.
This construction works in particular for the moduli scheme of polarized manifolds $F$ with $\omega_F^\delta={{\mathcal O}}_F$, and \[stthm\] implies that the variation of the morphism $Z^{\#} \to Y^{\#}$ is again maximal.
The rest of the arguments, as given on page 311–313 of [@VZ3] remain unchanged, and one obtains \[stthm2\], i), ii) and iii).
For families $f:X\to Y$ with $\omega_{X_\eta}$ semiample, and with $\mu(f)=n$ one can add to [@VZ3], 1.4, a statement similar to \[stthm2\], iii) and iv). Since the later will not be used, we omit it.
\[stthm3\] Assume $\omega_{X_\eta}$ is semiample, and ${\rm Var}(f)=\dim(f)$.
1. There exists a non-singular finite covering $\psi:Y'\to Y$ and a big coherent subsheaf ${{\mathcal P}}'$ of $\psi^*S^{m}(\Omega^1_Y(\log S))$, for some $m \leq \mu(f)$.
2. If $\mu(f)=n$, then one finds $m=n$ in i).
3. Let $Z$ be a submanifold of $Y$ such that $S_Z=S\cap Z$ remains a normal crossing divisor, and such that $W=X\times_YZ$ is non-singular. For the induced family $h:W \to Z$ assume that $\mu(f)=\mu(h)=n$. Then one can choose the covering $\psi$ such that $\psi^{-1}(Z)$ is non-singular, $\psi^{-1}(S_Z)$ a normal crossing divisor and such that the image of the sheaf ${{\mathcal P}}'$ from part i) in $\psi^*S^n(\Omega^1_Z(\log S_Z))$ is non trivial.
We keep the notations from the sketch of the proof of \[stthm2\]. For part i) we replaced in [@VZ3], page 309 and 310, the sheaves $F^{p,q}$ (in fact a twist of those by some invertible sheaf on $Y$) by some quotient sheaves. But then $\mu(f)$ remains an upper bound for the number $m$, used there, and one obtains \[stthm2\], i), as stated.
However, one has no control on the behavior of $m$ under restriction to subvarieties. So for part ii) and iii) we have to recall the construction in more detail. To get the weak positivity of the kernels ${{\mathcal K}}$ one has to replace (over some covering $Y'$ of $Y$ whose ramification divisor is in general position) ${{\mathcal A}}\otimes F^{n-m,m}$ by its image ${{\mathcal A}}\otimes \tilde F^{n-m,m}$ in some larger sheaf $E^{n-m,m}$. Here $$\big(\bigoplus_{p+q=n}E^{p,q},\bigoplus_{p+q=n}\theta_{p,q}\big)$$ is again a Higgs bundle, and $\theta_{p,q}$ is compatible with $\tau_{p,q}$.
As stated in the proof of [@VZ3], 4.4, iv) the kernel and cokernel of the map $${{\mathcal A}}\otimes F^{n-m,m} \to E^{n-m,m}$$ are direct images of the $n-m-1$-forms of a multicanonical divisor. If $n=m$ there are no such forms, and ${{\mathcal A}}\otimes F^{0,n}$ is a subsheaf of $E^{0,n}$. Hence $\tau^n\neq 0$ implies that the corresponding map for ${{\mathcal A}}\otimes \tilde F^{n-m,m}$ is non-zero. The compatibility with restrictions follows by the argument used in the proof of \[stthm2\], iii).
Subvarieties of the moduli stack of polarized manifolds of Kodaira dimension zero {#sectappl}
=================================================================================
Theorem \[stthm2\] has a number of geometric implication for manifolds $U$ mapping to moduli stacks of polarized manifolds, i.e. for morphisms $\varphi:U \to M_h$ induced by a family $f:V\to U$. Those had been shown in [@VZ3] for the moduli stack of canonically polarized manifolds. The proves are all based on vanishing theorems for logarithmic differential forms, and they do not refer to the type of fibres of $f$, once \[stthm2\], i) and ii), is established.
Using \[stthm\] we extended \[stthm2\], i) and ii), to a larger class of families of polarized manifolds. Hence the geometric implications carry over to this larger class, i.e. to the moduli stack of polarized manifolds with $\omega_F^\delta={{\mathcal O}}_F$, for some $\delta>0$. For the readers convenience we recall the statements below.
\[cor1\] Let $M_h$ be the moduli scheme of canonically polarized manifolds, or of polarized manifolds $F$ with $\omega_F^\delta$ trivial for some $\delta >0$.
1. Assume that $U$ satisfies one of the following conditions
1. $U$ has a non-singular projective compactification $Y$ with $S=Y\setminus U$ a normal crossing divisor and with boundary $T_Y(-\log S)$ weakly positive.
2. Let $H_1+ \cdots + H_\ell$ be a reduced normal crossing divisor in ${{\mathbb P}}^N$, and $\ell< \frac{N}{2}$. For $0\leq r\leq l$ define $$\begin{gathered}
H = \bigcap_{j=r+1}^\ell H_j, \ \ \ S_i = H_i|_H, \ \ \
S = \sum_{i=1}^r S_i,\end{gathered}$$ and assume $U= H \setminus S$.
3. $U={{\mathbb P}}^N\setminus S$ for a reduced normal crossing divisor $$S=S_1+ \cdots + S_\ell$$ in ${{\mathbb P}}^N$, with $\ell< N.$
Then a morphism $U \to M_h$, induced by a family, must be trivial.
2. For $Y={{\mathbb P}}^{\nu_1}\times \cdots \times
{{\mathbb P}}^{\nu_k}$ let $$D^{(\nu_i)}=D_0^{(\nu_i)}+\cdots +D_{\nu_i}^{(\nu_i)}$$ be coordinate axes in ${{\mathbb P}}^{\nu_i}$ and $$D=\sum_{i=1}^kD^{(\nu_i)}.$$ Assume that $S=S_1+\cdots S_\ell$ is a divisor, such that $D+S$ is a reduced normal crossing divisor, and $\ell < \dim(Y)$. Then there exists no morphism $\varphi:U=Y\setminus (D+S) \to M_h$ with $$\dim(\varphi(U)) > {\rm Max}\{\dim(Y)-\nu_i; \ i=1,\ldots ,k\}.$$
3. Let $U$ be a quasi-projective variety and let $\varphi:U \to
M_h$ be a quasi-finite morphism, induced by a family. Then $U$ can not be isomorphic to the product of more than $\mu(f)$ varieties of positive dimension.
Rigidity
========
Again, $f:V\to U$ denotes a smooth family of manifolds with $\omega_{V/U}$ $f$-semiample and with ${\rm Var}(f)=\dim(U)>0$. We say that $f$ is rigid, if there exists no non-trivial deformation over a non-singular quasiprojective curve $T$.
Here a deformation of $f$ over $T$, with $0\in T$ a base point, is a smooth projective morphism $$g:{{\mathcal V}}\to U\times T$$ for which there exists a commutative diagram $$\@CD
V \> \simeq >> g^{-1}(U\times\{0\}) \>\subset >> {{\mathcal V}}\\
{{\mathbb V}}f VV {{\mathbb V}}V V {{\mathbb V}}V g V \\
U \> \simeq >> U\times\{0\} \> \subset >> U\times T
\@endCD.$$ If the fibres $F$ of $f$ are canonically polarized, or if some power of $\omega_F$ is trivial, this says that morphisms from $U$ to the moduli stack do not deform.
\[proprigid\] Assume either that $\omega_{X_\eta}$ is ample, or that $\omega^\delta_{X_\eta}={{\mathcal O}}_{X_\eta}$, for some $\delta$. Assume that ${\rm Var}(f)=\dim(U) >0$. Let $T$ be a non-singular quasi-projective curve. Let $g:{{\mathcal V}}\to U\times T$ be a deformation of $f$. If $\mu(f)=\mu(g)$, then ${\rm Var}(g)=\dim(U)$.
Suppose that ${\rm Var}(g)>\dim (U).$ Then $$\dim (U)+1 = \dim U\times T \geq {\rm Var}(g)>\dim (U),$$ hence, $\dim(U\times T)={\rm Var}(g).$ Let $\bar T$ be a non-singular compactification of $T$, $S_{\bar T}=\bar T \setminus T$. Correspondingly we write $S_{Y\times \bar T}$ for the complement of $U\times T$ in $Y\times \bar T$.
By Theorem \[stthm2\], ii, one finds a big coherent subsheaf ${{\mathcal P}}$ of $$S^{\mu(g)}(\Omega^1_{Y\times \bar T}(\log S_{Y\times \bar T})),$$ and by \[stthm2\], iii) the image of ${{\mathcal P}}$ in $$S^{\mu(g)}(\Omega^1_{Y\times \{0\}}(\log S_{Y\times \{0\}}))=
pr_1^*(S^{\mu(g)}(\Omega^1_{Y}(\log S)))|_{Y\times\{0\}}$$ is non-zero. Then, the image of ${{\mathcal P}}$ in $$pr_1^*(S^{\mu(g)}(\Omega^1_{Y}(\log S)))$$ is non-zero, and for a point $y \in Y$ in general position, the image of ${{\mathcal P}}$ under $$S^{\mu(g)}(\Omega^1_{Y\times \bar T}
(\log S_{Y\times \bar T}))|_{\{y\} \times T}
\>>>
pr_1^*(S^{\mu(g)}(\Omega^1_Y(\log S)))|_{ \{y \}\times T}$$ is not zero. Note that any non-zero quotient of the coherent sheaf ${{\mathcal P}}|_{\{y\} \times T}$ for $y$ in general position must be big. In fact, if ${{\mathcal P}}$ is ample over some open dense subset $W_0$ of $Y\times \bar T$, one just has to make sure that $\{y\}\times T$ meets $W_0$. Since $pr_1^*(S^{\mu(g)}(\Omega^1_Y(\log S)))|_{\{y\}\times T}$ is a direct sum of copies of ${{\mathcal O}}_{\{y\} \times T}$ this is not possible.
Using \[stthm3\] instead of \[stthm2\] one obtains a similar result for families with $\omega_{V/U}$ $f$ semiample, whenever $\mu(f)=n$.
\[proprigidsa\] Assume that $\omega_{X_\eta}$ is semiample, that $${\rm
Var}(f)=\dim(U) >0$$ and that $$\mu(f)=\dim(X_\eta)=n.$$ Let $T$ be a non-singular quasi-projective curve, and let $g:{{\mathcal V}}\to U\times
T$ be a deformation of $f$. Then ${\rm Var}(g)=\dim(U)$.
If ${\rm Var}(g)>\dim (U)$, again one finds $\dim(U\times T)={\rm Var}(g).$ Let us keep the notations from the proof of \[proprigid\]. By Theorem \[stthm3\], ii, one finds a finite covering $\psi:Y'\to Y$ and a big coherent subsheaf ${{\mathcal P}}'$ of $$\psi^*(S^{\mu(g)}(\Omega^1_{Y\times \bar T}(\log S_{Y\times \bar T}))),$$ and by \[stthm3\], iii) the image of ${{\mathcal P}}'$ in $$\psi^*(S^{\mu(g)}(\Omega^1_{Y\times \{0\}}(\log S_{Y\times \{0\}})))=
\psi^*pr_1^*(S^{\mu(g)}(\Omega^1_{Y}(\log S))|_{Y\times\{0\}})$$ is non-zero. Then, for a point $y \in Y$, in general position, the image of ${{\mathcal P}}'$ under $$\begin{gathered}
\psi^*(S^{\mu(g)}(\Omega^1_{Y\times \bar T}
(\log S_{Y\times \bar T}))|_{\{y\} \times T})
\>>>\\
\psi^*(pr_1^*(S^{\mu(g)}(\Omega^1_{Y}(\log S)))|_{\{y\} \times T})\end{gathered}$$ is not zero. Again, since the sheaf on the right hand side is trivial, one obtains a contradiction.
Let $h:{{\mathcal X}}\to Z$ be a polarized family of manifolds $F$ with $\omega_F$ semiample and of maximal variation, over a non-singular quasi-projective manifold $Z$. Assume that $\bar Z$ is a projective compactification of $Z$, such that $\bar Z \setminus Z$ is a normal crossing divisor. Assume in addition, that there is an open dense subscheme $Z_0$ such that for all subvarieties ${{\mathcal U}}$ of $Z$ meeting $Z_0$ $${\rm Var}({{\mathcal X}}\times_{Z}{{\mathcal U}}\to {{\mathcal U}})=\dim({{\mathcal U}}).$$ Let $Y$ be a non-singular projective curve and let $U\subset Y$ be open and dense. Let us write $${\rm \bf H}={\rm\bf Hom}((Y,U),(\bar Z,Z ))$$ for the scheme parameterizing non-trivial morphisms $\psi:Y \to \bar Z$ with $\psi(U)\subset Z$ and $${\rm \bf H}_{Z_0} ={\rm\bf Hom}((Y,U),(\bar Z,Z );Z_0) \subset {\rm\bf H}$$ for those with $\psi(U)\cap Z_0 \neq \emptyset$. Based on the bounds obtained in [@VZ1] we have shown in [@VZ3] that ${\rm\bf H}_{Z_0}$ is of finite type.
\[corrigid\]
1. 1. Let $\psi:U\to Z$ be a morphism and $f:V\to U$ the pull back family. Assume that $\psi(U)\cap Z_0 \neq \emptyset$ and that $$\mu(f:V \to U)=\dim(F)=n$$ Then the point $[\psi:Y\to \bar Z]$ is isolated in ${\rm\bf H}_{Z_0}$.
2. Assume for all fibres $F$ of $h^{-1}(Z_0) \to Z_0$ and for all $\xi\in H^1(F,T_F)$ $$0\neq \wedge^n \xi \in
H^n (F,\omega_F^{-1}).$$ Then ${\rm\bf H}_{Z_0}$ is a finite set of points.
3. Assume that the fibres $F$ of $h:{{\mathcal X}}\to Z$ are either canonically polarized, or of Kodaira dimension zero.
4. Let $\psi:U\to Z$ be a morphism and $f:V\to U$ the pull back family. Assume that $\psi(U)\cap Z_0 \neq \emptyset$ and that $$\mu(f:V \to U)=\mu(h:{{\mathcal X}}\to Z).$$ Then the point $[\psi:Y\to \bar Z]$ is isolated in ${\rm\bf H}_{Z_0}$.
5. Assume there exists a constant $\mu$ such that for all fibres $F$ of $h^{-1}(Z_0) \to Z_0$ and for all $\xi\in H^1(F,T_F)$ $$0\neq \wedge^\mu \xi \in
H^\mu (F,T_F^{\mu})$$ but $$0 = \wedge^{\mu+1} \xi \in
H^{\mu+1} (F,T_F^{\mu+1}).$$ Then ${\rm\bf H}_{Z_0}$ is a finite set of points.
In both cases b) follows from a). For the latter assume that $[\psi]$ lies in a component of $\rm\bf H$ of dimension larger than zero. Let $T$ be a curve in $\rm\bf H$, containing the point $[\psi]$. Then one has a non-trivial deformation $\Psi:U\times T \to Z$ of $\psi$, hence a non-trivial deformation $g:{{\mathcal V}}\to U\times T$ of $f:V\to U$. By \[proprigid\] in case II) or by \[proprigidsa\] in case I) $${\rm Var}(g)={\rm Var}(f) < \dim(U\times T) = \dim(U)+1,$$ contradicting the assumption made on $Z_0$ and ${{\mathcal X}}\to Z$.
Corollary \[corrigid\], II), should imply certain finiteness results for curves in the moduli scheme $M_h$ of canonically polarized manifolds, or the moduli scheme of minimal models of Kodaira dimension zero meeting an open subscheme $W$ where the assumption corresponding to the one in \[corrigid\], II), b), holds true. However, one would have to show, that morphisms $\varphi$ which factor through the moduli stack, are parameterized by some coarse moduli scheme. Hopefully this can be done extending the methods used in [@Cap] for moduli of curves to moduli of higher dimensional manifolds.
Here we will show a slightly weaker statement, which coincides with \[corrigid3\] for $\mu=n$.
\[corrigid2\] Let $M_h$ be either the moduli scheme of canonically polarized manifolds or the moduli scheme of polarized manifolds $F$ with $\omega_F^\delta={{\mathcal O}}$ for some $\delta >0$. Let $0< \mu \leq
n=\dim(F)$ be a constant such that for all $(F,{{\mathcal L}})$, and for all $\xi\in H^1(F,T_F)$ $$0 = \wedge^{\mu+1} \xi \in H^{\mu+1}
(F,T_F^{\mu+1}).$$ Then for a quasi-projective non-singular curve $U$ there are only finitely many morphisms $\varphi:U\to M_h$ which are induced by a smooth family $f:V\to U$ with $\mu(f)=\mu$.
Let us choose any projective compactification $\bar M_h$ of $M_h$, and an invertible sheaf ${{\mathcal H}}$ on $\bar M_h$ which is ample with respect to $M_h$. As usual, $Y$ will be a non-singular projective curve containing $U$. We write $s$ for the number of points in $S=Y\setminus U$ and $g(Y)$ for the genus of $Y$.
To show that there are only finitely many components of the scheme $${\rm\bf Hom}((Y,U),(M_h,\bar M_h))$$ which contain a morphism $\varphi:U\to M_h$ factoring through the moduli stack, one has to find an upper bound for $\varphi^*{{\mathcal H}}$. To this aim one may assume that $\bar M_h$ is reduced. The proof for the boundedness follows the line of the proof of [@VZ3], 6.2.
Kollar and Seshadri constructed (see [@Vie], 9.25) a finite covering of $M_h$ which factors through the moduli stack.
Consider any finite morphism $\pi:Z \to M_h$ with this property. We choose a projective compactifications $\bar Z$ of $Z$ such that $\pi$ extends to $\pi:\bar Z \to \bar{M}_h$. So $\pi^*{{\mathcal H}}$ is again ample with respect to $\pi^{-1}(M_h)$. Let $M_0$ be a non-singular subvariety of $\pi(Z)\cap M_h$ with $Z_0=\pi^{-1}(M_0)$ non singular.
Recall that for a family of projective varieties we constructed in [@VZ3], 2.7, a good open subset of the base space. Applying this construction to the restriction of the universal family to $Z_0$, we may assume furthermore, that $Z_0$ coincides with this subset.
By induction on the dimension of $Z$, we may assume that we have found an upper bound for $\varphi^*({{\mathcal H}})$ whenever $\varphi(Y)\subset \pi(Z)\setminus M_0$. Hence it is sufficient to find such a bound under the assumptions that $\varphi(Y)\subset \pi(Z)$ and $\varphi(Y)\cap M_0 \neq
\emptyset$. There exists a finite covering $Y'$ of $Y$ of degree $d \leq \deg(Z/\pi(Z))$, such that $$Y' \>\sigma >> Y \> \varphi >> \pi(Z)$$ factors through $\varphi':Y' \to Z$, and it is sufficient to bound the degree of $\sigma^*\varphi^*{{\mathcal H}}$. For simplicity, we assume that $\varphi:Y \to \pi(Z)$ factors through $\varphi':Y\to Z$.
By [@VZ3], 2.6 and 2.7, blowing up $Z$ with centers in $Z\setminus Z_0$ we may assume that $Z$ is non singular, that there exists a certain invertible sheaf $\lambda_\nu$ on $Z$, and a constant $N_\nu >0$, such that $$\deg({\varphi'}^*\lambda_\nu) \leq
N_\nu\cdot \deg(\det(f_*\omega^\nu_{X/Y})),$$ where, as usual, $f:X\to Y$ is an extension of $V\to U$ to a projective manifold $X$. By the explicit description of $\lambda_\nu$ in [@VZ3], 2.6, d), and by [@VZ3], 3.4, the sheaf $\lambda_\nu$ is ample with respect to $Z_0$ for some $\nu >1$. Hence it is sufficient to give an upper bound for $\deg({\varphi'}^*\lambda_\nu)$, or for $\deg(\det(f_*\omega^\nu_{X/Y}))$.
By [@VZ1] (see also [@BV] and [@Kov]) there exists a constant $e$, depending only on the Hilbert polynomial $h$, with $$\deg(\det(f_*\omega_{X/Y}^\nu)) \leq
(n\cdot(2g(Y)-2+s)+s)\cdot\nu\cdot {\rm
rank}(f_*\omega_{X/Y}^\nu)\cdot e,$$ and we found the bound we were looking for.
It remains to show, that the points $$[\varphi:Y\to \bar M_h] \in
{\rm\bf Hom}((Y,U),(M_h,\bar M_h))$$ which are induced by a family $f:X\to Y$ with $\mu(f)=\mu$, are discrete. If not, one finds a positive dimensional manifold $T$ and a generically finite morphism to ${\rm\bf Hom}((Y,U),(M_h,\bar M_h))$ whose image contains a dense set of points where the corresponding morphism is induced by a family. Let us choose a smooth projective compactification $\bar T$ with $S_{\bar T} = \bar T \setminus T$ a normal crossing divisor.
The induced morphism $Y\times T \to \bar M_h$ is not necessarily factoring through the moduli stack, but using again the Kollár Seshadri construction again, we find a generically finite morphism $\pi:Z \to Y\times \bar T$ which over $\pi^{-1}(U\times T)$ is induced by a smooth family. Assume that $Z$ is non-singular and that $S_Z=Z\setminus U\times T$ is a normal crossing divisor. Write $p:Z \to \bar T$ for the induced morphism.
Applying \[stthm2\], ii), one obtains a big coherent subsheaf $${{\mathcal P}}\subset S^\mu(\Omega^1_Z(\log S_Z)).$$ By part iii), is image ${{\mathcal P}}'$ in $S^\mu(\Omega_{Z/T}(\log S_Z))$ is non zero, and iv) implies that for a dense set of points $t\in T$ the restriction ${{\mathcal P}}'|_{p^{-1}(t)}$ lies in $$\pi^*S^\mu(\Omega^1_{Y\times \{t\}}(\log (S\times \{t\}))).$$
This is only possible, if ${{\mathcal P}}'$ is a big subsheaf of $$\pi^*S^\mu(\Omega^1_{Y\times \bar T}(\log (S\times \bar T))).$$ Restricting to $\pi^{-1}(\{y\}\times \bar T$, for general $y\in Y$ one obtains as in the proof of \[proprigidsa\] a big subsheaf of a trivial sheaf, a contradiction.
Needless to say, Corollary \[corrigid2\] is sort of empty, as long as we do not know any answer to Problem \[problem2\].
[XXX]{} Bedulev, E., Viehweg, E.: On the Shafarevich conjecture for surfaces of general type over function fields. Invent. Math. [**139**]{} (2000) 603–615 Caporaso, L.: On certain uniformity properties of curves over function fields. Compos. Math. [**130**]{} (2002) 1–19 Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems. DMV-Seminar [**20**]{} (1992), Birkhäuser, Basel-Boston-Berlin Kovács, S.: Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties. preprint (AG/0003019), to appear in Comp. Math. Liu, K., Todorov A., Yau, S.-T., Zuo, K.: The Analogue of Shafarevich’s Conjecture for Some CY Manifolds. preprint 2002 Maehara, K.: A finiteness property of varieties of general type. Math. Ann. [**262**]{} (1983) 101–123. Mori, S.: Classification of higher dimensional varieties. In: Algebraic Geometry. Bowdoin 1985, Proc. Symp. Pure Math. [**46**]{} (1987) 269 - 331 Viehweg, E.: Quasi-projective Moduli for Polarized Manifolds. Ergebnisse der Mathematik, 3. Folge [**30**]{} (1995), Springer Verlag, Berlin-Heidelberg-New York Viehweg, E., Zuo K.: On the isotriviality of families of projective manifolds over curves. J. Alg. Geom. [**10**]{} (2001) 781–799 Viehweg, E., Zuo K.: Base spaces of non-isotrivial families of smooth minimal models. In: Complex Geometry (Collection of Papers dedicated to Hans Grauert), Springer Verlag (2002) 279–328 Zuo, K.: On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applications. Asian J. of Math. [**4**]{} (2000) 279–302
[^1]: This work has been supported by the “DFG-Schwerpunktprogramm Globale Methoden in der Komplexen Geometrie”. The second named author is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4239/01P)
|
---
author:
- 'M. Carmen Eliche-Moral, Alejandro Borlaff, John E. Beckman,'
- Leonel Gutiérrez
bibliography:
- 'antit.bib'
date: 'Received July 28, 2014; accepted May 18, 2015'
title: |
Photometric scaling relations of antitruncated stellar discs\
in S0-Scd galaxies
---
Introduction {#sec:introduction}
============
@2005ApJ...626L..81E introduced for the first time a definition of antitruncated or Type-III galaxies, as those in which the surface brightness of the disc does not follow the typical exponentially-decaying profile with the radius [@1940BHarO.914....9P; @1957AJ.....62...69D; @1958ApJ...128..465D; @1970ApJ...160..811F], but presents an up-bending profile, with the outer disc exhibiting a distinct shallower slope than the inner disc outside a given radius (known as the break radius, [$R_\mathrm{brk III}$]{}). This nomenclature was an extension of the classification defined by @1970ApJ...160..811F, who classified Type-I discs as those with single exponentially-decaying profiles and Type-II discs as those with down-bending profiles outside the break radius .
In edge-on systems, antitruncations tend to coincide with the superposition of a thin disc and a thick disc [@2012ApJ...759...98C]. However, while nearly all Type-II profiles are associated with galaxy subcomponents (such as rings, pseudorings, lenses, or strong star formation regions), only $\sim 1/3$ of Type-III profiles are related to distinct morphological substructures [see @2014MNRAS.441.1992L L14 henceforth]. The structural properties and frequencies of antitruncations seem to differ in S0 and spiral types. The percentage of antitruncations rises from in Sc–Sd galaxies to in S0–Sa types . In spirals, antitruncations are basically disc-related phenomena, with less than of them associated with the contribution of central spheroidal components to the galaxy outskirts [@2012MNRAS.420.2475M; @2015MNRAS.447.1506M]. However, this percentage rises to in S0–Sb galaxies [@2005ApJ...626L..81E] and up to if only S0s are considered [@2015MNRAS.447.1506M].
Type-II profiles are known to be related to bars in most cases [see, e.g., @2014ApJ...782...64K], but the origin of Type-III discs is still poorly constrained. Diverse mechanisms have been proposed to explain the formation of antitruncations. The majority of them are related to gravitational or tidal interactions, such as minor mergers [@2001MNRAS.324..685L; @2006ApJ...650L..33P; @2007ApJ...670..269Y], major mergers , interactions of the disc with dark matter subhaloes [@2009ApJ...700.1896K], high-eccentricity fly-by encounters [@2008ApJ...676L..21Y], or harassment [@2012ApJ...758...41R]. Other formation scenarios include the existence of different star formation thresholds as a function of the radius in the galaxy [@2006ApJ...636..712E], ram-pressure stripping [@2012ApJ...758...41R], ongoing gas accretion , and simple fading of stellar discs [@2015MNRAS.447.1506M]. @2015MNRAS.448L..99H have also proposed that the disc profile type of a galaxy may basically depend on the initial spin of its host halo. It seems that bars are unrelated to antitruncations, as derived from the observational fact that the relative frequency of Type-III profiles found in samples of barred and unbarred galaxies is similar [@2009IAUS..254..173S; @2008AJ....135...20E E08 hereafter; G11; L14]. However, this needs to be confirmed by other means.
Recently, have found that the structures of the inner and outer discs and the location of the break in Type-III S0 galaxies are strongly coupled, and this coupling seems to be independent of the existence of bars in the galaxies. These authors have shown that the characteristic photometric parameters of the inner and outer discs and the breaks in S0s satisfy several scaling relations, tighter in many cases if the scalelengths are normalized to the optical size of the galaxy. The question is whether or not these scaling relations (or similar ones) are also obeyed by Type-III discs of other morphological types. If not this would imply that antitruncations do form through diverse and independent mechanisms in different Hubble types (which seems reasonable, accounting for the wide variety of possible formation processes). However, if the antitruncated discs of spiral galaxies satisfy scaling relations similar to those observed in S0s, it becomes challenging to understand the physical processes underlying this coupling in galaxies spanning the whole Hubble Sequence. Analogously, if bars are relevant in determining the structure of some antitruncated discs or have triggered their formation in some cases, we should expect to find significant differences between the photometric trends followed by Type-III discs of barred and unbarred galaxies, whereas negligible differences would be expected if both phenomena are structurally unrelated.
Therefore, we have investigated whether the Type-III discs of spirals obey scaling relations as tight as those observed in antitruncated S0s and, in this case, whether the scaling relations can be considered similar or both galaxy types exhibit significant differences between them. The same analysis has been performed for barred and unbarred galaxies, to find out whether bars and antitruncations are structurally related or not.
For this purpose, we have used the data published by E08 and G11 in the $R$ band, and by L14 in the [3.6$\mu$m]{} Spitzer band. In Section\[sec:data\], we briefly comment on the galaxy samples of these authors, their data, and the procedures they followed to obtain and characterize the surface brightness profiles. Section\[sec:fits\] describes our fitting technique to the trends found in the studied photometric planes, as well as the tests performed to identify the correlations that were statistically significant. In Section\[sec:results\] we show the main trends and scaling relations that we have found involving the characteristic scalelengths of the inner and outer discs ([$h_{\mathrm{i}}$]{} and [$h_{\mathrm{o}}$]{}, respectively), [$R_\mathrm{brk III}$]{}, and [$R_\mathrm{25}$]{}. There we also statistically analyse the differences and similarities of the trends followed by S0 vs.spiral galaxies, by barred vs.unbarred galaxies, and of $R$ vs.[3.6$\mu$m]{} data. Finally, the discussion and main conclusions are provided in Sections\[sec:discussion\] and \[sec:conclusions\].
[l lll]{}\
\
\
& Barred$^\mathrm{b}$ & Unbarred$^\mathrm{c}$ & Total\
S0–S0$/$a & 9 (22.5%) & 12 (30%) & 21 (52.5%)\
Sa–Sbc & 7 (17.5%) & 12 (30%) & 19 (47.5%)\
All Hubble types & 16 (40%) & 24 (60%) & 40 (100%)\
\
\
\
& Barred & Unbarred & Total\
S0–S0$/$a & 13 (21%) & 18 (29%) & 31 (50%)\
Sa–Scd & 16 (26%) & 15 (24%) & 31 (50%)\
All Hubble types & 29 (47%) & 33 (53%) & 62 (100%)\
[ *Notes*:\
$^\mathrm{a}$ The percentages are given with respect to the total number of galaxies in the sample of each band.\
$^\mathrm{b}$ All barred galaxies in the $R$-band sample are from E08.\
$^\mathrm{c}$ All unbarred galaxies in the $R$-band sample are from G11.\
$^\mathrm{d}$ The data in the [3.6$\mu$m]{} band come from L14.\
]{}
Data {#sec:data}
====
We have analysed the possible correlations between the characteristic parameters of the breaks and the inner and outer discs of two samples of local galaxies with Type-III stellar discs, in the $R$ and [3.6$\mu$m]{} bands. The $R$-band dataset contains the photometric parameters derived for 16 Type-III barred nearby galaxies by E08 and for 24 Type-III unbarred ones by G11 (40, in total), with S0-Sbc types. The [3.6$\mu$m]{} dataset comprises the 62 Type-III (barred and unbarred) galaxies from the sample analysed by L14, with types spanning from S0 to Scd. Our study is exclusively centered on galaxies with pure Type-III profiles, i.e., the galaxies with hybrid profiles from the original samples (Type II$+$III) have been excluded in our subsamples to avoid a possible additional dispersion in the trends we were looking for (they were 4 galaxies in the E08 sample, 5 from G11, and 7 from L14). The E08 sample overlaps with the L14 sample in 4 galaxies, while G11 sample has 7 galaxies in common with L14.
Table\[tab:samples\] summarizes the statistics of the samples in terms of morphological types and barred/unbarred nature in both bands. The statistics of the two subsamples is not very high (40 S0–Sbc galaxies in $R$ and 62 S0–Scd’s in [3.6$\mu$m]{}), but the numbers are sufficiently large to allow us to look for photometric scaling relations in S0s and spirals separately, because the galaxies distribute nearly equally among the two types in both bands. A similar argument holds for barred and unbarred galaxies. The original data, reduction, and methodology to characterize the surface brightness profiles are extensively described in the original papers, so we provide just a brief description here.
The original samples were defined using different selection criteria for the radial velocities, angular sizes, galactic latitudes, and morphologies of the galaxies. The galaxies in the $R$-band sample have distances $<$30Mpc, while those of the L14 sample lie at $<$80Mpc, but both datasets present $<$$M_B$$<$magnitudes. E08 and G11 used data in the $r$ and $R$ bands taken with different telescopes, with PSF FWHM$\sim$0.7” and limiting surface brightness $\mu_\mathrm{lim}$$\sim$26–27magarcsec$^{-2}$ in $R$ (Vega system). L14 combined data obtained in the [3.6$\mu$m]{} IRAC band for Sa-Sd galaxies of the S$^4$G survey [FWHM$\sim$1.7”, see @2010PASP..122.1397S] with $K_s$-band images for S0-S0/a galaxies from the NIRS0S survey [FWHM$\sim$0.7”, see @2011MNRAS.418.1452L]. In L14, the distances of the galaxies coming from the S$^4$G sample are $<$40Mpc, and $<$80Mpc for those coming from NIRS0S.
L14 converted the $K$-band surface brightness profiles of the galaxies from the NIRS0S sample to AB magnitudes in the [3.6$\mu$m]{} band accounting for the color differences and magnitude offsets derived for the 93 galaxies that the two surveys have in common. These authors computed total magnitudes in elliptical apertures tracing the $\mu=22.5$magarcsec$^{-2}$ isophote in [3.6$\mu$m]{} in each galaxy. The median difference between the magnitudes obtained in the two surveys was derived, including a linear term to describe the dependence on colour. This conversion factor was then applied to the surface brightness profiles and total magnitudes of the NIRS0S data to transform them into [3.6$\mu$m]{}. L14 data finally presented $\mu_\mathrm{lim}\sim$26.4magarcsec$^{-2}$ for the Sa-Scd’s and $\mu_\mathrm{lim}\sim$24.7magarcsec$^{-2}$ for the S0-S0/a’s in [3.6$\mu$m]{} (AB magnitudes).
Both data samples are analogous in terms of depth for the spiral types, but the $R$-band sample is at least $\sim 1$mag deeper than the [3.6$\mu$m]{} sample for the S0 galaxies. L14 compared the limiting surface brightness of their sample with that of the $V$-band sample by @2012MNRAS.419..669M, finding that $V - [3.6] \sim 1.5$mag (AB system, see their Section6). Considering that $V-R$ ranges $\sim 0.2$–0.5 in the discs of Sa–Sd galaxies and $(V-R) = 0.5$– 0.65 in those of S0s [@1989ApJS...69..217G], we find that the limiting magnitudes of the [3.6$\mu$m]{} sample by L14 roughly correspond in the $R$ band to $\mu_\mathrm{lim}\sim 27.5$ for the spirals and $\mu_\mathrm{lim}\sim 25.5$ for the S0s (Vega system). Here, we have considered that $V(\mathrm{AB}) - V(\mathrm{Vega}) = 0.02$ [@2007AJ....133..734B]. Assuming that $\mu_\mathrm{lim} \sim 26.5$magarcsec$^{-2}$ on average in the E08 and G11 samples, this means that the [3.6$\mu$m]{} data sample is $\sim 1$magarcsec$^{-2}$ deeper than the $R$-band sample for the spirals. On the other hand, the E08 and G11 samples are $\sim 1$magarcsec$^{-2}$ deeper than the L14 sample for the S0s. However, some profiles in E08 and G11 achieve $\mu_\mathrm{lim}\sim 28$magarcsec$^{-2}$. So, the $R$-band sample may be reaching similar depths to the L14 sample in some specific cases.
E08 and G11 used the morphological types available in the RC3 catalog [@1991rc3..book.....D], based on the optical morphology of the galaxies, whereas the types in L14 were assigned according to the morphology in their $K$ or [3.6$\mu$m]{} images [@2011MNRAS.418.1452L; @2015arXiv150100454B]. E08 considered as barred galaxies those exhibiting strong (SB) or weak (SAB) bars according to the RC3 classification, but revised the classes according to their deep $R$ band images and rejected the galaxies without clear bars in them or involved in strong interactions. The barred/unbarred classification in L14 was, however, made on the basis of their deep $K$ and [3.6$\mu$m]{} images from the NIRS0S and S$^4$G surveys.
The surface brightness profiles were obtained by azimuthally averaging the light within ellipses fitted to the isophotes of the galaxies. The three studies (E08, G11, and L14) held the values of the centre, ellipticity, and position angle of isophotes fixed to the values of the outer discs in the fits.
E08 and L14 fitted the disc profiles using ”broken-exponential” functions, which describe the inner and outer discs through two exponentially-decaying profiles joined by a transition region, according to the following expression:
$$\label{eq:broken}
I(r) = S\, I_0\, \exp\left[\frac{-r}{{\ensuremath{h_{\mathrm{i}}}}}\right]\,\lbrace 1 + \exp\left[ \alpha\,(r - {\ensuremath{R_\mathrm{brk III}}}) \right] \rbrace^{\frac{1}{\alpha}\,(\frac{1}{{\ensuremath{h_{\mathrm{i}}}}} - \frac{1}{{\ensuremath{h_{\mathrm{o}}}}})},$$
where $I_0$ represents the central intensity of the inner exponential section, $\alpha$ parameterizes the sharpness of the break, and $S$ is a scaling factor, given by
$$\label{eq:Sbroken}
S = \left[ 1 + \exp(-\alpha\,{\ensuremath{R_\mathrm{brk III}}})\right]^{\frac{1}{\alpha}\,(\frac{1}{{\ensuremath{h_{\mathrm{i}}}}} - \frac{1}{{\ensuremath{h_{\mathrm{o}}}}})}.$$
On the other hand, G11 performed independent exponential fits to the inner and outer discs (”piecewise fits”), defining [$R_\mathrm{brk III}$]{} as the radius at which the fitted profiles cross. The surface brightness of the profile at $r = {\ensuremath{R_\mathrm{brk III}}}$ is defined as the break surface brightness ([$\mu_\mathrm{brk III}$]{}). E08 showed that the two fitting procedures provide very similar results (within 5% for the characteristic scalelengths in case of Type-III profiles). This allows the comparison of the characteristic parameters of the samples by E08 and G11 in the $R$ band.
Consequently, we have used the characteristic parameters of the breaks and the inner and outer discs of Type-III galaxies derived by E08, G11, and L14 to compare the trends of S0 and spiral types in several photometric planes, in the $R$ and [3.6$\mu$m]{} bands. We remark that the magnitudes of the $R$-band data are the Vega system and in AB for the [3.6$\mu$m]{} dataset.
-- --
-- --
-- --
-- --
Fits and correlation tests {#sec:fits}
==========================
We performed linear fits ($y = m\,x + C_0$) to the trends followed in each photometric plane by all galaxies, by spirals and S0s independently, by types of spirals (Sa–Sab, Sb–Sbc, Sc–Scd), as well as by barred and unbarred galaxies. The trends of the types in each sample have been fitted using ordinary least squares. The photometric parameters of the inner and outer discs characterized by E08, G11, and L14 had no errors assigned in their original papers, so no error weighting could be considered in the fits. In order to estimate realistic confidence intervals to the fitted regression coefficients, we adopted a bootstrapping method. We generated $n=10^5$ artificial data samples with the same size as the original one in each diagram with replacement. The final regression coefficients of each fit correspond to the median values of the probability distributions of each coefficient obtained with the $10^5$ results, in order to reduce the systematic bias introduced by outliers. The upper and lower errors considered for each coefficient are those enclosing 2.5% and 97.5% of the values in the corresponding probability distribution. The bootstrap distribution is closer to the real probability distribution of the coefficients than a simple Gaussian in general, so this method derives robust and conservative (asymmetric) confidence intervals for the regression coefficients, reducing the effects of possible outliers or high leverage points at the same time.
We tested the significance of each photometric trend using the Spearman rank correlation test, which measures whether two variables are monotonically related and the level of correlation between them, and it has the advantage of being non parametric. Only those trends with an associated probability of random correlation below 5% according to the test ($p_S<0.05$) are considered as statistically significant. The [3.6$\mu$m]{} dataset presents higher statistics than the $R$-band sample, but the Spearman rank correlation test accounts for the number of data pairs yielding the trend to derive $p_S$. Additionally, the Pearson coefficient ($\rho$) has been used to determine the level of linear correlation of each trend.
The slope ($m$) and $Y$-intercept ($C_0$) of the linear fits performed to the different trends analysed in each photometric plane, their asymmetric error intervals, as well as the values of $p_S$, $\rho$, and the number of available data pairs ($N_ \mathrm{pairs}$) for each trend, are listed in Tables\[tab:hihorbreak\]-\[tab:r25\].
Figures\[fig:withRbreak\_R\]-\[fig:r25\] show the trends followed by Type-III galaxies in these photometric planes. We have overplotted the obtained linear fits *only* when they fulfill the Spearman rank correlation test at 95% of significance level, i.e., only if there is a significant correlation in the diagram. Note that, although there may be a significant correlation between two parameters according to the Spearman test, it does not have to be significantly linear. In fact, some trends are significant according to it ($p_S < 0.05$), but they exhibit low values of the Pearson coefficient $\rho$ (e.g., the [$\mu_\mathrm{brk III}$]{} – [$R_\mathrm{brk III}$]{} trend in [3.6$\mu$m]{} in Fig.\[fig:withRbreak\_3.6\]). In these figures, we have written the results of the most relevant fits which are being compared at the top of each panel even when the correlations are not significant. For simplicity, we have symmetrized the error interval of $m$ and $C_0$ in the figures, but the asymmetrical upper and lower errors really obtained for the coefficients of the fits are available in Tables\[tab:hihorbreak\]-\[tab:r25\].
The characteristic scalelengths are plotted in logarithmic scales in all figures, because the correlations exhibit more defined linear trends in this way than using linear scales. In many photometric planes, we have normalized the characteristic scalelengths ([$h_{\mathrm{i}}$]{}, [$h_{\mathrm{o}}$]{}, [$R_\mathrm{brk III}$]{}) to the optical radius of each galaxy. We have defined this following E08 and G11, i.e., as the radius of the isophote with $\mu = 25$mag arcsec$^{-2}$ in the $B$ band ([$R_\mathrm{25}$]{}). These authors provide [$R_\mathrm{25}$]{} for each galaxy in their samples, so we have used their tabulated values directly. The values of [$R_\mathrm{25}$]{} for the galaxies in the L14 sample have been obtained from HyperLeda[^1], and include a correction for Galactic extinction and inclination effects.
Results {#sec:results}
=======
In Section\[sec:trends\], we discuss the trends and correlations found in several photometric planes for the different morphological types and for barred and unbarred galaxies in the two datasets ($R$ and [3.6$\mu$m]{}). In Section\[sec:comparison\], we compare the slopes and $Y$-intercepts of the linear trends fitted in each photometric plane for S0s and spirals, as well as for barred and unbarred. The fits obtained for the $R$-band and [3.6$\mu$m]{} data are only compared in the photometric relations exclusively relating characteristic scalelengths.
-- --
-- --
-- --
-- --
Trends and scaling relations {#sec:trends}
----------------------------
### Trends with [$R_\mathrm{brk III}$]{} {#sec:rbreak}
Figures\[fig:withRbreak\_R\] and \[fig:withRbreak\_3.6\] show the trends of several photometric parameters of the inner and outer discs of Type-III galaxies with [$R_\mathrm{brk III}$]{} and ${\ensuremath{R_\mathrm{brk III}}}/{\ensuremath{R_\mathrm{25}}}$ in the $R$ and [3.6$\mu$m]{} bands, respectively. The two top rows of the figures display the trends of the inner and outer disc scalelengths with [$R_\mathrm{brk III}$]{} by Hubble types in each band, in logarithmic scale. The distributions of spirals and S0s are similar in these planes and overlap.
The main result is that [$\log(h_\mathrm{i})$]{}, [$\log(h_{\mathrm{o}})$]{}, [$\mu_{0,\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, and [$\mu_\mathrm{brk III}$]{} correlate strongly with [$\log(R_\mathrm{brk III})$]{} in both spirals and S0s (all these trends have $p_S<0.05$), and furthermore, similar correlations are obeyed for the different spiral types surveyed by each sample (Sa–Sab and Sb–Sbc in both bands, and Sc–Scd in [3.6$\mu$m]{}) within the observational uncertainties. The dispersions around the fitted linear trends in S0s and spirals are similar in both bands, although the linear trends of [$\log(h_\mathrm{i})$]{} and [$\log(h_{\mathrm{o}})$]{} with [$\log(R_\mathrm{brk III})$]{} are better defined in [3.6$\mu$m]{} than in $R$ (i.e., they have higher values of the linear correlation coefficient $\rho$), whereas it is the opposite in the trends involving [$\mu_{0,\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, and [$\mu_\mathrm{brk III}$]{} (this is noticeable just by visual inspection of the trends).
The only diagrams in which spirals (globally or by types) show no significant correlation according to the Spearman rank correlation test are ${\ensuremath{h_{\mathrm{o}}}}$ – ${\ensuremath{R_\mathrm{brk III}}}$ in $R$ and ${\ensuremath{\mu_{0,\mathrm{i}}}}$ or ${\ensuremath{\mu_{0,\mathrm{o}}}}$ versus ${\ensuremath{R_\mathrm{brk III}}}$ (or ${\ensuremath{R_\mathrm{brk III}/{\ensuremath{R_\mathrm{25}}}}}$) in [3.6$\mu$m]{}. The first may be only a question of the lower statistics of the $R$-band sample as compared to the [3.6$\mu$m]{} subsample, because the analogous plane in [3.6$\mu$m]{} shows significant linear correlations for all spirals and by their (sub-)types, and the data distributions in both planes are quite similar. Correspondingly, the distributions of S0s and spirals in the diagrams of ${\ensuremath{\mu_{0,\mathrm{i}}}}$ (or ${\ensuremath{\mu_{0,\mathrm{o}}}}$) – [$R_\mathrm{brk III}$]{} and ${\ensuremath{\mu_{0,\mathrm{i}}}}$ (or ${\ensuremath{\mu_{0,\mathrm{o}}}}$) – ${\ensuremath{R_\mathrm{brk III}}}/{\ensuremath{R_\mathrm{25}}}$ in [3.6$\mu$m]{} are also similar to the analogous distributions in the same diagrams of the $R$ band (compare the panels corresponding to [$\mu_{0,\mathrm{i}}$]{} in both figures), so the lack of significance in the correlations in [3.6$\mu$m]{} might also be a question of small numbers.
The trends of these photometric parameters with ${\ensuremath{\log(R_\mathrm{brk III}/{\ensuremath{R_\mathrm{25}}})}}$ present similar or even higher values of linear correlation (as measured by $\rho$) than with [$\log(R_\mathrm{brk III})$]{}. In general, the correlations of [$\mu_{0,\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, and [$\mu_\mathrm{brk III}$]{} improve when [$R_\mathrm{brk III}$]{} is normalized to the optical size of the galaxy (in particular, compare the trends and the Pearson coefficients of [$\mu_\mathrm{brk III}$]{} – [$R_\mathrm{brk III}$]{} and [$\mu_\mathrm{brk III}$]{} – [$R_\mathrm{brk III}/{\ensuremath{R_\mathrm{25}}}$]{} at the bottom panels of Figs.\[fig:withRbreak\_R\] and \[fig:withRbreak\_3.6\]). The values of [$\mu_{0,\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, and [$\mu_\mathrm{brk III}$]{} in Type-III discs are fainter as the breaks are more external (see the corresponding panels in the figures). However, these values seem to be more closely linked to the relative location of [$R_\mathrm{brk III}$]{} with respect to the outer radius of the galaxy (as measured by [$R_\mathrm{25}$]{}) than to [$R_\mathrm{brk III}$]{}.
In Figs.\[fig:withRbreak\_R\_barred\] and \[fig:withRbreak\_3.6\_barred\] we plot the same photometric planes as in Figs.\[fig:withRbreak\_R\] and \[fig:withRbreak\_3.6\], but distinguishing between barred and unbarred galaxies. Again, the linear fits performed to the barred and unbarred galaxies have been overplotted only if they are significant. The majority of the photometric planes show significant scaling relations for both barred and unbarred galaxies in the two bands. The trends fitted to the barred galaxies look similar to those obtained for unbarred galaxies within the observational dispersion, as derived from the fact that the distributions for the two galaxy classes practically overlap in the diagrams. This suggests that bars seem to affect these scaling relations very little (at least, within the uncertainties implied by the data samples).
The only relations which are not significant in Figs.\[fig:withRbreak\_R\_barred\] and \[fig:withRbreak\_3.6\_barred\] are the trends involving [$h_{\mathrm{o}}$]{} and [$\mu_{0,\mathrm{o}}$]{} for barred galaxies in $R$ and the [$\mu_{0,\mathrm{i}}$]{} – [$\log(R_\mathrm{brk III}/{\ensuremath{R_\mathrm{25}}})$]{} trend in [3.6$\mu$m]{} for the unbarred galaxies. But again, the lack of correlation in each band may reflect the low statistics of the samples.
The linear trends for barred and unbarred galaxies are very well defined in the [3.6$\mu$m]{} dataset in the planes involving [$h_{\mathrm{i}}$]{} and [$h_{\mathrm{o}}$]{} while those relating [$\mu_{0,\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, and [$\mu_\mathrm{brk III}$]{} with [$\log(R_\mathrm{brk III})$]{} have higher $\rho$ values in the $R$ band (as also happened in Figs.\[fig:withRbreak\_R\] and \[fig:withRbreak\_3.6\] for S0s and spirals). In any case, the trends in the photometric planes described by the $R$-band dataset look similar to their [3.6$\mu$m]{} analogs taking into account the data dispersion. Again, we find that the linear correlation coefficients of the trends relating [$\mu_{0,\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, and [$\mu_\mathrm{brk III}$]{} with [$\log(R_\mathrm{brk III})$]{} tend to improve if [$R_\mathrm{brk III}$]{} is normalized to [$R_\mathrm{25}$]{}, for both barred and unbarred galaxies (see the three rows of panels at the bottom of Figs.\[fig:withRbreak\_R\_barred\] and \[fig:withRbreak\_3.6\_barred\]).
Summarizing, we have found that the inner and outer discs of antitruncated spirals obey tight photometric scaling relations with [$R_\mathrm{brk III}$]{}, as discovered for Type-III S0 galaxies. The trends for each type look similar among different morphological types and among barred and unbarred galaxies within the dispersion of the data in the planes. This result suggests that antitruncations and bars are structurally independent phenomena in galaxies.
### Trends with [$h_{\mathrm{i}}$]{} and [$h_{\mathrm{o}}$]{} {#sec:hoorhi}
In Figs.\[fig:withhiorho\_R\] and \[fig:withhiorho\_3.6\] we analyse the basic scaling relations obeyed by the inner and outer discs of Type-III galaxies in $R$ and [3.6$\mu$m]{} respectively. We show the photometric planes also normalizing [$h_{\mathrm{i}}$]{} and [$h_{\mathrm{o}}$]{} by [$R_\mathrm{25}$]{}. The different morphological types (S0s and spirals, as well as by Hubble types) yield significant linear relations in these photometric planes too, again similar among them within the observed data dispersion. The distribution in the planes of spirals and S0s overlap also in these diagrams.
L14 already reported that the two exponential sections of galaxy discs of Types II and III in their sample independently satisfied the basic scaling relation observed in pure exponential discs between their central surface brightness and their scalelengths, although they did not distinguish between different Hubble types in their Fig.11. The two top panels in the first column of Fig.\[fig:withhiorho\_3.6\] show that this result also applies for different morphological types (S0, Sa–Sab, Sb–Sbc, and Sc–Scd) and spirals in general, and that it can be extended to the $R$ band (see the corresponding panels in Fig.\[fig:withhiorho\_R\]).
Again, the linear correlations involving [$\mu_{0,\mathrm{i}}$]{} and [$\mu_{0,\mathrm{o}}$]{} improve noticeably when the disc scalelengths are normalized to [$R_\mathrm{25}$]{} for the two main galaxy types being considered (compare the left panels with the right panels in the two figures). This is more striking in the [3.6$\mu$m]{} trends, where this improvement can be noticed by visual inspection: the dispersion around the fitted linear trends in the [$\mu_{0,\mathrm{i}}$]{} – [$h_{\mathrm{i}}$]{} and [$\mu_{0,\mathrm{o}}$]{} – [$h_{\mathrm{o}}$]{} plots is significantly reduced in Fig.\[fig:withhiorho\_3.6\] when [$h_{\mathrm{i}}$]{} and [$h_{\mathrm{o}}$]{} are normalized to [$R_\mathrm{25}$]{}. Moreover, $\rho$ increases significantly for both S0s and spirals in the two planes after this normalization.
The bottom panels of Fig.\[fig:withhiorho\_3.6\] show that [$\log(h_\mathrm{i})$]{} and [$\log(h_{\mathrm{o}})$]{} correlate linearly in both S0s and spiral galaxies in the [3.6$\mu$m]{} band (in fact, this applies independently for Sa–Sab, Sb–Sbc, and Sc–Scd types). In contrast, no significant trends are found in $R$, except for the S0s (see the same panel in Fig.\[fig:withhiorho\_R\]). Note that the [$\log(h_{\mathrm{o}})$]{} – [$\log(h_\mathrm{i})$]{} trends in [3.6$\mu$m]{} do not improve if the scalelengths are normalized to [$R_\mathrm{25}$]{} (compare the bottom panels of Fig.\[fig:withhiorho\_3.6\]).
We have plotted the same photometric relations in Figs.\[fig:withhiorho\_R\_barred\] and \[fig:withhiorho\_3.6\_barred\], but now differentiating barred from unbarred galaxies. The linear fits obtained for each galaxy class (barred vs.unbarred) have been overplotted only if the correlations were significant according to the Spearman rank correlation test, as above. The figures show that barred and unbarred galaxies overlap in these diagrams and follow tight scaling relations in them, similar within the observational dispersion. Therefore, these scaling relations seem to be independent of the existence of a bar in the galaxy within the observational uncertainties, again suggesting that bars and antitruncations are structurally unrelated phenomena.
In conclusion, we have found that the inner and outer discs of Type-III spirals obey tight scaling relations too, as observed in Type-III S0 galaxies. Again, we find that the existence of bars in the galaxies affect negligibly to these scaling relations within the observational uncertainties and that the relations in the [$\mu_{0,\mathrm{i}}$]{} – [$h_{\mathrm{i}}$]{} and [$\mu_{0,\mathrm{o}}$]{} – [$h_{\mathrm{o}}$]{} planes significantly improve when the scalelengths are normalized by [$R_\mathrm{25}$]{}.
### Trends with [$R_\mathrm{25}$]{} {#sec:R25}
As shown above, the linear correlations between the characteristic surface brightness values and the scalelengths become better defined in many photometric planes after normalizing the relevant parameters to [$R_\mathrm{25}$]{}. We have analysed the trends between these characteristic scalelengths ([$h_{\mathrm{i}}$]{}, [$h_{\mathrm{o}}$]{}, and [$R_\mathrm{brk III}$]{}) and [$R_\mathrm{25}$]{} in Fig.\[fig:r25\] for several galaxy types. First, the values of [$R_\mathrm{brk III}$]{}, [$h_{\mathrm{i}}$]{}, and [$h_{\mathrm{o}}$]{} span similar ranges for a given [$R_\mathrm{25}$]{} in both $R$ and [3.6$\mu$m]{}, implying that both bands must be sampling the same type of breaks, but in different wavelength ranges.
In $R$, only the S0s exhibit significant trends of [$\log(h_\mathrm{i})$]{}, [$\log(h_{\mathrm{o}})$]{}, or [$\log(R_\mathrm{brk III})$]{} with [$\log(R_\mathrm{25})$]{}, despite the fact that S0s and spirals basically overlap in all planes (left panels in the figure). However, the [3.6$\mu$m]{} dataset shows well-defined linear correlations for S0s and spirals, as well as for spiral sub-types in these diagrams (right panels). The trends look similar among them, as observed in the photometric parameters analysed previously. Again, the distributions of the $R$-band and [3.6$\mu$m]{} data are similar in the same photometric planes, so the lack of correlations for spirals in $R$ might be due to the small numbers in the sample, as commented above.
The scaling relations in the right panels of Fig.\[fig:r25\] relate the size of the galaxy computed from an optical blue band ($B$) with the structure of the inner and outer discs observed in a NIR band ([3.6$\mu$m]{}), indicating that there is a clear size scaling in these galaxies, such that larger (Type-III) discs have larger inner and outer disc scalelengths, and hence larger break radii. These scaling relations may present higher dispersion because [$R_\mathrm{25}$]{} is measured in the $B$ band by definition, a band which is not a proxy of the galaxy stellar mass certainly, whereas [$h_{\mathrm{i}}$]{}, [$h_{\mathrm{o}}$]{}, and [$R_\mathrm{brk III}$]{} have been derived using data in much redder bands. However, the effects of dust in the $B$ band must be very limited at radial locations near [$R_\mathrm{25}$]{} in the discs, so we can assume that in practice the $B$ band does trace the stellar mass similarly to the $R$ or [3.6$\mu$m]{} bands at these external radii. Moreover, the systematic improvement that we have found in many scaling relations after normalizing the scalelengths by [$R_\mathrm{25}$]{} implies that it must provide a robust estimate of the size of the stellar distribution, despite being computed in a blue band.
Comparison of the trends {#sec:comparison}
------------------------
In Sections\[sec:trends\] we have seen that the scaling relations followed by S0s and spirals and by barred and unbarred galaxies look similar, both in the $R$ and [3.6$\mu$m]{} bands. Here we analyse if there is statistical significant evidence that these relations differ within the observational uncertainties.
In order to do so, we have considered the relative differences between the fitted values of the slopes and $Y$-intercepts for each trend in the two pair of datasets compared in each case (S0 – spirals, barred – unbarred). We define the relative difference of the slopes $m$ obtained for one photometric relation $i$ between S0s and spirals in a given band as follows:
$$\label{eq:delta}
\Delta(\mathrm{m, S0-Sp, band}) = \frac{m(\mathrm{Sp,band}) - m(\mathrm{S0,band})}{m(\mathrm{S0,band})},$$
The errors in $\Delta(\mathrm{m, S0-Sp, band})$ correspond to the error propagation of the expression above, assuming as the error of each parameter the maximum between the absolute values of its upper and lower errors. Analogously, we have also defined the relative differences of the $Y$-intercepts ($C_0$) for the trends fulfilled by two datasets being compared, $\Delta (\mathrm{C_0, S0-Sp, band})$, and their associated errors. These $\Delta$ values for $m$ and $C_0$ have only been defined when the two data samples being compared exhibit statistically significant correlations in the photometric relation separately, according to the Spearman rank correlation test.
Even if $\Delta(m)$ and $\Delta(C_0)$ in a given trend were nearly zero, this does not ensure that the trends can be considered similar, because it depends on their errors. However, if $\Delta(m) \sim 0$ and $\Delta(C_0) \sim 0$ with errors below a given (low) percentage, the trends of the two samples can be considered similar within these uncertainties. Obviously, we must keep in mind that deeper data can reveal differences in these trends that cannot be discriminated with the available datasets.
In Fig.\[fig:compareSpS0\], we compare the relative differences of the slopes and $Y$-intercepts of the linear fits performed to the S0s and the spirals for each one of the 19 photometric relations analysed in Figs.\[fig:withRbreak\_R\]–\[fig:r25\], in $R$ and [3.6$\mu$m]{} (left and right panels, respectively). Let us assume that two fitted linear trends can be considered similar if the differences in $m$ and $C_0$ and their errors are below 25%. The figure shows that, under this criterion, no linear trend followed by S0s can be considered similar to the analogous one in spirals, either in the $R$ band or [3.6$\mu$m]{}. Although $|\Delta(m)|$ and $|\Delta(C_0)|$ are lower than 0.25 in many trends in each band (i.e., their values are contained within the horizontal blue lines in the planes of Fig.\[fig:compareSpS0\]), their errors (in one case and/or another) exceed this limit. The statistics of the samples are too small to conclude that the trends are similar within some reasonable uncertainty level.
We have repeated the plot in Fig.\[fig:comparebarredunbarred\], but comparing $\Delta(m)$ and $\Delta(C_0)$ for the linear trends followed by barred and unbarred galaxies. Again, only those trends which are significant in both datasets are compared. The figure shows that the errors in these relative differences are too high again to conclude that the linear trends fitted to Type-III barred galaxies are similar to those of unbarred galaxies within uncertainties of 25%, both in $R$ and [3.6$\mu$m]{} (left and right panels, respectively).
The comparison of the trends between distinct bands involving physical scalelengths is also reasonable, because if the breaks correspond to a change in the projected stellar density, they should be observed at a similar physical location in the disc in several bands. Therefore, we have compared the linear fits performed to the trends relating two characteristic scalelengths in the $R$ and [3.6$\mu$m]{} bands in Fig.\[fig:compareRNIR\]. We have again considered only the trends which are significant according to the Spearman test in both bands. The left panels of the figure compare $\Delta(m)$ and $\Delta(C_0)$ for the linear trends fitted to the S0s in the two bands, while the right ones show the same for those fitted to spirals. Although the relative differences of $m$ and $C_0$ can be below 25% for many trends, the uncertainties are too high to assess that these scaling relations observed in $R$ and [3.6$\mu$m]{} are similar.
In summary, Figs.\[fig:compareSpS0\]–\[fig:compareRNIR\] show that the observational dispersion is too high to robustly discern whether the analysed scaling relations are similar (or not) in both S0s and spirals, in barred and unbarred galaxies, and in the $R$ and [3.6$\mu$m]{} bands, although we do not find either statistical evidence of significant differences between the compared samples. Deeper data and larger samples are thus required to robustly confirm whether these scaling relations of antitruncated discs are really independent of the morphological type and the presence (or absence) of bars.
-- -- --
-- -- --
Discussion {#sec:discussion}
==========
In Section\[sec:results\], we have shown that Type-III discs obey tight photometric scaling relations for galaxy types spanning the whole Hubble Sequence. We have found no statistical evidence of noticeable differences between the relations followed by S0s and spirals and barred and unbarred galaxies, a fact that suggests that the structure of antitruncated discs may be independent of the galaxy type and the existence, or not, of bars in the galaxy. Although the statistical evidence is not strong, considering the low numbers and the uncertainties of the available data samples, the structural independence of bars and antitruncations agrees pretty well with the similar relative frequency of Type-III profiles found in samples of barred and unbarred galaxies [E08; @2009IAUS..254..173S G11; L14]. Our results thus support the idea that bars and antitruncations are decoupled structures in all morphological types. This does not imply that the two structures have formed independently, as some mechanisms are known to trigger the formation of both kind of features, such as mergers or flybys [@1996ApJ...460..121W; @2014arXiv1405.5832L]. However, this is indicative for that bars cannot have induced the formation of Type-III discs, as opposed to their tight structural link with Type-II discs [see, e.g., @2014ApJ...782...64K].
The scaling relations of Type-III discs found in the present study impose strong constraints on any formation scenarios proposed to explain the formation of antitruncations, independently of whether the relations really depend (or not) on the morphological type or the hosting of a bar. Accounting for the wide diversity of mechanisms proposed to explain the formation of antitruncations (see Section\[sec:introduction\]), it is challenging to understand the physics underlying the scaling relations that we have found between Type-III discs across the whole Hubble Sequence.
The dependence of these features on the environment becomes a key to discriminate between these mechanisms. Many studies have reported traces of recent or ongoing interactions in the outskirts of many Type-III discs, supporting a merging- or interaction-related nature [@2005ApJ...626L..81E]. L14 found a positive correlation between the scalelengths of Type-III discs and the tidal interaction strength, also pointing to external mechanisms. Coherently, flat and/or positive age gradients prevail in galaxies of the three profile types (in particular, of Type-III) in the Virgo cluster, contrary to the expectations of scenarios in which the formation of these discs were mostly driven by secular inside-out disc growth and/or stellar migrations [@2012ApJ...758...41R]. So, all these results suggest external processes as the main drivers of the formation of antitruncations, probably as a result of the gravitational response of the disc to a tidal interaction.
However, in this case we should also expect to find some dependence of the structural properties of Type-III discs on the local galaxy density. But, on the contrary, the inner and outer disc scalelengths and the break strength[^2] of Type-III discs show no trends with the environment, either in spiral or S0 galaxies [@2012MNRAS.419..669M; @2015MNRAS.447.1506M]. Additionally, similar fractions of Type-III S0 galaxies are found in both cluster and field environments, suggesting that the environment hardly affects the outer structure of these galaxies [@2012ApJ...744L..11E; @2012ApJ...758...41R]. How can all these results be reconciled?
External processes may induce the formation of antitruncations as the result of gravitational-driven instabilities in the disc or through gas/stars accretion in the galaxy outskirts. The tight scaling relations found here strongly support mechanisms related to the dynamical response of discs to tidal forces rather than other scenarios. In fact, a gravitational-driven mechanism would have three advantages. The first is that this can be induced through a wide diversity of processes (such as those commented in Section\[sec:introduction\]). Secondly, it could provide a feasible explanation for the independence of these scaling relations of the Hubble type of the galaxy (although it must be confirmed more robustly, as explained above), since just a stellar disc and gravity are required to give rise to them. And finally, a gravitationally-induced mechanism could also explain some of the apparently contradictory results with the environment discussed previously. If the processes triggering antitruncations are mostly related to gravitational interactions, we expect to find them present in both groups and clusters and in similar fractions, because mergers and interactions can be equally relevant in both environments [@2007ApJ...671.1503M; @2009ApJ...692..298W; @2010AJ....139.2643P; @2010AJ....140..612P; @2011MNRAS.415.1783B; @2013MNRAS.435.2713V]. Therefore, it is reasonable to find a weak dependence of their properties on the tidal interaction field (as reported by L14), but we do not expect to find significant trends of the break properties with the local density at the same time, because the antitruncation can have formed through an interaction not related with the current environment of the galaxy.
In any case, these speculative suggestions need to be confirmed through numerical simulations. At the moment, only have shown that major mergers are capable of producing antitruncated S0 galaxies that obey these scaling relations using N-body simulations. Nevertheless, it is obvious that the role of major mergers in the formation of late-type spirals must have been quite limited, so at least the Type-III discs in Sbc-Sd galaxies require different mechanisms, which also have to predict these scaling relations. In particular, satellite accretions are known to produce antitruncations [@2001MNRAS.324..685L; @2006ApJ...650L..33P; @2007ApJ...670..269Y], inducing secular evolution in the disc that can couple the inner and outer galaxy structure at the same time . This makes them good candidates to form antitruncated stellar discs. However, additional studies demonstrating the feasibility of this and other mechanisms in reproducing the scaling relations found here are required.
Conclusions {#sec:conclusions}
===========
We have investigated whether the tight scaling relations recently observed by in Type-III S0s are satisfied by antitruncated galaxies of other Hubble types. We have used the samples of Type-III galaxies published by E08 and G11 in the $R$ band and by L14 in Spitzer [3.6$\mu$m]{} band, as well as the characterizations performed by these authors to the surface brightness profiles of these galaxies. The $R$-band dataset consists of 40 antitruncated galaxies with types spanning from S0 to Sbc, while the [3.6$\mu$m]{} sample has 62 galaxies of S0–Scd types. Nearly half of the galaxies in each sample are barred.
We have analysed the trends followed by S0s and spirals (all, Sa–Sab, Sb–Sbc, and Sc–Scd), as well as for barred and unbarred galaxies, in several planes relating the characteristic photometric parameters of the breaks ([$\mu_\mathrm{brk III}$]{}, [$R_\mathrm{brk III}$]{}) and of the inner and outer discs of these antitruncated galaxies ([$\mu_{0,\mathrm{i}}$]{}, [$h_{\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, [$h_{\mathrm{o}}$]{}), for the $R$ and [3.6$\mu$m]{} datasets separately. We have used the Spearman rank correlation test to select the correlations which are significant at 95% of confidence level. Linear fits have been performed to the trends followed by each galaxy type in each photometric plane and the Pearson’s coefficient has been used to measure the level of linear correlation.
We have obtained the following results:
1. The antitruncated discs of spirals (taking them all together, or dividing them into Hubble classes) obey tight photometric relations, like those observed in S0 galaxies, both in the $R$ and [3.6$\mu$m]{} bands.
2. The antitruncated discs of barred and unbarred galaxies also follow tight photometric relations, again both in $R$ and [3.6$\mu$m]{}.
3. The majority of these correlations have high statistical significance despite the relatively low numbers of the available datasets, showing clear linear trends when [$h_{\mathrm{i}}$]{}, [$h_{\mathrm{o}}$]{}, [$R_\mathrm{brk III}$]{}, and [$R_\mathrm{25}$]{} are plotted on a logarithmic scale. This implies the existence of strong scaling relations in the Type-III discs of all Hubble types between their characteristic parameters ([$h_{\mathrm{i}}$]{}, [$h_{\mathrm{o}}$]{}, [$\mu_{0,\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, [$\mu_\mathrm{brk III}$]{}) and [$R_\mathrm{brk III}$]{}, as well as between the parameters of the inner and outer discs ([$\mu_{0,\mathrm{i}}$]{} – [$h_{\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{} – [$h_{\mathrm{o}}$]{}, and [$h_{\mathrm{i}}$]{} – [$h_{\mathrm{o}}$]{}).
4. The correlations between [$\mu_{0,\mathrm{i}}$]{}, [$\mu_{0,\mathrm{o}}$]{}, or [$\mu_\mathrm{brk III}$]{} with the logarithm of the characteristic scalelengths ([$h_{\mathrm{i}}$]{}, [$h_{\mathrm{o}}$]{}, or [$R_\mathrm{brk III}$]{}) improve significantly when the scalelengths are normalized to [$R_\mathrm{25}$]{}.
5. The logarithm of the characteristic scalelengths of antitruncated discs ([$h_{\mathrm{i}}$]{}, [$h_{\mathrm{o}}$]{}, and [$R_\mathrm{brk III}$]{}) scale with $\log({\ensuremath{R_\mathrm{25}}})$ for all galaxy types in [3.6$\mu$m]{}. In $R$, the linear trends are less tight and lose significance in spiral types.
6. The observational uncertainties of the data samples are too high to discern robustly whether the analysed scaling relations are similar (or not) in S0s and spirals, barred and unbarred galaxies, and in the $R$ and [3.6$\mu$m]{} bands. However, no statistical evidence is either found of significant differences between the relations followed by S0s and spirals and by barred and unbarred galaxies within errors. This result suggests that the scaling relations of antitruncated discs are independent of the morphological type and the presence or absence of bars. Deeper data and larger samples are required to confirm these results robustly.
In conclusion, the tight scaling relations found in the present study for Type-III discs impose strong constraints on any formation scenarios proposed to explain the formation of antitruncations in stellar discs across the Hubble Sequence, independently of whether the relations really depend (or not) on the morphological type or the hosting of a bar.
The authors thank to the anonymous referee for the provided input that helped to improve this publication significantly. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr). This research has made use of the NASA’s Astrophysics Data System and NASA/IPAC Extragalactic Database (NED). Supported by the Ministerio de Economía y Competitividad del Gobierno de España (MINECO) under project AYA2012-31277, the Instituto de Astrofísica de Canarias under project P3/86, and the Consejo Nacional de Ciencia y Tecnología de México (CONACYT) under project 167236. JEB acknowledges financial support to the DAGAL network from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007- 2013/ under REA grant agreement number PITN-GA-2011-289313.
[*Comments*: For each photometric relation, we list the results obtained from the linear fits performed to the various galaxies subsamples in columns (all galaxies, spirals, S0s, Sa–Sab’s, Sb–Sbc’s, Sc–Scd’s, barred galaxies, and unbarred ones). The first row of results in each parameter (for each relation and galaxy subsample considered) corresponds to the fits performed to the data in the $R$ band, while the second row corresponds to the results in the [3.6$\mu$m]{} band. In each linear fit, we provide the slope ($m$), the $Y$-intercept value ($C_0$), the Pearson coefficient of linear correlation ($\rho$), the Spearman rank probability of random correlation ($p_S$), and the number of data pairs available for each fit (see Section\[sec:fits\] for more details). ]{}
[*Comments*: See the notes of Table\[tab:hihorbreak\].]{}
[*Comments*: See the notes of Table\[tab:hihorbreak\].]{}
[*Comments*: See the notes of Table\[tab:hihorbreak\].]{}
[*Comments*: See the notes of Table\[tab:hihorbreak\].]{}
[*Comments*: See the notes of Table\[tab:hihorbreak\].]{}
[^1]: HyperLeda database is available at: http://leda.univ-lyon1.fr/
[^2]: The break strength of Type-II and Type-III profiles is defined as the logarithm of the outer-to-inner scalelengths ratio.
|
---
abstract: |
Background
: Time-dependent density-functional theory (TDDFT) continues to be useful in describing a multitude of low-energy static and dynamic properties. In particular, with recent advances of computing capabilities, large-scale TDDFT simulations are possible for fission dynamics as well as isovector dipole (IVD) resonances.
Purpose
: Following a previous paper \[Y. Shi, Phys. Rev. C 98, 014329(2018)\], we present an extension of the density-functional theory to allow for dynamic calculations based on the obtained static Hartree-Fock results. We perform extensive benchmark calculations, by comparing the calculated results with that of an existing code Sky3D. To perform linear-response calculations using the TDDFT method, comparisons have been made with the finite-amplitude quasiparticle random-phase approximation (FAM-QRPA) method. We plan to apply the TDDFT method to a systematic description of the IVD resonances in the Zr, Mo, and Ru isotopes.
Methods
: The strengths of IVD resonances are calculated using two complementary methods: TDDFT and FAM-QRPA methods. For the TDDFT results, additional benchmark calculations have been performed using the well-tested code Sky3D. In these three models, the important ingredients which have major influence on the results, such as time-odd potentials, boundary conditions, smoothing procedures, spurious peaks etc., have been carefully examined.
Results
: The current TDDFT and the Sky3D codes yield almost identical response functions once both codes use the same time-odd mean fields and absorbing boundary conditions. The strengths of the IVD resonances calculated using the TDDFT and FAM-QRPA methods agree reasonably well with the same position of the giant dipole resonance. Upon seeing a reasonable accuracy offered by the implemented code, we perform systematic TDDFT calculations for spherical Zr and Mo isotopes near $N=50$, where experimental data exist. For neutron-rich Zr, Mo, and Ru isotopes where shape evolution exist we predict the photoabsorption cross sections based on oblate and triaxial minima.
Conclusions
: The TDDFT code provides reasonable description for IVD resonances. Applying it to the spherical Zr and Mo nuclei, a reasonable agreement with experimental data has been achieved. For neutron-rich Zr isotopes, the photoabsorption cross section based on the two coexisting minima reflects the feature of the deformation of the minima.
author:
- 'Yue Shi (石跃)[^1]'
- Nobuo Hinohara
- Bastian Schuetrumpf
title: 'An implementation of nuclear time-dependent density-functional theory and its application to the nuclear isovector electric dipole resonance'
---
[UTF8]{}[gbsn]{}
Introduction {#sec1}
============
Since its first numerical realizations in the late 70s [@engel75; @Bon76a; @Cus76a], the time-dependent density-functional theory (TDDFT) continues to be useful in describing a variety of low-energy nuclear static and dynamic properties, ranging from the linear response of nuclear density, to the large-amplitude motion of heavy nuclei [@nege82; @naka16; @Umar15; @Simenel12; @burr19].
The modern developments [@naka05; @Maruhn2005; @Umar06a] allow for the inclusion of the full original Skyrme density-functional theory (DFT). Hence, the same energy density functionals (EDFs) obtained from the knowledge of the static properties of nuclei can be applied in dynamic simulations without any further approximation.
With advances of computing capabilities, nowadays one can perform TDDFT simulations that were not possible even twenty years ago. For example, the linear-response properties of medium or heavy nuclei, fission dynamics of actinides, as well as nuclear reaction involving medium-heavy nuclei are within the reach of calculations with single-node computers.
However, there are still demanding applications. For instance, for the time-dependent Hartree-Fock-Bogoliubov simulations in three-dimensional (3D) real space [@stetcu11; @bulgac16; @magi17], one still needs supercomputers. In the linearized limit, harmonic-oscillator (HO) based time-dependent Hartree-Fock-Bogoliubov (TDHFB) calculations with finite-range Gogny force have become available [@hash12; @hash13], taking advantage of the fact that the oscillation extends only in a relatively small region. Recently, finite-amplitude method for quasi-particle random-phase approximation (FAM-QRPA) calculations in 3D Cartesian coordinate space have also emerged [@wash17].
It is the purpose of the current work to present a new development, enabling the time-dependent capabilities, based on an earlier DFT code [@shi18]. Currently, the code is represented in 3D Cartesian coordinate space, using a light-weighted finite-difference method for derivative operators. It features an interface with the [<span style="font-variant:small-caps;">hfodd</span>]{} code [@doba97a; @doba09; @schunck17], which is a Skyrme-Hartree-Fock-Bogoliubov (HFB) code in a 3D coordinate HO basis. Such a flexable code is expected to provide a reasonable alternative for future developments.
As the first application, the current work provides systematic calculations for the isovector (IV) electric dipole ($E1$) vibration motion for stable and neutron-rich Zr, Mo, and Ru isotopes. Although there exist a few systematic calculations for IV and isoscalar vibrational properties for nuclei across the nuclear chart [@inak11; @scamps13; @scamps14; @ebata14], we find that a detailed analysis of the shape evolutions and shape coexistence in the same nucleus, reflected by the different structures of the GDR cross sections, is particularly useful [@kvasil09].
In Sec. \[model\] we present a description about the main features of the current TDDFT framework. Section \[results\] contains two parts: first, a set of careful benchmark calculations, with the current TDDFT, Sky3D codes, and the FAM-QRPA calculations, have been presented. Second, systematic calculations have been performed for the photoabsorption cross section of the isovector dipole (IVD) vibration in the spherical and deformed Zr, Mo, and Ru nuclei. A summary is contained in Sec. \[summary\].
The model {#model}
=========
This section describes in detail the procedure for the time development in connection with the previous static calculation [@shi18]. Then, we briefly describe the Sky3D [@Schuetrumpf2018] calculation, with which the current code is benchmarked. For the applications in the linearized limit of the current TDDFT calculation, we provide formulae for describing the relevant properties associated with the $E1$ vibrational mode.
The static calculations
-----------------------
Before the single-particle wave functions are propagated in time, one has to obtain the static solution of the HF problem. In this stage of the calculation, the time-odd components of the densities and mean-fields vanish for even-even nuclei. The form of the Hamiltonian, the way how the operators of the Hamiltonian are constructed, and how the integrations are performed have been explained in Ref. [@shi18].
### The grid points arrangement
The grid points in the present implementation are moved away from the origin of the simulating box and differs from those of Ref. [@shi18]. Specifically, in the example of one dimension, instead of using a set of coordinates at $$[-nx_{\rm max}, ...,0,1,...+nx_{\rm max}]\times dx,$$ the current code represents the problem on grid points at the coordinates $$\label{newgrid}
[-nx_{\rm max}+0.5, ...,-0.5,0.5,...+nx_{\rm max}-0.5]\times dx,$$ where $nx_{\rm max}$ is an integer number numerating the points at the edge of the simulating box. The $dx$ denotes the grid spacing. Note, that the latter choice has an even number of grid points, whereas the former one has an odd number of grid points. This choice is guided by the fact that the inclusion of the grid point at the origin of the box results in numerical problems [@maru14]. Using the grid shown in Eq. (\[newgrid\]), the integration can be carried out by summation [*on*]{} the grid, without the interpolation as presented in Ref. [@shi18].
### The Bardeen-Cooper-Schrieffer (BCS) pairing
To demonstrate the influence of the pairing interaction on the properties of the IVD resonances, we include a simple BCS pairing [@bardeen57; @ring80; @bender00]. For BCS method, we attach each single-particle wave function a real number, $v_i$, whose square gives the occupation probability of the $i$th orbit.
After each HF iteration, the occupation amplitude $v_i$ is determined, in the current work, by the following BCS equations $$\label{occupation}
v_{i,q}^2=\frac{1}{2}\left[1-\frac{\epsilon_{i,q}-\lambda_{q}}{\sqrt{(\epsilon_{i,q}-\lambda_q)^2+\Delta_{i,q}^2}}\right],$$ where $\epsilon_{i,q}$’s are the HF single-particle energies; $\lambda_q$ is the Fermi energy for given nucleonic type, which is adjusted so that $2\sum_i v_{i,q}^2$ gives the correct nucleon number. In Eq. (\[occupation\]), the state-dependent single-particle pairing gaps, $\Delta_{i,q}$’s, are given by $$\Delta_{i,q} = \sum_{\sigma}\int d \bm{r} \Delta_q(\bm{r}) \psi_{i,q}^*(\bm{r},\sigma)\psi_{i,q}(\bm{r},\sigma),$$ where $$\begin{aligned}
\label{densities_bcs}
\Delta_q(\bm{r}) &= -\frac{1}{2}V_q\int d \bm{r}\Big[1-\frac{\rho(\bm{r})}{\rho_{\rm pair}}\Big] \tilde{\rho}_q(\bm{r}), \\
\rho_q(\bm{r}) &= \sum_{i,\sigma} v_{i,q}^2\, \psi_{i,q}^*(\bm{r},\sigma)\psi_{i,q}(\bm{r},\sigma), \\
\tilde{\rho}_q(\bm{r}) &= \sum_{i,\sigma} v_{i,q}\, \sqrt{1-v_{i,q}^2}\, \psi_{i,q}^*(\bm{r},\sigma)\psi_{i,q}(\bm{r},\sigma).\end{aligned}$$ with $q=n,p$ denoting the neutron and proton, respectively. The quantities without subscriptions denote the summed contributions from protons and neutrons, for example, $\rho=\rho_n+\rho_p$. We choose $\rho_{\rm
pair}=0.32$fm$^{-3}$ in this work.
When applied to the drip-line nuclei, using the BCS pairing tends to scatter the particles to the positive-energy levels which are non-local, resulting in the unphysical nucleon gas surrounding the nucleus. This problem can be cured by replacing the BCS theory with the HFB theory [@doba84].
The nuclear mean fields including the time-odd parts
----------------------------------------------------
In the earlier static code presented in Ref. [@shi18], it has been explained that the time-odd densities and mean fields vanish due to the time-reversal symmetry. When time propagation is discussed, the time-odd densities and mean fields appear [@engel75]. Due to computing limitations, historically, the earlier TDHF calculations contained a few serious approximations such as the schematic treatment of the spin-orbit and pairing interactions. Modern TDHF calculations [@naka05; @Maruhn2005; @Umar06a] include the full Skyrme interactions. Recent studies discuss the influence of the tensor interactions when applied to the description of GDR [@frac12] and nuclear collisions [@guo18].
The current paper adopts the frequently-used Skyrme EDF which contains, in addition to the time-even densities, time-odd densities $\bm{s}$, $\bm{j}$. The tensor interaction is not considered in this work. See Eq. (A.19) of Ref. [@engel75] for a detailed form of the Skyrme energy density $\mathcal{H}(\bm{r})$.
After variation of the total energy, $E=\int\mathcal{H}(\bm{r})d \bm{r}$, with respect to the proton and neutron single-particle wave functions, the resulting Skyrme mean fields also contain terms of the above-mentioned time-odd densities. Modern nuclear DFT allows for a free parametrization of coupling constants in front of each term in the Skyrme mean field. See Eq. (2.6) of Ref. [@doba95] for details. Assuming local gauge invariance of the energy density, one requires the terms contributing to the mean fields to be grouped in pairs [@doba95], specifically, ($\rho\tau-\bm{j}^2$) and ($\rho\bm{\nabla}\cdot \bm{J}+\bm{s}\cdot{\bm{\nabla}\times\bm{j}}$).
In the current implementation of the TDDFT code, the single-particle Hamiltonian reads $$\begin{aligned}
\label{hamiltonian}
\hat{h}_q = -\bm{\nabla}\cdot\frac{\hbar^2}{2m^*}\bm{\nabla}+U_q&
-i\bm{B}_q\cdot(\bm{\nabla}\times\bm{\sigma})
+\bm{\sigma}\cdot\bm{\Sigma}_q \nonumber \\
&+\frac{1}{2i}(\bm{\nabla}\cdot\bm{I}_q+\bm{I}_q\cdot\bm{\nabla}).\end{aligned}$$ For protons, one needs to add Coulomb potentials \[Eqs. (20,24) in Ref. [@shi18]\]. The detailed expression of $U_q$ can be found in Eq. (18) of Ref. [@shi18]. The time-odd potentials included in Eq. (\[hamiltonian\]) read $$\begin{aligned}
\label{time_odd_pot}
\bm{\Sigma}_q&=\frac{1}{3}(-b_0+2b_0')\bm{s}-\frac{1}{3}(2b_0-b_0')\bm{s}_q \nonumber \\
&~~~~ ~~~~~~~~~~ ~~~~~~~~ -b_4\bm{\nabla}\times\bm{j}-b_4'\bm{\nabla}\times\bm{j}_q,\\
\bm{I}_q &=-2b_1\bm{j}+2b_1'\bm{j}_q-b_4\bm{\nabla}\times\bm{s}-b_4'\bm{\nabla}\times\bm{s}_q.\end{aligned}$$ In the EDFs, the term containing $\bm{s}\cdot\Delta\bm{s}$ is ignored.
Time propagation
----------------
The nuclear non-relativistic time-dependent Schrödinger equation reads $$\label{tdhf}
i\hbar\pdv{\psi_{i,q}(t)}{t}=\hat{h}_q(t)\psi_{i,q}(t),$$ where $\hat{h}_q$ can be found in Eq. (\[hamiltonian\]). In this section, the subscription $q$ is ignored for simplicity. The equation (\[tdhf\]) has the formal solution $$\psi_i(t)=\hat{\mathscr{U}}(t)\psi_i(0)=\hat{T}\exp(-\frac{i}{\hbar}\int_0^t\hat{h}(t')\,dt')\psi_i(0),$$ where $\hat{\mathscr{U}}$ is the time-evolution operator, and $\hat{T}$ is the time-ordering operator. To solve the time-dependent problem, one breaks up the total time evolution into $N$ small increments of time $\Delta t$ $$\hat{U}(t,t+\Delta t)=\exp(-\frac{i}{\hbar}\int_t^{t+\Delta t}\hat{h}(t')\,dt').$$
The time-evolution operator $\hat{\mathscr{U}}(t)$ can be obtained by consecutive actions of $\hat{U}(t,t+\Delta t)$ $$\hat{\mathscr{U}}(t)=\prod_{n=0}^{N-1}\hat{U}(n\Delta t,(n+1)\Delta t).$$
For small $\Delta t$ one could approximate $\hat{U}(t,t+\Delta t)$ by Taylor expansion up to order $m$: $$\label{expansion}
\exp(-\frac{i}{\hbar}\hat{h}\Delta t)\approx\sum_{n=0}^m\frac{1}{n!}\left(\frac{-i\Delta t}{\hbar}\right)^n \hat{h}^n,$$ where $\hat{h}$ has been assumed to be time independent in the time interval of $\Delta t$. In the current work, $\Delta t$ is taken to be 0.2fm/$c$, and $m=4$. These choices are motivated by previous TDHF calculations.
In the realistic calculations, each time advance of single-particle wave functions $\psi_i$, from time $t$ to $t+\Delta t$, has been achieved by using the Crank-Nicholson method [@Bon76a]. Specifically, from a series of single-particle wave functions at $t$, $\psi_i(t)$, one first performs $$\label{time_progress}
\psi_i^{\rm temp}(t+\Delta t)=\hat{U}^{\rm t}(t,t+\Delta t) \psi_i(t).$$ Having $\psi_i^{\rm temp}(t+\Delta t)$, and $\psi_i(t)$, one assembles various densities using respective single-particle wave functions, obtaining the $\rho^{\rm temp}(t+\Delta t)$ and $\rho(t)$.
Using these densities, one obtains the densities at a “middle time”, $\rho^{\rm mid}(t+\frac{\Delta t}{2})=0.5[\rho^{\rm temp}(t+\Delta
t)+\rho(t)]$. Now, one constructs the Hamiltonian $\hat{h}^{\rm mid}$, using $\rho^{\rm mid}(t+\frac{\Delta t}{2})$ \[see Eq. (15) of Ref. [@shi18], and Eq. (\[hamiltonian\]) for the form of the Hamiltonian\]. A second time propagation operation $\hat{U}^{\rm mid}(t,t+\Delta t)$ with $\hat{h}^{\rm
mid}$ \[Eq. (\[expansion\])\] is performed on the single-particle levels, finally obtaining the wave functions at $t+\Delta t$ $$\psi_i(t+\Delta t)=\hat{U}^{\rm mid}(t,t+\Delta t)\psi_i(t).$$ Here, $\hat{U}^{\rm mid}$ differs from $\hat{U}^{\rm t}$ \[Eq. (\[time\_progress\])\] in that the former uses the single-particle Hamiltonian in its exponent \[Eq. (\[expansion\])\] at the time $t+\frac{\Delta
t}{2}$, whereas the latter refers to the operator $\hat{U}$, where the Hamiltonian is constructed using the quantities at the time $t$.
Note that in the above procedure, one has to perform the time propagation twice. The single-particle Hamiltonian does not contain time specifically. In realistic calculations, the unitarity of the operator $\exp(-\frac{i}{\hbar}\hat{h}\Delta t)$ needs to be checked as it is approximated using a Taylor expansion \[Eq. (\[expansion\])\]. For the chosen parameter, $\Delta t=0.2$fm/$c$ and $m=4$, we evaluate the matrix elements $$\mathcal{I}_{ij} \equiv \langle \psi_i(t)|\hat{U}|\psi_j(t)\rangle
\approx \langle \psi_i(t)|\psi_j(t+\Delta t)\rangle.$$ Both the diagonal and off-diagonal matix elements start to deviate from 1 and 0, respectively, at or after the 6th place after the decimal point. For a better approximation of the $\hat{U}$ operator, one could decrease $\Delta t$ and increase $m$.
When the BCS pairing is included, the occupation amplitudes, $v_{i,q}$’s in Eq. (\[occupation\]), are kept unchanged when calculating the densities during the time development. When evaluating the densities, the single-particle wave functions vary according to Eq. (\[tdhf\]). This is a coarse approximation of dynamical pairing, as the occupation probabilities should vary with time. A natural solution would be to solve the full time-dependent HFB problem [@stetcu11; @bulgac16; @magi17]. Since the HFB theory treats nuclear interactions in the particle-hole and pairing channels in one single variational process [@ring80], a time-dependent HFB treatment allows for the occupation amplitudes being determined dynamically by the upper and lower components at a given time. In 3D Cartesian coordinate space, such methods require a direct diagonalization of a large HFB matrix. Work in this direction is in progress.
Absorbing boundary conditions (ABC) {#ABC}
-----------------------------------
With Dirichlet boundary conditions, it has been known that the TDDFT calculations show the occurrence of non-physical particle densities at the boundary region. To cure this problem, it has been proposed [@naka05] to use the so-called absorbing boundary conditions. This is achieved by introducing an imaginary potential
\[eq:ABC\] $$\hat{h}({\mbox{\boldmath $r$}})\rightarrow \hat{h}({\mbox{\boldmath $r$}})+i\tilde{\eta}({\mbox{\boldmath $r$}})$$ at the boundary region of the form $$\tilde{\eta}({\mbox{\boldmath $r$}})=\left\{\begin{array}{ll}
0 & \mbox{for }0<|{\mbox{\boldmath $r$}}| \le R\\
\eta_0\frac{|{\mbox{\boldmath $r$}}|-R}{\Delta r} & \mbox{for } R<|{\mbox{\boldmath $r$}}|<R+\Delta r
\end{array}\right.\, .$$
Recently, there have been efforts using more involved boundary conditions [@schu15; @he19]. Based on these studies, we decide to use the ABC due to its simplicity and effectiveness.
IVD resonance calculations {#gdr_cal}
--------------------------
The IVD resonance is the most common vibrational mode in nuclear physics, where protons and neutrons vibrate against each other. This mode is responsible for the $E1$ resonant strengths in the energy range of $\sim$10$-$20 MeV. This broad peak is called giant dipole resonance (GDR) [@hara01]. The current work aims at a description of the IVD resonance in terms of the TDDFT in its linearized limit, which is equivalent to the random-phase approximation (RPA) [@ring80].
In the TDDFT description, the strength of this IVD vibrational mode can be obtained by applying the following small boost on the obtained single-particle wave functions, $$\label{boost}
\psi_{i,q}(\bm{r},\sigma;t=0+)\equiv\exp\Big[-i\epsilon \sum_{\mu=-1}^{+1}\mathcal{M}(E1,\mu)\Big]\psi_{i,q}(\bm{r},\sigma),$$ with the IV operator $\mathcal{M}(E1,\mu)$ defined as $$\label{ISV_operator}
\mathcal{M}(E1,\mu)=e^{(E1)}_q r Y_{1\mu}(\vu{r})~~~~~~\mu=0,\pm 1,$$ where $e^{(E1)}_p=Ne/A$, and $e^{(E1)}_n=-Ze/A$. When $\mathcal{M}(E1,\mu)$ acts on proton/neutron single-particle wave functions, its coefficient takes value of $e^{(E1)}_p$/$e^{(E1)}_n$. The real spherical harmonics are defined as $$\{Y_{1\mu}\}_{\mu=-1,0,1}=\{\sqrt{\frac{3}{4\pi}}\frac{\lambda}{r}\}_{\lambda=y,z,x}.$$ In Eq. (\[boost\]), the boosted single-particle wave functions differ from the static ones by including “$t=0+$”, indicating their time-dependency. This IV boost has to be small enough to ensure that the vibration is still within the linearized regime. The typical magnitude of $|\epsilon|$ is 10$^{-3}$($e$fm)$^{-1}$. In this work, we apply 3D boost which has been indicated by the summation over $\mu$ in the exponent in Eq. (\[boost\]). The boost is applied over the whole box, although a masking procedure works better confining its effect in the range of the nucleus [@paul20].
The time evolution of the dipole moment $$\expval{\mathcal{M}(E1,\mu)} \equiv \int e^{(E1)}_n \rho_n r Y_{1\mu} d \bm{r}
+ \int e^{(E1)}_p \rho_p r Y_{1\mu} d \bm{r}$$ is then recorded to certain length of time. The strengths are the Fourier transform of $\langle\mathcal{M}(E1,\mu)\rangle(t)$ $$\label{strengths}
S(E;E1)=
-\frac{1}{\pi \hbar \epsilon}{\rm Im}\sum_{\mu=-1}^{+1}\int \expval{\mathcal{M}(E1,\mu)}(t)\,dt\,e^{(iE-\Gamma/2) t/\hbar},$$ where $\Gamma$ is a smoothing parameter. The photoabsorption cross section associated with the IVD resonance is obtained as follows [@yoshida11; @oishi16] $$\label{cross_section}
\sigma_\mathrm{abs.}=\frac{4\pi^2}{\hbar c} E \times S(E;E1).$$
Calculation of energy-weighted sum rule (EWSR) for the IVD vibration
--------------------------------------------------------------------
Another important aspect of the vibration calculations is the evaluation of EWSR [@ring80], which is a useful check of the implementation of the TDDFT code. In the TDDFT code, the sum rule is calculated using $$\label{sum_rule_tdhf}
m_1 = \int E \times S(E;E1) d E.$$ Recently, the EWSR for the density functional theory has been systematically derived in Refs. [@hino15; @hino19]. For the current IVD operator, the sum rule using Eq. (98) of Ref. [@hino19] can be adapted as follows $$\begin{aligned}
\label{sum_rule}
&m_1 = \sum_{\mu=-1}^{+1} \int d\bm{r}\big[\grad{(rY_{1\mu})}\big]^2
\Big\{\frac{\hbar^2}{2m} \Big[{e^{(E1)}_n}^2 \rho_n \nonumber + {e^{(E1)}_p}^2\rho_p\Big]\\
&+ (C_0^{\tau}-C_1^{\tau})\Big(e^{(E1)}_n+e^{(E1)}_p\Big)^2\rho_n\rho_p \nonumber \\
&+ \sum_{k=0,1}(C_k^{\tau}+C_k^j) \Big[e^{(E1)}_n\rho_n + (-1)^{k+1}e^{(E1)}_p\rho_p\Big]^2\Big\}.\end{aligned}$$ The definition of the spherical harmonics can be found in Eq. (\[ISV\_operator\]). The coupling constants in terms of $C_{0,1}^{\tau, j}$ are related to $b_1,b_1'$ through $$\begin{aligned}
C_0^{\tau}&=-C_0^j=b_1-0.5b_1', \\
C_1^{\tau}&=-C_1^j=-0.5b_1'.\end{aligned}$$ If we define the kinetic-energy contribution $$\begin{aligned}
\label{TRK}
m_1^{\rm kin} &= \sum_{\mu=-1}^{+1} \int d\bm{r}[\grad{(rY_{1\mu})}]^2 \frac{\hbar^2}{2m} \Big[{e^{(E1)}_n}^2 \rho_n + {e^{(E1)}_p}^2\rho_p\Big] \nonumber \\
&=\frac{9}{4\pi}\frac{\hbar^2}{2m}\frac{NZ}{A}e^2,\end{aligned}$$ then the enhancement factor, $\kappa$, due to the contribution of interaction-energy term with respect to the kinetic part, can be calculated through $$\label{kappa}
m_1=m_1^{\rm kin}(1+\kappa).$$ The classical sum rule of the IVD operator, which is the Thomas-Reiche-Kuhn (TRK) sum rule [@bohr75] can be analytically expressed as shown in Eq. (\[TRK\]).
The EWSR value obtained from Eq. (\[sum\_rule\]) are related to various densities of the ground state. Thus, they can be determined rather precisely. To what extent the $m_1$ values obtained from TDDFT \[Eq. (\[sum\_rule\_tdhf\])\] and Eq. (\[sum\_rule\]) agree, forms a stringent testing ground for the TDDFT code.
Sky3D calculations {#sky3d}
------------------
To demonstrate the precision of the current code, it is necessary to benchmark it against an existing code with an identical calculation. In this work, this benchmark is done with a well established code Sky3D.
We use the Sky3D code as described in Refs. [@maru14; @Schuetrumpf2018]. An important difference to the implementation presented in this code is that derivatives are performed utilizing the fast Fourier transform and thus the natural boundary conditions are periodic boundary conditions. The difference is of special importance for time-dependent calculations, as it affects the quantization of unbound energy states. Furthermore, when evaporated material is leaving the box it is again introduced from the other side of the box and not reflected as with Dirichlet boundary conditions. The codes differ slightly in the way the density at middle time is approximated. In Sky3D the wave functions are propagated until middle time $t+\Delta t/2$. These densities are then directly taken to calculate the Hamiltonian at middle time $\hat{h}^\mathrm{mid}$.
For the benchmarks we implemented the same boost as described in Sec. \[gdr\_cal\] and also the imaginary potential for ABC from Sec. \[ABC\].
calculated results {#results}
==================
To complete the benchmark of the implemented TDDFT code, one has to include careful calculations and compare the calculated results with those of existing codes. The particularly useful testing cases for the TDDFT code is the calculation of IVD resonance for light spherical and deformed nuclei.
In Ref. [@naka05] careful comparative study has been done between the TDDFT code and the RPA calculations. Detailed dipole-moment response as a function of time, as well as the corresponding strengths results for $^{16}$O nucleus has been presented with the specific force being provided. In this section, we first present results of the current code, comparing them with those of Sky3D code and Ref. [@naka05]. The calculation is then extended to spherical nucleus $^{40}$Ca, as well as deformed magnesium isotopes $^{24,34}$Mg with conventional Skyrme EDF SkM\* [@bart82], and a more recent EDF [<span style="font-variant:small-caps;">unedf1</span>]{} [@kort12].
The [<span style="font-variant:small-caps;">unedf1</span>]{} EDF contains Lipkin-Nogami (LN) pairing [@kort12] in the parameter adjustment process. In principle, one has to include this part specifically. However, we decide to be more flexible in the pairing treatment for the current TDDFT calculations based on two considerations. First, the original [<span style="font-variant:small-caps;">unedf1</span>]{} parameter is determined in the HO basis and with specific cut-off on the Hartree-Fock-Bogoliubov problem. Whereas the current code is working in Cartesian coordinate space. Hence, the continuum is discretized differently from that of a HO code. Consequently, there is no way to make the pairing treatment identical in the two codes [@shi18]. Second, the observables we are interested in, namely, the strengths for IVD resonances are well known to be insensitive to pairing interactions [@piek06]. The strengths corresponding to pygmy dipole resonance (PDR) are only enhanced very marginally by including the pairing interaction, as will be shown in Sec. \[magnesium\].
Results for light nuclei {#light}
------------------------
### Benchmark calculations for $^{16}$O with Skyrme SIII EDF {#siii}
The nucleus $^{16}$O is of particular interest in theoretical benchmarking calculations, as the structure of the strength is sensitive to the included terms in the EDF [@naka05]. Hence, many theoretical methods [@inak09; @frac12; @wu18] took $^{16}$O as a testing case for the proposed method. In this section, we perform TDDFT calculations with Skyrme force parameter SIII [@liu76] with time-odd potentials in the form of Eq. (\[time\_odd\_pot\]) (SIII-full), as well as SIII without any time-odd contributions (SIII-even).
![The responses of $\expval{\mathcal{M}(E1,\mu=0)}(t)$ of $^{16}$O calculated with the current TDDFT code and the Sky3D code [@maru14]. The inset of panel (b) shows the difference between the $\expval{\mathcal{M}(E1,\mu=0)}$ values of the TDDFT and the Sky3D results.[]{data-label="figure1"}](Figure1.pdf)
Figure \[figure1\] displays a set of comparisons of responses of dipole moments between the currently implemented code and the Sky3D code. In these calculations, the time-odd potentials are identical with that of the Sky3D code \[Eqs. (8e,8f) of Ref. [@maru14]\]. Specifically, the time-odd potentials are in the form of Eq. (\[time\_odd\_pot\]), except that the terms including [*only*]{} ${\mbox{\boldmath $s$}}$ are left out.
![The calculated photoabsorption cross sections of $^{16}$O using Eq. (\[cross\_section\]). The left panel shows the results with SIII-full EDF, whereas the right panel shows those of SIII-even. The thinner black lines indicate results without smoothing procedure.[]{data-label="figure2"}](Figure2.pdf)
Figure \[figure1\](a) compares the response functions without any absorbing mechanism. We see that the magnitude agrees well for $t\le400$fm/$c$. However, the good agreement starts to deteriorate after $t\approx500$fm/$c$. This is due to the different boundary conditions used in the two codes, which results in the different treatment of the particle densities bounced back from the border of the box. Indeed, even within the same code, using a finer grid results in rather different response functions after certain time.
Figure \[figure1\](b) compares the response functions with the ABC \[Eq. (\[eq:ABC\])\] calculated with both codes. For both codes we use $\eta_0=10\,\mathrm{MeV}$, $R=10\,\mathrm{fm}$, and $\Delta r=12\,\mathrm{fm}$. It can be seen that with the same ABC, both codes give almost identical response functions. The difference of the dipole moment, shown in the inset, is at least an order of magnitude smaller than the original moment value. The agreement is remarkable. Indeed, one has to take into account the fact that there are many differences in the features as discussed in Sec. \[sky3d\].
Figure \[figure2\] shows the photoabsorption cross sections calculated with Eq. (\[cross\_section\]) for SIII-full and SIII-even, with $\Gamma$=0.5MeV and without the smoothing procedure ($\Gamma$=0). Again, a good correspondence can be seen between Fig. 6(b) of Ref. [@naka05] and Fig. \[figure2\] of the current work. Specifically, for SIII-full we see, for both results, that the single largest peak occurs at $E\approx21.2$MeV. For SIII-even, the two peaks occur at $E\approx19.4$ and 21.8MeV for both the current result and those shown in Fig. 6(b) of Ref. [@naka05]. Using a smoothing parameter of $\Gamma=0.5$MeV brings the general shape of the curves rather close to those given in Fig. 6(b) of Ref. [@naka05]. It has to be noted that the peak heights of the cross section curves in the current work differ from those shown in Ref. [@naka05] by $\sim$10mb.
In these calculations, we use the ABC as described in Sec. \[ABC\]. In Fig. \[figure2\](a), we also include the results without the ABC. It can be seen that the strength without the ABC differs from that with the ABC in that the former gives small peaks for excitation energies larger than that corresponding to the main peak. These small peaks are spurious which are removed by absorbing potential in the outer layer region.
### Comparing TDDFT with FAM-QRPA: $^{16}$O and $^{40}$Ca {#spherical}
In this section, we compare our TDDFT approach to the QRPA calculation based on the linear-response formalism, the finite-amplitude method (FAM) [@naka07; @avog11]. The FAM allows us to calculate the response function without constructing the QRPA matrices in the case of the nuclear DFT. The present implementation of the FAM-QRPA [@kort15] is based on the nuclear DFT solver [<span style="font-variant:small-caps;">hfbtho</span>]{} [@stoi05; @stoi13; @perez17], which allows to describe the superconducting axially deformed nuclei in the HO basis.
Before showing the cross-section results, we first present the calculated static properties using both codes. Table \[table1\] lists the calculated ground-state energy decomposition into various terms, as well as the root-mean-square radii. For a fixed box size, three different grid spacings have been used. It can be seen that the ground-state energy is overbound by $<$200keV using the coarsest grid with $dx=1.0$fm. Using finer grid spacings reduces the total energy differences to $\le$50keV. It should be noted that, the seemingly poor accuracy of a spacing of 1.0 fm does not drastically affect the dynamic calculation \[see Fig. \[figure1\](b)\].
---------------------------------- ------------------- --------------------- --------------------- ------------ ------------------- --------------------- --------------------- -------------
$\Delta x$=1.0 fm $\Delta x$=0.784 fm $\Delta x$=0.707 fm $\Delta x$=1.0 fm $\Delta x$=0.784 fm $\Delta x$=0.707 fm
$E_{\rm tot}$ (MeV) $-$121.139 $-$120.997 $-$120.986 $-$121.000 $-$340.873 $-$340.599 $-$340.571 $-$340.625
$E_{\rm Kin.}$ (MeV) 236.905 236.443 236.414 236.494 659.414 658.387 658.290 658.505
$E_{\rho}$ (MeV) $-$406.666 $-$405.978 $-$405.936 $-$406.055 $-$1137.525 $-$1135.918 $-$1135.749 $-$1136.071
$E_{\tau}$ (MeV) $-$0.890 $-$0.886 $-$0.886 $-$0.886 $-$3.218 $-$3.209 $-$3.207 $-$3.209
$E_{\Delta\rho}$ (MeV) 36.522 36.486 36.492 36.520 68.788 68.601 68.575 68.640
$E_{\rm SO}$ (MeV) $-$0.636 $-$0.671 $-$0.677 $-$0.681 $-$0.979 $-$1.046 $-$1.059 $-$1.077
$E_{\rm dir.}^{\rm Coul.}$ (MeV) 16.448 16.429 16.427 16.428 80.201 80.132 80.124 80.134
$E_{\rm exc.}^{\rm Coul.}$ (MeV) $-$2.823 $-$2.820 $-$2.820 $-$2.820 $-$7.554 $-$7.547 $-$7.546 $-$7.548
$r_{\rm rms}^{\nu}$ (fm) 2.666 2.669 2.669 2.668 3.360 3.362 3.362 3.362
$r_{\rm rms}^{\pi}$ (fm) 2.684 2.687 2.687 2.686 3.395 3.398 3.398 3.398
$r_{\rm rms}^{\rm tot.}$ (fm) 2.675 2.678 2.678 2.677 3.377 3.380 3.380 3.380
---------------------------------- ------------------- --------------------- --------------------- ------------ ------------------- --------------------- --------------------- -------------
![The calculated photoabsorption cross sections for $^{16}$O and $^{40}$Ca, using TDDFT of the present implementation and FAM-QRPA based on the [<span style="font-variant:small-caps;">hfbtho</span>]{} with the SkM\* and [<span style="font-variant:small-caps;">unedf1</span>]{} EDFs.[]{data-label="figure3"}](Figure3.pdf)
Figure \[figure3\] shows the photoabsorption cross sections for $^{16}$O and $^{40}$Ca calculated with the TDDFT and FAM-QRPA. The energy of the main peak and the low-energy side of the main peak agree well between the two approaches, while the high-energy tail part is more fragmented in the FAM-QRPA strength. This behavior found in the calculation using the HO basis is also found in the QRPA calculations in deformed nuclei using the HO basis [@peru08; @losa10].
![The calculated photoabsorption cross sections for $^{16}$O, using HFBTHO-FAM method with the [<span style="font-variant:small-caps;">unedf1</span>]{} EDF. Three different HO basis numbers are used to see the convergence of the results.[]{data-label="figure4"}](Figure4.pdf)
Figure \[figure4\] displays the HFBTHO-FAM results with increasing number of HO basis ($N_{\rm sh}$). We see that the peak at $E\sim25-35$MeV moves toward the main peak with increasing $N_{\rm sh}$. We note a slow convergence of the strength function for this nucleus with this particular EDF. For medium-heavy nuclei, the isoscalar and isovector multipole strength functions are found to be converged already at $N_{\rm sh}=20$ [@stoit11; @oishi16].
### Results for deformed nuclei: $^{24,34}$Mg {#magnesium}
-------------------------- ------------ ---------------------------------------------------------- ------------ ----------------------------------------------------------
Current [<span style="font-variant:small-caps;">hfbtho</span>]{} Current [<span style="font-variant:small-caps;">hfbtho</span>]{}
$E_{\rm tot}$ $-$197.123 $-$197.155 $-$189.881 $-$189.852
$E_{\rm Kin.+c.m.}$ 384.483 384.091 401.148 400.387
$E_{\rm Coul}$ 28.681 28.650 28.713 28.671
$E_{\rm Skyrme}$ $-$610.287 $-$609.896 $-$619.742 $-$618.910
$Q_{20}$ 1.072 1.072 1.126 1.137
$\epsilon^{\pi}_{1/2^+}$ $-$34.236 $-$34.249 $-$29.474 $-$29.480
$\epsilon^{\pi}_{1/2^-}$ $-$23.510 $-$23.528 $-$20.865 $-$20.898
$\epsilon^{\pi}_{3/2^-}$ $-$19.429 $-$19.396 $-$17.348 $-$17.285
$\epsilon^{\pi}_{1/2^-}$ $-$13.945 $-$13.973 $-$13.219 $-$13.196
$\epsilon^{\pi}_{1/2^+}$ $-$12.066 $-$12.075 $-$11.036 $-$11.064
$\epsilon^{\pi}_{3/2^+}$ $-$9.525 $-$9.519 $-$8.596 $-$8.585
$\epsilon^{\nu}_{1/2^+}$ $-$39.279 $-$39.290 $-$34.215 $-$34.218
$\epsilon^{\nu}_{1/2^-}$ $-$28.361 $-$28.377 $-$25.500 $-$25.529
$\epsilon^{\nu}_{3/2^-}$ $-$24.274 $-$24.235 $-$22.034 $-$21.964
$\epsilon^{\nu}_{1/2^-}$ $-$18.667 $-$18.694 $-$17.833 $-$17.806
$\epsilon^{\nu}_{1/2^+}$ $-$16.725 $-$16.729 $-$15.604 $-$15.626
$\epsilon^{\nu}_{3/2^+}$ $-$14.141 $-$14.131 $-$13.148 $-$13.132
-------------------------- ------------ ---------------------------------------------------------- ------------ ----------------------------------------------------------
: Calculated static DFT results for $^{24}$Mg using SkM\* and [<span style="font-variant:small-caps;">unedf1</span>]{} EDFs. A comparison is made between the results using the current code and the [<span style="font-variant:small-caps;">hfbtho</span>]{} code [@stoi05; @stoi13]. There is no center-of-mass correction for [<span style="font-variant:small-caps;">unedf1</span>]{} calculations. The quadrupole moments are defined as $Q_{20}=2\expval{\hat{z}^2}-\expval{\hat{x}^2}-\expval{\hat{y}^2}$. The single-particle levels are doubly degenerate labeled by $\Omega^{\rm parity}$, where $\Omega$ denotes the total angular momentum projection of the level onto the $z$-axis. All quantities are in units of MeV, except for $Q_{20}$ values which are in barn.[]{data-label="table2"}
The nucleus $^{24}$Mg is one of the lightest nuclei with large prolate deformation. Hence, the IVD vibration motion of this nucleus has been frequently used as a testing case for TDDFT or RPA codes. Another interesting system that has a prolately deformed ground state is $^{34}$Mg. The occurrence of non-zero strength below $E=10$MeV in $^{34}$Mg is a signature of the pygmy mode for neutron-rich Mg isotopes [@ebat10]. For neutrons, there is a pairing correlation which makes $^{34}$Mg particularly interesting. In this section, we focus on the description of $^{24,34}$Mg with both TDDFT and the HFBTHO-FAM methods.
![Evolution of the IV density $\rho_p(\vb{r})- \rho_n(\vb{r})$ (in $\mathrm{fm}^{-3}$) in the $x-z$ plane ($y=0$) for the IVD mode in $^{24}$Mg.[]{data-label="figure5"}](Figure5.pdf)
Table \[table2\] lists the calculated static information on $^{24}$Mg with SkM\* and [<span style="font-variant:small-caps;">unedf1</span>]{} EDFs. Figure \[figure5\] plots the IV densities, $\rho_p(\vb{r})-\rho_n(\vb{r})$, on the $x-z$ plane with $y=0$, at a few instances. As the protons and neutrons vibrate against each other, a fading and strengthening pattern of the color can be seen. Careful examination reveals the left-right and up-down asymmetry, which is due to the 3D boost that has been initiated in the current calculations.
Figure \[figure6\] compares the strengths calculated with SkM\* and [<span style="font-variant:small-caps;">unedf1</span>]{} EDFs. It can be seen that the two peaks calculated with [<span style="font-variant:small-caps;">unedf1</span>]{} EDF are considerably lower and broader compared to those calculated with SkM\* EDF. The positions of the two peaks are a few hundreds of keV higher for [<span style="font-variant:small-caps;">unedf1</span>]{} EDF compared to those of SkM\* EDF. In Fig. \[figure6\], we plot our HFBTHO-FAM results too. The strength functions are almost identical up to the first peak, after which the FAM-QRPA calculations show more fragmented second peak or sub-peaks compared to the TDDFT calculations.
For the strength function of $^{24}$Mg calculated with SkM\* EDF, there are a few calculations using different models. For example, in Ref. [@inak09], the photoabsorption cross section for $^{24}$Mg has been calculated with the FAM-QRPA method. In Ref. [@ebat10] a canonical-basis TDHF calculation is performed to calculate the $E1$ strength in $^{24}$Mg. In particular, the result is consistent with their FAM-QRPA results [@ebat10]. In Ref. [@losa10], the QRPA calculations using (transformed) HO basis has been performed for the $E1$ strengths in $^{24}$Mg.
![The calculated strength functions of $^{24}$Mg using SkM\* and [<span style="font-variant:small-caps;">unedf1</span>]{} EDFs, with TDDFT and HFBTHO-FAM methods.[]{data-label="figure6"}](Figure6.pdf)
Comparing these three existing results \[Fig. 8(g) of Ref. [@inak09], Fig. 2 of Ref. [@ebat10], and Fig. 15 of Ref. [@losa10]\] with that in the current work which is shown in Fig. \[figure6\], it can be summarized that, for all calculated results, there are unambiguously two peaks at $E\approx16$ and 22MeV. The structure or sub-peaks appearing between these two are susceptible to, presumably, either the box size, or the truncation in the single-particle levels, and HO shells used by the respective models. It is rewarding to see such a consistency among independent methods and implementations.
Figure \[figure7\] shows the calculated $E1$ strengths for $^{34}$Mg using both TDDFT+BCS and the FAM-QRPA calculations. For the TDDFT+BCS calculations, pairing exists only for neutrons. The pairing strength for neutrons is $V_n=-500$MeVfm$^3$. There are 44 single-neutron levels included in the BCS problem. The highest-energy single-particle level has $\epsilon=3.85$MeV. To make the two methods comparable, we have fine tuned the pairing strengths in the [<span style="font-variant:small-caps;">hfbtho</span>]{} calculation in such a way that both codes give similar pairing energies in the static calculations.
We see from Fig. \[figure7\] that both calculations yield two peaks at $E\approx15$ and 20MeV. Again, the second peak from FAM-QRPA calculation is slightly more fragmented compared to the TDDFT+BCS calculations. These results are consistent with the canonical-basis TDHFB results of Ref. [@ebat10].
For the neutron-rich oxygen, neon, and magnesium isotopes, the appearance of the $E1$ strength below $E=10$MeV are of particular interest [@cao05; @ebat10; @wang17], as they correspond to the pygmy mode of vibration. It has been shown [@ebat10] that the inclusion of pairing correlation would result in a small enhancement of the fraction of the strengths below $E_c=10$MeV, compared to a TDDFT result.
![The calculated strength functions of $^{34}$Mg with SkM\* EDF using TDDFT and FAM-QRPA methods.[]{data-label="figure7"}](Figure7.pdf)
We compute the following PDR fraction [@ebat10; @ebata14] $$\label{fraction}
f_{\rm PDR}=\frac{m_1(E_c)}{m_1}\equiv\frac{\int^{E_c}E \times S(E)dE}{\int E \times S(E)dE},$$ for the strength functions from TDDFT calculations with and without pairing. The $f_{\rm PDR}$ value is 2.3% for the $E1$ strength without pairing. When pairing is included, this quantity increases to 2.7%, which is consistent with the results in Ref. [@ebat10].
### Calculated EWSR
Table \[table3\] compares the $m_1$ values calculated with the ground-state expectation value \[Eq. (\[sum\_rule\])\], and those calculated with the strength function obtained from the TDDFT method \[Eq. (\[sum\_rule\_tdhf\])\]. We see that the $m_1$ values from TDDFT and those from Eq. (\[sum\_rule\]) are rather close. The TDDFT values are systematically smaller than those of Eq. (\[sum\_rule\]) by less than 1% of the $m_1$ values. This indicates the correctness and good precision of the current implementation of the TDDFT code.
The classical TRK sum-rules \[Eq. (\[TRK\])\] are 59.2, 148.0, and 88.8 $e^2$fm$^2$MeV for $^{16}$O, $^{40}$Ca, and $^{24}$Mg, respectively. We have computed the enhancement factor $\kappa$ using Eq. (\[kappa\]), which are roughly 0.15 and 0.30 for each nucleus using [<span style="font-variant:small-caps;">unedf1</span>]{} and SkM\* EDFs, respectively.
TDDFT g.s. value
---------------------------------------------------------------------- ------- ------------
$^{16}$O (SIII-even) 67.1 67.3
$^{16}$O (SIII-full) 75.0 75.3
$^{16}$O (SkM\*) 72.5 72.8
$^{16}$O ([<span style="font-variant:small-caps;">unedf1</span>]{}) 67.0 67.6
$^{40}$Ca (SkM\*) 194.0 194.9
$^{40}$Ca ([<span style="font-variant:small-caps;">unedf1</span>]{}) 171.4 172.8
$^{24}$Mg (SkM\*) 113.7 114.3
$^{24}$Mg ([<span style="font-variant:small-caps;">unedf1</span>]{}) 101.8 102.9
: The EWSR values of the IVD operator (in $e^2$fm$^2$MeV) calculated using the current TDDFT code, compared to the ground-state values.[]{data-label="table3"}
Results for Zr, Mo, and Ru nuclei {#zr_mo_ru}
---------------------------------
In the previous TDDFT calculations for light spherical and deformed nuclei, we have seen the usefulness of the newly developed code. In this section, we perform systematic calculations for the photoabsorption cross sections of Zr, Mo, and Ru nuclei.
### Results for spherical Zr and Mo isotopes
Spherical Zr and Mo isotopes are interesting systems to study because the experimental cross sections for IVD $E1$ resonances are available [@berman67; @beil74]. In particular, for $^{92-100}$Mo, the strengths below the neutron emission threshold have been observed [@rusev06; @rusev08; @erhard10].
Figure \[figure8\] displays the calculated photoabsorption cross sections for Zr and Mo isotopes for which experimental data exists for comparison. In these calculated strengths \[Eq. (\[strengths\])\], a relatively larger smoothing parameter, $\Gamma=2.0$MeV, has been used. This is motivated by the observation in a previous RPA calculation [@nest06; @kvasil09], where a 2.0MeV smoothing parameter was seen to produce reasonable descriptions for these cross-section data. Using a smaller smoothing parameter results in the peaks being more pronounced, and narrower. The centroids remain at the same position. In these TDDFT calculations, we do not include BCS pairing since the relevant observables associated with the IVD resonances are insensitive to pairing.
We see from Fig. \[figure8\] that both the widths and the centroid of the GDR peak are well reproduced by the current TDDFT calculations. For Mo isotopes, the low-energy part ($E\le10$MeV) of the cross section were observed using bremsstrahlung method [@rusev06; @rusev08; @erhard10]. In the third row of Fig. \[figure8\], the same cross sections for Mo isotopes are plotted in a logarithmic scale. Our calculated results well reproduce the low-energy part ($E<10$MeV) of the experimental data, especially for $^{96-100}$Mo.
Note that the linear-response theories, such as those adopted in the current work, contain correlations on the one-particle-one-hole (1p-1h) level. To fully account for the width of the GDR peak, it is important to include 2p-2h correlations [@hara01]. It is satisfactory to see that the positions of the centroids, and the general structures of the GDR peaks are reasonably well reproduced.
### Static potential energy surfaces for neutron-rich nuclei
The shape evolution of the ground states of neutron-rich zirconium isotopes are particularly interesting, with prolate and oblate minima competing to become ground states [@togashi16; @zhao17]. With recent advances in the rare isotope facilities, the interesting low-energy spectra for the most neutron-rich isotopes in the Zr, Mo, and Ru nuclei [@naka17; @wata11; @paul17; @doherty17] are becoming more and more available. From various models, the neutron-rich isotopes of Mo and Ru are calculated to have well-defined triaxially deformed ground states [@zhang15; @bhat16; @xiang16].
Before showing the calculated IV $E1$ photoabsorption cross section of these neutron-rich nuclei, it is necessary to have some idea about the potential-energy surfaces of quadrupole deformations. Figures \[figure9\], \[figure10\], and \[figure11\] display the potential-energy surfaces for even-even $^{98-108}$Zr, $^{100-110}$Mo, and $^{102-112}$Ru nuclei, calculated with [<span style="font-variant:small-caps;">unedf1</span>]{} EDF. The constraint HFB calculations for these potential-energy surfaces are performed with the [<span style="font-variant:small-caps;">hfodd</span>]{} code (version 2.68h). For these HFB+LN calculations, there are 1140 ($N=17$) spherical HO bases included; the original pairing strengths and energy cut-off on the quasi-particle spectra are used [@kort12].
For the Zr isotopes, the spherical ground states for $50\le N \le58$ are replaced by a situation where prolate and oblate minima coexist for $^{100,102}$Zr, with the prolate minimum being slightly lower energetically in $^{102}$Zr. For Zr isotopes with $N\ge64$, the prolate minima move to a static triaxial deformation, with the oblate minima staying slightly higher in energy.
For Mo and Ru isotopes with $N\ge60$, the ground states are dominated with a triaxial deformation. For all Mo isotopes, the shape is rather soft in the $\gamma$ direction. Whereas for the Ru isotopes, the $\gamma$-deformation is more rigid in the sense that the energy barriers separating the positive- and negative-$\gamma$ minima are in general higher for Ru than that of Mo isotopes. For Mo isotopes with $N\le58$ the ground states are calculated to be spherical.
One has to be careful in interpreting the results of the Mo isotopes. The softness of the total energy of a nucleus with respect to certain deformation degree of freedom indicates that the single-Slater description provided by the HF or HFB method may no longer be valid. One of the appropriate cures of such a problem is to solve the Hill-Wheeler equation [@hill53] using multiple Slater determinants representing HF/HFB solutions that are constrained at a range of deformation values (multi-reference DFT or Generator coordinate method) [@ring80]. Such a description is outside the scope of the current work.
### Dynamical results for deformed nuclei
A good description for the IV $E1$ photoabsorption cross section using the current TDDFT method gives confidence in the predictions for neutron-rich Zr, Mo, and Ru isotopes with $N=60-68$. The predictions are displayed in Figs. \[figure12\] and \[figure13\]. These dynamic calculation are performed without pairing interaction as the structures of the GDR peaks are essentially independent of pairing. To achieve fast and better convergence, we have imported single-particle wave functions from the HF calculations using the [<span style="font-variant:small-caps;">hfodd</span>]{} code. These HF calculations using the HO basis and the finite-difference method are both without deformation constraint. For triaxially deformed minima, the HF calculations give small $\gamma$ differences compared to that of the HFB+LN results using [<span style="font-variant:small-caps;">hfodd</span>]{}. Specifically, for the softest nucleus, the triaxially deformed $^{110}$Mo, the HFB+LN calculation using [<span style="font-variant:small-caps;">hfodd</span>]{} gives quadrupole moments $(Q^{\rm HFB+LN}_{20}, Q^{\rm HFB+LN}_{22})\approx(8.5,3.5)$b (see Fig. \[figure10\]). Without pairing, the [<span style="font-variant:small-caps;">hfodd</span>]{} calculation gives $(Q^\textrm{HF}_{20},
Q^\textrm{HF}_{22})\approx(9.4,4.0)$b. Before performing dynamic calculation, the static HF calculation in Cartesian coordinate space gives $(Q^\textrm{static}_{20}, Q^\textrm{static}_{22})\approx(9.5,4.0)$b.
The effect of the deformation on the GDR peak has been well known [@hara01]. For tin isotopes with extreme neutron excess, the QRPA calculations based on the relativistic mean field [@arteaga09] have been studied. Recent large-scale RPA calculations [@inak11; @ebata14] have been performed for light and medium-heavy nuclei. For both of these calculations, the effect of the deformation on the GDR peak, as well as the PDR contributions are carefully analyzed in connection with the neutron excess. The current dynamical calculations allow for analysis of IVD resonances in the context of nuclear shape evolution within the isotopic chains and shape coexistence in the same nucleus.
For a spherical nucleus the GDR peaks corresponding to the three vibrational modes are identical due to the same density profiles along the three axes. When the nucleus acquires a prolate deformation, the originally identical GDR peaks split into two groups: (1) a mode corresponding to a vibration along the symmetry axis ($K=0$ mode); and (2) two modes corresponding to the vibrations along the axes perpendicular to the symmetry axis ($K=\pm1$ modes). For a prolate shape, intuitively, because of the larger material extension, the potential is wider along the symmetry axis. Hence, the energy cost is lower for the $K=0$ mode, compared to the $K=\pm1$ modes. While the peak for the $K=0$ mode shifts to lower energy, the contribution of this mode to the total strength becomes higher than those of the $K=\pm1$ modes. Similar effects can be found from light to heavy spherical nuclei, where the total GDR peak shifts to a lower energy and becomes taller and/or broader.
Due to the above-mentioned reasons, for an oblate deformation, the peaks corresponding to the $K=\pm1$ modes become higher and shift to lower excitation energies compared to that corresponding to the $K=0$ mode. We find all these features of the GDR peaks for prolate and oblate deformed $^{100,102}$Zr as shown in Fig. \[figure12\].
For $^{104,106,108}$Zr, $^{100-110}$Mo, and $^{102-112}$Ru nuclei, the IVD vibrations are based on the triaxial minima, see Figs. \[figure12\] and \[figure13\]. Due to the large smoothing parameter, $\Gamma=2.0$MeV, the peaks at higher excitation energies appear to be one taller and broader peak compared to the lower-energy one. For $^{110}$Mo and $^{108}$Zr, the three peaks corresponding to $x$-, $y$-, and $z$-axis, merge into one broader peak.
![The fraction of the strengths for IVD resonances below $E_c=10$MeV \[Eq. (\[fraction\])\] for zirconium isotopes with spherical, prolate and oblate deformations.[]{data-label="figure14"}](Figure14.pdf)
Figure \[figure14\] plots the fraction of the IVD strength below $E_c=10$MeV for zirconium isotopes based on different deformations. With the same deformation, the $f_{\rm PDR}$ values increase with neutron excess. From spherical to deformed nuclei, we see a small decrease of the $f_{\rm PDR}$ value at $N=60$, which agrees with previous studies [@inak11; @ebata14]. This is a net result of (1) the decrease of energy of the $K=0$ mode, and the increase of the energy of the $K=\pm1$ modes, as well as (2) an enhanced contribution in the total strength from the $K=0$ mode, as pointed out in Ref. [@arteaga09]. For the oblate deformation, there is a plateau structure below $E=10$MeV, which is the main contribution to the $f_{\rm PDR}$ value. For spherical Mo isotopes, one can observe similar plateau structure at lower-energy part as shown in Fig. \[figure8\].
Summary
=======
Based on a previous computer code developed for the nuclear density-functional theory (DFT), we present a further development, enabling the time-dependent DFT (TDDFT) calculations. We benchmark the code by comparing its calculated response functions of dipole moment of $^{16}$O with that of an existing 3D TDDFT code, Sky3D. Although the response functions for $^{16}$O are sensitive to a few subtle factors (time-odd mean fields, treatment of boundary conditions, etc.), a remarkable agreement has been found between the two codes, as long as those factors are carefully considered.
To apply the TDDFT in its linearized limit and describe the isovector (IV) electric dipole ($E1$) observables, we carry out finite-amplitude method for quasiparticle random-phase approximation (FAM-QRPA) for a few light spherical ($^{16}$O, $^{40}$Ca) and axially deformed ($^{24,34}$Mg) nuclei, and compare the calculated IV $E1$ properties with those resulted from TDDFT calculations. The comparisons are acceptable up to the first peak at $E\approx20$MeV. Beyond that, the FAM-QRPA calculations give more fragmented peaks compared to that of the TDDFT calculations.
Using the [<span style="font-variant:small-caps;">unedf1</span>]{} energy density functional (EDF), the current TDDFT calculations provide reasonable description for both the giant dipole resonance and the low-energy part of the IV $E1$ photoabsorption cross section for spherical Zr and Mo nuclei, where experimental data exist.
For heavier Zr isotopes, the calculated potential-energy surfaces show coexisting minima. The predicted $E1$ photoabsorption cross sections reflect typical features depending on the local minima that they are based upon.
For heavier Mo and Ru isotopes, the ground states are triaxially deformed. The predicted cross sections show features that distinguish them from the spherical one. For Mo isotopes considered here, the predicted onset of the triaxial deformation which occurs in $^{102}$Mo ($N=60$), is only two neutrons larger than the isotope in which experimental data exist. The photo-nuclear experiments on these Mo isotopes are recommended.
Useful discussions with W. Nazarewicz and P. Stevenson are gratefully acknowledged. The current work is supported by National Natural Science Foundation of China (Grant No. 11705038), JSPS KAKENHI Grant No. 16K17680, the JSPS-NSFC Bilateral Program for the Joint Research Project on “Nuclear mass and life for unravelling mysteries of r-process”, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 279384907 - SFB 1245. YS thanks the HPC Studio at Physics Department of Harbin Institute of Technology for computing resources allocated through INSPUR-HPC@PHY.HIT. A part of the numerical calculations were performed at the Oakforest-PACS Systems through the Multidisciplinary Cooperative Research Program of the Center for Computational Sciences, University of Tsukuba.
[81]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop @noop @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{}
[^1]: corresponding author
|
---
abstract: 'This paper studies the problem of adaptively sampling from $K$ distributions (arms) in order to identify the largest gap between any two adjacent means. We call this the MaxGap-bandit problem. This problem arises naturally in approximate ranking, noisy sorting, outlier detection, and top-arm identification in bandits. The key novelty of the MaxGap bandit problem is that it aims to adaptively determine the natural partitioning of the distributions into a subset with larger means and a subset with smaller means, where the split is determined by the largest gap rather than a pre-specified rank or threshold. Estimating an arm’s gap requires sampling its neighboring arms in addition to itself, and this dependence results in a novel hardness parameter that characterizes the sample complexity of the problem. We propose elimination and UCB-style algorithms and show that they are minimax optimal. Our experiments show that the UCB-style algorithms require $6 \mhyphen 8$x fewer samples than non-adaptive sampling to achieve the same error.'
author:
- |
Sumeet Katariya [^1]\
University of Wisconsin\
Madison, WI 53706.\
`sumeetsk@gmail.com` Ardhendu Tripathy ^^\
University of Wisconsin\
Madison, WI 53706.\
`astripathy@wisc.edu` Robert Nowak\
University of Wisconsin\
Madison, WI 53706.\
`rdnowak@wisc.edu`
bibliography:
- 'References.bib'
title: 'MaxGap Bandit: Adaptive Algorithms for Approximate Ranking'
---
[^1]: Authors contributed equally and are listed alphabetically.
|
---
author:
- 'Sina Bittens[^1], Ruochuan Zhang[^2], Mark A. Iwen[^3]'
bibliography:
- 'SFFTrefs.bib'
title: A Deterministic Sparse FFT for Functions with Structured Fourier Sparsity
---
**Keywords.** Sparse Fourier Transform (SFT), Structured Sparsity, Deterministic Constructions, Approximation Algorithms **AMS Subject Classification.** 05-04, 42A10, 42A15, 42A16, 42A32, 65T40, 65T50, 68W25, 94A12
Acknowledgements {#acknowledgements .unnumbered}
================
Sina Bittens was supported in part by the DFG in the framework of the GRK 2088. Mark Iwen and Ruochuan Zhang were both supported in part by NSF DMS-1416752. The authors would also like to thank both Felix Krahmer for introducing them at TUM in the summer of 2016, as well as Gerlind Plonka for her ongoing support, and particularly for her generosity in providing resources that aided in the writing of this paper.
[^1]: University of Göttingen, Institute for Numerical and Applied Mathematics, Lotzestr. 16-18, 37083 Göttingen, Germany (sina.bittens@mathematik.uni-goettingen.de, +49 551 394515).
[^2]: Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA (zhangr12@msu.edu).
[^3]: Department of Mathematics, and Department of Computational Mathematics, Science, and Engineering (CMSE), Michigan State University, East Lansing, MI, 48824, USA (markiwen@math.msu.edu).
|
---
abstract: |
Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review three Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fourth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These four temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.
**Keywords:** Non-negative matrix factorization, Time series data, Gamma Markov chains, MAP estimation
author:
- |
Louis Filstroff$^{1}$ Olivier Gouvert$^1$ Cédric Févotte$^1$ Olivier Cappé$^2$\
$^1$ IRIT, Université de Toulouse, CNRS, France\
$^2$ DI ENS, CNRS, INRIA, Université PSL
bibliography:
- 'articlebib.bib'
title: 'A Comparative Study of Temporal Non-Negative Matrix Factorization with Gamma Markov Chains'
---
Introduction {#sec-intro}
============
Non-negative matrix factorization
---------------------------------
Non-negative matrix factorization (NMF) [@paatero1994positive; @lee1999learning] has become a widely used class of methods for analyzing non-negative data. Let us consider $N$ samples in $\mathbb{R}^{F}_{+}$. We can store these samples column-wise in a matrix, which we denote by $\mathbf{V}$ (therefore of size $F \times N$). Broadly speaking, NMF aims at finding an approximation of $\mathbf{V}$ as the product of two non-negative matrices: $$\mathbf{V} \simeq \mathbf{WH},
\label{eq-nmf-approx}$$ where $\mathbf{W}$ is of size $F \times K$, and $\mathbf{H}$ is of size $K \times N$. $\mathbf{W}$ and $\mathbf{H}$ are referred to as the dictionary and the activation matrix, respectively. $K$ is usually chosen such that $K \ll \min(F,N)$, hence producing a low-rank approximation of $\mathbf{V}$. This factorization is often retrieved as the solution of an optimization problem, which we can write as: $$\min_{\mathbf{W} \geq 0,~\mathbf{H} \geq 0} D(\mathbf{V}|\mathbf{WH}),
\label{eq-nmf-min}$$ where $D$ is a measure of fit between $\mathbf{V}$ and its approximation $\mathbf{WH}$, and the notation $\mathbf{A} \geq 0$ denotes the non-negativity of the entries of the matrix $\mathbf{A}$. One of the key aspects to the success of NMF is that the non-negativity of the factors $\mathbf{W}$ and $\mathbf{H}$ yields an interpretable, part-based representation of each sample: $\mathbf{v}_n \simeq \mathbf{Wh}_n$ [@lee1999learning].
Various measures of fit have been considered in the literature, for instance the family of $\beta$-divergences [@fevotte2011algorithms], which includes some of the most popular cost functions in NMF, such as the squared Euclidian distance, the generalized Kullback-Leibler divergence, or the Itakura-Saito divergence. As it turns out, for many of these cost functions, the optimization problem described in Eq. can be shown to be equivalent to the joint maximum likelihood estimation of the factors $\mathbf{W}$ and $\mathbf{H}$ in a statistical model, that is: $$\max_{\mathbf{W},\mathbf{H}} p(\mathbf{V}|\mathbf{W},\mathbf{H}).
\label{eq-nmf-max}$$ This leads the way to so-called *probabilistic* NMF, i.e., estimation or inference tasks in probabilistic models whose observation distribution may be written as: $$\mathbf{v}_{n} \sim p(~.~;\mathbf{Wh}_{n}, \boldsymbol{\Theta}), \quad \mathbf{W} \geq 0, \quad \mathbf{H} \geq 0,
\label{eq-prob-nmf}$$ that is to say that the distribution of $\mathbf{v}_n$ is parametrized by the dot product of the factors $\mathbf{W}$ and $\mathbf{h}_n$. Other potential parameters of the distribution are generically denoted by $\boldsymbol{\Theta}$. Most of the time these distributions are such that $\mathbb{E}(\mathbf{v}_n) = \mathbf{Wh}_n$.
This large family encompasses many well-known models of the literature, for example models based on the Gaussian likelihood [@schmidt2009bayesian] or the exponential likelihood [@fevotte2009nonnegative; @hoffman2010bayesian]. It also includes factorization models for count data, which are most of the time based on the Poisson distribution[^1] [@canny2004gap; @cemgil2009bayesian; @zhou2012beta; @gopalan2015scalable], but can also make use of distributions with a larger tail, e.g., the negative binomial distribution [@zhou2018nonparametric]. Finally, more complex models using the compound Poisson distribution have been considered [@simsekli2013learning; @basbug2016hierarchical; @gouvert2019recommendation], allowing to extend the use of the Poisson distribution to various supports $(\mathbb{N}, \mathbb{R}_+, \mathbb{R}, ...)$.
In the vast majority of the aforementioned works, prior distributions are assumed on the factors $\mathbf{W}$ and $\mathbf{H}$. This is sometimes referred to as *Bayesian* NMF. In this case, the columns of $\mathbf{H}$ are most of the time assumed to be independent: $$p(\mathbf{H}) = \prod_{n=1}^{N} p(\mathbf{h}_n).
\label{eq-h-idp}$$ The factors being non-negative, a standard choice is the Gamma distribution, which can be sparsity-inducing if the shape parameter is chosen to be lower than one. The inverse Gamma distribution has also been considered.
Temporal structure of the activation coefficients
-------------------------------------------------
In this work, we are interested in the analysis of specific matrices $\mathbf{V}$ whose columns cannot be treated as exchangeable, because the samples $\mathbf{v}_n$ are correlated. Such a scenario arises in particular when the columns of $\mathbf{V}$ describe the evolution of a process over time.
From a modeling perspective, this means that correlation should be introduced in the statistical model between successive columns of $\mathbf{V}$. This can be achieved by lifting the prior independence assumption of Eq. , thus introducing correlation between successive columns of $\mathbf{H}$. In this paper, we consider a Markov structure on the columns of $\mathbf{H}$: $$p(\mathbf{H}) = p(\mathbf{h}_1) \prod_{n \geq 2} p(\mathbf{h}_n | \mathbf{h}_{n-1}).
\label{eq-h-markov}$$ We will refer to such a model as a $\textit{dynamical}$ NMF model. Note that very recent works go beyond the Markovian assumption, i.e., assume dependency with multiple past time steps, and are labeled as “deep” [@gong2017deep; @guo2018deep].
Several works [@fevotte2013non; @schein2016poisson] assume that the transition distribution $p(\mathbf{h}_n | \mathbf{h}_{n-1})$ makes use of a transition matrix $\boldsymbol{\Pi}$ of size $K \times K$ to capture relationships between the different components. In this case, the distribution of $h_{kn}$ depends on a linear combination of all the components at the previous time step: $$p(\mathbf{h}_n | \mathbf{h}_{n-1}) = \prod_{k} p(h_{kn}|\sum_l \pi_{kl} h_{l(n-1)}).
\label{eq-trans-h-matrix}$$
In this work, we will restrict ourselves to $\boldsymbol{\Pi} = \mathbf{I}_K$. Equivalently, this amounts to assuming that the $K$ rows of $\mathbf{H}$ are a priori independent, and we have $$p(\mathbf{H}) = \prod_{k} p(h_{k1}) \prod_{n \geq 2}p(h_{kn}|h_{k(n-1)}).
\label{eq-h-markov-idp}$$ We will refer to such a model as a *temporal* NMF model.
A first way of dealing with the temporal evolution of a non-negative variable is to map it to $\mathbb{R}_+$. It is then commonly assumed that this variable evolves in Gaussian noise. This is for example exploited in the seminal work of @blei2006dynamic on the extension of latent Dirichlet allocation to allow for topic evolution[^2]. A similar assumption is made in @charlin2015dynamic, which introduces dynamics in the context of a Poisson likelihood (factorizing the user-item-time tensor). Gaussian assumptions allow to use well-known computational techniques, such as Kalman filtering, but result in loss of interpretability.
We will focus in this paper on naturally non-negative Markov chains. Various non-negative Markov chains have been proposed in the NMF literature [@cemgil2007conjugate; @fevotte2009nonnegative; @acharya15nonparametric]. They are all built in relation with the Gamma (or inverse Gamma) distribution. As a matter of fact, these models exhibit the same drawback: the chains all have a degenerate stationary distribution. This can lead to undesirable behaviors, such as the instability or the degeneracy of realizations of the chains. We emphasize that this is problematic from the probabilistic perspective only, since these prior distributions may still represent an appropriate regularization in a MAP setting.
Contributions and organization of the paper
-------------------------------------------
The contributions of this paper are 4-fold:
- We review the existing non-negative Markov chains of the NMF literature and discuss some of their limitations. In particular we show that these chains all have a degenerate stationary distribution;
- We present an overlooked non-negative Markov chain from the time series literature, the first-order autoregressive Beta-Gamma process, denoted as BGAR(1) @lewis1989gamma, whose stationary distribution is Gamma. To the best of our knowledge, this particular chain has never been considered to model temporal dependencies in matrix factorization problems;
- We derive majorization-minimization-based algorithms for maximum a posteriori (MAP) estimation in the NMF models with all presented prior structures on $\mathbf{H}$, including BGAR(1);
- We compare the performance of all these models on a prediction task on three real-world datasets.
The paper is organized as follows. Section \[sec-study\] introduces and compares non-negative Markov chains from the literature. Section \[sec-map\] presents MAP estimation in temporal NMF models. Experimental work is conducted in Section \[sec-exp\], before concluding in Section \[sec-ccl\].
Comparative study of Gamma Markov chains {#sec-study}
========================================
This section reviews existing models of Gamma Markov chains, i.e., Markov chains which evolve in $\mathbb{R}_{+}$ in relation with the Gamma distribution. We have identified three different models in the NMF literature:
1. Chaining on the rate parameter of a Gamma distribution (Section \[sec-rate\]);
2. Chaining on the rate parameter of a Gamma distribution with an auxiliary variable (Section \[sec-cd-rate\]);
3. Chaining on the shape parameter of a Gamma distribution (Section \[sec-shape\]).
As shall be discussed in these subsections, these three models are all built around the assumption $\mathbb{E}(h_{kn}|h_{k(n-1)}) \propto h_{k(n-1)}$ (which roughly means that the chain should not drift too far away from its previous value), but lack a well-defined stationary distribution, which leads to the degeneracy of the realizations of the chains. A fourth model from the time series literature, called BGAR(1), is presented in Section \[sec-bgar\]. It is built to have a well-defined stationary distribution (it is marginally Gamma distributed), but does not share the property $\mathbb{E}(h_{kn}|h_{k(n-1)}) \propto h_{k(n-1)}$. The realizations of the chain are not degenerate and exhibit some interesting properties. To the best of our knowledge, this kind of process has never been used in a probabilistic NMF problem to model temporal evolution.
Throughout the section, $(h_n)_{n \geq 1}$ denotes the (scalar) Markov chain of interest, where the index $k$ as in Eq. has been dropped for enhanced readability. It is further assumed that $h_1$ is set to a fixed, deterministic value.
Chaining on the rate parameter {#sec-rate}
------------------------------
### Model
Let us consider a general Gamma Markov chain model with a chaining on the rate parameter: $$h_{n}|h_{n-1} \sim \text{Gamma} \left( \alpha, \frac{\beta}{h_{n-1}} \right).
\label{eq-gmc-rate-def}$$ As it turns out, Eq. can be rewritten as a multiplicative noise model: $$h_{n} = h_{n-1} \times \phi_n,
\label{eq-gmc-rate-noise}$$ where $\phi_n$ are i.i.d. Gamma random variables with parameters $(\alpha, \beta)$. We have $$\mathbb{E}(h_{n}|h_{n-1}) = \frac{\alpha}{\beta}h_{n-1}, \quad
\text{var}(h_{n}|h_{n-1}) = \frac{\alpha}{\beta^2}h^2_{n-1}.$$
This model was introduced in @fevotte2009nonnegative to add smoothness to the activation coefficients in the context of audio signal processing. The parameters were set to $\alpha > 1$ and $\beta = \alpha - 1$, such that the mode would be located at $h_{n} = h_{n-1}$. A similar inverse Gamma Markov chain was also considered in @fevotte2009nonnegative and in @fevotte2011majorization.
### Analysis
From Eq. we can write: $$h_{n} = h_1 \prod_{i=2}^{n} \phi_i.$$
The independence of the $\phi_i$ yields: $$\begin{aligned}
\mathbb{E}(h_{n}) & = h_1 \left( \frac{\alpha}{\beta} \right)^{n-1}, \\
\text{var}(h_{n}) & = h_1^2 \left[ \left( \frac{\alpha^2}{\beta^2} + \frac{\alpha}{\beta^2} \right)^{n-1} - \left( \frac{\alpha^2}{\beta^2} \right)^{n-1} \right].
\label{eq-gmc-rate-moments}\end{aligned}$$
We enumerate all the possible regimes ($n \rightarrow +\infty)$, which all give rise to degenerate stationary distributions for different reasons:
- $\beta > \sqrt{\alpha(\alpha+1)}$: both mean and variance go to zero;
- $\beta = \sqrt{\alpha(\alpha+1)}$: variance converges to 1, however the mean goes to zero;
- $\beta \in \left]\alpha;\sqrt{\alpha(\alpha+1)}\right[$: variance goes to infinity, mean goes to zero;
- $\beta = \alpha$: mean is equal to 1, but the variance goes to infinity;
- $\beta < \alpha$: both mean and variance go to infinity.
Each subplot of Figure \[fig-ch4-gmc-rate\] displays ten independent realizations of the chain, for a different set of parameters $(\alpha,\beta)$. As we can see, the realizations of the chain either collapse to 0, or diverge.
Hierarchical chaining with an auxiliary variable {#sec-cd-rate}
------------------------------------------------
### Model
Let us consider the following Gamma Markov chain model introduced in @cemgil2007conjugate: $$\begin{aligned}
z_{n}|h_{n-1} & \sim \text{Gamma}(\alpha_z, \beta_z h_{n-1}), \label{eq-gmc-cd-def1} \\
h_{n}|z_{n} & \sim \text{Gamma}(\alpha_h, \beta_h z_{n}). \label{eq-gmc-cd-def2}\end{aligned}$$ As it turns out, this model can also be rewritten as a multiplicative noise model: $$h_{n} = h_{n-1} \times \tilde{\phi}_n,
\label{eq-gmc-cd-noise}$$ where $\tilde{\phi}_n$ are i.i.d. random variables defined as the ratio of two independent Gamma random variables with parameters $(\alpha_h, \beta_h)$ and $(\alpha_z, \beta_z)$. The distribution of $\tilde{\phi}_n$ is actually known in closed form, namely $$\tilde{\phi}_n \sim \text{BetaPrime} \left( \alpha_h, \alpha_z, 1, \tilde{\beta} \right),
\label{eq-gmc-cd-betaprime}$$ with $\tilde{\beta} = \frac{\beta_z}{\beta_h}$ (see Appendix \[app-a\] for a definition). We have $$\begin{aligned}
\mathbb{E}(h_{n}|h_{n-1}) & = \tilde{\beta} \frac{\alpha_h}{\alpha_z-1} h_{n-1} & \text{for~} \alpha_z > 1, \\
\text{var}(h_{n}|h_{n-1}) & = \tilde{\beta}^2 \frac{\alpha_h ( \alpha_h + \alpha_z - 1)}{(\alpha_z-1)^2 (\alpha_z-2)} h^2_{n-1} & \text{for~} \alpha_z > 2.\end{aligned}$$
This model is less straightforward in its construction than the previous one, as it makes use of an auxiliary variable $z_n$ (note that a similar inverse Gamma construction was proposed as well in @cemgil2007conjugate). There are two motivations behind the introduction of this auxiliary variable:
1. Firstly, it ensures what is referred to as “positive correlation” in @cemgil2007conjugate, i.e., $\mathbb{E}(h_n|h_{n-1}) \propto h_{n-1}$ (something the model described by Eq. does as well).
2. Secondly, it ensures the so-called conjugacy of the model, i.e., the conditional distributions $p(z_{n}|h_{n-1},h_{n})$ and $p(h_{n}|z_{n},z_{n+1})$ remain Gamma distributions. Indeed, these are the distributions of interest when considering Gibbs sampling or variational inference. This property is not satistfied by the model described by Eq. (i.e., $p(h_n|h_{n-1},h_{n+1})$ is neither Gamma, nor a known distribution).
This particular chain has been used in the context of audio signal processing in @virtanen2008bayesian (under the assumption of a Poisson likelihood, which does not fit the nature of the data), and also to model the evolution of user and item preferences in the context of recommender systems [@jerfel17dynamic; @do2018gamma].
### Analysis
From Eq. , we can write: $$h_{n} = h_1 \prod_{i=2}^{n} \tilde{\phi}_{i}.$$
We have by independence of the $\tilde{\phi}_i$: $$\begin{aligned}
\mathbb{E}(h_{n}) & = h_{1} \left( \tilde{\beta} \frac{\alpha_h}{\alpha_z - 1} \right)^{n-1} \qquad \qquad \qquad \text{for~}\alpha_z > 1, \\
\text{var}(h_{n}) & = h_1^2 \tilde{\beta}^{2(n-1)} \left[
\left( \frac{\alpha_h^2}{(\alpha_z - 1)^2} + \frac{\alpha_h ( \alpha_h + \alpha_z - 1)}{(\alpha_z-1)^2 (\alpha_z-2)} \right)^{n-1} \right. \notag \\
& \left. \qquad \qquad \qquad - \left( \frac{\alpha_h^2}{(\alpha_z - 1)^2} \right)^{n-1}
\right]~\text{for~} \alpha_z > 2.\end{aligned}$$
As in the previous model, we can show that either the expectation or the variance diverges or collapses as $n \rightarrow \infty$ for every possible choice of parameters, which means that they all give rise to a degenerate stationary distribution of the chain. Each subplot of Figure \[fig-ch4-gmc-cd\] displays ten independent realizations of the chain, for a different set of parameters $(\alpha_z,\beta_z,\alpha_h,\beta_h)$. As we can see, the realizations of the chain either collapse to 0, or diverge.
Chaining on the shape parameter {#sec-shape}
-------------------------------
### Model
Let us consider a general Gamma Markov chain model with a chaining on the shape parameter: $$h_{n}|h_{n-1} \sim \text{Gamma}(\alpha h_{n-1}, \beta).
\label{eq-gmc-shape-def}$$ We have $$\mathbb{E}(h_{n}|h_{n-1}) = \frac{\alpha}{\beta}h_{n-1}, \quad
\text{var}(h_{n}|h_{n-1}) = \frac{\alpha}{\beta^2}h_{n-1}.$$
In contrast with the two models presented previously, this model cannot be rewritten as a multiplicative noise model. This model is therefore more intricate to interpret. It was introduced in @acharya15nonparametric in the context of Poisson factorization. It is mainly motivated by a computational trick that can be used when working with a Poisson likelihood, hence making a Gibbs sampling feasible in the model. The authors set the value of $\alpha$ to 1 (though the same trick can be applied for any value of $\alpha$).
### Analysis
Using the law of total expectation and total variance, it can be shown that $$\mathbb{E}(h_{n}) = h_1 \left( \frac{\alpha}{\beta} \right)^{n-1}, \quad
\text{var}(h_{n}) = h_{1} \frac{1}{\beta} \left( \frac{\alpha}{\beta} \right)^{n-1} \sum_{i=1}^{n-1} \left( \frac{\alpha}{\beta} \right)^i.$$
The discussion is hence driven by the value of $r = \frac{\alpha}{\beta}$.
- If $r < 1$, mean and variance go to zero;
- If $r = 1$, mean is fixed but variance goes to infinity (linearly);
- If $r > 1$, mean and variance go to infinity.
This chain only exhibits degenerate stationary distributions. Each subplot of Figure \[fig-ch4-gmc-shape\] displays ten independent realizations of the chain, for a different set of parameters $(\alpha,\beta)$. As we can see, the realizations of the chain either collapse to 0, or diverge.
BGAR(1) {#sec-bgar}
-------
We now discuss the first order autoregressive Beta-Gamma process of @lewis1989gamma, a stochastic process which is marginally Gamma distributed. The authors referred to the process as “BGAR(1)”. However, to the best of our knowledge, no extension to higher-order autoregressive processes exists in the time series literature. As such, from now on, we will simply refer to it as “BGAR”.
### Model
Consider $\alpha > 0$, $\beta > 0$, $\rho \in [0,1[$. The BGAR process is defined as: $$\begin{aligned}
h_1 & \sim \text{Gamma}(\alpha,\beta), \label{eq-gmc-bgar-def1} \\
h_n & = b_n h_{n-1} + \epsilon_n \qquad \text{for~} n \geq 2 \label{eq-gmc-bgar-def2},\end{aligned}$$ where $b_n$ and $\epsilon_n$ are i.i.d. random variables distributed as: $$\begin{aligned}
b_n & \sim \text{Beta}(\alpha \rho, \alpha(1-\rho)), \label{eq-gmc-bgar-def3} \\
\epsilon_n & \sim \text{Gamma}(\alpha(1-\rho), \beta) \label{eq-gmc-bgar-def4}.\end{aligned}$$ $(h_n)_{n \geq 0}$ is called the BGAR process. It is parametrized by $\alpha$, $\beta$ and $\rho$. We emphasize that the distribution $p(h_n|h_{n-1})$ is not known in closed form. Only $p(h_n|h_{n-1}, b_n)$ is known; it is a shifted Gamma distribution. The generative model may therefore be rewritten as $$\begin{aligned}
h_1 & \sim \text{Gamma}(\alpha,\beta), \label{eq-b1} \\
b_n & \sim \text{Beta}(\alpha \rho, \alpha(1-\rho)) \quad \text{for~}n \geq 2, \label{eq-b2} \\
h_n | b_n, h_{n-1} & \sim \text{Gamma}(\alpha(1-\rho), \beta, \text{loc} = b_n h_{n-1}) \label{eq-b3} \\
& \qquad \qquad \qquad \qquad \qquad~\text{for~}n \geq 2, \notag\end{aligned}$$ where the distribution in Eq. is a shifted Gamma distribution with a location parameter “loc”.
We have $$\begin{aligned}
\mathbb{E}(h_n|h_{n-1}) & = \rho h_{n-1} + \frac{\alpha(1-\rho)}{\beta}, \label{eq-gmc-bgar-cond-moment1} \\
\text{var}(h_n|h_{n-1}) & = \frac{\rho(1-\rho)}{\alpha + 1}h^2_{n-1} + \frac{\alpha(1-\rho)}{\beta^2}. \label{eq-gmc-bgar-cond-moment2}\end{aligned}$$ As we can see, BGAR(1) already differs from the three previously presented models because the conditional expectation $\mathbb{E}(h_n|h_{n-1})$ is not proportional to $h_{n-1}$ (it is an affine transformation).
### Analysis
To study the marginal distribution of the process, we recall the following lemma.
If $X \sim \text{Beta}(a,b)$ and $Y \sim \text{Gamma}(a+b,c)$ are independent random variables, then $Z = XY$ is $\text{Gamma}(a,c)$ distributed. \[lemma-ch4\]
Let $(h_n)_{n \geq 1}$ be a BGAR process. Then $h_n$ is marginally $\text{Gamma}(\alpha,\beta)$ distributed.
Follows by induction. Consider $n$ such that $h_n$ is $\text{Gamma}(\alpha,\beta)$ distributed. Then, $\epsilon_{n+1}h_n$ is $\text{Gamma}(\alpha \rho,\beta)$ distributed (Lemma \[lemma-ch4\]). Finally, $h_{n+1} = \epsilon_{n+1}h_n + b_{n+1}$ is $\text{Gamma}(\alpha,\beta)$ distributed (sum of independent Gamma random variables), which concludes the proof.
Therefore the parameters $\alpha$ and $\beta$ control the marginal distribution. The parameter $\rho$ controls the correlation between successive values, as discussed in the following proposition.
Let $(h_n)_{n \geq 1}$ be a BGAR process. Let $n$ and $r$ be two integers such that $r > 1$. We have $\text{corr}(h_n, h_{n+r}) = \rho^{r}$. \[prp2\]
See Appendix \[app-b\] for $r=1$.
Proposition \[prp2\] implies that the BGAR(1) process admits a (second order) AR(1) representation. Two limit cases of BGAR can be exhibited:
- When $\rho = 0$, the $h_n$ are i.i.d. random variables;
- When $\rho \rightarrow 1$, the process is not random anymore, and $h_n = h_1$ for all $n$ (note that $\rho = 1$ is not an admissible value).
Finally, from Eq. , we have $$\bigg( \mathbb{E}(h_n|h_{n-1}) > h_{n-1} \bigg) \Leftrightarrow \bigg( h_{n-1} < \frac{\alpha}{\beta} \bigg).$$ If $h_{n-1}$ is below the mean of the marginal distribution ($\frac{\alpha}{\beta}$), then $h_n$ will be in expectation above $h_{n-1}$, and vice-versa.
Note that BGAR is not the only Markovian process with a marginal Gamma distribution considered in the literature. We mention the GAR(1) process (first-order autoregressive Gamma process) of @gaver1980first, which is also marginally Gamma distributed. However, this particular process is piecewise deterministic, and its parameters are “coupled”: the parameters of the marginal distribution also have an influence on other properties of the model. As such, it is less suited to our problem, and will not be considered here.
Figure \[fig-ch4-bgar\] displays three realizations of the BGAR process, with parameters fixed to $\alpha = 2$ and $\beta = 1$, and a different parameter $\rho$ in each subplot. The mean of the marginal distribution is displayed in red. When $\rho = 0.5$, the correlation is weak, and no particular structure is observed. However, as $\rho$ goes to 1, the correlation becomes stronger, and we typically observe piecewise constant trajectories.
MAP estimation in temporal NMF models {#sec-map}
=====================================
We now turn to the problem of maximum a posteriori (MAP) estimation in temporal NMF models. More precisely, we assume a Poisson likelihood, that is $$v_{fn} \sim \text{Poisson}([\mathbf{WH}]_{fn}),$$ and we also assume that $\mathbf{W}$ is a deterministic variable. We consider four different models corresponding to the four temporal structures on $\mathbf{H}$ presented in Section \[sec-study\]. As such, $\mathbf{V}$ and $\mathbf{H}$ define a hidden Markov model, as displayed on Figure \[figure-hmm\].
(0,0) circle (0.5); (1.5,0) circle (0.5); (3,0) circle (0.5); (0,-1.5) circle (0.5); (1.5,-1.5) circle (0.5); (3,-1.5) circle (0.5); (0.5,0) – (1,0); (2,0) – (2.5,0); (0, -1) – (0, -0.5); (1.5, -1) – (1.5, -0.5); (3, -1) – (3, -0.5); (3.5, 0) – (3.95, 0); (-0.5, 0) – (-0.95, 0); (0,0) node[[$\mathbf{h}_{n-1}$]{}]{}; (1.5,0) node[[$\mathbf{h}_{n}$]{}]{}; (3,0) node[[$\mathbf{h}_{n+1}$]{}]{}; (0,-1.5) node[[$\mathbf{v}_{n-1}$]{}]{}; (1.5,-1.5) node[[$\mathbf{v}_{n}$]{}]{}; (3,-1.5) node[[$\mathbf{v}_{n+1}$]{}]{};
(1.5,-2.5) node [$\bullet$]{}; (1.5,-2.8) node [[$\mathbf{W}$]{}]{}; (1.5,-2.5) – (1.5,-2); (1.5,-2.5) – (0,-2); (1.5,-2.5) – (3,-2); (1.5, -2.5) – (-0.75, -2.125); (1.5, -2.5) – (3.75, -2.125);
Generally speaking, joint MAP estimation in such models amounts to minimizing the following criterion $$\begin{aligned}
C(\mathbf{W},\mathbf{H}, \boldsymbol{\beta}) & = -\log p(\mathbf{V},\mathbf{H};\mathbf{W},\boldsymbol{\beta}) \label{eq-ccc} \\
& = -\log p(\mathbf{V}|\mathbf{H};\mathbf{W}) - \sum_k \left( \log p(h_{k1}) + \sum_{n \geq 2} \log p(h_{kn}|h_{k(n-1)};\beta_k) \right),\end{aligned}$$ that is to say that the factors $\mathbf{W}$ and $\mathbf{H}$, as well as the scale hyperparameters $\boldsymbol{\beta} = [\beta_1, \dotsc, \beta_K]^{\text{T}}$, are going to be estimated (shape hyperparameters $\alpha$ or $\rho$ will be treated as fixed).
The optimization of the function $C$ is carried out with a block coordinate descent scheme over the variables $\mathbf{W}$, $\mathbf{H}$, and $\boldsymbol{\beta}$.
For the first two steps, we resort to a majorization-minimization (MM) scheme, which consists in iteratively majorizing the function $C$ (by a so-called auxiliary function, tight for some $\tilde{\mathbf{W}}$ or $\tilde{\mathbf{H}}$), and minimizing this auxiliary function instead. We refer the reader to @hunter2004tutorial for a detailed tutorial. Under this scheme, the function $C$ is non-increasing. As it turns out, only the Poisson likelihood term $-\log p(\mathbf{V}|\mathbf{H};\mathbf{W})$ needs to be majorized. This is a well-studied issue in the NMF literature. As stated in @lee2000algorithms [@fevotte2011algorithms], the function $$G_1(\mathbf{H};\tilde{\mathbf{H}}) = - \sum_{k,n} {p}_{kn} \log (h_{kn}) + \sum_{k,n} q_{k} h_{kn},
\label{eq-g1}$$ with the notations $$p_{kn} = \tilde{h}_{kn} \sum_f w_{fk} \frac{v_{fn}}{[\mathbf{W\tilde{H}}]_{fn}}, \quad q_k = \sum_f w_{fk},$$ is a tight auxiliary function of $- \log p(\mathbf{V}|\mathbf{H};\mathbf{W})$ at $\mathbf{H} = \tilde{\mathbf{H}}$. Similarly the function $$G_2(\mathbf{W};\tilde{\mathbf{W}}) = - \sum_{f,k} p'_{fk} \log (w_{fk}) + \sum_{f,k} q'_{k} w_{fk},
\label{eq-g2}$$ with the notations $$p'_{fk} = \tilde{w}_{fk} \sum_n h_{kn} \frac{v_{fn}}{[\mathbf{\tilde{W}H}]_{fn}}, \quad q'_k = \sum_n h_{kn},$$ is a tight auxiliary function of $- \log p(\mathbf{V}|\mathbf{H};\mathbf{W})$ at $\mathbf{W} = \tilde{\mathbf{W}}$.
Finally, for all considered models the function $C$ can be minimized in closed form w.r.t. the variable $\beta_k$.
Minimization w.r.t. **W**
-------------------------
The optimization w.r.t. $\mathbf{W}$ is common to all algorithms, and amounts to minimizing $G_2(\mathbf{W};\tilde{\mathbf{W}})$ only. The scale of $\mathbf{W}$ must be however be fixed in order to prevent potential degenerate solutions such that $\mathbf{W} \rightarrow + \infty$ and $\mathbf{H} \rightarrow 0$. Indeed, consider $\mathbf{W}^{\star}$ and $\mathbf{H}^{\star}$ minimizers of Eq. , and let $\boldsymbol{\Lambda}$ be a diagonal matrix with non-negative entries. Then $$\begin{aligned}
C(\mathbf{W}^{\star} \boldsymbol{\Lambda}^{-1}, \boldsymbol{\Lambda} \mathbf{H}^{\star}) & = - \log p(\mathbf{V}|\boldsymbol{\Lambda} \mathbf{H}^{\star}; \mathbf{W}^{\star} \boldsymbol{\Lambda}^{-1}) - \log p(\boldsymbol{\Lambda} \mathbf{H}^{\star}) \\
& = - \log p(\mathbf{V}|\mathbf{H}^{\star}; \mathbf{W}^{\star}) - \log p(\boldsymbol{\Lambda} \mathbf{H}^{\star}),\end{aligned}$$ and depending on the choice of the prior distribution $p(\mathbf{H})$, we may obtain $C(\mathbf{W}^{\star} \boldsymbol{\Lambda}^{-1}, \boldsymbol{\Lambda} \mathbf{H}^{\star}) < C(\mathbf{W}^{\star}, \mathbf{H}^{\star})$, i.e., a contradiction. Therefore, in the following we impose that $||\mathbf{w}_k||_1 = 1$.
The constrained optimization is performed with the following update rule $$w_{fk} = \frac{p'_{fk}}{\sum_f p'_{fk}}, \label{eq-upw}$$ see Appendix \[app-c\] for the proof.
The following subsections detail the optimization w.r.t. $\mathbf{H}$ (and other variables when necessary), which amounts to the minimization of $G_1(\mathbf{H};\tilde{\mathbf{H}}) - \log p(\mathbf{H})$, as well as the minimization of $C$ w.r.t. $\beta_k$, for each considered model
Chaining on the rate parameter {#chaining-on-the-rate-parameter}
------------------------------
The transition distribution $p(h_{kn}|h_{k(n-1)})$ is given by Eq. . The optimization w.r.t. $h_{kn}$ amounts to solving an order-2 polynomial equation $$a_{2,{kn}} h_{kn}^2 + a_{1,{kn}} h_{kn} + a_{0,{kn}} = 0.
\label{eq-poly-1}$$ The coefficients of the polynomial equation are given in Table \[table-1\]. This bears resemblance with the methodology described in @fevotte2009nonnegative, where the authors aimed at retrieving MAP estimates with a EM-like algorithm (with an exponential likelihood).
[cccc]{} $n$ & $a_{2,{kn}}$ & $a_{1,{kn}}$ & $a_{0,{kn}}$\
$1$ & $q_{k}$ & $\alpha - p_{1k}$ & $-\beta_k h_{k2}$\
\
$2,\dotsc,N-1$ & $q_k + \frac{\beta_k}{h_{k(n-1)}}$ & $1 - p_{kn}$ & $- \beta_k h_{k(n+1)}$\
\
$N$ & 0 & $q_k$ + $\frac{\beta}{h_{k(N-1)}}$ & $1- \alpha - p_N$\
The update for the hyperparameter $\beta_k$ is given by $$\beta_k = \frac{\alpha (N-1)}{\sum_{n \geq 2} \frac{h_{kn}}{h_{k(n-1)}}}.$$
Hierarchical chaining with an auxiliary variable {#hierarchical-chaining-with-an-auxiliary-variable}
------------------------------------------------
In this case, since the transition distribution $p(h_{kn}|h_{k(n-1)})$ is not known in closed form, we resort to optimizing the slighlty more involved following criterion $$\begin{aligned}
& C(\mathbf{W},\mathbf{H},\mathbf{Z}, \boldsymbol{\beta}_h, \boldsymbol{\beta}_z) = \notag \\
& -\log p(\mathbf{V}|\mathbf{H};\mathbf{W}) - \sum_k \bigg[ \log p(h_{k1}) + \sum_{n \geq 2} \left( \log p(z_{kn}|h_{k(n-1)};\beta_{z,k}) + \log p(h_{kn}|z_{kn};\beta_{h,k}) \right) \bigg],\end{aligned}$$ where $\boldsymbol{\beta}_h = [\beta_{h,1}, \dotsc, \beta_{h,K}]^{\text{T}}$ and $\boldsymbol{\beta}_z = [\beta_{z,1}, \dotsc, \beta_{z,K}]^{\text{T}}$. We recall that $p(z_{kn}|h_{k(n-1)})$ and $p(h_{kn}|z_{kn})$ are given by Eq. and Eq. , respectively. Note that @cemgil2007conjugate proposed a Gibbs sampler and variational inference, and as such the development of the MAP algorithm is novel.
The update for $z_{kn}$ is given by $$z_{kn} = \frac{\alpha_{z} + \alpha_{h} - 1}{\beta_{z,k} h_{k(n-1)} + \beta_{h,k} h_{kn}}.$$ As for the updates for $h_{kn}$, they are given by $$\begin{aligned}
h_{k1} & = \frac{p_{k1} + \alpha_{z}}{q_k + \beta_{z,k} z_{k2}}, \\
h_{kn} & = \frac{p_{kn} + \alpha_{h} + \alpha_{z} - 1}{q_k + \beta_{h,k} z_{kn} + \beta_{z,k} z_{k(n+1)}} \qquad n \in \{ 2,\dotsc, N \}, \\
h_{kN} & = \frac{p_{kN} + \alpha_{h} - 1}{q_k + \beta_{h,k} z_{kN}}.\end{aligned}$$
As such, imposing $\alpha_{k} \geq 1$ is a sufficient condition to preserve the non-negativity of all the updates. Finally, the update for the parameters $\beta_{z,k}$ and $\beta_{h,k}$ are given by $$\beta_{z,k} = \frac{(N-1)\alpha_{z}}{\sum_{n \geq 2} h_{k(n-1)} z_{kn}}$$ and $$\beta_{h,k} = \frac{(N-1)\alpha_h}{\sum_{n \geq 2} h_{kn} z_{kn}}.$$
Chaining on the shape parameter {#chaining-on-the-shape-parameter}
-------------------------------
The transition distribution $p(h_{kn}|h_{k(n-1)})$ is given by Eq. . The optimization w.r.t. $h_{kn}$ amounts to solving the following equations $$-p_{k1} + (q_k - \alpha \log(\beta_k h_{k2}) + \alpha \Psi(\alpha h_{k1}) )h_{k1} = 0,$$ $$\begin{aligned}
(1-\alpha h_{k(n-1)}-p_{kn}) & + (q_k + \beta_k - \alpha \log(\beta_k h_{k(n+1)})) h_{kn} + \alpha \Psi(\alpha h_{kn}) h_{kn} = 0,\end{aligned}$$ where $\Psi$ denotes the digamma function. Solving such equations can be done numerically with Newton’s method. Finally the update for $h_{kN}$ is given by $$h_{kN} = \frac{p_{kn} + \alpha h_{k(N-1)} -1}{q_k + \beta_k}.$$ The update for $\beta_k$ is given by $$\beta_k = \alpha \frac{\sum_{n \geq 2} h_{k(n-1)}}{\sum_{n \geq 2} h_{kn}}.$$
Note that a Gibbs sampling procedure is proposed in @acharya15nonparametric [@schein2016poisson], and as such the development of the MAP algorithm is novel.
BGAR(1) {#alg-map-bgar}
-------
In this case, since the transition distribution $p(h_{kn}|h_{k(n-1)})$ is not known in closed form, we resort to optimizing the slightly more involved following criterion $$\begin{aligned}
& C(\mathbf{W},\mathbf{H},\mathbf{B}, \boldsymbol{\beta}) = \notag \\
& -\log p(\mathbf{V}|\mathbf{H};\mathbf{W}) - \sum_k \bigg( \log p(h_{k1})~+ \sum_{n \geq 2} \left( \log p(h_{kn}|h_{k(n-1)},b_{kn};\beta_k) + \log p(b_{kn}) \right) \bigg) \label{eq-map-bgar}.\end{aligned}$$
In the following, we will use the notations $\gamma_k = \alpha_k(1-\rho_k)$ and $\eta_k = \alpha_k \rho_k$.
### Constraints
By construction, the variables $h_{kn}$ and $b_{kn}$ must lie in a specific interval given the values of all the other variables. Indeed, as $h_{kn} = b_{kn} h_{k(n-1)} + \epsilon_{kn}$ (Eq. ), where $\epsilon_{kn}$ is a non-negative random variable, we obtain $h_{kn} \geq b_{kn} h_{k(n-1)}$, $b_{kn} \leq \frac{h_{kn}}{h_{k(n-1)}}$, and $h_{kn} \leq \frac{h_{k(n+1)}}{b_{k(n+1)}}$.
This leads to the following constraints $$\begin{aligned}
0 & \leq h_{k1} \leq \frac{h_{k2}}{b_{k2}}, \\
b_{kn} h_{k(n-1)} & \leq h_{kn} \leq \frac{h_{k(n+1)}}{b_{k(n+1)}} \qquad 2 \leq n < N, \\
b_{kN} h_{k(N-1)} & \leq h_{kN},\end{aligned}$$ and $$0 \leq b_{kn} \leq \min \left( 1, \frac{h_{kn}}{h_{k(n-1)}} \right).$$
We therefore introduce the notations $$\begin{aligned}
c_{kn} = b_{kn} h_{k(n-1)}, \quad d_{kn} = \frac{h_{k(n+1)}}{b_{k(n+1)}}, \quad x_{kn} = \frac{h_{kn}}{h_{k(n-1)}},\end{aligned}$$ as these quantities arise naturally in our derivations.
### Minimization w.r.t. $h_{kn}$
The optimization of Eq. w.r.t. $h_{kn}$ may give rise to intractable problems, due to the logarithmic terms in the objective function. To alleviate this issue, we propose to control the limit values of the auxiliary function, by restricting ourselves to certain values of the hyperparameters. In particular, choosing $(1-\gamma_k) < 0$ ensures the existence of at least one minimizer.
In all sub-cases, the optimization w.r.t. $h_{kn}$ amounts to solving an order-3 polynomial equation $$a_{3,{kn}} h_{kn}^3 + a_{2,{kn}} h_{kn}^2 + a_{1,{kn}} h_{kn} + a_{0,{kn}} = 0.
\label{eq-poly-2}$$ The coefficients of the polynomial equation are given in Table \[table-2\]. If several roots belong to the definition interval, we simply choose the root which gives the lowest objective value.
### Minimization w.r.t. $b_{kn}$
Similarly, logarithmic terms of the objective function may give rise to degenerate solutions. Using the same reasoning, we choose to impose $(1-\gamma_k) < 0$ and $(1-\eta_k) < 0$ to ensure the existence of at least one minimizer.
The minimization of the auxiliary function w.r.t. $b_{kn}$ amounts to solving the following order 3 polynomial over the interval $[0, \min(1, x_{kn})]$ $$a_{3,kn} b_{kn}^3 + a_{2,kn} b_{kn}^2 + a_{1,kn} b_{kn} + a_{0,kn} d_{kn},$$ where $$\begin{aligned}
a_{3,kn} & = -\beta_k h_{k(n-1)}, \\
a_{2,kn} & = 2(1-\gamma_k) + (1-\eta_k) + \beta_k h_{k(n-1)}(x_{kn}+1), \\
a_{1,kn} & = -(1-\gamma_k)(x_{kn}+1) - (1-\eta_k)(x_{kn} + 1) - \beta_k h_{k(n-1)} x_{kn}, \\
a_{0,kn} & = (1-\eta_k) x_{kn}.\end{aligned}$$
### Minization w.r.t. $\beta_k$
The minimization of $C$ w.r.t. $\beta_k$ can be done in closed form and results in the following update rule $$\beta_k = \frac{(N-1) \alpha (1-\rho)}{\sum_{n \geq 2} (h_{kn} - b_{kn}h_{k(n-1)})}.$$
### Admissible values of hyperparameters
To recap the discussion on admissible values of hyperparameters, to ensure the existence of minimizers of the auxiliary function, we have restricted ourselves to $$\left\{
\begin{array}{l}
\alpha_k(1-\rho_k) > 1 \\
\alpha_k \rho_k > 1
\end{array}
\right.$$ This set is graphically displayed on Figure \[fig-admv\]. As we can see, choosing the value of $\rho_k$ to be close to one (to ensure correlation) leads to high values of $\alpha_k$.
Experimental work {#sec-exp}
=================
We now compare the performance of all considered temporal NMF models on a prediction task on three real datasets. This task will consist in hiding random columns of the considered datasets and predicting those missing values. We will also include the performance of a naive baseline, which we detail in the following subsection. Adapting the MAP algorithms presented in Section \[sec-study\] in a setting with a mask of missing values only consist in a slight modification, presented in Appendix \[app-d\]. Python code will be made available upon acceptance.
Experimental protocol
---------------------
For each considered dataset, the experimental protocol is as follows.
First of all, a value of the factorization rank $K$ (which will be used for all considered methods) must be selected. To do so, we apply the standard KL-NMF algorithm [@lee2000algorithms; @fevotte2011algorithms] on 10 random training sets, which consist of $80 \%$ of the original data, with a pre-defined grid of values for $K$. We then select the value of $K$ which yields the lowest generalized Kullback-Leibler error (KLE) (see definition below) on the remaining $20 \%$ of the data.
For the prediction experiment itself, we create 5 random splits of the data matrix, where $80 \%$ corresponds to the training set, $10 \%$ to the validation set, and the remaining $10 \%$ to the test set. To do so, we randomly select non-adjacent columns of the data matrix (excluding the first one and the last one), half of which will make up the validation set and the other half the test set. We also consider 5 different random initializations.
Thus, for each split-initialization pair, all the algorithms are run from this initialization point on the training set until convergence (the algorithms are stopped when the relative decrease of the objection function falls under $10^{-5}$). For each method, a grid of shape hyperparameters is considered, and the selection of this parameter is based on the lowest KLE on the validation set. The predictive performance of each method is then computed on the test set by comparing the original value $v_{fn}$ and its associated estimate $\hat{v}_{fn} = [\mathbf{WH}]_{fn}$ with two different metrics. Denoting by $\mathcal{T}$ the test set, we consider
- the generalized Kullback-Leibler error (KLE) $$\text{KLE} = \sum_{(f,n) \in \mathcal{T}} \left[ v_{fn} \log \left( \frac{v_{fn}}{\hat{v}_{fn}} \right) - v_{fn} + \hat{v}_{fn} \right];$$
- the relative error, as in @schein2016poisson $$\text{RE} = \sum_{(f,n) \in \mathcal{T}} \frac{|v_{fn} - \hat{v}_{fn}|}{v_{fn} + 1}.$$
Finally, we compare the NMF-based approaches to the following naive baseline, based on a random guess. In this case, the values of the missing columns are simply estimated by drawing $\hat{v}_{fn}$ from the empirical distribution of the observed data coefficient in every row $f$.
Datasets
--------
The following datasets are considered
- The `NIPS` dataset[^3], which contains word counts (with stop words removed) of all the articles published at the NIPS[^4] conference between 1987 and 2015. We grouped the articles per year, yielding an observation matrix of size $11463 \times 29$. We obtained $K = 3$.
- The `ICEWS` dataset[^5], an international relations dataset, which contains the number of interactions between two countries for each day of the year 2003. The matrix is of size $6197 \times 365$. We obtained $K = 15$.
- The `last.fm` dataset, based on the so-called “last.fm 1K” users[^6], which contains the listening history of users with timestamps information. We preprocessed this dataset to obtain the monthly evolution of the listening counts of artists with at least 20 different listeners. This yields a dataset of size $7017 \times 53$. We obtained $K = 6$.
Experimental results
--------------------
The averaged KLE and RE over the 25 split-initialization pairs are reported on Table \[table-nips\] for the `NIPS` dataset, on Table \[table-icews\] for the `ICEWS` dataset, and on Table \[table-last\] for the `last.fm` dataset.
First of all, on all the considered datasets, the naive baseline yields the worst performance results, for all metrics, as expected. Moreover, all temporal models achieve comparable predictive performance, and the slight advantage of some methods over the others being data-dependent.
Model KLE RelE
--------------------- -------------------------------------------------- --------------------------------------------------
Baseline $14.8 \times 10^5 \pm 57.4 \times 10^4$ $7.45 \times 10^4 \pm 17.9 \times 10^3$
Rate (II.A) $\mathbf{1.07 \times 10^5 \pm 2.01 \times 10^4}$ $\mathbf{2.98 \times 10^4 \pm 2.53 \times 10^3}$
Hierarchical (II.B) $\mathbf{1.07 \times 10^5 \pm 2.02 \times 10^4}$ $\mathbf{2.98 \times 10^4 \pm 2.45 \times 10^3}$
Shape (II.C) $1.29 \times 10^5 \pm 2.75 \times 10^4$ $\mathbf{3.07 \times 10^4 \pm 4.07 \times 10^3}$
BGAR (II.D) $\mathbf{1.05 \times 10^5 \pm 1.74 \times 10^4}$ $\mathbf{3.02 \times 10^4 \pm 2.00 \times 10^3}$
Model KLE RelE
--------------------- -------------------------------------------------- --------------------------------------------------
Baseline $99.1 \times 10^4 \pm 44.2 \times 10^3$ $3.33 \times 10^4 \pm 4.06 \times 10^2$
Rate (II.A) $\mathbf{8.68 \times 10^4 \pm 2.95 \times 10^3}$ $\mathbf{2.65 \times 10^4 \pm 5.93 \times 10^2}$
Hierarchical (II.B) $\mathbf{8.80 \times 10^4 \pm 3.39 \times 10^3}$ $\mathbf{2.62 \times 10^4 \pm 7.80 \times 10^2}$
Shape (II.C) $\mathbf{8.79 \times 10^4 \pm 3.31 \times 10^3}$ $\mathbf{2.61 \times 10^4 \pm 6.90 \times 10^2}$
BGAR (II.D) $9.09 \times 10^4 \pm 4.16 \times 10^3$ $\mathbf{2.54 \times 10^4 \pm 5.30 \times 10^2}$
Model KLE RelE
--------------------- -------------------------------------------------- --------------------------------------------------
Baseline $66.5 \times 10^4 \pm 1790 \times 10^2$ $3.40 \times 10^4 \pm 89.9 \times 10^2$
Rate (II.A) $\mathbf{1.55 \times 10^4 \pm 7.13 \times 10^2}$ $\mathbf{1.18 \times 10^4 \pm 5.55 \times 10^2}$
Hierarchical (II.B) $\mathbf{1.57 \times 10^4 \pm 6.18 \times 10^2}$ $\mathbf{1.17 \times 10^4 \pm 6.06 \times 10^2}$
Shape (II.C) $2.01 \times 10^4 \pm 51.9 \times 10^2$ $\mathbf{1.19 \times 10^4 \pm 7.16 \times 10^2}$
BGAR (II.D) $\mathbf{1.59 \times 10^4 \pm 9.10 \times 10^2}$ $\mathbf{1.20 \times 10^4 \pm 6.86 \times 10^2}$
Conclusion {#sec-ccl}
==========
In this paper, we have reviewed existing temporal NMF models in a unified MAP framework and introduced a new one. These models differ by the choice of the Markov chain structure used on the activation coefficients to induce temporal correlation. We began by studying the previously proposed Gamma Markov chains of the NMF literature, only to find that they all share the same drawback, namely the absence of a well-defined stationary distribution. This leads to problematic behaviors from the generative perspective, because the realizations of the chains are degenerate (although this is not necessarily a problem in MAP estimation). We then introduced a Markovian process from the time series literature, called BGAR(1), which overcomes this limitation, and which, to the best of our knowledge, had never been exploited for learning tasks.
We then derived MAP estimation algorithms for all these models, in the context of a Poisson likelihood, which allowed for a comprehensive comparison on a prediction task on real datasets. As it turns out, we cannot claim that there is a single model which outperforms all the others. Their strengths and weaknesses appear to depend on the nature of the data at hand, which is reasonable.
Future work will focus on finding a way to perform inference with the BGAR prior for a less restrictive set of hyperparameters, which might increase the performance of this particular model. We will also work to derive similar algorithms in the context of different likelihoods, such as an exponential likelihood [@fevotte2009nonnegative], which can be easily done thanks to the MM framework.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under grant agreement No 681839 (project FACTORY).
The Beta-Prime distribution {#app-a}
===========================
Distribution for a continuous random variable in $[0,+\infty[$, with parameters $\alpha > 0$, $\beta > 0$, $p > 0$ and $q > 0$. Its p.d.f. writes, for $x \geq 0$: $$f(x;\alpha,\beta,p,q) = \frac{p \left( \frac{x}{q} \right)^{\alpha p -1} \left(1 + \left(\frac{x}{q}\right)^p \right)^{-\alpha - \beta}}{q \text{B}(\alpha, \beta)}.$$
BGAR(1) linear correlation {#app-b}
==========================
We have between two successive values $h_n$ and $h_{n+1}$: $$\begin{aligned}
& \text{corr}(h_n, h_{n+1}) \\
& = \frac{\mathbb{E}(h_n h_{n+1}) - \mathbb{E}(h_n)\mathbb{E}(h_{n+1})}{\sigma(h_n)\sigma(h_{n+1})} \\
& = \frac{\mathbb{E}(h_n(b_{n+1}h_n + \epsilon_{n+1})) - \mathbb{E}(h_n)\mathbb{E}(h_{n+1})}{\sigma(h_n)\sigma(h_{n+1})} \\
& = \frac{\mathbb{E}(b_{n+1})\mathbb{E}(h_n^2) + \mathbb{E}(h_n)\mathbb{E}(\epsilon_{n+1}) - \mathbb{E}(h_n)\mathbb{E}(h_{n+1})}{\sigma(h_n)\sigma(h_{n+1})} \\
& = \frac{\frac{\alpha \rho}{\alpha \rho + \alpha(1-\rho)} \frac{\alpha(\alpha+1)}{\beta^2} + \frac{\alpha}{\beta} \frac{\alpha(1-\rho)}{\beta} - \frac{\alpha}{\beta} \frac{\alpha}{\beta} }{\frac{\alpha}{\beta^2}} \\
& = \rho.\end{aligned}$$
Constrained optimization {#app-c}
========================
We want to optimize $G_2(\mathbf{W};\tilde{\mathbf{W}})$ w.r.t. $\mathbf{W}$ s.t. $\sum_f{w_{fk}} = 1$. Rewriting this with Lagrange multipliers $\boldsymbol{\lambda} = [\lambda_1,\dotsc,\lambda_{K}]^{\text{T}}$, this is tantamount to $$\min_{\mathbf{W}, \boldsymbol{\lambda}} G_2(\mathbf{W};\tilde{\mathbf{W}}) + \sum_k \lambda_k (||\mathbf{w}_k||_1 - 1).$$ Deriving w.r.t $w_{fk}$ yields $$w_{fk} = \frac{p'_{fk}}{q'_k + \lambda_k}. \label{eq-lagr}$$ We retrieve the constraint by summing this expression over $f$. This gives the expression of the Lagrange multiplier: $\lambda_k = \sum_f p'_{fk} - q'_k$. Substituting this expression into Eq. , we obtain the following update rule $$w_{fk} = \frac{p'_{fk}}{\sum_f p'_{fk}}.$$
Algorithms with missing values {#app-d}
==============================
In the context of missing values, let us consider a mask matrix $\mathbf{M}$ of size $F \times N$ such that $m_{fn} = 1$ if the entry $v_{fn}$ is observed and 0 otherwise. The likelihood term can then be written as $$-\log p(\mathbf{V}|\mathbf{H};\mathbf{W}) = - \sum_{f,n} m_{fn} \log p(v_{fn}|[\mathbf{WH}]_{fn}).$$ The auxiliary function $G_1$ of Eq. and $G_2$ of Eq. can then be written is the same way, with $$p_{kn} = \tilde{h}_{kn} \sum_f w_{fk} \frac{m_{fn} v_{fn}}{[\mathbf{W\tilde{H}}]_{fn}}, \quad q_{kn} = \sum_f m_{fn} w_{fk},$$ for $G_1$, and $$p'_{fk} = \tilde{w}_{fk} \sum_n h_{kn} \frac{m_{fn} v_{fn}}{[\mathbf{\tilde{W}H}]_{fn}}, \quad q'_{kn} = \sum_n m_{fn} h_{kn},$$ for $G_2$.
[^1]: These models are sometimes generically referred to as “Poisson factorization” or “Poisson factor analysis”.
[^2]: Note that this particular mapping is actually slightly more complex, as the $K$-dimensional real vector must be mapped to the $(K-1)$ simplex due to further constraints in the model.
[^3]: <https://archive.ics.uci.edu/ml/datasets/NIPS+Conference+Papers+1987-2015>
[^4]: Now called NeurIPS.
[^5]: <https://github.com/aschein/pgds>
[^6]: <http://ocelma.net/MusicRecommendationDataset/>
|
---
author:
- |
Petr Hořava and Cynthia A. Keeler\
Berkeley Center for Theoretical Physics and Department of Physics\
University of California, Berkeley, CA, 94720-7300\
and\
Theoretical Physics Group, Lawrence Berkeley National Laboratory\
Berkeley, CA 94720-8162, USA
bibliography:
- 'alpha2.bib'
title: |
Strings on $AdS_2$ and the High-Energy Limit\
of Noncritical M-Theory
---
Introduction
============
Noncritical string theories in $1+1$ dimensions (see [@Nakayama:2004vk; @Ginsparg:1993is; @Klebanov:1991qa; @Alexandrov:2003ut; @Martinec:2004td] for reviews) have long been a useful playground for studying stringy physics. Unlike their ten-dimensional cousins, two-dimensional string theories are exactly solvable; thus, questions which are difficult to study in full string theory prove themselves more approachable in the theater of noncritical strings.
In this paper, we use the setting of noncritical theories to examine some of the mysteries of M-theory. In the full critical string case, we know very little about full M-theory beyond the use of either nonperturbative dualities, or the low-energy limit as described by eleven-dimensional supergravity. Following the resurgence of interest in two-dimensional Type 0A and 0B string theories [@McGreevy:2003kb; @Douglas:2003up; @Takayanagi:2003sm], in [@Horava:2005tt; @Horava:2005wm] we proposed a nonperturbative definition of [*noncritical M-theory*]{} in $2+1$ dimensions, related to Type 0A and 0B strings in two dimensions by a string/M-theory duality. The definition of noncritical M-theory as given in [@Horava:2005tt] is in terms of a double-scaling limit of a nonrelativistic Fermi liquid on a rigid two-dimensional plane. In the double scaling limit, the number of fermions $N$ goes to infinity and the potential felt by the fermions becomes that of an an inverted harmonic oscillator. Various ways of filling a Fermi sea correspond to various classical solutions of the theory. This is also how two-dimensional Type 0A and 0B string vacua with a linear dilaton and RR flux are reproduced as solutions of noncritical M-theory: we recover their nonperturbative description as particular Fermi liquids of matrix-model eigenvalues. In this correspondence, the role of the “extra” dimension of M-theory is played by the angular coordinate on the plane populated by the fermions: the Type 0A D0-brane charge is identified with the KK momentum along the extra dimension ([*i.e.*]{}, the angular momentum on the plane), mimicking the well-known correspondence from the critical case.[^1]
In addition to the two-dimensional string vacua, noncritical M-theory also contains a natural ground state, which we term the ${|M\rangle}$ state [@Horava:2005tt]. This state is the noncritical analog of the eleven-dimensional M-theory vacuum solution. As was shown in [@Horava:2005tt; @Horava:2005wm], the ${|M\rangle}$ state exhibits many features expected of a $2+1$ dimensional spacetime solution. However, its description in terms of an effective gravity theory in a dynamical $2+1$-dimensional spacetime remains unknown. In the related case of two-dimensional strings, the relation between the physical spacetime and the space populated by the fermions is quite subtle [@Ginsparg:1993is]. The time dimension is the same between the two pictures. However, the spatial Liouville dimension $x$ of the linear dilaton background is related to the spatial eigenvalue dimension $\lambda$ by an intricate integral transform, which can be viewed as an early form of string duality. In noncritical M-theory, the situation is worse: so far, only the fermionic description has been developed, and how it maps to a physical spacetime picture is not yet understood. It is the purpose of this paper to remedy this situation, and provide further evidence that the ground state of noncritical M-theory does indeed correspond to spacetime physics in $2+1$ dimensions, with an effective gravity description. We will address this problem in the conformal limit of the theory, where our analysis will be facilitated by the larger symmetries of the system.
In string theory, the conformal limit [@Strominger:2003tm; @Ho:2004qp; @Aharony:2005hm] corresponds to sending $\alpha'\to\infty$, which can also be interpreted as a high-energy limit [@Gross:1987kz; @Gross:1987ar; @Witten:1988zd] (see also [@Amati:1987uf; @Amati:1987wq; @Amati:1988tn]). This fact provides another motivation for this paper: taking the high-energy limit of a mysterious theory in order to learn more about its underlying degrees of freedom is a classic strategy, pursued in string theory since its early days. It has been widely speculated that in the high-energy limit, the theory might reveal an “unbroken phase” in which the massive string modes become massless [@Gross:1988ue; @Witten:1988zd]. Alternatively, one could probe the underlying degrees of freedom by heating the system to high temperature [@Atick:1988si]. We applied this latter strategy to noncritical M-theory in [@Horava:2005wm], and found a surprising connection between thermal noncritical M-theory and the topological strings of the A-model on the resolved conifold. In this correspondence, the radius of the Euclidean time circle ([*i.e.*]{}, the inverse temperature) on the M-theory side plays the role of the A-model string coupling, a relation expected of topological M-theory [@Dijkgraaf:2004te]. In the present paper, we complement the analysis of [@Horava:2005wm] and begin to probe the ground state of noncritical M-theory in another extreme regime, of high energies.
This paper is organized as follows. After providing a quick review of noncritical M-theory in Section \[non\], and of the Type 0A conformal limit in Section \[0A\], we will explore the same limit in the M-theory case in Section \[mlimit\], and argue that it describes an $AdS_2\times S^1$ spacetime. The spectrum of propagating modes corresponds to the quanta of a single massless Dirac fermion on this background. In Section \[spacetime\], we address the question of an effective description of this system on the spacetime side. First we embed the $AdS_2\times S^1$ as a vacuum solution to conformal $SO(3,2)$ Chern-Simons gravity in $2+1$ dimensions. Then we extend the theory to a higher-spin Chern-Simons gauge theory, which not only incorporates the infinite symmetry of noncritical M-theory, but also allows a coupling to the propagating fermionic matter using the “unfolded formalism” of Vasiliev [*et al.*]{} [@Vasiliev:1992gr; @Vasiliev:1992ix; @Shaynkman:2001ip] (see also [@Bekaert:2005vh] for a review). Finally, we present our conclusions in Section 6, together with the amusing observation that from the spacetime point of view, the underlying Fermi liquid system can be viewed as living on twistor space associated with the conformal group $SO(3,2)$ of the $2+1$-dimensional dynamical spacetime.
Review of Noncritical M-Theory {#non}
==============================
Definition as a Fermi Liquid
----------------------------
Following [@Horava:2005tt], we define noncritical M-theory by starting with a regulated inverted harmonic oscillator potential on a two-dimensional plane ${{\bf R}}^2$, with coordinates $\lambda_i$, $i=1,2$, filled with $N$ fermions. The two-dimensional plane carries a fixed flat metric $$ds^2=d\lambda_1^2+d\lambda_2^2.$$ This metric is not dynamical. The dynamical spacetime with a fluctuating metric field emerges as an effective structure associated with a particular solution of the theory. This is exactly parallel to the case of two-dimensional string theory, wherein the eigenvalues of the matrix model live on a rigid space related to the spacetime Liouville dimension by an integral transform [@Ginsparg:1993is].
Then we take a double-scaling limit, simultaneously reducing the potential to an inverted harmonic oscillator while taking the number of fermions to infinity. When the result of this process is written in the second-quantized language, with $\Psi (\lambda_i,t)$ a spinless fermion field, the appropriate action takes the nonrelativistic form $$S=\int dt d^2\lambda \left(i\Psi^\dag \frac{\partial\Psi}{\partial t}
-\frac{1}{2}\sum_{i=1,2}\frac{\partial\Psi^\dag}{\partial\lambda_i}
\frac{\partial\Psi}{\partial\lambda_i}
+\frac{1}{2}\omega_0^2\sum_{i=1,2}\lambda_i^2\Psi^\dag\Psi+\ldots\right),$$ where $\omega_0$ is the fundamental frequency of the theory, and “$\ldots$” stand for nonuniversal regulating terms in the potential that are scaled away in the double-scaling limit.[^2] In the string-theory solutions of noncritical M-theory [@Horava:2005tt], $\omega_0$ is related to $\alpha'$ by $$\omega_0=\frac{1}{\sqrt{2\alpha'}}.$$ In order to simplify our terminology, we will frequently refer to $1/(2\omega_0^2)$ as $\alpha'$ in the case of the M-theory vacuum as well.
We will later also use the first quantized action, which is simply given by $$\label{Maction}
S=\frac{1}{2}\int dt
\sum_{i=1,2}\left(\dot{\lambda}_i^2+\omega_0^2\lambda_i^2\right).$$ As explored in [@Horava:2005tt], the richness of noncritical M-theory comes from the freedom to pick any $N$ states to fill with fermions. However, there is still a most natural second quantized ground state. We construct this state by filling every fermion with individual energy below $-\mu$, while everything with higher energy is kept empty. This is the natural M-theory vacuum solution, and we will call it ${|M\rangle}$. Its properties depend on the (double-scaled) value of the Fermi energy $\mu$, which plays the role of a coupling constant in the M-theory vacuum [@Horava:2005tt; @Horava:2005wm].
Embedding of the Type 0A String
-------------------------------
In addition to the M-theory state, the vacua of two-dimensional Type 0A and 0B theories string theories are also solutions of noncritical M-theory as defined via the Fermi liquid system. Here we will concentrate on the embedding of the Type 0A linear dilaton vacuum with RR flux (see [@Horava:2005tt; @Horava:2005wm] for 0B). In order to find the Type 0A state in noncritical M-theory, we first change variables from $\lambda_i$ to the polar coordinates $\lambda$ and $\theta$ (with $\lambda$ the radial coordinate). Elementary separation of variables allows us to label the fermion quanta by their angular momentum $q$, and then discuss only the dependence on the radial coordinate, $\lambda$. Thus, we are left with a set of one-dimensional fermions, labelled by $q$, each living in the following potential: $$\label{potential}
V(\lambda)=-\frac{1}{2}\omega_0^2\lambda^2
+\frac{M}{2\lambda^2},$$ where $M=q^2-1/4$. The ground state describing the Type 0A vacuum with $q_0$ units of RR flux simply corresponds to placing all $N$ fermionic quanta in the lowest $N$ states with $q=q_0$ and taking the double scaling limit. (In the Type 0A language, the individual fermion corresponds to the open-string mode of a D0-$\overline{\rm D0}$ pair, and $q_0$ is the excess DO-brane charge.) Since this solution of noncritical M-theory has been prepared such that all single-particle states with $q\neq q_0$ are kept empty in this ground state, all excitations in sectors with $q\neq q_0$ are infinitely energetic with respect to this ground state and therefore decouple, leaving precisely the excitations of the two-dimensional Type 0A vacuum at $q_0$ units of RR flux.
This embedding of the Type 0A vacua as solutions in noncritical M-theory sheds new light on the M-theory ground state solution ${|M\rangle}$. Indeed, we can think of ${|M\rangle}$ as a coherent sum of Type 0A vacua for all values of RR flux. All single-particle (or hole) excitations are now finitely energetic with respect to the Fermi surface of ${|M\rangle}$, and represent physical excitations of the ${|M\rangle}$ state.
This formal decomposition of the ground state of noncritical M-theory into Type 0A sectors is useful technically, for example in the evaluation of the vacuum energy of the solution [@Horava:2005tt; @Horava:2005wm]. It also leads to a crucial “correspondence principle”: because the finite-energy excitations of every individual Type 0A sector carry finite energy in the ${|M\rangle}$ state, the effective 0A physics of each sector must be reproduced by the properties of the ${|M\rangle}$ state as well. As one application of this correspondence principle, one can argue that since the vacua of Type 0A string theory can be described by an effective action containing two-dimensional gravity, the ${|M\rangle}$ state should have a gravitational description also. We will use this correspondence as one of our guiding principles throughout this paper.
Symmetries of the ${|M\rangle}$ and ${|0A\rangle}$ States
---------------------------------------------------------
Now that we have reviewed the basic noncritical M-theory setup, let us discuss a few implications of its symmetries. The theory has an underlying infinite-dimensional symmetry algebra ${{\cal W}}$, first discussed in Section 8.2 of [@Horava:2005tt]. This algebra is the M-theory analog of the $w_\infty$ symmetry algebras known from two-dimensional string theories [@Ginsparg:1993is]. ${{\cal W}}$ arises from four basic conserved charges, given in the classical limit by $${\mbox{\sl a}}_i=\frac{1}{\sqrt{2}}(p_i+\omega_0\lambda_i)e^{-\omega_0t},\qquad
{\mbox{\sl b}}_i=\frac{1}{\sqrt{2}}(p_i-\omega_0\lambda_i)e^{\omega_0t}.$$ Here $\lambda_i$ with $i=1,2$ are again the Cartesian coordinates on ${{\bf R}}^2$, and $p_i$ are the conjugate momenta. $a_i$ and $b_j$ satisfy commutation relations implied by the canonical Poisson brackets between the momenta $p_i$ and coordinates $\lambda_j$. The four charges $(a_i,b_j)$ form a natural coordinate system on the phase space ${{\cal T}}={{\bf R}}^4$ of the fermions.
The full algebra ${{\cal W}}$ has a basis consisting of the Weyl-ordered products of an arbitrary finite number of ${\mbox{\sl a}}_i$ and ${\mbox{\sl b}}_j$. We can assign a “degree” to the elements of this basis, simply defined as the total degree of the corresponding monomial in ${\mbox{\sl a}}_i$ and ${\mbox{\sl b}}_j$. Linear combinations of the ten independent charges of degree two, $${\mbox{\sl a}}_1^2,\ {\mbox{\sl a}}_2^2,\ {\mbox{\sl b}}_1^2,\ {\mbox{\sl b}}_2^2,\ {\mbox{\sl a}}_1{\mbox{\sl a}}_2,\ {\mbox{\sl b}}_1{\mbox{\sl b}}_2,\
{\mbox{\sl a}}_1{\mbox{\sl b}}_2,\ {\mbox{\sl a}}_2{\mbox{\sl b}}_1,\ \frac{1}{2}({\mbox{\sl a}}_1{\mbox{\sl b}}_1+{\mbox{\sl b}}_1{\mbox{\sl a}}_1),\
\frac{1}{2}({\mbox{\sl a}}_2{\mbox{\sl b}}_2+{\mbox{\sl b}}_2{\mbox{\sl a}}_2),$$ form a finite-dimensional subalgebra in the full infinite symmetry algebra ${{\cal W}}$. This algebra of quadratic charges is isomorphic to the Lie algebra of $SO(3,2)$ or, equivalently, of the noncompact version $Sp(4,{{\bf R}})$ of the symplectic group. Taking appropriate linear combinations of these charges, one can show that the full algebra ${{\cal W}}$ (and, in particular, the $SO(3,2)$ subalgebra of quadratic charges) is maintained in the $\alpha'\to\infty$ limit.
Even though the Fermi liquid theory exhibits this large symmetry ${{\cal W}}$, any given [*solution*]{} will typically break some of ${{\cal W}}$. In particular, those solutions that are described by a semiclassical Fermi surface will generally break ${{\cal W}}$ to the subalgebra that preserves the Fermi surface. As an example, consider again the Type 0A string theory background with RR flux $q_0$. For this solution of noncritical M-theory, the relevant symmetries in ${{\cal W}}$ are those that commute with the angular momentum generator $$\label{J}
J=\frac{1}{2\omega_0}\left({\mbox{\sl a}}_1{\mbox{\sl b}}_2-{\mbox{\sl a}}_2{\mbox{\sl b}}_1\right)$$ on the two-dimensional plane. This is dictated by the fact that the Type 0A solution of M-theory corresponds to filling all available states of $J=q_0$ up to $\mu$, while keeping states with $J\neq q_0$ empty. Out of the ten quadratic generators in ${{\cal W}}$, four survive; $J$ itself, plus the three diagonal combinations $${\mbox{\sl a}}_1^2+{\mbox{\sl a}}_2^2,\qquad {\mbox{\sl b}}_1^2+{\mbox{\sl b}}_2^2,\qquad
\frac{1}{2}({\mbox{\sl a}}_1{\mbox{\sl b}}_1+{\mbox{\sl b}}_1{\mbox{\sl a}}_1+{\mbox{\sl a}}_2{\mbox{\sl b}}_2+{\mbox{\sl b}}_2{\mbox{\sl a}}_2).$$ The four quadratic charges that commute with $J$ form an $SL(2,{{\bf R}})\times U(1)$ subalgebra in the $SO(3,2)$ algebra of quadratic charges in ${{\cal W}}$. From the perspective of Type 0A string theory, the $SL(2,{{\bf R}})$ factor of the surviving symmetry algebra corresponds precisely to the generators of the ground ring. Of course, the $SL(2,{{\bf R}})$ symmetry may be further broken by the level $\mu$ of the Fermi sea in the Type 0A vacuum. This embedding of the Type 0A symmetries into ${{\cal W}}$ will be important below.
Review of The Conformal Limit of the 0A Matrix Model {#0A}
====================================================
The conformal limit of two-dimensional Type 0A string vacua vith RR flux has been studied in [@Strominger:2003tm; @Ho:2004qp; @Aharony:2005hm]. Taking $\alpha'$ to infinity in the linear dilaton spacetime with RR flux $q$ leads to the $AdS_2$ geometry. This limit can also be viewed as a near-horizon limit of an extremally charged two-dimensional black hole [@Gukov:2003yp; @Davis:2004xb; @Danielsson:2004xf]. In [@Aharony:2005hm], Aharony and Patir provide further evidence for this behavior, by analyzing the spectrum of the model in this limit.
By examining the potential in (\[potential\]), we see that the limit of $\omega_0\to 0$ allows us to ignore the $\lambda^2$ term. We can view this limit alternatively as probing small $\lambda$. Either way, the quantum mechanics of the individual eigenvalues reduces in this limit to $$\label{actcf}
S=\frac{1}{2}\int dt
\left(\dot{\lambda}^2-\frac{M}{\lambda^2}\right).$$ This is a conformal field theory in 0+1 dimensions, studied a long time ago in [@deAlfaro:1976je]. The ground state of the second-quantized Fermi liquid consists of $N$ eigenvalues occupying all available states up to Fermi energy $\mu=0$, which ensures that conformal invariance is maintained. (Conformal invariance would also result from completely emptying or completely filling the entire Fermi sea.)
Two results are of importance here: first, the action (\[actcf\]) is invariant under the conformal symmetry $SO(2,1)\sim SL(2,{{\bf R}})$, which of course is equivalent to the isometries of an $AdS_2$ spacetime. The $SL(2,{{\bf R}})$ generators are $$\begin{aligned}
H&=&\frac{1}{2}\left(\dot{\lambda}^2+\frac{M}{\lambda^2}\right)\nonumber\\
D&=&-\frac{1}{4}\left(\lambda\dot{\lambda}+\dot{\lambda}\lambda\right)+tH\\
K &=& \frac{1}{2}\lambda^2+2tD-t^2H.\nonumber\end{aligned}$$ $H$ is the Hamiltonian following from the action (\[actcf\]), and it of course exhibits a continuous spectrum. The proposal of [@Strominger:2003tm] is to interpret this Hamiltonian on the $AdS_2$ dual, as the evolution operator in the Poincaré time. The duality to $AdS_2$ suggests that we should consider the evolution with respect to the compact operator[^3] $$\label{R}
\widetilde H=\frac{1}{2}\left(\frac{1}{{{\cal R}}}K+{{\cal R}}H\right),$$ where ${{\cal R}}$ is an arbitrary constant scale, to be identified with the curvature radius of the $AdS_2$. $\widetilde H$ represents the time evolution with respect to the [*global time*]{} on $AdS_2$ [@Strominger:2003tm; @Ho:2004qp; @Aharony:2005hm]. $\widetilde H$ has a discrete spectrum, with the $n$-th level eigenvalue given by $$\label{Rspectrum}
h_n=\frac{1}{2}+n+\frac{|q|}{2}.$$ On the $AdS_2$ side, this should be interpreted as the spectrum of propagating matter in global time. Aharony and Patir have shown that this spectrum exactly matches the spectrum of a free Dirac fermion on $AdS_2$ with mass $$\label{mss}
m=\frac{|q|}{2{{\cal R}}}.$$ In this sense, the conformal limit of the type $0A$ matrix model is dual to a theory on $AdS_2$ whose matter excitations are precisely those of a spinor field of mass given in (\[mss\]). In particular, the bosonic degrees of freedom of the tachyon do not survive in the conformal limit. Indeed, from the point of view of the compact generator $\widetilde H$, the ground state of the system is empty, and there is no macroscopic Fermi sea and thus no collective bosonic modes. In the near-horizon interpretation of this limit, this means that the propagating excitations of the Fermi sea do not make it to the near-horizon region of the black hole, leaving just the individual fermionic eigenvalues as the only propagating excitations in that regime.
Large $\alpha'$ Limit of Noncritical M-Theory {#mlimit}
=============================================
As we have discussed in Section 3, the ground state of noncritical M-theory can be viewed in polar coordinates as a coherent collection of ground states of Type 0A string theory, all filled to the common Fermi level $\mu$. We now take the conformal limit of this ${|M\rangle}$ state. Thus, we set $\mu=0$ in order to maintain conformal invariance of the vacuum. This leads to manifest $SL(2,{{\bf R}})$ invariance of the vacuum. On the dual spacetime side, it is thus natural to expect that $AdS_2$ will be a part of the spacetime geometry. The rest of the geometry can be inferred as follows. Invoking our “correspondence principle,” one can predict the spectrum of excitations of the ${|M\rangle}$ state in the conformal limit, from the knowledge of the Type 0A spectrum as reviewed in Section 4. As $\alpha'\to\infty$, each individual Type 0A sector with fixed $q$ will contribute one copy of a massive fermion, with mass $m$ given by (\[mss\]). In the M-theory vacuum, we thus get an infinite collection of Dirac fermions on $AdS_2$, with masses $$\label{mssall}
m=\frac{|q|}{2{{\cal R}}},\qquad q\in{{\bf Z}}.$$ This spectrum of masses represents a Kaluza-Klein tower, obtained from the reduction of a massess fermion in $2+1$ dimensions on $S^1$ of radius $2{{\cal R}}$. Thus, we expect that noncritical M-theory in the conformal limit corresponds to spacetime that is $AdS_2\times S^1$. The matching of the spectra of propagating modes will be one of the tests of this conjecture. In the rest of the paper, we will subject this conjecture to several additional tests.
Symmetry Generators
-------------------
Let us begin our discussion of the large $\alpha'$, or small $\omega_0$, limit of noncritical M-theory by considering the symmetry generators on the fermion side. As in the 0A case above, we find that the action (\[Maction\]) simplifies in the small $\omega_0$ limit: $$\label{Mlimitaction} S=\frac{1}{2}\int dt
\left(\dot{\lambda}_1^2+\dot{\lambda}_2^2\right)$$ Now, we would like to consider the ${|M\rangle}$ state with $\mu$ set exactly to zero. If we think of the ${|M\rangle}$ state as a coherent collection of Type 0A ground states of all possible values of RR flux $q$, each of these Type 0A sectors will be filled up to Fermi energy $\mu=0$. In the conformal limit, the entire $SL(2,{{\bf R}})$ respects the Type 0A vacuum, and the same will be true of the M-theory ground state. In the $\alpha'\to\infty$ limit of noncritical M-theory, the generators of this $SL(2,{{\bf R}})$ subalgebra of ${{\cal W}}$ are $$\begin{aligned}
H&=&\frac{1}{4}({\mbox{\sl a}}_1+{\mbox{\sl b}}_1)^2+\frac{1}{4}({\mbox{\sl a}}_2+{\mbox{\sl b}}_2)^2
=\frac{1}{2}\left(\dot{\lambda}_1^2+\dot{\lambda}_2^2\right)\nonumber\\
K&=&\frac{1}{4\omega_0^2}({\mbox{\sl a}}_1-{\mbox{\sl b}}_1)^2+\frac{1}{4\omega_0^2}({\mbox{\sl a}}_2
-{\mbox{\sl b}}_2)^2
=\frac{1}{2}\left(\lambda_1-\dot{\lambda}_1t\right)^2+\frac{1}{2}
\left(\lambda_2-\dot{\lambda}_2t\right)^2\label{gensm}\\
D&=&\frac{1}{4\omega_0}\left({\mbox{\sl a}}_1^2+{\mbox{\sl a}}_2^2-{\mbox{\sl b}}_1^2-{\mbox{\sl b}}_2^2\right)
=\frac{1}{4}\left(\lambda_1\dot{\lambda}_1+\dot{\lambda}_1\lambda_1
+\lambda_2\dot{\lambda}_2+\dot{\lambda}_2\lambda_2-2\dot{\lambda}_1^2t
-2\dot{\lambda}_2^2t\right).\nonumber\end{aligned}$$ They all commute with the angular momentum generator $J$, $$\label{genj}
J=\frac{1}{2\omega_0}({\mbox{\sl a}}_1{\mbox{\sl b}}_2-{\mbox{\sl a}}_2{\mbox{\sl b}}_1)=\frac{1}{2}\left(\lambda_1
\dot\lambda_2-\lambda_2\dot\lambda_1\right).$$ It turns out that, from the point of view of the first-quantized formulation, these $SL(2,{{\bf R}})\times U(1)$ generators enjoy a special status among all quadratic charges: they can be realized geometrically by a change of coordinates on the space of $(t,\lambda_1,\lambda_2)$ which preserves the foliation of this space by constant time slices. The action (\[Mlimitaction\]) is indeed symmetric under the following sets of transformations, generated by (\[gensm\]) and (\[genj\]): $$\begin{array}{rlll}
H:\quad&t'=t-\omega,\quad&\lambda_1'(t')=\lambda_1(t),\ \
&\lambda_2'(t')=\lambda_2(t)\\
D:\quad&t'=e^{-\omega}t,&\lambda_1'(t')=e^{-\omega/2}\lambda_1(t),
&\lambda_2'(t')=e^{-\omega/2}\lambda_2(t)\\
K:\quad&t'=\frac{t}{\omega t+1},&\lambda_1'(t')=(1+\omega t)^{-1}\lambda_1(t),
&\lambda_2'(t')=(1+\omega t)^{-1}\lambda_2(t) \\
J:\quad&t'=t,&\lambda_1'(t')=(\cos{\omega})\lambda_1-(\sin{\omega})
\lambda_2,\ \
&\lambda_2'(t')=(\cos{\omega})\lambda_2+(\sin{\omega})\lambda_1
\end{array}$$ These four generators have one algebraic relation: $$\frac{1}{2}(HK+KH)-D^2-J^2=\frac{1}{4}.$$ The quadratic charges in $SO(3,2)$ that are not in this $SL(2,{{\bf R}})\times U(1)$ are not realized geometrically on $(t,\lambda_i)$. This does not mean that they cannot survive as “hidden” symmetries, but we do not expect them to be realized by Killing symmetries of the gravitational background. We shall see in Section 5 that this picture is indeed correct.
As an example, let us consider what happens to a particular generator of $SO(3,2)$ that does not belong to the $SL(2,{{\bf R}})\times U(1)$ subalgebra: $$H'=\dot{\lambda}_1^2-\dot{\lambda}_2^2.$$ This generator acts on $(t,\lambda_i(t))$ via $$H':\ t=t',\qquad\lambda_1'(t')=\lambda_1(t-\omega),\qquad\lambda_2'(t')
=\lambda_2(t+\omega).$$ There is no coordinate transformation on the $\lambda_i$ and $t$ which can represent this symmetry. It is a symmetry of the action, but not one which can be represented geometrically. One can easily check that the same holds true for the other generators of $SO(3,2)$ not in the $SL(2,{{\bf R}})\times U(1)$ subalgebra. As such, we do not expect any of these generators to be associated with an isometry in the dual spacetime picture. $SL(2,{{\bf R}})\times U(1)$ of course matches the isometries of $AdS_2\times S^1$.
Spectrum Matching: Fermions on $AdS_2\times S^1$
------------------------------------------------
As in the Type 0A case, the spectrum of $H$ in (\[gensm\]) is continuous, and corresponds to the Hamiltonian evolution in the Poincaré time on $AdS_2\times S^1$. We again switch to the global time Hamiltonian $\widetilde
H$, defined by (\[R\]), now in terms of the M-theory operators $H$ and $K$ of (\[gensm\]). It is straightforward but reassuring to see that the spectrum of $\widetilde H$ matches that of a massless fermion on $AdS_2\times S^1$, as expected from our “correspondence principle”. If we transform our action from (\[Mlimitaction\]) via $$\widetilde\lambda_i(\tau)=\frac{\sqrt{{{\cal R}}}}{\sqrt{{{\cal R}}^2+t^2}}\lambda(t),\qquad
\tau=\arctan{(t/{{\cal R}})},$$ we find $$S=\frac{1}{2}\int d\tau
\left[(\partial_\tau\widetilde\lambda_1)^2+(\partial_\tau\widetilde\lambda_2)^2
-\widetilde\lambda_1^2-\widetilde\lambda_2^2\right].$$ Here $\tau$ corresponds to the global time, and the individual eigenvalues see a rightside-up planar harmonic oscillator potential. $\widetilde H$ generates translations along $\tau$. Its spectrum is $$h_{n,m}=\frac{1}{2}(n+m+1),$$ where $n$ and $m$ both are non-negative integers. This is equivalent to the spectrum for the $1+1$ dimensional $\widetilde H$ as given in (\[Rspectrum\]), if one allows $q$ to range over all integers. The second-quantized ground state of $\widetilde H$ with the rightside-up harmonic oscillator potential is again empty of all fermions, and there is no macroscopic Fermi surface with propagating bosonic excitations.
Now, let us consider the spectrum of a free fermion, as calculated in global coordinates, on an $S^1$ fibered over $AdS_2$, as suggested from the symmetry arguments above. As shown in Appendix \[appendix\], the only fibering which produces the same spectrum is the direct product spacetime, that is $AdS_2\times S^1$. Moreover, the spectrum matching requires the radius of the $S^1$ factor to equal $2{{\cal R}}$, where ${{\cal R}}$ is the curvature radius of $AdS_2$.
Thus, we conjecture that [*in the $\alpha'\to\infty$ limit, the natural ground state ${|M\rangle}$ of noncritical M-theory describes a theory on a dynamical $AdS_2 \times S^1$ spacetime, with propagating matter described by a single free massless Dirac fermion.*]{}
This conjecture leads to two remarkable phenomena: ([*i*]{}) We get a relativistic dual out of a nonrelativistic Fermi liquid,[^4] and ([*ii*]{}) the angular dimension on the flat plane populated by the nonrelativistic fermions corresponds under this duality to a fixed-radius circle fibered trivially over the $AdS_2$ base spacetime of string theory; this of course is the traditional behavior of the extra dimension in the simplest forms of string/M-theory duality.
A Family of Solutions with $SL(2,{{\bf R}})$ Symmetry
-----------------------------------------------------
In passing, we wish to point out that the conformal $AdS_2\times S^1$ vacuum belongs to an interesting multi-parameter family of solutions of noncritical M-theory, all of which share the $SL(2,{{\bf R}})$ symmetry of the Type 0A conformal vacua.
Consider the ground state of the global-time Hamiltonian $\widetilde H$ of (\[R\]) in the conformal limit of Type 0A theory. As reviewed above, in the Fermi liquid picture this ground state is empty, [*i.e.*]{}, all the single-particle states are unoccupied by the fermions. Another way of preparing a conformally invariant ground state would be to keep all single-particle states occupied, and treat the holes as elementary excitations. Of course, in Type 0A theory these two solutions are isomorphic by the particle-hole duality, and we gain nothing by switching from one description to the other.
In $2+1$ dimensions, the situation is more interesting. The simplest ground state of the global-time Hamiltonian $\widetilde H$ is empty, leading to the $AdS_2\times S^1$ solution which is the main focus of the present paper. Using particle-hole duality, it is again possible to switch between filled and empty states. Doing so simultaneously for all values $q$ of the angular momentum would result in an equivalent solution. However, unlike in $1+1$ dimensions, we now have the additional freedom of deciding separately for each value of $q$ whether all states are empty or full, without losing the $SL(2,{{\bf R}})$ symmetry of the state. This leads to an interesting multi-parameter family of solutions, parametrized as follows. We start with the ground state of $\widetilde H$ with all states empty, choose a sequence $$q_1<\ldots <q_n,$$ and prepare a new state such that all available states with angular momentum $q$ in the range $q_{2k+1}\leq q<q_{2k}$ for $k=0,1,\ldots$ are occupied while those in sectors with $q_{2k}\leq q<q_{2k+1}$ remain empty.
These solutions have $n$ sheets of the Fermi surface, located at $J=q_i$, $i=1,\ldots n$. All of them inherit the $SL(2,{{\bf R}})$ symmetry from the Type 0A decomposition. It would be very interesting to investigate the spacetime interpretation of this family of solutions. When the individual Fermi surfaces are far separated, the physics of their collective excitations is almost decoupled. However, since there is one common $SL(2,{{\bf R}})$ symmetry shared by the $n$ sheets of the Fermi surface, we expect the solution to have the structure of a multi-sheeted version of $AdS_2$, sharing the same boundary with a common conformal dual description. This structure is very reminiscent of the multi-sheeted $AdS_5$ with a common CFT dual, studied in [@Aharony:2006hz; @Kiritsis:2006hy].
Spacetime Effective Action in the $AdS_2\times S^1$ Background {#spacetime}
==============================================================
An observer in the $AdS_2\times S^1$ spacetime is not likely to describe the system in terms of the two-dimensional nonrelativistic Fermi liquid on the ${{\bf R}}^2$ flat plane. Instead, the physics of excitations that are sufficiently close to the ground state should be encoded in a spacetime effective action. This action should contain spacetime gravity, it should have $AdS_2\times S^1$ as a solution, and should reproduce the symmetries and spectrum of the ground state. It is the goal of this Section to propose a natural effective action that satisfies such constraints.
We are working in a coordinate system $(x^\mu$), $\mu=0,1,2$, given by global coordinates $(x^0,x^1)=(t,\rho)$ on $AdS_2$, and a periodic coordinate $x^2=y$ of periodicity $2\pi$ on $S^1$. The metric takes the following form, $$\label{adssc}
ds^2=-{{\cal R}}^2\cosh^2\rho\,dt^2+{{\cal R}}^2d\rho^2 +4{{\cal R}}^2dy^2.$$ The only nonzero component of the Einstein tensor in this spacetime is $$R_{yy}-\frac{1}{2}Rg_{yy}=4.$$ Thus, unlike an $AdS_3$ spacetime, $AdS_2\times S^1$ will not solve the vacuum Einstein equations for any value of the cosmological constant.
With the hindsight afforded by AdS/CFT correspondence, it might be tempting to postulate the existence of a propagating $U(1)$ gauge field, whose flux through $AdS_2$ (and the dual flux through $S^1$) could provide just the right energy-momentum tensor to turn $AdS_2\times S^1$ into a solution of the coupled Einstein-Maxwell equations.[^5] This is indeed possible, and leads to a solution of the coupled system $$\label{spaceequation}
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu} = -\Lambda g_{\mu\nu} +8\pi
G_NT_{\mu\nu},$$ where $$T_{\mu\nu}=F_{\mu\lambda}F_\nu^\lambda-\frac{1}{4}g_{\mu\nu}
F_{\lambda\sigma}F^{\lambda\sigma}$$ is the conventional energy-momentum tensor of the Maxwell Lagrangian.
Let us consider $U(1)$ flux $F$ along the $AdS_2$ factor of the geometry, with $F$ proportional to the area two-form on $AdS_2$: $$F_{t\rho}=-F_{\rho t}=f_0\cosh\rho$$ The energy-momentum tensor is $$T_{\mu\nu}=\frac{f_0^2}{2{{\cal R}}^2}\left(\begin{array}{ccc}
\cosh^2\rho & 0 & 0\\
0 & -1 & 0 \\
0 & 0 & 1
\end{array}
\right).$$ The Einstein equations are now satisfied by $AdS_2\times S^1$ if we pick $$\label{lambf}
\Lambda = -\frac{4}{5{{\cal R}}^2}, \qquad f_0=\frac{{{\cal R}}}{\sqrt{5\pi G_N}}.$$ Of course, the two form field strength $F$ is exact, with the gauge field given by $$A= \frac{{{\cal R}}\sinh\rho}{\sqrt{5\pi G_N}}dt;$$ the Maxwell equations for $A$ are trivially satisfied.
The fact that there there is an electric flux through $AdS_2$ implies that the $S^1$ carries a dual, magnetic flux. In $2+1$ dimensions, the dual to a $U(1)$ one-form gauge field $A$ is a scalar $\phi$, related to the field strength $F$ of $A$ by $\ast F=d\phi$. By tracking this duality for our background (\[lambf\]), one can easily see that $\phi$ has a flux through the $S^1$ factor of $AdS_2\times S^1$.
However, even though $AdS_2\times S^1$ is a solution to the coupled Einstein-Maxwell system with negative cosmological constant in $2+1$ dimensions, this theory cannot be a good approximation to the effective theory describing the conformal limit of noncritical M-theory, for a simple reason. The analysis of the spectrum in Section 4 has revealed the existence of a single propagating matter field, a massless Dirac fermion in $AdS_2\times S^1$. On the other hand, the spectrum of low-energy excitations of the Einstein-Maxwell theory would contain a propagating photon, an excitation of which there is no evidence on the noncritical M-theory side. Hence, we must look for another effective theory that has $AdS_2\times S^1$ as a solution, but with fewer propagating degrees of freedom.
It turns out that the correct starting point is the Chern-Simons theory with $SO(3,2)$ gauge symmetry, [*i.e.*]{}, conformal gravity in $2+1$ dimensions.
$AdS_2\times S^1$ in $SO(3,2)$ Chern-Simons Gravity
---------------------------------------------------
In [@Horne:1988jf], Horne and Witten extend the work of [@Witten:1988hc] and show that conformal gravity in $2+1$ dimensions can be rewritten as an $SO(3,2)$ Chern-Simons gauge theory of the conformal group. We will now show that our $AdS_2\times S^1$ spacetime is a solution of this theory.
Recall that conformal gravity on a $2+1$-dimensional manifold ${{\cal M}}$ (with coordinates $x^\mu$) can be described by $$S_{CS}=\frac{k}{4\pi}\int_{{\cal M}}{{\rm Tr}}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge
A\right),$$ where $A$ is an $SO(3,2)$ Lie algebra-valued one-form gauge field.[^6] We write $A$ in components as[^7] $$\label{CSaction}
A_\mu=e_\mu{}^a\mathcal P_a+\omega_\mu{}^a\mathcal J_a+
\zeta_\mu{}^a\mathcal K_a + \phi_\mu\mathcal D.$$ Here $a$ runs over $0,1,2$ for each of $\mathcal P_a$, $\mathcal
J_a$, and $\mathcal K_a$. In the interpretation of this theory as conformal gravity, $e_\mu{}^a$ are the components of the vielbein, while $\omega^a$ is the corresponding spin connection, Hodge-dualized in its internal Lorentz indices using the $\epsilon^{abc}$ tensor associated with $\eta_{ab}={\rm diag}(-1,1,1)$. The commutation relations are $$\begin{array}{rlrl}
[\mathcal J_a,\mathcal J_b]&=\epsilon_{abc}\mathcal J^c, &
[\mathcal P_a,\mathcal P_b]&=[\mathcal K_a,\mathcal K_b]=
[\mathcal J_a,\mathcal D]=0, \cr
[\mathcal P_a,\mathcal K_b]&=\eta_{ab}\mathcal D-\epsilon_{abc}\mathcal J^c,
& & \cr
[\mathcal P_a,\mathcal J_b]& =\epsilon_{abc}\mathcal P^c, &
[\mathcal K_a,\mathcal J_b]&=\epsilon_{abc}\mathcal K^c,\cr
[\mathcal P_a,\mathcal D]& =\mathcal P_a,&
[\mathcal K_a,\mathcal D]&=-\mathcal K_a.
\end{array}$$ The equations of motion of (\[CSaction\]) are just the flatness conditions $$\label{flatness}
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu]=0.$$ An interesting class of solutions to (\[flatness\]) can be constructed as follows. First we assume that our vielbein is invertible, and then we pick a gauge in which $\phi_\mu=0$. In such circumstances, the equations of motion (\[flatness\]) reduce to $$\begin{aligned}
de^a-e_b\wedge\omega^{ab}&=&0 \label{Pflat}\\
d\zeta^a-\zeta_b\wedge \omega^{ab}&=&0 \label{Kflat}\\
e^a\wedge\zeta_a&=&0 \label{Dflat}\\
-d\omega^{ab}-\omega^{ac}\wedge\omega_c^{\
b}+e^a\wedge\zeta^b-e^b\wedge\zeta^a&=&0. \label{Jflat}\end{aligned}$$ Before we discuss the embedding of our $AdS_2\times S^1$ spacetime into this framework, let us discuss how $AdS_3$, $dS_3$, and the Minkowski space can be interpreted as solutions of this theory. First, we note that Eqn. (\[Pflat\]) is simply the torsion-free condition. Setting $\zeta_\mu=
\zeta e_\mu$ where $\zeta$ is a constant, we find that Eqn. (\[Kflat\]) reduces to the torsion-free condition as well. Also, Eqn. (\[Dflat\]) is trivially satisfied. Using $R^a_{\ b}=d\omega^a_{\ b}+\omega^a_{\ c}
\wedge\omega^c_{\ b}$, Eqn. (\[Jflat\]) becomes $$\label{remain}
R^a_{\ b}-2\zeta e^a\wedge e_b=0.$$ Consequently, when (\[remain\]) is satisfied, the Einstein tensor reduces to $$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=-2\zeta g_{\mu\nu}.$$ In this way, the vacuum with cosmological constant $\Lambda=2\zeta$ is embedded as a solution to conformal Chern-Simons gravity in $2+1$ dimensions. This solution is described by the following gauge field, $$\label{allsolns}
A_\mu=e_\mu^{\ a}\left(\mathcal P_a
+\frac{\Lambda}{2}\mathcal K_a\right) +\omega_\mu^{\ a}\mathcal J_a.$$ The explanation of the existence of such solutions is very simple. The solution (\[allsolns\]) only excites gauge field components of a certain subalgebra of $SO(3,2)$. When $\Lambda<0$, the nonzero components in (\[allsolns\]) belong to $SO(2,2)\subset SO(3,2)$, for $\Lambda>0$ they span $SO(3,1)\subset SO(3,2)$, and if $\Lambda=0$ we get $ISO(2,1)\subset
SO(3,2)$. In all cases, the flatness of the $SO(3,2)$ connection reduces to the flatness in the corresponding subalgebra.
Now, let us return to consider the embedding of our $AdS_2\times
S^1$ spacetime. Our vielbein and spin connection components are $$\begin{array}{rl}
e^0=&{{\cal R}}\cosh \rho dt\\
e^1=&{{\cal R}}\,d\rho\\
e^2=&2{{\cal R}}\,dy \\
\omega^2=&-\sinh\rho\,dt.
\end{array}$$ For the $\zeta^a$, we choose $$\zeta^0=-\frac{1}{2{{\cal R}}^2}e^0,\qquad\zeta^1=-\frac{1}{2{{\cal R}}^2}e^1,\qquad
\zeta^2=\frac{1}{2{{\cal R}}^2}e^2.$$ Thus, the $SO(3,2)$ gauge field that desribes $AdS_2\times S^1$ can finally be written as $$A_\mu=e_\mu^{\ 0}\left(\mathcal P_0-\frac{1}{2{{\cal R}}^2}\mathcal K_0\right)
+e_\mu{}^1\left(\mathcal P_1-\frac{1}{2{{\cal R}}^2}\mathcal K_1\right)+e_\mu{}^2
\left(\mathcal
P_2+\frac{1}{2{{\cal R}}^2}\mathcal K_2\right)+\omega_\mu{}^2 J_2.\label{gaugefield}$$ Since we have again chosen $\phi_\mu=0$, the flatness conditions $F_{\mu\nu}=0$ reduce to Eqns. (\[Pflat\]) – (\[Jflat\]). Simple algebra will show that these equations are satisfied. Thus, the background $A_\mu$ of (\[gaugefield\]) is a solution of $SO(3,2)$ Chern-Simons gauge theory, and consequently of conformal gravity in $2+1$ dimensions.
The explanation for the existence of such a solution is again simple. (\[gaugefield\]) corresponds to the embedding of $SL(2,{{\bf R}})\times U(1)$, the isometry of $AdS_2\times S^1$, into $SO(3,2)$. In particular, the $U(1)$ factor is generated by $$\mathcal P_2+\frac{1}{2{{\cal R}}^2}\mathcal K_2,$$ which indeed commutes with the three generators of $SL(2,{{\bf R}})$ also excited by the background gauge field (\[gaugefield\]). In more generality, for any embedding of a direct product $G_1\times G_2$ into the Chern-Simons gauge group $G$, the flatness conditions for $G$ factorize into the flatness in the $G_1$ and $G_2$ factors if the gauge field belongs to $G_1\times G_2$.
The $SO(3,2)$ Chern-Simons gauge theory has the following good features, which make it a suitable starting point for constructing our effective action: ([*i*]{}) $AdS_2\times S^1$ is a solution of this Chern-Simons theory, and the gauge group $SO(3,2)$ naturally coincides with the group of all the quadratic charges of the symmetry algebra ${{\cal W}}$. ([*ii*]{}) The isometry $SL(2,{{\bf R}})\times U(1)$ of $AdS_2\times S^1$ is embedded in $SO(3,2)$ in precisely the manner expected from the Type 0A string theory and noncritical M-theory arguments of Section 4. ([*iii*]{}) Unlike the Einstein-Maxwell system considered at the beginning of Section 5, the $SO(3,2)$ Chern-Simons gravity has no propagating bosonic degree of freedom, just as noncritical M-theory in the conformal limit.
Extension to Higher-Spin Gauge Theory {#higherspin}
-------------------------------------
Despite its good properties, the $SO(3,2)$ Chern-Simons gauge theory cannot be the whole story, for two reasons: (1) $SO(3,2)$ is only a subalgebra of the infinite symmetry algebra of noncritical M-theory, and (2) we have seen evidence that in the conformal limit, the matter content of the noncritical M-theory vacuum is that of a propagating massless Dirac fermion. In the conventional approach to Chern-Simons gravity, it is unknown how to couple second-quantized matter to the gravity sector as described by the gauge connection. We shall now show that once the correct infinite symmetries of Problem (1) are properly taken into account, Problem (2) will acquire a natural solution as well.
In order to resolve Problem (1), we are in need of a Chern-Simons gravity theory based on an infinite-dimensional extension of $SO(3,2)$. Remarkably, this theory is already available. It is the bosonic version of a supersymmetric Chern-Simons theory of an infinite hierarchy of conformal higher-spin fields constructed in [@Shaynkman:2001ip].
Higher-spin gauge theories have a rich history, going back to the original work of Fradkin and Vasiliev [@Fradkin:1987ks] (see [@Bekaert:2005vh] for a review). In $2+1$ dimensions, higher-spin gauge theories as Chern-Simons gauge theories were first written down by Blencowe [@Blencowe:1988gj]. The higher-spin version of conformal Chern-Simons gravity in $2+1$ dimensions appeared first in the work of Pope and Townsend [@Pope:1989vj], and Fradkin and Linetsky [@Fradkin:1989xt]. We shall follow most closely the detailed construction by Shaynkman and Vasiliev [@Shaynkman:2001ip]; see also [@Didenko:2006zd]. The first thing we need is a convenient parametrization of the higher-spin symmetry algebra. In order to construct this algebra, Shaynkman and Vasiliev [@Shaynkman:2001ip] first define operators[^8] $\hat a_{\alpha}$ and $\hat a^{+ \alpha}$, where the index $\alpha=1,2$ parametrizes the spinor representation of the Lorentz group in $2+1$ dimensions, and subject them to the commutation relations $$[\hat a_\alpha,\hat a^{+\beta}]=\delta_\alpha{}^\beta,\quad\quad[\hat a_\alpha,
\hat a_\beta]=[\hat a^{+\alpha},\hat a^{+\beta}]=0.$$ Rather than use an operator realization, it is convenient to use techniques of noncommutative geometry on ${{\bf R}}^4$ parametrized by commuting coordinates $a_{\alpha}$ and $a^{+\alpha}$ and endowed with the star product: $$\label{star}
(f \star g)(a,a^+)=f(a,a^+)\exp\left\{\frac{1}{2}\left(
\frac{\overleftarrow{{\partial}}}{{{\partial}}a_\alpha}\frac{\overrightarrow{{\partial}}}{{{\partial}}a^{+\alpha}}
-\frac{\overleftarrow{{\partial}}}{{{\partial}}a^{+\alpha}}\frac{\overrightarrow{{\partial}}}{{{\partial}}a_\alpha}
\right)\right\}g(a,a^+).$$ This definition results in the following $\star$ commutators: $$\label{starcom}
[a_\alpha,a^{+\beta}]_\star=a_\alpha\star a^{+\beta}-a^{+\beta}\star
a_\alpha=\delta_\alpha^\beta, \qquad
[a_\alpha,a_\beta]_\star=[a^{+\alpha},a^{+\beta}]_\star=0.$$ The associative algebra of the $\star$-product defined by generators consisting of all powers of $a$ and $a^+$ is called $A_2$ in [@Shaynkman:2001ip]. On $A_2$, one can define the structure of a Lie superalgebra, by first assigning even (or odd) grading to the generators given by even-degree (or odd-degree) monomials in $a_\alpha$ and $a^{+\alpha}$, and then defining the (anti)commutation relations via the $\star$-product (anti)commutator. In the rest of the paper, we shall call this higher-spin superalgebra $\widetilde{{\cal W}}$.
The subalgebra of quadratic charges in $\widetilde{{\cal W}}$ is $Sp(4,{{\bf R}})$, which is isomorphic to the $SO(3,2)$ algebra. It has a $\star$-product realization by $$\begin{aligned}
\mathcal P_a&=&\frac{1}{2}\sigma_a{}^{\alpha\beta}a_\alpha a_\beta, \qquad
\qquad\mathcal J_a=\frac{1}{2}\sigma_a{}^\alpha{}_\beta a_\alpha a^{+\beta},\\
\mathcal K_a&=&-\frac{1}{4}\sigma_{a\alpha\beta}a^{+\alpha}a^{+\beta},\ \qquad
\mathcal D=\frac{1}{4}\left(a_\alpha a^{+\alpha}+a^{+\alpha}a_\alpha\right),\end{aligned}$$ where $\sigma_a{}^{\alpha\beta}$ are symmetric matrices given by $$\sigma_0{}^{\alpha\beta}=\pmatrix{1 &0\cr 0&1},\quad
\sigma_1{}^{\alpha\beta}=\pmatrix{0 &1\cr 1&0},\quad
\sigma_2{}^{\alpha\beta}=\pmatrix{1 &0\cr 0&-1},\quad$$ and the spinor indices are raised and lowered as $c^\alpha=\epsilon^{\alpha\beta}c_\beta$, $c_\beta=\epsilon_{\alpha\beta}
c^\alpha$, with $\epsilon_{\alpha\beta}=-\epsilon_{\beta\alpha}$, $\epsilon_{12}=\epsilon^{12}=1$.
Clearly, this construction of $\widetilde{{\cal W}}$ is very closely related to the construction of the ${{\cal W}}$ symmetry algebra in noncritical M-theory which we reviewed briefly in Section 4. More precisely, the infinite subalgebra ${{\cal W}}_0$ of all even-degree charges in ${{\cal W}}$ coincides with the maximal bosonic subalgebra in $\widetilde{{\cal W}}$. Our generators ${\mbox{\sl a}}_i,b_j$ are related to $a_\alpha,a^{+\beta}$ of [@Shaynkman:2001ip] by a linear transformation that preserves the commutation relations, [*i.e.*]{}, by an $Sp(4,{{\bf R}})$ symplectomorphism of the phase space ${{\cal T}}$ of the Fermi liquid.
The conformal higher-spin theory that will be relevant for the conformal limit of noncritical M-theory is based on the bosonic higher-spin Lie algebra ${{\cal W}}_0$. This algebra contains generators that correspond to all integer spins; there is no evidence, in the noncritical M-theory vacuum that we consider here, of the half-integer fermionic spins. We shall comment on a possible supersymmetric extension in Section 5.4.
Thus, we consider the bosonic higher-spin theory, described again by the Chern-Simons action, $$\label{hcs}
S_{HCS}=\frac{k}{4\pi}\int{{\rm Tr}}\left({{\cal A}}\wedge d{{\cal A}}+\frac{2}{3}{{\cal A}}\wedge{{\cal A}}\wedge {{\cal A}}\right).$$ with the gauge field one-form ${{\cal A}}$ now taking values in the infinite-dimensional higher-spin Lie algebra ${{\cal W}}_0$ of even-degree bosonic charges.[^9] In general, ${{\cal A}}$ can be expanded in components, $$\label{expaa}
{{\cal A}}(x|a,a^+)=\sum_{\ell=0}^\infty\sum_{m=0}^{2\ell}\frac{1}{m!(2\ell-m)!}
{{\cal A}}_{\alpha_1\ldots\alpha_m}{}^{\alpha_{m+1}\ldots\alpha_{2\ell}}(x)
a^{+\alpha_1}\ldots a^{+\alpha_m}a_{\alpha_{m+1}}\ldots a_{\alpha_{2\ell}}.$$ Each component ${{\cal A}}_{\alpha_1\ldots\alpha_m}{}^{\alpha_{m+1}\ldots
\alpha_{2\ell}}(x)$ is a one-form on ${{\cal M}}$.
The equations of motion of (\[hcs\]) yield the flatness condition, $$\label{oflat}
d{{\cal A}}+{{\cal A}}\star\wedge{{\cal A}}=0.$$ The gauge field $A$ of (\[gaugefield\]), which describes the $AdS_2\times
S^1$ solution of $SO(3,2)$ Chern-Simons gravity, can be embedded into the higher-spin theory by setting the components of ${{\cal A}}$ in the $SO(3,2)$ subalgebra equal to $A$, and all others to zero. In the $\star$-product language, our $AdS_2 \times S^1$ background is described by $${{\cal A}}=\frac{1}{2}\left[e^a_\mu\sigma_a^{\alpha\beta}\left(a_\alpha
a_\beta + \frac{1}{4{{\cal R}}^2}a_\alpha^+a_\beta^+\right)-e^2_\mu
\sigma_2^{\alpha\beta}(\frac{1}{2{{\cal R}}^2}a^+_\alpha
a^+_\beta)+\omega_\mu^2\sigma_2{}^\alpha{}_\beta(a_\alpha
a^{+\beta})\right]dx^\mu.$$ Thus, $AdS_2\times S^1$ is a solution of (\[oflat\]).
The theory (\[hcs\]) is invariant under the full set of higher spin conformal transformations given by $$\label{higherspinxforms}
\delta {{\cal A}}=d\varepsilon+[{{\cal A}},\varepsilon]_\star$$ with $\varepsilon$ a scalar function with values in the infinite-dimensional higher-spin Lie algebra. For any given solution ${{\cal B}}$ of the equations of motion (\[oflat\]), we are interested in its global symmetries, [*i.e.*]{}, gauge transformations $\varepsilon$ that preserve ${{\cal B}}$: $$\label{globsyms}
d\varepsilon+[{{\cal B}},\varepsilon]_\star=0.$$ On a topologically trivial spacetime ${{\cal M}}$, one global symmetry can be constructed for each element of the symmetry algebra, as follows. Consider a solution ${{\cal B}}$ of (\[oflat\]). Since ${{\cal B}}$ is flat, it can be written as $${{\cal B}}=g^{-1}(x)\star dg(x)$$ for some function $g(x)$ with values in the Lie group ${{\cal W}}_0$. Then, for any fixed, constant element $\xi$ from the Lie algebra of ${{\cal W}}_0$, $$\label{rigsymloc}
\varepsilon=g^{-1}(x)\star\xi\star g(x)$$ is a symmetry of the background ${{\cal B}}$, [*i.e.*]{}, a solution of (\[globsyms\]).
In the case of topologically nontrivial ${{\cal M}}$, there could be obstructions against defining (\[rigsymloc\]) globally over ${{\cal M}}$. Consequently, the actual symmetry can be reduced to a subalgebra. As an example, consider for simplicity the Minkowski space, described as a solution of Chern-Simons theory in (\[allsolns\]), and focus on the quadratic charges belonging to $SO(3,2)$. Clearly, the spacetime-independent transformations $$\varepsilon(\mathcal P)=\xi^a\mathcal P_a$$ solve (\[globsyms\]) for any constant $\xi^a$; they represent the rigid translations of the Minkowski space. Less trivially, one can show that $$\begin{aligned}
\varepsilon(\mathcal J)&=& \xi^a\left(\mathcal J_a+e_\mu{}^bx^\mu\epsilon_{abc}
\mathcal P^c\right),\nonumber\\
\varepsilon(\mathcal D)&=& \xi\left(\mathcal D-e_\mu{}^ax^\mu\mathcal P_a
\right),\label{glbsmm}\\
\varepsilon(\mathcal K)&=& \xi^a\left(\mathcal K_a-e_\mu{}^bx^\mu
\left(\epsilon_{abc}\mathcal J^c+\eta_{ab}\mathcal D\right)+\left(e_{\mu a}
e_\nu{}^bx^\mu x^\nu-\frac{1}{2}\delta_a^b x_\mu x^\mu\right)
\mathcal P_b\right)\nonumber\end{aligned}$$ are also solutions of (\[globsyms\]), and thus represent global symmetries of Minkowski space viewed as a solution of $SO(3,2)$ Chern-Simons gauge theory.[^10] Thus, on ${{\cal M}}={{\bf R}}^3$, we see that the entire $SO(3,2)$ is a symmetry. However, if we compactify (say) $x^2$ on $S^1$, and require the global symmetries to be well-defined on $S^1$, only the linear combinations of (\[glbsmm\]) that are independent of the $x^2$ coordinate survive the compactification. The global symmetry group of the flat ${{\bf R}}^2\times S^1$ background is reduced to $ISO(1,1)\times U(1)$.
On $AdS_2\times S^1$ we are in a very similar situation. Consider again the algebra of quadratic charges $SO(3,2)$. If we sent the radius of $S^1$ to infinity, the global symmetry would correspond to the entire $SO(3,2)$. The main difference compared to the Minkowski example is that on $AdS_2\times
{{\bf R}}$, the solutions to (\[globsyms\]) are in fact periodic along the coordinate $x^2$ on ${{\bf R}}$ with a fixed periodicity, set by the radius ${{\cal R}}$ of $AdS_2$. Which symmetries survive on $AdS_2\times S^1$ is thus determined by the radius of $S^1$ in units of ${{\cal R}}$. For a generic radius of $S^1$, ${{\cal W}}_0$ is broken to the subalgebra of global charges that are independent of $x^2$.
The quadratic charges that are independent of $x^2$ form the $SL(2,{{\bf R}})\times U(1)$ subalgebra of $SO(3,2)$. It turns out that the quadratic charges that do depend on $x^2$ are periodic on $S^1$ whose radius is ${{\cal R}}$ (or any integer multiple thereof). In order to check this, we can go back to the coordinates $(t,\rho,y)$ of (\[adssc\]), and find six solutions to (\[globsyms\]) that depend on $y$, such as for example $$\begin{aligned}
\varepsilon&=&{{\cal R}}\,\sinh\rho\,\cos(2y)\left(\mathcal P_2-\frac{1}{2{{\cal R}}^2}
\mathcal K_2\right)-{{\cal R}}\,\cosh\rho\,\sin(2y)\left(\mathcal P_1+\frac{1}{2{{\cal R}}^2}
\mathcal K_1\right)\nonumber\\
& &\qquad\qquad{}-\cosh\rho\,\cos (2y)\,\mathcal J_0+\sinh\rho\,\sin(2y)
\,\mathcal D,\end{aligned}$$ Noting that $y$ is periodic with periodicity $2\pi$ when the $S^1$ radius is $2{{\cal R}}$, we obtain the stated result. Thus, we see that if the radius of $S^1$ is an integer multiple of ${{\cal R}}$, all $SO(3,2)$ symmetries will be unbroken. Since in noncritical M-theory the radius of the $S^1$ factor is twice the radius of $AdS_2$, the entire $SO(3,2)$ symmetry survives the compactification, in accord with our expectations from the Fermi liquid side discussed in Section 4.1.
In fact, there is an interesting refinement of the story. The same $AdS_2\times S^1$ background would also be a solution of the supersymmetric extension of the theory, which would result from keeping both even- and odd-degree charges in the higher-spin algebra $\widetilde{{\cal W}}$. One can again ask what would be the periodicity of the odd-degree charges along $x^2$. It is intriguing that the odd-degree charges, and in particular the linear charges that correspond to the supercharges in the $OSp(1|4)$ supersymmetric extension of $SO(3,2)$, are all antiperiodic on $S^1$ of radius ${{\cal R}}$. This implies that the supercharges survive as global symmetries of $AdS_2\times S^1$ if the radius of $S^1$ is an [*even*]{} multiple of the $AdS_2$ radius. We see that the radius of $S^1$ in the conformal limit of noncritical M-theory is precisely given by the minimal value for which the $AdS_2\times S^1$ background would be supersymmetric, if embedded into the supersymmetrized version of the higher-spin theory. This suggests that our noncritical M-theory may be a simple ${{\bf Z}}_2$ orbifold of a supersymmetric theory, on which we comment further in Section 5.4.
Coupling the Fermion {#higherspinf}
--------------------
Our analysis of the spectrum in Section 4 revealed that noncritical M-theory in the high-energy limit is not purely topological; the vacuum has at least one type of a propagating excitation, described by a second-quantized massless Dirac fermion field on the $AdS_2\times S^1$ background. In the full effective action, this matter sector should couple consistently to the topological Chern-Simons sector of the theory. The existence of such a coupling will represent another check of the proposed picture.
The standard lore of topological gravity is that the system cannot be coupled to propagating matter. This is sometimes avoided by representing matter in the first-quantized framework, essentially via a collection of Wilson lines coupled to the topological gauge field. The difficulty essentially stems from the fact that in order to write down the equations of motion for a second-quantized propagating field, we must invert the vielbein; however, this is an unnatural procedure in Chern-Simons theory where we interpret the vielbein as a part of the Chern-Simons gauge field.
Remarkably, this standard lore may no longer be valid once the gauge group becomes infinite dimensional. Vasiliev [*et al.*]{} have shown [@Shaynkman:2001ip] (see [@Bekaert:2005vh] for a review) that propagating matter fields of low spins (in particular, a scalar and a spinor) can indeed be coupled the Chern-Simons higher spin gravity in $2+1$ dimensions.
Again following [@Shaynkman:2001ip], we first introduce the Fock vacuum ${|0\rangle}$ defined to satisfy $a_\alpha{|0\rangle}=0$. In the $\star$-product realization, this vacuum can be described by a projector $$|0\rangle\langle0|=4\exp{(-2a_\alpha a^{+\alpha})},$$ which satisfies $$a_\alpha\star|0\rangle\langle0|=|0\rangle\langle0|\star
a^{+\alpha}=0,\qquad |0\rangle\langle0|\star|0\rangle\langle0| =
|0\rangle\langle0|.$$ In other words it is properly normalized, and annihilated on the left by $a_\alpha$. The full set of Fock states on this vacuum can be created by action on the left with $a^{+\alpha}$. The matter fields on ${{\cal M}}$ will be represented by a section of the Fock bundle over ${{\cal M}}$, $$\label{matterf}
|\Phi(x|a^+)\rangle=\sum^\infty_{\ell=0}\frac{1}{\ell!}c_{\alpha_1...
\alpha_\ell}(x)a^{+\alpha_1}\ldots a^{+\alpha_\ell}\star |0\rangle\langle0|.$$ For future reference, it will be natural to split ${|\Phi\rangle}$ into an even and odd part, $$\begin{aligned}
|\Phi_0\rangle&=&\sum^\infty_{n=0}\frac{1}{2n!}c_{\alpha_1...
\alpha_{2n}}(x)a^{+\alpha_1}\ldots a^{+\alpha_{2n}}\star |0\rangle\langle0|,
\nonumber\\
|\Phi_1\rangle&=&\sum^\infty_{n=0}\frac{1}{(2n+1)!}c_{\alpha_1...
\alpha_{2n+1}}(x)a^{+\alpha_1}\ldots a^{+\alpha_{2n+1}}
\star |0\rangle\langle0|.\label{splitt}\end{aligned}$$ Each component $c_{\alpha_1\ldots\alpha_\ell}(x)$ is symmetric in all its indices. Moreover, it is natural to consider the component fields $c_{\alpha_1\ldots\alpha_\ell}(x)$ as bosonic if $\ell$ is even and fermionic if $\ell$ is odd.
The dynamics of the matter fields in a Chern-Simons background ${{\cal A}}$ is encoded in the equations of motion, $$\label{phiflat}
d|\Phi\rangle+{{\cal A}}\star|\Phi\rangle=0.$$ In terms of the components $c_{\alpha_1\ldots\alpha_\ell}$, these equations become $$\begin{aligned}
2\partial_\mu c_{\alpha_1 \ldots
\alpha_\ell}&=&c_{\alpha_1\ldots\alpha_\ell\beta_1\beta_2}
e_\mu{}^{\beta_1\beta_2}
-\frac{\ell(\ell-1)}{4{{\cal R}}^2}c_{\alpha_1\ldots\alpha_{\ell-2}}e_{\mu
\alpha_{\ell-1}\alpha_\ell}\nonumber\\
&&{}+\frac{\ell(\ell-1)}{2{{\cal R}}^2}c_{\alpha_1\ldots\alpha_{\ell-2}}\sigma_{2\,
\alpha_{\ell-1}\alpha_\ell}e_\mu{}^2-(\ell+1)\omega_\mu{}^2\sigma_2{}^\beta
{}_{\alpha_1}c_{\alpha_2\ldots\alpha_\ell\beta},\label{compeq}\end{aligned}$$ where $e_\mu{}^{\alpha\beta}=e_\mu^a\sigma_a{}^{\alpha\beta}$, and the symmetrization over $\alpha_1\ldots\alpha_\ell$ is kept implicit on the right-hand side of (\[compeq\]). Assuming that the vielbein is invertible, the infinite chain of equations (\[compeq\]) can be used to solve algebraically for all the higher components $c_{\alpha_1\ldots\alpha_\ell}$ in terms of only two independend fields, given by the two lowest components $c$ and $c_\alpha$. The full equation of motion (\[phiflat\]) thus reduces to two dynamical equations for these remaining fields, $$\begin{aligned}
\left(g^{\mu\nu}D_\mu D_\nu-\frac{1}{4{{\cal R}}^2}\right)c(x)&=&0,\label{kgeq}\\
e^\mu_a\sigma^{a\beta}_{\phantom{a\beta}\alpha}\left(\partial_\mu
c_\beta+\omega_\mu^{\phantom{\mu}2}
\sigma_{2\phantom{\gamma}\beta}^{\phantom{2}\gamma}c_\gamma\right)&=&0.
\label{direq}\end{aligned}$$ Here $g^{\mu\nu}$ is the inverse metric and $D_\mu$ is the covariant derivative, built from the spin connection and the vielbein.
Some comments are now in order:
- Eqns. (\[kgeq\]) and (\[direq\]) are the Klein-Gordon and the Dirac equation for a massless spinor and scalar on $AdS_2\times S^1$. All the components $c_{\alpha_1\ldots\alpha_\ell}$ with $\ell\geq 2$ are formed from derivatives of $c$ or $c_\alpha$, and thus do not constitute separate degrees of freedom themselves.
- The equations of motion (\[compeq\]) decouple components $c_{\alpha_1\ldots
\alpha_\ell}$ with $\ell$ even from those with $\ell$ odd. Thus, in our theory with ${{\cal W}}_0$ gauge symmetry, the matter multiplet ${|\Phi\rangle}$ of (\[matterf\]) is reducible, and can be split into two irreducible components given by ${|\Phi_0\rangle}$ and ${|\Phi_1\rangle}$ of (\[splitt\]). From the perspective of the subalgebra of quadratic charges $SO(3,2)\sim Sp(4,{{\bf R}})$, ${|\Phi_0\rangle}$ and ${|\Phi_1\rangle}$ correspond essentially to the two irreducible metaplectic representations of $Sp(4,{{\bf R}})$, sometimes called Di and Rac in the representation theory of this algebra.
In order to match our expected spectrum of noncritical M-theory in the conformal limit, as given in (\[Rspectrum\]), we keep ${|\Phi_1\rangle}$ which contains the propagating massless Dirac fermion, but throw away ${|\Phi_0\rangle}$ which would contain the scalar.
Second-quantized propagating matter can thus be coupled to Chern-Simons theory at the level of the equations of motion. All matter interactions are mediated by the coupling to topological Chern-Simons theory, which itself does not propagate any physical degrees of freedom. It is unclear, however, how to formulate an action principle for this system of equations of motion. Interestingly, Vasiliev [*et al.*]{} [@Prokushkin:1999gc] have suggested that such an action principle would have to be formulated not on ${{\cal M}}$ but on a space that also includes $a_\alpha$.
Relation to a Supersymmetric Higher-Spin Theory {#susy}
-----------------------------------------------
It is remarkable that the geometric properties of $AdS_2\times S^1$ that are required to match the Fermi liquid spectrum in the conformal limit, are precisely such that the radius of $S^1$ equals the minimum possible value compatible with unbroken supersymmetry of the solution. This suggests that the effective theory that we propose for the description of the conformal limit of the noncritical M-theory vacuum is a simple ${{\bf Z}}_2$ orbifold of a supersymmetric theory, with our $AdS_2\times S^1$ as a maximally supersymmetric solution.
Such a supersymmetic extension of the effective theory of higher-spin Chern-Simons plus propagating massless matter can be easily constructed. In fact, it has already been written down by Shaynkman and Vasiliev in [@Shaynkman:2001ip].[^11] The gauge symmetry of this supersymmetric theory is given by the higher-spin conformal superalgebra $\widetilde{{\cal W}}$. The Chern-Simons gauge field $\widetilde{{\cal A}}$ is now in the adjoint of $\widetilde{{\cal W}}$. It can be decomposed into components, $$\widetilde{{\cal A}}(x|a,a^+)=\sum_{\ell=0}^\infty\sum_{m=0}^{\ell}\frac{1}{m!
(\ell-m)!}\widetilde{{\cal A}}_{\alpha_1\ldots\alpha_m}{}^{\alpha_{m+1}\ldots
\alpha_{\ell}}(x)a^{+\alpha_1}\ldots a^{+\alpha_m}a_{\alpha_{m+1}}\ldots
a_{\alpha_{\ell}}.$$ The component one-forms $\widetilde{{\cal A}}_{\alpha_1\ldots
\alpha_m}{}^{\alpha_{m+1}\ldots\alpha_{\ell}}(x)$ are bosons if $\ell$ is even, and fermions for $\ell$ odd. The lowest fermionic component of the gauge field corresponds to terms linear in $a_\alpha$ and $a^{+\alpha}$, and they describe the massless spin-$3/2$ gravitino of ${{\cal N}}=1$ conformal supergravity in $2+1$ dimensions. The full theory is then a higher-spin extension of $OSp(1|4,{{\bf R}})$ Chern-Simons supergravity.
The Chern-Simons gauge sector is coupled to the massless matter supermultiplet, described by ${|\Phi\rangle}={|\Phi_0\rangle}+{|\Phi_1\rangle}$ of Section 5.3. As we have seen there, in the $AdS_2\times S^1$ background ${|\Phi_0\rangle}$ gives rise to a propagating boson and ${|\Phi_1\rangle}$ describes a propagating fermion. Due to the periodicity properties of the generators of global symmetries, $AdS_2\times S^1$ is a supersymmetric solution of this theory if the radius of $S^1$ is an even multiple of the $AdS_2$ radius.
On this ${{\cal N}}=1$ supersymmetric theory, we can define the action of the orbifold group ${{\bf Z}}_2=\{1,\Omega\}$ via $$\Omega:\quad x^\mu\to x^\mu,\quad a_\alpha\to-a_\alpha,\quad
a^{+\alpha}\to-a^{+\alpha},$$ and extend it to the action on the fields by $$\begin{aligned}
\Omega:\quad{{\cal A}}(x|a,a^+)&\to&{{\cal A}}(x|-a,-a^+),\\
\quad{|\Phi(x|a^+)\rangle}&\to&-{|\Phi(x|-a^+)\rangle}.\end{aligned}$$ The orbifold projection by $\Omega$ projects out the odd-degree part of the gauge field $\widetilde{{\cal A}}$ and the even-degree part ${|\Phi_0\rangle}$ of the matter multiplet. Thus, our effective theory describing the conformal limit of the noncritical M-theory vacuum is an orbifold of the supersymmetric theory under this ${{\bf Z}}_2$ action.
This fact can perhaps be seen as another check of our proposal, for the following reason. Just as in the critical spacetime dimension, two-dimensional Type 0A and 0B string theories are believed to be related to supersymmetric Type II cousins [@McGreevy:2003dn; @Verlinde:2004gt; @Takayanagi:2004ge; @Seiberg:2005bx] by an orbifold construction. Naturally, such supersymmetric two-dimensional theories could also be tied together into a supersymmetric version of noncritical M-theory, much like Type 0A and 0B backgrounds were in [@Horava:2005tt]. If such a supersymmetric extension of noncritical M-theory exists, one would again expect it to be related to the nonsupersymmetric version by an orbifold. It is intriguing that in the conformal limit, such a simple and natural extension does exist, at least at the level of the effective spacetime description, and that $AdS_2\times S^1$ extends naturally to a supersymmetric solution of it.[^12] It would certainly be interesting to investigate these issues further.
We conclude this Section with a few comments on the precise choice of the gauge group. In particular, the quadratic charges in ${{\cal W}}_0$ generate the Lie algebra of $SO(3,2)$, and the question is which of its covers should be chosen. Conformal gravity in $2+1$ dimensions arises naturally as the gauge theory of $SO(3,2)$. However, since our effective theory couples the gravity sector to a propagating fermion, the natural gauge group should be at least as large as ${\it Spin}(3,2)$, the double-cover of $SO(3,2)$. The ${\it Spin}(3,2)$ group is isomorphic to $Sp(4,{{\bf R}})$. Since $Sp(4,{{\bf R}})$ also naturally arises on the Fermi-liquid side of noncritical M-theory, it might be tempting to propose it as the correct gauge group. However, $Sp(4,{{\bf R}})$ itself has a nontrivial topological structure, with $\pi_1(Sp(4,{{\bf R}}))={{\bf Z}}$. As a result, it has a unique connected double cover, known as the “metaplectic group” ${\it Mp}(4,{{\bf R}})$. The metaplectic group is [*not*]{} a matrix group; its smallest faithful representation, known as the “metaplectic representation,” is infinite-dimensional. In fact, this metaplectic representation is equivalent to the Fock space representation of the $\hat a_\alpha$ and $\hat a^{+\alpha}$ operator algebra! Thus, the metaplectic representation plays an essential role in the construction of our propagating matter multiplet. Since it is only a projective representation of $Sp(4,{{\bf R}})$, it is natural to expect that the correct gauge group is not $Sp(4,{{\bf R}})$ but its double cover ${\it Mp}(4,{{\bf R}})$. Perhaps, in noncritical M-theory in $2+1$ dimensions, “M” stands for “metaplectic”!
Conclusions
===========
In this paper, we have presented evidence suggesting that the ground state of noncritical M-theory, in the conformal limit, describes $AdS_2\times S^1$ spacetime. The symmetries and spectrum of propagating modes in this limit are compatible with an effective theory given by the infinite higher-spin extension Chern-Simons gravity in $2+1$ dimensions, coupled to a propagating massless Dirac fermion. This correspondence represents the simplest M-theory analog of the duality relation between the Liouville spacetime dimension and the eigenvalue coordinate as known from two-dimensional string theory.
This correspondence leads to a nice matching of the “extra dimension” of noncritical M-theory in the $AdS_2\times S^1$ spacetime and in the Fermi liquid. In the Fermi liquid picture, the “extra dimension” corresponds to the orbits of the $U(1)$ rotations of the rigid plane populated by the eigenvalues. On the spacetime side, this $U(1)$ group becomes the group of translations along the $S^1$ factor of the $AdS_2\times S^1$ background, and the “extra dimension” acquires its traditional role as in critical string/M-theory. In particular, the radius of the $S^1$ as measured by the spacetime metric is constant everywhere along $AdS_2$. The fact that such a simple and intuitive picture emerges on the effective spacetime side lends further support to the original proposal of [@Horava:2005tt] that the extra dimension of M-theory indeed corresponds to the angular coordinate on the plane populated by the Fermi liquid.
The conformal limit of the theory involves sending $\alpha'\to\infty$, and one can think of it as a certain high energy limit of the theory. We have argued that in this limit, the effective spacetime theory is given by a higher-spin Chern-Simons gravity, coupled to a propagating fermionic degree of freedom. It is intriguing that such a connection to higher-spin gauge theories emerges in the high-energy limit of noncritical M-theory. Indeed, the possibility of a close relation between higher-spin theories and the high-energy limit of critical string theory has been suspected for a long time (see, [*e.g.*]{}, [@Bekaert:2005vh; @Sundborg:2000wp; @Witten:2001js; @Sezgin:2002rt; @Klebanov:2002ja; @Girardello:2002pp; @Sagnotti:2003qa; @Petkou:2004nu; @Sagnotti:2005ns; @Francia:2006hp] and references therein). We believe that the exactly solvable setting of noncritical M-theory in $2+1$ dimensions now provides an explicit testing ground for such ideas.[^13]
Having found a dual interpretation of the ground-state solution of the Fermi liquid system in terms of a gravitational $AdS_2\times S^1$ background, one can turn the relation around, and ask the following question: what is, from the perspective of an observer in $AdS_2\times S^1$, the interpretation of the dual space on which the Fermi liquid resides? Since the double-scaling limit of the Fermi liquid involves taking a semiclassical limit, it is even more natural to look for an interpretation of the full phase space ${{\cal T}}$, as parametrized either by $(\lambda_i,p_i)$ of Section 2 or equivalently by the conserved charges $(a_\alpha,a^{+\alpha})$ of Section 5. Amusingly, it turns out that this phase space is precisely the twistor space associated with the $2+1$ dimensional conformal group $SO(3,2)$. This relation can be made explicit by defining the twistor transform, which associates to a given element $(a_\alpha,a^{+\beta})$ of the twistor space a null vector $p^\mu$ at a spacetime point $x^\mu$, via $$p^\mu=\sigma^{\mu\,\alpha\beta}a_\alpha a_\beta,\qquad
a^{+\alpha}=x^\mu\sigma_{\mu}{}^{\alpha\beta}a_\beta.$$ Perhaps the relation between the physical spacetime and the space of the Fermi liquid is simply related to such a twistor transform? In any case, whether or not the twistor transform proves useful in this context, it is certainly true that the Fermi liquid lives on the twistor space associated with the conformal group $SO(3,2)$ of the $2+1$-dimensional spacetime.[^14] This twistor perspective might shed some new light on another mysterious aspect of noncritical M-theory. As shown in [@Horava:2005wm], noncritical M-theory at finite temperature is closely related to the A-model topological strings on the resolved conifold, and plays essentially the role expected of topological M-theory. In turn, topological M-theory has been conjecturally described by an effective gauge theory action in seven dimensions [@Dijkgraaf:2004te]. It is puzzling why noncritical M-theory in $2+1$ dimensions should reproduce results expected of this seven-dimensional theory. We do not have answers to this question, but we at least wish to point out one reason why seven dimensions should be relevant for noncritical M-theory. As mentioned in Section 5.3, the equations of motion for the unfolded fermion (\[direq\]) do not follow from any obvious action in three spacetime dimensions. Attempts by Vasiliev [*et al.*]{} [@Prokushkin:1999gc] to resolve this problem suggest that an action might exist, but it would be naturally formulated on a bigger space that also includes the twistor coordinates. In particular, one can view the Chern-Simons gauge field ${{\cal A}}(x|a,a^+)$ of (\[expaa\]) as living on the seven-dimensional space ${{\cal T}}\times{{\cal M}}$, the total space of the double fibration ${{\cal T}}\leftarrow{{\cal T}}\times{{\cal M}}\rightarrow{{\cal M}}$ familiar from twistor theory.
Various interesting open questions still remain. In particular, it would be interesting to understand how to restore finite $\alpha'$ and study the full dynamics of the theory away from the conformal limit. This should include the dynamics of other moduli, such as the ratio of the radii of $AdS_2\times S^1$. It would also be nice to understand what is the role, if any, of the Type 0A backgrounds with long strings and both values of the RR flux in the context of noncritical M-theory. Answering this last question may require an understanding of the matrix model from which our Fermi liquid picture would follow. Our effective description of the conformal limit of the theory in terms of $AdS_2\times S^1$ is a strong hint that a dual matrix model should exist, at least in this limit, in the form of a conformal quantum mechanics with a global $U(1)$ symmetry.
We wish to thank Ofer Aharony, Eric Gimon, Tommy Levi, Oleg Lunin and Tassos Petkou for useful discussions. This material is based upon work supported in part by NSF grants PHY-0244900 and PHY-0555662, DOE grant DE-AC03-76SF00098, an NSF Graduate Research Fellowship, and the Berkeley Center for Theoretical Physics.
Appendix A: Free Fermion Spectrum on $S^1$ Fibered over $AdS_2$ {#appendix}
===============================================================
We wish to consider the spectrum of a free massless Dirac fermion on an $S^1$ fibration over $AdS_2$, assuming that the fibration preserves the $SL(2,{{\bf R}})$ symmetries of the base. We work in global coordinates on $AdS_2$, and will study the spectrum with respect to the evolution in the global time on $AdS_2$. The most general metric that preserves the $SL(2,{{\bf R}})$ symmetry of $AdS_2$ is $$ds^2=-{{\cal R}}^2\cosh^2\rho\,dt^2+{{\cal R}}^2d\rho^2 +{{\cal R}}^2(\gamma dy+\alpha\sinh\rho\,
dt)^2.$$ The arbitrary constant $\alpha$ effectively measures the Kaluza-Klein flux of the off-diagonal metric components in the fibration. The other arbitrary constant $\gamma$ parameterizes the ratio of the radius of $S^1$ and the curvature radius of $AdS_2$ and $S^1$ components; we take the $S^1$ coordinate $y$ to run from 0 to $2\pi$. Setting $\alpha=0$ would reproduce the direct product metric on $AdS_2\times S^1$. Alternatively, the $AdS_3$ Hopf fibration would be obtained by setting $\alpha=\gamma=1$ and allowing $y$ to run over ${{\bf R}}$.
We will keep $\alpha$ and $\gamma$ arbitrary, in order to explore the full set of fibrations. Additionally we will not impose boundary conditions yet to preserve generality. The vielbein components $$\begin{aligned}
e^0&=&{{\cal R}}\cosh \rho dt\\
e^1&=&{{\cal R}}d\rho\\
e^2&=&{{\cal R}}\gamma dy+{{\cal R}}\alpha\sinh\rho dt\end{aligned}$$ imply the spin connection components $$\begin{array}{l}
\omega_{01}=\frac{\alpha\gamma}{2}dy+\sinh\rho\left(\frac{\alpha^2}{2}
-1\right)dt\\
\omega_{02}=\frac{\alpha}{2}d\rho\\
\omega_{12}=-\frac{\alpha\cosh\rho}{2}dt.
\end{array}$$ We will use the following $\gamma$ matrices, which have no explicit factors of $i$: $$\gamma^0=i\sigma^2,\qquad\gamma^1=\sigma^1,\qquad\gamma^2=\sigma^3.$$ Next, we need to calculate $\Gamma_\mu=\frac{1}{8}\omega_\mu{}^{ba}
\sigma^{ab}$: $$\begin{array}{l}
\Gamma_y= -\frac{\alpha\gamma}{4}\sigma^3\\
\Gamma_\rho= \frac{\alpha}{4}\sigma^1\\
\Gamma_t=
\frac{1}{2}\left[(1-\frac{\alpha^2}{2})\right]\sinh\rho\sigma^3
-\frac{\alpha}{2}\cosh\rho i\sigma^2
\end{array}$$ Finally, the massless Dirac equation $$i\gamma^\mu\nabla_\mu\psi=(i\gamma^\mu\partial_\mu
+i\gamma^\mu\Gamma_\mu)\psi=0$$ becomes $$\left(\begin{array}{cc}
\cosh\rho\frac{1}{\gamma}\partial_y+\frac{\alpha}{4}\cosh\rho &
\partial_t+\cosh\rho\partial_\rho-\frac{\alpha}{\gamma}\sinh\rho\partial_y
-\frac{1}{2}\sinh\rho\\
-\partial_t+\cosh\rho\partial_\rho+\frac{\alpha}{\gamma}\sinh\rho\partial_y
-\frac{1}{2}\sinh\rho
&-\cosh\rho\frac{1}{\gamma}\partial_y+\frac{\alpha}{4}\cosh\rho
\end{array}\right)\psi=0.$$ We will make the coordinate change $\cosh\rho=1/\cos\theta$ additionally stipulating $\sinh\rho=-\tan\theta$. This gives $$\left(\begin{array}{cc}
\sec\theta\frac{1}{\gamma}\partial_y+\frac{\alpha}{4}\sec\theta &
\partial_t-\partial_\theta
+\frac{\tan\theta}{2}+\frac{\alpha}{\gamma}\tan\theta\partial_y\\
-\partial_t-\partial_\theta
+\frac{\tan\theta}{2}-\frac{\alpha}{\gamma}\tan\theta\partial_y &
-\sec\theta\frac{1}{\gamma}\partial_y+\frac{\alpha}{4}\sec\theta
\end{array}
\right)\psi=0.$$ Now, let us choose $\psi$ such that $$\psi=e^{i\beta y}e^{-i\omega t}Y(\theta)\left(
\begin{array}{c}
e^{i\omega\theta}X_1(\theta)u(\theta)\\
e^{-i\omega\theta}X_2(\theta)v(\theta)
\end{array}
\right),$$ where $$\partial_\theta Y=\frac{1}{2}\tan\theta Y, \quad\partial_\theta
X_1=-i\frac{\alpha\beta}{\gamma}\tan\theta X_1, \quad\partial_\theta
X_2=i\frac{\alpha\beta}{\gamma}\tan\theta X_2.$$ This choice of $\psi$ reduces our Dirac equation to $$\begin{aligned}
\left(i\frac{\beta}{\gamma}+\frac{\alpha}{4}\right)\sec\theta
e^{2i\omega\theta}\frac{X_1}{X_2}u-\partial_\theta v&=&0,\\
-\frac{X_1}{X_2}e^{2i\omega\theta}\partial_\theta u
+\left(-i\frac{\beta}{\gamma}+\frac{\alpha}{4}\right)\sec\theta v&=&0.\end{aligned}$$ These coupled equations reduce to the single second-degree equation $$\left(\frac{\alpha^2}{16}+\left(\frac{\beta}{\gamma}\right)^2\right)u
=\left(2i\omega\cos^2\theta-(1+2i\frac{\alpha\beta}{\gamma})\sin\theta
\cos\theta\right)\partial_\theta u + \cos^2\theta\partial_\theta^2u.$$ Following [@Aharony:2005hm], we change variables to $z=(1+\tan\theta)/2$ which gives us $$\label{z}
\left(\frac{\alpha^2}{16}+\frac{\beta^2}{\gamma^2}\right)u+z(1-z)u''
+\left[\omega+\left(1-2i\alpha\frac{\beta}{\gamma}\right)\left(\frac{1}{2}
-z\right)\right]=0.$$ Note that the $\alpha=0$ case reduces to Eqn. (A.17) in [@Aharony:2005hm], provided we set $\frac{\beta}{\gamma}=m{{\cal R}}$. Although our Dirac norm contains instead a factor of $1/\cos^3\theta$, we still have the same requirement for $u$ to vanish at $\theta=\pm
\pi/2$, or $z\to\pm\infty$. Thus, for the particular case of $\alpha=0$, we can use the result of [@Aharony:2005hm] that $$|\omega|=|\frac{\beta}{\gamma}|+\frac{1}{2}+n.$$ Now, even in the $\alpha=0$ case, we must also consider the restrictions the boundary conditions for $\psi$ in $y$ put on $\beta$. Assuming periodicity of the fermions, we find $$\beta=\frac{q}{2},$$ which gives us exactly the spectrum (\[Rspectrum\]) if we also set $\gamma=2$.
We should also check that no other combination of $\alpha$ and $\gamma$ produces the same spectrum. Let us proceed by comparing Equation (\[z\]) to the generic form for a hypergeometric equation $$z(1-z)u''+\left[c-(a+b+1)z\right]u'-abu=0.$$ We can match Equation (\[z\]) to this equation by choosing $$a=-i\alpha\frac{\beta}{\gamma}-s,\qquad
b=-i\alpha\frac{\beta}{\gamma}+s,\qquad
c=\omega+\frac{1}{2}-i\alpha\frac{\beta}{\gamma}$$ with $$s=\sqrt{\frac{\alpha^2}{16}-\frac{\alpha^2\beta^2}{\gamma^2}
+\frac{\beta^2}{\gamma^2}}.$$ A similar analysis to that done in [@Aharony:2005hm] shows us the spectrum must be $$|\omega|=n+\frac{1}{2}+s$$ for $n$ a non-negative integer. Now, we would like to see if we can match $s$ to the set of half integers $q/2$. For non-compact $y$, we find a continuous spectrum; for compact $y$, presuming $y \in
[0,2\pi]$, we find $\beta=q/2$, where $q$ ranges over the integers. One can quickly check that only $\alpha=0$ and $\gamma=2$ will allow $s$ to range over the set of half integers given by $q/2$. Thus, only the direct product spacetime with equal characteristic size for $AdS_2$ and $S^1$ will produce the desired spectrum.
[^1]: The full Type 0A theory in two dimensions has two separate RR fluxes [@Douglas:2003up]. Making both nonzero simultaneously requires the presence of a nonzero number of long strings [@Maldacena:2005he]. Only one of the RR-fluxes – identified with a D0-charge – plays a role in our definition of noncritical M-theory. Whether or not the noncritical M-theory framework can be extended to incorporate the long strings and both RR fluxes is an interesting open question.
[^2]: The coordinates $\lambda_i$ before and after the double-scaling limit differ by an overall rescaling factor; in order to avoid notational clutter, we keep this factor implicit, and refer the reader to [@Horava:2005tt] for the exact technical details of the double-scaling limit.
[^3]: In calling this generator $\widetilde H$, we differ slightly from the commonly accepted convention in the literature.
[^4]: Of course, this phenomenon is already present in two-dimensional string theory, in the duality between its Fermi liquid and spacetime descriptions, and therefore does not represent a novelty of noncritical M-theory. What is perhaps new is that such a duality extends to a dimension higher than $1+1$.
[^5]: AdS/CFT correspondence for $AdS_n\times S^1$ spaces has been previously encountered in [@Klebanov:2004ya] in the connection with higher-dimensional noncritical superstrings.
[^6]: Here “Tr” is the trace defined via the unique quadratic invariant on the simple group $SO(3,2)$. The coupling $k$ is quantized because $\pi_3(SO(3,2))=\pi_3(SO(3))$ is nontrivial. The precise quantization condition for $k$ will depend on the exact choice of the gauge group, [*i.e.*]{}, on whether we choose the $SO(3,2)$ group itself or one of its covers. We shall briefly return to this point in Section 5.4 below.
[^7]: We use essentially the same notation as [@Horne:1988jf], with the only exception that we refer to the gauge fields associated to the special conformal transformations $\mathcal K_a$ as $\zeta_\mu^a$, while [@Horne:1988jf] used $\lambda_\mu^a$.
[^8]: Despite appearances, $\hat a$ and $\hat a^+$ are not Hermitian conjugates of each other [@Shaynkman:2001ip].
[^9]: The “${{\rm Tr}}$” in (\[hcs\]) is defined as the bosonic restriction to ${{\cal W}}_0$ of the natural supertrace defined on $\widetilde{{\cal W}}$ (see [@Vasiliev:1986qx] and also Section 3 of [@Prokushkin:1999gc]). Conversely, one could try to keep the odd-degree generators as bosonic symmetries, [*i.e.*]{}, replace their anticommutation relations with commutation relations, as defined again via the $\star$-product algebra. However, the hypothetical gauge theory of ${{\cal W}}$ would contain bosonic gauge field components of half-integer spins, leading to many conceptual difficulties; consequently, we will not consider this option in this paper.
[^10]: Note that only a smaller algebra corresponds to isometries of the background.
[^11]: In [@Shaynkman:2001ip], the focus is on ${{\cal N}}=2$ supersymmetric theory; the ${{\cal N}}=1$ version can be obtained by setting their $\hat k$ to zero.
[^12]: A ${{\bf Z}}_2$-twisted version of the ground state of noncritical M-theory was studied in Section 9.4 of [@Horava:2005tt], where it was shown that its vacuum energy vanishes to all orders in the expansion in the coupling constant $1/\mu$. Whether this feature is explained by some form of hidden supersymmetry in this state is not known.
[^13]: It also seems worth pointing out that our effective field theory for noncritical M-theory in $2+1$ dimensions is remarkably similar to the “holographic field theory” proposal of [@Horava:1997dd]. Indeed, the eleven-dimensional theory of [@Horava:1997dd] is a Chern-Simons gauge theory based on a higher-dimensional conformal superalgebra ${it OSp}(1|32)
\times {\it OSp}(1|32)$, coupled to propagating (fermionic) matter. It thus appears that noncritical M-theory in $2+1$ dimensions might be a baby version of “holographic field theory” in the sense of [@Horava:1997dd].
[^14]: This reference may need a bit of explanation. In his after-dinner remarks at a Strings conference at (K)ITP Santa Barbara in the mid-1990’s, Joe Polchinski proposed a “Fermi liquid on twistor space” as a half-joking answer to the question of “what is string theory?”.
|
= cmr9
astro-ph/0008107\
August 2000
0.5 cm
[**Planck-scale deformation of Lorentz symmetry as a\
solution to the UHECR and the TeV-$\gamma$ paradoxes**]{}
1.5 cm
[**Giovanni AMELINO-CAMELIA**]{}$^a$ and [**Tsvi PIRAN**]{}$^b$\
[*$^a$Dipart. Fisica, Univ. Roma “La Sapienza”, P.le Moro 2, 00185 Roma, Italy*]{}\
[*$^b$Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel*]{}
[**ABSTRACT**]{}
=0.6in =0.6in
One of the most puzzling current experimental physics paradoxes is the arrival on Earth of Ultra High Energy Cosmic Rays (UHECRs) with energies above the GZK threshold ($5 {\times} 10^{19}$eV). Photopion production by CMBR photons should reduce the energy of these protons below this level. The recent observation of 20TeV photons from Mk 501 (a BL Lac object at a distance of 150Mpc) is another somewhat similar paradox. These high energy photons should have disappeared due to pair production with IR background photons. A common feature of these two paradoxes is that they can both be seen as “threshold anomalies": energies corresponding to an expected threshold (pion production or pair creation) are reached but the threshold is not observed. Several (relatively speculative) models have been proposed for the UHECR paradox. No solution has yet been proposed for the TeV-$\gamma$ paradox. Remarkably, the single drastic assumption of a violation of ordinary Lorentz invariance would resolve both paradoxes. We present here a formalism for the systematic description of the type of Lorentz-invariance deformation (LID) that could be induced by non-trivial short-distance structure of space-time, and we show that this formalism is well suited for comparison of experimental data with LID predictions. We use the UHECR and TeV-$\gamma$ data, as well as upper bounds on time-of-flight differences between photons of different energies, to constrain the parameter space of the LID. A model with only two free parameters, an energy scale and a dimensionless parameter characterizing the functional dependence on the energy scale, is shown to be sufficient to solve both the UHECR and the TeV-$\gamma$ threshold anomalies while satisfying the time-of-flight bounds. The allowed region of the two-parameter space is relatively small, but, remarkably, it fits perfectly the expectations of the quantum-gravity-motivated space-time models known to support such deformations of Lorentz invariance: an integer value of the dimensionless parameter and a characteristic energy scale constrained to a narrow interval in the neighborhood of the Planck scale.
plus .5pt minus .5pt
Introduction
============
Significant evidence has accumulated in recent years suggesting that in two different regimes, Ultra High Energy Cosmic Rays (UHECRs) and multi-TeV photons, the universe is more transparent than what it was expected to be. UHECRs interact with the Cosmic Microwave Background Radiation (CMBR) and produce pions. TeV photons interact with the Infra Red (IR) photons and produce electron-positron pairs. These interactions should make observations of UHECRs with $E > 5 {\times}
10^{19}$eV (the GZK limit) [@GZK] or of gamma-rays with $E >
20$TeV from distant sources unlikely [@Nikishov62; @Gould67; @Stecker92]. Still UHECRs above the GZK limit and 20TeV photons from Mk 501 are observed.
Numerous solutions have been proposed for the UHECR paradox (see [@Olinto] for a recent review). Most of these solutions require new Physics. There are practically no proposals concerning the TeV-$\gamma$ paradox (see however, [@Harwit99]). It is striking that there are some common features in these otherwise apparently unrelated paradoxes. In both cases low energy photons interact with high energy particles. The reactions should take place because when Lorentz transformed to the CM frame the low energy photon have sufficient energy to overcome an intrinsic threshold. In both cases the CM energies are rather modest ($\sim 100 $ MeV for UHECRs and $\sim 1$ MeV for the TeV photons) and the physical processes involved are extremely well understood and measured in the laboratory. In both cases we observe particles above a seemingly robust threshold and the observations can be considered as a “threshold anomaly”. It is remarkable that in spite of these similarities at present there is only one mechanism that could resolve both paradoxes: a mechanism based on the single, however drastic, assumption of a violation of ordinary Lorentz invariance.
The possibility that the cosmic-ray threshold anomaly could be a signal of violation of ordinary Lorentz invariance had already been emphasized in Refs. [@Gonzalez; @colgla; @ita; @bertli; @sato]. In this work we combine these earlier points with the very recent suggestion [@kifu; @kluz; @Protheroe_Meyer] that Lorentz-invariance violation could be the origin of the TeV-$\gamma$ threshold anomaly. We analyze a general phenomenological framework for the description of the type of Lorentz-invariance deformation (LID) that could be induced by non-trivial short-distance structure of space-time, and we ask whether there are choices of LID parameters that would simultaneously solve the two threshold anomalies while satisfying the constraints imposed by the fact that the results of experimental searches [@schaef; @billetal] of energy-dependent relative delays between the times of arrival of simultaneously emitted photons are still consistent with ordinary Lorentz invariance. We obtain, under these assumptions, strict limits on the possible parameter space of LID. The fact that one is at all able to give a quantitative description of both threshold anomalies with a simple two-parameter LID model provides encouragement for the interpretation of the data as a sign of LID; moreover, it is quite remarkable that the values expected from quantum-gravity considerations (most notably the energy scale characterizing the deformation being given by the Planck scale) are in agreement with the strict limits we derive.
We review in Sections 2 and 3 the observational background and the theoretical problems related to the observations of UHECRs (Section 2) and TeV photons (Section 3). In Section 4 we describe a special (two parameter) model for LID and we obtain limits on these two parameters. In Section 5 we describe a more general five-parameter LID formalism and again we constrain the parameter space with the available data. In Section 6 we compare our formalism with the Coleman and Glashow [@colgla] formalism for Planck-scale-independent Lorentz invariance violations. We summarize our results in section 7. Appendix A is devoted to the $\kappa$-Minkowski space-time, which is an example of quantum-gravity motivated space-time that allows a simple illustration of some of the structures here considered.
UHECRs and the GZK paradox
==========================
The high energy cosmic rays (CR) spectrum depicts a clear break at $\sim 5 {\times} 10^{18}$eV. This break is accompanied by a transition in the CR composition from nuclei to protons. Above this break the spectrum behaves (with a decreasing statistical certainty due to the small number of events) as a single power law $N(E) \sim E^{-2.7}$ all the way up to $3.2 {\times} 10^{20}$eV [@flyseye], the highest energy CR observed so far.
A sufficiently energetic CMBR photon, at the tail of the black body thermal distribution, is seen in the rest frame of an Ultra High Energy (UHE) proton with $E >5 {\times} 10^{19}$eV as a $> 140$MeV photon, above the threshold for pion production. UHE protons should loose energy due to photopion production and should slow down until their energy is below the GZK energy[^1]. The process stops when CMBR photons energetic enough to produce pions are not sufficiently abundant [@GZK]. The proton’s mean free path in the CMBR decreases exponentially with energy (down to a few Mpc) above the GZK limit ($\sim 5 {\times} 10^{19}$eV). Yet more than 15 CRs have been observed with nominal energies at or above $10^{20} {\pm} 30\%$ eV[@AGASA98; @watson:HEPiN].
There are no astrophysical sources capable of accelerating particles to such energies within a few tens of Mpc from us (at least not in the direction of the observed UHECRs). Furthermore if the CRs are produced homogeneously in space and time, we would expect a break in the CR spectrum around the GZK threshold: below the threshold we would observe CRs from the whole universe; above the threshold we would observe CRs only from the nearest few Mpc. The corresponding jump by a factor of $\sim 30-100$ in the extrapolated number counts above and below the threshold, is not seen.
Numerous solutions have been proposed to resolve the GZK paradox (see [@Olinto] for a recent review). These solutions include, among others, new physics solutions like the decay of topological defects, weakly interacting messengers like $S_0$ or neutrinos with anomalous cross sections at high energies (the ‘Z-burst" model). Conventional astrophysics solutions like acceleration of UHECRs by GRBs or local AGNs require the ad hoc assumption that Earth is located in a not generic place in space-time (we should be nearer than average to a typical source by a factor of 5) as well as very strong intergalactic magnetic fields [@Farrar_Piran2]. Another conventional solution, the acceleration of Fe nuclei by magnetars in the galactic halo, requires a new, otherwise unobserved, population of galactic halo objects. Clearly there is no simple conservative solution to this puzzle.
From the point of view of our LID phenomenology it is important to notice that for a solution of the GZK paradox it would be necessary (and sufficient) for LID to push the threshold energy upwards by a factor of six. In fact, the mean free path of a $5 {\times} 10^{19}$eV proton is almost a Gpc, while the highest observed UHECR energy is $3.2 {\times} 10^{20}$eV.
TeV photons from Mk 501 and Mk 421
==================================
HEGRA has detected high-energy photons with a spectrum ranging up to 24 TeV [@Aharonian99] from Markarian 501 (Mk 501), a BL Lac object at a redshift of 0.034 ($\sim 157$ Mpc). This observation indicates a second paradox of a similar nature. A high energy photon propagating in the intergalactic space can interact with an IR background photon and produce an electron-positron pair if the CM energy is above $2m_ec^2$. The maximal wavelength of an IR photon that could create a pair with a 10 TeV photon is 40$\mu m$. As the cross section for pair creation peaks at a center of mass energy of about $3
m_e c^2$, 10 TeV photons are most sensitive to 30$\mu$M IR photon and the mean free path of these photons depends on the spectrum of the IR photons at the $\sim 15-40 \mu$M range. These wavelengths scale like 10TeV/$E$ for different energies.
There have been several attempts to model the IR background resulting from different cosmological evolutionary models [@Primack99; @Malkan98; @Dwek98; @Fall]. Recently, new data from DIRBE at 2.2$\mu M$ [@Wright00], at 60 and 100 $\mu$M [@Finkbeiner] and at 140 and 240 $\mu$M [@Hauser98], and from ISOCOM at 15$\mu$M [@Biviano99] suggest that the IR background is even higher. According to these data the flux of IR photons is $\sim 2.5 {\times}
10^{-5}$erg cm$^{-2}$sec$^{-1}$sr$^{-1}$ around 60-120 $\mu$M and falls off by an order of magnitude towards 15 $\mu$M. This decrease is important as it would lead to a much shorter mean free path for 20TeV photons as compared to the mean free path of 10TeV photons.
It was originally suggested that the expected break, corresponding to hard-photon disappearance in the IR background, in the GeV-to-TeV spectrum of AGNs could be used to determine the IR background spectrum. This would have been based on searches of a distance-dependent break in the spectrum of various AGNs. However, no apparent break is seen in the spectrum of MK 501 at $\sim 20$TeV range, where the optical depth seems to exceed unity. Using current IR background estimates Coppi and Aharonian [@Coppi99] find an optical depth of 5 for 20TeV photons from MK 501 (see also [@Protheroe_Meyer]). This optical depth increases rapidly with energy. Thus, photons at these energies are exponentially suppressed, unless they somehow evade the pair-production process.
Unlike the GZK paradox only a few solutions have been proposed for the TeV-$\gamma$ paradox. First, it is possible that there is an upturn in the intrinsic spectrum emitted by Mk 501. Such an upturn would compensate for the exponential suppression at this region. Clearly this is an extremely fine-tuned solution as the expected energy of this upturn should somehow be tuned to the energy at which the optical depth from MK 501 to Earth is unity. This energy scale is distance dependent and it puts us in a very special position relative to the source. It is of course possible that the IR intensity has been overestimated. A shift in the energy estimate of HEGRA would also explain the paradox. Finally Harwit, Protheroe and Biermann [@Harwit99] suggest that multiple TeV photons may be emitted coherently by Mk 501 and if they arrive at Earth very close in time and space they may be confused with a single photon event with higher energy.
With current data, $\sim 10$TeV photons from Mk 501 could reach Earth, while $\sim 20$TeV photons are exponentially suppressed. This happens mainly because of the rapid fall off of the IR spectrum below 60$\mu$m. We conclude that a LID upwards shift of the threshold energy by a factor of two would resolve this paradox.
Having discussed the relevance of Mk 501 for the emergence of the TeV-$\gamma$ threshold anomaly we turn now to TeV photons from Mk 421 (another BL Lac object at a redshift of 0.031, corresponding to $\sim 143$Mpc). It is not clear if the spectrum of this source extends high enough to pose a paradox comparable to the one indicated by Mk 501. However, we note here the simultaneous (within the experimental sensitivity) arrival of 1TeV photons and 2TeV from this source. This was used to limit the time-of-flight differences between photons of different energies to less than 200 seconds. This in turn allowed to establish, through an analysis of the type proposed in Ref. [@grbgac], an upper limit on Planck-scale-induced LID [@billetal] which will be a key element of our analysis. We call these constraints in the following time-of-flight constraints.
Lorentz-invariance-violating dispersion relation
================================================
We start by considering first, a class of dispersion relations (following [@aemn1; @gackpoinplb; @grbgac] for $\alpha=1$, and [@polonpap; @AmePiran00] for a general $\alpha$) which in the high-energy regime takes the form: $$E^2 - \vec{p}^2 - m^2 \simeq \eta E^2 \left({E \over
E_{p}}\right)^\alpha
\simeq \eta \vec{p}^2 \left({E \over
E_{p}}\right)^\alpha
~.
\label{dispone}$$ $m$, $E$ and $\vec{p}$ denote the mass, the energy and the (3-component) momentum of the particle, $E_{p}$ is the Planck energy scale ($E_{p} \sim 10^{22}$MeV), while $\alpha$ and $\eta$ are free parameters characterizing the deviation from ordinary Lorentz invariance (in particular, $\alpha$ specifies how strongly the magnitude of the deformation is suppressed by $E_{p}$). Clearly, in (\[dispone\]) the “speed-of-light constant" $c$ has been set to one. (Note however that in this framework $c$ is to be understood as the speed of low-energy massless particles [@grbgac].) Also notice that in (\[dispone\]) we wrote the deformation term in two ways, as a $E^2 (E /E_{p})^\alpha$ correction and as a $p^2 (E/E_{p})^\alpha$ correction, which are equivalent within our present analysis based exclusively on high-energy data, for which $E \sim p$, but would be different when studied with respect to low-energy data. (Of course, a given short-distance picture of space-time will have only one dispersion relation; for example, in “$\kappa$-Minkowski space-time", the space-time which we describe in Appendix A in order to illustrate in an explicit framework some of the structures relevant for our analysis, one encounters a deformation of type $p^2 (E/E_{p})^\alpha$.)
In previous works \[31-35,9\] a slightly different notation had been used to describe this same class of deformations, which in particular replaced our $\eta$ by two quantities: the scale $E_{QG} \equiv |\eta|^{-1/\alpha} E_{p}$ and a sign variable $\xi_{\pm} \equiv \eta/|\eta|$. The $\alpha$,$\eta$ notation turns out to be more suitable for the description of the technical aspects of the analysis discussed here, but it is useful to keep in mind that the scale of Lorentz-deformation is obtained as $|\eta|^{-1/\alpha} E_{p}$.
As hinted by the presence of the Planck scale, our interest in deformed dispersion relations of type (\[dispone\]) originates from the fact that such deformations have independently emerged in theory work on quantum properties of space-time. We postpone the discussion of this motivation to the next Section, where we also clarify which types of generalizations of (\[dispone\]) could also be motivated by Planck-scale physics.
While our analysis is motivated by the role that the deformed dispersion relation (\[dispone\]) might have in quantum gravity, one could of course consider (\[dispone\]) quite independently of quantum gravity[^2]. The quantum-gravity intuition would then be seen as a way to develop a theoretical prejudice for plausible values of $\alpha$ and $\eta$. In particular, corrections going like $(E/E_{p})^\alpha$ typically emerge in quantum gravity as leading-order pieces of some more complicated analytic structures [@grbgac; @gackpoinplb; @kpoinap; @gacmaj]. This provides, of course, a special motivation for the study of the cases $\alpha=1$ and $\alpha=2$. \[$f(E/E_{p}) \simeq 1 + a_1 (E/E_{p})^{n_1} +...$.\] Moreover, the fact that in quantum gravity the scale $E_{QG}$ is expected to be somewhere between the GUT scale and the Planck scale corresponds to the expectation that $\eta$ should not be far from the range $1 \le \eta \le 10^{3 \alpha}$.
Deformed thresholds from deformed dispersion relations
------------------------------------------------------
We intend to discuss the implications of Eq. (\[dispone\]) for the evaluation of threshold momenta. Before doing that let us briefly summarize the derivation of the equation describing the threshold in the ordinary Lorentz-invariant case. Relevant for our phenomenological considerations is the process in which the head-on collision between a soft photon of energy $\epsilon$ and momentum $q$ and a high-energy particle of energy $E_1$ and momentum $\vec{p}_1$ leads to the production of two particles with energies $E_2$,$E_3$ and momenta $\vec{p}_2$,$\vec{p}_3$. At threshold (no energy available for transverse momenta), energy conservation and momentum conservation imply $$E_1+\epsilon=E_2+E_3
~,
\label{econsv}$$ $$p_1-q=p_2+p_3~;
\label{pconsv}$$ moreover, using the ordinary Lorentz-invariant relation between energy and momentum, one also has the relations $$q=\epsilon~,~~~E_i = \sqrt{p_i^2+m_i^2}
\simeq p_i + {m_i^2 \over 2 p_i}
~,
\label{lirel}$$ where $m_i$ denotes the mass of the particle with momentum $p_i$ and the fact that $p_1$ (and, as a consequence, $p_2$ and $p_3$) is a large momentum has been used to approximate the square root.
The threshold conditions are usually identified by transforming this laboratory-frame relations into CM-frame relations and imposing that the CM energy be equal to $m_2+m_3$; however, in preparation for the discussion of deformations of Lorentz invariance it is useful to work fully in the context of the laboratory frame. There the threshold value $p_{1,th}$ of the momentum $p_1$ can be identified with the requirement that the solutions for $p_2$ and $p_3$ as a function of $p_1$ (with a given value of $\epsilon$) that follow from Eqs. (\[econsv\]), (\[pconsv\]) and (\[lirel\]) should be imaginary for $p_1 < p_{1,th}$ and should be real for $p_1 \ge p_{1,th}$. This straightforwardly leads to the threshold equation $$p_{1,th} \simeq {(m_2 + m_3)^2 - m_1^2 \over 4 \epsilon}
~.
\label{lithresh}$$
This standard Lorentz-invariant analysis is modified [@ita; @sato; @kifu; @kluz; @Protheroe_Meyer; @AmePiran00] by the deformations codified in (\[dispone\]). The key point is that Eq. (\[lirel\]) should be replaced by $$\epsilon= q + \eta {q^{1+\alpha} \over 2 E_p^\alpha}
~,~~~
E_i \simeq p_i + {m_i^2 \over 2 p_i}
+ \eta {p_i^{1+\alpha} \over 2 E_p^\alpha}
~.
\label{lv1rel}$$ Combining (\[econsv\]), (\[pconsv\]) and (\[lv1rel\]) one obtains a deformed equation describing the $p_1$-threshold: $$p_{1,th} \simeq {(m_2 + m_3)^2 - m_1^2 \over 4 \epsilon}
+ \eta {p_{1,th}^{2+\alpha} \over 4 \epsilon E_p^\alpha} \left(
{m_2^{1+\alpha} + m_3^{1+\alpha} \over (m_2 + m_3)^{1+\alpha}} -1 \right)
~.
\label{lithresh2}$$ where we have included only the leading corrections (terms suppressed by both the smallness of $E_{p}^{-1}$ and the smallness of $\epsilon$ or $m$ were neglected).
Phenomenology
-------------
Early phenomenological interest in the proposal (\[dispone\]) came from studies based on time-of-flight analyses [@grbgac; @schaef; @billetal] of photons associated with gamma-ray bursts or with Mk 421. According to (\[dispone\]) (and assuming that there is no leading-order deformation of the standard relation $v = dE/dp$) one would predict [@grbgac; @aemn1] energy-dependent relative delays between the times of arrival of simultaneously emitted massless particles: $${\Delta T \over T} =
\eta {(\alpha + 1) \over 2}{E'^\alpha - E^\alpha \over E_p^\alpha}
~,
\label{velox}$$ where $T$ is the (average) overall time of travel of simultaneously emitted massless particles and $\Delta T$ is the relative delay between the times of arrival of two massless particles of energies $E$ and $E'$. The fact that such time delays have not yet been observed allows us to set bounds on the $\alpha,\eta$ parameter space. In particular, data showing (approximate) simultaneity of arrival of TeV photons from Mk 421 were used [@billetal] to set the bound $|\eta| < 3 {\cdot} 10^2$ for $\alpha=1$. The same data were used in Ref. [@polonpap] to set a more general $\alpha$-dependent bound on $\eta$.
We combine these existing bounds with the assumption that indeed the UHECR and TeV-$\gamma$ threshold anomalies are due to LID (\[dispone\]). The fact that the scale $E_{p}$ is very high might give the erroneous impression that the new term going like $p_{1,th}^{2+\alpha} /E_{p}^\alpha$ present in Eq. (\[lithresh2\]) could always be safely neglected, but this is not the case [@ita; @sato; @kifu; @kluz; @Protheroe_Meyer; @AmePiran00]. For given values of $\alpha,\eta$ one finds values of $\epsilon$ that are low enough for the “threshold anomaly” [@AmePiran00] (displacement of the threshold) to be significant. For certain combinations $\alpha,\eta,\epsilon$ the threshold completely disappears, [*i.e.*]{} Eq. (\[lithresh2\]) has no solutions. Assuming (\[lithresh2\]) one would predict dramatic departures from the ordinary expectations of Lorentz invariance; in particular, if $\alpha \sim - \eta \sim 1$, according to (\[lithresh2\]) one would expect that the Universe be transparent to TeV photons. The corresponding result obtainable in the UHECRs context would imply that the GZK cutoff could be violated [@kifu] even for much smaller negative values of $\eta$. Positive values of $\eta$ would shift the thresholds in the opposite direction ([*e.g.*]{} they would imply an even stricter limit than the GZK one) and are therefore not consistent with the hypothesis that UHECR and TeV-$\gamma$ threshold anomalies be due to LID (\[dispone\]).
In Figure 1 we provide a quantitative description of the region of the $\alpha,\eta$ parameter space which would provide a solution to both the UHECR and TeV-$\gamma$ threshold anomalies while satisfying the time-of-flight constraints [@schaef; @billetal] that are still consistent with ordinary Lorentz invariance. The curve describing the time-of-flight constraints was obtained using the information that there is [@billetal] an upper bound of order 200 seconds to the difference in time of arrivals of 2TeV photons and 1TeV photons simultaneously emitted by Mk 421 (at redshift of 0.031). The two threshold-anomaly curves reported in Figure 1 were obtained using Eq. (\[lithresh2\]) with $m_1=0$ and $m_2=m_3 = 5 {\cdot} 10^5$eV (TeV photons $\gamma + \gamma \rightarrow e^+ + e^-$ threshold analysis) and with $m_1 = m_2 = 9.4 {\cdot} 10^8$eV and $m_3= 1.4
{\cdot} 10^8$eV (UHECR $p + \gamma \rightarrow p + \pi$ threshold analysis[^3]). In light of the analysis of the experimental situation provided in Sections 2 and 3, we obtained the UHECRs curve by requiring sufficient LID to explain the factor-6 threshold shift $5
{\cdot} 10^{19}$eV$ \rightarrow 3 {\cdot} 10^{20}$eV, while for the TeV photons curve we required a factor-2 threshold shift $10$TeV $\rightarrow 20$TeV. Even though the shift is more significant in the UHECR context, it is the requirement to explain the TeV-$\gamma$ threshold anomaly that provides a more stringent constraint, as one should expect since in our LID, which is motivated by Planck-scale physics, the violation of ordinary Lorentz invariance is suppressed by some power of the ratio $E/E_p$.
Considering the diverse origin and nature of the three relevant classes of experimental data that we are considering, the fact that there is a region of the $\alpha,\eta$ parameter space consistent with all these constraints is non-trivial, and this in turn provides encouragement for the interpretation of the threshold anomalies as manifestations of LID. Moreover, it is quite striking that this region of parameter space, in spite of being relatively small, does contain one of the two mentioned quantum-gravity-motivated scenarios: $\alpha =1$ and $1 < \eta < 10^{3}$. The other quantum-gravity-motivated scenario, the one with $\alpha =2$ and $1 < \eta < 10^{6}$, is outside the relevant region of parameter space, being consistent with the absence of relative time delays and the UHECR threshold anomaly but being inconsistent with threshold anomaly for multi-TeV photons.
Concerning the consistency of the interpretation of the threshold anomalies as manifestations of LID it is also important to observe that the modified dispersion relation (\[dispone\]), in spite of affecting so significantly the GZK and TeV-$\gamma$ thresholds, does not affect significantly the processes used for the detection of the relevant high-energy particles. For the significance of the threshold modification a key role is played, as evident from equation (\[lithresh2\]), by the smallness of the energy of the background photons. The effect of (\[dispone\]) on atmospheric interactions of the relevant high-energy particles is instead suppressed by the fact that in these atmospheric interactions the “targets", nuclei or electrons, have energies much higher than those of the background photons.
A more general LID formalism
============================
Having shown that the simple two-parameter family of Lorentz-invariance-violating dispersion relations (\[dispone\]) provides a solution of the UHECRs and TeV-$\gamma$ threshold paradoxes, we turn now to a more general five-parameter LID formulation. The motivation for this formulation comes primarily from theory work on short-distance (so called, “quantum gravity") properties of space-time, in which modifications of space-time symmetries are encountered quite naturally. In particular, quantum-gravity effects inducing some level of nonlocality or noncommutativity would affect even the most basic flat-space continuous symmetries, such as Lorentz invariance. This has been recently emphasized in various quantum-gravity approaches \[31-33,39-50\] based on critical or noncritical string theories, noncommutative geometry or canonical quantum gravity. While we must be open to the possibility that some symmetries are completely lost, it appears plausible that some of them are not really lost but rather replaced by a Planck-scale-deformed version. Some mathematical frameworks which could consistently describe such deformations have emerged in the mathematical-physics literature [@gackpoinplb; @kpoinap; @gacmaj; @lukipap; @majrue]. An example of these structures is discussed in Appendix A.
The five-parameter formalism
----------------------------
The fact that a simple two-parameter family of Lorentz-invariance-violating dispersion relations (\[dispone\]) is consistent with all available data is of course of encouragement for the LID hypothesis, but, especially since relevant data are expected to improve rapidly in the coming years, it is also important to establish how much room for generalizations of (\[dispone\]) is available in the general framework of Planck-scale-induced LID.
One way to generalize (\[dispone\]) would involve attributing different independent values of $\alpha,\eta$ to different particles. We shall not pursue this (however phenomenologically viable) possibility, since the focus of the present article is on deformations of Lorentz symmetry which could be induced by non-trivial space-time structure, and such deformations would most likely treat “democratically" all particles. In any case, it is clear that models attributing different independent values of $\alpha$ and $\eta$ to each particle end up having a very large number of free parameters and available data will not be very effective in constraining such models. We shall come back to this point in Section 6, where we consider the alternative (Planck-scale independent) Coleman and Glashow [@colgla] scheme for Lorentz-invariance-violation. In fact, that scheme corresponds to the choice $\alpha = 0$ and an independent value of $\eta$ for each particle.
Another way to generalize the dispersion relation (\[dispone\]) is to include other deformation terms. In a space-time with some non-trivial structure at distances of order $E_p^{-1}$ one could expect that probes with energy much smaller than $E_p$ should obey a dispersion relation of type: $$E^2 - p^2 - m^2 = F(E,p,m;E_p)
~,
\label{toogeneral}$$ where $F$ is some general function with units of mass (or energy) squared and such that $F \rightarrow 0$ for $E_p \rightarrow \infty$. Actually, in studies, such as ours, looking only for the leading correction, one of the arguments of $F$ can be suppressed: one makes a subleading error by using $E^2 - p^2 - m^2 = 0$ to express one of the variables in $F$ in terms of the other variables. One could for example express $F$ as a function of $p$ and $m$ only: $F(p,m;E_p)$. Moreover, the fact that we are only looking for the leading correction in the high-energy regime[^4] allows us to approximate $F$ with its leading (if any) power dependence on $E_p$ and (within a given power dependence on $E_p$) leading dependence on $p$: $F(p,m;E_p) \simeq \eta \, p^{2
+\alpha-\sigma} \, m^{\sigma} \, E_p^{-\nu}$. In the high-energy regime there is therefore scope for considering the three-parameter family of dispersion relations $$E^2 - p^2 - m^2 \simeq \eta {\cdot} p^{2+\alpha-\sigma} {\cdot}
m^{\sigma} {\cdot} E_p^{-\alpha}
~,
\label{dispfull}$$ where, of course, it is understood that $m^{\sigma}=1$ whenever $\sigma=0$, even when $m=0$. (The parameter $\sigma$ has been introduced to characterize the type of dependence of the deformation term on the mass $m$, and therefore in our notation there is the implicit prescription that $m^{\sigma} \rightarrow 1$ in the formally ambiguous combined limit $\sigma \rightarrow 0$, $m \rightarrow 0$.)
Besides the structure of the dispersion relation a LID can also affect the law of sum of momenta. Since our emphasis is here on the phenomenology of LIDs, rather than on their formal/mathematical analysis, we limit our discussion of the motivation for this type of effect to the example of non-commutative geometry (the “$\kappa$-Minkowski space-time") considered in Appendix A. As that example clarifies, it is natural to consider a two-parameter class of modifications of the law of sum of (parallel) momenta $K_1 + K_2 \rightarrow K_1 + K_2
+ \delta (K_1 K_2)^{(1+\beta)/2} E_{p}^{-\beta}$. For our threshold analyses this corresponds to $$p_1 - \epsilon \rightarrow
p_1 - \epsilon - \delta {(p_1 \epsilon)^{(1+\beta)/2} \over E_{p}^\beta}
\simeq p_1 - \epsilon
~,~~~
p_2 + p_3 \rightarrow
p_2 + p_3 + \delta {(p_2 p_3)^{(1+\beta)/2} \over E_{p}^\beta}
~.
\label{psumrule}$$
Overall we consider a five-parameter space: $\alpha,\eta,\sigma$ for the dispersion relation and $\beta,\delta$ for the description of possible deformations (\[psumrule\]) of the law of addition of momenta. The analysis reported in the preceding Section corresponds of course to the $\sigma \rightarrow 0$, $\delta \rightarrow 0$ limit of this more general five-parameter ($\alpha,\eta,\sigma,\beta,\delta$) phenomenology.
As appropriate for the present preliminary status of the experimental situation and the fact that the two-parameter phenomenology analyzed in the previous Section turned out to give a fully satisfactory description of the data, we shall only provide here a preliminary and partial exploration of the enlarged five-parameter space. Our exploration of this parameter space will also be more detailed in some directions and less detailed in others. In particular, we shall limit our analysis to two classes of scenarios, one with $\sigma = 0$ and one with $\sigma=2,\delta=0$. This will be sufficient for a qualitative understanding of how different portions of our five-parameter space compare with the present experimental situation.
Retaining the leading corrections in $E_{p}^{-1}$, the threshold analysis in the general five-parameter ($\alpha,\eta,\sigma,\beta,\delta$) LID scenario leads to the threshold equation:[^5] $$\begin{aligned}
p_{1,th} \!\!\!&\simeq&\!\!\! {(m_2 + m_3)^2 - m_1^2 \over 4 \epsilon}
+ \eta {p_{1,th}^{2-\sigma} \over 4 \epsilon}
\left( {m_2^{1+\alpha} + m_3^{1+\alpha} \over (m_2 + m_3)^{1+\alpha-\sigma}}
- m_1^\sigma \right) \left({p_{1,th}\over E_p}\right)^\alpha
\label{newmast} \\
\nonumber
& &- \delta {p_{1,th}^{2} \over
2 \epsilon}
\left( {\sqrt{m_2 m_3} \over m_2 + m_3} \right)^{1+\beta}
\left({p_{1,th} \over E_p}\right)^\beta ~. \end{aligned}$$ In the following sections we apply this equation to several specific cases. To simplify the discussion we provide here explicit expressions for the threshold for photopion production:
$$\begin{aligned}
E_{GZK,th} \!\!\!& \simeq &\!\!\! {7 {\times} 10^{19}~ {\rm eV} \over
\epsilon/ 0.001 {\rm eV} }
\left[ 1 + \eta ~
10^{22.2-10.9\sigma-8.15 \alpha}
\left( (0.87^{1+\alpha}+ 0.13^{1+\alpha}) 1.15^\sigma -1 \right)~~~~~~
%( (0.87^{1+\alpha}+ 0.13^{1+\alpha}) 1.15^\sigma -1)
\right. \nonumber \\ & &~~~~~~~~~~~~~~~~~~~~~~~~
~~\left({E_{GZK,th}\over 7 {\times} 10^{19} {\rm eV}}\right)^{2-\sigma}
\left({E_{GZK,th}/7 {\times} 10^{19} {\rm eV}
\over E_p/10^{19} {\rm GeV} }\right)^\alpha
\label{neweq1} \\ \nonumber
& &~~~~~~~~~~~~~~~~~~ - ~\delta~ \left. 10^{22.1-8.15\beta}
\left({E_{GZK,th}\over 7 {\times} 10^{19}{\rm eV} }\right)^2
\left({E_{GZK,th}/7 {\times} 10^{19} {\rm eV}\over E_p/10^{19}
{\rm GeV}}\right)^\beta
\right]
~, \end{aligned}$$
and for pair creation threshold: $$\begin{aligned}
E_{\gamma,th} \!\!\!& \simeq &\!\!\!
{25 ~ {\rm TeV} \over \epsilon/ 0.01 {\rm eV} }
\left[ 1 + \eta ~ 10^{14.8 -7.7 \sigma - 14.6 \alpha}
\left(2^{\sigma-\alpha} - \delta^K_{\sigma,0}\right)
\left({E_{\gamma,th}\over 25 {\rm TeV}}\right)^{2-\sigma}
\left({E_{\gamma,th}/25 ~ {\rm TeV}\over E_p/10^{19}
{\rm GeV}}\right)^\alpha \right. \label{neweq2} \nonumber \\
& &~~~~~~~~~~~~~~~ - \left. \delta ~ 10^{14.8 - 14.9 \beta}
\left({E_{\gamma,th}\over 25 {\rm TeV}}\right)^{2}
\left({E_{\gamma,th}/25 ~ {\rm TeV}\over E_p/10^{19}
{\rm GeV}}\right)^\beta
\right] ~,\end{aligned}$$ where we found convenient to introduce the “Kronecker delta", here denoted with $\delta^K$ to differentiate it from our parameter $\delta$, to compactly write this equation consistently with our conventions for the $m_1 \rightarrow 0$ limit. \[In deriving Eq. (\[neweq2\]) from Eq. (\[newmast\]) it is necessary to take into account that, consistently with the conventions and notations we introduced (see, in particular, the comments made immediately after Eq. (\[dispfull\])), in the limit $m_1 \rightarrow 0$ the term $m_1^\sigma$ must be handled according to $m_1^\sigma \rightarrow 0$ if $\sigma \ne 0$ and according to $m_1^\sigma \rightarrow 1$ if $\sigma = 0$. Of course, the reader can verify by direct calculation that this prescription gives the correct threshold conditions that follow from Eq. (\[dispfull\]) in the two cases $\sigma = 0$ and $\sigma \ne 0$, and reproduces the threshold condition (\[lithresh2\]) obtained in the preceding Section (which was devoted to the case $\sigma = 0, \delta=0$).\]
Phenomenology with $\sigma=2,\delta=0$
--------------------------------------
In the case $\sigma=2,\delta=0$ there is no deformed law of addition of momenta and the threshold equation takes the form $$p_{1,th} \simeq {(m_2 + m_3)^2 - m_1^2 \over 4 \epsilon}
+ \eta {p_{1,th}^{\alpha} \over 4 \epsilon E_p^{\alpha}}
\left({m_2^{1+\alpha} + m_3^{1+\alpha} \over (m_2 + m_3)^{\alpha-1}}
- m_1^2 \right)
~.
\label{newmastwithm}$$
For $\sigma \ne 0$ the LID term in (\[dispfull\]) vanishes for massless particles. Thus, in general in all $\sigma \ne 0$ cases (like the one we discuss in this Subsection) the time of flight constraints [@schaef; @billetal] do not limit the LID parameters.
The constraints obtainable by interpreting the UHECR and TeV-$\gamma$ threshold anomalies as manifestations of LID suggest that this interpretation is quite unnatural in the case $\sigma=2,\delta=0$. The condition that both threshold be pushed upwards leads to the constraints $\eta >0$, $\alpha < 1.195$. Moreover, in order to describe the threshold anomaly for multi-TeV photons one should also make the awkward requirement $\eta > 10^{15 \alpha}$. Having provided in the previous section an elegant solution of the threshold paradoxes using $\sigma=0$ we do not pursue further this scenario which appears to require a higher level of fine tuning.
Scenarios with $\sigma \ne 0$ might regain some interest if there are significant new developments in the understanding of the threshold anomalies that will point in this direction. In the present experimental and theoretical situation we find appropriate to make in the following the assumption that $\sigma=0$.
General aspects of the phenomenology with $\sigma=0$
----------------------------------------------------
For $\sigma=0$ one is left with a four-parameter space on which significant information can be gained by combining data on possible time of flight delays, which will only constrain, through the prediction (\[velox\]), the parameters $\alpha,\eta$, and data on the threshold anomalies, which, through (\[newmast\]), are relevant for all four parameters $\alpha,\eta,\beta,\delta$.
It is important to observe that positive (discovery) results on both the thresholds and the time delays would allow to determine the values of all four parameters. If eventually the mentioned time delays are actually observed, and if they are observed in signals from a collection of sources diverse enough to allow the determination of the energy dependence of the time delays, we would then be able to use (\[velox\]) to fix $\alpha$ and $\eta$. Then, knowing $\alpha$ and $\eta$, a determination of the thresholds could be used to fix $\beta$ and $\delta$.
While waiting for these eventual discoveries, one can use the present upper limits on LID in relative time delays and (preliminary evidence of) lower limits on LID in threshold anomalies to reduce the allowed portion of the four-parameter space. We subdivide the discussion of this type of phenomenological analysis in three cases: $\alpha < \beta$, $\alpha = \beta$ and $\alpha > \beta$.
Phenomenology with $\alpha < \beta$ (and $\sigma=0$)
----------------------------------------------------
The case $\alpha < \beta$ (and $\sigma=0$) is essentially analogous to the case considered in the preceding Section with the two-parameter $\alpha,\eta$ phenomenology. In fact, for $\alpha < \beta$ the threshold corrections associated with the deformation of the law of addition of momenta are suppressed by factors of order $(E/E_p)^{\beta - \alpha}$ with respect to the threshold corrections associated with the deformed dispersion relation. The constraints derived for $\alpha,\eta$ in the preceding Section would still be valid and, as long as we have only lower or upper limits (rather than definite discoveries), no constraint could be put on $\beta,\delta$.
Phenomenology with $\alpha = \beta$ (and $\sigma=0$)
----------------------------------------------------
For $\alpha = \beta$ (and $\sigma=0$) the upper limit on time-of-flight LID still constrains only $\alpha,\eta$, but the constraints on $\alpha,\eta$ obtainable by interpreting the UHECR and TeV-$\gamma$ threshold anomalies as manifestations of LID are weakened by allowing also a deformed law of addition of momenta. In practice the parameters $\alpha,\eta$ and $\beta,\delta$ can in a sense “share the burden" of explaining the threshold anomalies. To illustrate this mechanism we show in Figure 2 the constraints on $\eta,\delta$ that are obtained for $\alpha = \beta = 1$ (and $\sigma=0$).
Phenomenology with $\alpha > \beta$ (and $\sigma=0$)
----------------------------------------------------
For $\alpha > \beta$ (and $\sigma=0$) the threshold corrections associated with the deformed dispersion relation are suppressed by factors of order $(E/E_p)^{\alpha - \beta}$ with respect to the threshold corrections associated with the deformation of the law of addition of momenta. Therefore the interpretation of the UHECR and TeV-$\gamma$ threshold anomalies as manifestations of LID imposes constraints (lower bounds on LID) on the parameters $\beta,\delta$. As always, the upper limit on time-of-flight LID constrains only $\alpha,\eta$. It is worth noticing that if future data should indicate that there is no LID relative time-delay effect but there are LID threshold anomalies this scenario with $\alpha > \beta$ would become favored.
Figure 3 depicts the limits on $\beta,\delta$ that follow, when $\alpha > \beta$, from interpreting the UHECR and TeV-$\gamma$ threshold anomalies as manifestations of LID. The limits on $\alpha,\eta$ due to the upper limit on time-of-flight LID are still the same as in Figure 1 (but, as just mentioned, the two threshold-anomaly curves in Figure 1 do not apply when $\alpha >
\beta$).
Comparison with the Coleman-Glashow scheme
==========================================
Coleman and Glashow [@colgla] have recently introduced a different scheme (denoted CG scheme hereafter) for violation of Lorentz invariance. Modifying the elementary particles Lagrangian they suggest a scheme in which there is a different maximum attainable velocity, $c_a$, for each particle. The relevant dispersion relations take the form $$E^2 - p^2 c_a^2 = m^2 c_a^4 ~,
\label{colglaeq}$$ where the index $a$ labels the particle. In the language developed in Sections 4 and 5 these dispersion relations (\[colglaeq\]) involve two terms, one with $\alpha = 0$ and $\sigma=0$ and the other with $\alpha = 0$ and $\sigma=2$. The particle-dependence of $c_a$ could be described by allowing for a different independent value of $\eta$ for each fundamental particle. At high energies, in which we are interested, the $\alpha = 0,\sigma=0$ term dominates and $\eta_a = c^2 - c_a^2 \approx 2 c (c-c_a)$. The condition $\alpha = 0$ reflects the fact that the CG scheme is not motivated by Planck-scale physics. The possibility for each particle to get its own independent value of $\eta$ reflects the fact that this scheme is not intended as a description of deformations of Lorentz invariance due to non-trivial short-distance space-time structure. (If a deformation of Lorentz symmetry is induced by the structure of space-time we expect that it would affect all particles in the same way. Such a symmetry deformation might allow for a dependence of the correction terms on the mass and the spin of the particle but the parameters of the model should not depend on the mass, spin or other quantum numbers of the particles.)
Using again the language we developed in Sections 4 and 5 one can also give an intuitive characterization of the way in which the CG scheme and the scheme considered here are alternative to one another as strategies for obtaining threshold anomalies. In fact, in that language one could describe undeformed thresholds[^6] as associated with $\alpha = 0,\sigma=0,\delta=0$, independently of the value of $\eta$. This corresponds to the fact that in the CG scheme there are of course no threshold anomalies if all $c_a$’s take the same value ($c_a = c - \eta/(2 c)$). Threshold anomalies are generated in the CG scheme by deforming the threshold conditions in the direction that corresponds to keeping $\alpha = 0,\sigma=0,\delta=0$ but allowing different independent values of $c_a$ for each fundamental particle. On the contrary, in our scheme the threshold anomalies are obtained by allowing for deviations from $\alpha = 0,\sigma=0,\delta=0$ while keeping a single (particle-independent) $\eta$.
In light of these comments it is not surprising that threshold anomalies within the CG scheme take the characteristic “$c_a - c_b$" dependence. In particular, as already observed in Ref. [@colgla], the description of the UHECR threshold anomaly requires (together with conditions on $c_\Delta - c_p$) that $c_\pi - c_p > 10^{-24}$. ($c_\pi$ and $c_p$ are the $c_a$’s for pions and protons respectively.) We observe that a resolution of the TeV-$\gamma$ threshold anomaly within the CG scheme requires the additional condition $c_e - c_\gamma > 5 {\cdot} 10^{-16}$. This combines with the absence [@colgla] of vacuum Cerenkov radiation by electrons with energies up to 500GeV in such a way that $c_e - c_\gamma$ is bound to $5 {\cdot} 10^{-16} < c_e - c_\gamma < 5 {\cdot} 10^{-13}$. There is therefore a relatively narrow range of allowed values for $c_e - c_\gamma$ just like[^7] we found in Section 4 a relatively narrow allowed region of the $\alpha,\eta$ parameter space.
One important difference between the two schemes is that in our Planck-scale-motivated LID the allowed region of parameter space is found exactly where quantum-gravity intuition would have sent us searching for new physics, while in the CG scheme values of $c_e - c_\gamma$ in the range $5 {\cdot} 10^{-16} < c_e - c_\gamma < 5 {\cdot} 10^{-13}$ do not have any special significance. Another important difference between the two schemes is that while the same $\alpha,\eta$ parameters of our scheme for LID are also constrained by UHECR threshold data, in the CG scheme $c_e - c_\gamma$ does not play any role in the equation for the UHECR threshold and vice versa. Any future development in the UHECR threshold data would leave $c_e -
c_\gamma$ unaffected. On the contrary, the plausibility of the Planck-scale-motivated LID will be strongly affected by future UHECR threshold data: if the lower limit on the threshold continues to be pushed higher the overall consistency and appeal of the LID model would increase, while the discovery of the threshold not much higher than the present $3 {\cdot} 10^{20}$eV lower limit would (unless the TeV-$\gamma$ threshold anomaly is eventually understood as a result of systematic errors) rule out the model considered in Section 4.
While the scheme considered here is more tightly constrained by high-energy data (because all high-energy data set constraints to the same few space-time related parameters), the CG scheme is constrained more tightly than ours by low-energy data. The parameters we considered in the present Article, dealing exclusively with the high-energy regime, are practically unconstrained by low-energy data since, as discussed in Section 5, the LID we considered here might emerge in quantum gravity as the leading order in the high-energy expansion of an analytic function whose low-energy expansion looks quite different. On the contrary the CG scheme takes a fixed (energy-independent) value of its parameters $c_a$ and therefore high-energy and low-energy data can be combined to obtain stricter limits.
Summary and outlook
===================
In the present Article we took as working assumption that the UHECR and TeV-$\gamma$ threshold anomalies do not have a simple explanation (whereas, especially for the case of TeV photons, it might still be legitimate to explore the possibility that systematic experimental errors be responsible for the paradox, and other solutions exist for the GZK paradox) and we attempted to test the plausibility of a description of the anomalies in terms of a Planck-scale-induced deformation of Lorentz symmetry. The results reported in Section 4 certainly indicate that this description is plausible. Had we not been considering such a dramatic departure from conventional physics, we would have probably gone as far as stating that the LIDs we considered provide a compellingly simple description of the anomalies. We do feel that the results of Section 4, also taking into account that there is no other known common explanation of the two threshold paradoxes, provide motivation for additional theory work on the speculative idea of LID and for additional experimental studies aimed at testing the class of Planck-scale-induced LID here considered.
While presently-available data do not in any way invite one to look beyond the simplest two-parameter LID examined in Section 4, in preparation for future studies, especially the expected improvement of the experimental input, we have developed in Section 5 a general parameterization that may prove useful for future attempts to constrain (even rule out) Planck-scale-induced LID. We have emphasized the fact that Planck-scale-induced LID, since it should reflect the structure of space-time, can be characterized by a small number of parameters. In the high-energy regime we found that a very general description of (the leading effects of) Planck-scale-induced LID only requires five parameters and we described how the determination of a few thresholds together with measurements of the speed of very-high-energy particles could fix all five parameters. While we have considered a very general class of LID, it should be stressed that we postponed to future studies the analysis of an important class of further generalizations of Planck-scale-induced LID [@gampul]: deformation terms involving a dependence on polarization/spin of the particles.
In closing we would also like to emphasize the fact that the experimental data here considered represent an important sign of maturity for the general programme of “Planck-length phenomenology" [@polonpap; @gacPLph]. Whether or not Planck-scale-induced LID turns out to successfully describe future experimental data, the fact that at present we are confronted with experimental paradoxes whose solution could plausibly involve the Planck length, and that certainly the relevant class of observations will eventually be able to rule out various pictures of the short-distance (possibly quantum) structure of space-time, shows that, contrary to popular folklore, some experimental guidance can be obtained for the search of theories capable of unifying gravitation and quantum mechanics. This confirms the expectations, which were based on analyses of the sensitivity of various classes of experiments [@ehns; @grbgac; @gacgwi], that emerged from the general quantum-gravity studies reported in Refs. [@polonpap; @gacgwi; @ahlunature; @gacPLph] and from analogous studies, primarily focusing on the hypothesis that the unification of gravitation and quantum mechanics should involve non-critical strings, reported in Ref. [@emnreview].
We thank Daniele Fargion and Glennys Farrar for many informative discussions on UHECRs.
plus .5pt minus .5pt
[99]{}
K. Greisen. , 16:748, 1966. G. T. Zatsepin and V. A. Kuzmin. , 4:78, 1966.
Nikishov A.I., 1962, [Sov. Phys. JETP]{}, [14]{}, 393
Gould J., Schreder G., 1967, [PR]{}, [155.5]{}, 1404
Stecker F.W., De Jager O.C. and Salomon M.H., 1992, [ApJ]{}, [390L]{}, L49
A. V. Olinto, Phys. Rept. 333-334 (2000) 329-348
M. Harwit, P. J., Protheroe and P. L., Biermann, Ap. J., [**524**]{}, L91, 1999.
L. Gonzalez-Mestres, physics/9704017
S. Coleman, S.L. Glashow, Phys. Rev. D59 (1999) 116008.
R. Aloisio, P. Blasi, P.L. Ghia and A.F. Grillo, astro-ph/0001258.
O. Bertolami and C.S. Carvalho, Phys. Rev. D61 (2000) 103002.
H. Sato, astro-ph/0005218.
T. Kifune, astro-ph/9904164, Astrophys. J. Lett. 518 (1999) L21.
W. Kluzniak, astro-ph/9905308.
R. J. Protheroe and H. Meyer, astro-ph/0005349, 2000.
B.E. Schaefer, Phys. Rev Lett. 82 (1999) 4964.
S.D. Biller [*et al*]{}, Phys. Rev. Lett. 83 (1999) 2108.
D. J. Bird et al. , 441:144, 1995.
M. Takeda et al, , 81 (1998) 1163.
A. Watson. Proc. Snowmass Workshop, p 126, 1996.
G. Farrar and T. Piran, in preparation, 2000.
F. A., Aharonian, et al., A&A, [**349**]{}, 11A, 1999.
Primack J.R., Bullock J.S., Somerville R.S., MacMinn D., 1999, [AstroPart.]{}, [11, 1-2]{}, 93
Malkan M.A., Stecker F.W., 1998, [ApJ]{}, [496]{}, 13
Dwek E., Arendt R.G., 1998, [ApJL]{}, [508]{}, L9
S.M Fall, S. Charlot, and Y. C., Pei 1996, [ApJ]{}, [464]{}, L43
E. L. Wright, astro-ph/0004192, 2000.
D. P., Finkbeiner, M. Davis and D. J. Schlegel, astro-ph/0004175, 2000.
Hauser M.G., Arendt R.G., Kelsall T. et al., 1998, [ApJ]{}, [508]{}, 25
A. Biviano, et al., astro-ph/9910314, 1999.
P. S. Coppi and F. A. Aharonain, astro-ph/9903160, 1999
G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos and S. Sarkar, , astro-ph/9712103, Nature [393]{} (1998) 763.
G. Amelino-Camelia, J. Ellis, N.E. Mavromatos and D.V. Nanopoulos, Int. J. Mod. Phys. A12 (1997) 607.
G. Amelino-Camelia, Phys. Lett. B392 (1997) 283.
G. Amelino-Camelia,, gr-qc/9910089, in: “Towards Quantum Gravity”, ed: J. Kowalski-Glikman. Springer-Verlag Heidelberg 2000,
G. Amelino-Camelia and T. Piran, hep-ph/0006210, 2000.
S.M. Carroll, G.B. Field and R. Jackiw, Phys. Rev. D41 (1990) 1231.
R. Jackiw and V.A. Kostelecky, Phys. Rev. Lett. 82 (1999) 3572.
J. Lukierski, A. Nowicki, H. Ruegg, and V.N. Tolstoy, Phys. Lett. B264 (1991) 331; J. Lukierski, A. Nowicki and H. Ruegg, , Ann. Phys. 243 (1995) 90.
G. Amelino-Camelia and S. Majid, hep-th/9907110, Int. J. Mod. Phys. A (in press).
V.A. Kostelecky and R. Potting, Phys. Lett. B381 (1996) 89.
S. Frittelli, C.N. Kozameh, E.T. Newman, C. Rovelli and R.S. Tate, Class.Quant.Grav. 14 (1997) A143.
G. Amelino-Camelia, Mod. Phys. Lett. A13 (1998) 1319.
R. Gambini and J. Pullin, Phys. Rev. D59 (1999) 124021.
G. Amelino-Camelia, J. Lukierski and A. Nowicki, gr-qc/9903066,Int. J. Mod. Phys. A14 (1999) 4575.
J. Alfaro, H.A. Morales-Tecotl, L.F. Urrutia, Phys. Rev. Lett. 84 (2000) 2318.
J. Ellis, K. Farakos, N.E. Mavromatos, V.A. Mitsou and D.V. Nanopoulos, astro-ph/9907340.
L. Smolin, Physics World 12 (1999) 79. Hong-wei Yu and L.H. Ford, gr-qc/0004063.
T. Thiemann and O. Winkler, hep-th/0005235.
G.Z. Adunas, E. Rodriguez-Milla and D.V. Ahluwalia, Phys. Lett. B485 (2000) 215. S. Majid and H. Ruegg, Phys. Lett. B334 (1994) 348.
J. Ellis, J.S. Hagelin, D.V. Nanopoulos and M. Srednicki, Nucl. Phys. B241 (1984) 381.
G. Amelino-Camelia, gr-qc/9808029, Nature [398]{} (1999) 216.
D.V. Ahluwalia, gr-qc/9903074, Nature 398 (1999) 199.
G. Amelino-Camelia, gr-qc/0008010. J. Ellis, N.E. Mavromatos and D.V. Nanopoulos, Gen. Rel. Grav. 31 (1999) 1257.
Appendix A: $\kappa$-Minkowski space-time {#appendix-a-kappa-minkowski-space-time .unnumbered}
=========================================
In order to illustrate in an explicit framework some of the structures relevant for our analysis of LID, in this Appendix we give a brief descrition of the “$\kappa$-Minkowski" non-commutative space-time, which was developed in Refs. [@gackpoinplb; @kpoinap; @gacmaj; @lukipap; @majrue]. The simplicity of $\kappa$-Minkowski, which is basically an ordinary Minkowski space-time on which however one postulates that the time coordinate does not commute with the space coordinates ($[x_i,t] \sim x_i/E_p$), renders it very useful for the purpose of illustrating the new conceptual elements required by space-times with a nontrivial short-distance structure.
A first point that deserves being emphasized is the connection between flat nontrivial space-times and quantum gravity. In quantum gravity one has the general intuition [@polonpap; @gacgrf98] that ordinary classical commutative space-times should emerge from some more fundamental underlying picture. To very compact (Planckian-energy) probes space-time should look completely different from an ordinary classical space-times. On the contrary probes of very low energy should not be affected in any noticeable way by the nontrivial short-distance structure of space-time. In the intermediate regime (mid-energy probes [@polonpap; @gacgrf98]) one would expect to be able to use roughly the same language of ordinary classical space-times, but with the necessity to introduce some new concepts (such as the little element of noncommutativity of $\kappa$-Minkowski) reflecting the leading-order effects of quantum-gravity at low energies. This hierarchy of regimes is to be expected not only in high-curvature space-times (where classical-gravity effects are stronger), but also in space-times that appear to be trivial and flat to very-low-energies probes. $\kappa$-Minkowski is [@gacgrf98] a model (toy model?) of how a probe of relatively high energy could perceive a space-time that instead appears to be trivially Minkowski to probes of very low energy.
In $\kappa$-Minkowski a relation of type (\[dispone\]) can be obtained as a direct consequence of the $\kappa$-Poincaré invariance [@gackpoinplb; @kpoinap; @gacmaj; @lukipap; @majrue] of this space-time. $\kappa$-Minkowski therefore provides an example of the mentioned scenario in which an ordinary symmetry is violated but there is no “net loss of symmetries” (the 10-generator Poincaré symmetry is replaced by the 10-generator $\kappa$-Poincaré symmetry). It is in order to capture the essence of these situations that one introduces the terminology “symmetry deformation" (in alternative to “symmetry violation" which could be reserved for cases with a net overall loss of symmetries). In Sections 4 and 5 we denominate our scheme as a LID just to emphasize that the equations we use do not necessarily reflect a loss of symmetry (whether or not they do imply a net loss of symmetry depends on the underlying algebraic structures that lead to those equations in a given space-time picture).
Importantly, consistency with the non-commutative nature of $\kappa$-Minkowski space-time also requires [@gackpoinplb; @kpoinap; @gacmaj; @lukipap; @majrue] that the law of addition of momenta be accordingly modified. This modification emerges at the level of the $\kappa$-Poincaré (Hopf) algebra, and of course requires physical interpretation (particle momenta in a noncommutative space-time are a new concept). A prescription suitable for handling the ambiguities due to the non-commutative nature of $\kappa$-Minkowski space-time was given recently in Ref. [@gacmaj], and in the cases here of interest, which always involve the sum of parallel momenta of two particles (at threshold particles are produced at rest in the CM frame), it reduces (in leading order in $E_{p}^{-1}$) to the prescription that the sum of momenta $K_1$ and $K_2$ can be handled with ordinary algebraic methods upon the replacement $K_1 + K_2 \rightarrow K_1 + K_2 + \delta K_1 K_2/ E_{p}$, where $\delta$ is a parameter analogous to $\eta$. In the analyses reported in Sections 4 and 5 this would imply $p_1 - \epsilon \rightarrow p_1 - \epsilon - \delta p_1 \epsilon/E_{p}$ and $p_2 + p_3 \rightarrow p_2 + p_3 + \delta p_2 p_3/ E_{p}$, and actually, since of course we have been here only interested in the leading $E_{p}^{-1}$ effect and $\epsilon \ll p_2 \sim p_3 \sim p_1$, one can neglect the term of order $p_1 \epsilon/ E_{p}$ while retaining the term of order $p_2 p_3/ E_{p}$.
[^1]: The exact composition of UHECRs is unknown and it is possible that UHECRs are heavy nuclei rather than protons. In this case such nuclei would undergo photodisintegration when interacting with CMBR photons. The threshold energy for a photodisintegration of a nuclei is several MeV. It just happens to be true, purely as a result of a numerical coincidence, that the threshold is reached when the energy of a typical nuclei, say Fe, is $\sim 5 {\times} 10^{19}$eV. Thus the GZK paradox is insensitive to the question of what is the exact composition of UHECRs.
[^2]: Having mentioned that of course the deformation (\[dispone\]) could be considered independently of its quantum-gravity motivation, let us also mention in passing that even outside the quantum-gravity literature there is a large amount of work on the theory and phenomenology of violations of Lorentz invariance (see, [*e.g.*]{}, the recent Refs. [@Gonzalez; @colgla; @carjack; @jackost], which also provide a good starting point for a literature search backward in time).
[^3]: The dominant contribution to the GZK cutoff actually comes from the $\Delta$ resonance, so one might find appropriate to replace the sum of the proton mass and the pion mass with the mass of the $\Delta$ in the UHECR threshold formula. However, the difference between $m_\Delta$ and $m_p + m_\pi$ would only introduce a relatively small correction in our UHECR limit which is not our dominant lower limit (a much stricter limit comes from the TeV-$\gamma$ anomaly). Moreover, once the contribution to GZK from the $\Delta$ is avoided one would still have a (weakened) GZK cutoff from non-resonant photopion production and this would anyway lead to the limit we use.
[^4]: It is perhaps worth emphasizing that the low-energy expansion of $F(p,m;E_p)$ may look quite different from its corresponding high-energy expansion. In the high-energy regime ($p \gg m$) the premium is on the leading dependence on $p$ while in the low-energy regime ($p \ll m$) the leading dependence on $m$ is dominant.
[^5]: Note that actually the threshold is not necessarily anomalous; in particular, as we already observed in Ref. [@AmePiran00], when $\alpha=\beta=1$, $\sigma=0$ and $\eta=-\delta$ there is a cancellation and the deformed symmetries lead to the same threshold equation obtained with undeformed symmetries.
[^6]: The fact that there are no UHECR and TeV-$\gamma$ threshold anomalies in our scheme for $\alpha = 0,\sigma=0,\delta=0$ can be easily derived directly from the corresponding dispersion relation. This is also implicit in Figure 1, which shows that $|\eta| \rightarrow \infty$ as $\alpha \rightarrow 0$. Also notice that undeformed thresholds are not only obtained for $\alpha = 0,\sigma=0,\delta=0$: the thresholds become undeformed also, for example, in the limit $\alpha \rightarrow \infty$ approached keeping $\sigma=0,\delta=0$. However the undeformed-threshold point $\alpha = 0,\sigma=0,\delta=0$ is best suited for a comparison between our scheme and the CG scheme.
[^7]: Note however that, while the TeV-$\gamma$ threshold anomaly is used in both, not all the experimental constraints used by the two phenomenological analysis are the same. In particular, the time-of-flight upper bound on LID was not used to establish $5 {\cdot} 10^{-16} < c_e - c_\gamma < 5 {\cdot} 10^{-13}$.
|
---
abstract: 'We present a combined analysis of the new eta photoproduction data for total and differential cross sections, target asymmetry and photon asymmetry. Using basic assumptions, this allows a model-independent extraction of the $E_{2-}$ and $M_{2-}$ multipoles as well as resonance parameters of the $D_{13}(1520)$ state. At higher energy, we show that the photon asymmetry is extremely sensitive to small multipoles that are excited by photons in the helicity $3/2$ state. These could be, e.g., the $F_{15}(1680)$, the $F_{17}(1990)$, or the $G_{17}(2190)$ resonances.'
author:
- |
Lothar Tiator[^1] and Germar Knöchlein[^2]\
[*Institut für Kernphysik, Johannes Gutenberg-Universität, D-55099 Mainz, Germany*]{}\
Cornelius Bennhold[^3]\
[*Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, D.C., 20052*]{}
title: Eta Photoproduction
---
0.505cm 0.5cm
INTRODUCTION {#introduction .unnumbered}
============
Over the last several years, eta photoproduction has demonstrated its potential as a new, powerful tool to selectively probe certain resonances that are difficult to explore with pions. It is well known that the low energy behavior of the eta production process is governed by the $S_{11}(1535)$ resonance[@Benn91; @Tiat94; @Benm95]. The recent, precise measurements of total and differential cross sections for eta photoproduction at low energies[@Krus95; @Wilh93] have allowed determining the $S_{11}(1535)$ resonance parameters with unprecedented precision. However, it is because of the overwhelming dominance of the $S_{11}$ that the influence of other resonances in the same energy regime, such as the $D_{13}(1520)$, is difficult to discern. It has been pointed out[@Tiat94] that polarization observables provide a new doorway to access these non-dominant resonances which relies on using the dominant $E_{0+}$ multipole to interfere with a smaller multipole. Especially the polarized photon asymmetry was shown to be sensitive to the $D_{13}(1520)$. Recently, polarization data for the target and photon asymmetries in eta photoproduction were measured at ELSA[@Bock98] and GRAAL[@Hour98], respectively, for the first time. Taken together with the data for the unpolarized cross section from MAMI, they allow a determination of the $D_{13}(1520)$ contribution in eta photoproduction.
MULTIPOLE ANALYSIS {#multipole-analysis .unnumbered}
==================
In the following all considerations refer to the c.m. frame. The three measured observables have the following representation in terms of the response functions defined in [@Knoe95]: $$\begin{aligned}
\frac{d \sigma}{d \Omega}& = &\frac{|\vec k_{\eta}|}{|\vec q|}
R_T^{00}\, ,\\
T & = & \frac{R_T^{0y}}{R_T^{00}}\, ,\\
\Sigma & = & - \frac{^{c}R_{TT}^{00}}{R_T^{00}}\, .\end{aligned}$$
Because of the overwhelming dominance of the $S_{11}$ channel in eta photoproduction, the observables can be expressed in terms of $s$–wave multipoles and interferences of the $s$ wave with other multipoles. In the CGLN basis this leads to an $F_1$ dominance and the observables can simply be expressed as $$\begin{aligned}
\label{cgln}
R_T^{00} & = & |F_1|^2 - \mbox{Re} \left\{ 2 \cos\theta F_1^* F_2 -
\sin^2\theta F_1^*F_4 \right\},\\
R_T^{0y} & = & 3 \sin\theta\, \mbox{Im} \left\{ F_1^* F_3 +
\cos\theta F_1^*F_4 \right\},\\
^cR_{TT}^{00} & = & \mbox{Re} \left\{F_1^* F_4 \right\}.\end{aligned}$$ If we retain only interferences with $p$– and $d$–waves (an approximation that is valid at least up to 1 GeV photon lab energy) we obtain $$\begin{aligned}
\label{o1}
R_T^{00} & = &
|E_{0+}|^2 - \mbox{Re} \left[ E_{0+}^* \left( E_{2-} - 3 M_{2-} \right) \right]
\nonumber \\
& & + 2 \cos \theta \mbox{Re} \left[ E_{0+}^*
\left( 3 E_{1+} + M_{1+} - M_{1-} \right) \right]
\nonumber \\
& & + 3 \cos^2 \theta \mbox{Re} \left[ E_{0+}^* \left( E_{2-} - 3 M_{2-} \right)
\right]\, ,\\
\label{o2}
R_T^{0y} & = &
3 \sin \theta \mbox{Im} \left[ E_{0+}^* \left( E_{1+} - M_{1+} \right) \right]
\nonumber \\
& & - 3 \sin \theta \cos \theta
\mbox{Im} \left[ E_{0+}^* \left( E_{2-} + M_{2-} \right) \right]\, ,\\
\label{o3}
^{c}R_{TT}^{00}& = &
- 3 \sin^2 \theta \mbox{Re} \left[ E_{0+}^* \left( E_{2-} + M_{2-} \right) \right]
\, .\end{aligned}$$ With the following angle-independent quantities $$\begin{aligned}
a & = & |E_{0+}|^2- \mbox{Re} \left[ E_{0+}^*\left( E_{2-}-3 M_{2-} \right) \right]
\, ,\\
b & = & 2 \mbox{Re} \left[ E_{0+}^* \left( 3 E_{1+} + M_{1+} - M_{1-} \right) \right]
\, ,\\
c & = & 3 \mbox{Re} \left[ E_{0+}^* \left( E_{2-} - 3 M_{2-} \right) \right]\, ,\\
d & = &
\frac{1}{a + \frac{1}{3} c}
3 \mbox{Im} \left[ E_{0+}^* \left( E_{1+} - M_{1+} \right) \right]\, ,\\
e & = & - 3
\frac{1}{a + \frac{1}{3} c}
\mbox{Im} \left[ E_{0+}^* \left( E_{2-} + M_{2-} \right) \right]\, ,\\
f & = & 3 \frac{1}{a + \frac{1}{3} c}
\mbox{Re} \left[ E_{0+}^* \left( E_{2-} + M_{2-} \right) \right]\, ,\end{aligned}$$ we can express the observables in a series of $\cos\theta$ terms that can be fitted to the experimental data at various energies $E_{\gamma,lab}$ $$\begin{aligned}
\frac{d \sigma}{d \Omega}& = & \frac{|\vec k_{\eta}|}{|\vec q|}
\left(a + b \cos \theta + c \cos^2 \theta\right)\, ,\\
T & = & \sin \theta \left(d + e \cos \theta \right)\, ,\\
\Sigma & = & f \sin^2 \theta \, .\end{aligned}$$ It is remarkable that a combined analysis of the three above observables allows a determination of the $d$–wave contributions to eta photoproduction once the quantities $a$, $c$, $e$ and $f$ have been determined from experiment. Already with the knowledge of $e$ and $f$ the helicity $3/2$ multipole $B_{2-}$, defined below, and the phase relative to the $S_{11}$ channel can be determined: $$\begin{aligned}
| B_{2-} | \equiv | E_{2-} + M_{2-} | & = & \frac{\sqrt{e^2+f^2}}
{3\sqrt{a+c/3}}\, ,\\
\tan (\phi_{E_{0+}}-\phi_{B_{2-}}) & = & \frac{e}{f}\, .\end{aligned}$$ If one neglects electromagnetic effects from the background of eta photoproduction affecting the phase of the electric and magnetic multipoles differently ($\phi_{E_{l \pm}} = \phi_{M_{l \pm}} = \phi_{l \pm}$), one can write $$\begin{aligned}
E_{l \pm} & = | E_{l \pm} | e^{i \phi_{l \pm}} \, ,\\
M_{l \pm} & = | M_{l \pm} | e^{i \phi_{l \pm}}\, , \end{aligned}$$ and one finds the following representation for the real and imaginary parts of the $d$–wave multipoles: $$\begin{aligned}
\mbox{Re} E_{2-} & = & \frac{1}{4} \sqrt{a + \frac{1}{3} c}
\left(f \cos \phi_{0+} + e \sin \phi_{0+} \right)
\left(1 + \frac{c}{3 f}\right)\, ,\\
\mbox{Im} E_{2-} & = & \frac{1}{4} \sqrt{a + \frac{1}{3} c}
\left(f \sin \phi_{0+} - e \cos \phi_{0+} \right)
\left(1 + \frac{c}{3 f}\right)\, ,\\
\mbox{Re} M_{2-} & = & \frac{1}{12} \sqrt{a + \frac{1}{3} c}
\left(f \cos \phi_{0+} + e \sin \phi_{0+} \right)
\left(1 - \frac{c}{f}\right)\, ,\\
\mbox{Im} M_{2-} & = & \frac{1}{12} \sqrt{a + \frac{1}{3} c}
\left(f \sin \phi_{0+} - e \cos \phi_{0+} \right)
\left(1 - \frac{c}{f}\right)\, .\end{aligned}$$ We note that this determination of the $E_{2-}$ and $M_{2-}$ multipoles is rather model independent. To be more explicit we list the assumptions used to arrive at the above formulae:
- Phase difference between electric and magnetic multipoles neglected, $\phi_{E_{l \pm}} = \phi_{M_{l \pm}}
= \phi_{l \pm}$
- Restriction to the truncated multipole representation of Eqs. (\[o1\]), (\[o2\]), (\[o3\])
- Knowledge of the phase of the $E_{0+}$ multipole.
The last point deserves further discussion: From total cross section data [@Krus95] it is obvious that in the region of the $S_{11}(1535)$ resonance the cross section can be perfectly fitted by a Breit–Wigner resonance resulting in $s$–wave dominated differential cross sections. An investigation of the background from the Born terms [@Tiat94] yielded a very small eta–nucleon coupling constant. As a consequence, the $E_{0+}$ multipole can be treated as being completely dominated by the $S_{11}(1535)$ contribution, which, as shown in ref.[@Krus95], allows parametrizing it through a Breit–Wigner form. In principle, an arbitrary phase for the complex $E_{0+}$ multipole could be added which is set equal to 0 by convention. For the complex $E_{0+}$ multipole we use the Breit–Wigner parametrization $$E_{0+} = - \sqrt{\frac{a}{4 \pi}} \frac{\Gamma^* M^*}{{M^*}^2 - W^2 -
i M^* \Gamma(W)}\, ,$$ where $W$ is the c.m. energy. The energy dependence of the resonance width is given by $$\Gamma(W) = \Gamma^* \left(
b_{\eta} \frac{|\vec k|}{|\vec k^*|}
+ b_{\pi} \frac{| \vec k_{\pi}|}{|\vec k_{\pi}^*|} + b_{\pi\pi}
\right) \, .$$ The analysis of the $E_{0+}$ interference with the $E_{2-}$ and $M_{2-}$ multipoles determines the $d$ wave multipoles and therefore the difference $\phi_{2-}
- \phi_{0+}$. It does not yield direct information on $\phi_{2-}$. However, making the above assumptions for the $E_{0+}$ multipole and thus the phase $\phi_{0+}$ permits the determination of $\phi_{2-}$.
To perform a similar analysis of the $p$–wave multipoles more information from additional polarization observables is required; in particular, a measurement of the recoil polarization would be very helpful. As before we obtain $$\begin{aligned}
P & = & \frac{R_T^{y0}}{R_T^{00}}\, ,\\ & = & \sin \theta \left(g +
h \cos \theta \right)\, \\ \end{aligned}$$ with $$\begin{aligned}
g & = & - \frac{1}{a + \frac{1}{3} c} \mbox{Im}
\left[ E_{0+}^* \left( 2 M_{1-} + 3 E_{1+} + M_{1+} \right) \right]\, ,\\
h & = & 3 \frac{1}{a + \frac{1}{3} c}
\mbox{Im} \left[ E_{0+}^* \left( E_{2-} - 3 M_{2-} \right) \right]\, .\end{aligned}$$ After performing single-energy fits we used a polynomial fit to the energy dependence of the coefficients $a$, $b$, $c$, $d$, $e$ and $f$ in order to arrive at a global (energy dependent) solution for the multipoles. This has several advantages: First the experimental data have been obtained in different set–ups at different labs, thus their energy bins do not match each other. Second, except for quantity $a$ that is in principle determined already by the total cross section, all other quantities contain considerable error bars, therefore, a combined fit can reduce the uncertainty of individual measurements considerably. In a simple Taylor expansion in terms of the eta momentum with only 1-3 parameters in each coefficient we obtain good results for an energy region from threshold up to about 900 MeV.
RESULTS {#results .unnumbered}
=======
Fig. 1 shows 4 out of 10 angular distributions measured by the TAPS collaboration at Mainz [@Krus95] in the energy range between 716 and 790 MeV. While our isobar model falls a bit low close to threshold, a perfect fit is possible using the Ansatz in Eq. (16). Our results for the coefficients $a$, $b$ and $c$ agree perfectly with the results ontained in Ref. [@Krus95]. As mentioned before, the $a$ coefficient can be fitted to a Breit-Wigner form with an energy-dependent width leading, e.g., to parameters of $M^*=(1549\pm 8)MeV$, $\Gamma_R=(202\pm 35) MeV$ and an absolute value of the $s$-wave multipole at threshold, $|E_{0+}|=16.14
\cdot 10^{-3}/m_\pi^+$ (Fit 1, Ref. [@Krus95]). For our purpose here it is more convenient to use a general polynomial expansion as mentioned above.
Fig. 2 shows the target polarization with the preliminary data from Bonn[@Bock98]. Here our isobar model fails to reproduce the angular shape of the data. In particular there is no node in our calculation and the role of the $D_{13}$ resonance plays a very small and insignificant role. In our previous coupled channel analysis the $D_{13}$ resonance came out much stronger and a node developed, however, with a minus sign at forward and a positive sign at backward angles. This is opposite to the experimental observation and, as we will see later, indicates a drastically different relative phase between $s$- and $d$–waves. With the ansatz of Eq. (17) we can fit the data and obtain a node at low energies that disappears around 800 MeV.
In Fig. 3 we show our isobar calculations for the photon asymmetry. This observable has been measured recently at GRAAL [@Hour98], however, the data are still in the analysis. A preliminary comparison, however, shows general agreement for energies below 1 GeV. From our calculations the importance of the $D_{13}$ channel for the photon asymmetry becomes obvious. Without this nucleon resonance, the asymmetry would be almost zero up to about 900 MeV. Even as the experimental data for the photon asymmetry are not yet available we can already perform a preliminary analysis of the $D_{13}$ multipoles under the constraint of the photon asymmetries determined by our isobar model. In this case, all coefficients of Eqs. (10-15) are available and we can evaluate the $d$–wave multipoles using Eqs. (21-24). As mentioned before, the solution for the individual multipoles $E_{2-}$ and $M_{2-}$ requires the additional assumption for the phase of the $s$–wave amplitude. This is taken from the Breit-Wigner Ansatz Eqs. (27-28) with the parameters of fit 1 in Ref. [@Krus95]. Of course, this form is rather ad hoc, however, comparing with coupled channels calculations [@Kais97; @Feus97] we find that the results of these very different approaches agree very well not only for the absolute magnitude of the $s$–wave but also for the phase.
Fig. 4 shows the result of our multipole analysis and compares it with our isobar model calculation. The biggest difference occurs in the relative phase between the $s$- and $d$–waves. As shown in Eq. (20) this phase difference is model independent. If we consider two Breit-Wigner type resonances for both, $S_{11}(1535)$ and $D_{13}(1520)$ this phase difference would be rather constant as both resonances are very close in their energy position and, furthermore, have a similar resonance width. From the fact that the $S_{11}$ is a bit higher in energy, the phase difference $\Phi_0 - \Phi_2$ should be negative as is shown in the figure as the dotted line.
>From the above analysis we conclude that this completely unexpected discrepancy is directly connected to the node structure of the target asymmetry. Without a node or with a node but an $e$–coefficient of opposite sign, the phase difference would be much smaller and closer to our model calculations.
ETA PHOTOPRODUCTION AT HIGHER ENERGIES {#eta-photoproduction-at-higher-energies .unnumbered}
======================================
The most remarkable fact of eta photoproduction in the low energy region is the strong dominance of the $S_{11}$ channel. Whether it occurs from a $N^*$ resonance, which is the most likely case, or from different mechanism is a very interesting question and subject of many ongoing investigations. In the experiment it shows up as a flat angular distribution and only very precise data can observe some tiny angular modulation as found by the Mainz experiment [@Krus95]. At Bonn, angular distributions of the differential cross section have been measured up to $1.15$ GeV [@Bock97] with no evidence for a break-down of the $s$–wave dominance. Therefore, we can speculate that this dominance continues up to even higher energies. Theoretically, this could be understood in terms of very small branching ratios for nucleon resonances into the $\eta N$ channel. For all resonances except the $S_{11}(1535)$ the branching ratio is below 1%, or in most cases even below 0.1%. In the case of the $D_{13}(1520)$ resonance this ratio is also assumed around 0.1%, however, an average number is no longer quoted in the Particle Data Tables [@PDG96]. Only branching ratios for the two $S_{11}$ resonances remain. As we have shown in the last Section, the photon asymmetry is a very sensitive probe for even tiny branching ratios such as the $D_{13}$ resonance.
In the following, we demonstrate that this is especially the case for nucleon resonances with strong helicity $3/2$ couplings $A_{3/2}$. In Table 1 we list all entries for $N^*$ resonances with isospin $1/2$. From this table one finds the $D_{13}$ as the strongest candidate to show up in the photon asymmetry. However, other resonances include the $F_{15}(1680)$ which plays an important role in pion photoproduction and, furthermore, the $F_{17}(1990)$ and the $G_{17}(2190)$ that are less established in photoproduction reactions. Furthermore, since these numbers are determined from data in the pion photoproduction channel, surprises in the eta photoproduction channel are not only possible but indeed very likely.
$N^*$ Resonance $A_{3/2} [10^{-3}GeV^{-1/2}]$ Multipoles
----------------- ------------------------------- ------------------------
$D_{13}(1520)$ $+166\pm 5$ $B_{2-}=E_{2-}+M_{2-}$
$D_{15}(1675)$ $+15\pm 9$ $B_{2+}=E_{2+}-M_{2+}$
$F_{15}(1680)$ $+133\pm 12$ $B_{3-}=E_{3-}+M_{3-}$
$D_{13}(1700)$ $-2\pm 24$ $B_{2-}=E_{2-}+M_{2-}$
$P_{13}(1720)$ $-19\pm 20$ $B_{1+}=E_{1+}-M_{1+}$
$F_{17}(1990)$ $+86\pm 60$ $B_{3+}=E_{3+}-M_{3+}$
$D_{13}(2080)$ $+17\pm 11$ $B_{2-}=E_{2-}+M_{2-}$
$G_{17}(2190)$ $81 - 180$ $B_{4-}=E_{4-}+M_{4-}$
: Photon couplings and multipolarities for $N^*$ Resonances with helicity $3/2$ excitation. The numbers are taken from PDG96[@PDG96], average numbers above and single quoted numbers (less certain) below the horizontal line.
\[tab:res\]
Assuming $S$–wave dominance and therefore $F_1$–dominance in the amplitude we can derive a general expression for the photon asymmetry, $$\begin{aligned}
\Sigma(\theta) & = & -\sin^2\theta\,\,
\mbox{Re}\big[F_1^* F_4\big]/R_T^{00}\, ,\\
& = & \sin^2\theta\,\, \mbox{Re}\bigg[E_{0+}^* \sum_{\ell \ge 2}
(B_{\ell -}+B_{\ell +}) P_\ell''(\cos\theta) \bigg] /R_T^{00}\, \\\end{aligned}$$ with $B_{\ell-}=E_{\ell-}+M_{\ell-}$ and $B_{\ell+}=E_{\ell+}-M_{\ell+}$. Both multipole combinations are helicity $3/2$ multipoles and for resonance excitation they are proportional to the photon couplings $A_{3/2}$. The helicity $1/2$ couplings $A_{1/2}$ do not enter here, they appear in the differential cross section and in the recoil polarization, e.g. as $A_{2-}=(3M_{2-}-E_{2-})/2$. Explicitly, we obtain up to $\ell_{max}=4$ $$\begin{aligned}
\Sigma(\theta) & = & \frac{\sin^2\theta}{|E_{0+}|^2}\, \mbox{Re}
\Big\{E_{0+}^*\Big[3(B_{2-}+B_{2+})-\frac{15}{2}(B_{4-}+B_{4+})
\nonumber \\
& + & 15(B_{3-}+B_{3+})\cos\theta
+ \frac{105}{2}(B_{4-}+B_{4+})\cos^2\theta \Big] \Big\}\, .\end{aligned}$$ In Fig. 5 we demonstrate how such interferences of higher resonances with the $S_{11}$ channel could show up in the photon asymmetry. Even if two small resonances of different multipolarity are excited in the same energy region they will produce a clear signal that will eventually allow determining $\eta$ branching ratios down to values well below $0.1\%$.
SUMMARY {#summary .unnumbered}
=======
We have demonstrated that polarization observables are a powerful tool in analyzing individual resonances in the eta photoproduction channel. The strong dominance of the $S_{11}$ channel allows a much easier analysis compared to pion photoproduction. Furthermore, the nonresonant background in eta physics appears to be small due to a very weak coupling of the eta to the nucleon. A combined analysis of differential cross section, photon asymmetry and target polarization allows a determination of $s$– and $d$–wave multipoles. The target polarization measured at Bonn reveals an unexpected phase shift between the $S_{11}$ and $D_{13}$ resonances that could lead to the conclusion that either of these resonances, perhaps the $S_{11}$, is heavily distorted or is even a completely different phenomenon, as frequently speculated. The new experiments therefore add another piece to the eta puzzle that makes the field of eta physics so exciting.
[9]{} C. Bennhold and H. Tanabe, Nucl. Phys. [**A530**]{}, 625 (1991). L. Tiator, C. Bennhold and S.S. Kamalov, Nucl. Phys. [**A580**]{}, 455 (1994). M. Benmerrouche, N.C. Mukhopadhyay and J.-F. Zhang, Phys. Rev. [**D51**]{}, 3237 (1995). B. Krusche et al., Phys Rev. Lett. [**74**]{}, 3736 (1995). M. Wilhelm, Thesis (Bonn University, 1993). A. Bock et al., submitted for publication in Phys. Rev. Lett. (1998). J. Ajaka et al., contribution to this workshop. G. Knöchlein, D. Drechsel and L. Tiator, Z. Phys. [**[A 352]{}**]{}, 327 (1995). N. Kaiser, T. Waas and W. Weise, Nucl. Phys. [**A 25**]{} (1997) 297 and N. Kaiser, private communication. T. Feuster and U. Mosel, nucl-th/9708051 and T. Feuster, private communication. A. Bock, Thesis (Erlangen, 1997). Review of Particle Physics, Particle Data Group, Phys. Rev. [D 54]{} (1996) 1.
[^1]: E-mail: [tiator@kph.uni-mainz.de]{}
[^2]: E-mail: [knoechle@kph.uni-mainz.de]{}
[^3]: E-mail: [bennhold@gwis2.circ.gwu.edu]{}
|
---
bibliography:
- 'biblio.bib'
nocite: '[@langley00]'
---
Introduction
============
In light of the recent successes of ever-larger Deep Neural Networks (DNN) models and data sets [@dai2019transformerxl], the need for efficient distributed machine learning strategies on massively parallel systems is more significant than ever before. Various distributed deep learning approaches have been explored throughout the years ranging from Multiple Instruction Multiple Data (MIMD) programming in model-parallelism [@Dean:2012MP] to the Single Program Multiple Data (SPMD) used in data-parallelism, and most recently pipelining algorithms [@gpipe], parallel tensor contractions [@mesh-tensorflow], and task graph-based strategies [@hybrid-parallel]. Despite many of these advances, data-parallelism [@data-parallel] remains the most widely adopted distributed deep learning strategy. Data-parallelism is both broadly applicable, and its implementation is agnostic to a system’s architecture, by contrast to MIMD programming.
As a distribution strategy, data-parallelism is communication-heavy, requiring the execution of blocking communication collectives to synchronize DNN gradients throughout a training run. A sub-optimal overlap between computation and communication operations during a single training step introduces communication overheads or inefficiencies in data-parallel distributed deep learning. On small to moderate-scale systems, with 10’s - 100’s of GPU/TPU accelerators, these scaling inefficiencies can be difficult to detect and systematically optimize due to system noise and load variability. Note, however, that even moderate scaling inefficiencies on the order of 5-10% accumulate across many training steps and training runs and further increase the enormous carbon footprint of deep learning and its associated environmental impact [@strubell2019energy]. The scaling inefficiencies of data-parallel implementations are most readily apparent on large-scale systems such as supercomputers with 1,000’s-10,000’s of accelerators. Here, we show that supercomputers are ideal systems to develop and test new gradient reduction strategies to achieve near-linear scaling of data-parallelism.[^1]
Extending data-parallelism to the massive scale of supercomputing systems is also motivated by the latter’s traditional workload consisting of scientific numerical simulations [@Kent348]. In particular, infusing deep learning into scientific simulations to speed-up their execution and decrease their computational demands often requires approximating the solution of longstanding inverse problems with DNN. Here, we demonstrate the first step in this direction, made possible by our improved gradient reduction strategies.
Overview
========
System and Environment
----------------------
All measurements reported here were carried out on the Summit supercomputer at the Oak Ridge Leadership Computing Facility, a US Department of Energy Office of Science User Facility. Summit is a system dedicated to open science with access applications in the form of peer-reviewed scientific user proposals.
The Summit system consists of 256 racks populated by IBM Power System AC922 compute nodes ($\sim$ 4600 nodes in total), each equipped with 2 IBM POWER9 CPUs and 6 NVIDIA V100 GPUs. It is ideally suited for Deep Learning workloads due to its node-local NVMe (burst buffer) and the Tensor Cores on V100 for faster low-precision operations. Within a Summit node, CPU-GPU and GPU-GPU communications are carried out over NVIDIA’s NVLink interconnect, supporting a (peak) bi-directional bandwidth of 100 GB/s, where each 3 V100 GPUs are grouped in a ring topology with all-to-all connections to a POWER9 CPU. The CPU-CPU bridge consists of two NVLink connections, each with a (peak) bandwidth of 25 GB/s. Summit nodes are configured in a non-blocking fat-tree topology via a dual-rail Mellanox EDR 100G InfiniBand Interconnects. The IBM Alpine GPFS provides 2.5 TB/s aggregated I/O bandwidth, which is not enough to feed over 27,600 V100 GPUs each processing at over 0.5 GB/s, while NVMe offers a read bandwidth of 6 GB/s per node and provides a local I/O alternative which scales linearly with the numbers of compute nodes. All of the data we include here was collected (and reproduced) during normal operations of the Summit supercomputer and in the presence of other applications running on available computed nodes. As such, the performance we report is typical of the system.\
Distributed Deep Learning on Supercomputers
-------------------------------------------
\[subsec:summit-specs\]
We focus on a data-parallelism approach to the distributed training of DNN. To date, the largest distributed DNN training was carried out by [@Kurth:2018] to learn a segmentation task on climate simulation data. These authors used a modified DNN segmentation model (DeepLabV3 [@deeplabv3]) which achieved a per GPU computing performance of 38.45 TFLOP$_{16}$, equivalently 31% of the theoretical peak of the V100 GPU (the subscript 16 refers to float16 precision).
One of the key innovations introduced in [@Kurth:2018] is a hierarchical Allreduce strategy consisting of intra-node collectives with NCCL (v2.3) and inter-node collectives with IBM’s Spectrum-MPI. This communication strategy proved highly effective at reducing the ratio of communication time to compute time, and achieving a scaling efficiency of 90.7% on 4560 Summit nodes with a sustained (peak) performance of 990 PFLOPS$_{16}$ (1.13 EFLOPS$_{16}$), but at the expense of skipping gradient synchronization/reduction every other training step. The scaling efficiency used in this previous study and in other work using data-parallelism (including ours) is defined as the total number of inputs (i.e. images) processed during a training step as a function of computing resources (e.g. Summit nodes).\
In subsequent sections, we describe new orchestration strategies of collectives during the gradients reduction stage, which prove to be more efficient than a hierarchical allreduce, allowing us to achieve $0.93$ scaling efficiency on 4600 nodes, and near perfect scaling efficiency ($>0.97$) on compute resources on the order of 1000’s of GPUs or less.
Distributed Training with Horovod {#section:distributed}
---------------------------------
#### Horovod
The optimized implementation of DNN mathematical operations in `cuDNN` and their fast execution on state of the art GPUs such as the V100 Tensor Cores leads to small computation times, $t_{\textrm{comp}}$, during a training step (typically $t_{\textrm{comp}} \sim$ sub-second to seconds). The time required to perform gradient reduction using blocking collectives, $t_{\textrm{comm}}$, therefore, is the key quantity to optimize in a data-parallel approach to distributed deep learning. We used `Horovod` [@sergeev2018horovod], an open source library to perform gradient reduction across model replicas during distributed training. `Horovod` embeds allreduce operations into the `TensorFlow` computation graph of the DNN and employs efficient inter-GPU communication via the `MPI Allreduce` algorithm and/or by using the NVIDIA Collective Communications Library (`NCCL`) [@nccl-site], depending on the configuration selected during installation time. Note that `Horovod` supports multiple frameworks and can be used to carry out data-parallelism on `PyTorch` [@pytorch] and `MXNet` [@mxnet].
![The influence of different Gradient Reduction Strategies on the Scaling Efficiency. The DNN model used was a modified version of the fully-convolutional dense neural network (FC-DenseNet) [@tiramisu] with 40 million parameters. The reported performance is the sustained performance in peta floating point operations per second carried in 16-bit numerical precision. 1 Summit node = 6 NVIDIA V100 GPUs.[]{data-label="fig:commstrat"}](figures/perf_comm_strategies2.png)
The hierarchical allreduce strategy introduced in [@Kurth:2018] was originally implemented within `Horovod` but the publicly available code base does not contain all of the features described in [@Kurth:2018]. As such, a direct comparison between the hierarchical allreduce in [@Kurth:2018] and the one we use here is not meaningful. Furthermore, some of the features of the original implementation of hierarchical allreduce made assumptions regarding the network topology that were somewhat specific to Summit’s architecture.
#### Scaling of Distributed Training Strategies
In Figure \[fig:commstrat\], we measured the scaling efficiency of hierarchical allreduce up to 1024 Summit nodes. The sub-linear scaling is evident and was traced to poor overlap between communication and computation caused by inefficient worker coordination at large nodes. The newly released `NCCL` (v2.4) addresses latency issues of the systolic ring algorithm of `NCCL` (v2.3), using an implementation of double binary trees for full bandwidth utilization and logarithmic latency of allreduce operations [@Sanders:2009]. This new `NCCL` double binary trees implementation obviates the need for Horovod’s explicitly hierarchical allreduce altogether, as is seen from the 3$\times$ gain in performance between the green and blue lines in . At larger node counts ($> 1,000$ GPUs), the scaling inefficiency of data-parallelism as originally implemented in `Horovod` becomes apparent, necessitating the need for new strategies.
Contributions
=============
Our main contributions consist of:
- Implementing new gradient reduction strategies which produce optimal overlap between computation and communication, a decrease in $t_{\textrm{comm}}$ during execution of the computation graph, and achieving state of the art scaling efficiency and performance of distributed deep learning up to 27,600 GPUs.
- Harnessing these gradient reduction strategies in the distributed training of a DNN with over $10^8$ weights on a dataset with size of a 500 TB to approximate, for the first time, a solution to an inverse problem in scientific imaging.
#### Gradient Reduction Strategies
The gradient reduction strategies consist of: (1) a lightweight worker coordination technique (BitAllReduce) and (2) a gradient tensor grouping strategy (Grouping). These two orchestration strategies improve on different aspects of distributed deep learning as currently implemented in `Horovod`. The effects of BitAllReduce and Grouping on the scaling efficiency are shown in in black and red lines, respectively. In tandem, they lead to over $8\times$ better scaling efficiency (). These gradient reduction strategies are computing platform agnostic and do not make any assumptions regarding the interconnect network topology.
First, Bitvector Allreduce modifies how the coordination of gradient tensors reduction via collective is performed (see Figure \[fig:coordination\]). The main idea of Bitvector Allreduce is the use of cached meta-data, associated with each gradient tensor, and locally accessible to each `MPI`-rank to globally coordinate the execution of collective operations. In essence, we replace the original master-worker strategy of `Horovod` with a single collective (an `MPI_Allreduce` on a bitvector) (see Figure \[fig:coordination:b\]).
Second, we introduce a “grouping" scheme for the gradient tensors akin to a graph coloring algorithm. Essentially, each `MPI` rank locally colors the nodes of its computational dependency graph (node = gradient tensor), and groups of gradient tensors are formed from like colors (see Figure \[fig:grouping\]). Collective operations are then only issued for those groups which are ready across all ranks. One of the strengths of “grouping" is to grant the user with the flexibility to order collectives in a fashion that exploits the architecture of her DNN model, thereby achieving greater efficiency.
Finally, we note that both “Grouping” and “Bitvector Allreduce" can be used independently, but used in combination they provided the massive gains in performance we report here. In the next section we describe in detail the implementations of these novel orchestration strategies.
![Horovod Timeline to illustrate the improved orchestration with grouping and bitvector allreduce. The blue vertical lines are cycle markers.[]{data-label="fig:timeline"}](figures/hvd_timelines_crop.png)
#### Scientific Inverse Problems
Harnessing the well-known function approximation capabilities of modern Deep Neural Networks (DNN) to solve challenging inverse problems in imaging [@Lucas:2018ev] has been mostly explored within the field of medical imaging [@Adler_2017; @rivenson2018phase], though there have been a few notable exceptions within materials imaging [@cherukara2018real; @laanait2019reconstruction]. In contrast to other application domains, materials imaging, especially at the atomic scale, has the benefit of having access to highly-accurate and fully-quantitative forward simulation models and theories underpinned by quantum theory. The massive size of a single training example, which are often multi-dimensional arrays, can easily reach GBs and presents new challenges in the training of DNN. Most notably, the need for efficient I/O and the distributed training of large DNN models and consequently large message sizes. While large scale scientific simulation problems are a prevalent workload on supercomputers, to this date, however, no previous work has harnessed the capabilities of high-performance computing to produce a DNN-based solution to a scientific inverse problem. We show that our improvements to gradient reduction strategies now make it possible to approximate solutions to scientific inverse problems with deep learning and supercomputing.
Gradient Reduction Strategies
=============================
Worker Coordination via Bitvector Allreduce
-------------------------------------------
TensorFlow’s use of a graph-based scheduler permits the order of operations executed across workers to vary, even when running an identical DNN. However, collective operations which involve all workers must be performed in a globally consistent order to avoid deadlock. To solve this issue, `Horovod` introduces additional worker coordination logic to ensure all workers submit collective operations in a common order. The preexisting logic uses a master-worker coordination strategy in which a single coordinator rank is tasked with gathering *requests* from all workers, determining common requests across workers, forming *responses* for common requests, and then broadcasting an ordered list of responses to all workers for execution. *Requests*, $T_n$, are objects submitted by each worker to request a collective operation, containing basic meta-data about the tensor involved in the operation (name, shape, datatype), as well as the type of collective operation desired (allreduce, allgather, or broadcast). *Responses*, $R_n$, which are associated with a given request, contain aggregated meta-data from all workers submitting a common request (for example, all displacements for an allgather operation and the set of ranks that submitted this request), and are used for the execution of the collective operation (see Figure \[fig:coordination:a\]). This scheme is implemented using MPI collectives, in particular `MPI_Gatherv` and `MPI_Bcast`, on serialized representations of the various request and response objects.
This coordination process occurs at frequent regular intervals for the duration of training, where at each tic only common collective operation requests across workers are executed. While this coordination strategy works well up to moderate scales, its effectiveness breaks down once the node count is increased further. At these larger scales, the communication cost for this coordination strategy increases to non-negligible levels, resulting in severe degradation in scaling efficiency (green and blue lines in ).
To address this, a new lightweight coordination scheme was implemented in `Horovod`, replacing the master-worker strategy and related MPI collective communication with a global intersection of a bit vector, implemented using only a single `MPI_Allreduce` operation. One of the major overheads of the existing coordination strategy is that although identical collective operations are completed during every training iteration, requests for each operation are redundantly communicated to the coordinator rank in order to create new responses for execution. To avoid this, we implemented a caching scheme where the responses to execute collective operations are gathered and processed by the coordinator rank only once, with the broadcasted result of this process stored in a cache on every worker. On subsequent iterations, this cached response can be directly used by each worker, bypassing redundant communication of requests to the coordinator rank. Assuming the cache remains globally consistent, it also forms the basis for a simple global enumeration of the collective operations and leads naturally to a simple procedure for worker coordination. For a given set of requests across workers, the coordination process is as follows:
1. Each worker populates a bit vector, setting bits associated with its pending requests with bit positions determined from the cache.
2. The bit vectors are globally intersected using `MPI_Allreduce` with the binary `MPI_BAND` operation.
3. Each worker searches for set bits in the intersected bit vector and forms a list of associated cache entries. This list is the common set of collective operation requests for each worker to execute.
A depiction of this improved coordination strategy can be seen in Figure \[fig:coordination:b\]. This new coordination strategy greatly reduces communication overheads and resulted in significant improvements to scaling efficiency, shown in the black line in .
[ ![Comparison of coordination strategies. \[fig:coordination:a\]: In the original coordination strategy, Rank 0: (i) gathers requests, $T_n$, (ii) determines common requests across all ranks, (iii) forms associated responses, $R_n$, and (iv) broadcasts an ordered list of responses to all ranks for execution. \[fig:coordination:b\]: In the improved coordination strategy, each rank checks if its requests are in the cache and sets bits in the bitvector accordingly. An initial set of bits in the bitvector are reserved for status signaling. Each cache entry is keyed by a request and maps to an integer cache bit position and stored response object. The bitvectors are globally intersected and a list of responses associated with common set bits are obtained for execution in cache bit order.[]{data-label="fig:coordination"}](figures/sc_diag1_bb.pdf "fig:") \[fig:coordination:a\]]{}
[ ![Comparison of coordination strategies. \[fig:coordination:a\]: In the original coordination strategy, Rank 0: (i) gathers requests, $T_n$, (ii) determines common requests across all ranks, (iii) forms associated responses, $R_n$, and (iv) broadcasts an ordered list of responses to all ranks for execution. \[fig:coordination:b\]: In the improved coordination strategy, each rank checks if its requests are in the cache and sets bits in the bitvector accordingly. An initial set of bits in the bitvector are reserved for status signaling. Each cache entry is keyed by a request and maps to an integer cache bit position and stored response object. The bitvectors are globally intersected and a list of responses associated with common set bits are obtained for execution in cache bit order.[]{data-label="fig:coordination"}](figures/sc_diag2_bb.pdf "fig:") \[fig:coordination:b\]]{}
Grouping
--------
As noted in the previous section, worker coordination in `Horovod` occurs at a fixed tic rate, referred to in `Horovod` as the cycle time (see blue vertical lines in Figure \[fig:timeline\]). This cycle time is user configurable at run-time via an environment variable. This tic rate controls how often worker coordination occurs and pending collective requests are processed and executed. One of the major features of `Horovod` is the ability to fuse individual collective operations into single operations on larger message buffers for better network performance. Notably, the scope of this fusion is limited to the requests that are encountered during a single coordination cycle. This leads to a coupling between the cycle time and collective message sizes, where in any given iteration, a shorter cycle time will lead to a more responsive execution of many collective operations with small message sizes, while a larger cycle time will lead to a slower execution of fewer collective operations with larger message sizes. This leads to a tuning dilemma: for low-latency execution of collective operations, the cycle time should be reduced as much as possible; however, for efficient network utilization, the minimum message sizes cannot be too small. Due to this, it is challenging to find an optimal cycle time that effectively balances these requirements and achieves good scaling performance.
To weaken the coupling between the cycle time and message sizes, we implemented an additional feature into `Horovod` that enables explicit assignment of collective operations into groups. When using this feature, rather than executing all collective operation requests encountered during a given cycle, only requests forming a complete group are fused and executed. If multiple complete groups are encountered, they are fused together into larger messages. By enforcing a lower bound on fusion to complete groups only, a minimum message size independent of the cycle time is enforced. This enables the use of lower cycle time for low-latency execution with a constant minimum message size, maintaining efficient network utilization. Usage of this feature in tandem with the lightweight bitvector coordination scheme described previously, yielded the red performance curve in Figure \[fig:commstrat\], a significant improvement in scaling behavior.
![Illustration of Grouping. A task graph with nodes that generate requests $T_n$ is depicted on the left, with the dashed boxes indicating requests visible to `Horovod` at 3 subsequent cycles. The nodes are colored to depict assignment to two groups (blue/solid borders and green/dashed borders). By default, a worker will submit all requests observed in a cycle to be processed/executed which can yield unbalanced sets of requests. With grouping enforced, requests are only submitted when complete groups are available.[]{data-label="fig:grouping"}](figures/sc_group_diag.pdf)
Results
=======
Power Efficiency
----------------
A strong indicator of the efficiency of an application on a supercomputer is the measured power consumption. In particular, the use of blocking collectives such as Allreduce causes all operations executed on a GPU/CPU to cease until the result from the collectives are returned. For instance, in a case where the reduction of gradients stalls due to overheads introduced by an inefficient coordination strategy, this stalling would be reflected in the GPU power consumption via a cyclic increase and decrease in the power as a function of application run-time or equivalently, in our case, the training steps.
In , we present the measured power consumption of the main hardware components on Summit during a distributed training run using Bitvector Allreduce and Grouping. The DNN model used in that training run and throughout the rest of the presented results is modified version of the fully-convolutional dense neural network (FC-DenseNet) [@tiramisu] with 220 million parameters. This choice of model produces a message size large enough to ensure that our experiments tests the robustness of the new gradient reduction strategies. The distributed training run shown in was carried out on 4600 out of 4608 available Summit nodes and allows us to directly measure the efficiency of our application as a whole. We found that energy metrics collected on time scales similar to the duration to a training step, show that our application’s power usage is nearly constant, due to the absence of power usage fluctuations caused by GPU idleness in the presence of communication overheads.
Performance
-----------
![Profiling of Summit’s Power Consumption during Distributed Training on 4600 Nodes. Power profiles were collected for the main hardware components of Summit (GPU, CPU, etc...) during one of our distributed training runs. Despite the use of blocking collectives, our orchestration strategies ensure that communication and computation are optimally overlapped as reflected in a near-constant GPU power usage profile sampled at time intervals similar to the duration of a training step.[]{data-label="fig:power"}](figures/summit_power.png)
![Scaling efficiency and Sustained Performance of distributed Deep Learning using the improved gradient reduction strategies up to 27,600 V100 GPUs.[]{data-label="fig:scaling"}](figures/perf_custom_5.png)
In addition to power consumption, we also profiled the compute performance of distributed training with the new gradient reduction strategies. All of our reported performance measurements include: (1) I/O (reading of data and writing of model checkpoints), (2) computation performed for the DNN forward and backward propagation, and (3) communication operations embedded in the computation graph.
We measure the single GPU performance of our code using two distinct methods. First, we use an analytical calculation of mathematical operations performed by DNN convolution layers assuming direct convolution. We then augment that with the tracing of calls to `cuDNN` during execution of `TensorFlow`’s computation graph to eliminate any errors that arise from the availability of the multiple numerical implementations of the convolution operation in `cuDNN` (e.g. FFT vs. Winograd vs. direct convolution) [@cuDNN]. The computational complexity of these algorithms can vary substantially, and `TensorFlow` makes runtime decisions regarding which algorithm to use for each operation in the graph. As shown in (Table \[tab:cudnn-freq\]), our DNN implementation uses exclusively algorithms with a direct convolution implementation, for which the number of multiply-add operations for a direct (2-D) convolution is given by: $$OPS_{conv} = 2 \times H \times W \times C \times K \times R \times S,$$ where $H$,$W$ are the height and width dimensions of the inputs, $C$ and $K$ are the number of input and output channels respectively, $R$ and $S$ are the convolution kernel dimensions, and the factor of 2 accounts for “multiply" and “add" operations.
The execution time of the `TensorFlow` graph, $$t_{\textrm{exec}}= t_{\textrm{comm}} + t_{\textrm{comp}} + t_{\textrm{misc}},$$ is obtained through the use of Python’s built-in `time` module as well as a GPU hardware trace with `CUPTI`. The `CUPTI` trace provides the runtime of every operation individually for a single training step, whereas the application-level timing has sufficiently low overhead to be used throughout a training run. We denote the application time spent in I/O and memory copies between the host and the device as $t_{\textrm{misc}}$. $t_{\textrm{comm}}$ and $t_{\textrm{comp}}$ are the times spent on communication and computation, respectively.
The two performance numbers we report, namely sustained and peak are then given by, $$\begin{aligned}
\textrm{Sustained Performance} &= \frac{3 \times OPS_{conv}}{t_{\textrm{exec}}},\\
\textrm{Peak Performance} &= \frac{3 \times OPS_{conv}}{t_{\textrm{comp}}},
\end{aligned}$$ where the factor of 3 accounts for forward convolutions (Conv2D\_FWD), gradient backpropagation with respect to the convolution kernels (Conv2D\_BackpropKernel), and gradient backpropagation with respect to the inputs (Conv2D\_BackpropInput).
Scaling
-------
Performance measurements on multiple nodes are carried out in a similar fashion, with the minor addition of averaging $t_{\textrm{exec}}$ across all MPI-ranks. The sustained performances reported at each node count is averaged across a distributed training run lasting 1000 steps and the variance is reported as error bars. While our definition of the peak performance at a single node does not account for $t_{\textrm{comp}}$, when we report its value on multiple nodes (see below), we multiply its value by the measured scaling efficiency ( $< 1$ for Summit nodes $> 1024$). This scaling is performed to accurately reflect the synchronous execution of our application.
In Table \[tab:ops\_tensorcore\], we summarize the math operations, their timing, and the overall performance during the execution of our application (one training step) on a single Summit node using the performance measurement methodology described in the previous section. We also account for the speed-up in execution enabled by the hardware implementation of half-precision intrinsics in the V100’s Tensor Cores. This is done by making use of `TensorFlow`’s `TF_DISABLE_CUDNN_TENSOR_OP_MATH` environment variable. We find that execution with Tensor Cores produces an average speed-up of approximately $6\times$ of the computation times of the convolution operations than without (Table \[tab:ops\_tensorcore\]).
During DNN training, we attain sustained (peak) performance of 59.67 (83.92) TFLOPS$_{16}$ per GPU representing 49.7% (70%) of the theoretical peak of a V100 (120 TFLOPS$_{16}$), which to our knowledge, exceeds the single GPU performance of all other DNN trained on the same system to date.
Finally, using the communication strategies described in , we are able to achieve a scaling efficiency of 0.93 at 4600 nodes during distributed deep learning (Figure \[fig:scaling\]) and reach a sustained (peak) performance of 1.54(2) (2.15(2)) EFLOPS$_{16}$. Both our scaling efficiency and sustained performance improve significantly ($>~50\%$) on the record established by the 2018 ACM Gordon Bell prize winner [@Kurth:2018]. Note that in the results reported in [@Kurth:2018], synchronized gradient updates were skipped every other training step, which introduces a level of asynchronous execution, and reduces their communication overhead (at the expense of gradient staleness). Our reported performance comes from fully-synchronous training, making the two results not directly comparable.
Scientific Application
======================
Problem Definition
------------------
In a general inverse problem in imaging, we seek to reconstruct an image $x\in X$ from a set of measurements $y\in Y$ (typically also given by an image), where $X$ and $Y$ are (Banach) spaces. The forward operator, $F$, defined by $$\label{eq:operator}
F : X \rightarrow Y$$ maps the space of solutions to the space of measurements. The goal of any reconstruction method is to find $x$ by solving $$\textrm{argmin}_x ||Fx -y||_p + \lambda R(x),$$ where $||.||_p$ denotes the $p$-norm (typically, $p=1,2$), $\lambda$ is a parameter (typically $\lambda \ll 1$), and $R(.)$ is a regularization function to incorporate *a priori* knowledge about $x$ that the solution ought to obey.
![Reconstruction of a material’s local electron density with atomic resolution from diffraction data streams acquired in an electron microscope is a longstanding inverse problem without a general solution.[]{data-label="fig:1"}](figures/microscope.png)
In our inverse problem of interest, illustrated in , $x$ represents the local electron density of a material $\rho$, $y$ is a diffraction pattern $||\psi||^2$, and $F$ is the celebrated Schrödinger equation of quantum mechanics. The central difficulty of the above inverse problem lies almost entirely in the fact that experimentally, one can only measure image intensities (i.e. diffraction patterns) of the exiting probe electrons $||\psi||^2$ and not the full complex-valued $\psi$ needed to find $\rho$ from $F$. Consequently, *half of the information needed to directly invert the forward model is always missing*. A problem known as the phase problem [@born2013principles].
Here, we seek to learn the “inverse" operator $F^{-1}: ||\psi||^2 \rightarrow \rho$, represented by a DNN, and trained using the technique of supervised learning with training data sampled from the forward model given by the fast-electron Schrödinger equation [@kirkland2010advanced].
Simulation and the Deep Learning Model {#sec:deep learning}
--------------------------------------
#### Forward Model Simulation
Deep Neural Networks are notoriously data hungry. To simulate enough training and test data from the forward model in optimal time, we developed a custom multi-GPU, multi-node electron scattering simulation code called `NAMSA`, which implements the multi-slice algorithm (MSA)[@Cowley:1957ga], a nearly-exact solution of the fast-electron Schrödinger equation [@kirkland2010advanced].
Our simulation workflow is shown in Figure \[fig:worflow\]A and consists of A material supercell is built (with typical dimensions $\sim 10\times10\times20$ nm$^3$ and $\sim 100,000$ atoms), followed by a simulation of the probe electron wavefuntion interacting and propagating through all atomic planes of the supercell to produce the intensity of the exit wavefunction, $||\psi||^2$ ($512\times512$ pixels). This procedure is performed at each position on a 2-D grid (32x32) defined at the surface of the supercell. The stack of $1024 \times ||\psi||^2$ represents the inputs to our DNN,$||\psi||^2_N$, while the target outputs of the DNN is the 2-D projected electron density, $\rho_z$ ($512\times512$ pixels). The projected electron density is computed, after the scattering simulation, by integrating $\rho(\bf{r})$ along the thickness of the supercell ($z-$axis).
#### Data
Our simulations span over 60,000 solid-state materials crystal structure files accessible via the materials project database [@ong2013python]. For each material structure, multiple crystallographic orientations were simulated as they produce markedly different pico-diffraction patterns and projected electron densities. In total, 400,000 configuration files were generated and then partitioned into a 90/5/5 split for training, development, and test data sets.
Simulations of training and test data sets were generated on-the-fly and stored on the node-local burst-buffer. Given our highly-optimized simulation code `NAMSA`, we found it to be more time-effective to generate the data immediately before the start of DNN training than to stage upwards of 500 TB of data (to 4600 nodes) via the global parallel filesystem- a shared resource accessible to all users. Typically, a simulation with 0.5 hours of wall-time generates about a 200 GB data set per compute node. Note, that the number of unique samples the DNN model trains on grows linearly with the numbers of GPUs used during distributed training. The entire complement of 360,000 training configuration files are only used when distributed training reaches 4600 nodes. All data I/O (file-saving during simulation, DNN model checkpointing, and data reading during DNN training/testing) was carried out via the burst buffer and used `LMDB` (in addition to Python’s built-in `multiprocessing` module during the reading stage).
#### Neural Network Architecture
[ Encoder-Decoder networks are prevalent in computer vision tasks such as segmentation and denoising [@badrinarayanan2017segnet]. This style of DNN architecture learns an encoding of multidimensional input into a compressed latent representation, followed by learning a reconstruction of a multidimensional output from an encoding along the decoder path [@vincent2008extracting]. Encoder-decode architectures have many variations: our work adapts a fully-convolutional dense neural networks (FC-DenseNet) [@tiramisu], shown in B. The two main modifications we introduce in our model consist of: (1) Large growth rates ($k$= 256) of the number of channels of the 2-D convolution layers, and (2) replacing max pooling with average pooling. The former modification is needed to give the model enough capacity to represent our input with its 1024 channels; a smaller number of channels in the first few 2-D convolutional layers would decimate most of the information encoded in $||\psi||^2_N$. The latter modification was found in earlier work to produce substantially more accurate DNN models on atomically-resolved imaging[@Vasudevan:2018jd], due the inherent sparsity of these images. The output of each dense block was passed through a rectifier non-linearity (ReLU) to compute the activation, followed by a dropout layer (with probability $p=0.5$). In total, our DNN model has $22\times10^7$ weights (free parameters). ]{}
#### Model Implementation
We trained our DNN to learn a reconstruction of the (projected) electron density, $\rho_z$ by minimizing the following loss function, $\mathcal{L}$ given by $$\label{eq:loss}
\mathcal{L}(\rho_z,\bar\rho_z) = \mathcal{L}_{Huber}(\rho_z,\bar\rho_z) + \epsilon R(W_i) ,$$ where $\mathcal{L}_{Huber}$ is the Huber loss evaluated on the true and predicted electron densities, $\rho_z$ and $\bar{\rho}_z$, respectively. We use an $L_2$-based regularization,$R$, on the weight values $W_i$ of the model with (weight-decay) coefficient $\epsilon = 10^{-4}$. We initialized the Huber loss “cutoff" value with $\delta=10$ and decreased it during training using an exponential decay rate policy (decay rate of 0.99 every data epoch).
Due to the large DNN model and input sizes, the 16 GB memory of a V100 can only accommodate a training batch size of 1 ($||\psi||^2_N: (1024,512,512)$, $\rho_z: (1,512,512)$), even in a float16 implementation. This batch size, however, increases linearly with the scale of distributed training, reaching 27,600 at 4600 nodes. It is well established that large batch sizes adversely affect the learning dynamics of DNN trained with stochastic gradient descent. To mitigate such effects, we used a layer-wise adaptive learning rate scaling strategy (LARS), which computes a layer-wise weight update based on the $L_2$-norm of the gradients [@LARS]. We used LARS in conjunction with an adaptive stochastic gradient descent optimizer (Adam optimizer, $\beta_1=0.9, \beta_2=0.999$), and a staircase learning rate decay policy. Furthermore, the warm-up policy was used to linearly increase the learning rate $\eta$, from an initial value of 0.0001, to a maximum value of $ \approx N\eta$, where $N$ is the number of GPUs (MPI-ranks) participating in the distributed training of the DNN.
Mixed-precision training has been shown to produce similar convergence behavior and accuracy to training in pure single-precision across many DL applications [@Child:2019vl; @LARS], as long as judicious numerical scaling strategies are applied. Here, we performed adaptive scaling of the loss before gradient computation (and application of LARS) to avoid numerical values outside of the dynamic range of float16, using the loss scaling strategies implemented in `OpenSeq2Seq`[@openseq2seq]. All of our deep learning code was implemented using the `TensorFlow` (v1.13) framework [@tensorflow].
Model Training and Validation
-----------------------------
![Comparison between the DNN-based Reconstruction and the Ground Truth Electron Density. The number of training samples processed per step is equal to the number of `MPI` ranks (6 per Summit node). The reconstruction quality was found to improve as a function of compute nodes.[]{data-label="fig:predictions"}](figures/results.png)
We carried out multiple distributed training runs extending to 2,000 training steps. In each run, the DNN was initialized randomly and trained using the optimization strategies described in the preceding sections. We found that the training error converges reasonably well as shown in for runs spanning 128 nodes through 4096 nodes ($\approx 90\%$ of the full machine). These observations indicate that the learning strategies employed were effective in enabling good training behavior irrespective of computational scale or equivalently batch size.
In typical data-parallelism work, the total size of the training data set, given by the number of training samples is fixed regardless of the number of DNN model replicas or equivalently the number of `MPI` ranks used in distributed training. In our application, however, the total number of unique data samples the DNN encounters during each one of the training runs depends on and grows linearly as a function of GPUs used (as discussed in ). This linear growth in the training data set size is necessary given the finite capacity of the local node storage which can accommodate less than 1% of the total data and the massive performance hits ($~ \times10$) our application would incur if I/O is performed directly from the larger capacity global file system (see ).
The increase in the predictive efficacy of machine learning, deep learning in particular, as a function of growth in data is well-documented [@35179; @sun2017revisiting]. As our data size grows as a function of `MPI`-ranks used, we expect that the quality of the DNN reconstruction on an unseen sample drawn from the test data improves. We show one such example in . We find that the reconstruction of the projected electron density is visibly far closer to the ground truth for a model trained on 1024 nodes versus 128 nodes. Both DNN models, however, fail to faithfully reconstruct the true electron density of this material across the entire field of view of the image. In the case of the DNN trained on 1024 nodes it is plausible that its reconstruction capabilities will improve with additional training and hyper-parameter tuning.
We also report the reconstruction error evaluated on the entire test data for models trained on 128, 1024, and 4096 nodes (see inset in ). We find that this test error, averaged over all test samples, decreases as the number of compute (and data) increases, indicating an improving reconstruction quality on materials configurations unseen during training.
Discussion
==========
We have shown that by introducing new coordination strategies during gradient reductions we exceed the state of the art in scaling efficiency. This opens up, in particular, opportunities in exploiting the different levels of parallelism present in many systems (e.g. intra-node vs inter-node) such as Summit to train even larger models than we do here, for instance via the combination of model- and data-parallelism. In addition, the scaling efficiency results clearly indicate that with carefully chosen synchronized gradient reduction strategies we obtain greater utilization of the interconnect network.
In regards to our application, the promising results shown here are a first in using DNN to solve the phase problem in the atomic imaging of materials. Future research directions can target improving the reconstruction baseline achieved here and extending the DNN-based reconstruction approach to the full 3-D electron density. Higher-dimensional reconstructions would require the use of GPU-memory intensive 3-D convolution layers presenting an additional opportunity to further benchmark the effectiveness of the novel coordination strategies we introduced here as well as extending our gradient reduction strategies to model-parallelism.
In light of the ever-increasing data streams emanating from large scale scientific user facilities, we believe this is an opportune time to harness state of the art supercomputing and machine learning. The impact of exascale machine learning on accelerating scientific progress could be, in due time, of comparable magnitude to the advances made possible via large scale physics-based simulations currently enabled by high-performance computing.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was funded by a Lab Directed Research and Development project at Oak Ridge National Laboratory, a U.S. Department of Energy facility managed by UT-Battelle, LLC. An award of computer time was provided by the INCITE program. This research also used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.
This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
Performance {#appendix}
===========
cuDNN Function/Algorithm \# Calls DNN Operation Tensor Cores Implementation
---------------------------------------------------- ---------- --------------- -----------------------------
`CUDNN_CONVOLUTION_FWD_ALGO_IMPLICIT_PRECOMP_GEMM` 480 Forward Yes
`CUDNN_CONVOLUTION_FWD_ALGO_IMPLICIT_GEMM` 20 Forward No
`CUDNN_CONVOLUTION_BWD_DATA_ALGO_1` 480 Backprop Yes
`CUDNN_CONVOLUTION_BWD_FILTER_ALGO_1` 500 Backprop Yes
[^1]: The gradient reduction strategies we describe below have either been recently incorporated in the latest release of `Horovod` (https://github.com/horovod/horovod) (i.e. Bitvector Allreduce) or are currently in the pull-request review stage (i.e. Grouping).
|
---
abstract: 'We analyze the possibility of hadron Dark Matter carriers consisting of singlet quark and the light standard one. It is shown that stable singlet quarks generate effects of new physics which do not contradict to restrictions from precision electroweak data. The neutral and charged pseudoscalar low-lying states are interpreted as the Dark Matter particle and its mass-degenerated partner. We evaluated their masses and lifetime of the charged component, and describe the potential asymptotes of low-energy interactions of these particles with nucleons and with each other. Some peculiarities of Sommerfeld enhancement effect in the annihilation process are also discussed.'
author:
- Vitaly Beylin
- Vladimir Kuksa
title: |
Dark Matter in the Standard Model Extension\
with Singlet Quark
---
Introduction
============
The problem of Dark Matter (DM) explanation has been in the center of fundamental physics attention for a long time. The existence of the DM is followed from astrophysical data and remains the essential phenomenological evidences of New Physics’ manifestations beyond the Standard Model (SM) [@1; @2]. An appropriate candidates as DM carriers should be stable particles which weakly interact with ordinary matter (so called, WIMPs). Such particles usually are considered in the framework of supersymmetric, hypercolor or other extensions of the SM (see, for instance, review [@3]). The last experimental rigid restrictions on cross section of spin-independent WIMP-nucleon interaction [@4] exclude many variants of WIMPs as the DM carriers. So, another scenarios are discussed in literature, such as quarks from fourth generation, hyper-colour quarks, dark atoms, axions and so on [@3]. In spite of some theoretical peculiarities, the possibility of hadronic DM is not excluded and considered, for example, in Refs. [@5]-[@10]. The possibility of new hadrons existence, which can be interpreted as carriers of the DM, was analyzed in detail within the framework of the SM chiral-symmetric extension [@10].
Principal feature of the hadronic DM structure is that the strong interaction of new stable quarks with standard ones leads to the formation of neutral stable meson or baryon heavy states. Such scenario can be realized in the extensions of the SM with extra generation [@5]-[@9], in mirror and chiral-symmetric models [@10; @16] or in extensions with singlet quark [@11]-[@15]. The second variant was detally considered in Ref. [@10], where the quark structure and low-energy phenomenology of new heavy hadrons were described. It was shown that the scenario does not contradict to cosmochemical data, cosmological tests and known restrictions for new physics effects. However, the explicit realization of the chiral-symmetric scenario faces with some theoretical troubles, which can be eliminated with the help of artificial assumptions. The extensions of SM with fourth generation and their phenomenology were considered during last decades in spite of strong experimental restrictions which, for instance, follows from invizible Z-decay channel, unitary condition for CM-matrix, FCNC etc. The main problem of 4th generation is the contribution of new heavy quarks to the Higgs boson decays [@10.9]. The contribution of new heavy quarks to vector boson coupling may be compensated by the contribution of 50 GeV neutrino [@13a; @Ilyin; @Novikov], however, such assumption looks as artificial. In this paper, we analyze the hypothesis of hadronic Dark Matter which follows from the SM extension with singlet quark.
The paper is organised as follows. In the second section we describe the extension of the SM with singlet quark and consider the restrictions on its phenomenology, following from precision electroweak data. Quark composition and interaction of new hadrons with the standard ones at low energies is analyzed in the third section. The masses of new hadrons, decay properties of charged partner of the DM carrier and annihilation cross section are analyzed in the fourth section.
Standard Model Extension with Stable Singlet quark
==================================================
There is a wide class of high-energy extensions of the SM with singlet quarks which are discussed during many decades. Here, we consider the simplest extension of the SM with singlet quarks as the framework for description of the DM carrier. Singlet (or vector-like) quark is defined as fermion with standard $U_Y(1)$ and $SU_C(3)$ gauge interactions but it is singlet under $SU_W(2)$ transformations. The low-energy phenomenology of both down- and up-type quarks (D and U) was considered in detail in large number of works (see, for instance, [@9a], [@Eberhardt], [@Botella], [@Kumar] and references therein). As a rule, singlet quark is supposed as unstable due to the mixing with the ordinary ones. This mixing leads to the FCNC appearing at the tree level. As a consequences, we get an additional contributions into rare processes, such as rare lepton and semy-lepton decays, and mixing in the systems of neutral mesons ($M^0-\bar{M}^0$ oscillations). The current experimental data on New Fhysics phenomena give rigid restrictions for the angles of ordinary-singlet quark mixing. In this work, we consider alternative aspect of the extensions with singlet quark $Q$, namely, the scenario with the absence of such mixing. As a result, we get stable singlet quark which have no the decay channels due to absence of non-diagonal $Q$-quark currents. More exactly, due to confinement the singlet quark forms bound states with the ordinary ones, for instance $(Qq)$, and the lightest state is stable. In this work, we consider some properties of such particles and analyze the possibility to interpret the stable neutral meson $M^0=(\bar{Q}q)$ as the DM carrier.
Now, we examine the minimal variants of the SM extension with singlet quark $Q_A$, where subscript $A=U,D$ denotes up- or down- type with charge $q=2/3,-1/3$. According to the definition, the field $Q$ is singlet with respect to $SU_W(2)$ group and has standard transformations under abelian $U_Y(1)$ and color $SU_C(3)$ groups. So, the minimal additional gauge-invariant Lagrangian has the form: $$\label{2.1}
L_Q=i\bar{Q}\gamma^{\mu}(\partial_{\mu}-ig_1 \frac{Y}{2} V_{\mu} -ig_s \frac{\lambda_a}{2}G^a_{\mu})Q - M_Q \bar{Q} Q,$$ where $Y/2=q$ is charge in the case of singlet $Q$, and $M_Q$ denotes phenomenological mass of quark. Note, singlet quark (SQ) can not get mass term from the standard Higgs mechanism because the Higgs doublet is fundamental representation of $SU(2)$ group. Abelian part of the interaction Lagrangian (\[2.1\]), which will be used in further considerations, includes the interactions with physical photon $A$ and $Z$ boson: $$\label{2.2}
L_Q^{int}=g_1 q V_{\mu}\bar{Q}\gamma^{\mu} Q = q g_1 (c_w A_{\mu}- s_w Z_{\mu})\bar{Q}\gamma^{\mu} Q,$$ where $c_w=\cos\theta_w$, $s_w=\sin\theta_w$, $g_1 c_w=e$ and $\theta_w$ is Weinberg angle of mixing. Note, the left and right parts of the singlet field $Q$ have the same transformation properties, interaction (\[2.2\]) has vector-like (chiral-symmetric) form and singlet quark usually is named vector-like quark [@Botella; @Kumar].
First of all, we should take into account direct and indirect restrictions on New Physics (NF) manifestations which follow from the precision experimental data. The additional chiral quarks, for instance from standard fourth generation, are excluded at the $5\,\sigma$ level by LHC data on Higgs searches [@Eberhardt]. As the vector-like (non-chiral) singlet fermions do not receive their masses from a Higgs doublet, they are allowed by existing experimental data on Higgs physics. The last limits on new colored fermions follow from the jets data from the LHC [@Llorente]. The corresponding limits for effective colored factors $n_{eff}=2,3,6$ are about 200 GeV, 300 GeV, 400 GeV. Note, these limits are much less then the estimation of quark mass which follows from the DM analysis (see the fourth section). Theoretical and experimental situation for long-lived heavy quarks was considerably discussed in the review [@9a], where it was noted that vector-like new heavy quarks can elude experimental contraints from LHC.
Indirect limits on new fermions follow from precision electroweak measurements of the effects, such as flavor-changing neutral currents (FCNC) and vector boson polarizations, which take place at the loop level in the SM. Because we consider the case of stable singlet quark, there are no mixing with ordinary quarks and, consequently, FCNC effects are absent. The NF manifestations in polarization effects of gauge bosons $\gamma,\,Z,\,W$ are usually described by oblique Peskin-Takeuchi parameters [@Peskin] (PT parameters). From Eq. (\[2.2\]), it follows that the singlet quark gives non-zero contributions into polarizations of $\gamma$ and $Z$-bosons which are described by the values of $\Pi_{\gamma\gamma},\,\Pi_{\gamma Z},\,\Pi_{ZZ}$. As $W$-boson does not interact with the SQ, corresponding contribution into polarizaton operator is zero, $\Pi_{WW}=0$. These parameters are expressed in terms of vector bosons polarizations $\Pi_{ab}(p^2)$, where $a,b=W, Z, \gamma$. Here, we use the definition $\Pi_{\mu\nu}(p^2)=p_{\mu}p_{\nu}P(p^2) +
g_{\mu\nu}\Pi(p^2)$ and the expressions for PT oblique parameters from [@Burgess]. In the case under consideration, $\Pi_{ab}(0)=0$ and PT parameters can be represented by the following expressions: $$\begin{aligned}
\label{2.3}
\alpha S=&4s^2_w c^2_w[\frac{\Pi_{ZZ}(M^2_Z,M^2_U)}{M^2_Z}-\frac{c^2_w-s^2_w}{s_w c_w}\Pi^{'}_{\gamma Z}(0,m^2_U)-\Pi^{'}_{\gamma\gamma}(0,M^2_U)];\notag\\
\alpha U=&-4s^2_w[c^2_w\frac{\Pi_{ZZ}(M^2_Z,M^2_U)}{M^2_Z}+2s_w c_w \Pi^{'}_{\gamma Z}(0,M^2_U)+s^2_w\Pi^{'}_{\gamma\gamma}(0,M^2_U)];\notag\\
\alpha T=&-\frac{\Pi_{ZZ}(0,M^2_U)}{M^2_Z}=0;\,\,\, \alpha V=\Pi^{'}_{ZZ}(M^2_Z,M^2_U)-\frac{\Pi_{ZZ}(M^2_Z,M^2_U)}{M^2_Z};\notag\\
\alpha W=&0\,\,(W\sim \Pi_{WW}=0);\,\,\,\alpha X=-s_w c_w [\frac{\Pi_{\gamma Z}(M^2_Z,M^2_U)}{M^2_Z}-\Pi^{'}_{\gamma Z}(0,M^2_U)].\end{aligned}$$ In (\[2.3\]) polarizations $\Pi_{ab}(p^2,M^2_U)$, where $a,b=\gamma,Z$, in one-loop approach can be represented in simple form (for the case of SQ with $q=2/3$): $$\begin{aligned}
\label{2.4}
\Pi_{ab}(p^2,M^2_U)=&\frac{g_1^2}{9\pi^2}k_{ab}F(p^2,M^2_U);\,\,\,k_{ZZ}=s^2_w,\,k_{\gamma\gamma}=c^2_w,\,k_{\gamma Z}=-s_w c_w;\notag\\
F(p^2,M^2_U)=&-\frac{1}{3}p^2+2M^2_U+2A_0(M^2_U)+(p^2+2M^2_U)B_0(p^2,M^2_U).\end{aligned}$$ In Eqs. (\[2.4\]) the function $F(p^2,M^2_U)$ contains divergent terms in the one-point, $A_0(M^2_U)$, and two-point, $B_0(p^2,M^2_U)$, Veltman functions which are exactly compensated in physical parameters (\[2.3\]). Using standard definitions of the functions $A_0(M^2_U)$ and $B_0(p^2,M^2_U)$ and the equality $B^{'}_0(0,M^2_U)=M^2_U/6$, by straightforward calculations we get simple expressions for oblique parameters: $$\label{2.5}
S=-U=\frac{16s^4_w}{9\pi}[-\frac{1}{3}+2(1+2\frac{M^2_Q}{M^2_Z})(1-\sqrt{\beta}\arctan\frac{1}{\sqrt{\beta}})],$$ where $\beta=4M^2_Q/M^2_Z -1$. We check that in the limit $M^2_Q/M^2_Z\to \infty$ the values of $S$ and $U$ go to zero as $\sim M^2_Z/M^2_Q$ in accordance with well-known results for the case of vector-like interactions [@2; @Burgess]. From Eq. (\[2.5\]) it follows that beginning from $M_Q=500$ GeV the parameter $S<10^{-2}$ and the rest non-zero parameters have near the same values. These values significantly less the current experimental limits [@PDG]: $S=0.00 +0.11(-0.10),\,\,\,U=0.08\pm 0.11,\,\,\,T=0.02+0.11(-0.12)$, that is the scenario with up-type singlet quark satisfy to the restrictions on indirect manifestations of heavy new fermions. Note, parameters $V,\,W,\,X$ describe the contributions of new fermions with masses close to the electro-weak scale. In the case of down-type singlet quark, having charge $q=-1/3$, the contributions into all polarizations and, consequently, into PT parameters are four times smaller.
In the quark-gluon phase (QGP) of the Universe evolution, stable SQ interacts with standard quarks through exchanges by gluons $g$, $\gamma$ and $Z$ according to Eq. (\[2.1\]). So, we have large cross-section for annihilation into gluons and quarks, $Q\bar{Q} \to gg$ and $Q\bar{Q}\to q\bar{q}$ correspondingly, and also small additional contributions in electroweak channels $Q\bar{Q} \to \gamma\gamma,\,ZZ$. These cross sections can be simply derived from the known expressions for the processes $gg\to Q\bar{Q}$ and $q\bar{q}\to
Q\bar{Q}$ (see review in Ref. [@PDG]) by time inversion. Two-gluon cross section in the low-energy limit looks like: $$\label{2.6}
\sigma(U\bar{U}\to gg)=\frac{14\pi}{3}\frac{\alpha^2_s}{v_r M^2_U},$$ where $M_U$ is mass of $U$-quark and $\alpha_s=\alpha_s(M_U)$ is strong coupling at the corresponding scale. Two-quark channel in the massless limit $m_q\to 0$ is as follows: $$\label{2.7}
\sigma(U\bar{U}\to q\bar{q})=\frac{2\pi}{9}\frac{\alpha^2_s}{v_r M^2_U}.$$ So, the two-gluon channel dominates. We should note, that the cross section of SQ -annihilation is suppressed by large $M_U$ in comparison with the annihilation of standard quarks.
After the transition from quark-gluon plasma to hadronization stage, the singlet quarks having standard strong interactions (gluon exchange), form coupled states with ordinary quarks. New heavy hadrons can be constructed as coupled states which consist of heavy stable quark $Q$ and a light quark from the SM quark sector. Here, we consider the simplest two-quark states, neutral and charged mesons. The lightest of them, for instance neutral meson $M=(\bar{Q}q)$, is stable and can be considered as the carrier of cold Dark Matter. Possibility of existence of heavy stable hadrons was carefully analyzed in [@10], where it was shown that this hypothesis does not contradict to cosmochemical data and cosmological test. This conclusion was based on the important property of new hadron, namely, repulsive strong interaction with nucleons at large distances. The effect will be qualitatively analyzed for the case of $MM$ and $MN$ interactions in the next section.
Quark composition of new hadrons\
and their interactions with nucleons
====================================
At the hadronization stage, heavy SQ form the coupled states with the ordinary light quarks. Classification of these new heavy hadrons was considered in Ref. [@10], where quark composition of two-quark (meson) and three-quark (fermion) states was represented for the case of up- and down-types of quark $Q$. Stable and long-lived new hadrons are divided into three families of particles with characteristic values of masses M, 2M and 3M, where M is the mass of $Q$-quark. Quantum numbers and quark content of these particles for the case of up-type quark $Q=U$ are represented in Table 1.
[Table 1. Characteristics of $U$-type hadrons]{}
$J^P=0^-$ $T=\frac{1}{2}$ $M=(M^0\,M^-)$ $M^0=\bar{U}u$, $M^-=\bar{U}d$
----------------- ----------------- -------------------------------- ------------------------------------
$J=\frac{1}{2}$ $T=1$ $B_1=(B_1^{++}\,B_1^+\,B_1^0)$ $B_1^{++}=Uuu,B_1^+=Uud,B_1^0=Udd$
$J=\frac{1}{2}$ $T=\frac{1}{2}$ $B_2=(B^{++}_2\,B^+_2)$ $B^{++}_2=UUu,B^+_2=UUd$
$J=\frac{3}{2}$ $T=0$ $(B^{++}_3)$ $B^{++}_3=UUU$
Some states in Table 1 were also considered in Ref. [@9a] for the case of long-lived vector-like heavy quark and in Ref. [@10.8], where $U$-type quark belong to the sequential 4-th generation. In Ref. [@13d], there were considered an important property of suppression of hadronic interaction of heavy quark systems containing three new quarks, like $(UUU)$ states. This model has $SU(3)\times SU(2)\times SU(2)\times U(1)$ symmetry and offers a novel alternative for the DM carriers — they can be an electromagnetically bound states made of terafermions. The charged $M^-$ and neutral $M^0$ particles can manifest themselves in cosmic rays and as carrier of the DM. In Refs. [@7; @8; @9] a possibility is discussed that new stable charged hadrons exist but are hidden from detection, being bounded inside neutral dark atoms. For instance, stable particles with charge $Q=-2$ can be bound with primordial helium.
Interactions of the baryon-type particles $B_1$ and $B_2$ (the second and third line in Table 1) are similar to the nucleonic ones, and they may compose atomic nuclei together with nucleons. As it was demonstrated in Ref. [@10], this circumstance does not prevent the $B_1$ and $B_2$ burn out in the course of cosmochemical evolution. There are no problems also with interaction of $B_3$ isosinglet with nucleons which proceeds mainly through exchange by mesons, $\eta$ and $\eta^{'}$. The constants of such interactions, as it follows from the quark model of the mesonic exchange (see Ref. [@10]), is not a large one, i.e. $B_3N$ interaction is suppressed in comparison with the $NN$ interaction.
There is another type of hypothetical hadrons which possess analogous properties of strong interactions. They are constructed from stable quark of the down-type (D-quark) with $Q =
-1/3$ electric charge. Quantum numbers and quark content of these particles are represented in Table 2 (see the corresponding analysis and comments in Ref. [@10]).
[Table 2. Characteristics of $D$-type hadrons.]{}
$J^P=0^-$ $T=\frac{1}{2}$ $M_D=(M^+_D\,M^0_D)$ $M^+_D=\bar{D}u$, $M^0_D=\bar{D}d$
----------------- ----------------- ----------------------------------------- -------------------------------------
$J=\frac{1}{2}$ $T=1$ $B_{1D}=(B_{1D}^+\,B_{1D}^0\,B_{1D}^-)$ $B_{1D}^{+-}=Duu(Ddd),B_{1D}^0=Dud$
$J=\frac{1}{2}$ $T=\frac{1}{2}$ $B_{2D}=(B^0_{2D}\,B^-_{2D})$ $B^0_{2D}=DDu,B^-_{2D}=DDd$
$J=\frac{3}{2}$ $T=0$ $(B^-_{3D})$ $B^-_{3D}=DDD$
In this table, the states $M^+_D,\,B^0_{1D},\,B^0_{2D},\,B^-_{3D}$ are stable. Particles possessing a similar quark composition appear in various high-energy generalizations of SM, in which $D$-quark is a singlet with respect to weak interactions group. For example, each quark-lepton generation in $E(6)\times E(6)$ -model contains two singlet $D$-type quarks. This quark appears, also, from the Higgs sector in supersymmetric generalization of $SU(5)$ Great Unification model. As a rule, with a reference to cosmological restrictions it is assumed that new hadrons are unstable due to the mixing of singlet $D$-quarks with the standard quarks of the down type. Note, the consequences for cosmochemical evolution, caused by existence of the hypothetical stable $U$- and $D$-types hadrons, are very different.
Cosmochemical evolution of new hadrons at hadronization stage was qualitatively studied both for $U$ and $D$ cases in [@10]. A very important conclusion was arrived from this analysis - baryon asymmetry in new quark sector must exist and has a sign opposite to asymmetry in standard quark sector (quarks $U$ disappear but antiquarks $\bar{U}$ remain). This conclusion follows from the strong cosmochemical restriction for the ratio “anomalous/natural” hydrogen $C\leqslant 10^{-28}$ for $M_Q\lesssim 1\,\mbox{TeV}$ [@Smith] and anomalous helium $C\leqslant 10^{-12} - 10^{-17}$ for $M_Q\leq 10\,\mbox{TeV}$ [@Muller]. In our case, the state $B^+_1=(Uud)$ is heavy (anomalous) proton which can form anomalous hydrogen. At the stage of hadronization, $B^+_1$ can be formed by direct coupling of quarks and as a result of reaction $\bar{M}^0 + N \to B^+_1 + X$, where $X$ is totality of leptons and photons in the final state. The antyparticles $\bar{B}^+_1$ are burning out due to the reaction $\bar{B}^+_1 + N \to M^0 +X$. The states like $(pM^0)$ can be also manifest itself as anomalous hydrogen, but as it was shown in [@10], interaction of $p$ and $M^0$ has a potential barrier at large distances. So, formation of coupled states $(pM^0)$ at low energies is strongly suppressed. As it follows from the experimental restrictions on anomalous hydrogen and helium [@Smith; @Muller], baryon symmetry in extra sector of quarks is not excluded for the case of super-heavy new quarks with masses $M_Q\gg 1\,\mbox{TeV}$ (see, also, the fourth section). Further, we consider the interaction of new hadrons with nucleons and their self-interaction in more detail.
At low energies the hadrons interactions can be approximately described by a model of meson exchange in terms of an effective lagrangian. It was shown in [@18], that low-energy baryon-meson interactions are effectively described by $U(1)\times SU(3)$ gauge theory, where $U(1)$ is the group of semi-strong interaction and $SU(3)$ is group of hadronic unitary symmetry. Effective physical lagrangian which was used for calculation of $MN$ interaction potential is represented in [@10]. By straightforward calculations, it was demonstrated there that the dominant contribution is resulted from the exchanges by $\rho$ and $\omega$ mesons. This lagrangian at low energies can be applied for analysis both of $MN$ and $MM$ interactions. Here, we give the part of lagrangian with vector-meson exchange which will be used for evaluation of the potential: $$\begin{aligned}
\label{3.1}
L_{int}&=g_{\omega}\omega^{\mu}\bar{N}\gamma_{\mu}N + g_{\rho}\bar{N}\gamma_{\mu}\hat{\rho}^{\mu}N
+ig_{\omega M}\omega^{\mu}(M^{\dagger}\partial_{\mu}M-\partial_{\mu}M^{\dagger}M)\notag\\ &+ ig_{\rho M} (M^{\dagger}\hat{\rho}^{\mu}\partial_{\mu}M-
\partial_{\mu}M^{\dagger}\hat{\rho}^{\mu}M).\end{aligned}$$ In (\[3.1\]) $N=(p,n),\,M=(M^0,\,M^-),\,M^{\dagger}=(\bar{M}^0,\,M^+)$ and coupling constants are the following [@10]: $$\begin{aligned}
\label{3.2}
g_{\rho}&=g_{\rho M}=g/2,\,\,\,g_{\omega}=\sqrt{3}g/2\cos\theta ,\,\,\,g_{\omega M}=g/4\sqrt{3}\cos\theta,\notag\\
&g^2/4\pi\approx 3.16,\,\,\,\cos\theta =0.644.\end{aligned}$$ Note, the one-pion exchange which is dominant in $NN$ interaction is forbidden in the $MM\pi$ -vertex due to parity conservation.
In Born approximation, the potential of interaction and the non-relativistic amplitude of scattering for the case of non-polarized particles are connected by the relation: $$\label{3.3}
U(\vec{r})=-\frac{1}{4\pi^2\mu}\int f(q)\exp(i\,\vec{q}\, \vec{r})\, d^3q,$$ where $\mu$ is the reduced mass of scattering particles. For the case of $M$ scattering off nucleons, this potential was calculated in Ref. [@10], where it was utilized the relation $f(q)=-2\pi i\mu F(q)$ between nonrelativistic amplitude, $f(q)$, and Feynman amplitude, $F(q)$. As it was shown, contributions of scalar and two-pion exchanges are suppressed by the factor $\sim m_N/m_M$. Expressions for potentials of interaction of various pairs from doublets $(M^0,M^-)$ and $(p,n)$ have following form: $$\begin{aligned}
\label{3.4}
U(M^0,p;r)&=U(M^-,n;r)\approx U_{\omega}(r)+U_{\rho}(r),\notag\\
U(M^0,n;r)&=U(M^-,p;r)\approx U_{\omega}(r)-U_{\rho}(r).\end{aligned}$$ In Eqs.(\[3.4\]) the terms $U_{\omega}(r)$ and $U_{\rho}(r)$ are defined by the following expressions: $$\label{3.5}
U_{\omega}=\frac{g^2K_{\omega}}{16\pi\cos^2\theta}\,\frac{1}{r}\,\exp(-\frac{r}{r_{\omega}}),\,\,\,U_{\rho}=\frac{g^2K_{\rho}}{16\pi}\,\frac{1}{r}\,\exp(-\frac{r}{r_{\rho}}),$$ where $K_{\omega}=K_{\rho}\approx0.92,\,\,r_{\omega}=1.04/m_{\omega},\,\,r_{\rho}=1.04/m_{\rho}$. Taking into account these values and $m_{\omega}\approx m_{\rho}$, we rewrite expressions (\[3.4\]) in a form: $$\begin{aligned}
\label{3.6}
U(M^0,p;r)&=U(M^-,n;r)\approx 2.5\,\frac{1}{r}\,\exp(-\frac{r}{r_{\rho}}),\notag\\
U(M^0,n;r)&=U(M^-,p;r)\approx 1.0\,\frac{1}{r}\,\exp(-\frac{r}{r_{\rho}}).\end{aligned}$$ Two consequences can be deduced from the expressions (\[3.6\]). Firstly, all four pairs of particles have repulsive potential ($U>0$) of interaction at long distances, where Born approximation is valid. Secondly, due to potential barrier the DM particles at low energies can not interact with nucleons, i.e. they can not form the coupled states $(pM^0)$ which manifest itself as anomalous protons. So, they can not be directly detected. To overcome the barrier, nucleons should have energy $\sim 1\, GeV$ or more and this situation takes place in high energy cosmic rays.
Potential of $MM$ interaction can be also reconstructed with the help of above given method. Here, we determine only the sign of potential which define characteristic (attractive or repulsive) of interaction at long distances. This characteristic plays crucial role for low-energy collisions of the DM particles and nucleons. To determine the sign of potential we use the definition of lagrangian in the non-relativistc limit: $$\label{3.7}
L=L_0+L_{int}\,\longrightarrow W_k-U,$$ where $W_k$ is kinetic part and $U$ is potential. There is a relation between effective $L_{int}(q)$ and Feynman amplitude $F(q)$: $F(q)=ikL_{int}(q)$, where $k>0$ is real coefficient depending on the type of particles. As a result, we get equality $signum(U)=signum(iF)$, where amplitude of interaction is determined by one-particle exchange diagrams for the process $M_1M_2\to M^{'}_1M^{'}_2$. Here, $M=(M^0,\bar{M^0})$ and vertexes are defined by the low-energy lagrangian (\[3.1\]). With the help of this simple approach, one can check previous conclusion about repulsive character of $MN$ interactions. First of all it should be noted, that low-energy effective lagrangians of $NM^0$ and $N\bar{M}^0$ have opposite sign due to different sign of vertexes $\omega M^0 M^0$ and $\omega \bar{M}^0 \bar{M}^0$. This effect can be seen from the differential structure of corresponding part of Lagrangian (\[3.1\]) and representation of field function of the $M$ -particle in the form: $$\begin{aligned}
\label{3.8}
M(x)=&\sum_p \hat{a}^-_p (M)\exp(-ipx)+\hat{a}^+_p(\bar{M})\exp(ipx),\notag\\
M^{\dagger}(x)=&\sum_p \hat{a}^+_p (M)\exp(ipx)+\hat{a}^-_p(\bar{M})\exp(-ipx).\end{aligned}$$ In Eqs. (\[3.8\]), $a^{\pm}_p(M)$ and $a^{\pm}_p(\bar{M})$ are the operators of creation and destruction of particles $M$ and antiparticles $\bar{M}$ with momentum $p$. As a result, we get the vertexes $\omega(q) M^0(p) M^0(p-q)$ and $\omega(q) \bar{M}^0(p) \bar{M}^0(p-q)$ in momentum representation with opposite signs, $L_{int}=\pm g_{\omega M}(2p-q)$, respectively. This leads to the repulsive and attractive potentials of $N M$ and $N \bar{M}$ low-energy effective interactions via $\omega$ exchange. Thus, the absence of potential barrier in the last cases give rise to the problem of coupled states $p\bar{M}^0$ (the problem of anomalous hydrogen). As it was noted earlier, to overcome this problem we make the suggestion that the hadronic DM is baryon asymmetric ($\bar{M}^0$ is absent at low-energy stage of hadronization) or particles $\bar{M}^0$ are superheavy. Properties of interactions of baryons $B_1$ and $B_2$ are similar to nucleonic one (the main contribution give one-pion and vector meson exhanges) and together with nucleons they may compose an atomic nuclei. So, new baryons can form superheavy nuclears which in the process of evolution are concentrated due to gravitation in the center of massive planets or stars.
Further, we have checked that the potential of $M^0M^0$ and $\bar{M}^0\bar{M}^0$ interactions is attractive ($U<0$) for the case of scalar meson exchange and repulsive for the case of vector meson exchange. Potential of $M^0\bar{M}^0$ scattering has attractive asymptotes both for scalar and vector meson exchanges. Thus, the presence of potential barrier in the processes of $M^0M^0$ and $\bar{M}^0\bar{M}^0$ scattering depends on the relative contribution of scalar and vector mesons. In the case of $M^0\bar{M}^0$ scattering the total potential is attractive and this property can lead to increasing of annihilation cross section in an analogy with Sommerfeld effect [@SEeff].
Main properties of new hadrons as the DM carriers
=================================================
The mass of heavy quark $M_Q$ and the mass splitting of the charged $M^- $ and neutral $M^0$ mesons, $\delta m = m^- - m^0$, are significant characteristics of these states both for their physical interpretation and for application in cosmology. In this analysis, we take into consideration standard electromagnetic and strong interactions only. So, some properties of new mesons doublet $M=(M^0,M^-)$ are analogous to properties of standard mesons consisting of pairs of heavy and light quarks. From experimental data on mass splitting in neutral-charged meson pairs $K=(K^0,K^{\pm})$, $D=(D^0,D^{\pm})$ and $B=(B^0,B^{\pm})$, it is seen that for down-type mesons $K$ and $B$ the mass-splitting $\delta m <0$ while for up-type meson $D$ the value of $\delta m>0$. Such results can be explained by the fitting data on current masses of quarks, $m_d>m_u$, and binding energy of the systems $(\bar{Q}u)$ and $(\bar{Q}d)$, where Coulomb contributions have different signs. The absolute value of $\delta m $ for the case of $ K- $ and $ D- $ mesons is $O(\mbox{MeV})$, but for $B- $ mesons it is less. Taking into account these data, for the case of up SQ we assume: $$\label{4.1}
\delta m =m(M^-)-m(M^0)>0,\,\,\,\mbox{and}\,\,\, \delta m=O(MeV).$$ Then, we conclude that neutral state $M^0=(\bar{U}u)$ is stable and can play the role of the DM carrier. The charged partner $M^-=(\bar{U}d)$ has only one decay channel with very small phase space: $$\label{4.2}
M^-\to M^0e^-\bar{\nu}_e,\,\,\,(\mbox{if}\,\,\,\delta m>m_e).$$ This semileptonic decay is resulted from the weak transition $d\to u+W^-\to u+e^-\bar{\nu}_e$, where heavy quark $\bar{U}$ is considered as spectator. The width of decay can be calculated in a standard way and final expression for differential width is as follows (see also review by R. Kowalski in [@PDG]): $$\label{4.3}
\frac{d\Gamma}{d\omega}=\frac{G^2_F}{48\pi^3}|U_{ud}|^2(m_-+m_0)^2m_0^3(\omega^2-1)^{3/2}G^2(\omega).$$ In the case under consideration $m_-\approx m_0$, $\omega =k^0/m_0\approx 1$ and $G(\omega)\approx 1$ (HQS approximation). Here, $G(\omega)$ is equivalent to normalized formfactor $f_+(q)$, where $q$ is the transferred momentum. In the vector dominance approach this formfactor is defined as $f_+(q)=f_+(0)/(1-q^2/m^2_v)$, where $m_v$ is the mass of vector intermediate state. So, HQS approximation corresponds to the conditions $q^2\ll m^2_v$ and $f_+(0)\approx 1$ for the case $\omega =k^0/m_0\approx 1$. Using Eq.(\[4.3\]), for the total width we get: $$\label{4.4}
\Gamma\approx \frac{G^2_F|U_{ud}|^2 m^5_0}{12\pi^3}\int_1^{\omega_m}(\omega^2-1)^{3/2} d\omega;\,\,\,\omega_m=\frac{m^2_0+m^2_-}{2m_0m_-}.$$ After integration, the expression (\[4.4\]) can be written in the simple form: $$\label{4.5}
\Gamma\approx \frac{G_F^2}{60\pi^3}(\delta m)^5,$$ where weak coupling constant is taking at a low-energy scale because of small transferred momentum in the process. From the expression (\[4.5\]) one can see that the width crucially depends on the mass splitting, $\Gamma\sim (\delta m)^5$ and does not depend on the mass of meson $M$. For instance, in the interval $\delta m=(1-10)\,\mbox{MeV}$ we get following estimations: $$\label{4.6}
\Gamma\sim (10^{-29}-10^{-24})\,\mbox{GeV};\,\,\,\tau\sim (10^5-10^0)\,\mbox{s}.$$ Thus, charged partner of $M^0$, which is long-lived (metastable), can be directly detected in the processes of $M^0 N-$ collisions with an energetic nucleons, $N$. This conclusion is in accordance with the experimental evidence of heavy charged metastable particles presence in cosmic rays (see Ref. [@10] and references therein). Note olso, the models of DM with a long-lived co-annihilation partner are discussed in literature (see, for instance, Refs. [@9a; @Khoze]).
Experimental and theoretical premises of new heavy hadron existence were discussed in the Ref. [@10]. With the help of low-energy model of baryon-meson interactions, it was shown that the potential of $MN$ -interaction has repulsive asymptotics. So, the low-energy particles $M$ do not form coupled states with nucleon and the hypothesis of their existence does not contradict to the cosmochemical data.
Now, we estimate the mass of new hadrons which are interpreted as carriers of the DM. The data on Dark Matter relic concentration result to value of the cross section of annihilation at the level: $$\label{4.7}
(\sigma v_r)^{exp}\approx 10^{-10}\,\,GeV^{-2}.$$ Comparing the model annihilation cross section (which depends on the mass) to this value, we estimate the mass of the meson $M^0$. Note, the calculations are fulfilled for the case of hadron-symmetrical DM, that is, the relic abundance is suggested the same for $M^0$ and $\bar{M}^0$. To escape the contradiction with strong restriction on anomalous helium, we should expect the mass of $M^0$ above 10 TeV. Approximate evaluation of the model cross section $\sigma(M^0\bar{M}^0)$ can be fulfilled in spectator approach $\sigma(M^0\bar{M}^0)\sim\sigma(U\bar{U})$ considering the light $u$-quarks as spectators. Main contributions to this cross section result from sub-processes $U\bar{U}\to gg$ and $U\bar{U}\to q\bar{q}$, where $g$ and $q$ are standard gluon and quark. Corresponding cross sections are represented in the second section (Eqs. (\[2.6\]) and (\[2.7\])) and their sum is used for approximate evaluation of the full annihilation cross section of the processes $M^0\bar{M}^0\to \mbox{hadrons}$. Thus, we can estimate $M_U$ mass from the following approximate equation: $$\label{4.8}
(\sigma v_r)^{exp}\approx \frac{44\pi}{9}\frac{\alpha^2_s}{M^2_U}.$$ Now, from (\[4.7\]) and (\[4.8\]) we get: $m(M^0)\approx M_U\approx 20\,\mbox{TeV}$ at $\alpha_s=\alpha_s(M_U)$. Note, this value get into the range (10–100) TeV which was declared for the case of heavy WIMPonium states in Ref. [@Asadi].
As it was noted in the previous section, attractive potential of $M^0\bar{M}^0$ interaction at long distances can increase the cross section due to the light meson exchange. This effect leads to Sommerfeld enhancement [@SEeff] of the cross section: $$\label{4.9}
\sigma v_r=(\sigma v_r)_0S(\alpha/v),$$ where $(\sigma v_r)_0$ is initial cross section which is results from the left side of the expression (\[4.8\]); $\alpha=g^2/4\pi$ is defined by the effective coupling according to (\[3.2\]) and $v=v_r/2$. At $m\ll M\approx M_U$, where $m$ is mass of mesons (the light force carriers), Sommerfeld enhancement (SE) factor can be represented in the form [@SEeff]: $$\label{4.10}
S(\alpha/v)=\frac{\pi \alpha/v}{1-\exp(-\pi \alpha/v)}\,.$$
In our case, the light force carriers are $\omega$- and $\rho$ -mesons and $\alpha\sim 1$ (see (\[3.1\]) and (\[3.2\])), so from (\[4.10\]), we get $10^2 \lesssim S(\alpha/v)/\pi
\lesssim 10^3$ in the interval $10^{-2}>v>10^{-3}$ . In this case, from (\[4.8\])-(\[4.10\]) it follows that at $v\sim 10^{-2}$ the mass of new quark $M_U\sim 10^2\,\mbox{TeV}$, which agree with the evaluation of the mass of baryonic DM in [@RanHuo] ($M\sim 100$ TeV). Thus, we get too heavy $M^0$ which can not be detected in the searching for signals of anomalous hydrogen ($M_{max}\lesssim 1\,\mbox{TeV}$) and anomalous helium ($M_{max}\lesssim 10\,\mbox{TeV}$). Note, however, that in these calculations we take into account the light mesons only, ($m\ll M_U$), which act at long distances $r\sim m_{\rho}^{-1}$. At short distance, near the radius of coupling state $M^0=(\bar{U}u)$, i.e. at $r\sim M^{-1}_U$, it is possible the exchange by heavy mesons containing heavy quark $U$, for instance, by vector or scalar $M$ -mesons. In this case, the expression (\[4.10\]) is not valid because of $M_{\chi}\sim M_U$, where $M_{\chi}$ is the mass of heavy force carriers. To evaluate SE factor in this case, we use its numerical calculation from [@Cirelli], where iso-contours of the SE corrections are presented as functions of $y=\alpha M/M_{\chi}$ and $x=\alpha/v$. Then $y\approx 1$, and from Ref. [@Cirelli] (see Fig.1 there) it follows that $S\approx
10$ in the interval $10^{-1}>v>10^{-3}$. As a result, from (\[4.8\]) and (\[4.10\]) it follows $M_U\approx 60\,TeV$ which does not change situation crucially. It should be noted, full description of SE requires an account of weak vector bosons $Z,W$ which interact with light quarks only. Thus, SE effect is formed at various energy regions corresponding to various distances and has very complicated and vague nature (see, also, Ref. [@Blum]).
Conclusion
==========
We have analyzed a scenario of the hadronic DM based on the simplest extension of the SM with singlet quark. It was shown in a previous work that the existence of new heavy hadrons does not contradict to cosmological constraints. Here, we demonstrate that the scenario is in accordance with the precision electroweak restrictions on manifestations of New Physics. With the help of effective model Lagrangian, we describe the asymptotes of interaction potential at low energies for interactions of new hadrons with nucleons and with each other. These asymptotics occure both atractive and repulsive for different pairs of interacting particles $N,M$ and their antipaticles. The cosmochemical constrictions on anomalous hydrogen amd anomalous helium lead to the conclusion that abundance of particles $M$ and antiparticles $\bar{M}$ is strongly asymmetrical, or new hadrons $M$ are superheavy (with mass larger 10 TeV).
Approximate value of the mass-splitting for charged and neutral components was evaluated and lifetime of charged meta-stable hadron component was calculated, it occurs rather large, $\tau\gg 1$ s. Using the value of the DM relic concentration and the expression for the model cross section of annihilation, mass of the hadronic DM carrier is estimated. The value of mass without account of SE effect is near 20 TeV and the SE increases it up to an order of $10^2$ TeV. These results agree with the evaluations of mass of baryonic DM, which are represented in literature (see previous section). So, superheavy new hadrons can not be generated in the LHC experiments and detected in the searching for anomalous hydrogen and helium. Some peculiarities of Sommerfeld enhancement effect in the process of annihilation are analyzed. It should be underlined, that the model annihilation cross section was evaluated at the level of sub-processes. So, for the description of the hadronic Dark Matter in more detail it is necessary to clarify the mechanism of annihilation process at various energy scales.
Data Availability {#data-availability .unnumbered}
=================
The graphic data used to support the findings of this study are available from the corresponding author upon request.
Conflict of interest {#conflict-of-interest .unnumbered}
====================
The authors declare that they have no conflict of interest.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work was supported by Russian Scientific Foundation (RSCF) \[Grant No.: 18-12-00213\].
[00]{} F. Sannino, “Conformal Dynamics for TeV Physics and Cosmology”, [*Acta Physica Polonica B*]{}, vol. 40, no. 12, pp. 3533-3744, 2009.
R. Pasechnik, V. Beylin, V. Kuksa, and G. Vereshkov, “Vector-like technineutron Dark Matter: is a QCD-type Technicolor ruled out by XENON100?”, [*European Physical Journal C*]{}, vol. 74, no. 2, p. 2728, 2014.
M. Khlopov, “Cosmological Reflection of Particle Symmetry”, [*Symmetry*]{}, vol. 8, p. 81, 2016.
XENON Collab. (E. Aprile et al.), “First Dark Matter Search Results from the XENON1 Experiment”, [*Physical Review Letters*]{}, vol. 119, article 181301, 2017.
M. Maltoni, V.A. Novikov, L.B. Okun, A.N. Rozanov, and M.I. Vysotsky, [*Physical Letters B*]{}, vol. 476, no. 1-2, pp. 107-115, 2000.
K.M. Belotsky, D. Fargion, M.Yu. Khlopov et al., “Heavy hadrons of 4th family hidden in our Universe and close to detection”, [*Gravit. Cosmol. Suppl.*]{}, vol. 11, p. 3, 2005.
M.Yu. Khlopov, “PHYSICS OF DARK MATTER IN THE LIGHT OF DARK ATOMS”, [*Modern Physical Letters A*]{}, vol. 26, no. 38, pp. 2823-2839, 2011.
M.Yu. Khlopov, “Introduction to the special issue on indirect dark matter searches”, [*International Journal of Modern Physics A*]{}, vol. 29, article 1443002, 2014.
J.R. Cudell and M. Khlopov, “Dark atoms with nuclear shell: A status review”, [*International Journal of Modern Physics D*]{}, vol. 24, article 1545007, 2015.
M. Buchkremer and A. Schmidt, “Long-lived Heavy Quarks: A Review”, [*Advances in High Energy Physics*]{}, vol. 2013, Article ID 690254, 17 pages, 2013.
Yu.N. Bazhutov, G.M. Vereshkov, and V.I. Kuksa, “Experimental and Theoretical Premises of New Stable Hadron Existence”, [*International Journal of Modern Physsics A*]{}, vol. 2, article 1759188, 2017.
J.C. Pati and A. Salam, “Lepton number as the fourth colour”, [*Physics Review D*]{}, vol. 10, no. 1, pp. 275-289, 1974.
V. Barger, N.G. Deshpande, et al., “Extra fermions in $E_6$ superstring models”, [*Physical Review D*]{}, vol. 33, no. 7, pp. 1902-1924, 1986.
V.D. Angelopoulos, J. Ellis, H. Kowalski et al., “Search for new quarks suggested by superstring”, [*Nuclear Physics B*]{}, vol. 292, pp. 59-92, 1987.
P. Langacker and D. London, “Mixing between ordinary and exotic fermions”, [*Physical Review D*]{}, vol. 38, no. 3, pp. 886-906, 1988.
V.A. Beylin, G.M. Vereshkov and V.I. Kuksa, “Mixing of singlet quark with standard ones and the properties of new mesons”, [*Physics of Atomic Nuclei*]{}, vol. 55, no. 8, pp. 2186-2192, 1992.
R. Rattazzi, “Phenomenological implications of a heavy isosinglet up-type quark”, [*Nuclear Physics B*]{}, vol. 335, pp. 301-310, 1990.
M.Yu. Khlopov and R.M. Shibaev, “Probes for 4th Generation Constituents of Dark Atoms in Higgs Boson Studies at the LHC”, [*Advances in High Energy Physics*]{}, article ID 406458, 7 pages, 2014.
M. Maltoni, V.A. Novikov, L.B. Okun et al., “Extra quark-lepton generations and precision measurements”, [*Physics Letters B*]{}, vol. 476, no. 1-2, pp. 107-115, 2000.
V.A. Ilyin, M. Maltoni, V.A. Novikov et al., “On the search for 50 GeV neutrinos”, [*Physics Letters B*]{}, vol. 503, no. 1-2, pp. 126-132, 2001.
V.A. Novikov, L.B. Okun, A.N. Rozanov and M.I. Vysotsky, “Extra generations and discrepancies of electroweak precision data”, [*Physics Letters B*]{}, vol. 529, no. 1-2, pp. 111-116, 2002.
O. Eberhardt, G. Herbert, H. Lacker et al., “Impact of a Higgs Boson at a Mass of 126 GeV on the Standard Model with Three and Four Fermion Generations”, [*Physics Review Letters*]{}, vol. 109, no. 24, article 241802, 2012.
F.J. Botella, G.C. Branco, M. Nebot, “The Hunt for New Physics in the Flavor Sector with up vector-like quarks”, [*Journal of High Energy Physics*]{}, vol. 12, article 040, 2012.
A. Kumar Alok, S. Banerjee, D. Kumar, S.U. Sankar, D. London, “New-physics signals of a model with vector singlet up-type quark”, [*Physical Review D*]{}, vol. 92, no. 1, article 013002, 2015.
J. Llorente, B. Nachman, “Limites on new coloured fermions using precision data from Large Hadron Collider”, arXiv:1807.00894(hep-ph).
M.E. Peskin, T. Takeuchi, “Estimations of oblique electroweak corrections”, [*Physical Review D*]{}, vol. 46, no.1, p. 381, 1992.
C. P. Burgess, S. Godfrey, H. Konig, et. al., “A global Fit to Extended Oblique Parameters”, [*Phys. Lett. B*]{}, vol. 326, no. 3-4, pp. 276-281, 1994.
M. Tanabashi et al. (Particle Data Group), “The Review of Particle Physics (2018)”, [*Physical Review D*]{}, vol. 98, 2018.
K. Belotsky, M. Khlopov and K. Shibaev, “Stable quarks of the 4th family?”, arXiv:astro-ph/0806.1067 (28 pages).
S.G. Glashow, “A Sinister Extension of the Standard Model to $SU(3)\times SU(2)\times SU(2)\times U(1)$”, arXiv:hep-ph/0504287 (9 pages).
P.F. Smith, J.R.J. Bennet, G.J. Homer et al., “A search for anomalous hydrogen in enriched $D_2O$, using a time-of-flight spectrometer”, [*Nuclear Physics B*]{}, vol. 206, no. 3, pp. 333-348, 1982.
P. Muller, L.-B. Wang, J. Holt et al., “Search for Anomalously Heavy Isotopes of Helium in the Earth’s Atmosphere”, [*Physical Review Letters*]{}, vol. 92, article 022501, 2004.
G.M. Vereshkov and V.I. Kuksa, “$U(1)SU(3)$-gauge model of baryon-meson interactions”, [*Physics of Atomic Nuclear*]{}, vol. 54, no. 6(12), pp. 1700-1704, 1991.
J.L. Feng, M. Kaplinghat, and Hai-Bo Yu, “Sommerfeld Enhancement for Thermal Relic Dark Matter”, [*Physical Review D*]{}, vol. 82, article 083526, 2010.
V.V. Khoze, A.D. Plascencia, K. Sakurai, “Simplified models of dark matter with a long-lived co-annihilation partner”, arXiv: 1702.00750\[hep-ph\].
P. Asadi, M. Baumgart, P.J. Fitzpatric et al., “Capture and decay of EW WIMPonium”, [*arXiv: 1610.07617 \[hep-ph\]*]{}.
Ran Huo, S. Matsumoto, Y-L.S. Tsai, T.T. Yanagida, “A scenario of heavy but visible baryonic dark matter”, [*JHEP 1609*]{}, p. 162, 2016.
M. Cirelli, A. Strumia, and M. Tamburini, “Cosmology and Astrophysics of Minimal Dark Matter”, [*IFUP-TH/2007-12*]{}; arXiv:0706.4071\[hep-ph\].
K. Blum, R. Sato, and T.R. Slatyer, “Self-consistent calculation of the Sommerfeld enhancement”, [*Journal of Cosmology and Astrophysics*]{}, vol. 06, article 021, 2016.
|
---
abstract: |
We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. First, we write out formulas for moves $3\to 3$ and $2\leftrightarrow 4$ based on the results of our two previous works, and then we study in detail moves $1\leftrightarrow 5$. On this basis, we obtain the formula for a four-dimensional manifold invariant.
As an example, we present a detailed calculation of our invariant for sphere $S^4$; in particular, the complex turns out, indeed, to be acyclic.
author:
- |
I. G. Korepanov\
Southern Ural State University\
76 Lenin avenue\
454080 Chelyabinsk, Russia\
E-mail: kig@susu.ac.ru
title: 'Euclidean 4-simplices and invariants of four-dimensional manifolds: III. Moves $1\leftrightarrow 5$ and related structures'
---
-.5cm
Introduction {#sec IIIvvedenie}
============
This work is third in the series of papers started with papers [@I] and [@II]. We also use Roman numerals I and II, respectively, for references to those works; in particular, equation (II.1) is the equation (1) from paper [@II], or “from paper II”.
Our goal is to construct and investigate a new type of acyclic complexes, wherefrom we should be able to extract invariants of four-dimensional piecewise-linear manifolds. Note, however, that at the time when paper I was being written, it was not yet clear that acyclic complexes were exactly those structures that stood behind the considered algebraic formulas, as well as behind the invariants of [*three-dimensional*]{} manifolds from papers [@3dim1] and [@3dim2]. These complexes were written out in paper II — the whole complex for the three-dimensional case and a part of it — for the four-dimensional case. Concerning the four-dimensional case, it turned out that, in order to construct a value invariant under Pachner moves (i.e., rebuildings of the manifold triangulation) of type $3\to 3$ [*only*]{}, it is enough to consider only a small central part of the complex (two vector spaces and one linear mapping between them), whereas the addition of rebuildings $2\leftrightarrow 4$ brings into consideration one more space (namely, the space of edge deviations) and one more mapping.
Paper I was devoted to moves $3\to 3$, and paper II — to moves $2\leftrightarrow 4$. Therefore, the complex [*in full*]{} was not necessary in those papers, and its full form was only announced in paper II (formulas (II.6) and (II.7)).
The contents of the present “paper III” by sections is as follows. In Section \[sec IIIp2\] we revisit, once again, the dimensionality 3 and write out the acyclic complex from Section 2 of paper II in a slightly different (longer) form, which clarifies the structure of one of the involved spaces as a [*factor*]{} space. This simple improvement may simplify considerably our constructions when we pass to higher dimensions.
In Section \[sec IIIp3\] we present the space of vertex deviations, as we promised in paper II, and its mapping into the space of edge deviations. This enables us to write out in Section \[sec IIIp4\] our complex for the four-dimensional case in full; we also “make it longer” with respect to paper II in analogy with the three-dimensional case. We still have to prove, though, that our sequence of spaces and mappings is really a complex, and we do that in Section \[sec IIIp5\].
We would like our constructed complex to be acyclic (i.e., an exact sequence) which would enable us to get manifold invariants out of its torsion. We do not give the full proof of the acyclicity in the general case, limiting ourselves to some remarks in the end of Section \[sec IIIp5\] (see also a concrete example in Section \[sec IIIp9\]). Then we show that, [*assuming the exactness*]{}, the torsion of our complex does not change under Pachner moves (and the exactness itself is conserved under those moves). To this end, we study in Section \[sec IIIp6\] how the complex changes under moves $3\to 3$ and $2\leftrightarrow 4$, and in Section \[sec IIIp7\] — under moves $1\leftrightarrow 5$ (one can say that it was the need to study the moves $1\leftrightarrow 5$ that required the detailed consideration of the [*whole*]{} algebraic complex). In Section \[sec IIIp8\] we present the final formula giving the invariant of a four-dimensional piecewise-linear manifold (i.e., the invariant of [*all*]{} Pachner moves) in terms of the torsion of the complex and other Euclidean geometric values. In Section \[sec IIIp9\] we show how our machinery works for the sphere $S^4$ (and check explicitly that, at least for the sphere, our complex is indeed acyclic).
In the concluding Section \[sec IIIobsuzhdenie\] we discuss the obtained results.
Revisiting three dimensions once again: a longer form of the acyclic complex {#sec IIIp2}
============================================================================
We recall that the acyclic complex from Section 2 of paper II, corresponding to a triangulation of a three-dimensional manifold $M$, had the following form: $$0
\leftarrow (\cdots)
\stackrel{B^{\rm T}}{\leftarrow} (d\omega) \stackrel{A}{\leftarrow} (dl)
\stackrel{B}{\leftarrow} (dx
\hbox{ and } dg) \leftarrow 0.
\eqno\hbox{(II.1)}$$
Here $(d\omega)$ is the space of column vectors made of infinitesimal deficit angles at all edges of the complex, $(dl)$ is the space of columns of differentials of edge lengths, $(dx)$ is the space of columns of differentials of Euclidean coordinates of the edges of the complex (to be exact, of their lift-ups onto the universal cover), $(dg)$ is the space of columns of differentials of continuous paprameters on which the representation $f\colon \; \pi_1(M)\to E_3$ may depend. In particular, $(dg)$ vanishes for manifolds with a finite fundamental group.
We thus continue to use the notations in the style of paper II for linear spaces entering in algebraic complexes, hoping that the convenience of such notations pays for their certain looseness. Of course, from the formal standpoint, for instance, space $(dl)$ is the [*tangent space*]{} to the manifold consisting of all sets of positive numbers — “lengths” — put in correspondence to the edges of our triangulation; $(dx\hbox{ and } dg)$ (the space of columns of all $dx$ and all $dg$) is the [*direct sum*]{} $(dx)\oplus(dg)$.
Recall that $(dx)$ in sequence (II.1) is the space of differentials of coordinates taken up to those infinitesimal motions of the Euclidean space $\mathbb R^3$ which are compatible with the given representation $f$, i.e. motions commuting with its image $\Im f=f\bigl( \pi_1(M)\bigr)$. Such motions form a subalgebra in the Lie algebra $\mathfrak e_3$, which we denote $\mathfrak a$. Column $(dx)$ is described explicitly for various cases in Section 2 of paper II. In the three-dimensional case, such description can be made easily, but difficultuies may increase when we pass to higher dimensions. That is the reason for rewriting sequence (II.1) in the following, more elegant form.
We now permit ourselves to [*change notations*]{} and understand below by $(dx)$ the space of columns of differentials of Euclidean coordinates of the vertices in the complex, with no further conditions, that is, columns of values $(dx_1,dy_1,dz_1,\ldots,\allowbreak dx_N,dy_N,dz_N)$. Then our [*former*]{} space $(dx)$ is written as the factor $(dx)/\mathfrak a$. This suggests an idea of adding one more term to the right of $(dx\hbox{ and }dg)$ in sequence (II.1) (and a symmetric term in the left-hand part of the sequence — see below), understanding now $(dx)$ in the new sense: 0 () () (d) (dl) (dx dg) a 0. \[3 dlinnaya\] Recall that the whole sequence (this applies to both (II.1) and (\[3 dlinnaya\])) is symmetric in the following sense. Each term in it is considered as a vector space with a fixed basis; mappings between them are identified with matrices; and any two matrices at equal distances from the left-hand and right-hand ends of the sequence must be obtained from one another by matrix transposing. In particular, matrix $A$ that gives the mapping $(dl)\to (d\omega)$ is symmetric.
As for the notation “$(\cdots)$”, we are using it for [*different*]{} linear spaces whose specific geometrical sense we are not going to investigate (at this moment; still, we know the matrices of mappings between such spaces, for instance, from the symmetry described in the previous paragraph).
The torsion of an acyclic complex is the product of certain minors in the matrices of its linear mappings, taken in alternating degrees $\pm 1$. If we adopt a convention that the $+$ sign corresponds to the first nontrivial mapping (coming after the zero injection, i.e., for example, in sequence (II.1) that is the second arrow from the right), then the torsion will change to its inverse when we pass from (II.1) to (\[3 dlinnaya\]). To avoid this, we can [*agree*]{} to take the $+$ sign for mapping $B$ in sequence (\[3 dlinnaya\]), as before. Then [*the torsion of complex (\[3 dlinnaya\]) coincides with the torsion of complex (II.1)*]{} if we choose a basis in Lie algebra $\mathfrak e_3$ in a natural way (namely, three infinitesimal translations along mutually orthogonal axes and three rotations around these very axes).
The full proof of this statement must involve the analysis of all the particular cases from Section 2 of paper II. Here, we will limit ourselves to the case of a lens space $L(p,q)$, with a nontrivial homomorphism $f\colon \; \pi_1\bigl(L(p,q)\bigr) \to E_3$ whose image consists of rotations around the $z$ axis. To compose the minor of the matrix of mapping $\mathfrak a \to (dx)$, we choose two basis elements in space $(dx)$, namely, $dx_1$ and $dz_1$, where subscript $1$ corresponds to some vertex in the triangulation which we agree to call the “first” one. Now a simple straightforward calculation shows that, indeed, the torsions of complexes (\[3 dlinnaya\]) and (II.1) coincide, if we choose the basis of the “old” $(dx)$ in the latter complex according to formula (II.2) (it makes sense to remind here that we consider the torsion [*to within its sign*]{}).
Vertex deviations and their mapping into edge deviations {#sec IIIp3}
========================================================
We have already used Euclidean coordinates in our constructions a few times, namely, the coordinates of a vertex in the complex were necessary to define the mappings $(dx)\to (dl)$ in the three-dimensional and four-dimensional cases, and the coordinates of the vector of edge deviation — for the mapping $(d\vec v)\to (dS)$. We can remark that we did not need any connection between the coordinate systems for [*different*]{} vertices and/or edges while doing these constructions. Any individual coordinate system could be chosen arbitrarily, for instance, by fixing some angles between coordinate axes and adjacent edges.
We cannot work with no such coordinate systems at all if we want to [*fix the bases*]{} in all vector spaces entering in a complex; on the other hand, the torsion of our complexes, as one can check, does not depend on the choice of those systems.
In this Section we, first, define the [*vertex deviation*]{} as a tensor value which needs, for its components to be fixed, a coordinate system corresponding to the [*vertex*]{}. Next, it will be convenient for us to define how it generates edge deviations for the edges abutting on the given vertex, using the [*same*]{} (i.e. corresponding to the vertex) coordinate system for those edges. We [*imply*]{} that the components of each edge’s deviation are then transformed into [*its own*]{} coordinate system by using a proper orthogonal transformation.
So, we call vertex deviation a bivector (antisymmetric tensor) $d\sigma_{\alpha \beta }$, where $\alpha, \beta=1,\ldots, 4$. Let $AB$ be one of the edges abutting on vertex $A$. This edge can be also considered as a four-dimensional vector $\overrightarrow{AB}$, whose coordinates we denote as $l_{\alpha}$. By definition, a deviation of vertex $A$ equal to $d\sigma_{\alpha \beta }$ generates the deviation of edge $d\vec v_{AB}$ with components ( dv\_[AB]{} )\_ = \_ l\_ d\_, \[sigma->v\] where $L_{AB}=\sum_{\alpha} l_{\alpha}^2$ is the squared length of edge $AB$.
In general, $d\vec v_{AB}$ is the sum of expression (\[sigma->v\]) and a similar expression involving the deviation of vertex $B$ (where, of course, vector $\overrightarrow{BA}$ must be used instead of $\overrightarrow{AB}$). The reasonableness of definition (\[sigma->v\]) will be clear in Section \[sec IIIp5\], where we will prove, in particular, that the edge deviations of type (\[sigma->v\]) generate zero differentials $(dS)$ of two-dimensional face areas. This will be part of the statement that “composition of two neighboring arrows (i.e. linear mappings) in the sequence is zero”, which will justify the name “complex” for that sequence. Let us now pass on to the construction of our sequence in its full form.
The full sequence of spaces and mappings in the four-dimensional case {#sec IIIp4}
=====================================================================
We will write out two “conjugate” sequences, as we did in Section 3 of paper II. Compared with formulas (II.6) and (II.7), we will increase them in length both on the right and on the left, in analogy with the three-dimensional case (Section \[sec IIIp2\] of the present work), because both the leftmost nonzero space in formula (II.7) and the rightmost one in (II.6) can be represented naturally as factor spaces.
As for the first of mentioned spaces, denoted as $(dx\hbox{ and }{dg})$ in formula (II.7), here our elongation goes, in principle, the same way as in three dimensions. Thus we pass at once to the second one, denoted $(d\sigma)$ in formula (II.6), i.e. to the space of vertex deviations introduced in the previous Section. It turns out that there exists an easily defined space of “trivial” deviations generating zero edge deviations $d\vec v_a$, and it finds its natural place in our complex.
Consider formula (\[sigma->v\]). We recall that, in it, the tensor value $d\sigma_{\alpha \beta }$ pertains to point $A$, and $l_{\alpha}$ are the components of vector $\overrightarrow {AB}$. If we take into account $d\sigma_{\alpha \beta }$ in point $B$ as well, we get ( dv\_[AB]{} )\_ = \_ l\_ ( (d\_A)\_ - (d\_B)\_ ). \[4.1\] One can see from here that $d\vec v_{AB}=0$ if the difference $(d\sigma_A)_{\alpha \beta } - (d\sigma_B)_{\alpha \beta }$ has the form $\displaystyle \sum_{\gamma,\delta} \epsilon_{\alpha \beta \gamma \delta}\,
l_{\gamma}\, ds_{\delta}$, where $\epsilon_{\alpha \beta \gamma \delta}$ is the totally antisymmetric tensor, $\epsilon_{1234}=1$, and $d\vec s$ is any infinitesimal vector.
Passing on from considering one edge $AB$ to considering all edges in the complex, we take, first, the case of trivial representation $f\colon \; \pi_1 (M) \to E_4$ of the fundamental group of our manifold $M$ into the group of motions of the four-dimensional Euclidean space, i.e. the case $\Im f =\{ e \}$. This means that all inverse images of any given vertex in the complex get into one and the same point of space $\mathbb R^4$ (see Section 3 of paper I), that is, simply speaking, to each vertex $A$ its radius vector $\vec r_A$ corresponds unambiguously with components $(r_A)_{\alpha}$ in some Cartesian coordinate system common for the whole complex.
Choose some infinitesimal antisymmetric tensor $d\tau_{\alpha\beta}$ and vector $ds_{\alpha}$, and set for each vertex $A$ (d\_A)\_ = d\_ + \_ (r\_A)\_ ds\_. \[trivsigma\] It follows from the foregoing that such set of vertex deviations yields zero deviations for all edges. Note that such columns $d\sigma$ make up a ten-dimensional linear space.
In case representation $f$ is not trivial, any vertex has more than one inverse images in the universal covering, and these inverse images are placed in different points of space $\mathbb R^4$. The requirement that formula (\[trivsigma\]) must give equal results for all such inverse images (after transforming them into the coordinate system corresponding to the given vertex — see Section \[sec IIIp3\]), leads to linear restrictions on admissible $d\tau_{\alpha\beta}$ and $ds_{\beta}$. In the present paper, we do not write out explicitly these restrictions: it will suffice for us to begin with studying just simply connected manifolds, for which $\pi_1(M)=\{e\}$. Nevertheless, we introduce notation $(d\sigma)_0$ for the subspace of columns of those vertex deviations which are correctly determined by formula (\[trivsigma\]), for [*any*]{} manifold $M$.
Now we are ready to rewrite sequences (II.6) as (II.7) in the renovated (longer) form. Despite the fact that we are still using notation “$(\cdots)$” for vector spaces with whose geometric sense we are not concerned now, we have [*given the definitions*]{} to all [*matrices*]{} of mappings denoted by arrows. So, here are our mutually conjugate sequences: $$\begin{aligned}
0\leftarrow (\cdots) \leftarrow (\cdots) \leftarrow (d\Omega_a) \stackrel{(\partial
\Omega_a / \partial S_i)}{\longleftarrow} (dS_i) \leftarrow (d\vec v_a)
\leftarrow (d\sigma) \leftarrow (d\sigma)_0 \leftarrow 0\,,
\label{posled} \\
0 \rightarrow \mathfrak a \rightarrow (dx \hbox{ and } dg) \rightarrow (dL_a) \stackrel{(\partial
\omega_i / \partial L_a)}{\longrightarrow} (d\omega_i) \rightarrow
(\cdots) \rightarrow (\cdots) \rightarrow (\cdots) \rightarrow 0\,.
\label{posledT}\end{aligned}$$ Here, of course, $\mathfrak a$ is a subalgebra of Lie algebra $\mathfrak e_4$ of motions of Euclidean space $\mathbb R^4$. Namely, $\mathfrak a$ consists of those motions commuting with the image of group $\pi_1(M)$ in $E_4$, in full analogy with the three-dimensional case from Section \[sec IIIp2\].
The sequence is a complex {#sec IIIp5}
=========================
Now we will show that the composition of any two successive arrows in sequence (\[posled\]) or, equivalently, (\[posledT\]) equals zero. Thus, we will justify the name “complex” for each of these sequences.
We start with the two arrows of sequence (\[posled\]) adjacent to the term $(d\sigma)$. It follows directly from formulas (\[4.1\]) and (\[trivsigma\]) of the previous Section that their composition is zero.
Moving to the left, we consider two arrows around the term $(d\vec v_a)$. Consider a triangle $ABC$ — one of the two-dimensional faces of our simplicial complex — and check that the deviations of its edges, if they are given by formulas of type (\[4.1\]), lead to the zero area differential $dS_{ABC}$.
It is enough to consider the case where only vertex $A$ has a nonzero deviation $d\sigma$. According to formula (\[sigma->v\]), the lengths of vectors $d\vec v$ in Figure \[IIIris1\]
=0.75mm
(66.00,66.00) (45.00,35.00)[(-3,-2)[6.00]{}]{} (15.00,35.00)[(-3,1)[7.80]{}]{}
(15.00,38.50)[(0,0)\[rb\][$d\vec v_{AB}$]{}]{} (7.00,36.00)[(0,0)\[rt\][$C'$]{}]{} (42.00,34.00)[(0,0)\[rb\][$d\vec v_{AC}$]{}]{} (39.50,30.50)[(0,0)\[lt\][$B'$]{}]{} (35.00,4.00)[(0,0)\[ct\][$A'$]{}]{} (4.00,4.00)[(0,0)\[rt\][$B$]{}]{} (66.00,4.00)[(0,0)\[lt\][$C$]{}]{} (25.00,66.00)[(0,0)\[cb\][$A$]{}]{}
are inversely proportional to the trianle’s sides to which they belong. Thus, one can easily deduce that the area of triangle $A'B'C'$ multiplied by four [*coincides*]{} with the area of triangle $ABC$ and hence $dS_{ABC}=0$ according to formula (II.8).
Moving further to the left along sequence (\[posled\]), we must consider two arrows adjacent to the term $(dS_i)$ — but the vanishing of their product has been already proven in Section 5 of paper II.
To investigate the two remaining pairs of arrows, we switch to sequence (\[posledT\]). In terms of that sequence, these are the pairs of arrows around the terms $(dx \hbox{ and } dg)$ and $(dL_a)$. Now it remains to remark that the statements we need can be proved in full analogy to how it was done for the three-dimensional case in Section 2 of paper II and Section 2 of the present work. Moreover, there is even [*exactness*]{} in those terms.
As for the exactness in other terms, nothing is known about it as yet in the general case, save that it is clear from the construction that the exactness holds in the left- and rightmost terms $\mathfrak a$ and $(d\sigma)_0$ which appeared when we “made longer” our complex. Still, we will see in the following Sections \[sec IIIp6\] and \[sec IIIp7\] that if the sequence is exact, then this property is preserved under Pachner moves, that is, exactness does not depend on a triangulation. Besides, the example of sphere $S^4$ studied below in Section \[sec IIIp9\] shows that at least for the sphere the sequence [*is*]{} exact (i.e., is an acyclic complex).
How the algebraic complex changes under moves $3\to 3$ and $2\leftrightarrow 4$ {#sec IIIp6}
===============================================================================
We already know from papers I and II what happens under moves $3\to 3$ and $2\leftrightarrow 4$. Our task in this Section is to reformulate these results while holding strictly to the algebraic language of acyclic complexes.
Moves $3\to 3$ {#subsec 6.1}
--------------
In paper I, devoted to moves $3\to 3$, we were considering matrix $(\partial \Omega_a / \partial S_i)$ and its conjugate $(\partial \omega_i / \partial L_a)$. From our current viewpoint, they form the central part of sequence (\[posled\]) and its conjugate (\[posledT\]). Some of the results of paper I (namely, Theorem 4) do not deal with moves $3\to 3$ as such but only reproduce (by somewhat amateurish means) a part of statement that the torsion of a complex does not depend on a specific choice of minors through which it is expressed. It is Theorem 3 of paper I that deals with moves $3\to 3$ proper. It seems worthwhile to recall it here once again (with only a very slight reformulation):
*Choose in matrix $(\partial \omega_i/ \partial L_a)$ a largest square submatrix $\cal B$ with nonzero determinant. Let $\cal B$ contain a row corresponding to such face $i=ABC$ that belongs to exactly three 4-simplices. Then those latter can be replaced by three new 4-simplices by a move $3\to 3$. After such a replacement, take in the [*new*]{} matrix $(\partial \omega_i/ \partial L_a)$ the new submatrix $\cal B$ containing the same rows and columns, with the only following change: the row corresponding to the face $ABC$ is replaced by the row corresponding to the new face $DEF$.*
The expression \[I.17\] does not change under such a rebuilding.
We add here the following. The torsion of complex (\[posledT\]) is the alternated product of minors one of which is exactly $\det {\cal B}$. With the presented realization of move $3\to 3$, [*all other minors*]{} obviously [*remain unchanged*]{}. Thus we can say that formula (\[I.17\]) describes the behaviour not only of $\det {\cal B}$ but of the whole torsion $\tau$ under a move $3\to 3$. It is evident also that the acyclicity property as such, if the complex possessed it, is preserved.
Moves $2\leftrightarrow 4$ {#subsec 6.2}
--------------------------
For convenience, we speak about moves $2\to 4$, having in mind the obvious invertibility of the following reasoning and formulas to the case of moves $4\to 2$.
Under a move $2\to 4$, one edge and four two-dimensional faces are added to the simplicial complex. As for the algebraic complex (\[posled\]), the dimensionality of space of column vectors $(dS_i)$ increases by 4, the dimensionality of space $(d\vec v_a)$ — by 3 and the dimensionality of space $(d\Omega_a)$ — by 1. Hence, we can increase by 3 the sizes of the minor of matrix $(\partial S_i / \partial \vec v_a)$ and by 1 — those of the minor of matrix $(\partial \Omega_a / \partial S_i)$, and [*keep unchanged the minors*]{} of which the torsion is made up [*beyond the fragment*]{} $(d\Omega_a) \leftarrow (dS_i) \leftarrow (d\vec v_a)$.
As in Section 6 of paper II, we denote the added edge as $AB$, and the four added faces will be $ABC$, $ABD$, $ABE$ and $ABF$. We append the derivatives of areas of the first three faces to the minor of matrix $(\partial S_i / \partial \vec v_a)$, while the derivatives w.r.t. $S_{ABF}$ — to the minor of matrix $(\partial \Omega_a / \partial S_i)$. We denote those minors simply as $\minor (\partial S_i / \partial \vec v_a)$, etc.: we consider only a single minor for every matrix, and there is no risk of confusion.
As concerns the 4-simplices, we continue to use the notations of paper II for them as well. Namely, under the move $2\to 4$, two adjacent 4-simplices $ACDEF=\hat B$ and $BCDFE=-\hat A$ are replaced with four ones: $ABCDE=\hat F$, $ABCFD=-\hat E$, $ABCEF=\hat D$ and $ABDFE=-\hat C$.
Formula (II.10) shows that under the move $2\to 4$ the important for us minor of matrix $(\partial S_i / \partial \vec v_a)$ gets multiplied by . \[minor-S-v\] As concerns the minor of matrix $(\partial \Omega_a / \partial S_i)$, it gets multiplied, according to formula (II.15), by . \[minor-L-omega\] It can be easily derived from here that the following value is conserved under moves $2\to 4$: . \[3324\]
Comparing this with formula (\[I.17\]), we see that the value (\[3324\]) is conserved under moves $3\to 3$ as well (of course, $\minor (\partial \Omega_a / \partial S_i)$ is the very same thing as $\det {\cal B}$ in formula (\[I.17\])). Besides, it is again evident that the acyclicity property is conserved.
How the algebraic complex changes under moves $1\leftrightarrow 5$ {#sec IIIp7}
==================================================================
Like in the previous Section, we consider, for concreteness, only a move in one direction, namely $1\to 5$. Under such move, a new [*vertex*]{} is added to the simplicial complex — denote it $F$ — which brings about the decomposition of one 4-simplex — denote it $ABCDE$ — into five 4-simplices, which we denote in the style of the previous Section as $\hat A$, $\hat B$, $\hat C$, $\hat D$ and $\hat E$.
We will explain how we extend the minors that enter in the torsion of complex (\[posled\]) or (\[posledT\]), starting from the left, that is from the mapping $(dx\hbox{ and }dg)\to (dL_a)$ (the minor of the “preceding” mapping $\mathfrak a \to (dx\hbox{ and }dg)$ does not change, see the remark after formula (\[dL(dxdg)\])). The length of columns of coordinate differentials $dx$ increases by 4: $dx_F$, $dy_F$, $dz_F$ and $dt_F$ are added to them — the differentials of four coordinates of the new vertex $F$. Hence, we must choose four $dL$, corresponding to four new rows of the minor. Let those be $dL_{AF}$, $dL_{BF}$, $dL_{CF}$ and $dL_{DF}$. The following formula holds which can be proved by a direct calculation using some easy trigonometry (as well as its analogue in 3 dimensions, see [@3dim1 formulas (31) and (32)]): dx\_F dy\_F dz\_F dt\_F = . \[7.1\] Strictly speaking, here a $\pm$ sign should have been added, but we take interest in equalities of such kind [*to within their sign*]{} (as well as in the preceding Section).
It follows from formula (\[7.1\]) that the important for us [*minor of the matrix of mapping*]{} $(dx\hbox{ and }dg)\to (dL_a)$ [*gets multiplied by*]{} 384 V\_[E]{} . \[dL(dxdg)\] Note also that the parameters $dg$ responsible for continuous deformations of representation $f$ are “used up” on algebra $\mathfrak a$ (if they existed at all) in the sense that their corresponding rows are included in another minor, that of mapping $\mathfrak a \to (dx\hbox{ and }dg)$, which does not change under our move.
Now consider the mapping $(dL_a)\to (d\omega_i)$. Here, only one column is added to the minor, corresponding to the still “available” $dL$, namely $dL_{EF}$. On the side of space $(d\omega_i)$, we add the row corresponding to $d\omega_{DEF}$. In consequence, the [*minor of the matrix of mapping*]{} $(dL_a)\to (d\omega_i)$ [*gets multiplied by*]{} \[domegadL\] (cf. formula (\[minor-L-omega\])).
Now we switch from sequence (\[posledT\]) to sequence (\[posled\]). Nine rows and columns must be added to the minor of mapping $(d\vec v_a)\to (dS_i)$. The columns correspond to the nine “still free” $dS_i$, that is $dS_i$ for all added faces $i$ except $i=DEF$. We are going to consider them all, step by step, choosing for them on our way nine components of vectors $d\vec v_a$ (while their total number is 15: five edges, each having three deviation components).
Matrix $(\partial S_i / \partial \vec v_a)$ contains many zeroes (the same applies, by the way, to the other matrices with which we are occupied here): nonzero entries appear only where edge $a$ enters in the boundary of face $i$. In consequence, many of its submatrices have a block-triangular form so that their determinants (i.e., minors of $(\partial S_i / \partial \vec v_a)$) factorize. In particular, it will be convenient for us to include the derivatives w.r.t. [*all*]{} components of deviations of edges $DF$ and $EF$ in the minor with which we are occupied now: we will find out that some [*separate factors*]{} correspond to them as their contributions to the quantity by which our minor is multiplied. These factors are = \[dSdv1\] (cf. formula (\[minor-S-v\])) for edge $DF$ and = \[dSdv2\] for edge $EF$.
There remain three area differentials — $dS_{ABF}$, $dS_{ACF}$ and $dS_{BCF}$ — for which we must choose three from nine components of vectors $d\vec v_{AF}$, $d\vec v_{BF}$ and $d\vec v_{CF}$. To this end, we first choose three axes for [*each*]{} of these vectors, in order to take their projections onto these axes as their components. Certainly, our triples of axes will make up orthonormal coordinate systems in the three-dimensional spaces orthogonal to corresponding edges. Some easy reasoning shows that the torsion of complex does not depend on a specific choice of such coordinate systems.
We require that the $x$ axes (different!) for $d\vec v_{AF}$ and $d\vec v_{BF}$ lie in the plane $ABF$ (this fixes their directions, because they must also be orthogonal to the respective edges). The $y$ axis (common) for $d\vec v_{AF}$ and $d\vec v_{BF}$ will lie in the three-dimensional hyperplane $ABCF$ (and be, of course, orthogonal to plane $ABF$). The $z$ axis — [*common for all three*]{} $d\vec v_{AF}$, $d\vec v_{BF}$ and $d\vec v_{CF}$ — will be orthogonal to hyperplane $ABCF$.
It remains to choose the directions of axes $x$ and $y$ for $d\vec v_{CF}$. It is enough to fix the direction of the $x$ axis: we choose it to be orthogonal to plane $BCF$.
Now we choose three components of vectors $d\vec v$, in order to include the derivatives with respect to them into the minor of matrix $(\partial S_i / \partial \vec v_a)$. These will be $(dv_{BF})_x$, $(dv_{CF})_x$ and $(dv_{CF})_y$.
The determinant \[3dS3dv\] again factorizes: it equals the product of the quantity $\partial S_{ABF} / (\partial v_{BF})_x = l_{BF}$ by the determinant $\displaystyle \frac{dS_{ACF} \wedge dS_{BCF}}{(dv_{CF})_x \wedge (dv_{CF})_y}$. But this determinant, too, factorizes due to the fact that $dS_{BCF}$ does not depend on $(dv_{CF})_x$ (recall how we chose the direction of axis $x$ for this vector). Finally we find that expression (\[3dS3dv\]) equals l\_[BF]{} l\_[CF]{}\^2 , \[dSdv3\] where $\alpha$ is the angle between planes $ACF$ and $BCF$.
Here is the expression — the product of expressions (\[dSdv1\]), (\[dSdv2\]) and (\[dSdv3\]) — by which the minor of matrix $(\partial S_i / \partial \vec v_a)$ is multiplied as a result of the move $1\to 5$: l\_[BF]{} l\_[CF]{}\^2 . \[dSdv\]
It remains to consider one more minor that changes under the move $1\to 5$ — the minor of matrix $(\partial \vec v_a / \partial \sigma)$. It involves the components $(dv_{AF})_x$, $(dv_{AF})_y$, $(dv_{AF})_z$, $(dv_{BF})_y$, $(dv_{BF})_z$ and $(dv_{CF})_z$ of edge deviations and, on the other hand, all six components of bivector $d\sigma$ corresponding to vertex $F$.
Bivector $d\sigma$ can be thought of as an element of Lie algebra $\mathfrak s \mathfrak o (4)$, and its components — as infinitesimal rotation angles within the six coordinate planes. We have to choose a proper coordinate system for it. This time, we will denote its axes by numbers 1, 2, 3, 4 (rather than letters $x$, $y$, $z$, $t$).
We choose axes 1 and 2 to lie in the plane $ABF$: axis 1 will be orthogonal to vector $\overrightarrow {BF}$, and axis 2 — parallel to it. Axis 3 will lie in the hyperplane $ABCF$ (and, of course, orthogonal to $ABF$), and axis 4 — orthogonal to $ABCF$.
Now (the thing we are already accustomed to) the value by which the minor is multiplied again factorizes. First (cf. formula (\[sigma->v\])), the components $(dv_{AF})_z$, $(dv_{BF})_z$ and $(dv_{CF})_z$ are determined just by three rotation angles in the direction of axis 4, i.e.$d\sigma_{14}$, $d\sigma_{24}$ and $d\sigma_{34}$. Namely, they are connected by means of the following $3\times 3$ submatrix of matrix $(\partial \vec v_a / \partial \sigma)$ (see again (\[sigma->v\])): (
[ccc]{} & &\
& &\
& &
) , \[4z\] where, for instance, $(AF)_1$ is the component of vector $\overrightarrow{AF}$ along axis 1. The determinant of matrix (\[4z\]) is = , \[dvdsigma1\] where $V_{ABCF}$ is the [*three-dimensional*]{} volume, while $\alpha$ is the same angle as in formula (\[dSdv3\]).
There remain three rotations within hyperplane $ABCF$, and three components $(dv_{AF})_x$, $(dv_{AF})_y$ and $(dv_{BF})_y$. Here, too, the factorizability persists: the rotation within the plane $ABF$ affects only $(dv_{AF})_x$, and this gives the factor = . \[dvdsigma2\] The rotation within the plane orthogonal to $BF$ affects only $(dv_{AF})_y$, which gives the factor = , \[dvdsigma3\] $\beta$ being the angle between edges $AF$ and $BF$. Now the last rotation remains with the corresponding factor = . \[dvdsigma4\]
Taking the product of the right-hand sides of formulas (\[dvdsigma1\]), (\[dvdsigma2\]), (\[dvdsigma3\]) and (\[dvdsigma4\]), we find the factor by which the minor of matrix $(\partial \vec v_a / \partial \sigma)$ gets multiplied under our move $1\to 5$: , \[dvdsigma\] where we have taken into account that $\sin \beta = 2\, S_{ABF} / l_{AF}\, l_{BF}$.
In the same way as in Subsections \[subsec 6.1\] and \[subsec 6.2\], it is clear from the performed reasoning and calculations that the property of acyclicity of the complex is conserved under the considered moves.
The formula for the manifold invariant {#sec IIIp8}
======================================
In the two preceding Sections we have analyzed the behaviour of the minors whose alternated product makes up the torsion of the complex (\[posled\]) or (\[posledT\]) (if this complex is acyclic), under Pachner moves $3\to 3$, $2\leftrightarrow 4$ and $1\leftrightarrow 5$. We have now to unite these results to obtain the quantity which [*does not depend on a triangulation*]{}, i.e. an [*invariant of a four-dimensional piecewise-linear manifold*]{}.
We choose the signs in the alternated product in such way that our formula for the invariant look as similar as possible to the “three-dimensional” formula (II.5). Namely, we take the minor of the matrix of mapping $\mathfrak a \to (dx\hbox{ and }dg)$ raised in the power $-1$, then the minor of the matrix of mapping $(dx\hbox{ and }dg) \to (dL_a)$ raised in the power $+1$ and so on. We will get the factor by which the so defined torsion $\tau$ is multiplied under the move $1\to 5$ when we multiply the expressions (\[dL(dxdg)\]) and (\[dSdv\]) and divide by the product of (\[domegadL\]) and (\[dvdsigma\]). The result can be written as $$2^7 \cdot 3^4 \cdot \frac{V_{\hat A}\, V_{\hat B}\, V_{\hat C}\, V_{\hat D}\, V_{\hat E}}{V_{\hat F}}
\cdot \frac{\displaystyle \prod_{
\hbox{\scriptsize
\begin{tabular}{c}
\rm over new\\[-.5\baselineskip]
\rm edges
\end{tabular} } } l^5 }{\displaystyle \prod_{
\hbox{\scriptsize
\begin{tabular}{c}
\rm over new\\[-.5\baselineskip]
\rm 2-dim.~faces
\end{tabular} } } S } \; .$$
Comparing this with the results of Section \[sec IIIp6\] (formulas (\[I.17\]) and (\[3324\])), we find the following final expression for the invariant of a four-dimensional manifold in terms of the torsion of complex (\[posled\]) or (\[posledT\]) and other Euclidean geometric values: I=2\^[-16]{} 3\^[-12]{} ( 2\^8 3\^6 )\^[()]{} . \[i4\] The factor $2^{-16} \cdot 3^{-12}$ has been added in order that the invariant be equal to unity for the sphere $S^4$, see the following Section.
Example: sphere $S^4$ {#sec IIIp9}
=====================
The fundamental group of sphere $S^4$ is trivial, thus algebra $\mathfrak a$ in sequence (\[posledT\]) will be the whole Lie algebra $\mathfrak e_4$ of motions of the four-dimensional Euclidean space. There are obviously no continuous parameters $dg$ describing the deformations of representation $\pi_1(S^4) \to E_4$. The space of “trivial” vertex deviations is ten-dimensional and consists of deviations of type (\[trivsigma\]) with arbitrary $d\tau_{\alpha \beta}$ and $ds_{\beta}$.
We take the canonical triangulation of $S^4$ consisting of two 4-simplices, both with vertices $A$, $B$, $C$, $D$ and $E$. For such triangulation, all deficit angles $d\omega_i$ and $d\Omega_a$ are identically equal to zero, because any $\omega_i$ is obtained by summing two terms differing only in sign (the $+$ sign is ascribed to one of 4-simplices, while the $-$ sign to the other one, see I, Section 3). Thus, matrix $(\partial \Omega_a / \partial S_i)$ and its conjugate $(\partial \omega_i / \partial L_a)$ are zero. This means that the torsion of complex (\[posled\]) or (\[posledT\]) factorizes in the product over two following sequences: 0 e\_4 (dx) (dL\_a) 0 \[l\] and 0 (dS\_i) (dv\_a) (d) (d)\_0 0. \[pr\]
We fix a Euclidean coordinate system in the space $\mathbb R_4$, with axes $x,y,z,t$. We place the vertex $A$ of our triangulation into the origin of coordinates, and the remaining vertices — in points $B(1,0,0,0)$, $C(0,1,0,0)$, $D(0,0,1,0)$ and $E(0,0,0,1)$.
Calculations for sequence (\[l\])
---------------------------------
The ten-dimensional algebra $\mathfrak e_4$ consists of infinitesimal rotations and translations (of course, we measure the rotations within the six coordinate planes in radians, and the translations — in the units of coordinate axes), whereas the twenty-dimensional space $(dx)$ — of column vectors $\pmatrix{dx_A \cr \vdots \cr dt_E}$. We take the minor of matrix of mapping $\mathfrak e_4 \to (dx)$ corresponding to the following [*ten*]{} coordinate differentials: $dx_A, dy_A, dz_A,\allowbreak
dt_A, dy_B, dz_B,\allowbreak dt_B, dz_C, dt_C, dt_D.$
From this minor, a unity factor splits off at once corresponding to the mapping (translations) $\to$ (vertex $A$ coordinates); this splitting off is caused by the fact that [*rotations do not affect*]{} the coordinates of $A$. There remains the mapping of six rotations into sets $(dy_B,dz_B,dt_B,\allowbreak dz_C,dt_C,dt_D)$. Factorization still works here: rotations within the planes $xy$, $xz$ and $xt$ affect only point $B$ (from those remaining) and so on. The result is: [*the minor of mapping $\mathfrak e_4 \to (dx)$ equals unity*]{} (as usual, to within a sign).
Now we consider the minor of matrix of the mapping $(dx)\to (dL_a)$ corresponding to the set $(dx_B, dx_C, dy_C,\allowbreak
dx_D, dy_D, dz_D,\allowbreak dx_E, dy_E, dz_E, dt_E)$. It, too, factorizes into the product of determinants of mappings $$\begin{aligned}
(dx_E, dy_E, dz_E, dt_E) & \to & (dL_{AE}, dL_{BE}, dL_{CE}, dL_{DE}),
\label{9.1.1} \\
(dx_D, dy_D, dz_D) & \to & (dL_{AD}, dL_{BD}, dL_{CD}),
\label{9.1.2} \\
(dx_C, dy_C) & \to & (dL_{AC}, dL_{BC}),
\label{9.1.3} \\
dx_B & \to & dL_{AB}.
\label{9.1.4}\end{aligned}$$ Recall that $L$ is the squared length of a corresponding edge. A direct calculation shows that the determinants of mappings (\[9.1.1\]), (\[9.1.2\]), (\[9.1.3\]) and (\[9.1.4\]) equal $2^4$, $2^3$, $2^2$ and $2$, respectively.
[**Conclusion:** ]{}the multiplicative contribution of sequence (\[l\]) to the torsion is $2^{10}$.
Calculations for sequence (\[pr\])
----------------------------------
We start from the right, i.e. from the mapping $(d\sigma)_0 \to (d\sigma)$. The ten-dimensional space $(d\sigma)_0$ consists of the components of antisymmetric tensor $d\tau_{\alpha \beta}$ and vector $ds_{\beta}$, whereas the thirty-dimensional space $(d\sigma)$ — of the components of all five vertices’ deviations. The mapping is given by formula (\[trivsigma\]).
For composing the minor of matrix of the mapping $(d\sigma)_0 \to (d\sigma)$, we choose the following ten components of tensors $d\sigma_A,\ldots,d\sigma_E$: we take [*all six*]{} components of $d\sigma_A$ and [*one*]{} component for each of the remaining deviations, namely, $(d\sigma_B)_{zt}$, $(d\sigma_C)_{zt}$, $(d\sigma_D)_{xy}$ and $(d\sigma_E)_{xy}$. It makes sense to remind here that this choice depends on us (the only requirement is that minors be nonzero), and this exactly choice was motivated by the convenience of further calculations.
The vector $ds_{\beta}$ does not affect $d\sigma_A$, because the radius vector of point $A$ is zero. Hence, our minor factorizes, and the factor corresponding to mapping $d\tau \to d\sigma_A$ is unity (because this is an identical mapping: $d\sigma_A=d\tau$). The remaining $4\times 4$ minor factorizes into the product of four unit factors, because, for instance, $(d\sigma_B)_{zt}$ depends on $ds_y$ only (this can be seen from formula (\[trivsigma\]) if we replace in it the subscript $A$ with $B$, ignore $d\tau_{\alpha \beta}$ and observe that the single nonzero component of vector $\vec r_B$ is $(r_B)_x$) and so on.
Thus, our selected minor of matrix of the mapping $(d\sigma)_0 \to (d\sigma)$ turned out to equal unity.
We pass on to the mapping $(d\sigma)\to (d\vec v_a)$. Here one should fix at first a three-dimensional basis for each of ten vectors $d\vec v_{AB},\ldots,d\vec v_{DE}$. It is done most easily for those edges that begin at point $A$: edge $AB$ lies on the $x$ axis, so we choose as the coordinate axes for $d\vec v_{AB}$ the three remaining axes $y$, $z$ and $t$; then we choose axes in a similar way for the deviations of edges $AC$, $AD$ and $AE$.
Each of the remaining six edges lies within some coordinate [*plane*]{}, and we will treat them in the following way. Edge $BC$ lies in the plane $xy$; we choose the following three axes for $d\vec v_{BC}$: the bisector of the angle formed by axes $x$ and $y$, and also axes $z$ and $t$; we follow this model in choosing the axes for deviations of edges $BD$, $BE$, $CD$, $CE$ and $DE$. We will denote the components of vectors of edge deviations along the bisectors of coordinate angles as $\left( dv_{BC} \right)_{\rm bis}$, etc. (keeping in mind that every deviation has its own bisector for an axis).
We must choose twenty components of vectors of edge deviations (because we have twenty components of vertex deviations not included in our previous minor). We choose [*all*]{} components for edges $AB$, $AC$, $AD$ and $AE$, and also the following eight components: $(dv_{BD})_y$, $(dv_{BD})_t$, $(dv_{BE})_y$, $(dv_{BE})_z$, $(dv_{CD})_x$, $(dv_{CD})_t$, $(dv_{CE})_x$ and $(dv_{CE})_z$. We drew in Figure \[IIIris2\]
=1.00mm
(97.00,67.00) (3.00,35.00)[(0,0)\[rc\][$B$]{}]{} (50.00,3.00)[(0,0)\[ct\][$E$]{}]{} (97.00,35.00)[(0,0)\[lc\][$C$]{}]{} (50.00,67.00)[(0,0)\[cb\][$D$]{}]{} (50.00,63.50)[(0,0)\[ct\][$\scriptstyle z$]{}]{} (92.50,35.00)[(0,0)\[rc\][$\scriptstyle y$]{}]{} (50.00,6.50)[(0,0)\[cb\][$\scriptstyle t$]{}]{} (7.50,35.00)[(0,0)\[lc\][$\scriptstyle x$]{}]{} (5.00,29.00)[(0,0)\[cc\][$d\sigma_{yt}$]{}]{} (5.00,41.00)[(0,0)\[cc\][$d\sigma_{yz}$]{}]{} (41.00,5.00)[(0,0)\[cc\][$d\sigma_{zx}$]{}]{} (41.00,65.00)[(0,0)\[cc\][$d\sigma_{tx}$]{}]{} (26.00,15.00)[(0,0)\[cc\][$dv_z$]{}]{} (20.00,19.00)[(0,0)\[cc\][$dv_y$]{}]{} (59.00,5.00)[(0,0)\[cc\][$d\sigma_{zy}$]{}]{} (95.00,29.00)[(0,0)\[cc\][$d\sigma_{xt}$]{}]{} (95.00,41.00)[(0,0)\[cc\][$d\sigma_{xz}$]{}]{} (59.00,65.00)[(0,0)\[cc\][$d\sigma_{ty}$]{}]{} (20.00,51.00)[(0,0)\[cc\][$dv_y$]{}]{} (26.00,55.00)[(0,0)\[cc\][$dv_t$]{}]{} (74.00,55.00)[(0,0)\[cc\][$dv_t$]{}]{} (80.00,51.00)[(0,0)\[cc\][$dv_x$]{}]{} (80.00,19.00)[(0,0)\[cc\][$dv_x$]{}]{} (74.00,15.00)[(0,0)\[cc\][$dv_z$]{}]{}
(36.50,8.00)[(-3,2)[6.00]{}]{} (9.50,26.00)[(3,-2)[6.00]{}]{} (9.50,44.00)[(3,2)[6.00]{}]{} (36.50,62.00)[(-3,-2)[6.00]{}]{} (63.50,62.00)[(3,-2)[6.00]{}]{} (90.50,44.00)[(-3,2)[6.00]{}]{} (90.50,26.00)[(-3,-2)[6.00]{}]{} (63.50,8.00)[(3,2)[6.00]{}]{} (50.00,5.00) (95.00,35.00) (50.00,65.00) (5.00,35.00)
the four edges to which these components belong, and wrote out these components near the edges (for example, we wrote $dv_y$ and $dv_t$ near the edge $BD$ meaning $(dv_{BD})_y$ and $(dv_{BD})_t$).
Of course, our $20\times 20$ minor again greatly factorizes. For example, the minor corresponding, on the one hand, to the components $(d\sigma_B)_{xy}$, $(d\sigma_B)_{xz}$ and $(d\sigma_B)_{xt}$, and on the other hand — to the three components of $d\vec v_{AB}$, factors out (this is because all the components of $d\sigma_A$ are already used up, whereas only they of all the remaining components of $d\sigma$ could influence $d\vec v_{AB}$). From formulas of type (\[sigma->v\]) we can see that this factor equals unity. Similarly, three more minors factor out which are obtained from the above minor by changing $B\to C$, $x\leftrightarrow y$, or by changing $B\to D$, $x\leftrightarrow z$, or by changing $B\to E$, $x\leftrightarrow t$.
After this, there remain the eight already mentioned components of vectors $d\vec v$ depicted in Figure \[IIIris2\], and the eight components of $d\sigma$ also depicted in Figure \[IIIris2\], with the understanding that if, e.g., $d\sigma_{yz}$ and $d\sigma_{yt}$ are drawn near the vertex $B$, then they are $(d\sigma_B)_{yz}$ and $(d\sigma_B)_{yt}$. The corresponding minor factorizes in eight separate factors. The reason for this is that each component of $d\vec v$ depends on [*only one*]{} component of $d\sigma$. These dependencies are shown by arrows in Figure \[IIIris2\]. Note also that the small letters near the vertices denote the axes where these vertices lie.
We see from formulas of type (\[sigma->v\]) or (\[4.1\]) that all the eight factors equal $\frac{1}{2}$. Consequently, their contribution to the torsion is $2^8$.
It remains for us to consider the mapping $(d\vec v_a)\to (dS_i)$. As we remember, all the components of vectors $d\vec v$ corresponding to edges $AB$, $AC$, $AD$ and $AE$ are already used up. For the remaining six edges, we have the components of $d\vec v$ along the bysectors of coordinate angles, as well as $(dv_{BC})_z$, $(dv_{BC})_t$, $(dv_{DE})_x$ and $(dv_{DE})_y$ (these latter belong to the two edges [*absent*]{} from Figure \[IIIris2\]).
Each of the six differentials $dS_i$, where face $i$ contains vertex $A$, is influenced by only one component of only one of the remaining vectors $d\vec v$. In this way, six factors appear: $\partial S_{ABC} / (\partial v_{BC})_{\rm bis} =\, \root\of{2}$ and five other ones, all equal to it. So, here we have the contribution to the torsion equal to $2^3$.
There remains the minor whose rows correspond to componentÁs $dS_{BCD}$, $dS_{BCE}$, $dS_{BDE}$ and $dS_{CDE}$, while columns — to $(dv_{BC})_z$, $(dv_{BC})_t$, $(dv_{DE})_x$ and $(dv_{DE})_y$. Here, too, each component $dS$ turns out to depend on only one component $dv$. We get the product of four partial derivatives, all equal to each other; for example, one of them is $$\frac{\partial S_{BCD}}{(\partial
v_{BC})_z} = l_{BC} \cdot \cos \gamma =\, \root\of{2}
\cdot \,
\root\of{\frac{2}{3}} = \frac{2}{\root\of{3}},$$ where $\gamma$ is the angle between face $BCD$ and axis $z$. Thus, here the contribution to the torsion is $2^4\cdot 3^{-2}$.
[**Conclusion:** ]{}sequence (\[pr\]) makes the multiplicative contribution to the torsion, equal to $2^8 \cdot 2^3 \cdot 2^4
\cdot 3^{-2} = 2^{15} \cdot 3^{-2}$.
The result: invariant for sphere $S^4$
--------------------------------------
Combining the conclusions made in the end of two previous subsections, we find that the torsion for sphere $S^4$ is $$\tau (S^4) = 2^{25}\cdot 3^{-2}.$$ Now we calculate the products entering in formula (\[i4\]).
In our complex, there are six two-dimensional faces of area $1/2$ and four faces of area $\,\root\of{3}/2$. Next, $\prod S = 3^2 / 2^{10}$. Then, there are two 4-simplices, both of volume $1/24$, thus $\prod V = 2^{-6}\cdot 3^{-2}$. Finally, there are four edges of length $1$ and six edges of length $\,\root\of{2}$, thus $\prod 72 \, l^5 = 2^{45}\cdot 3^{20}$.
All this together leads to the formula announced in the end of Section \[sec IIIp8\]: I(S\^4) = 1.
Discussion {#sec IIIobsuzhdenie}
==========
So, in the case of sphere $S^4$ the complex turned out to be acyclic, and we managed to calculate its torsion (and our invariant). The largest determinant that we had to deal with was of sizes $20\times 20$, but, luckily, it factorized in a product of smaller determinants.
Hopefully, new properties of our invariants will be discovered with time, which will simplify the calculations, and some relevant techniques will be elaborated. This will give the real possibility to calculate the invariants for a large enough manifold zoo. At this moment, one of the interesting questions is what we will get for the product of two-dimensional spheres $S^2\times S^2$ and whether we will be able to do something if the corresponding complex turns out [*not*]{} to be acyclic.
One more problem is the generalization of our complexes and finding their possible quantum analogues. One can begin with constructing a complex based on the $SL(2)$-solution to the pentagon equation from paper [@KM2].
It looks quite plausible that our constructions can be generalized in such way that they include also the [*Reidemeister torsion*]{}. In prospect, one can think about the creation of a new general theory that will combine the ideas of the algebra of acyclic complexes and quantum topology.
[**Acknowledgements.** ]{}The work has been performed with the partial financial support from Russian Foundation for Basic Research, Grant no. 01-01-00059.
[99]{}
. Euclidean 4-simplices and invariants of four-dimensional manifolds: I. Moves $3\to 3$. Theor. Math. Phys., Volume 131 (2002), 765–774.
. Euclidean 4-simplices and invariants of four-dimensional manifolds: II. An algebraic complex and moves $2\leftrightarrow 4$. Theor. Math. Phys., Volume 133 (2002), 1338–1347.
. Invariants of PL manifolds from metrized simplicial complexes. Three-dimensional case. J. Nonlin. Math. Phys., Volume 8, number 2 (2001), 196–210.
. Distinguishing three-dimensional lens spaces $L(7,1)$ and $L(7,2)$ by means of classical pentagon equation. J. Nonlin. Math. Phys., Volume 9, number 1 (2002), 86–98.
. A Classical Solution of the Pentagon Equation Related to the Group $SL(2)$. Theor. Math. Phys., Volume 129 (2001), 1320–1324.
|
---
abstract: |
We introduce a variant of the watchman route problem, which we call the *quickest pair-visibility* problem. Given two persons standing at points $s$ and $t$ in a simple polygon $P$ with no holes, we want to minimize the distance they travel in order to see each other in $P$. We solve two variants of this problem, one minimizing the longer distance the two persons travel (min-max) and one minimizing the total travel distance (min-sum), optimally in linear time. We also consider a query version of this problem for the min-max variant. We can preprocess a simple $n$-gon in linear time so that the minimum of the longer distance the two persons travel can be computed in $O(\log^2 n)$ time for any two query positions $s,t$ where the two persons start.\
**Keywords:** visibility polygon $\cdot$ shortest path $\cdot$ watchman problems
author:
- 'Hee-Kap Ahn[^1]'
- 'Eunjin Oh[^2]'
- 'Lena Schlipf[^3]'
- 'Fabian Stehn[^4]'
- 'Darren Strash[^5]'
bibliography:
- 'romeojulietpaper.bib'
title: 'On Romeo and Juliet Problems: Minimizing Distance-to-Sight[^6]'
---
Introduction
============
In the watchman route problem, a watchman takes a route to *guard* a given region—that is, any point in the region is visible from at least one point on the route. It is desirable to make the route as short as possible so that the entire area can be guarded as quickly as possible. The problem was first introduced in 1986 by Chin and Ntafos [@Chin1986] and has been extensively studied in computational geometry [@Carlsson1999; @Mitchell2013]. Though the problem is NP-hard for polygons with holes [@Chin1986; @Chin1988; @Dumitrescu2012], an optimal route can be computed in time $O(n^3\log n)$ for simple $n$-gons [@Dror2003] when the tour must pass through a specified point, and $O(n^4\log n)$ time otherwise.
In this paper, we study a variant of the watchman route problem. Imagine two persons, Romeo and Juliet, travel in a region from their starting locations. They want to minimize the distance they travel in order to see each other. More precisely, given the region and the locations where Romeo and Juliet start, the objective is to compute their paths, one for Romeo and one for Juliet, such that they see each other after traveling along the paths and their travel distances are minimized. This problem can be formally defined as follows.
Given two points $s$ and $t$ in a simple polygon $P$, compute the minimum distance that $s$ and $t$ must travel in order to see each other in $P$.
In the *min-max* variant of the quickest pair-visibility problem, we want to minimize the longer distance that the two points travel to see each other. In the *min-sum* variant, we want to minimize the total travel distance that the two points travel to see each other.
This problem may sound similar to the shortest path problem between $s$ and $t$, in which the objective is to compute the shortest path $\pi(s,t)$ for $s$ to *reach* $t$. However, they differ even for a simple case: for any two points lying in a convex polygon, the distance in the quickest pair-visibility problem is zero while in the shortest path problem, it is their geodesic distance $|\pi(s,t)|$. We would like to mention that our algorithm to be presented later uses the shortest path as a guide in computing the quickest pair-visibility paths.
The quickest pair-visibility problem occurs in optimization tasks. For example, mobile robots that use a line-of-sight communication model are required to move to mutually-visible positions to establish communication [@Ganguli2007]. An optimization task here is to find shortest paths for the robots to meet the visibility requirement for establishing communication among them.
Wynters and Mitchell [@Wynters1993] studied this problem for two agents acting in a polygonal domain in the presence of polygonal obstacles and gave an $O(nm)$-time algorithm for the min-sum variant (where $n$ is the number of vertices of the polygonal domain, and $m$ is the number of edges of the visibility graph of all corners) and an $O(n^3 \log{n})$-time algorithm for the min-max variant.
A query version of the quickest visibility problem has also been studied [@Arkin2016; @Khosravi2005; @Wang2017]. In the query problem, a polygon and a source point lying in the polygon are given, and the goal is to preprocess them and construct a data structure that supports, for a given query point, finding the shortest path taken from the source point to see the query point efficiently. Khosravi and Ghodsi [@Khosravi2005] considered the case for a simple $n$-gon and presented an algorithm to construct a data structure of $O(n^2)$ space so that, given a query, it finds the shortest visibility path in $O(\log n)$ time. Later, Arkin et al. [@Arkin2016] improved the result and presented an algorithm for the problem in a polygonal domain. Very recently, Wang [@Wang2017] presented an improved algorithm for this problem for the case that the number of the holes in the polygon is relatively small. Figure \[fig:problems\](a) illustrates differences in these problems for a simple polygon and two points, $s$ and $t$, in the polygon.
Our results
-----------
In this paper, we consider both min-max and min-sum variants of the quickest pair-visibility problem for a simple polygon. That is, either we want to minimize the maximum length of two traveled paths (min-max) or we want to minimize the sum of the lengths of two traveled paths (min-sum). We give a sweep-line-like approach that “rotates” the lines-of-sight along vertices on the shortest path between the start positions, allowing us to evaluate a linear number of candidate solutions on these lines. Throughout the sweep, we encounter solutions to both variants of the problem. We further show that our technique can be implemented in linear time.
![(a) The quickest pair-visibility problem finds two paths $\pi(s,s_1)$ and $\pi(t,t_1)$ such that $\overline{s_1t_1}\subset P$ and $\max\{|\pi(s,s_1)|,|\pi(t,t_1)|\}$ or $|\pi(s,s_1)|+|\pi(t,t_1)|$ is minimized. The quickest visibility problem for query point $t$ finds a shortest $\pi(s,t_2)$ with $\overline{tt_2}\subset P$. (b) **min-max:** Every pair $(s',t^*)$, where $t^*$ is some point within the geodesic disk centered in $t$ with radius $\pi(s,s')$, is an optimal solution to the min-max problem. (c) **min-sum:** An instance where $|\pi(s,s')|+|\pi(t,v_4)|=|\pi(s,v_4)|+|\pi(t,v_5)|$. Therefore, both $(s',v_4)$ and $(v_4,v_5)$ are optimal solutions to the min-sum problem.[]{data-label="fig:problems"}](figs/problems.pdf){width="90.00000%"}
We also consider a query version of this problem for the min-max variant. We can preprocess a simple $n$-gon in linear time so that the minimum of the longer distance the two query points travel can be computed in $O(\log^2 n)$ time for any two query points.
Preliminaries
=============
Let $P$ be a simple polygon and [$\partial P$]{}be its boundary where [$\partial P$]{}$\subset P$. The vertices of $P$ are given in counter-clockwise order along [$\partial P$]{}. We denote the shortest path within $P$ between two points $p,q\in P$ by $\pi(p,q)$ and its length by $|\pi(p,q)|$. Likewise, we denote the shortest path within $P$ between a point $p\in P$ and a line segment $\ell\subset P$ by $\pi(p,\ell)$. We say a point $p\in P$ is *visible* from another point $q\in P$ (and $q$ is visible from $p$) if and only if the line segment $\overline{pq}$ is contained in $P$.
For two starting points $s$ and $t$, our task is to compute a pair $(s',t')$ of points such that $s'$ and $t'$ are visible to each other, where we wish to minimize the lengths of $\pi(s,s')$ and $\pi(t,t')$. In the min-max setting, we wish to minimize $\max\{|\pi(s,s')|,|\pi(t,t')|\}$. For the min-sum setting, we wish to minimize $|\pi(s,s')| + |\pi(t,t')|$. Note that, for both variants, the optimum is not necessarily unique; see [Figure]{} \[fig:problems\](b) and (c).
We say a segment $g$ is *tangent* to a path $\pi$ at a vertex $v$ if $v\in g\cap \pi$ and $v$’s neighboring vertices on $\pi$ are in a closed half-plane bounded by the line containing $g$. Let $ \langle v_{0},v_{1},\ldots, v_{k-1},v_{k}\rangle$ be the sequence of vertices on $\pi(s,t)$ with $s=v_0$ and $t=v_k$.
\[lemma:contains-vertex\] Unless $s$ and $t$ are visible to each other, there is an optimal solution $(s^*,t^*)$ such that $\overline{s^*t^*}$ is tangent to the shortest path $\pi(s,t)$ at a vertex $v$ of $\pi(s,t)$.
We first show that there is a vertex of $P$ lying on $\overline{s^*t^*}$. Without loss of generality, assume that $\overline{s^*t^*}$ is horizontal with $s^*$ lying to the left of $t^*$. Let $\ell=\overline{xx'}$ be the maximal segment contained in $P$ that contains $\overline{s^*t^*}$ with $x$ closer to $s^*$ than to $t^*$. If $s=s^*$ (or $t=t^*$), then the lemma holds immediately because $s$ (or $t)$ is an endpoint of $\overline{s^*t^*}$. Assume to the contrary that $\overline{s^*t^*}$ contains no vertex of $P$. Then there are points $p\in P$ in a neighborhood of $s^*$ and $q\in P$ in a neighborhood of $t^*$ such that $p$ and $q$ are visible to each other, and $\max\{|\pi(s,p)|, |\pi(t,q)|\}<\max\{|\pi(s,s^*)|,|\pi(t,t^*)|\}$ and $|\pi(s,p)|+|\pi(t,q)|<|\pi(s,s^*)|+|\pi(t,t^*)|$. This contradicts the optimality of $(s^*,t^*)$.
We now show that $\overline{s^*t^*}$ contains a vertex of $\pi(s,t)$. Let $s'$ be the vertex on $\pi(s,s^*)$ preceding $s^*$ and let $t'$ be the vertex on $\pi(t,t^*)$ preceding $t^*$. Consider first the case that both $s'$ and $t'$ lie below the line through $\ell$. See Figure \[fig:ShortestPathVertex\](a). Then [$\partial P$]{}touches $\overline{s^*t^*}$ at a vertex $v$ locally from below. Otherwise, $(s^*, t^*)$ is not optimal by the same argument as in the previous paragraph. Then $s^*\in\overline{xv}$ and $t^*\in\overline{vx'}$. The path $\pi(s,t)$ must cross $\overline{xv}$ at a point $y$ and $\overline{vx'}$ at a point $y'$. Since $y$ and $y'$ are visible to each other, and $\pi(s,t)$ is a shortest path, $\pi(s,t)$ contains $\overline{yy'}$, which in turn contains $v$. Thus $v$ lies on $\pi(s,t)$ and $\overline{s^*t^*}$ is tangent to $\pi(s,t)$ at $v$.
Consider now the case that $s'$ and $t'$ lie on different sides of the line through $\ell$. Without loss of generality, assume that $s'$ lies below the line and $t'$ lies above the line. [Then $\overline{s^*t^*}$ intersects $\pi(s,t)$. We first show that $\overline{s^*t^*}$ contains an edge of $\pi(s,t)$. Assume to the contrary that $\overline{s^*t^*}$ intersects $\pi(s,t)$ only at a point, say $u$. Then there is another line segment $\ell'\subset P$ containing $u$ and intersecting both $\overline{s^*s'}$ and $\overline{t^*t'}$. See Figure \[fig:ShortestPathVertex\](b). This contradicts that $(s^*,t^*)$ is an optimal solution because, for $s''=\ell'\cap \overline{s^*s'}$ and $t''= \ell'\cap\overline{t^*t'}$, $d(s,s'')<d(s,s^*)$ and/or $d(t,t'')<d(t,t^*)$ and $s''$ and $t''$ are visible to each other. If $u$ coincides with $s^*$ or $t^*$, only one of the distance inequalities above holds, we hence consider lexicographic smallest (max,min) solutions in the min-max setting to establish the contradiction. Therefore, $\overline{s^*t^*}$ contains an edge of $\pi(s,t)$, say $\overline{vv'}$. Moreover, one of $v$ and $v'$ touches $\overline{s^*t^*}$ from above, and the other touches $\overline{s^*t^*}$ from below since $s'$ and $t'$ are on different sides of $\ell$. ]{} See Figure \[fig:ShortestPathVertex\](c). Thus, we can assume that ${\ensuremath{\partial P}\xspace}$ touches $\overline{s^*v}$ at a vertex $v'$ locally from below. Then $\pi(s,t)$ must cross $\overline{xv'}$ at a point $y$, and $\overline{vx'}$ at a point $y'$. Since $y$ and $y'$ are visible to each other, and $\pi(s,t)$ is a shortest path, $\pi(s,t)$ contains $\overline{yy'}$, which in turn contains both $v'$ and $v$. Thus both $v'$ and $v$ lie on $\pi(s,t)$ and $\overline{s^*t^*}$ is tangent to $\pi(s,t)$ at both $v'$ and $v$.
![Illustrating cases for the proof of Lemma \[lemma:contains-vertex\]. The examples show the shortest (geodesic) path $\pi(s,t)$ (blue dashed) and the line $\ell$ tangent to $v\in\pi(s,t)$ that, among all lines tangent to $v$ minimizes shortest paths from $s,t$ to $\ell$ (red). (a) Both $s'$ and $t'$ lie on the same side of $\ell$ through $s^*$ and $t^*$. (b) $s'$ and $t'$ lie on different sides of $\ell$. (c) The shortest path $\pi(s,t)$ passes through $v$ and $v'$.[]{data-label="fig:ShortestPathVertex"}](figs/SPV-new){width="80.00000%"}
Computing All Events for a Sweep-Line-Like Approach {#section:ComputingEvents}
===================================================
In the remaining part of the paper, we use $(s^*, t^*)$ to denote the optimal solution pair from $s$ and $t$ to a given line (and not necessarily a global optimal solution for the quickest pair-visibility problem). For each vertex $v$ on $\pi(s,t)$ we compute a finite collection of lines through $v$, each being a configuration at which the combinatorial structure of the shortest paths $\pi(s,s^*)$ and/or $\pi(t,t^*) $ changes. To be more precise, at these lines either the vertices of $\pi(s,s^*)$ or $\pi(t,t^*)$ (except for $s^*$ and $t^*$) change or an edge of [$\partial P$]{}changes that is intersected by the extension of $\overline{s^*t^*}$. To explain how to compute these lines, we introduce the concept of a *line-of-sight*.
\[line-of-sight\] We call a segment $\ell$ a *line-of-sight* if (i) $\ell$ is a maximal segment contained in $P$, and (ii) $\ell$ is tangent to $\pi(s,t)$ at a vertex $v\in \pi(s,t)$.
The algorithm we present is in many aspects similar to a sweep-line strategy, except that we do not sweep over the scene in a standard fashion but rotate a *line-of-sight* $\ell$ in $P$ around the vertices of the shortest path $\pi(s,t)$ in order from $s$ while making use of Lemma \[lemma:contains-vertex\]. Recall that $ \langle v_{0},v_{1},\ldots, v_{k-1},v_{k}\rangle$ is the sequence of vertices on $\pi(s,t)$ with $s=v_0$ and $t=v_k$. The process will be initialized with a line-of-sight that contains $s$ and $v_1$ and is then rotated around $v_1$ (while remaining tangent to $v_1$) until it hits $v_2$, see [Figure]{} \[fig:BoundaryEvents\](a). In general, the line-of-sight is rotated around $v_{i}$ in a way so that it remains tangent to $\pi(s,t)$ at $v_{i}$ (it is rotated in the interior of $P$) until the line-of-sight contains $v_{i}$ and $v_{i+1}$, then the process is iterated with $v_{i+1}$ as the new rotation center. The process terminates as soon as the line-of-sight contains $v_{k-1}$ and $v_k=t$.
While performing these rotations around the shortest path vertices, we encounter all lines-of-sight. As for a standard sweep-line approach, we will compute and consider events at which the structure of a solution changes: this is either because the interior vertices of $\pi(s,s^*)$ or $\pi(t,t^*)$ change or because the line-of-sight starts or ends at a different edge of [$\partial P$]{}. These events will be represented by points on [$\partial P$]{} [ (actually, we consider the events as vertices on [$\partial P$]{}unless they are already vertices).]{} Between two consecutive events, we compute the local minima of the relevant distances for the variant at hand in constant time and hence encounter all local minima eventually.
There are three event-types to distinguish:
**Path-Events** are endpoints of lines-of-sight that contain two consecutive vertices of the shortest path $\pi(s,t)$. See [Figure]{} \[fig:BoundaryEvents\](a).
**Boundary-Events** are endpoints of lines-of-sight that are tangent at a vertex of $\pi(s,t)$ and contain at least one vertex of $P\setminus \pi(s,t)$ (potentially as an endpoint). See [Figure]{} \[fig:BoundaryEvents\](b).
**Bend-Events** are endpoints of lines-of-sight where the shortest path from $s$ (or $t$) to the line-of-sight gains or loses a vertex while rotating the line-of-sight around a vertex $v$. See [Figure]{} \[fig:BoundaryEvents\](c). Note that bend-events can coincide with path- or boundary-events.
![Path-, boundary-, and bend-events. (a) The endpoints of the line-of-sight through $\overline{sv_1}$ make up the first path-event. The line-of-sight rotates until it hits the next path-event: the endpoints of the line-of-sight through $\overline{v_1v_2}$. (b) Boundary-events that are not path-events. (c) A bend-event (marked with a cross) occurs between the two boundary-events. The shortest path from $s$ to the line-of-sight changes at the bend-event.[]{data-label="fig:BoundaryEvents"}](figs/LofS-new){width="80.00000%"}
We will need to explicitly know both endpoints of the line-of-sight on [$\partial P$]{}at each event and the corresponding vertex of $\pi(s,t)$ on which we rotate.
\[lemma:InitQueue\] For a simple polygon $P$ with $n$ vertices and points $s,t\in P$, the queue $\mathcal{Q}$ of all path- and boundary-events of the rotational sweep process, ordered according to the sequence in which the sweeping line-of-sight encounters them, can be initialized in $O(n)$ time.
Consider some line-of-sight $\ell$ that is tangent to a vertex $v_{i}\in \pi(s,t)$ for some $0<i<k$. Then $\ell$ subdivides $P$ into a number of subpolygons. Consider $\ell$ as the union of two (sub)segments $\ell^+$ and $\ell^-$ of $\ell$ induced by $v_i$ such that $\ell^+\cap \ell^- = \{v_i\}$ and $\ell^-$ is incident to the subpolygon of $P$ induced by $\ell$ containing $s$.
We will discuss the computation of all boundary- and path-events swept by $\ell^+$. The other events swept by $\ell^-$ can be computed in a second round by changing the roles of $s$ and $t$. We do not maintain a queue for the events explicitly; instead we will introduce new vertices on [$\partial P$]{}or label existing vertices of [$\partial P$]{}as events. Later the events will be considered by following two pointers to vertices on [$\partial P$]{}and hence by processing the vertices in the order of their occurrence on [$\partial P$]{}.
We start with computing all path-events swept by $\ell^+$. For this we compute the *shortest path map* $M_s$ of $s$ in $P$. The shortest path map of $s$ is a decomposition of $P$ in $O(n)$ triangular cells such that the shortest path from $s$ to any point within a cell is combinatorially the same. It can be obtained by extending every edge of the shortest path tree of $s$ towards its descendants until it reaches [$\partial P$]{}in linear time [@Guibasetal1987]. A path-event occurs when a line-of-sight contains two consecutive vertices of $\pi(s,t)$. Note that for each path-event, $\ell^+$ appears as an edge of $M_s$ and its endpoints appear as vertices of $M_s$ (see also Figure \[fig:SPM\](a)). For each index $i$ with $0< i< k$, we find the edge incident to $v_i$ and parallel to $\overline{v_{i-1}v_{i}}$ by considering every edge of $M_s$ incident to $v_i$. This takes $O(n)$ time in total since there are $O(n)$ edges of $M_s$ and we consider every edge at most once. Note that the path-event induced by $v_{k-1}$ and $t$ is an exception, but it can also be computed in $O(1)$ time during the process [by considering the triangle of $M_s$ that contains $t$.]{}
![(a) The shortest path map $M_s$. All path-events, swept by $\ell^+$, appear as endpoints of edges of $M_s$, except the one induced by $v_{k-1}$ and $t$; these events are marked with small disks. (b) The shortest path tree $T_s$. The boundary-events, swept by $\ell^+$, tangent to $\pi(s,t)$ at $v_1$ are marked. Clearly, the parent vertex of these vertices in $T_s$ is $v_1$.[]{data-label="fig:SPM"}](figs/ShortestPathMap){width="50.00000%"}
For computing the boundary-events, we use the following properties. While rotating around $v_{i}$ from the position where $\ell$ contains $v_{i-1}$ to the position in which $\ell$ contains $v_{i+1}$, let $A_i^+$ ($A_i^-$) be the region of $P$ that is swept over by $\ell^+$ ($\ell^-$). (See [Figure]{} \[fig:queue\_init\].) Observe that
all $A_i^+$ for $0<i<k$ are pairwise disjoint in their interior,
all $A_i^-$ for $0<i<k$ are pairwise disjoint in their interior,
for all $0<i<k$ and all points $p\in A_i^+$ the shortest path $\pi(s,p)$ contains $v_{i}$ (i.e., $v_i$ is the predecessor of $p$ on $\pi(s,p)$),
for all $0<i<k$ and all points $p\in A_i^-$ the shortest path $\pi(p,t)$ contains $v_{i}$ (i.e., $v_i$ is the successor of $p$ on $\pi(p,t)$).
To compute all boundary-events that are vertices of $P$ swept by $\ell^+$, we will make use of the shortest path tree $T_s$ for $s$ in $P$. A boundary-event $x$ is defined by a vertex $v_{i}\in\pi(s,t)$ such that the line-of-sight that contains $x$ (potentially as one endpoint) is tangent to $\pi(s,t)$ in $v_{i}$. It follows from [Property **P3**]{}, that $\overline{v_{i}x}$ is an edge of $T_s$ (and by that it cannot be obstructed by edges of $P$) and $x\notin \pi(s,t)$. So the vertices of $P$ whose parent vertex in $T_s$ is a vertex of $\pi(s,t)$ are possible boundary-events. In order to compute all boundary-events we consider all consecutive path-events and compute all corresponding boundary-events by following [$\partial P$]{}and checking the vertices within the candidate set (see Figure \[fig:SPM\](b)). We compute the boundary-events that are vertices of $P$ swept by $\ell^-$ in a similar way.
So far we have labeled all vertices $x$ on [$\partial P$]{}that are boundary-events. We still need to compute the other endpoint $\tilde{x}$ of the line-of-sight $\overline{x\tilde{x}}$ that is tangent in $v_i$. Let $\overline{x_i\tilde{x}_i}$ be the line-of-sight at the path-event $x_i$ so that $\tilde{x}_i, v_{i-1}, v_i, x_i \in \ell$. See [Figure]{} \[fig:queue\_init\]. While rotating $\ell$ around $v_i$, $\ell^+$ sweeps over $A_i^+$ until the next path-event is met. Let $E_i^+$ be the sequence of the path- and boundary-events in $A^+_i$ we obtained so far sorted in counter-clockwise order along [$\partial P$]{}. The order of events in $E^+_i$ is the same as the order in which $\ell^+$ sweeps over them. Our goal is to compute $\tilde{x}$ for every event in $E^+_i$ in order. To do this, we consider the (triangular) cells of the shortest path map $M_t$ of $t$ incident to $v_i$ one by one in counter-clockwise order around $v_i$ starting from the cell incident to $\tilde{x}_i$. Since every point in such cells is visible from $v_i$, we can determine if $\tilde{x}$ is contained in a cell in constant time for any event $x\in E^+_i$. Therefore, we can compute $\tilde{x}$ for every event $x$ in $E^+_i$ in time linear in the number of the cells of $M_t$ incident to $v_i$ and the number of events of $E^+_i$, giving us all path- and boundary-events in $O(n)$ total time.
![Let $E^+_i = \langle x_{i,1},\ldots,x_{i,j}\rangle$ for an index $1\leq j\leq n$. We start at $\tilde{x}_i$ and follow the (triangular) cells of $M_t$ incident to $v_i$ in counter-clockwise order around $v_i$ until we find $\tilde{x}_{i,1}$. Then we continue to follow such cells until we find $\tilde{x}_{i,2}$, and so on.[]{data-label="fig:queue_init"}](figs/queue_init_rotated)
Once we initialized the event queue $\mathcal{Q}$, we can now compute and process bend-events as we proceed in our line-of-sight rotations.
\[lemma:BendEvents\] All bend-events can be computed in $O(n)$ time, sorted in the order as they appear on the boundary of $P$.
Bend-events occur between consecutive path- and boundary-events; they can also coincide with these events. We assume that all path- and boundary-events are already computed. Additionally, we assume that all vertices of the boundary- and path-events (the endpoints of the corresponding lines-of-sight) are inserted on [$\partial P$]{}. Recall that, for each event, we know both endpoints of the line-of-sight $\ell$ on [$\partial P$]{}and the corresponding vertex of $\pi(s,t)$ on which we rotate. The path- and boundary-events define the area which is swept over by $\ell$. Thus, we know which positions for $\ell$ we have to consider in order to compute all bend-events.
As in the proof of Lemma \[lemma:InitQueue\], we consider the line-of-sight $\ell$ tangent to a vertex $v\in \pi(s,t)$ as the union of two (sub)segments $\ell^+$ and $\ell^-$ of $\ell$ induced by $v$ such that $\ell^+\cap \ell^- = \{v\}$ and $\ell^-$ is incident to the subpolygon of $P$ induced by $\ell$ containing $s$. We discuss the computation of all bend-events that are encountered by $\ell^-$. The bend-events that are swept over by $\ell^+$ can be computed in a second round by changing the roles of $s$ and $t$.
We start with the path-event defined by $s$ and $v_1$, and consider all events in the order they appear. Let $\ell$ be the line-of-sight rotating around a vertex $v$ and denote by $x$ the endpoint of $\ell^-$ other than $v$. To find the bend-events efficiently, we compute and maintain the shortest path $\pi(s,\ell)=\pi(s,\ell^-)$ over the events.
While $\ell$ rotates around $v$, the combinatorial structure of $\pi(s,\ell)$ may change. Specifically, let $e_\ell=\overline{uw}$ denote the edge of $\pi(s,\ell)$ incident to $\ell$ with $w$ on $\ell$. Note that during the rotation of $\ell$, all the edges of $\pi(s,\ell)$ are stationary, except that $e_\ell$ rotates around $u$. Therefore, a change in the combinatorial structure of $\pi(s,\ell)$ occurs only when $e_\ell$ hits a vertex $u'$ of $P$ (if $u'$ at this event is an endpoint of $e_\ell$, then this bend-event coincides with a previously computed boundary-event) and splits into two edges sharing $u'$ (an event of [type **T1**]{}) or the two edges of $\pi(s,\ell)$ incident to $u$ become parallel (an event of [type **T2**]{}). (Then they merge into one and $u$ disappears from $\pi(s,\ell)$.) See Figure \[fig:BendEvents\]. From any event of the two event types above, $e_\ell, u$, and $\pi(s,\ell)$ are updated accordingly. Additionally, $x$ is updated and its new position is inserted as a vertex on [$\partial P$]{}as it represents a bend-event.
An event of [type **T1**]{} occurs only when (i) $x$ reaches a vertex $u'$, or (ii) $e_\ell$ hits a vertex $u'$ of $\pi(s,t)$ in its interior. Moreover, for case (ii), $u$ and $u'$ are consecutive in $\pi(s,t)$.
Imagine $\ell$ is rotated around $v$ infinitesimally further from the current event. Then either $e_\ell$ is orthogonal to $\ell$ or not. If $e_\ell$ is not orthogonal to $\ell$, the closest point in $\ell$ from $s$ is $x$. Thus, the only way that $e_\ell$ hits a vertex of $P$ is that $x$ reaches $u'$. See Figure \[fig:BendEvents\](a).
Now consider the case that $e_\ell$ is orthogonal to $\ell$. Notice that the shortest path from a vertex $v$ to a segment within a simple polygon lies inside a *funnel*, a region bounded by the shortest paths from $v$ to both endpoints of the segment and the segment. For more details see [@Guibasetal1987]. Thus, $u'$ is contained in $\pi(u,v)$. See Figure \[fig:BendEvents\](b). Since $\pi(u,v)$ is a subpath of $\pi(s,t)$, $u'$ is a vertex of $\pi(s,t)$, and thus $u$ is the vertex of $\pi(s,t)$ previous to $u'$ from $s$.
![(a) A bend-event of [type **T1**]{} occurs when $x=u_\ell$ reaches $u'$. (b) A bend-event of [type **T1**]{} occurs when $e_\ell=\overline{uw}$ hits a vertex $u'$ of $\pi(s,t)$. (c) A bend-event of [type **T2**]{} occurs when two edges incident to $u$ are parallel.[]{data-label="fig:BendEvents"}](figs/Lemma3){width="90.00000%"}
\[lemma:vertices-shortestpath\] Once a vertex disappears from $\pi(s,\ell)$, it never appears again on the shortest path during the rotation of the line-of-sight $\ell$.
Assume to the contrary that there is a vertex $u$ that disappears from $\pi(s,\ell_1)$, but then appears again on $\pi(s,\ell_2)$ for two distinct lines-of-sight $\ell_1$ and $\ell_2$ during the rotation. [First note that if $u$ is an endpoint of $\pi(s,\ell_1)$ (or $\pi(s,\ell_2)$), it is a boundary- and bend-event, and would only appear once when rotating the line-of-sight. Therefore, both $\pi(s,\ell_1)$ and $\pi(s,\ell_2)$ must contain $u$ in their interiors, and both of them also contain $\pi(s,u)$ in their interiors.]{} Since $u$ disappears from $\pi(s,\ell_1)$, the edge of $\pi(s,\ell_1)$ incident to $u$ (on $\pi(u,\ell_1)$) is orthogonal to $\ell_1$. We claim that $u$ appears on $\pi(s,\ell_2)$ due to case (ii) of [type **T1**]{}, that is, the edge of $\pi(s,\ell_2)$ incident to $\ell_2$ hits $u$. Assume to the contrary that $u$ appears on $\pi(s,\ell_2)$ due to case (i) of [type **T1**]{}. However, $u$ (and its event vertex on [$\partial P$]{}) is already swept by a line-of-sight before we consider $\ell_2$ because it appears on $\pi(s,\ell_1)$. [By [Property **P2**]{}, $\ell^-$ sweeps a vertex only once.]{} Thus, $u$ appears on $\pi(s,\ell_2)$ due to case (ii) of [type **T1**]{}, and the edge of $\pi(s,\ell_2)$ incident to $u$ is orthogonal to $\ell_2$. This means that $\ell_1$ and $\ell_2$ are parallel.
Since $\ell_1$ and $\ell_2$ are parallel, they are tangent to $\pi(s,t)$ at two distinct vertices, say $u_1$ and $u_2$, respectively. [Without loss of generality, assume that $u_1$ is closer to $s$ than $u_2$.]{} We show that $\pi(p_1,p_2)$ contains $u_1$ for any two points $p_1 \in P_1$ and $p_2\in \ell_2$, where $P_1$ is the subpolygon bounded by $\ell^-_1$ containing $s$. [Since both $u_1$ and $u_2$ are vertices of $\pi(s,t)$, $\pi(s,u_2)$ contains $u_1$. Let $p$ be the point on $\ell^-_2$ farthest from $u_2$ such that $\pi(s,p)$ contains $u_1$. Since the boundary of $P$ intersect neither $\overline{u_1p}$ nor $\overline{u_2p}$, $\pi(u_1,u_2)$ is contained in the triangle with corners $u_1, u_2, p$. No line segment parallel to $\ell_2$ is tangent to $\pi(s,t)$ at $u_1$, which is a contradiction. Therefore, $\pi(s,p_2)$ contains $u_1$ for any point $p_2\in\ell_2$. Then since $\ell_1$ is tangent to $\pi(s,t)$, $\pi(p_1,p_2)$ contains $u_1$ for any two points $p_1 \in P_1$ and $p_2\in \ell_2$.]{} Thus, $\pi(s,\ell_2)$ contains $\pi(s,u_1)$, and no vertex in $P_1$ other than the vertices of $\pi(s,u_1)$ appears on $\pi(s,\ell_2)$. Since $u$ is contained in $P_1$, it cannot appear on $\pi(s,\ell_2)$, which is a contradiction.
Using the two lemmas, we can compute all bend-events as follows. For a line-of-sight $\ell$ rotating around a vertex $v$, we have three candidates for the next bend-event. Let $e$ be the edge of $P$ containing the endpoint of $\ell^-$ other than $v$, and let $u'$ be the neighboring vertex of $u$ in $\pi(u,t)$. The next bend-event is (1) the endpoint of $e$ not contained in $\pi(s,\ell)$ if it exists, (2) the intersection point between $e$ and the line through $v$ and orthogonal to $uu'$ if it exists, or (3) the intersection point between $e$ and the line through $v$ and orthogonal to $u''$ if it exists, where $u''$ is the neighboring vertex of $u$ in $\pi(s,\ell)$ closer to $s$. Note that the first two cases are [type **T1**]{} events and the last case is a [type **T2**]{} event. We can compute all of the three events in constant time. Also, we can update $u, e_\ell,x$ and $\pi(s,\ell)$ accordingly in constant time. Therefore, the time for computing all bend-events is linear in the amount of the combinatorial change on $\pi(s,\ell)$. By Lemma \[lemma:vertices-shortestpath\], the the amount of the combinatorial change is $O(n)$, and therefore, we can compute all bend-events in $O(n)$ time.
Algorithm Based on a Sweep-Line-Like Approach {#section:Algorithm}
=============================================
In this section, we present a linear-time algorithm for computing the minimum distance that two points $s$ and $t$ in a simple polygon $P$ travel in order to see each order. We compute all events defined in Section \[section:ComputingEvents\] in linear time. The remaining task is to handle the lines-of-sight lying between two consecutive events.
\[lemma:greenstarformula\] For any two consecutive events, the line-of-sight $\ell$ lying between them that minimizes the sum or the maximum of the distances from $s$ and $t$ to $\ell$ can be found in constant time.
Let $\mathcal{L}$ be the set of all lines-of-sight lying between the two consecutive events. We assume that $\mathcal{L}$ contains no vertical line-of-sight. Otherwise, we consider the subset containing all lines-of-sight with positive slopes, and then the subset containing all lines-of-sight with negative slopes.
These lines-of-sight share a common vertex $v$ of $\pi(s,t)$. We will give an algebraic function for $|\pi(s,\ell)|$ for $\ell\in\mathcal{L}$. An algebraic function for $|\pi(t,\ell)|$ can be obtained analogously. Observe that $\pi(s,u)$ is the same for all $\ell\in\mathcal{L}$, where $u$ is the second to the last vertex $u$ of $\pi(s,\ell)$ from $s$. Thus, we consider only the length of $\pi(u,\ell)$, which is a line segment. The length is either the Euclidean distance between $u$ and the line containing $\ell$, or the Euclidean distance between $u$ and the endpoint of $\ell$ closest to $u$. We show how to handle the first case only because the second case can be handled analogously.
Let $\ell(\alpha)$ denote the line of slope $\alpha$ passing through $v$ for $\alpha>0$, which is represented as $y= \alpha x+ f(\alpha)$, where $f(\alpha)$ is a function linear in $\alpha$. Then the distance between $u$ and $\ell(\alpha)$ can be represented as ${|c_1\alpha + c_2|}/{\sqrt{\alpha^2 + 1}}$, where $c_1$ and $c_2$ are constants depending only on $v$ and $u$. Thus, our problem reduces to finding a minimum of the function of the form ${(|c_1\alpha + c_2|+|c_1'\alpha + c_2'|)}/{\sqrt{\alpha^2 + 1}}$ and $\max(|c_1\alpha + c_2|,|c_1'\alpha + c_2'|)/{\sqrt{\alpha^2 + 1}}$, respectively, for four constants $c_1, c_2, c_1'$ and $c_2'$, and for all $\alpha$ such that $\ell(\alpha)$ contains a line-of-sight in $\mathcal{L}$. We can find a minimum in constant time using elementary analysis.
\[theorem:main\_result\] Given a simple $n$-gon $P$ with no holes and two points $s,t\in P$, a point-pair $(s^*,t^*)$ such that (i) $\overline{s^*t^*}\subset P$ and (ii) either $|\pi(s,s^*)|+|\pi(t,t^*)|$ or $max\{|\pi(s,s^*)|, |\pi(t,t^*)|\}$ is minimized can be computed in $O(n)$ time.
Our algorithm first computes all path- and boundary-events as described in Lemma \[lemma:InitQueue\]. The number of events introduced during this phase is bounded by the number of vertices of the shortest path maps, $M_s$ and $M_t$, respectively, which are $O(n)$. In the next step, it computes the bend-events on [$\partial P$]{}as described in Lemma \[lemma:BendEvents\], which can be done in $O(n)$ time. Finally, our algorithm traverses the sequence of events. Between any two consecutive events, it computes the respective local optimum in constant time by Lemma \[lemma:greenstarformula\]. It maintains the smallest one among the local optima computed so far, and returns it once all events are processed. Therefore the running time of the algorithm is $O(n)$.
For the correctness, consider the combinatorial structure of a solution and how it changes. The path-events ensure that all vertices of $\pi(s,t)$ are considered as being the vertex lying on the segment connecting the solution $(s^*,t^*)$ (Lemma \[lemma:contains-vertex\]). While the line-of-sight rotates around one fixed vertex of $\pi(s,t)$, either the endpoints of line-of-sight sweep over or become tangent to a vertex of [$\partial P$]{}. These are exactly the boundary-events. Or the combinatorial structure of $\pi(s,s^*)$ or $\pi(t,t^*)$ changes as interior vertices of $\pi(s,s^*)$ or $\pi(t,t^*)$ appear or disappear. These happen exactly at bend-events. Therefore, our algorithm returns an optimal point-pair.
\[corollary:modificationsToObjective\] By the same algorithm, one can also compute optimal pairs $(s^*,t^*)$ that minimize
$\max(\lambda|\pi(s, s^*)|,(1-\lambda)|\pi(t, t^*)|)$ for some $0\leq \lambda\leq 1$,
$\max(\alpha+|\pi(s, s^*)|,\beta+|\pi(t, t^*)|)$ for some $\alpha, \beta \in \mathbb{R}^+$.
The first modification introduced in Corollary \[corollary:modificationsToObjective\] models that Romeo and Juliet travel with different speeds. It is easy to see, that this formulation is equivalent to minimizing the objective $\max(\alpha|\pi(s, s^*)|,\beta|\pi(t, t^*)|)$ for some $\alpha, \beta \in \mathbb{R}^+$. The second variant can be motivated as follows: Imagine Romeo (and Juliet) is driving a car that before departing from $s$ (and $t$) already drove a distance of $\alpha$ (and $\beta$). The objective $\max(\alpha+|\pi(s, s^*)|,\beta+|\pi(t, t^*)|)$ minimizes the largest distance any of the two cars had to drive in order to establish a line-of-sight.
Quickest Pair-Visibility Query Problem
======================================
In this section, we consider a query version of the min-max variant of the quickest pair-visibility problem: Preprocess a simple $n$-gon $P$ so that the minimum traveling distance for two query points $s$ and $t$ to see each other can be computed efficiently. We can preprocess a simple $n$-gon in linear time and answer a query in $O(\log^2n)$ time by combining the approach in Section \[section:Algorithm\] with the data structure given by Guibas and Hershberger [@Guibas1989; @Hersh-shortest-1991]. For any two query points $s$ and $t$ in $P$, the query algorithm for their data structure returns $\pi(s,t)$, represented as a binary tree of height $O(\log n)$, in $O(\log n)$ time [@Hersh-shortest-1991]. Thus, we can apply a binary search on the vertices (or the edges) on $\pi(s,t)$ efficiently.
Imagine that we rotate a line-of-sight along the vertices of $\pi(s,t)$ for two query points $s$ and $t$ in $P$. Lemma \[lemma:contains-vertex\] implies that there is a line-of-sight containing $s^*$ and $t^*$, where $(s^*,t^*)$ is an optimal solution. We call it an *optimal line-of-sight*. We define the order of any two lines-of-sight as the order in which they appear during this rotational sweep process. By the following lemma, we can apply a binary search on the sequence of events along [$\partial P$]{}and find two consecutive events such that the respective local optimum achieved between them is a global optimal solution.
The geodesic distance between $s$ (and $t$) and the rotating line-of-sight increases (and decreases) monotonically as the line-of-sight rotates along the vertices of $\pi(s,t)$ from $s$.
Let $\ell$ be a line-of-sight that is tangent to $\pi(s,t)$ at a vertex $v$. Consider the subdivision of $P$ induced by $\ell$ and let $P_s$ be the subpolygon that contains $s$. Let $\ell'$ be a line-of-sight that comes after $\ell$ during the rotational sweep process. We claim that $\ell'$ does not intersect the interior of $P_s$. If $\ell'$ is tangent to $\pi(s,t)$ at $v$, it never intersects the interior of $P_s$ as shown in the proof of Lemma \[lemma:InitQueue\]. Assume that $\ell'$ is tangent to $\pi(s,t)$ at a vertex $u$ that comes after $v$ along $\pi(s,t)$ from $s$, but intersects the interior of $P_s$. Without loss of generality, assume that $\ell$ is horizontal and $P_s$ lies locally below $\ell$. Then $u$ must lie strictly above the line containing $\ell$. However, since both $v$ and $u$ are vertices of $\pi(s,t)$ and $\ell$ is tangent to $\pi(s,t)$ at $v$, there must be another vertex $u'$ of $\pi(s,t)$ that lies on or below the line containing $\ell$ and appears between $v$ and $u$ along $\pi(s,t)$. See Figure \[fig:Lemma10\].
![Let $\ell$ be a be a line-of-sight which is tangent to $\pi(s,t)$ at a vertex $v$. And let $\ell'$ be be a line-of-sight that comes after $\ell$ during the rotational sweep process. Clearly, $|\pi(s,\ell')|\geq|\pi(s,\ell)|$.[]{data-label="fig:Lemma10"}](figs/Lemma10){width="25.00000%"}
Thus, $u$ is not visible from any point on $\ell$, and $\ell'$ does not intersect the interior of $P_s$. Since $\pi(s,\ell')$ intersects $\ell$, we have $|\pi(s,\ell')|\geq|\pi(s,\ell)|$. The claim for $t$ and the rotating line-of-sight can be shown analogously.
Binary Search for the Path-Events
---------------------------------
We first consider the path-events, and find two consecutive path-events containing an optimal line-of-sight between them. Let $ \langle v_{0},v_{1},\ldots, v_{k-1},v_{k}\rangle$ be the sequence of vertices on $\pi(s,t)$ with $s=v_0$ and $t=v_k$. Due to the shortest-path data structure by Guibas and Hershberger, we can obtain $\pi(s,t)$ represented as a binary tree of height $O(\log n)$ in $O(\log n)$ time. Consider an edge $\overline{v_iv_{i+1}}$ of $\pi(s,t)$. We can determine whether or not an optimal line-of-sight is tangent to $\pi(s,t)$ at a vertex lying after $v_i$ along $\pi(s,t)$ in $O(\log n)$ time. To do this, we compute the line-of-sight $\ell$ containing $\overline{v_iv_{i+1}}$ in $O(\log n)$ time. We use the data structure for ray shooting given by Hershberger and Suri [@Hershberger1995] with linear preprocessing and logarithmic query time. Then, we compute the length of $\pi(s,\ell)$ and $\pi(t,\ell)$ in $O(\log n)$ time [using the data structure given by Guibas and Hershberger for computing the distance between a query point and a query line segment in $O(\log n)$ time [@Guibas1989].]{} An optimal line-of-sight is tangent to $\pi(s,t)$ at a vertex lying after $v_i$ if and only if $\pi(s,\ell)$ is shorter than $\pi(t,\ell)$. Therefore, we can compute the two consecutive path-events with an optimal solution lying between them in $O(\log^2 n)$ time.
Binary Search for the Boundary-Events
-------------------------------------
Now we have the vertex $v_i$ of $\pi(s,t)$ contained in an optimal line-of-sight. We find two consecutive boundary-events defined by lines-of-sight tangent to $\pi(s,t)$ at $v_i$ such that an optimal line-of-sight lies between them. Let $\tilde{x}_i$ and $x_i$ be the first points of [$\partial P$]{}hit by the rays from any point in $\overline{v_{i-1}v_{i}}$ towards $v_{i-1}$ and $v_i$, respectively. See Figure \[fig:queue\_init\]. Similarly, let $\tilde{x}_{i+1}$ and $x_{i+1}$ be the first points of [$\partial P$]{}hit by the rays from any point in $\overline{v_{i}v_{i+1}}$ towards $v_{i}$ and $v_{i+1}$, respectively. These four points of [$\partial P$]{}can be found in $O(\log n)$ time by the ray-shooting data structure [@Hershberger1995]. Without loss of generality, we assume that a line-of-sight rotates around $v_i$ in the counter-clockwise direction in the rotational sweep process. Let $\tilde{\gamma}$ be the part of [$\partial P$]{}lying between $\tilde{x}_i$ and $\tilde{x}_{i+1}$ in counter-clockwise order, and $\gamma$ be the part of [$\partial P$]{}lying between $x_i$ and $x_{i+1}$ in counter-clockwise order. An optimal line-of-sight $\ell^*$ has one endpoint on $\tilde{\gamma}$ and the other endpoint on $\gamma$.
We first find the edge of $\tilde{\gamma}$ (resp. $\gamma$) containing an endpoint of $\ell^*$ by applying a binary search on the vertices of $\tilde{\gamma}$ (resp. $\gamma$). This gives two consecutive boundary-events such that $\ell^*$ lies between them. We now show how to find the edge of $\gamma$ containing an endpoint of $\ell^*$. The edge on $\tilde{\gamma}$ can be found analogously.
We perform a binary search on the vertices in $\gamma$ as follows. Let $x^*$ be the endpoint of $\ell^*$ contained in $\gamma$. For any vertex $u$ of $\gamma$, we can determine which part of $\gamma$ with respect to $u$ contains $x^*$ in $O(\log n)$ time. To do this, we consider the line-of-sight $\ell$ containing the edge of $\pi(v_i,u)$ incident to $v_i$. Observe that $\ell$ intersects $\pi(v_i,u)$ only in the edge including its endpoints as $\pi(v_i,u)$ is a shortest path. See Figure \[fig:query-bend\](a). Since we can obtain the edge of $\pi(v_i,u)$ incident to $v_i$ in $O(\log n)$ time using the shortest-path data structure, we can obtain $\ell$ in the same time. Here, to obtain the endpoint of $\ell$ on $\gamma$, we use the ray-shooting data structure that supports $O(\log n)$ query time [@Hershberger1995]. Then we compare $|\pi(s,\ell)|$ and $|\pi(t,\ell)|$ in $O(\log n)$ time. The point $x^*$ comes after $u$ from $x_i$ if and only if $|\pi(s,\ell)|<|\pi(t,\ell)|$. Therefore, we can determine which part of $\gamma$ with respect to $u$ contains $x^*$ in $O(\log n)$ time, and thus the binary search is completed in $O(\log^2 n)$ time. In this way, we can compute two consecutive boundary-events such that an optimal line-of-sight lies between them in $O(\log^2 n)$ time.
Binary Search for the Bend-Events
---------------------------------
Now we have two consecutive events in the sequence of all path- and boundary-events that contain an optimal line-of-sight $\ell^*$ between them. Let $\ell_1$ and $\ell_2$ be two lines-of-sight corresponding to the two consecutive events such that $\ell_2$ comes after $\ell_1$. The remaining task is to handle the bend-events lying between them. For the bend-events, we perform a binary search on the edges of $\pi(s,\ell_1)\cup\pi(s,\ell_2)$ in $O(\log^2 n)$ time. Then we perform a binary search on the edges of $\pi(t,\ell_1)\cup\pi(t,\ell_2)$ in $O(\log^2 n)$ time. In the following, we describe the binary search on $\pi(s,\ell_1)\cup\pi(s,\ell_2)$. The other one can be done analogously.
We find the point $s'$ such that $\pi(s,s')$ is the maximal common subpath of $\pi(s,\ell_1)$ and $\pi(s,\ell_2)$ from $s$ in $O(\log n)$ time using the shortest-path data structure [@Hersh-shortest-1991]. See Figure \[fig:query-bend\](b). Then we obtain $\pi'=\pi(s',\ell_1)\cup\pi(s',\ell_2)$ represented as a binary tree of height $O(\log n)$ in $O(\log n)$ time. Notice that $\pi'$ is a path from $\ell_1$ to $\ell_2$, concatenating the two shortest paths from $\ell_1$ to $s'$ and from $s'$ to $\ell_2$.
For an edge $e$ of $\pi'$, we use $\ell(e)$ to denote the line-of-sight containing $v_i$ and orthogonal to the line containing $e$. Observe that $\ell(e)$ comes after $\ell(e')$ if and only if $e$ comes after $e'$ along $\pi'$ from $\ell_1$ (because the order of the edges of $\pi'$, as they appear on the path, are radially sorted around $v_i$). Also, given an edge $e$ of $\pi'$, we can compute $\ell(e)$ in constant time. Using these properties, we can find two consecutive edges $e$ and $e'$ of $\pi'$ such that $\ell^*$ lies between $\ell(e)$ and $\ell(e')$ in $O(\log^2
n)$ time by applying a binary search on $\pi'$ as we did for path- and boundary-events.
Now we have two consecutive events in the sequence of all path-, boundary- and bend-events that contain $\ell^*$ between them. Recall that the combinatorial structure of $\pi(s,\ell)$ (and $\pi(t,\ell)$) is the same for every line-of-sight lying between the two events. Let $(u_s,w_s)$ and $(u_t,w_t)$ be the edges of $\pi(s,\ell)$ and $\pi(t,\ell)$ incident to $\ell$ at $w_s$ and $w_t$, respectively, for any line-of-sight $\ell$ lying between the two events. Using the shortest-path data structure, we can obtain $u_s, u_t, |\pi(s,u_s)|$ and $|\pi(t,u_t)|$ in $O(\log
n)$ time. Then we apply the algorithm in Lemma \[lemma:greenstarformula\] to find an optimal line-of-sight in constant time. In this way, we can obtain an optimal line-of-sight in $O(\log^2n)$ time in total.
![(a) The line-of-sight intersecting $\pi(v_i,u)$ contains the edge of $\pi(v_i,u)$ incident to $v_i$. (b) The maximal common subpath of $\pi(s,\ell_1)$ and $\pi(s,\ell_2)$ from $s$ is $\pi(s,s')$; $\pi'=\pi(s',\ell_1)\cup\pi(s',\ell_2)$ (blue). []{data-label="fig:query-bend"}](figs/query-bend.pdf){width="80.00000%"}
Therefore, we can find two consecutive events with an optimal solution between them, and we can obtain an optimal solution in $O(\log^2 n)$ time in total.
\[theorem:query\] Given a simple $n$-gon $P$, we can preprocess it in $O(n)$ time to find the minimum of the longer distance that $s$ and $t$ travel in order to see each other in $P$ can be computed in $O(\log^2 n)$ time for any two query points $s,t\in P$.
Conclusions and Open Problems
=============================
We have presented a linear time algorithm that solves two variants of the quickest pair-visibility problem for a simple polygon: either we want to minimize the maximum length of a traveled path or we want to minimize the sum of the lengths of both traveled paths.
Additionally, we have considered a query version of the quickest-visibility problem for the min-max variant. We can preprocess a simple $n$-gon in linear time so that the minimum of the longer distance the two query points travel can be computed in $O(\log^2 n)$ time for any two query points.
We conclude this paper with some interesting open problems.
1. Is there a way to extend our algorithm to more than two query points? More precisely, given $k$ points in a simple polygon, compute the minimum distance that these points must travel in order to see each other (at the same moment).
2. Find an efficient algorithm for the query version of the quickest-visibility problem for the min-sum problem.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was initiated at the 19th Korean Workshop on Computational Geometry in Würzburg, Germany.
[^1]: Department of Computer Science and Engineering, POSTECH, Pohang, South Korea, [<heekap@postech.ac.kr>]{}
[^2]: Max Planck Institute for Informatics, Saarbrücken, Germany, [<eoh@mpi-inf.mpg.de>]{}
[^3]: Theoretische Informatik, FernUniversität in Hagen, Hagen, Germany, [<lena.schlipf@fernuni-hagen.de>]{}
[^4]: Institut für Informatik, Universität Bayreuth, Bayreuth, Germany, [<fabian.stehn@uni-bayreuth.de>]{}
[^5]: Department of Computer Science, Hamilton College, Clinton, New York, USA, [<dstrash@hamilton.edu>]{}
[^6]: This work by Ahn and Oh was supported by the MSIT (Ministry of Science and ICT), Korea, under the SW Starlab support program (IITP-2017-0-00905) supervised by the IITP (Institute for Information & Communications Technology Promotion). An extended abstract of this paper appeared at SWAT’18.
|
---
abstract: 'We study the role of asymptotic curves in supporting the spiral structure of a N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions on the unstable asymptotic curves of the main unstable periodic orbits follow the shape of the periodic orbits for an initial interval of time and then they are diffused outwards supporting the spiral structure of the galaxy. Chaotic orbits having small deviations from the unstable periodic orbits, stay close and along the corresponding unstable asymptotic manifolds, supporting the spiral structure for more than 10 rotations of the bar. Chaotic orbits of different Jacobi constants support different parts of the spiral structure. We also study the diffusion rate of chaotic orbits outwards and find that chaotic orbits that support the outer parts of the galaxy are diffused outwards more slowly than the orbits supporting the inner parts of the spiral structure.'
author:
- 'G. Contopoulos'
- 'M. Harsoula'
date: 'Received: date / Accepted: date'
title: Chaotic Spiral Galaxies
---
Introduction
============
It is well known that the spiral arms of galaxies are density waves. This means that the spiral arms are not always composed of the same matter but they only represent the maxima of the density along every circle around the center. The stars passing through the spiral arms stay longer close to them, thus producing an increase of the local density.
The linear theory of spiral density waves assumes that the potential V, the density $\rho$ (or the surface density $\sigma$) and the distribution function *f* (phase space density) have small deviations from their axisymmetric values ($V_0$,$\rho_0$,$f_0$).
This linear theory was initiated by Lindblad $(1940,1942)$ but it was developed in its modern form by Lin and Shu $(1964, 1966)$, Kalnajs $(1971)$, Lynden-Bell and Kalnajs $(1972)$ Toomre $(1977)$,and others.
However the deviations near the main resonances of a galaxy (inner and outer Lindblad resonances and corotation) are large, thus near these resonances a nonlinear theory is necessary (Contopoulos $1970,1973,1975$).
In both the linear and the nonlinear theory, whenever the perturbation is relatively small, chaos is unimportant. Although some chaotic orbits appear near all the unstable periodic orbits, their proportion and their effects are small. This is the case of most normal spirals, where the density perturbations are of order $2-10\%$ of the axisymmetric background.
On the other hand in barred galaxies the perturbations are large, of the order of $50-100\%$. In such cases the chaotic orbits play an important role in the dynamics of the galaxies. The main chaotic effects in a galaxy appear near corotation.
It is well known now that chaos is generated by the overlapping of resonances (Contopoulos $1966$, Rosenbluth et al. $1966$, Chirikov $1979$). In the region near corotation there are many resonances between the angular velocities of the stars $(\Omega-\Omega_s)$ (in the frame rotating with the angular velocity of the pattern $\Omega_s$) and the epicyclic frequence $\kappa$.
The ratio of these frequencies is $$q =\frac {\Omega-\Omega_s}{\kappa}$$
These resonances are congested in a relatively small interval of Jacobi constants $E_j$ around the Jacobi constant $E_{j0}$ at corotation. Some important resonances in an extended region around corotation are $ 1/q=4/1, 3/1, 2/1$ (inside corotation), $q=0$ (at corotation) and $-2/1$,$-1/1$ (outside corotation). Chaos produced by the interactions of resonances, extends around the envelope of the bar and along the spiral arms beyond the end of the bar. The first example of a chaotic orbit that fills an envelope of the bar and the inner parts of the spiral structure was provided by Kaufmann and Contopoulos (1996) (Fig.1).
![First example of a chaotic orbit in a galactic model of NGC 3992 in [@Kauf1996].[]{data-label="fig2"}](fig01.eps){width="8.cm"}
However the most surprising result came from N-body simulations, which indicated that most orbits along the spiral arms beyond the ends of the bar are chaotic (Voglis et al. $2006 a$).
Thus started a systematic study of the chaotic spiral arms outside corotation in strong barred galaxies (Voglis and Stavropoulos 2005, Voglis et al. $2006 a,b$, Romero-Gomez et al. $2006$,$2007$, Tsoutsis et al. $2008$, Athanassoula et al.$2009 a,b$, **2010**, Harsoula et al.$2009$, $2011$). In the present paper we present the most recent results of this study.
Chaotic Density Waves
=====================
Although the spiral arms beyond the ends of the bars are composed of chaotic orbits, nevertheless these spiral arms are density waves, i.e. the density maxima are populated by different stars at every time.
The density of the stars is maximum in areas where their velocities are minimum. This happens mainly near the apocentres and the pericentres of the orbits. And these apocentres and pericentres appear close to the asymptotic manifolds of the main unstable periodic orbits around corotation.
The most important families of unstable periodic orbits in the corotation region are the families $PL_1$,$PL_2$ around the unstable Lagrangian points $L_1$ and $L_2$.
In a simple model of a barred galaxy the Hamiltonian near corotation $(r_*)$ (Contopoulos $1978$) is given by the relation $$H=h_\ast+\kappa_\ast I_1+a_\ast I_1^2+2b_\ast I_1 I_2 +c_\ast I_2^2
+ A_\ast \cos 2\theta_2=h$$ where $h$ is the Jacobi constant, $I_1$ the epicyclic action, $I_2=J-J_\ast$ is the azimuthal action, (where $J$, $J_\ast$ are the angular momenta of the star and of corotation) $\theta_2$ is the azimuth of the star and $h_\ast$, $\kappa_\ast$, $a_\ast$, $b_\ast<0$, $c_\ast<0$, $A_\ast$ are constants. The action $I_1$ is also constant, because the conjugate angle **$\theta_1$**, does not appear in the Hamiltonian. The orbits $PL_1$, $PL_2$ are represented by the unstable equilibrium points of the system, namely when $$\partial H/\partial I_2=\partial H/\partial\theta_2=0$$ Therefore $$sin2\theta_2=0$$ and $$b_\ast I_1+c_\ast I_2=0$$ The orbits close to the Lagrangian points $L_1$, $L_2$ have $\theta_2=\pi/2$ and $\theta_2=3\pi /2$, (while the orbits close to the Lagrangian points $L_4$,$L_5$ have $\theta_2=0$, and $\theta_2=\pi$),
In the lowest approximation the orbits $PL_1$, $PL_2$ start with $\theta_2=\pi/2$, or $\theta_2=3\pi/2,$ and $$h_\ast+\kappa_\ast I_1- A_\ast =h$$ hence $$I_1=\frac{h-h_\ast +A_\ast}{\kappa_\ast}>0$$ Therefore these orbits exist whenever $h>h_\ast -A_\ast$.
After finding $I_1$ we can find $I_2$ from Eq.(5).
We have (Contopoulos $ 1975$) $$b_\ast =\frac{\Omega_\ast \kappa'_\ast}{r_\ast
\kappa^2_\ast},c_\ast=\frac{\Omega_\ast \Omega'_\ast}{r_\ast
\kappa^2_\ast}$$ where $\Omega$ is the angular velocity, and $\kappa$ is the epicyclic frequency. The subscript $\ast$ denotes the values at corotation and the accents mean derivatives with respect to the radius r. Thus Eq.(5) can be written $$\kappa '_{\ast} I_{1} +\Omega^{'}_{\ast} I_{2} = 0$$ We consider now a simple model of the form $$V'_o = \frac{c^2}{r^\rho}$$ where $V_o$ is the potential and $V'_o$ is the force as a function of $r$. Then we have $$\Omega = \sqrt\frac{V'_o}{r}= \frac{c}{\sqrt r^{\rho+1}}$$ and $$\kappa = \sqrt { V{''}_o + \frac{3V'_o}{r}} = \frac{c}{\sqrt
r^{\rho+1}} (3-\rho)$$ Thus Eq.(9) gives $$(3-\rho)I_1 + I_2=0$$ and if $\rho<3$ we find that $I_2$ is negative. In particular in a Keplerian model $\rho =2$ and the relation (13) becomes $$I_1 + I_2 = 0$$
Therefore $I_2<0$. The initial point of the orbit $PL_1$ (with $\theta=\pi/2$ and increasing $\theta$) is $\bar{PL_1}$ inside the corotation distance with $$\Delta r_o = \frac{2\Omega_\ast}{r_\ast \kappa^2_\ast} I_2 < 0$$ The orbit $PL_1$ is described counterclockwise (while the galaxy is rotating clockwise.) Thus the orbit $PL_1$ intersects the $\theta=\pi/2$ axis once more with $\Delta r_o>0$ and decreasing $\theta$ (Fig. 2).
![The asymptotic manifolds from the orbits $PL_1$, $PL_2$ on the configuration plane.[]{data-label="fig2"}](fig02.eps){width="6.cm"}
The asymptotic manifolds from the orbits $PL_1$, $PL_2$ on the configuration plane are shown in Fig.2. There are two unstable manifolds from $PL_1$ namely $U$ along a trailing spiral and $UU$ along the leading edge of the bar, and two stable manifolds, $SS$ (along a leading direction) and S (along the trailing edge of the bar). These manifolds contain successive apocentres ($\dot{r}$=0)) of the asymptotic orbits i.e. orbits starting on this manifold. Similar manifolds emanate from $\bar{PL_1}$, $\bar{PL_2}$, corresponding to the pericentres of the asymptotic orbits.
The orbits starting close to $PL_1$ but not exactly on the asymptotic manifold approach this point along the stable directions $S$ and $SS$ and then deviate along the unstable directions $U$ and $UU$. After a longer time these orbits form trailing spiral arms and the envelope of the bar. The forms of the asymptotic curves emanating from $PL_1$ and $PL_2$ are shown in Fig. 3. We see that the curve U from $PL_1$ reaches the neighborhood of $PL_2$, making oscillations around the asymptotic curve $SS'$ of larger and larger amplitude as it approaches $PL_2$, coming closer and closer to the asymptotic curves $U'$ and $UU'$ from $PL_2$, which are symmetric to $U$ and $UU$ with respect to the center of the galaxy. Thus the matter that starts close to $PL_1$ and moves along the asymptotic curve $U$, after reaching the neighborhood of $PL_2$ moves very close to the asymptotic curves $U'$ and $UU'$ and approaches again the neighborhood of $PL_1$. In a similar way matter moves along the asymptotic curves $UU$, $U'$ and $UU'$. Thus we have circulation of the material along the spiral arms that lasts for a substantial fraction of the Hubble time, until the spiral arms fade away.
![The forms of the asymptotic curves emanating from $PL_1$ and $PL_2$.[]{data-label="fig2"}](fig03.eps){width="5.cm"}
The individual orbits along the asymptotic curve $U$ or close to it, are of the form of Fig.4. The orbits start by making some rotations close to the periodic orbit $PL_1$ and they deviate away from $PL_1$, reaching the neighborhood of $PL_2$. Then they proceed either close to the spiral $U'$ (Fig.4a), or close to the envelope of the bar $UU'$ (Fig. 4b).
The unstable asymptotic curves of the various unstable periodic orbits for the same Jacobi constant cannot cross themselves or each other. Thus they are obliged to follow nearly parallel paths. The main unstable orbits inside corotation are the families $4/1$, $3/1$ and $2/1$ (inner Lindblad), while close to corotation the most important families are $Pl_1$ and $PL_2$ whose asymptotic manifolds are shown in Fig. 5a. On the other hand, all the manifolds of the unstable periodic orbits corresponding to the same Jacobi constant, are approximately parallel and contribute to the formation of spiral arms outside corotation (Fig. 5b). This is the phenomenon of coalescence that was described by Tsoutsis et al.(2008).
![Orbits starting close to $PL_1$ (around the Lagrangian point $L_1$) approach the Lagrangian point $L_2$ and then deviate (a) along $U'$, or (b) along $UU'$.[]{data-label="fig2"}](fig04.eps){width="12.cm"}
![(a) The projected unstable asymptotic manifolds from $PL_1$ and $PL_2$ in the configuration space (b) The “coalescence” of the invariant asymptotic manifolds from the $PL_1$, $PL_2$, $-1//1$, $-2/1$ and $-41$ families. []{data-label="fig2"}](fig05a.eps "fig:"){width="6.2cm"} ![(a) The projected unstable asymptotic manifolds from $PL_1$ and $PL_2$ in the configuration space (b) The “coalescence” of the invariant asymptotic manifolds from the $PL_1$, $PL_2$, $-1//1$, $-2/1$ and $-41$ families. []{data-label="fig2"}](fig05b.eps "fig:"){width="5.5cm"}
If we consider also similar figures for other values of the Jacobi constant (in which case overlapping of various curves is permitted) we see the formation of thick spiral arms. In fact the observed spiral arms in N-body simulations of barred galaxies are very close to the general form of the spiral arms produced by the superposition of the various asymptotic curves.
However orbits close to the particular unstable periodic orbits support particular parts of the spiral arms. Thus it is of interest to study the orbits close to every resonance.
Resonant orbits and diffusion times
===================================
The planar orbits in a time independent rotating model have a fixed Jacobi constant (that we call “energy in the rotating frame”) $$E_j=\frac{1}{2}u^2+V(r,\theta)-\Omega_s J$$ where u is the velocity in the rotating frame, $V(r,\theta)$ is the potential in polar coordinates, $J$ is the angular momentum and $\Omega_s$ is the angular velocity of the system.
If $E_j$ is larger that the energy $E_{jo}$ at the Lagrangian points $L_1,L_2$, the orbits inside and outside corotation can communicate. If, however, $E_j<E_{jo}$ the orbits inside corotation cannot get outside and those outside corotation cannot get inside.
We consider a particular N-body system that was studied by Voglis and Stavropoulos (2005) and Voglis et al. (2006a) simulating a barred spiral galaxy and we separate the orbits in particular intervals $E_j\pm 10000$.
We consider first the N-body orbits that have energies in the interval $E_j = - 1090000 \pm 10000$. In this energy level there are almost no regular orbits at all (see Fig. 3 of Harsoula et al. 2011). For these energies the orbits can move both inside and outside corotation. We integrate the orbits with initial conditions outside corotation for $100 T_{hmct}$ (half-mass crossing times) and find the distribution of their $q$ values (Fig. 6a).
The time interval $\Delta t =100 T_{hmct}$ is about one third of the Hubble time. During that time most resonant orbits are concentrated near the resonances $-2/1$ (outer Lindblad), $-1/1$ and $-2/3$ . Only a few orbits have moved inside corotation $(q>0)$.
However after about one and a half Hubble time (Fig. 6b) the above resonances have fewer stars. Some stars have escaped, but a substantial proportion of stars have reached the inner resonances $3/1$ and $2/1$ (inner Lindblad). These stars are trapped near these resonances for very long times before escaping again outwards. On the other hand stars starting inside corotation remain close to the resonances $3/1$ and $2/1$ for more than $5$ Hubble times before escaping outside the galaxy (without being trapped by the outer resonances). It must be pointed out here that the 2-body relaxation time for galaxies is of the order of $10^6-10^7$ Hubble times, a time exceedingly longer that the diffusion time of the chaotic orbits in our N-body model.
Then we consider the stars outside corotation for an energy interval $E_j=-1250 000 \pm 10000$, where again only chaotic orbits exist outside corotation. In this case the orbits with initial conditions outside corotation cannot enter inside corotation.
In Figs. 7a,b the distribution of the $q$-values of these stars is given, for the time intervals $0-100 T_{hmct}$ (Fig.$7a$) and $400-500 T_{hmct}$ (Fig.$7b$). In this case there are no $PL_1,
PL_2$ orbits, but we notice some important resonances like $-2/1$ (outer Lindblad), $-1/1,-3/4,-2/3$ and $-1/2$. After one and a half Hubble times (Fig.$7b$) the number of resonant stars has decreased and many stars have escaped beyond the ends of the galaxy. Nevertheless there is still an appreciable number of stars close to these resonances. This is due to the stickiness phenomenon that lasts for very long times, before the stars escape from the galaxy. Comparing Figs. 6 and 7 we notice that in the case of chaotic orbits that are restricted in the area outside corotation, stickiness to resonances lasts for longer times in smaller values of Jacobi constants than in greater ones. Thus in general, the diffusion of chaotic orbits supporting the outer parts of the spiral arms outwards is more slow than the diffusion of orbits supporting more inner parts of the spiral structure. An example is given in Fig. 8 where the percentage of orbits starting close to the $-1/ 1$ unstable periodic orbits that stay located inside a radius $R=3r_{hm}$ (which in fact confines the bound part of the galaxy, $r_{hm}$ being the half mass radius of the system) is plotted as a function of time in $T_{hmct}$, for four different energy levels (Jacobi constants), namely for $E_j=-1250000$ (black solid curve), $E_j=-1230000$ (black dotted curve), $E_j=-1210000$ (gray curve) and $E_j=-1150000$ (black dashed curve). After 1 Hubble time ($\approx 300 T_{hmct}$), $95\%$ of the orbits with $E_j=-1250000$ is still located inside $R=5r_{hm}$, while $\bf{90\%}$ of the orbits with $E_j=-1230000$, $40\%$ with $E_j=-1210000$ and only $0.1\%$ of the orbits with $E_j=-1150000$ is still located inside $R=3r_{hm}$. The functions $\Delta N_{R<3}$N versus $T_{hmpt}$ for the various energy levels are approximated by exponentials of the form $$\Delta N _{R<3}/N=\alpha exp(-\lambda T_{hmpt})$$ where $\alpha$ and $\lambda$ take the values given in Table 1.\
E $\alpha$ $\lambda$
---------- ---------- -----------
-1250000 1.17 0.0008
-1230000 1.50 0.0017
-1210000 1.10 0.0038
-1150000 1.00 0.0240
: The values of $\alpha$ and $\lambda$ in eq. (17)
The values of $\alpha$ are close to $\alpha=1$, while $\lambda$ can be given by the approximate formula $$\lambda=A exp(\Lambda E)$$ with $A=1.8$ x $10^{15}$ and $\Lambda =3.4$ x $10^{-5}$. Therefore the diffusion is much faster for larger Jacobi constants. On the other hand for smaller Jacobi constants the diffusion is slower, and most of the stars remain close to the outer parts of the spiral arms for more than a Hubble time.
In the case of chaotic orbits with initial conditions inside corotation or close to it, the diffusion happens quickly for an initial time interval corresponding to $\approx$ 1/3 of the Hubble time (during which the spiral structure survives), while later on it is very slow (see Fig. 24 of Harsoula et al. 2011). We therefore conclude that the outer parts of the spiral structure of our N-body model survive for longer times than the inner parts of the spiral structure.
The role of asymptotic orbits
=============================
![The isodensities of the N-body particles belonging to three different energy levels, namely (a) $E_j=-1090000\pm 10000$, where the areas inside and outside corotation can communicate (b) $E_j=-1150000\pm 10000$, where the areas inside and outside corotation cannot communicate (c) $E_j=-1250000\pm 10000$ where again the areas inside and outside corotation cannot communicate and (d) the isodensities of particles belonging to all previous three levels. The spiral structure is apparent here.[]{data-label="fig2"}](fig09.eps){width="12.cm"}
In previous papers (Harsoula and Kalapotharakos 2009, Harsoula et al. 2011) we have emphasized the role of stickiness of chaotic orbits along the unstable asymptotic manifolds of the unstable periodic orbits, in supporting the structure of the spiral arms. In what follows, we investigate the role of the 2-D asymptotic orbits, i.e. orbits having initial conditions on the unstable manifolds of the various unstable periodic orbits, in supporting the spiral structure of the model.
![(a) The density distribution of 10000 2-D asymptotic orbits (in color) belonging to the $2/1$ (or $x1$) family having initial conditions inside corotation and Jacobi constant $E_j=-1090000$. Superimposed is the orbit $x1$ plotted in black. (b) The density distribution of 2-D orbits having initial conditions on a grid close and around the unstable periodic orbit $2/1$ (in color). Superimposed, in black, are the isodensities of the real N-body particles belonging to the same energy level. The corresponding time of integration of the orbits is $\approx$ one and a half Hubble time.[]{data-label="fig2"}](fig10.eps){width="12.cm"}
Orbits in different energy levels support different parts of the spiral structure. This is obvious in Fig.9 where the isodensities of the N-body particles are plotted on the configuration plane of rotation, belonging to three different energy levels. More precisely, particles having values of Jacobi constant $E_j=-1090000
\pm 10000$ correspond to the envelope of the bar and the innermost parts of the spiral arms. Particles with values of Jacobi constant $E_j=-1150000 \pm 10000$ correspond to parts of the spiral arms that extend further beyond and finally particles with values of Jacobi constant $E_j=-1250000 \pm 10000$ correspond to the outermost part of the spiral arms. In Fig. 9d we plot the isodensities of particles belonging to all previous energy levels of Figs. 9a,b,c. The spiral structure of the galaxy is apparent.
In Harsoula et al. 2011 we studied the density distribution of $3-D$ orbits starting close to the various resonances. However we find similar results if we consider the projections of the various orbits on the plane of symmetry ($y-z$) of the galaxy (having the bar along the z-axis).
In Fig. 10 an example of the density distribution of 2-D asymptotic orbits of an unstable periodic orbit is plotted (in color) for a Jacobi constant $E_j=-1090000$, where the areas inside and outside corotation can communicate. More precisely in Fig.10a we plot (in color) 10000 asymptotic orbits of the $2/1$ (or $x1$) family, having initial condition inside corotation, together with the unstable periodic orbit (in black). The corresponding time of integration of the orbits is $\approx$ one and a half Hubble time. For an initial time interval equal to $\approx$ 1/5 of the Hubble time, the chaotic orbits stay close to the periodic orbit, following its shape, but later on they are diffused outwards modulating the inner parts of the spiral structure. If we take the same number of orbits, with initial conditions not on the unstable manifold, but on a grid close and around the unstable periodic orbit having small deviations from it, on the ($z,\dot{z}$) surface of section(Fig. 10b), we find that the distribution of the orbits (in color) follows the inner parts of the spiral pattern, derived from the isodensities of the real N-body particles of the corresponding energy level (in black). This is an example of stickiness of chaotic orbits along the unstable asymptotic manifolds of the unstable periodic orbits. In fact the stickiness of chaotic orbits delays their diffusion outwards and is responsible for the survival of the spiral structure of the galaxy for more that 10 rotations of the bar (see Harsoula et al. 2011).
Similar results are found for the $PL_1,PL_2$ orbits near $L_1,L_2$ at the end of the bar, for the $-1/1$ orbits outside corotation and for the $PL_4,PL_5$ orbits around $L_4$ and $L_5$.
![The density distribution of the orbits starting close to the resonances $2/1, PL_1, PL_2, -1/1$ and $-2/1$ for various Jacobi constants, superimposed with the isodensities of the real N-body particles belonging to the corresponding energy levels (black curves).[]{data-label="fig2"}](fig13.eps){width="8.cm"}
Finally, in Fig. 11 we present the density distribution of the sticky chaotic orbits near the resonances $2/1,PL_1, PL_2,-1/1,-2/1$ superimposed with the isodensities of the real N-body particles belonging to the corresponding energy levels. We conclude that by using a sample of sticky chaotic orbits around a number of unstable periodic orbits inside and outside corotation in different energy levels, we are able to reproduce quite well the outer envelope of the bar and the spiral structure of the galaxy.
Conclusions
===========
The main conclusions of our paper are the following:
1\) Stickiness of chaotic orbits close to the unstable asymptotic manifolds of various periodic orbits delays the diffusion of these orbits outwards and therefore modulates the shape of the spiral structure of the galaxy for more than 10 rotations of the bar, corresponding to 1/3 of the Hubble time.
2\) Chaotic orbits that are limited outside corotation modulate the outer parts of the spiral structure for smaller values of Jacobi constant while orbits with greater values of Jacobi constant modulate the inner parts of the spiral structure. Moreover, in our N-body model, stickiness to resonances for smaller values of Jacobi constants lasts for longer times than stickiness for greater values of Jacobi constants.
3\) Asymptotic orbits (having initial conditions on the unstable asymptotic curve of an unstable periodic orbit) stay located close to the periodic orbit for an initial interval of time, following the shape of this specific orbit, before diffusing from it and supporting the spiral structure. Chaotic orbits having initial conditions inside corotation modulate the envelope of the bar and the innermost spiral structure during a time interval of fast diffusion ($\approx 1/3$ of the Hubble time) and then they are diffused outwards with much slower rates.
4\) Using a sample of sticky chaotic orbits close to a number of unstable periodic orbits inside and outside corotation, in different energy levels, we are able to reproduce quite well the outer envelope of the bar and the spiral structure of the galaxy.
[14]{} Athanassoula, E., Romero-Gomez, M. and Masdemont, J.J.: Rings and spirals in barred galaxies - I. Building blocks, Mon. Not. Roy. Astron. Soc., **394**, 67-81 (2009a) Athanassoula, E., Romero-Gomez, M., Bosma, A. and Masdemont, J.J.: Rings and spirals in barred galaxies - II. Ring and spiral morphology. Mon. Not. Roy. Astron. Soc., **400**, 1706-1720, (2009b) Athanassoula, E., Romero-Gomez, M., Bosma, A. and Masdemont, J.J.: Rings and spirals in barred galaxies - III. Further comparisons and links to observations. Mon. Not. Roy. Astron. Soc., **407**, 1433-1448, (2010) Contopoulos, G.: Resonance Effects in Spiral Galaxies. Astrophys.J., **160**, 113-133, (1970) Contopoulos, G.: The Particle Resonance in Spiral Galaxies. Nonlinear Effects. Astrophys.J., **181**, 657-684 (1973) Contopoulos, G.: Inner Lindblad resonance in galaxies - Nonlinear theory. I. Astrophys.J., **201**, 566-584 (1975) Contopoulos,G.: Periodic orbits near the particle resonance in galaxies. Astron. Astrophys., **64**, 323-332 (1978) Harsoula, M. and Kalapotharakos, C.: Orbital structure in N-body models of barred-spiral galaxies. Mon. Not. Roy. Astron.Soc., **394**, 1605-1619, (2009) Harsoula, M., Kalapotharakos, C. and Contopoulos, G.: Asymptotic orbits in barred spiral galaxies. Mon. Not. Roy. Astron.Soc., **411**, 1111-1126 (2011) Kaufmann D.E., and Contopoulos G.: Self-consistent models of barred spiral galaxies. Astron.Astrophys., 309, 381 (1996). Romero-Gomez, M., Athanassoula, E.,Masdemont, J.J. and Garcia-Gomez, C.: The origin of rR1 ring structures in barred galaxies. Astron.Astrophys., **453**, 39-45 (2006) Romero-Gomez, M., Athanassoula, E., Masdemont, J.J. and Garcia-Gomez, C.: The formation of spiral arms and rings in barred galaxies. Astron.Astrophys., **472**, 63-75 (2007) Tsoutsis, P., Efthymiopoulos, Ch. and Voglis, N.: The coalescence of invariant manifolds and the spiral structure of barred galaxies. Mon. Not. Roy.Astron.Soc., **387**, 1264-1280 (2008) Voglis,N. and Stavropoulos,I.: Can chaotic motion be responsible for the formation of spiral arms? in Solomos,N.(ed) “Recent Advances in Astronomy and Astrophysics” AIP Conf. Proceedings **848**, 647-659 (2006) Voglis, N., Stavropoulos, I. and Kalapotharakos, C.: Chaotic motion and spiral structure in self-consistent models of rotating galaxies. Mon. Not. Roy. Astron. Soc., **372**, 901-922 (2006b) Voglis, N., Tsoutsis, P. and Efthymiopoulos, Ch.: Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies. Mon. Not. Roy. Astron. Soc., **373**, 280-294 (2006c)
|
---
abstract: 'In this paper we study Koszul cohomology and the Green and Prym-Green conjectures for canonical and Prym-canonical binary curves. We prove that if property $N_p$ holds for a canonical or a Prym-canonical binary curve of genus $g$ then it holds for a generic canonical or Prym-canonical binary curve of genus $g+1$. We also verify the Green and Prym-Green conjectures for generic canonical and Prym-canonical binary curves of low genus ($6\leq g\leq 15$, $g\neq 8$ for Prym-canonical and $3\leq g\leq 12$ for canonical).'
address:
- 'Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133, Milano, Italy '
- ' Dipartimento di Matematica, Università di Pavia, via Ferrata 1, I-27100 Pavia, Italy '
author:
- Elisabetta Colombo
- Paola Frediani
title:
-
- 'On the Koszul cohomology of canonical and Prym-canonical binary curves'
---
[^1]
Introduction
============
Let $C$ be a smooth curve, $L$ a line bundle and $\mathcal{F}$ a coherent sheaf on $C$. We recall that the Koszul cohomology group $K_{p,q}(C,\mathcal{F},L)$ is the middle term cohomology of the complex: $$\Lambda^{p+1}H^0(L)\otimes H^0(\mathcal{F}\otimes
L^{q-1})\stackrel{d_{p+1,q-1}}\rightarrow \Lambda^{p}H^0(L)\otimes
H^0(\mathcal{F}\otimes L^{q})\stackrel{d_{p,q}}\rightarrow
\Lambda^{p-1}H^0(L)\otimes H^0(\mathcal{F}\otimes L^{q+1})$$ where $$d_{p,q}(s_1\wedge...\wedge s_p\otimes u):=\sum_{l=1}^{p}(-1)^l s_1\wedge...\wedge \hat{s_l} \wedge...\wedge s_p\otimes
(s_lu).$$ If $\mathcal{F}=\mathcal{O}_C$ the groups $K_{p,q}(C,\mathcal{O}_C,L)$ are denoted by $K_{p,q}(C,L)$. The Koszul cohomology theory has been introduced in [@gr] and has been extensively studied in particular in the case of the canonical bundle. We recall that Green and Lazarsfeld ([@gr]) proved that for any smooth curve $C$ of genus $g$ and Clifford index $c$, $K_{g-c-2,1}(C,K_C)\neq 0$. Green’s conjecture says that this result is sharp i.e. $K_{p,1}(C,K_C)=0$ for all $p\geq
g-c-1$. The Clifford index for a general curve is $[\frac{g-1}{2}]$, so generic Green’s conjecture says that $K_{p,1}(C,K_C)=0$ for all $p\geq[\frac{g}{2}]$, or equivalently, by duality, $K_{p,2}(C,K_C)=0$, i.e. property $N_p$ holds, for all $p\leq[\frac{g-3}{2}]$. Generic Green’s conjecture has been proved by Voisin in [@v02],[@v05]. Green’s conjecture has also been verified for curves of odd genus and maximal Clifford index ([@v05], [@hr]), for general curves of given gonality ([@v02], [@te] [@sch]), for curves on $K3$-surfaces ([@v02], [@v05], [@af]), and in other cases (see [@an]).
Another interesting case is when the line bundle is Prym canonical, $L=K_C\otimes A$ where $A$ is a non trivial 2 torsion line bundle. This case has been studied in [@fl], where the Prym-Green conjecture has been stated. This is an analogue of the Green conjecture for general curves, namely it says that for a general Prym-canonical curve $(C,K_C\otimes A)$, we have $K_{p,2}(C,K_C\otimes A)=0$, i.e. property $N_p$ holds, for all $p\leq[\frac{g}{2}-3]$. Prop.3.1 of [@fl] shows that for any $(C,K_C\otimes A)$ and $p>[\frac{g}{2}-3]$, $K_{p,2}(C,K_C\otimes
A) \neq 0$.
Debarre in [@deba] proved that a generic Prym-canonical curve of genus $g \geq 6$ is projectively normal (property $N_0$) and for $g \geq 9$ its ideal is generated by quadrics (property $N_1$).
In [@cfes] the Prym-Green conjecture is proved for genus $g=10,12,14$ by degeneration to irreducible nodal curves and computation with Macaulay2. In a private communication Gavril Farkas told us that they could verify the conjecture also for $g=18,20$. The computations made in [@cfes] for genus 8 and 16 suggest that the Prym-Green conjecture may be false for genus which is a multiple of 8 or perhaps a power of 2. The possible failure of the Prym-Green conjecture in genus 8 is extensively discussed in the last section of [@cfes], where a geometric interpretation of this phenomenon is given.
In this paper we study Koszul cohomology and the Green and Prym-Green conjectures for canonical and Prym-canonical binary curves. Recall that a binary curve of genus $g$ is a stable curve consisting of two rational components $C_j$, $j=1,2$ meeting transversally at $g+1$ points. The canonical and Prym-canonical models of binary curves that we analyze are the one used in [@ccm] and [@cf] and described in the next section. The main result of the paper (Theorem ) says that if property $N_p$ holds for a Prym-canonical binary curve of genus $g$ then it holds for a generic Prym-canonical binary curve of genus $g+1$. In particular, if the Prym-Green conjecture is true for a Prym-canonical binary curve of genus $g = 2k$, then it is true for a general Prym-canonical binary curve of genus $g = 2k+1$.
Moreover we verify the conjecture by a direct computation for $g=6,9,10,12,14$ (see Corollary ).
As a consequence, we show that the generic Prym-canonical curve of genus $g$ satisfies property $N_0$ for $g \geq 6$, property $N_1$ for $g \geq 9$ (already shown by Debarre), property $N_2$ for $g
\geq 10$, property $N_3$ for $g \geq 12$ and property $N_4$ for $g \geq 14$ (Corollary ).
For $g =8$ and $g =16$ our computations on Prym-canonical binary curves also suggest that Prym-Green conjecture’s might fail, in fact in our examples we find that $K_{\frac{g}{2}-3,2}(C, K_C \otimes A) =1$ both for $g=8$ and $g=16$ (see Remark ).
An analogous result of Theorem is proven for canonically embedded binary curves (Theorem ), where we show that if property $N_{p}$ holds for a canonical binary curve of genus $g$, then the same property holds for a general canonical binary curve of genus $g+1$. In particular, if the Green conjecture is true for a canonical binary curve of genus $g = 2k-1$, then it is true for a general canonical binary curve of genus $g = 2k$.
Theorem and analogous computations with maple in genus $g =3,5,7, 9,11$, imply that for a general canonical binary curve, if $g \geq 3$, then propery $N_0$ holds (see also [@ccm] section 2), if $g \geq 5$, then propery $N_1$ holds, if $g \geq 7$, then propery $N_2$ holds, if $g \geq 9$, then property $N_3$ holds, and if $g \geq 11$, then property $N_4$ holds.
[**Acknowledgments.**]{} We thank Riccardo Murri for having been so kind to do for us the computer computations in $g=14,16$.
Canonical and Prym-canonical binary curves
==========================================
Construction of canonical binary curves
---------------------------------------
Recall that a binary curve of genus $g$ is a stable curve consisting of two rational components $C_j$, $j=1,2$ meeting transversally at $g+1$ points. Moreover, $H^0(C,\omega_C)$ has dimension $g$ and the restriction of $\omega_C$ to the component $C_j$ is $K_{C_j}(D_j)$ where $D_j$ is the divisor of nodes on $C_j$. Since $K_{C_j}(D_j)\cong \OO_{{{\mathbb P}}^{1}}(g-1)$ we observe that the components are embedded by the complete linear system $|\OO_{{{\mathbb P}}^{1}}(g-1)|$ in ${{\mathbb P}}^{g-1}$.
Following [@ccm], we assume that the first $g$ nodes are $P_i=(0,...,0,1,0,...0),$ with 1 at the $i$-th place, $i=1,...,g$. Then we can assume that $C_j$ is the image of the map $$\label{can}
\begin{gathered}\phi_j:{{\mathbb P}}^1 \rightarrow {{\mathbb P}}^{g-1}, \
j=1,2\\
\phi_j(t,u):= [\frac{M_j(t,u)}{(t-a_{1,j}u)},
...,\frac{ M_j(t,u)}{(t-a_{g-1,j}u)}]
\end{gathered}$$ with $M_j(t,u):= \prod_{r=1}^{g} (t-a_{r,j}u)$, $j=1,2$ and $\phi_j([a_{l,j},1]) = P_l$, $l=1,...,g$.
We see that the remaining node is the point $P_{g+1}:=[1,...,1]$ and it is the image of $[1,0]$ through the maps $\phi_j$, $j=1,2$. One can easily check that, for generic values of the $a_{i,j}$’s, $C=C_1\cup
C_2$ is a canonically embedded binary curve.
Construction of Prym-canonical binary curves
--------------------------------------------
Let $C$ be a binary curve of genus $g$, and $A\in Pic^0(C)$ a nontrivial line bundle. Then $H^0(C,\omega_C \otimes A)$ has dimension $g-1$ and the restriction of $\omega_C \otimes A$ to the component $C_j$ is $K_{C_j}(D_j)$ where $D_j$ is the divisor of nodes on $C_j$. Since $K_{C_j}(D_j)\cong \OO_{{{\mathbb P}}^{1}}(g-1)$, the components are embedded by a linear subsystem of $\OO_{{{\mathbb P}}^{1}}(g-1)$, hence they are projections from a point of rational normal curves in ${{\mathbb P}}^{g-1}$. Viceversa, let us take 2 rational curves embedded in ${{\mathbb P}}^{g-2}$ by non complete linear systems of degree $g-1$ intersecting transversally at $g+1$ points. Then their union $C$ is a binary curve of genus $g$ embedded either by a linear subsystem of $\omega_C$ or by a complete linear system $|\omega_C \otimes A|$, where $A\in Pic^0(C)$ is nontrivial (see e.g. [@capo], Lemma 10). In [@cf] (Lemma 3.1) we constructed a binary curve $C$ embedded in ${{\mathbb P}}^{g-2}$ by a linear system $|\omega_C \otimes A|$ with $A^{\otimes 2}\cong \OO_C$, and $A$ is non trivial. Let us now recall this construction and denote a binary curve with this embedding a Prym-canonical binary curve.
Assume that the first $g-1$ nodes, are $P_i=(0,...,0,1,0,...0)$ with 1 at the $i$-th place, $i=1,...,g-1$, the remaining two nodes are $P_g:=[t_1,...,t_{g-1}]$ with $t_i=0$ for $i
=1,...,[\frac{g}{2}]$, $t_i =1$, for $i =
[\frac{g}{2}]+1,...,g-1$. and $P_{g+1}:=[s_1,...,s_{g-1}]$ with $s_i=1$ for $i
=1,...,[\frac{g}{2}]$, $s_i =0$, for $i =
[\frac{g}{2}]+1,...,g-1$.
Then the component $C_j$ is the image of the map
$$\label{pcan}
\begin{gathered}\phi_j:{{\mathbb P}}^1 \rightarrow {{\mathbb P}}^{g-2}, \
j=1,2, \ \text{where} \\
\phi_1(t,u):= [\frac{tM_1(t,u)}{(t-a_{1,1}u)},..., \frac{tM_1(t,u)}{(t-a_{k,1}u)}, \frac{-M_1(t,u)d_1 a_{k+1,1}u}{A_1(t-a_{k+1,1}u)},..., \frac{-M_1(t,u)d_1a_{g-1,1}u}{A_1(t-a_{g-1,1}u)}]\\
\phi_2(t,u):= [\frac{tM_2(t,u)}{(t-a_{1,2}u)},..., \frac{tM_2(t,u)}{(t-a_{k,2}u)}, \frac{-M_2(t,u)d_2 a_{k+1,2}u}{A_2(t-a_{k+1,2}u)},..., \frac{-M_2(t,u)d_2a_{g-1,2}u}{A_2(t-a_{g-1,2}u)}]\\
\end{gathered}$$
with $k :=[\frac{g}{2}]$, $M_j(t,u):= \prod_{r=1}^{g-1} (t-a_{r,j}u)$, and $A_j= \prod_{i=1}^{g-1} a_{i,j}$, $j=1,2$, $d_2$ is a nonzero constant and $d_1 = \frac{-d_2 A_1}{A_2}$. Notice that we have $\phi_j([a_{l,j},1]) = P_l$, $l=1,...,g-1$, $\phi_j([0,1]) = P_g$, $\phi_j([1,0]) = P_{g+1}$, $j=1,2$. In Lemma 3.1 of [@cf] we proved that for a general choice of $a_{i,j}$’s, $C=C_1\cup
C_2$ is a binary curve embedded in ${{\mathbb P}}^{g-2}$ by a linear system $|\omega_C \otimes A|$ with $A^{\otimes 2}\cong \OO_C$ and $A$ nontrivial. In fact, recall that $Pic^0(C) \cong {{{\mathbb C}}^*}^g \cong {{{\mathbb C}}^*}^{g+1}/{{\mathbb C}}^*$, where ${{\mathbb C}}^*$ acts diagonally, and in Lemma 3.1 of [@cf] it is shown and our line bundle $A$ corresponds to the element $[(h_1,...,h_{g+1})] \in {{{\mathbb C}}^*}^{g+1}/{{\mathbb C}}^*$, where $h_i=1$, for $i< [\frac{g}{2}]+1$, $h_i = -1$, for $i
=[\frac{g}{2}]+1,...,g-1$, $h_g=-1$, $h_{g+1} = 1$, so in particular $A$ is of 2-torsion.
Property $N_p$ for Prym-canonical binary curves
===============================================
Let $C \subset {{\mathbb P}}^{g-2}$ be a Prym-canonical binary curve embedded by $\omega_C \otimes A$, with $A^{\otimes 2} \cong
\OO_C$, as in (\[pcan\]). In this section we study the Koszul cohomology for these curves, in particular we investigate property $N_p$, i.e. the vanishing of $K_{p,2}(C, K_C \otimes A)$. Since by duality ([@gr], see also [@fr] prop.1.4) we have $K_{p,2}(C, K_C \otimes A)
\cong K_{g-3-p,0}(C, K_C,K_C \otimes A)^{\vee} $, this vanishing is equivalent to the injectivity of the Koszul map $$\label{koszul}
F_{g-3-p}: \Lambda^{g-3-p}H^0(C,\omega_C \otimes A) \otimes H^0(C,\omega_C) \rightarrow \Lambda^{g-4-p}H^0(C,\omega_C \otimes A) \otimes H^0(C,\omega_C^2 \otimes A).$$
Our strategy is to compare this map with analogous Koszul maps for a partial normalization of the curve $C$ at one node and possibly use induction on the genus.
To this end, let us introduce some notation: set $k :=[\frac{g}{2}]$ and denote by $\tilde{C}_r$ the partial normalization of $C$ at the node $P_r$ with $r\leq k$ if $g =2k$, $r \geq k+1$ if $g = 2k+1$. This choice of the node is necessary in order to obtain the Prym-canonical model for the curve $\tilde{C}_r$. In fact, observe that in this way, for a general choice of the $a_{i,j}$’s, the projection from $P_r$ sends the curve $C$ to the Prym-canonical model of $\tilde{C}_r$ in ${{\mathbb P}}^{g-3}$ given by the line bundle $K_{\tilde{C}_r} \otimes A'_r$ where $A'_r$ corresponds to the point $(h'_1,...,h'_{g-1}, 1) \in {{{\mathbb C}}^*}^{g}/{{\mathbb C}}^*$, with $h'_i = 1$ for $i\leq [\frac{g-1}{2}]$, $h'_i = -1$ for $i=[\frac{g-1}{2}]+1,...,g-1$, as described above. In fact $(\tilde{C}_r,A'_r)$ is parametrized by $a'_{i,j} = a_{i,j}$ for $i
\leq r-1$, $j =1,2$, $a'_{i,j} = a_{i+1,j}$ for $i
\geq r$, $j =1,2.$ So if we set $d'_{j} := \frac{d_j}{a_{r,j}}$, $j =1,2$, we clearly have a pair $(\tilde{C}_r,A'_r)$ as in . For simplicity let us choose $d_2 =1$, so $d_1 = -\frac{A_1}{A_2}$, hence $d'_{2} := \frac{1}{a_{r,2}}$, $d'_{1} := -\frac{A_1}{A_2 a_{r,1}}$.
To simplify the notation, set $T_g:= H^0(C,\omega_C \otimes A)$, $H_g := H^0(C,\omega_C)$, $B_g:= H^0(C,\omega_C^2 \otimes A)$. Denote by $\{t_1,...,t_{g-1}\}$ the basis of $T_g$ given by the coordinate hyperplane sections in ${\mathbb P}^{g-2} \cong {\mathbb P}(T_g^{\vee})$ and by $\{s_1,...,s_{g}\}$ the basis of $H_g$ given by the coordinate hyperplane sections in ${\mathbb P}^{g-1} \cong {\mathbb P}(H_g^{\vee})$. $T_{g-1,r}:= H^0(\tilde{C}_r,\omega_{\tilde{C}_r} \otimes A'_r)$, $H_{g-1,r} := H^0(\tilde{C}_r,\omega_{\tilde{C}_r})$, $B_{g-1,r}:= H^0(\tilde{C}_r,\omega_{\tilde{C}_r}^2 \otimes A'_r)$. Denote by $\{t'_1,...,t'_{g-2}\}$ the basis of $T_{g-1,r}$ given by the coordinate hyperplane sections in ${\mathbb P}^{g-3} \cong {\mathbb P}(T_{g-1,r}^{\vee})$ and by $\{s'_1,...,s'_{g-1}\}$ the basis of $H_{g-1,r}$ given by the coordinate hyperplane sections in ${\mathbb P}^{g-2} \cong {\mathbb P}(H_{g-1,r}^{\vee})$.
We have the following injections: $$T_{g-1,r} \stackrel{I_r}\hookrightarrow T_g, \ t'_i \mapsto t_i \ \text{for} \ i \leq r-1, \ t'_i \mapsto t_{i+1} \ \text{for} \ i \geq r,$$ $$\label{h}
H_{g-1,r} \stackrel{J_r}\hookrightarrow H_g, \ s'_i \mapsto s_i \ \text{for} \ i \leq r-1, \ s'_{i}\mapsto s_{i+1} \ \text{for} \ i \geq r.$$
Clearly these maps induce an injective map $$B_{g-1,r} \stackrel{L_r}\hookrightarrow B_g,$$ which on the set of generators of $B_{g-1,r}$ given by $t'_i s'_j$, $i = 1,...,g-2$, $j = 1,...,g-1$ is given by $t'_i s'_j \mapsto I_r(t'_i)J_r(s'_j)$. We claim that this map is well defined and injective by the definition of the $t'_i$’s and $s'_j$’s. In fact the restriction of $\sum \alpha_{i,j} t'_i s'_j$ to the two rational components of $\tilde{C}_r$ yields two polynomials $Q_1$ and $Q_2$. On the other hand we have $(\sum \alpha_{i,j} I_r(t'_i) J_r(s'_j))_{|C_{i}} = (t-a_{r,i})^2 Q_i$, hence $L_r$ is well defined and injective. We finally have a map $$\Lambda^{l-1} T_{g-1,r} \stackrel{\wedge t_{r}} \longrightarrow \Lambda^{l} T_g,$$ where by $\wedge t_{r}$ we indicate the composition of the natural map induced by $I_r$ at the level of the $(l-1)$-th exterior power $\Lambda^{l-1} T_{g-1,r} \rightarrow \Lambda^{l-1} T_g$ composed by the wedge product with $t_{r}$, $ \Lambda^{l-1} T_g \stackrel{ \wedge t_{r}} \longrightarrow \Lambda^{l} T_g $.
As in , denote by $F_l : \Lambda^l T_g \otimes H_g \rightarrow \Lambda^{l-1} T_g \otimes B_g$ the Koszul map.
We have the following commutative diagram
$$\label{diagram1} \xymatrix{
\Lambda^{l} T_{g} \otimes H_{g} \ar[r]^{F_{l}} \ar[r]& \Lambda^{l-1} T_{g} \otimes B_{g} \ar[r]^{ \pi_{r}}
& \langle t_{r} \rangle \wedge \Lambda^{l-2} T_g \otimes B_g& \\
\Lambda^{l-1} T_{g-1,r} \otimes H_{g-1,r} \ar[r]^{\tilde{F}_{l-1}}
\ar[u]^{\wedge t_{r} \otimes J_r}& \Lambda^{l-2} T_{g-1,r}
\otimes B_{g-1,r} \ar[ur]^{\wedge t_{r} \otimes L_r}}$$
From now on, given a multi-index $I=(i_1,...,i_l)$ we denote by $t_I:=t_{i_1}\wedge ... \wedge t_{i_l}$.
To study the injectivity of the maps $F_l$, a preliminary reduction comes from the following
\[W\] Let $W \subset \Lambda^l T_g \otimes H_g$ be the subspace generated by the elements of the form $t_I \otimes s_j$, where $j \not \in I$. Then the kernel of the Koszul map $F_l : \Lambda^l T_g \otimes H_g
\rightarrow \Lambda^{l-1} T_g \otimes B_g$ is contained in $W$.
Assume that $ v \in \Lambda^l T_g \otimes H_g $, $ v = \sum_{I, |I|=l} \sum_{j=1...g} \lambda^I_j t_I \otimes s_j$ is such that $F_{l}(v) = 0$. $F_l(v) = \sum_{J, |J| = l-1} \sum_{ I =J \cup \{m\}} \sum_{j=1...g} \lambda^I_j \epsilon(I,J) t_J \otimes t_m s_j=0$, where $\epsilon(I,J) = \pm 1$, depending on the position of $m$ in the multi-index $I = J \cup \{m\}$. Then if we fix a multi-index $J$ with $|J| = l-1$, we must have $ \sum_{ m} \sum_{j=1...g} \lambda^{J \cup \{m\}}_j \epsilon(J \cup \{m\},J) t_J \otimes t_m s_j =0$ and therefore $$\sigma_J:= \sum_{ m} \sum_{j=1...g} \lambda^{J \cup \{m\}}_j \epsilon(J \cup \{m\},J) t_m s_j =0.$$ So we have ${\sigma_J}_{|C_1} \equiv 0$, namely, if we denote by $P_1(t) := t \cdot M_1(t,1)$, as in , we have $$\sum_{j=1...g} \sum_{ m \leq k} \lambda^{J \cup \{m\}}_j \epsilon(J \cup \{m\},J) \frac{ P_1(t)}{t-a_{m,1}} \frac{ P_1(t)}{t-a_{j,1}}$$ $$+\sum_{j=1...g} \sum_{ m \geq k+1} \lambda^{J \cup \{m\}}_j \epsilon(J \cup \{m\},J) \frac{ P_1(t) a_{m,1}}{A_2t(t-a_{m,1})} \frac{ P_1(t)}{t-a_{j,1}} =0$$ If we evaluate in $t = a_{m,1}$, there remains only one term in the sum, namely the one with $j =m$, and hence we have
$$\lambda^{J \cup \{m\}}_m \epsilon(J \cup \{m\},J) a^2_{m,1} \cdot \prod_{ r \neq m, r =1...g-1} (a_{m,1} - a_{r,1})^2 =0, \ \text{if } \ m \leq k,$$ $$\lambda^{J \cup \{m\}}_m \epsilon(J \cup \{m\},J) \frac{a^2_{m,1}}{A_2} \cdot \prod_{ r \neq m, r =1...g-1} (a_{m,1} - a_{r,1})^2 =0, \ \text{if } \ m \geq k+1,$$ hence we have $ \lambda^{J \cup \{m\}}_m =0$ for all $m$.
Since this holds for every multi-index $J$ of cardinality $l-1$, we have shown that we can write $v = \sum_{I, |I|=l} \sum_{j=1...g, j \not \in I} \lambda^I_j t_I \otimes s_j$.
We can now state and prove our main result.
\[indstep\] Assume that $g = 2k$, or $g = 2k+1$ and take an integer $p \leq k-3$. If property $N_p$ holds for a binary curve $\tilde{C}$ of genus $g-1$ embedded in ${{\mathbb P}}^{g-3}$ by $|\omega_{\tilde{C}} \otimes A'|$ as in for a generic choice of the parameters $a'_{i,j}$, then it holds for all binary curves $C$ of genus $g$ embedded in ${{\mathbb P}}^{g-2}$ by $|\omega_C \otimes A|$ as in for a generic choice of the $a_{i,j}$’s.
We want to prove that $K_{p,2}(C, K_C \otimes A) =0$ for a binary curve of genus $g$ and we know that $K_{p,2}(\tilde{C_r}, K_{\tilde C_r} \otimes A'_r) =0$, for the curve $\tilde{C_r}$ which is obtained from $C$ by projection from $P_{r}$ with $r \geq k+1$ if $g = 2k+1$, $r \leq k$ if $g = 2k$.
By duality, $K_{p,2}(C, K_C \otimes A) \cong K_{g-3-p,0}(C, K_C,K_C \otimes A)^{\vee} $, so the statement is equivalent to prove injectivity of the Koszul map $$F_{g-3-p}: \Lambda^{g-3-p}T_g \otimes H_g \rightarrow \Lambda^{g-4-p}T_g \otimes B_g.$$ By assumption we know injectivity of the map $$\tilde{F}_{g-4-p}: \Lambda^{g-4-p}T_{g-1,r} \otimes H_{g-1,r} \rightarrow \Lambda^{g-5-p}T_{g-1,r} \otimes B_{g-1,r}.$$ For simplicity let us denote by $l:=g-3-p$.
Assume first of all that $g = 2k+1$ and consider the projection of $C$ from $P_{g-1}$.
Recall that by Lemma we can reduce to prove injectivity of $F_{l}$ restricted the subspace $W$ generated by such $T_I \otimes s_j$ with $j \not \in I$. Note that we can decompose $W$ as $W:=X_{g-1} \oplus Y_{g-1}$, where $X_{g-1}$ is the intersection with $W$ of the image of the map $\wedge t_{g-1} \otimes J_{g-1}$ in diagram and $Y_{g-1}$ is the subspace of $W$ generated by such $t_I \otimes s_j$ with $g-1 \not \in I$ and $j \not \in I$: $$X_{g-1} = \langle t_{g-1} \wedge t_{J} \otimes s_{j} \ | \ j \not \in J \rangle, \ Y_{g-1} = \langle t_I \otimes s_{j} \ | \ j ,g-1 \not \in I \rangle.$$
Assume now that $F_{l}(x_{g-1} + y_{g-1}) =0$, where $x_{g-1} \in X_{g-1}$, $y_{g-1} \in Y_{g-1}$. Then we have $0=\pi_{g-1} \circ F_{l}(x_{g-1} + y_{g-1}) = \pi_{g-1} \circ F_{l}(x_{g-1})= (\wedge t_{g-1} \otimes L_{g-1}) \circ \tilde{ F}_{l-1}(x_{g-1})$, by the commutativity of diagram . Hence $x_{g-1}=0$, since by induction we are assuming that $\tilde{ F}_{l-1}$ is injective. So we have reduced to prove injectivity of $F_{l}$ restricted to $Y_{g-1}$.
Now consider the projection of $C$ from the point $P_{g-2}$.
Set $$Y'_{g-2} = \langle t'_{J} \otimes s'_{j} \ | \ j, g-2 \not \in J \rangle \subset \Lambda^{l-1} T_{g-1,g-2} \otimes H_{g-1,g-2}$$
Observe that the image $X_{g-2}:= (\wedge t_{g-2} \otimes J_{g-2}) (Y'_{g-2})$ is contained in $Y_{g-1}$ and in fact $$X_{g-2}= \langle t_{g-2} \wedge t_{J} \otimes s_{j} \ | \ j,g-1 \not \in J \rangle.$$ So we have $Y_{g-1} = X_{g-2} \oplus Y_{g-2}$, where $Y_{g-2}$ is the subspace of $Y_{g-1}$ generated by those elements of the form $t_I \otimes s_j$ where $g-2,g-1,j \not \in I$. We have the following commutative diagram
$$\label{diagram2} \xymatrix{
\Lambda^{l} Y_{g-1} \ar[r]^{F_{l}} \ar[r]& \Lambda^{l-1} T_{g} \otimes B_{g} \ar[r]^{ \pi_{g-2}}
& \langle t_{g-2} \rangle \wedge \Lambda^{l-2} T_g \otimes B_g& \\
{ \ \ \ \ \ \ \ \Lambda^{l-1}Y'_{g-2} \ \ \ \ \ \ \ } \ar[r]^{\tilde{F}_{l-1}} \ar[u]^{\wedge t_{g-2} \otimes J_{g-2}}& \ \ \ \ \Lambda^{l-2} T_{g-1,g-2} \otimes B_{g-1,g-2} \ar[ur]^{\wedge t_{g-2} \otimes L_{g-2}}}$$
Assume that $v=x_{g-2} + y_{g-2} \in Y_{g-1} = X_{g-2} \oplus Y_{g-2}$ is such that $F_{l}(v) =0$, then we have $0 = \pi_{g-2} \circ F_{l}(x_{g-2}+y_{g-2}) = \pi_{g-2} \circ F_{l}(x_{g-2})$. So $0= \tilde{F}_{l-1}(x_{g-2})$ by the commutativity of the diagram, and this implies $x_{g-2} = 0$ by induction. Therefore we can assume that $v \in Y_{g-2}$, hence $v$ is a linear combination of vectors of the form $t_I \otimes s_j$ where $g-2,g-1,j \not \in I$.
Repeat the procedure, i.e. project from the points $P_r$, $r=g-3...l$. This can be done since $l = g-3-p \geq k+1$. In this way we can reduce to prove injectivity for the restriction of the map $F_{l}$ to the subspace $Y_{l}$ of $W$ generated by the elements of the form $t_I \otimes s_j$ where $l,...,g-1,j \not \in I$. Observe that since $|I| = l$, we have $Y_{l} = 0$, so $F_{l}$ is injective and the theorem is proved.
If $g =2k$ the proof is analogous: we subsequently project from the points $P_1, P_{2},...,P_{g-l}$. As before note that this can be done since $g-l= p+3 \leq k$. In this way we reduce to prove injectivity for the restriction of the map $F_{l}$ to the subspace $Y$ of $W$ generated by the elements of the form $t_I \otimes s_j$ where $1,2,3,...,g-l, j \not \in I$ and since $|I| = l$, we have $Y= 0$, so $F_{l}$ is injective and the theorem is proved.
If the Prym-Green conjecture is true for a Prym-canonical binary curve of genus $g = 2k$ as in , then it is true for a Prym-canonical binary curve of genus $g = 2k+1$ as in for generic parameters $a_{i,j}$.
The conjecture for $g = 2k+1$ says that $K_{k+1,0}(C,K_C,K_C \otimes A) = 0$, or analogously that property $N_{k-3}$ holds for a generic $C$ embedded with $K_C \otimes A$. Hence the corollary immediately follows from Theorem with $i =k-3$.
\[corpg\] \[Np\]The generic Prym-canonical curve of genus $g$ satisfies property $N_0$ for $g \geq 6$, $N_1$ for $g \geq 9$, $N_2$ for $g
\geq 10$, $N_3$ for $g \geq 12$, $N_4$ for $g \geq 14$.
With a direct computation one verifies the Prym-Green conjecture for explicit examples of Prym-canonical binary curves as in for $g = 6,
9,10,12,14$, so the proof follows from Theorem for generic Prym-canonical binary curves, and then by semicontinuity for generic Prym-canonical smooth curves.
To do the computations we wrote a very simple maple code ( http://www-dimat.unipv.it/ frediani/prym-can) in which we explicitly give the matrix representing the Koszul map $F_l$: for every multi-index $J$ with $|J| = l-1$, we take the projection of the image of $F_l$ onto $t_J \otimes B_{g}$ and we restrict it to the rational components $C_j$. So we have two polynomials in one variable and we take their coefficients.
Once the matrix is constructed, for $g = 6,
9,10,12$, maple computed its rank modulo $131$, which turned out to be maximal. In the case $g =14$ the order of the matrices was too big, so Riccardo Murri made the rank computation using the Linbox ([@1]) and Rheinfall ([@2]) free software libraries. Two different rank computation algorithms were used: Linbox’ “black box” implementation of the block Wiedemann method ([@4; @5]), and Rheinfall’s Gaussian Elimination code([@6]). Results obtained by either method agree.
In both cases, the GNU GMP library ([@3]) provided the underlying arbitrary-precision representation of rational numbers and exact arithmetic operations.
\[g8-16\] For Prym-canonical curves of genus 8, the maple computation on specific examples of binary curves gives $dimK_{1,2}(C, K_C \otimes A)=1$. This result is compatible with the computations in [@cfes].
For Prym-canonical binary curves of genus 16, we constructed the matrix representing the Koszul map $F_8$ on examples using maple and Riccardo Murri computed its rank as explained in the proof of Corollary . Again it turned out that $dimK_{5,2}(C, K_C \otimes A)=1$, confirming the computations in [@cfes].
Property $N_p$ for canonical binary curves
==========================================
In analogy with the Prym-canonical case, we study now property $N_p$ for canonical binary curves with the same inductive method, projecting from a node. So, let $C \subset {{\mathbb P}}^{g-1}$ be a canonical binary curve and denote by $\tilde{C}_r$ the partial normalization of $C$ at the node $P_r$, $1\leq r \leq g$. As above, for a general choice of the $a_{i,j}$’s, the projection from $P_r$ sends the curve $C$ to the canonical model of $\tilde{C}_r$ in ${{\mathbb P}}^{g-2}$, where $\tilde{C}_r$ is parametrized by $a'_{i,j} = a_{i,j}$ for $i \leq r-1$, $j =1,2$, $a'_{i,j} = a_{i+1,j}$ for $i \geq r$, $j =1,2$.
Set $H_g := H^0(C,\omega_C)$, $D_g:= H^0(C,\omega_C^2)$, $F_l : \Lambda^l H_g
\otimes H_g \rightarrow \Lambda^{l-1} H_g\otimes D_g$ the Koszul map. Denote as before by $\{s_1,...,s_{g}\}$ the basis of $H_g$ given by the coordinate hyperplane sections in ${\mathbb P}^{g-1} \cong {\mathbb P}(H_g^{\vee})$ .
$H_{g-1,r} := H^0(\tilde{C}_r,\omega_{\tilde{C}_r})$, $D_{g-1,r}:= H^0(\tilde{C}_r,\omega_{\tilde{C}_r}^2)$. Denote by $\{s'_1,...,s'_{g-1}\}$ the basis of $H_{g-1,r}$ given by the coordinate hyperplane sections in ${\mathbb P}^{g-2} \cong {\mathbb P}(H_{g-1,r}^{\vee})$.
We have the injections $H_{g-1,r} \stackrel{J_r}\hookrightarrow H_g,$ as in and $D_{g-1,r} \stackrel{L_r}\hookrightarrow D_g,$ which on the set of generators of $B_{g-1,r}$ given by $s'_i s'_j$, $i,j = 1...g-1$, is given by $s'_i s'_j \mapsto J_r(s'_i)J_r(s'_j)$.
We finally have a map $$\Lambda^{l-1} H_{g-1,r} \stackrel{\wedge s_{r}} \longrightarrow
\Lambda^{l} H_g,$$ where by $\wedge s_{r}$ we indicate the composition of the natural map induced by $J$ at the level of the $l-1$-th exterior power $\Lambda^{l-1} H_{g-1,r}
\rightarrow \Lambda^{l-1} H_g$ composed by the wedge product with $s_{r}$, $ \Lambda^{l-1} H_g \stackrel{ \wedge s_{r}}
\longrightarrow \Lambda^{l} H_g $.
We are interested in property $N_p$ for these curves, hence by duality, in the vanishing of $K_{g-2-p,1}(C,K_C)$. Clearly the vanishing of $K_{l,1}(C,K_C)$ is equivalent to the injectivity of the map $$\frac{\Lambda^l H_g \otimes H_g}{ \Lambda^{l+1} H_g}\rightarrow
\Lambda^{l-1} H_g\otimes D_g$$ coming from the Koszul complex.
Notice that there is an isomorphism between $\frac{\Lambda^l H_g
\otimes H_g}{ \Lambda^{l+1} H_g}$ and the subspace $V_{g}$ of $\Lambda^l H_g \otimes H_g$ generated by the elements of the form $s_I \otimes s_j$, where $j \geq i_1$, so the above injectivity is equivalent to the injectivity of the restriction of $F_l$ to $V_{g}$.
We have the following commutative diagram
$$\label{diagram1can} \xymatrix{
V_g \ar[r]^{F_{l} } \ar[r]& \Lambda^{l-1} H_{g} \otimes D_{g} \ar[r]^{ \pi_{r}}
& \langle s_{r} \rangle \wedge \Lambda^{l-2} H_g \otimes D_g& \\
\ \ \ \ \ \ \ V_{g-1,r} \ \ \ \ \ar[r]^{ \tilde{F}_{l-1}} \ar[u]^{\wedge s_{r} \otimes
J_r}& \ \ \ \ \Lambda^{l-2} H_{g-1,r} \otimes D_{g-1,r} \ar[ur]^{\wedge
s_{r} \otimes L_r}}$$
where $V_{g-1,r}$ is the subspace of $\Lambda^{l-1} H_{g-1,r} \otimes H_{g-1,r}$ generated by the elements of the form $s'_J \otimes s'_j$, where $j \geq j_1$.
Let $W \subset V_{g,l}$ be the subspace generated by the elements of the form $s_I \otimes
s_j$, where $j \not \in I$ and $j \geq i_1$.
\[W\_i\] The map $F_l :V_{g} \rightarrow \Lambda^{l-1}
\otimes B_g$ is injective if and only if ${F_l}_{|W}$ is injective.
The proof is completely analogous to the proof of .
\[can\] If property $N_{p}$ holds for a canonical binary curve of genus $g
-1$ as in , then the same property holds for a canonical binary curve of genus $g $ as in for a generic choice of the parameters.
From the above discussion we know that the statement is equivalent to prove injectivity of the Koszul map $ F_{l}: V_{g} \rightarrow \Lambda^{l-1}H_g \otimes D_g$ for $l = g-2-p$, while by assumption we know injectivity of the map $\tilde{F}_{l-1}: V_{g-1,r} \rightarrow \Lambda^{l-2}H_{g-1,r} \otimes D_{g-1,r}.$
We first project from $P_{g}$. By Remark we can reduce to prove injectivity of $F_{l}$ restricted the subspace $W$ generated by such $s_I \otimes s_j$ with $j \not \in I, j >i_1$. Note that as before we can decompose $W$ as $W:=X_{g} \oplus Y_{g}$, where $X_{g}$ is the intersection with $W$ of the image of the map $\wedge s_{g} \otimes J_{g}$ in diagram and $Y_{g}$ is the subspace of $W$ generated by such $s_I \otimes s_j$ with $g \not \in I$ and $j \not \in I, j >i_1$: $$X_{g} = \langle s_{g} \wedge s_{J} \otimes s_{j} \ | \ j \not \in J, j > j_1\rangle, \ Y_{g} = \langle s_I \otimes s_{j} \ | \ j ,g \not \in I, j>i_1 \rangle$$
If $F_{l}(x_{g} + y_{g}) =0$, where $x_{g} \in X_{g}$, $y_{g} \in Y_{g}$, then $0=\pi_{g} \circ F_{l}(x_{g} + y_{g}) = \pi_{g} \circ F_{l}(x_{g})= (\wedge s_{g} \otimes L_{g}) \circ \tilde{ F}_{l-1}(x_{g})$. Hence $x_{g}=0$, since by induction $\tilde{ F}_{l-1}$ is injective. So we have reduced to prove injectivity of $F_{l}$ restricted to $Y_{g}$.
Repeat the procedure, i.e. project from the points $P_r$, $r=g-1...l$. In this way we can reduce to prove injectivity for the restriction of the map $F_{l}$ to the subspace $Y_{l}$ of $W$ generated by the elements of the form $s_I \otimes s_j$ where $l,...,g,j \not \in I, j>i_1$. Observe that since $|I| = l$, we have $Y_{l} = 0$, so $F_{l}$ is injective and the theorem is proved.
Notice that, by the theorem of Green and Lazarsfeld ([@gr]), if $p>g-[\frac{g}{2}]-2$, condition $N_p$ does not hold for any curve $\tilde{C}$ of genus $g-1$.
If the Green conjecture is true for a canonical binary curve of genus $g = 2k-1$ as in , then it is true for a canonical binary curve of genus $g = 2k$ as in for a generic choice of the parameters.
The conjecture for $g = 2k$ says that $K_{k,1}(C,K_C) = 0$, or analogously that property $N_{k-2}$ holds for $C$ embedded with $K_C$. By assumption we know that $K_{k-1,1}(\tilde{C},K_{\tilde{C}}) = 0$, namely that property $N_{k-2}$ holds for $\tilde{C}$ embedded with $K_{\tilde{C}} $, so the thesis immediately follows from .
With maple (http://www-dimat.unipv.it/ frediani/greenfinal.tar.gz) one verifies the conjecture for $g = 5,7, 9,11$, so one can prove with the same method that if $g \geq 3$, then propery $N_0$ holds (see also [@ccm] section 2), if $g \geq 5$, then propery $N_1$ holds, if $g \geq 7$, then propery $N_2$ holds, and if $g \geq 9$, then property $N_3$ holds, and if $g \geq 11$, then property $N_4$ holds.
[99]{}
M. Aprodu, G. Farkas. Green’s Conjecture for curves on arbitrary K3 surfaces. Compositio Math. 147 (2011) 839-851.
M. Aprodu and J. Nagel, Koszul cohomology and algebraic geometry, University Lecture Series, Vol. 52, American Mathematical Society 2010. Calabri, A., Ciliberto, C., Miranda, R., The rank of the 2nd Gaussian map for general curves, Michigan Math. J. 60 (2011), no. 3, 545Ð559. Caporaso, L., Brill-Noether theory of binary curves. Mathematical Research Letters - Volume 17 - Issue 2/ March 2010 pp. 243-262.
Alessandro Chiodo, David Eisenbud, Gavril Farkas, Frank-Olaf Schreyer, Syzygies of torsion bundles and the geometry of the level l modular variety over $M_g$. arXiv:1205.0661v1. To appear in Inventiones Math.
Colombo, E., Frediani, Prym map and second gaussian map for Prym-canonical line bundles. arXiv:1105.4472v1. Advances in Mathematics (2013), http://dx.doi.org/10.1016/j.aim.2013.02.009. O. Debarre, Sur le Probleme de Torelli pour les Varietes de Prym. American Journal of Mathematics, Vol. 111, No. 1 (1989), pp. 111-134. Farkas, Gavril; Ludwig, Katharina, The Kodaira dimension of the moduli space of Prym varieties. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 755-795. Franciosi, Marco, Adjoint divisors on algebraic curves. Appendix A by Fabrizio Catanese. Adv. Math. 186 (2004), no. 2, 317Ð333. GNU GMP web site: http://www.gmplib.org/ M. Green, Koszul cohomology and the cohomology of projective varieties, Journal of Differential Geometry 19 (1984), 125-171.
A. Hirschowitz, S. Ramanan. New evidence for Green’s conjecture on syzygies of canonical curves. Ann. Sci. École Norm. Sup. 31 (1998) 145-152. E. Kaltofen and B. D. Saunders. On WiedemannÕs method of solving sparse linear systems. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, volume 539 of LNCS, pages 29Ð38, Springer, 1991.
Linbox web site: http://linalg.org/ R. Murri, A novel parallel algorithm for Gaussian Elimination of sparse unsymmetric matrices, in: Parallel Processing and Applied Mathematics 9th International Conference (PPAM 2011), Torun, Poland, September 11-14, 2011. Revised Selected Papers, Part I, volume 7203 of LNCS, Springer 2012.
Rheinfall web site: http://rheinfall.googlecode.com/ F.-O. Schreyer, Green’s conjecture for general $p$-gonal curves of large genus. Algebraic curves and projective geometry (Trento, 1988), 254-260, Lecture Notes in Math., 1389, Springer, Berlin, 1989. M. Teixidor i Bigas, Green’s conjecture for the generic $r$-gonal curve of genus $g\geq 3r-7$. Duke Math. J. 111 (2002) 195-222. W. J. Turner, Black box linear algebra with the LINBOX library, Ph.D. Thesis, University of North Carolina, 2002. C. Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface. J.European Math. Soc. 4 (2002), 363-404. C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compositio Math. 141 (2005), 1163-1190.
[^1]: Partially supported by PRIN 2009: “Moduli, strutture geometriche e loro applicazioni” and by INdAM (GNSAGA). AMS Subject classification: 14H10, 14H40, 13D02.
|
---
author:
- 'M. Lohse$^{1,2}$, C. Schweizer$^{1,2}$, O. Zilberberg$^{3}$, M. Aidelsburger$^{1,2}$, I. Bloch$^{1,2}$'
title: A Thouless Quantum Pump with Ultracold Bosonic Atoms in an Optical Superlattice
---
[**More than 30 years ago, Thouless introduced the concept of a topological charge pump [@Thouless:1983] that would enable the robust transport of charge through an adiabatic cyclic evolution of the underlying Hamiltonian. In contrast to classical transport, the transported charge was shown to be quantized and purely determined by the topology of the pump cycle, making it robust to perturbations [@Thouless:1983; @Niu:1984]. On a fundamental level, the quantized charge transport can be connected to a topological invariant, the Chern number, first introduced in the context of the integer quantum Hall effect [@Klitzing:1980; @Thouless:1982]. A Thouless quantum pump may therefore be regarded as a ’dynamical’ version of the integer quantum Hall effect. Here, we report on the realization of such a topological charge pump using ultracold bosonic atoms that form a Mott insulator in a dynamically controlled optical superlattice potential. By taking in-situ images of the atom cloud, we observe a quantized deflection per pump cycle. We reveal the genuine quantum nature of the pump by showing that, in contrast to ground state particles, a counterintuitive reversed deflection occurs when particles are prepared in the first excited band. Furthermore, we were able to directly demonstrate that the system undergoes a controlled topological phase transition in higher bands when tuning the superlattice parameters.**]{}
Charge pumping in solid-state systems has received much attention, mainly due to its potential for realizing novel current standards [@Niu:1990; @Pekola:2013], but also for characterizing many-body systems [@Splettstoesser:2005; @Marra:2015]. Quantized transport of electrons without an external bias was observed in tunnel junctions with modulated gate voltages [@Pothier:1992], in 1D channels using surface acoustic waves [@Talyanskii:1997] and in quantum dots [@Blumenthal:2007]. While the latter was used to realize an adiabatic quantum pump [@Switkes:1999; @Brouwer:1998], topological charge pumping has so far remained out of reach in condensed matter experiments. In engineered bosonic systems, non-quantized topological pumping of edge states was observed in 1D quasicrystalline photonic waveguide arrays [@Kraus:2012; @Verbin:2015]. In cold atomic systems, topological Bloch bands have been realized, ranging from the Su-Schrieffer-Heeger (SSH) model [@Atala:2013], the Hofstadter model in real space [@Aidelsburger:2013; @Miyake:2013] and synthetic dimensions [@Mancini:2015; @Stuhl:2015] to the Haldane model [@Jotzu:2014]. Their geometric features have been probed using novel interferometric [@Atala:2013; @Duca:2015] and transport probes [@Jotzu:2014; @Aidelsburger:2015] that have e.g. enabled the direct measurement of the Chern number through bulk topological currents [@Aidelsburger:2015].
Due to their versatility, ultracold atoms in optical superlattices constitute an ideal system for the implementation of quantized topological charge pumps [@Romero-Isart:2007; @Qian:2011; @Wang:2013]. A superlattice is formed by superimposing two lattices with periodicities $d{_{\mathrm{l}}}$ and $d{_{\mathrm{s}}} = \alpha d{_{\mathrm{l}}}$, $\alpha < 1$. This creates a potential $V{_{\mathrm{s}}} \sin^2 \left( \pi x / d{_{\mathrm{s}}} + \pi/2 \right) + V{_{\mathrm{l}}} \sin^2 \left( \pi x / d{_{\mathrm{l}}} - \varphi/2 \right) $, where $V{_{\mathrm{s}}}$ ($V{_{\mathrm{l}}}$) denotes the depth of the short (long) lattice, respectively. The relative position of the lattices is determined by the variable phase $\varphi$. When applying Bloch’s theorem, the resulting single-particle Hamiltonian $\hat{\mathcal{H}} (k_x, \varphi)$ is periodic in both the quasi-momentum $k_x \in \left]-\pi/d{_{\mathrm{l}}},\pi/d{_{\mathrm{l}}}\right]$ and the superlattice phase $\varphi$.
A cyclic pumping scheme can be realized by adiabatically varying $\varphi$ where one cycle corresponds to a change of $2\pi$, i.e. moving the long lattice by $d{_{\mathrm{l}}}$. A particle which is initially in an eigenstate ${\left| u_n (k_x, \varphi_0) \right>}$ of $\hat{\mathcal{H}}(k_x, \varphi_0)$ in the $n$-th band follows the corresponding instantaneous eigenstate ${\left| u_{n} (k_x, \varphi) \right>}$. Even with perfect adiabaticity, however, the particle acquires a small admixture of states from other bands, proportional to $\partial_t \varphi$. This gives rise to an anomalous velocity $\dot{x}_n = \Omega _n \partial_t \varphi$, determined by the Berry curvature $\Omega_n (k_x, \varphi) = i \left( \langle \partial_{\varphi} u_n | \partial_{k_x} u_n \rangle - \langle \partial_{k_x} u_n | \partial_{\varphi} u_n \rangle \right)$ [@Karplus:1954; @Xiao:2010]. Hence, changing $\varphi$ induces a motion of the particle which, depending on the sign of $\Omega_n$, is either in the same or opposite direction as the moving lattice. The displacement after one cycle is obtained by integrating $\dot{x}_n$ and can in principle take any arbitrary value.
For a filled or homogeneously populated band, however, the average displacement of the entire cloud per cycle can be related to a 2D topological invariant, the Chern number $\nu_n$ of the pumping process:
$$\label{eq:ChernNumber}
\nu_n = \frac{1}{2 \pi} \int{_{\mathrm{FBZ}}} \int_0^{2 \pi} \Omega_n (k_x, \varphi) \ \mathrm{d} \varphi \mathrm{d}k_x$$
Here, FBZ denotes the first Brillouin zone of the superlattice. Such a case can be realized with both fermions, by placing the Fermi energy in a band gap, and bosons, by preparing a Mott insulator in the $n$-th band as in our experiment. During one cycle, the cloud’s center-of-mass (COM) position changes by $\nu_n d{_{\mathrm{l}}}$ and is quantized in units of $d{_{\mathrm{l}}}$ because $\nu_n$ can only take integer values. Unlike in classical systems, this motion can be faster ($\nu_n > 1$) or even opposite ($\nu_n < 0$) compared to the one of the underlying lattice [@Wei:2015]. As the displacement is proportional to a topological invariant, it neither depends on the pumping speed, provided adiabaticity still holds, nor on the specific lattice parameters as long as band crossings do not occur. Hence, the transport is highly robust against perturbations such as interaction effects in fermionic systems or disorder [@Niu:1984].
The connection to a Chern topological invariant shows that charge pumping in a 1D superlattice is closely related to the integer quantum Hall effect, where charged particles move in 2D in the presence of a perpendicular magnetic field. Indeed, the adiabatic variation of $\varphi$ is equivalent to the threading of magnetic flux through a cylinder [@Thouless:1983] that generates an electric field in the orthogonal direction and leads to the quantized Hall conductance [@Thouless:1982]. A direct analogy between these systems can be made in two limiting cases: the quantum sliding lattice with $V{_{\mathrm{s}}} \rightarrow 0$ and the deep tight-binding limit with $V{_{\mathrm{s}}} \gg V{_{\mathrm{l}}}^2/(4 E{_{\mathrm{r,s}}})$. The former matches with the free particle limit of Landau levels that are associated with Chern numbers $\nu_n = +1$. In this limit, one can understand the pumping by following Laughlin’s argument [@Laughlin:1981], where the threading of magnetic flux leads to a sliding of the localized oscillator centers. In the latter, one obtains the generalized 1D Harper model [@Harper:1955; @Roux:2008] which, using dimensional extension [@Kraus:2012a; @Kraus:2013], can be mapped onto the 2D Harper-Hofstadter-Hatsugai (HHH) model describing non-interacting particles on a 2D square lattice with a uniform magnetic flux $2 \pi \alpha$ and nearest as well as next-nearest-neighbor hopping [@Harper:1955; @Azbel:1964; @Hofstadter:1976; @Hatsugai:1990] (see Supplementary Material). In this mapping, $\varphi$ corresponds to the transverse quasi-momentum $k_y$. In the following, unless mentioned otherwise, we will focus on the Wannier tunneling regime $V{_{\mathrm{s}}} \gtrsim V{_{\mathrm{l}}}^2/(4 E{_{\mathrm{r,s}}})$ where – even though the direct mapping to the 1D Harper model breaks down – the pump is characterized by the same Chern number distribution.
![Topological charge pumping in an optical superlattice. **(a)** Superlattice potential created by superimposing two lattices with periodicities $d{_{\mathrm{l}}}$ and $d{_{\mathrm{s}}} = d{_{\mathrm{l}}}/2$. Here, $J_1$ and $J_2$ denote the tunnel couplings and $\Delta$ the energy offset between neighboring sites. **(b)** Evolution of the ground state Wannier function (red) during the first half of a pumping cycle. The long lattice is shifted to the right when increasing the superlattice phase $\varphi$. An atom initially localized in a symmetric superposition on a double well tunnels to the lower lying site and thereby follows the motion of the long lattice in a quantized fashion. A classical particle, on the other hand, stays at a fixed position as the individual sites do not move. **(c)** Pumping cycle in the $(J_1-J_2)$-$\Delta$ parameter space. Varying $\varphi$ from 0 to $2\pi$ corresponds to a closed path around the degeneracy point at the origin where $\Delta = 0$ and $J_1 = J_2$. **(d)** Berry curvature $\Omega_1$ of the lowest band as a function of $\varphi$ and the quasi-momentum $k_x$ for a lattice with $V{_{\mathrm{s}}} = 10\,E{_{\mathrm{r,s}}}$ and $V{_{\mathrm{l}}} = 20\,E{_{\mathrm{r,l}}}$, where $E_{r,i} = h^2/(2m{_{\mathrm{a}}} \lambda_i^2)$ denotes the corresponding recoil energy, $m{_{\mathrm{a}}}$ the mass of an atom and $\lambda_i$ the respective wavelength. The panel below shows the Berry curvature averaged along $k_x$, as seen by a particle localized in a single double well, which is peaked at the symmetric double well configurations $\varphi = l \pi, l \in \mathbb{Z}$. The graph on the right shows $\Omega_1(k_x)$ for $\varphi = 0$. \[fig:1\]](Figures/Fig1.pdf){width="\linewidth"}
In the experiment, we use a superlattice with $d{_{\mathrm{l}}} = 2 d{_{\mathrm{s}}}$ and two sites per unit cell ([Fig. \[fig:1\]]{}a). In the tight-binding limit, it is described by the Rice-Mele Hamiltonian [@Rice:1982]
$$\begin{aligned}
\label{eq:RiceMele}
\hat{H}(\varphi) = &- \sum_{m} \left( J_{1}(\varphi) \hat{b}_{m}^{\dagger} \hat{a}^{\phantom \dagger}_{m} + J_{2}(\varphi) \hat{a}_{m+1}^{\dagger} \hat{b}^{\phantom \dagger}_{m} + \mathrm{h.c.} \right) \\
&+ \frac{\Delta(\varphi)}{2} \sum_{m} \left( \hat{a}_{m}^{\dagger} \hat{a}^{\phantom \dagger}_{m} - \hat{b}_{m}^{\dagger} \hat{b}^{\phantom \dagger}_{m} \right)
\end{aligned}$$
where $\hat{a}_m^{\dagger}$ ($\hat{a}^{\phantom \dagger}_m$) and $\hat{b}_m^{\dagger}$ ($\hat{b}^{\phantom \dagger}_m$) are the creation (annihilation) operators acting on the even (left) and odd (right) site of the $m$-th unit cell, respectively. $J_1$, $J_2$ denote the tunnel couplings and $\Delta$ the energy offset between neighboring sites. For $\Delta = 0$, this Hamiltonian reduces to the SSH model [@Su:1979].
The mechanism underlying the pumping can also be understood on a microscopic level. Shifting the phase $\varphi$ changes the shape of the potential ([Fig. \[fig:1\]]{}b) and modifies $J_1$, $J_2$ and $\Delta$ periodically ([Fig. \[fig:1\]]{}c). At $\varphi=0$ ($\Delta = 0$, $J_1 > J_2$), a ground-state particle localized in a single double well is initially in a symmetric superposition of residing on both the left and right sites. With increasing $\varphi$, i.e. shifting the long lattice to the right, the double wells are tilted ($\Delta > 0$) and the atom tunnels to the lower lying site on the right. The sign of $J_1 - J_2$ is reversed at $\varphi = \pi/2$, where the tilt is largest, and at $\varphi = \pi$ the lattice forms symmetric double wells again, but shifted by one short lattice constant to the right. The atom, which remained on the lower site for large $\Delta$, delocalizes over the shifted double well as $\Delta$ becomes comparable to $J_2$ and therefore has moved by $d{_{\mathrm{l}}}/2$ during the first half of the pumping cycle. In the second half, the same procedure is repeated, but shifted by one site. After one cycle, the lattice configuration is identical to the starting point, but the atom ends up in the double well next to the initial one. In contrast to this, a classical particle would not move because the positions of the individual sites do not change. This illustrates the importance of quantum tunneling for the pumping.
During the pump cycle, the system moves along a closed trajectory in the ($J_1-J_2)$-$\Delta$ parameter space. It encircles the degeneracy point at $\Delta = 0 $ and $J_1 = J_2$, where the two bands of the Rice-Mele model touch, and smoothly connects the topologically distinct phases $J_1 < J_2$ and $J_1 > J_2$ of the SSH model ([Fig. \[fig:1\]]{}c). The Berry curvature and thus the motion of the atoms is peaked around $\varphi = l \pi, l \in \mathbb{Z}$ where the tilt changes sign and the atoms tunnel to the neighboring sites ([Fig. \[fig:1\]]{}d). The Chern numbers of the pump cycles in the two bands of this model are $\nu_1 = +1$ and $\nu_2 = -1$, giving the same Chern number distribution as the HHH model with $\alpha = 1/2$ [@Hatsugai:1990]. Their sum vanishes because these bands emerge from the topologically trivial lowest Bloch band of the short lattice.
The experimental sequence starts by preparing an $n=1/2$ Mott insulator of $^{87}$Rb atoms in a 3D optical lattice with at most one atom per unit cell in the ground state of symmetric double wells with $\varphi = 0$ and $J_1 \gg J_2$ (see Methods). Due to the large on-site interaction, each atom is localized on an individual double well, resulting in a homogeneous delocalization over the entire first Brillouin zone. The pumping is performed by adiabatically shifting the phase $\varphi$ of the long lattice (see Methods) and the resulting motion of the atoms is tracked by measuring the COM position of the cloud in situ. The displacement during one cycle is indeed quantized and occurs in steps ([Fig. \[fig:2\]]{}a) – unlike the underlying linear motion of the long lattice. The cloud moves by one lattice constant $d{_{\mathrm{l}}}$ per cycle as expected for $\nu_1 = +1$ and the steps appear around $\varphi = l \pi, l \in \mathbb{Z}$, where the atoms tunnel from one side of the double wells to the other. When performing multiple cycles, the cloud keeps moving to the right whereas it propagates in the opposite direction for the reversed pumping direction $\varphi < 0$ ([Fig. \[fig:2\]]{}b). The small deviation from the expected displacement for the motion of ideal Wannier functions can be attributed to a finite pumping efficiency due to non-adiabatic band transitions and the additional trapping potential.
![Transition from a quantum sliding lattice to the Wannier tunneling limit for the lowest band. Differential deflection $\Delta x$ after one pump cycle for positive and negative pumping direction for various lattice depths $V{_{\mathrm{s}}}$ at $V{_{\mathrm{l}}} = 25(1)\,E{_{\mathrm{r,l}}}$. Each point consists of ten data sets comparing the COM position of ten averaged images for both directions. The error bars depict the error of the mean. For the data points in the tight-binding regime, the insets show the corresponding pump cycles in the ($J_1-J_2)$-$\Delta$ parameter space. For $V{_{\mathrm{s}}} = 8\,E{_{\mathrm{r,s}}}$, the two-band model breaks down for large tilts such that $J_1$, $J_2$ and $\Delta$ are not well-defined. The dashed line therefore connects the points where the gap between the second and third band becomes smaller than $10J_1$ for $\varphi = 0$. \[fig:3\]](Figures/Fig3.pdf)
The step-like transport behavior can also be observed in site-resolved band mapping measurements (inset of [Fig. \[fig:2\]]{}b) which determine the number of atoms on even and odd sites. As for the COM position, a step occurs in the even-odd distribution whenever a symmetric double well configuration is crossed at $\varphi = l \pi, l \in \mathbb{Z}$. Using the measured even-odd fractions, one can estimate the transfer efficiency, i.e. the fraction of atoms transferred from site $i$ to $i+1$ at each step. This is equivalent to the fraction staying in the lowest band during one half of the pumping cycle and allows to quantify the adiabaticity of the pumping protocol. From our data we obtain an efficiency of 98.7(1)% (see Supplementary Material). With the same model, the ideal COM displacement can be fitted to the measured positions which yields an efficiency of $97.9(2)\%$ with the small additional reduction most likely being caused by the trap.
Due to the topological nature of the pumping, the displacement per cycle for the lowest band does not depend on the path in the $(J_1 - J_2)$-$\Delta$ plane as long as it encompasses the degeneracy point. Moreover, it is independent of $V{_{\mathrm{s}}}$ since the sliding lattice and the tight-binding Thouless pump are topologically equivalent for the first band and connected by a smooth crossover without closing the gap to the second band. To verify this, we measured the deflection of the cloud with $V{_{\mathrm{l}}} = 25(1)\,E{_{\mathrm{r,l}}}$ for various values of $V{_{\mathrm{s}}}$. For all parameters, the resulting displacements are consistent within the error bars ([Fig. \[fig:3\]]{}).
![Cloud displacement and site occupations for the first excited band with $V{_{\mathrm{s}}} = 10.0(3)\,E{_{\mathrm{r,s}}}$ and $V{_{\mathrm{l}}} = 20(1)\,E{_{\mathrm{r,l}}}$. The main plot shows the evolution of the COM position for up to 1.5 pump cycles. The data points are averaged over ten data sets with the errors being the error of the mean. The dashed black line indicates the motion of a localized Wannier function of the second band and the gray line is a fit to the data using the same model as in [Fig. \[fig:2\]]{}b, giving a transfer efficiency of 97(2)%. The inset illustrates the evolution of the Wannier functions of the first and second band during a pump cycle, showing a deflection in opposite directions. The lower plot shows the imbalance of the fraction of atoms on even ($n{_{\mathrm{e}}}$) and odd sites ($n{_{\mathrm{o}}}$) averaged over 3-6 measurements each with the error bars depicting the standard deviation. The gray line is a fit of the even-odd distribution of the corresponding Wannier functions with a pumping efficiency of 96.7(3)% (see Supplementary Material). \[fig:4\]](Figures/Fig4.pdf)
The excited band in the Rice-Mele model exhibits counter-propagating charge pumping with $\nu_2 = -1$, i.e. the atoms are expected to move in the opposite direction as the long lattice. This underlines the pump’s intrinsic quantum mechanical character as such a motion could not occur for classical particles. To study this, the atoms were prepared in the second band at $\varphi = 0$ (see Methods) and pumping was performed with identical parameters as for the first band in [Fig. \[fig:2\]]{}. When moving the lattice to the right ($\varphi > 0$), the cloud indeed shifts to the left with $x < 0$ ([Fig. \[fig:4\]]{}). This is further confirmed by the measured site occupations, showing that the behavior is exactly reversed compared to the lowest band. While the first band is localized on the lower site for large $\Delta$, atoms in the second band are found on the upper site and are therefore transported in the opposite direction. The slightly larger deviation from theory is mostly due to the finite lifetime of atoms in the higher band. This leads to a lower transfer efficiency of 96.7(3)% obtained from the site populations, in agreement with the value from the fit to the COM positions of 97(2)% (see Supplementary Material).
By varying $V{_{\mathrm{s}}}/V{_{\mathrm{l}}}^2$, one can study the transition between the topologically very different regimes of the sliding long lattice with $V{_{\mathrm{s}}} = 0\,E{_{\mathrm{r,s}}}$ and the Wannier tunneling limit for $V{_{\mathrm{s}}} > V{_{\mathrm{l}}}^2/(4 E{_{\mathrm{r,s}}})$. For the latter, the Chern numbers of the bands alternate between $\nu_{2n+1} = +1$ and $\nu_{2n} = -1$ for $d{_{\mathrm{l}}} = 2 d{_{\mathrm{s}}}$, like in the HHH model with a flux of $\pi$. This causes the opposite deflections for the first and second band observed in [Fig. \[fig:2\]]{} and [Fig. \[fig:4\]]{}. In the other limit $V{_{\mathrm{s}}} \rightarrow 0\,E{_{\mathrm{r,s}}}$, however, each band corresponds to a single Landau level with $\nu_n = +1$. Between these two limiting cases, as $V{_{\mathrm{s}}}$ is decreased, an infinite series of topological phase transitions occurs where two bands touch and exchange their Chern numbers ([Fig. \[fig:5\]]{}a). Thereby the negative $\nu_n$ are successively transferred to higher bands and they become positive for the lower bands.
![Topological phase transition in the first excited band. **(a)** Band structure at $\varphi = 0.5\pi$, where the band crossings occur, and Chern number distribution versus $V{_{\mathrm{s}}}$ with $V{_{\mathrm{l}}} = 25\,E{_{\mathrm{r,l}}}$. The shaded areas illustrate the width of the Bloch bands, which were calculated by numerical diagonalization of $\hat{\mathcal{H}}(k_x, \varphi)$. The color indicates the Chern number of the corresponding pump cycle with red being $+1$ and blue $-1$. Starting from alternating Chern numbers in the Wannier tunneling limit for large $V{_{\mathrm{s}}}$, the negative Chern numbers successively propagate towards higher bands in a series of topological phase transitions when lowering $V{_{\mathrm{s}}}$, giving rise to a uniform distribution with $\nu_n = +1$ in the Landau limit of a sliding long lattice. The circle highlights the crossing of the 2nd and 3rd band leading to the topological transition studied in b. **(b)** Differential deflection $\Delta x$ between single pump cycles in opposite directions for the second band as a function of the lattice depths $V{_{\mathrm{s}}}$ and $V{_{\mathrm{l}}}$. At small $V{_{\mathrm{s}}}$, $\nu_2 = +1$ and atoms move in the same direction as the lattice, but as $V{_{\mathrm{s}}}$ increases a topological transition occurs at $V{_{\mathrm{s}}} =V{_{\mathrm{l}}}^2/(4 E{_{\mathrm{r,s}}})$, indicated by the black line, where $\nu_2$ suddenly changes to $-1$ and the direction of motion is reversed. Each square corresponds to one data set averaged over 10-12 pairs of images. \[fig:5\]](Figures/Fig5.pdf)
While the deflection in the first band is independent of $V{_{\mathrm{s}}}$, there is a transition for the second band where the gap to the third band closes and $\nu_2$ changes from $-1$ to $+1$. This transition can be mapped out by measuring the displacement of the cloud as a function of the lattice depths $V{_{\mathrm{s}}}$ and $V{_{\mathrm{l}}}$ ([Fig. \[fig:5\]]{}b). The direction of the motion reverses when crossing the transition, which occurs at $V{_{\mathrm{s}}} =V{_{\mathrm{l}}}^2/(4 E{_{\mathrm{r,s}}})$ in the tight-binding limit. For small $V{_{\mathrm{s}}}$, atoms in the second band move in the same direction as the lattice, whereas they move in the opposite directionin the Wannier tunneling regime.
In conclusion, we have demonstrated the implementation of a topological charge pump using ultracold atoms. Unlike previous experiments studying non-quantized pumping of edge states, we observe a quantized response in the bulk. Combining such a pump with novel techniques for engineering optical potentials at the single-site level [@Weitenberg:2011; @Preiss:2015] would allow for a direct observation of edge states in finite systems [@Kitagawa:2012] and to study their transport properties [@Kraus:2012]. Furthermore, by adjusting the ratio of the lattice constants $\alpha = d{_{\mathrm{s}}}/d{_{\mathrm{l}}}$, one can realize a wide variety of commensurate and incommensurate superlattices where the Chern number of the lowest band can in principle take arbitrary integer values. In addition to the negative deflection for $\nu_n < 0$ shown here, another counterintuitive case of charge pumping can occur in these systems where the atoms move faster than the long lattice for $\nu_n > 1$ [@Wei:2015]. By adding a spin degree of freedom, the pumping scheme can be used to implement the $Z_2$ spin pump [@Shindou:2005; @Fu:2006] in a spin-dependent superlattice [@Lee:2007]. Moreover, extending the Thouless pump to 2D systems would enable the realization of an analogon of the 4D integer quantum Hall effect [@Zhang:2001; @Kraus:2013].\
#### Note
Recently, we became aware of similar work by S. Nakajima et al. [@Nakajima:2015] implementing the Thouless pump with fermionic atoms.\
We acknowledge insightful discussions with F. Grusdt and S. Kohler. This work was supported by NIM and the EU (UQUAM, SIQS). M. L. was additionally supported by ExQM and O. Z. by the Swiss National Science Foundation.
Methods {#methods .unnumbered}
=======
[**Initial state preparation in the first band**]{} All sequences started by loading an $n=1$ Mott insulator of about 3000 $^{87}$Rb atoms in the lowest band of a 3D optical lattice. The lattice was created by three orthogonal standing waves with wavelengths $\lambda{_{\mathrm{l}}} = 1534$nm for the long lattice along the $x$-direction and $\lambda_{y} = 767$nm and $\lambda_{z} = 844$nm along the $y$- and $z$-axes, respectively. They were ramped up in 150ms to a depth of $V_i = 30(1)\,E_{r,i}, i \in \{l,y, z \}$. The superlattice potential was created by adding another lattice with $\lambda{_{\mathrm{s}}} = \lambda{_{\mathrm{l}}}/2$ along the $x$-direction. To prepare the atoms in the lowest band of the superlattice, the lattice sites along the $x$-axis were split symmetrically with $\varphi = 0.00(1)\pi$ by ramping up the short lattice to $V{_{\mathrm{s}}} = 10.0(3)\,E{_{\mathrm{r,s}}}$ within 10ms while simultaneously lowering $V{_{\mathrm{l}}}$ to $20(1)\,E{_{\mathrm{r,l}}}$ such that $J_1 \gg J_2$. It was verified with direct band mapping measurements that the atoms are homogeneously distributed over the entire band.
[**Initial state preparation in the second band**]{} For the preparation in the second band, the splitting was instead performed at $\varphi = 0.11(1)\pi$ and $V{_{\mathrm{s}}}$ was ramped to 30(1)$E{_{\mathrm{r,s}}}$. The phase was then changed non-adiabatically to $\varphi = -0.08(1)\pi$ in 20ms to transfer all atoms to the excited band. After that, the lattices along $x$ were lowered to $V{_{\mathrm{s}}} = 10.0(3)\,E{_{\mathrm{r,s}}}$ and $V{_{\mathrm{l}}}$ to $20(1)\,E{_{\mathrm{r,l}}}$, respectively, in 2ms and the phase was moved to $\varphi = 0.00(1) \pi$ in 10ms. This brings the atoms into the excited state of symmetric double wells with almost perfect efficiency.
[**Sequence for pumping**]{} The pumping cycle was implemented experimentally by slightly changing the laser frequency and thereby shifting the phase $\varphi$ of the long lattice along the $x$-direction. For this, two separate lasers were used with one laser covering the range from $-0.50(1)\pi$ to $0.62(1)\pi$ and the second laser from $0.62(1)\pi$ to $1.50(1)\pi$ (see Supplementary Material).
**Supplementary Material for:**\
****\
M. Lohse$^{1,2}$, C. Schweizer$^{1,2}$, O. Zilberberg$^{3}$, M. Aidelsburger$^{1,2}$, I. Bloch$^{1,2}$\
\
\
Mapping to the Hofstadter model in the deep tight-binding regime
================================================================
In the tight-binding approximation, the dynamics of ultracold atoms in an optical superlattice potential with arbitrary lattice constants $d{_{\mathrm{l}}}$ and $d{_{\mathrm{s}}} = \alpha d{_{\mathrm{l}}}$, $\alpha < 1$ can be described by the following Hamiltonian:
$$\begin{aligned}
\label{eq:SLHamiltonian}
\hat{H}(\varphi) = &- \sum_{m} \left( \left(J_0 + \delta J^{\phantom \dagger}_m (\varphi) \right) \hat{a}_{m+1}^{\dagger} \hat{a}^{\phantom \dagger}_{m} + \mathrm{h.c.} \right) \\
&+ \sum_{m} \Delta^{\phantom \dagger}_m(\varphi) \hat{a}_{m}^{\dagger} \hat{a}^{\phantom \dagger}_{m}
\end{aligned}$$
Here, $\hat{a}_m^{\dagger}$ ($\hat{a}^{\phantom \dagger}_m$) is the creation (annihilation) operator acting on the $m$-th lattice site and $J$ denotes the hopping amplitude between neighboring sites in the short lattice. $\delta J_m$ is the modulation of the hopping amplitude and $\Delta_m$ the on-site energy of the $m$-th site, both of which are induced by the long lattice and depend on $\varphi$. For $V{_{\mathrm{s}}} \gg V{_{\mathrm{l}}}^2/(4 E{_{\mathrm{r,s}}})$, [Eq. \[eq:SLHamiltonian\]]{} reduces to a generalized 1D Harper model [@S:Harper:1955; @S:Roux:2008] with
$$\begin{aligned}
\label{eq:HarperParameters}
\delta J_m (\varphi) &= \frac{\delta J}{2} \cos \bigl( 2\pi \alpha m - \varphi \bigr) \\
\Delta_m (\varphi) &= -\frac{\Delta}{2} \cos \bigl( 2\pi \alpha (m-1/2) - \varphi \bigr)
\end{aligned}$$
![Square lattice in the presence of a uniform magnetic field with nearest-neighbor hopping $J_x = J_0, J_y = \Delta/4$ and next-nearest-neighbor hopping $J_d = \delta J /4$ using the gauge of [Eq. \[eq:3hHamiltonian\]]{}.[]{data-label="fig:S1"}](Figures/FigS1.pdf)
Following the approach of dimensional extension [@S:Kraus:2012a], the periodic parameter $\varphi$ can be regarded as an extra dimension, i.e. a realization for a given $\varphi$ corresponds to a 1D subset of a 2D system. The pumping then samples the full 2D parameter space which is the reason why the displacement can be related to a 2D topological invariant. The 2D Hamiltonian onto which the 1D superlattice maps in this limit can be found by thinking of the 1D Hamiltonian as a single Fourier component of the 2D one and performing an inverse Fourier transform. For this, we relabel the creation and annihilation operators as $\hat{a}^{\dagger}_{m, \varphi}$ and $\hat{a}^{\phantom \dagger}_{m, \varphi}$ and define a 2D Hamiltonian
$$\label{eq:2DHamiltonian}
\hat{H}{_{\mathrm{2D}}} = \frac{1}{2\pi} \int_0^{2\pi} \hat{H}(\varphi) \,\mathrm{d\varphi}$$
Expanding $\hat{a}^{\dagger}_{m, \varphi}$ and $\hat{a}^{\phantom \dagger}_{m, \varphi}$ into their Fourier components, $\hat{a}^{\dagger}_{m,\varphi} = \sum_n e^{i \varphi n} \hat{a}^{\dagger}_{m,n}$ and $\hat{a}^{\phantom \dagger}_{m,\varphi} = \sum_n e^{-i \varphi n} \hat{a}^{\phantom \dagger}_{m,n}$, gives
$$\begin{aligned}
\label{eq:3hHamiltonian}
\hat{H}{_{\mathrm{2D}}} = - &\sum_{m,n} J_0 \hat{a}_{m+1,n}^{\dagger} \hat{a}^{\phantom \dagger}_{m,n} \\
- &\sum_{m,n} \frac{\Delta}{4} e^{+i 2\pi \alpha (m - 1/2)} \hat{a}_{m,n+1}^{\dagger} \hat{a}^{\phantom \dagger}_{m,n} \\
- &\sum_{m,n} \frac{\delta J}{4} e^{+i 2\pi \alpha m} \hat{a}_{m+1,n+1}^{\dagger} \hat{a}^{\phantom \dagger}_{m,n} \\
- &\sum_{m,n} \frac{\delta J}{4} e^{-i 2\pi \alpha m} \hat{a}_{m+1,n}^{\dagger} \hat{a}^{\phantom \dagger}_{m,n+1} \\
+ &\mathrm{\,h.c.}
\end{aligned}$$
which is precisely the 2D Harper-Hofstadter-Hatsugai model describing non-interacting particles on a 2D square lattice in the presence of a uniform magnetic flux $2 \pi \alpha$ per square plaquette with nearest and next-nearest neighbor tunneling as illustrated in [Fig. \[fig:S1\]]{} [@S:Hatsugai:1990]. Under this mapping, the superlattice phase $\varphi$ corresponds to the transverse quasi-momentum $k_y$ and in this sense the pumping protocol is equivalent to performing Bloch oscillations in the 2D lattice by applying a gradient. For our particular case of $d{_{\mathrm{s}}} = d{_{\mathrm{l}}}/2$, one obtains a flux of $\pi$ per plaquette.
Mapping of the sliding lattice to Landau levels
===============================================
In the opposite limit of the sliding lattice with $V{_{\mathrm{s}}} \rightarrow 0$, a similar analogy can be made to the Landau levels of a free particle in an external magnetic field in 2D. The Hamiltonian of a single particle in the long lattice reads
$$\label{eq:LongLatticeHamiltonian}
\hat{H}{_{\mathrm{l}}} = \frac{\hat{p}_x^2}{2 m{_{\mathrm{a}}}} + V{_{\mathrm{l}}} \sin^2 \left( \pi \alpha \hat{x} / d{_{\mathrm{s}}} - \varphi/ 2 \right)$$
For a deep lattice and when only considering the lowest bands, the lattice potential can be expanded in orders of $\hat{x}- x_m$ around each lattice site $x_m$ (i.e. the minimum of the potential). In second order this gives:
$$\label{eq:LLHamiltonianExp}
\hat{H}{_{\mathrm{l}}} = \frac{\hat{p}_x^2}{2 m{_{\mathrm{a}}}} + V{_{\mathrm{l}}} \frac{\pi^2 \alpha^2}{d{_{\mathrm{s}}}^2} \sum_m (\hat{x} - x_m)^2$$
Neglecting the coupling between neighboring sites, this splits into a series of decoupled Hamiltonians for each lattice site, each of which is a 1D harmonic oscillator.
On the other hand, the motion of a particle with charge $q$ in an external magnetic field $\mathbf{B} = B \bf{e}_z$ along the $z$-direction is described by the Hamiltonian
$$\begin{aligned}
\label{eq:LandauHamiltonian}
\hat{H}{_{\mathrm{LL}}} &= \frac{1}{2m{_{\mathrm{a}}}} \left( \mathbf{\hat{p}} - q \hat{\mathbf{A}} \right)^2 \\
&= \frac{\hat{p}{_{\mathrm{x}}}^2}{2m{_{\mathrm{a}}}} + \frac{1}{2m{_{\mathrm{a}}}} \left( \hat{p}{_{\mathrm{y}}} - q B\hat{x} \right)^2
\end{aligned}$$
with the vector potential $\hat{\mathbf{A}} = B \hat{x} \bf{e}{_{\mathrm{y}}}$ given in the Landau gauge. As the above Hamiltonian commutes with the operator for the transverse momentum $\hat{p}_y$, one can define a common set of eigenstates of both $\hat{H}{_{\mathrm{LL}}}$ and $\hat{p}_y$. For a given state with momentum $\hbar k_y$, [Eq. \[eq:LandauHamiltonian\]]{} can also be rewritten as a 1D harmonic oscillator
$$\label{eq:ReducedLandauHamiltonian}
\hat{H}{_{\mathrm{LL}}} = \frac{\hat{p}{_{\mathrm{x}}}^2}{2m{_{\mathrm{a}}}} + \frac{1}{2} m{_{\mathrm{a}}} \omega^2{_{\mathrm{c}}} \left( \hat{x} - \frac{\hbar k{_{\mathrm{y}}}}{m{_{\mathrm{a}}} \omega{_{\mathrm{c}}}}\right)^2$$
where $\omega{_{\mathrm{c}}} = qB/m{_{\mathrm{a}}}$ is the cyclotron frequency. Comparing this with [Eq. \[eq:LLHamiltonianExp\]]{}, one can see that the state in the $n$-th band localized at the lattice site $x_m$ corresponds to the state in the $n$-th Landau level with a transverse momentum $\hbar k_y = m{_{\mathrm{a}}} \omega{_{\mathrm{c}}} x_m$. As in the tight-binding regime, the number of magnetic flux quanta per unit cell of the (non-existent) short lattice is given by the ratio of the lattice constants $\alpha = d{_{\mathrm{s}}}/d{_{\mathrm{l}}}$ in this mapping.
Topological phase transition
============================
For $d{_{\mathrm{s}}} = d{_{\mathrm{l}}}/2$, the phase transition to the Wannier tunneling regime, where the pump is characterized by the same Chern number distribution as the 2D HHH model, occurs when the second and third band cross at the staggered configuration $\varphi = \pi/2$ during the pumping cycle. In the tight binding limit, the crossing point can be determined by comparing the energy of the first excited state on the lower site with the one of the ground state on the higher site. For a sufficiently deep short lattice, they can be approximated by harmonic oscillator states with $E_{n} = (n+1/2) \hbar \omega$. The on-site trapping frequency $\omega$ is given by $$\omega=\sqrt{\frac{2 \pi^{2}V{_{\mathrm{s}}}}{m{_{\mathrm{a}}} d{_{\mathrm{s}}}^2}}=\frac{2}{\hbar}\sqrt{V{_{\mathrm{s}}}E{_{\mathrm{r,s}}}}
\label{eq:trappingfrequency}$$ At $\varphi = \pi/2$, the tilt is largest with $\Delta=V{_{\mathrm{l}}}$ and the bands touch if $\Delta=\hbar\omega$. Hence, the transition occurs at $$V{_{\mathrm{l}}}=2\sqrt{V{_{\mathrm{s}}}E{_{\mathrm{r,s}}}}
\label{eq:crossing1st2ndband}$$
A similar derivation can be made for $d{_{\mathrm{s}}} = d{_{\mathrm{l}}}/(2j), j \in \mathbb{N}$ which gives the same transition point. For all other ratios, [Eq. \[eq:crossing1st2ndband\]]{} gives a lower bound of $V{_{\mathrm{l}}} (V{_{\mathrm{s}}})$ and the transition will in general take place at slightly larger $V{_{\mathrm{l}}}$.
Experimental sequence
=====================
![Schematic of the experimental sequence with the lattice and phase ramps for the initial state preparation as well as for two pumping cycles. The light and dark lines indicate the two different lasers used sequentially to overcome tuning range limitations. The phase $\varphi$ is shown modulo $2\pi$.[]{data-label="fig:S2"}](Figures/FigS2.pdf){width="\linewidth"}
Figure \[fig:S2\] illustrates the detailed experimental sequence for the initial state preparation of the atoms in the lowest band of the superlattice potential and two exemplary pump cycles. As discussed in the Methods of the main text, the sequence started by loading an $n=1$ Mott insulator in $150$ms in a 3D optical lattice created by the long lattice and two orthogonal standing waves with depth $V_i = 30(1)\,E_{r,i}, i \in \{l,y,z\}$. Subsequently, the long lattice sites were split symmetrically by ramping up the overlapped short lattice with $\lambda{_{\mathrm{s}}}=\lambda{_{\mathrm{l}}}/2$ to $V{_{\mathrm{s}}} = 10.0(3)\,E{_{\mathrm{r,s}}}$ at $\varphi = 0$. Simultaneously, the long lattice was lowered to $V{_{\mathrm{l}}}=20(1)\,E{_{\mathrm{r,l}}}$ such that $J_1 \gg J_2$.
For the pumping, the phase of the long lattice was moved by changing the laser frequency slightly. Due to the limited tuning range of a single laser, a successive hand over between two independent laser systems (indicated with light and dark lines in Fig. \[fig:S2\] for the different lasers) was implemented in order to extend the pumping range to arbitrary values. Each pump cycle consists of four segments with $s$-shaped phase ramps $\varphi \in [0, 0.62\pi],[0.62\pi, \pi],[\pi, 1.5\pi]$ and $[1.5\pi, 2\pi]$ of 50ms duration to minimize the probability for non-adiabatic transitions to higher bands at the symmetric configurations $\varphi = l \pi, l \in \mathbb{Z}$. For the measurements in Fig. 3 and Fig. 5 of the main text, the ramp time was scaled with $V{_{\mathrm{s}}}$ as the tunneling rates and thus the band gap decrease with larger $V{_{\mathrm{s}}}$. The switching between the lasers at $0.62\pi$ and $1.5\pi$ was done instantaneously within a 3ms hold time between the ramps. It was confirmed experimentally that this switch does not lead to any measurable excitation to other bands. This scheme was used for all measurements with the exception of the points with $\varphi < 1.5\pi$ in Fig. 2a of the main text, where a single laser was used with one ramp for $\varphi\leq \pi$ and two ramps for $\varphi > \pi$, with a duration of 100ms for each ramp.
Models for finite pumping efficiency
====================================
When transitions to other bands occur during the pumping process, the deflection of the cloud can be changed due to the different Chern numbers of the bands. For the measurements in the Wannier tunneling limit, we model the band occupations in a simple two-band model assuming that a small fraction of atoms is excited non-adiabatically to the other band during each cycle. In the two-band model, these transitions will predominantly take place when ramping over the symmetric double well configuration where the gap between the two lowest bands is smallest and atoms tunnel from one site to the next. This situation occurs twice within one pump cycle and we define the transfer efficiency as the fraction of atoms staying in the initially populated band during one ramp over a symmetric double well, which corresponds to half a pump cycle and thus one step in the COM position.
The band populations after the $m$-th step can be expressed as
$$\boldsymbol{n}^{(m)}=\boldsymbol{\epsilon}^{m}\cdot \boldsymbol{n}^{(0)}
\label{eqS:BandOcc}$$
with $$\begin{aligned}
\boldsymbol{n}^{(m)}=\begin{pmatrix} n_1^{(m)} \\ n_2^{(m)} \end{pmatrix}\text{,}\quad
\boldsymbol{\epsilon}=\begin{pmatrix} 1 - \epsilon_1 & \epsilon_2 \\ \epsilon_1 & 1 - \epsilon_2 \end{pmatrix}\end{aligned}$$
Here, $n_1$ ($n_2$) denotes the fraction of atoms in the first (second) band, $\epsilon_1$ is the fraction of atoms excited from the first to the second band during one step and $\epsilon_2$ the one transferred from the second to the first. While the contribution from non-adiabatic transitions induced by the pumping is expected to be the same for $\epsilon_1$ and $\epsilon_2$, the finite lifetime of atoms in the second band generally leads to $\epsilon_2 > \epsilon_1$.
When both $\epsilon_1$ and $\epsilon_2$ are very small, one can expand $\boldsymbol{\epsilon}^{m}$ in powers of $\epsilon_1$ and $\epsilon_2$ and obtains to first order $$\boldsymbol{\epsilon}^m
\approx
\begin{pmatrix}
1 - m \epsilon_1 & m \epsilon_2 \\
m \epsilon_1 & 1 - m \epsilon_2 \\
\end{pmatrix}$$
Assuming a perfect preparation of the atoms in one band, i.e. $\boldsymbol{n}^{(0)}=(1,0)$ or $\boldsymbol{n}^{(0)}=(0,1)$, the resulting band occupations depend only on $\epsilon_1$ and $\epsilon_2$, respectively. This model was used to fit both the measured COM position as well as the site-resolved band mapping data with a single free parameter.
The expected displacement for both bands can be calculated by evaluating the COM evolution of the corresponding Wannier functions for one half of a pump cycle from $\varphi = 0$ to $\pi$. For each segment $\varphi \in \left](m-1) \pi, m \pi \right]$, the combined motion is obtained by adding up the contributions from the two bands weighted with the respective population fraction $n_1^{(m)}$ and $n_2^{(m)}$. After each half of a pump cycle ($\varphi =m \pi$), the COM position can be expressed in terms of the Chern numbers of the bands:
$$\begin{aligned}
x^{(m)}= \frac{d_l}{2}~\boldsymbol\nu \cdot \left(\sum\limits_{j=1}^{m} \mathrm{sign}(m) \boldsymbol{\epsilon}^{|j|} \boldsymbol{n}^{(0)}\right)
\label{eqS:TotalDisplacement}\end{aligned}$$
with $\boldsymbol{\nu}=(\nu_1,\nu_2)$.
For the site occupations, the even-odd distribution was determined by integrating the probability density of the Wannier functions over the even and odd site of the corresponding double well. These were then weighted in the same way with $n_1^{(m)}$ and $n_2^{(m)}$ to obtain the fractions on even and odd sites as a function of $\epsilon_{1}$ and $\epsilon_{2}$.
![Evolution of the COM position along the $x$- and $y$-axis when pumping is performed along the $x$-axis. The grey points depict the COM displacement in the $y$-direction for the data points of Fig.2b in the main text, which are illustrated in red. The positions along $y$ were evaluated using the same procedure as for the ones along $x$. Each point is averaged over ten data sets and the error bars depict the standard deviation.[]{data-label="fig:S3"}](Figures/FigS3.pdf){width="\linewidth"}
Center-of-mass position in the perpendicular direction
======================================================
In addition to determining the COM position along the pumping direction, we also verified that the pumping along $x$ does not lead to any measurable displacement of the cloud along the perpendicular $y$-direction. For the measurement depicted in Fig. 2b of the main text, the COM positions along both axes are shown in [Fig. \[fig:S3\]]{}. As expected, we do not observe any significant displacement of the cloud along $y$ over multiple cycles in both pumping directions.
|
---
abstract: 'The Galactic center Arches stellar cluster, detected and studied until now only in the near-infrared, is comprised of at least one hundred massive (M$_\star \geq$ 20 M$_\odot$) stars. Here we report the detection at centimeter wavelengths of radio continuum emission from eight radio sources associated with the cluster. Seven of these radio sources have rising spectral indices between 4.9 and 8.5 GHz and coincide spatially with the brightest stars in the cluster, as determined from JHK photometry and Br$\alpha$ and Br$\gamma$ spectroscopy. Our results confirm the presence of powerful ionized winds in these stars. The eighth radio source has a nonthermal spectrum and its nature is yet unclear, but it could be associated with a lower mass young star in the cluster.'
author:
- 'Cornelia C. Lang, W. M. Goss, Luis F. Rodríguez'
title: VLA Detection of the Ionized Stellar Winds Arising from Massive Stars in the Galactic Center Arches Cluster
---
To appear in The Astrophysical Journal Letters
Introduction
============
High-resolution observations of the Galactic center at near-infrared wavelengths first revealed the remarkable Arches stellar cluster located at $\ell$=012, b=002, 11 (or 27.5 pc) in projection from Sgr A$^*$ (Nagata et al. 1995; Cotera et al. 1996). Here, we assume a distance to the Galactic center of 8.0 kpc (Reid 1993). Nagata et al. (1995) identified the 14 brightest stars in this cluster with JHK photometry and Br$\gamma$ and Br$\alpha$ hydrogen recombination lines, showing that these stars had the characteristic colors and emission lines of Wolf-Rayet (WR) and He I emission-line stars. The strength of the Br$\alpha$ and Br$\gamma$ emission lines suggests that these stars are losing mass at rates of 2$\times$10$^{-5}$ . Using K-band spectroscopy, Cotera et al. (1996) also found 12 stars in the cluster with spectra consistent with late-type WN/Of types, and which have 1$-$20 $\times$ 10$^{-5}$ M$_{\sun}$ , and 800$-$1200 .
Recent observations of the Arches cluster indicate that the number of massive stars is much greater than suggested by the initial studies. Serabyn et al. (1998) estimate that at least 100 cluster members have masses $>$ 20 M$_{\sun}$, and Figer et al. (1999) calculate that the total cluster mass exceeds 10$^4$ M$_{\sun}$ with as many as 160 O-stars and a stellar density of $\rho$ 3$\times$10$^5$ M$_{\sun}$ pc$^{-3}$. These parameters make the Arches cluster one of the densest and most massive young stellar clusters in the Galaxy. In addition, the ionizing flux (estimated to be a few 10$^{51}$ s$^{-1}$) is sufficient to ionize the surrounding Arched Filaments H II complex (Serabyn et al. 1998; Lang et al. 2001). The relative placement of the Arches cluster and the filamentary nebular features is depicted in Figure 1.
Thermal radio emission from ionized stellar winds is detectable at radio wavelengths and arises from the outer parts of the wind envelope. The classic theory of Panagia & Felli (1975) and Wright & Barlow (1975) predicts that the spectrum of radio emission from a stellar wind is $\propto$ $\nu^{+0.6}$ for a spherically symmetric, isothermal, stationary wind expanding at a constant velocity. Previous surveys made with the Very Large Array (VLA) of the National Radio Astronomy Observatory have detected radio emission from OB supergiant and Wolf-Rayet stellar winds (Abbott et al. 1986; Bieging et al. 1989). Recently, Lang et al. (1999) detected six radio point sources in the vicinity of a similar Galactic center cluster, the Quintuplet, which have spectra consistent with the theoretical predictions and close positional correspondances with known near-infrared stellar sources. Here, we present multi-frequency VLA observations of the Arches cluster in an attempt to detect stellar wind emission from the individual stars known to be members of this dense cluster.
VLA Continuum Observations
==========================
VLA multi-frequency continuum observations were made of the Arches cluster in July 1999 (A-array) at 4.9 GHz and 8.5 GHz (4 hours each), and in October 1999 (BnA array) at 43.3 GHz (3 hours). The phase center of all observations was the position given for the bright, central source in Nagata et al. (1995): =17 45 50.41, $-$28 49 21.8. Standard procedures for calibration, editing, and imaging were carried out using the [*AIPS*]{} software of NRAO. At all wavelengths, 3C286 was used for flux density calibration, and at 8.5 and 4.8 GHz, 1751-253 was used as a phase calibrator. At 43 GHz, fast switching between the bright source Sgr A$^*$ and the phase center of the observations (an angular distance of 125) was carried out on a 90 second cycle in order to closely track the phase variations. The resulting spatial resolutions and rms noise levels in the images are 079 033, PA=128 and 45 $\mu$Jy at 4.9 GHz; 041 016, PA=28 and 25 $\mu$Jy at 8.5 GHz; and 029 007, PA=492 and 300 $\mu$Jy at 43.3 GHz.
Results
=======
Figure 2 shows the 8.5 GHz image of the radio sources within a 16 (0.7 pc) region centered on the bright source at the cluster center. The crosses represent the positions of 9 of the 14 stars in Nagata et al. (1995) which showed Br$\gamma$ and Br$\alpha$ recombination lines. These positions have been shifted by 005 (066) to correct for what appears to be a systematic offset to the West. At 8.5 GHz, eight sources are detected above a 6$\sigma$ level and are referred to as AR1$-$8. Seven of the sources (with the exception of AR6) appear to be associated with the stellar positions plotted in Figure 2. AR7 also shows a larger displacement ($\Delta$$\alpha$=05) from the corresponding stellar position than the rest of the radio sources. However, the absolute uncertainty in the near-infrared positions is 2 or less, so an association is quite possible within these limits. Since the a priori probability of finding a 8.5 GHz background source with flux density greater than 0.1 mJy in the region considered is 0.005, we conclude that all the radio sources detected are indeed associated with the Galactic center Arches cluster.
The positions and properties of AR1$-$8 are listed in Table 1. Four of these sources are detected in the 4.9 GHz observations (AR1, AR4, AR6, AR7), while the central source AR1 is additionally detected in the 43.3 GHz observations. At 43.3 GHz, AR1 has a flux density of 3.10.1 mJy, consistent with the 4.9/8.5 GHz spectral index ($\alpha$+0.35). Spectral indices between 4.9/8.5 GHz are calculated for the sources detected at 4.9 GHz. Upper limits for the flux densities at 4.9 GHz can be estimated for the remaining sources, with resulting lower limits for the spectral indices. Seven of the sources show rising spectra in the range of +0.3 to +0.9 with the exception of AR6, which has a nonthermal spectral index (=$-$0.7). The sources AR2$-$8 are point sources with angular sizes $\leq$ 025. The deconvolved size of AR1 at 8.5 GHz is 011 (0.005 pc), implying a brightness temperature of 3000 K, a reasonable value for an ionized wind.
Discussion
==========
The Nature of the Radio Sources
-------------------------------
The spectral indices of the detected sources are consistent with values expected from stellar wind emission. Several of the sources have spectra which are flatter ($\alpha$ +0.3) than the predicted $\alpha$=+0.6, and AR6 has a negative spectral index ($\alpha$ $-$0.7). However, these spectra do not rule out stellar wind detections. About 25% of stellar winds arising from massive stars have been found to have flat or nonthermal spectra in the radio regime (Bieging, Abbott & Churchwell 1989). The precise mechanism that produces these nonthermal spectra is not established, but internal shocks in the wind of a single star (White 1985) or the presence of interacting winds in close binary systems have been proposed (Contreras & Rodríguez 1999). For a typical late-type WN star at the Galactic center, the expected flux density at 8.5 GHz arising from a stellar wind is 250 $\mu$Jy based on the assumed values of 1000 , 5$\times$10$^{-5}$ and T10$^4$ K. This value is similar to the flux densities we measure at 8.5 GHz for AR2$-$8. Figure 2 also shows an excellent positional correspondence between the radio sources and the stellar sources detected by Nagata et al. (1995). Table 2 lists the spectral types of these stars as determined by Cotera et al. (1996). The stars have been classified as late-type WN and Of stars, all of which are expected to be losing mass at high rates from their surfaces. As noted above, such stars often exhibit flattened or nonthermal spectral indices due to the presence of a nonthermal component. Therefore, the range of spectra we detect is consistent with predictions for late-type WN stars.
The radio source AR6 is displaced by $\Delta$$\alpha$=1 from the corresponding stellar position of Nagata et al. (1995). At 8.5 GHz, there is a 75 $\mu$Jy source (3$\sigma$) closer to the stellar position, which may be the correct identifcation. The stellar source at this position has been classified as a WN8 star by Cotera et al. (1996). If this displacement is real, we have then to consider the possibility that AR6 is associated with another star in the cluster. T Tauri stars are sometimes found to have associated nonthermal emission, usually interpreted as gyrosynchrotron produced by electrons accelerated in situ by magnetic reconnection flares near the stellar surface (André et al. 1988; Rodríguez, Anglada, and Curiel 1999). If this is the case, we may be detecting for the first time a young, low mass member of the cluster.
Mass Loss Rates
---------------
The flux densities at 8.5 GHz were used to calculate a mass-loss rate based on Panagia & Felli (1975):
$$\frac{\dot{M}}{10^{-5} M_\odot~yr^{-1}} = 0.52 \left(\frac{S_{8.5}}{mJy}\right)^{3/4} \left(\frac{v_{\infty}}{10^3~km~s^{-1}}\right) \left(\frac{d}{kpc}\right)^{3/2}$$
where S$_{8.5}$ is the flux density of the source at 8.5 GHz, is the terminal velocity of the wind, and d is the distance to the source (8 kpc). We have assumed an electron temperature of =10$^4$ K, Z=1, and a mean molecular weight, $\mu$=2, due to the enrichment in heavy elements of the late-type WN stars (Leitherer et al. 1997). The mass loss rate of AR1 has the largest value, =1.7 10$^{-4}$ yr$^{-1}$, comparable only to stars with extreme values for , such as in the R136 stellar cluster at the center of 30 Doradus (de Koter et al. 1997). The values of for AR2$-$8 are in the range of 3$-$4.5 10$^{-5}$ , consistent with values found for for late type WN stars by Leitherer et al. (1997): 2.5$-$3.9 $\times$10$^{-5}$ yr$^{-1}$.
The Arches Cluster and the Interstellar Medium
----------------------------------------------
The rich population of massive stars in the Radio Arc region, which includes both the Arches and Quintuplet clusters (Figer et al. 1999), suggests that the interstellar environment should be strongly influenced by the energetic processes related to such massive star formation and evolution. Ionization of the molecular material in this region and formation of the unusually shaped Arched Filaments and Sickle H II regions can be accounted for by the ionizing fluxes of each cluster (Lang et al. 1997, 2001). In addition, interactions between the winds in the densely-packed Arches cluster may give rise to a shock-heated diffuse “cluster wind”, which could have a temperature as high as 10$^7$ K (Cantó, Raga & Rodríguez 2000). This emission is predicted to be detectable in the X-ray regime with the [*Chandra X-ray Observatory*]{}. Studies of the interaction between the massive stars in the Arches and the surrounding interstellar environoment are currently underway at a number of complementary wavelengths.
Conclusions
===========
We present the following results from our 4.9, 8.5, and 43.3 GHz observations of the Arches Cluster:
\(1) Eight radio sources are detected at 8.5 GHz, four sources at 4.9 GHz, and one of them is detected also at 43.3 GHz. Seven of the eight detections show rising spectral indices between 4.9 and 8.5 GHz in the range +0.3 to +0.6, while one source shows a nonthermal spectrum ($-$0.7). All of the detections are consistent with stellar wind emission from mass-losing stars. However, the nonthermal source may be tracing a young, low mass member of the cluster. This is the first detection of emission associated with the members of the Arches cluster at a band other than the near-infrared.
\(2) Seven of the eight radio sources are coincident in position with the Br$\gamma$ and Br$\alpha$ emission-line stellar sources of Nagata et al. (1995), which correspond to the late WN and Of stellar types as classified by Cotera et al. (1996). These positional correspondences further confirm that the ionized winds from stars in the Arches cluster are detected.
\(3) Based on the radio flux density, AR1 has a mass-loss rate of 2$\times$10$^{-4}$ yr$^{-1}$, comparable only to that of the stars in 30 Doradus (de Koter et al. 1997). Mass-loss rates for AR2$-$8 are found to be in the range 3$-$4.5$\times$10$^{-5}$ , similar to the large mass loss rates for other late-type WN systems (Leitherer et al. 1997).
Abbott, D., Torres, A., Biejing, J., & Churchwell, E. 1986, ApJ, 303, 239 André, P., Montmerle, T., Feigelson, E.D., Stine, P. C., & Klein, K. L. 1988, ApJ, 335, 940 Bieging, J., Abbott, D. & Churchwell, E. 1989, ApJ, 340,518 Cantó, J., Raga, A.C. & Rodríguez, L.F. 2000, ApJ 536, 896 Contreras, M. E. & Rodríguez, L. F. 1999, ApJ, 515, 762 Cotera, A., Erickson, E., Colgan, S., Simpson, J., Allen, D., & Burton, M. 1996, ApJ, 461, 750 de Koter, A., Heap, S. & Hubeny, I. 1997, ApJ, 477, 792 Figer, D.F., Kim, S, Morris, M., Serabyn, E., Rich, M., & McLean, I. 1999, ApJ, 525, 750 Lang, C. C., Goss, W.M., & Wood, D.O.S. 1997, ApJ, 474, 275 Lang, C.C., Figer, D.F., Goss, W.M. & Morris, M. 1999, AJ, 118, 2327 Lang, C.C., Goss, W.M., & Morris, M. 2001, AJ, [*in press*]{}, astroph e-print 0102130 Leitherer, C., Chapman, J., & Koribalski, B. 1997, ApJ, 481, 898 Nagata, T., Woodward, C.E., Shure, M., Kobayashi, N. 1995, AJ, 109, 1676 Panagia, N. & Felli, M. 1975, , 39, 1 Reid, M. 1993, A&A Review, 31, 345 Rodríguez, L. F., Anglada, G., & Curiel, S. 1999, ApJS, 125, 427 Serabyn, E., Shupe, D., & Figer 1998, Nature, 394, 448 White, R.L. 1985, ApJ, 289, 698 Wright, A.E. & Barlow, M.J. 1975, , 170, 41
[lcccccc]{} AR1&17 45 50.42&$-$28 49 22.0&1.700.05&1.400.03&+0.350.04&1.710$^{-4}$\
AR2&17 45 50.39&$-$28 49 21.0&0.230.02&$<$0.13&$>$+0.9&3.910$^{-5}$\
AR3&17 45 50.20&$-$28 49 22.0&0.170.02&$<$0.13&$>$+0.4&3.210$^{-5}$\
AR4&17 45 50.47&$-$28 49 19.2&0.230.02&0.160.03&+0.650.13&3.910$^{-5}$\
AR5&17 45 50.57&$-$28 49 17.2&0.160.02&$<$0.13&$>$0.3&3.010$^{-5}$\
AR6&17 45 49.76&$-$28 49 25.7&0.270.02&0.400.03&$-$0.710.08&4.510$^{-5}$\
AR7&17 45 50.83&$-$28 49 26.1&0.250.02&0.210.03&+0.310.05&4.210$^{-5}$\
AR8&17 45 50.45&$-$28 49 31.7&0.200.02&$<$0.13&$>$+0.7&3.610$^{-5}$\
[ccccc]{} AR1&8&8&Of/WN9\
AR2&7&6&WN8/9\
AR3&4&9&Of/WN9\
AR4&10&5&Of/WN9\
AR5&11&2&WN8\
AR6&1&13&WN8\
AR7&14&11&Of/WN9\
AR8&9&$-$&$-$\
|
---
abstract: 'We study the properties of ISM substructure and turbulence in hydrodynamic (AMR) galaxy simulations with resolutions up to 0.8 pc and $5\times 10^3$ M$_{\sun}$. We analyse the power spectrum of the density distribution, and various components of the velocity field. We show that the disk thickness is about the average Jeans scale length, and is mainly regulated by gravitational instabilities. From this scale of energy injection, a turbulence cascade towards small-scale is observed, with almost isotropic small-scale motions. On scales larger than the disk thickness, density waves are observed, but there is also a full range of substructures with chaotic and strongly non-isotropic gas velocity dispersions. The power spectrum of vorticity in an LMC-sized model suggests that an inverse cascade of turbulence might be present, although energy input over a wide range of scales in the coupled gaseous+stellar fluid could also explain this quasi-2D regime on scales larger than the disk scale height. Similar regimes of gas turbulence are also found in massive high-redshift disks with high gas fractions. Disk properties and ISM turbulence appear to be mainly regulated by gravitational processes, both on large scales and inside dense clouds. Star formation feedback is however essential to maintain the ISM in a steady state by balancing a systematic gas dissipation into dense and small clumps. Our galaxy simulations employ a thermal model based on a barotropic Equation of State (EoS) aimed at modelling the equilibrium of gas between various heating and cooling processes. Denser gas is typically colder in this approach, which is shown to correctly reproduce the density structures of a star-forming, turbulent, unstable and cloudy ISM down to scales of a few parsecs.'
author:
- |
Frédéric Bournaud$^{1}$[^1], Bruce G. Elmegreen$^{2}$, Romain Teyssier$^{1,3}$, David L. Block$^{4}$,Ivânio Puerari$^{5}$\
$^{1}$Laboratoire AIM Paris-Saclay, CEA/IRFU/SAp – CNRS – Université Paris Diderot, 91191 Gif-sur-Yvette Cedex, France.\
$^{2}$IBM T. J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, New York 10598 USA.\
$^{3}$Institute for Theoretical Physics, University of Zürich, CH-8057 Zürich, Switzerland.\
$^{4}$School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, WITS 2050, South Africa.\
$^{5}$Instituto Nacional de Astrofísica, Optica y Electrónica, Calle Luis Enrique Erro 1, 72840 Santa María Tonantzintla, Puebla, Mexico.
date: 'Accepted 2010 July 14 — Received 2010 May 8 — in original form 2010 May 8'
title: |
ISM properties in hydrodynamic galaxy simulations:\
Turbulence cascades, cloud formation, role of gravity and feedback
---
\[firstpage\]
galaxies: evolution – galaxies: ISM – galaxies: star formation – galaxies: structure – ISM: kinematics and dynamics – ISM: structure
Introduction
============
Observations of the interstellar medium (ISM) of nearby galaxies reveal a large variety of complex substructures, such as density waves, molecular clouds, filamentary structures along shocks, and supernova bubbles. ISM turbulence is a major driver of large-scale star formation: turbulent compression can increase the gas density and initiate star-forming instabilities [@E93; @E02]. This can happen in the spiral arms of modern disk galaxies [@klessen2001], and there is recent evidence that very active star formation relates to increased turbulence and associated instabilities in both primordial disk galaxies [@E05; @FS06; @BEE07; @DSC09] and starbursting mergers [@bournaud08a; @teyssier10]. On the other hand, gas turbulence can also disrupt gaseous structures faster than they gravitationally collapse, which could potentially stabilize gas disks and quench the star formation activity of some galaxies [@martig09; @DSC09].
Nevertheless, the large-scale properties of ISM turbulence are still poorly understood and the ability of numerical simulations to realistically model ISM substructures remains largely unexplored. Hydrodynamic simulations of ISM turbulence and fragmentation in whole galaxy models have been performed [@wada02; @wada07; @tasker08; @tasker09; @agertz09] but have not been compared to observational constraints yet, probably because of lacking the required level of realism, both in spatial resolution and in global dynamical modeling (no old stellar components and live dark halo were considered), leading to a restricted dynamical range of structure sizes. All these models are characterized by a probability distribution function (PDF) of the gas density that has a log-normal shape, resulting from the combination of various structures including density waves, dense gas clouds, supernovae bubbles. The levels of turbulent speed in these models seem realistic for both nearby and high-redshift galaxies [see @agertz09; @ceverino10 respectively]. Whether a realistic spectrum of structures of various scales is reproduced in modern hydrodynamic models remains however unknown.
Observations suggest that turbulent motions are initiated by gravitational instabilities on a relatively large scale (the Jeans scale, which is $1/k \sim 100$ pc for wavenumber $k$ in nearby disk galaxies, Elmegreen et al. 2003) and that a turbulent cascade develops towards smaller scales. This turbulence should be three-dimensional since the Jeans scale length is expected to set the gas disk scale height. Turbulent motions could be triggered by gravity even inside molecular clouds [@field].
Observations of the power spectra of ISM emission from nearby galaxies are consistent with this. On scales smaller than $\sim100$ pc, the power spectrum is steep, $\sim-3$, as it is for 3D Kolomogorov turbulence, while on scales larger than $\sim100$ pc, the power spectrum is flatter by about 1 in the slope, with a power-law exponent of $\sim-2$. This has been observed in the Large Magellanic Cloud (LMC) (Elmegreen, Kim, & Staveley-Smith 2001; Block et al. 2010), and in NGC 1058 (Dutta et al. 2009). This tentatively suggests a two-dimentional regime for turbulent motions on large scales, because the break in the density distribution occurs at the expected disk scale height. The nature of the large-scale motions is uncertain, however, because the three-dimensional velocity field in individual galaxies is not observable. In the Small Magellanic Cloud, the power spectrum is steep everywhere (Stanimirovic et al. 2000), presumably because there is no thin disk on the line of sight.
In this paper, we explore the properties of ISM turbulence in simulations of isolated disk galaxies, performed with an adaptive grid hydrodynamic code (AMR) with a maximal resolution of 0.8 pc resolution. We use a “pseudo-cooling” approach to model the equilibrium of gas between various heating and cooling processes, based on a barotropic equation of state (EoS), and perform models with and without supernovae feedback. Our main model has a mass distribution and rotation curve representative for the LMC disk, so as to compare with observations from Block et al. (2010). However we do not try to reproduce the detailed individual structures of the real LMC, and we also analyze models with higher disk masses and gas fractions. We show that a large range of substructures spontaneously form in the ISM, with a power spectrum quite consistent with observations of the LMC from very large scales ($> 1$ kpc) down to very small scales ($< 10$ pc). A break in the power-law density power spectrum is observed at the gas disk scale height. Studying the velocity structure on various scales, we show that this corresponds to a transition from a 3D regime to a (quasi-)2D regime in the gas motion, as initially proposed from observations (Elmegreen et al. 2001). Even if large-scale forcings are probably important, the two-dimensional regime on large scales cannot be interpreted just as density waves and streaming motions in a rotating disk, and evidence for an inverse cascade of turbulence in a quasi-2D disk is proposed. The same properties are recovered in a model of a massive high-redshift disk galaxy, which has a high gas fraction (50%) and strong turbulence ($V/\sigma \sim 4$), suggesting these properties are relatively universal for gaseous galactic disks.
Analyzing the vertical structure of the disk, compared to the properties of gas turbulence, we suggest that the turbulent motions at all scales are mainly powered by gravitational instabilities around the Jeans scale length, which sets the disk scale height. Feedback processes such as supernovae explosions do not significantly change the initial statistical properties of the ISM. They are nevertheless fundamental to maintain a realistic ISM structure over a large number of disk rotations. Supernovae bubbles do not appear to set the disk scale height, but they expand up to the disk scale height that is maintained by gravitationally-driven turbulence: this breaks apart the densest self-gravitating clouds to maintain a steady state cloud population.
Simulations
===========
![The “pseudo-cooling” Equation of State used in our simulations. Gas cooling and heating rates (in erg s$^{-1}$ cm$^{-3}$) as a function of temperature and density is shown for 1/3 solar metallicity, as computed with the CLOUDY code [@ferland see text]. The EoS used in our simulations is shown by the dashed line. A low-density regime corresponds to the Virial temperature of hot gas in a diffuse halo. Denser gas follows an equilibrium given by the minimal cooling rate: an isothermal regime accounts for warm HI in the disc at $10^4$ K, and denser phases corresponding to cool/cold atomic and molecular gas. The EoS follows the thermal equilibrium, and variations of metallicity and UV background do not largely shift this equilibrium, so the same EoS would remain sensible for higher metallicities or UV fluxes. A thermal support at very high density is added to this EoS to prevent artificial fragmentation (see text).[]{data-label="fig:eos"}](eos2.eps){width="7.5cm"}
![Rotation curve of our LMC-sized model, measured at $t=254$ Myr.[]{data-label="fig:model-rc"}](vcirc.eps){width="8cm"}
Simulation technique
--------------------
We perform closed-box model of isolated galaxies, with physical parameters described in section \[model:params\]. We use the AMR code RAMSES [@teyssier2002]. Stars and dark matter are described with collisionless particles, evolved with a particle-mesh (PM) technique. The gaseous component is described on the same adaptive grid; hydrodynamic equations being solved with a second-order Godunov scheme.
Our simulations use a box size of 26 kpc. The coarse level of the AMR grid, $l_{min}=9$, corresponds to a $(2^9)^3 = 512^3$ Cartesian grid, i.e. the minimal spatial resolution is $\epsilon_{min} = 51$ pc where $\epsilon = \frac{26}{l}$ kpc is the cell size. The maximal refinement level in dense regions is set by $l_{max}=15$, i.e. an effective resolution equivalent to $32768^3$ and a maximal spatial resolution of $\epsilon_{max} = 0.8$ pc. As we want to resolve turbulent motions in the whole disk, we use a refinement strategy that ensures that the mid-plane of the initial gas disk is entirely resolved with $l>11$ ($\epsilon < 13$pc), and with $l=12$ ($\epsilon=6$ pc) at the initial average surface density. The coarsest resolution levels are used only in the outer stellar bulge and halo. The resolution increases towards a cell size of 0.8 pc in the inner disk and in dense substructures forming in the outer disk. Effective resolution maps are shown on Figure \[fig:sim-density\]. The refinement is as follows: a cell of level $l<l_{max}$ is refined if its gas mass is larger than $m_{res}=5 \times 10^3$ M$_{\sun}$ and/or the number of particles (stars and dark matter) is larger than $n_{res}=8$.
Only the gas density is computed in the smallest cells of the AMR grid with $l=l_{max}=15$, the mass density of particles being restricted to $l_{max,part}=14$ in the PM scheme. This ensure a gravitational softening of at least 2 pc for particles. Treating the particle density at the $l_{max}$ level could result in a low number of particles per cell, and induce two-body relaxation, in regions that are refined because of a high gas density without necessarily having a high density of stars and/or dark matter. Only a few $l_{max}$ cells at each instant would actually contain less than 3 particles per cell, as measured in our simulation results, but we prefer avoiding any spurious effect.
Resolving the gas cooling and heating equation is numerically costly and would reduce the achievable resolution. In order to resolve small-scale structures and gas turbulence at the best possible resolution, we model the cooling and heating processes using an EoS that fixes the gas temperature $T$ as a function of its local atomic density $n$. Here we use an EoS modeling the equilibrium of gas between the main heating (UV heating and stellar feedback) and cooling (atomic and dust cooling) processes [see also @teyssier10]. This EoS is a fit to the average equilibrium temperature of gas at one third solar metallicity: for densities $10^{-3}<n<0.3$ cm$^{-3}$, $T(n)=10^4$ K. For densities $n>0.3$ cm$^{-3}$, $T(n)=10^4 (n/0.3)^{-1/2}$ K. Densities $n<10^{-3}$ cm$^{-3}$ correspond to gas in gravo-thermal equilibrium in the halo, at the Virial temperature $T(n)=4\times 10^6 (n/{10^{-3}})^{2/3}$ K.
Gravitational instabilities are a major driver of turbulence in galactic disks, both by directly triggering gas motions and by fueling star formation that can also pump turbulent energy through feedback processes. This is known from observations [@E02], and models similar to ours but scaled to massive spiral galaxies [@wada02; @tasker09; @agertz09]. A fundamental aspect in our models is thus to avoid numerical instabilities and artificial fragmentation: a gas medium that should theoretically be stable, tends to be unstable in numerical models when the Jeans length is described by a small number of spatial resolution elements [@truelove97]. It is generally admitted that this effect is avoided if the Jeans length is resolved by at least a few elements [@machacek; @agertz09 e.g.]. We then force the grid refinement to ensure that the thermal Jeans length of the gas is always resolved by at least 4 cells – and hence the actual thermal+turbulent Jeans length is resolved by an even larger number of cells). At the $l_{max}$ grid level, we impose a temperature threshold computed to keep the thermal Jeans length larger than $4 \times \epsilon_{max}$. This temperature floor, added to the EoS at very high density ($\sim 10^{5}$ M$_{\sun}$ pc$^{-3}$ and above) can be considered as a sub-grid model for the unresolved turbulent motions at scales smaller than the maximal grid resolution. We can then consider that artificial fragmentation is generally avoided: simulations of galactic disk fragmentation imposing higher numbers of resolution elements per Jeans length do not show significant differences in the properties of gas stability and turbulence [e.g., @ceverino10].
One of our models includes only disk self-gravity, but the other model also includes star formation and feedback. A threshold for star formation is implemented at a gas mass density $\rho = 5000$ atoms cm$^{-3}$ $\simeq 115$ M$_{\sun}$ pc$^{-3}$. This choice ensures that stars form only in the densest gas structures formed in our models but that the density PDF remains correctly sampled around and above this threshold (see Figure \[fig:sim-pdf\]). Above the threshold, the specific star formation rate (SFR per gas mass unit) is defined as the product of the efficiency and the local free-fall time$\sqrt{\frac{3 \pi}{32 G \rho}}$. We use an efficiency of 10%, which was chosen to give a gas consumption timescale of 2 Gyr on the results of the first simulation without star formation. Note that lower resolution models usually need to combine a lower efficiency with a lower density threshold for star formation (see discussion in Teyssier et al. 2010).
We use the kinetic feedback model described by [@dubois]. A 20% fraction of the mass of stars formed at a given timestep is assumed to be in stars that will explode into supernovae. 15% of the typical energy of each supernova, $10^{51}$ erg, is re-injected into the gas, assuming that the remaining energy is radiated away. The energy is re-injected in the form of radial velocity kicks around the supernova, in a gas bubble of radius 3 pc (see Dubois & Teyssier 2008 for details).
We use outflow boundary conditions for the fluid, and isolated boundary conditions for gravity
Model Parameters {#model:params}
----------------
Our simulations are not aimed at reproducing the exact morphology of the LMC, but at studying the properties of turbulence in a galactic disk with general properties similar to the LMC. To this aim, we initialize an exponential stellar disk containing $4 \times 10^6$ particles, of total mass $3\times 10^9$ M$_{\sun}$ radial scale-length of 1.5 kpc, and truncation radius of 3 kpc. A non-rotating bulge of $3\times 10^8$ M$_{\sun}$ with radius 500 pc, made-up of $4 \times 10^5$ particles, is added. The stellar velocity dispersions initially correspond to a stellar Toomre parameter of $Q_{\mathrm S}=1.5$, so that stars are stable against axisymmetric instabilities but unstable to bar forming instabilities – a bar and a pair of spiral arms form spontaneously in our model.
The dark matter halo contains $4 \times 10^6$ particles, has a total mass of $5\times 10^9$ M$_{\sun}$ and follows a Burkert profile of core radius 3.5 kpc, truncated at 10 kpc. The initial gas disk is exponential, with a scale-length of 3 kpc and truncation radius of 5 kpc, an exponential scale-height of 50 pc truncated at 500 pc, and a gas mass of $6\times 10^8$ M$_{\sun}$. The gas disk is initially purely rotating, with no macroscopic velocity dispersion. The initial densities in the disk lie within the isothermal part of our EoS, so a thermal support of $\sim 10$ km s$^{-1}$ ($T=10^4$ K) is present all over the initial disk, which can be considered as a model for the turbulent support of typically a few km s$^{-1}$ in real disks. This support ensures an initial Toomre parameter for the gas $Q_{\mathrm G}\simeq 1-1.5$. Once dense structures form in the course of the simulation, the temperature decreases (following the EoS) and the thermal support is gradually replaced by a turbulent support. Outside the initial disk truncation, the AMR grid is initialized with a background density of $10^{-6}$ atom per cm$^3$.
The rotation curve in this model is shown on Figure \[fig:model-rc\]. It is broadly similar to the LMC rotation curve (Kim et al. 1998), with almost solid body rotation up to radii of at least 2 kpc, and gradually flattening with a circular velocity $\simeq 70$ km s$^{-1}$ at large radii. We study the internal dynamics of the LMC; the interaction with the Milky Way and its hot gas halo is not modelled. Ram pressure was found to compress the leading edge of the HI and H$\alpha$ disk, with little effect on the inner star-forming disk (Mastropietro et al. 2009; see also Tonnesen & Bryan 2009 on the effect of ram pressure on a cloudy ISM).
![[**Top:**]{} Gas density maps of the LMC-sized gas disk with feedback, at $t=254$ Myr, in a face-on projection ($7\times4$ kpc snapshot) and a side-on projection ($7\times 0.8$ kpc snaphot). [**Bottom:**]{} Map of the AMR grid refinement level in the mid-plane, with $l=15$ in white and $l=11$ in dark blue.[]{data-label="fig:sim-density"}](sim1a.eps "fig:"){width="8cm"}\
![[**Top:**]{} Gas density maps of the LMC-sized gas disk with feedback, at $t=254$ Myr, in a face-on projection ($7\times4$ kpc snapshot) and a side-on projection ($7\times 0.8$ kpc snaphot). [**Bottom:**]{} Map of the AMR grid refinement level in the mid-plane, with $l=15$ in white and $l=11$ in dark blue.[]{data-label="fig:sim-density"}](sim1b.eps "fig:"){width="8cm"}\
![[**Top:**]{} Gas density maps of the LMC-sized gas disk with feedback, at $t=254$ Myr, in a face-on projection ($7\times4$ kpc snapshot) and a side-on projection ($7\times 0.8$ kpc snaphot). [**Bottom:**]{} Map of the AMR grid refinement level in the mid-plane, with $l=15$ in white and $l=11$ in dark blue.[]{data-label="fig:sim-density"}](sim1c.eps "fig:"){width="8cm"}
{width="17cm"}\
{width="17cm"}
![Probability distribution function (PDF) of the gas density $\rho$ at $t=254$ Myr in the model with feedback. The dashed curve is a log normal profile centered on $\rho_\mathrm{max}=53$ cm$^{-3}$ and of standard deviation $\Delta$=1.27 in a base-10 log PDF (see text for details). The dotted line is for the simulation with reduced resolution (section \[resol\]). This PDF was measured within a cylindrical box of radius 5 kpc and height 2 kpc. The peak at low densities corresponds to low density gas in the hot halo.[]{data-label="fig:sim-pdf"}](density-pdf-plot.eps){width="8cm"}
![Power spectrum of the face-on gas surface density in the model with feedback, at $t=254$ Myr (dark), 261 Myr (green), 268 Myr (blue) and 275 Myr (red). The wavenumber unit is $4.70 \times 10^{-4}$ pc$^{-1}$.[]{data-label="fig:sim-powspec"}](powspec.eps){width="8cm"}
![Velocity power spectrum for the three separate components $V_r$, $V_\theta$ and $V_z$, at $t=254$ Myr (LMC-sized model with stellar feedback). The conversion of wavenumbers into linear size is as on Fig. \[fig:sim-powspec\].[]{data-label="fig:sim-velspec"}](velspec.eps){width="8cm"}
Results
=======
The main simulation analyzed in Sections 3.1 and 3.2 is the LMC-sized model with stellar feedback. We later compare to the simulation without star formation and feedback in Section 3.3, to better highlight the role of feedback in the global structure of the ISM.
ISM structure and density power spectrum
----------------------------------------
We measured the velocity dispersion $\sigma$ in the gas disk, and found that both the mass-weighted average value $<\sigma>$ and the standard deviation $\Delta \sigma$ rapidly increase in the first 200 Myr and do not significantly evolve after $t\sim 200$ Myr, indicating that a steady state in the turbulent regime has been reached. We thus mostly analyze our models around $t$=250 Myr.
Spectral analysis of the gas distribution was performed on a sub-box of size 13 kpc (the outer regions of the full simulation box contain only the outer regions of the dark matter halo). Throughout the paper, the wavenumber $k=1$ is for a wavelength of 13 kpc, which results in a wavenumber unit $u_k = 4.70 \times 10^{-4}$ pc$^{-1}$.
Figure \[fig:sim-density\] shows the gas density distribution for this simulation at $t=254$ Myr. The response to a stellar bar, and some long spiral arms are seen, together with more chaotic structures such as shorter arms, shocks, dense clouds, bubbles, etc (Fig. \[fig:sim-density-zoom\]). We fit a sech$^2$ vertical profile to the gas at radii smaller than 5 kpc, integrated over the plane, and find a scale height of 207 pc. The scale height is relatively constant, with moderate flaring, and varies from 187 pc inside the central kpc to 226 pc for $3<r<5$ kpc.
The density PDF, integrated over the central disk with a radius of 5 kpc and height of 2 kpc, is shown on Figure \[fig:sim-pdf\]. This PDF has a log-normal shape, truncated at quite high density ($\sim 10^6$ cm$^{-3}$) which is enabled by our high spatial and mass resolution. If we write the probability distribution for the density $$P(\rho) \propto e^{-0.5 \frac{ \left(\ln \frac{\rho}{\rho_{\mathrm max}} \right) ^2 }{\Delta^2}} \; ,$$ then the Gaussian dispersion is $\Delta \simeq 2.92$.
The power spectrum of the face-on gas column density is shown on Figure \[fig:sim-powspec\] at the same instant and later ones. No significant time evolution is found. The power spectrum shows a double power-law with a break at wavelengths of about 150 pc, which is about the measured gas disk scale-height. The power law for large structures has a slope of $\simeq -1.9$ and the slope for small structures is $\simeq -3.1$. The slope steepens for the smallest scales below 5-10 pc, which is discussed later in section 3.3.
The simulation without star formation and feedback shows a globally similar density power spectrum (see Sect. 3.3 and Fig. 14) with a double power-law shape with slopes of $-2.9$ on small scales and $-1.8$ on large scales. The transition occurring around the gas disk scale-height. Only the smallest scales below 10 pc significantly differ from the model with star formation and feedback (see section 3.3).
Our simulations produce a density structure that has about the power spectrum observed for the ISM in the LMC and other nearby galaxies. The power spectrum shows a break between a $-2$ power law and a $-3$ power law, with the transition occurring at scales around the gas disk scale-height, as usually speculated in observations [e.g., @E01; @dutta]. In the following, we study the velocity structure to understand the nature of the motions in the two regimes of the density distribution.
Velocity structure
------------------
[>m[6.1cm]{} >m[1.25cm]{} >m[6.1cm]{}]{} $V_r$ & & $V_z$ {width="6cm"} & total & {width="6cm"} {width="6cm"} & $k<50$ & {width="6cm"} {width="6cm"} & $k>200$ & {width="6cm"}
![Face-on gas density snapshot at $t=268$ Myr, with a $7\times 4$ kpc field of view, in the LMC-sized model with stellar feedback.[]{data-label="fig:sim2-density"}](sim2dens.eps){width="8cm"}
[>m[6.1cm]{} >m[1.25cm]{} >m[6.1cm]{}]{} $V_r$ & & $V_z$ {width="6cm"} & total & {width="6cm"} {width="6cm"} & $k<50$ & {width="6cm"} {width="6cm"} & $k>200$ & {width="6cm"}
### Velocity fields and power spectra
Maps of the in-plane radial velocity component $V_r$ and perpendicular component $V_z$ are shown on Figure \[fig:sim-velocities\] for the same snapshot as Figures \[fig:sim-density\] and \[fig:sim-density-zoom\], all at $t=254$ Myr. Another instant ($t=268$ Myr) is shown on Figure \[fig:sim2-density\] for the gas density and Figure \[fig:sim2-velocity\] for the velocity maps. The velocity maps are shown for the total velocity components, and for the low wavenumbers $k<50$ ($\lambda>300$ pc) and the high wavenumbers $k>200$ ($\lambda<70$ pc) separately. This separates the two regimes identified on the density power spectrum, where the transition occurs at $k \simeq 100$ or $\lambda \simeq 150$ pc. These maps were built by zeroing the low-$k$ or high-$k$ components in the result of a Fourier transform of the velocity field, and recovering the velocities through an inverse Fourier transform.
The power spectrum of the three velocity components $V_r$, $V_{\theta}$ and $V_z$ is shown on Figure \[fig:sim-velspec\]. A single power law is found for the in-plane components. The perpendicular velocity $V_z$ follows the same power law on small scales, but its power spectrum flattens on large scales, with a transition at about the same scale length ($\sim$150 pc) as in the density structure spectrum.
These differences in velocity power spectra correspond to higher power for the long-wavelength in-plane components than the long-wavelength perpendicular components of velocity. There are large in-plane motions at long wavelengths from spiral and bar-driven gas flows and relatively little perpendicular motions at long wavelengths to accompany them.
### A 3D cascade of turbulence on small scales
On scales smaller than the gas disk scale height (100-200 pc), the ISM density has the density power spectrum expected for a 3D cascade of turbulence. The three-dimensional nature of the associated motions is confirmed by the high-wavenumber velocity maps, which show about the same amplitude for the in-plane motions and the perpendicular one. Typical average values of $<V_z/V_r>$ on various scales are given in Table \[tab-vz-vr\]. The small-scale motions are globally isotropic[^2]$^,$[^3].
These results are overall consistent with a 3D cascade regime on the small-scale part of the density spectrum, initiated at scales around 150 pc, which is the gas disk scale height and the average Jeans length. This is naturally the case if the most gravitationally unstable scale, namely the Jeans length, corresponds to the disk thickness. Indeed, the gas disk scale height is expected to be set by the gaseous Jeans length in nearby disk galaxies [@E01; @dutta]; this is also the case in high-redshift disk galaxies, which are more turbulent, have larger Jeans lengths, and have thicker gas disks at the same time [@EE06; @genzel06; @FS06; @bournaud08b]. We checked this in our simulations by computing a mass-weighted average of the velocity dispersion $\sigma_{av}$ (8 km s$^{-1}$), the average surface density $\Sigma_{av}$, and then an “effective” Jeans length $1/k_{\mathrm{Jeans,eff}}=\frac{\sigma_{av}^2}{\pi G \Sigma_{av}} = 213$ pc (locally, the Jeans length can significantly differ from this average value, as the density and velocity dispersion vary). This is consistent with the disk scale height being set by the Jeans length.
---------------------------- --------- ---------
Time 254 Myr 268 Myr
small scales ($k>200$) 1.13 1.07
large scales ($k<50$) 0.16 0.14
total velocities (all $k$) 0.92 0.89
---------------------------- --------- ---------
: \[tab-vz-vr\] Mass-weighted average values of $<V_z/V_r>$ for the large scale, low scale, and total velocities (LMC-sized model with feedback).
### A 2D inverse cascade on large scales?
![Vorticity map for the large-scale ($k<50$) components, at $t=254$ Myr (LMC-sized model with stellar feedback). The scale ranges from $-10^{10}$ pc$^2$ Myr$^{-1}$ (black) to $10^{10}$ pc$^2$ Myr$^{-1}$ (white) with the same color bar as velocity fields.[]{data-label="fig:vorticity"}](vorticity.eps){width="8cm"}
![Enstrophy spectrum at $t=254$ Myr for the LMC-sized model with stellar feedback. The wavenumbers are divided by a factor of 2 compared to the density and velocity power spectrum (Fig. \[fig:sim-powspec\] and \[fig:sim-velspec\]). Scales larger than 100-200 pc (here $k \simeq 2$) have a -1 power law spectrum: the amplitude of vortices of various sizes is just as expected for two-dimensional turbulence.[]{data-label="fig:enstrophy"}](enstro_spec.eps){width="7cm"}
The low-wavenumber (large-scale) structures have a flatter density power spectrum. They also have different kinematic properties. The large-scale component of the velocity fields is dominated by in-plane motions. Perpendicular motions are much weaker. The low-wavenumber component of $V_z$ shows only modest peaks in Figure \[fig:sim-velocities\], and these peaks are generally found in low-density regions (they generally correspond to gaseous fountains above the disk plan, rather than vertical motions in the mid-plane).
The motions corresponding to the large-scale regime of the density power spectrum are quasi-2D, highly anisotropic (Table \[tab-vz-vr\]). This is not because the disk rotation dominates these motions: this applies to $V_r$ as much as $V_{\theta}$, and the $<V_z/V_r>$ ratio is almost unchanged when we remove the lowest wavenumbers $k\leq3$ tracing global rotation. The power spectra of the three velocity components (Fig. \[fig:sim-velspec\]) actually shows that the perpendicular velocity is much lower than the in-plane components for all wavelengths larger than the disk scale-height.
The $\sim -2$ slope of the large-scale branch of the density power spectrum and the quasi-2D nature of the associated motions suggest that we are observing something like a 2D inverse cascade of turbulence [@2Dturb]. We nevertheless have to explore the nature of these motions more accurately, to ensure that these properties are not a conspiracy from density waves and associated gas flows, which would then not be turbulent motions.
The large-scale components of the in-plane motions (Figs \[fig:sim-velocities\] and \[fig:sim2-velocity\]) show a central $m=2$ mode, which is the characteristic response to a bar+arms system of density waves [e.g., @combes; @athanassoula]. This response is however quite asymmetric, and does not dominate the large-scale motions outside of the central kpc. Overall, outside the central kpc, chaotic motions dominate the large-scale velocity components (of course superimposed on the disk rotation pattern). In particular the comparison of the gas density and velocity maps shows large vortices around dense gas clumps: for instance, the densest clump seen to the left of Figure \[fig:sim-density\] is clearly associated with an in-plane eddy, with positive and negative peaks of $V_r$ surrounding the position of this dense clump. Such signatures are reminiscent of a Rosby Wave instability [@lovelace99; @varniere-tagger06], which is observed in pure 2D disks but also in thick disks or 3D systems [@meheut2010] and could thus take place in our quasi-2D system. Such motions would be quasi-2D turbulence, which is more than a large-scale flow in a density wave, but which may be induced by a density wave of other large-scale forcings.
The vorticity map of the large-scale velocity component ($k<50$), after subtraction of the $m=0$ rotation pattern, is shown on Figure \[fig:vorticity\]. It shows a number of vortices with various sizes and strengths throughout the disk. The power spectrum of the enstrophy is shown in Figure \[fig:enstrophy\]. Enstrophy is the integral of the square of the vorticity: $\mathcal{E} = \int _\mathcal{S} || \vec \nabla \times \vec v ||^2$. The enstrophy has a power-law power spectrum throughout the entire range of 2D scales, like the in-plane motions shown before. The slope of this power spectrum is -1, just as expected for 2D turbulence, as found in two-dimensional experiments [e.g., @petersen06; @petersen07]. This enstrophy spectrum means that the quasi-2D part of the density power spectrum is made-up of vortices with size and strength distributed as expected for 2D-turbulence: this is unlikely to simply result from non-turbulent flows along density waves.
The properties of the gaseous motions on scales larger than the disk scale height thus contain a component from turbulence in a quasi-2D medium. The origin of this turbulence could be a combination of long-range forcing from spiral waves and other global disturbances, and an inverse cascade from smaller scales, where energy is put into the ISM by gravitational instabilities on the Jeans length and stellar feedback. Energy injection at the Jeans length is consistent with our observation of vorticity around the gravitating clumps and with the quasi-2D power spectrum on scales larger than the Jeans length. @wada02 already mentioned that the properties of turbulent motions in their galaxy models were consistent with an inverse cascade of turbulence. Their models were purely two-dimensional, while we here suggest a similar inverse cascade in a three-dimensional gas disk, which is thin but has a well resolved scale height.
The large-scale motions probably cannot simply result from a single inverse cascade with energy input only at the Jeans scale. Indeed, the growth rate of gravitational perturbations in a disk decreases only as $1/\sqrt{\lambda}$ where $\lambda$ is the scale of the perturbations, so that long-range forcing at scales larger than the typical Jeans scale length remains significant. At least, the gravitational coupling of stellar and gaseous components and the bi-component stability [@jog] should result in a large range of unstable scales, ranging typically from the gaseous Jeans scale to the stellar Jeans scale [@E95]. There should then be a range of unstable scales injecting energy into the inverse turbulence cascade. While the expected properties of 2D turbulence, such as the enstrophy spectrum, are recovered in our simulations, one can note that the slope of the density power spectrum in this regime is slightly higher than expected from pure two-dimensional turbulence (a $-2$ power law, instead of the $-5/3$ exponent expected for purely two-dimensional and uncompressible turbulence). The non-zero thickness of the gas disk and the relative importance of long-range forcing in a gaseous+stellar disk might cause this.
The role of long-range gravitational forcing and the coupling of the gaseous and stellar components also appear by comparing our simulations with the AMR models in @tasker09. The stellar disk is modeled with a rigid analytical potential in their simulations. Gas clumps form everywhere in their disk with a relatively peaked size distribution and a constant separation of about one Jeans scale. The Jeans length is also a characteristic scale in our model: it sets the disk scale height and gas clouds along spiral arms are also often separated by $L_J$ (Fig. \[fig:sim-density-zoom\]), but gaseous structures overall form over a wider range of sizes in our models. The inclusion of both stellar and gaseous gravity on large scales appears crucial to realistically reproduce the full spatial range of the main ISM structures from long spiral arms and large vortices to smaller clumps and shocks, even if ISM structures on small scales can be realistically reproduced without stellar gravity [e.g., @avillez07].
We have analyzed above the LMC-sized simulation with star formation and stellar feedback. In this simulation, the properties of ISM turbulence seem mainly driven by gaseous and stellar gravity. However, the comparison to the model without stellar feedback below will highlight a fundamental role of feedback in maintaining a steady state in the density distribution of the ISM.
Role of star formation feedback in gas disk properties
------------------------------------------------------
The “gravity-only” simulation (without star formation and feedback) shows a relatively similar distribution of gaseous structures, and a relatively similar density power spectrum compared to the model with feedback (see Figures \[fig:gravity-map\] and \[fig:gravity-spec\]), at least at $t=238$ Myr and at scales larger than 5-10 pc. Besides suggesting that our earlier results do not crucially depend on a specific feedback scheme, it also indicates that the ISM structuring and the pumping of turbulent energy into the turbulent cascades results mainly from gravitational processes and is not primarily driven by supernovae explosions. The disk scale-height and the disk substructures are unchanged in this gravity-only model.
Feedback processes nevertheless play a fundamental role in maintaining this gravity-driven turbulence distribution. We can indeed note already at $t=238$ Myr in Figure \[fig:gravity-spec\] that the ISM power spectrum shows an excess of very small-scale structures (below 5-10 pc) and the gas density map shows the associated small and dense gas clumps. Over longer timescales the power spectrum becomes less realistic as the large scales are gradually emptied and the bump at very small scale increases: gas dissipates its energy and accumulates in tiny dense bullets from where it is never removed (Fig. \[fig:gravity-map\]).
The role of feedback appears when comparing the density power spectra in our two models: feedback disrupts the dense smallest-scale structures, and supernovae bubbles expand up to the disk scale height. Density maps show supernovae-driven bubbles, all of which are smaller than or comparable to the disk scale-height. But this does not mean that the size of supernovae bubbles regulates the scale height: the measured scale-height and the break in the density power spectrum are not changed by feedback; the scale height is mostly regulated by gravitational processes, and is about the Jeans length. But feedback disrupts structures on the smallest scales and expands them up to the Jeans length, where gravitational processes can initiate a new cycle of 3D turbulent cascade towards small scales and 2D inverse cascade towards large scales.
Overall, the gas disk structure and its scale height are largely regulated by gravitational instabilities, but gas structures and energy dissipation on the smallest scales are also regulated by star formation feedback. The break in the face-on density power spectrum and the scale height measured on edge-on projections both correspond to the average Jeans length. 2D and 3D motions are initiated by instabilities around this scale length and large-scale forcings. Feedback processes prevent the otherwise inevitable accumulation of gas in tiny and dense bullets, disrupt small-scale structures and refill the turbulent cascades initiated at the Jeans length. This cycle maintains the ISM density distribution in a steady state.
Observations have already suggested that turbulent motions are primarily driven by gravitational instabilities such as spiral arms and molecular clumps [@E03]. Based on the analysis of the density distribution, these authors pointed out the role of star formation feedback in breaking apart small dense structures to preserve a steady state. @agertz09 also proposed from simulations that the main driver of ISM turbulence could be gravitational forcing, with nevertheless a significant contribution from star formation (see also Dib, Bell & Burkert 2006).
Gravity-only models without star formation and feedback initially produce a realistic density distribution in the ISM over a couple of disk rotation periods. But on longer timescales and over large numbers of disk rotations, the inclusion of feedback in numerical models appears necessary to maintain a realistic density distribution. However, this applies only to high-resolution simulations where gas cooling is modeled down to $\sim 100$ K. Lower-resolution SPH simulations, and more generally models where the ISM is only modeled as a smooth stable gas at $\sim$$10^4$ K or more (e.g., the stabilizing EoS by Springel & Hernquist 2003 and Robertson et al. 2004), would not be affected by catastrophic gas dissipation if they do not include feedback sources, as they do not resolve the turbulent and unstable nature of the star-forming ISM.
![“Gravity-only” LMC-sized model without star formation and feedback: face-on view of the gas density at $t=343$ Myr.[]{data-label="fig:gravity-map"}](density-grav.eps){width="8cm"}
![Face-on gas density power spectrum of the “gravity-only” model without stellar feedback at $t=258$ (solid lines) and $343$ Myr (dashed, shown at small wavelengths only for clarity). []{data-label="fig:gravity-spec"}](powspec-grav-correctedaaa.eps){width="8cm"}
Extension to massive and gas-rich galaxies
------------------------------------------
The results described so far were for a low-mass disk galaxy, scaled to the LMC properties. To explore whether the highlighted properties can extend to more massive disk galaxies and/or disk galaxies with different gas fractions, we have applied the same analysis to a high-redshift galaxy model described in @bournaud10. This is a massive galaxy (baryonic half-mass radius of 10 kpc and circular velocity of 280 km s$^{-1}$) with a 50% gas fraction, modeled with a 2 pc resolution. As typical star-forming galaxies at $z \sim 2$, it has a high gas fraction, a very unstable disk with high turbulent speeds (a few tens of km s$^{-1}$) and giant complexes of star formation in a thick ($h_{z} \sim 1$ kpc) gas disk (see observations by @EE06 [@E07; @genzel06] and simulations by @semelin [@BEE07; @agertz10; @ceverino10]).
The density power spectrum is shown on Figure \[fig:highz\]. It is consistent with a double power law, with the break occurring at wavelength around 1 kpc which is the disk scale height and the typical Jeans length in such galaxies. We measured a total ratio of velocity dispersions of $<V_z/V_r> = 0.88$, with $<V_z/V_r> = 0.26$ on large scales and $<V_z/V_r> = 1.07$ on small scales. The large-scale motions are still significantly non-isotropic, even if the low wavenumber velocities are somewhat more isotropic than in our LMC-sized model, which could be caused by the disk being thicker. The properties in the density distribution and chaotic motions found in our LMC-sized models thus seem to apply, at least qualitatively, to galaxies with different masses, rotation curves and gas fractions.
![Gas surface density power spectrum in a massive gas-rich galaxy model.[]{data-label="fig:highz"}](highzps.eps){width="8cm"}
Resolution requirements {#resol}
-----------------------
Our fiducial LMC-sized model with star formation and feedback has been re-done with $l_{max}=11$, i.e. a spatial resolution of 13 pc, which means that the disk scale height is resolved with $\sim$10 resolution elements, instead of 100-200 resolution elements at the maximal refinement level $l_{max}=15$. The density threshold for star formation was reduced by a factor of 80 to keep the resulting star formation rate similar.
The global density distribution of this reduced-resolution model is not very different from the initial case, except that the highest density structures in the PDF are removed; otherwise the PDF is only moderately shifted towards lower densities and has about the same dispersion (Fig. \[fig:sim-pdf\]). However, the velocity structure is significantly different. The initial model had almost isotropic velocity dispersion $\sigma_z /\sigma_r \simeq 0.9$ (the smallest scales begin the most isotropic ones). Interestingly, the reduced-resolution model has much less isotropic motions, with $\sigma_z/\sigma_r =0.32$ (at $t=262$ Myr and over all wavenumbers; the ratio was $\sim$0.9 in the high-resolution run). We thus note that a very high resolution is required to resolve the isotropy or non-isotropy of turbulent motions in disk simulations, and that a few ($\sim$10) resolution elements per disk scale height are not sufficient. Our interpretation is as follows: vertical motions can be initiated or amplified when instabilities occur off the midplane at heights $z$$>$0 or $z$$<$0, when the Jeans length becomes locally smaller than the disk scale height (which happens in dense regions or low-velocity dispersion regions). This requires to resolve, locally, several Jeans length per scale height. Then, each Jeans length itself needs to be resolved by a least a few resolution elements: 4 resolution elements per thermal Jeans length is a strict minimum imposed in our model and our EoS generally results in 5-10 elements per thermal Jeans length, and the number of resolution per thermal+turbulent Jeans length would be even higher. Thus, capturing instabilities at $z > 0$ and the associated triggering of vertical motions can require a few tens of resolution elements across the gas disk scale height. When the disk thickness is marginally resolved, the non-isotropy of gas motions can be artificially increased at small scales. In this case, each gas clump is mostly spinning around its minor axis (see Agertz et al. 2009a) generally aligned with the spin axis of the entire galaxy (but see Tasker & Tan 2009).
Summary and conclusions
=======================
Properties of ISM turbulence, role of gravity and feedback
----------------------------------------------------------
This paper has presented an LMC-sized models for comparison with observations of the LMC (Block et al. 2010) and a high-redshift clumpy disk model. The LMC-sized model does not match all individual structures observed in the LMC – this is not the initial purpose and the LMC has many low-density ionised filaments, and other substructures in low-density regions, that are out of reach of our mass resolution.
The model is successful in reproducing the double power law shape of the ISM density power spectrum observed in the LMC (Elmegreen et al. 2001 and Block et al. 2010) in a system with a similar mass, gas fraction and rotation curve. We analyzed the 3D velocity structure of the models, which could not be done in observations, to understand the origin of this density substructures.
Our results indicate that a dual turbulence cascade could take place in our simulations, and by extension in the real ISM. The scale height of the gas disk is set by the Jeans scale length, which is the most gravitationally unstable scale. Gravitational instabilities at/around this scale length inject energy in turbulent motions. These motions follow a classical 3D cascade of isotropic turbulence towards smaller scales, on which they form a hierarchy of gas clouds, GMCs and substructures therein. Turbulent motions on scales down to and inside GMCs thus appear to be from self-gravity. Nevertheless, stellar feedback plays a major role in disrupting dense structures on the smallest scales and in maintaining the turbulence cascade in a steady state.
On scales larger than the gas disk scale height, very anisotropic turbulent motions with a shallower density power spectrum are observed. Long-range gravitational forces are probably not negligible, but the vorticity and enstrophy properties on these large scales are as expected for (quasi-)2D turbulence. The large scale motions are not limited to global disk rotation and streaming flows along density waves. This inverse cascade appears to be initiated mainly by the same gravitational instabilities at the Jeans length as the direct 3D cascade, because it is present with and without stellar feedback.
While models without self-gravity do not find a characteristic scale for energy injection into ISM turbulence [@joung-ml], we identify a major role of gravitational processes around the Jeans length. Large-scale forcing and stellar feedback at small scales are important processes in regulating the ISM properties too, but the Jeans length is the scale at which chaotic gas motions change from quasi-2D to 3D isotropic turbulence, and at which a break occurs in the density power spectrum. Log-normal gas density PDFs have been found in models of supernovae-driven ISM turbulence without self-gravity [e.g. @avillez02], but without having the observed power spectrum reproduced. The power spectrum analysis in our models and in LMC observations rather suggests that the energy injection in the ISM is largely from gravitational processes [such as gravitational instabilities and inward mass accretion, @EB10; @KH10] with a regulating role of feedback on small scales.
We have shown that similar properties for ISM substructures and turbulence, at least qualitatively, are found in models of galaxies with higher masses and higher gas fractions. This includes the extreme case of high-redshift ($z \sim 2$) galaxies with thick, highly turbulent and very clumpy gaseous disks. An interesting implication for observers is that the isotropy of the gas velocity dispersion is strongly dependent on the scale, and hence on the resolution of an observation.
Modeling ISM substructures in galaxy simulations
------------------------------------------------
We have explored ISM modeling with a barotropic equation of state (EoS) corresponding to gas in thermal equilibrium between a standard radiation field with the main atomic and fine-structure cooling processes. We have shown that this EoS employed in an AMR hydrodynamic code with a quasi-Lagrangian refinement strategy reproduces a realistic power spectrum for the ISM density distribution. It does not form a truly multiphase medium, since the pressure is only a function of density. It nevertheless a clumpy ISM, with cold and dense clouds embedded in a warmer diffuse phase, and the resulting density distribution is consistent with that of the real, multiphase ISM. This technique is efficient to reach high spatial and mass resolutions: the computation of cooling and heating rates is not numerically costly, but the absence of out-of-equilibrium, colder or warmer gas in this model prevents the timescale to become extremely short, which would typically happen in parsec-scale models with complete cooling calculations.
Hydrodynamic models with cooling calculations produce a log-normal PDF for the ISM [@wada07; @tasker09; @agertz09], which is retrieved in our model over a larger dynamical range. The PDF alone does not guarantee that the correct mass fraction is distributed in structures of various sizes and densities. Our model with a mass and rotation curve similar to the LMC matches the density spectrum observed by Block et al. (2010) in the LMC. While all thermal processes are not explicitely computed, this at least means that the ISM modeled this way has realistic density distribution, with a correct number of gas structures of various sizes and a correct mass fraction therein, from the global galaxy-sized scales down to 5-10 pc, i.e. molecular clouds and their main substructures.
The ISM substructures are mostly powered by gravity-driven turbulence and instabilities around the Jeans scale length. Star formation feedback is however an essential ingredient, not only by structuring the ISM into shells locally, but also by providing an extra source of energy is needed to disrupt the smallest and densest structures that tend to be steadily filled by gas dissipation into compact clumps. Gravity, hydrodynamics, and a realistic thermal model are initially enough to produce a realistic structure distribution in the ISM: feedback seems relatively negligible over a couple of disk rotations, but becomes important over longer timescales to balance systematic gas dissipation. The source of energy that disrupt the structures at very small scales and refills the 3D cascade and the inverse 2D cascade at the Jeans scale is, in our model is supernova feedback. Other sources such as HII regions, stellar winds, and radiation pressure from young massive stars [@murray; @kd10] could also contribute to maintain the ISM density distribution in a steady state. The global density structure of the ISM remains relatively unchanged in simulations where the disk scale height is marginally resolved, but modelling accurately the quasi-isotropic turbulent motions on small scales is found to require a very large number (at least a few tens) of resolution elements per disk scale height.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was granted access to the HPC resources of CCRT and CINES under the allocation 2010-GEN2192 made by GENCI (Grand Equipement National de Calcul Intensif). We are grateful to John Scalo and Sebastien Fromang for suggestions, and to Oscar Agertz, Daniel Ceverino, Françoise Combes and Elizabeth Tasker for discussions and comments on the manuscript. DLB is indebted to Mr F. Titi as well as the AVENG Group of Companies for their sponsorship. DLB thanks Roger Jardine, Kim Heller and the AVENG Board of Trustees. IP acknowledges support from the Mexican foundation Conacyt.
[99]{} Agertz, O., Lake, G., Teyssier, R., Moore, B., Mayer, L., & Romeo, A.B. 2009a, MNRAS, 392, 294
Agertz, O., Teyssier, R., & Moore, B. 2009b MNRAS, 397, L64
Athanassoula, E. 2000, in Stars, Gas and Dust in Galaxies: Exploring the Links, ASP Conference Series, Vol. 221 p. 243, Danielle Alloin, Knut Olsen & Gaspar Galaz Eds.
de Avillez, M. A., & Mac Low, M.-M. 2002, ApJ, 581, 1047
de Avillez, M. A., & Breitschwerdt, D. 2007, ApJL, 665, L35
Block D. L., Puerari I., Elmegreen B. G. & Bournaud F., ApJL, 718, L1
Bournaud, F., Elmegreen, B. G., & Elmegreen, D. M. 2007, ApJ, 670, 237
Bournaud, F., Duc, P.-A., & Emsellem, E. 2008, MNRAS, 389, L8
Bournaud, F., et al. 2008b, A&A, 486, 741
Delaye, L., et al. in preparation
Ceverino, D., Dekel, A., & Bournaud, F. 2010, MNRAS, 440
Combes, F., & Gerin, M. 1985, A&A, 150, 327
Dubois, Y., & Teyssier, R. 2008, A&A, 477, 79
Dutta, P., Begum, A., Bharadwaj, S., & Chengalur, J. N. 2009, MNRAS, 397, L60
Dekel, A., Sari, R., & Ceverino, D. 2009, ApJ, 703, 785
Dib, S., Bell, E., & Burkert, A. 2006, ApJ, 638, 797
Elmegreen, B. G. 1993, ApJl, 419, L29
Elmegreen, B. G. 1995, MNRAS, 275, 944
Elmegreen, B. G., Kim, S., & Staveley-Smith, L. 2001, ApJ, 548, 749
Elmegreen, B. G. 2002, ApJ, 577, 206
Elmegreen, B. G., Elmegreen, D. M., & Leitner, S. N. 2003, ApJ, 590, 271
Elmegreen, B. G., Elmegreen, D. M., Vollbach, D. R., Foster, E. R., & Ferguson, T. E. 2005, ApJ, 634, 101
Elmegreen, B. G., & Elmegreen, D. M. 2006, ApJ, 650, 644
Elmegreen, D. M., Elmegreen, B. G., Ravindranath, S., & Coe, D. A. 2007, ApJ, 658, 763
Elmegreen, B. G., & Burkert, A. 2010, ApJ, 712, 294
Ferland, G. J., Korista, K. T., Verner, D. A., Ferguson, J. W., Kingdon, J. B., & Verner, E. M. 1998, PASP, 110, 761
Field, G., Blackman, E., Keto, E. 2009, MNRAS submitted, arXiv:0904.4077
F[ö]{}rster Schreiber, N. M., et al. 2006, ApJ, 645, 1062
Genzel, R., et al. 2006, Nature, 442, 786
Jog, C. J., & Solomon, P. M. 1984, ApJ, 276, 114
Joung, M. K. R., & Mac Low, M.-M. 2006, ApJ, 653, 1266
Kim, S., Staveley-Smith, L., Dopita, M. A., Freeman, K. C., Sault, R. J., Kesteven, M. J., McConnell, D. 1998, ApJ, 503, 674
Klessen, R. S. 2001, ApJ, 556, 837
Klessen, R. S., & Hennebelle, P. 2010, A&A submitted (arXiv:0912.0288)
Kraichnan, R.H. 1967, Phys. Fluids, 10, 1417
Krumholz & Dekel 2010 MNRAS in press, arXiv:1001.0765
Lovelace, R. V. E., Li, H., Colgate, S. A., & Nelson, A. F. 1999, ApJ, 513, 805
Machacek, M. E., Bryan, G. L., & Abel, T. 2003, MNRAS, 338, 273
Martig, M., Bournaud, F., Teyssier, R., & Dekel, A. 2009, ApJ, 707, 250
Mastropietro, C., Burkert, A., & Moore, B. 2009, MNRAS, 399, 2004
Meheut, H., Casse, F., Varniere, P., & Tagger, M. 2010, A&A, 516, A31
Murray, N., Quataert, E., & Thompson, T. A. 2010, ApJ, 709, 191
Petersen, M.R., K. Julien, & J.B. Weiss 2006, Physics of Fluids, 18, 26601
Petersen, M.R., K. Julien, & G.R. Stewart: 2007, ApJ, 658, 1236
Robertson, B., Yoshida, N., Springel, V., & Hernquist, L. 2004, ApJ, 606, 32
Semelin, B., & Combes, F. 2005, A&A, 441, 55
Springel, V., & Hernquist, L. 2003, MNRAS, 339, 289
Stanimirovic, S., Staveley-Smith, L., van der Hulst, J. M., Bontekoe, T.J. R., Kester, D. J. M., & Jones, P. A. 2000, MNRAS, 315, 791
Tasker, E.J., & Bryan, G.L. 2006, ApJ, 641, 878
Tasker, E.J., & Tan, J.C. 2009, ApJ, 700, 358
Teyssier, R. 2002, A&A, 385, 337
Teyssier, R., Chapon, D., & Bournaud, F. 2010, ApJL submitted, arxiv:1006.4757
Tonnesen, S., & Bryan, G. L. 2009, ApJ, 694, 789
Truelove, J. K., Klein, R. I., McKee, C. F., Holliman, J. H., II, Howell, L. H., & Greenough, J. A. 1997, ApJl, 489, L179
Varni[è]{}re, P., & Tagger, M. 2006, A&A, 446, L13
Wada, K., Meurer, G., & Norman, C.A. 2002, ApJ, 577, 197
Wada, K., & Norman, C. A. 2007, ApJ, 660, 276
[^1]: E-mail: frederic.bournaud@cea.fr
[^2]: The slightly higher values for the vertical velocity component correspond to low-density gas flows ejected above the disk plane by supernovae winds; the simulation without feedback has values of $<V_z>$ even closer to $<V_r>$ for high wavenumbers.
[^3]: Note that $V_r$ is slightly larger than $V_{\theta}$, so the in-plane motions are not perfectly isotropic. This is expected for instance if the gas tends to follow epicyclic motions, for which the ratio of the radial to tangential velocity dispersions is $2\Omega/\kappa$.
|
---
abstract: 'We present the first principle calculations of the electrical properties of graphene sheet/h-BN heterojunction(GS/h-BN) and 11-armchair graphene nanoribbon heterojunction(11-AGNR/h-BN), which were carried out using the density functional theory(DFT) method and the non-equilibrium Green’s function(NEGF) technique. Since 11-AGNR belongs to the conductive (3n-1)-family of AGNR, both are metallic nanomaterials with two transverse arrays of h-BN, which is a wide-gap semi-conductor. The two h-BN arrays act as double barriers. The transmission functions(TF) and $I$-$V$ characteristics of GS/h-BN and 11-AGNR/h-BN are calculated by DFT and NEGF, and they show that quantum double barrier tunneling occurs. The TF becomes very spiky in both materials, and it leads to step-wise $I$-$V$ characteristics rather than negative resistance, which is the typical behavior of double barriers in semiconductors.'
author:
- Taiga Wakai
- Shoichi Sakamoto
- Mitsuyoshi Tomiya
title: '$I$-$V$ Characteristics of Graphene Nanoribbon/h-BN Heterojunctions and Resonant Tunneling'
---
Introduction
============
Graphene sheet(GS) is considered as a promising material for next-generation devices as it shows metallic conduction properties[@BZLL]. A graphene nanoribbon (GNR)[@HO; @BSLW] is a narrow strip of GS with nanometer-level width. GNR is expected to have metallic or semiconductor-like properties. It depends on its chirality and width[@YP; @NGP]. To make nanodevices from graphene and GNR, and to imitate the process of transistor construction, it is indispensable to join them to an insulator and/or a semiconductor. For this, junctions with chemically and mechanically inert materials are required.
Moreover, single-layered hexagonal crystal boron nitride (h-BN), which is a wide gap semiconductor, is also considered as a material for manufacturing electronic devices[@BM; @LFS; @L; @J]. The same amounts of boron and nitrogen atoms are used to form the boron-nitrogen nanostructure, h-BN. Its structure is almost the same as that of graphene. Roughly speaking, the carbon atoms of their A-sites are replaced by boron atoms and those of their B-sites are replaced by nitrogen atoms. Therefore, h-BN is an almost idealistic material for the above purpose, because it has the same honeycomb structure and a large band gap in its band structure. The heterojunction of GS and h-BN has already been realized[@L]. It has a high affinity to exchange a part of the GS as a thin transverse belt to make the nanodevice(FIG.1).
![ Ball-stick model of graphene sheet/h-BN heterojunction. Just one array of the honeycomb structure of graphene is replaced with h-BN (show in between dotted lines). The two dashed lines and the two dotted lines represent the length of one unit, though the actual distances of GS and h-BN are slightly different. In between the dashed lines for graphene, it is $1$U$=4.395$Å, and in between the dotted lines for h-BN, it is $1$U$=4.470$Å. \[FIG.1\]](Fig1.eps){width="8cm"}
Numerical Method
================
In this work, the general-purpose package SIESTA[@SP; @SA], which is based on the density functional theory(DFT) method, and the nonequilibrium Green function(NEGF) package TranSIESTA[@BMO1] are utilized for the first principle quantum calculation. The cut-off energy is set at 350Ry. This parameter is used for considering the set of plane wave bases that determines the quality of the calculation. The samples of the wave-number of the first Brillouin zone are selected on 128 ($=4\times4\times8$) points. In addition, the KB (Kleinman-Bylander) potential[@KB] is adopted for the pseudo-potential and the exchange correlation term for the PBE[@PBE] of the generalized gradient approximation(GGA).
For numerical calculation of the band structure, a GS has to be built inside a super cell in SIESTA. In this method, GS must not have periodicity in the normal direction of its surface, and GNR must not have periodicity in its normal and transverse(width) directions. Unfortunately, the super cell in SIESTA has inherent 3D periodicity. Therefore, the vacuum domain must be settled to [20Å]{}, which is large enough for the directions that should not be periodic, in order to eliminate the influence from adjacent super cells. In the transverse(width) direction, the size is set about two times larger than the ribbon width for GNR. Finally, the size of the graphene sheet or GNR in the longitudinal direction is set such that they become periodic.
The NEGF method can deal with the calculation of an open system: the non-periodic boundary condition along the current(length) direction. TranSIESTA is the numerical realization of the NEGF method. Both the ends of GS and GNR are connected to the electrodes at half infinity. It makes the calculation of electric current flowing through the GNR possible. To avoid unnecessary dispersion of electric current at the junction parts of the central domain and electrodes, the width of the electrodes is set the same as that of the center graphene (nanoribbon).
Adopting GGA as an exchange correlation term in the DFT calculation, the lattice constant is known to be estimated a little larger than the experimental value. The carbon atomic distance in the graphene sheet is experimentally found to be [1.42Å]{}; however, our numerical result for the atomic distance is [1.465Å]{}. At this distance, the total energy is minimized and the structure is the most stable in the SIESTA simulation. Therefore, in this study, the carbon atomic distance in graphene is set to the latter value: [1.465Å]{}. In addition, by using the same process, the boron - nitrogen atomic distance in the h-BN sheet is set to [1.490Å]{}. Then, the band structure can be calculated by using DFT and the current through it. In this work, we use the length of one honeycomb lattice of our simulation as a unit U. However, the actual length of h-BN is slightly different from that of graphene, i.e., $1$U$=4.395$[Å]{} for graphene and $1$U$=4.470$[Å]{} for h-BN (cf. FIG.1).
Electrical Properties of GS/h-BN
================================
Our purpose is to find out the possibility of constructing nanodevices. GS and 11-AGNR, which belongs to the (3n-1)-AGNR family, are chosen as the base materials to be manipulated. They are both known to be conductive[@YP; @NGP]. On the other hand, h-BN sheet is found to be a wide gap semiconductor[@LFS]. Therefore, their heterojunction is expected to have unique electrical properties.
The dispersion relation of graphene sheets is known to have a Dirac cone shape[@NGP], and it shows metalic properties. It is also numerically confirmed that the bandgap is zero(FIG.2a). On the other hand, the h-BN sheet is known to have a wide gap in its band structure. It is also confirmed that the bandgap is about 4.6eV in our simulation of DFT (FIG.2b). Then, it has to be tested numerically whether graphene can have semiconductor properties by joining the h-BNs. The conductance was simulated by NEGF using TranSIESTA, which can also derive the transmission function(TF) $T(E)$.
![ Calculated band structure of (a)GS, (b)h-BN sheet, (c)GS/h-BN heterojunction of one h-BN array in the middle and 15U of carbon honeycomb lattice in the length direction(Fig.1). Sold horizontal straight lines in the middle of the graphs represent the Fermi energy. \[FIG.2\]](Fig2.eps){width="9cm"}
First, one unit(1U) array of h-BN is placed on a graphene sheet(FIG.1). Including the h-BN array on the graphene sheet, they are collectively called graphene sheet/h-BN heterojunction(GS/h-BN). On extending the leads, the bandgap tends to become smaller. Then it becomes about 0.2eV(FIG.2(c)), making the carbon unit parts long enough to be almost 7U at both ends(FIG.1).
Finally, by TranSIESTA, the TF of graphene/h-BN heterojunction(GS/h-BN) shows that the bandgap converges to about 0.1 eV (FIG.3(b))) and is finite. In this method, the leads through which current comes in and goes out can be considered semi-infinite. It means that the graphene is transferred to the semiconductor by joining the array of h-BN to the pure GS, even though the gap is tiny. In FIG.3, the TF of GS/h-BN is compared with that of pure graphene. Since h-BN array acts as a semiconductor with a wide gap in the sheet, its electric conductivity is significantly reduced, compared to that of pure GS, and remarkably, the bandgap appears.
![ Numerical TFs of (a) GS, (b) GS/h-BN heterojunction, which has a transverse h-BN array in the middle of the sheet. \[FIG.3\]](Fig3.eps){width="7cm"}
In this work, two transverse arrays of h-BN are also embedded(cf. FIG.4). The structure is similar to the double barrier system of semi-conductors, which shows unique quantum tunneling phenomenon. In the following, $\Delta u$ express the distance between the two arrays(h-BN) in the unit U.
![ Schematic illustration of the simulation model for the NEGF method: GS/h-BN in the cases of (a) $\Delta u = 1$U, and (b) $\Delta u = 3$U. The red arrows indicate the intervals $\Delta u$ which are measured in the carbon honeycomb lattice as a unit. \[FIG.4\]](Fig4.eps){width="7cm"}
The relation between the current $I$ and voltage $V$ can be derived using the Landauer formula[@BIL] and its advanced form for the NEGF method[@HT; @BMO2]. Then, the current $I$ can be evaluated as $$\begin{split}
I=\frac{2e}{h}\int T(E)& [f_L(E)-f_R(E)]dE \\
=\frac{2e}{h}\int T(E)& [f(E-(E_F -\frac{eV}{2})) \\
&-f(E-(E_F +\frac{eV}{2}))]dE
\end{split}$$ where $f(E)$ represents the Fermi distribution function, $E_F$ is the Fermi energy, $f_L(E)$ and $f_R(E)$ are the function of the left and the right leads, $V$ the external bias voltage, and $e$ the elementary charge.
In GS/h-BN, the calculated TFs have sharp spikes(FIG.5). A clear exception is the TF of $\Delta u=0$, where the TF greatly decreases around the Fermi energy, as shown in FIG.5(a), compared to the pure GS(FIG.3(a)). This means that only very slight electric currents can flow up to the external bias voltage of about 5V. It is in clear contrast to the case in which only one array is replaced by h-BN(FIG.3(b)); the TF of the single h-BN array still has remnants of graphene’s step-wise behavior. On the contrary, consecutive double arrays of h-BN attain a strong resistance with narrow transition channels. It is also confirmed that, by doubling the width, the penetration of electrons is exponentially reduced and becomes almost zero.
![ Transmission functions of GS/h-BN. The arrows between the peaks indicate the energy gap $\Delta E$. The peaks with symbols $\bigcirc$, $\times$, and $\bigtriangleup$ correspond to the peaks of the one dimensional Dirac equation model shown in FIG.10. \[FIG.5\] ](Fig5.eps){width="7cm"}
The structure that is formed by joining h-BN arrays at two places on a GS with some interval can be considered as double barriers of exactly the same shape; the same width and height, etc. In particular, $T(E)$ also becomes negligible in the close neighborhood of the Fermi energy and shows semiconductor-like property. The current can hardly flow in this region(Fig.5).
Therefore, our numerical TF of the GS/h-BN double barriers actually shows just the resonant tunneling. Its transmission function has high and sharp transmission peaks with some gaps, even under the energy gap in the band structure of h-BN. The peaks are located symmetrically on both sides of the Fermi energy. By enlarging the joint space bwtween two h-BNs, it is also confirmed that the number of peaks increased(FIG.5). The energy gap $\Delta E$ of the peaks around the neighborhood of the Fermi energy can be considered to be equivalent to the bandgap. Then it becomes smaller, so as to enlarge the space of h-BNs.
Finally, the $I$-$V$ property of each structure is investigated(FIG.6). For $\forall \Delta u$, in the lower external voltage $V$, the current $I$ does not flow, or is almost negligible. Since h-BN has a large resistance in the case of $\Delta u=0$(FIG.5(a)), it is diffcult for the electric current to flow even if the external voltage is raised, until the voltage overcomes the height of the barriers, which is set at the bandgap of h-BN, 4.6eV. On the other hand, in the case of $\Delta u \neq 0$, the resonant tunneling makes the current penetrate the walls. For example, the sudden burst of current starts only beyond 2.0V for $\Delta u=1$, and it also confirms that it has semiconductor-like characteristics. The threshold voltage is assumed to be the voltage at which the current exceeds $1\mu A$. The threshold is almost the same as $\Delta E$ in the TF, because it is evaluated as the starting point of the current. In addition, the graph of $I$-$V$ properties becomes mildly step-wise (FIG.6), owing to the discrete and spiky transmission function(FIG.5). The structure of TF is complicated, and there remains a lot to be analyzed for the GS/h-BN double barrier system.
![ I-V characterrictics of GS/h-BN with two h-BN arrays at a distance of $\Delta u = 1 \sim 7$U. \[FIG.6\]](Fig6.eps){width="7cm"}
Electrical Properties of AGRN/h-BN
==================================
Then, AGRN is also simulated by TranSIESTA, terminating the dangling bonds of carbon atoms on the edges by hydrogen atoms. Since the number of carbon atoms in the transverse direction belongs to the(3n-1)-family, 11-AGNR(n=4 case) is known to have metalic properties. Then h-BNs are embedded at two places in the AGNR in the same manner as that of the GS case. The spacing between two h-BNs, $\Delta E$ is changed from 1 to 7U.
The heterojunction between the two h-BNs also breaks the quantization of the TF of 11-AGNR as in the GS case, and makes it discretized and spiky(FIG.7). Actually, it has a simpler structure of TF than that in GS. On enlarging the space between two h-BN arrays in AGNR, the energy gap of the TF, $\Delta E$, becomes significantly smaller as in the GS case. The reciprocal of $\Delta E$ is plotted against $\Delta u$ in FIG.8(a). It shows that the bandgap of the GS crucially depends on the distance $\Delta u$. The energy gap is found to be almost inversely proportional to $\Delta u$. The $I$-$V$ characteristics with the heterostructures have step-wise shapes(FIG.8(b)). As in the GS/h-BN case for $\forall \Delta u$ with lower external voltage, the $I$-$V$ characteristics of the AGNR/h-BN double barrier shows that the current $I$ does not flow until the voltage $V$ reaches the threshold value, which is essentially the same as $\Delta E$. The threshold voltages at which the current starts to flow can be obtained from its TF and $I$-$V$ property. It suggests that it is possible to control the threshold voltage by $\Delta u$. Furthermore, the voltage becomes larger and approaches the other peaks in the TF, and subsequently, the current also becomes larger in steps.
![ Transmission functions of 11-AGNR/h-BN. Their shapes are actually simpler than that of GS/h-BN. The peaks with symbols $\bigcirc$, $\times$, and $\bigtriangleup$ correspond to the peaks of the one dimensional Dircac equation model shown in FIG.10. \[FIG.7\] ](Fig7.eps){width="6cm"}
![ (a) Relation of the reciprocal of the gap between the peaks of the transmission function around the Fermi energy and the interval between h-BN arrays in11-AGRN/h-BN. (b) I-V characteristics of 11-AGNR/h-BN with two h-BN arrays（ $\Delta u = 1 \sim 5$U） \[FIG.8\]](Fig8.eps){width="8cm"}
In cases of both GS/h-BN and AGNR/h-BN, the spiky TF leads to step-wise $I$-$V$ characteristics. Spiky peaks in the TF means that it has very narrow channels of current $I$, this causes the $I$-$V$ characteristics to have a step-wise shape. The current undergoes a discrete jump-up, when one more channel is allowed to contribute to the current. It has to be emphasized that the TF structures of the AGNR/h-BN double bariers are simpler to be analyzed than those of GS/h-BN. The TF structures of the 11-AGNR/h-BN double barrier system are much simpler than those of GS/h-BN, and it is fairly assumed to follow some clear principle.
One-dimentional Dirac Equation Model
====================================
In order to analyze the electric behavior of our simulation, we adopt the Dirac equation approach, which is based on the tight-binding approximation. It is known as a successful method for the analysis of graphene[@W]. Due to the continuity of the wave function of electron in the entire material, h-BN arrays also have to be treated by the Dirac equation. However, its mass is estimated as half of the band gap $m_{\mathrm{h-BN}}=2.3$eV $=4.6$eV$/2$. In h-BN, the band structure shows that the dispersion relation is not massless; it has a large band gap of 4.6eV. Two massive electron regions of h-BN are embedded in the massless region of GS or AGNR. Apparently, this structure resembles a pseudo-potential well that can realize resonant tunneling[@CET]. The massive electron region strikingly prevents electrical conductivity.
Resonant tunneling is the quantum mechanical phenomenon by which a particle goes through double consecutive barriers, utilizing pseudo-eigenstates in the double barriers, even when the energy is lower than the top of the barriers. The finite mass in the h-BN regions effectively becomes the barrier(FIG.9). If its energy is close to a pseudo-eigenstate and its wavefunction endures the exponetial dumping in the barriers, then tunneling can happen as follows. FIG.10 shows that the transmission probability also becomes very spiky and the peaks line up evenly. This feature is essentially the same as our first principle calculation of GS/h-BN and GNR/h-BN.
![ Shematic illustration of the approach for the 1D model of our double barrier system. Here, massive electron(h-BN) regions play the role of barriers. \[FIG.9\]](Fig9.eps){width="8cm"}
![ Illustration of the examples of the transmission probability $\tau$ derived from the Dirac equation model, with $m_{h-BN}=2.3$eV, $w=1$U of h-BN, and $d=1$U, $3$U, and $5$U of AGNR. Apparently, the sharp peaks appear much lower than the barrier height $m_{h-BN}$. On widening the interval between h-BNs, the number of peaks increase. The peaks with symbols $\bigcirc$, $\times$, and $\bigtriangleup$ correspond to the peaks shown in FIG.7 and FIG.5. \[FIG.10\]](Fig10.eps){width="8cm"}
Furthermore, by dropping the transverse degree of freedom in the narrow width of AGNR, it is simplified as the one-dimensional(1D) Dirac equation $$(v_F \sigma_x p_x + m \beta + V(x) -E ) \Phi(x) = 0 ,$$ which is set along the $x$ axis (longitudinal current direction of GS and AGNR). Here, $p_x= \frac{\hbar}{i} \frac{d}{dx}$ and two matices are set as $$\sigma_x=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \beta=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} .$$ The wavefunction of the Dirac equation is defined as a two component vector $\Phi(x) = (\phi(x), \psi(x))$. Assuming the potential $V$ is constant, from Eq.(2), the second component $\psi$ is determined by the first component $\phi$ as $$%\psi(x)=\frac{-i\hbar v }{m-v_F+E} \frac{d\phi}{dx} .
\psi(x)=\frac{-i\hbar v_{F} }{m {v_{F}}^2-V+E} \frac{d\phi}{dx} .$$ The first component can also be derived from the second one using Eq.(2). The mass is set as $m=0$ for graphene(regions I, III, and V in FIG.9), $m=m_{\mathrm{h-BN}}=2.3$eV for h-BN(regions II and IV in FIG.9) and in our model, the Fermi velocity is set slightly slower ($v_F = c/350$) than the usual $c/300$( this will be explained later).
Then, the continuity conditions at the boundaries between regions I - V are required for the wave functions $$\begin{matrix}
\rm{I}: & \phi_{\rm{I}} (x) = e^{ikx}+r e^{-ikx} & x<-\frac{d}{2}-w \\
\rm{II}: &\phi_{\rm{II}} (x) = a e^{i\kappa x}+ b e^{-i\kappa x} & -\frac{d}{2}-w < x < -\frac{d}{2} \\
\rm{III}: &\phi_{\rm{III}} (x) = c e^{ikx}+d e^{-ikx} & -\frac{d}{2} < x < \frac{d}{2} \\
\rm{IV}: &\phi_{\rm{IV}} (x) = f e^{i\kappa x}+ g e^{-i\kappa x} & \frac{d}{2} < x < \frac{d}{2} + w \\
\rm{V}: &\phi_{\rm{V}} (x) = t e^{ikx} & x> \frac{d}{2} + w ,
%\end{eqnarray}
\end{matrix}$$ and their counter parts $\psi_{\rm{I}} (x), \cdots, \psi_{\rm{V}} (x)$(see Eq.(4)). Note that $k=\frac{E}{\hbar c}$, $\kappa = \frac{\sqrt{2m_{\mathrm{h-BN}} E}}{\hbar}$, and $w$ is the width of the wall of h-BN, $d$ is the distance of the walls, $r$ is the reflection coefficient, and $t$ is the transmission coefficient. Accordingly, we have eight undertermined coefficients: $r, a, b, c, d, f, g,t$, and eight independent equations at the boundaries also. Thus, the coefficients can be determined in general.
The above continuity conditions determine the transmission probability(TP) $\tau = | t |^{2}$. The numerical result of $\tau$ is obtained in FIG.10, and it exhibits a clear spiky property. In this work, two transverse h-BN arrays with exactly the same shape form the pseudo-well, which can have pseudo-eigenstates inside. This causes the TF to have several peaks and become discrete. In graphene, it is assumed that an electron acts as the 1D massless Dirac fermion, whereas in h-BN, a massive Dirac equation with a mass of 2.3eV is assumed. The potential itself is $V(x)=0$ everwhere in our model. Selecting the parameters as above, the shape of $\tau$ is quite similar to our numerical TF. In particular, around the Fermi energy, there are several spiky peaks arrayed (at almost equal intervals and) symmetrically along both sides of the Fermi energy.
With exactly two similar shaped walls, sharp peaks appear in TP. Each peak corresponds to a pseudo-eigenstate in the well, which is formed by the walls. The number of peaks increases, when the interval of the walls becomes broader and the shapes of the peaks get narrower(FIG.10).
On adopting the Landauer formula (1), the spiky transmission function (FIG.5,7) and the Fermi distribution function at room temperature cause the $I$-$V$ characteristics to become moderately step-wise. This is not the case in a semiconductor, and there is no conduction band bottom in graphene, which is essentially metallic. When the bias voltage is increased, more spiky peaks contribute to the current $I$, because a higher bias results in a wider energy range for the transmission function. Thus, the $I$-$V$ characteristics become mildly step-wise, rather than showing of negative resistance.
In the following, the 11-AGNR/h-BN double barrier system and the 1D Dirac equation model are compared, because 11-AGNR has a relatively simple TF in the neighborhood of the Fermi energy. The 1D Dirac model explains the presence of many spiky peaks in the TF shown in FIG. 7 and 10. The peak positions correspond to the energies of the pseudo-eigenstates inside the well. However, the Fermi velocity of an electron has to be adjusted to be silightly slower, that is , $v_F = c/350$(c is the velocity of light) instead of $v_F=c/300$, which is the well-known result of the tight binding model. The velocity of electron that we have chosen would be some kind of ensemble thoughout the first Brillouin zone. Actually, the original $v_F$ is the estimation at the Dirac cone structure where an electron act as a massless fermion. With arbitrary momentum, there is an electron band gap and effectively massive dispersion relation. Then, the averaged velocity should be a little slower.
With NEGF, the TF with $\Delta u \neq 0$ shows spiky and discrete behavior with respect to energy. Widening the interval between h-BNs causes an increase in spikes as in our double square potential model(FIG.7,10). Therefore, resonant tunneling, which permits electrons to penetrate the double barrier, must occur when the energy of electrons accords with the resonance level (pseudo-steady state) inside the double wall. In particular, the transmission probability of our Dirac fermion model well reproduces the TF of the 11-AGNR/h-BN double barrier system by the first principle result of SIESTA. It clearly explains why $\Delta E$ is propotional to $1/\Delta u$. If the height of the potential, or the mass of electron, is infinitely high in the double barriers, then the eigenvalues are $E_n=\frac{\pi}{\Delta u}(n+\frac{1}{2})$[@BM2]. The threshold energy corresponds to the ground state $n=0$, and its value is inversely proportional to $\Delta u$. Owing to the actual finite mass, there exists a slight deviation from the proportional relation.
Moreover, according to our analysis, the 1D Dirac fermion model can also identify a large number of peaks in the TFs of GS/h-BN, which are produced as a result of resonant tunneling(FIG.5). The other peaks must be the consequence of the transverse degree of freedom of GS, which has inifite width compared with AGNR.
Conclusion
==========
First, the electrical conduction properties of the GS/h-BN heterojunctions were investigated. GS with one h-BN array becomes a semiconductor with a bandgap of 0.2eV. On inserting two h-BN arrays into GS, a pseudo-potential well is formed in the system. Pseudo-resonant states are formed in between the walls, and it exhibits the resonant tunneling phenomenon. The first principle calculation shows that the threshold voltage in the $I$-$V$ characteristics is inversely proportional to $\Delta u$.
Next, the electrical conduction properties of the 11-AGNR/h-BN heterojunction were calculated. Pure 11-AGRN belongs to the (3n-1)-family and is known to be a conductor as GS[@YP; @NGP].
Finally, we calculated the properties of the 11-AGNR/h-BN double barrier systems, which was also expected to exhibit resonant tunneling. The TF and the $I$-$V$ characteristics are similar to, or even simpler than GS/h-BN, because they do not have much transverse degree of freeedom. Actually, 11-AGNR is just a narrow ribbon. It also has a threshold voltage that is propotional to $1/\Delta u$. Therefore, the threshold voltage can be controlled to a certain degree by changing the distance between the two h-BN arrays of 11-AGRN/h-BN double barrier system.
This proportionality is due to the shape of the spikes in the TFs, and the energy of the gound states in the pseudo-potential well are inversely proportional to $\Delta u$. Using the Landauer formula, the current is essentially determined by the number of peaks covered by the window, which is created by the Fermi density function and shaped like a flat finite support. Thus, it does not show negative resistance. GS and the (3n-1)-family of AGNR are metallic (not semiconductor), and does not have a conduction band bottom. Then, increasing the bias voltage for GS/h-BN double barriers and AGNR/h-BN double barriers increases the number of contributing channels inside the window that are actually spiky peaks in TF. This phenomenon can be well simulated as a transmission probability by our 1D Dirac fermion model.
On the other hand, it is well-known that TF of semi-conductor has a more continuous shape and a conduction band bottom. It exhibits a much smoother change in current and a negative resistance. The result of 11-AGNR/h-BN calculation is more similar to our 1D Dirac model.
The same investigation was also conducted for the 14-AGNR/h-BN heterojunction. 14-AGNR also belongs to the (3n-1)-family. The results are almost the same as that of 11-AGNR/h-BN. Finally, our study shows that the electronic properties of GS and GNR can be controlled by varying positions at which h-BNs are inserted in them.
[99]{}
|
---
abstract: 'For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a [*nonlocality of the ensemble.*]{} In this paper, we consider bipartite state discrimination in a composite system consisting of $N$ subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to $1$ as $N\rightarrow\infty$ if the entropy of the probability distribution associated with the ensemble is less than $1$, even if the success probability is less than $1$ for any finite $N$. In other words, the nonlocality of the $N$-[*fold*]{} ensemble asymptotically disappears if the probability distribution associated with each ensemble is concentrated. Furthermore, we show that the disappearance of the nonlocality can be regarded as a remarkable counterexample of a fundamental open question in theoretical computer science, called a [*parallel repetition conjecture*]{} of [*interactive games*]{} with two classically communicating players. Measurements for the discrimination task include a projective measurement of one party represented by stabilizer states, which enable the other party to perfectly distinguish states that are sampled with high probability.'
author:
- Seiseki Akibue
- Go Kato
bibliography:
- 'generic.bib'
nocite: '[@*]'
title: Bipartite discrimination of independently prepared quantum states as a counterexample to a parallel repetition conjecture
---
Introduction
============
Various aspects of nonlocal properties of quantum mechanics have been investigated by considering multipartite information-processing tasks undertaken by joint quantum operations called [*local operations and classical communication*]{} (LOCC). Indeed, considered not to increase quantum correlation between the parties, LOCC is widely used for characterizing entanglement measures [@VVedral; @Horodecki; @MBPlenio] and nonlocal properties of unitary operations [@Soeda1; @Stahlke; @Soeda2].
Another aspect of nonlocal properties is characterized by considering bipartite state discrimination. Bipartite state discrimination is a task where two parties, typically called Alice and Bob, perform a measurement implemented by LOCC to distinguish states sampled from an a priori known fixed ensemble of quantum states. By definition, the ability of the parties in bipartite state discrimination is more restricted than in single-party state discrimination as illustrated in Fig. \[fig:intro\](a) and (b). However, if each state constituting the ensemble is a [*classical state*]{}, namely a probabilistic mixture of the tensor products of two fixed mutually orthogonal states, there is no gap in bipartite and single-party distinguishability. Thus, when the gap exists, it can be interpreted as a [*nonlocality of the ensemble*]{}, which has been extensively studied.
![Graphical representations of three types of quantum state discrimination. Rounded rectangles represent quantum systems and dotted rectangles represent subsystems where joint quantum operations can be performed. The state of each system is randomly sampled from an a priori known ensemble $\{(p_i,\rho_i)\}$. (a) In bipartite state discrimination, a measurement is implemented by LOCC. (b) In single-party state discrimination, a measurement is implemented by a joint quantum operation. (c) Bipartite discrimination of quantum states sampled from a three-fold ensemble.[]{data-label="fig:intro"}](intro.eps){height=".15\textheight"}
Several studies have revealed the difference between the nonlocality of an ensemble and entanglement: for distinguishing any two entangled pure states, the ensemble is local, i.e., they can be optimally distinguished by LOCC as well as by joint measurement [@twostatediscrimination1; @twostatediscrimination2]; there exist nonlocal ensembles comprising only product states [@productstate1; @productstate2; @productstate3; @productstate4; @productstate5]; and increasing the number of entangled states constituting an ensemble can decrease the nonlocality [@localdiscrimination]. On the other hand, for distinguishing $M$ orthogonal maximally entangled states having local dimension $d$, any ensemble comprising such states (each state is sampled with non-zero probability) is nonlocal if $M>d$ [@optimalprob1; @optimalprob2; @optimalprob3; @optimalprob4]. When $M=d$, any ensemble with $M=d=3$ is local [@optimalprob2] whereas there exists a nonlocal ensemble with $M=d=4$, which also demonstrates a novel phenomenon of entanglement discrimination catalysis [@optimalprob5]. Furthermore, a novel application of the nonlocality of an ensemble was found in quantum data hiding [@datahiding1; @datahiding2]. Recently, connections to other fundamental issues, such as the monogamy of entanglement [@datahidingstate2], an area law [@datahidingstate3], and a characterization of quantum mechanics in general probabilistic theories [@GPTs], have also been found.
In this paper, we reveal a counterintuitive behavior of the nonlocality of an ensemble caused by entanglement, and show that it provides a remarkable counterexample of a fundamental open question in theoretical computer science, called a [*parallel repetition conjecture*]{} of [*interactive games*]{} with two players, which has been proven to be true in a classical scenario. We consider bipartite state discrimination in a composite system consisting of $N$ subsystems, where each subsystem is shared between Alice and Bob and the state of each subsystem is randomly sampled from a particular ensemble. In the discrimination task, they perform an LOCC measurement to distinguish states sampled from the $N$-[*fold*]{} ensemble formed by taking $N$ copies of one particular ensemble as illustrated in Fig. \[fig:intro\](c).
We consider that each ensemble comprises the Bell states, and show that the bipartite distinguishability approaches the single-party distinguishability as $N$ grows if the entropy of the probability distribution associated with each ensemble is less than a certain value. More precisely, we measure the distinguishability by the success probability of perfectly identifying a state used in the minimum-error discrimination [@statediscrimination]. We show that if the entropy condition is satisfied, the success probability in bipartite discrimination converges to $1$ as $N\rightarrow\infty$, even if the success probability in bipartite discrimination is less than $1$ for any finite $N$, namely, the nonlocality of the $N$-fold ensembles asymptotically disappears. Since such a disappearance of the nonlocality occurs only if the mixed state corresponding to each ensemble is entangled, it also demonstrates a difference between the nonlocality of an ensemble and entanglement.
An [*interactive proof system*]{} is a fundamental notion of (probabilistic) computation in computational complexity theory [@AM; @MIP; @QIP; @QMIP], with important applications to modern cryptography [@ZK; @QSZK] and hardness of approximation [@PCP1; @PCP2]. Its general description is based on an [*interactive game*]{} [@quantumproofs] involving an interaction between a referee and players. The referee makes a fixed probabilistic trial to judge players to [*win*]{} or [*lose*]{} the game, and the players try to maximize the winning probability. If the maximum winning probability of an interactive game is less than $1$, it is natural to expect a [*parallel repetition conjecture*]{} of the game holds, namely, the maximum winning probability of the repeated game, where the referee simultaneously repeats the game independently and judges the players to win the repeated game if the players win all the games, decreases exponentially. If the parallel repetition conjecture holds, efficient error reduction of the computation in interactive proof systems is possible without increasing the round of interactions. The conjecture has been proven for interactive games with a single player [@KW; @productrule3; @quantumproofs] and with two separated classical players [@Raz; @Holenstein]; however, it remains widely open as to whether the conjecture holds for interactive games with two quantum players, with several positive results for special cases [@parallelrep1; @parallelrep2].
Bipartite state discrimination can be regarded as an interactive game with two classically communicating quantum players, where the referee prepares the state of a composite system randomly sampled from an ensemble and judges the players to win the game if they guess the state correctly. To the best of our knowledge, it is unknown whether the parallel repetition conjecture of interactive games with two classically communicating players holds, which originates from an open problem posed in [@QMIPLOCC]. We show that the disappearance of the nonlocality in the state discrimination can be regarded as a remarkable counterexample of the conjecture, i.e., while the maximum winning probability of each game is less than $1$, that of the repeated game does not decrease; moreover, it asymptotically approaches $1$.
This paper is organized as follows: In Section II, we provide precise definitions of an $N$-fold ensemble and the distinguishability of states sampled from it and introduce some notations concerning the definitions. In Section III, we review some known upper bounds for the success probability of the identification in bipartite discrimination and apply them to show an asymptotic behavior of an upper bound of the success probability in our scenario. In Section IV, we construct an LOCC measurement for the success probability in bipartite discrimination to converge to $1$. In Section V, we review an interactive game and its parallel repetition conjecture and show that the disappearance of the nonlocality can be regarded as a counterexample of the parallel repetition conjecture of interactive games with two classically communicating players. The last section is devoted to conclusion and a discussion.
Definitions and notations
=========================
We denote the Hilbert space of Alice’s system and Bob’s system by $\mathcal{A}$ and $\mathcal{B}$, respectively. Suppose the state of the composite system $\mathcal{A}\otimes\mathcal{B}$ is randomly sampled from an a priori known ensemble of finite quantum states, $$\{(q_{\mathbf{m}},{| {\Psi_{\mathbf{m}}} \rangle}\in\mathcal{A}\otimes\mathcal{B})\}_{\mathbf{m}},$$ where $q_{\mathbf{m}}$ is an element of a probability vector. (Note that in general, an ensemble can comprise mixed states in state discrimination; however, it is sufficient to consider pure states in our scenario.)
Moreover, we consider that Alice’s system and Bob’s system consist of $N$ subsystems $\mathcal{A}=\otimes_{n=1}^N\mathcal{A}_n$ and $\mathcal{B}=\otimes_{n=1}^N\mathcal{B}_n$ respectively, where $\mathcal{A}_n=\mathcal{B}_n=\mathbb{C}^2$ for all $n$, and a state of each subsystem $\mathcal{A}_n\otimes\mathcal{B}_n$ is randomly sampled from a particular ensemble comprising the Bell states $\{(p_{m},{| {\Phi_{m}} \rangle}\in\mathcal{A}_n\otimes\mathcal{B}_n):m\in F_2^2\}$, where $F_2^k$ is the direct product of $k$ finite fields of two elements, $\mathbf{p}=(p_{00},p_{01},p_{10},p_{11})$ is a probability vector, and $$\begin{aligned}
{| {\Phi_{00}} \rangle}=\frac{1}{\sqrt{2}}({| {00} \rangle}+{| {11} \rangle}),\\
{| {\Phi_m} \rangle}=(\mathbb{I}\otimes\sigma_{m}){| {\Phi_{00}} \rangle},\\
\sigma_{00}=\mathbb{I},\sigma_{01}=X,\sigma_{10}=Z,\sigma_{11}=Y.\end{aligned}$$ Note that $\mathbb{I}$ represents the identity operator, $X,Y,Z$ represent Pauli operators, and $\{{| {0} \rangle},{| {1} \rangle}\}$ is a fixed orthonormal basis of $\mathbb{C}^2$ such that $X{| {x} \rangle}={| {1-x} \rangle}$, $Z{| {x} \rangle}=(-1)^x{| {x} \rangle}$ and $Y{| {x} \rangle}=(-1)^xi{| {1-x} \rangle}$ for $x\in \{0,1\}$.
The $N$-fold ensemble formed by taking $N$ copies of an ensemble $\{(p_{m},{| {\Phi_{m}} \rangle}\in\mathcal{A}_n\otimes\mathcal{B}_n):m\in F_2^2\}$ is represented by $\{(q_{\mathbf{m}},{| {\Psi_\mathbf{m}} \rangle}\in\mathcal{A}\otimes\mathcal{B}):\mathbf{m}=(m_1,\cdots,m_N)\in F_2^{2N}\}$ such that $$\begin{aligned}
q_{\mathbf{m}}&=&\prod_{n=1}^N p_{m_n}\label{eq:productprob}\\
{| {\Psi_{\mathbf{m}}} \rangle}&=&\otimes_{n=1}^N{| {\Phi_{m_n}} \rangle}\\
&=&(\mathbb{I}^{(\mathcal{A})}\otimes\sigma_{\mathbf{m}}^{(\mathcal{B})}){| {\Phi_{00}} \rangle}^{\otimes N},\end{aligned}$$ where $\sigma_{\mathbf{m}}^{(\mathcal{B})}=\otimes_{n=1}^N\sigma_{m_n}$, the superscript of a linear operator represents the Hilbert space it acts on, and the order of the Hilbert spaces is appropriately permuted in $\otimes_{n=1}^N{| {\Phi_{m_n}} \rangle}$ and ${| {\Phi_{00}} \rangle}^{\otimes N}$.
Alice and Bob’s measurement can be described by a [*positive-operator valued measure*]{} (POVM) $\{M_{\hat{\mathbf{m}}}\in P(\mathcal{A}\otimes\mathcal{B})\}_{\hat{\mathbf{m}}}$ satisfying $\sum_{\hat{\mathbf{m}}} M_{\hat{\mathbf{m}}}=\mathbb{I}$, where $P(\mathcal{A}\otimes\mathcal{B})$ represents a set of positive semidefinite operators on $\mathcal{A}\otimes\mathcal{B}$. When a state ${| {\Psi_{\mathbf{m}}} \rangle}$ is sampled, a measurement outcome $\hat{\mathbf{m}}$, corresponding to their estimation of $\mathbf{m}$, is obtained with probability given by ${\langle {\Psi_{\mathbf{m}}} |}M_{\hat{\mathbf{m}}}{| {\Psi_{\mathbf{m}}} \rangle}$. Thus, the success probability of perfectly identifying a state is given by $\sum_{\mathbf{m}} q_{\mathbf{m}}{\langle {\Psi_{\mathbf{m}}} |}M_{\mathbf{m}}{| {\Psi_{\mathbf{m}}} \rangle}$. The bipartite distinguishability of states sampled from an $N$-fold ensemble is measured by the [*maximum*]{} success probability of the identification, namely, $$\label{eq:LOCCprob}
\gamma=\sup\left\{\sum_{\mathbf{m}\in F_2^{2N}} q_{\mathbf{m}}{\langle {\Psi_{\mathbf{m}}} |}M_{\mathbf{m}}{| {\Psi_{\mathbf{m}}} \rangle}:\{M_{\hat{\mathbf{m}}}\}_{\hat{\mathbf{m}}}\in LOCC\right\},$$ where $LOCC$ represents a set of POVMs implemented by LOCC between Alice and Bob. Note that the single-party distinguishability, where the supremum is taken over all POVMs in Eq., is always $1$ for any $N$ and $\mathbf{p}$ since $\{{| {\Psi_{\mathbf{m}}} \rangle}\}_{\mathbf{m}\in F_2^{2N}}$ is a set of orthogonal states. As other measures of the distinguishability, the maximum success probability of unambiguous state discrimination [@unambiguous1; @unambiguous2; @unambiguous3] and the separable fidelity [@sepfid] have been also studied.
In our construction of an LOCC measurement given in Section \[sec:construction\], it is sufficient to consider an important subset of $LOCC$, a set of POVMs implemented by [*one-way*]{} LOCC from Alice to Bob. Indeed, in many cases, one-way LOCC is sufficient for perfect discrimination when perfect bipartite state discrimination is possible [@twostatediscrimination1; @localdiscrimination; @optimalprob2; @optimalprob5]. In one-way LOCC (from Alice to Bob), first Alice performs a measurement on her own system described by a POVM $\{A_{\mathbf{a}}\in P(\mathcal{A})\}_{\mathbf{a}}$ and sends the measurement outcome $\mathbf{a}$ to Bob. Then Bob performs a measurement on his own system described by a POVM $\{B_{\hat{\mathbf{m}}|\mathbf{a}}\in P(\mathcal{B})\}_{\hat{\mathbf{m}}}$ based on $\mathbf{a}$. Thus, the maximum success probability of the identification by one-way LOCC is given by $$\begin{aligned}
\gamma_1=\max\Big\{\sum_{\mathbf{m}\in F_2^{2N}}\sum_{\mathbf{a}} q_{\mathbf{m}}{\langle {\Psi_{\mathbf{m}}} |}A_{\mathbf{a}}\otimes B_{\mathbf{m}|\mathbf{a}}{| {\Psi_{\mathbf{m}}} \rangle}\nonumber\\
:\{A_{\mathbf{a}}\}_{\mathbf{a}}{\rm \ and\ }\{B_{\hat{\mathbf{m}}|\mathbf{a}}\}_{\hat{\mathbf{m}}}{\rm \ are\ POVMs}\Big\}.
\label{eq:1LOCCprob}\end{aligned}$$ By definition, $\gamma\geq\gamma_1$. Note that $\gamma_1$ is always achievable by some measurements implemented by one-way LOCC due to its compactness, in contrast to general LOCC [@productstate4; @CLMOW].
Since the parameters in an $N$-fold ensemble consist only of a probability vector $\mathbf{p}=(p_{00},p_{01},p_{10},p_{11})$ and the number of subsystems $N$, in the following sections, we denote the maximum success probabilities of the identification defined in Eq. and Eq. by $\gamma^{(N)}(\mathbf{p})$ and $\gamma_1^{(N)}(\mathbf{p})$, respectively.
Upper bound of LOCC measurements {#sec:upperbound}
================================
In [@optimalprob2] (simpler proof in [@optimalprob4]), it was shown that the success probability of identifying a state randomly sampled from an ensemble comprising $M$ equiprobable maximally entangled states having local dimension $d$ is at most $d/M$. Since ${| {\Psi_{\mathbf{m}}} \rangle}$ is a maximally entangled state having local dimension $2^N$ for any $\mathbf{m}\in F_2^{2N}$, by applying the result, we obtain $$\gamma^{(N)}\left(\left(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\right)\right)\leq\frac{1}{2^N}.
\label{eq:Nanthan}$$ The upper bound in the right-hand side is achievable by performing one-way LOCC measurements to each subsystem independently: for each subsystem $\mathcal{A}_n\otimes\mathcal{B}_n$, Alice and Bob measure their own subsystem with respect to the fixed basis $\{{| {0} \rangle},{| {1} \rangle}\}$, compare the measurement results by one-way classical communication from Alice to Bob, and Bob guesses $m_n$ as $00$ if the measurement results agree and $m_n$ as $01$ if they disagree.
By applying Theorem 4 in [@optimalprob6], an upper bound of $\gamma^{(N)}(\mathbf{p})$ for a non-uniform probability vector $\mathbf{p}$ is obtained: $$\label{eq:upperbound}
\gamma^{(N)}(\mathbf{p})\leq\max\left\{\sum_{\mathbf{m}\in X}q_{\mathbf{m}}:X\subseteq F_2^{2N},|X|=2^N\right\}.$$ Note that this upper bound can also be obtained by simply using Eq. as shown in Appendix \[appendix:upperbound\]. For $N=1$, the upper bound is tight, namely, $$\gamma^{(1)}(\mathbf{p})= p_{a_0}+p_{a_1},$$ where $\{a_0,a_1,a_2,a_3\}=F_2^2$ such that $p_{a_0}\geq p_{a_1}\geq p_{a_2}\geq p_{a_3}$. Since any set of two Bell states is locally unitarily equivalent to $\{{| {\Phi_{00}} \rangle},{| {\Phi_{01}} \rangle}\}$ [@LUequivalent], the upper bound is achievable by one-way LOCC.
Using Eq. , we can easily verify that the success probability $\gamma^{(N)}(\mathbf{p})$ is less than $1$ for any finite $N$ if and only if the number of non-zero elements in a probability vector $\mathbf{p}$ is greater than or equal to $3$. Furthermore, we can show a condition where the success probability $\gamma^{(N)}(\mathbf{p})$ converges to 0 as $N\rightarrow\infty$.
\[theorem:upperbound\] Let $H(\mathbf{p})=-\sum_{x\in F_2^{2}}p_x\log p_x$ be the entropy of a probability vector $\mathbf{p}=(p_{00},p_{01},p_{10},p_{11})$. If $H(\mathbf{p})>1$, $$\lim_{N\rightarrow\infty}\gamma^{(N)}(\mathbf{p})=0.$$
We define a set of [*typical sequences*]{} $T(\epsilon)\subseteq F_2^{2N}$ as $$T(\epsilon)=\left\{\mathbf{m}\in F_2^{2N}:\left|-\frac{1}{N}\log q_\mathbf{m}-H(\mathbf{p})\right|<\epsilon\right\}$$ for $\epsilon>0$. It is obvious that if $\mathbf{m}\in T(\epsilon)$, $$\label{eq:rangeqm}
2^{-N(H(\mathbf{p})+\epsilon)}<q_{\mathbf{m}}< 2^{-N(H(\mathbf{p})-\epsilon)},$$ and thus $$|T(\epsilon)|<2^{N(H(\mathbf{p})+\epsilon)}.$$ By the [*asymptotic equipartition property*]{}, $$\label{eq:AEP}
\sum_{\mathbf{m}\notin T(\epsilon)}q_{\mathbf{m}}\leq 2\exp\left(-2\frac{\epsilon^2}{\Delta^2}N\right),$$ where $\Delta$ is a non-negative real number defined by $\Delta=\log (\max\{p_x\})-\log (\min\{p_x\})$. An explicit derivation of Eq. is given in Appendix \[appendix:AEP\]. Therefore, for any $N$ and $\epsilon>0$ and for any $X\subseteq F_2^{2N}$ satisfying $|X|=2^N$, $$\begin{aligned}
\sum_{\mathbf{m}\in X}q_{\mathbf{m}}&\leq&\sum_{\mathbf{m}\in X\cap T(\epsilon)}q_{\mathbf{m}}+\sum_{\mathbf{m}\notin T(\epsilon)}q_{\mathbf{m}}\\
\label{eq:upperb}
&<&2^{N(1-H(\mathbf{p})+\epsilon)}+2\exp\left(-2\frac{\epsilon^2}{\Delta^2}N\right).\end{aligned}$$ Hence, if $1-H(\mathbf{p})<0$, there exists $\epsilon>0$ such that the right-hand side converges to $0$ as $N\rightarrow\infty$ since $\Delta$ is a constant when $N$ changes. Since the right hand side of Eq. is also bounded by Eq. , this completes the proof.
Note that this convergence condition is tight in the sense that there exists a probability vector $\mathbf{p}$ such that $H(\mathbf{p})\geq1$ but the success probability $\gamma^{(N)}(\mathbf{p})$ does not converge to $0$. Indeed, for $\mathbf{p}=\left(\frac{1}{2},\frac{1}{2},0,0\right)$, $H(\mathbf{p})=1$ and $\gamma^{(N)}(\mathbf{p})=1$ for any $N$.
Construction of an LOCC measurement {#sec:construction}
===================================
In this section, we show that the success probability of the identification, $\gamma^{(N)}(\mathbf{p})$, converges to $1$ if the entropy of a probability vector $H(\mathbf{p})$ is less than $1$ by constructing a one-way LOCC measurement. The one-way LOCC measurement consists two steps:
1. Alice performs a projective measurement described by $\{A_{\mathbf{a}}={| {\phi_{\mathbf{a}}} \rangle}{\langle {\phi_{\mathbf{a}}} |}\}_{\mathbf{a}}$, where $\{{| {\phi_{\mathbf{a}}} \rangle}\in\mathcal{A}\}_{\mathbf{a}}$ is an orthonormal basis of $\mathcal{A}$.
2. Bob performs a measurement on his system depending on Alice’s measurement outcome described by $\{B_{\hat{\mathbf{m}}|\mathbf{a}}:\hat{\mathbf{m}}\in F_2^{2N}\}$.
When a state ${| {\Psi_{\mathbf{m}}} \rangle}$ is sampled, the (unnormalized) state of Bob’s system after Alice’s measurement is given by $$\begin{aligned}
({\langle {\phi_{\mathbf{a}}} |}\otimes\mathbb{I}^{(\mathcal{B})}){| {\Psi_{\mathbf{m}}} \rangle}&=&({\langle {\phi_{\mathbf{a}}} |}\otimes\sigma_{\mathbf{m}}^{(\mathcal{B})}){| {\Phi_{00}} \rangle}^{\otimes N}\\
&=&\frac{1}{\sqrt{2^N}}\sigma_{\mathbf{m}}^{(\mathcal{B})}{| {\phi^*_{\mathbf{a}}} \rangle},\end{aligned}$$ where ${| {\phi^*_{\mathbf{a}}} \rangle}\in\mathcal{B}$ is the complex conjugate of ${| {\phi_{\mathbf{a}}} \rangle}$ with respect to the fixed basis. Note that $\{{| {\phi^*_{\mathbf{a}}} \rangle}\in\mathcal{B}\}_{\mathbf{a}}$ is an orthonormal basis if and only if $\{{| {\phi_{\mathbf{a}}} \rangle}\in\mathcal{A}\}_{\mathbf{a}}$ is. Therefore, the success probability $\gamma_1^{(N)}(\mathbf{p})$ defined by Eq. is bounded by $$\begin{aligned}
\label{eq:lowerbound}
\gamma_1^{(N)}(\mathbf{p})\geq\frac{1}{2^N}\sum_{\mathbf{m}\in F_2^{2N}}\sum_{\mathbf{a}}q_{\mathbf{m}}{\langle {\phi^*_{\mathbf{a}}} |}\sigma_{\mathbf{m}}B_{\mathbf{m}|\mathbf{a}}\sigma_{\mathbf{m}}{| {\phi^*_{\mathbf{a}}} \rangle}.\end{aligned}$$
We choose each state ${| {\phi^*_{\mathbf{a}}} \rangle}\in\mathcal{B}$ from a [*stabilizer state*]{}, which is widely used in quantum error correction [@GotPhD], quantum computation [@GotKnill], and measurement-based quantum computation [@MBQC]. Suppose $\mathcal{S}$ is a subgroup of an $N$-qubit Pauli group $\{\pm1,\pm i\}\times\{\sigma_{\mathbf{s}}:\mathbf{s}\in F_2^{2N}\}$. An $N$-qubit state ${| {\psi} \rangle}\in\mathbb{C}^{2^N}$ is [*stabilized*]{} by $\mathcal{S}$ if ${| {\psi} \rangle}$ is a simultaneous eigenstate of all elements of $\mathcal{S}$ with the eigenvalue $+1$: $$\forall S\in\mathcal{S},S{| {\psi} \rangle}={| {\psi} \rangle}.$$ It is known that stabilized state ${| {\psi} \rangle}$ is uniquely determined (up to a global phase) if and only if subgroup $\mathcal{S}$ is generated as a product of generators $\langle g_1,\cdots,g_N\rangle$, where each generator $g_n$ is taken from a subset of the Pauli group as $g_n\in\{\pm \sigma_{\mathbf{s}(n)}\}$, and the generators are commutative and [*independent*]{} in the sense that $\{\mathbf{s}(n)\in F_2^{2N}\}_{n=1}^N$ is linearly independent. Note that an orthonormal basis of $N$-qubit $\{{| {\psi_{\mathbf{a}}} \rangle}:\mathbf{a}=(a_1,\cdots,a_N)\in F_2^{N}\}$ can be constructed by taking each ${| {\psi_{\mathbf{a}}} \rangle}$ as a state stabilized by $\langle (-1)^{a_1}g_1,\cdots,(-1)^{a_N}g_N\rangle$ since two eigenspaces of the Pauli group corresponding to different eigenvalues are orthogonal. If we construct an orthonormal basis $\{{| {\phi^*_{\mathbf{a}}} \rangle}\in\mathcal{B}\}_{\mathbf{a}}$ using stabilizer states, Bob’s measurement can be significantly simplified using the following lemma:
\[lemma:unitary\] Let $\{{| {\psi_{\mathbf{a}}} \rangle}\in\mathbb{C}^{2^N}:\mathbf{a}=(a_1,\cdots,a_N)\in F_2^N\}$ be an orthonormal basis stabilized by $$\langle(-1)^{a_1}g_1,\cdots,(-1)^{a_N}g_N\rangle,
\label{eq:stabilizer}$$ where $\{g_n\}_{n=1}^N$ is a set of commutative and independent elements of $\{\sigma_{\mathbf{s}}:\mathbf{s}\in F_2^{2N}\}$. Then, for any $\mathbf{a}\in F_2^{N}$, there exists a unitary operator $U_{\mathbf{a}}\in U(\mathbb{C}^{2^N})$ such that for any $\mathbf{m}\in F_2^{2N}$, $$\sigma_{\mathbf{m}}{| {\psi_{\mathbf{a}}} \rangle}\propto U_{\mathbf{a}}\sigma_{\mathbf{m}}{| {\psi_{\mathbf{0}}} \rangle},$$ where $U(\mathbb{C}^{2^N})$ represents a set of $N$-qubit unitary operators.
Let $g_n=\sigma_{\mathbf{s}(n)}$, where $\{\mathbf{s}(n)\in F_2^{2N}\}_{n=1}^N$ is linearly independent. Let $G=(\mathbf{s}(1),\cdots,\mathbf{s}(N))^T$ be a $N\times 2N$ matrix over $F_2$. By straightforward calculation, we obtain $$\sigma_{\mathbf{m}}{| {\psi_{\mathbf{a}}} \rangle}\propto {| {\psi_{\mathbf{a}+GP\mathbf{m}}} \rangle},$$ where $P$ is a $2N\times 2N$ matrix over $F_2$ such that $$P=\oplus_{n=1}^N
\begin{pmatrix}
0&&1\\
1&&0
\end{pmatrix}.$$ Since $rank(G)=rank(GP)=N$, there exists linearly independent $N$ columns in $GP$. Thus, $$\exists U_{\mathbf{a}}\in U(\mathbb{C}^{2^N}),\forall\mathbf{m}\in F_2^{2N},{| {\psi_{\mathbf{a}+GP\mathbf{m}}} \rangle}\propto U_{\mathbf{a}}{| {\psi_{GP\mathbf{m}}} \rangle}$$ is equivalent to $$\exists U_{\mathbf{a}}\in U(\mathbb{C}^{2^N}),\forall\mathbf{c}\in F_2^{N},{| {\psi_{\mathbf{c}+\mathbf{a}}} \rangle}\propto U_{\mathbf{a}}{| {\psi_{\mathbf{c}}} \rangle}.$$ This is true since $\{{| {\psi_{\mathbf{c}+\mathbf{a}}} \rangle}\}_{\mathbf{c}\in F_2^N}$ is an orthonormal basis for any $\mathbf{a}\in F_2^N$.
Suppose $\{{| {\phi^*_{\mathbf{a}}} \rangle}\in\mathcal{B}\}_{\mathbf{a}\in F_2^N}$ is an orthonormal basis $\{{| {\psi_{\mathbf{a}}} \rangle}\}_{\mathbf{a}\in F_2^N}$ defined in Lemma \[lemma:unitary\], and Bob’s measurement is represented by $B_{\hat{\mathbf{m}}|\mathbf{a}}=U_{\mathbf{a}}B_{\hat{\mathbf{m}}}U_{\mathbf{a}}^{\dag}$, where $\{B_{\hat{\mathbf{m}}}\in P(\mathcal{B})\}_{\hat{\mathbf{m}}\in F_2^{2N}}$ is a POVM and $\{U_{\mathbf{a}}\}_{\mathbf{a}\in F_2^N}$ is a set of unitary operators defined in Lemma \[lemma:unitary\]. Due to Lemma \[lemma:unitary\] and Eq. , the success probability $\gamma_1^{(N)}(\mathbf{p})$ is bounded by $$\begin{aligned}
\label{eq:lowerbound2}
\gamma_1^{(N)}(\mathbf{p})&\geq& \eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})\\
\eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})&:=& \max\Big\{\sum_{\mathbf{m}\in F_2^{2N}}q_{\mathbf{m}}{\langle {\xi^{(N)}} |}\sigma_{\mathbf{m}}B_{\mathbf{m}}\sigma_{\mathbf{m}}{| {\xi^{(N)}} \rangle}:\nonumber\\
&&\{B_{\hat{\mathbf{m}}}\}_{\hat{\mathbf{m}}\in F_2^{2N}}{\rm\ is\ a\ POVM}\Big\},\end{aligned}$$ where ${| {\xi^{(N)}} \rangle}:={| {\phi^*_{\mathbf{0}}} \rangle}$ is a $N$-qubit state stabilized by $\langle g_1,\cdots, g_N\rangle$. Note that Bob’s measurement is optimal in the sense that the maximum of the right-hand side of Eq. over $\{B_{\hat{\mathbf{m}}|\mathbf{a}}\}_{\hat{\mathbf{m}}\in F_2^{2N}}$ and $\eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})$ are the same.
The probability $\eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})$ can be understood in the scenario of quantum error correction, i.e., Alice sends an $N$-qubits stabilizer state ${| {\xi^{(N)}} \rangle}$ to Bob via a noisy channel. In the noisy channel, an error described by a Pauli operator $\sigma_{m}$ occurs on each qubit with probability $p_m$ independently and identically, and Bob tries to detect what types of error occurred. The probability $\eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})$ is equal to the maximum success probability of the perfect error detection. The existence of quantum error correction code suggests that faithful error detection is possible if the probability of error is less than a certain value. In the following theorem, we show that the probability $\eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})$ converges to $1$ if the entropy of the probability distribution of error $H(\mathbf{p})$ is less than $1$.
\[theorem:revival\] If the entropy satisfies $H(\mathbf{p})<1$, there exist a set of stabilizer states $\{{| {\xi^{(N)}} \rangle}\in\mathbb{C}^{2^N}\}_{N\in\mathbb{N}}$ such that $\lim_{N\rightarrow\infty} \eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})=1$.
We show the existence of the set of stabilizer states using the idea of the [*random coding*]{}. For any subspace $C\subseteq F_2^{2N}$, the [*symplectic dual subspace*]{} is defined by $$C^{\bot}=\{\mathbf{u}\in F_2^{2N}:\forall \mathbf{v}\in C,\mathbf{u}\odot\mathbf{v}=0\},$$ where $\odot$ denotes the symplectic product: $\mathbf{u}\odot\mathbf{v}=\mathbf{u}^TP\mathbf{v}$. Note that $\dim C+\dim C^{\bot}=2N$. $N$-dimensional subspace $C$ is called [*symplectic self-dual*]{} if $C=C^{\bot}$, or equivalently, $$\forall \mathbf{u},\mathbf{v}\in C, \mathbf{u}\odot \mathbf{v}=0.$$
Suppose ${| {\xi^{(N)}} \rangle}$ is stabilized by $\langle \sigma_{\mathbf{s}(1)},\cdots,\sigma_{\mathbf{s}(N)}\rangle$, where $\{\mathbf{s}(n)\}_{n=1}^N$ is a basis of $N$-dimensional symplectic self-dual subspace $C$. Since $[\sigma_{\mathbf{s}(m)},\sigma_{\mathbf{s}(n)}]=0$ if and only if $\mathbf{s}(m)\odot\mathbf{s}(n)=0$, $\{\sigma_{\mathbf{s}(n)}\}_{n=1}^N$ is commutative and independent; thus, ${| {\xi^{(N)}} \rangle}$ is well defined.
Since the state with an error $\mathbf{m}$ is given by $$\sigma_{\mathbf{m}}{| {\xi^{(N)}} \rangle}\propto{| {\psi_{GP\mathbf{m}}} \rangle},$$ where ${| {\psi_{\mathbf{a}}} \rangle}$ is a state stabilized by $\langle (-1)^{a_1}\sigma_{\mathbf{s}(1)},\cdots,(-1)^{a_N}\sigma_{\mathbf{s}(N)}\rangle$, and $G=(\mathbf{s}(1),\cdots,\mathbf{s}(N))^T$ is a matrix over $F_2$ as defined in the proof of Lemma \[lemma:unitary\], and since two states corresponding to errors $\mathbf{m}$ and $\mathbf{m}'$ are distinguishable if and only if $GP\mathbf{m}\neq GP\mathbf{m}'$, the Bob’s optimal measurement detecting error is described by $\{B_{\hat{\mathbf{m}}(\mathbf{a})}={| {\psi_{\mathbf{a}}} \rangle}{\langle {\psi_{\mathbf{a}}} |}\}_{\mathbf{a}\in F_2^{N}}$, where $\hat{\mathbf{m}}:F_2^N\rightarrow F_2^{2N}$ is defined by $$\hat{\mathbf{m}}(\mathbf{a})=\arg\max_{\mathbf{m}\in F_2^{2N}}\{q_{\mathbf{m}}:GP\mathbf{m}=\mathbf{a}\},$$ and for $\hat{\mathbf{m}}'\notin range(\hat{\mathbf{m}})$, $B_{\hat{\mathbf{m}}'}=0$.
Then, the failure probability of the error detection is given by $$\begin{aligned}
&&1-\eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})\\
&\leq&\sum_{\mathbf{m}\in F_2^{2N}}q_{\mathbf{m}}\mathbf{I}\Big[\exists \mathbf{m}'\in F_2^{2N},\nonumber\\
&&\mathbf{m}\neq\mathbf{m}'\wedge q_{\mathbf{m}'}\geq q_{\mathbf{m}}\wedge GP(\mathbf{m}+\mathbf{m}')=\mathbf{0}\Big]\\
&\leq&\sum_{\mathbf{m}\in T(\epsilon)}q_{\mathbf{m}}\sum_{\mathbf{m}'\neq\mathbf{m}}\mathbf{I}\left[q_{\mathbf{m}'}\geq q_{\mathbf{m}}\right]\nonumber\\
&&\mathbf{I}\left[ GP(\mathbf{m}+\mathbf{m}')=\mathbf{0}\right]+2\exp\left(-2\frac{\epsilon^2}{\Delta^2}N\right),\end{aligned}$$ where $\mathbf{I}[L]$ is the [*indicator function*]{}, defined by $\mathbf{I}[L]=1$ if $L$ is true and $\mathbf{I}[L]=0$ if $L$ is false, and $T(\epsilon)$ is a set of typical sequences defined in the proof of Theorem \[theorem:upperbound\]. Note that we used Eq. to derive the second inequality.
We calculate the expectation value of the failure probability when $N$-dimensional subspace $C$ is randomly sampled from sample space $\Omega=\{C\subset F_2^{2N}:C=C^{\bot}\}$ with a uniform probability. For any $\mathbf{c}(\neq\mathbf{0})\in F_2^{2N}$, $$\label{eq:symplectic}
E\left[\mathbf{I}\left[ GP\mathbf{c}=\mathbf{0}\right]\right]
=E\left[\mathbf{I}\left[\mathbf{c}\in C\right]\right]
=\frac{1}{2^{N}+1}<\frac{1}{2^N}.$$ The last equation is obtained by calculating the number of symplectic self-dual subspaces as shown in Appendix \[appendix:symplectic\]. Thus, we obtain $$\begin{aligned}
&& E\left[1-\eta^{(N)}(\mathbf{p},{| {\xi^{(N)}} \rangle})\right]\nonumber\\
&< &2^{-N}\sum_{\mathbf{m}\in T(\epsilon)}q_{\mathbf{m}}\sum_{\mathbf{m}'\neq\mathbf{m}}\mathbf{I}\left[q_{\mathbf{m}'}\geq q_{\mathbf{m}}\right]\nonumber\\&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + 2\exp\left(-2\frac{\epsilon^2}{\Delta^2}N\right)\\
&<& 2^{N(H(\mathbf{p})+\epsilon-1)}\sum_{\mathbf{m}\in T(\epsilon)}q_{\mathbf{m}}+ 2\exp\left(-2\frac{\epsilon^2}{\Delta^2}N\right)\\
&\leq& 2^{N\left(H(\mathbf{p})+\epsilon-1\right)}+2\exp\left(-2\frac{\epsilon^2}{\Delta^2}N\right).\end{aligned}$$ Note that we used Eq. to derive the second inequality. Since there exists subspace $C$ bounded by the right-hand side for any $N$, this completes the proof.
With Eq. , this theorem implies the success probabilities of the identification, $\gamma_1^{(N)}(\mathbf{p})$ and $\gamma^{(N)}(\mathbf{p})$, converge to $1$ as $N\rightarrow\infty$ if $H(\mathbf{p})<1$. Note that if $H(\mathbf{p})<1$, the [*quantum state merging*]{} is possible without entanglement [@merging]; therefore, we can also construct a one-way LOCC measurement for the success probability $\gamma_1^{(N)}(\mathbf{p})$ to converge to $1$ by using the merging protocol proposed in [@merging]. However, our one-way LOCC measurement is easier to implement in the sense that the measurement is sampled from a finite set. Moreover, our measurement shows a closed connection between bipartite state discrimination and error correction.
Using the [*positive partial transpose*]{} (PPT) criterion [@PPT], the mixed state corresponding to each ensemble $\rho=\sum_{m\in F_2^2}p_m{| {\Phi_m} \rangle}{\langle {\Phi_m} |}$ is separable if and only if all the elements of probability vector $\mathbf{p}$ is less than or equal to $1/2$. We summarize properties of mixed state $\rho$ and success probability $\gamma^{(N)}(\mathbf{p})$ in Fig. \[fig:result\] when probability vector $\mathbf{p}$ is characterized by two parameters, $s$ and $t$, as $\mathbf{p}=(s,t,1-s-t,0)$. As shown in the figure, there exists a region of $\mathbf{p}$, the interior of the white region of Fig. \[fig:result\] (b), where the success probability of bipartite discrimination, $\gamma^{(N)}(\mathbf{p})$, is less than $1$ for any finite $N$ but converges to that of single-party discrimination as $N\rightarrow \infty$, i.e., the nonlocality of the $N$-fold ensembles asymptotically disappears. Note that mixed state $\rho$ is entangled in the region. On the other hand, if mixed state $\rho$ is separable and success probability $\gamma^{(N)}(\mathbf{p})$ is less than $1$ for some finite $N$, $\gamma^{(N)}(\mathbf{p})$ converges to $0$ as $N\rightarrow \infty$ as shown in Appendix \[appendix:mixed\].
![Properties of mixed state $\rho$ and success probability $\gamma^{(N)}(\mathbf{p})$ with probability vector $\mathbf{p}=(s,t,1-s-t,0)$. Probability vector $\mathbf{p}$ is represented by a point in a triangle or on its boundary in oblique coordinates. (a) Mixed state $\rho$ is separable if $\mathbf{p}$ is in the interior of the gray region or on its boundary, and entangled if $\mathbf{p}$ is in the exterior of the gray region. (b) Success probability $\gamma^{(N)}(\mathbf{p})$ is $1$ for any $N$ if $\mathbf{p}$ is on the boundary of the triangle and less than $1$ for any finite $N$ if $\mathbf{p}$ is in the triangle. The interior of the gray region corresponds to $H(\mathbf{p})>1$, where $\lim_{N\rightarrow\infty}\gamma^{(N)}(\mathbf{p})=0$. The exterior of the gray region corresponds to $H(\mathbf{p})<1$, where $\lim_{N\rightarrow\infty}\gamma^{(N)}(\mathbf{p})=1$.[]{data-label="fig:result"}](result.eps){height=".16\textheight"}
A similar result can be found in [@similar], where $N$-partite discrimination of three states sampled from an ensemble of $N$-copies of three unknown states was investigated; however, we investigate bipartite discrimination of $4^N$ states sampled from an $N$-fold ensemble in this paper.
Bipartite state discrimination as an interactive game
=====================================================
In general, an interactive game can be formulated by [*quantum combs*]{} [@comb] or [*quantum strategies*]{} [@strategy]. However, for our purpose, it is enough to use a normal quantum circuit description to introduce a two-turn interactive game between a referee and two classically communicating players as shown in Fig. \[fig:game\].
![A two-turn interactive game between a referee and two classically communicating players, Alice and Bob. The two-turn interaction consists of quantum communication from the referee to the players and vice versa. The referee’s operation (the shaded part) consists of preparing composite system $\mathcal{A}_1\otimes\mathcal{B}_1\otimes\mathcal{R}_1$ in mixed state $\rho$, sending subsystem $\mathcal{A}_1\otimes\mathcal{B}_1$ to the players, and performing a two-valued joint measurement on system $\mathcal{A}'_1\otimes\mathcal{B}'_1$ received from the players and his internal subsystem $\mathcal{R}_1$ to judge the players to win or lose the game. The players perform LOCC operation $\mathcal{M}$ to maximize the winning probability.[]{data-label="fig:game"}](game.eps){height=".14\textheight"}
The two-turn interactive game consists the three steps:
1. The referee prepares composite system $\mathcal{A}_1\otimes\mathcal{B}_1\otimes\mathcal{R}_1$ in mixed state $\rho \in D(\mathcal{A}_1\otimes\mathcal{B}_1\otimes\mathcal{R}_1)$, where $D(\mathcal{H}):=\{\rho\in P(\mathcal{H}):tr(\rho)=1\}$ represents a set of density operators on $\mathcal{H}$, and sends subsystems $\mathcal{A}_1$ and $\mathcal{B}_1$ to Alice and Bob, respectively.
2. Alice and Bob perform LOCC operations described by linear map $\mathcal{M}:P(\mathcal{A}_1\otimes\mathcal{B}_1)\rightarrow P(\mathcal{A}'_1\otimes\mathcal{B}'_1)$ on subsystem $\mathcal{A}_1\otimes\mathcal{B}_1$ and send the resulting systems, $\mathcal{A}'_1\otimes\mathcal{B}'_1$, to the referee.
3. The referee performs a measurement described by POVM $\{R,\mathbb{I}-R\}\subset P(\mathcal{A}'_1\otimes\mathcal{B}'_1\otimes\mathcal{R}_1)$ to judge Alice and Bob to win (corresponding to $R$) or lose (corresponding to $\mathbb{I}-R$) the game.
Then, the maximum winning probability of the players is given by $$\chi^{(1)}=\sup\{tr\left(R\mathcal{M}\otimes\mathcal{I}(\rho)\right):\mathcal{M}{\rm\ is\ LOCC}\},$$ where $\mathcal{M}\otimes\mathcal{I}:P(\mathcal{A}_1\otimes\mathcal{B}_1\otimes\mathcal{R}_1)\rightarrow P(\mathcal{A}'_1\otimes\mathcal{B}'_1\otimes\mathcal{R}_1)$ is a linear map satisfying $\mathcal{M}\otimes\mathcal{I}(V\otimes W)=\mathcal{M}(V)\otimes W$ for all $V\in P(\mathcal{A}_1\otimes\mathcal{B}_1)$ and $W\in P(\mathcal{R}_1)$.
The repeated game is an interactive game where the referee simultaneously repeats one particular game independently and judges the players to win the repeated game if the players win all the games. Thus, the $N$-times repeated game of the two-turn interactive game consists the three steps:
1. The referee prepares $N$-copies of composite system $\mathcal{A}_1\otimes\mathcal{B}_1\otimes\mathcal{R}_1$ and sends subsystems $\otimes_{n=1}^N\mathcal{A}_n$ and $\otimes_{n=1}^N\mathcal{B}_n$ to Alice and Bob, respectively, where each composite system is labelled $\mathcal{A}_n\otimes\mathcal{B}_n\otimes\mathcal{R}_n$ $(n=1,\cdots,N)$.
2. Alice and Bob perform LOCC operations described by linear map $\mathcal{M}:P((\otimes_{n=1}^N\mathcal{A}_n)\otimes(\otimes_{n=1}^N\mathcal{B}_n))\rightarrow P((\otimes_{n=1}^N\mathcal{A}_n')\otimes(\otimes_{n=1}^N\mathcal{B}_n'))$ and send the resulting systems to the referee, where each $\mathcal{A}_n'$ ($\mathcal{B}_n'$) has the same dimension as $\mathcal{A}'_1$ ($\mathcal{B}'_1$) in the single game.
3. The referee performs a measurement described by POVM $\{R^{\otimes N},\mathbb{I}-R^{\otimes N}\}$ to judge Alice and Bob to win (corresponding to $R^{\otimes N}$) or lose (corresponding to $\mathbb{I}-R^{\otimes N}$) the game, where $\{R,\mathbb{I}-R\}\subset P(\mathcal{A}'_n\otimes\mathcal{B}'_n\otimes\mathcal{R}_n)$.
Then, the maximum winning probability of the players is given by $$\chi^{(N)}=\sup\{tr\left(R^{\otimes N}\mathcal{M}\otimes\mathcal{I}(\rho^{\otimes N})\right):\mathcal{M}{\rm\ is\ LOCC}\},$$ where the order of the Hilbert spaces is appropriately permuted in $R^{\otimes N}$ and $\rho^{\otimes N}$.
A parallel repetition conjecture of an interactive game holds if $\chi^{(1)}<1$ implies $\chi^{(N)}<c^N$ with some constant $c<1$. It is easy to verify that the bipartite discrimination of states sampled from an ensemble comprising the Bell states task is a two-turn interactive game by setting $$\begin{aligned}
\rho&=&\sum_{m\in F_2^2}p_m{| {\Phi_m} \rangle}{\langle {\Phi_m} |}^{(\mathcal{A}_n\otimes\mathcal{B}_n)}\otimes{| {m} \rangle}{\langle {m} |}^{(\mathcal{R}_n)},\\
R&=&\sum_{m\in F_2^2} {| {m} \rangle}{\langle {m} |}^{(\mathcal{A}'_n)}\otimes {| {m} \rangle}{\langle {m} |}^{(\mathcal{B}'_n)}\otimes {| {m} \rangle}{\langle {m} |}^{(\mathcal{R}_n)},\end{aligned}$$ where each subsystem $\mathcal{R}_n$ stores the label of a Bell state the referee sampled. The maximum winning probability of the players, $\chi^{(1)}$, is equal to the success probability of the identification, $\gamma^{(1)}(\mathbf{p})$. The maximum winning probability of the $N$-times repeated game, $\chi^{(N)}$, is equal to the success probability of the identification of an $N$-fold ensemble, $\gamma^{(N)}(\mathbf{p})$. Theorem \[theorem:revival\] shows that there exist probability vectors $\mathbf{p}$ such that $\chi^{(N)}$ converges to $1$ as $N\rightarrow \infty$ while $\chi^{(1)}<1$, which is a remarkable counterexample of the parallel repetition conjecture.
Conclusion and Discussion
=========================
We have investigated bipartite discrimination of states sampled from an $N$-fold ensemble comprising the Bell states. We showed that the success probability of the perfect identification, $\gamma^{(N)}(\mathbf{p})$, converges to $1$ as $N\rightarrow\infty$ if the entropy of the probability distribution associated with each ensemble, $H(\mathbf{p})$, is less than $1$, even if $\gamma^{(N)}(\mathbf{p})<1$ for any finite $N$, namely, the nonlocality of the $N$-fold ensemble asymptotically disappears. Furthermore, the disappearance of the nonlocality can be regarded as a remarkable counterexample of the parallel repetition conjecture of interactive games with two classically communicating players.
Conversely, if $H(\mathbf{p})>1$, the quantum state merging is impossible without entanglement; moreover, we showed that $\gamma^{(N)}(\mathbf{p})$ converges to $0$ as $N\rightarrow\infty$. Therefore, our result also demonstrates a significant gap of the distinguishability with respect to $\gamma^{(N)}(\mathbf{p})$ between a mergeable ensemble and a non-mergeable ensemble. Note that there does not always exist such a gap for ensembles comprising more general states, e.g., $N$-fold ensemble formed by ensemble $\{(s,{| {01} \rangle}),(t,{| {10} \rangle}),(1-s-t,{| {\Phi_{00}} \rangle})\}$ is perfectly distinguishable for any probability vector $(s,t,1-s-t)$ and any $N$ while the ensemble is not mergeable for particular probability vectors as shown in Fig. \[fig:merging\]. There remains a future work in investigating the gap of the distinguishability between a mergeable ensemble and a non-mergeable ensemble comprising more general states.
![The possibility of quantum state merging for ensemble $\{(s,{| {01} \rangle}),(t,{| {10} \rangle}),(1-s-t,{| {\Phi_{00}} \rangle})\}$. In the gray region, the quantum state merging is impossible without entanglement, on the other hand, it is possible in the white region without entanglement[]{data-label="fig:merging"}](merging.eps){height=".15\textheight"}
We can also discuss our result in another context. Intuitively, the optimal distinguishability in $N$ independent subsystems can be achieved by performing measurements on each subsystem independently. Indeed, in the case of single-party discrimination of quantum states sampled from independent (but not necessarily identical) ensembles, independent measurements can extract as much information about the composite system as any joint measurement [@productrule1] as depicted in Fig. \[fig:discussion\] (a) and (b). The result was extended to the joint estimation of the parameters encoded in independent processes, where the optimal joint estimation can be achieved by estimating each process independently [@productrule2].
However, our result shows that the optimal distinguishability of an $N$-fold ensemble cannot be achieved by independent LOCC measurement as depicted in Fig. \[fig:discussion\] (c) but can be achieved by joint LOCC measurement, where Alice and Bob perform entangled measurement within their own system.
![Graphical representations of three types of state discrimination where the state of each subsystem is randomly sampled from independent ensemble $\{(p_i,\rho_i)\}$. Rounded rectangles represent quantum systems and dotted rectangles represent subsystems where joint quantum operations can be performed. Communication across the bold line is forbidden. (a) Single-party state discrimination by joint measurement. (b) Single-party state discrimination by independent measurement. (c) Bipartite state discrimination by independent LOCC measurement.[]{data-label="fig:discussion"}](discussion.eps){height=".15\textheight"}
We are greatly indebted to Seiichiro Tani, Kohtaro Suzuki, Hiroki Takesue, Mio Murao, Marco Tulio Quintino, Mateus Araujo, and Takuya Ikuta for their valuable discussions.
Upper bound of LOCC measurements {#appendix:upperbound}
================================
In this Appendix, we derive the upper bound in Eq. by applying the following elementary lemma:
\[lemma:satprob\] Let $\{a_k\in[0,1]\}_{k=1}^K$ be a set of real numbers, and $(\lambda_1,\cdots,\lambda_K)$ be a probability vector. If $\sum_{k=1}^K a_k\leq \tilde{K}$ for a non-negative integer $\tilde{K}\leq K$, then $$\sum_{k=1}^K \lambda_ka_k\leq\max\left\{\sum_{k\in X}\lambda_k:X\subseteq [K],|X|= \tilde{K}\right\},$$ where $[K]=\{1,2,\cdots,K\}$.
Suppose $X^*$ maximizes the right-hand side, and let $X^{*c}=[K]\setminus X^*$ be the complement of $X^*$. Let $\alpha=\sum_{k\in X^*}a_k$ and $\beta=\sum_{k\in X^{*c}}a_k$. Then $ \tilde{K}-\alpha\geq\beta\geq 0$. Let $\mu=\min\{\lambda_k:k\in X^*\}$ and $\nu=\max\{\lambda_k:k\in X^{*c}\}$. Then $\mu\geq \nu\geq 0$, and we obtain
$$\begin{aligned}
&& \sum_{k\in X^*}\lambda_k-\sum_{k=1}^K \lambda_ka_k\nonumber\\
&=&\sum_{k\in X^*}\lambda_k(1-a_k)-\sum_{k\in X^{*c}}\lambda_ka_k\\
&\geq&\sum_{k\in X^*}\mu(1-a_k)-\sum_{k\in X^{*c}}\nu a_k\\
&=&\tilde{K}\mu-\alpha \mu-\beta \nu\\
&\geq&(\tilde{K}-\alpha)(\mu-\nu)\geq0.\end{aligned}$$
Eq. implies that for any LOCC measurement $\{M_{\hat{\mathbf{m}}}\}_{\hat{\mathbf{m}}}$, $$\sum_{\mathbf{m}\in F_2^{2N}}{\langle {\Psi_{\mathbf{m}}} |}M_{\mathbf{m}}{| {\Psi_{\mathbf{m}}} \rangle}\leq 2^N.$$ Since ${\langle {\Psi_{\mathbf{m}}} |}M_{\mathbf{m}}{| {\Psi_{\mathbf{m}}} \rangle}\in [0,1]$, by applying Lemma \[lemma:satprob\], we obtain $$\begin{aligned}
&&\sum_{\mathbf{m}\in F_2^{2N}}q_{\mathbf{m}}{\langle {\Psi_{\mathbf{m}}} |}M_{\mathbf{m}}{| {\Psi_{\mathbf{m}}} \rangle}\nonumber\\
&\leq&\max\left\{\sum_{\mathbf{m}\in X}q_{\mathbf{m}}:X\subseteq F_2^{2N},|X|= 2^N\right\}\end{aligned}$$ for any LOCC measurement $\{M_{\hat{\mathbf{m}}}\}_{\hat{\mathbf{m}}}$. Hence, the upper bound in Eq. is derived.
Asymptotic equipartition property {#appendix:AEP}
=================================
In this Appendix, we derive Eq. . Define a set of indices of the Bell states associated with non-zero probability $F=\{x\in F_2^2:p_x>0\}$. Let $\Omega=F^N$ be a sample space and $p(\mathbf{m})=q_{\mathbf{m}}$ be the probability mass function. Define random variables $Y_n(\mathbf{m})=-\log p_{m_n}$ and $Y(\mathbf{m})=\frac{1}{N}\sum_{n=1}^N Y_n(\mathbf{m})$, which are well-defined for $\mathbf{m}=(m_1,\cdots,m_N)\in\Omega$. Then $\{Y_n\}_{n=1}^N$ are mutually independent random variables, and $$\begin{aligned}
E[Y]=E[Y_n]=H(\mathbf{p}),\\
\Delta:=\max\{\log p_{max}-\log p_{min},\delta\},\end{aligned}$$ where $p_{max}=\max\{p_x:x\in F\}$, $p_{min}=\min\{p_x:x\in F\}$ and $\delta$ is an arbitrary positive real number. By Hoeffding’s’s inequality, for any $\epsilon>0$, $$\begin{aligned}
Pr[|Y-E[Y]|\geq\epsilon]\leq 2\exp\left(-2\frac{\epsilon^2}{\Delta^2}N\right).\end{aligned}$$ Since $$\begin{aligned}
Pr[|Y-E[Y]|\geq\epsilon]&=&\sum_{\mathbf{m}\in \Omega\setminus T(\epsilon)}q_{\mathbf{m}}\\ &=&\sum_{\mathbf{m}\in F_2^{2N}\setminus T(\epsilon)}q_{\mathbf{m}},\end{aligned}$$ Eq. is derived.
Number of symplectic self-dual subspaces {#appendix:symplectic}
========================================
In this Appendix, we calculate the size of $\Omega=\{C\subset F_2^{2N}:C=C^{\bot}\}$ and $\Omega_{\mathbf{c}}=\{C\in \Omega:\mathbf{c}\in C\}$, and show that $|\Omega_{\mathbf{c}}|/|\Omega|=1/(2^N+1)$ for any $\mathbf{c}(\neq\mathbf{0})\in F_2^{2N}$, which implies the last equation in Eq. .
Any symplectic self-dual subspace of $F_2^{2N}$ can be constructed by the following procedure:
1. Set $C_0=\{\mathbf{0}\in F_2^{2N}\}$ and $n=0$.
2. Choose $\mathbf{s}(n+1)\in F_2^{2N}$ so that $\mathbf{s}(n+1)\in C_{n}^{\bot}$ and $\mathbf{s}(n+1)\notin C_{n}$.
3. Set $C_{n+1}=span\{\mathbf{s}(m)\}_{m=1}^{n+1}$ and increase $n$ by one.
4. Repeat step 2 to step 3 until no $\mathbf{s}\in F_2^{2N}$ satisfies the condition in step 2.
Since $\{\mathbf{s}(n)\}_n$ is linearly independent, $\dim C_n=n$. Since $\mathbf{s}(k)\odot\mathbf{s}(l)=0$ for any $k$ and $l$, $C_n\subseteq C_n^{\bot}$. Thus, using the procedure, we can obtain symplectic self-dual subspace $C_N$ and its basis $\{\mathbf{s}(n)\}_{n=1}^N$. Conversely, we can easily verify that any symplectic self-dual subspace and any its basis are constructed by the procedure.
In the procedure, we obtain $\prod_{n=0}^{N-1}(2^{2N-n}-2^{n})$ different families of linearly independent vectors $\{\mathbf{s}(n)\}_{n=1}^N$. For any $N$-dimensional subspace $C_N$, there exist $\prod_{n=0}^{N-1}(2^{N}-2^{n})$ different families $\{\mathbf{s}(n)\}_{n=1}^N$ each of which is a basis of $C_N$. Therefore, the number of symplectic self-dual subspaces is given by $$|\Omega|=\frac{\prod_{n=0}^{N-1}(2^{2N-n}-2^{n})}{\prod_{n=0}^{N-1}(2^{N}-2^{n})}.$$
If we choose $\mathbf{s}(1)=\mathbf{c}(\neq\mathbf{0})$, we obtain $\prod_{n=1}^{N-1}(2^{2N-n}-2^{n})$ different families $\{\mathbf{c},\mathbf{s}(2),\cdots,\mathbf{s}(N)\}$. For any $N$-dimensional subspace $C_N$ containing $\mathbf{c}(\neq\mathbf{0})$, there exist $\prod_{n=1}^{N-1}(2^{N}-2^{n})$ different families $\{\mathbf{c},\mathbf{s}(2),\cdots,\mathbf{s}(N)\}$ each of which is a basis of $C_N$. Therefore, the number of symplectic self-dual subspaces containing $\mathbf{c}(\neq\mathbf{0})$ is given by $$|\Omega_{\mathbf{c}}|=\frac{\prod_{n=1}^{N-1}(2^{2N-n}-2^{n})}{\prod_{n=1}^{N-1}(2^{N}-2^{n})}.$$
Identification in separable ensembles {#appendix:mixed}
=====================================
If the mixed state corresponding to each ensemble, $\rho=\sum_{m\in F_2^2}p_m{| {\Phi_m} \rangle}{\langle {\Phi_m} |}$, is separable, $2p_m\leq 1$ for all $m\in F_2^2$. Using an equation $$\label{eq:relativeentropy}
\sum_{m\in F_2^2}p_m\log (2p_m)=1-H(\mathbf{p}),$$ we obtain that if $\rho$ is separable, $H(\mathbf{p})\geq 1$ with equality occurring only when the number of non-zero elements in a probability vector $\mathbf{p}$ is $2$. Using Theorem \[theorem:upperbound\], we can verify that if $\rho$ is separable and the success probability of the identification, $\gamma^{(N)}(\mathbf{p})$, is less than $1$ for some finite $N$, $\gamma^{(N)}(\mathbf{p})$ converges to $0$ as $N\rightarrow \infty$.
[99]{} V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. [**78**]{}, 2275 (1997). M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. [**84**]{}, 2014 (2000). M. B. Plenio and S. Virmani, Quant. Inf. Comp. [**7**]{}, 1 (2007). A. Soeda, P. S. Turner, and M. Murao, Phys. Rev. Lett. [**107**]{}, 180510 (2011). D. Stahlke and R. B. Griffiths, Phys. Rev. A [**84**]{}, 032316 (2011). A. Soeda, S. Akibue, and M. Murao, J. Phys. A, Math. Theor. [**47**]{}, 424036 (2014). J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Phys. Rev. Lett. [**85**]{}, 4972 (2000). S. Virmani, M. F. Sacchi, M. B. Plenio, and D. Markham, Phys. Lett. A. [**288**]{}, p. 62 (2001). C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin and W. K. Wootters, Phys. Rev. A [**59**]{}, 1070 (1999). C.H. Bennett, D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin and B.M. Terhal, Phys. Rev. Lett. [**82**]{}, 5385 (1999). A. Peres and W. K. Wootters, Phys. Rev. Lett. [**66**]{}, 1119 (1991). E. Chitambar and M. H. Hsieh, Phys. Rev. A [**88**]{}, 020302 (2013). A. M. Childs, D. Leung, L. Mancinska, M. Ozols, Commun. Math. Phys. [**323**]{}, No. 3, 1121 (2013). M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki, Phys. Rev. Lett. [**90**]{}, 047902 (2003). S. Ghosh, G. Kar, A. Roy, A. Sen(De), and U. Sen, Phys. Rev. Lett. [**87**]{}, 277902 (2001). M. Nathanson, J. Math. Phys. [**46**]{}, 062103 (2005). M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, Phys. Rev. Lett. [**96**]{}, 040501 (2006). A. Cosentino, Phys. Rev. A [**87**]{}, 012321 (2013). N. Yu, R. Duan, and M. Ying, Phys. Rev. Lett. [**109**]{}, 020506 (2012). D. P. DiVincenzo, D. W. Leung and B. M. Terhal, IEEE Trans. Inf. Theory [**48**]{}, 580 (2002). T. Eggeling and R. F. Werner, Phys. Rev. Lett. [**89**]{}, 097905 (2002). F. G. S. L. Brandao, M. Christandl, and J. Yard, in [*Proceedings of the 43rd ACM Symposium on Theory of Computation, 2011*]{}, edited by L. Fortnow (Northwestern University, Chicago, 2011) and S. Vadhan (Harvard University, Cambridge, 2011), p. 343. M. B. Hastings, J. Stat. Mech. Theory Exp., P08024 (2007). L. Lami, C. Palazuelos, and A. Winter, arXiv: 1703.03392, 2017 (to be published). J. Bae, L. C. Kwek, J. Phys. A: Math. Theor. [**48**]{}, 083001 (2015). L. Babai. in [*Proceedings of the 17th ACM Symposium on Theory of Computing, 1985*]{}, edited by R. Sedgewick (Princeton University, Princeton, 1985), p. 421. M. Ben-Or, S. Goldwasser, J. Kilian, and A. Wigderson, in [*Proceedings of the 20th ACM Symposium on Theory of Computing, 1988*]{}, edited by J. Simon (University of Chicago, Illinois, 1988), p. 113. J. Watrous, in [*Proceedings of the 40th Annual Symposium on Foundations of Computer Science, 1999*]{}, p. 112. H. Kobayashi and K. Matsumoto, J. Computer and System Sciences [**66**]{}, 429 (2003). S. Goldwasser, S. Micali, and C. Rackoff, in [*Proceedings of the 17th ACM Symposium on Theory of Computing, 1985*]{}, edited by R. Sedgewick (Princeton University, Princeton, 1985), p. 291. J. Watrous, in [*Proceedings of the 43rd Annual Symposium on Foundations of Computer Science, 2002*]{}, p. 459. U. Feige, S. Goldwasser, L. Lovasz, S. Safra, and M. Szegedy, in [*Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, 1991*]{}, p. 2. S. Arora and M. Safra, Journal of the ACM [**45**]{}, 70 (1998). T. Vidick and J. Watrous, Foundations and Trends in Theoretical Computer Science [**11**]{}, 1 (2015). A. Kitaev and J. Watrous, in [*Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000*]{}, edited by F. Yao (City University of Hong Kong, Kowloon Tong, 2000) and E. Luks (University of Oregon, Eugene, 2000), p. 608. R. Mittal and M. Szegedy. in [*Proceedings of the 16th International Symposium on Fundamentals in Computation Theory, 2007*]{}, edited by E. Csuhaj-Varju (Eotvos Lorand University, Budapest, 2007) and Z. Esik (University of Szeged, Szeged, 2007), p. 435. R. Raz, SIAM Journal on Computing [**27**]{}, 763 (1998). T. Holenstein, in [*Proceedings of 39th Annual ACM Symposium on Theory of Computing, 2007*]{}, edited by D. Johnson (AT&T Labs, Florham Park, 2007) and U. Feige (Microsoft Research and Weizmann Institute, Israel, 2007), p. 411. R. Cleve, W. Slofstra, F. Unger and S. Upadhyay, in [*Proceedings of the 22nd Annual IEEE Conference on Computational Complexity, 2007*]{}, p.109. I. Dinur, D. Steurer, and T. Vidick, in [*Proceedings of the 29th Annual IEEE Conference on Computational Complexity, 2014*]{}, p. 197. M. Ben Or, A. Hassidim, and H. Pilpel, in [*Proceedings of the 49th Annual Symposium on Foundations of Computer Science, 2008*]{}, p. 467. I. D. Ivanovic, Phys. Lett. A [**123**]{}, 257 (1987). D. Dieks, Phys. Lett. A [**128**]{}, 303 (1988). A. Peres, Phys. Lett. A [**128**]{}, 19 (1988). M. Navascues, Phys. Rev. Lett. [**100**]{}, 070503 (2008). E. Chitambar, D. Leung, L. Mancinska, M. Ozols, and A. Winter, Commun. Math. Phys. [**328**]{}, 303 (2014). S. Bandyopadhyay and M. Nathanson, Phys. Rev. A [**88**]{}, 052313 (2013). J. Dehaene, M. Van den Nest, B. De Moor, and F. Verstraete, Phys. Rev. A [**67**]{}, 022310 (2003). D. Gottesman, Ph.D. thesis, California Institute of Technology (1997). D. Gottesman, in [*Proceedings of the 12th International Colloquium on Group Theoretical Methods in Physics, 1999*]{}, edited by S. P. Corney, R. Delbourgo (University of Tasmania, Hobart, 1999), and P. D. Jarvis, p. 32. R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. [**86**]{}, 5188 (2001). M. Horodecki, J. Oppenheim, and A. Winter, Nature [**436**]{}, 673 (2005). M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A [**223**]{}, 1 (1996). E. Chitambar, R. Duan, and M. H. Hsieh, IEEE Trans. Inf. Theory [**60**]{}, 1549 (2014). G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys. Rev. A [**80**]{}, 022339 (2009). G. Gutoski and J. Watrous, in [*Proceedings of 39th Annual ACM Symposium on Theory of Computing, 2007*]{}, edited by D. Johnson (AT&T Labs, Florham Park, 2007) and U. Feige (Microsoft Research and Weizmann Institute, Israel, 2007), p. 565. Chi-HangFred Fung and H. F. Chau, Phys. Rev. A [**78**]{}, 062308 (2008). G. Chiribella, New J. Phys. [**14**]{}, 125008 (2012).
|
---
abstract: 'Evolving out-of-equilibrium networks have been under intense scrutiny recently. In many real-world settings the number of links added per new node is not constant but depends on the time at which the node is introduced in the system. This simple idea gives rise to the concept of accelerating networks, for which we review an existing definition and – after finding it somewhat constrictive – offer a new definition. The new definition provided here views network acceleration as a time dependent property of a given system, as opposed to being a property of the specific algorithm applied to grow the network. The defnition also covers both unweighted and weighted networks. As time-stamped network data becomes increasingly available, the proposed measures may be easily carried out on empirical datasets. As a simple case study we apply the concepts to study the evolution of three different instances of Wikipedia, namely, those in English, German, and Japanese, and find that the networks undergo different acceleration regimes in their evolution.'
address:
- '$^1$Clarendon Laboratory, Physics Department, Oxford University, Oxford, U.K'
- '$^2$Laboratory of Computational Engineering, Helsinki University of Technology, Finland'
author:
- 'David M.D. Smith$^{1}$'
- 'Jukka-Pekka Onnela$^{1,2}$'
- 'Neil F. Johnson$^{1}$'
title: Accelerating networks
---
Introduction {#sec:introduction}
============
In many real-world networks the rate at which links are added to the network is different from the rate at which nodes are added to it, and this seems to be the case in particular in “functionally organized systems whose operation is reliant on the integrated activity of any or all of its component nodes" [@mattick:2005]. This results in accelerating or decelerating networks, of which the Internet is one example [@internet]. Other examples of such systems include the arrangement of components on an integrated chip, configuration of computers in clusters, structure of integrated production systems, configuration of human social and communication systems, and the organisation of regulatory proteins controlling gene expression in bacteria [@mattick:2005].
Consider an arbitrary evolving network that may explore any allowed trajectory in its phase space of node number $N(t)$ (number of nodes at time $t$) and link number $M(t)$ (number of links at time $t$). A constructed example of such a network is shown in Fig. \[fig:phase\]. In an undirected network of size $N(t)$, the total number of possible links is $N(t)(N(t)-1)/2$ of which $M(t)$ exist at some time $t$. The ratio of the two is described by
$$\begin{aligned}
\label{eqn:ratio1}
q(t)&=&\frac{2~M(t)}{N(t)(N(t)-1)}.\end{aligned}$$
![ \[fig:phase\] The trajectory of an evolving network might exist anywhere in the allowable region of the phase space corresponding to its number of nodes $N(t)$ and number of links $M(t)$. This might be a network evolving according to some algorithm or an empirically observed network. Whilst it is possible that at any stage in its evolution the network might have many components, the criterion $M(t)<(N(t)-1)$ imposes the condition. ](phase.eps){width="70.00000%"}
In many conventional non-equilibrium evolving (growing) networks this quantity is expected to decrease for large values of $t$. As time-stamped network data becomes increasingly available, we expect most networks to exhibit non-trivial behaviour with respect to their acceleration characteristics. The notion of accelerating networks has been considered before [@GM; @Sen], and here we consider the notions of network acceleration as discussed by Gagen and Mattick in [@GM]. Although interesting, the definitions in [@GM] seem constrictive in that they are applicable only to a specific algorithm, and the concept of acceleration is static in nature, i.e., a given network is considered to be either accelerating, non-accelerating, or decelerating throughout its evolution. Contrary to this, the notion of acceleration in physical systems refers to the nature of the evolution of a system at a specific moment in time, suggesting that acceleration in the context of complex networks should be reconsidered. In addition, it is imperative that network acceleration be measurable for empirical networks.
We review the model of Gagen and Mattick in Section \[sec:GM\] and critically examine the proposed concept of network acceleration in Section \[sec:approach\]. This is followed by a new definition of network acceleration in Section \[sec:definition1\]. This might be used to describe a network at any stage of its evolution (whether decaying, growing, or neither), and applies to both directed and undirected networks. We then extend the analysis to incorporate weighted links. We demonstrate the use of the new definition in a simple empirical case study in Section \[sec:case\], and conclude our discussion in Section \[sec:conclusion\].
The model of Gagen and Mattick {#sec:GM}
==============================
In the model of Gagen and Mattick (GM-model), as in many other conventional out-of-equilibrium evolving network models, the system evolves by introducing *exactly* one new node at each time step. The key feature of the model is that the new node connects to the existing single-component network through a time-dependent number of links, whereas in most evolving network models [@dorogovtsev] the new node attaches to the existing network with a fixed, time-independent number of links $m$, as demonstrated in Fig. \[fig:evolvntet2\].
![ \[fig:evolvntet2\] Evolution of a network. In most out-of-equilibrium network models a new node attaches to the existing network with $m$ links at each time step via some arbitrary attachment mechanism. These links might be directed or undirected, weighted or unweighted.](evolvntet2.eps){width="60.00000%"}
Here we shall consider the scenario whereby the new node attaches with undirected binary (unweighted) links. Specifically, we assume that each new node attaches with *on average* $m(t)$ links. Whether or not the number of links added per time step is a stochastic process, the actual evolution of the system will be if a stochastic attachment algorithm (random, preferential, or otherwise) is employed. Clearly, the maximum number of links with which the new node can connect with to the existing network, is equal to the number of nodes within the existing network. Gagen and Mattick actually specify the functional form describing the expected rate of link addition $m(t)$ as $$\begin{aligned}
~\label{eqn:GM}
m(t)&=& p\big(N(t)\big)^\alpha. \end{aligned}$$ Here $\alpha$ is an described as an acceleration parameter and $p$ as a probability constant with the constraint of $0 \le p \le 1$. Gagen and Mattick describe the type of a network in terms of $\alpha$ as either decelerating ($\alpha < 0$), non-accelerating ($\alpha=0$), accelerating ($0< \alpha<1$), or hyper-accelerating ($\alpha \ge 1$). Some examples are shown in Fig. \[fig:GM1\]. We shall revisit such definitions later.
![ \[fig:GM1\] The evolution of the expected number of links added per new node, $m(t) = p (N(t))^{\alpha}$, within the Gagen and Mattick model of accelerating networks. Three examples comprising non-acceleration ($\alpha=0$), acceleration ($\alpha=0.8$) and hyper-acceleration ($\alpha=1.5$) are displayed [@GM].](GM1.eps){width="60.00000%"}
We note that for a non-accelerating network ($\alpha = 0$) within this description, the expected number of new links added is simply $p \in [0,1]$ such that, on average, at most one link may be added per time step. Note also that the hyper-accelerating network has a finite time scale for which it can be applied in that the maximum number of links that can possibly be added to the system is $N(t)$. This sets an upper bound on $m(t)$ after which the network cannot hyper-accelerate, i.e., in order to continue hyper-acceleration, it would have to introduce more than $N(t)$ links per time step, which is impossible without allowing for multiple links between any pair of nodes.
We also note also that so far the actual attachment mechanism, random or otherwise, has not been discussed, meaning that the *accelerating nature of the network is simply related to the rate of link addition*. This is an important point to consider because although the expected number of links added for the new node is $m(t)$, the variance in this will differ between microscopic attachment mechanisms. Indeed, the variance could be zero for integer $m(t)$, corresponding to exactly $m(t)$ begin added every time step, or the number of links to add could be drawn from a probability distribution with mean $m(t)$. Even though the number of links to be added might be deterministic, one still needs to specify the algorithm to determine to which nodes within the existing network these links are attached.
Rethinking the accelerating network {#sec:approach}
===================================
The notion of network acceleration is applicable to situations whereby the rates of node and link addition are not stochastic. As such, in introducing the concepts key to their understanding, it is useful to concentrate on this conceptually simpler scenario.
Consider some arbitrary evolving network whose evolution we can observe. This might be a realisation of an algorithm or an empirically observed evolving network. At time $t=0$ we have an initial configuration of $N(0)$ nodes and $M(0)$ links. At each time step, a quantity of new nodes $n(t)$ are added and they are connected to the pre-existing network with some number of binary links $m(t)$. These are both integer quantities. In this scenario no new links are formed between existing nodes within the network, although this feature could be incorporated. For now, we shall assume that $n(t) = 1$ such that exactly one new node is added per time step[^1]. As such, at time $t=1$ the number of nodes is $N(1)=N(0)+1$. The maximum number of links that could have been introduced on attaching the $N(1)$th node is clearly $N(0)$ as the new node can link to each previous node with at most one link. Similarly, for a node added at time $t$, the total number of nodes is $N(t) = N(0)+t$ and the maximum number of links that could have been used to connect it is $N(t-1)$. For this process, by which one new node is added per time step, we know that $N(t-1)=N(t)-1$. This is the upper limit for the time-dependent number of added links, $m(t)$, at each time step. This region is depicted Fig. \[fig:shaded\].
![ \[fig:shaded\] The allowable number of links (shaded area) added with each new node is capped by the total number of nodes in the existing network (line). Here, as one new node is added per time step, the growth of the network in terms of number of nodes is linear.](shaded.eps){width="60.00000%"}
Clearly, the functional form of $m(t)$ could be any function that exists within the allowable region of $m(t)\le (N(t)-1)$ for the addition of one node per time step. For the addition of $n(t)$ nodes per time step, the constraint becomes
$$\begin{aligned}
\label{eqn:ntconstraint}
\frac{m(t)}{n(t)}&\le&N(t-1).\end{aligned}$$
One can envisage any number of such functions for the time-dependent number of links added. We note that once the function reaches the constraint, the network can no longer evolve according to such a process.
![ \[fig:NEWACC\] Left: A fairly standard example of a non-accelerating network in which more that one link is used to link the new node to the existing network. Right: Although this evolving network asymptotes to non-accelerating behaviour, clearly, initially, it is accelerating.](NEWACC1.eps "fig:"){width="49.00000%"} ![ \[fig:NEWACC\] Left: A fairly standard example of a non-accelerating network in which more that one link is used to link the new node to the existing network. Right: Although this evolving network asymptotes to non-accelerating behaviour, clearly, initially, it is accelerating.](NEWACC2.eps "fig:"){width="49.00000%"}
A simple example might be such that a single new node is added with $m(t)= N(0)$ links per time step with $N(0)>1$ as depicted in Fig. \[fig:NEWACC\]. This is a non-accelerating network that could not be described by the GM-model that only allows at most one new link per new node for non-accelerating networks, corresponding to $p=1$ and $\alpha=0$ in their model. Another example function is one that is initially increasing but asymptotes to a constant value as depicted in Fig. \[fig:NEWACC\]. This might be some empirically observed network growth or the behaviour of some growth algorithm. Would this be described an accelerating, decelerating or non-accelerating network? Clearly, different regimes of this particular network evolution might satisfy differing descriptions. As such, we must re-define the accelerating network to encompass this feature.
Defining accelerating networks {#sec:definition1}
==============================
One might expect that one could identify the regimes of accelerating, non-acceleration and deceleration with relative ease, writing these phenomena in terms of the mean degree $\langle k \rangle = 2M(t) / N(t)$ of the network. Specifically, one might expect that if the addition of a new node via some number of links results in the mean degree of the entire network to increase, the network would be described as accelerating. Likewise, if the mean degree remains constant, the network is not accelerating and, if the mean degree decreases with the addition of new nodes, the network could be considered decelerating.
![ \[fig:meanplot\] A schematic illustration of a network evolving over fourteen time steps in the link and node phase space (solid line). The mean degree of the network is given by $\langle k \rangle = 2M(t) / N(t)$, and the dashed line represents half the mean degree of the system at time step $8$ (and time step $4$). At time step $8$ the network accelerates but the mean degree actually decreases.](meanplot.eps){width="70.00000%"}
Although these ideas are intuitively appealing, one can envisage a scenario whereby a network might be accelerating without increasing its mean degree. This event can occur if a network has undergone rapid deceleration such that the rate of node addition is very low even though it might have been high previously. At some point the ratio of number of links added to the number of nodes added over a time step, $[M(t) - M(t-1)] / [N(t) - N(t-1)]$, might be less than that for the existing network as a whole, $M(t) / N(t)$, thereby decreasing the average degree $\langle k \rangle$ while still constituting network acceleration. This is evident in Fig. \[fig:meanplot\], in which at time step $t=8$ the network accelerates, although the mean degree of the system decreases as the ratio $M(t) / N(t)$ decreases. In order for the mean degree to increase the trajectory would have to exceed the dash line whose gradient represents half the mean degree of the system at time $t=8$.
In order to identify the different regimes of network acceleration, we must relate the rate of increase in the number of links with the rate of addition of new nodes, denoting the rate of link addition and node addition by the approximate derivatives
$$\begin{aligned}
\label{eqn:approximate}
m(t) = & \frac{d M(t)}{d t} & \approx M(t) - M(t-1) \nonumber\\
n(t) = & \frac{d N(t)}{d t} & \approx N(t) - N(t-1).\end{aligned}$$
We can then define the regimes of network acceleration. The important ratio is that of the rate of link addition to the rate of node addition, $m(t)/n(t)$, the evolution of which prescribes a network measure. We define network *acceleration* $a(t)$ as
$$\begin{aligned}
\label{eqn:ratio}
a(t) & \equiv & \frac{d}{d t}\left(\frac{m(t)}{n(t)}\right). \end{aligned}$$
We approximate the discrete values with continuous derivatives and define the following three regimes:
$$\begin{aligned}
\label{eqn:definitions}
\left\{ \begin{array}{lll}
a(t) & < 0 & \quad \textrm{decelerating} \nonumber\\
a(t) & = 0 & \quad \textrm{non-accelerating} \nonumber\\
a(t) & > 0 & \quad \textrm{accelerating}.
\end{array} \right. \end{aligned}$$
As such, a single evolving network might navigate all regimes. Note that the definition of $a(t)$ allows more than one node to be added per time step. It is interesting to note that within this definition of network acceleration, a decaying network (losing nodes) could accelerate. Also, we note that the definition holds for directed graphs. The above definition alludes to the notion of network *velocity* $v(t)$, which we define as
$$\begin{aligned}
\label{eqn:velocity}
v(t) \equiv \frac{d M(t)}{d N(t)} = \frac{m(t)}{n(t)}.\end{aligned}$$
This velocity is simply the gradient of the network trajectory in the link-node phase space as in Fig. \[fig:meanplot\].
Note on hyper-acceleration {#subsec:hyper}
--------------------------
We note the existence of a turning point in the accelerating $a(t)>0$ regime of network evolution. The acceleration regime $a(t)>n(t)$ cannot be sustained indefinitely as the number of added links per new node would have to exceed the number of existing nodes $N(t)$ which is not possible. As such, this behaviour is deemed [*hyper-acceleration*]{} if
$$~\label{eqn:hyper}
a(t) >n(t).$$
![ \[fig:NEWACC3\] Superimposing the appropriate contour lines, we observe hyper-acceleration between [**A**]{} and [**B**]{}, acceleration between [**B**]{} and [**C**]{} and non-acceleration between [**C**]{} and [**D**]{}.](NEWACC3.eps){width="70.00000%"}
If we reconsider the function $m(t)$ being an initially increasing function of time asymptoting to a constant value and superimpose the appropriate contour lines, i.e. $y = const$ and $y =n~x+const$, where coefficient $n$ corresponds to the constant rate of node addition $n(t)=n$, one can clearly identify the acceleration regimes for this particular evolving network. This evolution is depicted in Fig. \[fig:NEWACC3\]. We observe hyper-acceleration between [**A**]{} and [**B**]{}, acceleration between [**B**]{} and [**C**]{} and non-acceleration between [**C**]{} and [**D**]{}. These acceleration regimes are shown schematically in Fig. \[fig:Rdiag\].
![ \[fig:Rdiag\] Different regimes of network acceleration.](Rdiag.eps){width="70.00000%"}
Accelerating weighted networks {#subsec:weighted}
------------------------------
Having defined the accelerating unweighted evolving network, we now extend the concept to encompass weighted networks. This is a relatively simple process. The key components in the definition of the network acceleration for unweighted graphs were the rates of node and link addition, both of which are macroscopic properties of the system. Similarly, we can observe the macroscopic weight of the system, denoted $L(t)$, which reflects the total weight of all the links within the network expressed as
$$\begin{aligned}
\label{eqn:weights}
L(t)&=&\sum_{i=1}^{M(t)}w_i.\end{aligned}$$
The total weight of the evolving network at time $t$ is constrained by the total number of possible links, given by $L_{\max}(t) = N(t) (N(t)-1)/2$.
In a similar manner to the unweighted scenario (see Eq. \[eqn:approximate\] for comparison), we make use of the approximate derivatives and write $$\begin{aligned}
~\label{eqn:weightedderivs}
l(t) & = \frac{d L(t)}{d t} & \approx L(t) - L(t-1) \nonumber\\
n(t) & = \frac{d N(t)}{d t} & \approx N(t) - N(t-1).\end{aligned}$$ For useful comparison between networks, it in important to normalise the weights such that for any link $w\in[0,1]$ otherwise $l(t)$ would vary enormously according to the type of network under consideration[^2] The evolution of these rates for weighted graphs are then used to define velocity and acceleration as
$$\begin{aligned}
\label{eqn:ratioweighted}
\tilde{v}(t)&\equiv& \frac{l(t)}{n(t)} \nonumber\\
\tilde{a}(t)&\equiv&\frac{d}{d t}\left(\frac{l(t)}{n(t)}\right). \end{aligned}$$
Note that if the weights are restricted to be binary in nature, the above weighted definitions of Eq. \[eqn:ratioweighted\] recover the unweighted definitions of Eq. \[eqn:approximate\], i.e., $\tilde{v}(t) \to v(t)$ and $\tilde{a}(t) \to a(t)$ as weights are made binary, which is a desirable feature of any weighted network metric. The above definition is then possibly the most general definition in that it can be applied to both weighted and unweighted, as well as directed or undirected, evolving networks.
Stochastic accelerating networks {#subsec:stochastic}
--------------------------------
The definitons (weighted and unweighted) outlined in this section have been introduced for the scenario whereby the rates of node and link addition are not stochastic. They are, infact, easily applied to stochastic situations. In this case, the corresponding measures would be the expected velocity $\langle v(t) \rangle$ and expected acceleration $\langle a(t)\rangle$. In certain cases, it might be possible to achieve this by simply replacing $n(t)$, $m(t)$ and $l(t)$ by their expectation values although in general this will [*not*]{} be suitable. For this to be appropriate we require rather specific constraints on the evolution of the system, namely that the rate of node addition $n(t)$ is deterministic and the rate of link (weight) addition is not path dependent. That is, $m(t)$ is not dependent on the number of links added at the last time step (i.e. is independent of $M(t-1)$).
In general, to evaluate the required quantities properly, we must consider all possible contributing trajectories of the system’s evolution and their corresponding probabilities. We must incorporate the possibility that the rates of node and link addition at a given time step might not be independent of each other and that their outcomes might also influence the evolution of the network at the next time step. For the unweighted case, this would give
$$\begin{aligned}
\langle v(t) \rangle &=& \sum_{a,b} P[(m(t)=a)\cap(n(t)=c)]\left(\frac{a}{c}\right) \nonumber\\
\langle a(t) \rangle &=& \sum_{a,b,c,d} \Bigg( P[(m(t+1)=b)\cap(n(t+1)=d)\cap(m(t)=a)\cap(n(t)=c)] \nonumber\\
{}&{}& \left(\frac{b}{d}-\frac{a}{c}\right) \Bigg).\end{aligned}$$
Case study: Wikipedia {#sec:case}
=====================
As a simple example of application of the concepts above introduced, we look at the evolution of Wikipedia in three different languages, namely, in English, German and Japanese. Each of these is a distinct evolving network, such that the nodes correspond to the articles and the links correspond the the links between articles. The data, albeit imprecise, is available in the public domain [@wikidata]. The evolution through the macroscopic $M(t)$ - $N(t)$ phase space is shown in the upper panel of Fig. \[fig:wiki\]. All three networks appear to converge on the same non-accelerating behaviour.
 The evolution of the Wikepedia site network for different languages. This comprises articles (nodes) and internal links (links) for the English, German and Japanese sites. The evolution through the macroscopic $M(t)$ and $N(t)$ phase space of total node and link numbers is shown in the top plot and the network velocity in the lower. ](wiki2.eps "fig:"){width="80.00000%"}  The evolution of the Wikepedia site network for different languages. This comprises articles (nodes) and internal links (links) for the English, German and Japanese sites. The evolution through the macroscopic $M(t)$ and $N(t)$ phase space of total node and link numbers is shown in the top plot and the network velocity in the lower. ](wiki1.eps "fig:"){width="80.00000%"}
While the public domain data is somewhat imprecise, the plot of the evolution of the velocity $v(t)$ of the system does indicate that all networks show an initial accelerating trend before non-accelerating behaviour is reached as shown in the lower plot of Fig. \[fig:wiki\]. There are no negative velocities as the total number of links and the total number of nodes are increasing in time for all three networks. It is interesting to note that acceleration of the growing Japanese site far exceeded that of the English and German sites. This simple example demonstrates that it is reasonable to consider network velocity and acceleration as time-dependent properties of networks as opposed to considering them as static properties of networks as suggested in [@GM]. It is also very straightforward to measure the introduced characteristics, velocity and acceleration, for empirical networks, which further supports their role in network characterization.
Conclusion {#sec:conclusion}
==========
We have revisited the framework of network acceleration suggested by Gagen and Mattick [@GM]. We have explored the limits of the proposed definition of network acceleration and, based on our findings, have provided an alternative definition for accelerating networks. Perhaps most important is the conceptual difference between the two definitions: the concept of network acceleration as introduced in this paper refers to the properties of the network at a particular moment in time as opposed to an algorithm governing the evolution of the network as suggested in [@GM]. In addition to introducing the related concept of network velocity, we have augmented the definition of network acceleration to cover weighted networks as well. As such, the definition put forward in this paper holds for both weighted and unweighted, as well as directed and undirected graphs. We have demonstrated the utility of these concepts by their simple application to study the evolution of Wikipedia in three different languages. While the data obtained from public domain is not very accurate, the obtained results clearly support the conclusion that networks undergo different regimes of acceleration throughout their evolution. Since measurement of the proposed characteristics for empirical networks is very simple, we hope that the measures will find their use in the study of network evolution, in particular as time-stamped network data becomes increasingly available in the future.
**Acknowledgements:** D.M.D.S. acknowledges support from the European Union under the MMCOMNET program and J-P.O. by a Wolfson College Junior Research Fellowship (Oxford, U.K.).
[99]{} J. S. Mattick and M. J. Gagen. Accelerating networks. [*Science*]{}, 307:856, 2005. S. N Dorogovtsev and J. F. F. Mendes. Effect of the accelerating growth of communications networks on their structure. [*Phys. Rev. E*]{}, 63:2510, 2001. G.M Gagen and J. S. Mattick. Accelerating, hyperaccelerating and decelerating probabilistic networks. [*Phys. Rev. E*]{}, 72:16123, 2005. P. Sen. Accelerated growth in outgoing links in evolving networks: Deterministic versus stochastic picture. [*Phys. Rev. E*]{}, 69:46107, 2004. See “http://www.physics.ox.ac.uk/users/smithdmd/FDN.pdf" for details. S. N. Dorogovtsev and J. F. F. Mendes. [*Evolution of Networks: From Biological Nets to the Internet and WWW*]{}. Oxford University Press, 2003. E. Erdös and A. Rényi. On random graphs. [*Publ. Math. Debrecen*]{}, 6:290, 1959. P. L. Krapivsky, S. Redner, and F. Leyvraz. Connectivity of growing networks. [*Phys Rev. Lett.*]{}, 85:4629, 2000. S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin. Structure of growing networks with preferential linking. [*Phys Rev. Lett.*]{}, 85:4633, 2000. A. L. Barabási and R. Albert. Emergence of scaling in random networks. [*Science*]{}, 286:509, 1999. A. L. Barabási, R. Albert and H. Jeong. Mean-field theory for scale-free random networks. [*Physica A*]{}, 272:173, 1999. See “http://mathworld.wolfram.com/MarkovChain.html" for details on the Markov Chain. See “http://stats.wikimedia.org/EN/Sitemap.htm" for Wikipedia site statistics.
[^1]: Often, there is little merit in adding more than one node per time step as this might be equivalent to adding one node per time step over a longer period. However, situations might arise, virtual or empirical, in which several nodes are added per time step.
[^2]: The method employed with which to perform this normalisation is likely to be situation specific. A simple division by the largest weight in the network might suffice or some cumulative binning process might be appropriate. In either case, it is necessary to take care with respect to the statistical significance of outliers.
|
---
abstract: 'In a temporal network, the presence and activity of nodes and links can change through time. To describe temporal networks we introduce the notion of temporal quantities. We define the addition and multiplication of temporal quantities in a way that can be used for the definition of addition and multiplication of temporal networks. The corresponding algebraic structures are semirings. The usual approach to (data) analysis of temporal networks is to transform it into a sequence of time slices – static networks corresponding to selected time intervals and analyze each of them using standard methods to produce a sequence of results. The approach proposed in this paper enables us to compute these results directly. We developed fast algorithms for the proposed operations. They are available as an open source Python library TQ (Temporal Quantities) and a program Ianus. The proposed approach enables us to treat as temporal quantities also other network characteristics such as degrees, connectivity components, centrality measures, Pathfinder skeleton, etc. To illustrate the developed tools we present some results from the analysis of Franzosi’s violence network and Corman’s Reuters terror news network.'
author:
- Vladimir Batagelj
- Selena Praprotnik
title: An algebraic approach to temporal network analysis based on temporal quantities
---
Introduction
============
In a [*temporal network*]{}, the presence and activity of nodes and links can change through time. In the last two decades the interest for the analysis of temporal networks increased partially motivated by travel-support services and the analysis of sequences of interaction events (e-mails, news, phone calls, collaboration, etc.). The approaches and results were recently surveyed by Holme and Saramäki in their paper [@TNsur] and the book [@TNbook].
Most of temporal social networks data contain the information about activity time intervals of their links, sometimes augmented by the activity intensity. The usual approach to the (data) analysis of temporal networks is to transform it into a sequence of time slices – static networks corresponding to selected time intervals – see for example [@DNvis; @trends; @elDyn]. Afterward each time slice is analyzed using the standard methods for analysis of static networks. Finally the results are collected into a temporal sequence of results. In this paper we propose an alternative approach, based on the notion of temporal quantity, that bypasses explicit construction of time slices. The developed algorithms are transforming temporal networks directly into results in the form of temporal quantities, vectors, temporal vectors or partitions, and temporal networks.
In the paper, we first present the basic notions about temporal networks. In Section \[secTQ\] we introduce the temporal quantities and propose an algebraic approach, based on semirings, to the analysis of temporal networks. In the following sections we show that most of the traditional network analysis concepts and algorithms such as degrees, clustering coefficient, closeness, betweenness, weak and strong connectivity, PathFinder skeleton, etc. can be straightforwardly extended to their temporal versions.
Description of temporal networks \[desc\]
=========================================
For the description of temporal networks we propose an elaborated version of the approach used in Pajek [@ESNA]. In our approach we also consider values of links (in most cases measuring the intensity/frequency of the activity). Pajek supports two types of descriptions of temporal networks based on [*presence*]{} and on [*events*]{} (Pajek 0.47, July 1999). Here, we will describe only the approach to capturing the presence of nodes and links.
A [*temporal network*]{} ${\mathcal{N}}_T =({\mathcal{V}},{\mathcal{L}}, {\mathcal{T}},{\mathcal{P}},{\mathcal{W}})$ is obtained by attaching the [*time*]{}, ${\mathcal{T}}$, to an ordinary network, where ${\mathcal{T}}$ is a set of [*time points*]{}, $t \in {\mathcal{T}}$. ${\mathcal{V}}$ is the set of nodes, ${\mathcal{L}}$ is the set of links, ${\mathcal{P}}$ is the set of node properties, and ${\mathcal{W}}$ is the set of link properties or weights [@ency]. The time ${\mathcal{T}}$ is usually either a subset of integers, ${\mathcal{T}}\subseteq {\mathbb{Z}}$, or a subset of reals, ${\mathcal{T}}\subseteq {\mathbb{R}}$. In Pajek ${\mathcal{T}}\subseteq {\mathbb{N}}$. In a general setting it could be any linearly ordered set.
In a temporal network, nodes $v \in {\mathcal{V}}$ and links $l \in {\mathcal{L}}$ are not necessarily present or active at all time points. Let $T(v)$, $T \in {\mathcal{P}}$, be the activity set of time points for the node $v$; and $T(l)$, $T \in {\mathcal{W}}$, the activity set of time points for the link $l$. The following [*consistency*]{} condition is imposed: If a link $l(u,v)$ is active at the time point $t$ then its end-nodes $u$ and $v$ should be active at the time $t$. Formally we express this by $$T(l(u,v)) \subseteq T(u) \cap T(v) .$$ The activity set $T(e)$ of a node/link $e$ is usually described as a sequence of activity time intervals $([s_i,f_i))_{i=1}^k$, where $s_i$ is the [*start*]{}ing time and $f_i$ is the [*finish*]{}ing time.
We denote a network consisting of links and nodes active in the time $t \in {\mathcal{T}}$ by ${\mathcal{N}}(t)$ and call it the (network) [*time slice*]{} or [*footprint*]{} of $t$. Let ${\mathcal{T}}' \subset {\mathcal{T}}$ (for example, a time interval). The notion of a time slice is extended to ${\mathcal{T}}'$ by $${\mathcal{N}}({\mathcal{T}}') = \bigcup_{t\in {\mathcal{T}}'} {\mathcal{N}}(t) .$$
Examples
--------
Let us look at some examples of temporal networks.
**Citation networks** can be obtained from bibliographic data bases such as Web of Science (Knowledge) and Scopus. In a citation network ${\mathcal{N}} =({\mathcal{V}},{\mathcal{L}}, {\mathcal{T}},{\mathcal{P}},{\mathcal{W}})$, its set of nodes ${\mathcal{V}}$ consists of selected works (papers, books, reports, patents, etc.). There exists an arc $a(u,v) \in {\mathcal{L}}$ iff the work $u$ cites the work $v$. The time set ${\mathcal{T}}$ is usually an interval of years $[{\textit{year}}_{{\scriptsize\textit{first}}}, {\textit{year}}_{{\scriptsize\textit{last}}}]$ in which the works were published. The activity set of the work $v$, $T(v)$, is the interval $[{\textit{year}}_{{\scriptsize\textit{pub}}}(v), {\textit{year}}_{{\scriptsize\textit{last}}}]$; and the activity set of the arc $a(u,v)$, $T(a)$, can be set to the interval $[{\textit{year}}_{{\scriptsize\textit{pub}}}(u), {\textit{year}}_{{\scriptsize\textit{pub}}}(u)]$ (instances approach) or to the interval $[{\textit{year}}_{{\scriptsize\textit{pub}}}(u), {\textit{year}}_{{\scriptsize\textit{last}}}]$ (cumulative approach). An example of a property $p \in {\mathcal{P}}$ is the number of pages or the number of authors. Other properties, such as work’s authors and keywords, are usually represented as two-mode networks.
**Project collaboration networks** are usually based on some project data base such as Cordis. The set of nodes ${\mathcal{V}}$ consists of participating institutions. There is an edge $e(u\colon v) \in {\mathcal{L}}$ iff institutions $u$ and $v$ work on a joint project. The time set ${\mathcal{T}}$ is an interval of dates/days $[{\textit{day}}_{{\scriptsize\textit{first}}}, {\textit{day}}_{{\scriptsize\textit{last}}}]$ in which the collaboration data were collected. $T(v) = {\mathcal{T}}$ and $T(e) = \{ [s,f] : $ there exists a project $P$ such that $u$ and $v$ are partners on $P$; $s$ is the start and $f$ is the finish date of $P \}$.
**KEDS/WEIS networks** are networks registering political events in critical regions in the world (Middle East, Balkans, and West Africa) on the basis of daily news. Originally they were collected by KEDS (Kansas Event Data System). Currently they are hosted by Parus Analytical Systems. The set of nodes ${\mathcal{V}}$ contains the involved actors (states, political groups, international organizations, etc.). The links are directed and are describing the events: $$( date, actor_1, actor_2, action )$$ on a given $date$ the $actor_1$ made the $action$ on the $actor_2$. Different actions are determining different relations – we get a multirelational network with a set of links partitioned by actions ${\mathcal{L}} = \{ {\mathcal{L}}_a : a \in {\textit{Actions}} \}$. The time set is determined by the observed period ${\mathcal{T}}= [day_{{\scriptsize\textit{first}}},day_{{\scriptsize\textit{last}}}]$. Since most of the actors are existing during all the observed period their node activity time sets are $T(v) = {\mathcal{T}}$. Another option is to consider as their node activity time sets the period of their engagement in the region. The activity time set $T(l)$ of an arc $l(u,v) \in {\mathcal{L}}_a$ contains all dates – intervals $[day,day+1)$ – in which the actor $u$ made an action $a$ on the actor $v$. Another possibility is to base the description on a single relation network and store the information about the action $a$ as a structured value in a triple $(day,day+1,value)$ $$value = [(action_1,count_1), \ldots ,(action_k,count_k) ]$$ and introduce an appropriate semiring over such values (see Section \[secTQ\]).
There are many other examples of temporal networks such as: genealogies, contact networks, networks of phone calls, etc.
Temporal quantities {#secTQ}
===================
Besides the presence/absence of nodes and links also their properties can change through time. To describe the changes we introduce the notion of a [*temporal quantity*]{} $a$ with the activity set $T_a \subseteq {\mathcal{T}}$ $$a = \left\{\begin{array}{ll}
a'(t) & t \in T_a \\
{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}& t \in {\mathcal{T}}\setminus T_a
\end{array}\right.$$ where $a'(t)$ is the value of $a$ at an instant $t$, and denotes the value [*undefined*]{}.
We assume that the values of temporal quantities belong to a set $A$ which is a semiring $(A,\oplus,\odot,0,1)$ for binary operations $\oplus : A\times A \to A$ and $\odot : A\times A \to A$ [@GoMi; @semi]. This means that $(A,\oplus,0)$ is an Abelian monoid – the addition $\oplus$ is associative and commutative, and has 0 as its neutral element; and $(A,\odot,1)$ is a monoid – the multiplication $\odot$ is associative and has 1 as its neutral element. Also, multiplication distributes from both sides over the addition. Note that $0$ and $1$ denote the two elements of $A$ that satisfy the required properties. In expressions the precedence of the multiplication $\odot$ over the addition $\oplus$ is assumed. We can extend both operations to the set $A_{\scriptsize{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}} = A \cup \{{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}\}$ by requiring that for all $a \in A_{\scriptsize{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}}$ it holds $$a \oplus {\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}= {\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}\oplus a = a \quad \mbox{and} \quad
a \odot {\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}= {\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}\odot a = {\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}.$$ The structure $(A_{\scriptsize{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}},\oplus,\odot,{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}},1)$ is also a semiring.
![Semiring addition and multiplication in networks.\[semiG\]](semiring.pdf){width="75mm"}
The “default” semiring is the [*combinatorial*]{} semiring $({\mathbb{R}}_0^+,+,\cdot,0,1)$ where $+$ and $\cdot$ are the usual addition and multiplication of real numbers. In some applications other semirings are useful.
In applications of semirings in the analysis of graphs and networks the addition $\oplus$ describes the composition of values on parallel walks and the multiplication $\odot$ describes the composition of values on sequential walks – see Figure \[semiG\]. For the combinatorial semiring these two schemes correspond to basic principles of combinatorics: the [*Rule of Sum*]{} and the [*Rule of Product*]{} [@comb].
The semiring $(\overline{{\mathbb{R}}_0^+}, \min, +, \infty, 0)$, $\overline{{\mathbb{R}}_0^+}
= {\mathbb{R}}_0^+ \cup \{\infty\}$, is suitable to deal with the shortest paths problem in networks; and the semiring $(\{0,1\}, \lor,
\land, 0, 1)$ for reachability problems. The standard references on semirings are [@GaN] and [@GoMi].
Semiring of temporal quantities
-------------------------------
Let $A_{\scriptsize{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}}({\mathcal{T}})$ denote the set of all temporal quantities over $A_{\scriptsize{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}}$ in the time ${\mathcal{T}}$. To extend the operations to networks and their matrices we first define the [*sum*]{} (parallel links) $ a \oplus b $ as $$(a \oplus b)(t) = a(t) \oplus b(t)$$ and $T_s = T_a \cup T_b$; and the [*product*]{} (sequential links) $ a \odot b $ as $$(a \odot b)(t) = a(t) \odot b(t)$$ and $T_p = T_a \cap T_b$.
In these definitions and also in the following text, to avoid the ‘pollution’ with many different symbols, we use the symbols $\oplus$ and $\odot$ to denote the semiring operations. The appropriate semiring can be determined from the context. For example, in the definition of addition of temporal quantities the symbol $\oplus$ on the left hand side of the equation operates on temporal quantities and the symbol $\oplus$ on the right hand side denotes the addition in the basic semiring $A_{\scriptsize{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}}$.
Let us define the temporal quantities $\mathbf{0}$ and $\mathbf{1}$ with requirements $\mathbf{0}(t) = {\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}$ and $\mathbf{1}(t) = 1$ for all $t \in {\mathcal{T}}$. It is a routine task to verify that the structure $(A_{\scriptsize{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}}({\mathcal{T}}),\oplus,\odot,\mathbf{0},\mathbf{1})$ is also a semiring, and therefore so is the set of square matrices of order $n$ over it for the addition $\mathbf{A} \oplus \mathbf{B} = \mathbf{S}$ $$s_{ij} = a_{ij} \oplus b_{ij}$$ and multiplication $\mathbf{A} \odot \mathbf{B} = \mathbf{P}$ $$p_{ij} = \bigoplus_{k=1}^n a_{ik} \odot b_{kj} .$$ Again, the symbols $\oplus$ and $\odot$ on the left hand side operate on temporal matrices and on the right hand side in the semiring of temporal quantities.
The matrix multiplication is closely related to traveling on networks. Consider an entry $p_{ij}$ in an instant $t$ $$p_{ij}(t) = \bigoplus_{k=1}^n a_{ik}(t) \odot b_{kj}(t) .$$ For a value $p_{ij}(t)$ to be defined (different from ${\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}$) there should exist in the instant $t$ at least one node $k$ such that both the link $(i,k)$ and the link $(k,j)$ exist – the transition from the node $i$ to the node $j$ through a node $k$ is possible. Its contribution is $a_{ik}(t)\odot b_{kj}(t)$. This means that the matrix multiplication is taking into account only the links inside the time slice ${\mathcal{N}}(t)$.
Operationalization
------------------
In the following we shall limit our discussion to temporal quantities that can be described in the form of time-interval/value sequences $$a = ( (I_i, v_i) )_{i=1}^k$$ where $I_i$ is a time-interval and $v_i$ is a value of $a$ on this interval. In general, the intervals can be of different types: 1 – $[ s_i, f_i]$; 2 – $[ s_i, f_i)$; 3 – $( s_i, f_i]$; 4 – $( s_i, f_i)$. Also the value $v_i$ can be structured. For example $v_i = (w_i, c_i, \tau_i)$ – weight, capacity and transition time, or $v_i = (d_i,n_i)$ – the length of geodesics and the number of geodesics, etc. We require $s_i \leq f_i$, for $i=1,\ldots,k$ and $s_{i-1} < s_i$, for $i=2,\ldots,k$.
$b$ $a$ $c \gets [\ ]$ $(s_a,f_a,v_a) \gets {\textit{get}}(a)$; $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ $s_c \gets s_a$; $v_c \gets v_a$ $f_c \gets s_b$; $s_a \gets s_b$ $f_c \gets f_a$; $(s_a,f_a,v_a) \gets {\textit{get}}(a)$ $s_c \gets s_a$; $f_c \gets \min(f_a,f_b)$ $v_c \gets {\textit{sAdd}}(v_a,v_b)$ $s_a \gets s_b \gets f_c$; $f_d \gets f_a$ $(s_a,f_a,v_a) \gets {\textit{get}}(a)$ $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ $s_c \gets s_b$; $v_c \gets v_b$ $f_c \gets s_a$; $s_b \gets s_a$ $f_c \gets f_b$; $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ $c.{\textit{append}}((s_c,f_c,v_c))$ ${\textit{standard}}(c)$
To simplify the exposition we will assume in the following that all the intervals in our descriptions of temporal quantities are of type 2 – $[ s_i, f_i)$ and $f_{i-1} \leq s_i$, for $i=2,\ldots,k$. Therefore we can describe the temporal quantities with sequences of triples $$a = ( (s_i, f_i, v_i) )_{i=1}^k .$$ In the examples we will also assume that ${\cal T} = [t_{min}, t_{max}] \subset {\mathbb{N}}$.
To provide a computational support for the proposed approach we are developing in Python a library TQ (Temporal Quantities). In the examples we will use the Python notation for temporal quantities.
The following are two temporal quantities $a$ and $b$ represented in Python as a list of triples
a = [(1, 5, 2), (6, 8, 1), (11, 12, 3),
(14, 16, 2), (17, 18, 5), (19, 20, 1)]
b = [(2, 3, 4), (4, 7, 3), (9, 10, 2),
(13, 15, 5), (16, 21, 1)]
The temporal quantity $a$ has on the interval $[1,5)$ (i.e. in instances 1, 2, 3 and 4) value 2; on the interval $[6,8)$ value 1; on the interval $[11,12)$ value 3, etc. Outside the specified intervals its value is undefined, ${\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}$.
The temporal quantities can also be visualized as it is shown for $a$ and $b$ at the top half of Figure \[ops\].
For the simplified version of temporal quantities we wrote procedures ${\textit{sum}}$ (Algorithm \[sumTQ\]) for the addition and ${\textit{prod}}$ (Algorithm \[proTQ\]) for the multiplication of temporal quantities over the selected semiring. Because, by assumption, the triples in a description of a temporal quantity are ordered by their starting times, we can base both procedures on the ordered lists merging scheme. The basic semiring operations of addition and multiplication are provided by functions ${\textit{sAdd}}$ and ${\textit{sMul}}$.
The function ${\textit{length}}(a)$ returns the length (number of items) of the list $a$. The function ${\textit{get}}(a)$ returns the current item of the list $a$ and moves to the next item; if the list is exausted it returns a ‘sentinel’ triple $(\infty,\infty,0)$. The statement $(s,f,v) \gets e$ describes the unpacking of the item $e$ into its parts. The statement $c.{\textit{append}}(e)$ appends the item $e$ to the tail of the list $c$. The function ${\textit{standard}}(a)$ joins, in the list $a$, adjacent time intervals with the same value into a single interval.
$[\ ]$ $c \gets [\ ]$; $(s_a,f_a,v_a) \gets {\textit{get}}(a)$; $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ $(s_a,f_a,v_a) \gets {\textit{get}}(a)$ $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ $s_c \gets \max(s_a,s_b)$; $f_c \gets \min(f_a,f_b)$; $v_c \gets {\textit{sMul}}(v_a,v_b)$ $c.{\textit{append}}((s_c,f_c,v_c))$ $(s_a,f_a,v_a) \gets {\textit{get}}(a)$ $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ ${\textit{standard}}(c)$
[l]{} $a$ :\
\
$b$ :\
\
$a\oplus b$ :\
\
$a\odot b$ :\
[l]{} $a$ :\
\
$b$ :\
\
$a\oplus b$ :\
\
$a\odot b$ :\
The following are the sum $s$ and the product $p$ of temporal quantities $a$ and $b$. They are visually displayed at the bottom half of Figure \[ops\].
s = [(1, 2, 2), (2, 3, 6), (3, 4, 2),
(4, 5, 5), (5, 6, 3), (6, 7, 4),
(7, 8, 1), (9, 10, 2), (11, 12, 3),
(13, 14, 5), (14, 15, 7), (15, 16, 2),
(16, 17, 1), (17, 18, 6), (18, 19, 1),
(19, 20, 2), (20, 21, 1)]
p = [(2, 3, 8), (4, 5, 6), (6, 7, 3),
(14, 15, 10), (17, 18, 5), (19, 20, 1)]
Let $l_a = {\textit{length}}(a)$ and $l_b = {\textit{length}}(b)$. Then, assuming that the semiring operations take constant time each, the time complexity of both algorithms is $O(l_a+l_b)$. The example in Figure \[grops\] shows that in extreme cases the sum can be almost 4 times longer than each of its arguments, and the product almost twice as long as the arguments. If ${\cal T} = [t_{min}, t_{max}] \subset {\mathbb{N}}$ the length of a list describing a temporal quantity can not exceed $L = t_{max}-t_{min}$.
The aggregated value
--------------------
In some applications over the combinatorial semiring we shall use the [*aggregated value*]{} of a temporal quantity $a = ( (s_i, f_i, v_i) )_{i=1}^k$. It is defined as $$\Sigma a = \sum_{i=1}^k (f_i-s_i)\cdot v_i$$ and is computed using the procedure ${\textit{total}}(a)$. For example $\Sigma a = 23$ and $\Sigma b = 30$. Note that $\Sigma a + \Sigma b =
\Sigma (a+b)$.
Temporal partitions
-------------------
The description of temporal partitions has the same form as the description of temporal quantities $a = ( (s_i, f_i, v_i) )_{i=1}^k$. They differ only in the interpretation of values $v_i \in {\mathbb{N}}$. In case of partitions $v_i = j$ means that the unit described with $a$ belongs to a class $j$ in the time interval $[s_i,f_i)$. We shall use temporal partitions to describe connectivity components in Section \[conn\].
We obtain a more adequate description of temporal networks by using vectors of temporal quantities (temporal vectors and temporal partitions) for describing properties of nodes and making also link weights into temporal quantities. In the current version of the library TQ we use a representation of a network ${{\mathcal{N}}}$ with its matrix $\mathbf{A} = [a_{uv}]$ $$a_{uv} = \left\{\begin{array}{ll}
w(u,v) & (u,v) \in {\mathcal{L}} \\
{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}& \mbox{otherwise}
\end{array}\right.$$ where $ w(u,v)$ is a temporal weight attached to a link $(u,v)$.
Products of a temporal matrix and a temporal vector
---------------------------------------------------
In some applications the product of a temporal matrix with a temporal vector is useful. There are two products – left and right.
Let $\mathbf{A}$ be a temporal matrix of size $n\times m$, $\mathbf{v}$ a vector of size $n$, and $\mathbf{u}$ a vector of size $m$. The [*product from left*]{} of $\mathbf{A}$ with $\mathbf{v}$, denoted by $\mathbf{u} = \mathbf{v} \bullet \mathbf{A}$, is defined by $$u_j = \bigoplus_{i=1}^n v_i \odot a_{ij}, \qquad j = 1, \ldots, m$$ and the [*product from right*]{} of $\mathbf{A}$ with $\mathbf{u}$, denoted by $\mathbf{v} = \mathbf{A} \bullet \mathbf{u}$, is defined by $$v_i = \bigoplus_{j=1}^m a_{ij} \odot u_j, \qquad i = 1, \ldots, n .$$ In the TQ library both products are implemented as functions ${\textit{MatVecMulL}}(A,v)$ and ${\textit{MatVecMulR}}(A,v)$.
If a vector $\mathbf{v}$ of size $n$ is considered as a column vector – an $n \times 1$ matrix – it holds $\mathbf{v} \bullet \mathbf{A} = (\mathbf{v}^T \odot \mathbf{A})^T$ and $\mathbf{A} \bullet \mathbf{u} = \mathbf{A} \odot \mathbf{u}$. $T$ denotes the matrix transposition operation.
Node activities\[activ\]
========================
In this section we show how we can use the proposed operations with temporal quantities (the addition) for a simple analysis of temporal networks.
Assume that the values in temporal quantities $a_{uv}$ from a temporal network matrix $\mathbf{A}$ are positive real numbers measuring the intensity of the activity of the node $u$ on the node $v$. We define the [*activity*]{} of a group of nodes ${\mathcal{V}}_1$ on a group ${\mathcal{V}}_2$ (using the combinatorial semiring) as $$\mbox{act}({\mathcal{V}}_1,{\mathcal{V}}_2) =
\sum_{u\in{\mathcal{V}}_1} \sum_{v\in{\mathcal{V}}_2} a_{uv} .$$ To illustrate the notion of activity we applied it on Franzosi’s violence temporal network [@RF]. Roberto Franzosi collected from the journal news in the period January 1919 – December 1922 information about the different types of interactions between political parties and other groups of people in Italy. The violence network contains only the data about violent actions and counts the number of interactions per month.
-- --
-- --
We determined the temporal quantities $pol = \mbox{act}(\{ \mbox{police} \}, {\mathcal{V}}) +
\mbox{act}({\mathcal{V}}, \{ \mbox{police} \})$, ${\textit{fas}} = \mbox{act}(\{ \mbox{fascists} \}, {\mathcal{V}}) +
\mbox{act}({\mathcal{V}}, \{ \mbox{fascists} \})$ and $all = \mbox{act}({\mathcal{V}}, {\mathcal{V}})$. They are presented in Figure \[viol\]. Comparing the intensity charts of police and fascists activity with overall activity we see that most of the violent activities in the first two years 1919 and 1920 were related to the police. In the next two years (1921 and 1922) they were taken over by the fascists.
![First example network. All unlabeled links have a value of $[(1,9,1)]$.\[graph\]](G2pic.pdf){width="75mm"}
Temporal degrees \[degrees\]
============================
For an ordinary graph with a (binary) adjacency matrix $\mathbf{A}$ we can compute the corresponding indegree, $\mathbf{i}$, and outdegree, $\mathbf{o}$, vectors using (over the combinatorial semiring) the relations $$\mathbf{i} = \mathbf{e} \bullet \mathbf{A} \qquad \mbox{and} \qquad
\mathbf{o} = \mathbf{A} \bullet \mathbf{e}$$ where $\mathbf{e}$ is a column vector of size $n = |{\mathcal{V}}|$ with all its entries equal to 1. The same holds for temporal networks. In this case the vector $\mathbf{e}$ contains as values the temporal unit $\mathbf{1} = [(0,\infty,1)]$.
For a temporal network presented in Figure \[graph\] the corresponding temporal indegrees and outdegrees are given in Table \[iod\]. For example, the node 5 has in the time interval $[1,5)$ outdegree 2. Because the arc $(5,7)$ disappears at the time point 5 the outdegree of the node 5 diminishes to 1 in the interval $[5,9)$.
indegrees outdegrees
1 : [(1, 9, 1)] 1 : [(1, 9, 1)]
2 : [(1, 9, 2)] 2 : [(1, 3, 1),
3 : [] (3, 9, 2)]
4 : [(1, 3, 1), 3 : [(1, 9, 1)]
(3, 9, 2)] 4 : [(1, 9, 1)]
5 : [(1, 9, 1)] 5 : [(1, 5, 2),
6 : [(1, 9, 1)] (5, 9, 1)]
7 : [(1, 5, 1), 6 : [(1, 9, 1)]
(7, 9, 1)] 7 : [(1, 9, 3)]
8 : [(1, 9, 2)] 8 : [(1, 9, 2)]
9 : [(1, 9, 2)] 9 : [(1, 9, 2)]
10 : [(1, 9, 3)] 10 : [(1, 9, 1)]
11 : [(1, 9, 2)] 11 : [(1, 7, 1),
12 : [] (7, 9, 2)]
13 : [(2, 8, 2)] 12 : []
14 : [(2, 8, 2)] 13 : [(2, 8, 2)]
15 : [(2, 8, 2)] 14 : [(2, 8, 2)]
15 : [(2, 8, 2)]
We will use the simple temporal network from Figure \[graph\] also for the illustration of some other algorithms because it allows the users to manually check the presented results.
Temporal co-occurrence networks
===============================
Let the binary matrix $\mathbf{A}=[a_{ep}]$ describe a two-mode network on the set of events $E$ and the set of of participants $P$: $$a_{ep} = \left\{\begin{array}{ll}
1 & p \mbox{ participated in the event } e \\
0 & \mbox{otherwise}
\end{array}\right.$$ The function $d: E \to {\mathcal{T}}$ assigns to each event $e$ the date $d(e)$ when it happened. ${\mathcal{T}}= [{\textit{first}}, {\textit{last}}]$. Using these data we can construct two temporal affiliation matrices:
- **instantaneous** $\mathbf{Ai}=[ai_{ep}]$, where $$ai_{ep} = \left\{\begin{array}{ll}
[(d(e),d(e)+1,1)] & a_{ep} = 1 \\
\lbrack\ \rbrack & \mbox{otherwise}
\end{array}\right.$$
- **cumulative** $\mathbf{Ac}=[ac_{ep}]$, where $$ac_{ep} = \left\{\begin{array}{ll}
[(d(e),last+1,1)] & a_{ep} = 1 \\
\lbrack\ \rbrack & \mbox{otherwise}
\end{array}\right.$$
Using the multiplication of temporal matrices over the combinatorial semiring we get the corresponding instantaneous and cumulative co-occurrence matrices $$\mathbf{Ci} = \mathbf{Ai}^T \cdot \mathbf{Ai} \qquad \mbox{and}
\qquad \mathbf{Cc} = \mathbf{Ac}^T \cdot \mathbf{Ac}$$ A typical example of such a matrix is the papers authorship matrix where $E$ is the set of papers, $P$ is the set of authors and $d$ is the publication year [@bibnet].
ci[IDI/B,HCL/B] cc[IDI/B,HCL/B]
1 : (2003, 2004, 1) (2003, 2004, 1)
2 : (2004, 2005, 2) (2004, 2005, 3)
3 : (2005, 2006, 3) (2005, 2006, 6)
4 : (2006, 2007, 2) (2006, 2007, 8)
5 : (2007, 2008, 1) (2007, 2008, 9)
6 : (2008, 2009, 7) (2008, 2009, 16)
7 : (2009, 2010, 6) (2009, 2010, 22)
8 : (2010, 2011, 7) (2010, 2011, 29)
9 : (2011, 2013, 18) (2011, 2012, 47)
10 : (2012, 2013, 65)
ci[HCL/B,HCL/B] cc[HCL/B,HCL/B]
1 : (1997, 1998, 2) (1997, 1998, 2)
2 : (1998, 1999, 5) (1998, 1999, 7)
3 : (1999, 2000, 8) (1999, 2000, 15)
4 : (2000, 2001, 7) (2000, 2001, 22)
5 : (2001, 2002, 5) (2001, 2002, 27)
6 : (2002, 2003, 6) (2002, 2003, 33)
7 : (2003, 2004, 14) (2003, 2004, 47)
8 : (2004, 2005, 20) (2004, 2005, 67)
9 : (2005, 2006, 10) (2005, 2006, 77)
10 : (2006, 2007, 14) (2006, 2007, 91)
11 : (2007, 2008, 20) (2007, 2008, 111)
12 : (2008, 2009, 28) (2008, 2009, 139)
13 : (2009, 2010, 56) (2009, 2010, 195)
14 : (2010, 2011, 78) (2010, 2011, 273)
15 : (2011, 2012, 84) (2011, 2012, 357)
16 : (2012, 2013, 112) (2012, 2013, 469)
The triple $(s,f,v)$ in a temporal quantity $ci_{pq}$ tells that in the time interval $[s,f)$ there were $v$ events in which both $p$ and $q$ took part.
The triple $(s,f,v)$ in a temporal quantity $cc_{pq}$ tells that in the time interval $[s,f)$ there were in total $v$ accumulated events in which both $p$ and $q$ took part.
The diagonal matrix entries $ci_{pp}$ and $cc_{pp}$ contain the temporal quantities counting the number of events in the time intervals in which the participant $p$ took part.
For example, in a data set on the stem cell research during 1997–2012 in Spain collected by Gisela Cantos-Mateos [@stem] we get from the basic two-mode network, where $E$ is the set of papers and $P$ is the set of institutions, for selected two institutions (HCL/B $=$ University Hospital Clínic de Barcelona, Barcelona and IDI/B $=$ Institut d’Investigacions Biomèdiques August Pi i Sunyer, Barcelona) the collaboration temporal quantities presented in Table \[tec\].
The first column in the table contains the yearly collaboration (co-authorship) data and the second column contains the cumulative collaboration data. Let’s read the table:\
$ci[\texttt{IDI/B},\texttt{HCL/B}](2005, 2006) = 3$ — in the year 2005 researchers from both institutions published 3 joint papers;\
$ci[\texttt{IDI/B},\texttt{HCL/B}](2011, 2013) = 18$ — in the years 2011 and 2012 researchers from both institutions published 18 joint papers each year;\
$ci[\texttt{HCL/B},\texttt{HCL/B}](2010, 2011) = 78$ — in the year 2010 researchers from the institution `HCL/B` published 78 papers;\
$cc[\texttt{IDI/B},\texttt{HCL/B}](2008, 2009) = 16$ — till the year 2008 (included) researchers from both institutions published 16 joint papers.
Note that the violence network from Section \[activ\] is essentially a co-occurrence network that could be obtained from the more primitive instantaneous two-mode network about violent actions reported in journal articles and the involved political actors.
Clustering coefficients
=======================
Let us assume that the network ${{\mathcal{N}}}$ is based on a simple directed graph ${\mathcal{G}} = ({\mathcal{V}},{\mathcal{A}})$ without loops. From a simple undirected graph we obtain the corresponding simple directed graph by replacing each edge with a pair of opposite arcs. In such a graph the [*clustering coefficient*]{}, $C(v)$, of the node $v$ is defined as the proportion between the number of realized arcs among the node’s neighbors and the number of all possible arcs among the node’s neighbors $N(v)$, that is $$C(v) = \frac{|{\mathcal{A}}(N(v))|}{k(k-1)}$$ where $k$ is the number of neighbors of the node $v$. For a node $v$ without neighbors or with a single neighbor we set $C(v)=0$.
The clustering coefficient measures a local density of the node’s neighborhood. A problem with its applications in network analysis is that the identified densest neighborhoods are mostly very small. For this reason we provided in Pajek the [*corrected clustering coefficient*]{}, $C'(v)$, $$C'(v) = \frac{|{\mathcal{A}}(N(v))|}{\Delta(k-1)}$$ where $\Delta$ is the maximum number of neighbors in the network.
To count the number of realized arcs among the node’s neighbors we use the observation that each arc forms a triangle with links from its end-nodes to the node $v$; and that the number of triangles in a simple undirected graph can be obtained as the diagonal value in the third power of the graph matrix (over the combinatorial semiring).
----------------------------------------------------------------------- -- -- --------------------------------------------------------------- --
![Counting triangles.\[trik\]](tri1.pdf "fig:"){width="20mm"} ![Counting triangles.\[trik\]](tri5.pdf "fig:"){width="20mm"}
\[2mm\] ![Counting triangles.\[trik\]](tri2.pdf "fig:"){width="20mm"} ![Counting triangles.\[trik\]](tri6.pdf "fig:"){width="20mm"}
\[2mm\] ![Counting triangles.\[trik\]](tri3.pdf "fig:"){width="20mm"} ![Counting triangles.\[trik\]](tri7.pdf "fig:"){width="20mm"}
\[2mm\] ![Counting triangles.\[trik\]](tri4.pdf "fig:"){width="20mm"} ![Counting triangles.\[trik\]](tri8.pdf "fig:"){width="20mm"}
----------------------------------------------------------------------- -- -- --------------------------------------------------------------- --
For simple directed graphs the counting of triangles is slightly more complicated. Let us denote $\mathbf{T} = \mathbf{A}^T$ and $\mathbf{S} = \mathbf{A}+\mathbf{T}$. From Figure \[trik\] we see that each triangle (determined with a link opposite to the dark node) appears exactly once in $$\mathbf{AAA} + \mathbf{AAT} + \mathbf{TAT} + \mathbf{TAA} = \mathbf{AAS} + \mathbf{TAS} = \mathbf{SAS} .$$ This gives us a simple way to count the triangles which is used in Algorithm \[CCoef\]. The function ${\textit{nRows}}(\mathbf{A})$ returns the size (number of rows) of matrix $\mathbf{A}$. The function ${\textit{VecConst}}(n,v)$ constructs a vector of size $n$ filled with the value $v$. The function ${\textit{MatBin}}(\mathbf{A})$ transforms all values in the triples in the matrix $\mathbf{A}$ to 1. The function ${\textit{MatSetDiag}}(\mathbf{A},c)$ sets all the diagonal entries of the matrix $\mathbf{A}$ to the value $c$. The function ${\textit{MatSym}}(\mathbf{A})$ makes the transformation $\mathbf{S} = \mathbf{A}\oplus\mathbf{T}$. Functions ${\textit{VecSum}}$ and ${\textit{VecProd}}$ implement a component wise composition of temporal vectors: ${\textit{VecSum}}(a,b) = [a_i \oplus b_i,\ i = 1, \ldots, n]$ and ${\textit{VecProd}}(a,b) = [a_i \odot b_i,\ i = 1, \ldots, n]$. Similarly ${\textit{VecInv}}(a) = [{\textit{invert}}(a_i),\ i = 1, \ldots, n]$ in the combinatorial semiring; where ${\textit{invert}}(a) = [ (s,f,1/v) \mbox{ \bf for } (s,f,v) \in a ]$. The function ${\textit{MatProd}}(\mathbf{A},\mathbf{B})$ determines the product $\mathbf{A \odot B}$. Since we need only the diagonal values of the matrix $\mathbf{SAS}$ we applied a special function ${\textit{MatProdDiag}}(\mathbf{A},\mathbf{B})$ that determines only the diagonal vector of the product $\mathbf{A \odot B}$. Afterward, to get the clustering coefficient, we have to normalize the obtained counts. The number of neighbors of the node $v$ is determined as its degree in the corresponding undirected temporal skeleton graph (in which an edge $e=(v:u)$ exists iff there is at least one arc between the nodes $v$ and $u$). The maximum number of neighbors $\Delta$ can be considered either for a selected time point (${\textit{type}}=2$) or for the complete time window (${\textit{type}}=3$). Note that to determine the temporal $\Delta$ we used summing of temporal degrees over the [*maxmin*]{} semiring $({\mathbb{R}},\max,\min,-\infty,\infty)$.
The time complexity of Algorithm \[CCoef\] is $O(n^3 \cdot L)$.
In Table \[cct\] and Table \[ccct\] the ordinary and the corrected clustering coefficients are presented for the example network from Figure \[graph\] and its undirected skeleton.
\
\#[ [*type*]{} $= 1$ - standard CC]{}\
\#[ [*type*]{} $= 2$ - corrected CC / temporal degMax]{}\
\#[ [*type*]{} $= 3$ - corrected CC / overall degMax]{} ${\textit{SetSemiring}}({\textit{combinatorial}})$ $n \gets {\textit{nRows}}(A)$; $ve \gets {\textit{VecConst}}(n,[(0,\infty,-1)])$ $B \gets {\textit{MatSetDiag}}({\textit{MatBin}}(A),\mathbf{0})$ $S \gets {\textit{MatBin}}({\textit{MatSym}}(B))$ ${\textit{deg}} \gets {\textit{MatVecMulR}}(S,{\textit{VecConst}}(n,\mathbf{1}))$ ${\textit{fac}} \gets {\textit{VecProd}}({\textit{deg}},{\textit{VecSum}}({\textit{deg}},ve))$ ${\textit{SetSemiring}}({\textit{maxmin}})$; $\delta \gets \mathbf{0}$ $\delta \gets {\textit{sum}}(\delta,d)$ $\Delta \gets \max([v \mbox{ \bf for } (s,f,v) \in \delta])$ $\delta \gets [(0,\infty,\Delta)]$ ${\textit{SetSemiring}}({\textit{combinatorial}})$ ${\textit{degm}} \gets {\textit{VecSum}}({\textit{deg}},ve)$; ${\textit{fac}} \gets \mathbf{0}$ ${\textit{fac}}.{\textit{append}}({\textit{prod}}(\delta,d))$ ${\textit{tri}} \gets {\textit{MatProdDiag}}({\textit{MatProd}}(S,B),S)$ ${\textit{VecProd}}({\textit{VecInv}}({\textit{fac}}),{\textit{tri}})$
1 : []
2 : []
3 : []
4 : [(1, 3, 0.5), (3, 9, 0.1667)]
5 : [(1, 5, 0.1667), (5, 9, 0.5)]
6 : [(1, 9, 0.5)]
7 : [(1, 5, 0.25), (5, 9, 0.5)]
8 : [(1, 7, 0.4167), (7, 9, 0.5)]
9 : [(1, 7, 0.4167), (7, 9, 0.5)]
10 : [(1, 7, 0.4167), (7, 9, 0.5)]
11 : [(1, 9, 0.5)]
12 : []
13 : [(2, 8, 1.0)]
14 : [(2, 8, 1.0)]
15 : [(2, 8, 1.0)]
1 : []
2 : []
3 : []
4 : [(1, 3, 0.5), (3, 9, 0.25)]
5 : [(1, 5, 0.25), (5, 9, 0.5)]
6 : [(1, 9, 0.5)]
7 : [(1, 5, 0.5), (5, 7, 0.75),
(7, 9, 1.0)]
8 : [(1, 7, 0.8333), (7, 9, 1.0)]
9 : [(1, 7, 0.8333), (7, 9, 1.0)]
10 : [(1, 7, 0.8333), (7, 9, 1.0)]
11 : [(1, 7, 0.75), (7, 9, 1.0)]
12 : []
13 : [(2, 8, 0.5)]
14 : [(2, 8, 0.5)]
15 : [(2, 8, 0.5)]
Closures in temporal networks
=============================
When the basic semiring $(A,\oplus,\odot,0,1)$ is [*closed*]{} – an unary [*closure*]{} operation $\star$ with the property $$a^\star = 1 \oplus a\odot a^\star = 1 \oplus a^\star \odot a , \qquad
\mbox{for all } a \in A$$ is defined in it – this property can be extended also to the corresponding matrix semiring. When it exists, a standard closure is obtained as $$a^\star = \bigoplus_{i=0}^\infty a^i .$$ In some semirings different closures can exist. For computing the matrix closure we can apply the Fletcher’s algorithm [@closure]. The entry $c_{uv}$ in the matrix $\mathbf{C} = \mathbf{A}^\star$ is equal to the sum of values of all walks from the node $u$ to the node $v$. In most of the semirings, except the combinatorial, for which we are interested in determining the closures, also the [*absorption law*]{} holds $$1\oplus a=1, \qquad \mbox{for all } a \in A .$$ In these semirings $a^\star = 1$, for all $a \in A$, and therefore the Fletcher’s algorithm can be simplified and performed in place as implemented in Algorithm \[closeTQ\].
For a temporal quantity $a$ over a closed semiring it holds $T_{a^\star} = {\mathcal{T}}$.
The time complexity of Algorithm \[closeTQ\] is $O(n^3 \cdot L)$.
$n \gets {\textit{nRows}}(R)$ $C \gets R$ $C[u,v] \gets$
${\textit{sum}}(C[u,v], {\textit{prod}}(C[u,k],C[k,v]))$ $C[k,k] \gets {\textit{sum}}(\mathbf{1},C[k,k])$ $C$
Temporal node partitions
========================
In the previous sections, the nodes of temporal networks were considered as being present all the time. We can describe the presence of nodes through time using a temporal binary (single valued) node partition $T : {\mathcal{V}} \to A_{\scriptsize{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}}({\mathcal{T}})$, $$T(u) = ( (s_i, f_i, 1) )_{i=1}^k, \quad \mbox{for } u \in {\mathcal{V}}$$ specifying that a node $u$ is present in time intervals $[s_i, f_i)$, $i = 1, \ldots, k$.
The node partition $T_{Min}$ determined from the temporal network links by $$T_{Min}(u) = \bigcup_{l \in {\mathcal{L}}: u \in \mbox{\scriptsize ext}(l)} {\textit{binary}}(a_l) ,$$ for $ u \in {\mathcal{V}}$, is the smallest temporal partition of nodes that satisfies the consistency condition from Section \[desc\]. The term $\mbox{ext}(l)$ denotes the set of endnodes of the link $l$, $a_l$ is the temporal quantity assigned to the link $l$, and the function ${\textit{binary}}$ sets all values in a given temporal quantity to 1. In the library TQ the partition $T_{Min}$ can be computed using the function ${\textit{minTime}}$.
A temporal node partition $q$ can also be used to extract a corresponding subnetwork from the given temporal network described with a matrix $\mathbf{A}$. The subnetwork contains only the nodes active in the partition $q$ and the active links satisfying the consistency condition with respect to $q$.
To formalize the described procedure we first define the procedure ${\textit{extract}}(p,a) = b$, where $p$ is a binary temporal quantity and $a$ is a temporal quantity, as $$b(t) = \left\{\begin{array}{ll}
a(t) & t \in T_p \cap T_a \\
{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}& \mbox{otherwise}
\end{array}\right. .$$ Let $\mathbf{B}$ be a temporal matrix describing the links of the subnetwork determined by the partition $q$. Its entries for $l(u,v) \in {\mathcal{L}}$ are determined by $$b_l = {\textit{extract}}(q(u) \cap q(v),a_l) .$$ In TQ this operation is implemented as a procedure ${\textit{MatExtract}}(\mathbf{q},\mathbf{A})$.
Temporal reachability and weak and strong connectivity \[conn\]
===============================================================
For a temporal network represented with the corresponding binary matrix $\mathbf{A}$ its transitive closure $\mathbf{A}^\star$ (over the reachability semirings based on the semiring $(\{0,1\},\lor,\land,0,1)$) determines its [*reachability*]{} relation matrix. We obtain its [*weak connectivity*]{} temporal matrix $\mathbf{W}$ as $$\mathbf{W} = (\mathbf{A} \cup \mathbf{A}^T)^\star$$ and its [*strong connectivity*]{} temporal matrix $\mathbf{S}$ as $$\mathbf{S} = \mathbf{A}^\star \cap (\mathbf{A}^\star)^T .$$ The use of the strict transitive closure instead of a transitive closure in these relations preserves the inactivity value $\mathbf{0}$ on the diagonal for all isolated nodes.
Reachability degrees
--------------------
Let $\mathbf{R} = \overline{\mathbf{A}} = \mathbf{A} \odot \mathbf{A}^\star$ be the strict reachability relation of a given network. Then the temporal vectors ${\textit{inReach}} = {\textit{inDeg}}(\mathbf{R})$ and ${\textit{outReach}} = {\textit{outDeg}}(\mathbf{R})$ contain temporal quantities counting the number of nodes: from which a given node $v$ is reachable $( {\textit{inReach}}[v] )$ / which are reachable from the node $v$ $( {\textit{outReach}}[v] )$. The results for our example network are presented in Table \[ioReach\]. For example, 8 nodes $\{ 4,5,6,7,8,9,10,11 \}$ are reachable from node 6 in the time interval $[1,5)$, and 3 nodes $\{ 4,5,6 \}$ are reachable in the time interval $[5,9)$.
1 : [(1, 9, 3)]
2 : [(1, 9, 3)]
3 : []
4 : [(1, 3, 3), (3, 9, 6)]
5 : [(1, 3, 3), (3, 9, 6)]
6 : [(1, 3, 3), (3, 9, 6)]
7 : [(1, 3, 3), (3, 5, 6), (7, 9, 5)]
8 : [(1, 3, 8), (3, 5, 11), (5, 9, 5)]
9 : [(1, 3, 8), (3, 5, 11), (5, 9, 5)]
10 : [(1, 3, 8), (3, 5, 11), (5, 9, 5)]
11 : [(1, 3, 8), (3, 5, 11), (5, 9, 5)]
12 : []
13 : [(2, 8, 3)]
14 : [(2, 8, 3)]
15 : [(2, 8, 3)]
Temporal weak connectivity
--------------------------
The function ${\textit{weakConnMat}}(\mathbf{A})$ for a given temporal network matrix $\mathbf{A}$ determines the corresponding temporal weak connectivity matrix $\mathbf{W}$. Every time slice ${{\mathcal{N}}}(t)$, $t \in {\mathcal{T}}$, of the matrix $\mathbf{W}$ is an equivalence relation that can be compactly described with the corresponding partition.
To transform the temporal equivalence matrix $\mathbf{E}$ into the corresponding temporal partition $\mathbf{p}$ we use the fact that on a given time interval equivalent (in our case weakly connected) nodes get the same value on this interval in the product of the matrix $\mathbf{E}$ with a vector computed over the combinatorial semiring $({\mathbb{N}},+,\cdot,0,1)$. We take for the vector values randomly shuffled integers from the interval $1:n$. With a very high probability the values belonging to different equivalence classes are different. This is implemented as a procedure ${\textit{eqMat2Part}}(\mathbf{E})$ (see Algorithm \[eqparTQ\]). Maybe in the future implementations we shall add a loop with the check of the injectivity of this mapping. The classes of the obtained temporal partition are finally renumbered with consecutive numbers using the function $renumPart(p)$ (see Algorithm \[renumTQ\]). The variable $C$ in the description of the function $renumPart$ is a dictionary (data structure).
For our first example network we obtain the temporal weak partition presented on the left hand side of Table \[weakC\].
${\textit{SetSemiring}}({\textit{combinatorial}})$ $v \gets {\textit{shuffle}}([ [(0,\infty,i+1)] \mbox{ \bf for } i \in 1:{\textit{nRows}}(E)])$ $p \gets {\textit{MatVecMulR}}(E,v)$ ${\textit{renumPart}}(p)$
$C \gets \{\ \}$; $q = [\ ]$ $r \gets [\ ]$ $C[c_a] \gets 1+{\textit{length}}(C)$ $r.{\textit{append}}((s_a,f_a,C[c_a]))$ $q.{\textit{append}}(r)$ $q$
Temporal strong connectivity
----------------------------
The procedure ${\textit{strongConnMat}}(\mathbf{A})$ for a given temporal network matrix $\mathbf{A}$ determines the corresponding temporal strong connectivity matrix $\mathbf{S}$. To determine the intersection of temporal network binary matrices $\mathbf{A}$ and $\mathbf{B}$ we use the function ${\textit{MatInter}}(\mathbf{A},\mathbf{B})$. Again, to get the strong connectivity partition we have to apply the function ${\textit{eqMat2Part}}$ to the strong connectivity matrix.
The time complexity of algorithms for temporal weak and strong connectivity partitions is $O(n^3 \cdot L)$.
For our first example network we obtain the temporal strong partition presented on the right hand side of Table \[weakC\]. In the library TQ both matrices and partitions are based on the strict transitive closure.
Weak partition
1 : [(1, 3, 1), (3, 5, 2), (5, 9, 3)]
2 : [(1, 3, 1), (3, 5, 2), (5, 9, 3)]
3 : [(1, 3, 1), (3, 5, 2), (5, 9, 3)]
4 : [(1, 3, 4), (3, 5, 2), (5, 9, 3)]
5 : [(1, 3, 4), (3, 5, 2), (5, 9, 3)]
6 : [(1, 3, 4), (3, 5, 2), (5, 9, 3)]
7 : [(1, 3, 4), (3, 5, 2), (5, 9, 5)]
8 : [(1, 3, 4), (3, 5, 2), (5, 9, 5)]
9 : [(1, 3, 4), (3, 5, 2), (5, 9, 5)]
10 : [(1, 3, 4), (3, 5, 2), (5, 9, 5)]
11 : [(1, 3, 4), (3, 5, 2), (5, 9, 5)]
12 : []
13 : [(2, 8, 6)]
14 : [(2, 8, 6)]
15 : [(2, 8, 6)]
Strong partition
1 : [(1, 9, 1)]
2 : [(1, 9, 1)]
3 : []
4 : [(1, 9, 2)]
5 : [(1, 9, 2)]
6 : [(1, 9, 2)]
7 : [(7, 9, 3)]
8 : [(1, 7, 4), (7, 9, 3)]
9 : [(1, 7, 4), (7, 9, 3)]
10 : [(1, 7, 4), (7, 9, 3)]
11 : [(1, 7, 4), (7, 9, 3)]
12 : []
13 : [(2, 8, 5)]
14 : [(2, 8, 5)]
15 : [(2, 8, 5)]
Temporal closeness and betweenness
==================================
Closeness and betweenness are among the traditional social network analysis indices measuring the importance of nodes [@cent]. They are somehow problematic when applied to non (strongly) connected graphs. In this section we will not consider these questions. We will only show how to compute them for non-problematic temporal graphs.
Temporal closeness
------------------
The [*output closeness*]{} of the node $v$ is defined as $$ocl(v) = \frac{n-1}{\displaystyle\sum_{u \in {\mathcal{V}}\setminus \{v\}} d_{vu}} .$$
To determine the closeness we first need to compute the matrix $\mathbf{D} = [d_{uv}]$ of geodetic distances $d_{uv}$ between the nodes $u$ and $v$. It can be obtained as a closure of the network matrix $\mathbf{A}$ over the [*shortest paths*]{} semiring $(\overline{{\mathbb{R}}_0^+},\min,+,\infty,0)$. Note that the values in the matrix $\mathbf{A}$ can be any nonnegative real numbers.
![Second example network. All unlabeled arcs have the value $[(1,9,1)]$.\[graph2\]](Geo2pic.pdf){width="75mm"}
In Figure \[graph2\] we present our second example temporal network which is an extended version of the example given in Figure 3 from [@semi].
Because a complete strict closure matrix $\mathbf{D}$ is too large to be listed we present only some of its selected entries:
D[3,1] = [(3, 7, 3), (7, 9, 5)]
D[4,6] = [(1, 4, 1), (4, 6, 5), (6, 9, 1)]
D[6,3] = [(3, 5, 6), (5, 9, 4)]
D[7,6] = [(1, 9, 4)]
To compute the vector of closeness coefficients of nodes we have to sum the temporal distances to other nodes over the combinatorial semiring. See Algorithm \[closTQ\]. While summing we replace gaps (inactivity intervals inside ${\cal T}$) with time intervals with the value infinity, using the procedure ${\textit{fillGaps}}$. The time complexity of Algorithm \[closTQ\] is $O(n^3 \cdot L)$.
The temporal closeness coefficients for our second example network are given in Table \[cloC\].
\
\# [[*type*]{}: 1 - output, 2 - all, 3 - input]{} $s \gets {\textit{startTime}}(A)$; $f \gets {\textit{finishTime}}(A)$ $n \gets {\textit{nRows}}(A)$ ${\textit{SetSemiring}}({path})$ $D \gets {\textit{MatClosure}}(A,{\textit{strict}}=True)$ ${\textit{SetSemiring}}({\textit{combinatorial}})$ $k \gets (2-|{\textit{type}}-2|)\cdot (n-1)$; ${\textit{fac}} \gets [(0,\infty,k)]$ $d \gets \mathbf{0}$ $d \gets {\textit{sum}}(d,{\textit{fillGaps}}(D[v,u],s,f))$ $d \gets {\textit{sum}}(d,{\textit{fillGaps}}(D[u,v],s,f))$ $cl[v] \gets {\textit{prod}}({\textit{fac}},{\textit{invert}}(d))$ $cl$
1 : [(1, 9, 0.4375)]
2 : [(1, 3, 0.0000), (3, 5, 0.4375),
(5, 9, 0.5833)]
3 : [(1, 3, 0.0000), (3, 7, 0.4375),
(7, 9, 0.3889)]
4 : [(1, 3, 0.0000), (3, 4, 0.4375),
(4, 6, 0.3500), (6, 7, 0.4375),
(7, 9, 0.3500)]
5 : [(1, 3, 0.0000), (3, 7, 0.4375),
(7, 9, 0.3500)]
6 : [(1, 3, 0.0000), (3, 5, 0.2917),
(5, 9, 0.3500)]
7 : [(1, 3, 0.0000), (3, 7, 0.4375),
(7, 9, 0.3500)]
8 : [(1, 3, 0.0000), (3, 5, 0.3500),
(5, 9, 0.4375)]
Temporal betweenness
--------------------
The [*betweenness*]{} of a node $v$ is defined as $$b(v) = \frac{1}{(n-1)(n-2)} \sum_{u,w \in {\mathcal{V}} \atop |\{v,u,w\}| = 3}
\frac{n_{u,w}(v)}{n_{u,w}}$$ where $n_{u,w}$ is the number of $u$-$w$ geodesics (shortest paths) and $n_{u,w}(v)$ is the number of $u$-$w$ geodesics passing through the node $v$.
Suppose that we know the matrix $$\mathbf{C} = \lbrack ( d_{u,v}, n_{u,v} ) \rbrack$$ where $d_{u,v}$ is the length of $u$-$v$ geodesics. Then it is also easy to determine the quantity $n_{u,w}(v)$: $$n_{u,w}(v) = \left\{ \begin{array}{ll}
n_{u,v} \cdot n_{v,w} \qquad & d_{u,v} + d_{v,w} = d_{u,w} \\
0 & \mbox{\rm otherwise}
\end{array} \right. .$$ This gives the following scheme of procedure for computing the nontemporal betweenness coefficients $\mathbf{b}$
compute $\mathbf{C}$ $r \gets 0$ $r \gets r + n[u,v]\cdot n[v,w]/n[u,w]$ $b[v] \gets r / ((n-1)\cdot (n-2))$
In [@semi] it is shown that the matrix $\mathbf{C}$ can be obtained by computing the closure of the network matrix over the [*geodetic semiring*]{} $(\overline{{\mathbb{N}}}^2, \oplus, \odot,$ $ (\infty,0), (0,1))$, where $\overline{{\mathbb{N}}} = {\mathbb{N}}\cup \{ \infty \}$ and we define [*addition*]{} $\oplus$ with $$(a,i) \oplus (b,j) = ( \min(a,b), \left\{ \begin{array}{ll}
i & a < b \\ i+j \quad & a = b \\ j & a > b
\end{array} \right. )$$ and [*multiplication*]{} $\odot$ with: $$(a,i) \odot (b,j) = (a+b,i\cdot j) .$$
To compute the geodetic closure we first transform the network temporal adjacency matrix $\mathbf{A}$ to a matrix $\mbox{\bf G} = \lbrack (d,n)_{u,v} \rbrack$ which has for entries pairs defined by $$(d,n)_{u,v}(t) = \left\{ \begin{array}{ll}
(1,1) \quad & \exists l \in {\mathcal{L}}: l(u,v) \land t \in T(l) \\
{\raisebox{-.025em}{\includegraphics[height=.7em]{command.pdf}}}& \mbox{otherwise}
\end{array} \right.$$ where $d$ is the length of a geodesic and $n$ is the number of geodesics from $u$ to $v$. In temporal networks the distance $d$ and the counter $n$ are temporal quantities.
The presented scheme adapted for computing the temporal betweenness vector is implemented in TQ as the function ${\textit{betweenness}}(A)$. First we compute its strict geodetic closure $\mathbf{C}$ over the geodetic semiring. We present only some of its selected entries for our second example network:
C[1,7] = [(1, 9, (3, 4))]
C[2,2] = [(1, 3, (4, 4)), (3, 4, (4, 6)),
(4, 5, (4, 5)), (5, 9, (2, 1))]
C[4,6] = [(1, 4, (1, 1)), (4, 6, (5, 3)),
(6, 9, (1, 1))]
C[5,5] = [(1, 9, (1, 1))]
C[6,3] = [(3, 5, (6, 2)), (5, 9, (4, 1))]
C[7,6] = [(1, 3, (4, 2)), (3, 4, (4, 6)),
(4, 6, (4, 3)), (6, 7, (4, 6)),
(7, 9, (4, 2))]
For example, the value $\mathbf{C}[4,6]$ reflects the facts that an arc exists from node 4 to node 6 in time intervals $[1,4)$ and $[6,9)$; and in the time interval $[4,6)$ they are connected with 3 geodesics of length 5: $(4,7,8,2,5,6)$, $(4,7,1,3,5,6)$, $(4,7,1,2,5,6)$.
We continue and using the combinatorial semiring we compute the temporal betweenness vector $\mathbf{b}$. The specificity of temporal quantities $d[u,v]$ and $n[u,v]$ is considered in the auxiliary function ${\textit{between}}$ that implements the temporal version of the statement
[**if**]{} $ d[u,w]=d[u,v]+d[v,w]$ [**then**]{}\
$r \gets r + n[u,v]\cdot n[v,w]/n[u,w]$
from the basic betweenness algorithm. Again we apply the merging scheme. The time complexity of the procedure ${\textit{betweenness}}$ is $O(n^3 \cdot L)$.
The temporal betweenness coefficients for our second example network are presented in Table \[betwC\].
1 : [(3, 4, 0.2500), (4, 6, 0.2754),
(6, 7, 0.2500), (7, 9, 0.1429)]
2 : [(1, 3, 0.3452), (3, 4, 0.4048),
(4, 6, 0.4187), (6, 7, 0.4048),
(7, 9, 0.6071)]
3 : [(1, 3, 0.0595), (3, 4, 0.0952),
(4, 6, 0.1052), (6, 7, 0.0952),
(7, 9, 0.0595)]
4 : [(1, 3, 0.1667), (3, 4, 0.2500),
(4, 5, 0.1762), (5, 6, 0.1048),
(6, 9, 0.1786)]
5 : [(1, 3, 0.1667), (3, 4, 0.2500),
(4, 5, 0.3476), (5, 6, 0.2762),
(6, 9, 0.1786)]
6 : [(1, 3, 0.1190), (3, 4, 0.0952),
(4, 6, 0.0544), (6, 7, 0.0952),
(7, 9, 0.1786)]
7 : [(1, 3, 0.1190), (3, 4, 0.4048),
(4, 5, 0.4694), (5, 6, 0.3266),
(6, 7, 0.2619), (7, 9, 0.1786)]
8 : [(1, 3, 0.3095), (3, 4, 0.2500),
(4, 6, 0.2484), (6, 7, 0.2500),
(7, 9, 0.5238)]
Temporal PathFinder
===================
The Pathfinder algorithm was proposed in the eighties [@PF88; @PF90] for the simplification of weighted networks – it removes from the network all links that do not satisfy the (generalized) triangle inequality – if for a weighted link there exists a shorter path connecting its endnodes then the link is removed. The basic idea of the Pathfinder algorithm is simple. It produces a network $\mbox{PFnet}(\mathbf{W},r,q) = ({\mathcal{V}},{\mathcal{L}}_{PF})$ determined by the following scheme of procedure
compute $\mathbf{W}^{(q)}$; ${\mathcal{L}}_{PF} \gets \emptyset$; ${\mathcal{L}}_{PF} \gets {\mathcal{L}}_{PF} \cup \{ e \}$
where $\mathbf{W}$ is a network ***dissimilarity*** matrix and $\mathbf{W}^{(q)} = \bigoplus_{i=1}^q \mathbf{W}^i =
(\mathbf{1} \oplus \mathbf{W})^q$ is the matrix of the values of all walks of length at most $q$ computed over the [*Pathfinder*]{} semiring $(\overline{{\mathbb{R}}^+_0},\oplus,{\mathop{\raisebox{-1.5pt}{\mbox{$\Box$\kern-.55em\raisebox{2.5pt}{{\tiny $r$}}\kern2.9pt}}}},\infty,0)$ with $a {\mathop{\raisebox{-1.5pt}{\mbox{$\Box$\kern-.55em\raisebox{2.5pt}{{\tiny $r$}}\kern2.9pt}}}}b = \sqrt[r]{a^r+b^r}$ and $a \oplus b = \min(a,b)$. The value of $w_{uv}(q)$ in the matrix $\mathbf{W}^{(q)}$ is equal to the value of all walks of length at most $q$ from the node $u$ to the node $v$.
The scheme of Pathfinder is implemented as the function ${\textit{pathFinder}}$. The temporal version of the statement
[**if**]{} $\mathbf{W}^{(q)}[u,v] = \mathbf{W}[u,v]$ [**then**]{} ${\mathcal{L}}_{PF} := {\mathcal{L}}_{PF} \cup \{ e \}$
is implemented in the function ${\textit{PFcheck}}$ using the merging scheme.
The function ${\textit{MatPower}}(A,k)$ computes the $k$-th power of the matrix $\mathbf{A}$.
The time complexity of Algorithm \[pfTQ\]$+$\[pfcTQ\] is $O(L \cdot n^3 \cdot \log q)$ [@PFc].
$n \gets {\textit{nRows}}(W)$; ${\textit{SetSemiring}}({\textit{pathfinder}},r,q)$ $Z \gets {\textit{MatClosure}}(W)$ $\ Z \gets {\textit{MatPower}}({\textit{MatSetDiag}}(W,\mathbf{1}),q)$ $PF[u,v] \gets {\textit{PFcheck}}(W[u,v],Z[u,v])$ $PF$
$a$ $a$ $c \gets [\ ]$ $(s_a,f_a,v_a) \gets {\textit{get}}(a)$; $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ $(s_a,f_a,v_a) \gets {\textit{get}}(a)$ $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ $s_c \gets \max(s_a,s_b)$; $f_c \gets \min(f_a,f_b)$ $c.{\textit{append}}((s_c,f_c,v_a))$ $(s_a,f_a,v_a) \gets {\textit{get}}(a)$ $(s_b,f_b,v_b) \gets {\textit{get}}(b)$ ${\textit{standard}}(c)$
![Pathfinder example.\[PFex\]](PFa2.pdf "fig:"){width="75mm"}\
![Pathfinder example.\[PFex\]](PFc2.pdf "fig:"){width="75mm"}
The bottom network in Figure \[PFex\] presents the Pathfinder skeleton $\mbox{PFnet}({{\mathcal{N}}},1,\infty)$ of a network ${{\mathcal{N}}}$ presented in the top part of the same figure. Because $r=1$ a link $e$ is removed if there exists a path, connecting its initial node to its terminal node, with the value (sum of link values) smaller than the value of the link $e$. The arc $(1,2)$ is removed because $3 = v(1,2) > v(1,3)+v(3,2) = 2$. The arc $(1,6)$ is removed in the time interval $[5,9)$ because in this interval $5 = v(1,6) > v(1,3)+v(3,4)+v(4,5) +v(5,6) = 4$.
September 11th Reuters terror news
==================================
The Reuters terror news network was obtained from the CRA (Centering Resonance Analysis) networks produced by Steve Corman and Kevin Dooley at Arizona State University. The network is based on all the stories released during 66 consecutive days by the news agency Reuters concerning the September 11 attack on the U.S., beginning at 9:00 AM EST 9/11/01. The nodes of this network are important words (terms). There is an edge between two words iff they appear in the same utterance (for details see the paper [@CRA]). The weight of an edge is its frequency. The network has $n = 13332$ nodes (different words in the news) and $m = 243447$ edges, 50859 with value larger than 1. There are no loops in the network.
The Reuters terror news network was used as a case network for the Viszards visualization session on the Sunbelt XXII International Sunbelt Social Network Conference, New Orleans, USA, 13-17. February 2002.
We transformed the Pajek version of the network into the Ianus format used in TQ. To identify important terms we computed their aggregated frequencies and extracted the subnetwork of the 50 most frequently used (during 66 days) nodes. They are listed in Table \[terrF\].
Trying to draw this subnetwork it turns out to be almost a complete graph. To obtain something readable we removed all temporal edges with a value smaller than 10. The corresponding underlying graph is presented in Figure \[terrEx\]. The isolated nodes were removed.
For each of the 50 nodes we determined its temporal activity and drew it. By visual inspection we identified 6 typical activity patterns – types of terms (see Figure \[terrTy\]). For all charts in the figure the displayed values are in the interval $[0,200]$ – the largest activity value for the term Wednesday is larger than 200.
n term $\Sigma$freq n term $\Sigma$freq
---- ------------------- -------------- ---- ----------- --------------
1 united\_states 15000 26 terrorism 2212
2 attack 10348 27 day 2128
3 taliban 6266 28 week 2017
4 people 5286 29 worker 1983
5 afghanistan 5176 30 office 1967
6 bin\_laden 4885 31 group 1966
7 new\_york 4832 32 air 1962
8 pres\_bush 4506 33 minister 1919
9 washington 4047 34 time 1898
10 official 3902 35 hijack 1884
11 anthrax 3563 36 strike 1818
12 military 3394 37 afghan 1775
13 plane 3078 38 flight 1775
14 world\_trade\_ctr 3006 39 tell 1746
15 security 2906 40 terrorist 1745
16 american 2825 41 airport 1741
17 country 2794 42 pakistan 1714
18 city 2689 43 tower 1685
19 war 2679 44 bomb 1674
20 tuesday 2635 45 new 1650
21 pentagon 2620 46 buildng 1634
22 force 2516 47 wednesday 1593
23 government 2380 48 nation 1589
24 leader 2375 49 police 1587
25 world 2213 50 foreign 1558
: 50 most frequent terms in the Terror news network.\[terrF\]
The *primary* terms are the terms with a very high frequency of appearance in the first week after September 11th and smaller, slowly declining values in the following period. The representative of this group in Figure \[terrTy\] is **hijack** and other members are: airport, american, attack, city, day, flight, nation, New York, official, Pentagon, people, plane, police, president Bush, security, tower, United States, Washington, world, World Trade center. These are the terms describing the event.
The *secondary* terms are a reaction to the event. There are no big changes in their values. We identified three subgroups: a) *slowly declining* represented with **bin Laden** (country, foreign, government, military, minister, new, Pakistan, tell, terrorism, terrorist, time, war, week); b) *stationary* represented with **taliban** (afghan, Afghanistan, force, group, leader); and c) *occasional* with several peaks, represented with **bomb** (air, building, office, strike, worker).
There are three special patterns – two *periodic* **Wednesday** and Tuesday; and one *episodic* **anthrax**.
[l]{} hijack :\
\
bin Laden :\
\
taliban :\
\
bomb :\
\
Wednesday :\
\
anthrax :\
To consider in a measure of importance of the node $u \in {\mathcal{V}}$ also the node’s position in the network we constructed the attraction coefficient $\mbox{att}(u)$.
Let $\mathbf{A} = [ a_{uv}]$ be a network matrix of temporal quantities with positive real values. We define the [*node activity*]{} $\mbox{act}(u)$ as (see Section \[activ\]) $$\mbox{act}(u) = \mbox{act}(\{u\},{\mathcal{V}}\setminus \{u\}) = \sum_{v \in {\mathcal{V}}\setminus \{u\}} a_{uv} .$$ Then the [*attraction*]{} of the node $u$ is defined as $$\mbox{att}(u) = \frac{1}{\Delta} \sum_{v \in {\mathcal{V}}\setminus \{u\}}
\frac{a_{vu}}{\mbox{act}(v)} .$$ Note that the fraction $\frac{a_{vu}}{\mbox{act}(v)}$ is measuring the proportion of the activity of the node $v$ that is shared with the node $u$.
From $0 \leq \frac{a_{vu}}{\mbox{act}(v)} \leq 1$ and $\deg(v)=0 \Rightarrow a_{vu}=0$ it follows that $$\sum_{v \in {\mathcal{V}}\setminus \{u\}}
\frac{a_{vu}}{\mbox{act}(v)} \leq \deg(u) \leq \Delta$$ where $\Delta$ denotes the maximum degree. Therefore we have $0 \leq \mbox{att}(u) \leq 1$, for all $u \in {\mathcal{V}}$.
The maximum possible attraction value 1 is attained exactly for nodes: a) in an undirected network: that are the root of a star; b) in a directed network: that are the only out-neighbors of their in-neighbors – the root of a directed in-star.
We computed the temporal attraction and the corresponding aggregated attraction values for all the nodes in our network. We selected 30 nodes with the largest aggregated attraction values. They are listed in Table \[terrA\]. Again we visually explored them. In Figure \[terrAt\] we present temporal attraction coefficients for the 6 selected terms. For all charts in the figure the displayed attraction values are in the interval $[0,0.2]$.
Comparing on the common terms (taliban, bomb, anthrax) the activity charts in Figure \[terrTy\] with the corresponding attraction charts in Figure \[terrAt\] we see that they are “correlated” (obviously $\mbox{act}(a;t) = 0$ implies $\mbox{att}(a;t) = 0$), but different in details.
For example, the terms taliban and bomb have small attraction values at the beginning of the time window – the terms were disguised by the primary terms. On the other hand, the terms taliban and Kabul get increased attraction towards the end of the time window.
n term $\Sigma$att n term $\Sigma$att
---- ---------------- ------------- ---- ------------ -------------
1 united\_states 12.216 16 war 2.758
2 taliban 7.096 17 force 2.596
3 attack 7.070 18 new\_york 2.590
4 afghanistan 5.142 19 government 2.496
5 people 5.023 20 day 2.338
6 bin\_laden 4.660 21 leader 2.305
7 anthrax 4.601 22 terrorism 2.202
8 pres\_bush 4.374 23 time 2.182
9 country 3.317 24 group 2.072
10 washington 3.067 25 afghan 2.040
11 security 2.939 26 world 1.995
12 american 2.922 27 week 1.961
13 official 2.831 28 pakistan 1.943
14 city 2.798 29 letter 1.866
15 military 2.793 30 new 1.851
: 30 most attractive terms in the Terror news network.\[terrA\]
[l]{} pres Bush :\
\
Pakistan :\
\
taliban :\
\
Kabul :\
\
bomb :\
\
anthrax :\
Conclusions
===========
In the paper we proposed an algebraic approach to the “deterministic” analysis of temporal networks based on temporal quantities and presented algorithms for the temporal variants of basic network analysis measures and concepts. We expect that the support for many temporal variants of other network analysis notions can be developed in similar ways. Our results on temporal variants of eigen values/vectors based indices (Katz, Bonacich, hubs and authorities, page rank) are presented in a separate paper [@tcent].
The proposed approach is an alternative to the traditional cross sectional approach based on time slices. Its main advantages are:
- the data and the results are expressed using temporal quantities that are natural descriptions of properties changing through time;
- the user does not need to be careful about the intervals on which the time slices are determined – exactly the right intervals are selected by the merging (sub)operations. This also improves, on average, the efficiency of the proposed algorithms.
All the described algorithms (and some others) are implemented in a Python library TQ (Temporal Quantities) available at the page [@TQ]. We started to develop a program Ianus that will provide a user-friendly (Pajek like) access to the capabilities of the TQ library.
The main goal of the paper was to show: it can be done. Therefore we based the current version of the library TQ on a matrix representation of temporal networks as it is presented in the paper. For this representation most of the network algorithms have the time complexity of $O(n^3 \cdot L)$ and the space complexity of $O(n^2 \cdot L)$. This implies that their application is limited to networks of moderate size (up to some thousands of nodes). Large networks are usually sparse. On this assumption more efficient algorithms can be developed based on a graph (sparse matrix) representation – one of the directions of our future research.
In a description of a temporal network ${\mathcal{N}}$ we can consider also a transition time or latency $\tau \in {\mathcal{W}}$: $\tau(l,t)$ is equal to the time needed to traverse the link $l$ starting at the instant $t$. Problems considering latency are typical for operations research but could be important, when such data are available, also in social network analysis [@moody; @Xuan; @Rout; @TVGsur; @alge]. The analysis of temporal networks considering also the latency seems a much harder task – for example, in such temporal networks the strongly connected components problem is NP-complete [@SCC].
The results obtained from temporal procedures are relatively large. To identify interesting elements we used in the paper the aggregated values and the visualization of selected elements. Additional tools for browsing and presenting the results should be developed.
The work was supported in part by the ARRS, Slovenia, grant J5-5537, as well as by a grant within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation.
[99]{}
Batagelj, V., 1994. Semirings for social networks analysis. Journal of Mathematical Sociology, 19(1), 53-68.
Batagelj, V., 2009. Social Network Analysis, Large-Scale. R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer, 8245-8265
Batagelj, V, Cerinšek, M, 2013. On bibliographic networks. Scientometrics 96 (3), 845-864.
Batagelj, V, Praprotnik, S, 2014. TQ library home page. <http://vladowiki.fmf.uni-lj.si/doku.php?id=tq> .
Bhadra, S., Ferreira, A., 2003. Complexity of Connected Components in Evolving Graphs and the Computation of Multicast Trees in Dynamic Networks. In ADHOC-NOW, LNCS 2865, Springer, 259-270.
Cantos-Mateos, G., Zulueta, M.Á., Vargas-Quesada, B., Chinchilla-Rodríguez, Z., 2014. Estudio evolutivo de la investigación española con células madre. Visualización e identificación de las principales líneas de investigación. El Profesional de la Información, 23(3), 259-271.
Carré, B., 1979. Graphs and Networks. Clarendon, Oxford.
Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N., 2012. Time-varying graphs and dynamic networks. International Journal of Parallel, Emergent and Distributed Systems, 27(5), 387-408.
Corman, S.R., Kuhn, T., McPhee, R.D., Dooley, K.J., 2002. Studying complex discursive systems: Centering resonance analysis of communication. Human Communication Research, 28(2), 157-206.
Fletcher, J.G., 1980. A more general algorithm for computing closed semiring costs between vertices of a directed graph. CACM 23, 350-351.
Franzosi, R., 1997. Mobilization and Counter-Mobilization Processes: From the “Red Years” (1919-20) to the “Black Years” (1921-22) in Italy. A New Methodological Approach to the Study of Narrative Data. Theory and Society, 26(2-3), 275-304.
Freeman, L.C., 1978. Centrality in Social Networks; Conceptual Clarification. Social Networks 1, 215-239.
George, B., Kim, S., Shekhar, S., 2007. Spatio-temporal Network Databases and Routing Algorithms: A Summary of Results. in D. Papadias, D. Zhang, and G. Kollios (Eds.): SSTD 2007, LNCS 4605, Springer-Verlag, Berlin, Heidelberg, 460-477.
Gondran, M., Minoux, M., 2008. Graphs, Dioids and Semirings – New Models and Algorithms. Springer.
Guerrero-Bote, V.P., Zapico-Alonso, F., Espinosa-Calvo, M.E., Crisóstomo, R.G., de Moya-Anegón, F., 2006. Binary Pathfinder: An improvement to the Pathfinder algorithm. Information Processing and Management, 42 (6), 1484-1490.
Gulyás, L., Kampis, G., Legendi, R.O., 2013. Elementary models of dynamic networks. The European Physical Journal, Special Topics 222, 1311–1333.
Holme, P., Saramäki, J., 2012. Temporal networks. Physics Reports 519 (3), 97–125.
Holme, P., Saramäki, J. (Eds.), 2013. Temporal Networks. Understanding Complex Systems. Springer.
Kim, H., Yoon, J.W., Crowcroft, J., 2012. Network analysis of temporal trends in scholarly research productivity. Journal of Informetrics **6**, 97– 110
Kolaczyk, E.D., 2009. Statistical Analysis of Network Data: Methods and Models. New York: Springer.
Kontoleon, N., Falzon, L., Pattison, P., 2013. Algebraic structures for dynamic networks. Journal of Mathematical Psychology, **57**(6), 310–319.
Moody, J., 2002. The Importance of Relationship Timing for Diffusion. Social Forces, **81**(1), 25-56.
Moody, J., McFarland, D., Bender-deMoll, S., 2005. Dynamic Network Visualization. American Journal of Sociology, **110**(4), 1206-1241.
de Nooy, W., Mrvar, A., Batagelj, V., 2012. *Exploratory Social Network Analysis with Pajek* (Structural Analysis in the Social Sciences), revised and expanded second edition. Cambridge University Press, Cambridge.
Praprotnik, S., Batagelj, V., 2015. Spectral centrality measures in temporal networks. Accepted in Ars Mathematica Contemporanea.
Riordan, J., 1958. Introduction to combinatorial analysis. New York: Wiley.
Schvaneveldt, R. W., Dearholt, D. W., Durso, F. T., 1988. Graph theoretic foundations of Pathfinder networks. Comput. Math. Applic. **15**(4), 337-345.
Schvaneveldt, R.W. (Ed.), 1990. Pathfinder Associative Networks: Studies in Knowledge Organization. Norwood, NJ: Ablex.
Xuan, B.B., Ferreira, A., Jarry, A., 2003. Computing shortest, fastest, and foremost journeys in dynamic networks. International Journal of Foundations of Computer Science, **14**(2), 267–285.
|
---
abstract: 'We consider strangeness correlations of the EPR type in [$K^0 \bar K^0$]{} pairs created in a $J^{PC} = 1^{--}$ state as a function of time under the hypothesis that spontaneous decoherence takes place. We parameterize the degree of decoherence by a factor $(1-\zeta)$ which multiplies the quantum-mechanical interference terms occurring in the amplitudes for like and unlike strangeness events and discuss the dependence of this procedure on the basis chosen in the $K^0$–$\bar{K}^0$ space to which the interference terms correspond. Consequently, all statements about the “decoherence parameter” $\zeta$ inferred from experimental data are basis-dependent as well. We illustrate this point by estimating the value of $\zeta$ for the two bases $\{K_L, K_S\}$ and $\{K^0, \bar{K}^0\}$ with the help of recent data of the CPLEAR experiment.'
address: |
Institut für Theoretische Physik, Universität Wien\
Boltzmanngasse 5, A-1090 Vienna, Austria
author:
- 'R.A. Bertlmann, W. Grimus and B.C. Hiesmayr'
title: 'Quantum mechanics, Furry’s hypothesis and a measure of decoherence in the [$K^0 \bar K^0$]{} system'
---
Introduction
============
Since the first formulation of the EPR paradox in 1935 [@EPR] tests of quantum mechanics (QM) against local realistic (hidden variable) theories are of great interest. According to QM, if a pair of particles is created by any kind of interaction in an entangled state, the two particle wave function retains its non-separable character even if the particles are space-like separated. The famous inequality of J.S. Bell [@bell] made it possible to discriminate quantitatively between the predictions of QM and of local realistic theories. Numerous experiments have been performed since, in particular, using entangled photon states, which all confirmed QM (see, e.g., Refs. [@aspect; @kwiat]). On the particle physics side, entangled [$K^0 \bar K^0$]{} and $B^0 \bar{B}^0$ states are suitables objects for the study of EPR-like correlations [@lee; @six; @selleri; @datta]. The presence of the QM interference term can also be deduced from existing data on $B^0 \bar{B}^0$ systems produced in the decay of $\Upsilon(4S)$ as was demonstrated in Refs. [@BG97; @dass; @BG98]. Recently, an important experiment of the CPLEAR Collaboration using strangeness correlations of [$K^0 \bar K^0$]{}pairs created by annihilation of $p \bar p$ pairs at rest showed impressively the non-separability of the QM [$K^0 \bar K^0$]{} wave function in this situation [@CPLEAR98]. This experiment excluded a spontaneous wave function factorization immediately after the [$K^0 \bar K^0$]{} creation with a CL of more than 99.99%.
In this paper we want to draw the attention to the fact that the notion of spontaneous factorization depends on the basis chosen in the two systems which, after the creation of the particle pairs, are non-interacting. The reason for this is that a basis has to be fixed with respect to which the process of spontaneous factorization takes place [@furry] (see also Refs. [@dass; @BG98]). In our case of interest the two non-interacting systems are given by the two neutral kaons moving into opposite directions in the rest frame of the [$K^0 \bar K^0$]{} source. In this paper we discuss spontaneous factorization and decoherence of the [$K^0 \bar K^0$]{} wave function created by a $J^{PC} = 1^{--}$ state as a function of the basis chosen to describe the non-interacting neutral kaons on each side of the source. We modify the quantum-mechanical interference term of the entangled 2-kaon state by multiplying it with $(1-\zeta)$. Thus we change QM expressions in such a way that the decoherence parameter $\zeta$ [@eberhard] parameterizes the deviation from QM (corresponding to $\zeta = 0$) and gives a measure how far the total system is away from total decoherence or spontaneous factorization ($\zeta
= 1$). As an observable relevant in this discussion we use the time-dependent asymmetry $A(t_r,t_l)$ of like and unlike strangeness events measured by the CPLEAR Collaboration [@CPLEAR98] and discussed in their paper. The proper times at which the strangeness of the neutral kaons is measured at the right and the left of the [$K^0 \bar K^0$]{}source at rest are denoted by $t_r$ and $t_l$, respectively. We investigate the dependence of $A(t_r,t_l)$ on the basis chosen in the 2-dimensional $K^0$–$\bar{K}^0$ space and on the decoherence parameter $\zeta$. Then we use the results on $A(t_r,t_l)$ of the CPLEAR experiment [@CPLEAR98] to estimate the value of the decoherence parameter for two basis choices: first we consider the basis given by $\{K_L, K_S\}$, which was used in Ref. [@CPLEAR98] to compare spontaneous factorization of the wave function with the result of QM, and, secondly, the basis $\{K^0, \bar{K}^0\}$. As we will see, numerically the two estimates of $\zeta$ will be quite different and, therefore, the statement that spontaneous factorization is excluded with a CL of more than 99.99% is valid only for the $\{K_L$, $K_S\}$ basis in the factorization process. Also we will see that this basis is the only one which leads to a vanishing asymmetry upon spontaneous factorization.
Let us mention that strangeness in the [$K^0 \bar K^0$]{} system is a quantity analogous to polarization in the well-known entangled two-photon systems. However, strangeness is time-dependent due to $K^0 \leftrightarrow \bar K^0$ oscillations and it has been shown that because of the actual values of the decay constants of $K_L$ and $K_S$ it is not possible to find an experimental set-up for violating an inequality of the Bell type [@ghirardi91; @ghirardi92; @trixi] by using the strangeness of the kaons. This is different from situations where Bell inequalities for kaon systems can be tested with kaon decay products [@bellineq]. Finally, we want to stress that the [$K^0 \bar K^0$]{} system considered here is a further example exhibiting QM interference effects over macroscopic distances with the neutral kaons being separated several centimeters when their strangeness is measured in the CPLEAR experiment.
A formalism for the [$K^0 \bar K^0$]{} system in an arbitrary basis and with modified interference terms {#theoretical basis choice}
========================================================================================================
A [$K^0 \bar K^0$]{} pair, created in a $J^{PC}=1^{--}$ quantum state and thus antisymmetric under C and P, is described at proper times $t_r=t_l=0$ by the an entangled state: $$\label{1}
| \psi (0,0) \rangle =\frac{1}{\sqrt{2}}
\left\{ | K^0 \rangle_r \otimes\! | \bar K^0 \rangle _l -
| \bar K^0 \rangle _r \otimes | K^0 \rangle _l \right\} \,.$$ Here $r$ and $l$ denote the neutral kaons on the right and left side of the source. On the other hand, the physical states which decay are the long and short-lived states: $$\begin{aligned}
\label{2}
| K_S \rangle & = & p | K^0 \rangle - q | \bar K^0 \rangle \,,
\nonumber\\
| K_L \rangle & = & p | K^0 \rangle + q | \bar K^0 \rangle \,,\end{aligned}$$ where the normalization $|p|^2 + |q|^2 = 1$ is understood. Rewriting the initial state (\[1\]) in the basis $\{K_L, K_S\}$, we obtain $$\label{3}
| \psi (0,0) \rangle = \frac{1}{2 \sqrt{2}\, p q}
\left\{ | K_S \rangle_r \otimes | K_L \rangle_l -
| K_L \rangle_r \otimes | K_S \rangle_l \right\} \,.$$ Of course, with respect to QM the states (\[1\]) and (\[3\]) are identical and will lead to equal probabilities. But if we modify interference terms by introducing the decoherence parameter $\zeta$ the derived probabilities depend on the basis chosen [@furry], a feature discussed already for the analogously entangled $B^0 \bar B^0$ state in Refs. [@BG97; @dass; @BG98].
To develop a general formalism for the neutral kaons we take an arbitrary basis [@BG98] $$\begin{aligned}
\label{3.5}
| k_j \rangle = S_{1j} | K^0 \rangle + S_{2j} | \bar
K^0 \rangle \quad \mathrm{with} \quad j=1,2,\end{aligned}$$ where $S_{ij}$ are elements of an arbitrary invertible matrix $S$. Thus the two special cases considered above correspond to $S=\mathbf{1}$ and $S=M$ with $$M =
\left( \begin{array}{lr}
p & p\\
q & -q\\
\end{array} \right) \,,$$ respectively.
According to the Wigner–Weisskopf approximation the decaying states evolve exponentially in time: $$\begin{aligned}
| K_S (t)\rangle &=& g_S(t) | K_S \rangle \,, \nonumber\\
| K_L (t)\rangle &=& g_L(t) | K_L \rangle \end{aligned}$$ with $$g_{S,L}(t) = e^{-i \lambda_{S,L} t} \quad \mathrm{and} \quad
\lambda_{S,L} = m_{S,L} - \frac{i}{2} \Gamma_{S,L} \,.$$ The subsequent time evolution for $K^0$ and $\bar K^0$ is therefore given by $$\begin{aligned}
| K^0(t) \rangle &=&
g_{+}(t) | K^0 \rangle + \frac{q}{p} g_{-}(t) | \bar K^0 \rangle \,,
\nonumber\\
| \bar K^0(t) \rangle &=&
\frac{p}{q} g_{-}(t) | K^0 \rangle + g_{+}(t) | \bar K^0 \rangle \end{aligned}$$ with $$g_{\pm}(t)= \frac{1}{2} \left[ \pm e^{-i \lambda_S t} + e^{-i \lambda_L t}
\right] \,.$$ Defining the time evolution matrix by $$T(t) \equiv M \hat g(t) M^{-1} S \quad \mathrm{with} \quad
\hat g = \mathrm{diag} (g_L,g_S)$$ and $$M \hat g(t) M^{-1} =
\left( \begin{array}{cc}
g_{+}(t) & \frac{p}{q} g_-(t)\\
\frac{q}{p} g_-(t) & g_+(t)
\end{array} \right)$$ we can write the time evolution of the basis vectors (\[3.5\]) as $$| k_j(t) \rangle =
T_{1j}(t) | K^0 \rangle + T_{2j}(t) | \bar K^0 \rangle
\quad \mathrm{for} \quad j=1,2.$$ Then the initial state given by (\[1\]) or (\[3\]), written in terms of an arbitrary basis (\[3.5\]) as $$| \psi(0,0) \rangle = \frac{1}{\sqrt{2} \det S}
\left\{ | k_1 \rangle_r \otimes | k_2 \rangle_l -
| k_2 \rangle_r \otimes | k_1 \rangle_l \right\} \,,$$ has the time evolution $$\label{timeevo}
| \psi(t_r,t_l) \rangle = \frac{1}{\sqrt{2} \det{S}}
\left\{ T_{i1}(t_r) T_{j2}(t_l) - T_{i2}(t_r) T_{j1}(t_l) \right\}
| \hat k_i \rangle \otimes | \hat k_j \rangle \,,$$ where we have defined $$| \hat k_1 \rangle \equiv | K^0 \rangle , \quad
| \hat k_2 \rangle \equiv | \bar K^0 \rangle \,.$$ In Eq.(\[timeevo\]) a summation over equal indices is understood. At this point we want to stress that QM is invariant under the basis manipulations we have made up to now, as has been explicitely demonstrated in the case of the $B^0 \bar B^0$ system in Ref. [@BG98].
Now we are ready to modify QM. The class of observables we are interested in is the probability that the state $\psi$ evolves into final states $f_1$ and $f_2$, which are measured at proper times $t_r$ on the right side and $t_l$ on the left side of the [$K^0 \bar K^0$]{} source. This probability is given by $$\begin{aligned}
& &
\left| \langle f_1 \otimes f_2 |
\psi(t_r,t_l) \rangle \right|^2 = \frac{1}{2 | \det{S} |^2}
\nonumber\\
& & \hphantom{AA}
\times \left\{
\left| \langle f_1 | k_1(t_r) \rangle \right|^2
\left| \langle f_2 | k_2(t_l) \rangle \right|^2 +
\left| \langle f_1 | k_2(t_r) \rangle \right|^2
\left| \langle f_2 | k_1(t_l) \rangle \right|^2 \right. \nonumber\\
& & \hphantom{AA}
-2\: (1-\zeta) \left. \mathrm{Re} \left(
\langle f_1 | k_1(t_r) \rangle ^*
\langle f_2 | k_2(t_l) \rangle ^*
\langle f_1 | k_2(t_r) \rangle
\langle f_2 | k_1(t_l) \rangle \right) \right\} \,,
\label{ff}\end{aligned}$$ where the usual QM interference term has been modified by the factor $1-\zeta$. QM corresponds to $\zeta =0$ and for $\zeta = 1$ no interference term is present, corresponding to Furry’s hypothesis of spontaneous factorization [@furry] (called “method A” in his paper). The aim of our investigation is to find the range of $\zeta$ allowed by present experimental data. We will take the results of experiment of the CPLEAR Collaboration [@CPLEAR98] where they measured four final states of the neutral K-meson pairs $(f_1,f_2)=(K^0,K^0)$, $(\bar K^0,\bar K^0)$, $(K^0,\bar K^0)$, $(\bar K^0, K^0)$. With these states we obtain the following probabilities.\
**Like-strangeness events:** final states $(K^0, K^0)$ and $(\bar K^0, \bar K^0)$ $$\begin{aligned}
P^S_\zeta(K^0, t_r; K^0, t_l) &=&
\frac{1}{8} \left| \frac{p}{q} \right|^2 P_\mathrm{like}^\mathrm{QM}(t_r,t_l)
+ \zeta \frac{1}{| \det{S}|^2}
\mathrm{Re} \left\{
(T_{11}^*(t_r) T_{12}(t_r)) (T_{11}^*(t_l) T_{12}(t_l))^*
\right\} \,, \nonumber\\
P^S_\zeta(\bar K^0, t_r; \bar K^0, t_l) &=&
\frac{1}{8} \left| \frac{q}{p} \right|^2 P_\mathrm{like}^\mathrm{QM}(t_r,t_l)
+ \zeta \frac{1}{| \det{S}|^2}
\mathrm{Re} \left\{
(T_{22}^*(t_r) T_{21}(t_r))^* (T_{22}^*(t_l) T_{21}(t_l))
\right\} \,. \nonumber\\
&&\label{3.3}\end{aligned}$$ **Unlike-strangeness events:** final states $(K^0, \bar K^0)$ and $(\bar K^0, K^0)$ $$\begin{aligned}
P^S_\zeta(K^0, t_r;\bar K^0, t_l) &=&
\frac{1}{8} P_\mathrm{unlike}^\mathrm{QM}(t_r,t_l)
+ \zeta \frac{1}{| \det{S}|^2}
\mathrm{Re} \left\{
T_{11}^*(t_r) T_{12}(t_r) T_{22}^*(t_l) T_{21}(t_l)
\right\} \,, \nonumber\\
P^S_{\zeta}(\bar K^0, t_r; K^0, t_l) &=&
\frac{1}{8} P_\mathrm{unlike}^\mathrm{QM}(t_r,t_l)
+ \zeta \frac{1}{| \det{S}|^2}
\mathrm{Re} \left\{
T_{22}^*(t_r) T_{21}(t_r) T_{11}^*(t_l) T_{12}(t_l)
\right\} \,.
\label{3.4}\end{aligned}$$ The QM probabilities in Eqs.(\[3.3\]) and (\[3.4\]), apart from CP-violating effects, are given by [@six] $$\begin{aligned}
P_\mathrm{like}^\mathrm{QM} &=&
2\, e^{-\Gamma(t_r+t_l)}
\left\{ \cosh \left( {\textstyle \frac{1}{2}}
\Delta \Gamma \Delta t \right) -
\cos(\Delta m\, \Delta t) \right\} \,, \nonumber\\
P_\mathrm{unlike}^\mathrm{QM} &=&
2\, e^{-\Gamma(t_r+t_l)}
\left\{ \cosh \left( {\textstyle \frac{1}{2}}
\Delta \Gamma \Delta t \right) +
\cos(\Delta m\, \Delta t) \right\}
\label{QMterms}\end{aligned}$$ with $$\Delta t \equiv t_r - t_l, \quad \Delta m \equiv m_L-m_S, \quad
\Delta \Gamma = \Gamma_L - \Gamma_S
\quad \mathrm{and} \quad \Gamma = \frac{1}{2}(\Gamma_L +\Gamma_S) \,.$$ In contrast to the QM terms (\[QMterms\]), the “decoherence terms” proportional to $\zeta$ in Eqs.(\[3.3\]) and (\[3.4\]) depend on the matrix $S$, ie., on the choice of the basis. These terms can be characterized by the two functions $$\begin{aligned}
T_{11}^*(t) T_{12}(t) &=&
I_+(t) S_{11}^* S_{12} + I_-(t) \left| \frac{p}{q} \right|^2
S_{21}^* S_{22} +I_{+-}(t) \frac{p}{q} S_{11}^* S_{22} +
I_{-+}(t) \frac{p^*}{q^*} S_{21}^* S_{12} \,, \nonumber\\
T_{22}^*(t) T_{21}(t) &=&
I_+(t) S_{22}^* S_{21} + I_-(t) \left| \frac{q}{p} \right|^2
S_{12}^* S_{11} +I_{+-}(t) \frac{q}{p} S_{22}^* S_{11} +
I_{-+}(t) \frac{q^*}{p^*} S_{12}^* S_{21} \,,
\label{T}\end{aligned}$$ where the t-dependent functions are defined by $$\begin{aligned}
I_{\pm}(t) &=& | g_{\pm}(t) |^2 =
\frac{1}{2}\, e^{-\Gamma t}
\left\{ \cosh \left( {\textstyle \frac{1}{2}}
\Delta \Gamma\, t \right) \pm
\cos(\Delta m\, t) \right\} \,, \nonumber\\
I_{+-}(t) &=& g_{+}^*(t) g_{-}(t) =
-\frac{1}{2}\, e^{-\Gamma t}
\left\{ \sinh \left( \textstyle{ \frac{1}{2}}
\Delta \Gamma\, t \right) +
i \sin(\Delta m\, t) \right\} \,, \nonumber\\
I_{-+}(t) &=& \left( I_{+-}(t) \right)^* \,.
\label{I}\end{aligned}$$
The quantity which is directly sensitive to the interference terms is the asymmetry $$\label{5}
A(t_r,t_l) = \frac{P_\mathrm{unlike}(t_r,t_l) - P_\mathrm{like}(t_r,t_l)}
{P_\mathrm{unlike}(t_r,t_l) + P_\mathrm{like}(t_r,t_l)}$$ measured in the CPLEAR experiment [@CPLEAR98]. With the definition $$\label{def}
\rho = \frac{1}{2} \left( \left| \frac{p}{q} \right|^2 +
\left| \frac{q}{p} \right|^2 \right) \,,$$ the QM result for the asymmetry (\[5\]) is given by $$\label{QMA}
A^\mathrm{QM}(t_r,t_l) =
\frac{(1-\rho) \cosh (\frac{1}{2} \Delta \Gamma \Delta t) +
(1+\rho) \cos (\Delta m \Delta t)} {(1+\rho) \cosh (\frac{1}{2} \Delta \Gamma \Delta t) +
(1-\rho) \cos (\Delta m \Delta t)} \,,$$ whereas with the modified interference term the asymmetry depends on $S$ and $\zeta$ and has to be calculated with the full expressions (\[3.3\]) and (\[3.4\]), i.e., $$\label{5.5}
A^S_\zeta(t_r,t_l) =
\frac{P^S_\zeta (K^0,t_r; \bar K^0,t_l) +
P^S_\zeta (\bar K^0,t_r; K^0,t_l) -
P^S_\zeta (K^0,t_r; K^0,t_l) -
P^S_\zeta (\bar K^0,t_r; \bar K^0,t_l)} {P^S_\zeta (K^0,t_r; \bar K^0,t_l) +
P^S_\zeta (\bar K^0,t_r; K^0,t_l) +
P^S_\zeta (K^0,t_r; K^0,t_l) +
P^S_\zeta (\bar K^0,t_r; \bar K^0,t_l)} \,.$$
Discussion of the asymmetry
===========================
CP violation in [$K^0 \bar K^0$]{} mixing is small and proportional to $\mathrm{Re}\, \varepsilon \approx (|p/q|-1)/2$. In the strangeness asymmetry (\[QMA\]) calculated according to the rules of QM, CP violation enters through the coefficient $\rho$ (\[def\]) and appears therefore quadratically in the CP-violating parameter. One can easily check that the same suppression of CP-violating effects is true for the asymmetry (\[5.5\]) in the bases $\{K^0, \bar K^0\}$ and $\{K_L, K_S\}$ corresponding to $S=\mathbf{1}$ and $S=M$, respectively. Because of the smallness of CP violation in [$K^0 \bar K^0$]{} mixing, $\rho = 1$ or $(|p|^2-|q|^2)^2=0$ holds to an extremely good approximation. Though in general the CP-violating parameter appears linearly in Eq.(\[5.5\]) we will neglect CP violation in [$K^0 \bar K^0$]{} mixing from now on, which is sufficient for our purpose of estimating the allowed range of $\zeta$ for different bases. Without loss of generality we will use the phase convention $p = q$.
As a further simplification we will restrict ourselves to unitary matrices $S$. Since the expressions (\[3.3\]) and (\[3.4\]) do not depend a phase factor multiplying $S$ we simply assume that $$S =
\left( \begin{array}{cc} a & b \\ -b^* & a^* \end{array} \right)
\, \in \, \mathrm{SU(2)} \quad \mathrm{with} \quad
|a|^2 + |b|^2 = 1 \,.$$ Note that with this convention the basis $\{K_L, K_S\}$ is described by a matrix $S$ given by $a = b = i/\sqrt{2}$.
In conventional [$K^0 \bar K^0$]{} physics, a non-zero difference in the numbers of $K^0$ and $\bar K^0$ pairs measured at $t=t_r=t_l$ signifies CP violation. Since we assume CP conservation in [$K^0 \bar K^0$]{} mixing we want to ask the question if the artificial modification (\[ff\]) of the QM interference terms introduces such a difference. Indeed, in general this is the case as seen from the equation $$\begin{aligned}
&& P^S_\zeta (K^0,t; K^0,t) -
P^S_\zeta (\bar K^0,t; \bar K^0,t) \nonumber\\
&& = -2\, \zeta e^{-2 \Gamma t}
\sinh \left( \frac{1}{2} \Delta \Gamma\, t \right)
\left\{ (|a|^2-|b|^2)\, \mathrm{Re}\, (ab) \cos (\Delta m\, t) +
\mathrm{Im}\, (a^2 b^2) \sin (\Delta m\, t) \right\} \,.
\label{dif}\end{aligned}$$ However, for the bases $\{K_L, K_S\}$ with $|a|=|b|$ and $a^2$, $b^2$ being real and $\{K^0, \bar K^0\}$ with $b=0$ the right-hand side of Eq.(\[dif\]) is zero (see Ref. [@BG98] for an analogous consideration in the $B^0 \bar B^0$ system).
With the above simplifiying assumptions we can readily calculate the asymmetry (\[5.5\]) for any orthonormal basis given by a unitary matrix $S$ and for any decoherence parameter $\zeta$. For $\zeta = 0$ we obtain the QM result [@CPLEAR98] $$\label{QMAA}
A^\mathrm{QM}(t_r,t_l) =
\frac{\cos(\Delta m\, \Delta t)}{\cosh(\frac{1}{2} \Delta \Gamma \Delta t)} \,,$$ which is, of course, independent of $S$. For the basis $\{K_L,
K_S\}$ we arrive at the simple formula $$\label{6}
A^{K_LK_S}_\zeta (t_r,t_l)= (1-\zeta) A^\mathrm{QM}(t_r,t_l) \,,$$ whereas for $\{K^0, \bar K^0\}$ we obtain $$\begin{aligned}
\label{7}
A^{K^0 \bar K^0}_\zeta (t_r,t_l)
&=&\frac{\cos(\Delta m \Delta t) - \frac{1}{2} \zeta
\{ \cos(\Delta m \Delta t)-\cos(\Delta m (t_r+t_l)) \} }{\cosh(\frac{1}{2} \Delta \Gamma \Delta t) - \frac{1}{2} \zeta
\{ \cosh(\frac{1}{2} \Delta \Gamma \Delta t) -
\cosh(\frac{1}{2} \Delta \Gamma (t_r+t_l)) \} } \,.\end{aligned}$$ Eqs.(\[6\]) and (\[7\]) represent the two cases for which we will perform a numerical estimate of the decoherence parameter $\zeta$ in the next section.
In our formalism, Furry’s hypothesis (“method A” in Ref. [@furry]) corresponds to $\zeta = 1$. Looking at Eqs.(\[6\]) and (\[7\]) we read off in this case $$\begin{aligned}
A^{K_LK_S}_1 (t_r,t_l) & = & 0 \,, \label{klks} \\
A^{K^0 \bar K^0}_1 (t_r,t_l) & = &
\frac{\cos(\Delta m \Delta t) + \cos(\Delta m (t_r+t_l))}{\cosh(\frac{1}{2} \Delta \Gamma \Delta t) +
\cosh(\frac{1}{2} \Delta \Gamma (t_r+t_l)) }
\,. \label{k0k0}\end{aligned}$$ In the basis $\{K_L, K_S\}$ the asymmetry vanishes [@CPLEAR98], whereas in the basis $\{K^0, \bar K^0\}$ spontaneous factorization leads to a non-zero asymmetry (\[k0k0\]). One might ask the question for which bases the asymmetry (\[5.5\]) is zero at $\zeta = 1$ for arbitrary $t_r$, $t_l$. The answer to this question is given by the following proposition.\
**Proposition:** *With the assumptions $S \in$ SU(2) and $|p/q|=1$, Furry’s hypothesis leads to a zero asymmetry (\[5.5\]) if and only if spontaneous factorization of the [$K^0 \bar K^0$]{} wave function takes place in the $\{K_L, K_S\}$ basis.*\
*Proof:* We study the general expression $$\begin{aligned}
&&
P^S_\zeta (K^0,t_r; \bar K^0,t_l) + P^S_\zeta (\bar K^0,t_r; K^0,t_l) -
P^S_\zeta (K^0,t_r; K^0,t_l) - P^S_\zeta (\bar K^0,t_r; \bar K^0,t_l)
\nonumber\\
&& \hphantom{AAAAA}
= e^{-\Gamma (t_r+t_l)} \left\{ \vphantom{\frac{\alpha}{\alpha}}
\cos (\Delta m \Delta t) \right.
\nonumber\\
&& \hphantom{AAAAABBBBaaaaa} -\frac{1}{2} \zeta \left[
\left( |(a^*)^2 + b^2|^2 + 4|a|^2|b|^2 \right) \cos (\Delta m \Delta t)
\right. \nonumber\\
&&
\hphantom{AAAAABBBBaaai -\frac{1}{2}\, \zeta }
-\left( |(a^*)^2 + b^2|^2 - 4|a|^2|b|^2 \right) \cos (\Delta m (t_r+t_l))
\nonumber\\
&&
\hphantom{AAAAABBBBaaai -\frac{1}{2} \zeta }
\left. \left.
-4\, (|a|^2-|b|^2)\, \mathrm{Im}\, (ab) \sin (\Delta m (t_r+t_l))
\right] \vphantom{\frac{\alpha}{\alpha}} \right\} \,.\end{aligned}$$ The right-hand side of the equation is zero $\forall\: t_r, t_l$ at $\zeta=1$ if the system of equations $$\begin{aligned}
|(a^*)^2 + b^2|^2 + 4|a|^2|b|^2 & = & 2 \,, \nonumber\\
|(a^*)^2 + b^2|^2 - 4|a|^2|b|^2 & = & 0 \,, \nonumber\\
(|a|^2-|b|^2)\, \mathrm{Im}\, (ab) & = & 0\end{aligned}$$ is fulfilled. The general solution of this system is given by $a=e^{i\alpha}/\sqrt{2}=\epsilon b^*$ with $\epsilon = \pm 1$ and $\alpha$ being an arbitrary phase. It is then easy to show that a matrix $S$ with these coefficients represents the basis $\{K_L, K_S\}$ apart from trivial redefinitions. $\Box$
Estimation of the decoherence parameter $\zeta$ from the data of the CPLEAR Experiment
======================================================================================
In the CPLEAR experiment [@CPLEAR98] at CERN, [$K^0 \bar K^0$]{} pairs are produced by $p \bar p$ annihilation at rest. These pairs are predominantly in an antisymmetric state with quantum numbers $J^{PC}=1^{--}$ and the strangeness of the kaons is detected via strong interactions in surrounding absorbers. The experimental set-up has two configurations. In configuration C(0) both kaons have nearly equal proper times ($t_r \approx t_l$) when they interact in the absorber. This fulfills the conditions for an EPR-type experiment. In configuration C(5) the flight-path difference is 5 cm on average, corresponding to a proper time difference $| t_r-t_l | \, \approx 1.2 \,\tau_{S}$.
The asymmetry (\[5\]) is measured for these two configurations, giving the experimental results [@CPLEAR98] $A^\mathrm{exp}(0) = 0.81\pm 0.17$ for C(0) and $A^\mathrm{exp}(5) = 0.48\pm 0.12$ for C(5). On the other hand, the QM predictions for the asymmetry, when corrected according to the specific experimental design, yield $A^\mathrm{QM}_\mathrm{corr}(0) = 0.93$ and $A^\mathrm{QM}_\mathrm{corr}(5) = 0.56.$ These values are in agreement with the above experimental ones, demonstrating in this way the non-separability of the QM [$K^0 \bar K^0$]{} wavefunction.
Our approach is somewhat different. We estimate the amount of decoherence and show, as already emphasized, its dependence on the basis chosen. Specifically, we fit the decoherence parameter $\zeta$ by comparing the asymmetry for the two cases (\[6\]) and (\[7\]) with the experimental data of CPLEAR. For the fit of the parameter $\zeta$ we use the least squares method of Ref. [@orear] which is an effective variance method, taking into account not only the experimental error in $A(t_r,t_l)$ but also the variations in $x_{r,l} = 4.28\: \mathrm{cm}\: (t_{r,l}/\tau_{S})$, the space coordinates where the strangeness of the kaons is measured.
First we discuss the asymmetry in the $\{K_L,K_S\}$ basis where formula (\[6\]) is relevant. This case is rather easy to handle because $\zeta$ enters into the asymmetry linearly. Our fit result is $$\label{fit1}
\bar \zeta = 0.13
\begin{array}{cc}
+0.16\\
-0.15
\end{array} \,.$$ The CL for this fit is 97%.
In Fig. 1 we have plotted the asymmetry (\[6\]) for the values of $\zeta$ given in Eq.(\[fit1\]). The three solid lines represent the asymmetry (\[6\]) for the meanvalue $\bar \zeta$ and its one standard deviation values $\bar\zeta = 0.29$ and $-0.03$ (see Eq.(\[fit1\])), respectively. The dashed curve shows the QM prediction (\[QMAA\]) ($\zeta = 0$). The experimental results for the two configurations C(0) and C(5) are also indicated according to Fig. 9 in Ref. [@CPLEAR98], where in comparison to the experimental numbers quoted above some “background subtraction” has been performed [@CPLEAR98]. We have scaled the variable $\Delta t$ in the QM asymmetry (\[QMAA\]) in order to reproduce the QM curve in Fig. 9 of the CPLEAR Collaboration [@CPLEAR98].
The case of the asymmetry in the $\{K^0, \bar K^0 \}$ basis (\[7\]) is a bit more intricate than the previous one because formula (\[7\]) depends not only on the time difference $\Delta t = t_r - t_l$ but also on the sum $t_r + t_l$, and, moreover, the dependence on $\zeta $ is not linear. We estimate the values of the sum $t_r + t_l$ and their variations from the experimental configurations C(0) and C(5). With this additional but less accurate information we obtain the estimate $$\label{fit2}
\bar \zeta \sim \, 0.4 \pm 0.7 .$$ Here the quality of the fit is also good with a CL of about 67%.
In order to assess the validity of QM there are two measures associated with the parameter $\zeta$. They are given by the distance of the fitted mean value from 0 and from 1, each expressed in units of one standard deviation, corresponding to pure QM and to Furry’s hypothesis, respectively. Considering first the distance of $\bar \zeta$ from 0, we see that for both bases, $\{K_L,K_S\}$ and $\{K^0, \bar K^0 \}$, the results of our fit are very well compatible with QM. On the other hand, $\zeta = 1$ corresponds to Furry’s hypothesis, i.e., total decoherence or spontaneous factorization. In this case, with the $\{K_L,K_S\}$ basis the asymmetry vanishes identically (see Eq.(\[klks\])), which – as found by the CPLEAR Collaboration [@CPLEAR98] – has a probability of less than $10^{-4}$. In the $\{K^0,\bar K^0 \}$ basis, however, formula (\[k0k0\]) gives the values $$A^{K^0 \bar K^0}_1(0) = 0.90, \quad
A^{K^0 \bar K^0}_1(5) = 0.50 \nonumber$$ for the two configurations C(0) and C(5), respectively, which are within one standard deviation of the corresponding data and therefore not excluded at all. Note also that the two fit results for $\zeta$ (\[fit1\]) and (\[fit2\]) clearly disfavour negative values of the decoherence parameter.
Last but not least we want to mention that we also checked the effect of the experimental errors of the input quantities $\Delta m$ and $\Delta \Gamma$ on the asymmetries (\[6\]) and (\[7\]). The resulting error in the asymmetries is less than 1.5% for all times. Therefore, we can safely neglect these errors for our purpose.
Conclusions
===========
We reconsidered the results of the experiment of the CPLEAR Collaboration [@CPLEAR98] which investigated strangeness correlations of [$K^0 \bar K^0$]{} pairs produced by $p\bar p$ annihilation. We introduced the decoherence parameter $\zeta$ in order to measure quantitatively deviations from pure QM. This introduces a certain arbitrariness since the quantity we consider, the asymmetry (\[5.5\]), depends strongly on the basis chosen (pure QM, of course, not). This is not so surprising, it is merely the consequence of the freedom in QM to choose a basis in $K^0$–$\bar K^0$ space, which eventually leads to different interference terms for different basis choices [@dass; @BG98]. Thus the parameters $\zeta$ modifying these different interference terms will get in general different values when confronted with experiment. What is essential, however, is the existence of a basis where the [$K^0 \bar K^0$]{} system is far away from total decoherence and the corresponding $\zeta$ is close to 0 in agreement with QM. We have shown that the “best basis” in this respect is the $\{K_L,K_S\}$ basis (see proposition).
Furthermore, because of the introduction of the matrix S, which is arbitrary apart from $S\in$ SU(2), even for $|p/q| = 1$ we have in general $P^S_\zeta (K^0,t; K^0,t) - P^S_\zeta (\bar K^0,t; \bar K^0,t) \ne 0$, a sign for CP violation in [$K^0 \bar K^0$]{} mixing. Thus, in our ad hoc modification of QM by the decoherence parameter $\zeta$, the matrix $S$ mimics CP violation in general. However, as we have shown, for the two bases under consideration, $\{K_L, K_S\}$ and $\{K^0, \bar K^0\}$, the difference (\[dif\]) vanishes. This is analogous to considerations in the $B^0 \bar B^0$ system [@BG98].
Finally, we want to mention that our statistical analysis is not very refined and, in addition, further experimental data could diminish the errors of our fits. Nevertheless, long range interference effects, i.e., the presence of the QM interference term in the [$K^0 \bar K^0$]{}system, are quite well confirmed in agreement with QM and thus also interference effects of massive particles over macroscopic distances.
[10]{}
A. Einstein, B. Podolski and N. Rosen, Phys. Rev. **47**, 777 (1935).
J.S. Bell, *Speakable and Unspeakable in Quantum Mechanics* (Cambridge University Press, Cambridge, 1987).
A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. **49**, 1840 (1982).
P.G. Kwiat, K. Mattle, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett. **75**, 4337 (1995); G. Weihs, T. Jennewein, C. Simon, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett. **81**, 5039 (1998).
T.D. Lee and N.C. Yang, reported by T.D. Lee at Argonne National Laboratory, May 28, 1960 (unpublished); D.R. Inglis, Rev. Mod. Phys. **33**, 1 (1961); T.B. Day, Phys. Rev. **121**, 1204 (1961).
J. Six, Phys. Lett. B **114**, 200 (1982).
F. Selleri, Lett. Nuovo Cim. **36**, 521 (1983); P. Privitera and F. Selleri, Phys. Lett. B **296**, 261 (1992).
A. Datta and D. Home, Phys. Lett. A **119**, 3 (1986).
R.A. Bertlmann and W. Grimus, Phys. Lett. B **392**, 426 (1997).
G.V. Dass and K.V.L. Sarma, Eur. Phys. J. C **5**, 283 (1998).
R.A. Bertlmann and W. Grimus, Phys. Rev. D **58**, 034014 (1998).
A. Apostolakis *et al.*, CPLEAR Coll., Phys. Lett. B **422**, 339 (1998).
W.H. Furry, Phys. Rev. **49**, 393 (1936).
P.H. Eberhard, in *The Second Da$\Phi$ne Physics Handbook*, edited by L. Maiani, G. Pancheri and N. Paver (SIS–Pubblicazioni dei Laboratori di Frascati, Italy, 1995) Vol I, p. 99.
G.C. Ghirardi, R. Grassi und T. Weber, in Proc. of Workshop on *Physics and Detectors for DA$\Phi$NE, the Frascati $\Phi$ Factory*, April 9-12th, 1991, edited by G. Pancheri.
G.C. Ghirardi, R. Grassi and R. Ragazzon, in *The DA$\Phi$NE Physics Handbook*, edited by L. Maiani, G. Pancheri and N. Paver (Servizio Documentazione dei Laboratori Nazionale di Frascati, Italy, 1992), Vol. I, p. 283.
B. Hiesmayr, Diploma Thesis (1999).
P.H. Eberhard, Nucl. Phys. B **398**, 155 (1993); A. Di Domenico, Nucl. Phys. B **450**, 293 (1995); F. Benatti and R. Floreanini, Phys. Rev. D **57**, 1332 (1998) and hep-ph/9812353; B. Ancochea, A. Bramon and M. Nowakowski, preprint UAB-FT-459-98 (hep-ph/9811404); A. Bramon and M. Nowakowski, UAB-FT-458-98 (hep-ph/9811406).
J. Orear, Am. J. Phys. **50**, 912 (1982).
|
---
abstract: 'Real-time monitoring and responses to emerging public health threats rely on the availability of timely surveillance data. During the early stages of an epidemic, the ready availability of *line lists* with detailed tabular information about laboratory-confirmed cases can assist epidemiologists in making reliable inferences and forecasts. Such inferences are crucial to understand the epidemiology of a specific disease early enough to stop or control the outbreak. However, construction of such line lists requires considerable human supervision and therefore, difficult to generate in real-time. In this paper, we motivate [[**Guided Deep List**]{}]{}, the first tool for building automated line lists (in near real-time) from open source reports of emerging disease outbreaks. Specifically, we focus on deriving epidemiological characteristics of an emerging disease and the affected population from reports of illness. [[**Guided Deep List**]{}]{} uses distributed vector representations (ala word2vec) to discover a set of indicators for each line list feature. This discovery of indicators is followed by the use of dependency parsing based techniques for final extraction in tabular form. We evaluate the performance of [[**Guided Deep List**]{}]{} against a human annotated line list provided by HealthMap corresponding to MERS outbreaks in Saudi Arabia. We demonstrate that [[**Guided Deep List**]{}]{} extracts line list features with increased accuracy compared to a baseline method. We further show how these automatically extracted line list features can be used for making epidemiological inferences, such as inferring demographics and symptoms-to-hospitalization period of affected individuals.'
author:
- Saurav Ghosh
- Prithwish Chakraborty
- 'Bryan L. Lewis'
- 'Maimuna S. Majumder'
- Emily Cohn
- 'John S. Brownstein'
- 'Madhav V. Marathe'
- Naren Ramakrishnan
title: 'Guided Deep List: Automating the Generation of Epidemiological Line Lists from Open Sources'
---
Introduction {#sec:intro}
============
An epidemiological line list [@lau2014accuracy; @majumder2014estimation] is a listing of individuals suffering from a disease that describes both their demographic details as well as the timing of clinically and epidemiologically significant events during the course of disease. These are typically used during outbreak investigations of emerging diseases to identify key features, such as incubation period, symptoms, associated risk factors, and outcomes. The ultimate goal is to understand the disease well enough to stop or control the outbreak. Ready availability of line lists can also be useful in contact tracing as well as risk identification of spread such as the spread of Middle Eastern Respiratory Syndrome (MERS) in Saudi Arabia or Ebola in West Africa.
Formats of line lists are generally dependent on the kind of disease being investigated. However, some interesting features that are common for most formats include demographic information about cases. Demographic information can include age, gender, and location of infection. Depending on the disease being investigated, one can consider other addendums to this list, such as disease onset features (onset date, hospitalization date and outcome date) and clinical features (comorbidities, secondary contact, animal contact).
Traditionally, line lists have been curated manually and have rarely been available to epidemiologists in near-real time. Our primary objective is to automatically generate line lists of emerging diseases from open source reports such as WHO bulletins [@WHODONs] and make such lists readily available to epidemiologists. Previous work [@lau2014accuracy; @majumder2014estimation] has shown the utility in creating such lists through labor intensive human curation. We now seek to automate much of this effort. [**To the best of our knowledge, our work is the first to automate the creation of line lists.**]{}
{width="\linewidth"}
The availability of massive textual public health data coincides with recent developments in text modeling, including distributed vector representations such as word2vec [@mikolovefficient; @mikolovdistributed] and doc2vec [@le2014distributed]. These neural network based language models when trained over a representative corpus convert words to dense low-dimensional vector representations, most popularly known as word embeddings. These word embeddings have been widely used with considerable accuracy to capture linguistic patterns and regularities, such as vec(*Paris*) - vec(*France*) $\approx$ vec(*Madrid*) - vec(*Spain*) [@mikolovregular; @levyregular]. A second development relevant for line list generation pertains to semantic dependency parsing, which has emerged as an effective tool for information extraction, e.g., in an open information extraction context [@wu2010open], Negation Detection [@ou2015automatic; @sohn2012dependency; @ballesteros2012ucm], relation extraction [@bunescu2005shortest; @levy2014dependency] and event detection [@muthiah2015planned]. Given an input sentence, dependency parsing is typically used to extract its semantic tree representations where words are linked by directed edges called *dependencies*.
Building upon these techniques, we formulate [[**Guided Deep List**]{}]{}, a novel framework for automatic extraction of line list from WHO bulletins [@WHODONs]. [[**Guided Deep List**]{}]{} is guided in the sense that the user provides a seed indicator (or, keyword) for each line list feature to guide the extraction process. [[**Guided Deep List**]{}]{} uses neural word embeddings to expand the seed indicator and generate a set of indicators for each line list feature. The set of indicators is subsequently provided as input to dependency parsing based shortest distance and negation detection approaches for extracting line list features. As can be seen in Figure \[fig:ll\_overall\], [[**Guided Deep List**]{}]{} takes a WHO bulletin as input and outputs epidemiological line list in tabular format where each row represents a line list case and each column depicts the features corresponding to each case. The extracted line list provides valuable information to model the epidemic and understand the segments of population who would be affected.
Our main contributions are as follows.\
$\>\bullet$ **Automated:** [[**Guided Deep List**]{}]{} is fully automatic, requiring no human intervention.\
$\>\bullet$ **Novelty:** To the best of our knowledge, there has been no prior systematic efforts at tabulating such information automatically from publicly available health bulletins.\
$\>\bullet$ **Real-time:** [[**Guided Deep List**]{}]{} can be deployed for extracting line list in a (near) real-time setting.\
$\>\bullet$ **Evaluation:** We present a detailed and prospective analysis of [[**Guided Deep List**]{}]{} by evaluating the automatically inferred line list against a human curated line list for MERS outbreaks in Saudi Arabia. We also compare [[**Guided Deep List**]{}]{} against a baseline method.\
$\>\bullet$ **Epidemiological inferences:** Finally, we also demonstrate some of the utilities of real-time automated line listing, such as inferring the demographic distribution and symptoms-to-hospitalization period.
Problem Overview {#sec:prob}
================
In this manuscript, we intend to focus on Middle Eastern Respiratory Syndrome (MERS) outbreaks in Saudi Arabia [@majumder2014estimation] (2012-ongoing) as our case study. MERS was a relatively less understood disease when these outbreaks began. Therefore, MERS was poised as an emerging outbreak leading to good bulletin coverage about the infectious cases individually. This makes these disease outbreaks ideally suited to our goals. MERS is infectious as well and animal contact has been posited as one of the transmission mechanisms of the disease. For each line list case, we seek to extract automatically three types of epidemiological features as follows. (a) **Demographics:** Age and Gender, (b) **Disease onset:** onset date, hospitalization date and outcome date and (c) **Clinical features:** animal contact, secondary contact, comorbidities and specified healthcare worker (abbreviated as HCW).
In \[fig:ll\_block\], we show all the internal components comprising the framework of [[**Guided Deep List**]{}]{}. [[**Guided Deep List**]{}]{} takes multiple WHO MERS bulletins as input. The textual content of each bulletin is pre-processed by sentence splitting, tokenization, lemmatization, POS tagging, and date phrase detection using spaCy [@spacycite] and BASIS Technologies’ Rosette Language Processing (RLP) tools [@naren2014forecasting]. The pre-processing step is followed by three levels of modeling as follows. (a) Level 0 Modeling for extracting demographic information of cases, such as age and gender. In this level, we also identify the key sentences related to each line list case, (b) level 1 Modeling for extracting disease onset information and (c) level 2 Modeling for extracting clinical features. This is the final level of modeling in [[**Guided Deep List**]{}]{} framework. Features extracted at this level are associated with two labels: *Y* or *N*. Therefore, modeling at this level combines neural word embeddings with dependency parsing-based negation detection approaches to classify the clinical features into *Y* or *N*. In the subsequent section, we will discuss each internal component of [[**Guided Deep List**]{}]{} in detail.
![Block diagram depicting all components of the [[**Guided Deep List**]{}]{} framework. Given multiple WHO MERS bulletins as input, these components function in the depicted order to extract line lists in tabular form)[]{data-label="fig:ll_block"}](figs/GDL_block_diagram){width="\linewidth"}
Guided Deep List {#sec:methods}
================
Given multiple WHO MERS bulletins as input, [[**Guided Deep List**]{}]{} proceeds through three levels of modeling for extracting line list features. We describe each level in turn.
Level O Modeling
----------------
In level 0 modeling, we extract the age and gender for each line list case. These two features are mentioned in a reasonably structured way and therefore, can be extracted using a combination of regular expressions as shown in Algorithm \[al:level0\]. One of the primary challenges in extracting line list cases is the fact that a single WHO MERS bulletin can contain information about multiple cases. Therefore, there is a need to distinguish between cases mentioned in the bulletin. In level 0 modeling, we make use of the age and gender extraction to also identify sentences associated with each case. Since age and gender are the fundamental information to be recorded for a line list case, we postulate that the sentence mentioning the age and gender will be the starting sentence describing a line list case (see the textual block in Figure \[fig:ll\_overall\]). Therefore, the number of cases mentioned in the bulletin will be equivalent to the number of sentences mentioning age and gender information. We further postulate that information related to the other features (disease onset or critical) will be present either in the starting sentence or the sentences subsequent to the starting one not mentioning any age and gender related information ((see the textual block in Figure \[fig:ll\_overall\])). For more details on level 0 modeling, please see Algorithm \[al:level0\]. In Algorithm \[al:level0\], $\mathcal{N}$ represents the number of line list cases mentioned in the bulletin and $\mathcal{SC}_{n}$ represents the set of sentences mentioning the $n^{th}$ case.
n = 0;\
$\mathcal{SC}_{n}$ = Null;\
$\mathcal{R}_{1}$ = `\s+(?P<age>\d{1,2})(.{0,20})(\s+|-)(?P<gender>woman|man|male|female|boy|girl|housewife)`;\
$\mathcal{R}_{2}$ = `\s+(?P<age>\d{1,2})\s*years?(\s|-)old`;\
$\mathcal{R}_{3}$ = `\s*(?P<gender>woman|man|male|female|boy|girl|housewife|he|she)`;\
$\mathcal{N}$ = n;
WHO Template Learning
---------------------
Before presenting the details of level 1 modeling and level 2 modeling, we will briefly discuss the WHO template learning process which provides word embeddings as input to both these levels of modeling (see \[fig:ll\_block\]). In the template learning process, our main objective is to identify words which tend to share similar contexts or appear in the contexts of each other specific to the WHO bulletins (contexts of a word refer to the words surrounding it in a specified window size). For instance, consider the sentences $\mathcal{S}_{1} = $***The patient had no contact with animals*** and $\mathcal{S}_{2} =$***The patient was supposed to have no contact with camels***. The terms *animals* and *camels* appear in similar contexts in both $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$. Both the terms *animals* and *camels* are indicative of information pertaining to patient’s exposure to animals or animal products.
Similarly, consider the sentences $\mathcal{S}_{3} = $***The patient had an onset of symptoms on 23rd January 2016*** and $\mathcal{S}_{4} = $***The patient developed symptoms on 23rd January 2016***. The terms *onset* and *symptoms* are indicators for the onset date feature and both of them appear in similar contexts or contexts of each other in $\mathcal{S}_{3}$ and $\mathcal{S}_{4}$.
For the template learning process, neural network inspired word2vec models are ideally suited to our goals because these models work on the hypothesis that words sharing similar contexts or tending to appear in the contexts of each other have similar embeddings. In recent years, word2vec models based on the skip-gram architectures [@mikolovefficient; @mikolovdistributed] have emerged as the most popular word embedding models for information extraction tasks [@levyparam; @levydependency; @ghoshcikm]. We used two variants of skip-gram models: (a) the skip-gram model trained using the negative sampling technique ([[**SGNS**]{}]{} [@mikolovdistributed]) and (b) the skip-gram model trained using hierarchical sampling ([[**SGHS**]{}]{} [@mikolovdistributed]) to generate embeddings for each term in the WHO vocabulary $\mathcal{W}$. $\mathcal{W}$ refers to the list of all unique terms extracted from the entire corpus of WHO Disease Outbreak News (DONs) corresponding to all diseases downloaded from <http://www.who.int/csr/don/archive/disease/en/>. The embeddings for each term in $\mathcal{W}$ were provided as input to level 1 modeling and level 2 modeling as shown in \[fig:ll\_block\].
Level 1 Modeling {#sec:level1}
----------------
The level 1 modeling is responsible for extracting the disease onset features, such as symptom onset date, hospitalization date and outcome date for each linelist case, say the $n^{th}$ case. For extracting a given disease onset feature, the level 1 modeling takes three inputs: (a) seed indicator for the feature, (b) the word embeddings generated using [[**SGNS**]{}]{} or [[**SGHS**]{}]{} for each term in the WHO vocabulary $\mathcal{W}$ and (c) $\mathcal{SC}_{n}$ representing the set of sentences describing the $n^{th}$ case for which we are extracting the feature.
#### **Growth of seed indicator** {#growth-of-seed-indicator .unnumbered}
In the first phase of level 1 modeling, we discover the top-$K$ similar (or, closest) indicators in the embedding space to the seed indicator for each feature. The similarity metric used is the standard cosine similarity metric. Therefore, we expand the seed indicator to create a set of $K+1$ indicators for each feature. In Table \[fig:seed\] we show the indicators discovered by [[**SGNS**]{}]{} for each disease onset feature given the seed indicators as input.
[|c|c|c|]{} Features & Seed indicator & Discovered indicators\
Onset date & onset &
---------------------------
symptoms, symptom, prior,
days, dates
---------------------------
: Seed indicator and the discovered indicators using word embeddings generated by [[**SGNS**]{}]{}[]{data-label="fig:seed"}
\
Hospitalization date & hospitalized &
-----------------------------------
admitted, screened, hospitalised,
passed, discharged
-----------------------------------
: Seed indicator and the discovered indicators using word embeddings generated by [[**SGNS**]{}]{}[]{data-label="fig:seed"}
\
Outcome date & died &
----------------------------
recovered, passed, became,
ill, hospitalized
----------------------------
: Seed indicator and the discovered indicators using word embeddings generated by [[**SGNS**]{}]{}[]{data-label="fig:seed"}
\
#### **Shortest Dependency Distance** {#shortest-dependency-distance .unnumbered}
In the second phase, we use these $K+1$ indicators to extract the disease onset features. For each indicator $\mathcal{I}_{t} \forall t \in 1,2,\ldots,K+1$, we identify the sentences mentioning $\mathcal{I}_{t}$ by iterating over each sentence in $\mathcal{SC}_{n}$. Then, for each sentence mentioning $\mathcal{I}_{t}$, we discover the shortest path along the undirected dependency graph between $\mathcal{I}_{t}$ and the date phrases mentioned in the sentence. Subsequently, we calculate the length of the shortest path as the number of edges encountered while traversing along the shortest path. The length of the shortest path is referred to as the *dependency distance*. E.g., consider the sentence $\mathcal{S}_{5} = $ ***He developed symptoms on 4-June and was admitted to a hospital on 12-June.*** The sentence $\mathcal{S}_{5}$ containes the date phrases *4-June* and *12-June*. $\mathcal{S}_{5}$ also contains the indicator *symptoms* for onset date and *admitted* for hospitalization date (see Tables \[fig:seed\]). In Figure \[fig:undep\], we show the undirected dependency graph for $\mathcal{S}_{5}$. We observe that the *dependency distance* from *symptoms* to *4-June* is 3 (*symptoms* $\rightarrow$ *developed* $\rightarrow$ *on* $\rightarrow$ *4-June*) and *12-June* is 4 (*symptoms* $\rightarrow$ *developed* $\rightarrow$ *admitted* $\rightarrow$ *on* $\rightarrow$ *12-June*). Similarly, the dependency distance from *admitted* to *4-June* is 3 (*admitted* $\rightarrow$ *developed* $\rightarrow$ *on* $\rightarrow$ *4-June*) and *12-June* is 2 (*admitted* $\rightarrow$ *on* $\rightarrow$ *4-June*). Therefore, for each indicator we extract a set of date phrases and the dependency distance corresponding to each date phrase. The output value of the indicator is set to be the date phrase located at the shortest dependency distance. E.g., in $\mathcal{S}_{5}$, the output values of *symptoms* and *admitted* will be *4-June* and *12-June* respectively. The final output for each disease feature is obtained by performing majority voting on the outputs of the indicators. For more algorithmic details, please see Algorithm \[al:level1\].
![Undirected dependency graph corresponding to $\mathcal{S}_{5}$. The red-colored edges depict those edges included in the shortest paths between the date phrases (*4-June*, *12-June*) and the indicators (*symptoms*, *admitted*)[]{data-label="fig:undep"}](figs/Dependency-depTree.png){width="\linewidth"}
Growth of seed indicator using word embeddings to generate $K+1$ indicators represented as $\mathcal{I}_{t} \forall t \in 1,2,\ldots,K+1$;\
final output = majority voting on the outputs of each $\mathcal{I}_{t}$;\
Level 2 Modeling {#sec:level2}
----------------
The level 2 modeling is responsible for extracting the clinical features for each line list case. Extraction of clinical features is a binary classification problem where we have to classify each feature into two classes - *Y* or *N*. The first phase of level 2 modeling is similar to level 1 modeling. Seed indicator for each clinical feature is provided as input to the level 2 modeling and we extract the $K+1$ indicators for each such feature by discovering the top-$K$ most similar indicators to the seed indicator (in terms of cosine similarities) using the word embeddings generated during the WHO template learning process.
#### **Dependency based negation detection** {#dependency-based-negation-detection .unnumbered}
In the second phase, we make use of the $K+1$ indicators extracted in the first phase and a static lexicon of negation cues [@diaz2012ucm], such as *no*, *not*, *without*, *unable*, *never*, etc. to detect negation for a clinical feature. If no negation is detected, we classify the feature as *Y*, otherwise *N*. For each indicator $\mathcal{I}_{t} \forall t \in 1,2,\ldots,K+1$, we identify the first sentence (referred to as $\mathcal{S}_{\mathcal{I}_{t}}$) mentioning $\mathcal{I}_{t}$ by iterating over the sentences in $\mathcal{SC}_{n}$. Once $\mathcal{S}_{\mathcal{I}_{t}}$ is identified, we perform two types of negation detection on the directed dependency graph $\mathcal{D}_{\mathcal{I}_{t}}$ constructed for $\mathcal{S}_{\mathcal{I}_{t}}$.\
**Direct Negation Detection:** In this negation detection, we search for a negation cue among the neighbors of $\mathcal{I}_{t}$ in $\mathcal{D}_{\mathcal{I}_{t}}$. If a negation cue is found, then the output of $\mathcal{I}_{t}$ is classified as *N*.\
**Indirect Negation Detection.** Absence of a negation cue in the neighborhood of $\mathcal{I}_{t}$ drives us to perform indirect negation detection. In this detection, we locate those terms in $\mathcal{D}_{\mathcal{I}_{t}}$ for which $\mathcal{D}_{\mathcal{I}_{t}}$ has a directed path from each of these terms as source to $\mathcal{I}_{t}$ as target. We refer to these terms as the predecessors of $\mathcal{I}_{t}$ in $\mathcal{D}_{\mathcal{I}_{t}}$. Then, we search for negation cues in the neighborhood of each predecessor. If we find a negation cue around a predecessor, we assume that the indicator $\mathcal{I}_{t}$ is also affected by this negation and we classify the output of $\mathcal{I}_{t}$ as *N*. For example, consider the sentence $\mathcal{S}_{6} = $***The patient had no comorbidities and had no contact with animals.*** and the directed dependency graph corresponding to $\mathcal{S}_{6}$ is shown in Figure \[fig:negdep\]. Sentence $\mathcal{S}_{6}$ contains the seed indicators $\textit{comorbidities}$ for comorbidities and $\textit{animals}$ for animal contact. In Figure \[fig:negdep\], we observe direct negation detection for comorbidities as the negation cue *no* is located in the neighborhood of the indicator *comorbidities*. However, for animal contact, we observe indirect negation detection as the negation cue *no* is situated in the neighborhood of the term *contact* which is one of the predecessors of the indicator *animals*.
![Directed dependency graph corresponding to $\mathcal{S}_{6}$ showing direct and indirect negation detection[]{data-label="fig:negdep"}](figs/negation-depTree.pdf){width="\linewidth"}
Therefore, for a clinical feature we have $K+1$ indicators and the classification output *Y* or *N* from each indicator. The final output for a feature is obtained via majority voting on the outputs of the indicators.
Growth of seed indicator using word embeddings to generate $K+1$ indicators represented as $\mathcal{I}_{t} \forall t \in 1,2,\ldots,K+1$;\
final output = majority voting on the outputs of each $\mathcal{I}_{t}$;\
Experimental Evaluation {#sec:expts}
=======================
In this section, we first provide a brief description of our experimental setup, including the models for automatic extraction of line lists, human annotated line lists, accuracy metric and parameter settings.
WHO corpus
----------
The WHO corpus used in the template learning process (see \[fig:ll\_block\]) was downloaded from <http://www.who.int/csr/don/archive/disease/en/>. The corpus contains outbreak news articles related to a wide range of diseases reported during the time period 1996 to 2016. The textual content of each article was pre-processed by sentence splitting, tokenization and lemmatization using spaCy [@spacycite]. After pre-processing, the WHO corpus was found to contain 35,485 sentences resulting in a vocabulary $\mathcal{W}$ of 4447 words.
Models
------
We evaluated the following automated line listing models.
$\>\bullet$ [[**Guided Deep List (SGNS)**]{}]{}: Variant of [[**Guided Deep List**]{}]{} with [[**SGNS**]{}]{} used as the word2vec model in the WHO template learning process.\
$\>\bullet$ [[**Guided Deep List (SGHS)**]{}]{}: Variant of [[**Guided Deep List**]{}]{} with [[**SGHS**]{}]{} used as the word2vec model in the WHO template learning process.\
[[**Guidedlist**]{}]{}: Baseline model which does not use any word embedding model (absence of WHO template learning) to expand the seed indicator in order to generate $K+1$ indicators for each feature. Therefore, [[**Guidedlist**]{}]{} uses only a single indicator (seed indicator) to extract line list features.\
Human annotated line list
-------------------------
We evaluated the line list extracted by the automated line listing models against a human annotated line list for MERS outbreaks in Saudi Arabia. To create the human annotated list, patient and outcome data for confirmed MERS cases were collected from the MERS Disease Outbreak News (DONs) reports of WHO [@WHODONs] and curated into a machine-readable tabular line list. In the human annotated list, total number of confirmed cases were 241 curated from 64 WHO bulletins reported during the period October 2012 to February 2015. Some of these 241 cases have missing (null) features (see Figure \[fig:ll\_overall\]). In Figure \[fig:nonnull\], we show the distribution of non-null features in the human annotated list. We observe that majority of human annotated cases have at least 6 (out of 9) non-null features with the peak of the distribution at 8.
![Distribution of non-null features in the human annotated line list[]{data-label="fig:nonnull"}](figs/linelist_nonull){width="\linewidth"}
Accuracy metric
---------------
#### **Matching automated line list to human annotated list.** {#matching-automated-line-list-to-human-annotated-list. .unnumbered}
For evaluation, the problem is: we are given a set of automated line list cases and a set of human annotated cases for a single WHO MERS bulletin. Our strategy is to costruct a bipartite graph [@naren2014forecasting] where (i) an edge exists if the automated case and the human annotated case is extracted from the same WHO bulletin and (ii) the weight on the edge denotes the quality score (QS). Quality score (QS) is defined as the number of correctly extracted features in the automated case divided by the number of non-null features in the human annotated case. We then construct a maximum weighted bipartite matching [@naren2014forecasting]. Such matchings are conducted for each WHO bulletin to extract a set of matches where each match represents a pair (automated case, human annotated case) and is also associated with a QS. Once the matches are found for all the WHO bulletins, we computed the average QS by averaging the QS values across the matches.
Once the average QS and QS for each match are computed, we also computed the accuracy for each line list feature. For the demographic and disease onset features, we computed the accuracy classification score using scikit-learn [@scikit-learn] by comparing the automated features against the human annotated features across the matches. The clinical features are associated with two classes - *Y* and *N* (see Figure \[fig:ll\_overall\]). For each class, we computed the F1-score using scikit-learn [@scikit-learn] where F1-score can be interpreted as a harmonic mean of the precision and recall. F1-score reaches its best value at 1 and worst score at 0. Along with the F1-score for each class, we also report the average F1-score across the two classes.
Parameter settings
------------------
Each variant of [[**Guided Deep List**]{}]{} inherits the parameters of the word embedding models as shown in Table \[tab:ben\_param\]. Apart from the word embedding parameters, [[**Guided Deep List**]{}]{} also inherits the parameter $K$ which refers to the $K+1$ indicators for disease onset or clinical features (see Section \[sec:methods\]). In Table \[tab:ben\_param\], we provide the list of all parameters, the explored values for each parameter and the applicable models corresponding to each parameter. We selected the optimal parameter configuration for each model based on the maximum average QS value as well as maximum average of the individual feature accuracies across the matches.
Results {#sec:results}
=======
In this section we try to ascertain the efficacy and applicability of [[**Guided Deep List**]{}]{} by investigating some of the pertinent questions related to the problem of automated line listing.
[**Multiple indicators vs single indicator - which is the better method for automated line listing?**]{}
As mentioned in section \[sec:expts\], [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} uses multiple indicators discovered by word2vec, whereas the baseline [[**Guidedlist**]{}]{} uses only the seed indicator to infer line list features. We executed our automated line listing models taking as input the same set of 64 WHO MERS bulletins from which 241 human annotated line list cases were extracted. In Table \[tab:qs\], we observe that the number of automated line list cases (198) and the matches (182) after maximum bipartite matching is same for all the models. This is due to the reason that level 0 modeling (age and gender extraction) is the common modeling component in all the models and the number of extracted line list cases depends on the age and gender extraction (see section \[sec:methods\]). In Table \[tab:qs\], we also compared the average QS achieved by each model. We observe that [[**Guided Deep List (SGNS)**]{}]{} is the best performing model achieving an average QS of 0.74 over [[**Guided Deep List (SGHS)**]{}]{} (0.71) and [[**Guidedlist**]{}]{} (0.67). To further validate the results in Table \[tab:qs\], we also show the QS distribution for each model in Figure \[fig:coverage\] where x-axis represents the QS values and the y-axis represents the number of automated line list cases having a particular QS value. For [[**Guidedlist**]{}]{}, the peak of QS distribution is at 0.62. However, for [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{}, the peak of the distribution is at 0.75. We further observe that [[**Guided Deep List (SGNS)**]{}]{} extracts higher number of line list cases with a perfect QS of 1 in comparison to [[**Guidedlist**]{}]{}.
We also compared the models on the basis of individual accuracies of the line list features across the matches in Tables \[tab:acc\_table\] and \[tab:f1\]. In Table \[tab:acc\_table\], all the models achieve similar performance for the demographic features since level 0 modeling is similar for all the models (see section \[sec:methods\]). However, for the disease onset features, both [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} outperform the baseline achieving an average accuracy of $0.45$ and $0.43$ in comparison to [[**Guidedlist**]{}]{} ($0.12$) respectively. [[**Guided Deep List (SGNS)**]{}]{} is the best performing model for onset date. However, for hospitalization date and outcome date, [[**Guided Deep List (SGHS)**]{}]{} is the better performing model than [[**Guided Deep List (SGNS)**]{}]{}. In Table \[tab:f1\], for the clinical features, we observe that [[**Guided Deep List (SGNS)**]{}]{} performs better than [[**Guided Deep List (SGHS)**]{}]{} and [[**Guidedlist**]{}]{} for comorbidities and specified HCW on the basis of average F1-score. Specifically, for specified HCW, [[**Guided Deep List (SGNS)**]{}]{} outperforms [[**Guided Deep List (SGHS)**]{}]{} and [[**Guidedlist**]{}]{} for the minority class *Y*. For animal contact, [[**Guided Deep List (SGHS)**]{}]{} emerges out to be the best performing model in terms of average F1-score, specifically outperforming the competing models for the minority class *Y*. [[**Guidedlist**]{}]{} only performs better for secondary contact, even though the performance for the minority class *Y* is almost similar to [[**Guided Deep List (SGHS)**]{}]{} and [[**Guided Deep List (SGNS)**]{}]{}. Overall, we can conclude from Table \[tab:f1\] that [[**Guided Deep List**]{}]{} employing multiple indicators discovered via [[**SGNS**]{}]{} or [[**SGHS**]{}]{} shows superior performance than [[**Guidedlist**]{}]{} in majority of the scenarios, specifically for the minority class of each clinical feature. To further validate the results in Table \[tab:f1\], the confusion matrix for each model and each clinical feature can be found in <https://github.com/sauravcsvt/KDD_linelisting>.
Models Human lists Auto lists Matches Average QS
-------- ------------- ------------ --------- ------------
241 198 182 0.67
241 198 182 0.71
241 198 182 **0.74**
: Average Quality Score (QS) achieved by each automated line listing model for MERS line list in Saudi Arabia. As can be seen, [[**Guided Deep List (SGNS)**]{}]{} shows best performance achieving an average QS of 0.73[]{data-label="tab:qs"}
![Distribution of QS values for each automated line listing model corresponding to MERS line list in Saudi Arabia. X-axis represents QS values and Y-axis represents the number of automated line list cases having a particular QS value[]{data-label="fig:coverage"}](figs/MERS_SA_qs_dist){width="\linewidth"}
Features
-- ---------------------- ---------- ---------- ----------
Age 0.87 **0.91** 0.87
Gender **0.99** 0.98 0.97
Average 0.93 **0.95** 0.92
Onset date 0.01 0.01 **0.37**
Hospitalization date 0.11 **0.63** 0.62
Outcome date 0.48 **0.66** 0.36
Average 0.20 0.43 **0.45**
: Comparing the automated line listing models based on the accuracy score for the demographics and disease onset features. For the disease onset features, [[**Guided Deep List (SGNS)**]{}]{} emerges out to be the best performing model. However, for the demographic features, all the models achieve almost similar performance[]{data-label="tab:acc_table"}
[|c|c|c|c|c|]{} & Class & &
------------------------------
[[**Guided Deep List**]{}]{}
([[**SGHS**]{}]{})
------------------------------
: Comparing the performance of the automated line listing models for extracting clinical features corresponding to MERS line list in Saudi Arabia. We report the F1-score for class Y, class N and average F1-score across the two classes. For animal contact, [[**Guided Deep List (SGHS)**]{}]{} emerges out to be the best performing model. For comorbidities and specified HCW, [[**Guided Deep List (SGNS)**]{}]{} shows best performance. However, for secondary contact, [[**Guidedlist**]{}]{} achieve superior performance in comparison to [[**Guided Deep List**]{}]{}[]{data-label="tab:f1"}
&
------------------------------
[[**Guided Deep List**]{}]{}
([[**SGNS**]{}]{})
------------------------------
: Comparing the performance of the automated line listing models for extracting clinical features corresponding to MERS line list in Saudi Arabia. We report the F1-score for class Y, class N and average F1-score across the two classes. For animal contact, [[**Guided Deep List (SGHS)**]{}]{} emerges out to be the best performing model. For comorbidities and specified HCW, [[**Guided Deep List (SGNS)**]{}]{} shows best performance. However, for secondary contact, [[**Guidedlist**]{}]{} achieve superior performance in comparison to [[**Guided Deep List**]{}]{}[]{data-label="tab:f1"}
\
& Y & 0.33 & **0.68** & 0.37\
& N & 0.87 & **0.91** & 0.88\
& Average & 0.60 & **0.79** & 0.63\
& Y & **0.57** & 0.52 & 0.56\
& N & **0.86** & 0.70 & 0.72\
& Average & **0.71** & 0.61 & 0.64\
& Y & 0.52 & 0.52 & **0.81**\
& N & 0.56 & 0.54 & **0.61**\
& Average & 0.54 & 0.53 & **0.71**\
& Y & 0.26 & 0.35 & **0.44**\
& N & **0.95** & 0.93 & 0.90\
& Average & 0.61 & 0.64 & **0.67**\
[**What are beneficial parameter settings for automated line listing?**]{}
To identify which parameter settings are beneficial for automated line listing, we looked at the best parameter configuration (see Table \[tab:ben\_param\]) of [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} which achieved the accuracy values in Tables \[tab:qs\], \[tab:acc\_table\] and \[tab:f1\]. In Table \[tab:ben\_param\], we explored the standard settings of each word2vec parameter (dimensionality of word embeddings, window size, negative samples and training iterations) in accordance with previous research [@levyparam]. Regarding dimensionality of word embeddings, [[**Guided Deep List (SGHS)**]{}]{} prefers $600$ dimensions, whereas [[**Guided Deep List (SGNS)**]{}]{} prefers $300$ dimensions. For the window size, both the models seem to benefit from smaller-sized (5) context windows. The number of negative samples is applicable only for [[**Guided Deep List (SGNS)**]{}]{} where it seems to prefer a single negative sample. Finally, for the training iterations, both the models benefit from more than 1 training iteration. This is expected as the WHO corpus used in the template learning process (see section \[sec:expts\]) is a smaller-sized corpus with a vocabulary of only $\mathcal{W} = 4447$ words. In such scenarios, word2vec models ([[**SGNS**]{}]{} or [[**SGHS**]{}]{}) generate improved embeddings with higher number of training iterations. Finally, both the models are also associated with the parameter $K$ which refers to the number of indicators $K + 1$ used for extracting the disease onset and clinical features. As expected, the models prefer at least 5 indicators, along with the seed indicator to be used for automated line listing. Using higher number of indicators increases the chance of discovering an informative indicator for a line list feature.
[|c|c|c|c|c|c|]{} Models &
----------------
Dimensionality
(300:600)
----------------
: Parameter settings in [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} for which both the models achieve optimal performance in terms of average QS and individual feature accuracies corresponding to MERS line list in Saudi Arabia. Non-applicable combinations are marked by *NA*[]{data-label="tab:ben_param"}
&
-----------
Window
size
(5:10:15)
-----------
: Parameter settings in [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} for which both the models achieve optimal performance in terms of average QS and individual feature accuracies corresponding to MERS line list in Saudi Arabia. Non-applicable combinations are marked by *NA*[]{data-label="tab:ben_param"}
&
----------
Negative
samples
(1:5:15)
----------
: Parameter settings in [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} for which both the models achieve optimal performance in terms of average QS and individual feature accuracies corresponding to MERS line list in Saudi Arabia. Non-applicable combinations are marked by *NA*[]{data-label="tab:ben_param"}
&
------------
Training
Iterations
(1:2:5)
------------
: Parameter settings in [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} for which both the models achieve optimal performance in terms of average QS and individual feature accuracies corresponding to MERS line list in Saudi Arabia. Non-applicable combinations are marked by *NA*[]{data-label="tab:ben_param"}
&
---------------
Indicators
($K$ = 3:5:7)
---------------
: Parameter settings in [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} for which both the models achieve optimal performance in terms of average QS and individual feature accuracies corresponding to MERS line list in Saudi Arabia. Non-applicable combinations are marked by *NA*[]{data-label="tab:ben_param"}
\
------------------------------
[[**Guided Deep List**]{}]{}
([[**SGHS**]{}]{})
------------------------------
: Parameter settings in [[**Guided Deep List (SGNS)**]{}]{} and [[**Guided Deep List (SGHS)**]{}]{} for which both the models achieve optimal performance in terms of average QS and individual feature accuracies corresponding to MERS line list in Saudi Arabia. Non-applicable combinations are marked by *NA*[]{data-label="tab:ben_param"}
& 600 & 5 & *NA* & 5 & 7\
& 300 & 5 & 1 & 2 & 5\
[**Which indicator keywords discovered using word2vec contribute to the improved performance of [[**Guided Deep List**]{}]{}?**]{}
Next, we investigate the informative indicators discovered using word2vec which contribute to the improved performance of [[**Guided Deep List (SGNS)**]{}]{} or [[**Guided Deep List (SGHS)**]{}]{} in Tables \[tab:acc\_table\] and \[tab:f1\]. In Figure \[fig:features\], we show the accuracies (or, average F1-score) of individual indicators (including the seed indicator) corresponding to the best performing model for a particular line list feature. Regarding onset date (see Figure \[fig:onset\]), [[**Guided Deep List (SGNS)**]{}]{} is the best performing model and the seed indicator provided as input is *onset*. We observe that *symptoms* is the most informative indicator achieving an accuracy of 0.36 similar to the overall accuracy (see Table \[tab:acc\_table\]). Rest of the indicators (including the seed indicator) achieve negligible accuracies and therefore, do not contribute to the overall performance of [[**Guided Deep List (SGNS)**]{}]{}. Similary, for hospitalization date with the seed keyword *hospitalization* provided as input, [*admitted*]{} emerges out to be most informative indicator followed by the seed indicator, *hospitalised* and *treated* (see Figure \[fig:hospital\]). Finally, for the outcome date, *died* (seed indicator) and *passed* are the two most informative indicators as observed in Figure \[fig:outcome\].
Regarding the clinical features, we show the average F1-score of individual indicators. For animal contact, the seed indicator provided as input is *animals*. We observe in Figure \[fig:animal\] that the most informative indicator for animal contact is *camels* followed by indicators such as *animals* (seed), *sheep* and *direct*. This shows that contact with *camels* is the major transmission mechanism for MERS disease. The informative indicators found for comorbidities are *patient*, *comorbidities* and *history*. Finally, regarding specified HCW, the informative indicators discovered are *healthcare* (seed), *tracing* and *intensive*.
[0.33]{} {width="1.0\linewidth"}
[0.33]{} {width="1.0\linewidth"}
[0.33]{} {width="1.0\linewidth"}
\
[0.33]{} {width="1.0\linewidth"}
[0.33]{} {width="1.0\linewidth"}
[0.33]{} {width="1.0\linewidth"}
[**Does indirect negation detection play an useful role in extracting clinical features?**]{}
In level 2 modeling for extracting clinical features, both direct and indirect negation detection are used. For more details, please see section \[sec:methods\]. To identify if indirect negation detection contributes positively, we compared the performance of [[**Guided Deep List**]{}]{} with and without indirect negation detection for each clinical feature in Table \[tab:indirect\] by reporting the F1-score for each class as well as average F1-score. We observe that indirect negation detection has a positive effect on the performance for animal contact and secondary contact. However, for comorbidities and specified HCW, indirect negation detection plays an insignificant role.
Clinical Feature Class Direct Negation Direct + Indirect Negation
------------------ --------- ----------------- ----------------------------
Y 0.56 **0.63**
N 0.80 **0.90**
Average 0.68 **0.77**
Y **0.55** 0.54
N 0.65 **0.72**
Average 0.60 **0.63**
Y **0.86** 0.82
N **0.64** 0.62
Average **0.75** 0.72
Y **0.44** **0.44**
N **0.90** **0.90**
Average **0.67** **0.67**
: Comparing the performance of [[**Guided Deep List**]{}]{} on extraction of clinical features with or without indirect negation for MERS line list in Saudi Arabia. It can be seen that indirect negation improves the performance of [[**Guided Deep List**]{}]{} for animal contact and secondary contact.[]{data-label="tab:indirect"}
[**What insights can epidemiologists gain about the MERS disease from automatically extracted line lists?**]{}
Finally, we show some of the utilities of automated line lists by inferring different epidemiological insights from the line list extracted by [[**Guided Deep List**]{}]{}.\
**Demographic distribution.** In Figure \[fig:ll\_overall\], we show the age and gender distribution of the affected individuals in the extracted line list. We observe that males are more prone to getting infected by MERS rather than females. This is expected as males have a higher probability of getting contacted with infected animals (animal contact) or with each other (secondary contact). Also individuals aged between 40 and 70 are more prone to getting infected as evident from the age distribution.\
**Analysis of disease onset features.** We analyzed the symptoms-to-hospitalization period by analyzing the difference (in days) between onset date and hospitalization date in the extracted line list as shown in Figure \[fig:inho\]. We observe that most of the affected individuals with onset of symptoms got admitted to the hospital either on the same day or within 5 days. This depicts a prompt responsiveness of the concerned health authorities in Saudi Arabia in terms of admitting the individuals showing symptoms of MERS. In Figure \[fig:hoou\], we also show a distribution of the hospitalization-to-outcome period (in days). Interestingly, we see that the distribution has a peak at 0 which indicates that most of the infected individuals admitted to the hospital died on the same day indicating high fatality rate of MERS case.
[0.5]{} {width="1.0\linewidth"}
[0.5]{} {width="1.0\linewidth"}
Acknowledgements {#acknowledgements .unnumbered}
================
[Supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior National Business Center (DoI/NBC) contract number D12PC000337, the US Government is authorized to reproduce and distribute reprints of this work for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBC, or the US Government. ]{}
Supplementary Information {#supplementary-information .unnumbered}
=========================
Codes and data for this manuscript are available at <https://github.com/sauravcsvt/KDD_linelisting>.
[10]{} url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
*et al.* . ** ****, ().
, , & . ** ().
. (). <http://www.who.int/csr/don/archive/disease/coronavirus_infections/en/>.
, , & . ** **** (). <http://arxiv.org/abs/1301.3781>.
, , , & . In **, ().
& In **, vol. , ().
, & . In **, (). <http://aclweb.org/anthology/N/N13/N13-1090.pdf>.
& . In **, (). <http://aclweb.org/anthology/W/W14/W14-1618.pdf>.
& . In **, (, ).
& . ** ****, ().
, & . ** ****, ().
*et al.* . In **, (, ).
& . In **, (, ).
& In **, ().
*et al.* In **, ().
& . In **, (, , ). <https://aclweb.org/anthology/D/D15/D15-1162>.
*et al.* . In **, (, ).
, & . ** ****, (). <https://tacl2013.cs.columbia.edu/ojs/index.php/tacl/article/view/570>.
& . In **, (). <http://aclweb.org/anthology/P/P14/P14-2050.pdf>.
, , , & . In **, (, ).
, , & , ().
*et al.* . ** ****, ().
|
---
abstract: 'In this paper, we obtain optimal time hypercontractivity bounds for the free product extension of the Ornstein-Uhlenbeck semigroup acting on the Clifford algebra. Our approach is based on a central limit theorem for free products of spin matrix algebras with mixed commutation/anticommutation relations. With another use of Speicher’s central limit theorem, we may also obtain the same bounds for free products of $q$-deformed von Neumann algebras interpolating between the fermonic and bosonic frameworks. This generalizes the work of Nelson, Gross, Carlen/Lieb and Biane. Our main application yields hypercontractivity bounds for the free Poisson semigroup acting on the group algebra of the free group ${\mathbb{F}}_n$, uniformly in the number of generators.'
author:
- |
Marius Junge, Carlos Palazuelos,\
Javier Parcet, Mathilde Perrin and Éric Ricard
title: Hypercontractivity for free products
---
[**Introduction**]{} {#introduction .unnumbered}
====================
The two-point inequality was first proved by Bonami and rediscovered years later by Gross [@Bonami; @Gross1]. In the context of harmonic analysis, this inequality was central for Bonami’s work on the relation between integrability of a function and the decay properties of its Fourier coefficients. It was also instrumental in Beckner’s theorem on the optimal constants for the Hausdorff-Young inequality [@Beckner]. On the other hand, motivated by quantum field theory, Gross used it as a key step towards his logarithmic Sobolev inequalities [@Gross1]. More recently, the two-point inequality has also produced very important applications in computer science and in both classical and quantum information theory [@BRSW; @GKKRW; @KhVi; @KlRe]. If $1 < p \le q < \infty$ and $\alpha,\beta \in {\mathbb{C}}$, Bonami-Gross inequality can be rephrased for $r=e^{-t}$ as follows $$\Big( \sum_{\varepsilon = \pm 1}
\Big| \frac{(1+ \varepsilon r) \alpha + (1-\varepsilon r)
\beta}{2^{1+\frac{1}{q}}} \Big|^q \Big)^{\frac1q} \le \Big(
\frac{|\alpha|^p + |\beta|^p}{2} \Big)^{\frac1p} \ \Leftrightarrow \ r
\le \sqrt{\frac{p-1}{q-1}}.$$ It can be regarded —from Bonami’s viewpoint— as the optimal hypercontractivity bound for the Poisson semigroup" on the group ${\mathbb{Z}}_2$, while Gross understood it as the optimal hypercontractivity bound for the Ornstein-Uhlenbeck semigroup on the Clifford algebra with one generator $\mathcal{C}({\mathbb{R}})$. Although the two-point inequality can be generalized in both directions, harmonic analysis has developed towards other related norm inequalities in the classical groups —like $\Lambda_p$ sets in ${\mathbb{Z}}$— instead of analyzing the hypercontractivity phenomenon over the compact dual of other discrete groups. Namely, to the best of our knowledge only the cartesian products of ${\mathbb{Z}}_2$ and ${\mathbb{Z}}$ have been understood so far, see [@Weissler]. The first goal of this paper is to replace cartesian products by free products, and thereby obtain hypercontractivity inequalities for the free Poisson semigroups acting on the group von Neumann algebras associated to ${\mathbb{F}}_n = {\mathbb{Z}}* {\mathbb{Z}}* \cdots * {\mathbb{Z}}$ and $\mathbb{G}_n = {\mathbb{Z}}_2 * {\mathbb{Z}}_2 * \cdots * {\mathbb{Z}}_2$.
Let $\G$ denote any of the free products considered above and let $\lambda: \G \to \mathcal{B}(\ell_2(\G))$ stand for the corresponding left regular representation. The group von Neumann algebra $\mathcal{L}(\G)$ is the weak operator closure of the linear span of $\lambda(\G)$. If $e$ denotes the identity element of $\G$, the algebra $\mathcal{L}(\G)$ comes equipped with the standard trace $\tau(f) = \langle \delta_e, f \delta_e \rangle$. Let $L_p(\mathcal{L}(\mathrm{G}), \tau)$ be the $L_p$ space over the noncommutative measure space $(\mathcal{L}(\mathrm{G}), \tau_\G)$ —the so called noncommutative $L_p$ spaces— with norm $\|f\|_p^p =
\tau |f|^p$. We invite the reader to check that $L_p(\mathcal{L}(\G),\tau) = L_p(\mathbb{T})$ for $\G = {\mathbb{Z}}$ after identifying $\lambda_{{\mathbb{Z}}}(k)$ with $e^{2\pi i k \cdot}$. In the general case, the absolute value and the power $p$ are obtained from functional calculus for this (unbounded) operator on the Hilbert space $\ell_2(\G)$, see [@PX] for details. If $f = \sum_g \widehat{f}(g)
\lambda(g)$, the free Poisson semigroup on $\G$ is given by the family of linear maps $$\mathcal{P}_{\G,t} f \, = \, \sum_{g \in \G} e^{-t
|g|} \widehat{f}(g) \lambda(g) \quad \mbox{with} \quad t \in {\mathbb{R}}_+.$$ In both cases $\G \in \{ {\mathbb{F}}_n,\mathbb{G}_n \}$, $|g|$ refers to the Cayley graph length. In other words, $|g|$ is the number of letters (generators and their inverses) which appear in $g$ when it is written in reduced form. It is known from [@Ha] that $\mathcal{P}_\G =
(\mathcal{P}_{\G,t})_{t \ge 0}$ defines a Markovian semigroup of self-adjoint, completely positive, unital maps on $\mathcal{L}(\G)$. In particular, $\mathcal{P}_{\G,t}$ defines a contraction on $L_p(\mathcal L(\G))$ for every $1\leq p\leq
\infty$. The hypercontractivity problem for $1< p \le q < \infty$ consists in determining the optimal time $t_{p,q} > 0$ above which $$\|\mathcal{P}_{\G,t} f \|_q \, \le \, \|f\|_p \qquad \mbox{for
all} \qquad t \ge t_{p,q}.$$ In our first result we provide new hypercontractivity bounds for the free Poisson semigroups on those group von Neumann algebras. If $g_1, g_2, \ldots, g_n$ stand for the free generators of ${\mathbb{F}}_n$, we will also consider the symmetric subalgebra $\mathcal{A}_{sym}^n$ of $\mathcal{L}({\mathbb{F}}_n)$ generated by the self-adjoint operators $\lambda(g_j) + \lambda(g_j)^*$. In other words, we set $$\mathcal{A}_{sym}^n \, = \, \big\langle \lambda(g_1) +
\lambda(g_1)^*, \ldots, \lambda(g_n) + \lambda(g_n)^* \big\rangle''.$$
If $1 < p \le q < \infty$, we find$:$
1. Optimal time hypercontractivity for $\mathbb{G}_n$ $$\big\|
\mathcal{P}_{\mathbb{G}_n,t}: L_p(\mathcal{L}(\mathbb{G}_n)) \to
L_q(\mathcal{L}(\mathbb{G}_n)) \big\| \, = \, 1 \ \Leftrightarrow
\ t \ge \frac12 \log \frac{q-1}{p-1}.$$
2. Hypercontractivity for ${\mathbb{F}}_n$ over twice the optimal time $$\big\| \mathcal{P}_{{\mathbb{F}}_n,t} \hskip1pt : L_p( \hskip0.5pt
\mathcal{L}({\mathbb{F}}_n) \hskip0.5pt ) \to L_q( \hskip0.5pt
\mathcal{L}({\mathbb{F}}_n) \hskip0.5pt ) \hskip1pt \big\| \, = \, 1
\hskip13pt \mbox{if} \hskip13pt t \ge \log \frac{q-1}{p-1}.$$
3. Optimal time hypercontractivity in the symmetric algebra $\mathcal{A}_{sym}^n$ $$\big\| \hskip1pt \mathcal{P}_{{\mathbb{F}}_n,t}
\hskip1pt : L_p( \hskip1pt \mathcal{A}_{sym}^n \hskip1pt ) \hskip1pt
\to \hskip1pt L_q( \hskip1pt \mathcal{A}_{sym}^n \hskip1pt )
\hskip1pt \big\| \, = \, 1 \ \Leftrightarrow \ t \ge \frac12 \log
\frac{q-1}{p-1}.$$
Theorem A i) extends Bonami’s theorem for ${\mathbb{Z}}_2^n$ to the free product case with optimal time estimates. According to the applications in complexity theory and quantum information of Bonami’s result, it is conceivable that Theorem A could be of independent interest in those areas. These potential applications will be explored in further research. Theorem A ii) gives the first hypercontractivity estimate for the free Poisson semigroup on ${\mathbb{F}}_n$, where a factor 2 is lost from the expected optimal time. This is related to our probabilistic approach to the problem and a little distortion must be done to make ${\mathbb{F}}_n$ fit in. Theorem A iii) refines this, providing optimal time estimates in the symmetric algebra $\mathcal{A}_{sym}^n$. We also obtain optimal time $L_p \to L_2$ hypercontractive estimates for linear combinations of words with length less than or equal to one. Apparently, our probabilistic approach in this paper is limited to go beyond the constant $2$ in the general case. We managed to push it to $1+\frac 14 \log2 \sim 1.173$ in the last section. Actually, we have recently found in [@JPPP] an alternative combinatorial/numerical method which yields optimal $L_2 \to L_q$ estimates for $q \in 2{\mathbb{Z}}$ and also reduces the general constant to $\log \, 3 \sim 1.099$ for $1 < p \le q <
\infty$. The drawback of this method is the numerical part: the larger is the number of generators $n$, the harder is to implement and test certain pathological terms in a computer. In this respect, Theorem A ii) is complementary since —at the price of a worse constant— we obtain uniform estimates in $n$.
As we have already mentioned, it is interesting to understand the two-point inequality as the convergence between the *trigonometric point of view* outlined above and the *gaussian point of view*, which was developed along the extensive study of hypercontractivity carried out in the context of quantum mechanics and operator algebras. The study of hypercontractivity in quantum mechanics dates back to the work of Nelson [@Nelson1] who showed that semiboundedness of certain Hamiltonians $H$ associated to a bosonic system can be obtained from the (hyper)contractivity of the semigroup $e^{-tA_\gamma}:L_2({\mathbb{R}}^d,\gamma)\rightarrow
L_2({\mathbb{R}}^d,\gamma)$, where $A_\gamma$ is the Dirichlet form operator for the Gaussian measure $\gamma$ on ${\mathbb{R}}^d$. After some contributions [@Glimm; @HoSi; @Segal] Nelson finally proved in [@Nelson2] that the previous semigroup is contractive from $L_p({\mathbb{R}}^d,\gamma)$ to $L_q({\mathbb{R}}^d,\gamma)$ if and only if $e^{-2t}\leq\frac{p-1}{q-1}$; thus obtaining the same optimal time as in the two-point inequality. By that time a new deep connection was shown by Gross in [@Gross1], who established the equivalence between the hypercontractivity of the semigroup $e^{-tA_\mu}$, where $A_\mu$ is the Dirichlet form operator associated to the measure $\mu$, and the logarithmic Sobolev inequality verified by $\mu$. During the next 30 years hypercontractivity and its equivalent formulation in terms of logarithmic Sobolev inequalities have found applications in many different areas of mathematics like probability theory, statistical mechanics or differential geometry. We refer the survey [@Gross3] for an excellent exposition of the topic.
The extension of Nelson’s theorem to the fermonic case started with Gross’ papers [@Gross4; @Gross2]. Namely, he adapted the argument in the bosonic case by considering a suitable Clifford algebra $\mathcal
C({\mathbb{R}}^d)$ on the fermion Fock space and noncommutative $L_p$ spaces on this algebra after Segal [@Segal2]. In particular, hypercontractivity makes perfectly sense in this context by considering the corresponding Ornstein-Uhlenbeck semigroup $$\mathcal{O}_t \, := \, e^{-tN_0}: \, L_2(\mathcal
C({\mathbb{R}}^d),\tau) \rightarrow L_2(\mathcal C({\mathbb{R}}^d),\tau).$$ Here $N_0$ denotes the fermion number operator, see Section \[Section: preliminars\] for the construction of the Clifford algebra $\mathcal
C({\mathbb{R}}^d)$ and a precise definition of the Ornstein-Uhlenbeck semigroup on fermion algebras. After some partial results [@Gross2; @Lindsay; @LiMe], the optimal time hypercontractivity bound in the fermionic case was finally obtained by Carlen and Lieb in [@CaLi] $$\big\| \mathcal{O}_t: L_p(\mathcal C({\mathbb{R}}^d)) \rightarrow
L_q(\mathcal C({\mathbb{R}}^d)) \big\| \, = \, 1 \ \Leftrightarrow \ t \ge
\frac12 \log \frac{q-1}{p-1}.$$ The proof deeply relies on the optimal 2-uniform convexity for matrices from [@BCL].
Beyond its own interest in quantum mechanics, these contributions represent the starting point of hypercontractivity in the noncommutative context. This line was continued by Biane [@Biane], who extended Carlen and Lieb’s work and obtained optimal time estimates for the $q$-Gaussian von Neumann algebras $\Gamma_q$ introduced by Bozejko, Kümmerer and Speicher [@BKS]. These algebras interpolate between the bosonic and fermonic frameworks, corresponding to $q = \pm 1$. The semigroup for $q=0$ acts diagonally on free semi-circular variables —instead of free generators as in the case of the free Poisson semigroup— in the context of Voiculescu’s free probability theory [@VDN]. We also refer to [@Janson; @Kemp; @K1; @K2; @LeRi] for related results in this line. On the other hand, the usefulness of the two-point inequality in the context of computer science has motivated some other extensions to the noncommutative setting more focused on its applications to quantum computation and quantum information theory. In [@BRW], the authors studied extensions of Bonami’s result to matrix-valued functions $f:
{\mathbb{Z}}_2^n \rightarrow M_n({\mathbb{C}})$, finding optimal estimates for $q=2$ and showing some applications to coding theory. In [@MoOs], the authors introduced quantum boolean functions and obtained hypercontractivity estimates in this context with some consequences in quantum information theory, see also the recent work [@Montanaro].
The very nice point here is that, although our main motivation to study the Poisson semigroup comes from harmonic analysis, we realized that a natural way to tackle this problem is by means of studying the Ornstein-Uhlenbeck semigroup on certain von Neumann algebras. In particular, a significant portion of Theorem A follows from our main result, which extends Carlen and Lieb’s theorem to the case of free product of Clifford algebras. The precise definitions of reduced free products which appear in the statement will be recalled for the non-expert reader in the body of the paper.
Let $\M_\alpha = \mathcal{C}({\mathbb{R}}^{d_\alpha})$ be the Clifford algebra with $d_\alpha$ generators for any $1 \le \alpha \le n$ and construct the corresponding reduced free product von Neumann algebra $\M = \M_1
* \M_2 * \cdots * \M_n$. If $\mathcal{O}_\alpha =
(\mathcal{O}_{\alpha,t})_{t \ge 0}$ denotes the Ornstein-Uhlenbeck semigroup acting on $\M_\alpha$, consider the free product semigroup $\mathcal{O}_\mathcal{M} = (\mathcal{O}_{\M,t})_{t \ge 0}$ given by $\mathcal{O}_{\M,t} = \mathcal{O}_{1,t} * \mathcal{O}_{2,t} * \cdots *
\mathcal{O}_{n,t}$. Then, we find for $1 < p \le q < \infty$ $$\big\| \mathcal{O}_{\M,t}: L_p(\M) \to L_q(\M) \big\| \, = \, 1
\quad \Leftrightarrow \quad t \, \ge \, \frac12 \log
\frac{q-1}{p-1}.$$
It is relevant to point out a crucial difference between our approach and the one followed in [@Bonami; @CaLi; @Nelson2]. Indeed, in all those cases the key point in the argument is certain basic inequality —like Bonami’s two-point inequality or Ball/Carlen/Lieb’s convexity inequality for matrices— and the general result follows from an inductive argument due to the tensor product structure of the problem. However, no tensor product structure can be found in our setting (Theorems A and B). In order to face this problem, Biane showed in [@Biane] that certain optimal hypercontractive estimates hold in the case of spin matrix algebras with mixed commutation/anticommutation relations, and then applied Speicher’s central limit theorem [@Speicher]. In this paper we will extend Biane’s and Speicher’s results by showing that a wide range of von Neumann algebras can also be approximated by these spin systems. Namely, the proof of Theorem B will show that the same result can be stated in a much more general context. As we shall explain, we may consider the free product of Biane’s mixed spins algebras which in turn gives optimal hypercontractivity estimates for the free products of $q$-deformed algebras with $q_1, q_2, \ldots, q_n \in [-1,1]$.
**Acknowledgements.** The authors were supported by ICMAT Severo Ochoa Grant SEV-2011-0087 (Spain); ERC Grant StG-256997-CZOSQP (EU); NSF Grant DMS-0901457 (USA); MINECO Grants MTM-2010-16518 & MTM-2011-26912 and “Juan de la Cierva” program (Spain); and ANR-2011-BS01-008-11 (France).
[**Preliminaries**]{} {#Section: preliminars}
=====================
In this section we briefly review the definition of the CAR algebra and the Ornstein-Uhlenbeck semigroup acting on it. We also recall the construction of the reduced free product of a family of von Neumann algebras and introduce the free Ornstein-Uhlenbeck semigroup on a reduced free product of Clifford algebras.
The Ornstein-Uhlenbeck semigroup
--------------------------------
The standard way to construct a system of $d$ fermion degrees of freedom is by means of the antisymmetric Fock space. Let us consider the $d$-dimensional real Hilbert space $\mathcal H_{\mathbb{R}}={\mathbb{R}}^d$ and its complexification $\mathcal H_{\mathbb{C}}={\mathbb{C}}^d$. Define the Fock space $$\mathcal F(\mathcal H_{\mathbb{R}}) \, = \, {\mathbb{C}}\Omega \oplus
\bigoplus_{m=1}^\infty \mathcal H_{\mathbb{C}}^{\otimes_m}$$ for some fixed unit vector $\Omega \in \mathcal H_{\mathbb{C}}$ called the vacuum. If $S_m$ denotes the symmetric group of permutations over $\{1,2,\ldots,m\}$ and $i(\beta)$ the number of inversions of the permutation $\beta$, we define the hermitian form $\langle \cdot, \cdot \rangle$ on $\mathcal
F(\mathcal H_{\mathbb{R}})$ by $\langle \Omega,\Omega \rangle =1$ and the following identity $$\big\langle f_1\otimes\cdots\otimes f_m ,
g_1\otimes\cdots\otimes g_n \big\rangle \, = \, \delta_{mn}
\sum_{\beta \in S_m}^{\null} (-1)^{i(\beta)} \langle
f_1,g_{\beta(1)}\rangle \cdots \langle f_m, g_{\beta(m)} \rangle.$$ It is not difficult to see that the hermitian form $\langle \cdot, \cdot
\rangle$ is non-negative. Therefore, if we consider the completion of the quotient by the corresponding kernel, we obtain a Hilbert space that we will call again $\mathcal F(\mathcal H_{\mathbb{R}})$. Let us denote by $(e_j)_{j=1}^d$ the canonical basis of $\mathcal H_{\mathbb{R}}= {\mathbb{R}}^d$. Then, we define the $j$-th fermion annihilation operator acting on $\mathcal
F(\mathcal H_{\mathbb{R}})$ by linearity as $c_j(\Omega)=0$ and $$c_j(f_1\otimes\cdots \otimes f_m) \, = \, \sum_{i=1}^m(-1)^{i-1}
\langle f_i,e_j\rangle \, f_1 \otimes\cdots\otimes f_{i-1}\otimes
f_{i+1}\otimes\cdots\otimes f_m.$$ Its adjoint $c_j^*$ is called the $j$-th fermion creation operator on $\mathcal F(\mathcal H_{\mathbb{R}})$. It is determined by $c_j^*(\Omega)=e_j$ and $c_j^*(f_1\otimes\cdots \otimes
f_m)=e_j\otimes f_1\otimes\cdots\otimes f_m$. It is quite instrumental to observe that $c_ic_j + c_jc_i = 0$ and $c_i c_j^* + c_j^*c_i =
\delta_{ij} \mathbf{1}$. The basic free Hamiltonian on $\mathcal
F(\mathcal H_{\mathbb{R}})$ is the fermion number operator $$N_0 = \sum_{j=1}^d
c_j^*c_j.$$ It generates the fermion oscillator semigroup $(
\exp(-tN_0) )_{t \ge 0}$. Then, one defines the configuration operators $x_j=c_j+c_j^*$ for $1 \le j \le d$. Denote by $\mathcal
C({\mathbb{R}}^d)$ the unit algebra generated by them. Note that these operators verify the canonical anti-commutation relations (CAR) $$x_ix_j+x_jx_i=2\delta_{ij} \qquad \mbox{ and } \qquad
x_j^*=x_j.$$ It is well-known that $\mathcal C({\mathbb{R}}^d)$ can be concretely represented as a subalgebra of the matrix algebra $\mathbb{M}_{2^d}$ by considering $d$-chains formed by tensor products of Pauli matrices. The key point for us is that the $2^d$ distinct monomials in the $x_j$’s define a basis of $\mathcal C({\mathbb{R}}^d)$ as a vector space. Indeed, given any subset $A$ of $[d] := \{1,2,\ldots,d\}$ we shall write $x_A = x_{j_1} x_{j_2} \cdots x_{j_s}$ where $(j_1, j_2,
\ldots, j_s)$ is an enumeration of $A$ in increasing order. If we also set $x_\emptyset = \mathbf{1}$, it turns out that $\{ x_A \, | \ A
\subset [d] \}$ is a linear basis of $\mathcal C({\mathbb{R}}^d)$. In particular, any $X \in \mathcal C({\mathbb{R}}^d)$ has the form $$X \, = \, \alpha_\emptyset \mathbf{1} \, + \, \sum_{s=1}^d \, \sum_{1 \le j_1
< \cdots < j_s \le d} \alpha_{j_1, \ldots, j_s} x_{j_1} \cdots
x_{j_s}.$$ The vacuum $\Omega$ defines a tracial state $\tau$ on $\mathcal
C({\mathbb{R}}^d)$ by $\tau(X)=\langle X\Omega,\Omega\rangle$. We denote by $L_p(\mathcal C({\mathbb{R}}^d),\tau)$ or just $L_p(\mathcal C({\mathbb{R}}^d))$ the associated non-commutative $L_p$-space. The map $X\mapsto X\Omega$ defines a continuous embedding of $\mathcal C({\mathbb{R}}^d)$ into $\mathcal
F({\mathbb{R}}^d)$ which extends to a unitary isomorphism $L_2(\mathcal C({\mathbb{R}}^d))
\simeq \mathcal F({\mathbb{R}}^d)$. Then, instead of working on the Fock space $\mathcal F({\mathbb{R}}^d)$ and with the semigroup $\exp(-tN_0)$, we can equivalently consider $\mathcal C({\mathbb{R}}^d)$ and the Ornstein-Uhlenbeck semigroup on $\mathcal C({\mathbb{R}}^d)$ defined by $$\begin{aligned}
\mathcal{O}_t(X) \, = \, \alpha_\emptyset \mathbf{1} \, + \, \sum_{s=1}^d
e^{-ts} \, \sum_{1 \le j_1 < \cdots < j_s \le d} \alpha_{j_1, \ldots,
j_s} x_{j_1} \cdots x_{j_s}.\end{aligned}$$ If $1 < p \le q< \infty$, the main result in [@CaLi] yields $$\big\| \mathcal{O}_t: L_p(C({\mathbb{R}}^d))\rightarrow L_q(C({\mathbb{R}}^d))
\big\| \, = \, 1 \ \Leftrightarrow \ t \ge \frac12 \log
\frac{q-1}{p-1}.$$
Free product of von Neumann algebras
------------------------------------
Let $(A_j, \phi_j)_{j \in J}$ be a family of unital C$^*$-algebras with distinguished states $\phi_j$ whose GNS constructions $(\pi_j,
\mathcal H_j, \xi_j)$ with $\mathcal H_j=L_2(A_j, \phi_j)$ are faithful. Let us define $${\stackrel{\circ}{A_j}}\, = \, \Big\{ a \in A_j \, \big|
\ \phi_j(a)=0 \Big\} \qquad \mbox{and} \qquad {\stackrel{\circ}{\mathcal{H}_j}}=\xi_j^\perp$$ so that $A_j=\mathbb{C} \mathbf{1} \oplus {\stackrel{\circ}{A_j}}$ and $\mathcal
H_j=\mathbb{C}\xi_j \oplus {\stackrel{\circ}{\mathcal{H}_j}}$. Note that we have natural maps $i_j=A_j \rightarrow \mathcal H_j$ such that $\phi_j(a^*b)=\langle
i_j(a),i_j(b) \rangle_{\mathcal H_j}$ for every $j \in J$. Let us consider the full Fock space associated to the free product -6pt $$\mathcal F \ = \ \mathbb{C}\Omega \, \oplus
\bigoplus_{\substack{m \ge 1\\ j_1 \neq j_2 \neq \cdots \neq j_m}}
\stackrel{\circ}{\mathcal{H}}_{j_1} \otimes\cdots\otimes
\stackrel{\circ}{\mathcal{H}}_{j_m}$$ with inner product $$\big\langle
h_1 \otimes \cdots \otimes h_m, h_1' \otimes \cdots \otimes h_n'
\big\rangle \, = \, \delta_{mn} \prod_{j=1}^m \langle h_j,h_j'
\rangle.$$ Each algebra $A_j$ acts non-degenerately on $\mathcal F$ via the map $\omega_j: A_j\rightarrow \mathcal{B}(\mathcal F)$ in the following manner. Since we can decompose every $z \in A_j$ as $z =
\phi_j(z) \mathbf{1}+ a$ with $\phi_j(a)=0$, it suffices to define $\omega_j(a)$. Let $h_1 \otimes \cdots \otimes h_m$ be a generic element in $\mathcal{F}$ with $h_i \in \mathcal{H}_{j_i} \ominus {\mathbb{C}}\xi_{j_i}$. If $j \neq j_1$, we set $$\omega_j(a) \big( h_1\otimes
\cdots \otimes h_m \big) \, = \, i_j(a) \otimes h_1\otimes \cdots
\otimes h_m.$$ When $j = j_1$ we add and subtract the mean to obtain $$\begin{aligned}
\omega_j(a) \big( h_1\otimes\cdots \otimes h_m \big) & = & \big\langle
\xi_j, \pi_j(a)(h_1) \big\rangle_{\mathcal{H}_j}
h_2\otimes\cdots\otimes h_m \\ & + & \Big( \pi_j(a)(h_1) - \big\langle
\xi_j,\pi_j(a)(h_1) \big\rangle_{\mathcal{H}_j} \xi_j \Big) \otimes
h_2\otimes\cdots \otimes h_m.\end{aligned}$$ The faithfulness of the GNS construction of $(A_j, \phi_j)$ implies that the representation $\omega_j$ is faithful for every $j \in
J$. Thus, we may find a copy of the algebraic free product $$A \ =
\ \mathbb{C}\Omega \, \oplus \bigoplus_{\substack{m \geq 1\\ j_1\neq
j_2\neq\cdots\neq j_m}}
\stackrel{\circ}{A_{j_1}}\otimes\cdots\otimes
\stackrel{\circ}{A_{j_m}}$$ in $\mathcal{B}(\mathcal F)$. The reduced free product of the family $(A_j, \phi_j)_{j \in J}$ is the C$^*$-algebra generated by these actions. In other words, the norm closure of $A$ in $\mathcal{B}(\mathcal F)$. It is denoted by $$(A,\phi) \, = \, *_{j \in J} (A_j, \phi_j),$$ where the state $\phi$ on $A$ is given by $$\phi(\mathbf{1}) \, = \, 1 \qquad
\mbox{and} \qquad \phi(a_1\otimes\cdots\otimes a_m)=0$$ for $m\geq 1$ and $a_i \in \stackrel{\circ}{A_{j_i}}$ with $j_1\neq
j_2\neq\cdots\neq j_m$. Each $A_j$ is naturally considered as a subalgebra of $A$ and the restriction of $\phi$ to $A_j$ coincides with $\phi_j$. It is helpful to think of the elementary tensors above $a_1\otimes\cdots\otimes a_m$ as words of length $m$, where the empty word $\Omega$ has length $0$. In this sense, a word $a_1\otimes\cdots\otimes a_m$ can be identified with the product $a_1a_2\cdots a_m$ via the formula $a_1\cdots a_m\Omega=a_1\otimes
\cdots\otimes a_m$.
This construction also holds in the category of von Neumann algebras. Let $(\mathcal M_j, \phi_j)_{j \in J}$ be a family of von Neumann algebras with distinguished states $\phi_j$ whose GNS constructions $(\pi_j, \mathcal{H}_j, \xi_j)$ are faithful. Then, the corresponding reduced free product von Neumann algebra is the weak-$*$ closure of $*_{j \in J}(\mathcal M_j, \phi_j)$ in $\mathcal{B}(\mathcal F)$ which will be denoted by $(\M,\phi) \, = \,
\overline{*}_{j \in J}(\mathcal M_j,\phi_j).$ As before, the $\mathcal
M_j$’s are regarded as von Neumann subalgebras of $\M$ and the restriction of $\phi$ to $\mathcal M_j$ coincides with $\phi_j$. A more complete explanation of the reduced free product of von Neumann algebras can be found in [@VDN]. Let us now consider a family $(\Lambda_j: \mathcal M_j \rightarrow \mathcal M_j)_{j \in J}$ of normal, completely positive, unital and trace preserving maps. Then, it is known from [@BlDy Theorem 3.8] that there exists a map $\Lambda \, = \, *_{j \in J} \Lambda_j:\M\rightarrow \M$ such that $\Lambda (x_1 x_2 \cdots x_m) = \Lambda_{j_1}(x_1) \cdots
\Lambda_{j_m}(x_m)$, whenever $x_i \in \M_{j_i}$ is trace 0 and $j_i
\neq j_{i+1}$ for $1 \le i \le m-1$. This map is called the free product map of the $\Lambda_j$’s. In particular we may take $\M_j =
\mathcal C({\mathbb{R}}^d)$ for $1 \le j \le n$ and $\Lambda_j =
\mathcal{O}_{j,t}$, the Ornstein-Uhlenbeck semigroup on $\M_j$ at time $t$. The resulting free product maps $\mathcal{O}_{\M} =
(\mathcal{O}_{\M,t})_{t \ge 0}$ with $\mathcal{O}_{\M,t} =
\mathcal{O}_{1,t} * \mathcal{O}_{2,t} * \cdots * \mathcal{O}_{n,t}$ will be referred to as the *free Ornstein-Uhlenbeck semigroup* on the reduced free product von Neumann algebra $\M$.
[**The free Ornstein-Uhlenbeck semigroup**]{} {#Section: proof of theorem B}
=============================================
This section is devoted to the proof of Theorem B. Of course, we may and will assume for simplicity that $d_\alpha = d$ for all $1 \le \alpha \le
n$. The key idea is to describe the free product of fermion algebras and the corresponding Ornstein-Uhlenbeck semigroup as the limit objects of certain spin matrix models and certain semigroups defined on them. In this sense, we will extend Biane’s results [@Biane] by showing that these matrix models can be used to describe a wide range of operator algebra frameworks.
Note that the free Ornstein-Uhlenbeck semigroup restricted to a single free copy $\M_\alpha$ coincides with the fermion oscillator semigroup on $\M_\alpha$. In particular, we know from Carlen and Lieb’s theorem [@CaLi] that the optimal time in Theorem B must be greater than or equal to $\frac12 \log \frac{q-1}{p-1}$. This proves the necessity, it remains to prove the sufficiency. Given $1 \le \alpha \le n$ and recalling that $[d]$ stands for $\{1,2,\ldots,d\}$, we denote by $(x_i^\alpha)_{i \in [d]}$ the generators of $\M_\alpha=\mathcal
C({\mathbb{R}}^d)$. A reduced word in the free product $\M = \M_1 * \M_2 *
\cdots * \M_n$ is then of the form $$\label{general element free productI}
x \, = \, x_{A_1}^{\alpha_1}\cdots x_{A_\ell}^{\alpha_\ell}$$ with $A_j \subset [d]$ and $\alpha_j \neq \alpha_{j+1}$. The case $\ell = 0$ refers to the empty word $\mathbf{1}$. If we set $s_j =
|A_j|$ and write $A_j = \{i_{s_1+ \cdots +
s_{j-1}+1},\ldots,i_{s_1+\cdots+s_{j-1}+s_j}\}$ —labeling the indices in a strictly increasing order— $x$ can be written as follows $$\begin{aligned}
\label{general element free productII}
x \ = \ \overbrace{x_{i_1}^{\alpha_1}\cdots
x_{i_{s_1}}^{\alpha_1}}^{x_{A_1}^{\alpha_1}} \,
\overbrace{x_{i_{s_1+1}}^{\alpha_2}\cdots
x_{i_{s_1+s_2}}^{\alpha_2}}^{x_{A_2}^{\alpha_2}} \, \cdots \,
\overbrace{x_{i_{s_1+\cdots +s_{\ell-1}+1}}^{\alpha_\ell}\cdots
x_{i_{s_1+\cdots
+s_\ell}}^{\alpha_\ell}}^{x_{A_\ell}^{\alpha_\ell}}.\end{aligned}$$ In what follows, we will use the notation $|x|=|A_1|+\cdots+|A_\ell|=s_1+\cdots+s_\ell$.
Spin matrix model {#Subsection: matrix model}
-----------------
Given $m \geq 1$, we will describe a spin system with mixed commutation and anticommutation relations which approximates the free product of fermions $\M$ as $m \to \infty$. Let us first recall the construction of a spin algebra in general. In our setting, we will need to consider three indices. This is why we introduce the sets $\Upsilon=[n]\times [d]\times \mathbb{Z}_+$ and $\Upsilon_m =
[n]\times [d]\times [m]$ for $m\geq 1$. Let $\varepsilon:
\Upsilon\times \Upsilon\rightarrow \{-1,1\}$ be any map satisfying
- $\varepsilon$ is symmetric: $\varepsilon(x,y) = \varepsilon
(y,x)$,
- $\varepsilon \equiv -1$ on the diagonal: $\varepsilon (x,x) =
-1$.
Given $m \geq 1$, we will write $\varepsilon_m$ to denote the truncation of $\varepsilon$ to $\Upsilon_m \times
\Upsilon_m$. Consider the complex unital algebra $\mathcal
A_{\varepsilon_m}$ generated by the elements $(x_i^\alpha(k))_{(\alpha,i,k)\in \Upsilon_m}$ which satisfy the commutation/anticommutation relations $$\begin{aligned}
\label{relations free}
x_i^\alpha(k)x_j^\beta(\ell)-\varepsilon\big((\alpha,i,k),(\beta,j,\ell)\big)x_j^\beta(\ell)x_i^\alpha(k)
\, = \, 2\delta_{(\alpha,i,k),(\beta,j,\ell)}\end{aligned}$$ for $(\alpha,i,k),(\beta,j,\ell) \in \Upsilon_m$. We endow $\mathcal
A_{\varepsilon_n}$ with the antilinear involution such that $x_{i}^{\alpha}(k)^*=x_{i}^{\alpha}(k)$ for every $(\alpha,i,k)\in
\Upsilon_m$. If we equip $\Upsilon_m$ with the lexicographical order, a basis of the linear space $\mathcal A_{\varepsilon_m}$ is given by $x_\emptyset^{{\varepsilon}_m } = \mathbf{1}_{\mathcal A_{\varepsilon_m}}$ and the set of reduced words written in increasing order. Namely, elements of the form $$x_A^{{\varepsilon}_m} \, = \, x_{i_1}^{\alpha_1}(k_1)\cdots
x_{i_s}^{\alpha_s}(k_s),$$ where $A = \{(\alpha_1,i_1,k_1), \ldots,
(\alpha_s,i_s,k_s)\} \subset \Upsilon_m$ with $(\alpha_j,i_j,k_j)<(\alpha_{j+1},i_{j+1},k_{j+1})$ for $1 \le j \le
s-1$. For any such element we set $|x_A^{{\varepsilon}_m}|=|A|=s$. Define the tracial state on $\mathcal A_{\varepsilon_m}$ given by $\tau_{{\varepsilon}_m}(x_A^{{\varepsilon}_m})=\delta_{\emptyset, A}$ for $A \subset
\Upsilon_m$. The given basis turns out to be orthonormal with respect to the inner product $\langle x,y \rangle = \tau_{{\varepsilon}_m}(x^*y)$. Let $\mathcal A_{\varepsilon_m}$ act by left multiplication on the Hilbert space $\mathcal H_{\mathcal A_{\varepsilon_m}} = (\mathcal
A_{\varepsilon_m}, \langle \cdot,\cdot\rangle)$ to get a faithful $*$-representation of $\mathcal A_{\varepsilon_m}$ on $\mathcal
H_{\mathcal A_{\varepsilon_m}}$. We may endow $\mathcal
A_{\varepsilon_m}$ with the von Neumann algebra structure induced by this representation and denote by $L_p(\mathcal A_{\varepsilon_m},
\tau_{{\varepsilon}_m})$ the associated non-commutative $L_p$-space. At this point, it is natural to define the *$\varepsilon_m$-Ornstein-Uhlenbeck semigroup* on $\mathcal
A_{\varepsilon_m}$ by $$\label{Ptepsn}
\mathcal{S}_{{\varepsilon}_m,t}(x_A^{{\varepsilon}_m}) \, = \, e^{-t|x_A^{{\varepsilon}_m}|}
x_A^{{\varepsilon}_m}.$$ Biane extended hypercontractivity for fermions to these spin algebras in [@Biane] $$\label{Biane's theorem matrix model}
\big\| \mathcal{S}_{\varepsilon_m,t}: L_p(\mathcal A_{\varepsilon_m})
\rightarrow L_q(\mathcal A_{\varepsilon_m}) \big\| \, = \, 1
\ \Leftrightarrow \ t \, \ge \, \frac12 \log \frac{q-1}{p-1},$$ whenever $1 < p \le q < \infty$. We will also use the following direct consequence of Biane’s result. Namely, given $1 \le p < \infty$ and $r
\in {\mathbb{Z}}_+$ we may find constants $C_{p,r} > 0$ such that the following inequality holds uniformly for all $m \geq 1$ and all homogeneous polynomials $P$ of degree $r$ in $|\Upsilon_m|$ noncommutative indeterminates satisfying and written in reduced form $$\label{Biane 2-p}
\Big\|P\big((x_i^\alpha(k))_{(\alpha,i,k) \in \Upsilon_m} \big)
\Big\|_{L_p(\mathcal A_{\varepsilon_m})} \, \le \, C_{p,r} \Big\| P
\big((x_i^\alpha(k))_{(\alpha,i,k) \in \Upsilon_m} \big)
\Big\|_{L_2(\mathcal A_{\varepsilon_m})}.$$ According to , it is straightforward to show that we can take $C_{p,r}=(p-1)^{r/2}$.
A central limit theorem
-----------------------
In order to approximate the free product $\M$ of Clifford algebras, we need to choose the commutation/anticommutation relations randomly. More precisely, we consider a probability space $(\Omega,
\mu)$ and a family of independent random variables $$\varepsilon
\big((\alpha,i,k),(\beta,j,\ell) \big): \Omega \rightarrow \{-1,1\}
\quad \mbox{for} \quad (\alpha,i,k)<(\beta,j,\ell)$$ which are distributed as follows $$\begin{aligned}
\label{sign choice}
\mu \Big(\varepsilon\big(( \alpha,i,k),(\beta,j,\ell)\big)=-1\Big) \,
= \, \begin{cases} 1 & \mbox{if } \alpha=\beta, \\ 1/2 & \text{if }
\alpha\neq\beta. \end{cases}\end{aligned}$$ In particular, this means that all the generators $(x_i^\alpha(k))_{i\in [d],k\in[m]}$ anticommute for $\alpha \in [n]$ fixed and all $m \ge 1$. Therefore, the algebra $\A_{{\varepsilon}_m}^\alpha$ generated by them is isomorphic to $\mathcal C({\mathbb{R}}^{dm})$. Formally, we have a matrix model for each $\omega \in \Omega$. In this sense, the generators $x_i^\alpha(k)$ and the algebras $\A_{{\varepsilon}_m}^\alpha$ are also functions of $\omega$. In order to simplify the notation, we will not specify this dependence unless it is necessary for clarity in the exposition. Define also the algebra $$\tilde{\mathcal
A}^\alpha_{\varepsilon_m} \, = \, \Big\langle \tilde{x}_i^\alpha(m)
\, \big| \ i \in [d] \Big\rangle$$ with generators given by $$\tilde{x}_i^\alpha(m) \, = \, \frac{1}{\sqrt{m}}\sum_{k=1}^m
x_i^\alpha(k).$$
\[single algebra\] The von Neumann algebra $\tilde{\mathcal A}^\alpha_{\varepsilon_m}$ is canonically isomorphic to $\mathcal C({\mathbb{R}}^d)$.
[ [**Proof.** ]{}]{}It suffices to prove that the generators verify the CAR relations. All of them are self-adjoint since the same holds for the $x_i^\alpha$’s. Since $\alpha$ is fixed, our choice (\[sign choice\]) of the sign function $\varepsilon$ and give\
-20pt ${}$ $\displaystyle
\tilde{x}_i^\alpha(m)
\tilde{x}_j^\alpha(m)+\tilde{x}_j^\alpha(m)\tilde{x}_i^\alpha(m) \, =
\, \frac{1}{m}\sum_{k=1}^m\sum_{\ell=1}^m
x_i^\alpha(k)x_j^\alpha(\ell)+x_j^\alpha(\ell)x_i^\alpha(k) \, = \,
2\delta_{ij}.$ [ $\square$ 0.2cm]{} We will denote by $\Pi(s)$ the set of all partitions of $[s] =
\{1,2,\ldots,s\}$. Given $\sigma,\pi \in \Pi(s)$, we will write $\sigma\leq \pi$ if every block of the partition $\sigma$ is contained in some block of $\pi$. We denote by $\sigma_0$ the smallest partition, in which every block is a singleton. Given an $s$-tuple $\ibar=(i_1,\cdots, i_s)\in [N]^s$ for some $N$, we can define the partition $\sigma(\ibar)$ associated to $\ibar$ by imposing that two elements $j,k \in [s]$ belong to the same block of $\sigma(\ibar)$ if and only if $i_j=i_k$. We will denote by $\Pi_2(s)$ the set of all pair partitions. That is, partitions $\sigma=\{V_1,\cdots, V_{s/2}\}$ such that $|V_j|=2$ for every block $V_j$. In this case, we will write $V_j = \{e_j,z_j\}$ with $e_j< z_j$ so that $e_1< e_2 < \cdots <
e_{s/2}$. For a pair partition $\sigma\in \Pi_2(s)$ we define the set of crossings of $\sigma$ by $$I(\sigma) \, = \, \Big\{ (k,\ell) \,
\big| \ 1 \le k, \ell \le s, \ e_k < e_\ell < z_k < z_\ell \Big\}.$$ Moreover, given an $s$-tuple $\underline{\alpha} = (\alpha_1,\ldots,
\alpha_s)$ such that $\sigma \leq \sigma(\underline{\alpha})$, we can define the set of crossings of $\sigma$ with respect to $\underline{\alpha}$ by $I_{\underline{\alpha}}(\sigma)=\{(k,\ell) \in
I(\sigma) : \alpha_{e_k} \neq \alpha_{e_\ell}\}$. This notation allows us to describe the moments of reduced words in $\M$ with a simple formula. Indeed, the following lemma arises from [@Speicher Lemma 2] and a simple induction argument like the one used below to prove identity .
\[moments free pdct fermions\] If $\ibar \in [d]^s$ and $\underline{\alpha} \in [n]^s$ we have $$\tau\big(x_{i_1}^{\alpha_1} \cdots x_{i_s}^{\alpha_s}\big) \, =
\, \delta_{s \in 2 {\mathbb{Z}}} \sum_{\substack{\sigma\in \Pi_2(s)\\ \sigma
\leq \sigma(\ibar), \sigma(\underline{\alpha})
\\ I_{\underline{\alpha}}(\sigma)=\emptyset}}(-1)^{\#I(\sigma)}.$$
We can now prove that the moments of the free product von Neumann algebra $\M$ are the almost everywhere limit of the moments of our matrix model. More explicitly, we find the following central limit type theorem.
\[moments\] If $\ibar \in [d]^s$ and $\underline{\alpha} \in [n]^s$ we have $$\lim_{m\to \infty} \tau_{{\varepsilon}_m}
\Big(\tilde{x}_{i_1}^{\alpha_1}(m)(\omega) \cdots
\tilde{x}_{i_s}^{\alpha_s}(m)(\omega) \Big) \, = \, \tau
\big(x_{i_1}^{\alpha_1} \cdots x_{i_s}^{\alpha_s} \big) \quad
\mbox{a.e.}$$
[ [**Proof.** ]{}]{}We will first prove that the convergence holds in expectation. For $\omega\in \Omega$ fixed, by developing and splitting the sum according to the distribution we obtain $$\begin{aligned}
\label{sum with partitions}
\lefteqn{\hskip-15pt \tau_{{\varepsilon}_m} \Big(
\tilde{x}_{i_1}^{\alpha_1}(m)(\omega) \cdots
\tilde{x}_{i_s}^{\alpha_s}(m)(\omega) \Big)} \\ [3pt] \nonumber & =
& \frac{1}{m^{s/2}} \sum_{\k \in [m]^s} \tau_{{\varepsilon}_m}
\big(x_{i_1}^{\alpha_1}(k_1)(\omega) \cdots
x_{i_s}^{\alpha_m}(k_s)(\omega) \big) \\ [3pt] \nonumber &= &
\frac{1}{m^{s/2}} \sum_{\sigma \in \Pi(s)}
\underbrace{\sum_{\substack{\k \in [m]^s \\ \sigma (\k)=\sigma}}
\tau_{{\varepsilon}_m} \big( x_{i_1}^{\alpha_1}(k_1)(\omega)\cdots
x_{i_s}^{\alpha_s}(k_s)(\omega)\big)}_{\mu_{\sigma}(\omega)}.\end{aligned}$$ We claim that $$\lim_{m \to \infty} \frac{1}{m^{s/2}}
\mu_{\sigma}(\omega) \, = \, 0$$ for every $\sigma \in \Pi(s)
\backslash \Pi_2(s)$ and all $\omega \in \Omega$. Indeed, the upper bound $\mu_{\sigma}(\omega) \le m^r$ holds when $\sigma$ has $r$ blocks since $|\tau_{{\varepsilon}_m} ( x_{i_1}^{\alpha_1}(k_1)(\omega)\cdots
x_{i_s}^{\alpha_s}(k_s)(\omega))| \le 1$. Hence, the limit above vanishes for $r < s/2$. It then suffices to show that the same limit vanishes when $\sigma$ contains a singleton $\{j_0\}$. However, in this case we have $$\tau_{{\varepsilon}_m} \big(x_{i_1}^{\alpha_1}(k_1)(\omega)
\cdots x_{i_s}^{\alpha_s}(k_s)(\omega) \big) \, = \, 0$$ whenever $\sigma (\k)=\sigma$ since the $j_0$-th term can not be cancelled. This proves our claim. Hence, the only partitions which may contribute in the sum (\[sum with partitions\]) are pair partitions $\sigma=\{\{e_1,z_1\}, \ldots ,\{e_\frac{s}{2},z_\frac{s}{2}\}\}$. In particular, if $s$ is odd we immediately obtain that the trace converges to zero in (\[sum with partitions\]). Note that given such a pair partition $\sigma$, we must have that $\sigma \leq
\sigma(\underline{\alpha})$ and $\sigma \leq \sigma(\ibar)$. Indeed, if this is not the case we will have $i_{e_j}\neq i_{z_j}$ or $\alpha_{e_j}\neq \alpha_{z_j}$ for some $j=1,2,\ldots, s/2$. Now, for every $\k \in [m]^s$ such that $\sigma(\k)= \sigma$ we have $k_{e_j}=k_{z_j}\neq k_\ell$ for every $\ell \neq e_j,z_j$. Thus, the only way for the elements $$x_{i_{e_j}}^{\alpha_{e_j}}(k_{e_j})(\omega) \quad
\mbox{and} \quad x_{i_{z_j}}^{\alpha_{z_j}}(k_{z_j})(\omega)$$ to cancel is to match each other. Thus, we can assume that $(\alpha_{e_j},i_{e_j})=(\alpha_{z_j},i_{z_j})$.
We have seen that the letters of our word should match in pairs. We are now reduced to study the sign which arises from the commutation/anticommutation relations to cancel all elements. Assume that $\sigma$ has a crossing with respect to $\underline{\alpha}=(\alpha_1,\ldots, \alpha_s)$. That is, there exists $(k,\ell) \in I(\sigma)$ such that $\alpha_{e_k} \neq
\alpha_{e_\ell}$. Then we find that $$\mathbb{E}_\omega \tau_{{\varepsilon}_m}
\big(x_{i_1}^{\alpha_1}(k_1)(\omega) \cdots
x_{i_s}^{\alpha_s}(k_s)(\omega) \big) \, = \, 0$$ for every $(k_1,\ldots ,k_s)$ such that $\sigma(k_1,\ldots ,k_s) =
\sigma$. Indeed, define the sign function $$\varepsilon^{\underline{\alpha}}_{(k,\ell)}:=\varepsilon\big((\alpha_{e_\ell},i_{e_\ell},k_{e_\ell}),(\alpha_{z_k},i_{z_k},k_{z_k})
\big).$$ If $\sigma$ has such a crossing, we obtain (among others) this sign only once when canceling the letters associated to $(\alpha_{e_k}, i_{e_k}, k_{e_k})$ and $(\alpha_{z_k}, i_{z_k},
k_{z_k})$ as well as $(\alpha_{e_\ell}, i_{e_\ell}, k_{e_\ell})$ and $(\alpha_{z_\ell},i_{z_\ell}, k_{z_\ell})$. Furthermore, by independence and since $\mathbb{E}_\omega\varepsilon^{\underline{\alpha}}_{(k,\ell)}=0$ we get $$\begin{aligned}
\lefteqn{\hskip-20pt \mathbb{E}_\omega \tau_{{\varepsilon}_m}
\big(x_{i_1}^{\alpha_1}(k_1)(\omega)\cdots
x_{i_s}^{\alpha_s}(k_1)(\omega)\big)} \\ & = & \pm \,
\mathbb{E}_\omega \Big( \prod_{(k,\ell) \in
I_{\underline{\alpha}}(\sigma)}
\varepsilon^{\underline{\alpha}}_{(k,\ell)} \Big) \ = \ \pm
\prod_{(k,\ell) \in I_{\underline{\alpha}}(\sigma)} \mathbb{E}_\omega
\varepsilon^{\underline{\alpha}}_{(k,\ell)}=0,\end{aligned}$$ where $\pm$ denotes a possible change of signs depending on the crossings of $\sigma$. Then, we can also rule out these kind of partitions and we can assume that $\sigma \in \Pi_2(s)$ is such that $\sigma \le \sigma(\ibar), \sigma(\underline{\alpha})$ and $I_{\underline{\alpha}}(\sigma)=\emptyset$. In this case, we do not need to commute two letters $(\alpha,i,k)$ and $(\beta,j,\ell)$ with $\alpha \neq \beta$. Hence we will obtain deterministic signs coming from the commutations, which only depend on the number of crossings of $\sigma$. More precisely, given $\sigma \in \Pi_2(s)$ satisfying the properties above and $\k \in [m]^s$ such that $\sigma(\k) = \sigma$ we have $$\label{ByInduction}
\tau_{{\varepsilon}_m} \Big(x_{i_1}^{\alpha_1}(k_1)(\omega)\cdots
x_{i_s}^{\alpha_s}(k_s)(\omega) \Big)=(-1)^{\#I(\sigma)} \quad
\mbox{for every} \quad \omega.$$ Indeed, this can be proved inductively as follows. Using that $I_{\underline{\alpha}}(\sigma) = \emptyset$, there must exists a connected block of consecutive numbers in $[s]$ so that the following properties hold
- The letters in that block are related to a fixed $\alpha$.
- The product of the letters in that block equals $\pm
\mathbf{1}$.
- The block itself is a union of pairs of the partition $\sigma
\in \Pi_2(s)$.
If $\pi$ denotes the restriction of $\sigma$ to our distinguished block —well defined by the third property— the sign given by the second property equals $(-1)^{\#I(\pi)}$. After canceling this block of letters, we may start again by noticing that $I_{\underline{\beta}}(\sigma \setminus \pi) = \emptyset$ where $\underline{\beta}$ is the restriction of $\underline{\alpha}$ to the complement of our distinguished block. This allows to restart the process. In the end we obtain $(-1)^{\#I(\sigma)}$ as desired. We deduce that $$\begin{aligned}
\lefteqn{\lim_{m \to \infty} \mathbb{E}_\omega \tau_{{\varepsilon}_m} \Big(
\tilde{x}_{i_1}^{\alpha_1}(m)(\omega) \cdots
\tilde{x}_{i_s}^{\alpha_s}(m)(\omega) \Big)} \\ [3pt] & = & \lim_{m
\to \infty} \frac{1}{m^{s/2}} \mathbb{E}_\omega
\sum_{\substack{\sigma\in \Pi_2(s) \\ \sigma \le
\sigma(\ibar),\sigma(\underline{\alpha})
\\ I_{\underline{\alpha}}(\sigma)=\emptyset}} \sum_{\substack{\k
\in [m]^s \\ \sigma(\k)=\sigma}} (-1)^{\#I(\sigma)} \ =
\ \sum_{\substack{\sigma\in \Pi_2(s) \\ \sigma \le \sigma(\ibar),
\sigma\le \sigma(\underline{\alpha})
\\ I_{\underline{\alpha}}(\sigma)=\emptyset}} (-1)^{\#I(\sigma)}.\end{aligned}$$ Here we have used that $$\lim_{m \to \infty} \frac{|\{\k \in [m]^s :
\sigma(\k) = \sigma\}|}{m^{s/2}} \, = \, \lim_{m \to \infty}
\frac{m(m-1)\cdots (m-\frac{s}{2}+1)}{m^{s/2}} \, = \, 1.$$ By Lemma \[moments free pdct fermions\], this proves convergence in expectation and completes the first part of the proof. It remains to prove almost everywhere convergence in $\omega$. Let us define the random variables $$X_m(\omega) \, = \, \tau_{{\varepsilon}_m} \Big(
\tilde{x}_{i_1}^{\alpha_1}(m) \cdots \tilde{x}_{i_s}^{\alpha_s}(m)
\Big).$$ By the dominated convergence theorem, it suffices to show $$\lim_{m \rightarrow \infty} \mu \Big( \big\{\sup_{M\geq
m}\big|X_M-\mathbb{E}_\omega[X_M]\big|\geq \alpha\big\} \Big) \, =
\, 0$$ for every $\alpha> 0$. According to Tchebychev’s inequality, we find $$\mu\Big(\big\{\sup_{M\geq
m}\big|X_M-\mathbb{E}_\omega[X_M]\big|\geq \alpha\big\}\Big) \, \le
\, \frac{1}{\alpha^2}\sum_{M=m}^\infty V[X_M],$$ where $V[X_M]=\mathbb{E}_\omega[X_M^2]- (\mathbb{E}_\omega[X_M] )^2$ denotes the variance of $X_M$. We will prove the upper bound $V[X_M] \leq
C(s)/M^2$ for every $M$, for some contant $C(s)$ depending only on the length $s$. This will suffice to conclude the argument. To this end we write $$\label{variance}
V[X_M] \, = \, \frac{1}{M^s} \sum_{\substack{\sigma, \pi \in \Pi(s)}}
\, \sum_{\substack{\k \, : \, \sigma(\k) = \sigma \\ \l \, : \, \sigma
\hskip1pt (\l) \hskip1pt = \pi}} D_{\k,\l},$$ where $$\begin{aligned}
D_{\k,\l} & = & \mathbb{E}_\omega \Big[ \tau_{{\varepsilon}_m} \big(
x_{i_1}^{\alpha_1}(k_1)(\omega) \cdots
x_{i_s}^{\alpha_s}(k_s)(\omega)\big) \tau_{{\varepsilon}_m} \big(
x_{i_1}^{\alpha_1}(\ell_1)(\omega) \cdots
x_{i_s}^{\alpha_s}(\ell_s)(\omega) \big) \Big] \\ & - &
\mathbb{E}_\omega \Big[ \tau_{{\varepsilon}_m}
\big(x_{i_1}^{\alpha_1}(k_1)(\omega) \cdots
x_{i_s}^{\alpha_s}(k_s)(\omega) \big) \Big] \mathbb{E}_\omega \Big[
\tau_{{\varepsilon}_m} \big(x_{i_1}^{\alpha_1}(\ell_1)(\omega) \cdots
x_{i_s}^{\alpha_s}(\ell_s)(\omega) \big) \Big]\end{aligned}$$ for $\k = (k_1,\ldots, k_s)$ and $\l=(\ell_1,\ldots, \ell_s)$. Now, reasoning as above one can see that whenever $\sigma$ or $\pi$ has a singleton, all the corresponding terms in the sum are equal to zero. Thus, we may write $\sigma=\{V_1,\ldots,
V_{r_\sigma}\}$ and $\pi=\{W_1,\ldots, W_{r_\pi}\}$ with $r_\sigma,r_\pi \le \frac{s}{2}$. If neither $\sigma$ nor $\pi$ are pair partitions, we will have $r_\sigma,r_\pi \le \frac{s}{2}-1$ and the part of the sum in corresponding to these pairs $(\sigma,\pi)$ can be bounded above in absolute value by $C(s) / M^2$ as desired. Then, it remains to control the rest of the terms in . To this end, we assume that $\sigma$ is a pair partition. Actually, a cardinality argument as before allows us to conclude that $\pi$ must be either a pair partition or a partition with all blocks formed by two elements up to a possible four element block. In the following, we will explain how to deal with the case in which $\pi$ is a pair partition. The other case can be treated exactly in the same way, being actually even easier by cardinality reasons. Let us fix two pair partitions $\sigma$ and $\pi$ and let us consider $\k = (k_1,\ldots, k_s)$ and $\l = (\ell_1,\ldots, \ell_s)$ such that $\sigma(\k)=\sigma$ and $\sigma(\l)=\pi$. When rearranging the letters in the traces defining $D_{\k,\l}$, the deterministic signs —$\alpha = \beta$ in — do not have any effect in the absolute value of $D_{\k,\l}$. On the other hand, the random signs —$\alpha \neq \beta$ in — makes the second term of $D_{\k,\l}$ vanish. Thus, $D_{\k,\l} \neq 0$ if and only if $I_{\underline{\alpha}}(\sigma) \neq \emptyset \neq
I_{\underline{\alpha}}(\pi)$ and we obtain the same random signs coming from crossings in $I_{\underline{\alpha}}(\sigma)$ and $I_{\underline{\alpha}}(\pi)$. In particular, we should find at least two signs $${\varepsilon}\big( (\alpha_p, i_p, k_p),
(\alpha_q,i_q,k_q)\big)(\omega) \quad (\alpha_p \hskip0.5pt \neq
\hskip0.5pt \alpha_q) \quad \mbox{from} \quad
x_{i_1}^{\alpha_1}(k_1)(\omega)\cdots
x_{i_s}^{\alpha_s}(k_s)(\omega),$$ $${\varepsilon}\big((\alpha_u,i_u,\ell_u),
(\alpha_v,i_v,\ell_v)\big) (\omega) \quad (\alpha_u \neq \alpha_v)
\quad \mbox{from} \quad x_{i_1}^{\alpha_1}(\ell_1) \hskip0.5pt
(\omega)\cdots x_{i_s}^{\alpha_s}(\ell_s) \hskip0.5pt (\omega).$$ By independence, this implies that $$\big\{ (\alpha_p, i_p, k_p),
(\alpha_q,i_q,k_q) \big\} \, = \, \big\{ (\alpha_u, i_u, \ell_u),
(\alpha_v,i_v,\ell_v) \big\}.$$ Moreover, since we also need $\sigma
\leq \sigma(\underline{\alpha})$ for non-vanishing terms, we can conclude that $k_p \neq k_q$ and $\ell_u \neq \ell_v$. Therefore, the sets $\{k_1,\ldots, k_s\}$ and $\{\ell_1,\ldots, \ell_s\}$ must have four elements (corresponding to two different blocks) in common. This implies that the part of the sum in corresponding to pairs $(\sigma,\pi)$ of pair partitions is bounded above by $$C'(s)
\frac{M^{s/2} M^{(s-4)/2}}{M^s} \, = \, \frac{C'(s)}{M^2}$$ for a certain constant $C(s)'$ as we wanted. This completes the proof. [ $\square$ 0.2cm]{}
Let $x$ be a word in the reduced free product of Clifford algebras $\M$, which written in reduced form is given by . In what follows, we will associate to $x$ an element $\tilde{x}(m)$ in $\mathcal A_{\varepsilon_m}$ given by $$\label{general element free productI with n}
\tilde{x}(m) \, = \, \tilde{x}_{A_1}^{\alpha_1}(m) \cdots
\tilde{x}_{A_\ell}^{\alpha_\ell}(m).$$ If we develop $x$ as in , then we can write $\tilde{x}(m)$ as $$\overbrace{\tilde{x}_{i_1}^{\alpha_1}(m) \cdots
\tilde{x}_{i_{s_1}}^{\alpha_1}(m)}^{\tilde{x}_{A_1}^{\alpha_1}(m)}\overbrace{\tilde{x}_{i_{s_1+1}}^{\alpha_2}(m)
\cdots
\tilde{x}_{i_{s_1+s_2}}^{\alpha_2}(m)}^{\tilde{x}_{A_2}^{\alpha_2}}
\cdots \overbrace{\tilde{x}_{i_{s_1+\cdots
+s_{\ell-1}+1}}^{\alpha_\ell}(m) \cdots \tilde{x}_{i_{s_1+\cdots
+s_\ell}}^{\alpha_\ell}(m)}^{\tilde{x}_{A_\ell}^{\alpha_\ell}(m)}.$$
Hypercontractivity bounds
-------------------------
In this subsection we prove Theorem B. The result below can be obtained following verbatim the proof of [@Biane Lemma 4] just replacing Theorem 7 there by Theorem \[moments\] above.
\[approx norm\] If $p\geq 1$, we have $$\lim_{m \to \infty} \Big\| {\sum\nolimits}_j \rho_j
\tilde{x}_j(m) \Big\|_{L_p(\mathcal A_{\varepsilon_m})} \, = \, \Big\|
{\sum\nolimits}_j \rho_j x_j \Big\|_{L_p(\M)} \quad \mbox{a.e.}$$ for any finite linear combination $\sum_j \rho_j x_j$ of reduced words in the free product $\M$.
\[key lemma\] Given $x$ a reduced word in the free product $\M$, let $\tilde{x}(m)$ be the element in $\mathcal A_{\varepsilon_m}$ associated to $x$ as in . Then, there exists a decomposition $\tilde{x}(m) = \tilde{x}_1(m) + \tilde{x}_2(m)$ with the following properties
- $\langle \tilde{x}_1(m), \tilde{x}_2(m)\rangle=0$ a.e.,
- $\mathcal{S}_{\varepsilon_n,t}(\tilde{x}_1(m))=e^{-t|x|}\tilde{x}_1(m)$,
- $\displaystyle\lim_{m\rightarrow
\infty}\|\tilde{x}_1(m)\|_{L_2(\mathcal A_{\varepsilon_m})}=1$ a.e.
In particular, we deduce that $$\lim_{m\rightarrow \infty}
\|\tilde{x}_2(m)\|_{L_2(\mathcal A_{\varepsilon_m})} \, = \, 0 \quad
\mbox{a.e}.$$
[ [**Proof.** ]{}]{}If we set $s = |x|$ and $\sigma_0$ denotes the singleton partition, define $$\begin{aligned}
\tilde{x}_1(m)(\omega) & = & \frac{1}{m^{s/2}} \sum_{\substack{\k \in
[m]^s \\ \sigma(\k) = \sigma_0}} x_{i_1}^{\alpha_1}(k_1)(\omega)
\cdots x_{i_s}^{\alpha_s}(k_s)(\omega), \\ [5pt]
\tilde{x}_2(m)(\omega) & = & \frac{1}{m^{s/2}} \sum_{\sigma \in \Pi(s)
\setminus \{\sigma_0\}} \sum_{\substack{\k \in[m]^s \\ \sigma(\k) =
\sigma}} x_{i_1}^{\alpha_1}(k_1)(\omega) \cdots
x_{i_s}^{\alpha_s}(k_s)(\omega).\end{aligned}$$ Clearly $\tilde{x}(m) = \tilde{x}_1(m) + \tilde{x}_2(m)$ point wise and $\|\tilde{x}(m)\|_{L_2(\mathcal A_{\varepsilon_m})}=1$. Property i) is easily checked. Indeed, consider $\k,\underline{\ell}\in [m]^s$ with $\sigma(\k)=\sigma_0$ and $\sigma(\underline{\ell}) \in \Pi(s)
\setminus \{\sigma_0\}$. Since the $k_i$’s are all distinct and the $\ell_i$’s are not we must have $$\tau_{{\varepsilon}_m}
\Big(x_{i_1}^{\alpha_1}(k_1)(\omega)\cdots
x_{i_s}^{\alpha_s}(k_s)(\omega)x_{i_s}^{\alpha_s}(\ell_s)(\omega)\cdots
x_{i_1}^{\alpha_1}(\ell_1)(\omega)\Big) \, = \, 0.$$ The second property comes from the definition of the semigroup and the fact that for every $\k$ with $\sigma(\k)=\sigma_0$, we have no cancellations. Now it remains to show that $$\lim_{m \rightarrow
\infty} \frac{1}{m^s} \sum_{\substack{\k, \l \in [m]^s
\\ \sigma(\k)=\sigma(\l)=\sigma_0}} \tau_{{\varepsilon}_m}
\Big(\underbrace{x_{i_1}^{\alpha_1}(k_1)\cdots
x_{i_s}^{\alpha_s}(k_s)}_{x_{\ibar}^{\underline{\alpha}}(\k)}
\underbrace{x_{i_s}^{\alpha_s}(\ell_s)\cdots
x_{i_1}^{\alpha_1}(\ell_1)}_{x_{\ibar}^{\underline{\alpha}}(\l)^*}\Big)
\, = \, 1.$$ Indeed, if $\{k_1,\ldots, k_s\}\neq\{\ell_1,\ldots,
\ell_s\}$ the trace clearly vanishes and it suffices to consider the case $\{k_1,\ldots, k_s\}=\{\ell_1,\ldots, \ell_s\}$. Note that, the trace above is different from $0$ if and only if $(\alpha_j,i_j,k_j) =
(\alpha_{\beta(j)}, i_{\beta(j)}, \ell_{\beta(j)})$ for some permutation $\beta \in S_s$ and every $1 \le j \le s$. If we assume $k_s\neq \ell_s$, we get $(\alpha_j, i_j, k_j) = (\alpha_s, i_s,
\ell_s)$ for certain $j < s$. This means that $x_{i_j}^{\alpha_j}(k_j)$ and $x_{i_s}^{\alpha_s}(k_s)$ belong to different $\alpha$-blocks since the $i_j$’s are pairwise distinct in a fixed $\alpha$-block. Thus, to cancel these elements we must cross a $\beta$-block with $\beta \neq \alpha_s$. Since the $k$’s are all different, the ${\varepsilon}$-signs corresponding to these commutations appear just once. We can argue in the same way for every $1\leq j\leq s$ and conclude that $$\mathbb{E}_\omega \, \tau_{{\varepsilon}_m} \Big( x_{i_1}^{\alpha_1}(k_1)
\cdots x_{i_s}^{\alpha_s}(k_s) x_{i_s}^{\alpha_s}(\ell_s) \cdots
x_{i_1}^{\alpha_1}(\ell_1) \Big) \, = \, 0$$ unless $k_j = \ell_j$ for all $1\leq j\leq s$. Therefore $$\begin{aligned}
\lim_{m \rightarrow \infty}\mathbb{E}_\omega
\|\tilde{x}_1(m)\|_{L_2(\mathcal A_{\varepsilon_m})}^2 & = & \lim_{m
\rightarrow \infty} \frac{1}{m^s} \sum_{\substack{\k \in [m]^s
\\ k_i\neq k_j}} 1 \\ & = & \frac{m(m-1)\cdots (m-s+1)}{m^s} \ =
\ 1.\end{aligned}$$ Finally, arguing as in the proof of Theorem \[moments\] we see that the same limit holds for almost every $\omega\in \Omega$. This proves iii). The last assertion follows from i), iii) and the identity $\|\tilde{x}(m)\|_2=1$. The proof is complete. [ $\square$ 0.2cm]{}
\[approx normII\] If $p\geq 1$, we have $$\lim_{m \to \infty} \Big\|
\mathcal{S}_{{\varepsilon}_m,t} \Big( {\sum\nolimits}_j \rho_j \tilde{x}_j(m) \Big)
\Big\|_{L_p(\mathcal A_{\varepsilon_m})} \, = \, \Big\|
\mathcal{O}_{\M,t} \Big( {\sum\nolimits}_j \rho_j x_j \Big) \Big\|_{L_p(\M)}
\quad \mbox{a.e.}$$ for any finite linear combination $\sum_j
\rho_j x_j$ of reduced words in the free product $\M$.
[ [**Proof.** ]{}]{}According to Lemma \[key lemma\], we have $$\lim_{m \rightarrow
\infty} \Big\|\Big(\mathcal{S}_{\varepsilon_m,t}- e^{-t|x|}
\mathbf{1}_{\mathcal
A_{\varepsilon_m}}\Big)\big(\tilde{x}(m)\big)\Big\|_{L_2(\mathcal
A_{\varepsilon_m})} \, = \, 0 \quad \mbox{a.e.}$$ for any reduced word $x\in \M$ and the associated $\tilde{x}(m)$’s $\in \mathcal
A_{\varepsilon_m}$ given by . Thus $$\lim_{m \rightarrow \infty}
\Big\|\mathcal{S}_{\varepsilon_m,t} \Big({\sum\nolimits}_j \rho_j
\tilde{x}_j(m) \Big) - {\sum\nolimits}_j e^{-t|x_j|} \rho_j \tilde{x}_j(m)
\Big\|_{L_2(\mathcal A_{\varepsilon_m})} \, = \, 0 \quad \mbox{a.e.}$$ Then implies that the same limit vanishes in the norm of $L_p(\mathcal{A}_{{\varepsilon}_m})$. On the other hand, since $\mathcal{O}_{\M,t} (x_j)=e^{-t|x_j|}x_j$, the assertion follows from Lemma \[approx norm\]. [ $\square$ 0.2cm]{}
[[**Proof of Theorem B.** ]{}]{}Let $1<p\leq q <\infty$. By construction, the algebraic free product $A$ is a weak-$*$ dense involutive subalgebra of $\M$. In particular, it is dense in $L_p(\M)$ for every $p<\infty$. Given a finite sum $z = \sum_j \rho_j x_j \in A$, consider the corresponding sum $\tilde{z}(m) = \sum_j \rho_j
\tilde{x}_j(m)\in\mathcal A_{\varepsilon_m}$ following . Given any $t \ge \frac12 \log
(q-1/p-1)$, we may apply Lemmas \[approx norm\] and \[approx normII\] in conjunction with Biane’s theorem to conclude $$\begin{aligned}
\|\mathcal{O}_{\M,t}(z)\|_{L_q(\M)} & = & \lim_{m \to \infty} \big\|
\mathcal{S}_{{\varepsilon}_m,t}(\tilde{z}(m))
\big\|_{L_q(\mathcal{A}_{{\varepsilon}_m})} \\ & \le & \lim_{m \to \infty} \|
\tilde{z}(m)) \|_{L_p(\mathcal{A}_{{\varepsilon}_m})} = \|z\|_{L_p(\M)}.\end{aligned}$$ The necessity of the condition $t \ge \frac12 \log(q-1/p-1)$ was justified above. [ $\square$ 0.2cm]{}
Further comments {#remark generalization}
----------------
Note that the argument we have used in the proof of Theorem B still works in a more general setting. More precisely, we may replace the fermion algebras $\M_\alpha=\mathcal C({\mathbb{R}}^d)$ by spin system algebras $\mathcal{A}_\alpha$, where the generators $x_i^\alpha$ satisfy certain commutation and anticommutation relations given by a sign ${\varepsilon}^\alpha$ as follows $$x_i^\alpha
x_j^\alpha-{\varepsilon}^\alpha(i,j)x_j^\alpha x_i^\alpha=2\delta_{ij} \quad
\mbox{ for } 1\leq i,j\leq d.$$ Indeed, we just need to replace by $$\mu \Big(\varepsilon \big((
\alpha,i,k),(\beta,j,\ell)\big)=-1\Big)= \begin{cases}
{\varepsilon}^\alpha(i,j) & \mbox{if} \ \alpha=\beta, \\ 1/2 & \mbox{if}
\ \alpha \neq \beta. \end{cases}$$ This yields optimal time hypercontractivity bounds for the Ornstein-Uhlenbeck semigroup on the free product of spin matrix algebras. An additional application of Speicher’s central limit theorem allows us to obtain optimal hypercontractivity estimates for the Ornstein-Uhlenbeck semigroup on the free product of $q$-deformed algebras $\Gamma_q$, $-1\leq q \leq
1$.
*Slight modifications in lead to von Neumann algebras which are still poorly understood. For instance, let us fix a function $f:[1,n]\times [1,n]\to
[-1,1]$ which is symmetric and assume that $$\mu \big( \{{\varepsilon}((\alpha,i,k),(\beta,j,\ell)) = +1\} \big)
= \frac{1+f(\alpha,\beta)}{2} .$$ As usual we will assume that all the random variables ${\varepsilon}(x,y)$ are independent. Then it is convenient to first calculate expectation of the joint moments of $$\tilde{x}_i^{\alpha}(m) = \frac{1}{\sqrt{m}}
\sum_{k=1}^m x_i^{\alpha}(k) .$$ Again, only the pair partitions survive and we get $$\begin{aligned}
\lim_{m \to \infty}\mathbb{E}_\omega \tau_{\varepsilon_m}(\tilde{x}_{i_1}^{\alpha_1}(m)
\cdots \tilde{x}_{i_s}^{\alpha_s}(m)) &=&
\sum_{\substack{\sigma \in \Pi_2(s)\\ \sigma \leq \sigma(\underline{i}), \sigma(\underline{\alpha})}}\prod_{(k,\ell)\in I(\sigma)}f(\alpha_{e_k},\alpha_{e_\ell}).\end{aligned}$$ As above, we will have hypercontractivity with the optimal constant for the limit gaussian systems (they indeed produce a tracial von Neumann algebra). As an illustration, let us consider $n=2$, $q_1,q_2 \in
[-1,1]$, $f(1,1)= q_1q_2$ and $f(1,2)=f(2,1)=f(2,2)=q_2$. We deduce immediately that*
- *The von Neumann subalgebra generated by $$x_i^1 = \lim_m
\tilde{x}_i^1(m),$$ for $i=1, \ldots,\, d$ is isomorphic to $\Gamma_{q_1q_2}({\mathbb{R}}^d)$, generated by $d$ $q_1q_2$-gaussians.*
- *The von Neumann subalgebra generated by $$x_i^2 = \lim_m
\tilde{x}_i^2(m),$$ for $i=1, \ldots,\, d$ is isomorphic to $\Gamma_{q_2}({\mathbb{R}}^d)$, generated by $d$ $q_2$-gaussians.*
- *Let $A\subset [s]$ and let $y_i=x_{j_i}^1$ for $i\in A$ (and $\alpha_i=1$) and $y_i =
x_{j_i}^2$ ($\alpha_i=2$) otherwise. Let $\eta_0$ be the partition of $[s]$ defined by the possible values of $(j_i,\alpha_i)$. Then we get $$\tau(y_{1}y_{2}\cdots
y_{s})
= \sum_{\eta_0\geq \sigma \in \Pi_2(s)} q_1^{{\rm
\scriptsize inversion}(\sigma |A)} q_2^{{\rm
\scriptsize inversion}(\sigma)}
.$$ Here ${\sigma}|A$ is the restriction of ${\sigma}$ to $A$ where we count only inversions inside $A$. This construction is considered in [@JC] for constructing new Brownian motions.*
*We see that we can combine different $q$ gaussian random variables in one von Neumann algebra with a prescribed interaction behaviour. With this method we recover the construction from [@JC] of a non-stationary Brownian motion $B_t$. Indeed one can choose $0=t_0<t_1<\cdots < t_d$ such that $B_t$ is an abstract Brownian motion [@JC] and the random variables $s_t(j)=B_t-B_{t_j}$ are $q_0\cdots q_j$-Brownian motions. In this construction we needed a $q_1$-Brownian motion over a $q_2$-Brownian motion and hence the choice of the product $q_1q_2$ above. Although it is no longer trivial to determine the number operator, we see that hypercontractivity is compatible with non-stationarity. The algebras generated for arbitrary symmetric $f$ could serve as models for $q_1$-products over $q_2$-products, although in general there is no $q$-product of arbitrary von Neumann algebras.*
[**The free Poisson semigroup**]{} {#Section: proof of theorem A}
==================================
In this section we prove Theorem A and optimal hypercontractivity for linear combinations of words in ${\mathbb{F}}_n$ with length lower than or equal to 1. Let us start with a trigonometric identity, which follows from the binomial theorem and the identity $2\cos x = e^{ix} +
e^{-ix}$ $$(\cos x)^m \, = \, \frac{1}{2^{m-1}} \sum_{0 \le k \le
[\frac{m}{2}]} {m \choose k}
\frac{\cos((m-2k)x)}{2^{\delta_{m,2k}}}.$$ Let $g_j$ denote one of the generators of ${\mathbb{F}}_n$. Identifying $\lambda(g_j)$ with $\exp(2\pi i
\cdot)$, the von Neumann algebra generated by $\lambda(g_j)$ is $\mathcal{L}({\mathbb{Z}})$ and the previous identity can be rephrased as follows for $u_j=\lambda(g_j)$ $$\label{Trig-Fn}
\big( u_j+u_j^* \big)^m \, = \, \sum_{0\leq k\leq [\frac{m}{2}]}{m
\choose k} v_{j,m-2k},$$ with $v_{j,k}=u_j^k+(u_j^*)^k$ for every $k \geq 1$ and $v_0=\mathbf{1}$. We will also need a similar identity in $\mathbb{G}_{2n}$. Let $z_1, z_2, \ldots, z_{2n}$ denote the canonical generators of $\mathbb{G}_{2n}$, take $x_j=\lambda(z_j)$ for $1\leq j\leq 2n$ and consider the operators $a_{j,0} = \mathbf{1}, b_{j,0} = 0$ and $$\label{defab}
a_{j,k} \, = \,
\underbrace{x_{2j-1} x_{2j} x_{2j-1}\cdots}_k \qquad
\mbox{,} \qquad b_{j,k} \, = \,\underbrace{x_{2j} x_{2j-1}
x_{2j} \cdots}_k.$$ If we set $\zeta_j = u_j + u_j^*$ and $\psi_j = x_{2j-1} + x_{2j}$, let us consider the $*$-homomorphism $\Lambda: \mathcal{A}_{sym}^n \to
\mathcal{L}(\mathbb{G}_{2n})$ determined by $\Lambda(\zeta_j) =
\psi_j$. The result below can be proved by induction summing by parts.
\[lemma symmetric\] If $m \ge 0$, we find $$\big( x_{2j-1} + x_{2j} \big)^m \, = \,
\sum_{0 \leq k \leq [\frac{m}{2}]} {m \choose k} \big( a_{j,m-2k} +
b_{j,m-2k} \big).$$ Moreover, $v_{j,k} \in \langle u_j+u_j^* \rangle$ and we have $\Lambda(v_{j,k}) = a_{j,k} + b_{j,k}$ for every $k \geq
0$.
[[**Proof of Theorem A.** ]{}]{}As observed in the Introduction, the group von Neumann algebra $\mathcal{L}({\mathbb{Z}}_2)$ is $*$-isomorphic to the Clifford algebra $\mathcal{C}({\mathbb{R}})$. Moreover, the Poisson and Ornstein-Uhlenbeck semigroups coincide in this case. In particular, the first assertion follows from $\mathcal{L}(\mathbb{G}_n)=\mathcal{L}({\mathbb{Z}}_2) *\cdots *
\mathcal{L}({\mathbb{Z}}_2) \simeq \mathcal C({\mathbb{R}}) * \cdots * \mathcal C({\mathbb{R}})$, by applying Theorem B with $d=1$. To prove the second assertion, we consider the injective group homomorphism determined by $$\Phi: g_j
\in {\mathbb{F}}_n \, \mapsto \, x_{2j-1}x_{2j} \in \mathbb{G}_{2n}.$$ This map clearly lifts to an isometry $L_p(\mathcal{L}({\mathbb{F}}_n)) \to
L_p(\mathcal{L}(\mathbb{G}_{2n}))$ for all $p\geq 1$. Moreover, since $|\Phi(g)|=2|g|$, we see that $\Phi$ intertwines the corresponding free Poisson semigroup up to a constant $2$. More precisely, $\Phi
\circ \mathcal{P}_{{\mathbb{F}}_n,t} \, = \, \mathcal{P}_{\mathbb{G}_{2n}, t/2}
\circ \Phi$ for all $t > 0$. Hence, if $1<p\leq q <\infty$ and $f \in
L_p(\mathcal{L}({\mathbb{F}}_n))$, we obtain from the result just proved that $$\big\|\mathcal{P}_{{\mathbb{F}}_n,t} f \big\|_{L_q(\mathcal{L}({\mathbb{F}}_n))} =
\big\|(\mathcal{P}_{\mathbb{G}_{2n},t/2} \circ \Phi) f
\big\|_{L_q(\mathcal{L}(\mathbb{G}_{2n}))} \le \|\Phi
f\|_{L_q(\mathcal{L}(\mathbb{G}_{2n})}=\|f\|_{L_q(\mathcal{L}({\mathbb{F}}_n))},$$ whenever $t \ge \log (q-1/p-1)$. It remains to prove the last assertion iii). The necessity of the condition $t \ge \frac12 \log
(q-1/p-1)$ can be justified following Weissler argument in [@Weissler pp 220]. Therefore, we just need to prove sufficiency. According to [@NiSp], $\chi_{[-2,2]}(s)/\pi
\sqrt{4-s^2}$ is the common distribution of $\zeta_j$ and $\psi_j$. Moreover, since both families of variables are free, the tuples $(\zeta_1,\ldots, \zeta_n)$ and $(\psi_1,\ldots, \psi_n)$ must have the same distribution too. Therefore, for every polynomial $P$ in $n$ non-commutative variables we have $$\big\|P(\zeta_1,\ldots,\zeta_n)\big\|_{L_p(\mathcal A_{sym}^n)}
\, = \,
\big\|P(\psi_1,\ldots,\psi_n)\big\|_{L_p(\mathcal{L}(\mathbb{G}_{2n}))}$$ for every $1\leq p\leq \infty$. In particular, the $*$-homomorphism $\Lambda: \mathcal{A}_{sym}^n \to \mathcal{L}(\mathbb{G}_{2n})$ determined by $\Lambda(\zeta_j)=\psi_j$ for every $1\leq j\leq n$ extends to an $L_p$ isometry for every $1\leq p\leq \infty$. We claim that $$\Lambda \big( \mathcal{P}_{{\mathbb{F}}_n,t}(P(\zeta_1,\ldots, \zeta_n))
\big) \, = \, \mathcal{P}_{\mathbb{G}_{2n},t} \big(P(\psi_1,\ldots,
\psi_n) \big)$$ for every polynomial $P$ in $n$ non-commutative variables. It is clear that the last assertion iii) of Theorem A follows from our claim above in conjunction with the first assertion i), already proved. By freeness of the semigroups involved and the fact that $\Lambda$ is a $*$-homomorphism, it suffices to justify the claim for $P(X_1, X_2, \ldots, X_n) = X_j^m$ with $1 \le j \le n$ and $m \ge 0$. However, this follows directly from Lemma \[lemma symmetric\]. [ $\square$ 0.2cm]{}
In the lack of optimal time estimates for ${\mathbb{F}}_n$ through the probabilistic approach used so far —see [@JPPP] for related results— we conclude this paper with optimal hypercontractivity bounds for linear combinations of words with length lower than or equal to 1. We will use two crucial results, the second one is folklore and it follows from the invariance by rotation" of the CAR algebra generators.
- *The Ball*/*Carlen*/*Lieb convexity inequality* [@BCL] $$\nonumber \Big(\frac{\mathrm{Tr} |A+B|^p + \mathrm{Tr}
|A-B|^p}{2}\Big)^\frac{2}{p}\ge
\big(\mathrm{Tr}|A|^p\big)^\frac{2}{p}+(p-1)\big(\mathrm{Tr}|B|^p\big)^\frac{2}{p}$$ for any $1 \le p \le 2$ and any given pair of $m \times m$ matrices $A$ and $B$.
- *A Khintchine inequality for fermion algebras* $$\Big\|
\sum_{j=1}^d \rho_j x_j \Big\|_p \, = \, \Big( \sum_{j=1}^d
|\rho_j|^2 \Big)^{\frac12}$$ whenever $1 \le p < \infty$, $\rho_j \in {\mathbb{R}}$, $x_j=x_j^*$ and $x_ix_i + x_jx_i = 2
\delta_{ij}$.
\[ApplicationII\] Let us denote by $\mathcal W_1$ the linear span of all words in $\mathcal L ({\mathbb{F}}_n)$ of length lower than or equal to $1$. Then, the following optimal hypercontractivity bounds hold for $1 < p \leq 2$, every $t \ge -\frac12 \log (p-1)$ and all $f \in \mathcal
W_1$ $$\|\mathcal{P}_{{\mathbb{F}}_n,t} f \|_{L_2(\mathcal{L}({\mathbb{F}}_n))} \, \le \,
\|f\|_{L_p(\mathcal{L}({\mathbb{F}}_n))}.$$
[ [**Proof.** ]{}]{}The optimality of our estimate follows once again from Weissler argument in [@Weissler pp 220]. Moreover, it suffices to show the inequality for the extreme case $e^{-t}=\sqrt{p-1}$. The key point in the argument is the use of the $*$-homomorphism $\Phi:
\mathcal{L}({\mathbb{F}}_n) \to \mathcal{L}(\mathbb{G}_{2n})$ defined in the proof of Theorem A in conjunction with our characterization of $\mathcal L(\mathbb{G}_{2n})$ using a spin matrix model. Indeed, we will consider here exactly the same matrix model with $2n$ free copies and just one generator per algebra. More precisely, given $m \ge 1$ we will consider $x^\alpha(k)$ with $1 \le \alpha \le 2n$ and $1 \le k
\le m$ verifying the same relations as in depending on the corresponding random functions $\varepsilon
((\alpha,k),(\beta,\ell))$. We also set $$\tilde{x}^\alpha(m)=\frac{1}{\sqrt{m}}\sum_{k=1}^mx^\alpha(k)$$ as usual. Note that this model describes —in the sense of Theorem \[moments\]— the algebra $\mathcal
L(\mathbb{G}_{2n})$. In fact, according to Lemma \[approx norm\] we know that for every trigonometric polynomial $z = \sum_j \rho_j x_j
\in \mathcal L(\mathbb{G}_{2n})$ in the span of finite words, we can define the corresponding elements $\tilde{z}(m) = \sum_j \rho_j
\tilde{x}_j(m) \in \mathcal A_{\varepsilon_m}$ such that $$\lim_{m\rightarrow \infty}\|\tilde{z}(m)\|_{L_p(\mathcal
A_{\varepsilon_m})} \, = \, \|z\|_{L_p(\mathcal
L(\mathbb{G}_{2n}))}$$ almost everywhere. Furthermore, by dominated convergence we find $$\hskip-12pt \lim_{m\rightarrow
\infty}\mathbb{E}_\omega \|\tilde{z}(m)\|_{L_p(\mathcal
A_{\varepsilon_m})} \, = \, \|z\|_{L_p(\mathcal
L(\mathbb{G}_{2n}))}.$$ We first consider a function $f = a_0
\mathbf{1} + a_1 \lambda(g_1) + b_1 \lambda(g_1)^* + \ldots + a_n
\lambda(g_n) + b_n \lambda(g_n)^*$ in $\mathcal W_1$ such that $\arg(a_\alpha)=\arg(b_\alpha)$ for all $1\le \alpha \le n$. By the comments above, we have for every $1< p < 2$ $$\begin{aligned}
\|f\|_{L_p(\mathcal L({\mathbb{F}}_n))}^2 & = & \|\Phi f\|^2_{L_p(\mathcal
L(\mathbb{G}_{2n}))} \\ \nonumber & = & \lim_{m \rightarrow
\infty}\mathbb{E}_\omega \Big\| a_0 \mathbf{1} + a_1 \tilde{x}^1(m)
\tilde{x}^2(m) + b_1 \tilde{x}^2(m) \tilde{x}^1(m) \\ \nonumber & + &
\cdots \ + a_n \tilde{x}^{2n-1}(m) \tilde{x}^{2n}(m) + b_n
\tilde{x}^{2n}(m) \tilde{x}^{2n-1}(m)
\Big\|_{L_p(\mathcal{A}_{{\varepsilon}_m})}^2.\end{aligned}$$ Now, we claim that $\|f\|_{L_p(\mathcal L({\mathbb{F}}_n))}^2$ is bounded below by $$\lim_{m \rightarrow \infty} \mathbb{E}_\omega \Big(|a_0|^2 +
\frac{p-1}{m^2} \sum_{\begin{subarray}{c} 1\le \alpha \le n \\ 1\le k
\le m \end{subarray}} \Big\|\sum_{1 \le \ell \le m} \Big( a_\alpha
+ b_\alpha \varepsilon
\big((2\alpha-1,k),(2\alpha,\ell)\big)\Big)x^{2\alpha}(\ell)
\Big\|_p^2 \Big).$$ If this is true, we can apply Khintchine’s inequality for fixed $\alpha$ and $k$ to get $$\begin{aligned}
\lefteqn{\mathbb{E}_\omega \Big\|\sum_{1\leq \ell \leq m}
\Big( a_\alpha + b_\alpha
\varepsilon\big((2\alpha-1,k),(2\alpha,\ell) \big)\Big)
x^{2\alpha}(\ell) \Big\|_p^2} \\ & = & \mathbb{E}_\omega
\Big\|\sum_{1\leq \ell \leq m} \Big( |a_\alpha| + |b_\alpha|
\varepsilon\big((2\alpha-1,k),(2\alpha,\ell) \big)\Big)
x^{2\alpha}(\ell) \Big\|_p^2 \\ & =
&\sum_{1\le \ell \le m} \Big( |a_\alpha|^2 + |b_\alpha|^2 + 2
|a_\alpha b_\alpha| \mathbb{E}_\omega
\varepsilon\big((2\alpha-1,k),(2\alpha,\ell) \big) \Big) \, =
\, m(|a_\alpha|^2+ |b_\alpha|^2).\end{aligned}$$ Here, we have used that the $\varepsilon$’s are centered for $\alpha
\neq \beta$. Therefore, we finally obtain $$\|f\|_{L_p(\mathcal
L({\mathbb{F}}_n))}^2 \, \ge \, |a_0|^2 + (p-1)
\sum_{\alpha=1}^n(|a_\alpha|^2+|b_\alpha|^2) \, = \,
\|\mathcal{P}_{{\mathbb{F}}_n,t}f\|_{L_2(\mathcal{L}({\mathbb{F}}_n))}^2$$ for $e^{-t}=\sqrt{p-1}$. Therefore, it suffices to prove the claim. To this end, note that $$\|f\|_{L_p(\mathcal L({\mathbb{F}}_n))}^2 \, = \, \lim_{m
\rightarrow \infty} \mathbb{E}_\omega \big\|A_m + x^1(1) B_m
\big\|_{L_p(\mathcal{A}_{{\varepsilon}_m})}^2,$$ where $A_m$ and $B_m$ are given by $$\begin{aligned}
A_m & = & a_0 \mathbf{1} + \frac{1}{m} \sum_{\substack{2\le k \le m
\\ 1 \le \ell \le m}} \Big( a_1 + b_1 \varepsilon\big( (1,k),
(2,\ell) \big) \Big) x^1(k)x^2(\ell) \\ & + & \frac{1}{m} \sum_{1 \le
k,\ell \le m} \Big[ a_2 x^3(k) x^4(\ell) + b_2 x^4(k) x^3(\ell) +
\ldots + b_n x^{2n}(k) x^{2n-1}(\ell) \Big]\end{aligned}$$ and $B_m = \frac{1}{m} \sum_{1 \leq \ell \leq m} \big( a_1 + b_1
\varepsilon\big((1,1),(2,\ell) \big) \big) x^2(\ell)$. Then, since the spin matrix model is unaffected by the change of sign of one generator and $A_m,B_m$ do not depend on $x^1(1)$, we deduce $\|A_m+x^1(1)B_m\|_p=\|A_m-x^1(1)B_m\|_p$. Therefore, applying Ball/Carlen/Lieb inequality we conclude that $$\|f\|_{L_p(\mathcal
L({\mathbb{F}}_n))}^2 \, \ge \, \lim_{m\rightarrow \infty}\mathbb{E}_\omega
\Big( \|A_m\|_{L_p(\mathcal{A}_{{\varepsilon}_m})}^2 + (p-1)
\|B_m\|_{L_p(\mathcal{A}_{{\varepsilon}_m})}^2 \Big),$$ where we have used that $\|x^1(1)B_m\|_p=\|B_m\|_p$ for every $\omega$ and every $p$. If we apply the same strategy with $x^1(2),\ldots, x^1(m)$, it is not difficult to obtain the following lower bound $$\begin{aligned}
\|f\|_{L_p(\mathcal L({\mathbb{F}}_d))}^2 & \ge & \lim_{m \rightarrow \infty}
\mathbb{E}_\omega \Big\| a_0 \mathbf{1} + a_2 \tilde{x}^3(m)
\tilde{x}^4(m) + b_2 \tilde{x}^4(m) \tilde{x}^3(m) \\ [3pt] & + &
\cdots \ + a_n \tilde{x}^{2n-1}(m) \tilde{x}^{2n}(m) + b_n
\tilde{x}^{2n}(m) \tilde{x}^{2n-1}(m) \Big\|_p^2 \\ & + &
\frac{p-1}{m^2} \sum_{1 \le k \le m} \Big\| \sum_{1 \le \ell \le
m}\Big( a_1 + b_1 \varepsilon\big((1,k),(2,\ell)\big) \Big)x^2(\ell)
\Big\|_p^2.\end{aligned}$$ Our claim follows iterating this argument on $2 \le \alpha \le n$. It remains to consider an arbitrary $f = a_0 \mathbf{1} + a_1
\lambda(g_1) + b_1 \lambda(g_1)^* + \ldots + a_n \lambda(g_n) + b_n
\lambda(g_n)^* \in \mathcal W_1$. Let us set $(\theta_\alpha, \theta'_\alpha) = (\arg(a_\alpha), \arg(b_\alpha))$ and $(\nu_\alpha, \nu'_\alpha) = (\frac12 (\theta_\alpha+\theta'_\alpha),
\frac12 (\theta_\alpha-\theta'_\alpha))$ for each $1\le \alpha \le n$. Consider the 1-dimensional representation $\pi: {\mathbb{F}}_n \to {\mathbb{C}}$ determined by $\pi(g_\alpha) = \exp(i\nu'_\alpha )$ for the $\alpha$-th generator $g_\alpha$. According to the $L_p$-analog of Fell’s absorption principle [@PP], we have from the first part of the proof that $$\begin{aligned}
\|\mathcal{P}_{{\mathbb{F}}_n,t} f\|_2 & \le & \Big\| a_0 \mathbf{1} +
\sum_{\alpha=1}^n |a_\alpha|e^{i\nu_\alpha} \lambda(g_\alpha) +
|b_\alpha|e^{i\nu_\alpha} \lambda(g_\alpha)^*
\Big\|_{L_p(\mathcal{L}({\mathbb{F}}_n))} \\ & = & \Big\| a_0 \mathbf{1} +
\sum_{\alpha=1}^n |a_\alpha|e^{i\nu_\alpha} \pi(g_\alpha)
\lambda(g_\alpha) + |b_\alpha|e^{i\nu_\alpha}
\pi(g_\alpha^{-1})\lambda(g_\alpha)^* \Big\|_{L_p(\mathcal{L}({\mathbb{F}}_n))}
\\ & = & \Big\| a_0 \mathbf{1} + \sum_{\alpha=1}^n a_\alpha
\lambda(g_\alpha) + b_\alpha \lambda(g_\alpha)^*
\Big\|_{L_p(\mathcal{L}({\mathbb{F}}_n))} \ = \ \|f\|_{L_p(\mathcal{L}({\mathbb{F}}_n))}.\end{aligned}$$ The proof is complete. [ $\square$ 0.2cm]{}
We finish this section with further results on $L_p\to L_2$ estimates for the free Poisson semigroup. The key point here is to use a different model for Haar unitaries. In the sequel, we will denote by $\mathbb M_2$ the algebra of $2\times 2$ matrices.
\[haar\] If $u_j=\lambda(g_j)$ and $x_j=\lambda(z_j)$, the map $$u_j \mapsto \left[\begin{array}{cc} 0 & x_{2j-1} \\ x_{2j}&
0\end{array}\right]$$ determines a trace preserving $*$-homomorphism $\pi: \mathcal L(\mathbb F_n) \to \mathbb M_2{\overline{{\otimes}}}\mathcal L(\mathbb
G_{2n})$ such that $$\pi \circ \mathcal P_{{\mathbb{F}}_n, t} \, = \, \big( Id_{\mathbb M_2} \otimes \mathcal P_{\mathbb{G}_{2n}, t} \big) \circ \pi.$$
[ [**Proof.** ]{}]{}Since $\pi(u_j)$ is a unitary $w_j$ in $\mathbb M_2{\overline{{\otimes}}}\mathcal L(\mathbb{G}_{2n})$ and $\mathbb F_n$ is a free group, a unique $*$-homomorphism $\pi: \mathcal L(\mathbb F_n) \to \mathbb M_2{\overline{{\otimes}}}\mathcal L(\mathbb{G}_{2n})$ is determined by the $w_j$’s. Thus, it suffices to check that $\pi$ is trace preserving. The fact that $\pi(\lambda(g))$ has trace zero in $\mathbb M_2{\overline{{\otimes}}}\mathcal L(\mathbb G_{2n})$ for every $g \neq e$ follows easily from the equalities $$\pi(u_1)^{2k}= \left[\begin{array}{cc} a_{1,2k} &0 \\ 0 & b_{1,2k}\end{array}\right] \qquad \pi(u_1)^{2k+1} = \left[\begin{array}{cc} 0& a_{1,2k+1} \\ b_{1,2k+1} &
0 \end{array}\right]$$ and its analogous formulae for the product of different generators. Here, we have used the notations introduced in . The second assertion can be checked by simple calculations. The proof is complete. [ $\square$ 0.2cm]{}Biane’s theorem relies on an induction argument [@Biane Lemma 2] which exploits the Ball-Carlen-Lieb convexity inequality stated before Theorem \[ApplicationII\]. In fact, our proof of Theorem \[ApplicationII\] follows the same induction argument. We will now consider spin matrix models with operator coefficients. More precisely, given a finite von Neumann algebra $(\mathcal M,\tau)$, we will look at $\mathcal M{\overline{{\otimes}}}\mathcal A_{\varepsilon_m}$. In particular, following the notation in Section \[Section: proof of theorem B\] every $x\in\mathcal M{\overline{{\otimes}}}\mathcal A_{\varepsilon_m}$ can be written as $x=\sum_{A} \rho_A {\otimes}x_A^{\varepsilon_m}$ where $\rho_A \in \mathcal M$ for every $A \subset\Upsilon_m$. Then, the induction argument easily leads to the inequality below provided that $e^{-t}\leq \sqrt{p-1}$ $$\|x\|^2_{L_p(\mathcal M{\overline{{\otimes}}}\A_{\varepsilon_m})} \geq
\sum_{A\subset\Upsilon_m} e^{-2t|A|} \| \rho_A \|_{L_p(\mathcal M)}^2.$$ For our purpose we will consider $\mathcal M=\mathbb M_2$ with its normalized trace, so that $$\|a\|_p \geq 2^{\frac 1 2 - \frac 1 p} \|a\|_2$$ for every $a \in \mathbb M_2$. Let $x=\sum_{A} \rho_A {\otimes}x_A^{\varepsilon_m}$ be as above. Let us also define $\mathcal U$ as the (possible empty) set of the subsets $A$ of $\Upsilon_m$ such that $\rho_A$ is a multiple of a unitary. In particular, $\|\rho_A\|_{L_2(\mathcal M)}=\|\rho_A\|_{L_p(\mathcal M)}$ for every $A\in \mathcal U$. Then, letting $y=\sum_{A\in \mathcal U} \rho_A {\otimes}x_A^{\varepsilon_m}$, the following estimate holds provided $e^{-t}\leq \sqrt{p-1}$ $$\label{Bianecoef} \| x \|^2_{p} \geq \big\| Id_{\mathbb
M_2}{\otimes}\mathcal{S}_{\varepsilon_m,t}(y)\big\|^2_{2} +
2^{1-\frac 2 p } \big\| Id_{\mathbb M_2}{\otimes}\mathcal{S}_{\varepsilon_m,t}(x-y)\big\|^2_{2},$$ where the right hand side norms are taken in $\mathbb{M}_2{\overline{{\otimes}}}\A_{\varepsilon_m}$. Our first application of this alternative approach is that Weissler’s theorem [@Weissler] can be proved using probability and operator algebra methods.
\[Weissler\] If $1 < p \le q < \infty$, we find $$\big\| \mathcal{P}_{\mathbb{Z},t}: L_p(\mathcal{L}(\mathbb{Z})) \to
L_q(\mathcal{L}(\mathbb{Z})) \big\| \, = \, 1 \ \Leftrightarrow \ t \ge \frac12 \log \frac{q-1}{p-1}.$$
[ [**Proof.** ]{}]{}We will assume that $q=2$ since the optimal time for every $p,q$ can be obtained from this case by means of standard arguments involving log-Sobolev inequalities. We follow here the same approximation procedure of Lemmas \[approx norm\] and \[key lemma\] with $n=2$ and $d=1$. Consider a reduced word $x=x_{\alpha_1}\cdots x_{\alpha_s}$ in $\mathcal L(\mathbb G_2)$, so that $\alpha_j \in\{1,2\}$ and $\alpha_j\neq \alpha_{j+1}$. We then form the associated element $$\tilde{x}(m)(\omega) = \frac{1}{m^{s/2}} \sum_{\substack{\k \in
[m]^s \\ \sigma(\k) = \sigma_0}} x^{\alpha_1}(k_1)(\omega) \cdots
x^{\alpha_s}(k_s)(\omega)\in \A_{\varepsilon_m}
.$$ Note that restricting to $\sigma(\k) = \sigma_0$ implies that there will be no repetitions of the elements $x^{\alpha_j}(k_j)$, hence no simplifications in $\tilde{x}(m)$. As we showed in the proof of Lemma \[key lemma\], the terms with repetitions do not play any role. On the other hand, Lemma \[approx norm\] easily extends to operator coefficients so that for any $1\leq p\leq 2$, every $\rho_j \in \mathbb{M}_2$ and every reduced word $x_j\in \mathcal L(\mathbb G_2)$, we have $$\label{normcoef}
\lim_{m\to \infty} \Big\| \sum_j \rho_j {\otimes}\tilde x_j(m)\Big\|_{L_p(\mathbb M_2{\overline{{\otimes}}}\A_{\varepsilon_m})}
=\Big\|\sum_j \rho_j {\otimes}x_j\Big\|_{L_p(\mathbb M_2{\overline{{\otimes}}}\mathcal L(\mathbb G_2))} \quad
\mbox{a.e.}$$ Let us denote by $u=\lambda(g_1)$ the canonical generator of $\mathcal L (\mathbb Z)$. By the positivity of $\mathcal P_{{\mathbb{Z}},t}$ and a density argument, it suffices to show that $\|\mathcal P_{{\mathbb{Z}},t} f\|_{L_2(\mathcal L (\mathbb Z))}\leq \|f\|_{L_p(\mathcal L (\mathbb Z))}$ for every positive trigonometric polynomial $$\displaystyle{f=\rho_0 \mathbf{1}+\sum_{j=1}^d
(\rho_j u^j+\overline \rho_j u^{*j})}.$$ To this end, we use the map $\pi$ from Lemma \[haar\] and construct $$\begin{aligned}
x \ = \ \pi (f) & = & \left[\begin{array}{cc} \rho_0 & 0 \\ 0&
\rho_0\end{array}\right] {\otimes}\mathbf{1} \\ & + & \displaystyle{\sum_{\ell \ge 1}}
\left[\begin{array}{cc} \rho_{2\ell} & 0 \\ 0& \overline
\rho_{2\ell}\end{array}\right]{\otimes}a_{1,2\ell} +
\left[\begin{array}{cc} \overline \rho_{2\ell} & 0 \\ 0&
\rho_{2\ell}\end{array}\right]{\otimes}b_{1,2\ell} \\ & + &
\displaystyle{\sum_{\ell \ge 1}} \left[\begin{array}{cc} 0 &
\rho_{2\ell+1} \\ \overline \rho_{2\ell+1} &
0\end{array}\right]{\otimes}a_{1,2\ell+1} +
\left[\begin{array}{cc}0 & \overline \rho_{2\ell+1} \\ \rho_{2\ell+1}&
0\end{array}\right]{\otimes}b_{1,2\ell+1}.\end{aligned}$$ To use our approximation procedure, we consider the element $\tilde x(m)\in \mathbb M_2{\overline{{\otimes}}}\A_{\varepsilon_m}$ associated to $x$. We start noting that $\tilde x(m)$ is self-adjoint. Now, in order to use and make act $Id_{\mathbb M_2}\otimes \mathcal S_{\varepsilon_m,t}$, we must write $\tilde x(m)$ in reduced form. That is, for every $\k \in [m]^s$ with $\sigma(\k) = \sigma_0$ and $\underline{\alpha}=(\alpha_1,\cdots,\alpha_s)\in \{1,2\}^s$ with $\alpha_j\neq \alpha_{j+1}$, we want to understand the matrix coefficients $\gamma^{\underline{\alpha}}(\k)$ of $x^{\underline{\alpha}}(\k)=x^{\alpha_1}(k_1) \cdots x^{\alpha_s}(k_s)$, where the latter is an element in the basis of $\A_{\varepsilon_m}$. In fact, it suffices to show that these matrix coefficients are multiples of unitaries, so that all the subsets $A$ of $\Upsilon_m$ are in $\mathcal U$ and we do not loose any constant when applying . Let us first assume that $s=2\ell+1$ is odd. Since by definition there is no simplifications in $\tilde x(m)$, the term $x^{\underline{\alpha}}(\k)$ will only appear in the element in $\A_{\varepsilon_m}$ associated to either $a_{1,2\ell+1}$ or $b_{1,2\ell+1}$. By the commutation relations, we see that $x^{\underline{\alpha}}(\k)^*=\pm x^{\underline{\alpha}}(\k)$. Then its matrix coefficient must also satisfy $\gamma^{\underline{\alpha}}(\k)^*=\pm \gamma^{\underline{\alpha}}(\k)$. Moreover, one easily checks that it also has the shape $$\left[\begin{array}{cc} 0 & \delta \\ \mu
&0\end{array}\right]$$ from the above formula of $x$. Hence $\delta=\pm
\overline \mu$ and $\gamma^{\underline{\alpha}}(\k)$ is a multiple of a unitary (this can also be directly seen from the formula of $x$). If $s=2\ell$, the term $x^{\underline{\alpha}}(\k)$ will appear in the elements associated to the two reduced words $a_{1,2\ell}$ and $b_{1,2\ell}$. Since the commutation relations only involve signs, after a moment of thought we can conclude that $\gamma^{\underline{\alpha}}(\k)$ has the shape $$\left[\begin{array}{cc} \delta& 0 \\ 0 &\overline
\delta\end{array}\right].$$ Hence, it is a multiple of a unitary. Actually, we also know that $\delta$ is either real or purely imaginary. Once we have seen that the matrix coefficients of $\tilde x(m)$ written in reduced form are multiples of unitaries, we can conclude the proof as in Theorem B. Indeed, using Lemma \[haar\], and , we get $$\begin{aligned}
\| f\|_{L_p(\mathbb T)} & = & \| x\|_{L_p(\mathbb M_2{\overline{{\otimes}}}\mathcal L (\mathbb
G_2))} \\ &=& \lim_{m\to \infty} \|\tilde x(m)\|_{L_p(\mathbb M_2{\overline{{\otimes}}}\A_{\varepsilon_m})} \\ &\geq& \lim_{m\to
\infty} \|(Id_{\mathbb M_2}{\otimes}\mathcal S_{\varepsilon_m,t})\tilde x(m)\|_{L_2(\mathbb M_2{\overline{{\otimes}}}\A_{\varepsilon_m})} \\&= & \| (Id_{\mathbb M_2} {\otimes}\mathcal P_{\mathbb{G}_2,t}) (x)\|_{L_2(\mathbb M_2{\overline{{\otimes}}}\mathcal L(\mathbb
G_2))} =\|\mathcal P_{{\mathbb{Z}},t}(f)\|_{L_2(\mathbb T)},\end{aligned}$$where the limits are taken a.e. and $t\geq -\frac 1 2 \log(p-1)$. The proof is complete. [ $\square$ 0.2cm]{}A slight modification of the previous argument allows us to improve Theorem A ii) for $q=2$. In fact, by a standard use of log-Sobolev inequalities we may also improve the $L_p \to L_q$ hypercontractivity bound, see Remark \[pqHC\] below.
\[improvement A iii)\] If $1< p\leq 2$, we find $$\big\|\mathcal P_{\mathbb F_n, t} : L_p(\mathcal L (\mathbb F_n))\to L_2(\mathcal L (\mathbb F_n))\big\|=1 \quad {if }\quad t\geq \frac 1 2 \log \frac{1}{p-1} + \frac 1 2 \Big(\frac 1
p - \frac 1 2\Big) \log 2 .$$
[ [**Proof.** ]{}]{}Once again, by positivity and density it suffices to prove the assertion for a positive trigonometric polynomial $f\in \mathcal L (\mathbb F_n)$. If $\jbar=(j_1, \ldots, j_d)$, we will use the notation $|\jbar|=d$ and $u_{\jbar}=\lambda(g_{\jbar})$ with $g_{\jbar}=g_{j_1}\cdots g_{j_d}$ a reduced word in ${\mathbb{F}}_n$, so that $$f={\sum\nolimits}_{\jbar} \rho_{\jbar} u_{\jbar}.$$ Here we use the usual convention that $g_{-k}=g_k^{-1}$. We use again the trace preserving $*$-homomorphism $\pi: \mathcal L
(\mathbb F_n)\to \mathbb M_2 {\overline{{\otimes}}}\mathcal L (\mathbb G_{2n})$ coming from Lemma \[haar\]. This gives the identity $$\pi(u_{\jbar})=\left[\begin{array}{cc} 0 & x_{2j_1-1}
\\ x_{2j_1}& 0\end{array}\right] \left[\begin{array}{cc} 0 & x_{2j_2-1}
\\ x_{2j_2}& 0\end{array}\right] \cdots\left[\begin{array}{cc} 0 & x_{2j_d-1}
\\ x_{2j_d}& 0\end{array}\right]$$ with the convention that for $j>0$, $x_{-2j}=x_{2j-1}$ and $x_{-2j-1}=x_{2j}$. If $d=0$, we set $g_{\jbar} =e$ and $\pi(u_{\jbar})=Id_{\mathbb M_2}$. Hence with $x=\pi(f)$, summing up according to the length we obtain $$\label{formula x=pi(f)}
\begin{array}{rclcl} x & \!\!\! = \!\!\! & \left[\begin{array}{cc} \rho_0 & 0 \\ 0& \rho_0 \end{array} \right]
{\otimes}\mathbf{1} & \!\!\! + \!\!\! & \displaystyle{\sum_{\substack{|\jbar|=2\ell \\ \ell \geq 1}}}
\left[\begin{array}{cc} \rho_{\jbar} & 0 \\ 0&
\rho_{-\jbar}\end{array}\right] {\otimes}x_{2j_1-1}x_{2j_2} \cdots x_{2j_{2\ell}} \\
& & & \!\!\! + \!\!\! & \displaystyle{\sum_{\substack{|\jbar|=2\ell+1 \\ \ell \geq 0}}}
\left[\begin{array}{cc} 0 &\rho_{\jbar} \\
\rho_{-\jbar}&0\end{array}\right] {\otimes}x_{2j_1-1}x_{2j_2} \cdots x_{2j_{2\ell+1}-1}. \end{array}$$ We repeat the arguments used in the proof of Proposition \[Weissler\] to approximate $x$ by a spin model $\tilde x(m)(\omega)$ with operator coefficients. That is, $x_{\alpha_1}\cdots x_{\alpha_s}\in \mathcal{L}(\mathbb G_{2n})$ is associated to $$\tilde{x}(m)(\omega) = \frac{1}{m^{s/2}} \sum_{\substack{\k \in
[m]^s \\ \sigma(\k) = \sigma_0}} x^{\alpha_1}(k_1)(\omega) \cdots
x^{\alpha_s}(k_s)(\omega)\in \A_{\varepsilon_m}.$$ Note that the contribution to $x$ given by of words of length 0 and 1 is $$\begin{aligned}
\left[\begin{array}{cc} \rho_0 & 0 \\ 0& \rho_0\end{array}\right]
{\otimes}1 +\displaystyle{\sum_{j\in \mathbb Z \setminus \{0\}}}
\left[\begin{array}{cc} 0 & \rho_{j} \\ \rho_{-j} &
0\end{array}\right]{\otimes}x_{2j-1}. \end{aligned}$$ Since $f$ is self-adjoint, we have $\rho_{-j}=\overline \rho_j$ for $j\in {\mathbb{Z}}\setminus \{0\}$. Hence the matrix coefficients corresponding to the words of length 0 and 1 in the approximation are multiples of unitaries. We will have $\{A\subset \Upsilon_m \; :\; |A|\leq 1\}\subset \mathcal U$ with the notations of , and decompose $f=g+h$, where $g$ is the part of $f$ of degree less than 1 and $h$ is supported by the words of length greater or equal than 2. Observe that $g$ and $h$ are orthogonal. Let $t=t_0+t_1$ with $t_0=-\frac 1 2 \log(p-1)$. Since $h$ has valuation 2, we have $$\|\mathcal P_{{\mathbb{F}}_n,t_0+t_1}(h)\|_{2}\leq e^{-2t_1} \|\mathcal P_{{\mathbb{F}}_n,t_0}(h)\|_{2}.$$ Thus thanks to , as in the proof of Proposition \[Weissler\], we get by orthogonality $$\begin{aligned}
\|f\|_{p}^2 & \geq & \| \mathcal P_{{\mathbb{F}}_n,t_0} (g)\|_{2}^2 + 2^{1-\frac 2p}
\|\mathcal P_{{\mathbb{F}}_n,t_0} (h)\|_{2}^2 \\ & \geq & \| \mathcal P_{{\mathbb{F}}_n,t} (g)\|_{2}^2 + 2^{1-\frac 2p}e^{4t_1}
\|\mathcal P_{{\mathbb{F}}_n,t} (h)\|_{2}^2 \\ & \geq & \| \mathcal P_{{\mathbb{F}}_n,t} (g)\|_{2}^2 + \|\mathcal P_{{\mathbb{F}}_n,t} (h)\|_{2}^2
= \| \mathcal P_{{\mathbb{F}}_n,t} (f)\|_{2}^2,\end{aligned}$$ provided that $e^{-4t_1} 2 ^{\frac 2 p-1} \leq 1 \Leftrightarrow t_1 \ge \frac 1 2 (\frac 1
p - \frac 1 2) \log 2$. This completes the proof. [ $\square$ 0.2cm]{}
[ Let $\sigma$ be the involutive $*$-representation on $\mathcal L(\mathbb F_n)$ exchanging $u_j$ and $u_j^*=u_{-j}$ for all $j\geq 1$. So that if $f=\sum_{\jbar} \rho_{\jbar} u_{\jbar}$, then $\sigma(f)=\sum_{\jbar} \rho_{-\jbar} u_{\jbar}$. Denote by $\mathcal
L(\mathbb F_n)^\sigma$ the fixed point algebra of $\sigma$, it clearly contains $\mathcal A_{sym}^n$. The above arguments actually prove that $\mathcal P_{{\mathbb{F}}_n,t}$ is hypercontractive on $\mathcal
L(\mathbb F_n)^\sigma$ from $L_p$ to $L_2$ with optimal time. Indeed, under this symmetric condition for $f$ all the matrix coefficients will be multiples of unitaries. Then using the equivalence between hypercontractivity with optimal time and log-Sobolev inequality, one sees that Theorem A iii) can be extended to $\mathcal
L(\mathbb F_n)^\sigma$. The Gross’ argument to deduce general hypercontractive inequalities $L_p\to L_q$ from $L_p\to L_2$ estimates in this setting are recalled in [@JPPP].]{}
\[pqHC\] [ We claim that $$\frac 1 2 \log \frac{1}{p-1} + \frac 1 2 \Big(\frac 1 p - \frac 1 2\Big) \log 2
\leq \frac{\beta}{2} \log \frac{1}{p-1}$$ with $\beta=1+\frac{\log(2)}{4}$. This is not difficult to prove by using basic computations. Then in particular Theorem \[improvement A iii)\] proves that we have hypercontractive $L_p\to L_2$ estimates for $t\geq -\frac{\beta}{2} \log(p-1)$. Then Gross’ arguments relying on log-Sobolev inequality apply when the time has this shape, and the constant $2$ given by Theorem A ii) can be replaced by the better constant $\beta=1+\frac{1}{4} \log(2) \sim 1.17$. Hence for any $1 < p \le q < \infty$ we get $$\big\| \mathcal{P}_{{\mathbb{F}}_n,t} \hskip1pt : L_p( \hskip0.5pt
\mathcal{L}({\mathbb{F}}_n) \hskip0.5pt ) \to L_q( \hskip0.5pt
\mathcal{L}({\mathbb{F}}_n) \hskip0.5pt ) \hskip1pt \big\| \, = \, 1
\hskip13pt \mbox{if} \hskip13pt t \ge \frac{\beta}{2} \log \frac{q-1}{p-1}.$$ ]{}
[99]{}
K. Ball, E.A. Carlen and E.H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. **115** (1994), 463-482.
W. Beckner, Inequalities in Fourier analysis. Ann. of Math. **102** (1975), 159-182.
A. Ben-Aroya, O. Regev and R. de Wolf, A hypercontractive inequality for matrix-valued functions with applications to quantum computing. Proc. 49th FOCS. IEEE Computer Society Press, 2008.
P. Biane, Free hypercontractivity. Comm. Math. Phys. **184** (1997), 457-474.
E. Blanchard and K. Dykema, Embeddings of reduced free products of operator algebras. Pacific J. Math. **199** (2001), 1-19.
A. Bonami, Étude des coefficients de Fourier des fonctions de $L^p(\G)$. Ann. Inst. Fourier Grenoble **20** (1970), 335-402.
M. Bożejko, B. Kümmerer and R. Speicher, $q$-Gaussian processes: non-commutative and classical aspects. Comm. Math. Phys. **185** (1997), 129-154.
H. Buhrman, O. Regev, G. Scarpa and R. de Wolf, Near-Optimal and Explicit Bell Inequality Violations. IEEE Conference on Computational Complexity 2011, 157-166.
E.A. Carlen and E.H. Lieb, Optimal hypercontractivity for Fermi fields and related noncommutative integration inequalities. Comm. Math. Phys. **155** (1993), 27-46.
B. Collins and M. Junge, Noncommutative Brownian motion. Preprint 2012.
D. Gavinsky, J. Kempe, I. Kerenidis, R. Raz and R. De Wolf, Exponential separations for one-way quantum communication complexity, with applications to cryptography. Proc. 39th STOC. ACM Press, 2007.
J. Glimm, Boson fields with nonlinear self-interaction in two dimensions. Comm. Math. Phys. **8** (1968), 12-25.
L. Gross, Existence and uniqueness of physical ground states. J. Funct. Anal. **10** (1972), 52-109.
L. Gross, Logarithmic Sobolev inequalities. Amer. J. Math. **97** (1975), 1061-1083.
L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form. Duke Math. J. **43** (1975), 383-396.
L. Gross, Hypercontractivity, logarithmic Sobolev inequalities and applications: A survey of surveys. Diffusion, quantum theory, and radically elementary mathematics: a celebration of Edward Nelson�s contribution to science. Princeton University Press, 2006.
U. Haagerup, An example of a non nuclear $C^*$-algebra, which has the metric approximation property. Invent. Math. **50** (1979), 279-293.
R. Hoegh-Krohn and B. Simon, Hypercontractive semigroups and two dimensional self-coupled Bose fields. J. Funct. Anal. **9** (1972), 121-180.
S. Janson, On hypercontractivity for multipliers on orthogonal polynomials. Ark. Math. **21** (1983), 97-110.
M. Junge, C. Palazuelos, J. Parcet and M. Perrin, Hypercontractivity in group von Neumann algebras. Preprint 2012.
T. Kemp, Hypercontractivity in non-commutative holomorphic spaces. Comm. Math. Phys. **259** (2005), 615-637.
S. Khot and N. Vishnoi, The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into $\ell_1$. Proceedings of 46th IEEE FOCS, 2005.
B. Klartag and O. Regev, Quantum one-way communication can be exponentially stronger than classical communication. Proc. 43rd Annual ACM Symp. Theory of Computing, 2011. arXiv:1009.3640.
I. Królak, Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations. Math. Z. **250** (2005), 915-937.
I. Królak,Optimal holomorphic hypercontractivity for CAR algebras. Bull. Pol. Acad. Sci. Math. **58** (2010), 79-90.
H.H. Lee and É. Ricard, Hypercontractivity on the $q$-Araki-Woods algebras. Comm. Math. Phys. **305** (2011), 533-553.
J.M. Lindsay, Gaussian hypercontractivity revisited. J. Funct. Anal. **92** (1990), 313-324.
J.M. Lindsay and P.A. Meyer, Fermionic hypercontractivity. Quantum probability VII, Singapore: Wold Scientific, 1992.
A. Montanaro, Some applications of hypercontractive inequalities in quantum information theory. Preprint 2012. arXiv:1208.0161.
A. Montanaro and T.J. Osborne, Quantum boolean functions. Chicago Journal of Theoretical Computer Science **1** (2010).
E. Nelson, A quartic interaction in two dimensions. Mathematical Theory of Elementary Particles. M.I.T. Press, 1965, 69-73.
E. Nelson, The free Markov field. J. Funct. Anal. **12** (1973), 211-227.
A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability. Cambridge University Press, 2006.
J. Parcet and G. Pisier, Non-commutative Khintchine type inequalities associated with free groups. Indiana Univ. Math. J. **54** (2005), 531-556.
G. Pisier and Q. Xu, Non-commutative $L_p$-spaces. Handbook of the Geometry of Banach Spaces II (Ed. W.B. Johnson and J. Lindenstrauss) North-Holland (2003), 1459-1517.
I. Segal, Construction of non-linear local quantum processes I. Ann. of Math. **92** (1970), 462-481.
I. Segal, A non-commutative extension of abstract integration. Ann. Math. **57** (1953), 401-457.
R. Speicher, A noncommutative central limit theorem. Math. Z. **209** (1992), 55-66.
D.V. Voiculescu, K.J. Dykema, A. Nica, Free random variables. CRM Monograph Series. American Mathematical Society, 1992.
F. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle. J. Funct. Anal. **37** (1980), 218-234.
\
Department of Mathematics\
University of Illinois at Urbana-Champaign\
1409 W. Green St. Urbana, IL 61891. USA\
`junge@math.uiuc.edu`
**Carlos Palazuelos**\
Instituto de Ciencias Matem[á]{}ticas\
CSIC-UAM-UC3M-UCM\
Consejo Superior de Investigaciones Científicas\
C/ Nicolás Cabrera 13-15. 28049, Madrid. Spain\
`carlospalazuelos@icmat.es`
**Javier Parcet**\
Instituto de Ciencias Matem[á]{}ticas\
CSIC-UAM-UC3M-UCM\
Consejo Superior de Investigaciones Científicas\
C/ Nicolás Cabrera 13-15. 28049, Madrid. Spain\
`javier.parcet@icmat.es`
**Mathilde Perrin**\
Instituto de Ciencias Matem[á]{}ticas\
CSIC-UAM-UC3M-UCM\
Consejo Superior de Investigaciones Científicas\
C/ Nicolás Cabrera 13-15. 28049, Madrid. Spain\
`mathilde.perrin@icmat.es`
**Éric Ricard**\
Laboratoire de Mathématiques Nicolas Oresme\
Université de Caen Basse-Normandie\
14032 Caen Cedex, France\
`eric.ricard@unicaen.fr`
|
---
abstract: 'This paper follows from the work of [@che7] and [@che12]. In the former paper we give an explicit construction of opetopes, the underlying cell shapes in the theory of opetopic $n$-categories; at the heart of this construction is the use of certain trees. In the latter paper we give a description of trees using Kelly-Mac Lane graphs. In the present paper we apply the latter to the former, to give a construction of opetopes using Kelly-Mac Lane graphs.'
author:
- |
Eugenia Cheng\
\
Department of Pure Mathematics, University of Cambridge\
E-mail: e.cheng@dpmms.cam.ac.uk
bibliography:
- 'bib0209.bib'
date: October 2002
nocite:
- '[@kl1]'
- '[@js1]'
title: 'The theory of opetopes via Kelly-Mac Lane graphs'
---
Introduction {#introduction .unnumbered}
============
In [@bd1] Baez and Dolan give a definition of weak $n$-category in which the underlying shapes of cells are ‘opetopes’ and the underlying data are given by ‘opetopic sets’. In the present paper we give an alternative construction of opetopes using the language of closed categories. This is made possible by the results of [@che12], giving a precise correspondence between the trees involved in the construction of opetopes, and the allowable morphisms in certain closed categories.
The end result is a precise algebraic characterisation in place of the more geometric world of trees.
The idea is as follows. Recall ([@che7]) that a $k$-opetope is constructed as a ‘configuration’ for composing $(k-1)$-opetopes; this is expressed as a tree (see [@che7]) whose nodes are labelled by the $(k-1)$-opetopes in question, with the edges giving their inputs and outputs.
In order to express this more precisely, it is helpful to give a more formal description of trees, as a bijection between inputs and outputs of nodes subject to a condition ensuring that closed loops do not arise.
In fact, this leads to an abstract description of trees as certain Kelly-Mac Lane graphs; this is the subject of [@che12]. In the present paper we apply the results of [@che12] to the construction of opetopes.
First we need to generalise the work of [@che12] since we require the ‘labelled’ version of the trees presented in that paper, and this is the subject of Section \[label\].
In Section \[constr\] we give the actual construction. We use the result, from [@che12], that a tree is precisely an allowable Kelly-Mac Lane graph of a certain shape. We may therefore express the process of forming labelled trees more precisely, by seeking allowable morphisms in a certain closed category. The construction we use to build up dimensions in this way is what we call a ‘ladder’; the idea is to pick out precisely the allowable morphisms that satisfy two conditions. The first is that the morphism must be of the correct shape to correspond to a tree. For the second, recall that an arrow of a slice multicategory is a way of composing its source elements to give its target element; the second condition for the ‘ladder’ construction corresponds to this stipulation.
In Section \[opex\] we give some low-dimensional examples of the above construction to help elucidate this rather abstract approach.
Then, in Section \[compare\], we prove that this construction does give the same opetopes as defined in [@che7]. This is the main result of this work. We conclude, in Section \[loopex\], with a brief discussion about the category of opetopes. This category is constructed explicitly in [@che9], and the work in the present paper extends to the construction of this category. We do not explicitly give this construction here, but we give some low-dimensional examples of the ‘face’ maps of opetopes in the new framework.
### Terminology {#terminology .unnumbered}
1. Since we are concerned chiefly with [*weak*]{} $n$-categories, we follow Baez and Dolan ([@bd1]) and omit the word ‘weak’ unless emphasis is required; we refer to [*strict*]{} $n$-categories as ‘strict $n$-categories’.
2. In [@bd1] Baez and Dolan use the terms ‘operad’ and ‘types’ where we use ‘multicategory’ and ‘objects’; the latter terminology is more consistent with Leinster’s use of ‘operad’ to describe a multicategory whose ‘objects-object’ is 1.
3. As in [@che9], we work with opetopes not precisely the same as those given in [@bd1], but rather as given by the opetopic theory as explained in [@che7] using multicategories with a category rather than a set of objects; we refer the reader to [@che7] for the full details.
This work was supported by a PhD grant from EPSRC. I would like to thank Martin Hyland and Tom Leinster for their support and guidance.
Opetopes {#loopsope}
========
In this section we use the results of [@che12] to construct opetopes. We first need to introduce the notion of [*labelled*]{} Kelly-Mac Lane graphs.
Preliminaries {#label}
-------------
For the construction of opetopes we require the ‘labelled’ version of the theory presented in [@che12]: labelled shapes, labelled graphs and labelled trees.
Given a category we can form [*labelled shapes*]{} (in ), that is, shapes labelled by the objects of . A labelled shape is a shape $T$ with each 1 ‘labelled’ by an object $A_i$ of . We write this as $$|T|(A_1, \ldots, A_k).$$ The variable set is then defined to be the variable set of the underlying shape.
For example given $$T=[\ [1,1] \otimes 1\otimes 1 \ , \ I\ ] \otimes1$$ we have a labelled shape $$\alpha = |T|(A_1, \ldots, A_5) = [\ [A_1, A_2] \otimes A_3 \otimes
A_4 \ , \ I\ ] \otimes A_5$$ with underlying shape $T$, and $$v(\alpha) = v(T) = \{+,-,-,-,+\}.$$
Given a category we can form labelled graphs, that is, graphs whose edges are labelled by morphisms of as follows. Consider labelled shapes $\alpha$ and $ \beta$ with underlying shapes $T$ and $S$ respectively. A [*labelled graph*]{} $$\alpha \lra \beta$$ is a graph $$\xi:T \lra S$$ together with a morphism $x \lra y$ for each pair of labels $x,y$ whose underlying variables are mates under $\xi$, with $v(x)=-$ and $v(y)=+$ in the twisted sum. That is, the morphism is in the direction $$- \lra +.$$ For example, the following is a labelled graph pic51.lp with underlying graph and variances as below pic52.lp .
Observe that, since the variances of the domain are reversed in the twisted sum, the direction of morphisms is also reversed in the domain.
We write $K\bb{C}$ for the category of labelled shapes and labelled graphs in ; thus $G=K\cat{1}$ as mentioned in [@che12].
A labelled graph is called allowable if and only if its underlying graph is allowable. We write $A\bb{C}$ for the category of labelled shapes and allowable labelled morphisms. We observe immediately that the correspondence between trees and graphs exhibited in [@che12] generalises to a correspondence between labelled graphs and labelled trees.
A labelled tree in is precisely an allowable morphism $$\alpha_1, \tensordot \alpha_k \ \lra \ \alpha \ \in A\bb{C}$$ with underlying shape $$X_{m_1}\tensordot X_{m_k} \lra X_{(\sum_i m_i)-k+1}.$$
Recall ([@che7]) that a labelled tree gives a ‘configuration for composing’ arrows of a via object-morphisms, as used in the slice construction. By the above correspondence, a labelled graph as above may also be considered to give such a configuration; thus in Section \[loopslice\] we will use such graphs to give an alternative description of the slice construction. We will need the following construction.
Given categories and and a functor $$F:\bb{C} \lra K\bb{D}$$ we define a functor $$KF:K\bb{C} \lra K\bb{D}$$ as follows.
- on objects
An object in $K\bb{C}$ is a labelled shape $$\alpha = |T|(x_1, \ldots, x_n);$$ put $$KF(\alpha) = |T|(Fx_1, \ldots Fx_n) \in K\bb{D}.$$
- on morphisms
Given a morphism $$|T|(x_1, \ldots, x_n) \map{f} |S|(x_{n+1}, \ldots, x_m) \in
K\bb{C}$$ we define $KFf$ as follows. Suppose $f$ has underlying graph $\xi$, say. Consider a pair of mates $a$ and $b$ in $\xi$, with the edge joining them ‘labelled’ with morphism $$g:a\lra b \in \bb{C}.$$ This gives a morphism $$Fg:Fa\lra Fb \in K\bb{D}.$$ So $Fg$ is a graph labelled in $\bb{D}$. Then $KFf$ consists of all such graphs given by mates in $\xi$, positioned according to the positions in $\xi$.
Furthermore, if $F: \bb{C} \lra \bb{D}$ then we get $$AF: A\bb{C} \lra A\bb{D}$$ by restricting the functor $KF$.
The construction of opetopes {#constr}
----------------------------
We seek to define, for each $k\geq 0$, a category $\ope_k$ of $k$-opetopes. A $k$-opetope $\theta$ is to have a list of input $(k-1)$-opetopes $\alpha_1, \ldots, \alpha_m$, say, and an output $(k-1)$-opetope $\alpha$, say. This data is to be expressed as an object $$[\ \alpha_1 \tensordot \alpha_m\ ,\ \alpha\ ]\in A\ope_{k-1}$$ called the [*frame*]{} of $\theta$ (see [@bd1]). Each frame has shape $X_m = [1^{\otimes m},1]$ for some $m \geq 0$. So, for each $k$ we will have a functor $$\phi_k:\ope_k \lra A\ope_{k-1}$$ and thus $$A\phi_k:A\ope_k \lra A\ope_{k-1}.$$
$\ope_k$ is defined inductively; for $k\geq 2$ it is a certain full subcategory of the comma category $$(I\downarrow A\phi_{k-1})$$ with the following motivation. A $k$-opetope $\theta$ with frame $$[\ \alpha_1 \tensordot \alpha_m\ ,\ \alpha\ ]$$ is a configuration for composing $\alpha_1, \ldots, \alpha_m$ to result in $\alpha$. That is, it is an allowable morphism $$\ I \map{\theta} [\phi_{k-1}\alpha_1 \tensordot
\phi_{k-1}\alpha_m\ ,\ \phi_{k-1}\alpha\ ] \in A\ope_{k-2}$$ such that the composition does result in $\alpha$. Such a $\theta$ is clearly an object of $(I\downarrow A\phi_{k-1})$; so we take the full subcategory whose objects are all those of the correct form.
In fact we begin with a more general construction for building up dimensions.
A [*ladder*]{} is given by
- for each $k \geq 0$ a category $\bb{D}_k$
- for each $k \geq 1$ a functor $F_k:\bb{D}_k \lra
A\bb{D}_{k-1}$
such that for each $k\geq 2$, $F_k$ is of the form $$\bb{D}_k \lra (I\downarrow AF_{k-1}) \lra A\bb{D}_{k-1}$$ where the second morphism is the forgetful functor.
Note that given $F_k:\bb{D}_k \lra A\bb{D}_{k-1}$ we have a functor $$AF_k: A\bb{D}_k \lra A\bb{D}_{k-1}$$ and the comma category $(I\downarrow AF_{k-1})$ has as its objects pairs $(\theta, z)$ where $z \in A\bb{D}_{k-1}$ and $\theta$ is an allowable morphism $$I \map{\theta} AF_{k-1}(z) \ \in A\bb{D}_{k-1}.$$
\[opeladder\] The [*opetope ladder*]{} is given as follows.
**
- $\bb{D}_0 = 1 = \{x\}$, say
- $\bb{D} = 1 = \{u\}$, say, with $$\begin{array}{rccc}
\phi_1 : & \bb{D}_1 & \lra & A\bb{D}_0\\
& u & \longmapsto & [x,x] \\
\end{array}$$
- For $k\geq 2$, $\bb{D}_k$ is a full subcategory of $(I\downarrow
A\phi_{k-1})$. This comma category has objects $(\theta, z)$ where $z \in A\bb{D}_{k-1}$ and $$I \map{\theta} A\phi_{k-1}(z)$$ is an allowable morphism in $A\bb{D}_{k-2}$. Then the subcategory $\bb{D}_k$ by the following two conditions:
1. The objects of $\bb{D}_k$ are all $(\theta, z)$ such that $z$ has shape $X_m$ for some $m\geq 0$. So $z=[\alpha_1 \tensordot
\alpha_m, \alpha]$ for some $\alpha_i, \alpha \in \bb{D}_{k-1}$.
2. For $k\geq 3$ we require in addition that $$A\phi_{k-2}\bar{\theta} \circ (\alpha_1 \tensordot \alpha_m)
= \alpha$$ as morphisms in $A\bb{D}_{k-3}$.
- For $k\geq 2$ the functor $\phi_k: \bb{D}_k \lra
A\bb{D}_{k-1}$ is the following composite $$\bb{D}_k \hookrightarrow (I\downarrow A\phi_{k-1}) \lra
A\bb{D}_{k-1}$$ where the functors shown are the inclusion followed by the forgetful functor.
Note that the composition in condition B is possible: each $\alpha_i$ is an object of $\bb{D}_{k-1}$, so is by definition a morphism $$I \lra A\phi_{k-2}(\phi_{k-1}\alpha_i) \in A\bb{D}_{k-3}.$$ Now $\theta$ is a morphism $$\ I \lra [\phi_{k-1}\alpha_1 \tensordot \phi_{k-1}\alpha_m\
,\ \phi_{k-1}\alpha\ ]$$ so $$\ \bar{\theta}\ :\ \phi_{k-1}\alpha_1 \tensordot
\phi_{k-1}\alpha_m \ \lra\ \phi_{k-1}\alpha$$ so the domain of $A\phi_{k-2}\bar{\theta}$ is indeed the codomain of $(\alpha_1 \tensordot \alpha_m)$ and the composite in $A\bb{D}_{k-3}$ may be formed.
For each $k\geq 0$ the category $\bb{D}_k$ defined above is the category of [*$k$-opetopes*]{}. We write $\bb{D}_k=\ope_k$. If the frame of a $k$-opetope has shape $X_m$ we say $\theta$ is an [*$m$-ary opetope*]{}.
1. In general (that is for $k \geq 3$) the objects of $\bb{D}_k$ are those of $(I\downarrow A\phi_{k-1})$ satisfying the conditions A and B. Condition A restricts our scope only to those objects having the correct shape; condition B ensures that the ‘output’ of the opetope is indeed the composite given. For $k=2$ condition B does not apply; any configuration of composing identity maps gives the identity.
2. A morphism $(\theta,z) \map{f} (\theta',z')$ in $(I\downarrow
A\phi_{k-1})$ is a morphism $$f:z \lra z' \in A\bb{D}_{k-1}$$ such that the following diagram commutes: pic57.lp
so a morphism $\theta \map{f} \theta'$ in $\bb{D}_k$ is given as follows. Writing $$\phi_k \theta = [\alpha_1 \tensordot \alpha_m, \alpha]$$ $$\phi_k \theta'= [\beta_1 \tensordot \beta_j, \beta]$$ $f$ must be a morphism $$[\alpha_1 \tensordot \alpha_m, \alpha] \lra
[\beta_1 \tensordot \beta_j, \beta] \in A\bb{D}_{k-1}.$$ So we must have $m=j$ and $f$ has the form pic30.lp that is, a permutation $\sigma \in \cat{S}_m$ and morphisms $$g_i: \beta_i \lra \alpha_{\sigma(i)}, \ \ \mbox{ for each }1\leq i \leq m$$ $$g:\alpha \lra \beta$$ in $\bb{D}_{k-1}$, such that the following diagram commutes pic31.lp
Examples {#opex}
--------
We now give the first few stages of the construction explicitly, together with some examples.
- $k=0$
$\ope_0=\cat{1}$, that is, there is only one 0-opetope. We may think of this as an object $\cdot$ ; we write $x$ for convenience.
- $k=1$
$\ope_1=\cat{1}$, that is, there is only one 1-opetope, $u$, say. We have $$\phi_1(u) = [x,x] \in A\ope_0$$ that is, the unique 1-opetope $u$ has one input 0-opetope and one output 0-opetope. We may think of this as $$\lra$$ or, showing variances $$-\ \ +$$ and then we have pic58.lp ,an allowable morphism in $A\ope_0$. (We do not show arrowheads since all arrows in $\ope_0$ are identity arrows.)
- $k=2$
We seek to construct the category $\ope_2$. First we consider an object $\alpha \in \ope_2$. $\alpha$ has frame $$\phi_2 \alpha \in A\ope_1$$ where $\phi_2 \alpha$ has shape $X_m$ for some $m\geq 0$. So we have $$\phi_2 \alpha = [u^{\otimes m}, u] = [u \tensordot u, u].$$ Now $\alpha$ is an allowable morphism $$I \map{\alpha}[\ \phi_1 u \tensordot \phi_1 u\ ,\ \phi_1 u\ ] \in
A\ope_0 = A\cat{1}$$ that is $$I \map{\alpha}[\ [x,x] \tensordot [x,x]\ ,\ [x,x]\ ]$$ or equivalently a morphism $$[x,x] \tensordot [x,x] \lra [x,x] \in A\cat{1}.$$
For example, for $m=3$ we may have a graph pic32.lp which we will later see corresponds to the following pic59.lp where the numbers show the order in which the input 1-opetopes are listed.
For the nullary case $m=0$ we seek an allowable morphism $$I \lra [x,x].$$ There is precisely one such, given by the following graph pic13.lp and we will later see that this corresponds to the nullary 2-opetope pic60.lp .
We now consider a morphism $$\alpha \map{f} \alpha' \in \ope_2.$$ We must have $$\phi_2 \alpha = \phi_2 \alpha' = [u^{\otimes m}, u],$$ say. Then $f$ is a morphism $$[u^{\otimes m}, u] \lra [u^{\otimes m}, u] \in A\ope_1 =
A\cat{1}.$$ So $f$ must be a permutation $\sigma \in \cat{S}_m$, an isomorphism. So we have $$\ope_2(\alpha, \alpha') = \left\{ \begin{array} {l@{\extracolsep{3em}}l}
\cat{S}_m & \mbox{if $\alpha$ and $\alpha'$ are both $m$-ary} \\
\emptyset & \mbox{otherwise}
\end{array} \right.$$ and $\ope_2$ is equivalent to a discrete category whose objects are the natural numbers.
Note that the action of $\phi_2$ on morphisms is given as follows. Given a morphism $f$ as above, the morphism $$\phi_2 f: \phi_2\alpha \lra \phi_2 \alpha' \in A\ope_1$$ is given by the forgetful functor $$(I\downarrow A\phi_1) \lra A\ope_1$$ so is simply the graph given by the permutation $\sigma$.
- $k=3$
We now seek to construct the category $\ope_3$. We first consider an $m$-ary opetope $\theta \in \ope_3$ with frame $$[\alpha_1 \tensordot \alpha_m,\alpha] \in A\ope_2$$ such that $$\phi_2 \alpha_i = [u^{\otimes n_i},u] \ \ \mbox{ for each $1
\leq i \leq m$}$$ $$\phi_2 \alpha = [u^{\otimes n},u].$$ So $\theta$ is an allowable morphism $$I \map{\theta}[\ [u^{\otimes n_1},u] \tensordot [u^{\otimes
n_m},u], [u^{\otimes n},u]\ ]$$ or equivalently $$[u^{\otimes n_1},u] \tensordot [u^{\otimes n_m},u]
\ \map{\bar{\theta}}\ [u^{\otimes n},u] \in A\ope_1,$$ such that $$(A\phi_1) \bar{\theta} \circ (\alpha_1 \tensordot
\alpha_m) = \alpha$$ as morphisms in $A\ope_0$.
For example for $m=2$ consider pic33.lp\
pic34.lp\
pic35.lp so $$\phi_2 \alpha_1 = [u \otimes u \otimes u\ ,\ u]$$ $$\phi_2 \alpha_2 = [u \otimes u\ ,\ u]$$ $$\phi_2 \alpha = [u \otimes u \otimes u\otimes u\ ,\ u]$$
Then $\bar{\theta}$ may have the following graph in $A\ope_1$ pic36.lp The condition B is seen to be satisfied by the following diagram; we apply $\phi_1$ to each component, and compose with $\alpha_1 \otimes \alpha_2$:
pic37.lp
This corresponds to a 3-opetope of the form pic61.lp .
Note that we still do not need to label the edges of the graph since $\ope_1$ also only has identity arrows.
A morphism $$\theta \map{f} \theta' \in \ope_3$$ then has one of the following two forms pic38.lp or pic39.lp where $g_1, g_2, g$ are morphisms in $\ope_2$. Since all morphisms in $\ope_2$ are isomorphisms, it follows that all morphisms in $\ope_3$ are isomorphisms. In fact, since $\ope_2$ is equivalent to a discrete category, $\ope_3$ is also, and similarly $\ope_k$ for all $k \geq 0$; this is proved in Section \[compare\].
- $k=4$
Finally we give an example of a 4-opetope $\gamma \in \ope_4$, with $$\phi_4 \gamma = [\theta_1 \otimes \theta_2\ ,\ \theta]$$ where pic40.lp\
pic41.lp\
pic42.lp and we have $$\phi_3 \theta_1 = [\ [u^{\otimes 3}, u] \otimes [u^{\otimes
2},u]\ ,\ [u^{\otimes 4},u] \ ] = [U_3 \otimes U_2\ ,\ U_4], \mbox{
say}$$ $$\phi_3 \theta_2 = [\ [u^{\otimes 2}, u] \otimes [u^{\otimes
2},u] \ , \ [u^{\otimes 3},u]\ ] = [U_2 \otimes U_2\ ,\ U_3]$$ $$\phi_3 \theta = [U_2 \otimes U_2 \otimes U_2 \ , \ U_4].$$
Then $\bar{\gamma}$ may be given by the following graph in $A\ope_2$
pic43.lp
where each $\sigma_i$ is a morphism in $\ope_2$, that is, a permutation. We then check condition B by the following diagram:
pic44.lp\
pic45.lp
giving the composite $\theta$ as required. Note that the permutations $\sigma_i$ appear as permutations of the appropriate edges in the above diagram.
This corresponds to an opetope of the following form:
pic46.lp
Comparison with the multicategory approach {#compare}
==========================================
In [@bd1], opetopes are constructing using symmetric multicategories. Dimensions are built up using the slicing process. We compare this process with the use of closed categories as above.
The slice construction {#loopslice}
----------------------
Recall the slice construction for a symmetric multicategory. Let $Q$ be a symmetric multicategory. Then the slice multicategory $Q^+$ is given by
- Objects: $o(Q^+) = \elt{Q}$
- Arrows: $Q^+(f_1, \ldots, f_n;f)$ is given by the set of ‘configurations’ for composing $f_1, \ldots, f_n$ as arrows of $Q$, to yield $f$.
Recall further that such a configuration for composing is given by a labelled tree $(T, \rho, \tau)$ where the nodes give the positions for composing the $f_i$. So by the results of [@che12] we may restate this using allowable morphisms in $K\bb{C}$, where $\bb{C} =
o(Q)$.
Let $Q$ be a symmetric multicategory with category of objects . Given an arrow $f \in Q(x_1, \ldots, x_m; x)$ we write $$\phi f = [x_1 \tensordot x_m, x] \in A\bb{C}.$$ Then the slice multicategory $Q^+$ is given as follows.
- objects $o(Q^+) = \elt{Q}$
- an arrow $\theta \in Q^+(f_1, \ldots, f_j; f)$ is an arrow $$\theta \in A\bb{C}(\ I\ ,\ [\phi f_1 \tensordot \phi f_j\ ,\ \phi
f]\ )$$ such that composing the $f_i$ in this configuration gives $f$.
\[loopspropn\] $\phi$ extends to a functor $$\phi: \elt{Q} \lra A\bb{C}.$$
Let $$f \in Q(x_1, \ldots, x_m;x)$$ $$g \in Q(y_1, \ldots, y_j;y).$$ Then $\elt{Q}(f,g) = \emptyset$ unless $m=j$. If $m=j$ then a morphism $f \map{\gamma} g$ is given by a permutation $\sigma \in
\cat{S}_m$ together with morphisms $$t_i:y_i \lra x_{\sigma(i)}$$ $$t: x \lra y$$ satisfying certain conditions. This specifies a unique allowable morphism $$[x_1 \tensordot x_m, x] \lra [y_1 \tensordot y_m, y] \in
A\bb{C}$$ and we define $\phi\gamma$ to be this morphism. This makes $\phi$ into a functor.
We call $\phi$ the [*frame functor*]{} for $Q$. We now show how the slicing process corresponds to moving one rung up the ‘ladder’.
\[loopspropp\] Let $Q$ be a with category of objects . Then the category is isomorphic to a full subcategory of the comma category $(I\downarrow A\phi)$ and the frame functor for $Q^+$ is given by $$\elt{Q^+} \hookrightarrow (I\downarrow A\phi) \lra
A(\elt{Q})$$ where the functors shown are the inclusion followed by the forgetful functor.
Write $\bb{C}_1 = \elt{Q} = o(Q^+)$.
An object of $\elt{Q^+}$ is $(\theta,p)$ where $p\in
\free{\bb{C}_1}\times\bb{C}_1$ and $\theta \in Q^+(p)$.
Write $$p = (f_1, \ldots, f_m;f).$$ Then $\theta$ is an allowable morphism $$I \map{\theta} A\phi[f_1 \tensordot f_m, f]$$ that is, an object $$(\ \theta\ ,\ [f_1 \tensordot f_m, f]\ ) \in (I\downarrow A\phi)$$ such that composing the $f_i$ according to $\theta$ results in $f$.
A morphism $(\theta,p) \lra (\theta',p')$ in $\elt{Q^+}$ is a morphism $p \lra p'$ in $\free{\bb{C}_1}\times\bb{C}_1$ such that a certain commuting condition holds. Such a morphism is precisely an allowable morphism $$[f_1 \tensordot f_m, f] \lra [f'_1 \tensordot f'_m, f'] \in
A\bb{C}_1$$ and the commuting condition is precisely that ensuring that this is a morphism $\theta \lra \theta'$ in $(I\downarrow A\phi)$.
It is then clear that the frame functor is given by the inclusion followed by the forgetful functor as asserted.
\[corcor\] The category is the full subcategory of $(I\downarrow A\phi)$ whose objects are all $(\theta,p)$ satisfying the following two conditions:
1. $p$ has shape $X_m$ for some $m \geq 0$ so $p = [f_1
\tensordot f_m, f]$
2. the result of composing the $f_i$ according to $\theta$ is $f$.
If $Q$ is itself a slice multicategory then we can state the condition (ii) in the language of closed categories as well, since each $f_i$ is itself an allowable graph.
So we now consider forming $Q^{++}$, that is, the slice of a slice multicategory. Let $Q$ be a symmetric multicategory with category of objects $\bb{C}_0$. We write $$\bb{C}_1 = o({Q^+})$$ with frame functor $$\begin{array}{rccc}
\phi_1 : & \bb{C}_1 & \lra & A\bb{C}_0\\
& f \in Q(x_1, \ldots, x_m; x) & \longmapsto & [x_1 \tensordot x_m,x] \\
\end{array}$$
Also, we write $$\bb{C}_2 = \elt{Q^+}$$ with frame functor $$\begin{array}{rccc}
\phi_2 : & \bb{C}_2 & \lra & A\bb{C}_1\\
& \alpha \in Q^+(f_1, \ldots, f_m; f) & \longmapsto & [f_1 \tensordot f_m,f] \\
\end{array}$$
\[loopspropm\] Let $\theta$ be a configuration for composing $\alpha_1, \ldots,
\alpha_j \in \elt{Q^+}=\bb{C}_2$ expressed as an allowable morphism $$I \map{\theta} [\ \phi_2 \alpha_1 \tensordot \phi_2 \alpha_j\
,\ \phi_2 \alpha] \in A\bb{C}_1.$$ Then the result of composing the $\alpha_i$ in this configuration is $$(A\phi_1)\bar{\theta}\ \circ\ (\alpha_1 \tensordot \alpha_j)$$ composed as morphisms of $A\bb{C}_0$.
By definition, each $\alpha_i$ is a morphism in $A\bb{C}_0$ of shape $$I \lra [\ X_{im_1} \tensordot X_{im_i} \ , \ X \ ],$$ so is a tree labelled in $\bb{C}_0$. These trees are composed by node-replacement composition (see [@che12]) and the “composition graph” is given by $\bar{\theta}$.
\[loopsproppp\] An arrow $\theta \in Q^{++}(\alpha_1, \tensordot, \alpha_j;
\alpha)$ is precisely a morphism $$\theta \in A\bb{C}_1(\ I\ ,\ [\phi_2 \alpha_1 \tensordot \phi_2 \alpha_j\ ,
\ \phi_2 \alpha]\ )$$ such that $$(A\phi_1)\bar{\theta}\ \circ\ (\alpha_1 \tensordot \alpha_j) = \alpha\
\in A\bb{C}_0$$
\[loopspropq\] $\elt{Q^{++}}$ is the full subcategory of $(I\downarrow A\phi_2)$ whose objects are all $(\theta,p)$ satisfying the following two conditions:
1. $p$ has shape $X_m$ for some $m\geq 0$, so $p=[\alpha_1 \tensordot \alpha_m; \alpha] \in A\bb{C}_2$
2. $(A\phi_1)\bar{\theta} \circ(\alpha_1 \tensordot \alpha_j) =
\alpha$.
Finally we are ready to show that the opetopes constructed using symmetric multicategories correspond to those constructed in closed categories.
\[loopspropr\] Let $Q$ be the symmetric multicategory with just one object and one (identity) morphism. Then for all $k\geq 0$ $$o(Q^{k+}) \cong \ope_k$$ where $Q^{0+}=Q$.
For $k\leq 1$ the result is immediate by Definition \[opeladder\]. For $k=2$ we use Corollary \[corcor\] on $Q^+$; the result follows since condition (ii) is trivially satisfied. For $k \geq 3$ we use Corollary \[loopspropq\] on $Q^{(k-3)+}$; the result follow since the $\phi_2$ in the Corollary is $\phi_{k-2}$ in the case in question.
The category of opetopes {#loopex}
========================
Recall that in [@che9] we defined the category of opetopes. It is now possible to restate this definition in the framework of Kelly-Mac Lane graphs; we copy the definition exactly, using the fact that the bijection giving the formal definition of a tree gives the mates in the corresponding Kelly-Mac Lane graph.
Although we do not give the construction explicitly here, we give some examples of low-dimensional face maps. We use the example of a 3-opetope as given in Section \[opex\].
For the 2-opetopes we have face maps pic64.lp together with the isomorphic cases.
For 1-opetopes we then have $$s_1, s_2, s_3, t\ :\ u \lra \alpha_1$$ $$s_1, s_2, t\ :\ u \lra \alpha_2$$ $$s_1, s_2, s_3, s_4, t\ :\ u\lra \alpha$$ but by considering the generating relations, here given by mates in the graph $\theta$, we have $$\begin{array}{ccc}
s_1s_1 & = & s_2t\\
s_1s_2 & = & ts_2\\
s_1s_3 & = & ts_3\\
s_1t & = & tt\\
s_2s_1 & = & ts_1\\
s_2s_2 & = & ts_4;
\end{array}$$ note that $s_is_j$ give the $jth$ source of the $ith$ source of $\theta$.
For 0-opetopes we have in addition face maps $$x \lra u$$ and the relations on composites $$x \lra \theta$$ are generated by relations on composites $$x \lra \alpha_i$$ as well as by those on composites $$u \lra \theta.$$
For the former relations we are considering mates under graphs $\alpha_i \in A\ope_0$, and for the latter, mates under the graph $(A\phi_1) \bar{\theta} \in A\ope_0$. So in fact we are considering, in total, all objects connected in the composite graph $$(A\phi_1) \bar{\theta} \circ(\alpha \tensordot \alpha_m) \in
A\ope_1.$$ So we have $$\begin{array}{ccccccccccc}
ts_1s & = & s_2s_1s & = & s_2s_2t & = & ts_4t \\
ts_1t & = & s_2s_1t & = & s_2tt & = & s_1s_1t & = & s_1s_2s & = &
ts_2s\\
ts_2t & = & s_1s_2t & = & s_1s_3s & = & ts_3s \\
ts_3t & = & s_1s_3t & = & s_1tt & = & ttt \\
ts_4s & = & s_2s_2s & = & s_2ts & = & s_1s_1s & = & s_1ts & = &
tts
\end{array}$$
Note that since $$(A\phi_1) \bar{\phi} \circ(\alpha_1 \tensordot \alpha_m) =
\alpha$$ the 0-cell face maps for $\theta$ are precisely those of the form $tf$ where $f$ is a 0-cell face map of $\alpha = t(\theta)$. This reflects the fact that, when $2$-opetopes are composed along $1$-opetopes, the composite is formed by ‘deleting’ the boundary 1-opetopes, but no 0-cells are deleted. This result generalises to $k$-opetopes, but we do not prove this here.
|
---
abstract: 'This is the fourth and last paper in a sequence on Krull dimension for limit groups, answering a question of Z. Sela. In it we finish the proof, analyzing limit groups obtained from other limit groups by adjoining roots. We generalize our work on Scott complexity and adjoining roots from the previous paper in the sequence to the category of limit groups.'
address: |
Department of Mathematics\
University of Michigan\
Ann Arbor, MI 48109-1043\
USA
author:
- Larsen Louder
bibliography:
- 'krull.bib'
title: |
Krull dimension for limit groups IV:\
Adjoining roots
---
Introduction, Notation, Theorems {#sec:introduction}
================================
It will take a moment to establish the notation and define the objects needed to state our main theorem. Roughly, we are interested in solutions, in the category of limit groups, to equations of the form “adjoin a root to $g$.” We can give no specific characterizations of solutions, but under special circumstances arising in the second paper in this series, [@louder::stable], we are able to show that most of the time solutions are unique.
The notation $\cent_G(E)$ indicates the centralizer in $G$ of a subgroup $E$. The set of images of edge groups incident to a vertex group $V$ of a graph of groups decomposition is denoted by $\E(V)$. The phrase “‘$X$’ is controlled by ‘$Y$’” should be read as “there is a function $f,$ defined independently of ‘$X$’ and ‘$Y$’, such that $X\leq f(Y)$”.
Let $G$ be a group. A system of equations over $G$ is a collection of words in the alphabet $\set{x_i,g}:{g\in G},$ where the $x_i$ are variables distinct from the elements of $G$. The elements of $G$ are the coefficients, and the coefficients occuring in $\Sigma$ are the coefficients of $\Sigma$. If $\Sigma$ is a system of equations over $G$ there is a canonical group $G_{\Sigma}$ associated to $\Sigma$ with the presentation $\group{x_i,G}:{\Sigma},$ where the $x_i$ are the variables occuring in $\Sigma$. If the map $G\to G_{\Sigma}$ is injective then $\Sigma$ has a solution. If $G<H$ and the inclusion map extends to $G_{\Sigma}$ then $\Sigma$ has a solution in $H$. In analogy with field extensions, suppose $\Sigma$ is a system of equations over $G$. If $G<H$ and the inclusion map extends to a surjection $G_{\Sigma}\onto H$ then $H$ is a for $\Sigma,$ and $G$ is the . Splitting groups are partially ordered by the relation “maps onto.” Every pair $G<H$ is a ground-splitting pair for some (in general, many) system of equations $\Sigma(G,H)$. A tuple $(G,H,G')$ is if $H$ and $G'$ are both splitting groups over $G,$ and $H\onto G'$.
One may ask for splitting groups in a category $\mathcal{C}$ of groups. If $H\in\mathcal{C}$ is a splitting group, then $H$ is a splitting group in $\mathcal{C}$. If $\mathcal{C}$ is the class of all groups, then there are maximal $\mathcal{C}$–splitting groups, but this is not the case for general classes.
A sequence of inclusions $\G=(\G(0)<\G(1)<\dotsb)$ is a . A is a pair of sequences $(\G,\HH)$ such that $\G$ is a tower and $\G(i)$ is a splitting group for (some system) $\Sigma(\G(i-1),\HH(i)),$ that is, $(\G(i-1),\HH(i),\G(i))$ is a flight. All staircases considered in this paper have the property that all coefficients lie in $\G(0)$. The name staircase comes from the fact that a commutative diagram representing one looks like a staircase and walks up a tower.
\[def:adjunctionofroots\] Let $G$ be a finitely generated group, $\E$ a collection of nontrivial abelian subgroups of $G$. For each $E\in\E,$ let $\mathcal{F}(E)$ be a collection of finite index supergroups of $E,$ with an inclusion map $i_{E,F}\colon E\into F$ for each $F\in\mathcal{F}(E),$ and let $\mathcal{F}(\E)$ be the collection $\set{\mathcal{F}(E)}$. Let $$\adjoinroot{G}{\E}{\mathcal{F}(\E)} \define
\group{G,F}:{E=i_{E,F}(E)}_{F\in\mathcal{F}(E),E\in\E}$$
A finitely generated group $H$ is obtained from $G$ by *adjoining roots $\mathcal{F}(\E)$ to $\E$* if $G<H$ and the inclusion map extends to a surjection $$\adjoinroot{G}{\E}{\mathcal{F}(\E)}\onto H$$ Let $\Sigma=\Sigma(\E,\mathcal{F}(\E))$ be a system of equations corresponding to the identification of $E$ with $i_{E,F}(E)$ for all $E$ and $F\in\mathcal{F}(E)$. Then $H$ is a splitting group for $\Sigma$. We call $H$ a cyclic extension of $G$ because the relations are all of the form “adjoin a root to $G$.”
Most of the time the specific nature of $\mathcal{F}$ is immaterial, and we usually eliminate it from the notation. To further compress the language used, sometimes we simply write that $H$ is obtained from $G$ by adjoining roots.
A group is , or , if maximal abelian subgroups are malnormal. Let $\sim_Z$ be the relation “is conjugate into the centralizer of”. This is an equivalence relation as long as the group is . Two important consequences of are commutative transitivity and that every nontrivial abelian subgroup is contained in a unique maximal abelian subgroup.
Commutative transitivity can occasionally be used to simplify systems of equations. Suppose $H$ is obtained from $G$ by adjoining roots $\mathcal{F}(\E)$ to $\E$. Let $\eta$ be the inclusion map. We remove some redundancy by singling out a subcollection of each of $\E$ and $\mathcal{F}(\E),$ and replacing each subcollection by a single element. Fix some $\sim_Z$ equivalence class $[E]$. By conjugating we may assume that each element of $[E]$ is a subgroup of $\cent_G([E])$. Replace $[E]$ by $\set{\cent_G([E])},$ and replace $\cup_{B\in[E]}\mathcal{F}(B)$ by $$\group{\cent_G([E]),F}:{B=i_{B,F}(B)}_{B\in[E],F\in\mathcal{F}(B)}^{\text{ab}}$$ Then by commutative transitivity $H$ is a quotient of $$\adjoinroot{G}{\E}{\mathcal{F}(\E)}$$ Since limit groups are we make this reduction without comment. Since $\mathcal{F}(E)$ has a single element after this simplification, we will generally use the less ostentatious notation $F(E)$ or just $\sqrt{E}$. We will call a system of equations without any such redundancy .
\[def:seqadjunctionofroots\] A is a staircase, with tower $\G,$ equipped with a family families $\E$ of subgroups $\E_i$ of $\G(i),$ $(\G,\HH,\E),$ such that
- $(\G(i-1),\HH(i),\G(i))$ is a flight; $\HH(i)$ is obtained from $\G(i-1)$ by adjoining roots to $\E_{i-1}$
- Each $E'\in\E_{i}$ in $\G(i)$ centralizes, up to conjugacy, the image of an element $E$ of $\E_{i-1}$. If $E\in\E_{i-1}$ is mapped to $E'\in\E_{i}$ then we require that the image of $\cent_G(E)$ in $\cent_{G'}(E')$ be finite index.
To fix notation, the maps $\G(i)\into\G(i+1),$ $\G(i)\into\HH(i+1),$ and $\HH(i+1)\onto\G(i+1)$ are denoted by $\eta_i,$ $\nu_i,$ and $\pi_{i+1},$ respectively. The length of $\G$ is denoted $\Vert\G\Vert$.
It will be handy to have a rough description of a staircase. A staircase of limit groups is
- if all $\G(i)$ are freely decomposable
- if all $\G(i)$ are freely indecomposable
- if no $\G(i)$ has a subgroup
- if it has both freely decomposable and freely indecomposable groups, or, if freely indecomposable, has both groups with and without subgroups. Otherwise it is .
Let $(i_j)$ strictly increasing sequence of indices. A staircase $(\mathcal{V},\mathcal{W}),$ such that $\mathcal{V}(j)=\G(i_j)$ and $\mathcal{W}(j)=\HH(i_j),$ with maps obtained by composing maps from $(\G,\HH),$ is a of $(\G,\HH),$ and is $(i_j)$.
To see that a contraction of a cyclic staircase is a staircase consider the following diagram:
Each $E\in\E_i$ has finite index image in its counterpart in $\E_{i+1},$ the image of $E$ in its counterpart in $\E_{i_{j+1}}$ is finite index. Extending an abelian group by a finite index super-group multiple times can be accomplished by extending once.
The need for contractions explains the restriction that each $E\in\E_i$ contain a conjugate of the image of some $E'\in\E_{i-1}$. If this is not the case, then there is no hope for the existence of contractions; we can’t adjoin a root to an element that isn’t there.
A of a staircase is a contraction whose indices are consecutive, that is $i_{j+1}-i_j=1$ for all $j$.
Let $\E$ be a collection of elements of a group $G$. We denote by $\Vert\E\Vert$ the number of $\sim_Z$ equivalence classes in $\E$. The of $(\G,\HH,\E)$ is the triple $\comp((\G,\HH,\E))\define(\betti(\G),\depthpc(\HH),\Vert\E\Vert)$. Complexities are not compared lexicographically: $(b',d',e')\leq
(b,d,e)$ if $b'\leq b,$ $d'\leq d,$ and $e'\leq e+2(d-d')b$. That this defines a partial order follows easily from the definition. The inequality is strict if one of the coordinate inequalities is strict. See Definition \[def:depth\] and the material thereafter for a discussion of depth. Another immediate consequence of the definition of $\leq$ is that it is locally finite.[^1]
Let $(\G,\HH,\E)$ be a staircase. The quantity $\ninj((\G,\HH,\E))$ is the number of indices $i$ such that $\HH(i)\onto\G(i)$ is *not* an isomorphism.
\[maintheorem\] Let $(\G,\HH,\E)$ be a staircase. There is a function $\ninj(\comp((\G,\HH,\E)))$ such that $$\ninj((\G,\HH,\E))\leq\ninj(\comp((\G,\HH,\E)))$$
Although it would be nice to assign a complexity $c()$ to a limit group such that if, in a flight $(G,H,G'),$ $c(G)=c(G'),$ then $H\onto G'$ is an isomorphism, this doesn’t seem possible, and the approach taken in this paper requires that complexities be computed and compared in context.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author thanks Mladen Bestvina, Mark Feighn, and Zlil Sela for many discussions related to this paper.
Complexities of sequences
=========================
The main object which enables this analysis of adjoining roots is the decomposition, a device for encoding families of splittings of groups. This exposition borrows from [@bf::lg; @sela::jsj]. A , or of a group $G$ is a finite graph of groups decomposition over abelian edge groups such that every vertex group is marked as one of , , or , where by we mean is the fundamental group of a compact surface with boundary possessing two intersecting essential simple closed curves. Moreover, edge groups adjacent to a vertex group must be conjugate to boundary components of the surface. If $A$ is an abelian vertex group, the subgroup of $A$ is the subgroup of $A$ which dies under every map $A\to\zee$ killing all incident edge groups.
We say that two ’s of a limit group are if they have the same elliptic subgroups. A splitting is in a $\Delta$ if it corresponds to cutting a vertex group along a simple closed curve, a one-edged splitting of an abelian vertex group in which the peripheral subgroup is elliptic, or is a one edged splitting corresponding to an edge from an equivalent decomposition. If $\Delta$ is a , then $g\in G$ is $\Delta$–elliptic if elliptic in every one-edged splitting of $G$ visible in $\Delta$. Let $\elliptic(\Delta)$ be the set of $\Delta$–elliptic elements.
Let $G$ be a freely indecomposable finitely generated group, and let $\mathcal{C}$ be a family of one-edged splittings of $G$ such that
- edge groups are abelian,
- noncyclic abelian subgroups are elliptic.
The main construction of theory is that given a family of splittings $\mathcal{C}$ satisfying these conditions, there is a $\Delta$ such that $\elliptic(\Delta)=\cap_{C\in\mathcal{C}}\elliptic(C)$.
An abelian decomposition of $G$ is a $\ajsj(G)$ such that the set of [–elliptic]{} elements corresponds to the collection of all one-edged splittings satisfying the bullets above. The existence of a decomposition is somewhat subtle, as one needs to bound the size of a arising in this way [@sela::dgog1 Theorem 3.9]. If $G$ is a nonelementary freely indecomposable limit group then $G$ has a nontrivial decomposition. If $G$ is elementary, the is a point.
In this paper we are interested in the principle cyclic decomposition, which is the associated to the family of principle cyclic splittings.
\[def:principlecyclicsplitting\] A one-edged splitting over a cyclic subgroup is if at least one vertex group is cyclic, and is otherwise. A splitting of a limit group is an essential one-edged splitting $G\cong A*_CB$ or $G\cong A*_C,$ over a cyclic subgroup $C,$ such that either $\cent_G(C)$ is cyclic or $A$ is abelian.
The of a freely indecomposable limit group is the decomposition corresponding to the family of principle cyclic splittings. We denote the principle cyclic by $\jsj(G)$.
Let $\E\subset G$. The is a decomposition corresponding to the family of all principle cyclic splittings of $G$ such that each member of $\E$ is elliptic. We denote the relative by $\jsj(G;\E)$. A is simply a relative principle cyclic for some collection $\E$.
The , or for short, of a freely indecomposable limit group $G$ *with* subgroups is the relative decomposition associated to the set of principle cyclic splittings whose edge groups are hyperbolic in some other principle cyclic splitting. It is obtained from the by collapsing all edges not adjacent to some vertex group. If $G$ doesn’t have vertex groups, then the restricted is just the principle cyclic . The restricted principle cyclic is denoted by $\rjsj(G)$
That limit groups have principle cyclic splittings is [@sela::dgog1 Theorem 3.2]. It need not be the case that every splitting visible in the principle cyclic is principle; for instance, a boundary component of a vertex group may be the only edge attached to a cyclic vertex group. The splitting corresponding to the boundary component is not essential, but is certainly visible in the principle cyclic .
In this paper we work primarily with the principle cyclic of $G,$ indicated by $\jsj(G),$ and the . If $\Delta$ is a graph of groups decomposition then $T_{\Delta}$ is the Bass-Serre tree corresponding to $\Delta$.
We can give a more explicit description of the principle cyclic . Consider the abelian of a limit group $G$. Clearly all vertex groups of $\ajsj(G)$ appear as vertex groups of $\jsj(G)$. If $A$ is an abelian vertex group of $\ajsj(G)$ with noncyclic peripheral subgroup, since there is no principle cyclic splitting of $G$ over a subgroup of $A,$ the subgroup of $G$ generated by $A$ and conjugates of rigid vertex groups having nontrivial intersection with $A$ must be elliptic in $\jsj(G)$. If $R$ is a rigid vertex group of $G$ and an edge group $E$ incident to $R$ has noncyclic centralizer in $R,$ then the subgroup of $G$ generated by $R$ and any conjugate of a rigid vertex group $R'$ intersecting $R$ in a nontrivial subgroup of $E$ is also elliptic. From this we see that the principle cyclic of $G$ must have the following form:
- Every abelian vertex group has cyclic peripheral subgroup. If $R$ is adjacent to an abelian vertex group $A,$ $E$ the edge group, then $R$ does not have an essential one-edged splitting over $E$ in which each element of $\E(R)$ is elliptic.
- If an edge $e$ incident to a rigid vertex group $R$ has noncyclic centralizer in $R,$ then the edge is attached to a boundary component of a vertex group.
- If two edges incident to a rigid vertex group $R$ have the same centralizer in $R,$ then they are both incident to vertex groups, and their centralizer in $R$ is noncyclic.
The decomposition of a limit group, be it abelian or principle cyclic, is only unique up to morphisms of graphs of groups preserving elliptic subgroups. Some principle cyclic ’s are more convenient to work with than others, and we assume throughout that
- Edge groups not adjacent to vertex groups are closed under taking roots, and edge maps of edge groups into vertex groups are isomorphisms with the corresponding boundary components.
- There are no inessential splittings visible in the , other than from valence one cyclic vertex groups attached to boundary components of vertex groups.
Let $R$ be a rigid vertex group of the full abelian of a limit group $G,$ and let $\bar{R}$ be the subgroup of $G$ generated by $R$ and all elements with powers in $R$. A decomposition with the properties above can be thought of as the decomposition associated to the family of principle cyclic splittings in which all $\bar{R},$ $R$ a vertex group of the abelian , are elliptic.
In general, there are infinitely many principle cyclic decompositions of a limit group, all obtained from the principle cyclic by folding, cutting vertex groups along simple closed curves, and collapsing subgraphs.
\[lem:numberofdecompositions\] Let $G$ be a limit group and $\E\subset G$ a collection of elements of $G$. Then there are at most $2^{\Vert\E\Vert}$ equivalence classes of principle cyclic decompositions in which some elements of $\E$ are elliptic.
If $E\in\E$ is elliptic, then so is any $E'\in\E$ such that $E\sim_Z E'$.
We need to adapt the definition of the analysis lattice of a limit group given in [@sela::dgog1 §4] to the inductive proof given in section \[finishargument\]. A limit group is if it is abelian, free, or the fundamental group of a closed surface.
The of a limit group $G$ is the rooted tree of groups whose levels are defined as follows:
- $G$
- The free factors of a Grushko decomposition of $G$.
- The vertex groups at level $1$ are the vertex groups of the .
- Rinse and repeat, incrementing the index by one each time.
If an elementary limit group is encountered, it is a terminal leaf of the tree.
\[def:depth\] The of a limit group $H$ is the number of levels in its principle cyclic analysis lattice, and is denoted $\depthpc(H)$.
The of a staircase $(\G,\HH,\E)$ is $\max\set{\depthpc(\HH(i))},$ and is is denoted $\depthpc((\G,\HH,\E))$. The of $(\G,\HH,\E)$ is the first betti number of $\G(1)$.
It is not always necessary to refer to the family $\E,$ so we suppress it from the notation when its size is irrelevant. That the depth is well defined is a consequence of Theorem \[analysislatticebound\].
\[analysislatticebound\] The depth of the principle cyclic analysis lattice of a limit group $L$ is controlled by its rank.
We only need to worry about the possibility that the principle cyclic analysis lattice contains a long branch of the form $G_0>G_1>\dotsb,$ where each $G_i$ is freely indecomposable, has no vertex groups, no noncyclic abelian vertex groups, and $\jsj(G_i)$ has only one nonabelian vertex group $G_{i+1}$. After observing [@louder::strict; @houcine-2008] that $L$ has a strict resolution of length at most $6\rk(L),$ the proof is identical to [@louder::stable Theorem \[STABLE-lem:depthbound\]].
We motivate our proof of Theorem \[maintheorem\] and the previous definition with an example.
Suppose that $\G(i)$ has a one-edged decomposition with two nonabelian vertices for all $i$. Since a limit group has a principle cyclic splitting, the one-edged splitting of $\G(i)$ must be of the form $\G_1(i)*_{\group{e_i}}\G_2(i)$. By Lemma \[actshyperbolically\], if $H$ is obtained from $G$ by adjoining roots, then $G$ acts hyperbolically in all splittings of $H$. In particular, every vertex group of $H$ contains a vertex group of $G$. If the of $H$ was a loop, then the map to the underlying graph kills $G,$ but since the map $G\to H$ is almost onto on homology, this cannot happen. Thus each $\HH(i)$ has a one-edged decomposition $\HH_1(i)*_{\group{f_i}}\HH_2(i)$.
The triple $\G(i-1)\into\HH(i)\onto\G(i)$ has the following form: The pairs $(\G_j(i-1),\group{e_{i-1}})$ map to the pairs $(\G_j(i),\group{e_{i}})$ and $(\HH_j(i),\group{f_i}),$ and the maps $\eta_i,$ $\nu_i,$ and $\pi_i$ respect these one-edged splittings. We’ll show later that in fact $\HH_j(i)$ is obtained from $\G_j(i-1)$ by adjoining roots. By work from [@louder::stable] $\G_j(i)$ is obtained from the image of $\HH_j(i)$ by iteratively adjoining roots to the incident edge group (See Appendix \[appendix\_strict\]). Let $\G'_j(i)$ be the image of $\HH_j(i)$ in $\G_j(i)$. Now consider the staircase $(\G'_j(i),\HH_j(i))$. The sequence $\HH_j$ has strictly lower depth than $\HH$.
By induction on $\comp,$ there is an upper bound on the number of indices such that $\HH_j(i)\to\G(i)$ is not injective, and for at most twice that bound, both maps $\HH_j(i)\to\G_j(i)$ are injective. Then $\HH(i)\onto\G(i)$ is strict for such indices. Since every Dehn twist in $\group{f_i}$ pushes forward to a Dehn twist in $\group{e_{i+1}},$ $\pi_i$ is an isomorphism.
Aligning decompositions
========================
Let $G$ be a finitely generated group acting on a simplicial tree $T$ minimally and without inversions. It is a standard fact that the quotient $T/G$ is the underlying graph of a graph of groups decomposition of $G$. If $H<G$ is a finitely generated subgroup, there is a minimal subtree $S\subset T$ fixed (setwise) by $H,$ and the action of $H$ on $S$ endows $H$ with a graph of groups decomposition. Additionally, there is an induced map of quotient graphs $S/H\to T/G$.
We are interested in the following problem: Suppose $G,$ $H,$ $T,$ and $S$ are as above, $G$ and $H$ freely indecomposable limit groups, $T$ the Bass-Serre tree corresponding to the principle cyclic of $H$. We say that $G$ and $H$ are aligned if $S/H\to T/G$ is an isomorphism of graphs and $S/H$ is the underlying graph of the principle cyclic of $H$. Give a simple computable criterion which guarantees that $G$ and $H$ are aligned.
As long as $G^{ab}\to H^{ab}$ is virtually onto, we are able to answer this question in a reasonable way, constructing a (monotonically decreasing) complexity, equality of which will guarantee alignment of s. The properties of the alignment are then used to construct graphs of spaces and maps between them which resemble Stallings’ immersions. The main idea of this section is that an inclusion as above must either “tighten up” the Grushko/ becoming simpler in a quantifiable way, or can be written as a map of graphs of groups respecting the decompositions.
Let $T$ be the Bass-Serre tree corresponding to the principle cyclic of $H,$ let $S$ be the minimal subtree for $G$. The quotient $S/G$ is finite and it follows from the definitions that the induced graphs of groups decomposition of $G$ is principle. For convenience, we usually conflate underlying graphs and graphs of groups decompositions. Let $\Delta_H=T/H$ and $\Delta_G=S/G$ be the underlying graphs, and let $\eta_{\#}$ be the induced map. We label a vertex $v$ of $\Delta_G$ by the corresponding label on $\eta_{\#}(v),$ unless $G_v$ is abelian, in which case we label it abelian anyway. The map $\eta_{\#}$ is well behaved:
- If $v$ is rigid then the edge groups adjacent to $v$ have nonconjugate centralizers in $G_v$ unless they are all attached to boundary components of vertex groups.
- Let $B$ be a maximal connected subgraph of $\Delta_G$ such that every vertex is abelian. Commutative transitivity implies that $G_B$ is abelian, and the fact that all noncyclic abelian subgroups of $H$ are elliptic in $T_H$ implies that $B$ is a tree.
- If $v$ is abelian and $\eta_{\#}(v)$ is nonabelian, then $\eta_{\#}(v)$ is rigid.
- A valence one cyclic $v$ is adjacent to a $w$. This follows from the assumption that the only edge groups of $H$ allowed to be not closed under taking roots are adjacent to vertices.
\[actshyperbolically\] Let $\eta\colon G\to H$ be a homomorphism of freely indecomposable limit groups such that $H^1(H,\zee)\to H^1(G,\zee)$ is injective. Then $G$ is hyperbolic in every essential one-edged abelian splitting of $H$.
If $R$ is a nonabelian vertex group of a $\Delta_H$ of $H,$ then $G$ intersects a conjugate of $R$ in a nonabelian subgroup. If $\Delta_G$ is the induced decomposition of $G,$ and there is only one nonabelian vertex group $R'$ of $\Delta_G$ mapping to $R,$ then the map on underlying graphs is a submersion at $R'$.
Claim: If $G$ acts elliptically in some essential one-edged splitting then there is a map $H\onto\zee$ which kills $G$. If the one-edged splitting is an extension the claim is clear. If not, then both vertex groups of the amalgam have a map onto $\zee$ which kills the incident edge group.
To see the second half, suppose not, and let $\Delta'_H$ be the decomposition of $H$ obtained by conjugating edge maps to $R$ so that all incident edges either have the same or nonconjugate centralizers, and folding together edges of the conjugated decomposition which have the centralizers. Then pull all the centralizers of incident edge groups across the edge they centralize. If $T$ is the tree for $\Delta'_H,$ and $S$ is the minimal $G$–invariant subtree, then the map $S/G\to T/H$ clearly misses the vertex corresponding to $R$. Let $\Delta''$ be the decomposition of $H$ obtained by collapsing all edges not adjacent to $R$. Then $G$ is elliptic in $\Delta''$. The first part provides a contradiction.
If the map is not a submersion on the level of $\Delta_H,$ then the of graphs of groups $\Delta'_G\to\Delta'_H$ is not a submersion onto $R$ either, and there is an edge incident to $R$ missed by $\Delta'_G$. This edge represents an essential splitting of $H,$ and so we again have a contradiction.
\[def:compjsj\] Let $G$ be a finitely generated freely indecomposable limit group with principle cyclic decomposition $G=\Delta(\mathcal{R},\mathcal{Q},\mathcal{A},\E),$ where each $R\in\mathcal{R}$ is rigid, each $Q\in\mathcal{Q}$ is , each $A\in\mathcal{A}$ is finitely generated abelian, and each $E\in\E$ is an infinite cyclic edge group. Let
- $c_q(G)\define \vert\sum_{Q\in\mathcal{Q}}\chi(Q)\vert$ is the total Euler characteristic of subgroups.
- $c_{bq}(G)\define \sum_{Q\in\mathcal{Q}}\#\partial Q$ is the total number of boundary components of vertex groups
- $\mathcal{Z}(G)$ is the collection of conjugacy classes of centralizers of edge groups of $G$. Warning: *not* the center of $G$.
- For a given rigid vertex $R$ of the principle cyclic decomposition, let $v(R)$ be the valence of $R$. This is the same as the number of conjugacy classes of centralizers of incident edge groups *in* $R$.
- $c_a(G)\define\sum_{A\in\mathcal{A}}(\rk(A)-1)$
- $c_b(G)=c_a(G)+\betti(\Delta)$
The of $G$ with respect to $\Delta$ is the ordered tuple $$\jcomp(G,\Delta)=(c_q(G),-c_{bq}(G),\vert\mathcal{Z}\vert,c_b(G),\betti(\Delta),\vert\mathcal{R}\vert,\sum_{R\in\mathcal{R}}v(R))$$ The “,$\Delta$” is suppressed from the notation if $\Delta$ is the principle cyclic of $G$.
Complexities are compared lexicographically. The complexity $\jcomp_i$ is the restriction of $\jcomp$ to the first $i$ coordinates.
*Throughout this section $G$ and $H$ are freely indecomposable limit groups, $\eta\colon G\into H,$ and $\eta^{\#}\colon
H^1(H,\zee)\to H^1(G,\zee)$ is injective.*
We need to be able to compare the complexity of a principle decomposition to the complexity of the .
\[lem:identitymap\] Let $G$ be a freely indecomposable limit group with principle cyclic $\Delta_G,$ let $\E$ be a fixed family of subgroups of $G,$ and let $\Delta$ be the principle cyclic decomposition of $G$ associated to the family of principle cyclic splittings in which each $E\in\E$ is elliptic. Then $\jcomp(G,\Delta)\leq\jcomp(G),$ with equality if and only if $\Delta$ is the .
We can construct $\Delta$ by cutting vertex groups of $\Delta_G$ along simple closed curves, folding, and collapsing subgraphs. To handle $c_b,$ observe that any collection of disjoint simple closed curves on vertex groups of $\Delta$ can be completed to a collection which achives at most $c_b(G)$.
The inequalities on $c_q$ and $c_{bq}$ are obvious, and if they are equal, then the identity map simply identifies vertex groups. The remaining inequalities are obvious.
We spread the proof of Theorem \[thm:alignment\] across the next two lemmas.
\[surfaceinequality\] $c_q(G)\geq c_q(H)$. If equality holds then $c_{bq}(G)\leq c_{bq}(H)$.
Let $T$ be the Bass-Serre tree for the restricted of $H$. Since $\eta$ is injective, $G$ inherits a graph of groups decomposition $\Delta$ from its action on $T$. Let $Q$ be a vertex group of $\Delta$ conjugate into some element $Q'$ of $\mathcal{Q}(H)$. There are two possibilities: $Q$ either has finite or infinite index in $Q'$. If $Q$ has infinite index and is nontrivial then $G$ must be freely decomposable, contrary to hypothesis. Thus $Q$ is either trivial or finite index.
Let $c$ be a simple closed curve on some element $Q'$ of $\mathcal{Q}(H)$ giving a essential one-edged splitting $\Delta_c$ of $H$. By Lemma \[actshyperbolically\] $G$ acts hyperbolically in $\Delta_c,$ hence there is some $Q$ which maps to a finite index subgroup of a conjugate of $Q'$. The graph of groups decomposition $\Delta$ of $G$ is obtained by slicing vertex groups of $G$ along simple closed curves, folding, and collapsing subgraphs of the resulting decomposition. This immediately gives $c_q(G)\geq
c_q(H)$.
Suppose equality holds. Let $c_k$ be the simple closed curves cutting the vertex groups of $\jsj(G),$ and let $Q'_1,\dotsc,Q'_m$ be the complementary components which don’t map to vertex groups of $H$. Since $c_q(G)=c_q(H),$ each component $Q'_j$ has Euler characteristic $0$. Any such complementary component cannot be boundary parallel, thus if there are any then $c_{bq}(H)>c_{bq}(G)$. If equality holds then the subgroups of $G$ and those of $H$ are in one to one correspondence and the respective maps are isomorphisms.
An inclusion $G\into H$ as above is if it is a one-to-one correspondence on vertex groups and the maps are isomorphisms. If $H$ has an inessential one-edged splitting $\Delta,$ then $\Delta$ corresponds to an edge connecting a valence one cyclic vertex group of $\jsj(H)$ to a vertex group. If $G\into H$ is –preserving then it is necessarily bijective on such valence one vertex groups.
It follows immediately from Lemma \[actshyperbolically\] that if $G\into H$ is –preserving then $\vert\mathcal{Z}(G)\vert\geq\vert\mathcal{Z}(H)\vert$.
\[underlyinginequality\] \[lem:abelians\] \[centralizerinequality\] $\jcomp_5(G)\geq\jcomp_5(H)$. If equality holds then there is an induced bijection $\mathcal{A}(G)\to\mathcal{A}(H),$ and for each $A,$ $A/P(A)\to\eta_{\#}(A)/P(\eta_{\#}(A))$ is virtually onto.
We first handle $c_b$.
Let $\Delta_H$ be the principle cyclic of $H,$ and let $\Delta_G$ be the decomposition $G$ inherits from its action on $T_{\Delta_H}$. We may assume that $G\into H$ is –preserving, is bijective on conjugacy classes of centralizers of edge groups. Let $$H_1(H;\Delta^{(0)}_H\setminus
\mathcal{A}(H))=H_1(\Delta_H)\oplus\bigoplus_{A\in\mathcal{A}(H)}A/P(A)$$ Similarly, define $H_1(G;\Delta^{(0)}_G\setminus \mathcal{A}(G))$. The composition $G\to H\to H_1(H;\Delta^{(0)}_H\setminus
\mathcal{A}(H))$ factors through $H_1(G;\Delta^{(0)}_G\setminus
\mathcal{A}(G))$. Since $H^1(H,\zee)\into H^1(G,\zee)$ the map $H_1(G;\Delta^{(0)}_G\setminus \mathcal{A}(G))\to
H_1(H;\Delta^{(0)}_H\setminus \mathcal{A}(H)) $ must be virtually onto. But $\rk(H_1(H;\Delta^{(0)}_H\setminus
\mathcal{A}(H)))=c_b(H)$ and $\rk(H_1(G;\Delta^{(0)}_G\setminus
\mathcal{A}(G)))\leq c_b(G)$.
Let $\Delta$ be an essential one-edged splitting of $H$ in which all subgroups are elliptic. Let $T$ be the corresponding Bass-Serre tree. By Lemma \[actshyperbolically\] $G$ doesn’t fix a point in $T$ and it inherits an essential splitting $\Delta'$ from this action. Since $\eta$ is bijective of the sets of subgroups, and restricts to isomorphisms between them, every vertex group of $G$ acts elliptically in $T$. Thus there is an edge group $E'$ of $\jsj(G)$ which maps to a conjugate of the edge group of $\Delta$. Furthermore, $E'$ is an essential splitting, otherwise $G$ acts elliptically in $\Delta$.
Let $A\in\mathcal{A}(G)$ be a noncyclic abelian vertex group. If no element of $\mathcal{A}(H)$ contains the image of $A,$ then $c_b(G)>c_b(H)$. If equality holds there is a well defined map $\mathcal{A}(G)\to\mathcal{A}(H)$.
Let $A$ be an abelian vertex group of $G,$ and $\eta_{\#}(A)$ the associated vertex group of $H$. Since $H^1(H)\to H^1(G)$ is injective, the map $A/P(A)\oplus H_1(\Gamma_G)\to
\eta_{\#}(A)/P(\eta_{\#}(A))\oplus H_1(\Gamma_H)$ must be virtually onto. This map sends $A/P(A)$ to $\eta_{\#}(A)/P(\eta_{\#}(A))$ hence $\betti(\Delta_G)\geq\betti(\Delta_H),$ and if $\betti(\Delta_G)=\betti(\Delta_H)$ then $A/P(A)\to\eta_{\#}(A)/P(\eta_{\#}(A))$ must be virtually onto.
\[thm:alignment\]
$\jcomp_6(G)\geq\jcomp_6(H)$. If $\jcomp_6(G)=\jcomp_6(H)$ then $$\sum_{R\in\mathcal{R}(G)}v(R)\geq\sum_{R\in\mathcal{R}(H)}v(R),$$ i.e., $\jcomp(G)\geq\jcomp(H)$. If $\eta\colon G\into H,$ $\jcomp(G)=\jcomp(H),$ then $\eta$ is bijective on vertex and edge groups, maps abelian vertex, edge, and peripheral subgroups to finite index subgroups of their respective images. The map from the underlying graph of the of $G$ to the underlying graph of the of $H$ is an isomorphism.
The number of values the complexity can take is controlled by $\betti$.
Assume $\jcomp_5(G)=\jcomp_5(H)$. By Lemma \[centralizerinequality\], the inclusion is a one-to-one correspondence on noncyclic abelian vertex groups.
Let $\Delta_H$ be the principle cyclic of $H,$ let $\pi\colon\Delta_G\to\Delta_H$ be the induced map of underlying graphs , and let $R$ be nonabelian non- vertex group of $\Delta_H$. By Lemma \[actshyperbolically\] there is a nonabelian vertex group of $\Delta_G$ which maps to $R$. Since $\eta$ is bijective on subgroups, there is a rigid vertex group $R'$ of $G$ which maps to $R$. If $\jcomp_5(G)=\jcomp_5(H)$ then $R'$ is the unique such vertex group.
Again, by Lemma \[actshyperbolically\], since there is only one vertex group $R'$ mapping to $R,$ the map $\E(R')\to\E(R)$ is onto and $v(R')\geq v(R)$.
Let $Z$ be an essential cyclic abelian vertex group of $\Delta_H,$ and let $Z_1,\dotsc,Z_k$ be the vertex groups of $\Delta_G$ mapping to $Z$. Since $\eta$ is bijective on nonabelian vertex groups, and since all vertex groups adjacent to $Z$ are nonabelian, the induced map $\eta_{\#}\colon\sqcup\E(Z_i)\to\E(Z)$ is bijective. Arguing as in Lemma \[actshyperbolically\], $k=1$ and the map $\E(Z_1)\to\E(Z)$ is bijective. The same observation shows that if $A$ is noncyclic abelian, then there is a unique $A'$ mapping to $A$ and that the map on the link is onto. The map is also injective, again because $\eta$ is a bijective on nonabelian vertex groups.
Thus, if the complexities are equal, then the inclusion must induce a homeomorphism of underlying graphs. By construction, the map is label preserving, and it automatically respect all incidence and conjugacy data from the respective decompositions.
This shows that $\jcomp(G,\Delta_G)\geq\jcomp(H),$ and if equality holds, then the morphism $\Delta_G\to\Delta_H$ is of the correct form. By Lemma \[lem:identitymap\] $\jcomp(G)\geq\jcomp(G,\Delta_G),$ and if $\jcomp(G)=\jcomp(H),$ then $\Delta_G$ is just the principle cyclic of $G$. This gives the first half of the theorem.
The bound on the number of values the complexity can take follows from either acylindrical accessibility [@sela::acyl] plus the bound on the rank of a limit group with complexity $b_0,$ or [@louder::stable Lemma \[STABLE-lem:rankheightbound\]], which gives a bound on the complexity of the principle cyclic in terms of the first betti number. Those arguments bound the number of essential vertex groups. Adjoining roots doesn’t increase the first betti number, so if $b_1$ and $b_2$ are boundary components of a vertex group adjacent to inessential vertex groups, then a simple closed curve cutting off a pair of pants with $b_1,b_2$ as the two other boundary components makes a contribution of one to $\betti(G)$; $n$ nonintersecting simple closed curves as above make a contribution of $n$ to $\betti(G),$ thus each vertex group is attached to at most $2\betti(G)$ inessential vertex groups. Since $\betti(G)$ controls the number of vertex groups, there are boundedly many inessential abelian vertex groups.
In light of Theorem \[thm:alignment\], if $\jcomp(G)=\jcomp(H),$ then we say that $G$ and $H$ are . Before representing injections of limit groups topologically, we devote a section to proving Theorem \[maintheorem\], assuming the material from section \[hyptoell\].
Proof of Theorem \[maintheorem\] {#sec:mainproof}
================================
The bound implicitly computed in the proof of Theorem \[maintheorem\] can be made slightly better if we show that nonabelian limit groups with first betti number $2$ are free. The next lemma is not necessary, but we record it here for lack of a better place to put it. In [@rankthreeclassification], Fine, et al., classify limit groups with rank at most three. The next lemma shows that in rank two the rank can be relaxed to first betti number.
Let $G$ be a limit group with first betti number $2$. Then $G\cong\free_2$ or $\zee^2$.
We may assume $G$ is nonabelian and freely indecomposable. If $G$ is abelian it satisfies the theorem trivially, and if freely decomposable, the free factors are limit groups with first betti number one, and must be infinite cyclic.
The proof is by induction on the depth of the cyclic analysis lattice. All essential cyclic splittings of $G$ are extensions, otherwise there is a one-edged cyclic splitting such that each vertex group has betti number at least two, and $G$ therefore has first betti number at least three. By a simple variation of the proof of Theorem \[analysislatticebound\] the depth of the cyclic analysis lattice of $G$ is finite. Suppose that $G$ has a vertex group $Q$. Then any essential simple closed curve on $Q$ must correspond to an extension of $G$: $G=G'*_E$. Since the splitting comes from a vertex group, $G'$ must be freely decomposable, hence is $\free_2$. If $G$ has no vertex groups it’s principle cyclic decomposition must be a bouquet of circles. Let $G=G_0>G_1>G_2>\dotsb>G_n$ be a sequence of vertex groups of cyclic decompositions such that $G_i,$ $i<n-1,$ is freely indecomposable and has a bouquet of circles as its principle cyclic , terminating at the first index $n$ such that such that $G_n$ is freely decomposable, hence free. This chain must have finite length since the cyclic analysis lattice is finite. We argue that $G_n$ free implies that $G_{n-1}$ is free.
Let $f\colon G_{n-1}\to\free$ be a homomorphism such that $f(G_n)$ has nonabelian image. Since $G_{n-1}$ is an extension of $G_n,$ by Corollary 1.6 of [@louder::scott], the images of the incident edge groups in $G_n$ can be conjugated to a basis for $G_n$ and $G_{n-1}$ is freely decomposable, contrary to hypothesis.
An of a pure staircase $(\G,\HH)$ is a staircase $(\G,\HH')$ such that the diagrams in Figure \[fig:extension\] commute. An extension is if one of the following mutually exclusive conditions holds.
- $\G$ is freely decomposable, and the freely indecomposable free factors of $\HH'(i)$ embed in $\HH(i)$ under $\sigma_i$.
- $\G$ is freely indecomposable, has subgroups, and the vertex groups of the decomposition of $\HH'(i)$ obtained by collapsing all edges not adjacent to vertex groups embed in $\HH(i)$ for all $i$. (This is just the restricted principle cyclic .)
- $\G$ is freely indecomposable, –free, and for all $i,$ vertex groups of the (restricted) principle cyclic of $\HH'(i)$ embed in vertex groups of $\HH(i)$ under $\sigma_i$.
An admissible extension has the property that each $\sigma_i$ is strict, surjective, and maps elliptic subgroups of a decomposition of $\HH'(i+1)$ to elliptic subgroups of $\HH(i+1)$. The relation “mapps onto” partially orders the collection of extensions, and if $\HH''$ is an extension of $\HH'$ then $\HH''\geq_{\sqrt{}}\HH'$. For some $i,$ if $\sigma_i$ is not one-to-one on the sets of vertex groups or edge groups then the inequality is strict. The envelope of a rigid vertex group of the principle cyclic is just the vertex group, hence if $\sigma_i$ is one-to-one on the sets of vertex groups and edge groups then it is an isomorphism. If $(\G,\HH'')\onto(\G,\HH')\onto(\G,\HH)$ is a pair of admissible extensions then $(\G,\HH'')\onto(\G,\HH)$ is an admissible extension.
We work with staircases which are maximal with respect to $\geq_{\sqrt{}},$ rather than arbitrary staircases. To do this we have to pay a penalty, but not too large of one.
\[existresolutions\] For all $K$ there exists $C=C(K,\comp(\text{ }))$ such that if $(\G,\HH,\E)$ is a staircase and $\ninj(\comp((\G,\HH,\E)))=C(K,\comp((\G,\HH,\E))),$ then there is a $\geq_{\sqrt{}}$–maximal extension of a contraction $(\G',\HH',\E')$ of $(\G,\HH,\E)$ with $\ninj((\G',\HH',\E'))\geq K$ and $\comp((\G',\HH',\E'))\leq\comp((\G,\HH,\E))$.
The constants in this lemma do not depend on $\Vert\E\Vert,$ and its proof is formally identical to the proof of [@louder::stable Theorem \[STABLE-thr:nostrict\]]. To adapt the proof, we need to show that the strict resolutions arising in an extension have bounded length. This follows from [@louder::stable Lemma \[STABLE-lem:rankheightbound\]], bounding the rank of $\HH(i)$ from above by a function of $\comp((\G,\HH)),$ but a proof more in the spirit of this paper goes as follows: If $\HH^{(n)}(i)\onto\dotsb\onto\HH(i)$ is a strict resolution appearing in a sequence of extensions, then $\jcomp(G)\geq\jcomp(\HH^{(m)}(i))$ (See Lemma \[thm:alignment\] and Definition \[def:compjsj\].), moreover, if $\HH^{(m+1)}(i)\onto\HH^{(m)}(i)$ is not injective on sets of vertex or edge spaces, or collapses subsurface groups of vertex groups, the complexity must decrease. By Theorem \[thm:alignment\] the number of values the complexity takes is controlled by $\betti,$ and the resolutions have length controlled by $\comp((\G,\HH,\E))$.
Each pure kind of staircase is handled in turn over the next three subsections. In all cases the strategy is the same: either there is compatibility between (collapses of) decompositions/Grushko factorizations, the complexity decreases, or proper extensions exist.
Freely decomposable
-------------------
This is the most singular case in that the arguments work for nearly all finitely generated groups, not just limit groups.
The complexity for freely indecomposable groups is used to show that base sequences of freely indecomposable staircase can be divided into segments such that the base groups of a segment have the same decompositions, in the sense of Theorem \[thm:alignment\]. There is a similar complexity for freely decomposable groups which accomplishes the same thing but with regard to Grushko decompositions. The following theorem from [@louder::scott] shows how the complexity for freely decomposable groups is useful.
Let $G$ be a finitely generated group with Grushko decomposition $G=G_1*\dotsb*G_p*\free_q$. The of $G$ is the lexicographically ordered pair $\scott(G)\define(q-1,p)$.
The number of Scott complexities of limit groups with $\betti=b$ is bounded by $b^3$.
\[roots::freelydecomposable\] Suppose that $\phi\colon G\into H$ and $H$ is a quotient of $G'=\adjoinroot{G}{\gamma_i}{k_i},$ $\boldsymbol{\gamma}_i$ a collection of distinct conjugacy classes of indivisible elements of $G$ such that $\boldsymbol{\gamma}_i\neq\boldsymbol{\gamma}_j^{-1}$ for all $i,j$ and $\gamma_i\in\boldsymbol{\gamma}_i$. Then $\scott(G)\geq\scott(H)$. If equality holds and $H$ has no $\zee_2$ free factors, then there are presentations of $G$ and $H$ as $$G\cong\grushko{G}{p}{q}^G,\quad H\cong\grushko{H}{p}{q}^H$$ a partition of $\set{\boldsymbol{\gamma}_i}$ into subsets $\boldsymbol{\gamma}_{j,i},$ $j=0,\dotsc,p,$ $i=1,\dotsc,i_p,$ representatives $\gamma_{j,i}\in G_j\cap\boldsymbol{\gamma}_{j,i},$ $i\geq 1,$ $\gamma_{0,i}\in\free_q^G\cap\boldsymbol{\gamma}_{0,i},$ such that with respect to the presentations of $G$ and $H$:
- $\phi(G_i)<H_i$
- $\adjoinroot{G_j}{\gamma_{j,i}}{k_{j,i}}\onto H_j$
- $\phi(\free_q^G)<\free_q^H$
- $\free^G_q=\group{\gamma_{0,1}}*\dotsb*\group{\gamma_{0,i_0}}*F$
- $\free^H_q=\group{\sqrt{\gamma_{0,1}}}*\dotsb*\group{\sqrt{\gamma_{0,i_0}}}*F$
- $G'\cong\adjoinroot{G_1}{\gamma_{1,i}}{}*\dotsb*\adjoinroot{G_p}{\gamma_{p,i}}{}*\group{\sqrt{\gamma_{0,1}}}*\dotsb*\group{\sqrt{\gamma_{0,i_0}}}*F$
All homomorphisms are those suggested by the presentations, and the maps on $F$ are the identity.
This is [@louder::scott Theorem 1.2].
Theorem \[roots::freelydecomposable\] is stated in terms of adjoining roots to cyclic subgroups of a group, whereas Definition \[def:adjunctionofroots\] refers to collections of abelian subgroups. This difference is immaterial to the discussion here since adjoining roots to a noncyclic abelian group can be accomplished by adjoining roots to a suitable collection of cyclic subgroups. By passing from a noncyclic abelian subgroup to cyclic subgroups, the measure $\Vert\text{ }\Vert$ is unchanged.
Let $(\G_i,\HH_i)$ be a collection of staircases on the same index set $I$. Then the graded free product $((*_i\G_i),(*_i\HH_i)),$ with the obvious maps, is also a sequence of adjunctions of roots.
\[maintheorem::freelydecomposable\] Suppose Theorem \[maintheorem\] holds for all staircases with complexity less than $(b_0,d_0,e_0)$. Then Theorem \[maintheorem\] holds for pure freely decomposable staircases of complexity $(b_0,d_0,e_0)$.
Let $(\G,\HH,\E)$ be a staircase with complexity $(b_0,d_0,e_0)$. Since limit groups are torsion free, no $\G(i)$ has a $\zee_2$ free factor, and by Theorem \[roots::freelydecomposable\] for all but $\betti(\G(1))^3$ indices $i_j,$ the subsequences $\G(i_j)\into\G(i_j+1)\into\dotsb\into\G(i_{j+1}-1)$ can be decomposed into free products of freely indecomposable groups staircases. Moreover, elements of $\E_i$ are either part of a basis of a free free factor of $\G(i)$ or are conjugate into a freely indecomposable free factor of $\G(i)$. Write $\G(i)$ as the free product $$\G(i)_1*\dotsb*\G(i)_p*F_i$$ given by the lemma, where $F_i$ is a free group of rank $q$ and $\scott(\G(i))=(q-1,p)$ for all $i$. Let $\E^j_i$ be the subset of $\E_i$ consisting of elements conjugate into $\G(i)_j,$ and rearrange indices so that $\G(i)_j$ maps to $\G(i+1)_j$ for all $j$. Let $\E^0_{i}$ be the elements of $\E$ which are conjugate into $F_i$. By Theorem \[roots::freelydecomposable\] there are decompositions $$\adjoinroot{\G(i)}{\E_i}{}\cong\left(\adjoinroot{\G(i)_1}{\E^1_{i}}{}*\dotsb*\adjoinroot{\G(i)_p}{\E^p_{i}}{}\right)*\adjoinroot{F_i}{\E^0_{i}}{}$$ where the last factor is free. Let $\HH(i+1)_j\define\img_{\HH(i+1)}(\adjoinroot{\G(i)_j}{\E^j_{i}}{})$ The sequence $\HH'$ defined by $$\HH'(i+1)\define\left(*_j\HH(i+1)_j\right)*\adjoinroot{F_i}{\E^0_{i}}{}$$ is an extension of $\HH$. Then $(\G,\HH',\E)$ splits as a free product, the freely indecomposable free factors of which are $(\G_j,\HH'_j,\E^j)$. These free factors have strictly lower $\betti$ than $\G,$ depth at most $d_0=\depthpc(\HH),$ hence have $\ninj((\G_j,\HH'_j,\E^j))\leq
\ninj(\comp(b_0-1,d_0,e_0))=:B$. If $\Vert\G\Vert>B\cdot
\betti(\G)\geq B\cdot p,$ then, for some index $l,$ the map $\HH'(l)\onto\G(l)$ is visibly an isomorphism. Since this map factors through $\HH(l),$ $\HH(l)\onto\G(l)$ is an isomorphism as well.
We finish this subsection by proving the base case of the induction. Let $(\G,\HH,\E)$ be a maximal staircase of complexity $(b,2,e)$. By the proof of Lemma \[roots::freelydecomposable\], the staircase splits as a free product of freely indecomposable staircases $(\G_i,\HH_i,\E^i),$ and such that each $\HH_i(j)$ is elementary. If $\G_i$ is abelian, then clearly $\HH_i(j)\onto\G_i(j)$ is an isomorphism, and if nonabelian, $\HH_i(j)$ is the fundamental group of a closed surface. Since $\G_i$ is freely indecomposable, it is also the fundamental group of a closed surface. Divide $\G_i$ into segments such that the Euler characteristic is constant on each segment. Then $\G_i(j)\to\G_i(j+1)$ is an isomorphism on each segment and $\HH_i(j)$ is a trivial extension of $\G_i(j-1)$ for all $j$ on each segment, thus $\HH_i(j)\onto\G_i(j)$ is an isomorphism.
Freely indecomposable,
-----------------------
Lemma \[onetooneonedges\] allows us to handle injections $G\into H,$ $\jcomp(G)=\jcomp(H),$ and such that $G$ has a subgroup.
\[alignedandqh\] Suppose Theorem \[maintheorem\] holds for all staircases with strictly lower complexity than $(b_0,d_0,e_0)$. Then Theorem \[maintheorem\] holds for staircases with subgroups and complexity $(b_0,d_0,e_0)$.
The strategy is to find an extension $(\G,\HH',\E)$ of $(\G,\HH,\E)$ such that the $\qh$ subgroups of $\HH'$ are the “same” as those from $\G$. See Figure \[qhexample\]. The group $\HH(i)$ may be a total mess, but luckily it is a homomorphic image of a limit group which shares its restricted with $\G(i)$ and $\G(i+1)$.
To do this an auxiliary lemma which follows immediately from Lemma \[onetooneonedges\] is needed.
\[dontpassthroughqh\] Let $G'$ be obtained from $G$ by adjoining roots to a collection of abelian subgroups $\E$. If $\jcomp(G)=\jcomp(G')$ then every element $E\in\E$ such that $[E:F(E)]>1$ is conjugate into a non- vertex group of $\rjsj(G)$.
We use the immersion representing $G\into G'$ constructed in subsection \[subsec:graphsandimmersions\].
Fix $E$ as in the statement of the lemma. We are done if we show that $E$ is elliptic in every one edged splitting of $G$ obtained by cutting a subgroup along an essential simple closed curve which doesn’t cut off a band.[^2] Start with an immersion representing the decompositions of $G$ and $G',$ and let $\Sigma_Q$ be the surface which contains $c$. There is a unique element $\eta_{\#}(Q)$ containing the image of $Q,$ and the map $Q\to\eta_{\#}(Q)$ is surjective. Since $Q\to\eta_{\#}(Q)$ is represented by a homeomorphism $\Sigma_Q\to\Sigma_{\eta_{\#}(Q)}$ there is a simple closed curve $\eta_{\#}(c)$ contained in $\Sigma_{\eta_{\#}(Q)}$ and a closed annular closed neighborhood $A$ of $c$ mapping homeomorphically to a neighborhood of $\eta_{\#}(c)$. Use these neighborhoods to construct new graphs of spaces $Y_G$ and $Y_{G'}$ representing $G$ and $G'$ by regarding the annulus as a new edge space and collapsing all but the newly introduced edges. By construction, the map $Y_G\into Y_{G'}$ is an immersion. By Lemma \[onetooneonedges\], if some element of $\E$ crosses $c,$ then $c$ maps to a power of $\eta_{\#}(c)$.
![Illustration of Lemma \[alignedandqh\][]{data-label="qhexample"}](roots-figures/qhexample.eps)
Suppose $(\G,\HH)$ is a staircase such that $\scott(\G(i))$ is the constant sequence and $c_q(\G(1))\neq 0$. Let $\Delta_i$ be the $\rjsj$ of $\G(i)$. Every edge of $\Delta_i$ is infinite cyclic and connects a vertex group to a boundary component of a vertex group. Since the inclusions $\G(i)\into\G(i+1)$ respected graphs of spaces, by the first part of Lemma \[dontpassthroughqh\], every element of $\E_i$ is conjugate into some non- vertex group of $\Delta_i$. Let $V^i_1,\dotsc,V^i_n$ be the non- vertex groups of $\Delta_i$. We regard $\G(i)$ as a graph of groups $\Gamma(V^i_j,Q_k,E_l),$ where $\G(i)\into\G(i+1)$ is compatible with the decomposition $\Gamma$ in the sense that $V^i_j$ maps to a conjugate of $V^{i+1}_j,$ the map respects edge group incidences, and the inclusion is the identity on the vertex groups $Q_k$.
Let $\E^j_i$ be the elements of $\E_i$ conjugate into $V^i_j,$ and arrange that each $E\in\E^j_i$ is contained in $V^i_j$ by conjugating if necessary. Let $W^{i+1}_j$ be the image of $\adjoinroot{V^i_j}{\E^j_i}{}$ in $\HH(i+1)$ and let $\HH'(i+1)=\Gamma(W^{i+1}_j,Q_k,E_l)$. Then $(\G,\HH',\E)$ is an extension of $(\G,\HH,\E)$: The map implicit map $\sigma_i\colon\HH'(i)\to\HH(i)$ is clearly strict, therefore the sequence $\mathcal{H}'$ consists of limit groups. By definition, $V^{i+1}_j$ is obtained from $V^i_j$ by adjoining roots. Let $\mathcal{V}_j$ be the sequence $\mathcal{V}_j(i)=V^i_j$ and let $\mathcal{W}_j(i)=W^i_j$.
The staircases $(\mathcal{V}_j,\mathcal{W}_j,\E^j)$ all have lower first betti number than $(\G,\HH,\E)$. Let $B(b_0)$ be the maximal number of vertex groups of a limit group with first betti number $b_0$ [@louder::stable Lemma \[STABLE-lem:rankheightbound\]]. If $\Vert\G\Vert>\ninj((b_0-1,d_0,e_0))\cdot B(b_0)$ then for at least one index $l$ all $\mathcal{W}_j(l)\onto\mathcal{V}_j(l+1)$ are injective. Thus $\pi'_l$ is $\Mod(\HH'(l),\rjsj)$ strict. Since all modular automorphisms of $\HH'(l)$ are either inner, Dehn twists in boundary components of vertex groups, or induced by boundary respecting homeomorphisms of surfaces representing vertex groups, by construction, every element of $\Mod(\HH'(l),\rjsj)$ pushes forward to a modular automorphism of $\G(l)$. An easy exercise shows that $\HH'(l)\onto\G(l)$ is an isomorphism. Since $\pi'_l=\pi_l\circ\sigma_l,$ $\pi_l$ is an isomorphism.
Freely indecomposable, no
--------------------------
The neighborhood of a vertex group $V$ of a graph of groups decomposition is the subgroup generated by $V$ and conjugates of adjacent vertex groups which intersect $V$ nontrivially, and is denoted $\nbhd(V)$. Let $(G,H,G')$ be a flight and suppose $G$ is freely indecomposable, has no vertex groups, and that $\jcomp(G)=\jcomp(G')$. Let $\eta\colon G\to G'$ be the inclusion map. An abelian vertex group $A$ of $G$ is if $H$ doesn’t have a principle cyclic splitting over a subgroup of $\cent_H(\nu(A))$.
Let $\mathcal{A}_H$ be the collection of abelian vertex groups of $G$ which are $H$–elliptic. Suppose that $H$ is obtained from $G$ by adjoining roots to the collection $\E$. Let $\E^{ell}_H$ be the sub-collection of $\E$ consisting of elements of $E$ which are hyperbolic in the principle cyclic of $G$ but which have elliptic image in the principle cyclic of $H$. Let $\jsj_H(G)$ be the decomposition of $G$ with respect to the collection of principle cyclic splittings in which all $\nbhd(A),$ $A\in \mathcal{A}_H,$ and $E\in\E^{ell}_H$ are elliptic: $$\jsj_H(G)\define \jsj(G;\set{\nbhd(A),E}:{A\in \mathcal{A}_H,E\in\E^{ell}_H})$$ Let $\jsj^*_H(G')$ be the decomposition of $G'$ associated to the collection of all principle cyclic splittings of $G'$ in which all $\eta_{\#}(A_H),$ $A\in \mathcal{A}_H,$ and $\eta_{\#}(E),$ $E\in\E^{ell}_H,$ are elliptic. That is $$\jsj^*_H(G')\define \jsj(G';\set{\nbhd(\eta_{\#}(A)),\eta_{\#}(E)}:{A\in \mathcal{A}_H, E\in\E^{ell}_H} )$$
The main lemma is that the decompositions of $G$ and $G'$ induced by $H$ are intimately related to the principle cyclic of $H$ as long as the flight admits no proper extensions. Let $V$ be a vertex group of $\jsj_H(G)$. There is a vertex group $\eta_{\#}(V)$ of $\jsj^*_H(G')$ which contains the image of $V$. Let $\E_V$ be the collection of elements of $\E$ which are conjugate into $V,$ along with the collection of incident edge groups. Likewise for $\eta_{\#}(V),$ let $\E(\eta_{\#}(V))$ be the set of centralizers of images of elements of $\E_V$.
\[mainlemma\] Let $(G,H,G')$ be a flight without any proper extensions, and suppose $G$ is freely indecomposable, has no vertex groups, and that $\jcomp(G)=\jcomp(G')$. Let $\eta\colon
G\to G'$ be the inclusion map. Then the following hold:
- For each vertex group $W$ of the principle cyclic of $H$ there are unique vertex groups $V$ and $\eta_{\#}(V)$ of $\jsj_H(G)$ and $\jsj^*_H(G'),$ respectively, such that $\nu(V)<W,$ $\pi(W)<\eta_{\#}(V)$.
- $W$ is obtained from $V$ by adjoining roots to $\E_V$ and $\Vert\E_V\Vert\leq\Vert\E\Vert+2\betti(G).$
- $\eta_{\#}(V)$ is obtained from $\pi(W)$ by adjoining roots to the images of $\E(V)$ (the edge groups incident to $V$)
- If $\pi$ is injective on vertex groups then it is an isomorphism.
The proof of Lemma \[mainlemma\] is contained in section \[hyptoell\], where graphs of spaces $X_G,$ $X_H,$ representing $\jsj_H(G)$ and $\jsj(H),$ respectively, and an immersion $X_G\to X_H$ representing $G\into H,$ such that if the immersion is not one-to-one on edge spaces, then there must be a nontrivial extension, are constructed. The remainder of the lemma is largely formal, and relies on a simplification of the construction of strict homomorphisms from [@louder::stable].
Finishing the argument {#finishargument}
----------------------
In this section we prove Theorem \[maintheorem\], postponing the proofs of lemmas used in the previous section until section \[hyptoell\]. Let $(\G,\HH)$ be a staircase with complexity $(b_0,d_0,e_0),$ such that no contraction has any proper extensions, and suppose that Theorem \[maintheorem\] holds for staircasess with complexity less than $(b_0,d_0,e_0)$. By Theorem \[thm:alignment\] there is some constant $B(b_0)$ such that $(\G,\HH)$ can be divided into $B(b_0)$ staircases of constant Scott complexity: (To maintain uniformity of the exposition, some sequences are allowed to be empty.)
$$(\G,\HH)\mapsto\set{(\G_i,\HH_i)}_{i=1,\dotsc,B(b_0,d_0)}$$ $$\G_i(1)=\G(j_i),\dotsc\quad\HH_i(2)=\HH(j_i+1),\dotsc$$
Only the last of these can consist of freely indecomposable groups. Each staircase $(\G_i,\HH_i),$ $i<B(b_0),$ by Theorem \[maintheorem::freelydecomposable\], has $\ninj$ bounded above by $b_0\cdot \ninj(b_0-1,d_0,e_0),$ since there are at most $b_0$ freely indecomposable free factors. Thus we may confine our analysis to freely indecomposable staircases. By Theorem \[thm:alignment\], we may divide the staircase $(\G,\HH,\E)$ into boundedly many segments, the number depending only on the complexity of $\betti(\G),$ exhausting the tower, such that $\jcomp$ is constant on each segment. By Lemma \[existresolutions\] we may assume that each segment is maximal with respect to $\leq_{\sqrt{}}$.
Like the case when each $\G(i)$ is freely decomposable, if $\G(i)$ has a vertex group, by Lemma \[alignedandqh\] such staircases have bounded $\ninj$.
The only possibility left is that the contractions of $(\G,\HH)$ are –free. Let $I$ be the index set for $\G,$ and color the triple $i<j<k$ red if $\jsj^*_{\HH(j)}(\G(j))\cong\jsj_{\HH(k)}(\G(j)),$ and blue otherwise. Then by Ramsey’s theorem for hypergraphs, for all $K$ there exists an $L$ such that if $\Vert\G\Vert>L$ then there is a subset $I'\subset I$ of size at least $K$ such that all triples whose elements are in $I'$ have the same color.
\[lem:boundblue\] There is an upper bound to the size of blue subsets which depends only on $\betti(\G)$ and $\Vert\E\Vert$.
By Lemma \[lem:numberofdecompositions\], there are at most $2^{\Vert\E\Vert}$ equivalence classes of principle cyclic decomposition of $G$ in which some element of $\E$ is elliptic. (There may be none.) Suppose $\vert
I'\vert>2^{\Vert\E\Vert},$ and consider the collection of principle cyclic decompositions $\set{\jsj_{\HH(l)}(\G(i))}$. Thus, for some $i<j<k,$ $\jsj_{\HH(j)}(\G(i))$ and $\jsj_{\HH(k)}(\G(i))$ have the same elliptic subgroups. Then $\jsj_{\HH(k)}(\G(j))\cong\jsj^*_{\HH(j)}(\G(j))$ since a decomposition is determined up to equivalence solely by its elliptic subgroups.
We are now on the home stretch. Suppose again that $(b_0,d_0,e_0)$ is the lowest complexity for which Theorem \[maintheorem\] doesn’t hold. By Lemma \[lem:boundblue\] and the prior discussion, there must be staircases $(\G,\HH,\E)$ of arbitrary $\ninj,$ which have complexity $(b_0,d_0,e_0),$ are maximal, pure, and have no vertex groups.
Let $(\G,\HH)$ be such a staircase. Let $\V'_j(i)$ be the nonabelian vertex groups of $\G(i),$ indexed such that $\V'_j(i)$ maps to $\V'_j(k)$ for all $k>i$. Let $\W_j(i)$ be the corresponding rigid vertex group of $\HH(i)$. By the second bullet of Lemma \[mainlemma\], $\W_j(i+1)$ is obtained from $\V'_j(i)$ by adjoining roots to $\E^{j,\prime}_i,$ the set of elements of $\E_i$ which are conjugate into $\V'_j(i),$ along with the incident edge groups.
Let $\V_j(i)<\V'_j(i)$ be the image of $\W_j(i)$ in $\G(i)$. By the third bullet of Lemma \[mainlemma\], $\V'_j(i)$ is obtained from $\V_j(i)$ by adjoining roots to the images of the edge groups incident to $\W_j(i)$. Let $$\E^{j}_i\define\\
\set{E\cap \V_j(i)}:{E\in \E^{j,\prime}_i}\cup\\
\set{E\cap\V_j(i)}:{E\in\E(\V_j(i))}$$ The incident edge groups are cyclic, and we can build $\W_j(i)$ by simply adjoining roots to $\E^{j}_i$ in $\V_j(i)$. Then $\E^{j}_i$ is larger than $\E$ by at most the number of edge groups incident to $\V'_j(i),$ which is at most $2\betti(\mathcal{G})$. That is, $$\Vert\E^{j}_i\Vert\leq\Vert\E\Vert+2\betti(\G)
\leq
\Vert\E\Vert+2\betti(\G)(\depthpc(\HH)-\depthpc(\W_j))$$
Given a sufficiently long –free staircase $(\G,\HH,\E),$ we passed to a maximal extension (which we will also call $(\G,\HH,\E)$) of a substaircase of prescribed length, such that the sequences of vertex groups $(\V_j,\W_j,\E^{j})$ of the extension were cyclic staircases. The vertex groups of the extension are subgroups of the vertex groups of $\HH,$ hence the depth of $\W_j(i)$ is strictly less than the depth of $\HH$. Moreover, the first betti number of $\W_j(i)$ is at most $\betti(\HH)$ and by Lemma \[mainlemma\], $\comp((\G,\HH,\E))>\comp((\V_j,\W_j,\E^j))$. There is an upper bound $B(b_0)$ to the number of vertex groups of the principle cyclic of a limit group with first betti number $b_0$. If $\Vert\G\Vert>B(b_0)\cdot \ninj(b_0,d_0-1,e_0+2b_0)$ there is some index $l$ such that $\HH(l)\onto\G(l)$ is injective on all vertex groups. By the last bullet of Lemma \[mainlemma\], $\HH(l)\onto\G(l)$ is an isomorphism.
Hyperbolic to elliptic {#hyptoell}
======================
Graphs of spaces and immersions {#subsec:graphsandimmersions}
-------------------------------
In this section we are given a fixed flight $(G,G',H)$ of limit groups. By a of a limit group $G$ we mean a graph of spaces of the following form:
- For each rigid vertex group $R$ a space $X_R$. Let $\E(R)$ be the edge groups incident to $R,$ and for each $E\in\E(R)$ let $\sqrt{E}$ be the maximal cyclic subgroup of $R$ containing the image of $E$. For each $E\in\E$ there is an embedded copy $S_E$ of $S^1$ in $X_R$ representing the conjugacy class of $\sqrt{E}$.
- For each edge $E,$ a copy $T_E$ of $S^1,$ with basepoint $b_E$ and an edge space $T_E\times\mathrm{I}$. On occasion we confuse $T_E$ with $T_E\times\frac{1}{2},$ and sometimes refer to $T_E$ as the edge space. The interval $b_E\times I$ is denoted $t_E,$ and we choose an arbitrary orientation for $t_E$. The end of the edge space associated to $E$ is attached via the covering map $T_E\immerses S_E$ representing $E\into\sqrt{E}$.
- For each abelian vertex group a torus $T_A$. If $A$ is infinite cyclic then $T_A$ has a basepoint $b_A$ and the incident edge maps are simply covering maps which send $b_E$ to $b_A$. These covering maps are isomorphisms unless the edge is adjacent to a vertex group, in which case they may be proper. For each edge space edge $E$ adjacent to $A,$ an edge space $T_E\times I$ and an embedded copy of $T_E$ in $T_A$. This assumes edge groups not adjacent to vertex groups are primitive. Though there may be vertex groups, the cases which this definition is designed to handle do not, and we let this inconsistency slide. Unlike the rigid case, the embedded $T_E$ need not be disjoint, though if they meet, they coincide. We require that any two embedded $T_E,$ differ by an element of $T_A,$ treated now as a group.
- For each vertex group $Q$ a surface with boundary $\Sigma_Q$.
- If an edge group $E$ is incident to a vertex group $Q$ then $T_E$ is identified with a boundary component of $\Sigma_Q$.
- The resulting graph of spaces has the fundamental group of $G$.
Let $\eta\colon G\into H$ be an inclusion of limit groups, and let $\Pi_G$ and $\Pi_H$ be principle cyclic decompositions of $G$ and $H,$ respectively, such that $\eta$ maps vertex groups to vertex groups, edge groups to edge groups, and respects edge data, i.e., if $E\into V,$ $\eta_{\#}(E)\into\eta_{\#}(V),$ then the obvious square commutes. If this is the case then $\eta$ $\Pi_G$ and $\Pi_H$. Let $X_H$ be a graph of spaces representing $\Pi_H$. Then there is a principle cyclic decomposition $\Pi_G$ of $G,$ a space $X_G$ representing $\Pi_G,$ and an $\psi\colon X_G\to
X_H,$ inducing $\eta,$ of the following form:
- For each abelian vertex group $A$ there is a finite sheeted covering map $\psi\vert_{T_A}\colon T_A\immerses
T_{\eta_{\#}(A)}$. The inclusions of incident edge spaces are respected by $\eta$: $$\psi\vert_{\img(T_E)}=(T_{\eta_{\#}(E)}\into T_{\eta_{\#}(A)})\circ\psi\vert_{T_E}$$
- For each $E$ there is a finite sheeted product-respecting covering map $T_E\times\mathrm{I}\immerses
T_{\eta_{\#}(E)}\times\mathrm{I}$ which maps $t_E$ to $t_{\eta_{\#}(E)}$. If $E$ is adjacent to a vertex group then the degree of the covering map is one.
- For each $R$ there is a map $X_R\to X_{\eta_{\#}(R)}$ such that for each edge group $E$ incident to $R$ the following diagram commutes:
Likewise for $\times\set{1}$.
- For each $\Sigma_Q$ a homeomorphism $\Sigma_Q\to\Sigma_{\eta_{\#}(Q)}$. The maps $X_R\to
X_{\eta_{\#}(R)}$ (similarly for $T_A$’s) respect attaching maps of boundary components of surfaces.
- If $E_1$ and $E_2$ are incident to $A$ and $T_{E_1}$ and $T_{E_2}$ have the same image in $T_A,$ then $\eta_{\#}(E_1)\neq\eta_{\#}(E_2)$.
The existence of immersions as above is an easy variation on Stallings’s folding. One way to construct immersions of graphs representing subgroups is to pass to the cover of a graph representing a subgroup and trimming trees. There is an analogous construction in this context.
Roots, immersions, and resolving
--------------------------------
We need to be able to represent conjugacy classes of elements of limit groups as nice paths in graphs of spaces.
Let $X_G$ be a graph of spaces representing a principle cyclic decomposition of $G$. The of $X_G,$ denoted $X^0_G,$ is the union of vertex spaces.
An in a graph of spaces $X_G$ is a map $p\colon\left[0,1\right]\to X_G$ such that $p^{-1}(X^0_G)$ contains $\set{0,1}$ and is a disjoint collection of closed subintervals. Let $\left[a,b\right]$ be the closure of a complementary component of $p^{-1}(X^0_G)$. Then $p$ maps $\left[a,b\right]$ homeomorphically to some $t_E$.
Let $X_R$ be a vertex space. Set $\partial X_R$ be the union of copies of edge spaces contained in $X_R$. An edge path $p$ is reduced if every restriction $p\vert_{\left[a,b\right]}(\left[a,b\right];\set{a,b})\to(X_R,\partial
X_R)$ does not represent the relative homotopy group $\pi_1(X_R,\partial X_R)$
A continuous map $\gamma\colon\mathrm{S}^1\to X_G$ is if all edge-path restrictions of $\gamma$ to subintervals $\mathrm{I}\subset\mathrm{S}^1$ are reduced edge paths.
The following lemma is standard and follows easily from Stallings folding [@stallings1; @stallings0] and the definitions.
If $g\in G$ then there is a cyclically reduced edge path $\gamma\colon\mathrm{S}^1\to X_G$ representing the conjugacy class $\left[g\right]$.
Let $\psi\colon X_G\immerses X_H$ be an immersion representing $G\into H$. If $\gamma\colon\mathrm{S}^1\to X_G$ is a reduced edge path then $\psi\circ\gamma$ is a reduced edge path in $X_H$.
For each edge $E$ of $X_G,$ we introduced a subset $t_E$ of the edge space $T_E\times I$. We think of $t_E$ as a formal element representing the path $I\to b_E\times\mathrm{I}$ with a fixed but arbitrary orientation. Let $\tau(t_E)$ be the image of the basepoint of $T_E$ in the vertex space of $X_G$ at the terminal end of $T_E\times\mathrm{I},$ and let $\iota(t_E)$ be the image of the basepoint in the vertex space at the initial end of $T_E$. Then every nonelliptic element represented by a cyclically reduced path can be thought of as a composition $t_E$’s, their inverses, and elements of relative homotopy groups of vertex spaces. Moreover, if the subword $t_Egt^{-1}_E$ appears then $g$ is not contained in the image of $E$.
Let $\gamma\in G$ be represented by a cyclically reduced edge path $\gamma$; $\psi\circ\gamma$ is an edge path in $X_H,$ and if it is not cyclically reduced, then for some sub-path $t_E h t_E^{-1}$ of $\gamma$ (we may need to reverse the orientation of $t_E$), the image of this subpath is homotopic into $\eta_{\#}(T_E),$ which means that $\left[h\right]\in\eta_{\#}(E)$. Since $\gamma$ is reduced, $\left[h\right]\notin E,$ and since the image of $E$ in $\eta_{\#}(E)$ is finite index, for some $l>0,$ $\left[h\right]^l\in E$. Since edge groups are primitive unless adjacent to vertex groups, $E$ must be attached to a boundary component of a vertex. This implies that $\eta_{\#}(E)$ is also attached to a boundary component of a vertex group, but this means $E\to\eta_{\#}(E)$ is an isomorphism, contradicting the fact that $\left[h\right]\notin E$.
Let $G$ and $H$ be freely indecomposable limit groups, $H$ obtained from $G$ by adjoining roots to $\E,$ $\eta\colon
G\into H$. Let $\Pi_G$ and $\Pi_H$ be principle cyclic decompositions and suppose that if $K$ is elliptic in $\Pi_G$ if and only if $\eta(K)$ is elliptic in $\Pi_H$. Let $\psi\colon X_G\to X_H$ be an immersion representing the inclusion.
Without loss of generality, suppose that all elements of $\E$ are self-centralized and nonconjugate. Let $\E_e$ be the elements of $\E$ which are elliptic in $\Pi_G$ and let $\E_h$ be the elements of $\E$ which are hyperbolic in $\Pi_G$.
For each $E\in\E$ let $T_E$ be a torus representing $E,$ $T_{F(E)}$ a torus representing $F(E),$ and let $T_E\to T_{F(E)}$ be the covering map corresponding to the inclusion $E\into F(E)$. Let $M_E$ be the mapping cylinder of the covering map. If $\group{\gamma}\in\E$ we abuse notation and refer to $M_{\group{\gamma}}$ as $M_{\gamma}$. The copy of $T_{F(E)}$ in $M_E$ is the of $M_E,$ and if $E$ is infinite cyclic, it is the . The copy of $T_E$ in $M_E$ is the , and is denoted $\partial M_E$.
For each element $E\in\E_e,$ let $f_E\colon T_E\to X_G$ be a map representing the inclusion $E\into G$ which has image in a vertex space of $X_G$. If $E$ is an abelian vertex group of $\Pi_G$ then we identify $T_E$ with the torus $T_A\subset X_G$. For each $\group{\gamma}\in\E_h,$[^3] represent $\gamma$ by a reduced edge path, abusing notation, $\gamma\colon\partial
M_{\gamma}\to X_G$.
Build a space $X'_G$ by attaching the $M_E$ and $M_{\gamma}$ to $X_G$ along $T_E$ and $\img(\gamma)$ by the maps $f_E$ and $\gamma,$ respectively.
By hypothesis there is a $\pi_1$–surjective map $\psi'\colon
X'_G\to X_H$. We choose this map carefully: For $E\in\E_e,$ $F(E)$ has elliptic image in $\Pi_H$. Choose a map $T_{F(E)}\to X_H$ with image contained in the appropriate vertex space of $X_H,$ and extend the map across $M_E$ so that $M_E$ also has image contained in the vertex space of $X_H$. For $\group{\gamma}\in\E_h,$ the core curve of $M_{\gamma}$ is a $k_{\gamma}$–th root of $\gamma$. Choose a cyclically reduced representative of $\sqrt[k_{\gamma}]{\gamma}\colon S^1\to X_H$ and let the map on the core curve agree with this representative.
The restriction of $\psi',$ defined thus far, to the disjoint union of $X_G$ and the core curves of the $M_{\gamma},$ is transverse to the subsets $T_{\eta_{\#}(E)}\times\set{\frac{1}{2}}$. Extend $\psi'$ to $X_G$ so the composition $M_{\gamma}\into X'_G\xrightarrow{\psi'}X_H$ is transverse to all $T_{\eta_{\#}(E)}\times\frac{1}{2}$. Let $\Lambda$ be the preimage
$$\psi'^{-1}\left(\sqcup_{E\in\E(G)}\left(T_{\eta_{\#}(E)}\times\set{\frac{1}{2}}\right)\right)$$
Suppose some component of $\Lambda$ is a circle which misses the boundary and core of some $M_{\gamma}$. By transversality this component of $\Lambda$ is a one manifold without boundary, and is therefore a circle. If this circle bounds a disk then there is a map homotopic $\psi',$ which agrees with $\psi'$ on the core curves and $X_G$ such that the number of connected components of the preimage is strictly lower. If the circle doesn’t bound a disk in $M_{\gamma}$ then it is boundary parallel. If this is the case then $\gamma$ is elliptic and we have a contradiction.
Fix a mapping cylinder $M_{\gamma}$ and consider the preimage of $\Lambda$ under the map $M_{\gamma}\to X'_G$. The preimage is a graph all of whose vertices are contained in the core curve of $M_{\gamma}$ or in the boundary of $M_{\gamma}$. If any component of the preimage of $\Lambda$ doesn’t connect the boundary of $M_{\gamma}$ to the core curve, then it is an arc and there is an innermost such arc which can be used to show that one of either $\gamma$ or $\gamma'$ is not reduced. Thus the preimages of arcs connect the core curve to the boundary.
Let $b$ be a point of intersection of $\Lambda$ with the core curve of $M_{\gamma}$. There are $k_{\gamma}$ arcs, where $k_{\gamma}$ is the degree of the root added to $\gamma,$ $s_1,\dotsc,s_{k_{\gamma}}$ (cyclically ordered by traversing $\partial M_{\gamma}$) in $\Lambda$ connecting $b$ to $\partial
M_{\gamma}$. Now consider the arcs as paths $s_j\colon\left[0,1\right]\to M_{\gamma}$. The composition $p_{\gamma}\define s^{-1}_2s_1$ is a path in $M_{\gamma}$ from $\partial M_{\gamma}$ to $\partial M_{\gamma}$. Let $D_{\gamma}$ be the sub-path of $\gamma$ obtained by traversing $\partial M_{\gamma}$ from $*\define s_{1}\cap\partial M_{\gamma}$ to $*_2\define
s_{2}\cap\partial M_{\gamma}$. The path $D_{\gamma}p_{\gamma}$ is homotopic, relative to $*,$ to $s_1^{-1}\sqrt[k_{\gamma}]{\gamma}s_1$. In particular, $$(D_{\gamma}p_{\gamma})^{k_{\gamma}}\simeq\gamma$$
A possible neighborhood of a component of $\Lambda$ is illustrated in Figure \[trackneighborhood\].
![A neighborhood of a component of $\Lambda$ in $X'_G$[]{data-label="trackneighborhood"}](roots-figures/lambda.eps)
Three interrelated lemmas.
\[onetooneonedges\] Suppose $\eta\colon G\into H,$ $H$ obtained from $G$ by adjoining roots to $\E,$ $G$ freely indecomposable. Let $\Pi_H$ be a one-edged splitting of $G$ over a cyclic edge group $E_H$. Let $\Pi_G$ be the splitting $G$ inherits from its action via $\eta$ on the Bass-Serre tree for $\Pi_H$. Represent $\Pi_H$ by a graph of spaces $X_H,$ and choose a graph of spaces $X_G$ and an immersion $\psi\colon X_G\immerses X_H$ representing $\eta$. Suppose that $\Pi_G$ is one-edged, and that the edge group is $E$. If $\E_h$ is nonempty then $E\into \eta_{\#}(E)$ is a *proper* finite index inclusion.
Let $\group{\gamma}\in\E_h,$ and represent $\gamma$ by a reduced edge path crossing $t_E$. Since $\psi$ is one-to-one on edge spaces, $p_{\gamma}$ is a closed path. As such, it represents an element of the fundamental group of $X'_G$. Then $\left[\psi\circ
p_{\gamma}\right]\in\eta_{\#}(E)$. If $\left[\psi\circ
p_{\gamma}\right]\in\img(E)$ then there is a path $p'_{\gamma}$ in $T_E$ which is homotopic in $T_{\eta_{\#}(E)},$ relative to the image of $*,$ to $\psi\circ p_{\gamma}$. Let $\alpha=D_{\gamma}p'_{\gamma}$. Then $\psi\circ\alpha$ is homotopic rel the image of $*$ to $\psi\circ D_{\gamma}p_{\gamma}$. But then $\left[\alpha\right]^{k_{\gamma}}=\gamma$ contradicting indivisibility of $\gamma$.
\[abeliansonto\] Let $G\into G'$ be an adjunction of roots. Let $\Pi_{G'}$ be a principle cyclic splitting of $G'$ with one abelian vertex group $A,$ let $\Pi_G$ be the associated splitting of $G,$ and represent $G\into G'$ by an immersion $\eta\colon X_G\immerses X_{G'}$ reflecting $\Pi_G$ and $\Pi_{G'}$. Suppose there is a unique vertex group $A'$ of $\Pi_G$ mapping to $A,$ and that there is at most one element of $\E$ conjugate into $A'$. If $\eta$ is one-to-one on edges adjacent to $A'$ then the induced map $F(A')\to
A/P(A)$ is onto.
Let $E_1,\dotsc,E_n$ be the edges adjacent to $A',$ and set $F(E_i)=\eta_{\#}(E_i)=P(A)$. Let $H$ be the limit group defined as follows: Let $\Delta=\Delta(R_j,E_i,A')$ be a graphs of groups representation of $\Pi_G$. Let $\E_{R_j}$ be the subcollection of $\E$ consisting of elements conjugate into $R_j$. Let $$S_j\define\img_{G'}\left(\group{\adjoinroot{R_j}{\E_{R_j}}{},gF(E_i)g^{-1}}_{gE_ig^-1<R_j}\right)$$ and $$A''\define\img_{G'}(F(A'),P(A))$$ Let $H\define\Delta(S_j,F(E_i),A'')$ There are maps $G\into H\into
G'$. We now show that $H\into G'$ is actually surjective. To do this we need to show that every element $\group{\gamma}$ of $\E_h$ has a $k_{\gamma}$–th root in $H$. This is precisely the argument given at the end of Lemma \[onetooneonedges\]. Let $G'\onto A/P(A)$ be the map which kills all vertex, edge groups, and stable letters, other than $A$. The quotient map clearly kills everything except $A$ and $F(A'),$ giving the desired surjection.
\[noextensions\] \[vertexgroupsobtainedbyadjoiningroots\] Let $(G,H,G')$ be a flight without any proper extensions. Suppose $G$ is freely indecomposable, has no vertex groups, and $\jcomp(G)=\jcomp(G')$. Represent the $G\into H$ by an immersion $X_G\immerses X_H,$ representing $\jsj_H(G),$ and $\rjsj(H),$ respectively. Then $\nu$ is one-to-one on edge spaces.
Every vertex group $W$ of $H$ is obtained from a vertex group $V$ of $\jsj_H(G)$ by adjoining roots to the elements of $\E$ which are conjugate into $V,$ along with edge groups incident to $V$.
Represent $G\into H$ by an immersion $X_G\to X_H,$ where $X_G$ represents $\jsj_H(G)$ and $X_H$ represents the principle cyclic of $H$. For each edge $E_i$ of $X_G$ let $e_i$ be a generator, let $k_i$ be the largest degree of a root of $e_i$ in $H,$ let $F(E)=\group{f_i},$ and let $E\into F(E)$ be the map which sends $e_i$ to $f_i^{k_i}$. Let $\E'$ be the collection of elements of $\E$ which are elliptic in $H$ along with all edge groups of $\jsj_H(G)$.
Consider the group $\adjoinroot{G}{\E'}{}$. Let $\Delta=\Delta(R_i,A_j,E_k)$ be a graph of groups representation of $\jsj_H(G)$. Let $\E_{R_i}$ be the set of elements of $\E'$ which are conjugate into $R_i$. Likewise, let $\E_{A_j}$ be the set of elements of $\E'$ which are conjugate into $A_j$. Let $$S_l=\img_H(\group{S_l,gBg^{-1}}_{gBg^{-1}<S_l,B\in\E_{S_l}})$$ where $gBg^{-1}<S_l,$ and where $S_l$ is either some rigid vertex group $R_i$ or abelian vertex group $A_j$. Let $$H'=\Delta(R'_l,A'_j,F(E_k))$$ and choose a graph of spaces $X_{H'}$ representing this decomposition of $H'$. There is a pair of maps of graphs of spaces $X_G\to X_{H'},$ $X_{H'}\to X_H,$ and there is an epimorphism $\adjoinroot{G}{\E'}{}\onto H'$. The map $\psi'\colon X_{G}\to X_{H'}$ is one-to-one on edge spaces. Moreover, $H'$ is a limit group since the map $H'\to H$ is clearly strict.
The proof of the lemma will be complete if we can show that $\psi'$ extends to $X'_G,$ that is, if $H'$ contains all roots of elements adjoined to $\E$. Then the image of $\adjoinroot{G}{\E}{}$ (with the induced graph of groups decomposition) in $H'$ is a nontrivial extension of $H$.
Consider the paths $D_{\gamma}$ and $p_{\gamma}$ defined previously through resolving. We defined $p_{\gamma}\define s_2^{-1}s_1$ and set $*=s_1\cap\partial M_{\gamma}$. Let $*_2\define s_2\cap\partial
M_{\gamma}$. To show that $H'$ has a $k_{\gamma}$–th root of $\gamma$ we need to show that $X_{H'}$ has a path $p'_{\gamma}$ from the image of $*_2$ to the image of $*$ whose image under $X_{H'}\to
X_{G'}$ is homotopic rel endpoints to the image of $p_{\gamma}$.
Suppose that $T_{E_1}\times\frac{1}{2}$ and $T_{E_2}\times\frac{1}{2}$ are the midpoints of edge spaces containing $*$ and $*_2,$ respectively, and suppose, without loss of generality, that $D_{\gamma}$ starts and ends by traversing the second and first halves of $T_{E_1}$ and $T_{E_2},$ respectively, in the positive direction. The first key observation to make is that we can choose the orientations of $t_{E_i}$ so that the terminal endpoints of $t_{E_1}$ and $t_{E_2}$ are both contained in some $T_A$: $E_1$ and $E_2$ are conjugate in $H,$ must therefore be conjugate in $G$ since $\jcomp(G')=\jcomp(G),$ and cannot both be adjacent to a rigid vertex group of $G,$ otherwise there is a rigid vertex group $R$ of $G$ such that $v(R)>v(\eta_{\#}(R))$. The only other possibility is that they are both adjacent to an abelian vertex group $A,$ as claimed.
Let $t^+_{\varphi_{\#}(E_i)}$ be the half of $t_{\varphi_{\#}(E_i)}$ obtained by traversing $t_{\varphi_{\#}(E_i)}$ from the midpoint to the terminal endpoint. By Lemma \[abeliansonto\], $H'\to H$ is surjective on abelian vertex groups, and by construction, the terminal endpoints of $t^+_{\varphi_{\#}(E_i)}$ agree. Let $p''_{\gamma}\define t^+_{\varphi_{\#}(E_2)}
(t^+_{\varphi_{\#}(E_1)})^{-1}$. Then $p''_{\gamma}$ is a path from $\varphi(*_2)$ to $\varphi(*)$ whose image in $X_H$ is homotopic rel endpoints into $\psi_{\#}(E_1)(=\psi_{\#}(E_2))$. Since $H'\onto H$ is surjective on edge groups, there is a closed path $h_{\gamma}$ in $(T_{\varphi_{\#}(E_1)},\varphi(*))$ which maps to the image of $p_{\gamma}$. Set $p'_{\gamma}\define h_{\gamma}p''_{\gamma}$. The image of $p'_{\gamma}$ is homotopic rel endpoints to the image of $p_{\gamma}$ in $X_H$. Arguing as in Lemma \[onetooneonedges\], $(\varphi\circ D_{\gamma})p'_{\gamma}$ is a $k_{\gamma}$–th root of $\varphi\circ\gamma$ and the map $X'_G\to X_H$ factors through $X_{H'}$.
Thus there is a map $X'_G\to X_{H'}$. Since $H$ has no proper extensions, $\img_{H'}\left(\adjoinroot{G}{\E}{}\right)\to H$ is an isomorphism. In particular, $X_G\to X_H$ is one-to-one on edges and the situation above never occurs.
Consider the construction of $H'$. Now that we know that $H'\cong
H,$ $\Delta$ *must* be the principle cyclic of $H$. If there is a principle cyclic splitting of $H$ not visible in $\Delta$ then it must be a cyclic splitting inherited from the relative (to incident edge groups) principle cyclic decomposition of some vertex group of $\Delta$. On the other hand, all vertex groups of $\Delta$ must be elliptic in the principle cyclic of $H$ since they are obtained by adjoining roots to subgroups of $G$ which are guaranteed to be elliptic in the principle cyclic of $H$.
This nearly completes the proof of Lemma \[mainlemma\]. We need to prove that the vertex groups of $\jsj^*_H(G')$ are obtained from the images of the vertex groups of $\jsj(H)$ by adjoining roots, and that $\pi$ is injective if its restrictions to vertex groups are injective.
Let $\Delta=\Delta(R_i,A_j,E_k)$ be a graph of groups decomposition representing the principle cyclic of $H$. We know that all vertex and edge groups of $\Delta$ map to vertex and edge groups of $\jsj^*_H(G')$. Let $\Phi_s(\pi)\colon \Phi_s(H)\onto G'$ be the strict homomorphism constructed in [@louder::stable § \[STABLE-sec:constructstrict\]], and also in the appendix of this paper, and let $\Phi_s(\Delta)$ be the principle cyclic decomposition of $\Phi_s(H)$ in which all images of vertex groups of $\Delta$ are elliptic. Clearly $\Phi_s(\pi)$ maps elliptic subgroups of $\Phi_s(\Delta)$ to elliptic subgroups of $\jsj^*_H(G')$. Moreover, if $A$ is a noncyclic abelian vertex group of $H,$ then by construction, $\Phi_s(\pi)$ maps $A/P(A)$ onto $\pi_{\#}(A)/P(\pi_{\#}(A))$. Thus all modular automorphisms of $\Phi_s(H)$ supported on abelian vertex groups of $\Delta$ push forward to modular automorphisms of $G'$. Another consequence of the hypothesis $\jcomp(G)=\jcomp(G')$ is that $\Phi_s(H)\to G'$ is one to one on the set of edge groups adjacent to every vertex group, hence every Dehn twist of $\Phi_s(H)$ pushes forward to a Dehn twist of $G'$. A strict map which allows all modular automorphisms to push forward is an isomorphism, therefore $\Phi_s(\pi)$ is an isomorphism.
The third bullet follows immediately from the construction of $\Phi_s$.
Constructing strict homomorphisms {#appendix_strict}
=================================
We give here a description of the process of constructing strict homomorphisms of limit groups. Let $G$ be a group with a one-edged splitting $\Delta$ with nonabelian vertex groups of the form $G\cong
R*_{\group{e}}S,$ and suppose there is a map $\varphi\colon G\to L,$ $L$ a limit group, which embeds $R$ and $S$. Then $R$ and $S$ are limit groups. Suppose further that $\varphi$ embeds $R*_{\group{e}}\cent_S(\group{e})$ and $\cent_R(\group{e})*_{\group{e}} S$. Then $\varphi$ is , and $G$ is a limit group. There is a process, whose output is a limit group $\Phi_s(G),$ which takes the data $(G,\Delta,L,\varphi)$ and produces a triple $G\to\Phi_S(G)\to
L,$ such that the composition is $\varphi,$ $\Phi_S(G)$ splits over the centralizer of $\group{e},$ and $\Phi_S(G)\to L$ is strict.
The process is one of pulling centralizers and passing to images of vertex groups in a systematic way. The reader should compare this to the more general construction detailed in [@louder::stable], and a formally identical version in the proof of [@bf::lg Lemma 7.9]. Let $G=G_0$. Define for
- odd $i$: $G_i=R_{i-1}*_{\cent_{R_{i-1}}(\group{e})}S_i,$ where $$S_i\define\img_L(\cent_{R_{i-1}}(\group{e})*_{\cent_{S_{i-1}}(\group{e})}S_{i-1})$$
- even $i$: $G_i=R_i*_{\cent_{S_{i-1}}(\group{ e})}S_{i-1},$ where $$R_i\define\img_L(R_{i-1}*_{\cent_{S_{i-1}}(\group{ e})}\cent_{S_i}(\group{ e}))$$
We claim that this process terminates in finite time. The sequence of quotients $G_0\onto G_1\onto\dotsc$ embeds edge groups at every step. Since abelian subgroups of limit groups are finitely generated and free, and since finitely generated free abelian groups satisfy the ascending chain condition the assertion holds. The direct limit $G_{\infty}$ is called $\Phi_s(G)$.
This discussion is relevant to the proof of Lemma \[noextensions\], but we must vary the construction a little. Let $\bar{H}$ be the quotient of $H$ obtained by passing to the images in $G'$ of vertex groups of the (restricted) principle cyclic of $H,$ with the induced graph of groups decomposition $\Delta(\bar{R_i},A_j,E_k)$. The of $\bar{H},$ $\core(\bar{H})$ is the group obtained by replacing each abelian vertex group $A$ by its peripheral subgroup. Consider the situation in Lemma \[noextensions\]. There is a homomorphism $\core(\bar{H})\to G',$ and each group is equipped with a principle cyclic decomposition $\Delta_{\core(\bar{H})}$ and $\Delta_{G'},$ respectively. Moreover, the nonabelian vertex groups of $\Delta_{\core(\bar{H})}$ map to nonabelian vertex groups of $G',$ and the edge groups of $\core(\bar{H})$ map to edge groups of $\Delta_{G'}$. The centralizers of edges incident to nonabelian vertex groups of $G'$ are infinite cyclic, and this implies that in the process of pulling centralizers in $\core(\bar{H})_i,$ the pulled group is always infinite cyclic. Each vertex group of $\core(\bar{H})_i$ has elliptic image in $G',$ and since $G'$ is principle, centralizers are cyclic in the relevant vertex groups of $G'$. Iteratively adjoining roots to an infinite cyclic subgroup and passing to quotients multiple times can be accomplished in one step, thus the vertex groups of $\core(\bar{H})_{\infty}$ are obtained from the vertex groups of $\core(\bar{H})$ by adjoining roots to incident edge groups. There are surjective maps $H\onto
\Phi_s(H)\define\core(\bar{H})_{\infty}*_{Z(P(A_j))}(Z(P(A_j))\oplus
A/P(A_j))\onto L$.
[^1]: Fix $a$ and $b$. Then $\set{x}:{a\leq x\leq b}$ is finite.
[^2]: We could have instead redefined an essential curve as one which gives a principle cyclic splitting and isn’t boundary parallel.
[^3]: All elements of $\E_h$ are infinite cyclic.
|
---
abstract: 'In (100)-oriented GaAs illuminated at normal incidence by a laser and its second harmonic, interference between one- and two-photon absorption results in ballistic current injection, but not modulation of the overall carrier injection rate. Results from a pump-probe experiment on a transversely biased sample show that a constant electric field enables coherent control of the carrier injection rate. We ascribe this to the nonlinear optical Franz-Keldysh effect and calculate it for a two-band parabolic model. The mechanism is relevant to centrosymmetric semiconductors as well.'
author:
- 'J. K. Wahlstrand'
- 'H. Zhang'
- 'S. B. Choi'
- 'S. Kannan'
- 'D. S. Dessau'
- 'J. E. Sipe'
- 'S. T. Cundiff'
title: |
Electric field-induced quantum interference control in a semiconductor:\
A new manifestation of the Franz-Keldysh effect
---
Enabled by the accelerating development of ultrafast lasers, coherent control using interference between quantum pathways [@shapiro_coherent_2003] has become an increasingly important tool for the optical control of matter. Experiments using multiphoton pathways have included directional ionization in atoms [@yin_asymmetric_1992], photodissociation [@sheehy_phase_1995] and alignment [@de_field-free_2009] in molecules, and injection of ballistic charge currents [@dupont_phase-controlled_1995; @hache_observation_1997] and pure spin currents [@stevens_quantum_2003] in semiconductors, to list just a few examples. The symmetry of the system to be controlled constrains the processes that can occur, and an important but largely neglected extension is the use of constant or nearly constant external fields to change a coherent control process from forbidden to allowed. The simplest such field, and likely the most useful for potential applications in the solid state, is an electric field. While it has been shown that a constant (dc) electric field enables the control of the ionization rate of an atom by a two-color process [@manakov_dc_1999; @gunawardena_atomic_2007], the use of such a simple control parameter has not yet been exploited in other systems. Here we show that a dc field can enable new coherent control processes in semiconductors.
Interference between one- and two-photon pathways in semiconductors leads to the injection of a ballistic charge or spin current, the direction of which is related to the polarization of the light and the relative phase between superimposed optical beams with frequencies $\omega$ and $2\omega$. Referred to as $1+2$ quantum interference control (QUIC), this process has been used to create a carrier-envelope phase sensitive photodetector [@fortier_carrier-envelope_2004] and to study subpicosecond current dynamics [@kerachian_dynamics_2007; @zhao_dynamics_2008]. It is also possible to control the carrier *population* via $1+2$ QUIC [@fraser_quantum_1999], in a process arising from the imaginary part of the second-order nonlinear susceptibility $\chi^{(2)}$. This process may be observed in a zincblende semiconductor such as GaAs when the optical fields are aligned along certain crystal axes. One could easily imagine that, just as in electric field-induced second harmonic generation [@lee_nonlinear_1967], a dc field could combine with the always present third-order susceptibility $\chi^{(3)}$ to produce an effective $\chi^{(2)}$ that enables $1+2$ population control. Here, we present experimental evidence that such a process does in fact occur, and we develop a simple theory relating it to the Franz-Keldysh effect [@franz__1958; @keldysh__1958; @aspnes_electric_1967], a modification of the optical properties of a semiconductor due to the acceleration of photoexcited carriers.
We performed an all-optical measurement on (100) GaAs samples at room temperature. A diagram of the experimental apparatus and the results are shown in Fig. \[expt\]. The technique is similar to what was used to observe conventional population control in (111) GaAs [@fraser_quantum_1999], with the addition of a bias across the sample. A two-color pump pulse injects carriers through one- and two-photon absorption and QUIC. The transmission of a probe pulse depends on the photo-excited carrier density [@lee_room-temperature_1986], and we look for changes that depend on the phase parameter $\phi_{21} = \phi_{2\omega}-2\phi_{\omega}$. The light source for the pump beam is an optical parametric oscillator (OPO) using cesium titanyl arsenate (CTA), synchronously pumped by a mode-locked Ti:sapphire laser with repetition rate $\frep = 76$ MHz. The signal output of the OPO, centered at 1580 nm, is doubled using a $\beta$-barium borate (BBO) crystal, and the two harmonics are split by a prism into two paths, the relative delay of which can be adjusted using mirrors mounted on piezoelectric transducers. The harmonics are recombined using the same prism. We use the residual pump light, centered at 827 nm, as a probe beam. The pulse widths of the OPO fundamental, second harmonic, and probe pulses are approximately 160 fs, 220 fs, and 120 fs, respectively.
![(color online) Measurement of field-induced QUIC. (a) Schematic of the experimental apparatus, including the two-color interferometer and the radio frequency biasing scheme. OPO: optical parametric oscillator, BBO: $\beta$-barium borate crystal, BP: band pass filter, DDS: direct digital synthesis board. (b) The modulation of the transmitted probe power at the dither frequency as the phase parameter $\phi_{21} = \phi_{2\omega}-2\phi_{\omega}$ is swept by the slow ramp piezo. The measured signal for $\bEDC \parallel \langle 001 \rangle$ is shown for a positive bias (solid black), negative bias (dashed red), and zero bias (blue dotted). (c) The amplitude of the differential transmission signal is shown as a function of the bias voltage for $\bEDC \parallel \langle 001 \rangle$ (blue dots) and $\bEDC \parallel \langle 011 \rangle$ (red squares). The lines are linear fits to the data points. []{data-label="expt"}](fig2){width="8.5cm"}
We use a 1 $\mu$m thick, undoped GaAs epilayer grown by molecular beam epitaxy. The GaAs sample is attached to the sapphire disk using transparent epoxy, and the substrate is then removed by chemical etching. A 100 nm thick insulating SiO$_2$ layer is deposited on the surface. Using photolithography, Au electrodes separated by 100 $\mu$m are then patterned onto the sample. To apply an effective dc electric field $\EDC$, we use a radio frequency (rf) bias synchronized to $\frep$. This technique results in an electric field in the plane of the sample [@wahlstrand_uniform-field_2010] while avoiding highly nonuniform fields encountered when carriers are injected from electrodes into semi-insulating samples [@ralph_trap-enhanced_1991]. We derive a harmonic of $\frep$ from a photodiode and use it as a clock for direct digital synthesis (DDS) of a synchronized, variable phase bias waveform at $\frep$. A resonant $LC$ circuit passively enhances the voltage across the electrodes. The phase of the waveform is optimized so that the peak of the waveform occurs when the optical pulse arrives at the sample. The sign of the field may be reversed by changing the phase of the rf signal by $\pi$ using the DDS. Linear electroabsorption studies on the sample studied here, at the same bias voltage and optical power, show that we achieve a field strength of approximately 20 kV/cm. Two samples were studied: one with the rf field pointing along a $\langle 001 \rangle$ crystal direction and the other with the field along $\langle 011 \rangle$.
The pump and probe beams are focused on a common spot between the electrodes. The $\omega$ beam average power is 160 mW and the $2\omega$ beam average power is 6 mW. Part of the probe beam is split off before the sample and used as a reference for balanced detection. The spot size of the pump beams is roughly 25 $\mu$m, and the spot size of the probe beam half that. To detect only the contribution from QUIC processes, we dither $\phi_{21}$ using a mirror mounted on a piezoelectric transducer. A lock-in amplifier referenced to the dither frequency measures only the changes in the probe transmission $T$ due to changes in $\phi_{21}$; the signal is proportional to $dT/d\phi_{21}$. Using another mirror mounted on a piezoelectric transducer, we ramp the phase parameter over 0.5 s and average the data over approximately 100 ramp cycles. Results are shown in Fig. \[expt\]b with a positive and a negative bias, as well as with no bias. For light at $\omega$ and $2\omega$ normally incident on the (100) GaAs sample, for which population control enabled by the crystal symmetry is forbidden [@fraser_quantum_1999], we observe modulation of the transmitted probe beam *only* when the bias is applied.
The magnitude of the bias can be controlled by adjusting the output amplitude of the DDS. Results are shown in Fig. \[expt\]c. The signal is consistent with a linear dependence on bias, for both positive and negative bias (the latter is not shown). The dependence on the polarization of the optical fields was also studied by rotating the waveplates. Because of the polarization dependent reflection at the interfaces of the prisms in the interferometer, it is not feasible to continuously adjust the polarization. However, in a study of a few different polarization configurations we found that the signal drops by more than a factor of 4 when either or both of the $\omega$ or $2\omega$ beams are polarized perpendicular to the dc field direction. As described later, this result is consistent with the theory.
For one-photon absorption, the change in optical properties of a bulk semiconductor is typically dominated by the Franz-Keldysh effect (FKE) [@franz__1958; @keldysh__1958; @aspnes_electric_1967]. To calculate QUIC in the presence of a static electric field, we extend a theory of the one-photon FKE [@wahlstrand_theory_2010] to multiphoton processes. In the limit of long optical pulses, one can derive a Fermi Golden Rule expression for the rate of carrier injection due to QUIC. This is of the form $\dot{n}^{\mathrm{I}} = \eta^{jlm} (\omega) E^i_{2\omega} E^l_\omega E^m_\omega e^{i(\phi_{2\omega}-2\phi_{\omega})}+c.c.$, where $\eta^{jlm} (\omega)$ is a tensor that describes the efficiency of the process as a function of $\omega$. For two parabolic bands separated by a direct band gap $\hbar \omega_g$, $\eta^{jlm} (\omega)$ can be found analytically. We assume a $\bk$-independent interband velocity matrix element $\bV_{cv}$. The conduction band and valence band effective masses are $m_c$ and $m_v$, respectively, and the reduced effective mass $\mu = m_c m_v/(m_c+m_v)$. For a constant field pointing along $\hat{\mathbf{z}}$, we find $$\begin{gathered}
\eta^{zzz} (\omega) = \frac{e^3 \EDC^2}{2\hbar^3\omega^4\Omega} |V^z_{cv}|^2 \left(\frac{\mathrm{Ai}^2 (-\frac{2\omega-\omega_g}{\Omega})}{2\Omega} \right. \\ \left.+ \frac{-\frac{2\omega-\omega_g}{\Omega} \mathrm{Ai}^2 (-\frac{2\omega-\omega_g}{\Omega})-[\mathrm{Ai}'(-\frac{2\omega-\omega_g}{\Omega})]^2}{\omega} \right),
\label{lowtemp}\end{gathered}$$ where $\mathrm{Ai}(x)$ is the Airy function and $\Omega=(e^2 \EDC^2/2\mu\hbar)^{1/3}$ is the electro-optic frequency. This parabolic band approximation (PBA) result is shown in Fig. \[theory\] for the parameters of GaAs. Because the tensor is real, the injection rate is proportional to $\cos(\phi_{2\omega}-2\phi_{\omega})$.
![Calculation of field-induced $1+2$ quantum interference control of carrier injection in GaAs in the parabolic band approximation, with no lifetime broadening (solid line) and in the limit of large broadening (dashed line). The optical fields are linearly polarized parallel to the constant field $\EDC=20$ kV/cm.[]{data-label="theory"}](fig1){width="8cm"}
As with the one-photon absorption spectrum [@aspnes_electric_1967], the spectrum of the field-induced QUIC injection tensor displays Franz-Keldysh oscillations. The theory assumes no decoherence or lifetime broadening; we expect that damping tends to wash out the oscillations, as it does for one-photon absorption [@aspnes_electric_1967]. Using the asymptotic form of the Airy function for large $\omega$, which gives the spectrum in terms of $\sin^2(x)$ and $\cos^2(x)$, and replacing them with the average value 0.5, we find $$\begin{gathered}
\eta^{zzz} (\omega) = \\ \frac{\sqrt{2} e^4 \mu^{1/2}}{2\pi\hbar^{9/2} \omega^4 } |V^z_{cv}|^2 \left( \frac{1}{4\sqrt{2\omega-\omega_g}} - \frac{\sqrt{2\omega-\omega_g}}{ \omega} \right) \EDC
\label{qic_w_damping}\end{gathered}$$ for $2\omega>\omega_g$. This theoretical expression, more relevant than Eq. (\[lowtemp\]) for a sample at room temperature, is plotted as a dashed line in Fig. \[theory\]. We find that the injection rate due to QUIC changes sign at $(4/7)\hbar \omega_g$ in the PBA. Using analytical expressions for the one-photon and two-photon absorption in the PBA, neglecting field-induced changes in those (which are of order $\EDC^2$), one can show that $$\frac{\dot{n}^I}{\dot{n}} = \frac{6e^2 \EDC E_\omega^2 E_{2\omega} \omega \left(4\omega_g-7\omega\right)}{16e^2 E_\omega^4 (2\omega-\omega_g)^2+3E_{2\omega}^2 \mu \hbar \omega^4 (2\omega-\omega_g)}
\label{percent}$$ for $2\omega>\omega_g$.
A realistic model for the band structure displays a dependence on the direction of the fields with respect to the crystal axes, as well as more subtle effects due to nonparabolicity [@wahlstrand_theory_2010]. A preliminary calculation using a 14-band $\mathbf{k}\cdot\mathbf{p}$ model has the same qualitative spectral shape as the PBA result, with the zero in the injection rate occurring at a lower energy. Since the process we describe arises from a fourth-rank tensor (to lowest order in the DC field), the polarization dependence can be predicted knowing the zincblende crystal symmetry. The nonzero components are $\chi_{xxxx}$ (all fields parallel), $\chi_{xxyy}$, and $\chi_{xyxy}$. The largest effect is expected for all fields parallel, as observed in the experiment. The modulation of the pump-probe differential transmission due to QUIC in the $\langle 001 \rangle$ sample for the probe arriving 300 fs after the pump is approximately 2.0%. The expected carrier density modulation from Eq. (\[percent\]), using the fluences of the $\omega$ and $2\omega$ pulses in the experiment, is 0.6%. In an optically thin sample, the fractional change in the pump-probe transmission would equal the fractional change in the carrier density. Because the thickness of our sample is comparable to the coherence length between the $\omega$ and $2\omega$ beams ($\approx$1.2 $\mu$m [@fraser_quantum_2003]), the amplitude of the modulation varies as a function of depth, complicating the analysis. There are other complicating factors: The QUIC signal decays for longer pump-probe time delays. We attribute this to the relaxation of a non-uniform distribution of carriers due to the combination of dispersion, multiple reflections, and nonuniform absorption for the $2\omega$ pulse. A cascaded process [@stevens_enhanced_2005], arising in this case from electric field-induced second harmonic generation (EFISH) [@lee_nonlinear_1967] followed by optical interference, also contributes to the signal. Considering these experimental issues and the simplifying assumptions made in the theory, the agreement is reasonable. We have observed additional evidence of field-induced QUIC in experiments on low-temperature grown GaAs and Er-doped GaAs that use electrical [@hache_observation_1997; @wahlstrand_electrical_2006] rather than optical detection. Biasing the electrodes used to read out the injected photocurrent results in an enhancement in the signal that is inconsistent with current injection alone.
In summary, we have shown that a constant electric field enables control of the carrier injection rate by interference of one- and two-photon absorption in a semiconductor, in a new manifestation of the Franz-Keldysh effect. While the use of slowly-varying electric fields along with optical pulses has not been much explored in coherent control, the results here indicate that the experiments are feasible and the results for absorption in a semiconductor are interpretable in terms of an extension of Franz-Keldysh theory. We note that similar processes have been predicted [@manakov_dc_1999] and observed [@gunawardena_atomic_2007; @bolovinos_one-_2008] in atomic systems. The process described here is very different, because it relies on the acceleration of the photo-excited carriers (Franz-Keldysh effect) rather than the perturbation of bound states (Stark effect) by the electric field. While the experiment here showed a linear dependence of the coherent control efficiency on the dc field strength, experiments at low temperature or for energies near the half-band gap should show a highly nonlinear response characteristic of the nonperturbative nature of the Franz-Keldysh effect.
Because it results in exotic carrier distributions such as ballistic charge and spin currents, coherent control via QUIC is a promising strategy for studying transport across metal-semiconductor interfaces, of great importance in improving electronic devices. Since DC fields are often encountered in such structures, understanding their effect on QUIC is essential in realizing the potential of this diagnostic tool. In this paper we have studied the electro-optic coherent control of carrier population, perhaps the simplest solid state coherent control process, but also one of the weakest: even for crystals and polarization schemes where coherent control of population is allowed, it is small. But coherent control processes such as those for charge and spin current injection are more robust effects, and this work suggests that their modulation by DC fields are promising directions for future studies, from both fundamental and technological perspectives.
We thank R. Smith, J. Pipis, and R. Snider for assistance in the early stages of the experiment, R. Mirin for providing the GaAs sample, and T. Reber for technical assistance with the OPO. S.T.C. is a staff member in the NIST Quantum Physics division.
[27]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, , , , ****, ().
, , , ****, ().
, , , , , , , , , , and , ****, ().
, , , , , ****, ().
, , , , , , ****, ().
, , , , , , ****, ().
, , , ****, ().
, ****, ().
, , , , , , ****, ().
, , , , ****, ().
, , , , ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , , , , , , , and , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , , , ****, ().
, , , , , ****, ().
, Phys. Rev. B **82**, 075206 (2010).
, , , ****, ().
|
---
abstract: 'We deal with the virtual element method (VEM) for solving the Poisson equation on a domain $\Omega$ with curved boundaries. Given a polygonal approximation $\Omega_h$ of the domain $\Omega$, the standard order $m$ VEM [@hitchVEM], for $m$ increasing, leads to a suboptimal convergence rate. We adapt the approach of [@BDT] to VEM and we prove that an optimal convergence rate can be achieved by using a suitable correction depending on high order normal derivatives of the discrete solution at the boundary edges of $\Omega_h$, which, to retain computability, is evaluated after applying the projector $\Pi^\nabla$ onto the space of polynomials. Numerical experiments confirm the theory.'
address:
- 'IMATI “E. Magenes”, CNR, Pavia (Italy)'
- 'IMATI “E. Magenes”, CNR, Pavia (Italy)'
- 'IMATI “E. Magenes”, CNR, Pavia (Italy)'
author:
- Silvia Bertoluzza
- Micol Pennacchio
- Daniele Prada
title: 'High order VEM on curved domains.'
---
[10]{}
P. F. Antonietti, L. [Beirão]{} da Veiga, D. Mora, and M. Verani, *A stream virtual element formulation of the [Stokes]{} problem on polygonal meshes*, SIAM Journal on Numerical Analysis **52** (2014), no. 1, 386–404.
P. F. Antonietti, L. [Beirão]{} da Veiga, S. Scacchi, and M. Verani, *A [$C^1$]{} virtual element method for the [Cahn-Hilliard]{} equation with polygonal meshes*, SIAM Journal on Numerical Analysis **54** (2016), no. 1, 34–56.
P. F. Antonietti, L. Mascotto, and M. Verani, *A multigrid algorithm for the $p$-version of the virtual element method*, Math. Model. Numer. Anal. **52** (2018), no. 1, 337––364.
L. [Beirão]{} da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, *Basic principles of virtual element methods*, Mathematical Models and Methods in Applied Sciences **23** (2013), no. 1, 199–214.
L. [Beirão]{} da Veiga, F. Brezzi, and L. Marini, *Virtual elements for linear elasticity problems*, SIAM Journal on Numerical Analysis **51** (2013), no. 2, 794–812.
L. [Beirão]{} da Veiga, F. Brezzi, L. D. Marini, and A. Russo, *The [Hitchhiker]{}’s guide to the virtual element method*, Mathematical Models and Methods in Applied Sciences **24** (2014), no. 8, 1541–1573.
, *Mixed virtual element methods for general second order elliptic problems on polygonal meshes*, ESAIM: M2AN **50** (2016), no. 3, 727–747.
L. [Beirão]{} da Veiga, Lovadina C., and A. Russo, *[Stability Analysis for the Virtual Element Method]{}*, 2016, arXiv:1607.05988.
L. [Beirão]{} da Veiga, A. Chernov, L. Mascotto, and A. Russo, *[Basic principles of $hp$ virtual elements on quasiuniform meshes]{}*, Mathematical Models and Methods in Applied Sciences **26** (2016), no. 8, 1567–1598.
L. [Beir[ã]{}o da Veiga]{}, C. [Lovadina]{}, and G. [Vacca]{}, *Divergence free virtual elements for the [Stokes]{} problem on polygonal meshes*, ESAIM: M2AN **51** (2017), no. 2, 509–535.
, *[Virtual Elements for the Navier-Stokes problem on polygonal meshes]{}*, arXiv e-prints (2017).
L. [Beir[ã]{}o da Veiga]{}, A. [Russo]{}, and G. [Vacca]{}, *[The Virtual Element Method with curved edges]{}*, ArXiv e-prints (2017).
M. F. Benedetto, S. Berrone, and S. Scialó, *A globally conforming method for solving flow in discrete fracture networks using the virtual element method*, Finite Elements in Analysis and Design **109** (2016), 23 – 36.
J. H. Bramble, T. Dupont, and V. Thomée, *Projection methods for dirichlet’s problem in approximating polygonal domains with boundary-value corrections*, Mathematics of Computation **26** (1972), no. 120, 869–879.
L. [Beirão]{} da Veiga, C. Lovadina, and D. Mora, *A virtual element method for elastic and inelastic problems on polytope meshes*, Computer Methods in Applied Mechanics and Engineering **295** (2015), 327 – 346.
T. Dupont, *${L}_2$ error estimates for projection methods for parabolic equations in approximating domains*, Mathematical Aspects of Finite Elements in Partial Differential Equations (Carl de Boor, ed.).
Brezzi F., Lipnikov K., and Shashkov M., *Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces*, Math. Models Methods Appl. Sci. **16** (2006), no. 2, 275–297.
M. [Frittelli]{} and I. [Sgura]{}, *[Virtual Element Method for the Laplace-Beltrami equation on surfaces]{}*, arXiv e-prints (2016).
A. L. Gain, C. Talischi, and G. H. Paulino, *On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes*, Computer Methods in Applied Mechanics and Engineering **282** (2014), 132–160.
Botti L. and Di Pietro D., *Assessment of hybrid high-order methods on curved meshes and comparison with discontinuous galerkin methods*, J. Comput. Phys. **370** (2018), 58––84.
K. Lipnikov, *On shape-regularity of polyhedral meshes for solving pdes*.
L. [Mascotto]{}, L. [Beir[ã]{}o da Veiga]{}, A. [Chernov]{}, and A. [Russo]{}, *Exponential convergence of the hp [Virtual Element Method]{} with corner singularities*, Numer. Math. (2018), 138–581.
J.A. Nitsche, *[Ü]{}ber ein variationsprinzip zur [L]{}ösung von [D]{}irichlet-[P]{}roblemen bei [V]{}erwendung von [T]{}eilr äumen, die keinen [R]{}andbedingungen unterworfen sind*, Abhandlungen aus dem Mathematischen Seminar der Universit ät Hamburg **36** (1970), 9–15.
I. Perugia, P. Pietra, and A. Russo, *A plane wave virtual element method for the [Helmholtz]{} problem*, ESAIM: M2AN **50** (2016), no. 3, 783–808.
Sevilla R., Fernández-Méndez S., and Huerta A., *Comparison of high-order curved finite elements*, Internat. J. Numer. Methods Engrg. **87** (2011), no. 8, 719–734.
Vidar Thomée, *Polygonal domain approximation in dirichlet’s problem†*, IMA Journal of Applied Mathematics **11** (1973), no. 1, 33–44.
G. Vacca and L. [Beirão]{} da Veiga, *[Virtual element methods for parabolic problems on polygonal meshes]{}*, Numerical Methods for Partial Differential Equations **31** (2015), no. 6, 2110–2134.
|
---
abstract: 'A charged droplet can be electrodynamically levitated in air using a quadrupole trap by typically applying sinusoidal electric field. When a charged drop is levitated it exhibits surface oscillations simultaneously building charge density due to continuous evaporation and subsequently undergoes breakup due to Rayleigh instability. In this work, we examined large-amplitude surface oscillations of a sub-Rayleigh charged drop and its subsequent breakup, levitated by various applied signals such as sine, square and ramp waveform at various imposed frequencies, using high-speed imaging (recorded at 100-130 thousand Frames Per Second (fps)). It is observed that the drop surface oscillates in sphere-prolate-sphere-oblate (SPSO) mode and seldom in the sphere-prolate-sphere (SPS) mode depending on the intricate interplay of various forces due to charge(q), the intensity of applied field ($\Lambda$) and shift of the droplet from the geometric center of the trap ($z_{shift}$). The Fast Fourier Transformation (FFT) analysis shows that the droplet oscillates with the forced frequency irrespective of the type of the applied waveform. While in the sinusoidal case, the nonlinearities are significant, in the square and ramp potentials, there is an admittance of all the harmonic frequencies of the applied potential. Interestingly, the breakup characteristics of a critically charged droplet is found to be unaffected by the type of the applied waveform. The experimental observations are validated with an analytical theory as well as with the Boundary Integral (BI) simulations in the potential flow limit and the results are found to be in a reasonable agreement.'
author:
- Mohit Singh
- Neha Gawande
- Rochish Thaokar
title: Influence of the trap potential waveform on surface oscillation and breakup of a levitated charged drop
---
Introduction
============
The understanding of how inherently charged droplets oscillate, deform and break is vital in many technological applications such as ink-jet printing [@choi2008drop], fuel injection [@shrimpton1999], crop spraying [@bailey1986] and electrospray atomization for ion mass spectroscopy of biomolecules [@fenn1989]. Also, oscillating droplets or bubbles have been recognized an innovative experimental platform for understanding a wide variety of natural, analytical and biological applications such as interfacial reactions and rheology, thin-film, biosensors, and biophysical simulations. The experimental techniques involving oscillating droplet/bubble have several advantages such as a high rate of surface reaction and more substantial interfacial activity due to high surface-area-to-volume ratio, reduced consumption of valuable chemicals, and a well-defined environmental control due to miniaturized droplet platform. The controlled oscillations of a surface can be recognized as a basis for many analytical studies. For example, the sinusoidal oscillations in the presence of constant amplitude and imposed frequency of an applied force can be used to measure the surface dilational modulus of a molecular (surfactants or macromolecules) monolayer by estimating the time-varying interfacial tension.
Numerous methods such as pendent drop[@yeung2006; @mashayek1998], sessile drops [@von2003; @ravera2010; @chang2015], acoustic levitation [@marston1980; @shen2010], electromagnetic levitation [@hill2012] and electrodynamic levitation [@singh2018surface] have been practised to investigate droplet oscillations. For understanding the oscillation response of a small-sized ($\sim$ 10-300$\mu$m) charged droplet, the electrodynamic levitation is considered to be one of the most efficient techniques. One of the potential reasons is that as the droplet size is small the gravity plays a negligible role in the steady-state droplet deformation and the droplet shape remains nearly symmetric throughout oscillations.
The capillary oscillations of an electrodynamically levitated charged droplet could be considered as an idealized system because the droplet gets levitated in a contact-free environment, thereby removing any solid boundary effect on the droplet oscillation characteristics. Thus, suitable modeling and development of analytical theory for capillary oscillations of a charged droplet in a quadrupole trap, together with relevant experiments, can make this system a reliable and accurate tool for characterizing the fluid properties such as surface tension and viscosity.
Towards this, many analytical studies [@feng1997; @hasse1975; @tsamopoulos1985] on the droplet oscillations have been reported. Very recently @singh2018surface reported theoretical analysis, in the small electro capillary limit, of surface oscillations of a charged liquid droplet levitated in an electrodynamic balance in the presence of sinusoidal applied waveform. The oscillations were ascertained to be governed by a potential flow description with viscous corrections. The effect of viscosity was exercised into consideration by including it in the normal stress balance condition of a potential flow limit. The analytical model was validated through numerous experiments involving the recording of surface oscillation dynamics at a very high frame rate of $\sim$ 100-130 thousand fps. An interesting question one can ask here is: what will be the effect of non-sinusoidal waveforms such as square and ramp on the characteristics of charged droplet oscillations.
The droplet levitation using sinusoidal applied electric potential is reasonably well-established. The mechanism at play is the same as that in the classical Mathieu equation (see ref. [@singh2017levitation]) to describe dynamics under oscillatory forcing. A time average net ponderomotive force acting towards the center of the trap overcomes destabilizing forces such as gravity and charge repulsion (in case of two drops). It is then worthwhile to explore the robustness of the mechanism, by subjecting it to non-sinusoidal applied potentials. It is known that sinusoidal potentials of frequency $\omega$ can lead to responses in the deformation of the droplet in $\omega$ or $2 \omega$, which can be attributed to the quadrupole potential acting on the net charge or the square of the Maxwell stress tensor of the applied potential, respectively. From the requisite analysis of different waveforms, it is also known that non-sinusoidal potentials such as square and ramp consist of individual harmonics (odd, even or integer order multiplication of fundamental frequency) of the applied frequency. Thus it is interesting to see the impressions of these harmonics in the drop deformation pattern and establish the deformation response to arbitrary potentials. Undertaking a systematic investigation of this issue forms another motivation for the present study.
In the present manuscript, we have explored the surface oscillation characteristics in the presence of different forcing waveforms such as sine, square, and ramp. Unlike @singh2018surface, the surface oscillation characteristics are examined by the Fast Fourier transform (FFT) analysis of surface dynamics. The experimental observations are also compared with the asymptotic theory and boundary integral (BI) simulation in the potential flow limit. After a successful analysis of surface oscillations of a sub-Rayleigh charged (above Rayleigh charge the droplet surface becomes unstable even for small infinitesimal perturbations, see ref [@duft03]) drop in the presence of non-sinusoidal applied potential we allowed the droplet to evaporate further, thereby, droplet attains Rayleigh critical charge and breaks via ejecting a thick jet. The breakup of a charged droplet is captured in the presence of sine, square and ramp applied waveforms.
Materials and method
====================
The experiments presented in this work involve the levitation of ethylene glycol (EG) charged droplets. A small amount of NaCl was added to increase the electrical conductivity ($\sigma$) of the droplet, and the conductivity was measured using a conductivity meter (Hanna instruments). The value of $\sigma$ was obtained in the range of $\sim$ 50-80 $\mu$S/$cm$. The viscosity ($\mu_i$) of EG was measured using Ostwald’s viscometer, and the corresponding measured value was 0.016 Ns/$m^2$. The surface tension ($\gamma$) of the EG droplet was measured using the pendant drop (DIGIDROP, model DS) method and confirmed with the spinning drop (dataphysics, SVT 20) apparatus. The value of $\gamma$ was obtained as 47 mN/m. The experiments were carried out at normal atmospheric conditions (1 atm pressure and $25^0$ C temperatures).
![Detailed schematic of setup used for droplet charging and corresponding levitation of charged droplet.[]{data-label="fig:setup"}](setup_new.jpg){width="\linewidth"}
The experimental set-up consists of two end-cap electrodes (separation distance, $2z_0$= 12mm) and a ring electrode (diameter, $2r_0$= 12mm) to realise a quadrupole trap such that the electrical potential is given as $\phi=\Lambda(r^2-2z^2)$, where, $\Lambda$ (=$\frac{\phi_0}{r_0^2+2z_0^2}$) is the intensity of quadrupole field, $\phi_0$(=11 $kV_{pp}$) is the applied potential, $r$ and $z$ are the radial and axial directions respectively. A detailed schematic of the experimental setup used for droplet charging and levitation is shown in figure \[fig:setup\]. A function generator (33220A Function /Arbitrary Waveform Generator, 20 MHz), used to apply the potential of the desired waveform, was connected to a high-voltage amplifier source (Trek, model 5/80, high-voltage power amplifier). The peak to peak AC potential applied in our experiments was 11$kV_{pp}$ with frequency varying from 100 Hz to 500 Hz. The voltage was kept highest and constant to ensure the high center of mass stability of a levitated charged droplet. In a typical experiment, charged EG droplets were generated by an electrospray (in dripping mode) setup, realized by applying high positive DC potential on the tip of a stainless steel needle. A charged droplet was further injected between the endcap and ring electrodes and was suspended using the quadrupolar AC electric field. The field was applied between the end caps and the ring electrodes of the trap where the end caps were connected to the live power supply, and the ring was kept grounded, as shown in figure \[fig:setup\]. The levitated single droplet was observed using a high-speed CMOS camera (Phantom V 12, Vision Research, USA), which was connected with a stereo zoom microscope (SMZ1000, Nikon Instruments Inc.), see figure \[fig:setup\]. The camera can record up to 180 thousand fps at 128$\times$128 resolution with 2s recording time and was kept inclined at $30^0$-$40^0$ for visualization of the phenomenon. Nikon halogen light (150 W) was used as a light source to illuminate the levitated droplet. The maximum frame rate which we can achieve with the camera at 128$\times$128 resolution is 130-150 hundred thousand fps, indicating a 130-150 kHz of frequency. The frequencies discussed in this work are of the order 100-500 Hz. Therefore, it suffices to say that the analysis is carried out for the frequencies lower than the frequency cut-off of the camera speed.
Results and discussion
======================
Surface oscillations of a charged drop:
---------------------------------------
In a typical levitation experiment, the gravitational force associated with the mass of the drop is balanced by imposing an additional DC bias voltage superimposed on the AC voltage (see ref. @duft03 [@duft02]). It should be noted that in the present experiments, only the AC field is applied for levitation of the charged droplet. However, if one levitated the droplet at the center of the trap by applying a DC bias voltage, the presence of an applied field will have a negligible effect on the droplet deformation. Since in most practical situations, such as electrospray, the droplet experiences an asymmetric electric field and non-zero gravity, the scenario is easily simulated in a more controlled way in quadrupole trap, if we do not have any additional DC voltage. It was observed that in the presence of gravity, the droplet gets levitated at an off-centered location in the downward z-direction (at a distance $z_{shift}$, as shown in figure \[fig:setup\]) irrespective of the type of the applied waveform. An off-centered droplet experiences a local uniform field ($E$=$4\Lambda z_{shift}$) along with quadrupole field ($\Lambda$), and this local uniform field causes asymmetry of force on the droplet surface in the upward and downward directions due to finite body effect, i.e., $E \sim (z_{shift}\pm R)$, where, $R$ is the droplet radius. Thus, the presence of $z_{shift}$ modifies the stress distribution over the surface of the droplet and affects the oscillatory response.
![[]{data-label="fig:spso_sps"}](spso_sps){width="0.8\linewidth"}
It was observed in some of our experiments that the droplet surface oscillates in the sphere-prolate-sphere-oblate (SPSO) mode. However, the magnitude of the prolate deformation is observed to be higher than that of the oblate deformation. This experimental observation is different from the experiments of @duft02 wherein they observed symmetric SPSO mode and reported an equal magnitude of the prolate and oblate deformations in each oscillation cycle.A schematic of symmetric SPSO mode of droplet oscillations is shown in figure \[fig:spso\_sps\]a. In some experiments, moreover, it was observed that the droplet exhibited a sphere-prolate-sphere (SPS) mode of oscillations, as shown in figure \[fig:spso\_sps\]b. Such oscillation pattern can be attributed to a complex interplay between the electrical parameters such as charge ($q$) on the droplet, intensity applied quadrupole field ($\Lambda$), local uniform field ($E$) as well as fluid properties such as viscosity, surface tension and density of the levitated droplet, and is discussed later.
Deformation under sinusoidal and other time periodic waveform
-------------------------------------------------------------
When a charged droplet is levitated using a quadrupole AC electric field with sine waveform as an applied signal, the droplet surface oscillates with the applied frequency. However, the surface oscillations can admit harmonic frequencies along with the fundamental applied frequency due to nonlinear interaction of several terms such as $q$, $\Lambda$ and $E$. In order to characterize experimental droplet surface oscillations, the high-speed video is processed using software ImageJ where the surface dynamics are obtained by tracking the variation in the outline of the droplet surface in each frame of the video. The accuracy of the analysis and noise in the data depends upon the quality of the images in each frame. It is reported by @singh2018surface that the voltage applied for levitation affects the amplitude of droplet surface oscillations. Thus, at a lower voltage, the amplitude of fundamental and harmonic frequencies becomes small, and thereby lower amplitude harmonics become undetectable. Hence, in the present experiments, the droplet is levitated at a constant and maximum AC voltage, i.e., 11$kV_{pp}$, and different types of waveforms are applied between the electrodes. Since droplet levitation is a result of dynamic stability and all the parameters are interdependent, we cannot apply a very high voltage or frequency. The experimental or theoretical analysis presented here is performed within the stability limit of the center of mass motion. The oscillation characteristics are examined by evaluating the rate of change of the Taylor deformation parameter ($DD$=$\frac{L-B}{L+B}$, where, $L$ is the major axis, and $B$ is the minor axis. The magnitude of $L$ and $B$ were obtained by tracking the boundary of the droplet using software ImageJ). The fundamental frequency and other harmonics are identified by performing the FFT analysis using the software OriginLab. The software automatically chooses the correct sample interval of $\vartriangle$$t$ from the data. A triangular type of window is used to suppress frequency leakage, and the mean square amplitude method is used for power density normalization. The FFT analysis gives the amplitude of all kinds of frequencies present in the data. Since the high-speed video recording of $\sim$ 100-200 $\mu$m droplet diameter is performed at a low resolution, the blurriness in the image gives uncertainty in the exact drop size measurement. Additionally, the video was further processed using ImageJ software, where image greyscale and blur thresholding cause additional uncertainty in the measurement of various droplet dimensions such as major/minor axis and centroid calculations. The $DD$ data contains around 10-15% average uncertainty. In the FFT analysis, the uncertainty in the DD vs time data can cause a slight reduction in the corresponding peak intensity while order and occurrence of the peaks are expected to remain the same. Hence in the present manuscript, the FFT analysis is found to be an accurate tool for characterization of experimental droplet surface oscillations and its comparison with theoretical/numerical results.
@singh2018surface reported that a droplet surface oscillates only with the fundamental frequency that is the applied frequency of the electric potential. However, it is apparent from the experimental data of DD vs time (figure 7 of @singh2018surface) as well as from figure \[fig:sine\_exp\_fit\] of the present work that even with a perfect harmonic drive the droplet oscillation generates higher harmonics that are visible as a clear slanted deviation of the deformation signal from the fitted harmonic. A sine waveform of equal magnitude and frequency is fitted to the DD vs time data (see figure \[fig:sine\_exp\_fit\]) using the software OriginLab for better depiction of the slanted nature of the curve. The detailed discussion of various factors responsible for slanted nature thereby harmonic oscillations is given the preceding section. Thus, to identify the harmonic frequencies, at first, a sinusoidal signal was applied between the endcap electrodes and a ring electrode to levitate a moderately charged droplet. Since at high value of applied frequency but at a fixed charge and applied voltage, the drop exhibits poor center of mass stability, thereby restricting the use of very high applied frequency close to the natural frequency ($O(2 kHz)$) of the system. Hence, the droplet can be stably levitated for a specific range of applied potential ($O(10 kV_{pp})$) and frequency ($O(200 Hz)$) that is determined by the levitation dynamics and critical frequency thereof. Thus, the first analysis is performed for a typical value of applied AC voltage of 11 k$V_{pp}$, and the corresponding applied frequency of 255 Hz. The data of the applied sinusoidal signal was acquired with the help of an oscilloscope (Tektronix) and is plotted in figure \[fig:applied\_sin\].
[0.3]{} {width="\linewidth"}
[0.32]{} {width="\linewidth"}
[0.32]{} {width="\linewidth"}
[0.32]{} {width="\linewidth"}
[0.33]{} {width="\linewidth"}
[0.32]{} {width="\linewidth"}
[0.31]{} {width="\linewidth"}
[0.3]{} {width="\linewidth"}
The corresponding surface oscillations of a charged droplet is plotted in terms of DD vs time, as shown in figure \[fig:sine\_exp\_fit\]. It can be observed from figure \[fig:sine\_exp\_fit\] that the primary oscillation frequency of the droplet surface is the fundamental applied frequency, i.e., 225 Hz. However, an unmistakable noise at the peaks of the deformation cycle and slanting in the bottom curve can also be observed. This is an indication of the presence of higher harmonic frequencies. Hence, to confirm the presence of harmonic frequencies, an FFT analysis is performed for both applied waveform and the surface oscillation data. The FFT analysis of the applied signal is shown in figure \[fig:Applied\_sin\_fft\], and it can be observed that there exists a single peak at 255Hz which is the fundamental frequency of the applied signal. The FFT of surface oscillation on the other hand (shown in figure \[fig:exp\_sin\]) shows the existence of higher harmonics. Along with the fundamental frequency $f_1$=255Hz, the second harmonic frequency ($f_2$) at 2$f_1$ and the third harmonic frequency ($f_3$) at 3$f_1$ are clearly seen with diminishing magnitude. We hypothesized here that, while the fundamental ($f_1$) and second harmonics (2$f_1$) frequencies can be attributed to the effect of $\Lambda$ on $q$ and the $\Lambda^2$ term due to quadrupolar dependence of Maxwell stress on electric potential, the third harmonic could be due to the quadrupole field ($\sim$ $f_1$) acting on the charge induced on the deformed sphere ($\sim$ 2$f_1$) due to the quadrupole field itself. To re-confirm the presence of second and third harmonic frequencies, the droplet is levitated at a different frequency, i.e., at 150 Hz, and the FFT analysis is carried out for both applied signal and the surface oscillation data as shown in figures \[fig:applied\_sin\_new\], \[fig:exp\_sin\_new\]. It can be observed from figures that while the FFT of the applied signal has only fundamental frequency (150 Hz) peak, as shown in figure \[fig:applied\_sin\_new\], the deformation admits harmonic frequencies, as shown in figure \[fig:exp\_sin\_new\]. Additionally, the peaks in figure \[fig:exp\_sin\_new\] seem to be much sharper than that of peaks observed in figure \[fig:exp\_sin\]. This is due to the better quality of the video thereby less noise in the DD vs time data. Also, it is believed that the non-linearities in the droplet oscillations due to large deformation can lead to the admittance of the $3^{rd}$ harmonic frequency. To confirm the results obtained at a lower applied frequency, i.e., 150 Hz, the droplet is levitated at a still lower frequency, i.e., 115 Hz and corresponding FFT analysis of both applied signal and DD vs time data are shown in figure \[fig:applied\_sin\_new115\], \[fig:exp\_sin\_new115\]. At a very low frequency, for a fixed value of charge and applied voltage, the droplet becomes unstable via spring oscillations of center of mass motion (see ref. @singh2017levitation), whereby the freuency cannot be lowered any further. It can be observed that the FFT analysis of the applied signal, as shown in figure \[fig:applied\_sin\_new115\], shows the presence of a single peak, which corresponds to the applied frequency. On the other hand, the FFT analysis of droplet surface oscillation shows the presence of several harmonic frequencies along with fundamental frequency, i.e., 115 Hz. Its higher magnitude at 150 Hz as compared to that at 225 Hz is in agreement with the large deformation seen at the lower frequency.
[0.38]{} {width="\linewidth"}
[0.38]{} {width="\linewidth"}
[0.42]{} {width="\linewidth"}
[0.38]{} {width="\linewidth"}
In the previous section, we have seen the levitation of a charged droplet in the presence of a sine waveform. Therefore, two types of non-sinusoidal waveforms, namely square and ramp waveform, are applied for examining the controlled droplet surface oscillation characteristics. Firstly, to examine the case of the square waveform, a charged droplet is levitated at 11 k$V_{pp}$ applied potential and 220 Hz imposed frequency. The characteristics of the applied square wave signal, obtained from the oscilloscope, are shown in figure \[fig:applied\_square\]. The corresponding droplet surface oscillations, obtained from the image processing of high-speed video, are plotted in terms of DD vs time and shown in figure \[fig:square\_exp\]. The apparent noise observed at the peaks of the applied signal (figure \[fig:applied\_square\]) as well as in the DD vs time plot (figure \[fig:square\_exp\]) are a consequence of higher odd-harmonics typically associated with square waveforms [@weisstein2004fourier]. Additionally, the noise in the deformation data is a consequence of the noise in the applied signal as well as the noise associated with image processing of the high-speed video. For the preliminary confirmation of the surface oscillations in the presence of a square waveform, a square wave function of the same amplitude and frequency is fitted to the DD vs time data using the software OriginLab. It can be observed from figure \[fig:square\_exp\] that the surface oscillation characteristics of a droplet follow the applied waveform, and the droplet oscillates with fundamental applied frequency. Further confirmation of droplet oscillation behavior is done by performing the FFT analysis of the applied waveform and the surface oscillation data. The FFT analysis of the applied signal is shown in figure \[fig:applied\_square\_fft\], and it can be observed that the applied signal has several harmonic frequencies in odd multiples of the fundamental frequency ($f_1$=220Hz). This response is expected because of the characteristic behavior of the square waveform. The magnitude of the fundamental frequency is observed to be high as compared to the harmonic frequencies. The corresponding FFT analysis of surface oscillation behavior is shown in figure \[fig:exp\_square\], and it can be observed that the FFT analysis of the surface oscillation behavior is similar to that of the applied signal. The order of occurrence of the harmonic frequencies is similar to the applied signal. Unlike the sinusoidal waveform, the droplet surface oscillates with applied frequency in the presence of square waveform, and the $2^{nd}$ harmonic is also significantly marked. The dominant $3^{rd}$ and $4^{th}$ harmonic in surface oscillations here is actually the response of the droplet to the presence of $3^{rd}$ and $4^{th}$ harmonic in the applied voltage.
[0.38]{} {width="\linewidth"}
[0.4]{} {width="\linewidth"}
[0.4]{} {width="\linewidth"}
[0.4]{} {width="\linewidth"}
Similar to the square waveform, another non-sinusoidal waveform, a ramp waveform, is applied to levitate a charged droplet in an ED balance. The voltage applied across the ring, and the end cap electrode is 11 k$V_{pp}$ at 205 Hz applied frequency. The rate of change of applied voltage with time for ramp waveform is shown in figure \[fig:apllied\_ramp\]. The surface oscillations in the form of DD vs time are shown in figure \[fig:triangular\_exp\]. Unlike the case of sine and square waveforms, it can be observed that the droplet oscillates in the SPS mode of oscillations. To confirm that the oscillations follow the applied waveform a ramp waveform of equal magnitude and frequency is fitted onto the DD vs time data using the software OriginLab and a good fit is observed, as shown in figure \[fig:triangular\_exp\]. Figure \[fig:applied\_ramp\_fft\] shows the FFT analysis of the applied ramp waveform, and it can be observed that there exist integer harmonic frequencies in the applied signal. This is because the ramp waveform is theoretically an infinite series of sine integer harmonic frequencies [@weisstein1999fourier]. Although the magnitude of the fundamental frequency is higher than that of harmonics frequencies, all harmonic frequencies do show a significant contribution to the ramp signal. This is evident in figure \[fig:exp\_ramp\], where the FFT of the deformation shows a significant presence of the fundamental and all higher harmonics. These results conclusively prove that an unmistakable signature of the applied waveform is seen on the deformation dynamics.
Theoretical model:
------------------
The surface oscillation dynamics of a charged droplet levitated in an ED balance is governed by both electrostatics and hydrodynamics. Recently, @singh2018surface conducted a linear stability analysis for surface oscillations of a levitated charged droplet in the presence of a sinusoidal applied potential. In the present work, we extended the theoretical analysis for non-sinusoidal waveforms, analyzed the non-linearity within the system, and compared the results with the experimental observations. The details of the model are omitted here, only boundary conditions and governing equations for the surface dynamics are shown for self-consistency of the manuscript. The value of $\sigma$ is high (>30 $\mu$S/cm), and the surrounding medium (air) is a perfect dielectric medium while the droplet is considered as a perfect conductor drop. Thus, the non-dimensional equations and the boundary conditions governing the electrostatics of the droplet surface dynamics are:
$${\bf{\nabla}}^2 \phi =0, \quad 0 \leq r \leq r_s(\theta,t)
\label{Epot}$$
$$\phi\rightarrow \phi_\infty, \quad r\rightarrow \infty$$
$${\bf{n}}\cdot {\bf{\nabla}} \phi= -q(\theta,t), \quad r=r_s(\theta,t)
\label{norm_E}$$
$${\bf{t}}\cdot {\bf{\nabla}} \phi= 0, \quad r=r_s(\theta,t)
\label{eqn:tang_E}$$
where, the droplet surface is defined with a small amplitude of perturbation around a spherical shape ($r_s(\theta,t)$) as,
$$r_s(\theta,t)=[1+\sum_{l}\delta \thinspace\alpha_{l}(t)\thinspace P_{l}\cos(\theta)]$$
where, $\delta$ is a small perturbation parameter, $P_{l}$ is the $l^{th}$ Legendre mode and $\alpha$ is the amplitude of corresponding Legendre mode as well as measures the oscillation amplitude. The $l$=1 mode contributes to the center of mass translation of the drop and the other modes such as $l$=2, 3, 4 indicate the deformation of the drop shape from a sphere, e.g., $l$=2 gives dipolar shape deformation (dumbbell shape), $l$=3 gives asymmetric shape deformation (pear shape), $l$=4 gives quadrupolar shape deformations.
The typical range of Ohnesorge number ($Oh$=$\frac{\mu_i}{\sqrt{\gamma \frac{D_d}{2}\rho_i }}$), which relates the viscous and surface tension forces, for our experimental parameters is about 0.15-0.35. and is governed by the following non-dimensional equations:
$$\bnabla \cdot {\bf{v}}_{i,e}=0$$
$$\frac{\beta}{Oh^2}
\Biggr{(}\frac{\partial v_{i,e}}{\partial t}+ v_{i,e}\cdot \bnabla v_{i,e}\Biggr{)}=-\bnabla p_{i,e}+\lambda_{i,e}\bnabla^2 v_{i,e}$$
In the potential flow limit ($Oh\ll1$), the dynamics of the irrotational and incompressible fluid having velocity potential $\psi$ and the drop shape $r_s(\theta,t)$ is given by the following non dimensional equations, $${\bf{\nabla}}^2 \psi_i =0, \quad 0 \leq r \leq r_s(\theta,t)
\label{velpot}$$ $${\bf{\nabla}}^2 \psi_e =0, \quad r \geq r_s(\theta,t)$$ and the pressure is given by the Bernoulli equation, $$\triangle p+\Biggr{(}\frac{\partial \psi_i}{\partial t}+\frac{1}{2}\Biggr{[}{\frac{\partial \psi_i}{\partial r}}^2+{\frac{1}{r}\frac{\partial \psi_i}{\partial \theta}}^2\Biggr{]}\Biggr{)}-
\beta\Biggr{(}\frac{\partial \psi_e}{\partial t}+\frac{1}{2}\Biggr{[}{\frac{\partial \psi_e}{\partial r}}^2+{\frac{1}{r}\frac{\partial \psi_e}{\partial \theta}}^2\Biggr{]}\Biggr{)}=0
\label{bernoulli}$$ The boundary conditions are normal velocity continuity, $$\frac{\partial \psi_i}{\partial r}=\frac{\partial\psi_e}{\partial r},
\label{vel_conti}$$ and the normal stress balance, $$\begin{gathered}
\triangle p-2( Oh \frac{\partial v_{r_i}}{\partial r}+\lambda_e Oh \frac{\partial v_{r_e}}{\partial r})\\=\kappa-\frac{1}{32} (8 X+12 \sqrt{\text{Ca}}\thinspace \zeta \cos (\theta )+5 \sqrt{\text{Ca}_Q}\thinspace \zeta (3 \cos (2 \theta )+1))^2,
\label{stress_bal}\end{gathered}$$ at the interface $r=r_s(\theta,t)$. The equation \[stress\_bal\] is the non dimensional stress balance equation and $\kappa$ is the surface curvature and defined as: $$\kappa=2+\delta\sum_{l=2,3,4}(l(l+1)-2)\alpha_l P_l(\cos \theta)$$
The length is scaled by unperturbed droplet radius $R$(=$D_d/2$), velocity by $\sqrt{(\gamma/\rho R)}$, time by $\sqrt{(\rho R^3/\gamma)}$, $\zeta$ is a time-periodic function which depends on the type of applied waveform, e.g., $\zeta$=$\cos\omega t$ for the sinusoidal applied waveform. $X$ is the fissility, which is the ratio of the actual charge on the droplet to its Rayleigh charge ($q_R$=$\sqrt{(64 \pi^2 \epsilon_e R^3 \gamma)}$), where $\epsilon_e$ is the electrical permittivity of the surrounding fluid. $Ca_Q$=$(R^3 \epsilon_e \Lambda_{0}^{2}/\gamma)$ is the capillary number based on the strength of the applied quadrupole field and $Ca$=$(R\epsilon_e E^2/\gamma)$ is the electrical capillary number based on the uniform field, where $E$=$4\Lambda z_{shift}$. $$\alpha_1''(t)+\frac{1}{2 \beta +1}\Bigr{(}12 \text{Oh}\thinspace \alpha_1'(t)-12 (\sqrt{\text{Ca}} \sqrt{\text{Ca}_Q}\thinspace \zeta^2+X \sqrt{\text{Ca}}\thinspace \zeta)\Bigr{)}=0 \label{eqn:pot_diff1}$$ $$\alpha_2''(t)+\frac{6}{3 \beta +2} \Big{(}2 (\lambda +4) \text{Oh} \thinspace \alpha_2'(t)+4 \alpha_2(t)\\-3 \text{Ca}\thinspace \zeta^2-10 X \sqrt{\text{Ca}_Q}\thinspace \zeta-\frac{25 \text{Ca}_Q\thinspace \zeta^2}{7}\big{)}=0 \label{eqn:pot_diff2}$$ $$\alpha_3''(t)+\frac{1}{4 \beta +3}\Bigr{(}24 (2 \lambda +5) \text{Oh}\thinspace \alpha_3'(t)\\+120 \alpha_3(t)-108 \sqrt{\text{Ca}} \sqrt{\text{Ca}_Q}\thinspace \zeta^2\Bigr{)}=0 \label{eqn:pot_diff3}$$ $$\alpha_4''(t)+\frac{1}{5 \beta +4}\Bigr{(}840 (\lambda +2) \text{Oh} \thinspace\alpha_4'(t)\\+2520 \alpha_4(t)-900 \text{Ca}_Q\thinspace \zeta^2\Bigr{)}=0 \label{eqn:pot_diff4}$$ Equation \[eqn:pot\_diff1\] is based on the $P_1$ Legendre mode and primarily contributes in the translational motion of the droplet. The equation \[eqn:pot\_diff2\] is based on the $P_2$ Legendre mode and similarly, equation \[eqn:pot\_diff3\] and \[eqn:pot\_diff4\] are obtained as coefficients of the $P_3$ and $P_4$ Legendre modes. This infers that equations \[eqn:pot\_diff2\], \[eqn:pot\_diff3\], \[eqn:pot\_diff4\] govern the surface oscillation dynamics of the drop. From equation \[eqn:pot\_diff2\], it can be observed that for a highly charged drop, the value of X$\sqrt{Ca_Q}$ term dominates over other terms, and the droplet oscillates with the fundamental frequency. On the other hand, for mildly or uncharged droplets levitated at the center of the trap, the term $Ca_Q\zeta^2$ dominates over other terms, and the droplet oscillates with twice the frequency of fundamental applied frequency. Hence, a lower oblate deformation as compared to the prolate deformation during the negative and positive cycle of endcap potential in the SPSO mode of surface oscillation of a positively charged drop levitated in a quadrupole field depends upon the relative magnitude of $Ca\zeta^2$, X$\sqrt{Ca_Q}\zeta$, $Ca_Q\zeta^2$. The equation \[eqn:pot\_diff3\] shows that at a very high value of $z_{shift}$, the droplet shape can become highly asymmetric. The equation \[eqn:pot\_diff4\] shows that at high value of the applied field, the droplet oscillates with twice the imposed frequency. The relative magnitudes of the shape coefficients such as $\alpha_2$, $\alpha_3$, $\alpha_4$ are obtained by solving equations \[eqn:pot\_diff2\], \[eqn:pot\_diff3\] and \[eqn:pot\_diff4\] and these equations, finally, governs the characteristics of the droplet surface oscillations. For our experimental parameters, the magnitude of $\alpha_2$ found to be dominating over $\alpha_3$ and $\alpha_4$. Thus, the dynamics of a charged droplet oscillation in the presence of sinusoidal and non-sinusoidal waveform is explained using equation \[eqn:pot\_diff2\]. In the presence of gravitational force, it is observed that all the four differential equations (equations \[eqn:pot\_diff1\]-\[eqn:pot\_diff4\]) get coupled at higher order. Thus, the surface dynamics of the droplet which is governed by equations \[eqn:pot\_diff2\], \[eqn:pot\_diff3\] and \[eqn:pot\_diff4\] is significantly affected by the equation \[eqn:pot\_diff1\] which gives the COM motion of the droplet. This indicates that the shape deformation dynamics in the presence of gravity is the result of nonlinear interaction of the gravitational force with the electro-capillary forces, which cannot be explained using linear order theory. Thus to capture these nonlinear effects, the problem is also solved using the boundary integral method in the potential flow limit.
Boundary integral equations
---------------------------
As the flow is assumed to be irrotational inside the drop, the velocity is given by the gradient of the velocity potential $\tilde{\v}=\tilde{\bnabla}\tilde{\psi}$ such that velocity potential follows the Laplace equation and the pressure fields are related to flow fields through unsteady Bernoulli equation as follow: $$\begin{aligned}
%\tilde{\bnabla}^2 \tilde{\psi}&=0 \label{eq:pot_lap}\\
\frac{\del \tilde{\psi}}{\del \tilde{t}}+\frac{1}{2}(\tilde{\bnabla} \psi \cdot \tilde{\bnabla} \tilde{\psi})+\frac{\tilde{p}}{\rho_i}&=0 \label{eq:pot_bernoulli}\end{aligned}$$ The pressure in the equation \[eq:pot\_bernoulli\] can be obtained from the normal stress jump across the surface of the drop as shown below, $$\tilde{p}=\tilde{p}^{ext}+\gamma \tilde{\bnabla}_s \cdot \n$$ where $\n$ is the outward unit normal and $\bnabla_s \cdot \n$ gives the curvature of the drop denoted by $\kappa$. For electrified drops under the action of gravitational force, $$\tilde{p}^{ext}=\rho_i \thinspace g \thinspace \tilde{z}+\frac{1}{2} \epsilon_e \tilde{E_n}^2
\label{eqn:pressure}$$ where, first term is the force acting on the drop due to gravity and the second term is the electric stress acting on the surface of the drop.
Note that the dimensional quantities are indicated by tilde and non-dimensional quantities are without tilde. Here the time is non-dimensionalized by the characteristic inertial timescale, $\sqrt{\rho_i R^3/\gamma}$, the pressure by $\gamma/R$ and the total surface charge is non-dimensionalized by $\sqrt{\gamma R^3 \epsilon_e}$ such that the non-dimensional Rayleigh charge is $8\pi$. Thus, the non-dimensional governing equations for the flow field inside the drop are given by, $$\begin{aligned}
\bnabla^2\psi&=0,\\
\frac{\partial \psi}{\partial t}+\frac{1}{2}\v \cdot \v&=-\kappa-Bo \thinspace z+\frac{1}{2}E_n^2\end{aligned}$$ where, $Bo=\rho_i g R^2/\gamma$ is the gravitational bond number. Here, only the hydrodynamic integral equations in the potential flow limit are discussed. According to classical potential flow theory the velocity potential can be expressed as a surface distribution of dipole density per unit area $\mu$. This is also known as double layer potential representation and is given as, $$\psi({\xo})=\int \mu(\x) \n \cdot \bnabla \G dA$$ The potential is discontinuous across the surface of the drop.When $\x$ approaches the surface from inside the drop, $$\psi_1({\xo})=\frac{1}{2}\mu ({\xo})+\int_{PV} \mu (\x) \n \cdot \bnabla \G dA,
\label{eqn:psi1}$$ and when $\x$ approaces the surface from outside the drop, $$\psi_2({\xo})=-\frac{1}{2}\mu ({\xo})+\int_{PV} \mu (\x) \n \cdot \bnabla \G dA,
\label{eqn:psi2}$$ such that the dipole density $\mu=\psi_1-\psi_2$. The normal component of the velocity is continuous across the drop surface while tangential component of the velocity is discontinuous. Thus the surface distribution of dipoles is equivalent to a vortex sheet and the circulation density $\gamma_s$ can be written as, $$\gamma_s=-\n \times \bnabla_s \mu$$
As introduced by Lundgren and Monsour (1988), a vector potential $\mathcal{A}$ is related to the velocity by, $$\v=\bnabla\times \mathcal{A}$$ such that, $$\begin{aligned}
\mathcal{A}&=-\int\gamma_s \G dA\\
\mathcal{A}&=-\int_{PV} \mu(\x) \n \times \bnabla_s \G dA
\label{eqn:vectorPot}\end{aligned}$$ Eq. \[eqn:vectorPot\] is used to calculate the vector potential $\mathcal{A}$ when the point $\x$ is precisely on the drop surface. Thus the normal component of the velocity is then calculated using, $$\v \cdot \n=(\n \times \bnabla) \cdot \mathcal{A}$$ For axisymmetric problem ($\n \times \bnabla$) operator has surface values only, thus the normal component of the velocity can be obtained once the vector potential $\mathcal{A}$ is computed. For initially known scalar potential on the drop surface the tangential velocity is obtained by calculating the surface gradient of $\psi$. Thus the tangential and normal component of velocity are given by, $$\begin{aligned}
v_n&=\frac{1}{r}\frac{\del (r \mathcal{A}_\theta)}{\del s}\\
v_t&=\frac{\del \psi}{\del s}\end{aligned}$$ Since the velocity of the drop surface is computed, the surface can be evolved in time using kinematic condition, $$\frac{d \x}{d t}=\v$$ and the surface values of $\psi$ at next time iteration are obtained by using unsteady Bernoulli equation in terms of the material derivative. $$\frac{D \psi}{Dt}=\frac{1}{2} \v \cdot \v-\kappa-Bo \thinspace z+\frac{1}{2}E_n^2
\label{eqn:bernoulli}$$
The other details of the integral method, regularization of the kernel in equations \[eqn:psi1\], \[eqn:psi2\] and \[eqn:vectorPot\] and their numerical implementation can be found in the supplementary file.
![[]{data-label="sin_thoery"}](cos_fft){width="0.7\linewidth"}
![[]{data-label="square_thoery_exp"}](sqr_fft){width="0.7\linewidth"}
Validation of experimental results with theory
==============================================
Since the detailed experimental validation of surface oscillations in terms of the degree of deformation with time for a sine waveform was reported in one of our recent publications, i.e., ref. [@singh2018surface]. In the present study, we have experimentally, theoretically, and numerically validated the frequency response of surface dynamics in terms of FFT analysis of surface oscillations, as shown in figure \[sin\_thoery\]. In order to obtain the theoretical surface oscillation characteristics for a sinusoidal applied waveform, equation \[eqn:pot\_diff2\], \[eqn:pot\_diff3\] and \[eqn:pot\_diff4\] are solved for the experimental parameters. Since the oscillations are recorded before the droplet breakup, the charge is considered as sub-Rayleigh ($X$=0.9), and the $z_{shift}$ of the droplet is kept as a fitting parameter. The $DD$ value from the theory is obtained by using the following expression, $$D=\frac{3\alpha_2}{4}+\frac{\alpha_3}{2}+\frac{5\alpha_4}{16}$$ Similar to the experimental analysis, the embedded harmonics are resolved by performing the FFT analysis of theoretically obtained surface oscillation data with the help of the software OriginLab. It can be observed from figure \[sin\_thoery\] that there exist harmonic frequencies (in experiments) for fundamental applied frequency ($f_1$=255 Hz). The first two harmonics are expected because of the significant magnitude of the coefficient of $\zeta^2$=$\cos^2(\omega t)$ due to off-center position of the droplet, in addition to $\zeta$=$\cos(\omega t)$ due to charge on the droplet (term X$\sqrt{Ca_Q}$ in equation \[eqn:pot\_diff2\]). The resultant droplet oscillation dynamics contains both the frequencies. Similar observations are made in the theoretical oscillations, as shown in figure \[sin\_thoery\]. However, it can be also observed in the figure \[sin\_thoery\] that the experimental FFT exhibit 3$f_1$ and $4f_1$. The presence of these higher-order harmonic frequencies suggests non-linear interaction between the charges on the deformed droplet and the quadrupole field acting on these charges. Since the theory presented here is based on linear order approximation, it captures only the lower order frequencies i.e., $\omega$ and $2\omega$. The BEM simulations carried out in the potential flow limit capture these non-linear interactions and show the presence of higher harmonic frequencies along with the fundamental $f$ and $2f$ frequencies. The FFT analysis of the oscillations obtained from BEM simulations also indicates a peak at a frequency equal to $10f$. This peak corresponds to the natural inertial oscillations of the drop surface, which are not present in both the experiments and the theory as these natural oscillations are damped out by the viscous forces.
The analysis is further extended to surface dynamics for square and ramp waveforms. In the case of a square waveform, the time-varying function $\zeta$ can be given as: $$\zeta=2 \tan^{-1}\frac{[\frac{2 \pi ft}{\delta}]}{\pi},$$ Thus, if the value of $\delta$ is high the curve will be smooth at the corners. Figure \[square\_thoery\_exp\] shows the FFT analysis of theoretical and experimental surface oscillations. It can be observed that fundamental frequency and harmonic frequencies occur at the same order. Along with harmonic frequencies, other inter-harmonic frequencies can be observed in figure \[square\_thoery\_exp\]. It can be observed that, in the case of the square waveform, the oscillation pattern exhibits fundamental frequency ($f=230Hz$) and higher-order harmonic frequencies in terms of odd multipliers of the fundamental frequency i.e., $3f$, $5f$, $7f$, etc. The FFT analysis of the experimental observation shows peaks at $f$, $3f$, and $5f$, while the peaks corresponding to higher frequencies are insignificant. However, in both linear order theory and BEM simulations, the peaks at $7f$, $9f$, and $11f$ are clearly visible. Along with harmonic frequencies, other inter-harmonic frequencies can also be observed in figure \[square\_thoery\_exp\]. The presence of inter-harmonic frequencies shows the complex coupling between various terms discussed in the previous section. Additionally, it is also observed that if the magnitude of $Ca_\Lambda$ and $Ca_E$ terms is higher, the FFT of theoretical surface oscillations contains even multipliers (i.e., $2f_1$, $4f_1$, $6f_1$, etc.) of the fundamental frequencies. The results of BEM simulations show higher amplitudes of the inter-harmonic frequencies near the peak of $9f$. As we have seen in the case of the sinusoidal waveform, the natural frequency due to the inertia of the system is at $10f$. Thus the presence of higher amplitudes of the peaks corresponding to inter harmonic frequencies between $7f$ and $9f$ can be attributed to the resonance between the natural and applied oscillations.
![[]{data-label="ramp_thoery_exp"}](ramp_fft){width="0.7\linewidth"}
Similar to the case of a square waveform the case of a ramp waveform is examined theoretically for $\zeta$, given as: $$\zeta=\frac{(t-\tau t_c[\frac{t}{\tau}])-\frac{\tau}{2}}{\frac{\tau}{2}}$$ Where, $\tau=1/f$, $f$ is the applied frequency, $t_c$ is a defined function which accept only integer part. The theoretical validation of surface oscillations in the presence of ramp waveform is done by performing the FFT analysis of theoretical surface oscillations, as shown in figure \[ramp\_thoery\_exp\]. Unlike the case of a square waveform, no inter-harmonic frequencies are observed in the case of ramp waveform. This is because, while in the case of a square waveform, the harmonic frequencies occur in the multiple of an odd number of the fundamental frequency, the interactions of various terms can generate even order inter-harmonic frequencies along with odd-order harmonic frequencies. On the other hand, in the case of ramp waveform, all integer harmonic frequencies are present in the applied signal itself. Like the previous case, the theoretically obtained FFT, as shown in figure \[ramp\_thoery\_exp\], is in fair agreement with the experimental observations.
Droplet breakup characteristics
-------------------------------
![Breakup of levitated charged droplet in the presence of different waveform a) sine waveform b) square waveform c) ramp waveform. []{data-label="droplet_breakup"}](droplet_breakup){width="0.7\linewidth"}
To the best of our knowledge, there exist only two experimental evidence of breakup of charged droplet levitated in electrodynamic balance[@duft03; @singh2019subcritical]. In both the studies, the droplet is levitated in the presence of a sinusoidal applied potential. In this work, for the first time, we report the droplet breakup in the presence of a non-sinusoidal waveform. Thus, the charged droplet is levitated in the presence of sine, square, and ramp waveform. Unlike the surface oscillations of a sub-Rayleigh charged drop, for the breakup studies, we allowed the droplet to evaporate and build a Rayleigh critical charge at which droplet undergoes breakup. The droplet breakup is recorded at one hundred thousand fps. The experimental observations of the droplet breakup in the presence of various waveform is shown in figure \[droplet\_breakup\], where it can be observed that droplet breaks in the upward direction and in an asymmetric manner. Since the droplet is levitated in the presence of the AC field without any superimposed DC voltage to balance the mass of the drop, the droplet levitates away from the geometric center of the trap. At this off-centered location, it experiences asymmetric electrical stress, which causes an asymmetric breakup. It is also observed that the droplet breaks in the upward direction due to the high initial $P_2$ perturbation and higher curvature (+ve $P_3$) in the upward direction. The magnitude of asymmetry and droplet DD depends upon the $z_{shift}$ and $\Lambda$. As reported by @singh2019lang, the droplet breaks with more asymmetry at a higher value of $z_{shift}$ even for a fixed value of $\Lambda$. The thickness of the jet depends on the droplet diameter (or $z_{shift}$) and the applied voltage. The detailed explanations of such droplet breakup characteristics are discussed by @singh2019lang. The important part to note here is that there is no significant difference in the breakup mode in the presence of different waveforms.
Conclusions:
============
The surface oscillation characteristics are investigated by performing FFT analysis and it is found that the FFT is an appropriate characterizing tool for identification of the presence of different frequencies in the droplet deformation data. The surface oscillation behavior is also compared with the viscous-potential flow theory and a fair agreement is observed. The droplet breakup characteristics in the presence of sinusoidal and non-sinusoidal waveform are shown and it is observed that the application of a different waveform does not alter the breakup characteristics. The extension of linear stability analysis to higher-order analysis can make the system suitable for a contact-free and accurate surface tension measurement device. The work demonstrates that the mechanism of droplet oscillations and breakup is robust and is admitted as long as there is a time periodic driving potential applied to the trap.
Data availability statement {#data-availability-statement .unnumbered}
---------------------------
The data that support the findings of this study are available from the corresponding author upon reasonable request.
[25]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} in @noop [**]{}, Vol. (, ) pp. @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ()]{} @noop [ ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ()]{} @noop [****, ()]{}
|
---
abstract: |
Suppose $\alpha$ is a nonzero cardinal number, $\mathcal I$ is an ideal on arc connected topological space $X$, and ${\mathfrak P}_{\mathcal I}^\alpha(X)$ is the subgroup of $\pi_1(X)$ (the first fundamental group of $X$) generated by homotopy classes of $\alpha\frac{\mathcal I}{}$loops. The main aim of this text is to study ${\mathfrak P}_{\mathcal I}^\alpha(X)$s and compare them. Most interest is in $\alpha\in\{\omega,c\}$ and $\mathcal
I\in\{\mathcal P_{fin}(X),\{\varnothing\}\}$, where $\mathcal
P_{fin}(X)$ denotes the collection of all finite subsets of $X$. We denote ${\mathfrak P}_{\{\varnothing\}}^\alpha(X)$ with ${\mathfrak P}^\alpha(X)$. We prove the following statements:\
$\bullet$ for arc connected topological spaces $X$ and $Y$ if ${\mathfrak P}^\alpha(X)$ is isomorphic to ${\mathfrak P}^\alpha(Y)$ for all infinite cardinal number $\alpha$, then $\pi_1(X)$ is isomorphic to $\pi_1(Y)$;\
$\bullet$ there are arc connected topological spaces $X$ and $Y$ such that $\pi_1(X)$ is isomorphic to $\pi_1(Y)$ but ${\mathfrak P}^\omega(X)$ is not isomorphic to ${\mathfrak P}^\omega(Y)$;\
$\bullet$ for arc connected topological space $X$ we have ${\mathfrak P}^\omega(X)\subseteq{\mathfrak P}^c(X)
\subseteq\pi_1(X)$;\
$\bullet$ for Hawaiian earring $\mathcal X$, the sets ${\mathfrak P}^\omega({\mathcal X})$, ${\mathfrak P}^c({\mathcal X})$, and $\pi_1({\mathcal X})$ are pairwise distinct.\
So ${\mathfrak P}^\alpha(X)$s and ${\mathfrak P}_{\mathcal I}^\alpha(X)$s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.
author:
- 'Fatemah Ayatollah Zadeh Shirazi, Mohammad Ali Mahmoodi'
title: On special subgroups of fundamental group
---
[ 55Q05\
[**Keywords:**]{}]{} $\alpha-$arc, $\alpha\frac{\mathcal I}{}$arc, $\alpha\frac{\mathcal I}{}$loop, fundamental group, Hawaiian earring.
Introduction
============
The main aim of algebraic topology is “classifying the topological spaces”. One of the first concepts introduced in algebraic topology is “fundamental group”. As it has been mentioned in [@M84 page1], fundamental groups are introduced by Poincar$\acute{\rm e}$. In this text we consider special subgroups of fundamental group. Explicitly we pay attention to path homotopy classes induced by loops which are “enough one to one”. We have the following sections:
- Introduction
- What is an $\alpha\frac{\mathcal I}{}$arc?
- New subgroups
- A useful remark
- Primary properties of ${\mathfrak P}_{\mathcal I}^\alpha(X)$s
- Some preliminaries on Hawaiian earring
- $\mathfrak P^c(\mathcal X)$ is a proper subset of $\pi_1(\mathcal X)$
- $\mathfrak P_{{\mathcal P}_{fin}(\mathcal Y)}^c(\mathcal Y)$ is a proper subset of $\pi_1(\mathcal Y)$
- Main examples and counterexamples
- Main Table
- Two spaces having fundamental groups isomorphic to Hawaiian earring’s fundamental group
- A distinguished counterexample
- A diagram and a hint
- A strategy for future and conjecture
- Conclusion
Our main conventions located in section 2, although there are conventions in other sections too. Briefly, we introduce our new subgroups in Section 3 and obtain their primary properties in Section 5. Sections 6, 7 and 8 contain basic lemmas for our counterexamples in Section 9. Regarding these three sections 7, 8, and 9 we see $\mathfrak P^\omega (\mathcal
X)\subset\mathfrak P^c (\mathcal X)\subset\pi_1(\mathcal X)$ where $\mathcal X$ is Infinite or Hawaiian earring and “$\subset$” means strict inclusion; also we see $\mathfrak P_{{\mathcal
P}_{fin}(\mathcal Y)}^\omega (\mathcal Y) \subset\mathfrak
P_{{\mathcal P}_{fin}(\mathcal Y)}^c (\mathcal
Y)\subset\pi_1(\mathcal Y)$ ($\mathcal Y$ is introduced in Section 2). However Counterexamples of Section 9 are essential for Main Table in Section 10, which shows probable inclusion relations between different $\mathfrak P_{\mathcal I}^\alpha(X)$ for a fix $X$ (arc connected locally compact Hausdorff topological space), $\alpha\in\{\omega,c\}$ and ${\mathcal
I}\in\{\{\varnothing\},{\mathcal P}_{fin}(X), {\mathcal P}(X)\}$ where ${\mathcal P}(X)$ is the power set of $X$ and $\mathfrak
P_{{\mathcal P}(X)}^\alpha(X)$ is just $\pi_1(X)$ (the fundamental group of $X$) by Section 5. We continue to discover the properties of “our new subgroups” in Sections 12 and 13, as a matter of fact in Sections 11 and 12 we see $\pi_1(\mathcal
X)\cong\pi_1(\mathcal W)$ and $\mathfrak P^\omega(\mathcal
X)\ncong\mathfrak P^\omega(\mathcal W)$ ($\mathcal W$ is introduced in Section 2), consequently we have a diagram and two problems in Section 13. As a matter of fact using the diagram of Section 13 and “Distinguished Example” in Section 12, we try to show “these new subgroups” can make meaningful subclasses of a *class of arc connected locally compact Hausdorff topological spaces with the isomorphic fundamental groups*.\
Remembering all the conventions during reading the text is highly recommended.
A topological space $X$ is an arc connected space, if for all $a,b\in X$ with $a\neq b$ there exists a continuous one to one map $f:[0,1]\to X$ with $f(0)=a$ and $f(1)=b$. In this text all spaces assumed to be Hausdorff, locally compact, and arc connected with at least two elements.
Let $X$ be an arbitrary set. We call $\mathcal I\subseteq\mathcal
P(X)$, an ideal on $X$, if:
- $\mathcal I\neq\varnothing$,
- If $A,B\in\mathcal I$, then $A\cup B\in\mathcal I$,
- If $B\subseteq A$ and $A\in\mathcal I$, then $B\in\mathcal I$.
The collection of all finite subsets of $X$, ${\mathcal P}_{fin}(X)$, is one of the most famous ideals on $X$.
In this text ZFC+GCH (we recall that GCH or *Generalized Continuum Hypothesis* indicates that for transfinite cardinal number $\beta$, there is not any cardinal number $\gamma$ with $\beta<\gamma<2^\beta$, i.e. $2^\beta=\beta^+$ [@Ho99]) is assumed and by “$\subset$” we mean strict inclusion. Whenever $G$ is a group isomorphic to group $H$, we write $G\cong H$. Also $G \ncong H$ means that $G$ is not isomorphic to $H$. Whenever $g\in G$ and $A\subseteq G$, then $<A>$ denotes the subgroup of $G$ generated by $A$, denote $<\{g\}>$ simply by $<g>$. We recall that $\omega$ is the cardinality of ${\mathbb N}$ (the set of all natural numbers $\{1,2,\ldots\}$) and $c$ is the cardinality of ${\mathbb R}$ (the set of all real numbers). We denote the cardinality of $A$ by $|A|$. For cardinal numbers (real numbers) $\alpha,\beta$ we denote the maximum of $\{\alpha,\beta\}$ by $\max(\alpha,\beta)$ also we denote the minimum of $\{\alpha,\beta\}$ by $\min(\alpha,\beta)$.\
In addition for $n\in\mathbb N$, consider ${\mathbb R}^n$ under Euclidean norm. Also consider ${\mathbb
S}^1=\{(x,y)\in{\mathbb R}^2:x^2+y^2=1\}$ as a subspace of ${\mathbb R}^2$ (or $\{e^{i\theta}:\theta\in[0,2\pi]\}$ as a subspace of $\mathbb C$, the set of all complex numbers).
What is an $\alpha\frac{\mathcal I}{}$arc?
==========================================
The concept of $\alpha\frac{\mathcal I}{}$arc is a generalization of $\alpha-$arc which is originated from [@A07]. However a $1-$arc or briefly arc is a one to one map $f:[0,1]\to X$.
For nonzero cardinal number $\alpha$, and ideal ${\mathcal I}$ on $X$, the continuous map $f:Y\to X$ is called an $\alpha\frac{\mathcal I}{}$map if there exists $A\in\mathcal I$ such that for all $x\in X\setminus A$, $|f^{-1}(x)|<\alpha+1$ . In particular for infinite cardinal number $\alpha$, the continuous map $f:Y\to X$ is an $\alpha\frac{\mathcal I}{}$map if there exists $A\in\mathcal I$ such that for all $x\in X\setminus
A$, $|f^{-1}(x)|<\alpha$.\
We call $\alpha\frac{\mathcal I}{}$map $f:[0,1]\to X$, $\alpha\frac{\mathcal I}{}$arc. We call $\alpha\frac{\mathcal
I}{}$map $f:[0,1]\to X$ with $f(0)=f(1)=a$, an $\alpha\frac{\mathcal I}{}$loop with base point $a$.\
We use briefly terms $\alpha-$map, $\alpha-$arc, and $\alpha-$loop respectively instead of $\alpha\frac{\{\varnothing\}}{}$map, $\alpha\frac{\{\varnothing\}}{}$arc, and $\alpha\frac{\{\varnothing\}}{}$loop.
We want to study subgroups of $\pi_1(X)$ generated by path homotopy equivalence classes of $\alpha-$loops and $\alpha\frac{\mathcal I}{}$loops for nonzero cardinal number $\alpha$ and ideal $\mathcal I$ on $X$. We pay special attention to $\alpha\frac{\mathcal I}{}$loops for $\alpha\in\{\omega,c\}$ and ${\mathcal I}\in\{{\mathcal
P}_{fin}(X),\{\varnothing\}\}$. We use the following spaces and loops in most counterexamples in this text.
\[convention-asli\] Suppose $p\in\mathbb N$, let $$\begin{aligned}
\mathcal X & := & \left\{\dfrac1ne^{2\pi i\theta}+\dfrac{i}{n}:n\in{\mathbb N},\theta\in[0,1]\right\} \\
& (= & {\displaystyle\bigcup_{n\in\mathbb N}
\left\{(x,y)\in\mathbb R^2:x^2+(y-\dfrac1n)^2=\dfrac1{n^2}\right\}}){\: \: \: \: \:}{\rm (Hawaiian \: earring)}\\
\mathcal Y & := & \bigcup\left\{\dfrac{1}{2^{n+1}} \mathcal X+\dfrac1n:n\in\mathbb N\right\}\cup[0,1] \\
\mathcal Z & := & \left\{\dfrac1ke^{2\pi i(x-k-\frac14)}+\dfrac{i}{k}:k\in\{1,...,p\},x\in[0,1]\right\} \\
\mathcal W & := & \left\{\dfrac{1}{2^{n+1}}e^{2\pi i\theta}+\dfrac{1}{n}+\dfrac{i}{2^{n+1}}
:n\in\mathbb N,\theta\in[0,1]\right\}\cup[0,1] \\
C_n & := & \left\{\frac1ne^{2\pi it}+\frac{i}{n}:t\in[0,1]\right\} \\
& (= & \left\{(x,y)\in\mathbb R^2:x^2+(y-\dfrac1n)^2=\dfrac1{n^2}\right\})
\\
& & {\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\rm (circle \: with \: radius}\:\frac1n\:{\rm and \: center}\:\frac{i}{n}(n\in\mathbb N)) \\
\mathcal V & := & {\displaystyle\bigcup_{n\in\mathbb N}\left\{(x,y,z)\in\mathbb R^3:y^2+(z-\dfrac1n)^2=\dfrac{1}{n^2}
\wedge0\leq x\leq\dfrac1n\right\}}\end{aligned}$$ moreover define $f_{\mathcal X}:[0,1]\to\mathcal X$, $f_{\mathcal Y}:[0,1]\to\mathcal Y$ and $f_{\mathcal Z}:[0,1]\to\mathcal Z$ with: $$f_{\mathcal X}(x)=\left\{\begin{array}{lc}
\dfrac1ne^{2\pi i(n(n+1)x-n-\frac14)}+\dfrac{i}{n} &
\dfrac1{n+1}\leq x\leq\dfrac1n,n\in{\mathbb N}\: , \\
& \\
0 & x=0 \:,
\end{array}\right.$$ $$f_{\mathcal Y}(x)=\left\{\begin{array}{lc}
\dfrac{f_{\mathcal X}(4xn(n+1)-(2n+1))}{2^{n+1}}+\dfrac1n &
\dfrac{2n+1}{4n(n+1)}\leq x\leq\dfrac1{2n},n\in{\mathbb N} \: ,\\
& \\
2(n+1)(2n-1)x+(2-2n) & \dfrac1{2(n+1)}\leq x\leq\dfrac{2n+1}{4n(n+1)},n\in{\mathbb N}\:, \\
& \\
2-2x & \dfrac12\leq x\leq1 \:,\\
&\\
0 & x=0\:,
\end{array}\right.$$ and $$f_{\mathcal Z}(x)=\frac1ke^{2\pi i(px-k-\frac14)}+\dfrac{i}{k}{\: \: \: \: \:}(
\dfrac{k-1}{p}\leq x\leq\dfrac{k}{p},k\in\{1,...,p\})\:.$$ $\:$\
[*Note: Consider 0 as the base point of all spaces in this convention*]{}
---------------------------------------------- ----------------------------------------------
 
(Figure of $\mathcal X=f_{\mathcal X}[0,1]$) (Figure of $\mathcal Y=f_{\mathcal Y}[0,1]$)
 
(Figure of $\mathcal W$) (Figure of $\mathcal V$)
---------------------------------------------- ----------------------------------------------
\[taha10\] 1) The map $f_{\mathcal X}:[0,1]\to\mathcal X$ is an $\alpha-$loop if and only if $\alpha>\omega$, since: $$|f_{\mathcal X}^{-1}(x)|=\left\{\begin{array}{lc} 1 & x\neq0 \: , \\ \omega & x=0 \: .
\end{array}\right.$$ In addition for each nonzero cardinal number $\alpha$ and ideal $\mathcal I$ on $\mathcal X$ with $\{0\}\in\mathcal I$, $f_{\mathcal X}:[0,1]\to\mathcal X$ is an $\alpha\frac{\mathcal I}{}$loop.\
2) The map $f_{\mathcal Y}:[0,1]\to\mathcal Y$ is an $\alpha\frac{\mathcal I}{}$loop if and only if “$\alpha>\omega$” or “$\alpha\geq2$ and $\{\frac1n:n\in{\mathbb N}\}\in\mathcal I$”, since: $$|f_{\mathcal Y}^{-1}(x)|=\left\{\begin{array}{lc}
\omega & x\in\{\frac1n:n\in{\mathbb N}\} \: , \\
2 & {\rm otherwise}\:. \end{array}\right.$$ In particular $f_{\mathcal Y}:[0,1]\to\mathcal Y$ is an $\alpha\frac{{\mathcal P}_{fin}(Y)}{}$loop if and only if $\alpha\geq c$.\
3) The map $f_{\mathcal Z}:[0,1]\to\mathcal Z$ is an $\alpha-$loop if and only if $\alpha>p$. In addition for all nonzero cardinal number $\alpha$ and ideal $\mathcal I$ on $\mathcal X$ with $\{0\}\in\mathcal I$, $f_{\mathcal X}:[0,1]\to\mathcal X$ is an $\alpha\frac{\mathcal I}{}$loop.
New subgroups
=============
In this section we introduce $\mathfrak P^\alpha_\mathcal I(X)$ as a subgroup of $\pi_1(X)$.\
We recall that for continuous maps $f,g:[0,1]\to X$ with $f(1)=g(0)$, we have $f*g:[0,1]\to X$ with $f*g(t)=f(2t)$ for $t\in[0,\frac12]$ and $f*g(t)=g(2t-1)$ for $t\in[\frac12,1]$. If $f:[0,1]\to X$ is a continuous map, $[f]$ denotes its path homotopy equivalence class, where loops $f,g:[0,1]\to X$ with same base point $a$ are path homotopic (or $[f]=[g]$) if there exists continuous map $F:[0,1]\times[0,1]\to X$ with $F(s,0)=f(s)$, $F(s,1)=g(s)$ and $F(0,s)=F(1,s)=a$ for all $s\in[0,1]$.\
**In the rest of this paper simply we use term “*homotopy*” or “*homotopic*” respectively instead of “*path homotopy*” or “*path homotopic*”.**\
In addition for two loops $f,g:[0,1]\to X$ with same base point $a$, we define $[f]*[g]$ as $[f*g]$. The class of all homotopy equivalence classes of loops with base point $a$ under operation $*$ is a group which is denoted by $\pi_1(X,a)$. Whenever $X$ is arc connected and $a,b\in X$ we have $\pi_1(X,a)\cong\pi_1(X,b)$ so $\pi_1(X,a)$ is denoted simply by $\pi_1(X)$.
For nonzero cardinal number $\alpha$ and ideal $\mathcal I$ by $\mathfrak P^\alpha_\mathcal I(X,a)$ we mean subgroup of $\pi_1(X,a)$ generated by homotopy classes of $\alpha\frac{\mathcal I}{}$loops with base point $a$.
\[Narges1\] For infinite cardinal number $\alpha$ and ideal $\mathcal I$ on $X$, if $f,g:[0,1]\to X$ are $\alpha\frac{\mathcal I}{}$arcs with $f(1)=g(0)$, then $f*g:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$arc. Moreover $\overline f:[0,1]\to X$ with $\overline f(t)=f(1-t)$ is an $\alpha\frac{\mathcal I}{}$arc too.
Use the fact that for all $x\in X$, $(f*g)^{-1}(x)=(\frac12f^{-1}(x))\cup (\frac12g^{-1}(x)+\frac12)$, thus $|(f*g)^{-1}(x)|\leq|f^{-1}(x)|+|g^{-1}(x)|$. Also note to the fact that for all $x\in X$ we have ${\overline f}^{-1}(x)=\{1-t:t\in f^{-1}(x)\}$, hence $|{\overline f}^{-1}(x)|=|f^{-1}(x)|$.
\[Narges01\] For infinite cardinal number $\alpha$, $a\in X$ and ideal $\mathcal I$ on $X$, we have:
$\mathfrak P^\alpha_\mathcal
I(X,a)=\{[f]:\:f:[0,1]\to X$ is an $\alpha\frac{\mathcal
I}{}$loop with base point $a\}$.
Choose $b\in X\setminus\{a\}$. There exists a continuous one to one map $g:[0,1]\to X$ with $g(0)=a$ and $g(1)=b$. Using Theorem \[Narges1\], $g*\overline g:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$arc. Thus $[g*\overline
g]\in\{[f]:\:f:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$loop with base point $a\}$, and $\{[f]:\:f:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$loop with base point $a\}\neq\varnothing$. Using Theorem \[Narges1\], $\{[f]:\:f:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$loop with base point $a\}$ is a subgroup of $\pi_1(X,a)$ which completes the proof.
\[Narges2\] Using Theorem \[Narges01\] for $a\in X$ and infinite cardinal number $\alpha$, for the loop $g:[0,1]\to X$ with base point $a$, $[g]\in\mathfrak P^\alpha_\mathcal I(X,a)$ if and only if there exists an $\alpha\frac{\mathcal I}{}$loop $f:[0,1]\to X$ with base point $a$ homotopic to $g:[0,1]\to X$.
\[jaleb\] For all $a,b\in X$, ideal $\mathcal I$ on $X$ and infinite $\alpha$, $\mathfrak P^\alpha_\mathcal I(X,a)$ and $\mathfrak P^\alpha_\mathcal I(X,b)$ are isomorphic groups.
For $a\neq b$, suppose $f:[0,1]\to X$ is a continuous one to one map (1-arc) such that $f(0)=a$ and $f(1)=b$, and $\overline
f:[0,1]\to X$ is $\overline f(t)=f(1-t)$ for all $t\in[0,1]$. Using Theorem \[Narges1\], $g:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$arc if and only if $\overline f*g*f:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$arc too, which leads to the desired result (note: $\varphi:
{\mathfrak P}_{\mathcal I}^\alpha(X,a)\to{\mathfrak P}_{\mathcal
I}^\alpha(X,b)$, with $\varphi([g])=[ \overline f*g*f]$ is an isomorphism).
By the following counterexample the infiniteness of $\alpha$ in Theorem \[jaleb\] is essential.
Consider $X=\mathbb S^1\cup[1,2]$ as a subspace of $\mathbb R^2$) ($X$ and are homeomorph). If $a\in \mathbb S^1$ and $b\in(1,2]$, then:
- $\mathfrak P^1_{\mathcal P_{fin}(X)}(X,a)=\pi_1(X,a)\cong\mathbb Z$,
- $\mathfrak P^1_{\mathcal P_{fin}(X)}(X,b)=\{e\}$ (where $e$ is the identity of $\pi_1(X,b)$).
In particular $\mathfrak P^1_{\mathcal P_{fin}(X)}(X,a)$ and $\mathfrak P^1_{\mathcal P_{fin}(X)}(X,b)$ are nonisomorphic (although $X$ is linear connected).
\(1) By definition $\mathfrak P^1_{\mathcal
P_{fin}(X)}(X,a)\subseteq\pi_1(X,a)(=\mathbb Z)$. On the other hand $f:\mathop{[0,1]\to X}\limits_{{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}t\mapsto e^{2\pi it}}$ is a $1\frac{\mathcal
P_{fin}(X)}{}$arc and $\pi_1(X,a)=<[f]>\subseteq\mathfrak
P^1_{\mathcal P_{fin}(X)}(X,a)$, which completes the proof.\
(2) Suppose $f:[0,1]\to X$ with $f(0)=f(1)=b$ is a continuous map. If $f\neq b$, then there exists $c\in [1,2]\setminus\{b\}$ with $c=\inf[0,1]$. Let $s:=\min(c,b)$ and $t:=\max(c,b)$. For all $y\in(s,t)$ we have $|f^{-1}(y)|\geq2$, and $(s,t)\notin{\mathcal
P_{fin}(X)}$ (since $(s,t)$ is infinite). Therefore $f$ is not a $1\frac{\mathcal P_{fin}(X)}{}$loop, and the constant loop $b$ is the unique $1\frac{\mathcal P_{fin}(X)}{}$loop with base point $b$, thus $\mathfrak P^1_{\mathcal P_{fin}(X)}(X,b)=\{[b]\}=\{e\}$
\[salam\] Regarding Theorem \[jaleb\] for infinite cardinal number $\alpha$ and ideal $\mathcal I$ on $X$, we denote $\mathfrak
P^\alpha_\mathcal I(X,a)$ simply by $\mathfrak P^\alpha_\mathcal
I(X)$ (subgroup of $\pi_1(X)$ generated by homotopy classes of $\alpha\frac{\mathcal I}{}$loops). We denote $\mathfrak
P^\alpha_{\{\varnothing\}}(X)$ by $\mathfrak P^\alpha(X)$ (subgroup of $\pi_1(X)$ generated by homotopy classes of $\alpha-$loops).\
So for infinite cardinal number $\alpha$ we have (use Note \[Narges2\] and above discussion):
$\mathfrak P^\alpha_\mathcal I(X)=\{[f]:\:f:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$loop$\}$,
and
$\mathfrak P^\alpha(X)=\{[f]:\:f:[0,1]\to X$ is an $\alpha-$loop$\}$.
A useful remark
===============
For the remain of this text we use the following useful convention.
\[good10\] Suppose $X$ and $Y$ are closed subspaces of $Z$ such that $X\cap Y=\{t\}$. For $f:[0,1]\to X\cup Y$ define: $$f^X(x)=\left\{\begin{array}{lc}
f(x) & f(x)\in X \: , \\
t & f(x)\in Y \: .
\end{array}\right.$$
\[useful\] Suppose $X$ and $Y$ are closed (linear connected) subspaces of $Z$ such that $X\cap Y=\{t\}$. For loops $g,h:[0,1]\to X\cup Y$ with base point $t$ we have:
- If $g,h:[0,1]\to X\cup Y$ are homotopic loops, then $g^X,h^X:[0,1]\to X$ are homotopic loops (therefore $g^X,h^X:[0,1]\to X\cup Y$ are homotopic too).
- Let $g[0,1]\subseteq X$ and $h[0,1]\subseteq Y$. $g,h:[0,1]\to X\cup Y$ are homotopic if and only if they are null-homotopic.
- Let $g[0,1]\cup h[0,1]\subseteq X$. $g,h:[0,1]\to X\cup Y$ are homotopic if and only if $g,h:[0,1]\to X$ are homotopic.
- $\pi_1(X,t)$ and $\pi_1(Y,t)$ are subgroups of $\pi_1(X\cup Y,t)$ and $\pi_1(X,t)\cap\pi_1(Y,t)=\{[t]\}$ where $t$ denotes the constant arc with value $t$ (as a matter of fact the map $\mathop{\pi_1(X,t)\to\pi_1(X\cup Y,t)}\limits_{[f] \mapsto [f]{\: \: \: \: \:}{\: \: \: \: \:}}$ is a group embedding).
\(A) Suppose $g,h:[0,1]\to X\cup Y$ are homotopic loops, then there exists a continuous map $F:[0,1]\times[0,1]\to X\cup Y$ such that $F(s,0)=g(s)$, $F(s,1)=h(s)$ and $F(0,s)=F(1,s)=t$ for all $s\in[0,1]$. Define continuous map $P:X\cup
Y\to X$ with $P(z)=z$ for $z\in X$ and $P(z)=t$ for $z\in Y$. The map $P\circ F:[0,1]\times[0,1]\to X$ is continuous, moreover $P\circ F(s,0)=g^X(s)$, $P\circ F(s,1)=h^X(s)$ and $P\circ F(1,s)=P\circ F(0,s)=t$ for all $s\in[0,1]$, thus $g^X,h^X:[0,1]\to X\cup Y$ are homotopic.\
(B) If $g,h:[0,1]\to X\cup Y$ are homotopic, then by (A), $g^X,h^X:[0,1]\to X\cup Y$ are homotopic. On the other hand $g^X=t$ (constant function $t$) and $h^X=h$, since $g[0,1]\subseteq X$ and $h[0,1]\subseteq Y$. Therefore $h:[0,1]\to X\cup Y$ is null homotopic which leads to the desired result.
Primary properties of ${\mathfrak P}_{\mathcal I}^\alpha(X)$s
=============================================================
In this section we study primary properties of $\mathfrak P^\alpha_\mathcal I(X)$s. It is wellknown that $\Phi:\pi_1(X,x_0)\times\pi_1(Y,y_0)\to\pi_1(X\times Y,(x_0,y_0))$ with $\Phi([f],[g])=[(f,g)]$ is an isomorphism (for example see [@M07 Theorem 60.1]) where for $f:[0,1]\to X$ and $g:[0,1]\to Y$ we have $(f,g):[0,1]\to X\times Y$ with $(f,g)(t)=(f(t),g(t))$ ($t\in[0,1]$). For transfinite cardinal numbers $\alpha,\beta$, ideal ${\mathcal I}$ on $X$ and ideal $\mathcal J$ on $Y$ we prove $\Phi({\mathfrak P}_{\mathcal I}^\alpha(X,x_0)\times
{\mathfrak P}_{\mathcal J}^\beta(Y,y_0))
\subseteq{\mathfrak P}_{{\mathcal I}
\times{\mathcal J}}^{\max(\alpha,\beta)}(X\times
Y,(x_0,y_0))$, hence ${\mathfrak P}_{\mathcal I}^\alpha(X,x_0)\times
{\mathfrak P}_{\mathcal J}^\beta(Y,y_0)$ is isomorphic to a subgroup of ${\mathfrak P}_{{\mathcal I}\times{\mathcal J}}^{\max(\alpha,\beta)}(X\times
Y,(x_0,y_0))$.
\[Narges3\] For topological spaces $X$ and $Y$ we have (we recall that $X$ and $Y$ are arc connected locally compact Hausdorff topological spaces with at least two elements, moreover consider $x_0\in X$, and $y_0\in Y$):\
1. For all $\alpha>c$, nonzero $\beta$ and ideal $\mathcal I$ on $X$ we have $\mathfrak P^\alpha_{\mathcal I}(X)=\pi_1(X)=\mathfrak P^\beta_{{\mathcal P}(X)}(X)$.\
2. For nonzero cardinal numbers $\alpha,\beta$, $x_0\in X$, and ideals $\mathcal I,\mathcal J$ on $X$ we have:
- If $\alpha\leq\beta$, then $\mathfrak P_\mathcal I^\alpha(X,x_0)\subseteq\mathfrak P_\mathcal I^\beta(X,x_0)$.
- If ${\mathcal I}\subseteq{\mathcal J}$, then $\mathfrak P_{\mathcal I}^\alpha(X,x_0)\subseteq\mathfrak P_{\mathcal J}^\alpha(X,x_0)$.
- Therefore for infinite $\alpha$ we have (base point is $x_0$, whenever it is necessary):
- If $\alpha\leq\beta$, then $\mathfrak P_\mathcal I^\alpha(X)\subseteq\mathfrak P_\mathcal I^\beta(X)$.
- If ${\mathcal I}\subseteq{\mathcal J}$, then $\mathfrak P_{\mathcal I}^\alpha(X)\subseteq\mathfrak P_{\mathcal J}^\alpha(X)$;
- $\mathfrak P_{\mathcal I\cap \mathcal J}^\alpha(X)
\subseteq \mathfrak P_{\mathcal I}^\alpha(X)
\cap\mathfrak P_{\mathcal J}^\alpha(X)$.
3\. For infinite cardinal numbers $\alpha$, $\beta$ and ideals $\mathcal I$ on $X$ and $\mathcal J$ on $Y$ we have $$\Phi({\mathfrak P}_{\mathcal I}^\alpha(X,x_0)\times{\mathfrak P}_{\mathcal J}^\beta(Y,y_0))
\subseteq{\mathfrak P}_{{\mathcal I}\times{\mathcal J}}^{\max(\alpha,\beta)}(X\times
Y,(x_0,y_0)),$$ where ${\mathcal I}\times{\mathcal J}$ is ideal on $X\times Y$ generated by $\{A\times
B:A\in{\mathcal I},B\in{\mathcal J}\}$ and $\Phi([f],[g])=[(f,g)]$ for loops $f:[0,1]\to X$ and $g:[0,1]\to Y$. Hence ${\mathfrak P}_{\mathcal I}^\alpha(X)\times{\mathfrak P}_{\mathcal J}^\beta(Y)$ is isomorphic to a subgroup of ${\mathfrak P}_{{\mathcal I}\times{\mathcal J}}^{\max(\alpha,\beta)}(X\times
Y)$.\
4. For infinite cardinal numbers $\alpha$, $\beta$, ideal $\mathcal I$ on $X$, and isomorphism $\Phi:\pi_1(X)\times\pi_1(Y)\to\pi_1(X\times Y)$ with $\Phi([f],[g])=[(f,g)]$, we have:
- $\Phi({\mathfrak P}_{\mathcal I}^\alpha(X,x_0)\times\pi_1(Y,y_0))\subseteq
{\mathfrak P}_{{\mathcal I}\times{\mathcal P}(Y)}^\alpha(X\times Y,(x_0,y_0)),$
- $\Phi({\mathfrak P}_{\mathcal I}^\alpha(X,x_0)\times{\mathfrak P}^\beta(Y,y_0))
\subseteq{\mathfrak P}^\beta(X\times Y,(x_0,y_0))$;
- $\Phi({\mathfrak P}^\alpha(X,x_0)\times{\mathfrak P}^\beta(Y,y_0))
\subseteq{\mathfrak P}^{\min(\alpha,\beta)}(X\times Y,(x_0,y_0))$.
5\. For infinite cardinal numbers $\alpha$, $\beta$, ideal $\mathcal I$ on $X$, ideal $\mathcal J$ on $Y$, $\mathcal K:=\{A\cup B:A\in\mathcal I,B\in\mathcal J\}$, if $X\cap Y=\{t\}$ and $X,Y$ are (linear connected) closed subspaces of $Z$, then we have (note that $\mathcal K$ is an ideal on $X\cup Y$) (see Convention \[good10\] (D)):
- ${\mathfrak P}_{\mathcal I}^\alpha(X,t){\mathfrak P}_{\mathcal J}^\beta(Y,t)\subseteq
{\mathfrak P}_{\mathcal K}^{\max(\alpha,\beta)}(X\cup Y,t)$,
- ${\mathfrak P}^\alpha(X,t){\mathfrak P}^\beta(Y,t)\subseteq
{\mathfrak P}^{\max(\alpha,\beta)}(X\cup Y,t)$.
\(1) and (2) are clear by definition.\
(3) If $f:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$arc with base point $x_0$ and $g:[0,1]\to Y$ is a $\beta\frac{\mathcal J}{}$arc with base point $y_0$, then there exist $A\in{\mathcal I}$ and $B\in{\mathcal J}$ such that for all $x\in X\setminus A$ and $y\in Y\setminus B$ we have $|f^{-1}(x)|<\alpha$ and $|g^{-1}(y)|<\beta$. For $h=(f,g):[0,1]\to X\times Y$ with $h(t)=(f(t),g(t))$ and $(z,w)\in(X\times Y)\setminus(A\times B)$ we have: [ $$\begin{aligned}
(z,w)\in(X\times Y)\setminus(A\times B) & \Rightarrow &
z\in X\setminus A\vee w\in Y\setminus B \\
& \Rightarrow & |f^{-1}(z)|<\alpha\vee|g^{-1}(w)|<\beta \\
& \Rightarrow & |h^{-1}(z,w)|\leq\min(|f^{-1}(z)|,| g^{-1}(w)|)<\max(\alpha,\beta)
\end{aligned}$$]{} therefore $(f,g):[0,1]\to X\times Y$ is a $\max(\alpha,\beta)\frac{{\mathcal I}
\times{\mathcal J}}{}$arc, and $$\Phi([f],[g])=[(f,g)]\in{\mathfrak P}^{\max(\alpha,\beta)}_{{\mathcal I}
\times{\mathcal J}}(X\times Y,(x_0,y_0))\:.$$ (4) (a) is a special case of item (3), since $\pi_1(Y,y_0)=
{\mathfrak P}_{{\mathcal P}(Y)}^\alpha(Y,y_0)$.\
For rest note that for all $(x,y)\in X\times Y$, continuous maps $f:[0,1]\to X$, and $g:[0,1]\to Y$ we have $(f,g)^{-1}(x,y)=f^{-1}(x)\cap g^{-1}(y)$, thus $|h^{-1}(x,y)|\leq\min(|f^{-1}(x)|,|g^{-1}(y)|)$.
- If $f:[0,1]\to X$ is an $\alpha\frac{\mathcal I}{}$arc and $g:[0,1]\to Y$ is a $\beta-$arc, then for all $(x,y)\in X\times Y$ we have $|(f,g)^{-1}(x,y)|\leq\min(|f^{-1}(x)|,|g^{-1}(y)|)\leq|g^{-1}(y)|<\beta$. Therefore $(f,g):[0,1]\to X\times Y$ is a $\beta-$arc.
- If $f:[0,1]\to X$ is an $\alpha-$arc and $g:[0,1]\to Y$ is a $\beta-$arc, then for all $(x,y)\in X\times Y$ we have $|(f,g)^{-1}(x,y)|\leq\min(|f^{-1}(x)|,|g^{-1}(y)|)<\min(\alpha,\beta)$. Therefore $(f,g):[0,1]\to X\times Y$ is a $\min(\alpha,\beta)-$arc.
\(5) Since ${\mathfrak P}_{\mathcal I}^\alpha(X,t)\subseteq
{\mathfrak P}_{\mathcal K}^\alpha(X,t)\subseteq
{\mathfrak P}_{\mathcal K}^{\max(\alpha,\beta)}(X,t)
\subseteq{\mathfrak P}_{\mathcal K}^{\max(\alpha,\beta)}(X\cup Y,t)$ and similarly ${\mathfrak P}_{\mathcal J}^\beta(Y,t)\subseteq
{\mathfrak P}_{\mathcal K}^{\max(\alpha,\beta)}(X\cup Y,t)$.
If $\pi_1(X)$ is a group such that any minimal generator set of $\pi_1(X)$ has at most $n\in\mathbb N$ elements and $\{\mathcal
I_\zeta:\zeta\in\Gamma\}$ is a chain of ideals on $X$ such that for $\zeta_1\neq\zeta_2$ we have $\mathfrak P_{\mathcal
I_{\zeta_1}}^\alpha(X)\neq \mathfrak P_{\mathcal
I_{\zeta_2}}^\alpha(X)$, then $|\Gamma|<n+2$.
Since $\{\mathfrak P_{\mathcal
I_\zeta}^\alpha(X):\zeta\in\Gamma\}$ is a chain of subgroups of $\pi_1(X)$ (use Theorem \[Narges3\] (2)).
We may find the following easy examples:
- It’s evident that for any contractible space $X$, nonzero cardinal number $\alpha$ and ideal $\mathcal I$ on $X$, we have $\mathfrak
P_{\mathcal I}^\alpha(X)=\{e\}$.
- Let $X=\{e^{2\pi i\theta}:\theta\in[0,1]\}(={\mathbb S}^1)$. Then $\mathfrak P^\alpha_\mathcal I(X)=\pi_1(X)$, for all $\alpha\geq2$ and ideal $\mathcal I$ on $X$ (since for $f:[0,1]\to{\mathbb S}^1$ with $f(t)=e^{2\pi i t}$ we have $[f]\in{\mathfrak
P}^\alpha({\mathbb S}^1) \subseteq{\mathfrak
P}^\alpha_{\mathcal I}({\mathbb S}^1)\subseteq\pi_1({\mathbb S}^1)$ and $[f]$ is a generator of $\pi_1({\mathbb S}^1)$, thus ${\mathfrak
P}^\alpha_{\mathcal I}({\mathbb S}^1)=\pi_1({\mathbb S}^1)\cong{\mathbb Z}$).
- With a similar method described in item (2), for all $\alpha\geq2$ and ideal $\mathcal I$ on ${\mathbb
R}^2\setminus\{0\}$ (punctured space), we have ${\mathfrak
P}^\alpha_{\mathcal I}({\mathbb R}^2\setminus\{0\})=\pi_1({\mathbb
R}^2\setminus\{0\})\cong{\mathbb Z}$.
- Using (2) and a similar method described in Theorem \[Narges3\] for all $\alpha\geq2$ and ideal $\mathcal I$ on ${\mathbb T}={\mathbb S}^1\times{\mathbb S}^1$ (Torus) we have ${\mathfrak P}^\alpha_{\mathcal I}({\mathbb T})=\pi_1({\mathbb
T})$.
Some preliminaries on Hawaiian earring
======================================
In this section we bring some useful properties of Infinite earring (Hawaiian earring) (see [@M07 page 500, Exercise 5] too)
\[lem\] If loop $f:[0,1]\to\mathbb S^1$ is not null-homotopic and $f(0)=f(1)=1$, then there exist $a,b\in[0,1]$ such that $f(a,b)=\mathbb S^1\setminus\{1\}$ and $f(a)=f(b)=1$.
In the following proof for $g:[u,v]\to X$ with $g_{[u,v]}:[0,1]\to X$ we mean $g_{[u,v]}(t)=g(t(v-u)+u)$. Since $f:[0,1]\to\mathbb S^1$ is uniformly continuous,there exists $\varepsilon>0$ such that for all $s,t\in[0,1]$ with $|s-t|<\varepsilon$ we have $|f(s)-f(t)|<\frac12$.\
Let $T=\{t\in[0,1]:f(0)=f(t)=1$ and $f_{[0,t]}:[0,1]\to\mathbb
S^1$ is not null-homotopic$\}$. We have $T\neq\varnothing$, since $1\in T$. Suppose $\tau=\inf(T)$. Since $f$ is continuous and $T\subseteq f^{-1}(1)$, thus $f(\tau)=1$. We claim that $\tau\in
T$.There exists $t\in T$ such that $0\leq t-\tau<\varepsilon$, if $\tau=t\in T$ we are done, otherwise since $f_{[\tau,t]}([0,1])=f[\tau,t]\subseteq \{x\in\mathbb
S^1:|x-1|=|x-f(\tau)|<\frac12\}\subseteq\mathbb
S^1\setminus\{-1\}$, thus $f_{[\tau,t]}$ is null-homotopic. On the other hand $[f_{[0,t]}]=[f_{[0,\tau]}]*[f_{[\tau,t]}]=[f_{[0,\tau]}]$ and $f_{[0,\tau]}$ is not null-homotopic, which indicates $\tau\in T$.\
Let $S=\{s\in[0,\tau]:f(s)=f(\tau)=1$ and $f_{[s,\tau]}:[0,1]\to\mathbb
S^1$ is not null-homotopic$\}$, so $0\in S$ and $S\neq\varnothing$. let $\sigma=\sup(S)$. Similar to first part of proof, $\sigma\in
S$. It is clear that $\sigma<\tau$. Moreover $[f_{[0,\tau]}]=[f_{[0,\sigma]}]*[f_{[\sigma,\tau]}]$ and using the way of choose of $\tau$, $f_{[0,\sigma]}:[0,1]\to\mathbb
S^1$ is null-homotopic, thus $[f_{[0,\tau]}]=[f_{[\sigma,\tau]}]$ and $f_{[\sigma,\tau]}:[0,1]\to\mathbb
S^1$is not null-homotopic.\
Since $f_{[\sigma,\tau]}:[0,1]\to\mathbb S^1$ is not null-homotopic, $f[\sigma,\tau]=f_{[\sigma,\tau]}([0,1])=\mathbb S^1$.\
On the other hand if there exists $\zeta\in(\sigma,\tau)$ such that $f(\zeta)=1$. Respectively using the way of choose of $\tau$ and $\sigma$, two maps $f_{[0,\zeta]}:[0,1]\to\mathbb
S^1$ and $f_{[\zeta,\tau]}:[0,1]\to\mathbb
S^1$ are null-homotopic. Using $[f_{[0,\tau]}]=[f_{[0,\zeta]}]*[f_{[\zeta,\tau]}]$, $f_{[0,\sigma]}[0,1]\to\mathbb
S^1$ is null-homotopic, which is a contradiction. Therefore for all $\zeta\in(\sigma,\tau)$ we have $f(\zeta)\neq1$, which shows $f(\sigma,\tau)=\mathbb S^1\setminus\{1\}$.
\[bazlem\] If $X=(\mathbb S^1-1)\cup(\mathbb S^1+1)$ ($X$ and Figure are homeomorph), $\rho:[0,1]\to X$ with $\rho(t)=e^{4\pi
it}-1$ for $t\in[0,\frac12]$ and $\rho(t)=-e^{4\pi it}+1$ for $t\in[\frac12,1]$, and loop $f:[0,1]\to X$ with $f(0)=f(1)=0$ is homotopic to $\rho:[0,1]\to X$, then there exist $a,b,c,d\in[0,1]$ such that $a<b\leq c<d$, $f(a)=f(b)=f(c)=f(d)=0$, $f(a,b)=(\mathbb
S^1-1)\setminus\{0\}$ and $f(c,d)=(\mathbb S^1+1)\setminus\{0\}$.
Let: $$f^{\mathbb S^1-1}(t)=\left\{\begin{array}{lc}
f(t) & f(t)\in{\mathbb S^1-1} \\
0 & {\rm otherwise}
\end{array}\right.{\: \: \: \: \:},{\: \: \: \: \:}\rho^{\mathbb S^1-1}(t)=\left\{\begin{array}{lc}
\rho(t) & \rho(t)\in{\mathbb S^1-1} \\
0 & {\rm otherwise}
\end{array}\right.$$ $$f^{\mathbb S^1+1}(t)=\left\{\begin{array}{lc}
f(t) & f(t)\in{\mathbb S^1+1} \\
0 & {\rm otherwise}
\end{array}\right.{\: \: \: \: \:},{\: \: \: \: \:}\rho^{\mathbb S^1+1}(t)=\left\{\begin{array}{lc}
\rho(t) & \rho(t)\in{\mathbb S^1+1} \\
0 & {\rm otherwise}
\end{array}\right.$$ Two maps $f^{\mathbb S^1-1},\rho^{\mathbb S^1-1}:[0,1]\to\mathbb S^1-1$ are homotopic, since $f,\rho:[0,1]\to X$ are homotopic. Since $\rho^{\mathbb S^1-1}:[0,1]\to\mathbb S^1-1$ is not null-homotopic, by Lemma \[lem\] there exists $a,b\in[0,1]$ with $f^{\mathbb S^1-1}(a,b)=(\mathbb S^1-1)\setminus\{0\}$ and $f^{\mathbb S^1-1}(a)=f^{\mathbb S^1-1}(b)=0$. For all $t\in(a,b)$ we have $f^{\mathbb S^1-1}(t)\neq0$, therefore $f(t)=f^{\mathbb S^1-1}(t)$. Thus $f(a,b)=f^{\mathbb S^1-1}(a,b)=
{\mathbb S^1-1}\setminus\{0\}$. Moreover $f^{\mathbb S^1-1}(a)=f^{\mathbb S^1-1}(b)=0$, thus $f(a),f(b)\in {\mathbb S^1+1}$. Using the continuity of $f$ we have $f(a),f(b)\in\overline{f(a,b)}={\mathbb S^1-1}$, therefore $f(a),f(b)\in{\mathbb S^1-1}\cap{\mathbb S^1+1}=\{0\}$ and $f(a)=f(b)=0$. Let: $$\Gamma_1:=\{(a,b)\in[0,1]\times[0,1]:f(a,b)=(\mathbb S^1-1)\setminus\{0\},f(a)=f(b)=0\}\:,$$ $$\Gamma_2:=\{(a,b)\in[0,1]\times[0,1]:f(a,b)=(\mathbb S^1+1)\setminus\{0\},f(a)=f(b)=0\}\:.$$ By the above discussion, $\Gamma_1\neq\varnothing$. It is evident that for all distinct $(a,b),(a',b')\in\Gamma_1$ we have $(a,b)\cap(a',b')=\varnothing$. Since $f:[0,1]\to X$ is uniformly continuous there exists $\delta>0$ such that: $$\forall u,v\in[0,1]{\: \: \: \: \:}(|u-v|<\delta\Rightarrow|f(u)-f(v)|<1)$$ which leads to: $$\forall u,v\in[0,1]{\: \: \: \: \:}(|u-v|<\delta\Rightarrow f(u,v)\neq {\mathbb S^1-1})$$ so for all $(a,b)\in\Gamma_1$ we have $b-a\geq\delta$.\
Thus $\Gamma_1$ is finite, since $\Gamma_1$ is a nonempty collection of disjoint subintervals of $[0,1]$ with $b-a\geq\delta$ for all $(a,b)\in\Gamma_1$.\
In a similar way $\Gamma_2$ is a nonempty finite collection of disjoint subintervals of $[0,1]$.\
It is evident that for all $(a,b)\in\Gamma_1$ and $(c,d)\in\Gamma_2$ we have $(a,b)\cap(c,d)\neq\varnothing$ (since $f(a,b)\cap f(c,d)=
((\mathbb S^1-1)\cap(\mathbb S^1+1))\setminus\{0\}=\varnothing$), therefore $a<b\leq c<d$ or $c<d\leq a<b$.\
If there exist $(a,b)\in\Gamma_1$ and $(c,d)\in\Gamma_2$ with $a<b\leq c<d$, we are done, otherwise suppose for all $(a,b)\in\Gamma_1$ and $(c,d)\in\Gamma_2$ we have $c<d\leq a<b$. Let $$\Gamma_1=\{(a_1,b_1),\ldots,(a_n,b_n)\}\:,\:
\Gamma_2=\{(c_1,d_1),\ldots,(c_m,d_m)\}\:.$$ and suppose $$c_1<d_1\leq c_2<d_2\leq\cdots\leq
c_m<d_m\leq a_1<b_1\leq a_2<b_2\leq\cdots\leq a_n<b_n\:$$ Using the same notations as in Lemma \[lem\], if $d_1<c_2$, then $f_{[d_1,c_2]}:[0,1]\to X$ is null-homotopic (use Lemma \[lem\] and consider $\Gamma_1,\Gamma_2$). if $p\in[0,1]$ suppose $f_{[p,p]}:[0,1]\to X$ is constant 0 function. So $$f_{[0,c_1]},f_{[d_1,c_2]},f_{[d_2,c_3]},\ldots,f_{[d_{m-1},c_m]},
f_{[d_{m},a_1]},f_{[b_1,a_2]},f_{[b_{n-1},a_n]},f_{[a_n,1]}:[0,1]\to X$$ are null-homotopic. Thus $$[f]=[f_{[c_1,d_1]}]*\cdots*[f_{[c_m,d_m]}]*[f_{[a_1,b_1]}]*\cdots*[f_{[a_n,b_n]}]\:$$ For all $i,j$ we have $f_{[c_i,d_i]}\subseteq\mathbb S^1+1$ and $f_{[a_j,b_j]}\subseteq\mathbb S^1-1$. thus there exist $q_1,\ldots,q_m,p_1,\ldots,p_n\geq0$ with
$[f_{[c_i,d_i]}]=[\rho_{[\frac12,1]}]^{q_i}$ ($1\leq i\leq m$) and $[f_{[a_j,b_j]}]=[\rho_{[0,\frac12]}]^{p_j}$ ($1\leq j\leq n$)
(we recall that $\pi_1(X)=\pi_1(S^1-1)*\pi_1(S^1+1)=
<[\rho_{[0,\frac12]}]>*<[\rho_{[\frac12,1]}]>$, by van Kampen Theorem). Thus $$[\rho_{[0,\frac12]}]*[\rho_{[\frac12,1]}]=[\rho]=[f]
=[\rho_{[\frac12,1]}]^{q_1+\cdots+q_m}*[\rho_{[0,\frac12]}]^{p_1+\cdots+p_n}$$ which is a contradiction since $\pi_1(X)$ is nonabelian free group over two generators $[\rho_{[0,\frac12]}]$ and $[\rho_{[\frac12,1]}]$.
\[bazbazlem\] If loop $f:[0,1]\to {\mathcal Z}$ with $f(0)=f(1)=0$ is homotopic to $f_{\mathcal Z}:[0,1]\to {\mathcal Z}$, then there exist $s_1,t_1,s_2,t_2,\ldots,s_p,t_p\in[0,1]$ such that $s_1<t_1\leq s_2<t_2\leq\ldots\leq s_p<t_p$, $f(s_j)=f(t_j)=0$, $f(s_j,t_j)=
\{\frac1je^{2\pi i(x-j-\frac14)}+\frac{i}{j}:x\in[0,1]\}\setminus\{0\}$ for all $j\in\{1,\ldots,p\}$.
Use the same method described in Lemma \[bazlem\] and note to the fact that $\pi_1(\mathcal Z)$ is nonabelian free group over $p $ generators $[h_{[\frac{k-1}p,\frac{k}p]}]$ for $k=1,\ldots,p$ where $h:=f_{\mathcal Z}$ and using the notations of Lemma \[lem\].
\[Narges4\] Consider loops $f,g:[0,1]\to\mathcal X$ such that $f(0)=f(1)=g(0)=g(1)=0$. For nonempty subset $\Gamma$ of $\mathbb N$ and $h:[0,1]\to \mathcal X$ let: $$h^\Gamma(x)=\left\{\begin{array}{lc} h(x) &
h(x)\in\bigcup\{C_n:n\in\Gamma\}\:, \\
0 & h(x)\in ({\mathcal X}\setminus\bigcup\{C_n:n\in\Gamma\})\cup\{0\}\:.
\end{array}\right.$$ (As a matter of fact we denote $h^{\bigcup\{C_n:n\in\Gamma\}}$ (see Convention \[good10\]) briefly by $h^\Gamma$)
- If $f,g:[0,1]\to\mathcal X$ are homotopic, then $f^\Gamma,g^\Gamma:[0,1]\to \bigcup\{C_n:n\in\Gamma\}$ are homotopic (equivalently $f^\Gamma,g^\Gamma:[0,1]\to\mathcal X$ are homotopic).
- For loop $h:[0,1]\to\mathcal X$ with $h(0)=h(1)=0$ define: $$A(h):=\{n\in\mathbb N: h^{\{n\}}:[0,1]\to C_n\:{\rm is \: not \: null-homotopic}\}\:.$$ Then $A(h)$ is a subset of: $$\{n\in{\mathbb N}:\exists a,b\in[0,1]\:
(h(a,b)=C_n\setminus\{0\}\wedge h(a)=h(b)=0)\}\:.$$ Moreover if $f,g:[0,1]\to\mathcal X$ are homotopic, then $A(f)=A(g)$.
- For loop $h:[0,1]\to\mathcal X$ with $h(0)=h(1)=0$, we have $|h^{-1}(0)|\geq|A(h)|$.
- If $[f]\in\mathfrak P^\omega(\mathcal X)$, then $|A(f)|<\omega$ and $A(f)$ is finite.
$\:$
- Note to the fact that $A=\bigcup\{C_n:n\in\Gamma\}$ and $B=(\mathcal X\setminus A)\cup\{0\}$ are closed (linear connected) subsets of $\mathcal X$. Moreover $A\cap B=\{0\}$. Now use the same argument as in Convention \[good10\].
- If $n\in A(h)$, then $h^{\{n\}}:[0,1]\to C_n$ is not null-homotopic. By Lemma \[lem\] there exist $a,b\in[0,1]$ with $h(a)=h(b)=0$ and $h(a,b)=C_n\setminus\{0\}$. Use item (1) to complete the proof.
- By (2) for all $n\in A(h)$ there exists $a_n,b_n\in[0,1]$ with $h(a_n,b_n)=C_n\setminus\{0\}$ and $h(a_n)=h(b_n)=0$. We claim that $\mathop{A(h)\to h^{-1}(0)}\limits_{n\mapsto a_n{\: \: \: \: \:}}$ is one to one. Suppose $n\neq m$ and $n,m\in A(h)$. By $$h(a_n,b_n)\cap h(a_m,b_m)=(C_n\setminus\{0\})\cap(C_m\setminus\{0\})
=\varnothing$$ we have $(a_n,b_n)\cap(a_m,b_m)=\varnothing$, thus $a_n\neq a_m$.
- If $[f]\in\mathfrak P^\omega(\mathcal X)$, then by Note \[Narges2\] there exists $\omega-$loop $k:[0,1]\to\mathcal X$ with $k(0)=k(1)=0$ homotopic to $f:[0,1]\to\mathcal X$. By (3) we have $|A(k)|\leq|k^{-1}(0)|<\omega$. By item (2) we have $A(f)=A(k)$ which leads to $|A(f)|=|A(k)|\leq|k^{-1}(0)|<\omega$.
\[Narges5\] For $(m,n)\in\mathbb N\times\mathbb N$ and loop $h:[0,1]\to\mathcal Y$ with base point $0$, define: $$h^{(m,n)}(t)=\left\{\begin{array}{lc} h(t) & h(t)\in\dfrac1{2^{m+1}}C_n+\dfrac1m \: , \\
& \\
\dfrac1m & h(t)\notin\dfrac1{2^{m+1}}C_n+\dfrac1m \: . \end{array}\right.$$ (As a matter of fact we denote $h^{\frac1{2^{m+1}}C_n+\frac1m}$ (see Convention \[good10\]) briefly by $h^{(m,n)}$)\
Moreover for loop $h:[0,1]\to\mathcal Y$ we define: [$$B(h):=\left\{(m,n)\in{\mathbb N}\times{\mathbb
N}:h^{(m,n)}:[0,1]\to\dfrac1{2^{m+1}}C_n+\dfrac1m\:{\rm is \: not \:
null-homotopic}\right\}\:,$$]{} then $B(h)$ is a subset of: [$$\left\{(m,n)\in{\mathbb N}\times{\mathbb
N}:\exists a,b\in[0,1]\:
(h(a,b)=(\dfrac1{2^{m+1}}C_n+\dfrac1m)\setminus\{\dfrac1m\}\wedge
h(a)=h(b)=\dfrac1m)\right\}$$]{} and for loops $f,g:[0,1]\to\mathcal Y$ with $f(0)=f(1)=g(0)=g(1)=0$, we have:\
\
1. If $f,g:[0,1]\to\mathcal Y$ are homotopic, then $B(f)=B(g)$.\
2. For $m\in\mathbb N$, we have:
- $|f^{-1}(\frac1m)|\geq|\{n\in\mathbb N:(m,n)\in B(f)\}|$.
- If $[f]\in\mathfrak P^\omega(\mathcal Y)$, then $|\{n\in\mathbb N:(m,n)\in B(f)\}|<\omega$.
- If $\mathcal I$ is an ideal on $\mathcal Y$ and $[f]\in\mathfrak P^\omega_{\mathcal I}(\mathcal Y)$, then there exists $F\in\mathcal I$ such that for all $k\in\mathbb N$ with $\frac1k\in\mathcal Y\setminus F$, we have $|\{n\in\mathbb N:(k,n)\in B(f)\}|<\omega$.
If $(m,n)\in B(h)$, then $h^{(m,n)}:[0,1]\to\frac1{2^{m+1}}C_n+\frac1m$ is not null-homotopic, thus by Lemma \[lem\] there exist $a,b\in[0,1]$ with $h(a,b)=(\frac1{2^{m+1}}C_n+\frac1m)\setminus\{\frac1m\}$ and $h(a)=h(b)=\frac1m$.\
1) Suppose $f,g:[0,1]\to\mathcal Y$ are homotopic. For $m,n\in\mathbb N$, two sets $\frac1{2^{m+1}}C_n+\frac1m$ and $({\mathcal
Y}\setminus(\frac1{2^{m+1}}C_n+\frac1m))\cup\{\frac1m\}$ are closed (linear) subsets of $\mathcal Y$ with $(\frac1{2^{m+1}}C_n+\frac1m)\cap(({\mathcal
Y}\setminus(\frac1{2^{m+1}}C_n+\frac1m))\cup\{\frac1m\})=\{\frac1m\}$. Thus using the same argument as in Convention \[good10\] (note to the fact that base point in the proof of Convention \[good10\] is not important) two maps $f^{(m,n)},g^{(m,n)}:[0,1]\to\frac1{2^{m+1}}C_n+\frac1m$ are homotopic, therefore $(m,n)\in B(f)$ if and only if $(m,n)\in
B(g)$. So $B(f)=B(g)$.\
2-a) For all $(m,n)\in B(f)$ there exists $a_{(m,n)},b_{(m,n)}\in[0,1]$ with $f(a_{(m,n)},b_{(m,n)})=(\frac1{2^{m+1}}C_n+\frac1m)\setminus\{\frac1m\}$ and $f(a_{(m,n)})=f(b_{(m,n)})=\frac1m$. We claim that $$\mathop{\{n\in{\mathbb N}:(m,n)\in B(f)\}\to f^{-1}(\dfrac1m)}\limits_{
{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}n\mapsto a_{(m,n)}}$$ is one to one. Suppose $n\neq k$ and $(m,k),(m,n)\in B(f)$. By $$f(a_{(m,n)},b_{(m,n)})\cap
f(a_{(m,k)},b_{(m,k)})$$ $$=((\dfrac1{2^{m+1}}C_n+\dfrac1m)\setminus\{\dfrac1m\})
\cap((\dfrac1{2^{m+1}}C_k+\dfrac1m)\setminus\{\dfrac1m\})
=\varnothing$$ we have $(a_{(m,n)},b_{(m,n)})\cap(a_{(m,k)},b_{(m,k)})=\varnothing$, thus $a_{(m,n)}\neq a_{(m,k)}$. Therefore $|f^{-1}(\frac1m)|\geq|\{n\in{\mathbb N}:(m,n)\in B(f)\}|$\
2-b) This item is a special case of (c) for ${\mathcal I}=\{\varnothing\}$.\
2-c) If $[f]\in\mathfrak P^\omega_{\mathcal I}(\mathcal Y)$, then by Note \[Narges2\] there exists $\omega\frac{\mathcal I}{}$loop $h:[0,1]\to\mathcal Y$ homotopic to $f:[0,1]\to\mathcal Y$ also we may suppose $h(0)=h(1)=0$. There exists $F\in\mathcal I$ such that for all $z\in{\mathcal Y}\setminus
F$ we have $|h^{-1}(z)|<\omega+1$. In particular for all $k\in\mathbb N$ with $\frac1k\in{\mathcal Y}\setminus F$ we have $|h^{-1}(\frac1k)|<\omega$, which leads to $|\{n\in\mathbb N:(k,n)\in B(h)\}|\leq|h^{-1}(\frac1k)|<\omega$ by (a). Using (1) we have $B(f)=B(h)$, thus $|\{n\in\mathbb N:(k,n)\in B(h)\}|=|\{n\in\mathbb N:(k,n)\in B(f)\}|<\omega$.
$\mathfrak
P^c(\mathcal X)$ is a proper subset of $\pi_1(\mathcal X)$
==========================================================
Here we want to prove $\pi_1(\mathcal X)\setminus \mathfrak
P^c(\mathcal X)\neq\varnothing$ step by step.\
Consider the following conventions in this section:\
Usually in order to construct Cantor set, one may remove the following intervals step by step from $[0,1]$:
So $M=[0,1]\setminus\bigcup\{(c_n^i,d_n^i):n\in\mathbb
N,i\in\{1,...,2^{n-1}\}\}$ is Cantor set. Now suppose:
**Define $g:[0,1]\to{\mathcal X}$ with: $$g(x)=\left\{\begin{array}{lc}
\dfrac1ne^{2\pi i\frac{x-a_n}{b_n-a_n}}+\dfrac{i}{n}& x\in(a_n,b_n),n\in\mathbb N \\
&\\
0 & {\rm otherwise} \end{array}\right.$$ Suppose the loops $g,f:[0,1]\to\mathcal X$ are homotopic with $f(0)=f(1)=0$. Consider the above mentioned $f$ and $g$ in this section.**\
It is well-known that (see [@Rudin]): $$M=\left\{{\displaystyle\sum_{n=1}^\infty\frac{x_n}{3^n}}:
\forall n\in\mathbb N{\: \: \: \: \:}x_n\in\{0,2\}\right\}\:.$$ For $x={\displaystyle\sum_{n=1}^\infty\frac{x_n}{3^n}}\in M$ with $x_n\in\{0,2\}$ ($n\in\mathbb N$). For $m\in\mathbb N$ choose $n^x_m\in\mathbb N$ such that: $$a_{n^x_m}=\left\{\begin{array}{lc}
\min\{c_m^i:1\leq i\leq2^{m-1},x\leq c_m^i\} & x_m=0 \\
\max\{c_m^i:1\leq i\leq2^{m-1},c_m^i\leq x\} & x_m=2
\end{array}\right.$$ also let $$E^x:=\{n\in{\mathbb N}:x_n=0\}\:,\:F^x:=\{n\in{\mathbb N}:x_n=2\}\:.$$ Finally consider: $$K:=\{x\in M: E^x\:{\rm and}\: F^x\:{\rm are \: infinite}\}\:.$$ We have the following sequel of lemmas and notes.
\[note-cantor\] For $x={\displaystyle\sum_{n=1}^\infty\frac{x_n}{3^n}}\in M$ with $x_n\in\{0,2\}$ , we have: $$a_{n_k^x}=\left\{\begin{array}{lc}
{\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}+\dfrac1{3^k} & x_k=0 \: , \\
{\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}-\dfrac1{3^k} & x_k=2 \: ,
\end{array}\right.
{\: \: \: \: \:}{\rm and}{\: \: \: \: \:}b_{n_k^x}=\left\{\begin{array}{lc}
{\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}+\dfrac2{3^k} & x_k=0 \: , \\
{\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}} & x_k=2 \: .
\end{array}\right.\tag{*}$$ And: $$|a_{n_k^x}-x|\leq\dfrac2{3^k}{\: \: \: \: \:}{\rm and}{\: \: \: \: \:}|b_{n_k^x}-x|\leq\dfrac2{3^k}\:
({\rm for\: all\:}k\in\mathbb {N})\:.\tag{**}$$
For each $k\in\mathbb N$ suppose $$A_k=\left\{{\displaystyle\sum_{n=1}^k\frac{y_n}{3^n}}:y_1,\ldots,y_k\in\{0,2\}\right\}\:,$$ then we may suppose $A_k=\{w_k^1,\ldots,w_k^{2^k}\}$ with $w_k^1<w_k^2<\cdots<w_k^{2^k}$. It is easy to see that: $$\begin{array}{lcl}
c^1_k=w_k^1+{\displaystyle\sum_{n\geq k+1}\frac{2}{3^n}}=w_k^1+\dfrac1{3^k} & , & d^1_k=w_k^2 \\
c^2_k=w_k^3+{\displaystyle\sum_{n\geq k+1}\frac{2}{3^n}}=w_k^3+\dfrac1{3^k} & , & d^2_k=w_k^4 \\
\vdots && \\
c^i_k=w_k^{2i-1}+{\displaystyle\sum_{n\geq k+1}\frac{2}{3^n}}=w_k^{2i-1}+\dfrac1{3^k} & ,
& d^i_k=w_k^{2i} \\
\vdots && \\
c^{2^{k-1}}_k=w_k^{2^k-1}+{\displaystyle\sum_{n\geq k+1}\frac{2}{3^n}}=w_k^{2^k-1}+\dfrac1{3^k}
& , & d^{2^{k-1}}_k=w_k^{2^k}
\end{array}$$ so $(c^i_k,d^i_k)=(w_k^{2i-1}+\frac1{3^k},w_k^{2i})$.\
Now for $x={\displaystyle\sum_{n\in\mathbb N}\frac{x_n}{3^n}}\in M$ with $x_1,x_2,\ldots\in\{0,2\}$ we have:
- For $p\in\mathbb N$ we have $x_p=0$ and $x_{p+1}=x_{p+2}=\cdots=2$ if and only if there exists $i\in\{1,\ldots,2^{p-1}\}$ with $x=c^i_p$.
- For $p\in\mathbb N$ we have $x_p=2$ and $x_{p+1}=x_{p+2}=\cdots=0$ if and only if there exists $i\in\{1,\ldots,2^{p-1}\}$ with $x=d^i_p$.
- $x\in K$ if and only if for all $p\in\mathbb N$ we have $p\notin\{c^i_p:1\leq i\leq2^{p-1}\}\cup\{d^i_p:1\leq i\leq2^{p-1}\}$ (and $x\in M$).
In particular if $x_k=0$, then $a_{n_k^x}={\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}+\dfrac1{3^k}$, in other words if $w_k^i={\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}$, then $i=2j-1$ is odd and $a_{n_k^x}=w_k^{2j-1}+\dfrac1{3^k}=
{\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}+\dfrac1{3^k}=c^i_k$. Also if $x_k=2$, then ${\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}\in A_k$ and there exists even $i=2j$ such that ${\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}=w_k^{2j}$, moreover $b_{n_k^x}=w_k^{2j}$. So we have (\*), moreover considering the following inequalities will complete the proof: $$|a_{n_k^x}-x| \leq \left|a_{n_k^x}- {\displaystyle\sum_{n=1}^k\frac{x_n}{3^n}}\right|+
{\displaystyle\sum_{n=k+1}^\infty\frac{x_n}{3^n}}
\leq \dfrac1{3^k}+{\displaystyle\sum_{n=k+1}^\infty\frac2{3^n}}=\dfrac2{3^k}\:,$$ and: $$\begin{aligned}
|b_{n_k^x}-x| & = & \left|{\displaystyle\sum_{n=1}^{k-1}\frac{x_n}{3^n}}+\dfrac2{3^k}-x\right| \\
& = & \left|\dfrac{2-x_k}{3^k}-{\displaystyle\sum_{n=k+1}^\infty\frac{x_n}{3^n}}\right| \\
& = & \left\{\begin{array}{lc}
{\displaystyle\sum_{n=k+1}^\infty\frac{x_n}{3^n}}\leq
{\displaystyle\sum_{n=k+1}^\infty\frac{2}{3^n}}=\dfrac1{3^k} & x_k=2 \\
\dfrac{2}{3^k}-{\displaystyle\sum_{n=k+1}^\infty\frac{x_n}{3^n}}\leq\dfrac2{3^k} & x_k=0
\end{array}\right.\end{aligned}$$ which shows (\*\*).
\[sara1-20\] Let $x={\displaystyle\sum_{n\in\mathbb N}\frac{x_n}{3^n}}\in M$ with $x_n\in\{0,2\}$ ($n\in\mathbb N$), then we have:
- ${\displaystyle\lim_{k\to\infty}a_{n_k^x}}=
{\displaystyle\lim_{k\to\infty}b_{n_k^x}}=x$.
- For $i<j$ if $x_i=x_j=0$, then $x\leq a_{n^x_j}<b_{n^x_j}<a_{n^x_i}<b_{n^x_i}$.
- For $i<j$ if $x_i=x_j=2$, then $a_{n^x_i}<b_{n^x_i}<a_{n^x_j}<b_{n^x_j}\leq x$.
Use (\*\*) in Lemma \[note-cantor\] in order to prove (1).\
2) Suppose $i<j$ and $x_i=x_j=0$, then by (\*) in Lemma \[note-cantor\] we have: $$\begin{aligned}
b_{n^x_i}& > & a_{n^x_i}={\displaystyle\sum_{n=1}^i\frac{x_n}{3^n}}+\frac1{3^i}
={\displaystyle\sum_{n=1}^{i}\frac{x_n}{3^n}}+
{\displaystyle\sum_{n=i+1}^\infty\frac{2}{3^n}}
> {\displaystyle\sum_{n=1}^{i}\frac{x_n}{3^n}}
+{\displaystyle\sum_{n=i+1}^{j}\frac{2}{3^n}} \\
& > & {\displaystyle\sum_{n=1}^{i}\frac{x_n}{3^n}}+
{\displaystyle\sum_{n=i+1}^{j-1}\frac{x_n}{3^n}}+\frac2{3^j}
={\displaystyle\sum_{n=1}^{j}\frac{x_n}{3^n}}+\frac2{3^j}
=b_{n^x_j}>a_{n^x_j}={\displaystyle\sum_{n=1}^j\frac{x_n}{3^n}}+\frac1{3^j} \\
& = & {\displaystyle\sum_{n=1}^j\frac{x_n}{3^n}}+
{\displaystyle\sum_{n=j+1}^\infty\frac{2}{3^n}}\geq
{\displaystyle\sum_{n=1}^j\frac{x_n}{3^n}}+
{\displaystyle\sum_{n=j+1}^\infty\frac{x_n}{3^n}}
=x\end{aligned}$$ 3) Use a similar method described in the proof of (2), to prove (3).
\[sara10\] There exists a sequence $((p_n,q_n):n\in{\mathbb N})$ such that for all $n,m\in\mathbb N$ we have:
- $0\leq p_n<q_n\leq1$, $f(p_n,q_n)=C_n\setminus\{0\}$ and $f(p_n)=f(q_n)=0$;
- if $a_n<b_n<a_m<b_m$, then $p_n<q_n<p_m<q_m$.
For all $n\in{\mathbb N}$, by Note \[Narges4\] we have $f^{\{n\}},g^{\{n\}}:[0,1]\to C_n$ are homotopic loops, therefore $f^{\{n\}}:[0,1]\to C_n$ is not null-homotopic. By Lemma \[lem\] there exist $a,b\in[0,1]$ with $f^{\{n\}}(a,b)=C_n\setminus\{0\}$ and $f^{\{n\}}(a)=f^{\{n\}}(b)=0$, therefore $f(a,b)=C_n\setminus\{0\}$ and $f(a)=f(b)=0$. On the other hand $f:[0,1]\to\mathcal X$ is uniformly continuous, thus $$\Gamma_n:=\{(a,b)\in[0,1]\times[0,1]:
f(a,b)=C_n\setminus\{0\},f(a)=f(b)=0\}$$ is a finite nonempty set. For $k\in\mathbb N$, by considering $f^{\{1,\ldots,k\}}:[0,1]\to C_1\cup\cdots\cup C_k$, Note \[Narges4\] and Lemma \[bazbazlem\] there exist $(u_1,v_1)\in\Gamma_1,\ldots,(u_k,v_k)\in\Gamma_k$ such that if $a_i<b_i<a_j<b_j$, then $u_i<v_i\leq u_j<v_j$ for all $i,j\in\{1,\ldots,k\}$.\
Using the above mentioned note and finiteness of $\Gamma_1$, there exists $(p_1,q_1)\in\Gamma_1$ such that $\sup\{k\in{\mathbb N}:$ there exist $u_2,v_2,u_3,v_3,\ldots,u_k,v_k\in[0,1]$ such that for $u_1=p_1$ and $v_1=q_1$ and all $i,j\in\{1,\ldots,k\}$ we have $(u_i,v_i)\in\Gamma_i$ and if $a_i<b_i<a_j<b_j$, then $u_i<v_i\leq u_j<v_j\}=\infty$.\
For $m\in\mathbb N$ if $(p_1,q_1)\in\Gamma_1,...,(p_m,q_m)\in\Gamma_m$ are such that $\sup\{k\in{\mathbb N}:$ there exist $u_{m+1},v_{m+1},u_{m+2},v_{m+2},
\ldots,u_k,v_k\in[0,1]$ such that for $u_1=p_1,v_1=q_1,u_2=p_2,v_2=q_2,\ldots,u_m=p_m,v_m=q_m$ for all $i,j\in\{1,\ldots,k\}$ we have $(u_i,v_i)\in\Gamma_i$ and if $a_i<b_i<a_j<b_j$, then $u_i<v_i\leq u_j<v_j\}=\infty$. Since $\Gamma_{m+1}$ is finite, there exists $(p_{m+1},q_{m+1})\in\Gamma_{m+1}$ such that $\sup\{k\in{\mathbb N}:$ there exist $u_{m+2},v_{m+2},u_{m+3},v_{m+3},
\ldots,u_k,v_k\in[0,1]$ such that for $u_1=p_1,v_1=q_1,u_2=p_2,v_2=q_2,\ldots,u_{m+1}=p_{m+1},v_{m+1}=q_{m+1}$ for all $i,j\in\{1,\ldots,k\}$ we have $(u_i,v_i)\in\Gamma_i$ and if $a_i<b_i<a_j<b_j$, then $u_i<v_i\leq u_j<v_j\}=\infty$.\
The sequence $((p_n,q_n):n\in\mathbb N)$ is our desired sequence.
\[sara20\] Let $x={\displaystyle\sum_{n\in\mathbb N}\frac{x_n}{3^n}}\in K(\subset M)$ with $x_n\in\{0,2\}$ ($n\in\mathbb N$), and $$E^x=\{n\in{\mathbb N}:x_n=0\}=\{u_k:k\in\mathbb N\}\:,$$ $$F^x=\{n\in{\mathbb N}:x_n=2\}=\{v_k:k\in\mathbb N\}$$ such that $u_1<u_2<\cdots$ and $v_1<v_2<\cdots$, and consider the sequence $((p_n,q_n):n\in\mathbb N)$ as in Lemma \[sara10\], then we have:\
1. The sequences $\{a_{n^x_{u_k}}:k\in\mathbb N\}$ and $\{b_{n^x_{u_k}}:k\in\mathbb N\}$ are strictly decreasing to $x$.\
2. The sequences $\{a_{n^x_{v_k}}:k\in\mathbb N\}$ and $\{b_{n^x_{v_k}}:k\in\mathbb N\}$ are strictly increasing to $x$.\
3. The sequences $\{p_{n^x_{u_k}}:k\in\mathbb N\}$ and $\{q_{n^x_{u_k}}:k\in\mathbb N\}$ are strictly decreasing.\
4. The sequences $\{p_{n^x_{v_k}}:k\in\mathbb N\}$ and $\{q_{n^x_{v_k}}:k\in\mathbb N\}$ are strictly increasing.\
5. ${\displaystyle\lim_{k\to\infty}p_{n^x_{v_k}}
=\lim_{k\to\infty}q_{n^x_{v_k}}}\leq
{\displaystyle\lim_{k\to\infty}p_{n^x_{u_k}}
=\lim_{k\to\infty}q_{n^x_{u_k}}}$.
$\:$\
Use Lemma \[sara1-20\] in order to prove (1) and (2).\
3) By Lemma \[sara1-20\] (2), for all $k\in\mathbb N$ we have $$a_{n^x_{u_{k+1}}}<b_{n^x_{u_{k+1}}}
<a_{n^x_{u_k}}<b_{n^x_{u_k}}\:,$$ which leads to $p_{n^x_{u_{k+1}}}<q_{n^x_{u_{k+1}}}
<p_{n^x_{u_k}}<q_{n^x_{u_k}}$.\
4) Lemma \[sara1-20\] (3), for all $k\in\mathbb N$ we have $$a_{n^x_{v_k}}<b_{n^x_{v_k}}
<a_{n^x_{v_{k+1}}}<b_{n^x_{v_{k+1}}}\:,$$ which leads to $p_{n^x_{v_k}}<q_{n^x_{v_k}}
<p_{n^x_{v_{k+1}}}<q_{n^x_{v_{k+1}}}$.\
5) Using (3) and (4), we have ${\displaystyle\lim_{k\to\infty}p_{n^x_{u_k}}
=\lim_{k\to\infty}q_{n^x_{u_k}}}$ and ${\displaystyle\lim_{k\to\infty}p_{n^x_{v_k}}
=\lim_{k\to\infty}q_{n^x_{v_k}}}$. On the other hand for all $k\in\mathbb N$ we have $a_{n^x_{v_k}}<b_{n^x_{v_k}}<x<a_{n^x_{u_k}}<b_{n^x_{u_k}}$, thus $$p_{n^x_{v_k}}<q_{n^x_{v_k}}<p_{n^x_{u_k}}<q_{n^x_{u_k}}\:,$$ which leads to ${\displaystyle\lim_{k\to\infty}p_{n^x_{v_k}}
\leq\lim_{k\to\infty}p_{n^x_{u_k}}}$.
\[sara30\] For $x={\displaystyle\sum_{n\in\mathbb N}\frac{x_n}{3^n}}\in K$ with $x_n\in\{0,2\}$ and $E^x=\{n\in{\mathbb N}:x_n=0\}=\{u_k:k\in\mathbb N\}$ with $u_1<u_2<\cdots$ under the same notations as in Lemma \[sara10\], by Lemma \[sara20\], $\{p_{n^x_{u_k}}:k\in\mathbb N\}$ is an strictly decreasing sequence (in $[0,1]$). Let $\eta(x)={\displaystyle\lim_{k\to\infty}p_{n^x_{u_k}}}$, then $\eta:K\to[0,1]$ is strictly increasing, and for all $x\in K$ we have $f(\eta(x))=0$.
Consider $x,y\in K$ with $x<y$. Suppose $x={\displaystyle\sum_{n\in\mathbb N}\frac{x_n}{3^n}}$ and $y={\displaystyle\sum_{n\in\mathbb N}\frac{x_n}{3^n}}$ with $x_n,y_n\in\{0,2\}$ for all $n\in\mathbb N$. Let $$E^x=\{n\in{\mathbb N}:x_n=0\}=\{u_k:k\in\mathbb N\}\:,$$ $$E^y=\{n\in{\mathbb N}:y_n=0\}=\{u'_k:k\in\mathbb N\}\:,$$ with $$u_1<u_2<\cdots{\: \: \: \: \:}{\rm and}{\: \: \: \: \:}u'_1<u'_2<\cdots{\: \: \: \: \:}.$$ By Lemma \[sara20\] (1), $\{a_{n_{u_k}^x}:k\in\mathbb N\}$ is a strictly decreasing sequence to $x$, and $\{a_{n_{u'_k}^x}:k\in\mathbb N\}$ is a strictly decreasing sequence to $y$. Since $x<y$, there exists $m\in\mathbb N$ such that $$x\leq\cdots<a_{n_{u_{m+2}}^x}<a_{n_{u_{m+1}}^x}<a_{n_{u_m}^x}<y
\leq\cdots<a_{n_{u'_{m+2}}^y}<a_{n_{u'_{m+1}}^y}<a_{n_{u'_m}^y}\:.$$ Thus $$\begin{aligned}
x & \leq & \cdots<a_{n_{u_{m+2}}^x}<b_{n_{u_{m+2}}^x}<a_{n_{u_{m+1}}^x}<
b_{n_{u_{m+1}}^x}<a_{n_{u_m}^x}<b_{n_{u_{m}}^x} \\
& < & y\leq\cdots<a_{n_{u'_{m+2}}^y}<b_{n_{u'_{m+2}}^y}<a_{n_{u'_{m+1}}^y}
<b_{n_{u'_{m+1}}^y}<a_{n_{u'_m}^y}<b_{n_{u'_{m}}^y}\:.\end{aligned}$$ Using Lemma \[sara10\] we have: $$\begin{array}{c}
\cdots<p_{n_{u_{m+2}}^x}<q_{n_{u_{m+2}}^x}<p_{n_{u_{m+1}}^x}<
q_{n_{u_{m+1}}^x}<p_{n_{u_m}^x}<q_{n_{u_{m}}^x} \\
<\cdots<p_{n_{u'_{m+2}}^y}<q_{n_{u'_{m+2}}^y}<p_{n_{u'_{m+1}}^y}
<q_{n_{u'_{m+1}}^y}<p_{n_{u'_m}^y}<q_{n_{u'_{m}}^y}\:.
\end{array}$$ Therefore $$\eta(x)={\displaystyle\lim_{k\to\infty}p_{n^x_{u_k}}}\leq p_{n^x_{u_{m+1}}}
< p_{n^x_{u_m}}\leq {\displaystyle\lim_{k\to\infty}p_{n^y_{u'_k}}}=\eta(y)\:,$$ and $\eta:K\to[0,1]$ is strictly increasing. Since $f(p_{n^x_{u_k}})=0$ for all $k\in\mathbb N$ and $f$ is continuous, we have $f(\eta(x))=0$.
\[sara40\] $|f^{-1}(0)|\geq c$ and $f$ is not a $c-$arc.
Consider $\eta:K\to[0,1]$ as in Lemma \[sara30\]. By Lemma \[sara30\] we have $|f^{-1}(0)|\geq|\eta(K)|$ and by Lemma \[sara20\] $\eta$ is one to one, therefore $|\eta(K)|=|K|=c$. Thus $|f^{-1}(0)|\geq c$ and $f$ is not a $c-$arc.
\[taha20\] We have $${\mathfrak P}^c(\mathcal X)\subset\pi_1(\mathcal X)\:, \:
\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)\not\subseteq{\mathfrak P}^c(\mathcal X)\:,$$ $$\mathfrak P^\omega
(\mathcal X)\subset\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)\:{\rm and}\:\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)\not\subseteq\mathfrak P^c
(\mathcal X)\:.$$
Using Note \[Narges2\], and Lemma \[sara40\], $[g]\notin\mathfrak P^c(\mathcal X)$, thus ${\mathfrak P}^c(\mathcal X)\subset\pi_1(\mathcal X)$. Using $[g]\in \mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)$ shows $\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)\not\subseteq{\mathfrak P}^c(\mathcal X)$. Also $[g]\in\mathfrak P^\omega
(\mathcal X)\setminus \mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)$, thus $\mathfrak P^\omega
(\mathcal X)$ is a proper subgroup of $\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)$. Using $[g]\in\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)\setminus\mathfrak P^c
(\mathcal X)$ will complete the proof.
$\mathfrak
P_{{\mathcal P}_{fin}(\mathcal Y)}^c(\mathcal Y)$ is a proper subset of $\pi_1(\mathcal Y)$
===========================================================================================
In this section we prove $\pi_1(\mathcal Y)\setminus \mathfrak
P_{{\mathcal P}_{fin}(\mathcal Y)}^c(\mathcal Y)\neq\varnothing$. We use the same notations as in Section 7.\
Define $G:[0,1]\to\mathcal Y$ with: $$G(x)=\left\{\begin{array}{lc}
\dfrac{g(4xn(n+1)-(2n+1))}{2^{n+1}}+\dfrac1n &
\dfrac{2n+1}{4n(n+1)}\leq x\leq\dfrac1{2n},n\in{\mathbb N} \: ,\\
& \\
2(n+1)(2n-1)x+(2-2n) & \dfrac1{2(n+1)}\leq x\leq\dfrac{2n+1}{4n(n+1)},n\in{\mathbb N}\:, \\
& \\
2-2x & \dfrac12\leq x\leq1 \:,\\
&\\
0 & x=0\:,
\end{array}\right.$$ where $g:[0,1]\to{\mathcal X}$ as in section 7 is: $$g(x)=\left\{\begin{array}{lc}
\dfrac1ne^{2\pi i\frac{x-a_n}{b_n-a_n}}+\dfrac{i}{n}& x\in(a_n,b_n),n\in\mathbb N \:,\\
&\\
0 & {\rm otherwise}\:. \end{array}\right.$$
\[bahar10\] Let $K,G:[0,1]\to\mathcal Y$ are homotopic and $m\in\mathbb N$, then $|K^{-1}(\frac1m)|=c$.
Choose $\theta\in(\frac1{m+1}+\frac1{2^{m+2}},\frac1m-\frac1{2^{m+1}})$. Consider $h,\overline h:[0,1]\to\mathcal Y$ with $h(x)=\theta x$ and $\overline h(x)=\theta(1-x)$. Since $K,G:[0,1]\to\mathcal Y$ are path homotopic with base point 0, $\overline h*K*h,\overline h*G*h:
[0,1]\to\mathcal Y$ are path homotopic with base point $\theta$. Using Convention \[good10\] two maps $$(\overline h*K*h)^{\{(x,y)\in\mathcal Y:x\geq\theta\}},
(\overline h*G*h)^{\{(x,y)\in\mathcal Y:x\geq\theta\}}:
[0,1]\to\{(x,y)\in\mathcal Y:x\geq\theta\}$$ are path homotopic with base point $\theta$. Let $$K_1=(\overline h*K*h)^{\{(x,y)\in\mathcal Y:x\geq\theta\}}
\:{\rm and}\:G_1=(\overline h*G*h)^{\{(x,y)\in\mathcal Y:x\geq\theta\}}\:.$$ If $m=1$ let $K_2=K_1$ and $G_2=G_1$.\
If $m>1$, choose $\mu\in(\frac1m+\frac1{2^{m+1}},\frac1{m-1}-\frac1{2^m})$. Consider $h_1,\overline h_1:[0,1]\to\mathcal Y$ with $h_1(x)=
(\mu-\theta)x+\theta$ and $\overline h_1(x)=(\mu-\theta)(1-x)+\theta$. Since $K_1,G_1:[0,1]\to\{(x,y)\in\mathcal Y:x\geq\theta\}$ are path homotopic with base point $\theta$, $\overline h_1*K_1*h_1,\overline h_1*G_1*h_1:
[0,1]\to\{(x,y)\in\mathcal Y:x\geq\theta\}$ are path homotopic with base point $\mu$. Using Convention \[good10\] two maps $(\overline h_1*K_1*h_1)^{\{(x,y)\in\mathcal Y:\theta\leq x\leq\mu\}}$ and $(\overline h_1*G_1*h_1)^{\{(x,y)\in\mathcal Y:\theta\leq x\leq\mu\}}$ from $[0,1]$ to $\{(x,y)\in\mathcal Y:\theta\leq x\leq\mu\}$ are path homotopic with base point $\mu$. Let $h_2(x)=(\frac1m-\mu)x+\mu$ and $\overline h_2(x)=(\frac1m-\mu)(1-x)+\mu$ for $x\in[0,1]$.\
Now let: $$K_2=\left\{\begin{array}{lc}
\overline h_2*
(\overline h_1*K_1*h_1)^{\{(x,y)\in\mathcal Y:\theta\leq x\leq\mu\}}*h_2
& m>1 \: , \\
K_1 & m=1 \: , \end{array}\right.$$ and $$G_2=\left\{\begin{array}{lc}
\overline h_2*
(\overline h_1*G_1*h_1)^{\{(x,y)\in\mathcal Y:\theta\leq x\leq\mu\}}*h_2
& m>1 \: , \\
G_1 & m=1 \: , \end{array}\right.$$ also in order to be more convenient, whenever $m=1$ let $\mu=1$. Then $K_2,G_2:[0,1]\to\{(x,y)\in\mathcal Y:\theta\leq x\leq\mu\}$ ($\subseteq
(\frac1{2^{m+1}}\mathcal X+\frac1m)\cup[\theta,\mu]$) are path homotopic with base point $\frac1m$. Hence there exists a continuous map $F:[0,1]\times[0,1]\to\{(x,y)\in\mathcal Y:\theta\leq x\leq\mu\}$ such that $F(0,s)=F(1,s)=\frac1m$, $F(s,0)=K_2(s)$ and $F(s,1)=G_2(s)$ for all $s\in[0,1]$.\
Define $\mathcal K , \mathcal G :[0,1]\to\mathcal X$ and $\mathcal F:[0,1]\times[0,1]\to \mathcal X$ with: $$\mathcal K(x)=\left\{\begin{array}{lc}
2^{m+1}(K_2(x)-\frac1m) & K_2(x)\in\frac1{2^{m+1}}\mathcal X+\frac1m \: , \\
-ie^{\frac{i\pi(K_2(x)-\frac1m)}2}+i & \theta\leq K_2(x)\leq \mu \: ,
\end{array}\right.$$ $$\mathcal G(x)=\left\{\begin{array}{lc}
2^{m+1}(G_2(x)-\frac1m) & G_2(x)\in\frac1{2^{m+1}}\mathcal X+\frac1m \: , \\
-ie^{\frac{i\pi(G_2(x)-\frac1m)}2}+i & \theta\leq G_2(x)\leq \mu \: ,
\end{array}\right.$$ $$\mathcal F(s,t)=\left\{\begin{array}{lc}
2^{m+1}(F(s,t)-\frac1m) & F(s,t)\in\frac1{2^{m+1}}\mathcal X+\frac1m \: , \\
-ie^{\frac{i\pi(F(s,t)-\frac1m)}2}+i & \theta\leq F(s,t)\leq \mu \: .
\end{array}\right.$$ Using the gluing lemma, $\mathcal K$, $\mathcal G$ and $\mathcal F$ are continuous, moreover by the above definition, for all $s\in[0,1]$ we have:
- the equality $F(0,s)=F(1,s)=\frac1m$, implies $$\mathcal F(0,s)=\mathcal F(1,s)=-ie^{\frac{i\pi(\frac1m-\frac1m)}2}+i=0\:,$$
- two equalities $F(s,0)=K_2(s)$ and $F(s,1)=G_2(s)$, imply $\mathcal F (s,0)=\mathcal K (s)$ and $\mathcal F (s,1)=\mathcal G (s)$.
So $\mathcal K, \mathcal G ;[0,1]\to\mathcal X$ are path homotopic with base point 0. One could verify that $\mathcal G, g:[0,1]\to\mathcal X$ are homotopic, thus $\mathcal K, g:[0,1]\to\mathcal X$ are homotopic and by Lemma \[sara40\] in which we proved $|f^{-1}(0)|=c$ whenever $f,g:[0,1]\to\mathcal X$ are homotopic, we have $|\mathcal K^{-1}(0)|\geq c$. Since $\mathcal K^{-1}(0)$ and $K^{-1}(\frac1m)$ differs in a finite set, we have $|K^{-1}(\frac1m)|= c$.
\[bahar20\] We have $[G]\in\pi_1(\mathcal Y)\setminus \mathfrak
P_{{\mathcal P}_{fin}(\mathcal Y)}^c(\mathcal Y)$.
If $[G]\in\mathfrak
P_{{\mathcal P}_{fin}(\mathcal Y)}^c(\mathcal Y)$, then by Note \[Narges2\], there exists $c\frac{{\mathcal P}_{fin}(\mathcal Y)}{}$loop $K:[0,1]\to\mathcal Y$ with $K(0)=K(1)=0$ and $[F]=[G]$. Since $G:[0,1]\to\mathcal Y$ is not null-homotopic, $K:[0,1]\to\mathcal Y$ is not constant. Thus there exists $k\in\mathbb N$ such that for all $m\geq k$ we have $\frac1m\in K[0,1]$. By Lemma \[bahar10\], for all $m\geq k$ we have $|K^{-1}(\frac1m)|=c$, thus $\{x\in{\mathcal Y}:|K^{-1}(x)|\not<c\}$ is infinite, which is a contradiction, since $K:[0,1]\to\mathcal Y$ is a $c\frac{{\mathcal P}_{fin}(\mathcal Y)}{}$loop. Therefore $[G]\notin\mathfrak
P_{{\mathcal P}_{fin}(\mathcal Y)}^c(\mathcal Y)$.
Main examples and counterexamples
=================================
Now we are ready to present examples.
\[example1\] Using Note \[Narges4\] (4), since $A(f_{\mathcal X})(=\mathbb N)$ is infinite, thus $[f_{\mathcal X}]\notin\mathfrak P^\omega(\mathcal X)$. On the other hand, using Example \[taha10\] (1), $f_{\mathcal X}:[0,1]\to\mathcal X$ is a $c-$loop, thus $[f_{\mathcal X}]\in{\mathfrak P}^c(\mathcal X)\setminus\mathfrak P^\omega(\mathcal X)$ and $\mathfrak P^\omega(\mathcal X)$ is a proper subgroup of $\mathfrak P^c(\mathcal X)$. Therefore by Theorem \[taha10\], we have: $${\mathfrak P}^\omega(\mathcal X)\subset{\mathfrak P}^c(\mathcal X)\subset\pi_1(\mathcal X)\:.$$ Also using Theorem \[taha10\] again we have $\mathfrak P^\omega
(\mathcal X)\subset\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)$, which leads to $\mathfrak P^\omega
(\mathcal X)\subset\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^c
(\mathcal X)$, since $\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)\subseteq\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^c
(\mathcal X)$. We recall that according to Theorem \[taha10\], $\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)\not\subseteq\mathfrak P^c
(\mathcal X)$, which leads to $\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^c
(\mathcal X)\not\subseteq\mathfrak P^c
(\mathcal X)$ since $\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^\omega
(\mathcal X)\subseteq\mathfrak P_{\mathcal P_{fin}(\mathcal X)}^c
(\mathcal X)$.
The following Example deal with Theorem \[Narges3\]. We again recall that $\Phi:\pi_1(X)\times\pi_1(Y)\to\pi_1(X\times
Y)$, with $\Phi([f],[g])=[(f,g)]$ (where $(f,g)(t)=(f(t),g(t))$ (for $t\in[0,1]$ and loops $f:[0,1]\to X$, $g:[0,1]\to Y$)) is a group isomorphism. Moreover as it was proved in Theorem \[Narges3\] (4c), for infinite cardinal number $\alpha$ we have $\Phi({\mathfrak P}^\alpha(X)\times{\mathfrak
P}^\alpha(Y))
\subseteq{\mathfrak P}^\alpha(X\times Y)$. In the following we bring an example in which $\Phi({\mathfrak
P}^\alpha(X)\times{\mathfrak P}^\alpha(Y))
\neq{\mathfrak P}^\alpha(X\times Y)$, in particular we prove that $\Phi\restriction_{{\mathfrak P}^\omega(\mathcal X)
\times{\mathfrak P}^\omega(\mathcal X)}:
{\mathfrak P}^\omega(\mathcal X)
\times{\mathfrak P}^\omega(\mathcal X)\to
{\mathfrak P}^\omega({\mathcal X}\times{\mathcal X})$ is a group monomorphism but it is not an isomorphism.
\[example6\] Define $\overline f_{\mathcal X}:[0,1]\to {\mathcal X}$ with $\overline f_{\mathcal X}(t)=f_{\mathcal X}(1-t)$. $(f_{\mathcal X},\overline f_{\mathcal X}):[0,1]\to {\mathcal X}\times{\mathcal X}$ is an $\omega-$arc since for all $(x,y)\in{\mathcal X}\times{\mathcal X}$, if $|(f_{\mathcal X},\overline f_{\mathcal X})^{-1}(x,y)|>1$, then $x=y=0$. Moreover $(f_{\mathcal X},\overline
f_{\mathcal X})^{-1}(0,0)\subseteq\{t\in[0,1]:t,1-t\in\{\frac1n:n\in{\mathbb
N}\}\}\cup\{0,1\}=\{0,1,\frac12\}$. Therefore for all $(x,y)$ we have $|(f_{\mathcal X},\overline f_{\mathcal X})^{-1}(x,y)|\leq3<\omega$ and $(f_{\mathcal X},\overline
f_{\mathcal X}):[0,1]\to X\times X$ is an $\omega-$arc. Thus $\Phi([f_{\mathcal X}],[\overline
f_{\mathcal X}])=[(f_{\mathcal X},\overline f_{\mathcal X})]\in
{\mathfrak P}^\omega({\mathcal X}\times{\mathcal X})$. Since $\Phi:\pi_1({\mathcal X})\times\pi_1({\mathcal X})\to\pi_1
({\mathcal X}\times{\mathcal X})$ is a group isomorphism, there exist unique $([g],[h])\in\pi_1({\mathcal X})\times\pi_1({\mathcal X})$ with $\Phi([g],[h])=[(f_{\mathcal X},\overline f_{\mathcal X})]$ therefore $[g]=[f_{\mathcal X}]$ and $[h]=[\overline f_{\mathcal X}]$. Using Example \[example1\], $[f_{\mathcal X}]\notin{\mathfrak P}^\omega({\mathcal X})$, so $([g],[h])=
([f_{\mathcal X}],[\overline f_{\mathcal X}])\notin{\mathfrak
P}^\omega({\mathcal X})\times{\mathfrak P}^\omega({\mathcal X})$. So (note: $\Phi$ is one to one): $$[(f_{\mathcal X},\overline f_{\mathcal X})]=\Phi([g],[h])
=\Phi([f_{\mathcal X}],[\overline f_{\mathcal X}])
\notin\Phi({\mathfrak P}^\omega({\mathcal X})\times{\mathfrak P}^\omega({\mathcal X}))$$ which shows $\Phi({\mathfrak P}^\omega({\mathcal X})\times
{\mathfrak P}^\omega({\mathcal X}))\neq
{\mathfrak P}^\omega({\mathcal X}\times{\mathcal X})$.
\[example3\] Using the same notations as in Note \[Narges5\] we have $B(f_{\mathcal Y})=\mathbb N\times\mathbb N$, therefore for all $m\in\mathbb N$, $\{n\in\mathbb N: (m,n)\in B(f_{\mathcal
Y})\}(=\mathbb N)$ is infinite. If $F\in\mathcal P_{fin}(\mathcal
Y)$, then $F$ is finite and there exists $k\in\mathbb N$ with $\frac1k\in{\mathcal Y}\setminus F$. Using infiniteness of $\{n\in\mathbb N: (k,n)\in B(f_{\mathcal Y})\}$ and Note \[Narges5\] (c) we have $[f_{\mathcal Y}]\notin \mathfrak
P_{\mathcal P_{fin}(\mathcal Y)}^\omega(\mathcal Y)$. On the other hand using Example \[taha10\] (2), $f_{\mathcal
Y}:[0,1]\to\mathcal Y$ is a $c-$loop, thus $[f_{\mathcal
Y}]\in{\mathfrak P}^c({\mathcal Y})$. So $\mathfrak P_{\mathcal
P_{fin}(\mathcal Y)}^\omega(\mathcal Y)$ is a proper subgroup of ${\mathfrak P}^c({\mathcal Y})$ and $\pi_1(\mathcal Y)$ (Hint: We can prove $\mathfrak P^\omega(\mathcal Y)$ is a proper subgroup of $\mathfrak P_{\mathcal P_{fin}(\mathcal
Y)}^\omega(\mathcal Y)$, thus $\mathfrak P^\omega(\mathcal
Y)\subset \mathfrak P_{\mathcal P_{fin}(\mathcal
Y)}^\omega(\mathcal Y)\subset\pi_1(\mathcal Y)$).
\[example4\] Map $f_{\mathcal Z}:[0,1]\to\mathcal Z$ is a $p+1-$arc and it is not homotopic with any $k-$arc $g:[0,1]\to{\mathcal Z}$ for $k<p+1$. However for all $\alpha\geq2$ and ideal $\mathcal I$ on ${\mathcal Z}$ we have $\mathfrak P^\alpha_{\mathcal I}(\mathcal
Z)=\pi_1(\mathcal Z)$. For this aim, for all $k\in\{1,...,p\}$, define $f_k:[0,1]\to{\mathcal Z}$ with $f_k(t)=\frac1ke^{2\pi
i(t-\frac14)}+\frac{i}{k}$. For all $\alpha\geq2$ and ideal $\mathcal I$ on ${\mathcal Z}$, we have $[f_k]\in\mathfrak
P^2({\mathcal Z}) \subseteq\mathfrak P^\alpha_{\mathcal
I}({\mathcal Z}) \subseteq\pi_1({\mathcal Z})$. Since $\{[f_1],...,[f_n]\}$ generates $\pi_1({\mathcal Z})$, thus $\mathfrak P^\alpha_{\mathcal I}({\mathcal Z}) =\pi_1({\mathcal
Z})$.
\[example8\] We recall that $\pi_1(\mathcal Y)\setminus \mathfrak P_{\mathcal
P_{fin}({\mathcal Y})}^c(\mathcal Y)\neq \varnothing$ by Theorem \[bahar20\]. However $[f_{\mathcal Y}]\in \mathfrak
P_{\mathcal P_{fin}({\mathcal Y})}^c(\mathcal Y)$ (since $f_{\mathcal Y}:[0,1]\to\mathcal Y$ is a $c-$loop, thus $[f_{\mathcal Y}]\in\mathfrak
P_{\mathcal P_{fin}({\mathcal Y})}^c(\mathcal Y)$) One may show $[f_{\mathcal Y}]\notin\mathfrak
P_{\mathcal P_{fin}({\mathcal Y})}^\omega(\mathcal Y)$, thus: $$\mathfrak
P_{\mathcal P_{fin}({\mathcal Y})}^\omega(\mathcal Y) \subset \mathfrak
P_{\mathcal P_{fin}({\mathcal Y})}^c(\mathcal Y) \subset\pi_1(\mathcal Y)\:.$$
Main Table
==========
\[jadval\] We have the following Table:
$\dfrac{{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}K}{H{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}}$ $\mathfrak P^\omega(X)$ $\mathfrak P_{\mathcal P_{fin}(X)}^\omega(X)$ $\mathfrak P^c(X)$ $\mathfrak P_{\mathcal P_{fin}(X)}^c(X)$ $\pi_1(X)$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------- ----------------------------------------------- -------------------- ------------------------------------------ -------------
$\mathfrak P^\omega(X)$ $\subseteq$ $\subseteq$ $\subseteq$ $\subseteq$ $\subseteq$
$\mathfrak P_{\mathcal P_{fin}(X)}^\omega(X)$ \[example1\] $\subseteq$ \[example1\] $\subseteq$ $\subseteq$
$\mathfrak P^c(X)$ \[example1\] \[example3\] $\subseteq$ $\subseteq$ $\subseteq$
$\mathfrak P_{\mathcal P_{fin}(X)}^c(X)$ \[example1\] \[example3\] \[example1\] $\subseteq$ $\subseteq$
$\pi_1(X)$ \[example1\] \[example3\] \[example1\] \[example8\] $\subseteq$
\
$\:$\
In the above table “$\subseteq$” means that in the corresponding case we have $H\subseteq K$.\
In addition the number [*i.j*]{} means that in Example [*i.j*]{} there exists an example such that $H\not\subseteq K$ in the corresponding case.
Two spaces having fundamental groups isomorphic to Hawaiian earring’s fundamental group
=======================================================================================
In this section we prove in a sequel of Lemmas, that $\mathcal X$ (Hawaiian earring) and $\mathcal W$ are homeomorph with two deformation retracts of $\mathcal V$. Thus we have $\pi_1({\mathcal X})\cong\pi_1({\mathcal V})\cong\pi_1({\mathcal
W})$, which is important for our main counterexamples in next section.\
We recall sign map ${\rm sgn}:{\mathbb R}\to\{\pm1,0\}$ with ${\rm sgn}(x)=\frac{x}{|x|}$ for $x\neq0$ and ${\rm sgn}(0)=0$.\
Note: In a connected topological space $A$, we call $x\in A$ a cut point of $A$ if $A\setminus\{x\}$ is disconnected. It is evident that $\mathcal X$ and $\mathcal W$ are not homeomorphic since $\mathcal X$ has just one cut point and $\mathcal W$ has infinitely many cut points.
\[salam1\] For $x\in[0,1]$, the map $\Phi_x:[0,\frac12]\to\{w\in[-1,1]:x+w\leq0\}=[-1,1]\cap(-\infty,-x]=[-1,-x]$ with: $$\Phi_x(t)=\left\{\begin{array}{lc}
(1-\sin(\pi t))(1-\frac{x}{1-2t})-1 & t\in[0,\frac12) \\
-1 & t=\frac12 \end{array}\right.$$ is a homeomorphism.
Suppose $z\in(-1,1]$ and $z+x\leq0$. The map $\varphi:[0,\frac12)\to {\mathbb R}$ with $\varphi(t)=(1-\sin(\pi t))(1-\frac{x}{1-2t})-1$ is continuous, moreover $\varphi(0)=-x$ and ${\displaystyle\lim_{{\: \: \: \: \:}t\to{\frac12}^-}\varphi(t)}=-1$. By $-1< z\leq -x$ and the mean value theorem there exists $t\in[0,\frac12)$ with $\varphi(t)=z$. In addition $\Phi_x\restriction_{[0,\frac12)}=\varphi:[0,\frac12)\to {\mathbb R}$ is strictly decreasing, therefore $\Phi_x:[0,\frac12]\to[-1,-x]$ is a bijective continuous map which completes the proof.
\[salam3\] Using the same notations as in Lemma \[salam1\], $\widehat{\Phi}:\{(x,w)\in[0,1]\times[-1,0]:x+w\leq0\}\to[0,\frac12]$ with $\widehat{\Phi}(x,w)=\Phi_x^{-1}(w)$ is continuous.
Using Lemma \[salam1\], $\widehat{\Phi}:\{(x,w)\in[0,1]\times[-1,0]:x+w\leq0\}\to[0,\frac12]$ is well-defined. Let $A:=\{(x,w)\in[0,1]\times[-1,0]:x+w\leq0\}$. Consider $(x,w)\in A$, $s\in[0,\frac12]$, and sequence $\{(x_n,w_n):n\in{\mathbb N}\}$ such that ${\displaystyle\lim_{n\to\infty}x_n}=x$, ${\displaystyle\lim_{n\to\infty}w_n}=w$, ${\displaystyle\lim_{n\to\infty}\widehat{\Phi}(x_n,w_n)}=s$. Let $t=\widehat{\Phi}(x,w)$ and $t_n=\widehat{\Phi}(x_n,w_n)$ ($n\in\mathbb N$). We show $s=t$, i.e. ${\displaystyle\lim_{n\to\infty}\widehat{\Phi}(x_n,w_n)}=\widehat{\Phi}(x,w)$.\
We have the following cases:\
[*Case 1*]{}. $s\neq\frac12$. In this case we may suppose for all $n\in\mathbb N$ we have $t_n\neq\frac12$. For all $n\in\mathbb N$ we have $w_n=\Phi_{x_n}(t_n)=(1-\sin(\pi t_n))(1-\frac{x_n}{1-2t_n})-1$ moreover: $$\begin{aligned}
\Phi_x(t) & = & w = {\displaystyle\lim_{n\to\infty}w_n}
= {\displaystyle\lim_{n\to\infty}\Phi_{x_n}(t_n)} \\
& = & {\displaystyle\lim_{n\to\infty}(1-\sin(\pi t_n))(1-\frac{x_n}{1-2t_n})-1} \\
& = & (1-\sin(\pi s))(1-\frac{x}{1-2s})-1 =\Phi_x(s)\end{aligned}$$ and $s=t$ since $\Phi_x$ is one to one according to Lemma \[salam1\].\
[*Case 2*]{}. $s=\frac12$ and for infinitely many of $n$s we have $t_n=\frac12$. In this case we may suppose for all $n\in\mathbb N$ we have $t_n=\frac12$. Thus we have: $$\begin{aligned}
\Phi_x(t) & = & w = {\displaystyle\lim_{n\to\infty}w_n}
= {\displaystyle\lim_{n\to\infty}\Phi_{x_n}(t_n)} \\
& = & {\displaystyle\lim_{n\to\infty}\Phi_{x_n}(\frac12)}
={\displaystyle\lim_{n\to\infty}-1}=-1=\Phi_x(\frac12)\end{aligned}$$ and $s=\frac12=t$ since $\Phi_x$ is one to one according to Lemma \[salam1\].\
[*Case 3*]{}. $s=\frac12$ and for infinitely many of $n$s we have $t_n\neq\frac12$. In this case we may suppose for all $n\in\mathbb N$ we have $t_n\neq\frac12$. Thus we have: $$\begin{aligned}
\Phi_x(t) & = & w = {\displaystyle\lim_{n\to\infty}w_n}
= {\displaystyle\lim_{n\to\infty}\Phi_{x_n}(t_n)} \\
& = & {\displaystyle\lim_{n\to\infty}(1-\sin(\pi t_n))(1-\frac{x_n}{1-2t_n})-1} \\
& = & {\displaystyle\lim_{n\to\infty}\frac{(1-\sin(\pi t_n))}{1-2t_n}
\lim_{n\to\infty}(1-2t_n-x_n)}-1 \\
& = & 0\times(1-s-x)-1=-1=\Phi_x(\frac12)\end{aligned}$$ and $s=\frac12=t$ since $\Phi_x$ is one to one according to Lemma \[salam1\].\
Using the above cases $s=t$ and $\widehat{\Phi}:\{(x,w)\in[0,1]\times[-1,0]:x+w\leq0\}\to[0,\frac12]$ is continuous (otherwise since $[0,\frac12]$ is compact, there exists $(x,w)\in A$ and sequence $\{(x_n,w_n):n\in\mathbb N\}$ converging to $(x,w)$ such that the sequence $\{\widehat{\Phi}(x_n,w_n):n\in\mathbb N\}$ converges to a point $s\in[0,\frac12]\setminus\{\widehat{\Phi}(x,w)\}$).
\[salam5\] Consider $X=\{(x,y,z)\in{\mathbb R}^3:y^2+z^2=1,0\leq x\leq1\}$ and $\widehat{\Phi}$ as in Lemma \[salam3\]. Let $M_1=\{(x,y,z)\in X:x+z\leq0\}$, the map $F_1:[0,1]\times M_1\to X$ with $F_1(\mu,(x,y,z))=(x',y',z')$ for: $$\left\{\begin{array}{l} x'=x+(1-2(1-x)\widehat{\Phi}(x,z)-x)\mu \: , \\
z'=(1-\mu)z-\mu \: , \\
y'={\rm sgn}(y)\sqrt{1-z'^2} \: ,
\end{array}\right.$$ is continuous.
Let $(\mu,(x,y,z))\in[0,1]\times M_1$, since $\widehat{\Phi}(x,z)\in[0,\frac12]$, we have $1-2\widehat{\Phi}(x,z)\in[0,1]$ which leads to (use $x,\mu\in[0,1]$): $$\begin{aligned}
0\leq x(1-\mu)&=&x+(0-x)\mu\leq x'= x+(1-2(1-x)\widehat{\Phi}(x,z)-x)\mu \\
&\leq& x+(1-x)\mu\leq x+(1-x)=1\end{aligned}$$ thus $x'\in[0,1]$. Moreover using $\mu\in[0,1]$ and $z\in[-1,0]$ we have: $$-1 =(1-\mu)(-1)-\mu\leq (1-\mu)z-\mu\leq(1-\mu)0-\mu=-\mu\leq0\: ,$$ thus $z'\in[-1,0]$ using $y'^2+z'^2=1$, $F_1:[0,1]\times M_1\to X$ is well-defined.\
Using Lemma \[salam3\], $F_1:[0,1]\times M_1\to X$ is continuous.
\[salam7\] For $x\in[0,1]$, the map $\Psi_x:[0,\frac12]\to\{z\in[-1,1]:x+z\geq0\}=[-1,1]\cap[-x,+\infty)=[-x,1]$ with $\Psi_x(t)=\sin(\pi t)-(1+\sin(\pi t)-4t)x$ is a homeomorphism.
Suppose $z\in[-1,1]$ and $z+x\geq0$. Since $\Psi_x(0)=-x$ and $\Psi_x(\frac12)=1$ by the mean value theorem there exists $t\in[0,\frac12]$ with $\Psi_x(t)=z$. Thus $\Psi_x:[0,\frac12]\to[-x,1]$ is a bijection continuous map which completes the proof.
\[salam9\] Using the same notations as in Lemma \[salam7\], $\widehat{\Psi}:\{(x,w)\in[0,1]\times[-1,1]:x+w\geq0\}\to[0,\frac12]$ with $\widehat{\Psi}(x,w)=\Psi_x^{-1}(w)$ is continuous.
Using Lemma \[salam7\], $\widehat{\Psi}:\{(x,w)\in[0,1]\times[-1,1]:x+w\geq0\}\to[0,\frac12]$ is well-defined. Let $B:=\{(x,w)\in[0,1]\times[-1,1]:x+w\geq0\}$. Consider $(x,w)\in B$, $s\in[0,\frac12]$, and sequence $\{(x_n,w_n):n\in{\mathbb N}\}$ such that ${\displaystyle\lim_{n\to\infty}x_n}=x$, ${\displaystyle\lim_{n\to\infty}w_n}=w$, ${\displaystyle\lim_{n\to\infty}\widehat{\Psi}(x_n,w_n)}=s$. Let $t=\widehat{\Psi}(x,w)$ and $t_n=\widehat{\Psi}(x_n,w_n)$ ($n\in\mathbb N$). We have: $$\begin{aligned}
\Psi_x(t) & = & w= {\displaystyle\lim_{n\to\infty}w_n}
={\displaystyle\lim_{n\to\infty}\Psi_{x_n}(t_n)} \\
& = & {\displaystyle\lim_{n\to\infty}\sin(\pi t_n)-(1+\sin(\pi t_n)-4t_n)x_n} \\
& = & \sin(\pi s)-(1+\sin(\pi s)-4s)x=\Psi_x(s)\end{aligned}$$ and $s=t$ since $\Psi_x$ is one to one according to Lemma \[salam7\]. Using the above discussion and the compactness of $[0,\frac12]$, $\widehat{\Psi}:\{(x,w)\in[0,1]\times[-1,1]:x+w\geq0\}\to[0,\frac12]$ is continuous.
\[salam11\] Consider $X=\{(x,y,z)\in{\mathbb R}^3:y^2+z^2=1,0\leq x\leq1\}$ and $\widehat{\Psi}$ as in Lemma \[salam9\]. Let $M_2=\{(x,y,z)\in X:x+z\geq0\}$, the map $F_2:[0,1]\times M_2\to X$ with $F_2(\mu,(x,y,z))=(x',y',z')$ for: $$\left\{\begin{array}{l} x'=x+(1-x)\mu \: , \\
z'=(1-\mu)z+(4\widehat{\Psi}(x,z)-1)\mu \: , \\
y'={\rm sgn}(y)\sqrt{1-z'^2} \: ,
\end{array}\right.$$ is continuous.
Let $(\mu,(x,y,z))\in[0,1]\times M_2$, since $x,\mu\in[0,1]$ we have $0\leq x\leq x+(1-x)\mu\leq x+(1-x)=1$ and $x'\in[0,1]$. Since $\widehat{\Psi}(x,z)\in[0,\frac12]$ we have $1-4\widehat{\Psi}(x,z)\in[-1,1]$. Now using $\mu\in[0,1]$ and $1-4\widehat{\Psi}(x,z),z\in[-1,1]$ we have $$-1=(1-\mu)(-1)+(-1)\mu\leq(1-\mu)z+(4\widehat{\Psi}(x,z)-1)\mu\leq1-\mu+\mu=1$$ therefore $z'\in[-1,1]$ using $y'^2+z'^2=1$, $F_2:[0,1]\times M_2\to X$ is well-defined.\
Using Lemma \[salam9\], $F_2:[0,1]\times M_2\to X$ is continuous.
\[construction1\] Consider $X=\{(x,y,z)\in{\mathbb R}^3:y^2+z^2=1,0\leq x\leq1\}$, $Y=\{(x,y,z)\in X:x=1\vee z=-1\}$, $M_1=\{(x,y,z)\in X:x+z\leq0\}$ and $M_2=\{(x,y,z)\in X:x+z\geq0\}$. $F:[0,1]\times X\to X$ with $F\restriction_{M_1}=F_1$ as in Lemma \[salam5\] and $F\restriction_{M_2}=F_2$ as in Lemma \[salam11\]. Then we have:
- $F:[0,1]\times X\to X$ is continuous.
- For all $(x,y,z)\in X$ we have $F(0,(x,y,z))=(x,y,z)$ and $F(1,(x,y,z))\in Y$
- For all $(x,y,z)\in Y$ and $\mu\in[0,1]$ we have $F(\mu,(x,y,z))=(x,y,z)$.
\(1) For all $x\in[0,1]$ we have $\widehat{\Phi}(x,-x)=\widehat{\Psi}(x,-x)=0$, so using Lemma \[salam5\], Lemma \[salam11\] and gluing lemma the map $F:[0,1]\times X\to X$ is continuous.\
(2) For $(x,y,z)\in X$, $F(0,(x,y,z))=(x,y,z)$ is clear by definition of $F_1$ and $F_2$. Suppose $F(1,(x,y,z))=(x_1,y_1,z_1)$. If $(x,y,z)\in M_1$, then $z_1=(1-1)z-1=-1$ and $F(1,(x,y,z))=(x_1,y_1,z_1)\in Y$. If $(x,y,z)\in M_2$, then $x_1=x+(1-x)1=1$ and $F(1,(x,y,z))=(x_1,y_1,z_1)\in Y$.\
(3) Suppose $(x,y,z)\in Y$, $\mu\in[0,1]$ and $F(\mu,(x,y,z))=(x',y',z')$. We have the following cases:\
[*Case 1*]{}. $z=-1$. In this case $y=0$, $(x,y,z)\in M_1$ and $\widehat{\Phi}(x,z)=\widehat{\Phi}(x,-1)=\frac12$. Thus $x'=x+(1-2(1-x)\frac12-x)\mu=x$, $z'=(1-\mu)(-1)-\mu=-1=z$ and $y'={\rm sgn}(y)\sqrt{1-z'^2}={\rm sgn}(y)\sqrt{1-1}=0=y$\
[*Case 2*]{}. $x=1$. In this case $y=0$, $(x,y,z)\in M_2$ and $\widehat{\Psi}(x,z)=\widehat{\Psi}(1,z)=t$ implies $z=\Psi_1(t)=4t-1$, i.e. $\widehat{\Psi}(1,z)=\frac{z+1}4$. Thus $x'=1+(1-1)\mu=1=x$, $z'=(1-\mu)z+(4\times\frac{z+1}4-1)\mu=z$, and $y'={\rm sgn}(y)\sqrt{1-z'^2}={\rm sgn}(y)\sqrt{1-z^2}={\rm sgn}(y)|y|=y$.\
Considering the above cases we are done.
\[construction2\] For $n\in\mathbb N$ let $$X_n=\{(x,y,z)\in{\mathbb R}^3:y^2+(z-\frac1{n})^2=\frac1{n^2},0\leq x\leq\frac1n\}\:,$$ and $X_0=\bigcup\{X_n:n\in\mathbb N\}$, in this construction we want to define a map $F_0:[0,1]\times X_0\to X_0$.\
Considering the same notations as in Construction \[construction1\] for $m\in\mathbb N$ and $(x,y,z)\in X_m$ we have $(mx,my,mz-1)\in X$. For $\mu\in[0,1]$ if $$F(\mu,(mx,my,mz-1))=(x'_m,y'_m,z'_m)\in X\:,$$ then $0\leq x'_m\leq1$ and $y'^2_m+z'^2_m=1$, thus $0\leq\frac{x'_m}{m}\leq\frac1m$ and $$\left(\frac{y'_m}{m}\right)^2+\left(\frac{z'_m+1}{m}-\frac1{m}\right)^2=\frac1{m^2}\:,$$ therefore $(\frac{x'_m}{m},\frac{y'_m}{m},\frac{z'_m+1}{m})\in X_m$, let $F_m(\mu,(x,y,z))=(\frac{x'_m}{m},\frac{y'_m}{m},\frac{z'_m+1}{m})$, i.e. $$F_m(\mu,(x,y,z))=\frac1{m}F(\mu,(mx,my,mz-1))+(0,0,\frac1m)\:.$$ It is clear that $F_m:[0,1]\times X_m\to X_m$ is continuous. Suppose $s,t\in\mathbb N$, $s<t$, $\mu\in[0,1]$ and $(x,y,z)\in F_s\cap F_t$, then: $$0\leq x\leq\min(\frac1t,\frac1s)\wedge y^2+(z-\frac1s)^2=\frac1{s^2}\wedge
y^2+(z-\frac1t)^2=\frac1{t^2}$$ which leads to $0\leq x\leq\frac1t(<\frac1s)$ and $y=z=0$. Now, since $(sx,0,-1),(tx,0,-1)\in Y$ (in Construction \[construction2\]), we have: $$F(\mu,(sx,0,-1))=(sx,0,-1)\:,\:F(\mu,(tx,0,-1))=(sx,0,-1)\:,$$ and: [$$\begin{aligned}
F_s(\mu,(x,y,z))& = & F_s(\mu,(x,0,0))=\frac1{s}F(\mu,(sx,0,-1))+(0,0,\frac1s) \\
& = & \frac1s(sx,0,-1)+(0,0,\frac1s)=(x,0,0)=\frac1s(tx,0,-1)+(0,0,\frac1t) \\
& = & \frac1{t}F(\mu,(tx,0,-1))+(0,0,\frac1t)
=F_t(\mu,(x,0,0))=F_t(\mu,(x,y,z))\end{aligned}$$]{} Therefore for $F_0=\bigcup\{F_n:n\in\mathbb N\}$, $F_0:[0,1]\times X_0\to X_0$ is well-defined.
Note: We recall that for $A\subseteq B$, we call $A$ a deformation retract of $B$ if there exists a continuous map $\nu:[0,1]\times B\to A$ with $\nu(0,b)=b$, $\nu(1,b)\in A$, and $\nu(t, a)=a$ (for all $b\in B,a\in A,t\in[0,1]$). It is well-known that if $A$ is a deformation retract of $B$ (and $a_0\in A$), then $\mathop{\Upsilon:\pi_1(A,a_0)\to\pi_1(B,a_0)}\limits_{{\: \: \: \: \:}{\: \: \: \: \:}[k]\mapsto[k]}$ is a group isomorphism, in particular $\pi_1(A)\cong\pi_1(B)$ [@M07 Theorem 58.3].
\[salam13\] For $n\in\mathbb N$ let $$X_n=\{(x,y,z)\in{\mathbb R}^3:y^2+(z-\frac1{n})^2=\frac1{n^2},0\leq x\leq\frac1n\}\:,$$ and $$Y_n=\{(x,y,z)\in X_n:x=\frac1{n}\vee z=0\}\:,$$ then $Y_0=\bigcup\{Y_n:n\in\mathbb N\}$ is a deformation retract of $X_0=\bigcup\{X_n:n\in\mathbb N\}$.
Consider $F_0:[0,1]\times X_0\to X_0$ as in Construction \[construction2\]. We prove the following claims:
- Claim 1. $F_0:[0,1]\times X_0\to X_0$ is continuous.
- Claim 2. $\forall(x,y,z)\in X_0{\: \: \: \: \:}(F_0(0,(x,y,z))=(x,y,z)\wedge F_0(1,(x,y,z))\in Y_0)$.
- Claim 3. $\forall(x,y,z)\in Y_0\:\forall\mu\in[0,1]{\: \: \: \: \:}F_0(\mu,(x,y,z))=(x,y,z)$.
[*Proof of Claim 1*]{}. Since for all $n\in\mathbb N$, $F_n:[0,1]\times X_n\to X_n$ is continuous, using the gluing lemma, $\bigcup\{F_i:1\leq i\leq n\}:[0,1]\times\bigcup\{X_i:1\leq i\leq n\}
\to\bigcup\{X_i:1\leq i\leq n\}$ is continuous.\
If $(x,y,z)\in X_0\setminus\{(0,0,0)\}$, then there exist $n\in\mathbb N$ and open neighborhood $V$ of $(x,y,z)$ in $X_0$ such that $V\subseteq\bigcup\{X_i:1\leq i\leq n\}$. Since $\bigcup\{F_i:1\leq i\leq n\}\restriction_{[0,1]\times V}
:[0,1]\times\bigcup\{X_i:1\leq i\leq n\} \to\bigcup\{X_i:1\leq
i\leq n\}$ is continuous, $\bigcup\{F_i:1\leq i\leq
n\}:[0,1]\times V \to X_0$ is continuous, i.e. $F_0\restriction_{[0,1]\times V}:[0,1]\times V \to X_0$ is continuous, therefore $F_0$ is continuous in all points of $[0,1]\times\{(x,y,z)\}$.\
In order to show the continuity of $F_0:[0,1]\times X_0\to X_0$, we should prove that it is continuous in all points $(\mu,(0,0,0))$ ($\mu\in[0,1]$). Consider $\varepsilon>0$ there exists $n\in\mathbb N$ such that $\frac{\sqrt6}n<\varepsilon$ for all $(x,y,z)\in X_0$ and $\mu,\lambda\in[0,1]$ we have (consider $[0,1]\times X_0$ and $X_0$ respectively under Euclidean norm of ${\mathbb R}^4$ and ${\mathbb R}^3$):
$||(\mu,(0,0,0))-(\lambda,(x,y,z))||<\frac1n $ $$\begin{aligned}
& \Rightarrow & x<\frac1n \\
& \Rightarrow & (x,y,z)\in\bigcup\{X_i:i\geq n\} \\
& \Rightarrow & F_0(\lambda,(x,y,z))=\bigcup\{F_i:i\geq n\}(\lambda,(x,y,z)) \\
& \Rightarrow & F_0(\lambda,(x,y,z))\in
\bigcup\{F_i:i\geq n\}([0,1]\times\bigcup\{X_i:1\leq i\geq n\}) \\
& \Rightarrow & F_0(\lambda,(x,y,z))\in
\bigcup\{F_i([0,1]\times X_i):i\geq n\}=\bigcup\{X_i:i\geq n\} \\
& \Rightarrow & ||F_0(\lambda,(x,y,z))||
\leq\max\{\frac{\sqrt6}{i}:i\geq n\}=\frac{\sqrt6}{n} \\
& \Rightarrow & ||F_0(\lambda,(x,y,z))-F_0(\mu,(0,0,0))||=||F_0(\lambda,(x,y,z))||\leq
\frac{\sqrt6}{n}<\varepsilon\end{aligned}$$ (note to the fact that $X_n\subseteq[0,\frac1n]\times[-\frac1n,\frac1n]\times[0,\frac2n]$, thus for all $(u,v,w)\in X_n$ we have $||(u,v,w)||\leq\sqrt{\frac1{n^2}+\frac1{n^2}+\frac4{n^2}}=\frac{\sqrt6}{n}$) therefore $F_0:[0,1]\times X_0\to X_0$ is continuous in $(\mu,(0,0,0))$ as well as it is continuous in other points of $[0,1]\times X_0$.\
[*Proof of Claim 2*]{}. Suppose $(x,y,z)\in X_0$, there exists $n\in\mathbb N$ such that $(x,y,z)\in X_n$, using Construction \[construction1\] (2), we have: $$\begin{aligned}
F_0(0,(x,y,z)) & = & F_n(0,(x,y,z))=\frac1{n}F(0,(nx,ny,nz-1))+(0,0,\frac1n) \\
& = & \frac1{n}(nx,ny,nz-1)+(0,0,\frac1n)=(x,y,z)\end{aligned}$$ and $F_0(1,(x,y,z))=F_n(1,(x,y,z))=\frac1{n}F(1,(nx,ny,nz-1))+(0,0,\frac1n)$, by Construction \[construction1\] (2) we have $F(1,(nx,ny,nz-1))\in Y$ which leads to $F_0(1,(x,y,z))\in\frac1{n}Y+(0,0,\frac1n)=Y_n\subseteq Y_0$.\
[*Proof of Claim 3*]{}. Suppose $\mu\in[0,1]$ and $(x,y,z)\in X_0$, there exists $n\in\mathbb N$ such that $(x,y,z)\in Y_n\subseteq X_n$, now we have (use Construction \[construction1\] (3)): $$\begin{aligned}
(x,y,z)\in Y_n & \Rightarrow &
((x,y,z)\in X_n\wedge x=\frac1n)\vee ((x,y,z)\in X_n\wedge z=0) \\
& \Rightarrow & (y^2+(z-\frac1n)^2=\frac1{n^2}\wedge x=\frac1n)
\vee(0\leq x\leq\frac1n\wedge y=z=0) \\
& \Rightarrow & (((ny)^2+(nz-1)^2=1\wedge nx=1) \\
& & {\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}\vee(0\leq nx\leq1\wedge ny=0\wedge nz-1=-1)) \\
& \Rightarrow & (nx,ny,nz-1)\in Y \\
& \Rightarrow & F(\mu,(nx,ny,nz-1))=(nx,ny,nz-1)\end{aligned}$$ thus $$\begin{aligned}
F_0(\mu,(x,y,z)) & = & F_n(\mu,(x,y,z))=\frac1{n}F(\mu,(nx,ny,nz-1))+(0,0,\frac1n) \\
& = & \frac1{n}(nx,ny,nz-1)+(0,0,\frac1n)=(x,y,z)\end{aligned}$$ Which completes the proof of Claim 3.\
Using Claims 1, 2, and 3, $Y_0$ is a deformation retract of $X_0$.
\[retract\] Under the same notations as in Construction \[construction2\] and Lemma \[salam13\], $Z_0=\{(0,y,z):\exists x\:(x,y,z)\in X_0\}$ is a deformation retract of $X_0$. In particular $\pi_1(Y_0)\cong\pi_1(X_0)\cong\pi_1(Z_0)$.
The map $\mathop{[0,1]\times X_0\to Z_0}\limits{(\mu,(x,y,z))\mapsto
((1-\mu)x,y,z)}$ shows that $Z_0$ is a deformation retract of $X_0$ too. Now use [@M07 Theorem 58.3] to complete the proof.
\[retract10\] Two sets $\mathcal X$ and $\mathcal W$ are homeomorphic with deformation retracts of $\mathcal V$, therefore $\pi_1(\mathcal X)\cong\pi_1(\mathcal V)\cong\pi_1(\mathcal W)$.
Under the same notations as in Theorem \[retract\], $\mathcal X$ and $Z_0$ are homeomorph, moreover $Y_0$ and $\mathcal W$ are homeomorph too, also $X_0=\mathcal V$. Now by Theorem \[retract\] we have $\pi_1({\mathcal X})\cong\pi_1({\mathcal V})\cong\pi_1({\mathcal W})$.
A distinguished counterexample
==============================
In Section 11 we have proved $\pi_1(\mathcal X)\cong\pi_1(\mathcal W)$, in this section we prove $\mathfrak P^\omega(\mathcal X)\ncong\mathfrak
P^\omega(\mathcal W)$.
\[tir10\] We have $|{\mathfrak P}^\omega(\mathcal X)|=\omega$.
For $n\in\mathbb N$ consider $\rho_n:[0,1]\to C_n$ with $\rho_n(t)=\frac1ne^{2\pi it-\frac{\pi i}2}+\frac{i}{n}$, then $\omega-$loops $\rho_n,\rho_m:[0,1]\to{\mathcal X}$ are homotopic if and only if $n=m$. Therefore $\{[\rho_n]:n\in\mathbb N\}$ is an infinite subset of ${\mathfrak P}^\omega(\mathcal X)$ which leads to $|{\mathfrak P}^\omega(\mathcal X)|\geq\omega$. On the other hand as it has been mentioned in Note \[Narges4\] (4), if $[f]\in{\mathfrak P}^\omega(\mathcal X)$, then $|A(f)|<\omega$, which leads to ${\mathfrak P}^\omega(\mathcal X)\subseteq*\{\pi_1(C_n):n\in\mathbb N\}$, thus $$\begin{aligned}
|{\mathfrak P}^\omega(\mathcal X)| & \leq & |*\{\pi_1(C_n):n\in\mathbb N\}| \\
& = & |\{\rho_{i_1}^{j_1}*\rho_{i_2}^{j_2}*\cdots*\rho_{i_m}^{j_m}:m\in\mathbb N,
i_1,j_1,i_2,j_2,\ldots,i_m,j_m\in\mathbb Z\}| \\
& \leq & |{\displaystyle\bigcup_{m\in\mathbb N}\{(i_1,j_1,\cdots,i_m,j_m):
i_1,j_1,\ldots,i_m,j_m\in\mathbb Z\}}|=\omega
\end{aligned}$$ Hence $|{\mathfrak P}^\omega(\mathcal X)|=\omega$.
\[tir20\] We have $|{\mathfrak P}^\omega(\mathcal W)|=c$.
It is well-known that for all Hausdorff separable space $A$, $|C(A,\mathbb R^2)|\leq c$ where $C(A,B)$ denotes the collection of all continuous maps $\phi:A\to B$. Therefore $$|{\mathfrak P}^\omega(\mathcal W)|\leq
|C([0,1],\mathcal W)|\leq|C([0,1],\mathbb R^2)|=c\:.$$ Now for all $a=(a_n:n\in\mathbb N)\in\{0,1\}^{\mathbb N}$ define $f_a:[0,1]\to\mathcal W$ with:
then $f_a:[0,1]\to\mathcal W$ is an $\omega-$loop, thus $[f_a]\in{\mathfrak P}^\omega(\mathcal W)$. We claim that $\psi:\{0,1\}^{\mathbb N}\to{\mathfrak P}^\omega(\mathcal X)$ with $\psi(a)=[f_a]$ ($a\in\{0,1\}^{\mathbb N}$) is one to one. Let $a=(a_n:n\in\mathbb N),b=(b_n:n\in\mathbb N)\in\{0,1\}^{\mathbb N}$ and $a\neq b$, then there exists $m\in\mathbb N$ such that $a_m\neq b_m$. Suppose $a_m=0$ and $b_m=1$. Let $W:=\{\frac{1}{2^{m+1}}e^{2\pi i\theta}+\frac{1}{m}+\frac{i}{2^{m+1}}:
\theta\in[0,1]\}$. Since $f_a^W$ is constant map $\frac1m$, $[f_a^W]$ is null-homotopic. However $[f_b^W]$ is not null-homotopic, thus $[f_a^W]\neq[f_b^W]$ which leads to $[f_a]\neq[f_b]$ according to Convention \[good10\]. Hence $\psi:\{0,1\}^{\mathbb N}\to{\mathfrak P}^\omega(\mathcal X)$ is one to one which leads to $|{\mathfrak P}^\omega(\mathcal X)|\geq|\{0,1\}^{\mathbb N}|=c$ and completes the proof.
\[example5\] Two groups $\pi_1(\mathcal X)$ and $\pi_1(\mathcal W)$ are isomorphic and two groups $\mathfrak P^\omega(\mathcal X)$ and $\mathfrak P^\omega(\mathcal W)$ are non-isomorphic. Briefly $\pi_1(\mathcal X)\cong\pi_1(\mathcal W)$ and $\mathfrak P^\omega(\mathcal X)\ncong\mathfrak P^\omega(\mathcal W)$ (use Lemma \[tir10\], Lemma \[tir20\], and Corollary \[retract10\]).
A diagram and a hint
====================
Consider the following diagram:
Arrows (III) and (IV) are valid regarding Theorem \[Narges3\] (1). However by Counterexample \[example5\], there exist $X, Y$ such that $\pi_1(X)\cong\pi_1(Y)$ and ${\mathfrak
P}^\omega(X)\cong{\mathfrak P}^\omega(Y)$, thus: $$\pi_1(X)\cong\pi_1(Y)\wedge\neg(\forall\alpha\geq\omega\:{\mathfrak P}^\alpha(X)
\cong{\mathfrak P}^\alpha(Y))$$ Hence the above diagram is valid. We have the following arising problems:
Find a counterexample for arrow (I), i.e. find $X$, $Y$ such that $\pi_1(X)\cong\pi_1(Y)$, ${\mathfrak P}^c(X)\cong{\mathfrak P}^c(Y)$ and ${\mathfrak P}^\omega(X)\ncong{\mathfrak P}^\omega(Y)$ (Hint: is it true that ${\mathfrak P}^c(\mathcal X )\cong{\mathfrak P}^c(\mathcal W )$).
Find a counterexample for arrow (II), i.e. find $X$, $Y$ such that $\pi(X)_1\cong\pi_1(Y)$ and ${\mathfrak P}^c(X)\ncong{\mathfrak P}^c(Y)$.
A Strategy for Future and Conjecture
====================================
Let’s extend of the idea of this text to homotopy group of order $n$. Let $b\in \mathbb S^n$ be a fixed point. For infinite cardinal number $\alpha$ and ideal $\mathcal I$ on $X$ which contains all finite subsets of $X$, if $f,g:\mathbb S^n\to X$ are $\alpha\frac{\mathcal I}{}$maps, with $f(b)=g(b)$, then it is easy to see that $f\vee
g:\mathbb S^n\to X$ is $\alpha\frac{\mathcal I}{}$map too. So we may have the following definition.
For $a\in X$, by $\mathfrak P_{(n,\mathcal I)}^\alpha(X,a)$ we mean subgroup of $\pi_n(X,a)$ generated by $\alpha\frac{\mathcal I}{}$maps with base point $a$.
It’s evident by the definition that for ideals ${\mathcal I}$, ${\mathcal J}$ on $X$ containing finite subsets, transfinite cardinal number $\alpha$, and $a\in X$, we have:
- If ${\mathcal I}\subseteq{\mathcal J}$, then $\mathfrak P_{(n,\mathcal I)}^\alpha(X,a)\subseteq\mathfrak
P_{(n,\mathcal J)}^\alpha(X,a)$;
- $\mathfrak P_{(n,\mathcal I\cap \mathcal J)}^\alpha(X,a)
\subseteq \mathfrak P_{(n,\mathcal I)}^\alpha(X,a)
\cap\mathfrak P_{(n,\mathcal J)}^\alpha(X,a)$.
Now we are ready to the following conjecture:\
[**Conjecture.**]{} Arc connected spaces $X$ and $Y$ are homeomorph if and only if there exists a bijection $f:X\to Y$ such that for all nonzero cardinal number $\alpha$ and all ideal $\mathcal I$ on $X$, $\mathfrak P_{\mathcal I}^\alpha(X)
\cong\mathfrak P_{f({\mathcal I})}^\alpha(Y)$.
Conclusion
==========
In this paper, for arc connected locally compact Hausdorff topological space $X$ (with at least two elements), $a\in X$, nonzero cardinal number $\alpha$, and ideal $\mathcal
I$ on $X$ we introduce ${\mathfrak P}_{\mathcal I}^\alpha(X,a)$ as a subgroup of $\pi_1(X,a)$. We prove that for transfinite $\alpha$ and $a,b\in X$ two groups ${\mathfrak P}_{\mathcal
I}^\alpha(X,a)$ and ${\mathfrak P}_{\mathcal I}^\alpha(X,b)$ are isomorphic, therefore for transfinite $\alpha$ we denote ${\mathfrak P}_{\mathcal I}^\alpha(X,a)$ simply by ${\mathfrak
P}_{\mathcal I}^\alpha(X)$ and ${\mathfrak
P}_{\{\varnothing\}}^\alpha(X)$ simply by ${\mathfrak
P}^\alpha(X)$. Moreover for $\alpha\geq2^c$ we have ${\mathfrak
P}_{\mathcal I}^\alpha(X)=\pi_1(X)$, hence the most interest is in $\omega\leq\alpha<2^c$ using GCH we prefer to study $\alpha\in\{\omega,c\}$. We obtain that for Hawaiian earring (infinite earring) $\mathcal X$, three groups ${\mathfrak
P}^\omega(\mathcal X)$, ${\mathfrak P}^c(\mathcal X)$, and ${\mathfrak P}^{2^c}(\mathcal X)(=\pi_1(\mathcal X))$ are pairwise distinct. Also we introduce $\mathcal Y$ such that ${\mathfrak P}_{{\mathcal P}_{fin}({\mathcal Y})}^\omega(\mathcal
Y)$, ${\mathfrak P}_{{\mathcal P}_{fin}({\mathcal Y})}^c(\mathcal
Y)$, and ${\mathfrak P}_{{\mathcal P}_{fin}({\mathcal
Y})}^{2^c}(\mathcal Y) (=\pi_1(\mathcal Y))$ are pairwise distinct. We find $\mathcal W$ such that $\pi_1(\mathcal
X)\cong\pi_1(\mathcal W)$ and ${\mathfrak P}^\omega(\mathcal
X)\ncong{\mathfrak P}^\omega(\mathcal W)$, this example leads us to the fact that we can classify spaces with isomorphic first homotopy groups using the concept of ${\mathfrak P}^\alpha(-)$s (*first homotopy groups with respect to $\alpha\geq\omega$*). However investigating the structure of our examples and specially Section 12, shows remarkable role of the number of (locally) cut points their and order in $\alpha-$arcs, $\alpha\frac{\mathcal I}{}$arcs, and our constructed subgroups of first fundamental group.
Acknowledgement {#acknowledgement .unnumbered}
===============
With special thanks to our friends A. Hosseini, P. Mirzaei, and M. Nayeri for their helps and comments.
[AA]{}
Ayatollah Zadeh Shirazi, F., [*Linear connectivity*]{}, Advanced Studies in Contemporary Mathematics, 2007 ([**14**]{}, 2) 317-323.
Holz, M.; Steffens, K.; Weitz, E., [*Introduction to cardinal arithmetic*]{}, Birkh$\ddot{a}$user-Verlag, 1999.
Munkres, J. R., [*Elements of algebraic topology*]{}, Addison-Wesley Publishing Company, 1984.
Munkres, J. R., [*Topology*]{}, Prentice-Hall of India (2nd. Ed.), 2007.
Rudin, W., [*Principles of mathematical analysis*]{}, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co. (3rd. Ed.), 1976.
$$\underline{{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}{\: \: \: \: \:}}$$ [, Faculty of Mathematics, Statistics and Computer Science, College of Science, University of Tehran , Enghelab Ave., Tehran, Iran\
([*e-mail*]{}: fatemah@khayam.ut.ac.ir)\
[**Mohammad Ali Mahmoodi**]{}, Faculty of Mathematics, Statistics and Computer Science, College of Science, University of Tehran , Enghelab Ave., Tehran, Iran\
([*e-mail*]{}: m.a.mahmoodi@ut.ac.ir)]{}
|
---
address: 'Department of Mathematics and Statistics, University of Ottawa, Canada'
author:
- Sanghoon Baek
---
Introduction
============
Let $G$ be a split simple simply connected group of rank $n$ over a field $F$. Fix a maximal split torus $T$ of $G$ and a Borel subgroup $B$ containing $T$. We denote by $W$ the Weyl group of $G$ with respect to $T$. Let $\Lambda$ be the weight lattice of $G$ (hence, $T^{*}=\Lambda$).
We denote by $\omega_{1},\cdots, \omega_n$ the fundamental weights of $\Lambda$. We let $I_{K}:={\operatorname{Ker}}({\mathbb{Z}}[\Lambda]\to {\mathbb{Z}})$ and $I_{CH}:={\operatorname{Ker}}(S^{*}(\Lambda)\to {\mathbb{Z}})$ be the augmentation ideals, where ${\mathbb{Z}}[\Lambda]\to {\mathbb{Z}}$ (respectively, $S^{*}(\Lambda)\to {\mathbb{Z}}$) is the map from the group ring ${\mathbb{Z}}[\Lambda]$ (respectively, the symmetric algebra) of $\Lambda$ to the ring of integers by sending $e^{\lambda}$ to $1$ (respectively, any element of positive degree to $0$).
For any $i\geq 0$, we consider the ring homomorphism $$\phi^{(i)}:{\mathbb{Z}}[\Lambda]\to{\mathbb{Z}}[\Lambda]/I_{K}^{i+1}\to S^{*}(\Lambda)/I_{CH}^{i+1}\to S^{i}(\Lambda),$$ where the first and the last maps are projections and the middle map sends $e^{\sum_{j=1}^{n}a_j\omega_j}$ to $\prod_{j=1}^{n}(1-\omega_j)^{-a_j}$. The *$i$th-exponent of $G$* (denoted by $\tau_{i}$), as introduced in [@BNZ], is the gcd of all nonnegative integers $N_{i}$ satisfying $$N_i\cdot (I_{CH}^W)^{(i)} \subseteq \phi^{(i)}(I_{K}^W),$$ where $I_{K}^W:=\langle {\mathbb{Z}}[\Lambda]^W\cap I_{K}\rangle$ (respectively, $I_{CH}^W:=\langle S^{*}(\Lambda)^W\cap I_{CH}\rangle$) denotes the $W$-invariant augmentation ideal of ${\mathbb{Z}}[\Lambda]$ (respectively, $S^{*}(\Lambda)$) and $(I_{CH}^W)^{(i)}=I_{CH}^W\cap S^i(\Lambda)$. Informally, these numbers $\tau_i$ measure how far is the ring $S^{*}(\Lambda)^{W}$ from being a polynomial ring in basic invariants.
For any $i\leq 4$, it was shown that the $i$th-exponent of $G$ divides the Dynkin index in [@BNZ] and this integer was used to estimate the torsion of the Grothendieck gamma filtration and the Chow groups of $E/B$, where $E/B$ denotes the twisted form of the variety of Borel subgroups $G/B$ for a $G$-torsor $E$.
In this paper, we show that all the remaining exponents of spinor groups divide the Dynkin index $2$.
#### **Acknowledgments.**
The work has been partially supported from the Fields Institute and from Zainoulline’s NSERC Discovery grant 385795-2010.
Exponent
========
Let $G$ be ${\operatorname{\mathbf{Spin}}}_{2n+1}$ ($n\geq 3$) or ${\operatorname{\mathbf{Spin}}}_{2n}$ ($n\geq 4$). The fundamental weights are defined by $$\begin{aligned}
\label{fundamental weights}
\omega_{1}&=e_{1}, \omega_{2}=e_{1}+e_{2}, \cdots, \omega_{n-1}=e_{1}+\cdots+e_{n-1}, \omega_{n}=\frac{e_{1}+\cdots+e_{n}}{2},\\
\omega_{1}&=e_{1}, \omega_{2}=e_{1}+e_{2}, \cdots, \omega_{n-1}=\frac{e_{1}+\cdots+e_{n-1}-e_{n}}{2}, \omega_{n}=\frac{e_{1}+\cdots+e_{n}}{2},\end{aligned}$$ respectively, where the canonical basis of $\mathbb{R}^{n}$ is denoted by $e_{i}$ ($1\leq i \leq n$).
For $1\leq i \leq n$, let $$\label{basicinv}
q_{2i}:=e_{1}^{2i}+\cdots +e_{n}^{2i}$$ be the basic invariants of the group $G$, i.e., be algebraically independent homogeneous generators of $S^*(\Lambda)^{W}$ as a ${\mathbb{Q}}$-algebra (see [@Hum §3.5 and §3.12]), together with $$\label{Dndegn}
q'_{n}:=e_{1}\cdots e_{n}$$ if $G={\operatorname{\mathbf{Spin}}}_{2n}$.
For any $\lambda\in \Lambda$, we denote by $W(\lambda)$ the $W$-orbit of $\lambda$. For any finite set $A$ of weights, we denote $-A$ the set of opposite weights.
The Weyl groups of ${\operatorname{\mathbf{Spin}}}_{2n+1}$ and ${\operatorname{\mathbf{Spin}}}_{2n}$ are $({\mathbb{Z}}/2{\mathbb{Z}})^{n}\rtimes S_{n}$ and $({\mathbb{Z}}/2{\mathbb{Z}})^{n-1}\rtimes S_n$, respectively. Hence, by the action of these Weyl groups, one has the following decomposition of $W$-orbits: if $G={\operatorname{\mathbf{Spin}}}_{2n+1}$ (respectively, $G={\operatorname{\mathbf{Spin}}}_{2n}$), then for any $1\leq k \leq n-1$ (respectively, $1\leq k\leq n-2$) $$\label{orbitwk}
W(\omega_{k})=W_{+}(\omega_{k})\cup -W_{+}(\omega_{k}),$$ where $W_{+}(\omega_{k})=\{e_{i_1}\pm \cdots \pm e_{i_{k}}\}_{i_{1}<\cdots <i_{k}}$. If $n$ is even, then the $W$-orbits of the last two fundamental weights of ${\operatorname{\mathbf{Spin}}}_{2n}$ are given by $$\label{orbitwn}
W(\omega_{n-1})=W_{+}(\omega_{n-1})\cup -W_{+}(\omega_{n-1}) \text{ and } W(\omega_{n})=W_{+}(\omega_{n})\cup -W_{+}(\omega_{n}),$$ where $W_{+}(\omega_{n-1})$ (respectively, $W_{+}(\omega_{n})$) is the subset of $W(\omega_{n-1})$ (respectively, $W(\omega_{n})$) containing elements of the positive sign of $e_1$.
For any $\lambda=\sum_{j=1}^{n}a_j\omega_{j}\in \Lambda$ and any integer $m\geq 0$, we set $\lambda(m)=\sum_{j=1}^{n}a_j\omega_{j}^{m}$. For example, $\lambda(0)=\sum_{j=1}^{n}a_j$ and $\lambda(1)=\lambda$. We shall need the following lemma:
\[elemenlemma\] $(i)$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n+1}$ $($respectively, ${\operatorname{\mathbf{Spin}}}_{2n}$$)$, then for any odd integer $p$, any nonnegative integers $m_1,\cdots, m_p$ and, any $1\leq k\leq n-1$ $($respectively, any $1\leq k\leq n-2$$)$, we have $$\sum_{\lambda\in W(\omega_{k})}\lambda(m_1)\cdots \lambda(m_{p})=0.$$
$(ii)$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n}$ with odd $n$, then for any even integer $p$ and any nonnegative integers $m_1,\cdots, m_p$, we have $$\sum_{\lambda\in W(\omega_{n})}\lambda(m_1)\cdots \lambda(m_{p})=\sum_{\lambda\in W(\omega_{n-1})}\lambda(m_1)\cdots \lambda(m_{p}).$$
$(iii)$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n}$, then for any odd integer $p<n$ and any nonnegative integers $m_1,\cdots, m_p$, we have $$\sum_{\lambda\in W(\omega_{n})}\lambda(m_1)\cdots \lambda(m_{p})=\sum_{\lambda\in W(\omega_{n-1})}\lambda(m_1)\cdots \lambda(m_{p})=0.$$
$(i)$ It follows from (\[orbitwk\]) that $$\begin{aligned}
\sum_{\lambda\in W(\omega_{k})}\lambda(m_1)\cdots \lambda(m_{p})&=\sum_{\lambda\in W_{+}(\omega_{k})}\lambda(m_1)\cdots \lambda(m_{p})+\sum_{\lambda\in -W_{+}(\omega_{k})}\lambda(m_1)\cdots \lambda(m_{p})\\
&=\sum_{\lambda\in W_{+}(\omega_{k})}\lambda(m_1)\cdots \lambda(m_{p})-\sum_{\lambda\in W_{+}(\omega_{k})}\lambda(m_1)\cdots \lambda(m_{p})\\
&=0.\end{aligned}$$
$(ii)$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n}$ with odd $n$, then we have $W(\omega_{n})=-W(\omega_{n-1})$. Hence, the result immediately follows from the assumption that $p$ is even.
$(iii)$ If $n$ is even, then the result follows from (\[orbitwn\]) by the same argument as in the proof of $(i)$. In general, note that for any $\lambda_{i_{1}},\cdots, \lambda_{i_{p}}\in W_{+}(\omega_{1})$ the term $\lambda_{i_{1}}(m_1)\cdots \lambda_{i_{p}}(m_{p})/2^{p}$ (respectively, -$\lambda_{i_{1}}(m_1)\cdots \lambda_{i_{p}}(m_{p})/2^{p}$) appears $2^{n-2}$ times (respectively, $2^{n-2}$) in both sums in $(iii)$.
Let $p$ be an even integer and $q\geq 2$ an integer. For any nonnegative integers $m_1,\cdots, m_p$, we define $$\Lambda(p,q)(m_1,\cdots,m_p):=\sum \lambda_{j_1}(m_1)\cdots \lambda_{j_p}(m_{p}),$$ where the sum ranges over all different $\lambda_{i_1},\cdots,\lambda_{i_{q}}\in W_{+}(\omega_1)$ and all $\lambda_{i_1},\cdots,\lambda_{i_{p}}\\\in \{\lambda_{i_1},\cdots,\lambda_{i_{q}}\}$ such that the numbers of $\lambda_{i_1},\cdots,\lambda_{i_{q}}$ appearing in $\lambda_{i_1},\cdots,\lambda_{i_{p}}$ are all nonnegative even solutions of $x_1+\cdots+x_q=p$. If $p<2q$, then we set $\Lambda(p,q)(m_1,\cdots,m_p)=0$. Given $m_1,\cdots,m_p$, we simply write $\Lambda(p,q)$ for $\Lambda(p,q)(m_1,\cdots,m_p)$.
For instance, $\Lambda(4,2)$ is the sum of $\lambda_{j_{1}}(m_1)\lambda_{j_{2}}(m_2)\lambda_{j_{3}}(m_3)\lambda_{j_{4}}(m_4)$ for all $j_{1}, j_{2}, j_{3}, j_{4}\in \{i,j\}$ and all $1\leq i\neq j\leq n$ such that two $i$’s and two $j$’s appear in $j_{1}, j_{2}, j_{3}, j_{4}$.
\[omega2eg\] We observe that $$\label{simpleob}
(x_1+x_2)(x'_1+x'_2)+(x_1-x_2)(x'_1-x'_2)=2(x_1x'_1+x_2x'_2).$$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n+1}$ or ${\operatorname{\mathbf{Spin}}}_{2n}$, then by (\[orbitwk\]) and (\[simpleob\]) we have $$\sum_{W_{+}(\omega_2)}\lambda(m_1)\lambda(m_2)=2(n-1)\sum_{W_{+}(\omega_1)}\lambda(m_1)\lambda(m_2)$$ for any nonnegative integers $m_1$ and $m_2$ as we have $(n-1)$ choices of such pairs in the left hand side of (\[simpleob\]) from $W_{+}(\omega_2)$, which implies that $$\sum_{W(\omega_2)}\lambda(m_1)\lambda(m_2)=2(n-1)\sum_{W(\omega_1)}\lambda(m_1)\cdots\lambda(m_2),$$ (cf. [@BNZ Lemma 5.1(ii)]). For any even $p\geq 4$, we apply the same argument with the expansion of $(x_1+x_2)\cdots(x_{1}^{(p)}+x_{2}^{(p)})+(x_1-x_2)\cdots(x_{1}^{(p)}-x_{2}^{(p)})$. Then, we have $$\sum_{W_{+}(\omega_2)}\lambda(m_1)\cdots\lambda(m_p)=2(n-1)\sum_{W_{+}(\omega_1)}\lambda(m_1)\cdots\lambda(m_p)+2\Lambda(p,2),$$ which implies that $$\sum_{W(\omega_2)}\lambda(m_1)\cdots\lambda(m_p)=2(n-1)\sum_{W(\omega_1)}\lambda(m_1)\cdots\lambda(m_p)+2^{2}\Lambda(p,2).$$
We generalize Example \[omega2eg\] to any $\omega_{k}$ as follows.
\[generallem\] If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n+1}$ $($respectively, ${\operatorname{\mathbf{Spin}}}_{2n}$$)$, then for any $1\leq k \leq n-1$ $($respectively, $1\leq k\leq n-2$$)$, any even $p$, and any nonnegative integers $m_1,\cdots m_p$ we have $$\sum_{W(\omega_{k})}\lambda(m_1)\cdots \lambda(m_p)=2^{k-1}{{n-1}\choose{k-1}}\sum_{W(\omega_{1})}\lambda(m_1)\cdots \lambda(m_p)+\sum_{j=2}^{k}2^{k}{{n-j}\choose{k-j}}\Lambda(p,j).$$
For any $\lambda \in W(\omega_1)$, there are $2^{k}{{n-1}\choose{k-1}}$ choices of the element containing $\lambda$ in $W(\omega_k)$, thus we have the term $2^{k-1}{{n-1}\choose{k-1}}\sum_{W(\omega_{1})}\lambda(m_1)\cdots \lambda(m_p)$ in $\sum_{W(\omega_{k})}\lambda(m_1)\cdots \lambda(m_p)$.
If an element $\lambda\in W(\omega_1)$ appears odd times in a term $\lambda_{i_1}(m_1)\cdots \lambda_{i_p}(m_p)$ of $\sum_{W(\omega_{k})}\lambda(m_1)\cdots \lambda(m_p)$, where $\lambda_{i_1},\cdots,\lambda_{i_p}\in W(\omega_1)$, then by the action of Weyl group this term vanishes in $\sum_{W(\omega_{k})}\lambda(m_1)\cdots \lambda(m_p)$. Hence, the remaining terms in $\sum_{W(\omega_{k})}\lambda(m_1)\cdots \lambda(m_p)$ are a linear combination of $\Lambda(p,j)$ for all $2\leq j\leq k$ such that $p\geq 2k$. As each term $\Lambda(p,j)$ appears $2^{k}{{n-j}\choose{k-j}}$ times in $\sum_{W(\omega_{k})}\lambda(m_1)\cdots \lambda(m_p)$, the result follows.
For any $\lambda\in \Lambda$, we denote by $\rho(\lambda)$ the sum of all elements $e^{\mu}\in {\mathbb{Z}}[\Lambda]$ over all elements $\mu$ of $W(\lambda)$. Let $i!\cdot \phi^{(i)}(e^\lambda)=\lambda^{i}+S_{i}$ for any $i\geq 1$, where $S_i$ is the sum of remaining terms in $i!\cdot \phi^{(i)}(e^\lambda)$ and $\lambda=\sum a_{j}\omega_j$, $a_{j}\in {\mathbb{Z}}$. Hence, for any fundamental weight $\omega_k$ we have $$\label{bsifactorial}
i!\cdot \phi^{(i)}(\rho(\omega_{k}))=\sum_{W(\omega_{k})}\lambda^{i}+\sum_{W(\omega_{k})}S_{i}.$$ We view $i!\cdot \phi^{(i)}(e^\lambda)$ as a polynomial in variables $\lambda, \lambda(m_1),\cdots ,\lambda(m_j)$ for some nonnegative integers $m_1,\cdots, m_j$. Let $T_{i}$ be the sum of monomials in $S_i$ whose degrees are even.
If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n+1}$ $($respectively, ${\operatorname{\mathbf{Spin}}}_{2n}$$)$, then by Lemma \[elemenlemma\](i) the equation (\[bsifactorial\]) reduces to $$\label{bsifactorial2}
i!\cdot \phi^{(i)}(\rho(\omega_{k}))=\sum_{W(\omega_{k})}\lambda^{i}+\sum_{W(\omega_{k})}T_{i}.$$ for any $1\leq k\leq n-1$ $($respectively $1\leq k\leq n-2$$)$.
Given $p$ and $q$, we define $$\Omega(p,q):=\sum\Lambda(p,q)(m_1,\cdots,m_p),$$ where the sum ranges over all $m_1,\cdots,m_p$ which appear in all monomials of $T_i$.
\[deg6phi\]
$(i)$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n+1}$ or ${\operatorname{\mathbf{Spin}}}_{2n}$ and $i=4$, then by (\[bsifactorial2\]) and Lemma \[generallem\] we have $$\begin{aligned}
4!\phi^{(4)}(\rho(\omega_{1}))&=\sum_{W(\omega_{1})}\lambda^{4}+\sum_{W(\omega_{1})}T_{4},\\
4!\phi^{(4)}(\rho(\omega_{2}))&=\sum_{W(\omega_{2})}\lambda^{4}+\sum_{W(\omega_{2})}T_{4}\\
&=\sum_{W(\omega_{2})}\lambda^{4}+2(n-1)\sum_{W(\omega_{1})}T_{4},\end{aligned}$$ which implies that $$4!(\phi^{(4)}(\rho(\omega_{2}))-2(n-1)\phi^{(4)}(\rho(\omega_{1})))=\sum_{W(\omega_{2})}\lambda^{4}-2(n-1)\sum_{W(\omega_{1})}\lambda^{4}.$$ By Lemma \[generallem\], the right-hand side of the above equation is equal to $$4\Lambda(4,2)=4\cdot \frac{4!}{2!2!}\sum_{i<j}e_{i}^{2}e_{j}^{2}.$$ Hence, we have $$\phi^{(4)}(\rho(\omega_{2}))-2(n-1)\phi^{(4)}(\rho(\omega_{1}))=\sum_{i<j}e_{i}^{2}e_{j}^{2}.$$
$(ii)$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n+1}$ ($n\geq 4$) or ${\operatorname{\mathbf{Spin}}}_{2n}$ ($n\geq 5$) and $i=6$, then by (\[bsifactorial2\]) and Lemma \[generallem\] we have $$\begin{aligned}
6!\phi^{(6)}(\rho(\omega_{1}))&=\sum_{W(\omega_{1})}\lambda^{6}+\sum_{W(\omega_{1})}T_{6},\\
6!\phi^{(6)}(\rho(\omega_{2}))&=\sum_{W(\omega_{2})}\lambda^{6}+2(n-1)\sum_{W(\omega_{1})}T_{6}+4\Omega(4,2),\\
6!\phi^{(6)}(\rho(\omega_{3}))&=\sum_{W(\omega_{3})}\lambda^{6}+4{{n-1}\choose{2}}\sum_{W(\omega_{1})}T_{6}+8(n-2)\Omega(4,2),\end{aligned}$$ which implies that $$\phi^{(6)}(\rho(\omega_{3}))-2(n-2)\phi^{(6)}(\rho(\omega_{2}))+2(n-1)(n-2)\phi^{(6)}(\rho(\omega_{1}))=\sum_{i<j<k}e_{i}^{2}e_{j}^{2}e_{k}^{2}.$$
\[Dnqprime\] $(i)$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n}$, then we have $$\sum_{W(\omega_n)}\lambda^{n} - \sum_{W(\omega_{n-1})}\lambda^{n}=n!e_1\cdots e_n.$$
$(ii)$ If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n}$, then for any $1\leq p\leq n-1$ and any nonnegative integers $m_1,\cdots,m_p$ we have $$\sum_{W(\omega_n)}\lambda(m_1)\cdots \lambda(m_p)=\sum_{W(\omega_{n-1})}\lambda(m_1)\cdots \lambda(m_p).$$
$(i)$ First, assume that $n\geq 4$ is even. We show that $$\sum_{W_{+}(\omega_n)}\lambda^{n} - \sum_{W_{+}(\omega_{n-1})}\lambda^{n}=(n!/2)e_1\cdots e_n.$$ As $|W_{+}(\omega_n)|=|W_{+}(\omega_{n-1})|=2^{n-2}$, we have $$(n!/2^{n})2^{n-2}e_1\cdots e_n - (-(n!/2^{n})2^{n-2}e_1\cdots e_n)=(n!/2)e_1\cdots e_n$$ in $\sum_{W(\omega_n)}\lambda^{n} - \sum_{W(\omega_{n-1})}\lambda^{n}$. If one of the exponents $i_{1},\cdots,i_{n}$ in $e_1^{i_1}\cdots e_n^{i_n}$ (except the case $i_1=\cdots=i_n=1$) is odd, then from the orbits $W_{+}(\omega_n)$ and $W_{+}(\omega_{n-1})$ this monomial vanishes in each sum of $\sum_{W_{+}(\omega_n)}\lambda^{n} - \sum_{W_{+}(\omega_{n-1})}\lambda^{n}$. Otherwise, the terms $2^{n-2}\sum_{j=1}^{n}e_{j}^{n} ,\Lambda(n,2)\cdots, \Lambda(n,n/2)$ with $m_1=\cdots=m_n=1$ are in both $\sum_{W_{+}(\omega_n)}\lambda^{n}$ and $\sum_{W_{+}(\omega_{n-1})}\lambda^{n}$.
Now, we assume that $n\geq 4$ is odd. As $|W(\omega_n)|=|W(\omega_{n-1})|=2^{n-1}$, we have $$(n!/2^{n})2^{n-1}e_1\cdots e_n - (-(n!/2^{n})2^{n-1}e_1\cdots e_n)=n!e_1\cdots e_n$$ in $\sum_{W(\omega_n)}\lambda^{n} - \sum_{W(\omega_{n-1})}\lambda^{n}$. By the same argument, if one of the exponents $i_{1},\cdots,i_{n}$ in $e_1^{i_1}\cdots e_n^{i_n}$ (except the case $i_1=\cdots=i_n=1$) is odd, then this monomial vanishes in each sum of $\sum_{W(\omega_n)}\lambda^{n} - \sum_{W(\omega_{n-1})}\lambda^{n}$. This completes the proof of $(i)$.
$(ii)$ By Lemma \[elemenlemma\](ii)(iii), it is enough to consider the case where both $n$ and $p$ are even. For any $p$ and any $n\geq p+2$, we have $2^{n-2}(\sum_{W_{+}(\omega_1)}\lambda(m_1)\cdots \lambda(m_{p}))$ in both $\sum_{W_{+}(\omega_n)}\lambda(m_1)\cdots \lambda(m_{p})$ and $\sum_{W_{+}(\omega_{n-1})}\lambda(m_1)\cdots \lambda(m_{p})$. By the action of Weyl group, any term $\lambda_{i_1}(m_1)\cdots \lambda_{i_p}(m_p)$, where an element $\lambda\in W(\omega_1)$ appears odd times in either $\sum_{W_{+}(\omega_n)}\lambda(m_1)\cdots \lambda(m_{p})-2^{n-2}(\sum_{W_{+}(\omega_1)}\lambda(m_1)\\\cdots \lambda(m_{p}))$ or $\sum_{W_{+}(\omega_{n-1})}\lambda(m_1)\cdots \lambda(m_{p})-2^{n-2}(\sum_{W_{+}(\omega_1)}\lambda(m_1)\cdots \lambda(m_{p}))$, vanishes. As each term of $\Lambda(p,2),\cdots, \Lambda(p,p/2)$ appears in both $\sum_{W_{+}(\omega_n)}\lambda(m_1)\cdots\\ \lambda(m_{p})$ and $\sum_{W_{+}(\omega_{n-1})}\lambda(m_1)\cdots \lambda(m_{p})$, this completes the proof.
\[mainthm\] If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n+1}$ $($respectively, ${\operatorname{\mathbf{Spin}}}_{2n}$$)$, then for any $i\geq 3$ and any $n\geq [i/2]+1$ $($respectively, $n\geq [i/2]+2$$)$ the exponent $\tau_i$ divides the Dynkin index $\tau_2=2$.
As $B_2=C_2$ and $D_3=A_3$, we have $1=\tau_3\mid 2$ by [@BNZ Theorem 5.4]. If $G$ is ${\operatorname{\mathbf{Spin}}}_{2n}$ for any $n\geq 4$, then by Lemma \[Dnqprime\](i)(ii) we have $$q'_n=\phi^{(n)}(\rho(\omega_{n}))-\phi^{(n)}(\rho(\omega_{n-1})),$$ which implies that the invariant $q'_n$ is in the ideal generated by the image of $\phi^{(n)}$. As there are no invariants of odd degree except $q'_n$, we have $$\tau_{2i+1}\mid \tau_{2i}$$ for all $i\geq 1$. Therefore, it suffices to show that $\tau_{2i}\mid \tau_{2}$ for any $i\geq 2$.
By Lemma \[generallem\] together with the same argument as in Example \[deg6phi\] we have $$\label{complicatedfor}
\phi^{(2i)}(\rho(\omega_{i}))+\sum_{j=1}^{i-1}a_{j}\phi^{(2i)}(\rho(\omega_{i-j}))=\sum_{j_{1}<\cdots<j_{i}}e_{j_1}^{2}\cdots e_{j_i}^{2},$$ where the integers $a_1,\cdots, a_{i-1}$ satisfy $$\Big(\sum_{j=k}^{i-2}2^{j+1}{{n-1-k}\choose{j-k}}a_{j+1}\Big)+2^{i}{{n-1-k}\choose{i-1-k}}=0,$$ for $0\leq k\leq i-2$. Let $p_i$ be the right-hand side of (\[complicatedfor\]). Then this equation implies that $p_i$ is in the image of $\phi^{(2i)}$.
We show that the invariant $q_{2i}$ is in the ideal $\phi^{(2i)}(I_K^W)$ for any $i\geq 2$. We proceed by induction on $i$. As $q_{2}=\phi^{(2)}(\rho(\omega_1))$, the case $i=2$ is obvious. By Newton’s identities we have $$\label{newton}
(-1)^{i-1}q_{2i}=ip_{i}-\sum_{j=1}^{i-1}(-1)^{j-1}p_{i-1-j}q_{2j}$$ with $p_{0}=1$. By the induction hypothesis, the sum of (\[newton\]) is in $\phi^{(2i)}(I_K^W)$. Hence, $q_{2i}$ is in $\phi^{(2i)}(I_K^W)$.
For any nonnegative integer $n$, we denote by $v_{2}(n)$ the $2$-adic valuation of $n$. For a smooth projective variety $X$ over $F$, we denote by $\Gamma^{*}K(X)$ the gamma filtration on the Grothendieck ring $K(X)$. We let $c_{CH}:S^{*}(\Lambda)\to CH(G/B)$ be the characteristic map.
\[gammatorsion\] Let $G$ be ${\operatorname{\mathbf{Spin}}}_{2n}$ $($respectively, ${\operatorname{\mathbf{Spin}}}_{2n+1}$$)$. If $2^{m(i)}(\ker c_{CH})^{(i)}\subseteq(I_{CH}^W)^{(i)}$ for some nonnegative integer $m(i)$, then for any $i\geq 3$ and any $n\geq [i/2]+2$ $($respectively, $n\geq [i/2]+1$$)$ the torsion of $\Gamma^{i}K(G/B)/\Gamma^{i+1}K(G/B)$ is annihilated by $2^{g(i)}$, where $g(i)=1+m(i)+v_{2}((i-1)!)$.
It is shown that $m(3)=0$ and $m(4)=1$ in [@BNZ Lemma 6.4].
The proof of [@BNZ Theorem 6.5] still works with Theorem \[mainthm\].
Let $G$ be ${\operatorname{\mathbf{Spin}}}_{2n}$ $($respectively, ${\operatorname{\mathbf{Spin}}}_{2n+1}$$)$. If $2^{m(i)}(\ker c_{CH})^{(i)}\subseteq(I_{CH}^W)^{(i)}$ for some nonnegative integer $m(i)$, then for any $G$-torsor $E$, any $i\geq 3$ and any $n\geq [i/2]+2$ $($respectively, $n\geq [i/2]+1$$)$ the torsion of ${\operatorname{CH}}^{i}(E/B)$ is annihilated by $2^{t(i)}$, where $t(i)=1+\sum_{j=3}^{i}g(j)+v_{2}((i-1)!)$.
By [@Panin Theorem 2.2(2)], we have $$\Gamma^{i}K(G/B)/\Gamma^{i+1}K(G/B)\simeq \Gamma^{i}K(E/B)/\Gamma^{i+1}K(E/B).$$ As the torsion of ${\operatorname{CH}}^{i}(E/B)$ is annihilated by $$(i-1)!\prod_{j=1}^{i}e(\Gamma^{i}K(E/B)/\Gamma^{i+1}K(E/B)),$$ where $e(\Gamma^{i}K(E/B)/\Gamma^{i+1}K(E/B))$ denotes the finite exponent of its torsion subgroup (see [@BNZ p.149]), the result follows from Corollary \[gammatorsion\].
[10]{}
S. Baek, E. Neher, K. Zainoulline, *Basic polynomial invariants, fundamental representations and the Chern class map*, Doc. Math. **17** (2012), 135–150.
J. Humphreys, *Reflection groups and Coxeter groups*. Cambridge studies in Advanced Math. [**29**]{}, Cambridge Univ. Press (1990).
I. A. Panin, *On the algebraic K-theory of twisted flag varieties*, K-Theory [**8**]{} (1994), no. 6, 541–585.
|
---
abstract: 'Third party tracking allows companies to identify users and track their behaviour across multiple digital services. This paper presents an empirical study of the prevalence of third-party trackers on 959,000 apps from the US and UK Google Play stores. We find that most apps contain third party tracking, and the distribution of trackers is long-tailed with several highly dominant trackers accounting for a large portion of the coverage. The extent of tracking also differs between categories of apps; in particular, news apps and apps targeted at children appear to be amongst the worst in terms of the number of third party trackers associated with them. Third party tracking is also revealed to be a highly trans-national phenomenon, with many trackers operating in jurisdictions outside the EU. Based on these findings, we draw out some significant legal compliance challenges facing the tracking industry.'
author:
- 'Reuben Binns, Ulrik Lyngs, Max Van Kleek, Jun Zhao, Timothy Libert, Nigel Shadbolt'
bibliography:
- 'bibliography.bib'
title: Third Party Tracking in the Mobile Ecosystem
---
<ccs2012> <concept> <concept\_id>10002978.10003029.10003031</concept\_id> <concept\_desc>Security and privacy Economics of security and privacy</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002978.10003022.10003465</concept\_id> <concept\_desc>Security and privacy Software reverse engineering</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010405.10010455.10010458</concept\_id> <concept\_desc>Applied computing Law</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003083.10003014.10003017</concept\_id> <concept\_desc>Networks Mobile and wireless security</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction
============
Billions of people use smartphones every day, generating vast amounts of data about themselves. Much of the functionality afforded by these devices comes in the form of applications which derive revenue from monetising user data and displaying behaviourally targeted advertising. Firms with the ability to collect such data have become a significant part of the digital economy [@acquisti2016economics], with the online advertising industry earning \$59.6 billion per year in the U.S. alone [@iab2016].
This business model is primarily enabled through ‘third-party’ trackers [@montes2015value], which track users via ‘first-party’ mobile applications, whose developers embed their technology into application source code. Such networks link activity across multiple apps to a single user, and also link to their activities on other devices or mediums like the web. This enables construction of detailed profiles about individuals, which could include inferences about shopping habits, socio-economic class or likely political opinions. These profiles can then be used for a variety of purposes, from targeted advertising to credit scoring and targeted political campaign messages.
This paper aims to provide a high-level empirical overview of the extent of third party tracking on the mobile ecosystem. In particular, we aim to answer the following:
1. How are third party trackers distributed across apps on the Google Play Store?[^1]
2. Which companies ultimately own these tracking technologies, and in which jurisdictions are they based?
3. Do different trackers prevail amongst different genres of apps?
Our motivation is to shed light on the status quo, in order that future efforts to address and mitigate third party tracking can be more informed and targeted.
Background
==========
We begin by introducing previous work on tracker detection methods, and on large-scale field studies of tracking on the web and mobile. Then, to motivate some of the present analysis, we provide an overview of existing approaches to addressing mobile tracking, including end-user controls, OS provider rules, and legal regulation. The shortcomings of the first two approaches have driven a renewed focus on the latter; by surveying the existing state of mobile tracking, we aim to provide insights into the extent to which current tracking activities may be affected by certain key data protection regulations.
Detecting third party tracking at scale in the wild
---------------------------------------------------
The third party tracking ecosystem has been studied on both the web and mobile using a variety of methods. Large scale web tracking studies detect third-party trackers by inspecting network traffic associated with a website. Some approaches use crowd-sourcing (e.g. [@vallina2016tracking; @yu2016tracking]) while others use automated web crawlers (e.g. [@englehardt2016census; @roesner2012detecting; @libert2015exposing; @yu2016tracking]. In all cases, a small number of dominant trackers are observed.
Several studies of third-party tracking have also been conducted on mobile platforms [@vallina2016tracking; @book2015empirical], using both dynamic and static detection methods. Dynamic methods, as in web-based tracking studies, involve inspecting network traffic from the browser / device and identifying any third party destinations that relate to tracking. One common approach has been OS-level instrumentation, such as those of TaintDroid [@enck2014taintdroid], and AppTrace [@qiu2015apptrace]. An alternative to low-level OS instrumentation is to analyse all communications traffic transmitted by an app whilst it is in use [@ren2016recon]. Other methods involve unpacking an application’s source code (on Android systems, this comes as an Android Application Package (APK)) and detecting use of third-party tracking libraries [@arzt2014flowdroid; @batyuk2011using; @egele2011pios; @lin2014privacygrade].
Other aspects of tracking have been studied, including the variety of techniques that are used, from cookies [@aziz2015cookie; @englehardt2015cookies; @englehardt2016census] to fingerprinting [@acar2013fpdetective]. A more recent field study by Yu et al. provided a finer-grained view into tracker behaviour, by classifying data being transmitted to trackers as either ‘safe’ or ‘unsafe’ [@yu2016tracking]. Another factor is the permissions requested by an app, which constrain the kinds of data a third party can obtain; longitudinal research has found that Android apps request additional privacy-risking permissions on average every three months [@taylor2017].
The crossover between the mobile and web tracking ecosystem has also attracted attention in recent research. Various comparisons have shown that web and mobile tracking are different, both in terms of the companies that operate on each environment [@vallina2016tracking], and the specific kinds of personal information that are shared by web and mobile versions of the same service [@leung2016recon]. In previous work comparing 5,000 apps and 5,000 websites, it was found that while certain companies dominate both environments, the overlap between top trackers is only partial, even for web and mobile versions of the same service [@binns2018measuring].
Existing approaches to addressing risks of tracking
---------------------------------------------------
There are three main approaches for addressing the risks of tracking; end-user privacy controls, industry self-regulation, and traditional legal regulation.
### End-user privacy controls
Tracking exists on both the web and on mobile apps, but web browsers have traditionally enabled end-users to control tracking via default browser settings or through third party plugins. By contrast, no major smartphone platform OS currently gives end-users the ability to block or otherwise control third party tracking by apps (although tracker blocking is available on mobile web browsers). The privacy settings are primarily focused on app-by-app permissions, or permissions regarding certain data types (e.g. location, contacts, etc.). While various changes have been introduced like run-time permissions, and advertising identifier controls [@nauman2010apex], these do not address the distinction between first party apps and third party trackers. More recently, awareness-raising tools have been proposed which do reveal the presence of third-parties. They make use of techniques including reverse-engineering of app source code and network traffic analysis [@batyuk2011using; @qiu2015apptrace; @egele2011pios; @arzt2014flowdroid; @enck2014taintdroid; @zang2015knows; @gordon2015information], allowing identification of personal data flows from apps to first and third parties. These tools have been used to map data flows and display them to end-users [@balebako2013little; @van2017better; @srivastava2017privacyproxy; @chitkara2017does]. Such focus on third-party data collection, rather than app-level permissions, may be a more meaningful way to enact privacy choices. However, until such controls are enabled by the OS providers, third party tracking via apps remains largely invisible to end-users. This is in contrast to the web, where millions of users make use of tracker protection tools such as uBlock Origin or Ghostery.
### Self-regulation by platforms
In response to the development and proliferation of trackers, and the lack of wide-scale deployment of effective end-user tracker controls, various efforts have been made by mobile OS platform developers to address the risks. Mobile application developers are required to follow the rules of the app market providers in order for their apps to be listed [@anderson2010inglorious]. Since few consumers use multiple app stores on a single smartphone, these platforms are in a stronger position to impose industry self-regulation than browser vendors, because they have the ability to effectively kick an application off the platform entirely.
Industry-led self-regulatory initiatives have thus far attempted to strike a balance between protecting users from malicious behaviour and creating a relatively permissive environment. With respect to smartphone operating systems, Apple and Google have the power to exert varying degrees of control over the behaviour of apps appearing in their default app stores. Thus far, both of their respective developer agreements permit third-party tracking, although certain user-protective practices are required, such as collecting a replaceable advertising identifier (IDFA / AAID) rather than the permanent device identifier.
More stringent action against third party tracking may also have been held back by vested interests of the OS providers. Both Google and Apple have historically had a stake in the digital advertising industry. Google own several tracker companies such as DoubleClick and others. Apple used to take a cut of advertising revenue from ad network trackers in iPhone apps, through the iADs program, but this scheme ended in 2016.
### Legal regulation {#legalbackground}
These self-regulatory efforts, such as they are, sit alongside a variety of specific legal regulations with varying levels of enforcement in different countries around the world. Perhaps the most stringent and far-sighted of these is the data protection legal regime in Europe. With updated rules incoming this year in the form of the European Union’s General Data Protection Regulation, new enforcement powers including the issuing of larger fines and scope for indefinitely suspending processing may substantially curtail the activities of third party trackers.
For instance, the specific identities and purposes of third party trackers will have to be made transparent to the data subject (i.e. the user of the app); and special safeguards must be applied in the case of children. While profiling of children is not outright prohibited by the GDPR, the Article 29 Working Party (the EU body responsible for providing guidance on data protection), advise that organisations should ‘refrain from profiling them for marketing purposes’.
Regarding transfer of data across borders, while existing requirements are not fundamentally different under the GDPR, transnational data transfer is likely to receive additional scrutiny in light of the introduction of stronger enforcement powers. Under the existing regime, personal data is permitted to flow from one jurisdiction to another, subject to compliance with certain conditions. The least onerous condition is if the recipient organisation is based in a country whose existing data protection regime has been assessed by the European Commission and deemed ‘adequate’. Otherwise, special arrangements such as standard contractual clauses and binding agreements between organisations in both jurisdictions may be necessary in order to make cross-jurisdictional data flows legitimate. Similar data flow agreements exist between other countries. In some cases these are reciprocal (such as between the EU and Andorra), while others are not (e.g., the Russian privacy regulator allows personal data to flow from Russia to EU countries[^2], but the reverse is not true).
Such cross-border rules and data ‘trade blocs’ have consequences for the legal basis for third party tracking when tracking companies, app developers, app stores and end-users are located in different jurisdictions. While the transfer of data from people residing in the EU to countries whose data protection regime is deemed inadequate could be legitimate in principle, more onerous conditions would need to be met. As such, any efforts to assess the legality of current practices must consider the extent to which tracking occurs across borders.
Data Collection & Methodology
=============================
Play Store Indexing and App Discovery
-------------------------------------
The first step was to identify available apps. We programmatically identified popular search terms in the Play Store by autocompleting all character strings of up to a length of five, and then issued each search term to get a list of apps, ranked by popularity [@google2017Autocomplete]. The identified apps were then downloaded using the `gplaycli` [@gplaycli], a command line tool for interacting with the Play Store.
### Static analysis method
An Android Package Kit (APK) is an Android file format that contains all resources needed by an app to run on a device. Upon download, each APK was unpacked and decoded using APKTool [@tumbleson2017APKTool] to obtain the app’s assets, in particular its icon, bytecode (in the DEX format) and metadata (in XML format). Finally, permission requests were parsed from the XML and hosts were found in the bytecode using a simple regex[^3].
### Mapping hostnames to known tracker companies
While this static analysis process effectively identified references to hosts in the APKs, it did not provide a means of mapping them to companies, let alone selecting only those companies who are in fact engaged in tracking. A large number of the hostnames found in the static code analysis refer to a wide range of benign external resources which are not necessarily engaged in tracking. In order to isolate only those engaged in tracking, we combined two lists of trackers derived from previous research. One list is compiled by the Web X-Ray project [@libert2015exposing]. It maps third party web tracking domains to companies that own them, as well as parent-subsidiary relationships. The second list is compiled from previous research by the authors of the present paper [@binns2018measuring; @VanKleek:2017:BDY:3025453.3025556], which also maps domains to companies, and companies to their owners, but incorporates mobile app-centric trackers which are missing from web-oriented tracker lists. An example of domain-company ownership in the resulting aggregated list is shown in Figure \[fig:domain-company\], and parent-subsidiary relationship in Figure \[fig:parent-subsidiary\].
Host names in the tracker lists were shortened to 2-level domains using the python library `tldextract`[^4] (e.g. for ‘subdomain.example.com’, the domain name ‘example’ and top-level domain suffix ‘.com’ were kept and any subdomains were omitted). Tracker hosts were then matched to hosts identified in app bytecode with a regular expression which excluded matches that was followed by a dot or an alphabetic character (matching ‘google.com’ to ‘google.com/somepath’ but not ‘google.com.domain’ or ‘google.coming’).
{width="10cm" height="3cm"}
\[fig:domain-company\]
{width="\textwidth" height="3cm"}
\[fig:parent-subsidiary\]
Data analysis
-------------
Most of the data analysis was conducted in R, using RStudio[^5].
Results
=======
Numbers of tracker hosts in apps
--------------------------------
The distribution of number of tracker hosts per app was highly right-skewed (see Figure \[fig:trackerRefsAcrossAllApps\]). Gini inequality coefficient was 0.44. Across all analyzed apps (n = 959,426), the median number of tracker hosts included in the bytecode of an app was 10. 90.4% of apps included at least one, and 17.9% more than twenty.
\[fig:trackerRefsAcrossAllApps\]
Numbers of distinct tracker companies behind hosts
--------------------------------------------------
The distribution of number of distinct tracker companies (at the lowest subsidiary level) behind the hosts in an app was similarly right-skewed (see Figure \[fig:numCompaniesReferred\]). The median number of companies was 5, 90.4% of apps included hosts associated with at least one company, and 17.4% with more than ten companies.
\[fig:numCompaniesReferred\]
There were 13 apps for which our analysis identified 30 or more different tracking companies referred to via hosts in the bytecode. In some cases, these high numbers can be explained by the particular function of the app; for instance, some of these apps integrate multiple different services into one app (e.g. ‘Social Networks All in One’); in such cases, any tracking domains associated with those integrated services will be identified by our method. For others, mostly gaming apps, the high numbers of trackers serve no obvious function other than the usual kinds of behaviourally targeted advertising and analytics.
Rather than simply counting number of companies, we can query the proportion of apps containing hosts associated with specific companies. As illustrated in Figure \[fig:parent-subsidiary\], however, many companies have been acquired by larger parent or holding companies, such as Alphabet. The result of grouping by ’root parent’ the percentages of apps which include hosts associated with specific companies is shown in Table \[tab:topCompanies\].
[>p[2cm]{} r >p[2cm]{} r c]{} [*Root parent*]{} & [*% apps*]{} & [*Subsidiary*]{} & [*% apps*]{} & [*Country*]{}\
Alphabet & 88.44 & Google & 87.57 & US\
& & Google APIs & 67.51 & US\
& & DoubleClick & 60.85 & US\
& & Google Analytics & 39.42 & US\
& & Google Tag Manager & 33.88 & US\
& & Adsense & 30.12 & US\
& & Firebase & 19.20 & US\
& & Admob & 14.67 & US\
& & YouTube & 9.51 & US\
& & Blogger & 0.46 & US\
Facebook & 42.55 & Facebook & 42.54 & US\
& & Liverail & 1.03 & US\
& & Lifestreet & <0.01 & US\
Twitter & 33.88 & Twitter & 30.94 & US\
& & Crashlytics & 5.10 & US\
& & Mopub & 2.51 & US\
Verizon & 26.27 & Yahoo & 20.82 & US\
& & Flurry & 6.28 & US\
& & Flickr & 1.37 & US\
& & Tumblr & 1.22 & US\
& & Millennialmedia & 0.71 & US\
& & Verizon & 0.11 & US\
& & AOL & 0.06 & US\
& & Intowow & <0.01 & US\
& & One By AOL & <0.01 & US\
& & Brightroll & <0.01 & US\
& & Gravity Insights & <0.01 & US\
Microsoft & 22.75 & Microsoft & 22.11 & US\
& & Bing & 0.12 & US\
& & LinkedIn & 20.62 & US\
Amazon & 17.91 & Amazon Web Services & 11.57 & US\
& & Amazon & 7.72 & US\
& & Amazon Marketing Services & 1.73 & US\
& & Alexa & <0.01 & US\
Unitytechnologies & 5.78 & Unitytechnologies & 5.78 & US\
Chartboost & 5.45 & Chartboost & 5.45 & US\
Applovin & 3.95 & Applovin & 3.95 & US\
Cloudflare & 3.85 & Cloudflare & 3.85 & US\
Opera & 3.20 & Adcolony & 3.12 & US\
& & Admarvel & 0.09 & US\
& & &
\[tab:topCompanies\]
Company prevalence by genre
---------------------------
The Google Play store metadata divides apps into 49 different genres (no less than 17 of these are subcategories of games, e.g. ’Casino Games’ and ’Adventure Games’). To provide a high-level analysis, we grouped these genres into 8 more succinct ’super genres’ (by e.g. clustering all game genres, plus the genres ’Comics’, ’Entertainment’, ’Sports’ and ’Video Players’ into a single ’Games & Entertainment’ category[^6]). In addition, given concern of in particular tracking of children[@Livingstone2010], we created a super genre consisting of apps included in one of the Google Play store’s ‘family’ categories.[^7] For each super genre, we reran the company analysis, which revealed some important differences between the nature of tracking by genre.
First, there are differences in the number of distinct tracking companies associated with apps from different genres. Figure \[fig:byGenreCompanyRefs\] shows the number of apps in each super genre, and descriptive statistics of number of distinct tracker companies associated with apps within each. *News* and *Family* apps have the highest median number of tracker companies associated with them, and over 20% of apps in the *News*, *Family*, and *Games & Entertainment* super genres are linked to more than ten tracker companies. Meanwhile, the lowest median number of trackers are found within *Productivity & Tools*, *Education*, *Communication & Social*, and *Health & Lifestyle* apps, and over 10% of *Productivity & Tools*, *Education* and *Communication & Social* apps have no trackers at all.
[*Genre*]{} [*$ K$*]{} [ *$\sum{K}$*]{}
------------------------ ------------ ------------------ -- --
Productivity & Tools 0.14 5.5
Games & Entertainment 0.13 5.41
Health & Lifestyle 0.1 5.5
Communication & Social 0.09 5.29
Art & Photography 0.09 5.12
Family 0.04 4.33
News 0.03 4.5
Education 0.03 5.42
Music 0.02 5.24
: K distances between tracker rankings for each genre compared to all apps (K), and sum of pairwise distances between each genre and every other genre ($\sum{K}$).
\[tab:genreDistances\]
\[fig:byGenreCompanyRefs\]
Second, there are differences in which particular trackers are associated with apps from each super genre. By comparing rankings for each, we can see the extent to which different trackers dominate each super genre. In addition to comparing the difference in rankings for any given tracker, we use an overall distance metric, the Kendall tau distance, in order to measure the extent to which rankings differ between super genres [@kendall1938new].
The Kendall Tau distance may be defined as:
K(\_1,\_2) =
\_[{i,j}P]{} |[K]{}\_[i,j]{}(\_1,\_2)
where:
1. ”P” is the set of unordered pairs of distinct elements in $ \tau_1 $ and $\tau_2$
2. |[K]{}\_[i,j]{}(\_1,\_2)
= 0 if ”i” and ”j” are in the same order in $\tau_1$ and $\tau_2$
3. |[K]{}\_[i,j]{}(\_1,\_2)
= 1 if ”i” and ”j” are in the opposite order in $\tau_1$ and $\tau_2.$
In this context, ”P” is the set of unordered pairs of trackers (e.g. ‘DoubleClick’ and ‘AdChina’), in one genre ranking $ \tau_1 $ (e.g. ‘Games’) and another genre ranking $ \tau_2 $ (e.g. ‘News’). $ K $ is based on the number of discordant pairs between $ \tau_1 $ and $ \tau_2 $, where a higher $ K $ indicates greater distance.
We find that the Productivity & Tools and Games & Entertainment categories exhibit the biggest differences in ranking of trackers compared to the overall ranking of trackers across the whole Play Store, while the ranking of trackers in the Music category is the closest to the overall ranking (see Table \[tab:genreDistances\]).
In addition to calculating the distance between the rankings of each genre and the rankings for the entire Play Store, we also calculated the distances between each distinct pair of genres and summed them to get an idea of the overall distance of a single genre from every other genre. When considering the distance in tracker rankings from the tracker rankings of all other categories, Productivity & Tools and Health & Lifestyle appear to be the biggest outliers; the top 20 trackers in the former include companies not present in the top 20 for all apps, like Mapbox (rank \#64 across all apps) as well as Chinese companies Alibaba and Baidu.
Country differences
-------------------
We also analysed the prevalence of countries in which the tracker companies are based (including both subsidiary and root parent level; see Table \[tab:CountryPrev\]). Just over 90% of all apps contained at least one tracker owned by a company based in the United States. China, Norway, Russia, Germany, Singapore, and the United Kingdom were the next most common destinations. The median number of unique countries associated with the companies referred to in an app was 1 (see Figure \[fig:numCountriesReferred\]).
{width="0.9\columnwidth"}
\[fig:numCountriesReferred\]
We also calculated the country prevalence figures on a genre-by-genre basis. While the US remained the most prevalent in every case, (between 86-96%), the prevalence rankings for other countries differed by super genre. For instance, UK-based trackers were the second-most prevalent in ‘Art & Photography’, despite being only 7th overall.
[*Country*]{} [*\# apps present*]{} [*% apps*]{}
--------------- ----------------------- --------------
U.S. 865369 90.2%
China 48451 5.1%
Norway 30674 3.2%
Russia 24889 2.6%
Germany 24773 2.6%
Singapore 19323 2.0%
UK 14451 1.5%
Austria 4754 0.5%
South Korea 3366 0.4%
Japan 1801 0.2%
: Apps including at least one tracker associated with a subsidiary or root parent within a given country.
\[tab:CountryPrev\]
Discussion
==========
We begin by discussing the limitations of our data collection methods. Next we consider some differences between tracking on websites and on mobile apps, and finally we draw out implications for the regulatory approaches outlined in section \[legalbackground\].
Limitations of data collection methods
--------------------------------------
There are several limitations to our tracker detection methods. First, it is incomplete; our knowledge base of tracker domain to company mappings is limited to those trackers which have been discovered in the course of previous research (namely [@VanKleek:2017:BDY:3025453.3025556; @binns2018measuring; @libert2015exposing]). While these lists were compiled in a systematic way, focusing on the most prevalent tracking domains, including the entire long tail of less prevalent domains might change the results reported. The inclusion and exclusion criteria for what constitutes a ‘tracker’ are also open to debate; the list compiled in prior works, and relied on here, defines a third-party tracker as ‘an entity that collects data about users from first-party websites and / or apps, in order to link such data together to build a profile about the user’, but the definition and its application are debateable.[^8] Another issue is that without dynamic network traffic analysis of all apps, including successful man-in-the-middle proxying and ability to interpret the data payloads, we cannot confirm precisely what data is sent to each tracker. Finally, different trackers serve different purposes; some facilitate targeted advertising, while others are used for analytics. Without further fine-grained distinctions between such purposes, the figures presented here do not represent the full nuance and variety of third party tracking and its impacts.
Web vs. Mobile
--------------
Previous large-scale studies of tracking have largely focused on the web. The distribution model of the web allows measurement of tracking to scale in a way that the model for smartphone app distribution does not; web services are delivered in a standardised way through a browser which can easily be automated. As a result, large-scale web tracking studies typically include millions of sites. By contrast, the largest smartphone app tracking study to our knowledge at the time of writing is derived from network traffic detected by the Lumen app, which includes the data flows of 14,599 apps installed on Lumen user’s devices [@razaghpanah2018apps]. While such crowdsourced methods have many advantages in terms of the granularity of the data flows and ecological validity, at best they scale to tens of thousands of apps. By contrast, our method is scalable to hundreds of thousands of apps (indeed, our dataset of apps is close to a million).
Implications for tracker regulation
-----------------------------------
While the distribution of trackers across apps is of general interest from a privacy and data protection regulation perspective, we focus here on several particular regulatory implications arising from our findings.
### Cross-jurisdictional data flow
As explained in Section \[legalbackground\], the rules regarding transfers of data outside the EU under the GDPR are similar to the previous regime (under the Data Protection Directive), but with some new details as well as larger associated fines. In so far as these developments result in more investigation and enforcement by authorities, the impact will be different for companies depending on their jurisdiction. There will be no impact on those based in the EU, such as Germany (the fifth-most prevalent country in which trackers are based), who benefit from rules permitting the free flow of data within the Union. Some third countries such as Canada also benefit from being on the EU Commission’s list of legal regimes that are deemed ‘adequate’ and therefore data transfers to trackers in those jurisdictions are legitimate without further measures in place.
However, amongst the top-10 most prevalent countries there are several which lie outside the E.U. and are not deemed adequate, such as China, Russia, Singapore, South Korea and Japan. In order for transfers to these countries to be legitimate, additional safeguards must be in place as explained in Section \[legalbackground\]. We cannot determine whether such arrangements have been put in place by the identified companies based in non-approved jurisdictions, but our figures give an indication of the volume of companies to whom these more onerous rules apply. While the percentages of apps which include trackers from such jurisdictions are small compared to the US—China (5.1%), Russia (2.6%), Singapore (2%) versus US (90%)—they are still significant, numbering in the tens of thousands.
### Profiling
The GDPR uses the term ‘profiling’ to describe any fully or partly automated processing of personal data with the objective of evaluating personal aspects of a natural person (Article 4(4)). Many of the tracking companies included in our knowledge base engage in data processing activity that would likely constitute ‘profiling’ under this definition. For instance, the purpose of many of the most common trackers is behaviourally targeted advertising, whereby individuals are evaluated along demographic and behavioural dimensions to determine their propensity to respond to certain marketing messages. Profiling is prohibited if it has ‘legal or significant’ effects on the data subject. While the definition of ‘significant effects’ is not entirely clear, the Article 29 Working Party has advised that even profiling for marketing purposes could potentially give rise to significant effects, including if it is: intrusive; targets vulnerable, minority groups, or those in financial difficulty; involves differential pricing; or deprives certain groups of opportunities.[^9] Trackers which enable such activities without consent of the data subject could therefore be in breach of Article 22 (unless such profiling is necessary for entering or performing a contract, or it is authorised by another member state law). Many of the most prevalent trackers observed in our study have the capacity to be used in such ways, and evidence of such practices is beginning to emerge. For instance, DoubleClick (present on 60% of apps analysed) has been shown to target adverts for higher-paid jobs to men at a higher rate than to women [@datta2015automated]; while web-based price discrimination has also been documented by numerous studies in recent years [@mikians2012detecting; @hannak2014measuring].
### Rights and obligations regarding children
Like the old Directive, the GDPR defines certain additional rights and obligations regarding processing the personal data of children (defined as anyone under the age of 16, and for certain additional protections, 13). If a tracker is relying on consent as a legitimating ground for processing, then such consent would not be valid from a child under 13; instead a parent or guardian would need to consent. Furthermore, as discussed above, Recital 38 states that special protections should be in place if children’s data are being processed for marketing and user profiling. This description would likely cover many of the trackers which are embedded in apps from the Family and Games & Entertainment genre categories, which are clearly targeted at children. Problematically, apps from these two genres are especially exposed to third party tracking, with the average app including hosts associated with 7 distinct tracker companies for Family apps, and 6 for Games & Entertainment apps (only News apps are more exposed). Given the relatively higher level of protection set in the law regarding profiling children for marketing, it seems that tracking is most rampant in the very context in which regulators are most concerned to constrain it.
Conclusion
==========
We believe that by undertaking analysis of the distribution of tracking technology on close to 1 million smartphone apps, we gain insight into the breadth and scale of this highly important phenomenon. Unlike previous studies whose coverage of apps numbers in the tens of thousands, and may be skewed towards the app choices of the users from whom data is gathered, our study is a systematic analysis of apps on the Play Store.
Our genre-by-genre analysis suggests that there are differences in the behaviour and distribution of trackers depending on the functionality or purpose the app provides. News and Games apps appear amongst the worst in terms of the number of tracker companies associated with them. Tracking is also a substantially trans-national phenomenon; around 100,000 apps we analysed send data to trackers located in more than one jurisdiction.
These findings suggests that there are challenges ahead both for regulators aiming to enforce the law, and for companies who intend to comply with it. Full audits of mobile app stores such as this could help regulators identify areas to focus on. Previous privacy enforcement ‘sweeps’[^10] have focused on the most popular apps, and their terms of service and privacy policies. But the analysis here suggests that apps may not necessarily be the most efficient point of analysis; rather, identifying and investigating the most prevalent trackers might be a better target. Some of the practices likely to be involved - such as allowing profiling of children without attempting to obtain parental consent - may be downright unlawful. It remains to be seen how and if regulators will attempt to detect and prevent behavioural targeting that has ‘significant effects’ on data subjects.
The governance of these activities is complex, involving many stakeholders, including: users, smartphone operating system developers, equipment manufacturers, alternative app market operators, app developers, and tracking companies (who also operate multi-sided markets with advertisers and therefore have the ability to impose constraints on what ads can be served). Effective regulation will require collaboration between regulators and these myriad other actors.
All authors are supported under *SOCIAM: The Theory and Practice of Social Machines*, funded by the UK Engineering and Physical Sciences Research Council (EPSRC) under grant number EP/J017728/2 and comprises the University of Oxford, the University of Southampton, and the University of Edinburgh. Reuben Binns and Max Van Kleek are also supported by *ReTiPS: Repectful Things in Private Spaces*, a project funded through the PETRAS IoT Hub Strategic Fund, which, in turn, was funded by the EPSRC under grant number N02334X/1. Timothy Libert is also supported by the Google Digital News Project at the Reuters Institute for the Study of Journalism. Jun Zhao is also supported by KOALA (http://SOCIAM.org/project/koala): Kids Online Anonymity & Lifelong Autonomy, funded by EPSRC Impact Acceleration Account Award, under the grant number of EP/R511742/1.
[^1]: We did not study the Apple iOS App Store because there are no equivalently scalable iOS app collection and analysis methods
[^2]: [https://www.huntonprivacyblog.com/2017/08/16/russian-privacy-regulator-\\adds-countries-list-nations-sufficient-privacy-protections/](https://www.huntonprivacyblog.com/2017/08/16/russian-privacy-regulator-\adds-countries-list-nations-sufficient-privacy-protections/)
[^3]: We note that this method has the inherent problem that we cannot confirm if bytecode relating to or referencing such hosts is ever called. More sophisticated static analysis methods might better distinguish but this is left for future work. The regex used to identify hosts in the bytecode is available on [osf.io/4nu9e](osf.io/4nu9e)
[^4]: <https://github.com/john-kurkowski/tldextract>
[^5]: Analysis scripts plus data are available via the Open Science Framework at [osf.io/4nu9e](osf.io/4nu9e). For access to the full data set, contact the authors.
[^6]: See [osf.io/4nu9e](osf.io/4nu9e) for details of this grouping.
[^7]: All apps on the Google Play store have an ordinary genre classification, but some apps are in classified into one of the Play store’s family genres.
[^8]: The principles behind the criteria used here are discussed in the aforementioned prior works
[^9]: Article 29 Working Party: Guidelines on Automated individual decision-making and Profiling for the purposes of Regulation 2016/679 <http://ec.europa.eu/newsroom/article29/item-detail.cfm?item_id=612053>
[^10]: See [https://www.privacyenforcement.net/node/906
](https://www.privacyenforcement.net/node/906
)
|
---
abstract: |
Let $G$ be a finite group. The bipartite divisor graph $B(G)$ for the set of irreducible complex character degrees ${\mathrm{cd}}(G)$ is the undirected graph with vertex set consisting of the prime numbers dividing some element of ${\mathrm{cd}}(G)$ and of the non-identity character degrees in ${\mathrm{cd}}(G)$, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. The graph $B(G)$ is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees.
The scope of this paper is two-fold. We draw some attention to $B(G)$ by outlining the main results that have been proved so far, see for instance [@H; @Hregular; @mus; @moo4; @moo5]. In this process we improve some of these results.
address:
- 'Roghayeh Hafezieh, Department of Mathematics, Gebze Technical University, P.O.Box 41400, Gebze, Turkey'
- 'Pablo Spiga, Dipartimento di Matematica Pura e Applicata, University of Milano-Bicocca, Via Cozzi 55, 20126 Milano, Italy'
author:
- Roghayeh Hafezieh
- Pablo Spiga
title: An overview on the bipartite divisor graph for the set of irreducible character degrees
---
[^1]
0.2 true cm
Introduction
============
An active line of research studies the relations between structural properties of groups and sets of invariants. There is a large number of examples of this and we name only four which might be considered as the genesis of this type of investigations; these four examples should also clarify our interest in this paper. Given a finite group $G$, we may associate the prime graph based on the conjugacy class sizes: the vertices are the prime numbers dividing the cardinality of some conjugacy class of $G$ and the prime numbers $p$ and $q$ are declared to be adjacent if and only if $pq$ divides the cardinality of some conjugacy class of $G$. On the set of conjugacy class sizes, we may also associate the divisor graph: the vertices are the cardinalities of the non-central conjugacy classes of $G$ and the numbers $m$ and $n$ are declared to be adjacent if and only if they are not relatively prime. It is well known that a great deal of information on $G$ is encoded in both of these graphs and this establishes a beautiful flow of information between the algebraic structure of $G$ and the combinatorial properties of the graphs. Two entirely similar constructions can be done replacing the set of conjugacy class sizes with the set of irreducible complex character degrees. Again, the prime graph and the divisor graph on the character degrees encode interesting information about the group. Considering the “duality” between conjugacy classes and irreducible characters there are also some remarkable connections among all four of these graphs.
In [@L], Mark L. Lewis has generalized in a very natural and useful way these graphs. Given a subset $X\subseteq \mathbb{N}\setminus\{0,1\}$, Lewis has considered the [*prime graph*]{} $\Delta(X)$ and the [*divisor graph*]{} $\Gamma(X)$. The vertices of $\Delta(X)$ are the prime numbers dividing some element of $X$ and two distinct prime numbers are declared to be adjacent if and only if their product divides some member of $X$. The vertex set of $\Gamma(X)$ is $X$ and two distinct elements of $X$ are declared to be adjacent if and only if they are not relatively prime. Then, Lewis has shown (for arbitrary sets $X$) some remarkable general connections between $\Delta(X)$ and $\Gamma(X)$. By taking $X$ the set of conjugacy class sizes or the set of irreducible complex characters, one recovers the graphs introduced in the previous paragraph and rediscovers some of their basic relations.
There is a gadget that can be used to study simultaneously $\Delta(X)$ and $\Gamma(X)$. Inspired by the remarkable connections between the common divisor graph $\Gamma(X)$ and the prime degree graph $\Delta(X)$ discussed in [@L] by Lewis, Iranmanesh and Praeger [@IP] introduced the notion of [*bipartite divisor graph*]{} $B(X)$, and proved that most of these connections follow immediately from $B(X)$. The vertex set of $B(X)$ is the disjoint union of the set of prime numbers dividing some element of $X$ and the set $X$ itself, where a prime number $p$ is declared to be adjacent to an element $x$ of $X$ if and only if $p$ divides $x$. For instance, with this new tool, Iranmanesh and Praeger (re)established the links between the number of connected components and the diameters of $\Delta(X)$ and $\Gamma(X)$ simply working with $B(X)$. They were also able to classify the graphs $\Gamma$ with $\Gamma\cong B(X)$, for some set $X$.
Before continuing our discussion, it is very important to observe that $B(X)$ brings more information than $\Delta(X)$ and $\Gamma(X)$. In other words, the graph $B(X)$ cannot be recovered only from $\Delta(X)$ and $\Gamma(X)$. For instance, when $\Delta(X)\cong \Gamma(X)$ is the complete graph $K_3$ on three vertices, the graph $B(X)$ can be isomorphic to one of the two graphs in Figure \[fig1fig1\].
\[scale=1,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m1) at (-3,1) ; (m2) at (-4,1) ; (m3) at (-5,1) ; (m4) at (-3,-1) ; (m5) at (-4,-1) ; (m6) at (-5,-1) ; (n1) at (3,1) ; (n2) at (4,1) ; (n3) at (2,1) ; (n4) at (3,-1) ; (n5) at (4,-1) ; (n6) at (2,-1) ; (m1) to (m4); (m1) to (m5); (m2) to (m4); (m2) to (m6); (m3) to (m6); (m3) to (m5); (n1) to (n6); (n1) to (n5); (n1) to (n4); (n3) to (n4); (n2) to (n4);
These graphs arise by taking $p,q$ and $r$ three distinct primes and by taking, for instance, $X=\{pq,pr,qr\}$ or $X=\{pqr,p,p^2\}$. (There are other isomorphism classes of $B(X)$ yielding $\Delta(X)\cong \Gamma(X)\cong K_3$, here we just presented two.) It goes without saying that the extra information brought by $B(X)$ asks for a finer investigation.
The first application of the bipartite divisor graph in group theory is for the set of conjugacy class sizes, that is, for a given finite group $G$, $X:=\{m\in\mathbb{N}\mid m \textrm{ is the conjugacy class size of some non-central element of }G\}$. (There is no standard notation for this graph and, in this section, we denote it by $B(Cl(G))$.) In [@BDIP], the authors considered this graph and studied various properties. Among other things, they proved that the diameter is at most $6$ and they classified the groups attaining the upper bound. Moreover, when the graph has no cycles, the diameter is actually at most $5$ and they classified the groups for which the graph is a path of length $5$.
The classification of Dolfi and Jabara [@DJ] of the finite groups with only two non-trivial conjugacy class sizes has spurred more interest in $B(Cl(G))$ and has proved useful in studying $B(Cl(G))$. For instance, it follows immediately from [@DJ] that there is no group $G$ with $B(Cl(G))\cong C_{4}$, where $C_4$ is the [*cycle*]{} of length $4$. Therefore, it is interesting to see, if one minded so, whether there exists a group $G$ with $B(Cl(G))$ isomorphic to a cycle. Taeri in [@T] has answered this question and has proved that $B(Cl(G))$ is a cycle if and only if it is the cycle $C_6$ of length six. Moreover, Taeri has classified the groups $G$ with $B(Cl(G))\cong C_6$; indeed, $G\cong A\times \mathrm{SL}_2(q)$ where $A$ is an abelian group and $q\in \{4,8\}$. Since $C_4$ is also the [*complete bipartite*]{} graph $K_{2,2}$ and since there is no finite group $G$ with $B(Cl(G))\cong K_{2,2}$, Taeri [@T Question 1] has asked whether $B(Cl(G))$ can be isomorphic to some complete bipartite graph. In [@HSpiga], we answered this question and we constructed infinitely many groups $G$ with $B(Cl(G))\cong K_{2,5}$. However, as far as we are aware, it is not known for which positive integers $n$ and $m$ there exists a finite group $G$ with $B(Cl(G))\cong K_{n,m}$ (let alone a meaningful classification of the groups $G$ with $B(Cl(G))\cong K_{n,m}$).
We conclude this brief discussion on $B(Cl(G))$ recalling that the first author and Iranmanesh [@HI Theorem $4.1$] have classified the groups $G$ where $B(Cl(G))$ is isomorphic to a [*path*]{}. This classification was obtained by investigating the combinatorial properties of the bipartite divisor graphs constructed from the product of subsets of positive integers [@HI].
In this paper we are concerned with the [*bipartite divisor graph for the set of irreducible complex character degrees*]{}. Given a finite group $G$, we let ${\mathrm{Irr}}(G)$ be the set of the irreducible complex characters of $G$, we let ${\mathrm{cd}}(G):=\{\chi(1)\mid \chi\in {\mathrm{Irr}}(G)\}$ and we let ${\mathrm{cd}}(G)^*:={\mathrm{cd}}(G)\setminus\{1\}$. Finally, we let $B(G)$ denote the bipartite divisor graph for the set of integers ${\mathrm{cd}}(G)^*$. We recall that the vertex set is the disjoint union of the set of prime numbers dividing some element of ${\mathrm{cd}}(G)^*$ and ${\mathrm{cd}}(G)^*$ itself, where we declare the prime $p$ to be adjacent to the character degree $m$ if and only if $p$ divides $m$.
The scope of this paper is to outline some main results on $B(G)$. We feel that future research on $B(G)$ might benefit from this because these results are scattered over a number of papers (cf. [@H; @Hregular; @mus; @moo4; @moo5]). During this process, we are able to improve some of these results. Moreover, along the way, we leave some problems and questions.
In Section \[sec:special graphical shapes\], we investigate the groups $G$ where $B(G)$ is in a certain class of graphs (paths, union of paths, cycles and complete bipartite). In Section \[sec:reg\], we study the groups $G$ where $B(G)$ is a regular graph, that is, all vertices of $B(G)$ have the same valency. Finally, in Section \[sec:bounded\], we study the groups $G$ with $B(G)$ having at most $6$ vertices. We proceed by discussing the main results that have been proved already and (in several occasions) by improving some of this work.
Notation
--------
All groups and graphs in our paper are finite. We denote by $\g{\mathrm{cd}}(m,n)$ the [*greatest common divisor*]{} of the integers $m$ and $n$. Given a prime number $p$, we let $n_p$ be the [*$p$-part*]{} of $n$, that is, the largest power of $p$ dividing the integer $n$. Similarly, we denote by $n_{p'}$ the [*$p'$-part*]{} of $n$, that is, $n_{p'}:=n/n_p$. We let $\pi(n)$ denote the set of all prime divisors of the natural number $n$.
Given a graph $\mathcal{G}$, we let $V(\mathcal{G})$ denote the [*vertex set*]{}, we let $E(\mathcal{G})$ denote the [*edge set*]{}, we let $n(\mathcal{G})$ denote the number of [*connected components*]{} and we let $o(\mathcal{G})$ denote the cardinality of $V(\mathcal{G})$. The diameter of $\mathcal{G}$, denoted by $\operatorname{{\rm diam}}(\mathcal{G})$, is the maximum of the diameters of the connected components of $\mathcal{G}$. If $\mathcal{G}$ is disconnected and $\mathcal{G}_{1},\ldots,\mathcal{G}_{n}$ are the connected components of $\mathcal{G}$, then we write $\mathcal{G}:=\mathcal{G}_{1}+\cdots +\mathcal{G}_{n}$. By [*length of a path*]{} or [*a cycle*]{}, we mean the number of edges in the path or in the cycle. Also, by $P_{n}$ and $C_{n}$, we mean a path of length $n$ and a cycle of length $n$, respectively. A complete graph on $n$ vertices and a complete bipartite graph on $(m,n)$ vertices are denoted by $K_{n}$ and $K_{m,n}$, respectively.
Given a finite group $G$, we let $\pi(G)$ be the set of all [*prime divisors of the order*]{} of $G$. As usual, we write $dl(G)$ and $h(G)$ to denote the [*derived length*]{} and the [*Fitting height*]{} of $G$, respectively. We denote the [*first and second Fitting subgroups*]{} of $G$ by $\F G$ and ${\bf F}_2(G)$, respectively. Other notations throughout the paper are standard and should cause no confusion.
We let $\rho(G)$ be the set of all [*prime numbers dividing some element of*]{} ${\mathrm{cd}}(G)^*$. The graphs that we use in this paper are:
Prime graph $\Delta(G)$
: $$\begin{aligned}
V(\Delta(G))&:=\rho(G),\\
E(\Delta(G))&:=\{\{p,q\}\mid p,q\in \rho(G), p\ne q, pq \textrm{ divides some element of }{\mathrm{cd}}(G)\};\end{aligned}$$
Common divisor graph $\Gamma(G)$
: $$\begin{aligned}
V(\Gamma(G))&:={\mathrm{cd}}(G)^*,\\
E(\Gamma(G))&:=\{\{m,k\}\mid m,k\in {\mathrm{cd}}(G)^{*}, m\ne k, \g{\mathrm{cd}}(m,k)\neq 1\};\end{aligned}$$
Bipartite divisor graph $B(G)$
: $$\begin{aligned}
V(B(G))&:=\rho(G)\amalg {\mathrm{cd}}(G)^{*}\, (\textrm{disjoint union}),\\
E(B(G))&:=\{\{p,m\}\mid p\in\rho(G),m\in {\mathrm{cd}}(G)^{*}, p\textrm{ divides }m\}.\end{aligned}$$
Notation in the figures of this paper
-------------------------------------
We have consistently drawn all figures of this paper so that the vertices in the lower part of the picture are light blue and are the elements of $\rho(G)$ and the vertices in the upper part of the picture are blue and are the elements of ${\mathrm{cd}}(G)^{*}$.
Groups whose bipartite divisor graphs have special shapes {#sec:special graphical shapes}
=========================================================
One of the main questions that naturally arises in this area of research is classifying those groups whose bipartite divisor graphs have special shapes. In [@H], the first author of this paper discussed the cases where the bipartite divisor graph for the set of irreducible character degrees is
- a path (see Theorem \[thm:990\]),
- a union of paths for non-solvable groups (see Theorem \[thm:99\]), or
- a cycle (see Theorems \[thm:55\]).
In this section, one the one hand, we review and we improve the results in [@H], on the other hand, we discuss the algebraic structure of a solvable group whose bipartite divisor graph is a union of paths.
In our analysis we use the classification into [*six types*]{} of Mark Lewis [@ML2] of the solvable groups whose degree graph is disconnected. Lewis has named these classes Type 1–6 and for each of these types he has given a detailed description in [@ML2 Lemmas $3.1$–$3.6$]. Except for the proof of Theorem \[thm:P4\], we do not need this full classification here. We assume that the reader is broadly familiar with these types, however we highlight below the following properties tailored to our needs.
\[rem:20\]
Let $X$ be a solvable group with $\Delta(X)$ disconnected. Then $\Delta(X)$ has two connected components. Moreover, the following hold.
- If $X$ is of type $1$, $2$, $3$, or $5$, then at least one of the connected components of $\Delta(X)$ has cardinality $1$. Thus, if each connected component of $\Delta(X)$ has at least two vertices, then $X$ is a group of type $4$ or $6$. The converse is not true: there are some groups of type $4$ having prime graph consisting of two isolated vertices. For example, the group $$\texttt{SmallGroup(168,43)}\cong \mathrm{A}\Gamma\mathrm{L}(1,8)$$ in the “SmallGroup” library in GAP [@GAP4] is of type $4$ and its prime graph has vertex set $\{3,7\}$. Moreover, if $X$ is a group of type $6$, then $X$ has a normal Sylow $p$-subgroup and $\Delta(X)$ has a connected component consisting of $\pi([{\bf F}_{2}(X):\F X])\cup\{p\}$ and this set has cardinality greater than $1$.
- If $X$ is a group of type $1$, then $h(X)=2$, while for all the other types $h(X)\ge 3$, see [@ML2 Lemma 4.1].
- The group $X$ has a normal non-abelian Sylow subgroup if and only if $X$ is of type $1$ or $6$.
- If $X$ is a group of type $5$, then $\{1,2,2^{a}+1\}\subseteq {\mathrm{cd}}(X)$, for some positive integer $a$.
- If $X$ is a group of type $2$, then ${\mathrm{cd}}(X)=\{1,2,3,8\}$ and if $X$ is of type $3$, then ${\mathrm{cd}}(X)=\{1,2,3,4,8,16\}$.
The case where the bipartite divisor graph is a path
----------------------------------------------------
Let $G$ be a finite group. In [@H] it is proved that $B(G)$ has diameter at most seven and this upper bound is the best possible. In the special case that $B(G)$ is a path of length $n$, [@H Proposition 2] improves this bound by showing that $n\leq 6$; moreover, $G$ is solvable and $dl(G)\leq 5$. The following theorem gives a more detailed description of $G$.
\[thm:990\] Let $G$ be a finite group with $B(G)$ a path of length $n$. Then, one of the following occurs:
- $G$ has an abelian normal subgroup $N$ such that ${\mathrm{cd}}(G)=\{1,[G : N]\}$ and $G/N$ is abelian. Furthermore, $n\in\{1,2\}$.
- There exist normal subgroups $N$ and $K$ of $G$ and a prime number $p$ with the following properties:
- $G/N$ is abelian;
- $\pi(G/K)\subseteq\rho(G)$;
- either $p$ divides all the non-trivial irreducible character degrees of $N$ (this implies that $N$ has a normal $p$-complement), or ${\mathrm{cd}}(N)=\{1, l, k,h/m\}$, where ${\mathrm{cd}}(G)=\{1, m, h, l, k\}$.
Furthermore, $n\in\{4,5,6\}$.
- ${\mathrm{cd}}(G)=\{1,p^{\alpha},q^{\beta},p^{\alpha}q^{\beta}\}$, where $p$ and $q$ are distinct primes and $\alpha,\beta$ are positive integers. Thus $n=4$.
- There exists a prime $s$ such that $G$ has a normal $s$-complement $H$. Either $H$ is abelian and $n\in\{1,2\}$ or $H$ is non-abelian and
- ${\mathrm{cd}}(G)=\{1,h,hl\}$, for some positive integers $h$ and $l$, and $n=3$;
- $n=4$ and $G/H$ is abelian. Either ${\mathrm{cd}}(H)=\{[H:\F H]\}\cup {\mathrm{cd}}(\F H)$ or ${\mathrm{cd}}(H)=\{1, [{\bf F}_{2}(H):\F H], [H:\F H]\}$. Also $[G:\F G]\in {\mathrm{cd}}(G)$ and ${\mathrm{cd}}(\F G)=\{1,h_{s^{'}}\}$, where $[G:\F G]\neq h\in {\mathrm{cd}}(G)$;
- $n=3$, $G/H$ is abelian, $h:=[G:\F G]\in {\mathrm{cd}}(G)$, $\F G=P\times A$, where $P$ is a $p$-group for some prime number $p$, $A\leq \Z G$, ${\mathrm{cd}}(G)={\mathrm{cd}}(G/A)$, and ${\mathrm{cd}}(P)=\{1, m_{s^{'}}\}$ for $h\neq m\in {\mathrm{cd}}(G)$.
In Theorem \[thm:990\], the description of the groups $G$ with $B(G)\cong P_n$ and $n\le 3$ is rather good. (For instance, when $n=1$, or $n=2$ and $|\rho(G)|=2$, the group $G$ has a unique non-trivial character degree. These groups are classified by the work in [@IS Chapter 12] and [@BH]. Similarly, when $|{\mathrm{cd}}(G)^*|=2$, a great deal of information is in [@N].) However, when $n\ge 4$, the information on $G$ is not yet very satisfactory.
For the time being, we focus on the case $n=4$, that is, $B(G)\cong P_{4}$. This situation may arise from two different cases: either $|\rho(G)|=2$ or $|\rho(G)|=3$. Both cases are possible as it is shown in Table \[tab:p4\]. Examples of the first case are rather elementary, it suffices to take the direct product $G:=A\times B$, where $A$ and $B$ are both groups having a unique non-trivial character degree $p_a^\alpha$ and $p_b^\beta$, respectively, where $p_a$ and $p_b$ are distinct prime numbers. It is rather more intriguing to construct examples of the second kind, as witnessed by the fact that the smallest finite group $G$ with $B(G)\cong P_4$ and $\rho(G)=3$ has cardinality $960$. Therefore, we now look more closely to this case.
$G$ $|\rho(G)|$ ${\mathrm{cd}}(G)$
----------------------------------------- ------------- --------------------
$\mathrm{Sym}(3)\times \mathrm{Alt}(4)$ $2$ $\{1,2,3,6\}$
$\texttt{SmallGroup(960,5748)}$ $3$ $\{1,12,15\}$
: Examples of $B(G)=P_{4}$
\[tab:p4\]
Assume that $G$ is a finite group with $B(G)\cong P_{4}$ and $|\rho(G)|=3$. Let $\rho(G):=\{p,q,r\}$ and let $\alpha,\beta,\gamma,\delta$ be positive integers such that ${\mathrm{cd}}(G):=\{1,p^\alpha q^\beta,r^\gamma q^\delta\}$. Since every non-linear character degree of $G$ is divisible by the prime $q$, we deduce (from a celebrated theorem of Thompson [@IS (12.2)]) that $G$ has a normal $q$-complement $L$. Let $Q$ be a Sylow $q$-subgroup of $G$. Thus $G$ equals the semidirect product $L\rtimes Q$. Let $\theta\in {\mathrm{Irr}}(L)$ and let $\chi\in{\mathrm{Irr}}(G)$ with $\langle \chi_L,\theta\rangle\ne 0$. Then, from Clifford theory, $\chi_L=e(\theta_1+\cdots+\theta_t)$, where $\theta_1,\ldots,\theta_t$ are the conjugates of $\theta$ under $G$. As $\chi(1)\in \{1,p^\alpha q^\beta,r^\gamma q^\delta\}$ and $e,t$ are divisors of $|G:L|=|Q|$ by [@IS (11.29)], we deduce $$\label{eq:2}
{\mathrm{cd}}(L)=\{1,p^\alpha,r^\gamma\}.$$ Therefore $\Delta(L)$ is a disconnected graph with two isolated vertices. As $L$ is solvable, Remark \[rem:20\] implies that $L$ is a group of type $1$, $4$ or $5$ in the sense of Lewis. In particular, if $L$ has a non-abelian normal Sylow subgroup, then $L$ is of type $1$. An example of this case is in Table \[tab:p4\], which we now discuss.
\[exam: P4\]
*Let $G=\texttt{SmallGroup}(960,5748)$. Then ${\mathrm{cd}}(G)=\{1,12,15\}$ and $$G\cong (\mathbb{Z}_{2}\times\mathbb{Z}_{2}).(\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{15}$$ Indeed, $G$ has a normal Sylow $2$-subgroup $P$ with $|P|=2^6=64$, $\Z P=P'=\Phi(P)$ and $|\Z P|=4$.*
Using the notation that we have established above, $q=3$, $L$ is the normal $3$-complement of $G$ and ${\mathrm{cd}}(L)=\{1,4,5\}$. As $\Delta(L)\cong K_{1}+K_{1}$ and $L$ has a normal non-abelian Sylow subgroup, Remark \[rem:20\] implies that $L$ is of type $1$.
Motivated by Example \[exam: P4\], the following theorem verifies that $L$ is always a group of type $1$ and explains in part the elusiveness of the groups $G$ with $B(G)\cong P_4$ and $|\rho(G)|=2$.
\[thm:P4\] Suppose that ${\mathrm{cd}}(G_0)=\{1,p^\alpha q^\beta,r^\gamma q^\delta\}$. Then there exists $A\le \Z {G_0}$ such that for the factor group $G:=G_0/A$ the following holds (replacing $r$ with $p$ if necessary):
- $G$ contains a normal Sylow $p$-subgroup $P$;
- $P=G'=\F G$;
- $P$ is semiextraspecial and $G/P$ is cyclic ($P$ is called semiextraspecial if, for all maximal subgroups $N$ of $\Z P$, the factor group $P/N$ is extraspecial);
- $G/P'$ is a Frobenius group with Frobenius kernel $P/P'$;
- $P<\cent G{P'}<G$;
- $G/\cent G{P'}$ acts as a Frobenius group on $P'$;
- ${\mathrm{cd}}(G)={\mathrm{cd}}(G_0)$, where $p^{2\alpha}=|P:P'|$ and $q^{\beta}=|G:\cent G {P'}|$;
- if $L$ is the $q$-complement of $G$, then $L$ is of type $1$ in Lewis sense.
The proof follows applying Theorem $5.6$ in [@N] in our context.
Given a finite group $G$, it is easy verify that, if $B(G)$ is a path, then both $\Delta(G)$ and $\Gamma(G)$ are paths. In particular, if $B(G)\cong P_{6}$, then $\Gamma(G)\cong P_{3}$ and hence $\Gamma(G)$ has diameter three. The converse is not always true, as we show in the following example.
*We recall a construction from [@shafiei]. Let $p$, $q$, and $r$ be three, not necessarily distinct, primes with $q\neq r$ such that $q$ and $r$ do not divide $p^{qr}-1$, let $V$ be the additive group of the field $\mathbb{F}_{p^{qr}}$ of order $p^{qr}$, let $S$ be the Galois group of the field extension $\mathbb{F}_{p^{qr}}/\mathbb{F}_p$, let $C$ be the cyclic subgroup of order $\frac{p^{qr}-1}{p^{r}-1}$ of the multiplicative group of $\mathbb{F}_{p^{qr}}^*$ and let $G:=(V\rtimes C)\rtimes S$. Then $${\mathrm{cd}}(G)=\left\{1,q,qr,\frac{p^{qr}-1}{p^{r}-1},r\frac{p^{qr}-1}{p^{r}-1}\right\}.$$ Thus $B(G)$ is a path if and only if $\frac{p^{qr}-1}{p^{r}-1}$ is a prime power. Lemma $3.1$ in [@shafiei] yields that, if $\frac{p^{qr}-1}{p^{r}-1}$ is a prime power, then either $r$ is a power of $q$, or $p=q=2$ and $r=3$.*
The first case is not possible because by hypothesis $r$ and $q$ are distinct primes. In the second case, $r=3$ is a divisor of $p^{qr}-1=63$, which contradicts our hypothesis. Thus $\frac{p^{qr}-1}{p^{r}-1}$ is not a prime power and $B(G)$ is not a path.
Next, we prove the existence of groups $G$ with $B(G)\cong P_{n}$, for $n\in\{5,6\}$. This answers Question 1 in [@H].
\[ex:5\]
When $n=5$, we give infinitely many groups $G$ with $B(G)\cong P_5$. Our example is due to Péter Pál Pálfy, and we gratefully acknowledge his contribution.
Let $r$ be an odd prime, let $p$ be a prime with $p\equiv 1\pmod {2r}$ and let $P$ be an extraspecial group of order $p^{3}$ with exponent $p$. (Observe that the existence of infinitely many primes $p$ follows, for instance, from Dirichlet theorem on primes in arithmetic progression.) The group $P$ has the following presentation: $$P=\langle x_{1},x_{2}\mid x_{1}^{p}=x_{2}^{p}=[x_1,x_2]^p=[x_1,[x_1,x_2]]=[x_2,[x_1,x_2]]=1\rangle.$$ Since $2r$ divides $p-1$, there exists $\alpha\in \mathbb{Z}$ such that $\alpha $ has order $2r$ in the multiplicative group of $(\mathbb{Z}/p\mathbb{Z})^*$. Set $\beta:=\alpha^{r-1}$ and observe that $\beta$ has order $r$ in the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ because $\gcd(2r,r-1)=2$. From the presentation of $P$, we see that the mapping $$x_{1}\mapsto x_{1}^{\alpha},\quad x_{2}\mapsto x_{2}^{\beta}$$ defined on the generators of $P$ extends to an automorphism of $P$ of order $2r$, which we denote by $y$. Set $G:=P\rtimes \langle y\rangle$ with respect to the above action. We have $$[x_1,x_2]^y=[x_1^y,x_2^y]=[x_1^\alpha,x_2^\beta]=[x_1,x_2]^{\alpha\beta}=[x_1,x_2]^{\alpha\alpha^{r-1}}=[x_1,x_2]^{\alpha^r}.$$ Since $\alpha^r$ has order $2$ modulo $p$, we obtain $[x_1,x_2]^y=[x_1,x_2]^{-1}$ and hence $y$ acts on $\Z P$ by inverting its elements. From this, it is easy to see that ${\mathrm{cd}}(G)=\{1,r,2r,2p\}$, so $B(G)$ is a path of length five.
\[ex:6\]
When $n=6$, our construction is based on some preliminary theoretical work and then its implementation in a computer. Here, we report only the outcome of our computations because we are not able to give a general construction.
Let $P$ be the group $\texttt{SmallGroup}(256,3679)$. One can check that the group $P$ has nilpotency class $3$, $|P:\gamma_2(P)|=|\gamma_2(P):\gamma_3(P)|=2^3=8$ and $|\gamma_3(P)|=4$. Moreover, each section of the lower central series has exponent $2$. Let $T$ be a Hall $2'$-subgroup in the automorphism group of $P$. A simple computation yields that $T$ is non-abelian and has cardinality $21$. Let $G$ be the semidirect product $P\rtimes T$. Then $G$ is a solvable group having cardinality $5\,376=2^8\cdot 3\cdot 7$ and a computation yields ${\mathrm{cd}}(G)=\{1,3,7,14,24\}.$ Therefore $B(G)$ is a path of length $6$. Furthermore, considering $G$ as a permutation group of degree $32$, it is generated by $\alpha_{1}$, $\alpha_{2}$, and $\alpha_{3}$, where $$\begin{aligned}
&\text{\small$\alpha_{1}:=(1,2)(3,14,9,20)(4,15,19,27)(5,7)(6,8)(10,21,18,26)(11,22,12,23)(13,16)(17,31,25,29)(24,32,28,30)$},\\
&\text{\small$\alpha_{2}:=(2,27,25,19,21,10,31)(3,29,8,22,17,11,14)(4,20,9,30,16,15,24)(7,23,28,12,26,18,32)$},\\
&\text{\small$\alpha_{3}:=(3,12,30)(4,29,18)(5,13,6)(7,16,8)(9,11,32)(10,19,31)(14,23,24)(15,17,26)(20,22,28)(21,27,25)$}.\end{aligned}$$ This is the smallest group we managed to construct having $B(G)$ a path of length $6$.
Now that we have established the existence of a group $G$ with $B(G)\cong P_n$ for $n\in\{5,6\}$, it would be interesting to give a classification of this family of groups.
Give structural information on the finite groups $G$ with $B(G)\cong P_{n}$ for $n\in\{5,6\}$.
0.4 true cm
Union of paths {#unionpaths}
--------------
As it is explained in [@N Example 3.4], given a prime $p$ and two positive integers $k$ and $m$ with $k$ dividing $p^m\pm 1$, there exists a solvable group $PH$ such that ${\mathrm{cd}}(PH)=\{1,k,p^{m}\}$. (More information on the structure of $G$ is in [@N], but this is of no concern here.) In particular, when $|\pi(k)|=2$, $B(PH)$ is the first graph in Figure $\ref{fig: 5}$, and hence $B(PH)$ is a union of paths. On the other hand, ${\mathrm{cd}}(M_{10})=\{1,9,10,16\}$ and ${\mathrm{cd}}(\mathrm{PSL}_2(25))=\{1,13,24,25,26\}$ and hence $B(M_{10})$ and $B(\mathrm{PSL}_2(25))$ have two connected components which are both paths: $B(M(10))=P_2+P_3$ and $B(\mathrm{PSL}_2(25))=P_2+P_5$.
These examples stimulate the curiosity of investigating those finite groups $G$ such that each connected component of $B(G)$ is a path. For this aim, first we let $G$ be a finite non-solvable group. In the following theorem, we refine the main result of [@H], cf. [@H Theorem 6].
\[thm:99\] Let $G$ be a finite non-solvable group with $B(G)$ a union of paths. Then $B(G)$ is disconnected, $B(G)$ has $2$ or $3$ connected components and $|{\mathrm{cd}}(G)|\in \{4,5\}$.
Moreover, we have one of the following cases:
- $n(B(G))=2$, $|{\mathrm{cd}}(G)|=4$, $G$ has a normal subgroup $U$ such that $U\cong \mathrm{PSL}_2(q)$ or $U\cong \mathrm{SL}_2(q)$ for some odd $q\ge 5$ and, if $C:=\cent G U$, then $C\le \Z G$ and $G/C\cong \mathrm{PGL}_2(q)$. Thus ${\mathrm{cd}}(G)=\{1,q,q-1,q+1\}$.
- $n(B(G))=2$, $|{\mathrm{cd}}(G)|=4$, $G$ has a normal subgroup of index $2$ that is a direct product of $\mathrm{PSL}_2(9)$ and a central subgroup $C$. Furthermore, $G/C\cong M_{10}$ and ${\mathrm{cd}}(G)=\{1,9,10,16\}$.
- $n(B(G))=2$, $|{\mathrm{cd}}(G)|=5$, $G$ has a solvable normal subgroup $V$ such that $G/V$ is almost simple and isomorphic to either $\mathrm{PSL}_2(q)$, or $\mathrm{PGL}_2(q)$, or $\mathrm{P}\Gamma\mathrm{L}_2(2^r)$ (for some prime number $r$), $M_{10}$, $\mathrm{P}\Sigma\mathrm{L}_2(9)$, or $\mathrm{PGL}_2(3^{2f'}).2$ (for some $f'\ge 1$).
- $n(B(G))=3$, ${\mathrm{cd}}(G)=\{1,2^n,2^n-1,2^n+1\}$ and $G\cong \mathrm{PSL}_2(2^{n})\times A$, where $A$ is an abelian group and $n\geq 2$.
From Theorem \[thm:990\], the non-solvability of $G$ implies that $B(G)$ is disconnected. By [@L Theorem 6.4], we have $n(\Delta(G))\le 3$ and, by [@IP], we have $n(B(G))=n(\Delta(G))$. Thus $B(G)$ is disconnected with at most three connected components.
When $n(B(G))=3$, the result follows from [@H] and we obtain part (iv). Suppose then $n(B(G))=2$. From [@H], we deduce ${\mathrm{cd}}(G)\in \{4,5\}$. When ${\mathrm{cd}}(G)=4$, the result follows from [@GA Theorem A] and we obtain parts (i) and (ii). Finally, suppose that ${\mathrm{cd}}(G)=5$. The finite groups having $5$ (or $6$) character degrees are classified in [@LZ Corollary C]. From this classification, we see that $G$ contains a normal solvable subgroup $V$ with $G/V$ almost simple with socle $\mathrm{PSL}_2(p^f)$ (with $p^f\ge 4$), or $\mathrm{PSL}_3(4)$, or $^{2}B_2(2^{2m+1})$ (with $m\ge 1$). The cases $\mathrm{PSL}_3(4)$ and $^2{B}_2(2^f)$ do not arise here because any almost simple group having socle one of these two groups has $6$ irreducible complex character degrees. In particular, $G/V$ is almost simple with socle $\mathrm{PSL}_2(p^f)$. Now, the actual structure of $G/V$ can be inferred from the work of White [@white Theorem A]. Indeed, White computes explicitly the character degrees of each almost simple group $X$ having socle $\mathrm{PSL}_2(p^f)$. In part (iii), we have selected the groups $X$ with $B(X)$ a union of paths.
Observe that the converse of Theorem \[thm:99\] does not hold; for instance, $B(\textrm{PSL}_2(2^n))$ consists of three connected components for each value of $n$, however these connected components are not necessarily paths (this depends on number-theoretic questions concerning the factorization of $2^n+1$ and $2^n-1$).
0.3 true cm
Now, we let $G$ be a [*solvable*]{} group with $B(G)$ disconnected and union of paths (the connected case was discussed in the previous section). As $G$ is solvable, we have $n(B(G))=2$ and $|\rho(G)|\geq 2$. Here we describe the structure of $G$ using the types introduced by Lewis. By Remark \[rem:20\], type $3$ does not occur. As $\Delta(G)$ is triangle-free, the main result in [@Tong-viet Lemma $2.2$] yields $|\rho(G)|\leq 4$. Using $n(B(G))=2$ and $2\le |\rho(G)|\le 4$, a simple case-by-case analysis gives that $B(G)$ is one of the graphs drawn in Figures \[fig: 4\], \[fig: 5\] or \[fig: 6\]. We have tabulated some information on these graphs and the groups yielding these graphs inTable \[tab:TTCR\]. This table consists of three columns: the first column is one of the graphs $\Gamma$ in Figures \[fig: 4\], \[fig: 5\] or \[fig: 6\], the second column are the Lewis types $X\subseteq\{1,2,3,4,5,6\}$ of the groups $G$ with $\Gamma\cong B(G)$ and in the third column we exhibit (for each $x\in X$) a group of Lewis type $x$ and with $\Gamma\cong B(G)$. The task in the rest of this section is proving the correctedness of Table \[tab:TTCR\].
\[lemma:new\] Let $G$ be a group with $B(G)$ isomorphic to the second graph in Figure $\ref{fig: 4}$. Then $G$ is not of Lewis type $4$.
We argue by contradiction and we suppose that $G$ is of type $4$. We use the notation established in [@ML2] for the groups of type $4$ and we suppose that the reader is familiar with basic properties about groups in this class (in particular with Example $2.4$ and Lemma $3.4$ in [@ML2]). Since ${\mathrm{cd}}(G)={\mathrm{cd}}(G/\Z G)$ by [@ML2 Lemma $3.4$], we may suppose $\Z G=1$. In particular, $G=V\rtimes H$ and $H$ acts irreducibly as a linear group on the elementary abelian $p$-group $V$.
Recall, from [@ML2], that $K:=\F H$, $m:=|E:K|>1$, $K$ is cyclic, $|V|=q^m$ where $q$ is a power of the prime $p$, and $(q^m-1)/(q-1)$ divides $|K|$. Now, ${\mathrm{cd}}(G|V)=\{|K|\}$ and ${\mathrm{cd}}(G/V)$ consists of $1,m$ and eventually some other divisors of $m$, see [@ML2 Lemma $3.4$].
As $B(G)$ is isomorphic to the second graph in Figure \[fig: 4\], we deduce that $|K|$ is a prime power (say $|K|=r^\ell$ for some prime number $r$ and some positive integer $\ell$), $m$ is a prime power (say $m=s^t$ for some prime number $s$ and some positive integer $t$) and ${\mathrm{cd}}(G/V)=\{1,s^t,s^{t'}\}$ for some $0<t'<t$. In particular, $t\ge 2$.
Recall that, given two positive integers $x$ and $y$, the prime number $z$ is said to be a primitive prime divisor of $x^y-1$ if $z$ divides $x^y-1$, but (for every $i\in \{1,\ldots,y-1\}$) $z$ does not divide $x^i-1$.
Let $z$ be a primitive prime divisor of $q^{m}-1=q^{s^t}-1$. As $(q^m-1)/(q-1)$ divides $|K|=r^\ell$, we deduce $z=r$. As $(q^{s^{t-1}}-1)/(q-1)$ divides $(q^{s^t}-1)/(q-1)$ and $t\ge 2$, we deduce that $r$ divides $q^{s^{t-1}}-1$, contradicting the fact that $r$ is a primitive prime divisor of $q^m-1=q^{s^t}-1$. Therefore $q^{s^t}-1$ has no primitive prime divisors. From a celebrated theorem of Zsigmondy [@zsigmondy], we deduce that either $q=2$ and $s^t=6$, or $s^t=2$ and $q$ is a Mersenne prime, however in both cases we obtain a contradiction.
\[lemma:new1\] Let $G$ be a group with $B(G)$ isomorphic to the second graph in Figure $\ref{fig: 5}$, to the first or to the third graph in Figure $\ref{fig: 6}$, or to the third graph in Figure \[fig: 2\]. Then $G$ is not of Lewis type $6$.
We argue by contradiction and we suppose that $G$ is of type $6$. We use the notation established in [@ML2] for the groups of type $6$ and we suppose that the reader is familiar with basic properties about groups in this class (in particular with Example $2.6$ and Lemma $3.6$ in [@ML2]). From [@ML2 Lemma $3.6$ (v),(vi)], ${\mathrm{cd}}(G)={\mathrm{cd}}(G/A')\cup{\mathrm{cd}}(G|A')$, where $G/A'$ is a group of Lewis type $4$ and ${\mathrm{cd}}(G|A')$ consists of degrees that divide $|P||E:F|$ and are divisible by $p|B|$, (observe that $p$ and $|B|$ are relatively prime). These facts together imply that $B(G/A')$ is a union of paths and is obtained by deleting some blue vertices of $B(G)$ having at least two light blue neighbors. If $B(G)$ is the third graph in Figure \[fig: 6\], then we can delete only one vertex from $B(G)$ in order to obtain $B(G/A')$. However, the resulting graph has three connected components, contradicting the fact that $G/A'$ is solvable. If $B(G)$ is the first graph in Figure \[fig: 6\], then (again) we can delete only one vertex from $B(G)$ in order to obtain $B(G/A')$. Therefore, the resulting graph is the second graph in Figure \[fig: 4\]. However, Lemma \[lemma:new\] excludes this possibility. If $B(G)$ is the second graph in Figure \[fig: 5\], then we can delete only one vertex from $B(G)$ in order to obtain $B(G/A')$. Now, the resulting graph is connected, contradicting the fact that it is a graph of Lewis type $4$ and hence disconnected. Finally, if $B(G)$ is the third graph in Figure \[fig: 2\], then we can delete only one vertex from $B(G)$ to obtain $B(G/A')$. The result is a connected graph which is impossible.
\[exa:new\]
Let $P$ be the group $\mathtt{SmallGroup}(2\,187,9\,308)$. A computation shows that $\mathrm{Aut}(P)$ contains a cyclic subgroup $H$ of order $10$ with the property that ${\mathrm{cd}}(P\rtimes H)=\{1,2,10,27\}$ and $P\rtimes H$ has Lewis type $1$. In particular, $G:=P\rtimes H$ is a finite soluble group of order $21\,870$ of Lewis type $1$ with ${\mathrm{cd}}(G)=\{1,2,10,27\}$ and with $B(G)$ isomorphic to the third graph in Figure \[fig: 5\].
This example was constructed with the help of a computer after deducing some preliminary theoretical properties. Incidentally, this is the smallest example we managed to find, but we were not able to prove that it is indeed the example of smallest cardinality with $B(G)$ isomorphic to the third graph in Figure \[fig: 5\].
\[exa:newnew\]
Let $G$ be the polycyclic group with presentation $G:=\langle x_1,\ldots,x_{15}\mid R\rangle$, where the set of polycyclic relations $R$ are given by $$\begin{aligned}
&x_1^2 = x_{15},\,
x_2^2 = x_{15},\,
x_3^3 = 1,\,
x_4^{11} = 1,\,
x_5^2 = x_{15},\,
x_6^2 = 1,\,
x_7^2 = 1,\,
x_8^2 = 1,\,
x_9^2 = 1,\,
x_{10}^2 = x_{15},\,
x_{11}^2 = 1,\\
& x_{12}^2 = 1,\,
x_{13}^2 = 1,\,
x_{14}^2 = 1,\,
x_{15}^2 = 1,\,
x_2^{x_1} = x_2 \cdot x_{15},\,
x_3^{x_2} = x_3^2,\,
x_4^{x_2} = x_4^{10},\,
x_5^{x_2} = x_5 \cdot x_{15},\\
&x_5^{x_3} = x_5 \cdot x_7 \cdot x_{10} \cdot x_{11} \cdot x_{12},\,
x_5^{x_4} = x_{10},\,
x_6^{x_3} = x_6 \cdot x_8 \cdot x_{11} \cdot x_{12} \cdot x_{13},
x_6^{x_4} = x_{11},\,
x_7^{x_3} = x_7 \cdot x_9 \cdot x_{12} \cdot x_{13} \cdot x_{14},\\
& x_7^{x_4} = x_{12},\,
x_8^{x_3} = x_5 \cdot x_7 \cdot x_8 \cdot x_{10} \cdot x_{12} \cdot x_{13} \cdot x_{14},\,
x_8^{x_4} = x_{13},\,
x_9^{x_3} = x_6 \cdot x_8 \cdot x_9 \cdot x_{10} \cdot x_{11} \cdot x_{12} \cdot x_{13} \cdot x_{14},\\
& x_9^{x_4} = x_{14},\,
x_{10}^{x_2} = x_7 \cdot x_8 \cdot x_{10} \cdot x_{15},\,
x_{10}^{x_3} = x_5 \cdot x_6 \cdot x_7 \cdot x_{12} \cdot x_{15},\,
x_{10}^{x_4} = x_5 \cdot x_{12} \cdot x_{13},\,
x_{10}^{x_5} = x_{10} \cdot x_{15},\,\\
& x_{10}^{x_6} = x_{10} \cdot x_{15}, \,
x_{10}^{x_7} = x_{10} \cdot x_{15}, \,
x_{10}^{x_8} = x_{10} \cdot x_{15}, \,
x_{11}^{x_2} = x_8 \cdot x_9 \cdot x_{11}, \,
x_{11}^{x_3} = x_6 \cdot x_7 \cdot x_8 \cdot x_{13} \cdot x_{15},\,\\
& x_{11}^{x_4} = x_6 \cdot x_{13} \cdot x_{14}, \,
x_{11}^{x_5} = x_{11} \cdot x_{15}, \,
x_{11}^{x_6} = x_{11} \cdot x_{15}, \,
x_{11}^{x_7} = x_{11} \cdot x_{15}, \,
x_{12}^{x_2} = x_5 \cdot x_7 \cdot x_9 \cdot x_{12}, \,\\
& x_{12}^{x_3} = x_7 \cdot x_8 \cdot x_9 \cdot x_{14} \cdot x_{15}, \,
x_{12}^{x_4} = x_7 \cdot x_{10} \cdot x_{12} \cdot x_{14}, \,
x_{12}^{x_5} = x_{12} \cdot x_{15}, \,
x_{12}^{x_6} = x_{12} \cdot x_{15}, \,\\
& x_{13}^{x_2} = x_5 \cdot x_6 \cdot x_7 \cdot x_8 \cdot x_{13} \cdot x_{15}, \,
x_{13}^{x_3} = x_5 \cdot x_7 \cdot x_8 \cdot x_9 \cdot x_{10} \cdot x_{12}, \,
x_{13}^{x_4} = x_8 \cdot x_{10} \cdot x_{11} \cdot x_{12} \cdot x_{13} \cdot x_{15}, \,\\
& x_{13}^{x_5} = x_{13} \cdot x_{15}, \,
x_{13}^{x_9} = x_{13} \cdot x_{15}, \,
x_{14}^{x_2} = x_6 \cdot x_7 \cdot x_8 \cdot x_9 \cdot x_{14} \cdot x_{15}, \,
x_{14}^{x_3} = x_5 \cdot x_6 \cdot x_7 \cdot x_8 \cdot x_9 \cdot x_{11} \cdot x_{13} \cdot x_{15}, \,\\
& x_{14}^{x_4} = x_9 \cdot x_{11} \cdot x_{12} \cdot x_{13} \cdot x_{14} \cdot x_{15}, \,
x_{14}^{x_8} = x_{14} \cdot x_{15}, \,
x_{14}^{x_9} = x_{14} \cdot x_{15}.\end{aligned}$$ The group $G$ has order $270\, 336=2^{13}\cdot 3\cdot 11$; moreover, $G$ contains a Hall $2'$-subgroup $K$ with $K$ cyclic and $G$ contains a normal $2$-subgroup $Q$ with $|G:QK|=2$. Furthermore, $Q/\gamma_2(Q)$ is elementary abelian of order $2^{10}=1\,024$, $\gamma_2(Q)$ is elementary abelian of order $2^2=4$ and $K$ centralizes $\gamma_2(Q)$. One might check that $G$ has Lewis type $5$. Finally, ${\mathrm{cd}}(G)=\{1,2,33,64\}$ and hence $B(G)$ is isomorphic to the second graph in Figure \[fig: 5\].
As in Example \[exa:new\], this example was constructed (with some luck) with the help of a computer after deducing some preliminary theoretical properties.
\[exa:newnewnew\] [Let $G$ be the polycyclic group with presentation $G:=\langle x_1,\ldots,x_{16}\mid R\rangle$, where the set of polycyclic relations $R$ are given by $$\begin{aligned}
&x_1^3 = 1,
x_2^{11} = 1,
x_3^2 = x_{13},
x_4^2 = x_{13} \cdot x_{16},
x_5^2 = x_{13},
x_6^2 = x_{13},
x_7^2 = 1,
x_8^2 = 1,
x_9^2 = x_{16},
x_{10}^2 = x_{13},
x_{11}^2 = x_{13},\\
& x_{12}^2 = x_{16},
x_{13}^2 = 1,
x_{14}^2 = 1,
x_{15}^2 = 1,
x_{16}^2 = 1,
x_3^{x_1} = x_4 \cdot x_6 \cdot x_8,
x_3^{x_2} = x_4 \cdot x_6 \cdot x_7 \cdot x_8 \cdot x_{11} \cdot x_{12},\\
& x_4^{x_1} = x_5 \cdot x_7 \cdot x_9,
x_4^{x_2} = x_3 \cdot x_4 \cdot x_6 \cdot x_7 \cdot x_{12},
x_5^{x_1} = x_6 \cdot x_8 \cdot x_{10},
x_5^{x_2} = x_3 \cdot x_6 \cdot x_7 \cdot x_9,
x_5^{x_3} = x_5 \cdot x_{13},\\
& x_6^{x_1} = x_7 \cdot x_9 \cdot x_{11},
x_6^{x_2} = x_4 \cdot x_7 \cdot x_8 \cdot x_{10},
x_6^{x_3} = x_6 \cdot x_{13} \cdot x_{16},
x_6^{x_4} = x_6 \cdot x_{16},
x_6^{x_5} = x_6 \cdot x_{16},
x_7^{x_1} = x_8 \cdot x_{10} \cdot x_{12},\\
& x_7^{x_2} = x_5 \cdot x_8 \cdot x_9 \cdot x_{11} \cdot x_{13} \cdot x_{16},
x_7^{x_3} = x_7 \cdot x_{13} \cdot x_{16},
x_7^{x_4} = x_7 \cdot x_{16},
x_7^{x_5} = x_7 \cdot x_{13},
x_7^{x_6} = x_7 \cdot x_{13}, \\
& x_8^{x_1} = x_3 \cdot x_4 \cdot x_5 \cdot x_6 \cdot x_8 \cdot x_{11},
x_8^{x_2} = x_6 \cdot x_9 \cdot x_{10} \cdot x_{12} \cdot x_{16},
x_8^{x_3} = x_8 \cdot x_{16},
x_8^{x_4} = x_8 \cdot x_{16},
x_8^{x_6} = x_8 \cdot x_{13} \cdot x_{16},\\
& x_8^{x_7} = x_8 \cdot x_{13},
x_9^{x_1} = x_4 \cdot x_5 \cdot x_6 \cdot x_7 \cdot x_9 \cdot x_{12},
x_9^{x_2} = x_3 \cdot x_4 \cdot x_5 \cdot x_6 \cdot x_7 \cdot x_8 \cdot x_9 \cdot x_{10} \cdot x_{11} \cdot x_{13},
x_9^{x_3} = x_9 \cdot x_{16}, \\
& x_9^{x_4} = x_9 \cdot x_{13},
x_9^{x_5} = x_9 \cdot x_{16},
x_9^{x_6} = x_9 \cdot x_{13} \cdot x_{16},
x_9^{x_7} = x_9 \cdot x_{13} \cdot x_{16},
x_9^{x_8} = x_9 \cdot x_{16},
x_{10}^{x_1} = x_3 \cdot x_4 \cdot x_7 \cdot x_9 \cdot x_{10},\\
& x_{10}^{x_2} = x_4 \cdot x_5 \cdot x_6 \cdot x_7 \cdot x_8 \cdot x_9 \cdot x_{10} \cdot x_{11} \cdot x_{12} \cdot x_{16},
x_{10}^{x_3} = x_{10} \cdot x_{16},
x_{10}^{x_4} = x_{10} \cdot x_{13} \cdot x_{16},
x_{10}^{x_5} = x_{10} \cdot x_{13} \cdot x_{16},\\
& x_{10}^{x_6} = x_{10} \cdot x_{16},
x_{10}^{x_7} = x_{10} \cdot x_{13},
x_{10}^{x_9} = x_{10} \cdot x_{13},
x_{11}^{x_1} = x_4 \cdot x_5 \cdot x_8 \cdot x_{10} \cdot x_{11},
x_{11}^{x_2} = x_3 \cdot x_4 \cdot x_7 \cdot x_{10} \cdot x_{11} \cdot x_{12} \cdot x_{16}, \\
& x_{11}^{x_3} = x_{11} \cdot x_{13} \cdot x_{16},
x_{11}^{x_4} = x_{11} \cdot x_{13},
x_{11}^{x_6} = x_{11} \cdot x_{13} \cdot x_{16},
x_{11}^{x_8} = x_{11} \cdot x_{13} \cdot x_{16},
x_{11}^{x_9} = x_{11} \cdot x_{13},
x_{11}^{x_{10}} = x_{11} \cdot x_{13} \cdot x_{16},\\
& x_{12}^{x_1} = x_5 \cdot x_6 \cdot x_9 \cdot x_{11} \cdot x_{12},
x_{12}^{x_2} = x_3 \cdot x_6 \cdot x_9 \cdot x_{11} \cdot x_{12} \cdot x_{13},
x_{12}^{x_3} = x_{12} \cdot x_{13},
x_{12}^{x_4} = x_{12} \cdot x_{13},
x_{12}^{x_5} = x_{12} \cdot x_{13} \cdot x_{16},\\
& x_{12}^{x_6} = x_{12} \cdot x_{13} \cdot x_{16},
x_{12}^{x_8} = x_{12} \cdot x_{13},
x_{12}^{x_9} = x_{12} \cdot x_{13},
x_{12}^{x_{10}} = x_{12} \cdot x_{16},
x_{12}^{x_{11}} = x_{12} \cdot x_{13} \cdot x_{16},
x_{15}^{x_{14}} = x_{15} \cdot x_{16}.\end{aligned}$$ The group $G$ has order $540\, 672=2^{14}\cdot 3\cdot 11$; moreover, $G$ contains a Hall $2'$-subgroup $K$ with $K$ cyclic and $G$ contains a normal Sylow $2$-subgroup $P$. Furthermore, $P/\gamma_2(P)$ is elementary abelian of order $2^{12}=2\,048$, $\gamma_2(P)$ is elementary abelian of order $2^2=4$ and $K$ centralizes $\gamma_2(P)$. One might check that $G$ has Lewis type $1$. Finally, ${\mathrm{cd}}(G)=\{1,32,33,64\}$ and hence $B(G)$ is isomorphic to the second graph in Figure \[fig: 5\]. ]{}
If $|{\mathrm{cd}}(G)^{*}|=4$, then $B(G)$ is either the first or the second graph in Figure \[fig: 6\] or the third graph in Figure \[fig: 4\]. Except for the second graph in Figure \[fig: 6\], as $\Gamma(G)$ has no isolated vertices, by [@ML2 Theorem 5.2], we deduce that $G$ has a normal non-abelian Sylow subgroup. Now Remark \[rem:20\] implies that $G$ is a group of type $1$ or $6$ in the sense of Lewis. For the first graph in Figure \[fig: 6\] the case of Lewis type $6$ is excluded by Lemma \[lemma:new1\]. If $B(G)$ is the last graph in Figure \[fig: 4\], then both connected components of $\Delta(G)$ are isolated vertices; so by Remark \[rem:20\] and the previous results we conclude that $G$ is a group of type $1$ (see also [@LL Theorem $3.1$]).
If $B(G)$ is the second graph in Figure \[fig: 6\], then $\Gamma(G)\cong K_{1}+P_{2}$ consists of one isolated vertex and one edge and hence, by [@LL Theorem 3.3], we deduce that $G$ is either a group of type $1$ or $4$ in the sense of Lewis. If $G$ has no non-abelian normal Sylow subgroup, then [@ML2 Theorem 5.2] implies that the prime divisors of the isolated vertex of $\Gamma(G)$ gives the larger component of $\Delta(G)$, which is not the case. Thus $G$ has a non-abelian normal Sylow subgroup. This implies that $G$ is not a group of type $4$ and so it is a group of type $1$. 0.3 true cm
Suppose $B(G)$ is one of the first two graphs in Figure \[fig: 4\]. As $\Delta(G)$ has two isolated vertices, from Remark \[rem:20\], we conclude that $G$ is neither a group of type $3$ nor of type $6$ in the sense of Lewis. If $B(G)$ is the first graph in Figure \[fig: 4\], then it is a $1$-regular bipartite graph. The structure and the Lewis type of such a group is explicitly explained in Theorem \[thm: reg\] below (and we refer the reader to this theorem for a detailed description). Finally, if $B(G)$ is the second graph in Figure \[fig: 4\], then $G$ is a group of type $1$, $2$ or $5$ in the sense of Lewis (type $4$ does not arise because of Lemma \[lemma:new\]). 0.3 true cm
Suppose $B(G)$ is either the first or the third graph in Figure \[fig: 5\]. By Remark \[rem:20\], $G$ is not a group of type $2$ or $3$. If $B(G)$ is the third graph in this figure and $G$ has no non-abelian normal Sylow subgroup, then by [@ML2 Theorem 5.2] we conclude that the prime divisors of the isolated vertex of $\Gamma(G)$ lie in a larger component of $\Delta(G)$ which is not the case for this graph. Hence $G$ has a non-abelian normal Sylow subgroup which implies that $G$ is of Lewis type $1$ or $6$.
Suppose $B(G)$ is the first graph in Figure \[fig: 5\]. As $|cd(G)^{*}|$ consists of co-prime degrees, with respect to its Fitting height which is either $2$ or $3$, $G$ has one of the structures explained in [@L1998 Lemma 4.1]. By [@ML2 Lemma 4.1, Theorem 4.5] we have $h(G)=2$ if and only if $G$ is a group of type $1$. While $h(G)=3$, [@L1998 Lemma 4.1(a-iii)] implies that $\F G$ is abelian and in particular $G$ has no non-abelian normal Sylow subgroup. Hence by Remark \[rem:20\] $G$ is either of Lewis type $4$ or $5$. Suppose $G$ is of Lewis type $5$. Considering the notations in [@ML2 Lemma 3.5], we deduce that $\{1,2,2^{a}+1\}\subseteq{\mathrm{cd}}(G)$, ${\mathrm{cd}}(G|Q')\neq\emptyset$ as $Q$ is non-abelian, and ${\mathrm{cd}}(G|Q')$ contains powers of $2$ that are divisible by $2^a$. Hence $a=1$, ${\mathrm{cd}}(G|Q')=\{2\}$, and ${\mathrm{cd}}(G)=\{1,2,3\}$ which is not the case. Thus in this case $G$ is not of type $5$, so it is of type four.
If $B(G)$ is the last graph in Figure \[fig: 6\], then $\Gamma(G)$ has no isolated vertices and hence [@ML2 Theorem 5.2] implies that $G$ has a non-abelian normal Sylow subgroup. Now Remark \[rem:20\] verifies that $G$ is either a group of type $1$ or $6$. The case of Lewis type $6$ is excluded by Lemma \[lemma:new1\].
Finally, if $B(G)$ is the second graph in Figure \[fig: 5\], then $G$ is either a group of type $1$, $4$, $5$ or $6$. The case of Lewis type $6$ is excluded by Lemma \[lemma:new1\].
We have summarized this remark in Table \[tab:TTCR\].
Graph Types Examples
------------------------- ------- ----------------------------------------------------------------------------------------------------------------
nr 1 Figure \[fig: 4\] 1,4 $\mathtt{SmallGroup}(24,3)$ has type $1$
$\mathtt{PrimitiveSolvablePermGroup}(2,2,1,1)$ has type $4$
nr 2 Figure \[fig: 4\] 1,2,5 $\mathtt{SmallGroup}(288,860)$ has type $1$
$\mathtt{PrimitiveGroup}(9,6)$ has type $2$
$\mathtt{SmallGroup}(48,28)$ has type $5$
nr 3 Figure \[fig: 4\] 1 A family of examples are constructed in [@LL Theorem 2.3], the smallest arises by taking
(using the notation in [@LL Theorem $2.3$]) $p=3$, $a=2$ and $b=4$
nr 1 Figure \[fig: 5\] 1,4 The group $PH$ in [@N Example 3.4] where $|\pi(k)|= 2$ is of Lewis type $1$
$\mathtt{PrimitiveSolvablePermGroup}(4,2,2,2)$ has type $4$
nr 2 Figure \[fig: 5\] 1,4,5 See Example \[exa:newnewnew\] for a group of Lewis type $1$
$\mathtt{PrimitiveSolvablePermGroup}(4,2,1,4)$ has type $4$
See Example \[exa:newnew\] for a group of Lewis type $5$
nr 3 Figure \[fig: 5\] 1,6 See Example \[exa:new\] for a group of Lewis type $1$
$\mathtt{SmallGroup}(1344,816)$ has type $6$
nr 1 Figure \[fig: 6\] 1 A family of examples of type $1$ are constructed in [@LL Theorem 2.3], the smallest arises
by taking (using the notation in [@LL Theorem $2.3$]) $p=5$, $a=12$ and $b=24$
nr 2 Figure \[fig: 6\] 1 $\mathtt{SmallGroup}(1920,240059)$ has type $1$ and ${\mathrm{cd}}(G)^*=\{3,5,8,15\}$
nr 3 Figure \[fig: 6\] 1 For each three distinct primes $q$, $r$, and $s$, where $q\equiv 3\pmod 4$ and $q\equiv 1 \pmod{rs}$,
there exists a solvable group $G$ with ${\mathrm{cd}}(G)=\{1,r,s,rs,q^{4},q^{5}\}$, see [@Benjamin Section 4].
This group has cardinality $q^{12}rs$.
: Lewis types when $B(G)$ is a union of paths
\[tab:TTCR\]
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\]
(m2) at (2,1) ; (m3) at (3,1) ; (m4) at (2,3) ; (m5) at (3,3) ;
/in [m2/m4,m3/m5]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (3,3) ; (m4) at (1,3) ; (m5) at (3,1) ; (m6) at (4,3) ;
/in [m2/m3,m2/m4,m5/m6]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (3,3) ; (m4) at (1,3) ; (m5) at (5,1) ; (m6) at (4,3) ; (m7) at (6,3) ;
/in [m2/m3,m2/m4,m5/m6,m5/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (3,1) ; (m4) at (3,3) ; (m5) at (4,3) ; (m6) at (5,1) ;
/in [m2/m4,m3/m5,m5/m6]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (1,3) ; (m4) at (3,3) ; (m6) at (3,1) ; (m7) at (4,3) ; (m8) at (5,1) ;
/in [m2/m3,m2/m4,m6/m7,m7/m8]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (8,1) ; (m3) at (7,3) ; (m4) at (9,3) ; (m5) at (10,1) ; (m6) at (11,1) ; (m7) at (10,3) ;
/in [m2/m3,m2/m4,m5/m4,m6/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (8,1) ; (m3) at (7,3) ; (m4) at (9,3) ; (m5) at (4,3) ; (m6) at (5,1) ; (m7) at (6,3) ; (m8) at (7,1) ;
/in [m2/m3,m2/m4,m6/m5,m6/m7,m7/m8]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (6,3) ; (m3) at (7,3) ; (m4) at (9,3) ; (m5) at (11,3) ; (m6) at (7,1) ; (m7) at (8,1) ; (m8) at (10,1) ;
/in [m2/m6,m7/m4,m7/m3,m8/m4,m5/m8]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (6,3) ; (m6) at (7,1) ; (m9) at (8,3) ; (m4) at (9,3) ; (m5) at (11,3) ; (m7) at (10,1) ; (m8) at (12,1) ; (m10) at (13,3) ;
/in [m2/m6,m6/m9,m7/m4,m8/m5,m5/m7,m8/m10]{} () – ();
In analyzing the graphs in this section, the reader should observe how the investigation of $B(G)$ requires the techniques developed for studying the graphs $\Gamma(G)$ and $\Delta(G)$ and [*also*]{} some number theoretic (or arithmetic) considerations.
We conclude this section proposing the following problem, which generalizes Question 1 in [@H]. (Recall that we have solved [@H Question 1] in Examples \[ex:5\] and \[ex:6\].)
[Determine all graphs $X$ with no cycles such that there exists a group $G$ with $X\cong B(G)$.]{}
Cycles and complete bipartite graphs
------------------------------------
Recall (from the introductory section) that Taeri [@T] has proved that the bipartite divisor graph for the set of conjugacy class sizes of a finite group $G$ is a cycle if and only if it is a cycle of length six; moreover this happens if and only if $G\cong A\times \mathrm{SL}_2(q)$, for some abelian group $A$ and some $q\in \{4,8\}$. The situation is very different and much more rich for irreducible character degrees. From [@LM Section 6] we know that, for every pair of odd primes $p$ and $q$ such that $p$ is congruent to $1$ modulo $3$ and $q$ is a divisor of $p+1$, there exists a solvable group $G$ such that ${\mathrm{cd}}(G)=\{1,3q,p^{2}q,3p^{3}\}$. This gives an example of a solvable group $G$ with $B(G)$ a cycle of length $6$. On the other hand, among groups of order $588$, there are exactly two groups $G$ with $B(G)$ a cycle of length four. These groups have ${\mathrm{cd}}(G)=\{1,6,12\}$.
In [@H], it is shown that, if $G$ is a finite group with $B(G)$ a cycle of length $n\geq 6$, then $\Delta(G)$ and $\Gamma(G)$ are cycles. This fact yields the following theorem.
\[thm:55\] Let $G$ be a finite group with $B(G)$ a cycle of length $n$. Then $n\in\{4,6\}$, $G$ is solvable, and $dl(G)\leq |{\mathrm{cd}}(G)|\leq 4$. In particular, if $B(G)$ is a cycle of length $4$, then there exists a normal abelian Hall subgroup $N$ of $G$ such that ${\mathrm{cd}}(G)=\{[G:I_{G}(\lambda)] : \lambda\in {\mathrm{Irr}}(N)\}$.
Since the cycle of length four is also the complete bipartite graph $K_{2,2}$, it seems natural to discuss here also the case $B(G)\cong K_{m,n}$, for some positive integers $m\geq 2$ and $n\geq 2$. When $B(G)$ is complete bipartite, the graphs $\Delta(G)$ and $\Gamma(G)$ are both complete. Therefore, by [@L Theorem 7.3] or [@BCLP Main Theorem], we deduce that $G$ is solvable. The best structural result on $G$ is given by Moosavi [@mus].
Let $G$ be a finite group with $B(G)$ complete bipartite. Then $G=AH$, where $A$ is an abelian normal Hall subgroup of $G$ and $H$ is either abelian or a non-abelian $p$-group for some prime $p$.
Here we observe that there exist groups $G$ where $B(G)$ is an arbitrary complete bipartite graph. The analogous problem for the bipartite divisor graph for the set of conjugacy class sizes seems considerably harder; it is widely open and it is stated in [@HSpiga].
\[prop:2\^m-1\]For every positive integers $m$ and $k$, there exists a group $G$ with $B(G)\cong K_{m,k}$.
Let $m$ be a positive integer and let $p_1,\ldots,p_m$ be $m$ distinct prime numbers. Set $n:=p_1\cdots p_m$. From Dirichlet’s theorem on primes in arithmetic progression, there exists a prime $p$ with $p\equiv 1\pmod n$. Let $p$ be one of these primes and let $P$ be a cyclic group of order $p$. Next, let $\alpha$ be an automorphism of $P$ of order $n$ and set $H:=\langle P,\alpha\rangle$. Clearly, $H$ is a Frobenius group of order $np$, with cyclic Frobenius complement $\langle \alpha\rangle$, with cyclic Frobenius kernel $P$ and with ${\mathrm{cd}}(H)=\{1,n\}$.
Let $k$ be a positive integer and let $G:=H^k$ be the Cartesian product of $k$ copies of $H$. Clearly, $${\mathrm{cd}}(G)=\{1,n,n^2,\ldots,n^k\}$$ and hence $B(G)$ is the complete bipartite graph $K_{m,k}$.
Regular Bipartite Divisor Graph {#sec:reg}
===============================
A graph is said to be $k$-regular if each of its vertices has valency $k$. Since cycles are $2$-regular connected graphs, the investigation of groups $G$ with $B(G)$ a cycle has inspired the investigation [@Hregular] of groups $G$ where $B(G)$ is $k$-regular. It is clear that $0$-regular graphs (that is, empty graphs) play no role in the study of bipartite divisor graphs. So we start by discussing the influence of $1$-regularity of $B(G)$ (that is, $B(G)$ is a complete matching) on the group structure of $G$. (Theorem \[thm: reg\] is a refinement of [@Hregular Theorem $2.1$], where we have improved its statement by taking into account [@BH].)
\[thm: reg\] Let $G$ be a finite group with $B(G)$ $1$-regular. Then one of the following occurs:
- $G$ is non-solvable, $B(G)=K_2+K_2+K_2$, $G\cong A\times \mathrm{PSL}_2(2^{n})$, where $A$ is abelian and $n\in\{2,3\}$;
- $G$ is solvable and one of the following cases holds:
- $B(G)\cong K_2$ and ${\mathrm{cd}}(G)=\{1,p^\alpha\}$, for some prime $p$ and some positive integer $\alpha$. Moreover, either
- $G\cong P\times A$, where $P$ is a non-abelian $p$-group and $A$ is abelian, or
- $\alpha=1$, $\F G$ is abelian and $|G:\F G|=p$, or
- $G'\cap \Z G=1$ and $G/\Z G$ is a Frobenius group with kernel $(G'\times \Z G)/\Z G$ and cyclic complement of order $p^\alpha=|G:G'\times \Z G|$.
- $B(G)\cong K_2+K_2$, $h(G)\in\{2,3\}$ and $G$, with respect to its Fitting height, has one of the two structures mentioned in [@L1998 Lemma 4.1]. In particular:
- If $h(G)=3$, then ${\mathrm{cd}}(G)=\{1,[G:{\bf F}_{2}(G)],[{\bf F}_{2}(G):\F G]\}$, where $[G:{\bf F}_{2}(G)]$ is a prime $s$ and ${\bf F}_{2}(G)/\F G$ is a cyclic $t$-group for some prime $t\neq s$. Moreover, $G$ has Lewis type $4$.
- If $h(G)=2$, then ${\mathrm{cd}}(G)=\{[G:\F G]\}\cup {\mathrm{cd}}(\F G)$, where $G/\F G$ is a cyclic $t$-group for some prime $t$ and $|{\mathrm{cd}}(\F G)|=2$. Moreover, $G$ has Lewis type $1$.
Except for the fact that the groups in (2iia) are of Lewis type $4$ and the groups in (2iib) are of Lewis type $1$, the result follows immediately from [@Hregular Theorem $2.1$] and using the main result of [@BH] (when $n(B(G))=1$).
Suppose than that $G$ satisfies $B(G)=K_2+K_2$, $h(G)=3$, ${\mathrm{cd}}(G)=\{1,[G:{\bf F}_{2}(G)],[{\bf F}_{2}(G):\F G]\}$, where $[G:{\bf F}_{2}(G)]$ is a prime $s$ and ${\bf F}_{2}(G)/\F G$ is a cyclic $t$-group for some prime $t\neq s$. If follows readily from the description of the Lewis types and Remark \[rem:20\] that $G$ has type $4$ or $5$. Suppose that $G$ has type $5$. (We use the notation in [@ML2 Lemma $3.5$].) From [@ML2 Lemma $3.5$ (iii)], we deduce $2,2^a+1\in{\mathrm{cd}}(G)$. Moreover, from [@ML2 Lemma $3.5$ (iv)], we deduce that either ${\mathrm{cd}}(G|Q')=\emptyset$ or ${\mathrm{cd}}(G|Q') $ contains powers of $2$ that are divisible by $2^a$. Assume first that ${\mathrm{cd}}(G|Q')=\emptyset$. This means that every irreducible character of $G$ contains $Q'$ in its kernel, but this is clearly a contradiction because $Q'\ne 1$. Assume now that ${\mathrm{cd}}(G|Q')\ne\emptyset$. As $|\rho(G)|=2$, we must have $\rho(G)=\{2,2^a+1\}$ and hence ${\mathrm{cd}}(G|Q')=\{2\}$ and $a=1$. Therefore, ${\mathrm{cd}}(G)=\{1,2,3\}$. At this point to conclude we invoke [@N Theorem $3.5$], which classifies the groups $X$ with ${\mathrm{cd}}(X)=\{1,m,n\}$ and $\gcd(m,n)=1$. Since $|G:\F G|=2\cdot 3=6$, we deduce that part (1) of [@N Theorem $3.5$] holds. We infer that $\F G$ is abelian and hence so is $Q$, but this contradicts the description of the groups of type $5$.
Finally suppose that $G$ satisfies $B(G)=K_2+K_2$ and $h(G)=2$. Then $G$ is of Lewis type $1$ by Remark \[rem:20\].
The groups described in (1) and in (2i) are clear and, for each of these cases, there exists a group $G$ with $B(G)$ a complete matching. Now, $\mathtt{SmallGroup}(320,1012)$ provides an example satisfying (2iib). The groups in (2iia) must be of type $4$ in Lewis’ sense and examples occur plentiful ($\mathrm{Sym}(4)$ has type $4$ and $B(\mathrm{Sym}(4))=K_2+K_2$).
Let $G$ be a finite group with $B(G)$ a connected $2$-regular graph. As a connected $2$-regular graph is a cycle, by Theorem \[thm:55\], $G$ is solvable with $dl(G)\leq 4$ and $B(G)$ is a cycle of length four or six. The following theorem shows that $B(G)$ cannot be a disconnected $2$-regular graph.
\[thm: 51\] Suppose that $G$ is a group with $B(G)$ $2$-regular. Then $G$ is solvable, $B(G)$ is connected and $B(G)$ is a cycle of length four or six. In particular, if $\operatorname{{\rm diam}}(B(G))=2$, then there exists a normal abelian Hall subgroup $N$ of $G$ such that ${\mathrm{cd}}(G)=\{[G:I_{G}(\lambda)] : \lambda\in \mathrm{Irr}(N)\}$.
Theorem \[thm: 51\] verifies that the union of two cycles is not the bipartite divisor graph of any finite group.
Finally, in the following two theorems, we consider the case where $B(G)$ is $3$-regular.
\[thm: 4\] Let $G$ be a group with $B(G)$ $3$-regular. Then $B(G)$ is connected. Moreover, if $\Delta(G)$ is $n$-regular for $n\in\{2,3\}$, then $G$ is solvable and $\Delta(G)\cong K_{n+1}\cong \Gamma(G)$.
\[cor: 1\] Let $G$ be a solvable group with $B(G)$ $3$-regular. Then:
- If at least one of $\Delta(G)$ or $\Gamma(G)$ is not complete, then $\Delta(G)$ is neither $2$-regular, nor $3$-regular.
- If $\Delta(G)$ is regular, then it is a complete graph. Furthermore, if $\Gamma(G)$ is not complete, then $\Delta(G)$ is isomorphic with $K_{n}$, for $n\geq 5$.
As a complete bipartite divisor graph $K_{m,m}$ is an $m$-regular graph, Proposition \[prop:2\^m-1\] applied with $m=k$ yields infinitely many solvable groups whose bipartite divisor graph is $K_{m,m}$. In particular, we obtain an example of a group whose bipartite divisor graph is a $3$-regular graph.
In Table \[tab:Tcr\] we give some examples of $n$-regular bipartite divisor graphs for $n\in\{1,2,3\}$.
connected disconnected
------------- ---------------------------------------------------- ---------------------
$1$-regular $\mathrm{Sym}(3)$ $\mathrm{PSL}_2(8)$
$2$-regular $\mathtt{SmallGroup}(588,41)$ Does Not Exist
$3$-regular $G$ as in Proposition \[prop:2\^m-1\] with $m=k=3$ Does Not Exist
: Examples of $n$-regular $B(G)$
\[tab:Tcr\]
As the reader can see, we know very little on groups $G$ with $B(G)$ a regular graph and on the possible bipartite regular graphs that might arise.
[Construct (if possible) groups $G$ with $B(G)$ an $n$-regular graph with $n\ge 3$ and with $B(G)\ncong K_{n,n}$.]{}
Bounded order bipartite divisor graph of a finite group {#sec:bounded}
=======================================================
One of the questions that has been largely discussed by different authors is the classification of graphs that can occur as $\Delta(G)$, for some finite group $G$. To build confidence into this problem researchers have first considered graphs of bounded order. The first family of graphs that cannot occur as $\Delta(G)$ was discovered in [@BL]; later, this family was generalized in [@BJL]. These families contain graphs with arbitrarily many vertices, however they provide a great help for the problem of classifying the graphs that do occur as $\Delta(G)$, when $\Delta(G)$ has at most six vertices. For instance, in [@BJLL; @L3], the authors undertake a systematic investigation on the prime degree graphs of solvable groups with six vertices and they classify the disconnected graphs with six vertices.
Following these footsteps, in this section we study bipartite divisor graphs having at most $6$ vertices. When $B(G)$ has only two vertices, $B(G)=K_2$ and the group $G$ has only two character degrees and a great deal is known on these groups, see [@BH] and the references therein (see also Theorem \[thm: reg\] (2i)). When $B(G)$ has three vertices, the classification of $G$ boils down to the understanding of groups having only two character degrees, or of groups with ${\mathrm{cd}}(G)=\{1,p^\alpha,p^\beta\}$ (which in turn is a problem on $p$-groups).
Let $G$ be a finite group with $B(G)$ connected and having at most four vertices. Then $G$ is solvable, $B(G)$ is one of the graphs in Figure $\ref{fig: 11}$, and we have the following properties:
- if $B(G)$ has two vertices, then $G'$ is abelian, $G=AP$, where $P\in Syl_{p}(G)$ and $A$ is an abelian normal $p$-complement;
- if $B(G)$ has three vertices, then
- $G=AP$, where $P\in Syl_{p}(G)$ and $A$ is an abelian normal $p$-complement, or
- $G'$ is abelian, $G'\cap \Z G=1$ and $\frac{G}{\Z G}$ is a Frobenius group with cyclic complement;
- if $B(G)$ has four vertices, then
- $G=AH$ is the semidirect product of an abelian normal subgroup $A$ and a Hall subgroup $H$ which is either a Sylow $p$-subgroup of $G$ or an abelian $\{p,q\}$-subgroup, or
- $G'$ is abelian, $G'\cap \Z G=1$ and $\frac{G}{\Z G}$ is a Frobenius group with cyclic complement.
This theorem shows that when $B(G)$ has at most four vertices the structure of the graph $B(G)$ and [*also* ]{} the structure of the group $G$ is well-understood. (If $B(G)$ is disconnected, then $B(G)=K_2+K_2$ and this case was dealt with in the previous section.) The same behavior occurs when $B(G)$ has five vertices.
Let $G$ be a finite group with $B(G)$ connected and having five vertices. Then $B(G)$ is one of the graphs in Figure $\ref{fig: 8}$, $\ref{fig: 9}$ or $\ref{fig: 10}$. Furthermore, we have the following properties:
- If $|\rho(G)|=1$, then $G=AP$, where $P$ is a Sylow $p$-subgroup for some prime $p$ and $A$ is a normal abelian $p$-complement.
- If $|\rho(G)|=2$, then $G$ is solvable and $G=HN$, where $H$ is either a Sylow $p$-subgroup or a Hall $\{p,q\}$-subgroup of $G$ and $N$ is a normal complement.
- If $|\rho(G)|=3$, then $G$ is solvable and one of the following cases occurs:
- $G=HN$ where $H$ is a Sylow $p$-subgroup or a Hall $\{p,q\}$-subgroup or a Hall abelian $\{p,q,r\}$-subgroup of $G$ and $N$ is its normal complement.
- $G=QN$, where $Q$ is an abelian Sylow $q$-subgroup of $G$ and $N$ is its normal complement.
- If $|\rho(G)|=4$, then $G'$ is abelian, $G'\cap \Z G=1$ and $\frac{G}{\Z G}$ is a Frobenius group with cyclic complement.
The following example will be a useful tool to construct most of the groups in Table \[tab:BOB\].
\[exam: mi\][Let $1<m_1<m_2<\cdots<m_r$ be distinct positive integers such that $m_i$ divides $m_{i+1}$ for all $i\in\{1,\ldots,r-1\}$. Then by [@N Theorem 4.1] there exists a group $G$ such that ${\mathrm{cd}}(G)=\{1,m_1,\ldots,m_r\}$. We denote this group by $G_{m_1,\ldots,m_r}$. Indeed, our Proposition \[prop:2\^m-1\] is a very special case of this general result.]{}
0.3 true cm
Graph Examples Graph Examples
-------------------------- --------------------------------------------- -------------------------- ------------------------------------
nr 1 Figure \[fig: 8\] $Q_{8}\times Q_{8}\times Q_{8}\times Q_{8}$ nr 1 Figure \[fig: 11\] $\mathtt{Sym}(3)$
nr 2 Figure \[fig: 8\] $G_{210}$ as in Example \[exam: mi\] nr 2 Figure \[fig: 11\] $\mathtt{SmallGroup}(32,6)$
with normal Sylow $2$-subgroup
nr 1 Figure \[fig: 9\] $\mathtt{SmallGroup}(108,17)$ nr 2 Figure \[fig: 11\] $\mathtt{SmallGroup}(96,13)$
with non-normal Sylow $2$-subgroup
nr 2 Figure \[fig: 9\] $G_{2,6,12}$ as in Example \[exam: mi\] nr 3 Figure \[fig: 11\] $\mathtt{SmallGroup}(42,1)$
nr 3 Figure \[fig: 9\] $G_{6,12,24}$ as in Example \[exam: mi\] nr 4 Figure \[fig: 11\] $\mathtt{SmallGroup}(930,1)$
nr 4 Figure \[fig: 9\] $\mathtt{SmallGroup}(72,15)$ nr 5 Figure \[fig: 11\] $\mathtt{SmallGroup}(384,20)$
with non-normal Sylow $2$-subgroup
nr 1 Figure \[fig: 10\] $G_{5,30}$ as in Example \[exam: mi\] nr 5 Figure \[fig: 11\] $\mathtt{SmallGroup}(128,71)$
with normal Sylow $2$-subgroup
nr 2 Figure \[fig: 10\] $G_{15,30}$ as in Example \[exam: mi\] nr 6 Figure \[fig: 11\] $\mathtt{SmallGroup}(588,38)$
nr 3 Figure \[fig: 10\] $G_{30,60}$ as in Example \[exam: mi\] nr 7 Figure \[fig: 11\] $\mathtt{SmallGroup}(96,70)$
nr 4 Figure \[fig: 10\] $\mathtt{SmallGroup}(960,5748)$
: Examples of $G$ with $o(B(G))\leq 5$, where it is connected
\[tab:BOB\]
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (1,3) ;
/in [m2/m3]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (1,3) ; (m4) at (3,3) ;
/in [m2/m3,m2/m4]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (3,1) ; (m4) at (2,3) ;
/in [m2/m4,m3/m4]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (3,1) ; (m6) at (2,3) ;
/in [m2/m6,m3/m6,m4/m6]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (1,3) ; (m4) at (2,3) ; (m5) at (3,3) ;
/in [m2/m3,m2/m4,m5/m2]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (1,3) ; (m6) at (2,3) ;
/in [m6/m3,m6/m2,m3/m4,m2/m4]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (1,3) ; (m6) at (2,3) ;
/in [m6/m3,m3/m4,m2/m4]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2.5,1) ; (m3) at (1,3) ; (m4) at (2,3) ; (m5) at (3,3) ; (m6) at (4,3) ;
/in [m2/m3,m2/m4,m5/m2,m2/m6]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (3,1) ; (m5) at (4,1) ; (m6) at (2.5,3) ;
/in [m6/m3,m6/m2,m6/m4,m6/m5]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (1,3) ; (m4) at (3,1) ; (m5) at (2,3) ; (m7) at (3,3) ;
/in [m2/m3,m2/m5,m2/m7,m4/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (1,3) ; (m4) at (3,1) ; (m5) at (2,3) ; (m7) at (3,3) ;
/in [m2/m3,m2/m5,m2/m7,m4/m7,m4/m5]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2.5,1) ; (m3) at (2,3) ; (m4) at (3,3) ; (m6) at (3.5,1) ; (m7) at (4,3) ;
/in [m2/m3,m2/m4,m2/m7,m7/m6,m6/m3,m6/m4]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1.5,1) ; (m3) at (2.5,1) ; (m5) at (1,3) ; (m6) at (2,3) ; (m7) at (3,3) ;
/in [m2/m5,m2/m6,m6/m3,m3/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (3,1) ; (m5) at (2,3) ; (m7) at (3,3) ;
/in [m2/m5,m3/m5,m5/m4,m4/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (3,1) ; (m5) at (2,3) ; (m7) at (3,3) ;
/in [m2/m5,m3/m5,m5/m4,m4/m7,m3/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (3,1) ; (m5) at (1.5,3) ; (m7) at (2.5,3) ;
/in [m2/m5,m3/m5,m5/m4,m4/m7,m3/m7,m7/m2]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (3,1) ; (m5) at (1.5,3) ; (m7) at (2.5,3) ;
/in [m2/m5,m3/m5,m4/m7,m3/m7]{} () – ();
[Let $G$ be a finite group with $B(G)$ disconnected and having five vertices. A case-by-case analysis yields that $B(G)$ is one of the graphs in Figure \[fig: 1\]. In particular, $B(G)$ is a union of two paths and hence the structure of $G$ is described in Section \[unionpaths\]. ]{}
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (1,3) ; (m4) at (3,3) ; (m5) at (3,1) ; (m6) at (4,3) ;
/in [m2/m3,m2/m4,m5/m6]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (3,1) ; (m4) at (4,1) ; (m5) at (3,3) ; (m6) at (2,3) ;
/in [m6/m3,m6/m2,m5/m4]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (1,3) ; (m4) at (3,3) ; (m5) at (3,1) ; (m6) at (2,3) ; (m7) at (4,3) ;
/in [m2/m3,m2/m4,m2/m6,m5/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (5,1) ; (m3) at (3,3) ; (m4) at (2,1) ; (m5) at (1,3) ; (m6) at (4,3) ; (m7) at (6,3) ;
/in [m4/m3,m2/m6,m5/m4,m2/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (3,1) ; (m5) at (2,3) ; (m6) at (4,1) ; (m7) at (3,3) ;
/in [m2/m5,m3/m5,m4/m5,m6/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (1,3) ; (m4) at (2,1) ; (m5) at (2,3) ; (m6) at (3,1) ; (m7) at (3,3) ;
/in [m2/m3,m4/m5,m5/m6,m6/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (1,3) ; (m4) at (2,1) ; (m5) at (2,3) ; (m6) at (3,1) ; (m7) at (3,3) ;
/in [m2/m3,m4/m5,m6/m7,m5/m6,m4/m7]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (2,1) ; (m3) at (1,3) ; (m4) at (3,3) ; (m6) at (3,1) ; (m7) at (4,3) ; (m8) at (5,1) ;
/in [m2/m3,m2/m4,m6/m7,m7/m8]{} () – ();
\[scale=.8,auto=left,every node/.style=[circle,fill=blue!20]{}\] (m2) at (1,1) ; (m3) at (2,1) ; (m4) at (3,1) ; (m5) at (1,3) ; (m6) at (2,3) ; (m7) at (3,3) ;
/in [m2/m5,m3/m6,m4/m7]{} () – ();
0.4 true cm
Finally in the last theorem of this paper, we look at disconnected bipartite graphs with six vertices, and we attempt to determine whether each graph can or cannot occur as the bipartite divisor graph of a solvable group.
\[thm: six\] Let $G$ be a finite group with $B(G)$ disconnected and with six vertices. Then $B(G)$ is one of the graphs in Figure $\ref{fig: 2}$ or $\ref{fig: 3}$. Furthermore we have the following properties:
- $n(B(G))=3$ if and only if $G\cong A\times \mathrm{PSL}_2(2^{n})$, where $n\in\{2,3\}$ and $A$ is an abelian group.
- If $|\rho(G)|=4$, then $B(G)$ is the third graph in Figure $\ref{fig: 2}$, $G$ is solvable and it is of Lewis type one or four with the structure explained in [@L1998 Lemma 4.1].
- If $|\rho(G)|=2$, then $G$ is solvable and $B(G)$ is one the first two graphs in Figure $\ref{fig: 2}$. If $B(G)$ is the second graph in Figure $\ref{fig: 2}$, then $G$ is a group of Lewis type one. If $B(G)$ is the first graph in Figure $\ref{fig: 2}$, then $G$ is of Lewis type one or five.
- If $|\rho(G)|=3$ and $n(B(G))=2$, then $B(G)$ is one of the first three graphs in Figure $\ref{fig: 3}$. If $G$ is non-solvable, then one of the following cases holds:
- $G$ has a normal subgroup $U$ such that $U\cong \mathrm{PSL}_{2}(q)$ or $\mathrm{SL}_{2}(q)$ for some odd $q\geq 5$ and if $C=\cent G U$, then $C\leq \Z G$ and $G/C\cong \mathrm{PGL}_{2}(q)$; or
- $G$ has a normal subgroup of index $2$ that is a direct product of $\mathrm{PSL}_{2}(9)$ and a central subgroup $C$. Furthermore, $G/C\cong M_{10}$.
If $G$ is solvable, then it is of Lewis type one or six when $B(G)$ is one of the first two graphs in Figure $\ref{fig: 3}$ and of Lewis type one, four, or five when $B(G)$ is the third graph in Figure $\ref{fig: 3}$.
As $n(B(G))>1$ and $B(G)$ has six vertices, we have $|\rho(G)|+|{\mathrm{cd}}(G)^{*}|=6$, where $|\rho(G)|\in\{2,3,4\}$. So we consider three cases with respect to $|\rho(G)|$.
If $|\rho(G)|=4$, then $|{\mathrm{cd}}(G)|=3$. Now [@IS Theorem 12.15] and [@L Corollary 4.2] imply that $G$ is solvable of derived length at most $3$ and each connected component of $\Delta(G)$ is a complete graph. On the other hand, [@L Theorem 4.3] verifies that $K_{2}+K_{2}$ is not the prime degree graph of a solvable group, so we conclude that $B(G)$ is the third graph in Figure \[fig: 2\]. Now by Remark \[rem:20\] we conclude that $G$ is of Lewis types one, four, five, or six. Similar to the proof of Lemma \[lemma:new1\], we can see that $G$ is not a group of Lewis type six. If $G$ is of Lewis type five, then considering the notations in [@ML2 Lemma 3.5] we conclude that either $2^{a}=2$ and ${\mathrm{cd}}(G)=\{1,2,3\}$ which contradicts the structure of $B(G)$ or ${\mathrm{cd}}(G|Q^{'})=\emptyset$ which is not possible as $Q$ is non-abelian. Therefore, $G$ is of Lewis type one or four.
If $|\rho(G)|=2$, then $B(G)$ has two connected components. It is easy to see that $B(G)$ is one of the first two graphs in Figure \[fig: 2\]. First suppose that $B(G)$ is the second graph in Figure \[fig: 2\]. By [@L Theorem 7.1], the case $\Gamma(G)\cong K_{2}+K_{2}$ is impossible for a non-solvable group, therefore $G$ is solvable and, by Table \[tab:TTCR\], $G$ is of Lewis type one. Now consider the first graph in Figure \[fig: 2\]. As every character degree of $G$ is a power of some prime and $|\rho(G)|=2$, by [@Hu1 Theorem 30.3], we conclude that $G$ is solvable with $\Delta(G)\cong K_{1}+K_{1}$ and, by Remark \[rem:20\], we deduce that $G$ is of Lewis type one, four, or five. Similar to the proof of Lemma \[lemma:new\], we see that the case where $G$ has Lewis type four does not occur. Hence $G$ is either of Lewis type one or five.
Finally consider the case where $|\rho(G)|=|{\mathrm{cd}}(G)^{*}|=3$. If $n(B(G))=3$, then $B(G)$ is $1$-regular and, by Theorem \[thm: reg\] we can observe that $G\cong A\times \mathrm{PSL}_{2}(2^{n})$, where $n\in\{2,3\}$ and $A$ is an abelian group. Assume that $n(B(G))=2$ and $G$ is non-solvable. Since $|{\mathrm{cd}}(G)|=4$, [@GA Theorem A] implies that $G$ has one of the following structures:
- $G$ has a normal subgroup $U$ such that $U\cong \mathrm{PSL}_{2}(q)$ or $\mathrm{SL}_{2}(q)$ for some odd $q\geq 5$ and if $C=\cent G U$, then $C\leq \Z G$ and $G/C\cong \mathrm{PGL}_2(q)$; or
- $G$ has a normal subgroup of index $2$ that is a direct product of $\mathrm{PSL}_{2}(9)$ and a central subgroup $C$. Furthermore, $G/C\cong M_{10}$.
In particular by [@GA Corollary B], ${\mathrm{cd}}(G)=\{1,q-1,q,q+1\}$ for some odd prime power $q>3$ or ${\mathrm{cd}}(G)=\{1,9,10,16\}$. Consequently, $\Delta(G)$ has an isolated vertex and $B(G)$ is one of the first two graphs in Figure \[fig: 3\]. When $G$ is solvable, by Remark \[rem:20\] we can see that $G$ is a group of Lewis type one, four, five, or six. Suppose $G$ is a group of Lewis type four or five. As $G$ has no non-abelian normal Sylow subgroup, [@ML2 Theorem 5.2] implies that one connected component of $\Gamma(G)$ contains only one degree $a$ where the prime divisors of $a$ lie in the larger connected component of $\Delta(G)$. None of the first two graphs in Figure \[fig: 3\] satisfies this property, so in these cases $G$ is not a group of Lewis type four or five. Hence it is of Lewis type one or six.
If $B(G)$ is the third graph in Figure \[fig: 3\], then, by Table \[tab:TTCR\], it is a group of Lewis type one, four, or five.
Here we give some examples of solvable groups whose bipartite divisor graphs have six vertices, are disconnected, but are not a union of paths.
- For $G=\mathtt{SmallGroup}(320,1581)$, we have ${\mathrm{cd}}(G)=\{1,2,4,8,5\}$, $B(G)$ is the first graph in Figure \[fig: 2\], and $G$ is of Lewis type five.
- Let $p$ be an odd prime and let $a$ be a positive integer with $|\pi(p^{a}-1)|\geq 3$ (the smallest case is $p=31$ and $a=1$). Let $b$ be a divisor of $p^{a}-1$ with exactly three prime divisors. Let $E$ be the extraspecial group of order $p^{2a+1}$ with exponent $p$. Then $E$ has an automorphism $\varphi$ of order $b$ which centralizes $\Z E$. Let $G:=E\rtimes \langle\varphi\rangle$. Then ${\mathrm{cd}}(G)=\{1,b,p^{a}\}$ and $B(G)$ is the third graph in Figure \[fig: 2\], and $G$ is of Lewis type one.
We were not able to find a group $G$ with $B(G)$ isomorphic to the second graph in Figure \[fig: 3\]. We leave this as an open question.
\[question:11111\] Is there a finite group $G$ with $B(G)$ isomorphic to the second graph in Figure $\ref{fig: 3}$?
[99]{} D. Benjamin, Coprimeness among irreducible character degrees of finite solvable groups, *Proc. Amer. Math. Soc.* **125**(10) (1997), 2831–2837. M. Bianchi, A. Berta Mauri Gillio, M. Herzog, G. Qian, W. Shi, Characterization of non-nilpotent groups with two irreducible character degrees, *J. Algebra* **284** (2005), 326-332.
M. Bianchi, D. Chillag, M. L. Lewis, E. Pacifici, Character degree graphs that are complete graphs, *Proc. Amer. Math. Soc.* **135** (2007), 671–676.
M. W. Bissler, J. Laubacher, M. L. Lewis, Classifying character degree graphs with 6 vertices, *Beitr Algebra Geom* (2019), https://doi.org/10.1007/s13366-019-00437-y.
M. W. Bissler, J. Laubacher, Classifying families of character degree graphs of solvable groups, *Int. J. Group Theory* **8** (2019), 27–46.
M. W. Bissler, M. L. Lewis, A family of graphs that cannot occur as character degree graphs of solvable groups. arXiv:1707.03020, 2017.
D. Bubboloni, S. Dolfi, M. A. Iranmanesh, C. E. Praeger, On bipartite divisor graphs for group conjugacy class sizes, *Journal of Pure and Applied Algebra* **213** (2009), 1722–1734.
S. Dolfi, E. Jabara, The structure of finite groups of conjugate rank $2$, *Bull. London Math. Soc* **82** (2010), 167–183. The GAP Group, *GAP – Groups, Algorithms, and Programming, Version 4.10.1*; 2019, `(https://www.gap-system.org)`.
R. Hafezieh, Bipartite divisor graph for the set of irreducible character degrees, *Int. J. Group Theory* **6** (2017), 41–51.
R. Hafezieh, On regular bipartite divisor graph for the set of irreducible character degrees, *Hacettepe J. Mathematics and Statistics*, Doi: 10.15672/HJMS.2018.594.
R. Hafezieh, M. A. Iranmanesh, Bipartite divisor graph for the product of subsets of integers, *Bull. Aust. Math. Soc.* **87** (2013), 288–297.
R. Hafezieh, P. Spiga, Groups having complete bipartite divisor graphs for their conjugacy class sizes, *Rend. Sem. Mat. Univ. Padova* **133** (2015), 117–123.
L. He, M. L. Lewis, Common divisor character degree graphs of solvable groups with four vertices, *Comm. Algebra* **43** (2015), 4916–4922.
L. He, G. Zhu, Nonsolvable subgroups and irreducible character degrees, *J. Algebra* **372** (2012), 68–84.
M. A. Iranmanesh, C. E. Praeger, Bipartite divisor graphs for integer subsets, *Graphs and Combinatorics* **26** (2010), 95–105.
B. Huppert, Character Theory of Finite Groups, *de Gruyter, Berlin*, 1998.
I. M. Isaacs, Character theory of finite groups, *Academic Press, New York*, 1976.
M. L. Lewis, Determining group structure from sets of irreducible character degrees, *Journal of Algebra* **206** (1998), 235–260.
M. L. Lewis, Solvable groups whose degree graphs have two connected components, *Journal of Group Theory* **4** (2001), 255–275.
M. L. Lewis, Classifying character degree graphs with $5$ vertices. In: Finite groups 2003, 247-265, Walter de Gruyter GmbH & Co. KG, Berlin, 2004.
M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, *Rocky Mountain Journal of Math.* **38** (2008), 175–211.
M. L. Lewis, A. Moretó, T. R. Wolf, Nondivisibility among character degrees, *J. Group Theory* **8** (2005), 561–588.
G. Malle, A. Moretó, Nonsolvable groups with few character degrees, *Journal of Algebra* **294** (2005), 117–126.
S. A. Moosavi, On bipartite divisor graph for character degrees, *International Journal of Group Theory* **6** (2017), 1–7.
S. A. Moosavi, Groups whose bipartite divisor graph for character degrees has four or fewer vertices, *Asian-European Journal of Mathematics*, DOI: 10.1142/S1793557120500394.
S. A. Moosavi, Groups whose bipartite graph for character degrees has five vertices, *Iranian Journal of Mathematical Sciences and Informatics*, to appear.
T. Noritzsch, Groups having three complex irreducible character degrees, *Journal of Algebra* **175** (1995), 767–798.
F. Shafiei, A. Iranmanesh, M. L. Lewis, K. Aziziheris, Finite groups with a given set of character degrees, *Algebras and Representation Theory* **19** (2016), 335–354.
B. Taeri, Cycles and bipartite graph on conjugacy class of groups, *Rend. Sem. Matematico della Università di Padova* **123** (2010), 233–247.
H. P. Tong-Viet, Groups whose prime graph has no triangles, *Journal of Algebra* **378** (2013), 196–206.
D. L. White, Character degrees of extensions of $\mathrm{PSL}_2(q)$ and $\mathrm{SL}_2(q)$, *J. Group Theory* **16** (2013), 1–33.
K. Zsigmondy, Zur Theorie der Potenzreste, *Monatsh. Math. Phys.* **3** (1892), 265–284.
[^1]: [-0.4 true cm MSC(2010): Primary 05C25; secondary 05C75. Keywords: Bipartite divisor graph, character degrees, conjugacy class sizes, prime graph, divisor graph\
$*$Corresponding author: roghayeh@gtu.edu.tr]{}
|
---
abstract: |
We have measured the kinematic profile of the early-type (E/S0) lens galaxy in the system 0047$-$281 ($z$=0.485) with the [*Echelle Spectrograph and Imager*]{} (ESI) on the W.M. Keck–II Telescope, as part of the [*Lenses Structure and Dynamics (LSD) Survey*]{}. The central velocity dispersion is $\sigma=229\pm 15$ [km$\,{\rm s}^{-1}$]{}, and the dispersion profile is nearly flat to beyond one effective radius ($R_e$). No significant streaming motion is found. Surface photometry of the lens galaxy is measured from [*Hubble Space Telescope*]{} images. From the offset from the local Fundamental Plane (FP), we measure an evolution of the effective mass-to-light ratio of $\Delta \log
M/L_B=-0.37\pm0.06$ between $z=0$ and $z=0.485$, consistent with the observed evolution of field E/S0 galaxies. (We assume $h_{65}=1$, $\Omega_{\rm m}$=0.3 and $\Omega_\Lambda$=0.7 throughout.) Gravitational lens models provide a mass of M$_{\rm
E}=(4.06\pm0.20)\times 10^{11}\,h_{65}^{-1}$ M$_\odot$ inside the Einstein radius of $R_{\rm E}=(8.70\pm0.07)\,h_{65}^{-1}$kpc. This allows us to break the degeneracy between velocity anisotropy and density profile, typical of dynamical models for E/S0 galaxies. We find that constant M/L model, even with strongly tangential anisotropy of the stellar velocity ellipsoid, are excluded at $>$99.9% CL. The total mass distribution inside $R_{\rm E}$ can be described by a single power-law density profile, $\rho_t\propto r^{-\gamma'}$, with an effective slope $\gamma'=1.90^{+0.05}_{-0.23}$ (68% CL; $\pm0.1$ systematic error). Two-component models yield an upper limit (68% CL) of $\gamma\le 1.55(1.12)$ on the power-law slope of the dark-matter density profile and a projected dark-matter mass fraction of $0.41(0.54)^{+0.15}_{-0.05}\left(^{+0.09}_{-0.06}\right)$ (68% CL) inside $R_{\rm E}$, for Osipkov–Merritt models with anisotropy radius $r_i=\infty(R_e)$. The stellar $M_*/L$ values derived from the FP agrees well with the maximum allowed value from the isotropic dynamical models (i.e. the “maximum-bulge solution”).
The fact that both lens systems 0047$-$281 ($z$=0.485) and MG2016+112 ($z$=1.004) are well described inside their Einstein radii by a constant $M_*/L$ stellar mass distribution embedded in a nearly logarithmic potential – with an isotropic or a mildly radially anisotropic dispersion tensor – could indicate that E/S0 galaxies underwent little [*structural evolution*]{} at $z \la 1$ and have a close-to-isothermal total mass distribution in their inner regions. Whether this conclusion can be generalized, however, requires the analysis of more systems. We briefly discuss our results in the context of E/S0 galaxy formation and cold-dark-matter simulations.
author:
- 'Léon V.E. Koopmans'
- Tommaso Treu
title: 'The Structure and Dynamics of Luminous and Dark-Matter in the Early-Type Lens Galaxy of 0047$-$281 at z=0.485$^1$'
---
Introduction
============
To understand the evolution and internal structure of luminous and dark matter in early-type galaxies (E/S0), we have started the [*Lenses Structure and Dynamics*]{} (LSD) Survey. From an observational point of view, the LSD Survey aims at measuring – using the Keck-II Telescope – the internal (stellar) kinematics of a relatively large sample of E/S0 gravitational-lens galaxies in the redshift range $z=0-1$ (see Treu & Koopmans 2002, hereafter TK02, for a description of the survey and its scientific rationale). The stellar kinematic profiles are combined with constraints from a gravitational-lensing analysis – in particular the mass enclosed within the Einstein radius – in order to break degeneracies inherent to each method individually. In this way, we are uniquely able to constrain the luminous and dark mass distribution and the velocity ellipsoid of the luminous (i.e. stellar) component.
Several of the main questions about E/S0 galaxies that we aim to answer with the LSD Survey are: (i) What is the amount of dark matter within the inner few effective radii ($R_e$)? (ii) Does the dark matter density profile agree with the universal profiles inferred from CDM simulations (Navarro, Frenk & White 1997, hereafter NFW; Moore et al. 1998; Ghigna et al. 2001)? (iii) Is there a universal total (luminous plus dark-matter) density distribution that well describes the inner regions of E/S0 galaxies? If so, is it isothermal, as observed in the local Universe and often assumed in lensing analyses? (iv) How does the stellar mass-to-light ratio of E/S0 galaxies evolve with redshift, and does it agree with the evolution of the stellar populations of field early-type galaxies as inferred from Fundamental Plane (hereafter FP) measurements (Treu et al. 1999, 2001a, 2002, hereafter T99, T01a, T02; Kochanek et al. 2000; van Dokkum et al.2001)? (v) Does the structure of E/S0 galaxies evolve between $z=0$ and 1, or is the evolution of the FP purely an evolution of the stellar population? (vi) Is the stellar velocity dispersion tensor isotropic or anisotropic?
First results from the LSD Survey were presented in two recent papers (Koopmans & Treu 2002, hereafter KT02; TK02), where we combined our measurement of the luminosity-weighted stellar velocity dispersion of the lens galaxy in MG2016+112 at $z=1.004$ with the mass enclosed by the Einstein radius as determined from gravitational lens models (Koopmans et al. 2002). A robust constraint was found on the slope of the total density profile of the lens galaxy, i.e. $\gamma'=2.0\pm0.1\pm0.1$ (random/systematic errors) for $\rho_t\propto r^{-\gamma'}$. In addition, we were able to determine the stellar mass-to-light ratio and constrain the slope of the dark-matter halo, leading to a relatively simple self-consistent picture of the lens galaxy: an old and metal-rich stellar component embedded in a logarithmic (i.e. isothermal) potential observed at a look-back time of $\sim 8$ Gyrs – remarkably similar to many present-day E/S0 galaxies. Constant M/L models were ruled out at a very high confidence level and a significant mass fraction ($\sim$75%) of dark matter was shown to be present inside the Einstein radius of about 13.7 kpc. Unfortunately, given the faintness of the galaxy at $z$=1, only a luminosity-weighted velocity dispersion could be obtained and only minimal constraints could be set on the anisotropy of stellar orbits.
In this paper, we present Keck and HST observations, a gravitational-lens model, and a dynamical model of the lens galaxy in 0047$-$281 at $z=0.485$ (Warren et al. 1996, 1998, 1999). This galaxy is bright and sufficiently extended that we were able to measure a spatially resolved velocity dispersion profile, thus setting unprecedented constraints on the orbital structure of a galaxy at a look-back time of $\sim 5$ Gyrs.
The paper is organized as follows. In Section 2, we describe archival HST observations and spectroscopic observations with ESI on the Keck–II telescope. In Section 3, we analyze these observations and determine the luminosity evolution of the lens galaxy with respect to the local Fundamental Plane. In Section 4, we present a gravitational lens model from which we determine the mass enclosed by the Einstein radius. In Section 5, we present a model for the luminous and dark-matter distributions of the lens galaxy. In Section 6, the results from the dynamical models are presented. Section 7 summarizes and discusses the results.
In the following, we assume that the Hubble constant, the matter density, and the cosmological constant are H$_0=65h_{65}$ kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}=0.3$, and $\Omega_{\Lambda}=0.7$, respectively. Throughout this paper, $r$ is the radial coordinate in 3-D space, while $R$ is the radial coordinate in 2-D projected space.
Observations
============
Hubble Space Telescope Imaging
------------------------------
Wide Field and Planetary Camera 2 (WFPC2) images of the system are available from the HST archive, through filters F555W and F814W[^1]. In particular, the system has been imaged for 9500s in F555W on the Wide-Field Camera, and 2700s in F814W on the Planetary Camera.
The images were reduced using a series of [iraf]{} scripts based on the [iraf]{} package [drizzle]{} (Fruchter & Hook 2002), to align the independent pointings and perform cosmic ray rejection. A subsampled (pixel scale $0\farcs05$) image was produced for the F555W image. The final “drizzled” image in F555W is shown in Fig.\[fig:HST\]. Note that for the redshift of the source, $z=3.595$, Ly-$\alpha$ is redshifted to 5589 Å (Warren et al.1998) and the F555W magnitude therefore includes the bright Ly-$\alpha$ emission. The F814W image is significantly shallower than the F555W image and although adequate for measuring the structural parameters, it is not particularly useful for the lens modeling, since the multiple images are faint and their positions cannot be accurately determined.
Surface photometry was performed on the F555W and F814W images as described in T99 and Treu et al. (2001b; hereafter T01b). The galaxy brightness profiles are well represented by an [$R^{1/4}\,$]{} profile, which we fit – taking the HST point spread function into account – to obtain the effective radius ($R_e$), the effective surface brightness (SB$_e$), and the total magnitude. The relevant observational quantities of galaxy G in 0047$-$281 and their errors are listed in Table 1. Note that errors on SB$_e$ and $R_e$ are tightly correlated and that the uncertainty on the combination $\log R_e - 0.32 SB_e$ that enters the Fundamental Plane (see Section \[sec:FP\]) is very small ($\sim 0.015$; see Kelson et al. 2000; T01b; Bertin, Ciotti & del Principe 2002). In the right hand panel of Fig.\[fig:HST\] we show the HST image after removal of a smooth model for the lens galaxy. Notice the nearly circular structure of the lensed image configuration around the lens galaxy.
Rest frame photometric quantities listed in Table 1 – computed as described in T01b – are corrected for galactic extinction using E(B–V)=0.016 from Schlegel, Finkbeiner & Davis (1998).
Keck Spectroscopy {#keck}
-----------------
We observed 0047$-$281 using the Echelle Spectrograph and Imager (ESI; Sheinis et al. 2002) on the W.M. Keck–II Telescope during four consecutive nights (23–26 July, 2001), for a total integration time of 20,700s (7x1800s+3x2700s). The seeing was good ($0\farcs6<$FWHM$<0\farcs8$) and three out of four nights were photometric. Between each exposure, we dithered along the slit to allow for a better removal of sky residuals in the red end of the spectrum. The slit (20$''$ in length) was aligned with the major axis of the galaxy. The slit width of $1\farcs25$ yields an instrumental resolution of 30[km$\,{\rm s}^{-1}$]{} which is adequate for measuring the stellar velocity dispersion and removing narrow sky emission lines. The centering of the galaxy in the slit was constantly monitored by means of the ESI viewing camera (the galaxy was bright enough to be visible in a few seconds exposure) and we estimate the centering perpendicular to the slit to be accurate to $\la0\farcs1$.
--------------------------------- -------------------
Redshift (G) 0.485$\pm$0.001
$F814W$ (mag) 18.67$\pm$0.09
$F555W$ (mag) 20.61$\pm$0.09
SB$_{e,F814W}$ (mag/arcsec$^2$) 20.23$\pm$0.22
SB$_{e,F555W}$ (mag/arcsec$^2$) 22.59$\pm$0.07
$R_{e,F814W}$ (arcsec) 0.82$\pm$0.12
$R_{e,F555W}$ (arcsec) 0.99$\pm$0.04
$b/a$=$(1-e)$ 0.80$\pm$0.10
Major axis P.A. ($^\circ$) $67\pm5$
$\sigma$ (kms$^{-1}$) 229$\pm$15
$M_{V}-5\log h_{65}$ (mag) $-$22.90$\pm$0.04
$M_{B}-5\log h_{65}$ (mag) $-$22.22$\pm$0.11
$R_{e,V}$ ($h_{65}^{-1}$kpc) 5.21$\pm$0.72
$R_{e,B}$ ($h_{65}^{-1}$kpc) 5.82$\pm$0.58
SB$_{e,V}$ (mag/arcsec$^2$) 19.29$\pm$0.22
SB$_{e,B}$ (mag/arcsec$^2$) 20.39$\pm$0.11
--------------------------------- -------------------
[Table 1.— Observed spectro-photometric quantities of the lens galaxy (G) in Q0047-281. The second part of the table lists rest-frame quantities, derived from the observed quantities as described in Section 2. Note that $\sigma$ is the central velocity dispersion corrected to a circular aperture of radius R$_e/8$. All quantities in this table assume H$_0=65$ kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}=0.3$, and $\Omega_{\Lambda}=0.7$. \[tab:HST\]]{}
Data reduction was performed using the [iraf]{} package EASI2D[^2] as described in KT02. To preserve most of the spatially resolved information and to achieve an adequate signal-to-noise ratio at the largest distance for the center of the galaxy, we defined 5 apertures along the spatial direction of the spectrum and summed the signal within each aperture. The apertures correspond approximately to angular dimensions along the slit: $0\farcs6\times1\farcs25$, $0\farcs6\times1\farcs25$, $0\farcs4\times1\farcs25$, $0\farcs6\times1\farcs25$, $0\farcs6\times1\farcs25$ (centered respectively at $-1\farcs1,-0\farcs5,0'',0\farcs5,1\farcs1$ along the major axis). A velocity dispersion profile was measured with the Gauss–Hermite Pixel-Fitting Software (van der Marel 1994) and the Gauss Hermite Fourier Fitting Software (van der Marel & Franx 1993) on the spectral region including the G-band ($\sim 4304$ Å), using as kinematic templates spectra of G–K giants observed at twilight with a $0\farcs3$ slit width, appropriately smoothed to match the instrumental resolution of the $1\farcs25$ slit. The two codes provide consistent measurements. The total error on velocity dispersion was estimated by adding in quadrature the formal uncertainty given by the codes, the scatter in the results obtained with different templates and the semi-difference of the results obtained with the two codes. The fit to the spectrum from the aperture including part of the brightest lensed images (A and B) was poor, with severe mismatch and unstable measurements. We interpret this as due to contamination by emission from the lensed images. We discarded this measurement from the analysis. The velocity dispersion profile was then folded around the center, as determined by fitting the centroid of the light distribution at the wavelength of the G band, and the velocity dispersion averaged in the corresponding apertures (symmetric apertures provided results within the errors). The final results are listed in Table 2.
Aperture ($\sq''$) $\sigma$ ([km$\,{\rm s}^{-1}$]{}) $\Delta\sigma$ ([km$\,{\rm s}^{-1}$]{})
----------------------- ----------------------------------- -----------------------------------------
(0.0–0.2)$\times$1.25 219 12
(0.2–0.8)$\times$1.25 212 14
(0.8–1.4)$\times$1.25 205 13
[Table 2.— Kinematic data along the major axis of 0047$-$281. The adjacent rectangular apertures are indicated. The seeing was 07 during the observations.\[tab:sigma\]]{}
Using the procedure detailed in T01b, the value in the central aperture can be corrected to standard central velocity dispersion of $\sigma=229\pm15$ [km$\,{\rm s}^{-1}$]{} within a circular aperture of radius $R_e/8$ (see also Sec.6.1). No evidence for significant streaming motions (e.g. rotation) was found, with an upper limit of 50 [km$\,{\rm s}^{-1}$]{} relative radial velocity between the center and the outermost aperture.
The Fundamental Plane and the evolution of stellar populations {#sec:FP}
==============================================================
Early-type galaxies in the local Universe occupy approximately a plane in the three-dimensional space defined by the parameters effective radius ($\log R_e$), effective surface brightness (SB$_e$) and central velocity dispersion ($\log \sigma$), $$\label{eq:FP}
\log R_{{\textrm{e}}} = \alpha \log~\sigma + \beta~SB_{{\textrm{e}}} + \gamma_{\rm FP}$$ known as the Fundamental Plane (hereafter FP; Dressler et al. 1987; Djorgovski & Davis 1987).
In recent years, it has been shown that a similar correlation between those observables exists in clusters out to $z\sim0.8$ (e.g. van Dokkum & Franx 1996; Kelson et al. 1997; Bender et al. 1998; Pahre 1998; van Dokkum et al. 1998; J[ø]{}rgensen et al. 1999; Ziegler et al. 2001). The observed evolution of the intercept $\gamma_{\rm FP}$ of the FP with redshift is consistent with the expectations of passive evolution models for an old stellar population (redshift of formation $z_f\ga
2$; see e.g. van Dokkum et al. 1998). No evidence for a dramatic evolution of the slopes $\alpha$ and $\beta$ of the FP with redshift is found with the available data (see J[ø]{}rgensen et al. 1999, Kelson et al. 2000, and T01a for discussion). The correlation is observed to be tight also in intermediate redshift field samples, although a faster evolution of the intercept is found in the highest redshift field samples (to $z\sim0.7$) and interpreted as evidence for secondary episodes of star formation in the field population at $z<1$ (T02; see also T99, T01a, Kochanek et al. 2000, van Dokkum et al.2001, Trager et al. 2000).
Assuming that galaxy G in 0047$-$281 lies on a FP with slopes similar to those in the local Universe – and pure luminosity evolution – we can determine the offset of its effective mass-to-light ratio ($M/L
\propto \sigma^2 10^{-0.4SB_e}/R_e$) with respect to the local relation, which is related to the evolution of the intercept by $\Delta \log M/L = - 0.4 \Delta \gamma_{\rm FP} / \beta$. As the local FP in the B band we adopt the relation found by Bender et al. (1998), i.e. $\alpha=1.25$, $\beta=0.32$, and $\gamma_{\rm FP}=-8.895-\log h_{50}$. In this way, we obtain $\Delta \log M/L_B=-0.37\pm0.06$. The error is dominated by the observed FP parameters of galaxy G and dominates uncertainties on the local FP relation. In Fig.\[fig:FP\] we plot the evolution of the effective M/L for cluster and field E/S0 galaxies as function of redshift (dashed and solid line; from T02), together with the value obtained for galaxy G (large filled square). The effective M/L evolution for galaxy G is consistent with what is observed for field galaxies, i.e. faster than for the cluster sample, possibly indicating younger luminosity-weighted stellar populations (see, e. g., T01a, T02). As described and discussed in TK02, we can use this measurement to infer the stellar mass-to-light ratio ($M_*/L_B$) of galaxy G assuming that $$\log (M_*/L_B)_{z}= \log (M_*/L_B)_{0} + \Delta \log (M/L_B),
\label{eq:FPev}$$ where the second term on the right hand side of the equation is measured from the evolution of the FP, and the first term on the right hand side of the equation can be measured for local E/S0 galaxies. Note that Equation \[eq:FPev\] uses the non-trivial assumption that the stellar mass is a redshift-independent function of the combination of observables used to define the effective mass $\sigma^2R_e$ (for a full discussion see T02, and TK02). Using the local value of $(7.3\pm2.1) h_{65}$ [M$_\odot$/L$_{B,\odot}$]{} determined from data by Gerhard et al. (2001) as in TK02, we infer $M_*/L_B=(3.1\pm1.0)
h_{65}$ [M$_\odot$/L$_{B,\odot}$]{} for galaxy G.
Gravitational–Lens Model {#sec:lensing}
========================
The Einstein radius ($R_{\rm E}$) and mass enclosed by the Einstein radius, i.e. $M_{\rm E}\equiv M(<R_{\rm E})$ are quantities required in the dynamical models that will be discussed in Sec.\[sect:dyn\]. Both $R_{\rm E}$ and $M_{\rm E}$ are very insensitive to the assumed mass profile (see e.g. Kochanek 1991), especially for highly symmetric cases such as 0047$-$281 (see Fig.\[fig:HST\]). For consistency – to ensure uniform modeling throughout the LSD survey – we model the lens galaxy as a Singular Isothermal Ellipsoid (SIE; Kormann et al. 1994; see also Chen, Kochanek & Hewitt 1995; Kochanek 1995; Grogin & Narayan 1996; Koopmans & Fassnacht 1999; Cohn et al. 2001; Mu[ñ]{}oz, Kochanek & Keeton 2001; Rusin et al. 2002). Although the assumption of a SIE mass distribution might not be accurate enough for some applications (see, e. g., Saha 2000, Rusin & Tegmark 2001, Wucknitz 2002), we again stress that this choice of the mass profile does [*not*]{} significantly bias the determination of the quantities used in our analysis, i.e. $M_{\rm E}$ and R$_E$. The reason is that the image deflection angles in four-image systems like 0047$-$281 are not only nearly the same, but also that the deflection angle is only a function of the enclosed mass and therefore is little affected by either the radial mass profile or the ellipticity of the lens (see §3 in Kochanek 1991 for a more detailed discussion). Constraints on the mass density profile obtained in Sec.\[sect:dyn\] are therefore virtually [*independent*]{} of the choice of lens mass model.
Image $\Delta x$ ($''$) $\Delta y$ ($''$) $\delta(x,y)$ (mas)
------- ------------------- ------------------- ---------------------
G $\equiv$0.000 $\equiv$0.000 –
A $+$1.048 $+$0.726 10,10
B $+$0.896 $-$0.802 10,10
C $-$0.126 $-$1.165 10,10
D $-$1.011 $+$0.263 10,10
[Table 3.— Centroid image positions of 0047$-$281 with respect to the centroid of the primary lens galaxy (G). \[tab:lensimages\]]{}
The centroids of the four lensed images are used as constraints on the lens model (Table 3), assuming errors of 10 mas, i.e. a fifth of a drizzled pixel size. We do not use the flux ratios in our analysis since they are not only difficult to measure accurately for extended images, but also change as function of position along the arcs and could be affected by differential dust extinction (e.g. Surpi & Blandford 2002). The position and position angle of the lens-galaxy mass distribution are set equal to those of its surface brightness distribution (Table 1) and the presence of an external shear is allowed for. Hence, there are six free parameters (i.e. the ellipticity and velocity dispersion of the lens-galaxy mass distribution, the source position, and the external shear), eight constraints (i.e. the four image positions) and consequently two degrees of freedom. The velocity dispersion of the SIE mass model is defined such that the mass enclosed by the critical curve is equal to that inside the Einstein radius of a Singular Isothermal Sphere with the same velocity dispersion (Kormann et al. 1994).
The best solution for a single SIE mass distribution yields $\sigma_{\rm SIE}=253$ [km$\,{\rm s}^{-1}$]{} for the lens galaxy, whereas the required external shear is 0.13. The $\chi^2$ is rather high, 306 for two degrees of freedom. It is possible to device more elaborate models (for example with a dwarf companion) that fit the observational constraints much better. However – as mentioned before – the mass enclosed by the Einstein radius is very insensitive to the precise lensing model. As an illustration of the uncertainty related to the mass modeling we considered including a dwarf companion galaxy with $\sigma_{SIE}\sim60$ [km$\,{\rm s}^{-1}$]{}. Although the fit is greatly improved ($\chi^2=0$ because there are more free parameters than constraints), the velocity dispersion of the primary lens changes by only $-$4[km$\,{\rm s}^{-1}$]{}. We will therefore not elaborate further on the details of the lens model – which is not the intention of this paper – and use the results from SIE mass model in the rest of the analysis. More detailed lens models are being constructed based on the structure in the lensed arcs (R. Webster, private communication).
We note that $\sigma_{\rm SIE}$ is close to the central stellar velocity dispersion (Table 1). However, we emphasize that $\sigma_{\rm
SIE}$ is a model-dependent expression of the well-determined total enclosed mass, while the central stellar velocity dispersion depends on the precise total mass profile, on the luminous mass profile, and on the dynamical state of the luminous component. Hence, the two quantities do not have to agree in principle. Their agreement is rather an indication of a regular behavior in the physical properties of early-type galaxies, as discussed in Sections 5, 6 and 7 (see also Kochanek 1994 and Kochanek et al. 2000).
The adopted SIE velocity dispersion corresponds to a circular Einstein radius of $R_{\rm E}=(8.70\pm 0.07) h_{65}^{-1}$ kpc (i.e. $\theta_{\rm E}=1\farcs34\pm0\farcs01$) and an enclosed mass of $M_{\rm E}=4.06\times 10^{11}h_{65}^{-1}$ M$_\odot$. The errors on $R_{\rm E}$ and $M_{\rm E}$ are correlated (both depend on $\sigma_{\rm SIE}$): for fixed $R_{\rm E}$ one finds that $\delta
M_{\rm E}/M_{\rm E}= 2\,(\delta\sigma_{\rm SIE}/\sigma_{\rm
SIE})$. This error corresponds to about 3% for $\delta\sigma=4$ [km$\,{\rm s}^{-1}$]{}, i.e. the difference between the model with and without an additional companion galaxy, which we adopt as systematic error (the random error on $\sigma_{\rm SIE}$ is a negligible 0.5%). By adopting a wide range of mass profiles and/or ellipticities the enclosed mass changes by $\la 4\%$ for symmetric four-image systems like 0047$-$281 (e.g. Kochanek 1991). Hence, adding the two contributions in quadrature, the total error becomes 5% on $M_{\rm E}$ inside a radius of $R_{\rm E}\equiv 8.70 h_{65}^{-1}$ kpc.
Dynamical Model {#sect:dyn}
===============
Following TK02, we model the galaxy mass distribution as a superposition of two spherical components, one for the luminous stellar matter and one for the dark-matter halo[^3] . The luminous mass distribution is described by a Hernquist (1990) model $$\rho_L(r)=\frac{M_* r_*}{2\pi r(r+r_*)^3},
\label{eq:HQ}$$ where $M_*$ is the total stellar mass. This profile well reproduces the [$R^{1/4}\,$]{}surface brightness profile for $r_* = R_e / 1.8153$. In Sec.\[sec:slope\], we also examine the effect of a steeper inner core, $\rho_L(r)\propto r^{-2}$, using the Jaffe (1983) model. We find the effect to be negligible within the errors. The dark-matter distribution is modeled as $$\rho_d(r)=\frac{\rho_{d,0}}{(r/r_b)^{\gamma}(1+(r/r_b)^2)^{(3-\gamma)/2}}
\label{eq:DM}$$ which closely describes a NFW profile for $\gamma=1$, and has the typical asymptotic behavior at large radii found from numerical simulations of dark matter halos $\propto r^{-3}$ (e.g. Ghigna et al.2000). See TK02 for further discussion of this mass profile and dynamical model.
According to the CDM simulations given in Bullock et al. (2001), a galaxy with the virial mass of galaxy G at $z=0.485$ has $r_b\approx
50$ kpc $\gg R_{e} \sim R_{\rm E}$. Hence, in the light of a comparison with CDM models, we can safely assume that the dark matter profile in the region of interest (i.e. inside $\sim R_{\rm E}$) is well described by a power-law $\rho_{\rm d}\propto
r^{-\gamma}$. Throughout this study, we will set $r_b=50$ kpc (effectively equal to $r_b=\infty$). The effects of changing $r_b$ are discussed in TK02.
In addition, we assume an Osipkov-Merritt (Osipkov 1979; Merritt 1985a,b) parametrization of the anisotropy $\beta$ of the luminous mass distribution $$\beta(r)=1-\frac{\sigma^2_{\theta}}{\sigma_{r}^2}=\frac{r^2}{r^2+r^2_i},
\label{eq:OM}$$ where $\sigma_{\theta}$ and $\sigma_{r}$ are the tangential and radial component of the velocity dispersion and $r_i$ is called the anisotropy radius . Note that $\beta>0$ by definition, not allowing tangentially anisotropic models. A brief discussion of tangentially anisotropic models with negative constant values of $\beta$ is given in Sec.6.2. As a further caveat, note that at infinite radii, Osipkov-Merrit models become completely radial. Although this behavior is not commonly found within the inner regions of E/S0 probed by observations (e. g. Gerhard et al. 2001; but see van Albada 1982 and Bertin & Stiavelli 1993 for theoretical grounds), it has little impact in the case considered here, since the pressure tensor only becomes significantly radial well outside the Einstein Radius and in projection is significantly down-weighted by the rapidly falling luminosity–density profile.
The line-of-sight velocity dispersion is obtained solving the three dimensional spherical Jeans equation (e.g. Binney & Tremaine 1987 Eq. 4-30) for the luminous component in the total gravitational potential and computing the luminosity-weighted average along the line of sight (Binney & Tremaine 1987; Eq. 4-60; see also, e. g. Ciotti, Lanzoni & Renzini 1996). We correct for the average seeing of 0.7$''$, during the observations, and average the velocity dispersion – weighted by the surface brightness – inside the appropriate rectangular apertures. For completeness, we rescale the apertures (Table 2) by 0.9 and 1.1 in the directions of the major and minor axes, respectively, such that their projection on the axisymmetric model is equivalent to their projection on an elliptical galaxy with an axial ratio of 0.8, even though this has minimal effects on the model velocity dispersions ($<1$%), much smaller than the observational errors. The uncertainties on seeing, aperture size, and galaxy centering are taken into account as systematic errors in the following discussion.
Luminous and Dark Matter in 0047$-$281
======================================
The unknown parameters of our dynamical model are $M_*$, $\gamma$, $r_i$ and $\rho_{d,0}$ (we note again that $r_*=R_e/1.8153$ and $r_b=50$ kpc). We can eliminate one of these parameters using $M_{\rm
E}$, the mass inside the Einstein radius, which is the most accurately known constraint on the lens mass model. We choose to eliminate $\rho_{d,0}$. In addition, we transform $M_*$ into the stellar mass-to-light ratio $M_*/L_{B}$, fixing the model luminosity exactly to the value $1.2\times 10^{11}$$h_{65}^{-2}$L$_{\rm B,
\odot}$. Hence, values of $M_*/L_{B}$ that we derive from the dynamical model, bears an additional uncertainty of 11%, i.e. the observational error on $L_{B}$, whereas $M_*$, which is used in the model calculation, does not have this error. We use the average $R_e=(5.52\pm0.55) h_{65}^{-1}$ kpc of the rest-frame V– and B–band values.
For any given set $\{M_*/L_{B}, \gamma \}$ the dynamical model is completely determined and the luminosity-weighted velocity dispersions for each of the three apertures (Tab. 2) can be computed. The likelihood is determined assuming Gaussian error distributions and the confidence contours using the likelihood ratio statistic[^4].
Power-Law Models {#sec:slope}
----------------
Before studying the luminous and dark-matter profiles individually, we determine the effective slope ($\gamma'$) of the total (luminous [*plus*]{} dark) matter density profile ($\rho_t$) inside the Einstein radius. We emphasize that an effective slope of $\gamma'$ does not imply that the density profile follows $\rho_t\propto r^{-\gamma'}$ exactly, but only [*effectively*]{}.
For MG2016+112 (TK02) we measured an effective slope $\gamma'=2.0\pm
0.1 \pm 0.1$ (random and systematic errors), assuming $\rho_t\propto r^{-\gamma'}$. Based on the data in Warren et al.(1998) and Kochanek et al. (2000), we found a similar effective slope for 0047$-$281. However, the errors were relatively large due to a lack of an extended kinematic profile and the large uncertainty on the velocity dispersion given by Warren et al. (1998). With the data presented here we can perform a more accurate measurement, and set constraints on the anisotropy of the velocity ellipsoid.
In Fig. \[fig:lhplot\](c), we show the likelihood contours of $\gamma'$ versus the anisotropy radius $r_i$, based on our extended kinematic profile of 0047$-$281. Three main conclusions can be drawn: (i) for a spherical isotropic stellar distribution function ($r_i\rightarrow \infty$) $\gamma'=1.90\pm0.05$ (68% CL) (see also Fig.\[fig:dispersion\]); (ii) a lower limit of $r_i/R_e\ga 0.7$ (68% CL) can be set, implying that the velocity distribution function is isotropic in the inner regions of the galaxy; (iii) independent of $r_i$, one finds $\gamma'=1.90^{+0.05}_{-0.23}$ (68% CL).
To assess systematic errors on $\gamma'$, we varied $M_{\rm E}$ and $R_{\rm e}$ by the total uncertainties $\pm$5% and $\pm$10%, respectively. The value of $\gamma'$, required to fit the data, changes by $\pm$0.05 and $\pm$0.03, respectively. Other potential sources of errors, such as seeing, aperture corrections, aperture offsets, etc., were found to be negligible. We add both contributions and we conservatively round up to a total systematic error of $\pm$0.1, to account for all potential minor sources of error.
Although the outer region of galaxy G is well fit by a Hernquist (1990) model, the inner region with $\rho_L(r)\propto r^{-1}$ is less constrained due to the finite resolution of the HST images. We therefore examined power-law models with steeper inner luminosity density profiles, $\rho_L(r)\propto r^{-2}$, using the model from Jaffe (1983). For both $r_i=\infty$ and $r_i=R_e$, we find that the stellar velocity dispersions change by $\le$6 [km$\,{\rm s}^{-1}$]{}. In general, Jaffe models give slightly lower velocity dispersions for a fixed value of $\gamma'$ and the best-fit models therefore yield slightly higher values of $\gamma'$ (few hundredths) compared with the Hernquist models. However, the differences are not significant, given the errors on dispersion profile, and therefore we conclude that our results are insensitive to the precise shape of the inner luminosity density profile.
For completeness, we note that our model with $\gamma'=1.90$ results in a central stellar velocity dispersion of $\sigma$=221 [km$\,{\rm s}^{-1}$]{} inside $R_e/8$, in excellent agreement with the empirically derived value of $\sigma$=229$\pm$15 [km$\,{\rm s}^{-1}$]{} in Sec.\[keck\], confirming the self-consistency of our models.
In conclusion, the total mass distribution of galaxy G is well-matched by a single power-law density profile, that is isothermal (i.e. $\rho_t\propto r^{-2}$) to within $\sim$5%, and the velocity ellipsoid of the luminous component is isotropic at least inside $\sim70\%$ of the effective radius. Note that this limit is generally consistent with the limits set by other physical considerations. In fact, strongly radial orbits would result in radial instability (e.g. Merritt & Aguilar 1985 find that Osipkov-Merritt models with initial Jaffe density profile are unstable for $r_i\la 0.3r_0$ where $r_0\approx R_e/0.763$ is the Jaffe half-mass radius; for the effects of a dark matter halo see, e.g., Stiavelli & Sparke 1991) or negative values of the distribution function (e.g. Ciotti 1999, and references therein).
Constant M/L Models
-------------------
A stellar mass-to-light ratio of [M$_*$/L$_B$]{}= $(5.4\pm0.8) h_{65}$ [M$_\odot$/L$_{B,\odot}$]{}is required to account for the mass $M_E$, enclosed by the Einstein radius. This is larger than the value [M$_*$/L$_B$]{}= $(3.1\pm1.0)
h_{65}$ [M$_\odot$/L$_{B,\odot}$]{} derived from the FP evolution. If we had no kinematic information this could only be interpreted as marginal evidence for dark matter inside the Einstein radius.
However, the velocity dispersion profile changes the situation dramatically: no model where mass follows light can be found to fit the data. For example, in Fig.\[fig:dispersion\] we show the velocity dispersion profile for an isotropic velocity distribution and constant M/L model (open squares). The dispersion falls too sharply with radius and this model can be excluded at the $>99.9\%$ CL. Even increasing $R_e$ by 10% and decreasing $M_{\rm E}$ by 10% (twice its error) – both resulting in a smaller stellar velocity dispersion – the fit (open stars) can still be excluded at the $>$99.9% CL. Setting $r_i=R_e$ for the latter model (not shown) worsens the fit and the model can be excluded at $>99.9\%$ CL.
Strong tangentially anisotropic models for lens galaxies (see e.g. Romanowsky & Kochanek 1999), although probably not very likely, can lead to flatter velocity dispersion profiles, even if M/L is constant. We therefore tested models with constant negative values of $\beta$. For $\beta\la -1.5$, we find that the dispersion profile indeed becomes flat. However, the predicted stellar velocity dispersion is much higher (i.e. 240–260 [km$\,{\rm s}^{-1}$]{}) than the observed values and the model is excluded at $>99.9\%$ CL. Models with $\beta\la-1.5$ can only fit the data when $M_{\rm E}$ is lowered by $\sim$30%. This is at least six times the error on $M_{\rm E}$ and is incompatible with any acceptable lens model of 0047$-$281. We therefore conclude that mass does not follow light in the lens galaxy G of 0047$-$281, but that M/L must increase with radius.
This is a key illustration that knowledge of the enclosed mass, $M_E$, inferred from the gravitational lens models, breaks the mass–anisotropy degeneracy.
Stellar Mass and Dark-Matter Slope
----------------------------------
We now turn to the two-component models described in Section 5, to assess the density profile of the halo. In Fig.\[fig:lhplot\] we show the likelihood contours (dashed lines) as a function of the stellar mass-to-light ratio ([M$_*$/L$_B$]{}) and inner slope of the dark-matter halo ($\gamma$). We choose two representative values for the anisotropy radius, $r_i=R_e$ and $r_i=\infty$. Note that both the mass within the Einstein Radius (M$_E$) and $r_i$ are fixed for all models in Figures 5(a) and 5(b). Hence, for each panel we effectively have two free parameters and three data points corresponding to the velocity dispersion profile (a fourth is given by the $M_*/L_B$ measured by the evolution of the FP, Section \[sec:FP\]). We find that: (i) For [M$_*$/L$_B$]{}$\rightarrow 0$, the slope of the dark-matter halo $\gamma$ approaches $\sim$2, consistent with the findings of Sec.\[sec:slope\] (panel c). (ii) For increasing values of [M$_*$/L$_B$]{}, the total density profile and the velocity dispersion profile become steeper. Hence, $\gamma$ has to decrease to fit the data.
When we include constraints on [M$_*$/L$_B$]{} from the Fundamental Plane (Sec.3), the limits tighten on both [M$_*$/L$_B$]{} and $\gamma$ (solid contours in Fig.\[fig:lhplot\]). Remarkably, we find that the stellar mass-to-light ratio inferred from the FP agrees well with the maximum stellar mass-to-light ratio allowed by our isotropic dynamical models (as for MG2016+112; TK02). Such a “maximum-bulge solution” is in that sense equivalent to the “maximum-disk solution” for spiral galaxies (e.g. van Albada & Sancisi 1986). We note however that the uncertainties are still considerable. In addition, the following limits are found: (i) [M$_*$/L$_B$]{}=$3.2^{+0.2}_{-0.9}\, h_{65}$ [M$_\odot$/L$_{B,\odot}$]{} and $\gamma$$<$1.55 (68% CL) for $r_i=\infty$ and (ii) [M$_*$/L$_B$]{}=$2.5^{+0.3}_{-0.5}\,h_{65}$ [M$_\odot$/L$_{B,\odot}$]{} and $\gamma<1.12$ (68% CL) for $r_i=R_e$. Because the velocity dispersion profile steepens with increasing radial anisotropy, a smaller $\gamma$ is required to retain a relatively flat dispersion profile. This explains the mild degeneracy between $r_i$ and $\gamma$.
In the context of adiabatic contraction (AC) models (e.g. Blumenthal et al. 1986; Mo, Mao and White 1998), the initial slope ($\gamma_i$) of the dark-matter halo, i.e. before baryons assembled in the dark-matter potential well, is in general expected to be shallower than the observed DM slope ($\gamma$). We find that the difference between $\gamma_i$ and $\gamma$ is relatively small, because the stellar mass in the case of 0047$-$280 is quite extended (i.e. large effective radius) and therefore affects the dark-matter slope less than found in MG2016+112 (TK02), for example. Even in the absence of AC, however, we find that $\gamma_i=1.5$ (Moore et al. 1998; Ghigna et al. 2001) is inconsistent with the results from 0047$-$281 at the 90%(68%) CL for $r_i=R_e(\infty)$, whereas the NFW profile ($\gamma_i=1$; Navarro, Frenk & White 1997) is consistent at the 68% CL for $r_i>R_e$. However, any mechanism, including AC, that steepens the initial slope by more that $\Delta\gamma=\gamma-\gamma_i\approx$0.6, would imply that the results from 0047$-$281 are inconsistent with CDM simulations.
Summary & Discussion
====================
We have presented HST and Keck observations of the gravitational lens system 0047-281. In particular, HST images have been used to measure the surface photometry of the lens galaxy G and to build a simple lens model of the quadruple-image system. Keck-ESI data have been used to measure a spatially resolved velocity dispersion profile extended beyond the effective radius, with exquisite accuracy ($\sim 5$%). We have combined all these measurements to study the internal structure and dynamics of the lens galaxy at $z=0.485$, finding the following:
[**(i)**]{} The offset of galaxy G from the local Fundamental Plane, $\Delta \log M/L_B=-0.37\pm0.06$ between $z=0$ and $z=0.485$, is consistent with what is observed for field E/S0 galaxies at similar redshift (T02), i.e. somewhat larger than for cluster E/S0 galaxies. In terms of pure luminosity evolution this could be explained with intermediate age single stellar populations, or – more likely – with secondary episodes of star formation contributing a fraction of young stars to an old underlying stellar population (see discussion in T02).
[**(ii)**]{} No dark-matter or constant M/L models are excluded at $>99.9\%$ CL. Also constant M/L models with strongly tangential anisotropy of the stellar velocity ellipsoid are excluded at $>$99.9% CL.
[**(iii)**]{} The stellar mass-to-light ratio [M$_*$/L$_B$]{}=$(3.1\pm 1.0)\,h_{65}$ [M$_\odot$/L$_{B,\odot}$]{} obtained from the offset of the FP is inconsistent with the required [M$_*$/L$_B$]{}=$5.4\,h_{65}$ [M$_\odot$/L$_{B,\odot}$]{} to fully account for $M_E$. This suggest the presence of dark matter. The FP value is consistent with what is obtained with our two-component dynamical models, and combining the two constraints we find [M$_*$/L$_B$]{}=$3.2^{+0.2}_{-0.9}\,h_{65}$ [M$_\odot$/L$_{B,\odot}$]{} (68% CL) for an isotropic velocity ellipsoid. Hence, dark matter comprises a fraction of $0.41^{+0.05}_{-0.15}$ of the total mass enclosed by the Einstein radius of 8.70$\,h_{65}^{-1}\,$kpc for $r_i=\infty$. For $r_i=R_{e}$ this fraction increases to $0.54^{+0.09}_{-0.06}$. The data are also consistent with an Osipkov-Merritt (OM) radial anisotropy with anisotropy radius $r_i\ge 0.7 R_e$ (68 % CL).
[**(iv)**]{} The total (luminous plus dark) mass distribution inside the Einstein radius can be described by a single power-law density distribution, $\rho_t\propto r^{-\gamma'}$, with $\gamma'=1.90\pm0.05$ (68% CL) for isotropic models, i.e. $r_i=\infty$. In general, $\gamma'=1.90^{+0.05}_{-0.23}$ (68% CL) is found. The systematic error is estimated at 0.1.
[**(v)**]{} An upper limit $\gamma\la 1.55$ (68 % CL) is found on the slope on the dark-matter halo inside the Einstein radius for an isotropic model. This limit tightens to $\gamma\la 1.12$ for mildly anisotropic models with $r_i=R_e$. Initial dark-matter profiles with $\gamma_i=1.5$ (Moore et al. 1998; Ghigna et al. 2001) are therefore only marginally acceptable, especially since the profile is expected to be less steep before the galaxy assembled. If $\gamma$ steepens by $\Delta\gamma>0.6$ during galaxy formation all CDM simulations are inconsistent with our results.
In summary, the lens galaxy in 0047$-$281 appears to convey the same picture formulated for MG2016+112 that early-type galaxies at significant look-back times can be effectively described by a [$R^{1/4}\,$]{}luminous component (modeled in this paper as either a Hernquist or Jaffe profile) embedded in a nearly-isothermal total mass distribution and that their stellar velocity dispersion is relatively isotropic, in particular inside the effective radius. In fact, both lens galaxies in MG2016+112 (KT02, TK02) and 0047$-$281 show that the total mass distribution is well approximated to within 5% by a simple power law density profile $\rho_t\propto r^{-2}$ (i.e. isothermal). Whether this conclusion can be generalized, however, requires the analysis of more systems.
Even more so, we have shown that deviations from isothermality or isotropy in the lens galaxies of 0047$-$281 and MG2016+112 quickly lead to inconsistencies with constraints from either the FP, the observed stellar kinematics, the stellar mass-to-light ratio, observations of local E/S0 galaxies, the gravitational-lens models, etc., whereas the models that fit all constraints are internally consistent, appear to agree with all observational constraints available, and indicate both isothermality and near-isotropy. Constant M/L or steep mass profiles inside the Einstein radius are excluded at very high confidence levels.
A physical explanation is required [*if*]{} isotropy and the almost perfect isothermality are confirmed to be generic features of early-type galaxies. In particular, this regularity might suggest that luminous and dark-matter were strongly coupled at some point during galaxy assembly. Whereas adiabatic contraction has been suggested as a mechanism that can lead to near-isothermal mass profiles (e.g. Keeton 2001), it is not clear why such a process should [*only*]{} stop when the density profile is isothermal to better than apparently a few percent (see also TK02). Adiabatic contraction also leads to a slope of the inner density profile, inconsistent with the observed absence of lensed images in the centers of lens galaxies (e.g. Keeton 2001), if either the central black holes are not very massive or the inner density profiles do not steepen through some other process. Violent relaxation could be a natural and viable explanation for this regularity, although it should also be explained why luminous and dark matter have [*different*]{} density profiles (see discussion in TK02 and references therein). A combination of the two processes during some period in the formation of the galaxy can not be excluded.
The striking similarity of the internal structure of E/S0 galaxies at large look-back times with the internal structure of local E/S0 galaxies (e.g. Franx et al. 1994; Bertin et al. 1994; Rix et al. 1994; Gerhard et al. 2001; see also Kochanek 1995 and reviews by de Zeeuw & Franx 1991; Bertin & Stiavelli 1993; Merritt 1999) suggests that little structural evolution occurred during the past 8 Gyrs (although again a larger sample is needed to make a quantitative and general statement). The lack of significant structural evolution is also suggested by the remarkable agreement between the stellar [M$_*$/L$_B$]{} obtained with our dynamical models and the stellar [M$_*$/L$_B$]{} estimated using the FP evolution. This fact adds further evidence in favor of a scenario where the general population of massive (field) E/S0 galaxies changed little in the past 8 Gyrs (from $z\sim1$) – as indicated for example by the modest evolution in their number density (Schade et al. 1999; Im et al. 2002; Cohen 2002; McCarthy et al.2002) and by the little evolution in the scatter of the FP (T02) – with most of the evolution being driven by ageing of old stars and secondary episodes of star formation (Jimenez et al. 1999; Trager et al. 2000; Menanteau, Abraham & Ellis 2001; T02).
[We thank Eric Agol, Andrew Benson, Giuseppe Bertin, Roger Blandford, Richard Ellis, Chris Kochanek, and Massimo Stiavelli for useful comments on this manuscript and stimulating conversations. We thank the referee for the comments that helped clarify the presentation of our results. The use of the Gauss-Hermite Pixel Fitting Software and Gauss-Hermite Fourier Fitting Software developed by R. P. van der Marel and M. Franx is gratefully acknowledged. The ESI data were reduced using software developed in collaboration with D. Sand. We acknowledge the use of the HST data collected by the CASTLES collaboration. LVEK and TT acknowledge support by grants from NSF and NASA (AST–9900866; STScI–GO 06543.03–95A; STScI-AR-09222). We thank J. Miller, M. Bolte, R. Guhathakurta, D. Zaritsky and all the people who worked to make ESI such a nice instrument. Finally, the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. ]{}
Bender R., Saglia R. P., Ziegler B., Belloni P., Greggio L., Hopp U., Bruzual G., 1998, ApJ, 493, 529 Bertin, G., Ciotti, L., del Principe, M., 2002, A&A, 386, 149 Bertin, G., & Stiavelli, M., 1993, Rep. Prog. Phys, 56, 493 Bertin, G. et al. 1994, A&A, 292, 381 Binney, J. & Tremaine, S, 1987, Galactic Dynamics, Princeton University Press, Princeton Blumenthal, G. R., Faber, S. M., Flores, R., Primack, J. R. Bullock, J. S., Kolatt T. S., Sigad, Y., Somerville, R. S., Kravtsov, A. V., Klypin, A. A., Primack, J. R., & Dekel, A., 2001, MNRAS, 321, 598 Chen, G. H., Kochanek, C. S., & Hewitt, J. N. 1995, , 447, 62 Ciotti, L., 1999, , 520, 574 Cohen, J. G. 2002, , 567, 672 Cohn, J. D., Kochanek, C. S., McLeod, B. A., & Keeton, C. R. 2001, , 554, 1216 de Zeeuw, T., & Franx, M., 1991, ARA&A, 29, 239 Djorgovski S. G., Davis M., 1987, ApJ, 313, 59 Dressler, A., Lynden-Bell, D., Burstein, D., Davies, R. L., Faber, S. M., Terlevich, R, Wegner G. 1987, ApJ, 313, 42 Falco, E. E., Kochanek, C. S., & Munoz, J. A. 1998, , 494, 47 Franx, M., van Gorkom J. H., & de Zeeuw, P.T. 1994, , 436, 642 Fruchter, A. S. & Hook R. N., 2002, PASP, 114, 144 Gerhard, O., Kronawitter, A., Saglia, R. P., & Bender, R., 2001, , 121, 1936 Ghigna, S., Moore, B., Governato, F., Lake, G., Quinn, T., Stadel, J., 2000, , 544, 616 Grogin, N. A. & Narayan, R. 1996, , 464, 92 Helbig, P., Marlow, D., Quast, R., Wilkinson, P. N., Browne, I. W. A., & Koopmans, L. V. E. 1999, , 136, 297. Hernquist, L., 1990, , 356, 359 Im, M., Faber, S. M., Koo, D. C., Phillips, A. C., Schiavon, R. P., Simard, L. & Willmer, C. N. A., 2002, ApJ, 571, 136 Jaffe, W. 1983, , 202, 995 Jimenez, R., Friaca, A., Dunlop, J. S., Terlevich, R.J., Peacock J. A., Nolan, L. A., 1999, MNRAS, 305, L16 J[ø]{}rgensen I., Franx M., Hjorth J., van Dokkum P. G., 1999, MNRAS, 308, 833 Keeton, C. R. 2001, ApJ, 561, 46 Kelson D. D., van Dokkum P. G., Franx M., Illingworth G. D., Fabricant D., 1997, ApJ, 478, L13 Kelson D. D., Illingworth G. D., van Dokkum P. G., Franx M., 2000a, ApJ, 531, 137 Kochanek, C. S., 1991, , 371, 289 Kochanek, C. S., 1994, , 436, 56 Kochanek, C. S., 1995, , 445, 559 Kochanek, C. S. et al. 2000, ApJ, 543, 131 Koopmans, L. V. E., 2001, PASA, 18, 179 Koopmans, L. V. E. & Fassnacht, C. D., 1999, , 527, 513 Koopmans, L. V. E. & Treu, T., 2002, , 568, L5 (KT02) Koopmans, L. V. E., Garrett, M. A., Blandford, R. D., Lawrence, C. R., Patnaik, A. R., Porcas, R. W., 2001, , 334, 39 Kormann, R., Schneider, P., Bartelmann, M., 1994, A&A, 284, 285 Kronawitter, A., Saglia, R. P., Gerhard, O., & Bender, R. 2000, A&AS, 144, 53 McCarthy, P. J. et al. 2002, ApJ, 560, L131 Menanteau, F., Abraham, R. G., Ellis, R. S. 2001, MNRAS, 322, 1 Merritt, D. 1985a, , 90, 1027 Merritt, D. 1985b, , 214, 25 Merritt, D., 1999, PASP, 111, 129 Merritt, D. & Aguilar, L. A. 1985, , 217, 787 Moore, B., Governato, F., Quinn, T., Stadel, J. & Lake, G., 1998, , 499, L5 Mu[ñ]{}oz, J. A., Kochanek, C. S., & Keeton, C. R. 2001, , 558, 657 Navarro, J, Frenk, C. S., & White S. D. M, 1997, , 490, 493 \[NFW\] Osipkov L. .P., 1979, Pis’ma Astron. Zh., 5, 77 Pahre, M. 1998, PhD Thesis, California Institute of Technology Romanowsky, A. J. & Kochanek, C. S., 1999, , 516, 18 Rix, H. W., de Zeeuw, P. T, Cretton, N., van der Marel, R. P., & Carollo, C. M. 1997, , 488, 702 Rusin, D., Norbury, M., Biggs, A. D., Marlow, D. R., Jackson, N. J., Browne, I. W. A., Wilkinson, P. N., & Myers, S. T. 2002, , 330, 205 Saha, P. 2000, , 120, 1654 Saglia, R. P., Bertin, G. & Stiavelli, M. 1992, , 384, 433 Schade D. et al., 1999, ApJ, 525, 31 Sheinis, A. I., Bolte, M., Epps, H. W., Kibrick, R. I., Miller, J. S., Radovan, M. V., Bigelow, B.C. & Sutin, B.M. 2002, PASP, 114, 851 Schlegel, D. J., Finkbeiner D. P., Davis M., 1998, ApJ, 500, 525 Stiavelli, M., & Sparke, L., 1991, , 382, 466 Surpi, G., & Blandford, R. D., 2002, , submitted, astro-ph/0111160 Trager S. C., Faber S. M., Worthey G., Gonzalez J. J., 2000a, AJ, 119, 1645 Treu, T. & Koopmans, L. V. E. 2002, , 575, 87 \[TK02\] Treu, T., Stiavelli, M., Bertin G., Casertano, C., & M[ø]{}ller, P. 2001a, , 326, 237 (T01a) Treu, T., Stiavelli, M., Casertano, C., M[ø]{}ller, P., & Bertin G. 1999, , 308, 1037 (T99) Treu, T., Stiavelli, M., Casertano, C., M[ø]{}ller, P., & Bertin, G. 2002, , 564, L13 (T02) Treu, T., Stiavelli, M., M[ø]{}ller, P., Casertano, C., & Bertin G. 2001b, , 326, 221 (T01b) van Albada, T. S. & Sancisi, R. 1986, Royal Society of London Philosophical Transactions Series A, 320, 447 van der Marel, R. P. 1994, MNRAS, 270, 271 van der Marel R. P., Franx M., 1993, ApJ, 407, 525 van Dokkum, P. G., Franx, M., Kelson D. D. & Illingworth G. D., 1998, ApJ, 504, L17 van Dokkum, P. G., Franx, M., Kelson D. D. & Illingworth G. D., 2001, ApJ, 553, L39 Warren, S. J., Hewett, P. C., Lewis, G. F., M[ø]{}ller, P., Iovino, A., Shaver P. A., 1996, MNRAS, 278, 139 Warren, S. J., Iovino, A., Hewett, P. C., Shaver P. A., 1998, MNRAS, 299, 1215 Warren, S. J., Lewis, G. F., Hewett, P. C., M[ø]{}ller, P., Shaver P. A., & Iovino, A. 1999, A&A, 343, L35 Ziegler, B. L., Bower, R. G., Smail, I. R., Davies, R. L. & Lee, D. 2001, , 325, 1571
[^1]: Obtained as part of the CASTLeS Survey
[^2]: developed by D. Sand and T. Treu; Sand et al. (2002), in prep.
[^3]: Spherical dynamical models provide an adequate approximate description for computing kinematic quantities of E/S0 galaxies with axis ratio $b/a\sim0.8$ like 0047-281 (e. g. Saglia, Bertin & Stiavelli 1992; Kronawitter et al. 2000), even though clearly they are not appropriate for the lensing analysis, since, e. g., they do not produce quadruply lensed images.
[^4]: Note that the likelihood ratio statistic is distributed as a $\chi^2$ only asymptotically, hence the interpretation of the likelihood ratio contours as confidence contours is an approximation. However, because the confidence contours (i.e. Fig.5) on which we base our results only mildly deviate from true ellipses – i.e. the limiting case for large numbers of constraints –, this approximation should work well.
|
---
abstract: 'We present a general picture of the ongoing formation and evolution of early-type galaxies via a specific evolutionary sequence starting in the blue cloud and ending in the low-mass end of the red sequence. This evolutionary sequence includes a Seyfert AGN phase in the green valley, but this phase occurs too late after the shutdown of star formation to be responsible for it. Thus, the bulk of black hole accretion in low-redshift early-type galaxies occurs in post-starburst objects, and not concurrent with star formation. On the other hand, a low-luminosity AGN phase switching on at an earlier stage when some star formation activity remains may be responsible for destroying the molecular gas reservoir fueling star formation.'
author:
- Kevin Schawinski
title: 'The role of AGN in the migration of early-type galaxies from the blue cloud to the red sequence'
---
[ address=[Einstein Fellow]{}, altaddress=[Department of Physics, Yale University, New Haven, CT 06511, U.S.A.]{}, altaddress=[Yale Center for Astronomy and Astrophysics, Yale University, P.O. Box 208121, New Haven, CT 06520, U.S.A.]{} ]{}
4[$L[\mbox{O\,{\sc iii}}]$/$\sigma^4$]{}
Introduction
============
When selected purely by morphology, the population of early-type galaxies in the local Universe includes a substantial subset of objects that are not passively evolving. These active early-type galaxies may host current or recent star formation as well as ongoing black hole growth. Due to their nature, these active early-type galaxies represent a window onto the ongoing formation and evolution of spheroidal galaxies and the connection between star formation and black hole growth ([@2006Natur.442..888S; @2007MNRAS.382.1415S; @2009ApJ...690.1672S]).
Migration from the blue cloud to the red sequence
=================================================
![Color-mass diagrams for the MOSES sample of early-type galaxies (@2007MNRAS.382.1415S). In each panel, the gray points are the non-early-type galaxy population while the orange points are quiescent early-type galaxies without detectable emission lines. The colored points in each panel are early-type galaxies with emission lines, separated by diagnostic diagrams ([@1981PASP...93....5B; @2001ApJ...556..121K; @2007MNRAS.382.1415S]). The star-forming early-types (blue) reside in the blue cloud. the AGN+SF composites (TRO) have similar masses, but redder optical colors. The Seyfert AGN are redder yet. Some LINERs have similar masses and even redder colors, while the majority of them reside in massive red sequence early-type galaxies. \[fig:ur\_cmstellar\]](ur_cmstellar_lowres){width="\textwidth"}
SDSS reveals a strong association between the dominant source of gas ionization determined via emission line ratio diagrams and galaxy properties (e.g., star formation, Seyfert AGN, LINERs, AGN+SF composites; e.g., @1981PASP...93....5B [@2001ApJ...556..121K]). @2007MNRAS.382.1415S showed that the active early-type galaxy population whose nebular emission lines are dominated by star formation only cluster strongly in the blue cloud at relatively low stellar masses (see Figure \[fig:ur\_cmstellar\]). Early-type galaxies where AGN and star formation have roughly equal impact on the ionized gas occupy roughly the same mass range, but have redder optical colors. Redder yet, at the same mass, are the Seyfert AGN – they perfectly occupy the green valley. LINERs cluster predominantly in massive red sequence early-type galaxies, but at the low masses characteristic of the other active early-types, they straddle the low mass end of the red sequence.
If we take $u-r$ optical color as a proxy for stellar age, then this observational picture implies an *evolutionary time sequence* for low-mass early-type galaxies starting from star formation in the blue cloud followed by the commencement of black hole accretion accompanied by a decline in star formation rate resulting in an optically ‘green’ AGN host galaxy. As the underlying stellar population ages further, the galaxy then settles onto the low-mass end of the red sequence, exhibiting a LINER phase driven most likely by post-AGB stars (e.g., [@2008MNRAS.391L..29S; @sarzi_gaspaper2]) for a while before settling into passive evolution on the red sequence.
![The evolutionary sequence for low-mass early-type galaxies with velocity dispersions less than 100. Early-type galaxies pass through these phases on their way from the blue cloud to the red sequence. Note that the pure Seyfert AGN phase occurs several hundred Myr after the end of star formation (from [@2007MNRAS.382.1415S]). \[fig:sequence\]](timeline_low){height=".3\textheight"}
However, the interpretation of $u-r$ optical color as an age proxy is not unique. There are many plausible star formation histories that could give rise, for example, to the green optical colors seen in Seyfert AGN host galaxies that are in no way connected to either the star-forming blue early-types on the one hand, or the most recent arrivals on the low mass end of the red sequence on the other. More detailed work on the star formation histories of objects along this putative sequence is required to establish this evolutionary link, e.g. studies as performed by @2007MNRAS.382.1415S. By taking into account information from the UV-optical-NIR broad-band SED (from *GALEX*, SDSS and 2MASS, respectively) and the stellar absorption (Lick) indices, they showed that the active early-type galaxies at fixed (low) mass (excluding the high-mass LINERs) all share the same recent star formation history: they all experienced a substantial recent burst of star formation in which 1–10% of the stellar mass was formed; the only difference in objects with different emission line classifications is the time elapsed since this burst. Marginalizing over all nuisance parameters to recover the typical age of each phase yields the time sequence shown in Figure \[fig:sequence\].
Do AGN suppress star formation?
===============================
The evolutionary sequence presented in Figure \[fig:sequence\] challenges scenarios where a luminous AGN phase is responsible for the shutdown of star formation by destroying the gas reservoir fueling the burst of star formation, as the Seyfert AGN phase occurs several hundred million years *after* the end of substantial star formation (see also [@2009ApJ...692L..19S]). Seyfert AGN in early-type galaxies always occur in galaxies with post-starburst stellar populations. This observation rules out the Seyfert AGN as the agent that transforms blue early-type galaxies into red sequence objects. However, the AGN appears as a low-luminosity object at earlier times in the AGN+SF phase. Is this the phase along the evolutionary sequence where star formation is shut down?
@2009ApJ...690.1672S observed the CO ($1\rightarrow 0$) and CO ($2 \rightarrow 1$) transition using the IRAM 30m telescope in a sample of objects along the evolutionary sequence to measure the amount of cold molecular gas present at each phase. They find that the star-forming early-types at the start of the sequence have ample molecular gas reservoirs of $\sim 10^{9}$, while all four Seyfert AGN do not yield any detections. It is within the AGN+SF phase that the molecular gas reservoirs are destroyed. In fact, the drop in molecular gas mass is so rapid that it cannot be accounted for by gas consumption due to star formation alone assuming the Schmidt law. The evidence thus points to an additional process active during the AGN+SF phase destroying the molecular gas reservoir and thus suppressing star formation. If we identify this process with the action of the AGN, then perhaps the low-luminosity AGN phase in the AGN+SF composites is radiatively inefficient, and therefore allows the remaining star formation to be visible in the emission lines, and does most of its work in a kinetic mode as a RIAF/ADAF (radiatively inefficient accretion disk/advection dominated accretion flow; [@1994ApJ...428L..13N]).
Summary
=======
We have presented a general picture of the ongoing formation and evolution of early-type galaxies via a specific evolutionary sequence starting in the blue cloud and ending on the low-mass end of the red sequence. This evolutionary sequence includes a Seyfert AGN phase in the green valley, but this phase occurs too late after the shutdown of star formation to be responsible for it. Thus, the bulk of black hole accretion in low-redshift early-type galaxies occurs in post-starburst objects, and not concurrent with star formation. On the other hand, a low-luminosity AGN phase switching on at an earlier stage when some star formation activity remains may be responsible for destroying the molecular gas reservoir fueling star formation.
Support for the work of KS was provided by NASA through Einstein Postdoctoral Fellowship grant number PF9-00069 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. KS gratefully acknowledges support from Yale University.
[10]{} natexlab\#1[\#1]{}\[1\][“\#1”]{} url \#1[`#1`]{}urlprefix\[2\]\[\][[\#2](#2)]{}
K. [Schawinski]{}, S. [Khochfar]{}, S. [Kaviraj]{}, S. K. [Yi]{}, A. [Boselli]{}, T. [Barlow]{}, T. [Conrow]{}, K. [Forster]{}, P. G. [Friedman]{}, D. C. [Martin]{}, P. [Morrissey]{}, S. [Neff]{}, D. [Schiminovich]{}, M. [Seibert]{}, T. [Small]{}, T. K. [Wyder]{}, L. [Bianchi]{}, J. [Donas]{}, T. [Heckman]{}, Y.-W. [Lee]{}, B. [Madore]{}, B. [Milliard]{}, R. M. [Rich]{}, and A. [Szalay]{}, *[[Nature]{}]{}* **442**, 888–891 (2006), .
K. [Schawinski]{}, D. [Thomas]{}, M. [Sarzi]{}, C. [Maraston]{}, S. [Kaviraj]{}, S.-J. [Joo]{}, S. K. [Yi]{}, and J. [Silk]{}, *[[MNRAS]{}]{}* **382**, 1415–1431 (2007), .
K. [Schawinski]{}, C. J. [Lintott]{}, D. [Thomas]{}, S. [Kaviraj]{}, S. [Viti]{}, J. [Silk]{}, C. [Maraston]{}, M. [Sarzi]{}, S. K. [Yi]{}, S.-J. [Joo]{}, E. [Daddi]{}, E. [Bayet]{}, T. [Bell]{}, and J. [Zuntz]{}, *[[ApJ]{}]{}* **690**, 1672–1680 (2009), .
K. [Schawinski]{}, C. [Lintott]{}, D. [Thomas]{}, M. [Sarzi]{}, D. [Andreescu]{}, S. P. [Bamford]{}, S. [Kaviraj]{}, S. [Khochfar]{}, K. [Land]{}, P. [Murray]{}, R. C. [Nichol]{}, M. J. [Raddick]{}, A. [Slosar]{}, A. [Szalay]{}, J. [Vandenberg]{}, and S. K. [Yi]{}, *[[MNRAS]{}]{}* **396**, 818–829 (2009), .
K. [Schawinski]{}, S. [Virani]{}, B. [Simmons]{}, C. M. [Urry]{}, E. [Treister]{}, S. [Kaviraj]{}, and B. [Kushkuley]{}, *[[ApJ]{}]{}* **692**, L19–L23 (2009), .
J. A. [Baldwin]{}, M. M. [Phillips]{}, and R. [Terlevich]{}, *[[PASP]{}]{}* **93**, 5–19 (1981).
L. J. [Kewley]{}, M. A. [Dopita]{}, R. S. [Sutherland]{}, C. A. [Heisler]{}, and J. [Trevena]{}, *[[ApJ]{}]{}* **556**, 121–140 (2001), .
G. [Stasi[ń]{}ska]{}, N. [Vale Asari]{}, R. [Cid Fernandes]{}, J. M. [Gomes]{}, M. [Schlickmann]{}, A. [Mateus]{}, W. [Schoenell]{}, and L. [Sodr[é]{}]{}, Jr., *[[MNRAS]{}]{}* **391**, L29–L33 (2008), .
, M., [Shields]{}, J. C., [Schawinski]{}, K., [Jeong]{} H. J., [Shapiro]{}, K., [Bacon]{}, R., [Bureau]{}, M.,[Cappellari]{}, M., [Davies]{}, R. L., [de Zeeuw]{}, P. T., [Emsellem]{}, E., [Falc[ó]{}n-Barroso]{}, J., [Krajnovi[ć]{}]{}, D., [Kuntschner]{}, H., [McDermid]{}, R. M., [Peletier]{}, R. F., [van den Bosch]{}, R. C. E., [van de Ven]{}, G., & [Yi]{}, S. K. 2009, *[[MNRAS]{}]{}*, submitted
R. [Narayan]{}, and I. [Yi]{}, *[[ApJ]{}]{}* **428**, L13–L16 (1994), .
|
---
abstract: |
Many algorithms have been developed for enumerating various combinatorial objects in time exponentially less than the number of objects. Two common classes of algorithms are dynamic programming and the transfer matrix method. This paper covers the design and implementation of such algorithms.
A host of general techniques for improving efficiency are described. Three quite different example problems are used for examples: 1324 pattern avoiding permutations, three-dimensional polycubes, and two-dimensional directed animals.
For those new to the field, this paper is designed to be an introduction to many of the tricks for producing efficient enumeration algorithms. For those more experienced, it will hopefully help them understand the interrelationship and implications of a variety of techniques, many or most of which will be familiar. The author certainly found his understanding improved as a result of writing this paper.
address: |
ARC Centre of Excellence for Mathematics and Statistics of Complex Systems,\
Department of Mathematics and Statistics,\
The University of Melbourne, Victoria 3010, Australia
author:
- 'Andrew R. Conway'
bibliography:
- 'combinatorics.bib'
title: The design of efficient algorithms for enumeration
---
Introduction
============
There are many problems in combinatorics where one wants to know the number of some type of object for a given size $n$ (e.g the number of polygons of perimeter $n$ on a given lattice[@enting1980sap]), but no formula is known. In this case computer enumeration is helpful to get the first few terms of the series, which can then be analysed to estimate their asymptotic behaviour. For many combinatorial problems the series terms grow rapidly - frequently exponentially - and it rapidly becomes computationally intractable to examine each object individually. An algorithm that groups multiple objects together can be more efficient, and can enable enumeration of more terms, which is usually highly desirable. Many such algorithms have been designed for many different problems. Many of them can be described as dynamic programming algorithms or as transfer matrix algorithms. This paper gives ways ot thinking about and optimizing such algorithms.
The paper does not intend to be exhaustive - that would be impractical - but rather to describe a coherent group of techniques in a common language.
Enumeration problems as a set of equations
------------------------------------------
Ignoring efficiency for the moment, consider how one could count a combinatorial object of size $n$. Often this involves starting with some base state $S_b$ (which typically includes $n$) and adding something to it - in the case of a geometrical object, this could be the presence or absence of a bond or a site. This then produces a set of other states $S_i$ which consist of the base state plus the information about the new site. For each of these states $S_i$, one adds the next possible item to get a new set of (more complex) states. Continue doing this recursively until one gets to an end state, at which point one increments a counter and then continues with the next substate (or [*child*]{} state) in the recursive algorithm. At the end the answer is what is left in the counter.
The set of states and connections to their child states defines a directed acyclic graph, called the [*call graph*]{} as it represents the calls in the recursive evaluation of the set of equations. Example call graphs for different formulations of the same problem are given in figures \[fig:FullDirAnTree\], \[fig:PartialDirAnTree\] and \[fig:PartialDirAnTree2\].
This can be written out more formally as a set of equations. Let $f(S)$ be the number of objects with state $S$. So what one wants to calculate is $f(S_b)$. For each state $S$ define a set of next-possible-states (or [*child*]{} states) $N(S)$ as the states one can get to by doing the next thing to $S$. This may be empty for an end state. Let $k(S)$ be a constant (typically 0, or 1 if $S$ is an end state). Then one can say $$f(S)=k(S)+\sum_{s\in N(S)}f(s).$$
We require that there be no loops in the directed graph implied by $N(S)$ in order to make the above equation, in principle, straightforward to evaluate without having to solve simultaneous equations. That is, there is no set of $k>1$ states ${s_i}$, such that $s_{i+1}\in N(s_i)$ and $s_1=s_k$. This is normally obvious, and one can often define a [*hierarchy*]{} function from states to integers (or other well ordered values), $h(s)$, with the property $\forall s,\forall t\in N(s), h(t)<h(s)$. Clearly if there exists a hierarchy function, there cannot be any loops. A hierarchy function is called [*ideal*]{} if $\forall s,\forall t\in N(s), h(t)=h(s)-1$.
An example is given in figure \[fig:FullDirAnTree\] which builds up all directed animals on a square lattice of size 4. Each rectangle is a state, with arrows pointing to its child states. The hierarchy function of a state in this case is a pair of integers; the number of sites left to add, and the number of grey sites. Ordering on the tuples is primarily by the number of sites to add, and secondarily by the number of grey sites.
![Generation of all directed animals of 4 steps. Each box contains, at top, the animal so far. Black sites are occupied; grey sites are potentially occupied, and white sites are unoccupied or not reached yet. At each step, the rightmost grey site in the uppermost row containing a grey site is processed. It is determined to be either unoccupied (left child) or occupied (right child). If it becomes occupied, then the sites accessible from it are set to grey. If the site is the last grey, then it may not be unoccupied as such a child would never be able to grow to the desired size, so there is only one child. If the animal is the desired size, it is finished and has no children. Below the picture of the animal so far is derived information on the border; the grey sites (ignoring surrounding, unreachable sites), and the number of sites left to be occupied to reach the desired size. Above and to the left of each box is the number of directed animals descending from that state; it is one for a leaf box and otherwise equal to the sum of its direct children. The root node’s number, 13, is the number of four site directed animals.[]{data-label="fig:FullDirAnTree"}](FullDirAnTree.pdf){width="\linewidth"}
Frequently an ideal hierarchy function is obvious — the number of elements to add to the partially constructed object, or the number of sites left to consider when passing over a finite lattice.
Sometimes there are weights associated with the substates - this is particularly common when the value of $f$ is a partial generating function (typically of integers) rather than an integer. Let $w(s,S)$ be the (easy to compute) weight associated with state $s$ coming from state $S$, giving us a more general formula $$f(S)=k(S)+\sum_{s\in N(S)}w(s,S)f(s)$$
Sometimes the series term number, $n$, is not explicitly in the state but is implicitly stored and is represented in the structure of $f$ (e.g. the size of the lattice when enumerating on a finite lattice) and/or in the size of the partial generating function that is the value of $f$.
We call a definition of $N$ [*clean*]{} if there is no path from the base state to a state $s$ where $f(s)=0$. If $N$ is not clean then extra, useless, work will be done. Unfortunately it can be difficult to be clean.
It turns out that a large class of combinatorial enumeration problems fall into this formalism, although the notation may be somewhat different. Sometimes the formalism is slightly different, requiring modifications of the techniques below (e.g. section \[sec:Factorizations\])
As an example, consider enumerating the set of well-formed bracket expressions with $n$ open and $n$ closing brackets[^1]. Define a state as a pair of integers, $S=(o,c)$, meaning the state where one has $o$ open and $c$ closing brackets left to add. So the stating state would be $S_b=(n,n)$, the ending state would be $(0,0)$, and adding an open or closing bracket would decrease $o$ or $c$ respectively. Overall one has $$f(o,c)=
\left\{
\begin{array}{ll}
1 & \mbox{if } o=0 \mbox{ and } c=0 \\
0 & \mbox{if } o>c \\
f(o-1,c) & \mbox{if } o=c>0 \\
f(o-1,c)+f(o,c-1) & \mbox{otherwise}
\end{array}
\right.$$ Note that this definition is redundant. The second line could be omitted, as such states will never be reached as the definition is clean. Alternatively, the third line could be omitted, but then the system would not be clean and the second line would be used. Also note that there are many other ways of enumerating well formed bracket expressions, with different state definitions.
The straightforward recursive implementation of this function typically takes time proportional to $f(S_b)$ as the only terminal state is $1$. In the case of the well-formed bracket expressions (and many others of interest), this grows exponentially with $n$.
Dynamic programming
-------------------
Recursive functions of this kind often have a simple optimization - whenever one has calculated $f(S)$ for some $S$, one adds $S\rightarrow f(S)$ in some table. Whenever one try to compute $f(s)$, one checks whether $s$ is present in the table already, and if so use the stored value rather than recomputing it. This is generally called [*dynamic programming*]{} (DP), [*memoization*]{}, or [*caching*]{}
DP takes up memory, but can vastly reduce the running time. Of course it relies on the same state being reachable multiple ways - otherwise the table will never be used.
As an easy to analyze case study, consider the well formed bracket expressions from above, and suppose one is trying to compute $f(10,10)$. The first step is forced, $f(10,10)=f(9,10)$. The next step has two legal choices, $f(9,10)=f(8,10)+f(9,9)$. Now comes the interesting bit. $f(8,10)=f(7,10)+f(8,9)$ and $f(9,9)=f(8,9)$. Both use $f(8,9)$. Using dynamic programming, the second time the program attempts to evaluate $f(8,9)$ it would just use the value previously computed. The total number of legal signatures needed to compute $f(n,n)$ is the number of pairs of integers $(o,c)$ where $n\geq c \geq o \geq 0$. This is clearly $O(n^2)$ which means the total number of states visited (and thus time and memory use) of the algorithm is $O(n^2)$. This compares with the naive recursive function which is $O(4^n n^{-1.5})$ which is clearly much worse.
In this case dynamic programming reduced the complexity of the algorithm from exponential to polynomial, and this sometimes happens in practice. When it does, the problem becomes computationally easy, a long series is generated, and people don’t bother enumerating it again! Frequently indeed it means that it is plausible to get a closed form solution. In other problems DP only reduces the exponential growth rate, which can significantly extend the number of terms one can get, although it rapidly becomes computationally intractable again. People tend to spend most of their time on such problems, as they are the unsolved ones.
Transfer matrix
---------------
Another approach is to keep a set of states and associated values (or [*multiples*]{}), starting off with $S_b$, multiple $1$. Then, in each iteration of the algorithm one takes each state $S$ and associated value $v$, and applies the function $f$ to it, adding each of the states $s \in N(S)$ with multiple $m$ (or $m \cdot w(s,S)$ if weights are present) to the new set of states and associated multiples. Also one keeps a counter and whenever applying $f$ to a state $S$ produces a constant, one adds the associated multiple times the constant to the counter. Then one replaces the old set of states and multiples with the new set of states and multiples and iterates until the set of states becomes empty.
So far nothing has been gained; one is effectively doing a breadth first search of the recursive algorithm rather than the obvious depth first search. But crucially, when one ends up putting the same state into the new set of states multiple times, one can merge them together and sum the associated multiples. Now, in a very similar manner to DP, one has bunched together a variety of ways to get to the same state, preventing redundant computation. This technique is often called the [*transfer matrix technique*]{} (TM).
Again going through the well formed brackets example, we have $$\begin{aligned}
f(10,10) &= f(9,10) \\
&= f(8,10)+f(9,9) \\
&= f(7,10)+2f(8,9) \\
&= f(6,10)+3f(7,9)+2f(8,8) \\
&= \dots
\end{aligned}$$
Typically a TM algorithm reduces the total amount of computation to the number of valid signatures reachable from the initial state, the same as DP[^2]. The memory use is somewhat less than DP, as one is effectively ordering computation so that one has some states that one can discard information about once one knows that one can never revisit them. The memory use is the maximum number of states one gets in a breadth first search of the structure of $f$[^3]. This is bounded above by the number of states from DP, and bounded below by said number of states divided by the number of iterations of $f$ before reaching a final state, which is typically $n$ or at least polynomial in $n$. This means that the memory use is better than DP, but still growing with the same exponent if it is exponential.
Comparison and interconversion of DP and TM
-------------------------------------------
The computational complexity of DP and TM are generally the same, but TM generally has somewhat lower memory requirements. So why would anyone ever use DP?
One reason is algorithm design complexity. TM is somewhat more complex to think about, implement, and test and debug than DP, while useful tools like Maple sometimes have direct support for DP (Maple calls it memoization). The resources used to implement TM could possibly be better spent making other changes to the algorithm that give a better return on the implementer’s time.
Another reason is lack of awareness of the trick of doing TM on DP problems. The main purpose of this paper is to raise awareness of such tricks, and introduce a way of thinking about the algorithms such that the interconversion is conceptually straightforward.
A third reason is that DP can be used to solve non-linear equations for $f$ - see section \[sec:Factorizations\]. This can be the real deciding factor.
A TM algorithm can almost always be converted to a DP algorithm, possibly requiring adding some information to the state - undoing the effect of section \[sec:TMItNoSig\]. But there is rarely a point in doing so.
Finite Lattice Method
---------------------
For enumeration of geometrical objects on a lattice, it is common to split up the problem into enumeration on each possible finite lattice. Then the TM method is used, where each step consists of dealing with one extra element of the lattice (usually site). The ideal hierarchy function is the number of elements left to be added. The desired series is then produced by combining the series from the finite lattices. This is commonly referred to as the Finite Lattice Method.
Typically the sites are added in a systematic way that minimises the size of the boundary between processed and unprocessed sites, and the state then describes that boundary and connectivity information. On a rectangular lattice[^4] the boundary would be a cross sectional line through the lattice, possibly with a kink in it if the number of sites processed is not an even multiple of the lattice width. This is shown in figure \[fig:2Dboundary\].
![The boundary of a finite lattice. The grey sites are processed, the white sites unprocessed. The light grey sites are processed sites on the boundary, the dark grey ones are sites whose status is now irrelevant other than in so far as they contribute to connectivity information on the boundary. The boundary may be sites, in which case it is usually the grey sites, or bonds, in which case it is usually the dotted bonds. Note the kink as sites are processed one at a time; the next site to be processed is the white site adjacent to two grey sites. []{data-label="fig:2Dboundary"}](2Dboundary.pdf)
Doing it on multiple lattices has several advantages
- It makes it easy to deal with uniqueness under translation invariance
- One can commonly require that the object on the finite lattice touches all edges of the lattice. While keeping track of the information required to track this increases the total number of legal states, in practice it often decreases the number of states that will be visited in a clean implementation restricted to a particular size.
- One can use lattice symmetries. Enumerating on the $n$ by $m$ lattice is usually the same as the $m$ by $n$ lattice. The orientation is chosen to minimize the length of the boundary, and thus the number of states.
The finite lattice method has been used to great effect on two dimensional lattices, e.g. [@conway1995polyonimo; @jensen2001polyonimo; @jensen2003polyonimo; @conway1993saw; @conway1996saw; @jensen2013saw; @enting1980sap; @jensen1999sap; @jensen2003parallelsap; @clisby2012sap]. When enumerating size $n$ objects, one generally has a maximum boundary length of roughly $n/2$. The number of states usually grows exponentially in the boundary length, although pruning can mean that the peak number of states actually encountered is highest at a smaller portion of $n$.
On $d>2$ dimensional lattices, the boundary is now a $d-1$ dimensional surface, typically of size $(n/d)^{d-1}$. The number of states can grow exponentially in this cross section, which can, in a unclean system, easily produce a number of states far greater than the number of objects being enumerated, making it a dreadfully inefficient algorithm. Making a clean system can be difficult, and even if it is clean the performance advantage over direct enumeration is much smaller than in two dimensions.
A technique occasionally used in the finite lattice method (e.g. [@conway1993saw]) is to enumerate subportions of the objects being enumerated and construct the objects being enumerated from the subportions. This enables smaller lattices to be used, producing smaller boundary sizes and vast reductions in the number of states.
This has been used for a long time [@deNeef1977finite].
Examples
--------
There are three significantly different examples that will be described in detail. They are used throughout the rest of this paper to provide specific examples of some techniques and trade-offs.
### 2D directed animals
A directed animal on some directed lattice is a set of sites such that each site apart from a single root site is directly downstream of another site. We will deal here with directed animals on a square lattice $\mathbb{Z}^2$ with the direction constraint being in the positive direction of either the $x$ or $y$ axes (north or east) Often the lattice will be shown rotated clockwise by 135 degree, so that the directed constraint becomes down and left or down and right. This is a well solved problem [@dhar1982dirAns; @gouyou1988dirAns; @betrema1993dirAns]; it is included as it is a simple example. Similar techniques can be used in many other two dimensional problems and on other lattices. It is solved for the square and triangular lattice[^5], but not for the hexagonal or most other lattices. Hereafter we will assume the square lattice.
There is one size one directed animal, the root. There are two size two directed animals, the root and one of the two downstream sites. There are five size three directed animals: four coming from the root, one of the two downstream sites, and also one of the two further downstream sites; the fifth is the root and both of the two downstream sites. There are thirteen size four directed animals; their construction is show in figure \[fig:FullDirAnTree\].
These can be solved in a straightforward manner using both the dynamic programming and transfer matrix techniques, although the dynamic programming method is more obvious. One starts from the root, and add sites along a line perpendicular to the preferred direction (north east). Connectivity is then not an issue, as everything reached on this line must have come from the root, meaning the presence or absence of the sites on this line, along with the number of sites used, is sufficient information. The line will end up having a kink in it at the site being added, and will be partly the line being processed and partly the subsequent line. In practice one can do even better by using the (kinked) line one beyond the line to which sites have just been added, and record which sites are reachable rather than which sites have been reached. This is because there can be multiple sets of reached sites that can produce the same set of reachable sites. The call graph of such a formulation is given in figure \[fig:PartialDirAnTree\], with corresponding equations in table \[tab:PartialDirAnTree\]. An alternate formulation is given in figure \[fig:PartialDirAnTree2\] and the transfer matrix algorithm applied to it is presented in table \[tab:TMDirAn\].
![A simplification of figure \[fig:FullDirAnTree\] just showing the border information, which is the only thing that can affect children. This allows coalescing of now identical states together to produce a smaller tree, which will be able to take advantage of dynamic programming. Note there is an arrow skipping the fifth row. []{data-label="fig:PartialDirAnTree"}](PartialDirAnTree.pdf)
![An alternative algorithm for enumeration, like figure \[fig:PartialDirAnTree\], except instead of processing one grey site at each step, the algorithm goes through grey sites until one is determined to be occupied. This is a more complex algorithm, but is better suited to the transfer matrix method as there are no row skipping arrows - see section \[sec:TMItNoSig\]. []{data-label="fig:PartialDirAnTree2"}](PartialDirAnTree2.pdf){width="\linewidth"}
-------------- --- --------------------------
$f(?,?,0)$ = $1$, otherwise
$f(1,0,n)$ = $f(11,0,n-1)$
$f(11,0,n)$ = $f(1,0,n)+f(10,11,n-1)$
$f(10,11,n)$ = $f(11,0,n)+f(111,0,n-1)$
$f(111,0,n)$ = $f(11,0,n)+f(110,11,n)$
-------------- --- --------------------------
: The equations corresponding to the graph in figure \[fig:PartialDirAnTree\]. The first two arguments to the function are binary numbers representing the pattern of grey sites, with a 1 meaning a grey site. Leading zeros are not shown. The third argument is the number of sites remaining to be added. The three arguments together define the state.[]{data-label="tab:PartialDirAnTree"}
------------ --- --------------------------------------------------
$f(1,0,4)$ = $f(11,0,3)$
= $f(10,11,2)+f(11,0,2)$
= $f(111,0,1)+2f(11,0,1)+2f(10,11,1)$
= $f(110,11,0)+5f(11,0,0)+5f(10,11,0)+2f(111,0,0)$
= $1+5+5+2$
= $13$
------------ --- --------------------------------------------------
: Transfer matrix style evaluation of the graph in figure \[fig:PartialDirAnTree2\], with function arguments as described in table \[tab:PartialDirAnTree\].[]{data-label="tab:TMDirAn"}
### 1324 Pattern avoiding permutations
A permutation $P$ of the integers $1 \dots n$ is said to avoid the pattern $p$ where $p$ is itself a permutation of $1 \dots l$ if there does not exist an $l$ length subsequence (consecutive or not) of $P$ that has the same relative order of elements as in $p$. For instance the permutation $15342$ contains the pattern $123$ through the subsequence $134$. The enumeration problem is to compute the number of such permutations for a given $n$.
Surprisingly, all pattern avoiding permutations grow roughly exponentially, rather than the factorial growth of the permutations.[@marcus2004excluded]
The series are well understood for all 2, 3, and 4 length patterns except for $1324$ and its complement $4231$. This has been enumerated using a dynamic programming algorithm. Conceptually the permutations are built up one element at a time, and the state consists of the remaining integers and constraints upon their order. This was done in [@johansson20141324PAPs] enumerating to 31 terms and improved on in [@conway20151324PAPs] with a more efficient state definition extending the series to 36 terms. The latter algorithm will be used in this paper.
The state definition consists of a series of numbers separated by brackets like $4(5(6)3)5$. Each number represents a set of consecutive integers yet to be chosen, and the value is the length of this set. The sum of all the numbers is the number of elements left to add. This sum is an ideal hierarchy function. One may not use an element inside a bracketed expression until all numbers after the bracketed expression have been dealt with. That is, in the example, there are 4 low numbers and 5 high numbers that are legal as the next number to be chosen for the permutation being built up. That is, $N$ of the example will have 9 elements. There may also be a comma at the start of the state indicating that at least one number lower than all remaining numbers has been already taken. See [@conway20151324PAPs] for a more precise definition and rules for computing $N$. The start state is the simple integer $n$, and the end state is the empty state. The system is clean.
### 3D polycubes
A polycube is a connected set of unit length cubes in $\mathbb{Z}^3$. An example is in figure \[fig:polycube\]. This is the 3D equivalent of a polyonimo. The objective is determining the number of polycubes consisting of $n$ cubes. This is an open problem in two or higher dimensions, with a roughly exponential growth rate.
![A size (volume) 4 polycube[]{data-label="fig:polycube"}](polycube.pdf)
There is one size 1 polycube - the single cube. There are three size two polycubes - two adjacent blocks, pointing along one of the three axes.
Polycubes are difficult to count efficiently. There has been extensive direct enumeration based on [@redelmeijer1981polyonimoes]. As the series grows rapidly, the most recent [@luther2011highDlatticeAnimals] has only got to 18 terms. The techniques in this paper have generally not been used for this or other three dimensional problems, as the number of intermediate states is large, and if one is not sufficiently careful, can be larger than the number of objects being enumerated.
Here is presented a transfer matrix method as a demonstration of the difficulty but not uselessness of TM on three dimensional problems. It is not clean, and will be a dreadful algorithm asymptotically, but may be reasonable for intermediate values. Enumeration will be done on finite lattices, adding one site at a time, with a two dimensional cross section as shown in figure \[fig:PolycubeBoundary\]. Initial evidence indicates it may be possible[^6] to get another couple of terms using this algorithm with comparable computational resources to [@luther2011highDlatticeAnimals].
![The boundary of a polycube finite lattice enumeration. This is the bottom of a finite lattice two wide, three deep, and at least three tall. Sites are processed starting with the bottom back left cube. Subsequent cubes are processed left to right. When the right boundary is reached (quickly in this case), the row starts again one step closer to the front. When the whole layer is done, the next layer starts again at the back left. The cubes shown are processed. The six light grey cubes are on the boundary; the nine dark grey cubes are behind the boundary and only matter in so far as they affect connectivity of the boundary cubes and which edge faces have been touched at least once. The next to be processed is the one in the top middle right indentation, adjacent to three light grey cubes labelled P (prior plane), R (prior row) and C (prior column).[]{data-label="fig:PolycubeBoundary"}](PolycubeBoundary.pdf)
The two dimensional equivalent, polyonimoes, has been well studied and enumerated using the TM method in [@conway1995polyonimo], significantly improved with pruning in [@jensen2001polyonimo], tweaked by Knuth to add a symmetry (public talk), and parallelized in [@jensen2003polyonimo].
Techniques common to both algorithms
====================================
The algorithms are very similar in the sense that the number of states is the primary driver of efficiency and memory use. The efficiency of these techniques comes from the conflation of different states when it can be recognised that they produce the same result. The more that states can be conflated, the more efficient the algorithm.
One can think of starting with a simple algorithm that builds up the objects being enumerated one element at a time, with the state being the entire history (as in figure \[fig:FullDirAnTree\]). There is no gain from dynamic programming so far as these states are never reached twice. Then one works out a way of abstracting some reduced information about a state called a [*signature*]{} which is sufficient to identify the state sufficiently precisely to compute downstream computations (as in figure \[fig:PartialDirAnTree\]). The expectation is that many states map to one signature. One then redefines the signatures to be the states and one has a more efficient algorithm.
So a signature is just a state, although with the connotation of being a summary of many other original states.
Signature design
----------------
Design of a signature is usually the most important part of the algorithm. The more states that map into a signature, the more efficient the algorithm. The specifics depend very much on the problem, but there are a couple of approaches that can be tried.
One approach is to think about different ways of describing the state. This can be exemplified by 1324 PAPs. In [@conway20151324PAPs] the same basic state is used as in the earlier work [@johansson20141324PAPs], but a different way of describing it is used. The information is basically what numbers in the permutation are left and what restrictions there are on their use. In [@johansson20141324PAPs] these restrictions are, for each available number, an index pointing back at which prior numbers are unlocked after all numbers after this one are dealt with. In [@conway20151324PAPs] it is noticed that this can be represented by a set of nested brackets. Looking at things this way, it becomes clear that consecutive brackets can be simplified, reducing the number of states. It also makes some state factorizations more obvious.
Another approach is to think about storing different information in the state. The following sections give several examples of this.
### Signature design on a finite lattice
For the frequent case of enumerating objects on a finite lattice, where each operation consists of considering one extra site to be processed, the state generally consists of the boundary of the processed sites. This is generally one dimension smaller than the lattice being processed. The state then consists of the elements on the boundary, plus any needed connection information. The boundary consists of the set of points in the processed set that have at least one neighbour in the unprocessed set. Connectivity information is necessary in order to prevent counting disconnected or otherwise invalid objects.
In the 3D polycube case, this connectivity information is basically which sites on the boundary are connected to which other sites. This can be practically implemented by assigning an (arbitrary) integer to each site. Sites with the same integer are connected. It is important to [*canonicalize*]{} such information to prevent the same state being referred to in two different ways by different choices of integers. A simple canonicalization in this case is to order the sites, and define the canonical choice of integers to be the one that would come first in lexicographic order if the integers were written out as a series in site order.
For 2D lattices, the connectivity information is simpler as there are frequently restrictions on crossings. So for the 2D version of polyonimoes, the boundary is a line (possibly with a kink in it). It is impossible to have a series of sites A,B,C,D where A is connected to C, and B is connected to D, but A is not connected to B. This means that connectivity information can often be defined as “connected to the next appropriate thing” or “connected to the last appropriate thing” [@conway1995polyonimo]. This does not intrinsically improve the algorithm, but it does improve implementation as it removes the need for canonicalization, and means that the state can be stored in a small number of bits, which is good for speed and memory usage of the big hash map typically used.
For the 2D directed animals case, the connectivity information is trivial... everything on the line perpendicular to the preferred direction must have come from the root, so is all connected, so no connectivity information is needed, just the state of the sites on the boundary.
For the enumeration of objects where bonds between sites have a meaning, it is worth considering the boundary along either sites or bonds - usually one will be significantly more efficient than the other.
Another common technique is to move the boundary forward one step to be just in front of the processed sites. That is, the sites that are in the unprocessed set but have a neighbour in the processed set. Instead of recording what is on the processed sites, one records what could be in the unprocessed site. Directed animals are an excellent example. Consider the following two states when recording what is present on the boundary. State $A$ has three consecutive sites occupied - subsequent growth can come from any of the four sites in the next row (figure \[fig:DirAnAB\], left). State $B$ is the same, except the middle of those three sites is not occupied (figure \[fig:DirAnAB\], right). But subsequent growth for state $B$ is the same as state $A$, as the same four sites are reachable from state A. So $f(A)=f(B)$, and $A$ and $B$ should be considered the same state. This can be done by storing which sites in the next row could be occupied - the new state replacing $A$ and $B$ would be four occupiable sites in a row.
![Two different starts of a directed animal that have different last sites occupied (the second last row), but the same sites occupiable (the last row). Black sites are occupied; grey sites are occupiable.[]{data-label="fig:DirAnAB"}](DirAnAB.pdf)
A similar technique can be used with 3D polycubes. The kink in the boundary will usually (depending on which site one is currently processing) have a single place where there are three sites adjacent to the next-to-be-added site. These three sites are in the prior plane $P$, the prior row $R$, and the prior column $C$ (see figure \[fig:PolycubeBoundary\]). Sites $R$ and $C$ have multiple sites still to be considered that they are adjacent to, but $P$ is only adjacent to the site about to be processed. If $P$ is occupied, and connected to at least one of $R$ or $C$, then the $P$’s presence does not affect anything, as any site added will be connected to the group containing $P$ by dint or $R$ and/or $C$. So the number of states could be reduced slightly by removing $P$ in such a circumstance. This is a minor optimization as it only affects one site.
Instead of storing what is connected to what in the past, one could store the information about what should be connected to what. This is different information with a significantly different set of child states. In [@clisby2012sap] this reduced the number of states slightly, but more importantly made the trimming (see later) much faster.
### Touching boundaries on a finite lattice
Suppose one wants to enumerate polycubes of size 15. This can be done by enumerating all polycubes in a 15 by 15 by 15 finite lattice. Of course smaller polycubes will appear multiple times differing only by translation. This can be accounted for by enumerating polycubes on all subsets of the 15 by 15 by 15 lattice and doing appropriate subtractions[^7].
There are ways of improving this. For many symmetric lattices the result is independent of the order of the dimensions. For instance, the number of 3D polycubes on a 3 by 5 by 7 lattice is the same as on a 7 by 3 by 5 lattice. This clearly reduces the number of lattices that must be enumerated upon, but, more importantly, one can choose the largest dimension to be in the direction perpendicular to the boundary. The size of the border (which is often the main driver of algorithm complexity) is determined by all but that dimension, so choosing it as the longest reduces the maximum size of the border.
This means the biggest border that one will come across to enumerate polycubes of size 15 will be 5 by 6, a much more tractable problem.
Furthermore, if one is counting polycubes on an $A$ by $j$ lattice, where $A$ is the boundary area and $j$ the depth, then instead of enumerating separately for each $j$, one can just work up to the maximum $j$ and one will pass through all lower values in the process, and whenever an object is finished in a layer $l\leq j$, one stores for the $A$ by $l$ lattice. One generally also wants to add the requirement that, after the first layer (of size $A$) is done, the empty boundary is not accepted. This provides uniqueness in the $j$ dimension, avoiding having to explicitly canonicalize, but, more importantly, it reduces the number of states that will be processed, improving efficiency.
Another way to get translational uniqueness is to require each edge of the lattice to be touched. Intuitively, this seems a bad idea, as it makes the state more complex as the state will now contain flags indicating whether each lattice edge[^8] has been touched. This increases the total number of legal states, which is an upper bound on the number of states that will be reached, and a useful heuristic for algorithm complexity. However, in practice [@jensen2001polyonimo] adding the boundary requirement often significantly reduces the number of states that [*will*]{} be reached, as it will take too many sites or bonds to get to that state and still be in a position to finish the object (see next section on trimming).
Signature trimming
------------------
Trimming is the process of making sure that the algorithm is clean. That is, it never produces a state $Z$ such that $f(Z)=0$. The number of states processed, which is the main driver of efficiency, is bounded above for a clean algorithm by the number of objects being enumerated times the maximum length of a chain from the start state to an end state (which is typically linear in the length of the sequence being computed). An unclean algorithm can be much worse. So trimming is of comparable importance to state design.
An example is in two dimensional polyonimo enumeration where the TM algorithm with trimming in [@jensen2001polyonimo] is much more efficient than the prior algorithm [@conway1995polyonimo], even though the states are more complex, as they involve keeping track of whether the boundaries have been reached. Similarly it has been very effective in 2D polygon enumeration with [@jensen1999sap] improving on [@enting1980sap], and in many other problems.
The trimming algorithm is used on each state processed to discard useless next possible states. Sometimes this is obvious, and not even worth mentioning. In other cases it is quite complex [@jensen1999sap], which can be problematic if it is time consuming, as it is executed for each state. In [@clisby2012sap] trimming is made faster by changing the state definitions from describing the connectivity in the processed space to describing the required connectivity in the unprocessed states.
In the finite lattice case, trimming basically consists of determining the minimum number of sites or bonds needed to complete the object, usually by resolving connectivity issues including to the boundaries. If this minimum number is too high the state can be rejected. Typically “too high” means that the lowest term of the generating function associated with the state plus the minimum completion number is greater than the desired length of the series.
Sometimes it is difficult to come up with a perfect trimming algorithm, in which case an imperfect algorithm may be used which leaves the algorithm still not clean, but better than with no trimming algorithm. An imperfect (or conservative) trimming algorithm generally produces a lower bound on the number of elements needed to finish rather than the number itself. The enumeration algorithm will then still produce the correct answer, but will waste time dealing with states that will go to zero. This is the case for the 3D polycube algorithm presented here as an example of a difficult trim.
### Polycube trimming
Specific details of the algorithm for trimming polycubes is presented here. It is not provided as an example of a good algorithm; to the contrary, it is a horrible algorithm. It is slow, complex (and therefore error prone), and imperfect. Rather it is presented to show how this is difficult, and the why it is sometimes worth changing the signature design to make trimming easier. Bad as it is, without it the algorithm would be totally impractical. Hopefully a reader will be able to improve it!
The finite lattice method applied to 3D polycubes adds one site at a time. So there are three nested loops, the outermost iterating over layer ($z$) then row ($y$) and then column ($x$). The boundary is a slice through the finite lattice, with a kink - some sites are in the layer currently being processed ($z$), some are in the prior layer ($z-1$, or empty if $z=0$). The state consists of the occupied sites on this border, and their connectivity. This is defined as a number for each site. Zero means unoccupied; a positive integer means occupied and connected to all other sites with the same number. For the rest of this section, these numbers will be called [*colors*]{}. Flags are also kept for which of the four sides of the finite lattice are attached (the $z=0$ side is connected by construction - no zero states are allowed after the last site of the $z=0$ layer is processed, and so the last $z$ layer touched is always the one currently being processed).
The task for the trimmer is to compute what the minimum number of sites is that needs to be added to finish the polycube. This requires adding cubes to:
- connect each distinct color,
- connect to the unconnected sides,
- connect to the minimum value of $z$ that ending is allowed[^9].
A fast precise answer is unknown to the author. A reasonably fast, conservative algorithm is presented here.
Ignore for the moment everything other than connecting all the colors. The only sites that need to be considered for this are the ones one layer beyond the boundary, with an extra row above the kink to allow connectivity between above and below the kink. Any minimal connection using other sites could be done equivalently using just this set of sites. Call this set of sites the [*grid*]{} $G$.
For any color $c$, we can define $table(c)$ to be a number for each site in $G$, being the minimum number of sites needed to get to that cell from any cell of color $c$ on the boundary. This can be computed in polynomial time using depth first search. Alternatively, define a consistency function $C(t)$ which takes a table $t$ and makes it consistent, that is, reduces any value to no more than one more than any of its neighbours. Then $table(c)$ is $C$ of the table which is infinite other than neighbours of $c$ which are $1$.
In the one color case, the color connectivity cost is trivially zero.
In the two color ($a$ and $b$) case, the cost is the minimum in $table(a)+table(b)-1$, alternatively the minimum cost in $table(a)$ of anything adjacent to $b$. Indeed, the minimum cost of connecting $a$ and $b$, going via a given grid element, is $bitable(a,b)=C(table(a)+table(b))$
In the three color ($a$, $b$, and $c$) case the connectivity cost is the minimum value of $bitable(a,b)+table(c)-1$, alternatively the minimum cost in $bitable(a,b)$ of anything adjacent to $c$.
Indeed, we can now define $tritable_c(a,b,c)=C(bitable(a,b)+table(c)-1)$. Then $tritable(a,b,c)=\min{\left\lbrace tritable_c(a,b,c),tritable_b(a,c,b),tritable_a(b,c,a)\right\rbrace}$ is now the cost of connecting $a$, $b$ and $c$ going via a particular site.
This lets us now do the four color case ($a$,$b$,$c$,$d$), in which case the connectivity cost is the minimum value of $tritable(a,b,c)$ adjacent to $d$.
For the five or more color case, the cost of continuing this is becoming prohibitive. A conservative lower bound is used, being just the cost to connect four of the colors[^10].
Now consider the cost to connect to the edges as well. Ignoring connection costs, one could find the sites with minimum and maximum $x$ and $y$ values, and take the distance to the edges from them. The connection cost will be that, probably with some extra as one also usually[^11] need to go up a site in order to build out. If one adds in the color connection cost, frequently the sites used in that will provide the extra height. Working out exactly when extra sites are needed is difficult. The sites in the color connection cost will almost never[^12] detour outside the minimum and maximum $x$ and $y$ values, so the color connection cost can be safely added to this edge connection cost[^13] to get a lower bound.
Similarly, to get to the maximum $z$ one has to have sites going from the current layer up to the minimum $z$ layer. One of these may be counted in the color connection cost if it is non-zero. So this distance can be added, with a 1 discount in the case of a non-zero color connection cost.
This ends up with a somewhat conservative trimming function which produces an algorithm that performs vastly better than with no trimming. Asymptotically it will be a dreadful algorithm as the number of states that end up being zero will become huge, but appears to be reasonable for currently calculable polycube sizes.
Binning
-------
A process, sometimes described as binning, can be occasionally used to reduce memory use at the cost of execution time. One chooses some intermediate set of states, divides them into groups, and, for each group, one does the enumeration repeatedly, requiring that group to be passed through. Then one adds up the results. This is rarely useful as the time penalty is usually significantly greater than the memory advantage.
Techniques particular to dynamic programming {#sec:TechDP}
============================================
Dynamic programming is generally inferior to a transfer matrix algorithm, but is still used extensively as it is conceptually easier, and has some optimizations not available to the transfer matrix method.
Factorizations {#sec:Factorizations}
--------------
Sometimes it is possible to factorize a state, that is say $f(S)=f(S_1)f(S_2)$, for some states $S_1$ and $S_2$. Generally $S_1$ and $S_2$ will be significantly smaller than $S$, vastly reducing the work needed to compute them. For instance, in the 1324 PAPs, the state $S=(S_1)S_2$ has this property, as the brackets mean that all things to the right of the bracket must be processed before anything inside the bracket[^14].
As far as dynamic programming is concerned, multiplication is insignificantly different from addition. But multiplication of two states does not fit into the transfer matrix paradigm of cumulative sums at all. This ability is the main advantage of the dynamic programming method over the transfer matrix method.
It is possible to use a hybrid algorithm, where one basically uses a transfer matrix technique, but when a factorization is encountered, the simpler state is evaluated using a dynamic programming algorithm, and then becomes a constant multiplier for the more complex state. A constant multiplier is then fine from a transfer matrix perspective, apart from frequently having skipped ahead some steps in the transfer matrix. This skip makes one have to store it and merge it back in when other states have caught up. The advance storage of states plus the extra overhead of dynamic programming storage as well somewhat undoes the smaller memory use advantage of the transfer matrix method. It turned out to look promising but not actually help significantly for 1324 PAPs (unpublished work). Such hybrid algorithms are almost twice as complex to implement as either base algorithm, increasing the likelihood of programming errors.
Probabilistic caching
---------------------
When memory is a greater issue than speed, it is possible to reduce memory use by only caching results probabilistically. That is, after computing $f(S)$, instead of storing $S\rightarrow f(S)$ in some table, only store it with some probability $p$. If $f(S)$ is never needed again (as often happens), then there is no cost in not storing it. If $f(S)$ is needed frequently, then eventually it will be stored, and there will be no subsequent penalty. A smaller $p$ produces more memory savings, but a greater time penalty, allowing some tuning of the algorithm to just fit in the memory available. A high quality random number generator is not needed; one fast and simple approach is to use a simple accumulator. Each time one wants to see if something should be stored, add $p$ to the accumulator. If the result is at least one, subtract one and store the value.
Superficially, this may sound as if it reduces the memory use to a factor of $p$, and increases the time by a factor of $1/p$, but it is not that simple. The total number of calls made will be increased, and so the amount stored will be $p$ times a larger number. So memory is reduced to a factor between $p$ and $1$. Execution time increase is not as bad as may be expected because the number of times each particular state is referenced changes the effect of $p$ on the running time, with large and small numbers both improving matters. It is difficult to theoretically determine this factor, as it depends on the call graph, so empirical results are needed.
Empirical results show that this works surprisingly well and was used in the memory constrained enumeration of 1324 PAPs [@conway20151324PAPs] to get an extra term. Figures \[fig:PC1324\] and \[fig:PCDirAn\] show the empirical effects of $p$ on directed animals and 1324 PAPs[^15]. The patterns are surprisingly similar given that the directed animals have a maximum of two children while the 1324 PAPs can have dozens. A $p$ of $0.3$ gives roughly a 40 percent reduction in memory and takes roughly twice as long in the two cases shown.
This optimization cannot be used in transfer matrix algorithms as the data store is a cumulative sum, rather than a cache.
![Probabilistic caching in 27 step PAPs as a function of the probability of caching a newly computed result. Memory use is roughly proportional to the bottom line; time is roughly proportional to the top line.[]{data-label="fig:PC1324"}](ProbabilisticCaching27Paps1324.pdf){width="\linewidth"}
![Probabilistic caching in 70 site directed animals on the square lattice as a function of the probability of caching a newly computed result. Memory use is roughly proportional to the bottom line; time is roughly proportional to the top line.[]{data-label="fig:PCDirAn"}](ProbabilisticCaching70DirAn.pdf){width="\linewidth"}
Cache inspection
----------------
One is trying to canonicalize states such that two states that can be proven to produce the same result are actually encoded as one state. In the absence of infinite wisdom, a method that can be used to check that nothing obvious has been missed is to inspect the cache after the algorithm has run to see if there are multiple states with the same value. To reduce the chance of coincidences, one can run the algorithm until there are a hundred thousand or so entries in the cache, and look at those entries with associated values over a million. If there are multiple states producing the same large value, an inspection of those states will hopefully inspire a realisation of some equivalence of states, producing a more efficient definition of states.
This has been recently applied by the author to the 1324 PAPs algorithm in [@conway20151324PAPs] to notice that signatures starting with a 1 or a comma allow commutation of some of the ending terms. Proving this then led to other useful realisations leading to a more efficient algorithm (not yet published).
This is easy to do for a dynamic programming algorithm; for a TM algorithm a similar technique can be used, but is less directly associated with state equivalence.
Techniques particular to transfer matrix
========================================
There are also some techniques primarily applicable to the transfer matrix method.
Part of the signature is the iteration number {#sec:TMItNoSig}
---------------------------------------------
For directed animals, the obvious state is the combination of the sites on the boundary eligible for occupancy, and the number of sites left to include. For children there are two main implementation choices:
- One can take a processing step to consider a certain eligible site on the boundary, and have one child if it is occupied, and another if it is not occupied. In the first case the number of sites left to include will have decreased, in the second the number of available sites on the boundary will have reduced. One can define a hierarchy function based primarily upon the number of sites left to include and secondarily on the number of available sites; this hierarchy function will be lower for both child states, so loops are impossible. The call graph for this algorithm is shown in figure \[fig:PartialDirAnTree\].
- One can take a processing step to mean the next used site is chosen. There can be many substates in this case (one for each available site in the boundary). The number of sites left to include is decreased by one each time; it makes an ideal hierarchy function; loops are impossible. The call graph for this algorithm is shown in figure \[fig:PartialDirAnTree2\].
For dynamic programming implementations the two are comparable. The latter is slightly more complex to implement but will be slightly faster and use a little less memory. However for a transfer matrix style implementation they are significantly different. Both algorithms can be implemented with each iteration processing one value of the hierarchy function. For the first algorithm this is somewhat fiddly and has the issue that a child may have a value of the hierarchy function a few steps further on; these need to be tracked. For the second algorithm, each child will go directly into the list to be processed at the next iteration as the hierarchy function is ideal.
In both cases, the number of sites to be included is implicit in the current TM iteration number. Therefore one does not have to store it as part of the signature, saving memory in the states, a large number of which will be stored. This is shown in table \[tab:TMDirAn\] where the third argument of each state is determined by the row number in the table, and therefore does not need to be stored with each state.
This implicit storage of part of the state is obvious for the finite lattice methods where each iteration corresponds to moving the boundary out by one site. The shape of the boundary is identical for each element of a particular iteration, and so does not have to be stored with each state.
Signature Invariants {#sec:siginvariant}
--------------------
Sometimes it is possible to divide the states into groups that have the property that there is no state that is a child of a member of two different groups (or bins). For instance, in the 3D polycubes case, define a group by the occupied/unoccupied status of each site other than the one about to be covered. Adding (or not) a new site may affect connectivity, but it won’t change other sites’ occupancy. These are signature invariants
This division means that instead of processing each signature sequentially, putting all the results in a giant hash map, one can do each group separately, using a small hash map (with better cache locality), and then, when a group is finished, extract the states and associated multiples, store them in some more compact format, and reuse the hash table for the next group. This enables the use of a sparsely filled hash map, improving speed, without the massive memory hit of having a huge sparse hash map. One does have to be very careful with the overhead of small groups and cache clearing.
Processing this way, it is often possible to make the construction of the next set of groups implicit. In the 3D polycube case, it makes sense to assign each group a number whose binary representation contains the occupied status of each site, with the most recent site added as the most significant bit. Then the output from the processing of each group will be in one of two groups, identified by shifting the current groups’ number down one bit and adding a new 0 or 1 as the most significant bit. These can be serialized into two piles based on the most significant bit, and then when all groups are processed, the two piles are concatenated. If the groups started off in order, they will now again be in order, this time for the next set of groups. Since the initial null state is by definition in order, the groups will always be in order.
This can be used to efficiently use disk as storage instead of memory. Current low latency SSDs are too slow to be used as swap space for a giant hash map (2016 unpublished tests), but using them as storage for the serialized group outputs requires many fewer random accesses and is reasonable in some situations.
Of probably greatest importance, this can be very useful on multiprocessor systems, as each group can be assigned to a node, and the node can process that group knowing it will not have to share the hash map with any other node. Examples of such use include 2D self avoiding walks [@conway1996saw; @jensen2013saw], polygons [@jensen2003parallelsap] and polyonimoes [@jensen2003polyonimo].
This sort of construction is often possible for geometric entities on finite lattices; it is less clear for other cases like 1324 PAPs which has proven difficult to parallelize.
Finite lattice techniques
-------------------------
Finite lattice enumerations generally involve enumerating objects on a set of finite lattices individually, and reconstructing the total number of objects from the results on finite lattices.
Finite lattice enumerations are ideal for the TM algorithm as each iteration is well defined (add one more portion, typically site, of the lattice). The techniques described here can be used for DP algorithms as well, but there is rarely any reason to use DP rather than TM for such algorithms.
### Finite lattice symmetries
As mentioned before, when enumerating all the objects on a symmetric lattice of a certain size, the result is often identical to enumerating it on other sizes determined by the lattice symmetry. For instance, enumerating 3D polycubes on a 3\*8\*2 lattice is the same as a 2\*3\*8 lattice (and four others). This reduces the number of finite lattices that need to be computed.
This is not always possible - if one of the sides of the lattice is special (e.g enumeration of paths that are attached to one side of the lattice) or if the dimensions are different (e.g. enumeration of polygons on the square lattice by both horizontal and vertical bonds).
Generally it is better to make the boundary go across the shorter dimension(s), as this makes the states simpler, and probably less numerous, although with good trimming this can be less important than one might expect.
### State reflection
When the iteration is such that the kink does not preclude symmetry (e.g. when finishing a row or a plane), frequently the boundary condition is equivalent to its mirror image. At this point a consolidation can be done, declaring one of these arbitrarily to be the canonical one, and merging the two.
This can in principle halve the number of states to process, although this is somewhat misleading as the number of states will continue to grow up to almost the number it would have been anyway during the subsequent iterations where the kink prevents this consolidation. It also breaks many of the invariants (section \[sec:siginvariant\]).
### Reconstruction from fragments
Some of the objects being enumerated can be split up into fragments and then reconstructed by enumerating those fragments. This can reduce the number of items being enumerated. The fragments being enumerated are often called [*irreducible*]{}.
For instance, a bridge on a lattice $L$ is a path on $L$ that has one end at one side of $L$, and the other at the opposite side of $L$. The bridge is irreducible if there is no plane slicing through L parallel to the attached sides of $L$ that intersects only one bond. For many problems one can reconstruct all bridges from just the reducible bridges by chaining them with single bonds connecting the irreducible bridges. That is, if $b(x)$ is the bridge generating function, and $b_i(x)$ is the irreducible bridge generating function, then $$b(x)=b_i(x)+xb_i(x)^2+x^2b_i(x)^3+\dots=\frac{b_i(x)}{1-xb_i(x)}$$
Enumerating irreducible objects can potentially be faster than enumerating all objects as there are fewer of them, and therefore there will probably be fewer states reached. Also this can be used to reduce the size of the signature as irreducible objects are more compact and fit onto narrower lattices. This was critical to [@conway1993saw] but later advances in understanding of pruning made this less useful.
Associated values
=================
The associated values for dynamic programming usually are integers as that fits in well with the paradigm. The state is then the boundary condition and number of elements yet to be included.
With the transfer matrix, one could do the same thing, although it is often more efficient to have the state be the boundary condition, and the associated value is then an array of integers; the polynomial coefficients of a generating function multiple. This is more efficient as the state only needs to be stored once, and trimming calculations only need to be done once for each state. This is difficult to do with a dynamic programming algorithm as one usually does not know in advance how big the generating function will need to be, which is essential information to pass to the child state evaluator.
The number of non-zero entries in a generating function is usually quite small. This is because most of the states are complex, as there are only small numbers of simple states, and complex states typically take a lot of elements to produce, leaving few spare elements for the series. To take advantage of this, avoid storing zero elements. In the majority of cases, the non-zero elements are consecutive, so that means each generating function can be represented by a start index, a length, and an array of coefficients of that length.
Of course if most generating functions are only length 1, then that is a lot of overhead to store one integer, but the total overhead is generally less storing generating functions, especially if the alternative of storing the number of elements left with the state just makes it have similar extra overhead instead. See figure \[fig:GFLength\] for an example distribution of lengths, and the effect of even imperfect trimming on reducing not just the total number of states, but the size of the associated generating functions as well.
![Number of generating functions of each length encountered midway through the enumerating of up to 16 element polycubes on and spanning the 3 by 5 by 5 lattice. Computations were done with and without using the (imperfect) trimming.[]{data-label="fig:GFLength"}](GFLength.pdf){width="\linewidth"}
Sometimes only even (or odd) terms can be non-zero, in which case only those terms should be stored. This is particularly common with enumeration by bonds.
Multi variable series and moments
---------------------------------
It is straight forward to have more than one variable in the generating function. Suppose instead of counting polycubes just by number of cubes, one also wants to count by surface area (3D perimeter). Let $A_{i,j}$ be the number of polycubes with $i$ cubes and perimeter $j$. Instead of having a 1D array as the generating function associated with each state, one would have a 2D generating function, and update it in the same way. Trimming becomes a little more complex to describe, as does the highest term of the enumeration.
Sometimes however, that can take up too much memory, and having a moment series is useful. The $m$-th moment of surface area series, counting by volume, would then be $$M_m(x) = \sum_{i,j} A_{i,j}x^i j^m$$ Note that $M_0(x)$ is just the normal enumeration by volume (number of cubes).
In a finite lattice computation, when a state is being processed for a site, the new generating function needs to be modified by adding $c$ cubes and surface area $s$. With a 2D generating function, this is straightforward, one just shifts it $c$ in one dimension and $s$ in another dimension. For $M_0(x)$ it is also straightforward, just shift $c$. For other moments it is not quite as obvious, but it turns out to be straightforward. Let $M^*_m$ be the new desired moment, and $M_m$ be the existing moment. Then $$M^*_m(x) = \sum_{i,j} A_{i,j}x^{i+c} (j+s)^m = x^c \sum_{i,j} A_{i,j}x^i \sum_{k=0}^m \binom{m}{k} j^{m-k}s^k = x^c \sum_{k=0}^m \binom{m}{k}s^k M_{m-k}$$ which means if one has existing moments from $0$ to $m$, one can easily compute the new moments from $0$ to $m$.
Given moments, one can compute, say, the mean surface area for size $i$ polycubes by dividing the $i$-th coefficient of $M_1$ by the $i$-th coefficient of $M_0$. Different variables allow different properties of the object to be studied.
This is described in detail in [@conway1995polyonimo] for the 2D polyonimo case. By a similar process one can compute percolation series, where the generating function is or the form $\sum A_{i,j} p^i(1-p)^j$ storing just a single generating function for each state.
Implementation issues
=====================
Generally, an efficient algorithm is more powerful than an efficient implementation. An efficient algorithm can be many orders of magnitude better than a competitor, whereas an efficient implementation is typically only a couple of orders of magnitude better than a simpler, less efficient implementation. With that said, the implementation does matter.
Chinese remainder theorem
-------------------------
The numbers being computed often are larger than the native size of an integer. Many languages have libraries for big integers (data structures representing large integers). However these tend to be slow and memory consuming.
One can use the Chinese remainder theorem to deal with this problem. One does the enumeration with small integers by doing everything modulo some number (typically a prime). Do this for several different, coprime, moduli, and this is enough information to be able to simply regenerate the number modulo the product of the original moduli. This is generally faster, and more memory efficient than using a big integer library. It also has the advantage of offering some redundancy; a concern in long running computations can be an error somewhere in the computer (e.g. from gamma ray strikes). Such an error in one of the moduli will usually produce a very large, noticeable, inconsistency in the results.
This is probably the most commonly used trick in implementation.
Knuth[^16], implementing TM algorithms, faced with the inefficiency of repeating the computations multiple times, separated out the state part from the value part. He made two programs, the first of which would carry around all the state information, and would produce as output a long stream of instructions for another program to execute, to actually do the additions and trimming of generating functions. This is more complex and introduces significant disk IO, but has two advantages:
- The first program only needs to be run once, while the second program can be run for each modulus, rather than recomputing the states for each modulus. This is particularly useful if trimming is computationally expensive and therefore a majority of the time.
- The memory consumption can be slightly lower (in principle up to a factor of 2) than doing both at the same time.
Parallelization
---------------
Multi-processor machines are the norm, and to use a big computer effectively, a parallel algorithm is usually needed. This is usually quite difficult, as distributed hash maps take a large amount of inter-processor communication which can be the bottleneck, even on shared memory systems due to the expense of cache synchronization.
Most parallelized success stories with transfer matrices have used signature invariants (see section \[sec:siginvariant\]). If speed rather than memory is the constraint, then the problem can be partitioned cleanly by different moduli and lattice sizes.
Writing parallelized code is very difficult and error prone. Writing efficient communication code is even more difficult and requires a deep understanding of the architecture used. One can use distributed libraries, although they generally don’t do exactly what one wants and have an associated overhead. There are no clear general answers.
TM algorithms are usually easier to parallelize than DP algorithms, as the distributed hash map lookup of DP is particularly unfriendly to parallel architectures.
Hash map
--------
Both TM and DP require a large amount of storage for a map from state to value. For DP this is the cache; it must be a map as the operations are [*look up value for a state*]{} and [*store calculated value for a state*]{}. For TM the operations are [*iterate over state,value pairs*]{} and [*store value for a state, adding to existing if available*]{}. These are most obviously performed by a map, although the map-reduce framework (such as Apache Hadoop) is also possible. Also if invariants are used, a small hash and long list may be used. But most frequently a map (usually a hash map) is used.
This hash map is usually very large. This makes some of its properties affect the performance of the program significantly. The load factor of the hash map is important - higher values mean a slower program using less memory.
Having a good hash function of the state is very important as the state tends to have many similar values, and a poor hash function can lead to a very large number of collisions.
Most languages come with support for hash maps built into a standard library. However this is generally optimized for convenience and flexibility rather than performance, and a less general purpose one can give order of magnitude improvements particularly to memory but also time.
For instance, in Java, the standard hash map is from one object to another object. It is often possible to encode the state as a bit string and therefore as an integer. Similarly the values are often integers[^17]. When the generic map is used from integers to integers, then each integer has to be wrapped in an object, causing a large overhead. Similarly, many standard library hash maps handle collisions by storing at each entry a list of values that mapped to that entry. This causes another level of wrapping for each entry. There exist much more memory efficient (and fast) hash maps, such as the [*GNU Trove*]{} library which includes multiple versions for each combination of primitives as key or value.
It is frequently useful to have multiple maps instead of one big one. Some function of the key determines which map to use, and some other function of the key is used in that map. Reasons for doing this include:
- Many hash map libraries have limited size hash maps. Many use 32 bit integers in the indexing (and hash values). On current computers this can be a serious limit to the size of the map.
- If one doesn’t set the size of a map in advance, it will typically automatically increase in size when needed. This operation is usually implemented by creating a new data store of larger size and copying values across. During this process both the new and old datastore are in memory, which can exceed available memory. By splitting up the maps, this situation causes less transient memory increase.
- In a multithreaded, shared memory program, it can reduce contention for write locks.
- Some of the key can be implicitly stored in the map selection stage. In [@conway20151324PAPs] the keys for 1324 enumeration were encoded as 128 bit integers. The first 64 bits were used to determine which hash map to use (in a reversible manner). The remaining 64 bits were used as the key in the map. This reduced the memory used for the keys in the maps.
Another way of reducing memory use at the cost of time is by using multiple hash maps with different size values depending upon the actual size of the value. Suppose the values are 64 bit integers. Many of the actual values may fit in a 32 bit integer. This is because a large portion of the states produced will have very restricted paths to endpoints resulting in modest actual values. This means one uses two hash maps, one mapping to 64 bit values and one to 32 bit values. Reads have to check both maps; writes just go to the best fitting map. This is easy for DP; for TM it may require changing which map a value is stored in when another value is added to it. Of course one could get an even greater effect by using 32 bit moduli in the Chinese remainder theorem, but the multiple maps method can achieve much the same effect with a smaller performance hit. Profiling of the bit length in the 1324 PAPs algorithm found it looked useful to have 24 bit, 40 bit, and 64 bit results (see figure \[fig:Bits1324\])
![Bits used in cache for 1324 PAPs, length 20, 25 and 30, as a proportion of the total cache entries. The full answer for length 30 is 78 bits[]{data-label="fig:Bits1324"}](BitLength1324.pdf){width="\linewidth"}
Conclusion
==========
The art of efficient enumeration algorithms is, like most skills, enhanced by knowledge of a host of techniques, only a few of which will be appropriate for any specific problem. This paper has described many such techniques with consistent terminology and a discussion of their relative merits and applicability. This will hopefully help the reader to realise when given tricks can be used. The exercise of writing this paper certainly helped me realise how I could have improved various algorithms I came up with in the past.
In particular, the transfer matrix technique is well known and used for two dimensional lattice enumeration problems where it seems natural, but it is rarely used outside of that domain. However, totally unrelated problems with an ideal hierarchy function can often use the transfer matrix technique instead of the dynamic programming that has been typically used. This can provide a reduction in memory usage and significant advantages for parallelization. This paper has emphasised the similarities of the two techniques and the issues in converting from dynamic programming to transfer matrix.
Each enumeration problem will have its own special requirements and peculiarities, but many tricks can be reused.
Special thanks to Tony Guttmann for introducing me to enumeration, being a fantastic supervisor, and many comments on this manuscript.
[^1]: A well formed bracket(or parenthesis) expression has the same number of open and closing brackets, and there are never more closing brackets than open brackets for any prefix of the expression. So ()() is OK, as is (()), but not ())(. This is a well known (and solved) problem whose series is the Catalan numbers.
[^2]: If one can reach the same state via two different routes [*of different lengths*]{} through $f$, then one will end up with the same state in multiple iterations and the amount of computation will be higher - worse than DP. This can usually be avoided. In particular, if one has a hierarchy function $h$ defined on states, then one can, at each pass, process only the states with the highest hierarchy value.
[^3]: Actually roughly twice this value, as one needs to have two sets of states in memory at once; the set being processed and the set being produced
[^4]: A rectangular subset of the square lattice $\mathbb{Z}^2$
[^5]: The generating function $f(x)$ for the square lattice satisfies $(3x-1)f^2+(3x-1)f+x=0$, the triangular similar.
[^6]: Work in progress.
[^7]: Define $U_L$ to be the total number of objects on a lattice $L$ (or a generating function of them), and $C_L$ to be a canonicalized version only counting objects that fit into $L$ but not any smaller lattice. Then $C_L$ can be generally computed as $U_L$ minus some multiples of $C_l$ for lattices $l$ smaller than $L$. For the zero size lattice, typically $C_l=0$. Then by induction given all $U_l$ for $l$ a subset of $L$, one can compute $C_l$. One can then add up all $C_l$ to get the number of objects that fit on lattice $L$ without multiple counting. In practice it is usually simpler than this due to constraints on touching boundaries
[^8]: Not including the two edges dealt with in the prior paragraph
[^9]: By symmetry, one can always have $z$ be the biggest dimension. So polyonomoes can be required to reach to a certain minimum $z$, being the maximum of the width and height
[^10]: In practice this was done by computing $tritable(a,b,c)$ and then finding for each remaining color the minimum value adjacent to it, and taking the maximum of these values
[^11]: The exception is when an edge site is in the kink
[^12]: Exception: sometimes the connection cost can be done in the row below the kink at the same cost as the row above the kink.
[^13]: With a 1 discount for the case of the previous footnote, when one needs to get to the bottom and the kink is up to the last or second last row.
[^14]: There is another similar but not quite as effective optimization used... if $S=S_1(S_2)S_3$ then $f(S)=\sum_i m_i f(s_i S_2)$ where integers $m_i$ and prefix states $s_i$ are functions of $S_1$ and $S_3$ but not $S_2$.
[^15]: The actual time and memory use for the 1324 PAPs is a little more complex due to factorizations.
[^16]: Public talk
[^17]: Integer is used as a generic name for a natively handled data type. It may be called long or something other than integer.
|
---
abstract: 'We report twelve new transit observations of the exoplanet WASP-4b from the Transit Monitoring in the South Project (*TraMoS*) project. These transits are combined with all previously published transit data for this planet to provide an improved radius measurement of $R_p= 1.395 \pm 0.022~R_{jup}$ and improved transit ephemerides. In a new homogeneous analysis in search for Transit Timing Variations (TTVs) we find no evidence of those with [*RMS*]{} amplitudes larger than [$20$ seconds]{} over a 4-year time span. This lack of TTVs rules out the presence of additional planets in the system with masses larger than about $2.5 ~M_{\earth}$, $2.0~M_{\earth}$ and $1.0~M_{\earth}$ around the 1:2, 5:3 and 2:1 orbital resonances. Our search for the variation of other parameters, such as orbital inclination and transit depth also yields negative results over the total time span of the transit observations. Finally we perform a simple study of stellar spots configurations of the system and conclude that the star rotational period is about 34 days.'
author:
- |
S. Hoyer$^{1,5,6}$, M. López-Morales$^{2}$, P. Rojo$^1$, V. Nascimbeni$^{3,4}$, S. Hidalgo$^{5,6}$, N. Astudillo-Defru$^{1,7}$, F. Concha$^1$, Y. Contreras$^1$, E. Servajean$^1$, T.C. Hinse$^{8,9}$.\
$^1$ Astronomy Department, Universidad de Chile, Casilla 36-D, Santiago de Chile, Chile.\
$^2$ Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA.\
$^3$ Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy.\
$^4$ INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy.\
$^5$ Instituto de Astrofísica de Canarias, Via Láctea s/n, E38200 La Laguna, Tenerife, Canary Islands, Spain.\
$^6$ Department of Astrophysics, University of La Laguna, Via Láctea s/n, E38200 La Laguna, Tenerife, Canary Islands, Spain.\
$^{7}$UJF-Grenoble 1/CNRS-INSU, Institut de Planétologie et d’Astrophysique de Grenoble (IPAG) UMR 5274, Grenoble, F-38041, France\
$^8$ Korea Astronomy and Space Science Institute, 776 Daedukdae-ro, Yuseong-gu, 305-348 Daejeon, Republic of Korea.\
$^9$ Armagh Observatory, College Hill, BT61 9DG, United Kingdom.
bibliography:
- 'refs.bib'
title: 'TraMoS project III: Improved physical parameters, timing analysis, and star-spot modelling of the WASP-4b exoplanet system from 38 transit observations.'
---
exoplanets: general – transiting exoplanets: individual(WASP-4b)
Introduction
============
Since the discovery of the first extrasolar planet around the Sun-like star 51-Peg via radial velocities [@mayorandqueloz95], a number of systematic extrasolar planet searches have spread adopting a wide variety of techniques, of which the radial velocity method is still the most prolific approach. The transit technique is currently the second most successful, with the detection of over 290 systems with confirmed planet detection[^1]. Of these, most are Hot-Jupiters (Jupiter-mass objects with orbital periods of a few days). Just recently, space missions such as Kepler and Corot have started to expand transit discoveries to planets smaller than $50~M_{\earth}$.
Transiting planets provide a wealth of information about their systems. For instance, transits are currently the only tool to measure the planet-to-star radius ratio and orbital inclination. Combined with radial velocity results, those parameters allow the determination of the absolute mass of the planet and its mean density. Another type of study that can be conducted via transiting planets is the search of [*unseen*]{} companions in the system. Those companions introduce variations in the orbital period of the transiting planet [@miralda02; @Agol05; @holman05], which can be detected by monitoring the systems in search for Transit Timing Variations (TTVs). This TTV technique has the potential of finding planets in the Earth-mass regime and even exomoons [@Kipping09a].
In addition, [@Silva-Valio2008] and [@Sanchis-Ojeda2011] have pointed out that observations of star-spot occultations during closely-spaced transits can be used to not only estimate the rotation period of the host star, but also to measure alignment differences between the rotation axis of the star and the orbital axis of the planet.
As part of the Transit Monitoring in the South (*TraMoS*) project [@Hoyer-ogletr111-2011; @Hoyer-wasp5-2012], we are conducting a photometric monitoring survey of transits observable from the Southern Hemisphere. The aim of this project is to perform a careful and homogeneous monitoring of exoplanet transits trying to minimize systematics and reduce uncertainties in the transit parameters, such as the transit mid-time, following the approach of using high-cadence observations and the same instruments and setups.
In the framework of the *TraMoS* project we present twelve new transit observations of the exoplanet WASP-4b. This was the first exoplanet detected by the WASP-South survey in 2008. A that time, @Wilson08 [hereafter W08] reported a Hot-Jupiter ($P=1.34$ days) with a mass of $M_{P}=1.22^{+0.09}_{-0.08}~M_J$ and a planetary radius of $R_{p}=1.42^{+0.07}_{-0.04} ~R_{J}$ orbiting a G7V southern star. This discovery paper included WASP photometry, two additional transit epochs (observed in September 2007), and radial velocities measurements.
@Gillon.WASP4WASP5.2009 [hereafter G09], added to the follow-up of this exoplanet a *VLT/FORS2* light curve observed in October 2007 with a *z-GUNN* filter. Using a reanalysis of the W08 data they found no evidence of period variability. @Winn.WASP4.2009 [hereafter W09], presented two new high-quality transits observed in 2008 with the Baade Telescope (one of the twin 6.5-m Magellan telescopes at Las Campanas Observatory) using a *z-band* filter. Four new transit epochs were reported shortly after by @Southworth.WASP4.2009 [hereafter S09], with the 1.54-m Danish Telescope at La Silla Observatory using a *Cousins-R* filter during 2008. @Sanchis-Ojeda2011 [hereafter S011], using four new transit light curves observed during 2009, interpreted two anomalies in the photometry as starspot occultations by the planet and concluded from that result that the stellar rotation axis is nearly aligned with the planet’s orbital axis. This result agrees with the observations of the Rossiter-McLaughlin effect for this system by [@Triaud2010]. Later, two new transits of WASP-4b were reported by @Dragomir11 [hereafter D11], with data from the 1-m telescope at Cerro Tololo Inter-American Observatory (*V-band* and *R-band* filter). Most recently, Nikolov et al. (2012, hereafter N12) observed three transits simultaneously in the [ *Sloan g’, r’, i’ and z’*]{} filters with the Gamma Ray burst Optical and Near-infrared Detector (GROND) at the MPG/ESO-2.2 m telescope at La Silla Observatory.
In this work we present twelve new transits observations. We combined these new light curves with all the previously available light curves (twenty-six additional light curves) and reanalyzed them to provide a new homogeneous timing analysis of the transits of WASP-4b and to place stronger constraints to the mass of potential perturbers in the main orbital resonances with this planet. We also search the entire dataset for signs of stellar spots that would help improve the conclusions of S011.
In Section \[observaciones\] and \[reduccion\] we describe the new observations and the data reduction. Section \[fiteo\] details the modelling of the light curves and in Section \[ttv\] we present the timing analysis and discuss the mass limits for *unseen* perturbers. In section \[spots\] we discuss the occultations of stellar spots by the planet. Finally, we present our conclusions in Section \[conclusiones\].
Observations {#observaciones}
============
The instruments
---------------
As mentioned before, the *TraMoS* project has undertaken a photometric campaign to follow-up transiting planets observable from the Southern Hemisphere. Our goal is to use high cadence observations minimizing change of instruments to reduce systematics and therefore, based on an homogeneous analysis, obtain the most precise values of the light curve parameters, such as the central time of the transit, the orbital inclination and the planet radius, among others.
The observations we present in this work were performed with the Y4KCam on the SMARTS 1-m Telecope at Cerro Tololo Inter-American Observatory (CTIO) and with the SOAR Optical Imager (SOI) at the 4.2-m Southern Astrophysical Research (SOAR) telescope in Cerro Pach[ón]{}. The epochs of four of the transits we present here coincide with previous published data (see Table \[tabla-obs\]).
We have taken advantage of the $20\times20$ squared arcminute of Field of View (FoV) of the Y4KCam, which is a $4064\times4064$ CCD camera with a pixel scale of 0.289 arcsec pixel$^{-1}$, which despite its large dimensions allows to use a readout time of only $\sim16/5$ sec when using the 2x2/4x4 binning mode (compared with the $46$ sec of the unbinned readout time). The SOI detector is composed of two E2V mosaics of $4096\times2048$ pixels with a scale of 0.077 arcsec pixel$^{-1}$. The SOI has a FoV of $5.2\times5.2$ squared arcminutes and allows a readout time of only $\sim11$ sec after binning 2x2 ($20.6$ sec is its standard readout time).
As part of the *TraMoS* project we have observed a total of 12 transits of WASP-4b, between August 2008 and September 2011[^2]. Three transits were observed with the SOI at the SOAR telescope and the remaining nine were observed with the Y4KCAM at the 1-m CTIO telescope. The first ten transits were observed using a *Bessell I* or *Cousins I* filter. For the 2011 transit observations we use the 4x4 binning mode of the Y4KCam and a *Cousins R* filter. The observing log is summarized in Table \[tabla-obs\]. All the transits were fully covered by our observations except for some portions of the $E=754$, $482$ and $-62$ transits that were lost due to technical failures (see Figure \[lc.miscurvas\]). The before-transit and ingress portions of the $E=-71$ transit were not observed, but after $phase=-0.034$ (where $phase=0$ is defined as the phase of the mid-transit) the observation of the transit was done without interruption.
[lccccccc]{} UT date & Epoch$^{b}$ & Telescope/Instrument & Filter & Binning & Average & Detrending &$K_{\textit{RMS}}$\
& & & & & exptime (sec) & &\
2008-08-23$^{a}$ & -91& 1-mt SMART/Y4KCam & Cousins I & 2x2 & 20 & 3 & 0.78\
2008-08-23$^{a}$ & -91& SOAR/SOI & Bessell I & 2x2 & 7 & – & –\
2008-08-27 & -88& 1-mt SMART/Y4KCam & Cousins I & 2x2 & 20 & 1+3 & 0.90\
2008-09-19 & -71& 1-mt SMART/Y4KCam & Cousins I & 2x2 & 20 & – & –\
2008-09-23$^{a}$ & -68& 1-mt SMART/Y4KCam & Cousins I & 2x2 & 20 & – & –\
2008-10-01$^{a}$ & -62& 1-mt SMART/Y4KCam & Cousins I & 2x2 & 26 & –& –\
2009-07-29 & 163& SOAR/SOI & Bessell I & 2x2 & 8 & 2 & 0.91\
2009-09-22 & 204& SOAR/SOI & Bessell I & 2x2 & 10 & 2 & 0.87\
2009-10-28 & 231& 1-mt SMART/Y4KCam & Bessell I & 2x2 & 14 & – & –\
2010-09-29 & 482& 1-mt SMART/Y4KCam & Cousins I & 2x2 & 14 & 1+3 & 0.96\
2011-09-24 & 751& 1-mt SMART/Y4KCam & Cousins R & 4x4 & 20 & 1+3 & 0.95\
2011-09-28 & 754& 1-mt SMART/Y4KCam & Cousins R & 4x4 & 20 & – & –\
\
\
Data Reduction {#reduccion}
==============
The trimming, bias and flatfield correction of all the collected data was performed using custom-made pipelines specifically developed for each instrument.
The *Modified Julian Day* (JD-2400000.5) value was recorded at the start of each exposure in the image headers. In the SMARTS telescope, the time stamp recorded in the header of each frame is generated by a *IRIG-B GPS time synchronization protocol* connected to the computers that control the instrument. The SOAR telescope data use the time values provided by a time service connected to the instrument. We confirmed that these times coincide with the time of the Universal Time (UT) reference clock within 1 second. The time stamp assigned to each frame corresponds to the *Julian Day* at the start of the exposure plus 1/2 of the integration time of each image.
For the photometry and light curve generation we used the same procedure described in [@Hoyer-wasp5-2012]. That is, we performed a standard aperture photometry where the optimal sky-aperture combination was chosen based on the *RMS* minimization of the differential light curves of the target and the reference stars. For this analysis we excluded the ingress and egress portions of the light curves, i.e. we used only the out-of transit and in-transit data. The final light curves were generated computing the ratio between the target flux and the average flux of the best 1 to 3 comparison stars.
In order to detrend the light curves, we searched for correlations of the out-of-transit flux ($F_{OOT}$) with the X-Y CCD coordinates of the target, airmass and/or time. Trends are identified as such when the correlation coefficient values are larger than the *RMS* of the out-of-transit points. To remove the trends, we applied linear or quadratic regressions fits, which were subtracted from the light curve only when we detect that [*RMS*]{} of the $F_{OOT}$ is improved. Otherwise no subtraction is applied. The [ *RMS*]{} after detrending are $3-22\%$ lower than on the raw light curves. A posteriori we have checked that doing this detrending previous to modelling our light curves does not introduce any bias or differences in the estimated parameters and its uncertainties. In the last two columns of Table \[tabla-obs\] we show which detrending was applied and the value $K_{{\it RMS}}$, which corresponds to the ratio of the $F_{OOT}$ [*RMS*]{} after and before removing the trends to illustrate the improvement in the [*RMS*]{} by the detrending. Therefore, the values of $K_{RMS}$ indicate how much the dispersion of the light curves changes after detrending. When no numerical values of $K_{{\it RMS}}$ is shown means that no significant systematic trend with any of the parameters searched was found. Also, to illustrate the amount of detrending applied for each light curve in left panels of Figure \[rawvsdetrended\] we show the detrended over the raw light curves and in the right-side of the same figure we also show the differences in flux (magnified by 100 times) between the light curves before and after detrending.
Light Curve Modelling {#fiteo}
=====================
In a first step we performed a modelling of the 12 new light curves we present in this work and the 26 available light curves of the WASP-4 system from other authors. The goal of this first analysis is to determine transit parameters single values from each light curve. With these values we search for variations in 4 years time span in parameters such as the central time of the transits, the orbital inclination and/or the transit’s depth. Variations in these parameters can be indicative of the presence of a perturbing body in the system. In a second step, we take advantage of the information contained in all the light curves by fitting simultaneously the transit parameters. With this global analysis we determined the properties of the system.
Individual Analysis: searching for parameter variations {#individual}
-------------------------------------------------------
We used the TAP package [@TAP] to fit all the available light curves of WASP-4. This package allows to fit the analytical models of [@MandelAgol02] based on the Markov Chain Monte Carlo (MCMC) method which have proved to give the most reliable results compared with other approaches, particularly in the case of uncertainty estimations of the fitted parameters [see @Hoyer-wasp5-2012 section 3.1 and 3.2]. This point is critical especially for the timing analysis of the transits. TAP also implements a wavelet-based method [@Carter09] to account for the red-noise in the light curve fitting. This method helps impose more conservative uncertainty estimates compared to other red noise calculations such as the ‘*time averaging*’ or ‘*residual permutation*’ methods, in cases where the noise has a power spectral density (PSD) that varies as $1/f^{\gamma}$ (where $f$ is the frequency and $\gamma$ is a spectral index $> 0$). For all other types of noise the [@Carter09] method gives uncertainty estimations as good as the other methods [see e.g. @Hoyer-ogletr111-2011; @Hoyer-wasp5-2012].
We fitted the twelve new transit light curves presented in this work and the other 26 available light curves from W08, G09, W09, S09, D11, SO11 and N12[^3]. To perform the modelling we grouped all the light curves observed with the same filter in a given telescope (we treated all our light curves as observed with the same filter/telescope). As we described below, this allows us to fit for the linear limb-darkening coefficient ($\mu_1$) using all the light curves simultaneously as was done in [@Hoyer-wasp5-2012]. Therefore, the first group is composed by our ten *I-band* light curves (from 2008 to 2010) and the second by our two *Cousins-R* light curves (the 2011’s transits). A third group is formed by the six SO11 *z-band* light curves (which includes the two transits observed by W09) and the S09 light curves form the fourth group (*Cousins-R*). We fit the [*V*]{}- and [ *R*]{}-light curves of D11 separately. Finally, we fit the twelve light curves of N12 in four groups (three light curves for each of the four filters).
We fit each group independently, leaving as free parameters for each light curve the orbital inclination ($i$), the planet-to-star radius ratio ($R_p/R_s$) and the central time of the transit ($T_c$). The orbital period, the eccentricity, and the longitude of the periastron were fixed to the values $P= 1.33823326 ~days$ (from D11), $e=0$ and $\omega = 0$. We searched for possible linear trend residuals in the light curves by fitting for the out-of-transit flux ($F_{OOT}$) and for a flux slope ($F_{slope}$), but did not find any. The ratio between the semi-major axis and the star radius, $a/R_{s}$, usually presents strong correlations with $i$ and $R_p/R_s$. To break those correlations and their effects in our results (and because we are searching for relative variations in $i$ and $R_p/R_s$), we fixed $a/R_s$ in all the light curves to 5.53 (from D11). We checked that doing this we are not introducing any bias/effect on the rest of the parameters fitted and its errors (for details, see @Hoyer-wasp5-2012). In particular we are not affecting the mid-times of the transits due to its weak correlation with $a/R_s$. We also fit for a white and red noise parameter, $\sigma_w$ and $\sigma_r$, respectively, as defined by [@Carter09].
The coefficients of a quadratic limb-darkening law ($\mu_1$ and $\mu_2$) in our light curves are strongly correlated, therefore we fixed $\mu_2$ to 0.32 (based in the tabulated values by @Claret2000) and let $\mu_1$ as free parameter on each group of light curves.
Therefore, for each light curve fit we obtained a value of $i$, $R_p/R_s$, $\sigma_w$, $\sigma_r$ and $T_c$ and a single value of $\mu_1$ for each group.
To fit the transits and derive errorbars for all parameters we ran 10 MCMC chains of $10^5$ links each, discarding the first 10% of the results from each chain to minimize any bias introduced by the parameter initial values. The resulting values for each parameter of the thirty-eight light curves together with their 1$\sigma$ errors are shown in Table \[tabla-parametros-mias\] and \[tabla-parametros-otras\]. The data and the best models fits for all twelve light curves presented in this work are illustrated in Figure \[lc.miscurvas\]. The same is shown in Figure \[lc.otrascurvas\] for the twenty-six light curves from the literature.
Variations in transit parameters, in particular in $i$ and $R_p/R_s$, can be attributed to the perturbations produced by an additional body in the system. In Figure \[parametros\], we plot $R_p/R_s$ and $i$ as a function of the transit epoch, based in the results of the 38 transit fits. We do not find any significant variations in those parameters. As reference, in Figure \[parametros\] the weighted average values and the $\pm1\sigma$ errors of $i$ and $R_p/R_s$ based on all the light curves results are represented by the solid and dashed horizontal lines, respectively. Our analysis of the transit timing is described in detail in section \[ttv\].
Is worthy to noticing that including the detrending parameters in the MCMC during the modelling is the most appropriate approach to estimate uncertainties of the transit parameters. We have checked that in our data there is no significant differences by doing the detrending before the MCMC fitting. The TAP version we use (v2.104) does not allow to detrend against airmass or X-Y position of the centroid or neither perform quadratic regression fits of the flux against the time, that is why we decided to check for correlations against these parameters before modelling with TAP. Also, most of the literature light curves we used to search for variations in transit parameters have been already detrended and therefore the uncertainties we estimate from them can be compared with those obtained from our data when we detrended it before the MCMC modelling.
[lccccccccc]{} Epoch & Filter &$T_{c}-2450000$ & $i$ & $R_{p}/R_{s}$ & $\mu_{1}$ & $\sigma_{red}^{a}$& $\sigma_{white}^{a}$ & [*RMS*]{} & Spot\
& & ($BJD_{TT}$) & $(\degr)$ & & & & & (residuals) & Detection$^{b}$\
-91 & I & $4701.81280^{+0.00022}_{-0.00023}$ & $89.52^{+0.33}_{-0.44}$ & $0.1558^{+0.0012}_{-0.0012}$ & $0.217 ^{+0.019}_{-0.020}$& $0.0066$ & $0.0017$& $0.0023$ & $\surd$\
&&&&&&\
-91 & ” & $4701.81303^{+0.00018}_{-0.00019}$ & $89.53^{+0.33}_{-0.44}$ & $0.1575^{+0.0016}_{-0.0016}$ & ”& $0.0101$ & $0.0012$ & $0.0015$ & $\surd$\
&&&&&&\
-88 & ” & $4705.82715^{+0.00029}_{-0.00030}$ & $89.29^{+0.48}_{-0.59}$ & $0.1506^{+0.0016}_{-0.0017}$ & ”& $0.0066$ & $0.0022$ & $0.0024$ & $\cdots$\
&&&&&&\
-71 & ” & $4728.57767^{+0.00043}_{-0.00042}$ & $89.18^{+0.57}_{-0.75}$ & $0.1497^{+0.0013}_{-0.0013}$ &” & $0.0046$ & $0.0021$ & $0.0023$ & $\cdots$\
&&&&&&\
-68 & ” & $4732.59197^{+0.00050}_{-0.00051}$ & $88.39^{+0.96}_{-0.76}$ & $0.1510^{+0.0031}_{-0.0030}$ & ”& $0.0165$ & $0.0034$ & $0.0049$ & $\cdots$\
&&&&&&\
-62 & ” & $4740.62125^{+0.00036}_{-0.00035}$ & $88.64^{+0.82}_{-0.63}$ & $0.1545^{+0.0018}_{-0.0017}$ &” & $0.0058$ & $0.0019$ & $0.0021$ & $\cdots$\
&&&&&&\
163 & ” & $5041.72377^{+0.00019}_{-0.00018}$ & $89.59^{+0.29}_{-0.40}$ & $0.1525^{+0.0014}_{-0.0014}$ &” & $0.0095$ & $0.0012$ & $0.0014$ & $\surd$\
&&&&&&\
204 & ” & $5096.59148^{+0.00023}_{-0.00022}$ & $89.53^{+0.32}_{-0.45}$ & $0.1533^{+0.0016}_{-0.0016}$ & ”& $0.0108$ & $0.0022$ & $0.0013$ & $\surd$\
&&&&&&\
231 & ” & $5132.72310^{+0.00041}_{-0.00041}$ & $89.25^{+0.51}_{-0.66}$ & $0.1569^{+0.0018}_{-0.0019}$ & ”& $0.0082$ & $0.0021$ & $0.0034$ & $\cdots$\
&&&&&&\
482 & ” & $5468.61943^{+0.00046}_{-0.00046}$ & $89.26^{+0.52}_{-0.70}$ & $0.1562^{+0.0023}_{-0.0025}$ & ”& $0.0166$ & $0.0034$ & $0.0033$ & $\cdots$\
&&&&&&\
751 & R & $5828.60375^{+0.00042}_{-0.00041}$ & $88.85^{+0.75}_{-0.78}$ & $0.1489^{+0.0028}_{-0.0029}$ & $0.212^{+0.066}_{-0.067}$& $0.0167$ & $0.0021$ & $0.0033$ & $\cdots$\
&&&&&&\
754 & ” & $5832.61815^{+0.00041}_{-0.00042}$ & $88.96^{+0.69}_{-0.76}$ & $0.1546^{+0.0024}_{-0.0023}$ & ”& $0.0141$ & $0.0023$ & $0.0033$ & $\cdots$\
\
\
[lccccccccc]{}
Epoch & Author$^{a}$, &$T_{c}-2450000$ & $i$ & $R_{p}/R_{s}$ & $\mu_{1}$ & $\sigma_{red}^{b}$& $\sigma_{white}^{b}$ & [*RMS*]{} & Spot\
& Filter & ($BJD_{TT}$) & $(\degr)$ & & & & & (residuals) & Detection$^{c}$\
-340 & W08, [*R*]{} &$4368.59279^{+0.00033}_{-0.00032}$ & $89.19^{+0.57}_{-0.73}$ & $0.1507^{+0.0020}_{-0.0020}$ & $0.219^{+0.073}_{-0.078}$ & $0.0036$ & $0.0015$ & $0.0016$ & $\cdots$\
-319 & G09, [*z*]{} &$4396.69576^{+0.00012}_{-0.00012}$ & $88.29^{+0.47}_{-0.49}$ & $0.15279^{+0.00094}_{-0.00085}$ & $0.253^{+0.030}_{-0.037}$ & $0.0027$ & $0.0005$ & $0.0006$ & $X$\
-94 & S09, [*R*]{} & $4697.79788^{+0.00013}_{-0.00013}$ & $89.53^{+0.32}_{-0.41}$ & $0.15533^{+0.00072}_{-0.00072}$ & $0.333^{+0.017}_{-0.017}$& $0.0010$ & $0.0006$ & $0.0008$ & $X$\
&&&&&&\
-91 & ” & $4701.81234^{+0.00026}_{-0.00026}$ & $89.41^{+0.41}_{-0.53}$ & $0.1534^{+0.0016}_{-0.0015}$ &” & $0.0037$ & $0.0007$ & $0.0008$ & $\surd$\
&&&&&&\
-68 & ” & $4732.59188^{+0.00027}_{-0.00027}$ & $89.51^{+0.34}_{-0.49}$ & $0.1532^{+0.0017}_{-0.0020}$ & ”& $0.0025$ & $0.0005$ & $0.0007$ & $\surd$\
&&&&&&\
-62 & ” & $4740.62118^{+0.00016}_{-0.00016}$ & $89.59^{+0.29}_{-0.41}$ & $0.1530^{+0.0010}_{-0.0011}$ & ”& $0.0020$ & $0.0005$ & $0.0006$ & $X$\
-94& W09, [*z*]{} &$4697.79817^{+0.00008}_{-0.00009}$ & $89.73^{+0.19}_{-0.28}$ & $0.15560^{+0.00077}_{-0.00079}$ & $0.2027^{+0.0076}_{-0.0076}$ & $0.0027$ & $0.0005$ & $0.0007$& $\cdots$\
&&&&&&\
-56& ” & $4748.65111^{+0.00007}_{-0.00007}$ & $89.72^{+0.20}_{-0.28}$ & $0.15369^{+0.00057}_{-0.00058}$ &” & $0.0024$ & $0.0003$ & $0.0005$ & $X$\
-53& D11, [*V*]{} &$4752.66576^{+0.00067}_{-0.00069}$ & $87.4^{+1.6}_{-1.1}$ & $0.1562^{+0.0053}_{-0.0059}$ & $0.50^{+0.18}_{-0.16}$& $0.0104$ & $0.0022$ & $0.0029$ & $\cdots$\
&&&&&&\
196& D11, [*R*]{}&$5085.88418^{+0.00084}_{-0.00086}$ & $88.6^{+0.79}_{-0.99}$ & $0.1418^{+0.0092}_{-0.0099}$ & $0.42^{+0.20}_{-0.19}$ & $0.0132$ & $0.0012$ & $0.0026$ & $\cdots$\
166 & SO11, [*z*]{} &$5045.73853^{+0.00008}_{-0.00008}$ & $89.8^{+0.14}_{-0.22}$ & $0.15441^{+0.00053}_{-0.00055}$ & $0.2027^{+0.0076}_{-0.0076}$ & $0.0023$ & $0.0004$ & $0.0005$ & $X$\
&&&&&&\
169 & ” & $5049.75325^{+0.00007}_{-0.00007}$ & $89.65^{+0.24}_{-0.29}$ & $0.15347^{+0.00049}_{-0.00047}$ & ”& $0.0018$ & $0.0004$ & $0.0005$ & $X$\
&&&&&&\
172 &” &$5053.76774^{+0.00009}_{-0.00009}$ & $89.72^{+0.19}_{-0.28}$ & $0.15346^{+0.00058}_{-0.00058}$ & ” & $0.0026$ & $0.0004$ & $0.0005$ & $\surd$\
&&&&&&\
207 & ”&$5100.60595^{+0.00012}_{-0.00012}$ & $89.66^{+0.23}_{-0.32}$ & $0.15318^{+0.00086}_{-0.00087}$ & ” & $0.0043$ & $0.0004$ & $0.0007$ & $\surd$\
184 & N12, [*g’*]{} & $5069.82676^{+0.00031}_{-0.00030}$ & $89.22^{+0.54}_{-0.64}$ & $0.1550^{+0.0019}_{-0.0020}$ & $0.598^{+0.029}_{-0.031}$ & $0.0061$ & $0.0020$ & $0.0026$ & $\cdots$\
&&&&&&\
187 & ” &$5073.84108^{+0.00028}_{-0.00029}$ & $89.34^{+0.47}_{-0.57}$ & $0.1548^{+0.0020}_{-0.0020}$ & ” & $0.0056$ & $0.0017$ & $0.0023 $ & $\cdots$\
&&&&&&\
216& ” & $5112.65009^{+0.00032}_{-0.00033}$ & $89.13^{+0.58}_{-0.68}$ & $0.1593^{+0.0020}_{-0.0020}$ & ” & $0.0001$ & $0.0015$ & $0.0018 $ & $\cdots$\
184& N12, [*i’*]{} & $5069.82617^{+0.00038}_{-0.00038}$ & $89.42^{+0.41}_{-0.60}$ & $0.1574^{+0.0026}_{-0.0029}$ & $0.238^{+0.037}_{-0.040}$ & $0.0$ & $0.0021$ & $0.0028$ & $\cdots$\
&&&&&&\
187&” & $5073.84128^{+0.00025}_{-0.00026}$ & $89.52^{+0.34}_{-0.58}$ & $0.1562^{+0.0019}_{-0.0019}$ & ” & $0.0$ & $0.0018$ & $0.0025$ & $\cdots$\
&&&&&&\
216&” & $5112.65005^{+0.00048}_{-0.00049}$ & $89.25^{+0.52}_{-0.70}$ & $0.1550^{+0.0034}_{-0.0033}$ & ” & $0.0020$ & $0.0008$ & $0.0019$ & $\cdots$\
184 & N12, [*z’*]{} & $5069.82670^{+0.00028}_{-0.00027}$ & $89.41^{+0.42}_{-0.55}$ & $0.1580^{+0.0018}_{-0.0018}$ & $0.218^{+0.034}_{-0.035}$ & $0.0$ & $0.0023 $ & $0.0029$ & $\cdots$\
&&&&&&\
187 & ”&$5073.84111^{+0.00023}_{-0.00023}$ & $89.25^{+0.51}_{-0.58}$ & $0.1576^{+0.0017}_{-0.0017}$ & ” & $0.0$ & $0.0016$ & $0.0022$ & $\cdots$\
&&&&&&\
216 & ” &$5112.64986^{+0.00036}_{-0.00039}$ & $89.18^{+0.56}_{-0.68}$ & $0.1583^{+0.0025}_{-0.0025}$ & ” & $0.0$ & $0.0016$ & $0.0020$ & $\cdots$\
184 & N12, [*r’*]{} & $5069.82661^{+0.00029}_{-0.00029}$ & $89.34^{+0.46}_{-0.59}$ & $0.1564^{+0.0019}_{-0.0020}$ & $0.390^{+0.025}_{-0.027}$ & $0.0003$ & $0.0019$ & $0.0025$ & $\cdots$\
&&&&&&\
187 & ”&$5073.84114^{+0.00018}_{-0.00018}$ & $89.56^{+0.30}_{-0.44}$ & $0.1543^{+0.0013}_{-0.0014}$ & ” & $0.0002$ & $0.0011$ & $0.0017$ & $\cdots$\
&&&&&&\
216 & ” & $5112.65005^{+0.00031}_{-0.00031}$ & $89.20^{+0.55}_{-0.67}$ & $0.1587^{+0.0019}_{-0.0019}$ & ” & $0.0$ & $0.0014$ & $0.0018$ & $\cdots$\
\
\
\
\
Global Analysis {#global}
---------------
Since we find no evidence of significant variations in the parameters fitted for each individual light curve, we can model all light curves simultaneously to improve the determination of $i$, $R_p/R_s$ and $a/R_s$. For this, we fitted simulaneausly these parameters in the 38 light curves, while letting to vary on each light curve the transit mid-times, $F_{slope}$, $F_{OOT}$, $\sigma_{W}$ and $\sigma_{red}$. We fixed *P* and the linear and quadratic limb-darkening coefficients for each filter to the values obtained in the previous section. We used 10 chains of $10^{5}$ links each in the MCMC. The resulting values of the MCMC analysis for the simultaneuosly fitted parameters are shown in Table \[tabla-final\] and the resulting distributions in Figure \[histogramas\]. Using the resulting value for $R_p/R_s$ and the value of the star radius ($R_S = 0.907 \pm
0.014~R_{\sun}$) derived by SO11, we obtained an improved radius measurement for the planet of $R_p= 1.395 \pm 0.022~R_{jup}$, which is consistent with the most recent radius estimation reported by SO11 and N12.
Timing analysis and limits to additional planets {#ttv}
================================================
The times of our twelve transits and the D11 transits were initially computed in Coordinated Universal Time (UTC) and then converted to Barycentric Julian Days, expressed in Terrestrial Time, BJD(TT), using the [@Eastman2010] online calculator[^4]. The time stamps of the S09 light curves were initially expressed in HJD(UT) and have also been converted to BJD(TT). The same was done with the transit midtimes obtained from W08 and G09 light curves. No conversion was necessary for the light curves reported by SO11 (which includes the two transits of W09) and the ones reported by N12, since they are already expressed in BJD(TT). Finally, we did not include the $T_c$ derived from the 2006 and 2007 WASP data since that value is based on the folded transits of the entire WASP observational seasons, and therefore lacks the precision required for our timing analysis.
We used the D11 ephemeris equation to calculate the residuals of the mid-times of the 38 transits of WASP-4b analyzed in this work. Panel A in Figure \[o-c\] shows the $Observed$ $minus$ $Calculated$ ($O-C$) diagram of the times for our twelve transits. In panel B we combined the $O-C$ values of these twelve transits with the new values derived for the W08, G09, W09, S09, SO11, D11 and N12 (shown as open circles). A linear trend is evident in the residuals of all the transits (represented by a dashed line in panel B). That trend is caused by the accumulated error over time in the transit ephemerides. Once those ephemerides are updated, that trend is removed (panel C), and the RMS of the transit times residuals is only of $29$ seconds. The [*reduced-chi-squared*]{} of the linear regression is $\chi^2_{red}=1.25$ ($\chi^2=45$ for 36 degrees of freedom). If we removed the D11’s transit with the largest uncertainties (E=196, which corresponds to an incomplete transit observation) and re-calculated the linear trend, the RMS of the residuals is only $20$ seconds after updated the ephemerides ($\chi^2_{red}=1.18$, $\chi^2=41.3$ for 35 degrees of freedom). Once the linear trend is removed (using the linear regression with $\chi^2_{red}=1.18$) the updated ephemeris equation is: $$T_{c}=2454823.591924(28)[BJD_{TT}] + 1.33823204(16) \times E,$$ where $T_{c}$ is the central time of a transit in the epoch $E$ since the reference time $T_{0}$. The errors of the last digits are shown in parenthesis.
Panel C in Figure \[o-c\] shows the resulting $O-C$ values of all available transits using the updated ephemeris equation. Almost all the $T_{c}$ coincide with it within the $\pm1~\sigma$ errors represented by the point-dashed lines. Despite there is still non negligible residuals in the timing of $E=751,754$ transits (panel C) those epochs do not deviate from the updated ephemeris equation by more than $2.5\sigma$. These points have relative large uncertainties compare with other epochs and therefore have less weight in the calculated linear regression. We found no evidence of a quadratic function in the $O-C$ values. We show in panels D and E a close-up of the $O-C$ diagram around the -70 and 200 epochs, were the transit observations are more clustered. All the $T_c$ of the common transits analyzed in this work are in excellent agreement within the errors.
This newly obtained precision permits to place strong constraints in the mass of an hypothetical companion, particularly in MMR’s, as we discuss below.
We used the *Mercury* N-body simulator [@Chambers99] to place upper limits to the mass of a potential perturber in the WASP-4 system, based on our timing analysis of the transits. A detailed description of the setup we used for running the dynamical simulations can be found in [@Hoyer-ogletr111-2011] and [@Hoyer-wasp5-2012]. As a summary, for the simulated perturber bodies we used circular ($e=0$) and coplanar orbits with WASP-4b. We explored a wide range of perturber masses ($0.1M_{\earth}~\leq
M_{pert} \leq 5000 M_{\earth}$) and distances between $0.1 ~AU$ and $0.06 ~AU$ in steps of $0.001 ~AU$. The semi-major axis steps were reduced to $0.0005~AU$ near MMRs with the respective transiting body. The density of the perturber body corresponds to the mean density of the Earth (for $M_{pert}\leq1M_{\earth}$) or Jupiter (for $\geq 300
M_{\earth}$). The density was obtained from a linear function that varies from Earth’s to Jupiter’s density for all the other $M_{pert}$.
We identified a region of unstable orbits where any orbital companion experimented close encounters with the transiting body in the time of the integrations we studied (10 years). For all the other stable orbits we recorded the central times of the transits, which were compared with predicted times assuming an average constant orbital period for each system. This period did not deviate by more than $3\sigma$ from the derived period of each transiting body. When the calculated TTV *RMS* approached to 60 seconds we reduced the mass sampling in order to obtain high precision values ($\leq
1~M_{\earth}$) in the mass of the perturber.
The results of our model simulations are illustrated in Figure \[wasp4.mvsa\], where we show the perturber mass, $M_{pert}$ ($M_{\earth}$), versus orbital semi-major axis, $a$ ($AU$), diagram that places the mass limits to potential perturbers in the system. The solid line in the diagram indicates the derived upper limits to the mass of the perturbers that would produce TTV RMS of [$20$ seconds]{} at different orbital separation. The dashed line shows the perturber mass upper limits imposed by the most recent RV observations of the WASP-4 system, for which we have adopted a precision of $15~m~s^{-1}$ [@Triaud2010].
In the same figure, we also show the result of calculating the MEGNO (Mean Exponential Growth of Nearby Orbits) factor $\langle Y \rangle$ [@cincottasimo1999; @cincottasimo2000; @cincottasimo2003] which measures the degree of chaotic dynamics of the potential perturber. This technique found widespread application within dynamical astronomy in studies of stability and orbital evolution, in particular in extrasolar planetary systems and Solar system bodies [@Gozd2001a; @Gozd2001b; @Hinse2010]. Here we use MEGNO to identity phase space regions of orbital instability in the WASP-4 system. We calculated $\langle Y \rangle$ on a grid considering 450x400 initial conditions with the perturber initially placed on a $e=0$ orbit in all integrations. The transiting planet is located at $a=0.02312~AU$ from the host star. Each test orbit was integrated for $10^6$ days. For quasi-periodic orbits $\langle Y \rangle \rightarrow 2.0$ for $t
\rightarrow \infty$ and for chaotic orbits $\langle Y \rangle \ne 2$ (represented by the gray region in the figure). In theses cases $\langle Y \rangle$ usually diverges quickly away from 2.0 in the beginning of the orbit integration. The region close to the transiting planets is highly chaotic due to strong gravitational interactions resulting in collisions and/or escape scenarios and coincides with the unstable region we have identified with *Mercury* code. We identify the locations of MMRs by the chaotic time evolution at certain distances from the host star. These coincide with the TTV sensitivity for smaller masses of the perturber.
For WASP-4b, we found that the upper limits in the mass of an unseen orbital companion are 2.5, 2.0 and 1.0 $M_{\earth}$ in the 1:2, 5:3 and 2:1 MMRs respectively (vertical lines in Figure \[wasp4.mvsa\]). These limits are more strict than the radial velocities constraints, specially in the MMRs, although by using our approach we can not reproduce the mass limits derived by N12.
Starspot Occultations {#spots}
=====================
[lll]{}
Parameter & Derived Value & error\
P ($days$) & $1.33823204$ & $\pm0.00000016$\
$T_{0}$ ($BJD_{TT}$) & $2454823.591924$ & $\pm0.000028$\
$i$ (deg) & $88.52$ & $+0.39,-0.26 $\
$R_{p}/R_{s}$ & $ 0.15445$ & $\pm0.00025 $\
$a/R_{s}$ & $5.463$ & $+0.025,-0.020$\
$R_{p}$ ($R_{jup}$) & $1.395$ & $\pm0.022$\
$e$ & $0$ $^{a}$ & $\cdots$\
\
Epoch [*RMS*]{} $A_{occ}$ (ppm) $T_{c}-T_{occ}$ (days) $\sigma$ (days)
------------ ----------- ----------------------- ------------------------ -----------------------
-91 (CTIO) 0.0023 $ 0.0037 \pm 0.0037 $ $ -0.0211 \pm 0.0023 $ $0.0048 \pm 0.0027 $
-91 (SOAR) 0.0015 $ 0.0024 \pm 0.0006 $ $ -0.0189 \pm 0.0008 $ $0.0018 \pm 0.0009 $
-91 (S09) 0.0008 $ 0.0022 \pm 0.0006 $ $ -0.0190 \pm 0.0019 $ $0.0047 \pm 0.0018 $
-68 (S09) 0.0007 $ 0.0021 \pm 0.0020 $ $ 0.024 \pm 0.003 $ $0.0024 \pm 0.0019 $
163 (SOAR) 0.0014 $ 0.0019 \pm 0.0003 $ $ 0.0268 \pm 0.0006 $ $0.0022 \pm 0.0008 $
172 (SO11) 0.0005 $ 0.0014 \pm 0.0004 $ $ -0.0016 \pm 0.0008 $ $0.0021 \pm 0.0006 $
204 (SOAR) 0.0014 $ 0.0022 \pm 0.0003 $ $ -0.0201 \pm 0.0006 $ $0.0039 \pm 0.0006 $
207 (S011) 0.0007 $ 0.0017 \pm 0.0007 $ $ 0.0232 \pm 0.0005 $ $0.0026 \pm 0.0009 $
As SO11 previously noted, in some of the light curves during the transit there are bump-like features or anomalies that can be interpreted as star-spot occultations by the exoplanet while it crosses in front the star’s disk. SO11 detected evidence of these occultations in two of the light curves they presented and in the two light curves of S09. Depending on the *SNR* and the sampling of the data, we also identified these [*bumps*]{} in four of our light curves (during three different epochs).
Using the residuals of the detrended light curves after modelling the transits (see section \[fiteo\]), we fit a gaussian function around each bump feature, initially identified by eye and which we assume are caused by spots. In particular we fit for the amplitude ($A_{occ}$), central time of the spot occultation ($T_{occ}$) and the width ($\sigma$) of the Gaussian which can be used to described the duration of the occultation event. To estimate the uncertainties of these values, before fitting we added to the observational points an amount of random noise proportional to the *RMS* of the residuals (without bumps). We repeated this procedure 10000 times for each light curve and calculated the errors from the width of the resulting distributions. In Figure \[wasp4.spots.all\] we show the results of those fits for the light curves that present signs of spots in the stars. No spots were apparent in the remaining light curves. Our data of the -91 transit epoch confirms the detection of a spot occultation in the S09 light curve by SO11. Given the better sampling and signal-to-noise ratio (SNR) of our SOAR light curve, we can improve the central timing of the spot by a factor of three, as illustrated by the gray vertical bands in Figure \[wasp4.spots.all\], which indicate the error obtained in the central timings of each spots. However, our CTIO light curve of the transit epoch -68 shows no sign of spots and we cannot confirm the spot in that epoch in the S09 light curve. This can be attributed to the worse SNR of our light curve, but notice also that the amplitude of the spot occultation that we obtain in the S09 data is $A_{occ} = 0.0021 \pm 0.0020$. Therefore that detection could render spurious. Among our new data we also detect two new spots during transit epochs 163 and 204.
In Table \[tabla-spots\] we show the results of the gaussian fits. Taking into account the values obtained for $A_{occ}$ and the *RMS* values of the residuals, no starspot occultations were detected in the light curves of the epochs: -88, -71, 166 and 169. These non-detections (especially the transits closer to those with positive detections) can be used to constrain the spot’s location and/or lifetime. For example, it can be argued that during those transits the spot is located in the non-visible hemisphere of the star, or that the spot have migrated to different latitudes that those occulted by the exoplanet’s path.
SO11 found two possible values for the rotation period of the star, $22.2 ~days$ (based on the spots of their new light curves), and $25.5 ~days$, based on the S09 light curves. Both results are consistent with the constraint of the radial velocities measurements [@Triaud2010].
We used a simple linear model to estimate the rotational period of WASP-4 relying only in the timing of the occultations. We assumed an aligned system (an assumption consistent with the RM effect observations and with the SO11’s previous analysis of the spots) since this geometry increases the probability of detecting occultations of the same spots by the exoplanet.
We assigned angular coordinates for the location of the spots, such that at ingress (first contact) and at egress (fourth contact) of the transit a spot will have a relative angle of $0\degr$ and $180 \degr$, respectively. Relative angles in the non-visible hemisphere will extent between $180\degr$ and $360\degr$. For the epochs where no spot occultations were detected we assigned an aleatory angle for the location of the spot between 180 and 360 degrees and assumed larger errors in its timing ($\sim 0.05 ~days$) to compensate for the non-detection during transit. Later, as explained below, we checked if the rotational period we obtained is consistent with the corresponding non-detections.
Then, we fit the following linear function for the displacement of the spots:
$$\label{eq_prot}
\delta\Theta = \Theta_0 + \Omega \cdot (T-T_0) ,$$
where $\Theta_0$ is the relative angle of the spot in an arbitrary reference time $T_0$, and $\Omega$ is the *displacement rate* of the spots due to stellar rotation (assuming no migration), in degrees per days. To test our model we fit the same spot pairs analyzed by SO11 (E=172,207), using our newly computed $T_{occ}$ and obtain a stellar rotation period $P_{rot}=22.7 \pm 0.2~days$, consistent with the $22.2~days$ period derived by SO11 but inconsistent with the non-detections of E=166,169. Our new occultation central times for the S09 spots (E=-91,-68) give a $P_{rot}$ of $27.5 \pm 0.5~days$, slightly longer than the $25.5 ~days$ period obtained by SO11 but consistent with the no-detections in E=-62,-56. We conclude that, to first order approximation, our linear model provides a good estimate of the star’s rotational period, given the available data.
Next we tried to improve the $P_{rot}$ of the star by adding, first, the new 204 transit epoch to the 172 and 207 epochs (we assumed the spot was the same in the 47 days covered by those epochs). The first minimum of the fit to those three epochs gives a $P_{rot}$ of $44 \pm 1
~days$ ($\chi^2=10^2$). That period is almost twice the value derived by SO11 and their value of the RV constraints estimated to be $\sim
(21.5 \pm 4.3~days)\times sin(i_{s})$, where $i_s$ is the inclination of the stellar rotation axis with respect to the line of sight. The second minimum of the fit gives a $P_{rot}$ of $22.4 \pm 0.2~days$ but with a $\chi^{2}\sim10^3$. This high values of the $\chi^{2}$ mean that none of the obtained values of $P_{rot}$ are consistent with the the locations of the spot in the three epochs. Both solutions deviate by more than $10\degr$ in the location of the spot in E=204 event. The second minimun is also inconsistent with the location of the spot in E=207 while the first minimum is consistent with it within the errors. Furthemore, these solutions are inconsistent with the no-detection of the spot during E=166,169. Probably this is an indication that the events occurring in the 172 and 204-207 epoch do not correspond to occultations of same star-spot and therefore this is a constraint for the lifetime of the spot. Using only the spot $T_{occ}$ of the 204 and 207 epochs we obtained a $P_{rot}$ of $34
\pm 2~days$. However, this period do not match the occultation of the 172 epoch event, supporting the idea that the occultations in E=172 and E=204,207 are over different spots ($\sim43~days$ have elapse between the E=204 and E=172 transits). Nevertheless, is very likely that WASP-4b have occulted the same spot in E=204,207 due to time span between this events is only of $\sim 4 ~days$.
To explain why the $P_{rot}=34~days$ is above the limits of the RV’s measurements we can use the observed decrease in the amplitude and duration of spots between the 204 and 207 epochs. One possibility is that we are evidencing a change in the rotational speed of the spots over the average rotation of the star. The spot can be migrating to higher latitudes in the star surface, decreasing by this way the projected area that is being occulted by the planet, and if WASP-4 presents differential rotation (like the observed in the Sun), the relative displacement rate of the spot over the stellar surface could vary. Therefore the increment in the rotation period we observed compared with the values of the RV’s constraints and the rotational period we estimated using -91 and -68 data, can be indicative of the spot migration to latitudes with lower rotational periods. To fully support this argument we would need a more dense sample of transit observations using the same filters and observing configurations. A summary of the results of this section is presented in Table \[tabla-prot\].
With the same analysis, if we fit the 163 and 172 epoch’s occultations, the minimum we found corresponds to a rotational period of $\sim13\pm0.2~days$ (also consistent with the no-detections of the 166 and 169 epochs) but again this period is far below the range indicated by the RV’s measurements and is a evidence that those occultation events are over different spots.
Of course, a more detailed model is necessary to constrain strictly the rotation period of the star. This model has to take into account other parameters such the amplitude and duration of the spot occultations, the relative angle between spin axis of the star axis and the orbital axis of the planet, the lifetime of the spots, etc.
------------------------------------ ----------- ------------------ ------------ ----------
Epochs time span $P_{rot}$ $\chi^{2}$ No-D?
$(days)$ $(days)$
166, 169, **172**, **207** 55 $22.7\pm 0.2$ $10^{-5}$ no
**-91**, **-68** , -62, -56 47 $ 27.5 \pm 0.5 $ $10^{-7}$ yes
166, 169,**172**, **204**, **207** 55 $44 \pm 1$ $10^{2}$ no
166, 169,**172**, **204**, **207** 55 $22 \pm 0.4$ $10^{3}$ no
**204**, **207** 4 $34 \pm 1$ $10^{-8}$ $\cdots$
**163**, 166, 169, **172** 12 $ 13 \pm0.2 $ $10^{-8}$ yes
------------------------------------ ----------- ------------------ ------------ ----------
: Results of the fitting of the rotational period of WASP-4 using different spot occultation events (indicated with bold numbers) and no-detection of occultations during transit. As reference, in the second column we indicate the time span between the first and last epoch used in the minimization. In the last column is indicated if the fitted $P_{rot}$ is consistent with the no-detections (No-D).[]{data-label="tabla-prot"}
Summary {#conclusiones}
=======
We present twelve new transit epoch observations of the WASP-4b exoplanet. These new transits observed by the *TraMoS* project were combined with all the light curves available in the literature for this exoplanet. It is worth noticing that the analysis and modelling we perform in Section \[fiteo\] was done over detrended light curves (both for the TraMoS and literature data). This does not significantly affect the results in this paper because the system does not show TTVs. However, the light curve analysis should be done, whenever possible, including detrending coefficients as part of a global parameter fit. Therefore, we encourage authors to provide the raw light curves data in the publications to allow future homogeneous analyses of different datasets of a given planet. With an homogeneous modelling of all this data (the new presented here and the previously published) we performed a timing analysis of the transits. Based in the *RMS* of the $O-C$ diagram of about [$20$ seconds]{} we confirmed that WASP-4b orbits its host star with a linear orbital period. We updated the ephemeris equation of this planet and also refined the values of the inclination of the orbit and the planet-to-star radii ratio. Also, during the transits we detected small anomalies in the relative flux of four of the transits (of three different epochs) presented in this work. As [@Sanchis-Ojeda2011] previously noted we identified these anomalies as stellar spot occultations by the planet. With a simple modelling using the timing of these occultations we estimated the rotational period of the star. With the timing of the events we are more confident correspond to occultations of a same spot allowed to propose the rotational period is about 34 days. Since this value deviates from the limits imposed by the radial velocities measurements a further modelling that include spot migration and star differential rotation is needed. A monitoring of more consecutive transits will be necessary to allow for new spots detections and to do a better constrain in the rotational period of WASP-4. High cadence light curves with relative small dispersion are critical on this matter, as can be seen in our SOAR light curves.
Acknowledgements
================
The authors would like thank D. Dragomir and S. Kane for providing TERMS light curves, M. Gillon for the VLT data and the anonymous referee for the useful and accurate comments which helped to improve this manuscript. S.H. and P.R. acknowledge support from Basal PFB06, Fondap \#15010003, and Fondecyt \#11080271 and \#1120299. S.H, received support from ALMA-CONICYT FUND \#31090030 and from the Spanish Ministry of Economy and Competitiveness (MINECO) under the 2011 Severo Ochoa Excellence Program MINECO SEV-2011-0187 at the IAC. TCH acknowledges support from KRCF via the KRCF Young Scientist Fellowship program and financial support from KASI grant number 2013-9-400-00. TCH wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. V.N. acknowledge partial support by the Università di Padova through the *”progetto di Ateneo”* \#CPDA103591. We thanks the staff of CTIO and SOAR for the help and continuous support during the numerous observing nights, and R. Sanchis-Ojeda for helpful comments and discussions. Based on observations made with the SMARTS 1-m telescope at CTIO and the SOAR Telescope at Cerro Pachon Observatory under programmes ID CNTAC-08B-046,-09A-089,-09B-050,-10A-089 and -10B-066.
[^1]: The extrasolar planet encyclopaedia: http://www.exoplanet.eu
[^2]: In the remaining of the text we refer to each individual transit by the epoch number transit, using the ephemeris equation of D11. Transit epoch numbers are listed in the second column of Table \[tabla-obs\].
[^3]: The W08 data are available in the on-line version of the article in the ApJL website. The W09 and S011 data are available in the on-line material from the S011 publication on ApJ. The S09 and N12 data are available at the CDS (http://cdsweb.u-strasbg.fr). The D11 and G09 data were provided by the author (private communication).
[^4]: http://astroutils.astronomy.ohio-state.edu/time/utc2bjd.html
|
---
abstract: 'Estimates of the quantum accuracy threshold often tacitly assume that it is possible to interact arbitrary pairs of qubits in a quantum computer with a failure rate that is independent of the distance between them. None of the many physical systems that are candidates for quantum computing possess this property. Here we study the performance of a concatenated error-detection code in a system that permits only nearest-neighbor interactions in one dimension. We make use of a new message-passing scheme that maximizes the number of errors that can be reliably corrected by the code. Our numerical results indicate that arbitrarily accurate universal quantum computation is possible if the probability of failure of each elementary physical operation is below approximately $10^{-5}$. This threshold is three orders of magnitude lower than the highest known.'
author:
- 'Ashley M. Stephens$^{}\footnote[1]{Electronic address: a.stephens@physics.unimelb.edu.au}$ and Zachary W. E. Evans'
title: Accuracy threshold for concatenated error detection in one dimension
---
Introduction
============
For a quantum computer to reliably outperform a classical computer, it must be robust against the effects of decoherence and imprecise quantum control. There are many ideas on how to achieve such fault tolerance, including topological quantum computing [@Kitaev1; @Nayak1; @Raussendorf2], surface codes [@Kitaev1; @Bravyi1; @Dennis1], color codes [@Bombin1], self-correcting codes [@Bacon1], and concatenated codes [@Shor2; @Steane1]. The threshold theorem indicates that, under certain conditions, scalable quantum computing is possible in principle [@Aharonov1; @Aliferis1]. The theorem asserts that if the probability of failure of each elementary physical operation in a quantum computer is below some threshold then arbitrarily accurate quantum computation can be performed efficiently given sufficient time and qubits. The actual value of the threshold for a given error-correction code depends on a number of parameters that describe the quantum computer and the noise that affects it [@Gottesman1].
In most estimates of the accuracy threshold, the assumption is made that it is possible to interact arbitrary pairs of qubits in a quantum computer with a failure rate that is independent of the distance between them. This property is desirable, since it allows higher failure rates to be tolerated, but it is unrealistic in the limit of many qubits. In all of the physical systems that are candidates for quantum computing the range of controllable interactions is constrained such that at least some pairs of qubits will need to be transported before they can undergo logic gates such as ${\textsc{cnot}}$ and ${\textsc{cphase}}$. Moreover, we expect that in many systems, especially trapped-ion, superconducting, and solid-state systems, hardware limitations will require that the qubit array is two- or one-dimensional [@Kielpinski1; @Taylor1; @Hollenberg1; @Fowler1].
The combination of local interactions and low coordination number is not a problem in principle as it is known that the threshold theorem still applies [@Aharonov1; @Gottesman4]. The value of the threshold has been quantified for systems that permit only nearest-neighbor interactions in two dimensions [@Raussendorf1; @Svore1; @Roychowdhury1] and in various quasi one-dimensional settings [@Szkopek1; @Stephens1], and the threshold is also known for superconducting [@Fowler1] and ion-trap [@Metodi1] architectures with similar properties. However, there are a large number of systems under development that permit only nearest-neighbor interactions in one dimension [@Kane1; @Loss1; @Vrijen1; @Tian1; @Hollenberg2; @Feng1; @Pachos2; @Vandersypen1; @Solinas1; @Jefferson1; @Petrosyan1; @Vyurkov1; @Kamenetskii1], for which the threshold is unknown. The threshold in one dimension is expected to be significantly lower than in all other cases.
Here we find the accuracy threshold for a system that permits only nearest-neighbor interactions in one dimension. Where required, qubits are transported via nearest-neighbor ${\textsc{swap}}$ gates. To minimize this overhead we use a small error-detection code. To correct errors we make use of a new message-passing method that uses classical information gathered during error detection to maximize the number of errors that can be reliably corrected [@Evans2]. Our numerical results indicate that arbitrarily accurate quantum computation is possible if the probability of failure of each elementary physical operation is below approximately $10^{-5}$.
\[\[4,1,2\]\] subsystem code
============================
The code that we have chosen to use is the \[\[4,1,2\]\] subsystem code, a stabilizer CSS quantum code that encodes one logical qubit into four physical qubits [@Bacon1; @Bacon2]. Its stabilizer, $\mathcal{S}$, is $$\begin{aligned}
X_1X_2X_3X_4, Z_1Z_2Z_3Z_4,
\end{aligned}
\label{stabilizers:bs}$$ where $X_i$ and $Z_i$ represent the Pauli operators $\sigma_X$ and $\sigma_Z$ applied to the $i^{th}$ qubit respectively. Identity operators and tensor products between operators are omitted. Although there are two degrees of freedom in which to store encoded information, we choose to ignore one of these encoded qubits. Then, elements in the non-Abelian group, $\mathcal{T}$, generated by the operators $$\begin{aligned}
X_1X_2, X_3X_4, Z_1Z_3, Z_2Z_4,
\end{aligned}
\label{gauge}$$ act trivially on the sole protected qubit. The encoded $Z$ and $X$ operators for this qubit are $$X_1X_3, Z_1Z_2.$$ Because the elements in $\mathcal{S}$ are products of elements in $\mathcal{T}$, and since all elements in $\mathcal{T}$ commute with all elements in $\mathcal{S}$ and the encoded operators, to determine the eigenvalues of the elements of $\mathcal{S}$ it is sufficient to measure the eigenvalues of the elements in $\mathcal{T}$. This may change the state of the system but it will not affect the state of the protected qubit. This property allows us to use only two ancilla qubits to simultaneously perform operator measurements of the operators in $\mathcal{T}$ [@Aliferis3]. We decode the syndrome by taking the parity of the two measurement outcomes in each basis. This is used to infer the presence or absence of errors in each basis—if the parity is even there is no error, if the parity is odd there is an error. For the \[\[4,1,2\]\] subsystem code, ${\textsc{cnot}}$ is a valid transversal gate and $H$ is a valid transversal gate up to a permutation of qubits.
Using concatenated error detection to correct errors
====================================================
In our scheme the \[\[4,1,2\]\] subsystem code is concatenated such that physical qubits form encoded qubits which in turn form higher-level encoded qubits and so on. However, while concatenating the code $l$ times results in a final code that has distance $2^{l+1}$, if each level of error detection operates independently of every other level, then the code cannot reliably correct even a single error—a single physical failure may cause an encoded error at any level. To do more than simply detect errors classical messages must be passed from each level to the level above. These messages serve to indicate the location of potential errors, thereby removing the ambiguity in the cause of any odd parity syndrome that is observed, which allows errors that are detected to be located and corrected.
The message-passing method applied in this paper is similar to those of Refs. [@Knill1; @Evans1; @Aliferis5] but, unlike those, it will only fail if the number of concurrent errors is greater than or equal to half of the distance of the final concatenated code—in the case of the \[\[4,1,2\]\] subsystem code, the leading-order exponent of the probability of failure scales with the number of levels of concatenation as $1 $(physical)$,1,2,4,8,16,32$, and so on. This is also a property of the noisy-channel method of Ref. [@Poulin1]. Although we will only describe the message-passing method in the case where it is applied to the \[\[4,1,2\]\] subsystem code, it can be applied to codes with greater distance. A detailed description of the method is contained in Ref. [@Evans2].
Messages in the method consist solely of classical information which indicates our confidence that each location in the circuit has not failed given what is known about it. We call these messages flags. The probability that any given flag represents an actual error is described by the weight, $w$, carried by the flag such that $\textrm{Pr}(\textrm{error}) = \mathcal{O}(p^w)$, where $p$ is the probability of failure of an elementary physical location in the quantum computer. Since we assume that errors at each location are independent of all other locations, the probability of a set of flags representing actual errors is $\mathcal{O}(p^{\sum{w}})$, where the sum is over the entire set. Flags are raised at every elementary physical and encoded location. All flags at the physical level are given weight equal to one by definition and at all other levels weights are determined during error detection at the level below. Flags in the $X$ and $Z$ bases are separate.
As flags are raised they are classically propagated through the error-detection circuit to determine the effect that an error at the location at which the flag originated would have on the data and ancilla qubits at the point of syndrome extraction. Note that ${\textsc{cnot}}$ copies $X$ errors from the control qubit to the target qubit and $Z$ errors in the opposite direction, and that we define the point of syndrome extraction to be immediately after the ancilla qubits are measured. Since there are significantly fewer effects than there are locations at which flags are raised, flags are binned by effect. Each bin is assigned a single weight which is only updated if a flag is raised with a weight lower than its existing value.
Figure \[figure:bi\] illustrates the binning system where, for simplicity, non-local circuits are used and only $X$ errors are considered. Note that the $X$-error syndrome is obtained using the second half of the circuit in Fig. \[figure:bi\], so we are interested in the effect of $X$ errors at the second set of measurements. $X$ errors do not affect the $Z$-error syndrome, but since they can occur during the circuit to obtain the $Z$-error syndrome this part of the circuit must be considered. Because of the degeneracy of the code, we need only consider two pairs of data qubits rather that four individual data qubits. For example, since $X_1X_2$ acts trivially on the encoded qubit, both $X_1$ and $X_2$ give the same syndrome and can be corrected by the operator $X_1$.
Since the failure of a single ${\textsc{cnot}}$ or ${\textsc{swap}}$ gate can introduce a pair of correlated errors, these gates require special treatment. The weights of the two flags that are raised immediately following each two-qubit gate are used to update the bin which corresponds to the two-qubit correlated error. Note that both of the bins corresponding to single errors following the two-qubit gate are also updated. The weight of the correlated error is taken to be the maximum of the two single-qubit weights, as opposed to the sum, which would describe the probability of the pair of errors occurring without correlation.
Once the syndrome is measured its most likely cause is identified by finding the bin that has the lowest weight while still being consistent with, or matching, the syndrome. In the case of the \[\[4,1,2\]\] subsystem code, in each error-correction cycle we consider only three bins, $AG1$, $AG2$, and $A$. The most likely cause of an odd syndrome is a single error occurring in the bin with the lowest weight, as each of these bins corresponds to a change in the parity of the ancilla. The most likely cause of an even syndrome is always no error at all. If the match implies an error on a data qubit then the appropriate correction is applied.
Although we correct for the most likely error it is possible that the true error is the complement of the correction—that is, the true error and the correction that we apply combine to form an encoded operator. To determine the weight of the flag that is raised in the error-correction circuit at the level above, what we will call the encoded weight, we calculate the difference between the weight of the match on which we act and the weight of its complement. For example, if we have an odd syndrome and the $AG1$ is the match, then $AG2$ is the complement match and the encoded weight is the difference between the weights of the these two matches, $AG2-AG1$.
There is the possibility that a correction will result in a state that is outside the code space so that it is neither correct nor affected by an encoded operator. Just as the complement match is used to determine the probability of an encoded error, the conditional probability of single errors on the pairs of data qubits can be updated by considering other matches. At the end of each error-correction cycle the weights of $AG1$ and $AG2$ are updated to the minimum of the weight of any previous locations that have not yet had an opportunity to affect the ancilla, given by $G1$ and $G2$, and a weight obtained during the preceding circuit, calculated in an analogous way to the encoded weight based on the syndrome that is observed.
Table I shows a list of all possible flag matches along with the various flag-weight updates that would result from each of them being acted on, where we have retained the notation of Fig. \[figure:bi\]. By the careful consideration of every possible cause of every syndrome in this way we always have accurate weights for every qubit at every level. We can always, therefore, apply corrections based on the most likely set of errors.
synd. match corr. $C_{1}$ $C_{2}$ $C_{e}$
------- ------------- ----------------------- --------- --------- -----------
None none $AG1+A$ $AG2+A$ $AG1+AG2$
$AG1+A$
$AG2+A$
$AG1+AG2$
$AG1$ $X_1/Z_1$ $A-AG1$ $AG2+A$ $AG2-AG1$
$AG2$ $X_3/Z_2$ $AG1+A$ $A-AG2$ $AG1-AG2$
$A$ none $AG1-A$ $AG2-A$ $AG1+AG2$
$AG1+AG2+A$ \[tab: flag matches\]
Accuracy threshold in one dimension
===================================
To estimate the threshold for universal quantum computation in one dimension we construct circuits for error detection and for the encoded operations ${\textsc{cnot}}$, ${\textsc{swap}}$, Hadamard, state preparation, and measurement. These operations, which we will refer to as CSS operations, are sufficient to concatenate error detection and to perform state distillation following the ideas presented in Refs. [@Bravyi3; @Aliferis4]. State distillation involves preparing the ancillary state $\vert{\textrm{+}i}\rangle=\vert0\rangle+i\vert1\rangle$ to enable the logical phase gate, $S$, and preparing the ancillary state $\vert\textrm{Toffoli}\rangle=\vert000\rangle+\vert010\rangle+\vert100\rangle+\vert111\rangle$ to enable the logical Toffoli gate. These gates together with the CSS operations complete a universal set for quantum computation. Accurate states can be distilled from many noisy states provided the noisy states can be made with a failure rate lower than some distillation threshold, which is typically above one percent. We determine the threshold for CSS operations by numerically simulating the circuit for an error-corrected logical ${\textsc{cnot}}$ under a stochastic error model. The ${\textsc{cnot}}$ is chosen because it has the highest failure rate of the CSS operations. As the threshold we find is well below the distillation threshold it is the threshold for universal computation under our scheme.
Figure \[figure:se\] shows a circuit for syndrome extraction for the \[\[4,1,2\]\] subsystem code where only nearest-neighbor interactions on a linear array are permitted. Note that the encoded Hadamard can be achieved by transversal application of the Hadamard gate in addition to removing a single ${\textsc{swap}}$ gate from this circuit to permute the qubits. This syndrome-extraction circuit has the same depth as the non-local circuit in Fig. \[figure:bi\] and the two circuits differ only by a rearrangement of qubits and the addition of two ${\textsc{swap}}$ gates. This difference is significant, however, since each of the additional ${\textsc{swap}}$ gates involves two data qubits. New pairs of correlated errors are introduced with probability $\mathcal{O}(p)$, where $p$ is the probability of failure of an elementary physical location. Without the ${\textsc{swap}}$ gates these particular pairs of errors occur with probability $\mathcal{O}(p^2)$. Since these errors include the encoded operators, this means that an undetected encoded error occurs with probability $\mathcal{O}(p)$.
Figure \[figure:cn\] shows the encoded ${\textsc{cnot}}$ circuit. A pair of correlated errors caused by the failure of one of the ${\textsc{swap}}$ gates prior to the transversal ${\textsc{cnot}}$ results in a pair of errors on the encoded control qubit and a pair of errors on the encoded target qubit. This is because the transversal ${\textsc{cnot}}$ copies errors. This means that a pair of correlated, undetected encoded errors occurs with probability $\mathcal{O}(p)$. The encoded ${\textsc{swap}}$ gate, in which the transversal ${\textsc{cnot}}$ is replaced by a transversal ${\textsc{swap}}$, does not possess this property. Two gates must fail during the encoded ${\textsc{swap}}$ for both encoded qubits to fail undetected.
Here we make an important observation: to leading order, the relative probabilities of undetected failure of first-level encoded locations in the one-dimensional case mimic the relative probabilities of failure of physical locations in the non-local case. Specifically, all single-qubit locations fail with probability $\mathcal{O}(p)$, a pair of correlated errors after a ${\textsc{cnot}}$ occurs with probability $\mathcal{O}(p)$, and a pair of correlated errors after a ${\textsc{swap}}$ occurs with probability $\mathcal{O}(p^2)$. A pair of correlated errors can occur after a ${\textsc{swap}}$ gate because of a single fault, but they will always be detected. A corollary of this is that after the first level of concatenation our linear scheme will mimic a non-local scheme. In the non-local case we expect to succeed whenever the number of concurrent errors is less than half of the distance of the final concatenated code. This implies that in the one-dimensional case the first-order exponent of the probability of failure should scale with the number of levels of concatenation as $1 $(physical)$, 1,1,2,4,8,16$, and so on.
Note that Figs. \[figure:se\] and \[figure:cn\] show circuits constructed from physical gates. To generate circuits for error correction and encoded operations at higher encoded levels we replace all physical gates with encoded gates. Like in other concatenated schemes, after every encoded operation we perform error correction on the encoded qubits involved in that operation [@Aliferis1]. All circuits are designed so that ancilla qubits are always available to perform error correction using the circuit in Fig. \[figure:se\].
To attempt to verify that our scheme performs as expected and to determine the threshold, we simulate a logical ${\textsc{cnot}}$ under a stochastic error model. The simulated circuit consists of a ${\textsc{cnot}}$ extended rectangle [@Aliferis1], the failure rate of which is meant to approximate the failure rate of a ${\textsc{cnot}}$ in some algorithm or at a higher level of concatenation. Because the circuit only contains gates from the Clifford group, we need only simulate the propagation of errors that occur during the circuit [@Steane3] rather than store the complete state of the quantum computer. We have written our own simulator for this purpose. We simply assume the state begins in an arbitrary valid codeword state and is stochastically perturbed by errors during the circuit. The circuit is defined to succeed if measurement of the data qubits at the end of the circuit in either the $X$ or $Z$ basis would give the correct outcome. Equivalently, a circuit is defined to have succeeded if an errorless error-correction cycle applied to its output state would produce the correct state.
We generate data in two different ways. Where $n$ is the number of levels of error correction, for $n\leq3$, instead of applying an error at each location with some probability, we simulate the full concatenated circuit with exactly $i$ errors placed randomly. For all single-qubit locations (state preparation, memory, measurement) the error is a randomly selected Pauli error and for all two-qubit locations (${\textsc{cnot}}$, ${\textsc{swap}}$) the error is a randomly selected two-qubit Pauli error. This is repeated many times to generate the probability that the circuit fails given $i$ errors, $r_i$. These conditional probabilities can be combined to give the failure rate of the circuit as a function of $p$, $$\sum^N_{i=0}{r_i {N\choose i}p^i(1-p)^{N-i}},
\label{eq: expansion}$$ where $N$ is the number of locations in the entire circuit, 172, 11992, and 864496 for $n=1,2,3$ respectively. The series is truncated after $i=6$, 10, and 21 for $n=1,2,3$ respectively. These numbers are chosen so that the approximation is valid in the region of the threshold.
For $n=4$, too much time is required to generate statistically significant results in this way. Instead, after each elementary physical location in the circuit we apply an error with probability $p$, where errors at all locations are independent. We study the failure rate of the circuit as we vary $p$ between $8\times10^{-6}$ and $2\times10^{-5}$, again, simulating the full concatenated circuit. The time required to generate statistically significant results for $n>4$ is prohibitive using our methods. In our simulator we use the SIMD-oriented Mersenne Twister pseudorandom number generator [@Saito1]. The message-passing method is simulated along side the error-correction circuit so that it operates in the same way as it would in a real quantum computer.
Our results are summarized in Fig. \[figure:re\]. The gradients of the lines in Fig. \[figure:re\], which are related to the minimum number of errors that cause the code to fail, are as expected. Since the first two levels of error correction are unable to reliably correct errors, the failure rate for any less than three levels of error correction is always greater than $p$. Note that the lines for $n=3$ are truncated at $p>10^{-5}$. This is because for $p>10^{-5}$ the approximation that $r_i=0$ for $i>21$ breaks down as failure due to more than 21 errors becomes common.
Our results suggest that the threshold is approximately equal to $10^{-5}$. For elementary physical failure rates below this threshold, we expect that arbitrarily accurate CSS operations can be performed efficiently given sufficient time and qubits. This threshold is well below the threshold for distillation of ancilla states that enable universal computation [@Bravyi3; @Aliferis4]. A failure rate much lower than $10^{-5}$ would, however, be required to achieve a sufficiently low encoded failure rate using a practical amount of resources. For example, we estimate that to achieve an encoded failure rate of $10^{-15}$ using $\mathcal{O}(10^3)$ physical qubits per encoded qubit, a physical failure rate of approximately $10^{-8}$ is necessary. Furthermore, if there is a non-zero probability of defective qubits in the linear array then the threshold will cease to exist. In contrast, a constant density of defective qubits in a two-dimensional array can be tolerated and will merely lower the threshold.
Conclusions and further work
============================
That the accuracy threshold for universal quantum computation in a system that permits only nearest-neighbor interactions in one dimension may be $10^{-5}$ or higher is somewhat surprising. In two dimensions the highest proven threshold for concatenated error correction is $2\times10^{-5}$ [@Roychowdhury1]. There is strong evidence that the threshold in two dimensions can be as high as $7\times10^{-3}$ [@Raussendorf1] but this relies on techniques that are not expected to be useful in only one dimension. In quasi one-dimensional settings the highest proven threshold is $2\times10^{-6}$ [@Stephens1]. Since the threshold presented in this paper is based on numerical simulations, it would be useful to obtain a rigorous bound on its value. Due to the unconventional error-correction method that we use, it is unclear if this can be done using the established level-reduction procedure of Ref. [@Aliferis1].
By adding ${\textsc{swap}}$ gates where necessary, any quantum algorithm can be implemented on a linear nearest-neighbor array with an overhead that is, at most, polynomial in the number of qubits. This has been done explicitly for Shor’s algorithm [@Shor1] in Ref. [@Fowler2]. Our result strengthens the notion that one-dimensional architectures are viable candidates for quantum computing, although how to achieve defect tolerance remains an open question. It will be interesting to see if a higher threshold can be achieved by adapting the postselection scheme of Ref. [@Knill1] to a system that permits only nearest-neighbor interactions in one dimension. In this scheme ancilla states are protected by an error-detection code and postselected. We expect that the new methods of message passing presented in Ref. [@Evans2] will help in improving the efficiency of such a scheme, as the weights outputted from error correction may be useful in moderating the amount of postselection. These ideas are the subjects of further work.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Charles Hill and Magdalena Carrasco for their helpful suggestions. We acknowledge financial support from the Australian Research Council (ARC), the US National Security Agency (NSA), and the Army Research Office (ARO) under contract number W911NF-04-1-0290.
[10]{}
A. Kitaev. **303**, 2 (2003).
C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma. arXiv:0707.1889 (2007).
R. Raussendorf, J. Harrington, and K. Goyal. **9**, 199 (2007).
S. Bravyi and A. Kitaev. arXiv:quant-ph/9811052 (1998).
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill. **43**, 4452 (2002).
H. Bombin and M. A. Martin-Delgado. **97**, 180501 (2006).
D. Bacon. **73**, 012340 (2006).
P. W. Shor. **37**, 56 (1996).
A. M. Steane. **78**, 2252 (1997).
D. Aharonov and M. Ben-Or. **29**, 176 (1997).
P. Aliferis, D. Gottesman, and J. Preskill. **6**, 97 (2006).
D. Gottesman. arXiv:quant-ph/0701112 (2007).
D. Kielpinski, C. Monroe, and D. Wineland. **417**, 709 (2002).
J. M. Taylor, H. -A. Engel, W. Dür, A. Yacoby, C. M. Marcus, P. Zoller, and M. D. Lukin. **1**, 177 (2005).
L. C. L. Hollenberg, A. D. Greentree, A. G. Fowler, and C. Wellard. **74**, 045311 (2006).
A. G. Fowler, W. Thompson, Z. Yan, A. M. Stephens, B. Plourde, and F. Wilhelm. **76**, 174507 (2007).
D. Gottesman. **47**, 333 (2000).
K. Svore, D. DiVincenzo, and B. Terhal. **7**, 297 (2007).
R. Raussendorf and J. Harrington. **98**, 190504 (2007).
F. M. Spedalieri and V. P. Roychowdhury. arXiv:0805.4213v1 (2008).
T. Szkopek, P. O. Boykin, H. Fan, V. Roychowdhury, E. Yablonovitch, G. Simms, M. Gyure, and B. Fong. **5**, 42 (2006).
A. M. Stephens, A. G. Fowler, and L. C. L. Hollenberg. **8**, 330 (2008).
T. S. Metodi, D. D. Thaker, A. W. Cross, F. T. Chong, and I. L. Chuang. arXiv:quant-ph/0509051 (2005).
B. E. Kane. **393**, 133 (1998).
D. Loss and D. P. DiVincenzo. **57**, 120 (1998).
R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, and D. DiVincenzo. **62**, 012306 (2000).
L. Tian and P. Zoller. **68**, 042321 (2003).
L. C. L. Hollenberg, A. S. Dzurak, C. Wellard, A. R. Hamilton, D. J. Reilly, G. J. Milburn, and R. G. Clark. **69**, 113301 (2004).
M. Feng, I. DÕAmico, P. Zanardi, and F. Rossi. **66**, 14 (2004).
J. K. Pachos and P. L. Knight. **91**, 107902 (2003).
L. M. K. Vandersypen, R. Hanson, L. H. Willems van Beveren, J. M. Elzerman, J. S. Greidanus, S. De Franceschi, and L. P. Kouwenhoven. Kluwer Academic / Plenum Publishers, New York (2003).
P. Solinas, P. Zanardi, N. Zanghi, and F. Rossi. **67**, 121307 (2003).
J. H. Jefferson, M. Fearn, D. L. J. Tipton, and T P. Spiller. **66**, 042328 (2002).
D. Petrosyan and G. Kurizki. **89**, 207902 (2002).
V. Vyurkov and L. Y. Gorelik. **1**, 77 (2000).
E. O. Kamenetskii and O. Voskoboynikov. , 221 (2006).
Z. W. E. Evans and A. M. Stephens. arXiv:0902.4506 (2009).
D. Bacon and A. Casaccino. **44**, arXiv:quant-ph/0610088 (2006).
P. Aliferis and A. W. Cross. **98**, 220502 (2007).
E. Knill. **434**, 39 (2005).
Z. W. E. Evans and A. M. Stephens. **78**, 062317 (2008).
P. Aliferis and J. Preskill. **79**, 012332 (2009).
D. Poulin. **74**, 052333 (2006).
S. Bravyi and A. Kitaev. **71**, 022316 (2005).
P. Aliferis. Ph.D. thesis, California Institute of Technology, Pasadena, CA. arXiv:quant-ph/0703230 (2007).
A. M. Steane. **68**, 042322 (2003).
M. Saito and M. Matsumoto. **2**, 607 (2006).
P. Shor. **26**, 1484 (1997).
A. G. Fowler, S. J. Devitt, and L. C. L. Hollenberg. **4**, 237 (2004).
|
---
address:
- 'Applications Department, Poznań Supercomputing and Networking Center, Poland'
- 'Center for Computation & Technology, Louisiana State University, Baton Rouge, USA'
- 'Department of Computer Science, Louisiana State University, Baton Rouge, USA'
- 'Department of Physics & Astronomy, Louisiana State University, Baton Rouge, USA'
- 'Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, USA'
- 'Perimeter Institute for Theoretical Physics, Waterloo, Canada'
- 'Department of Physics, University of Guelph, Guelph, Canada'
author:
-
-
-
-
-
-
-
-
bibliography:
- 'CCT\_CS.bib'
- 'marqs.bib'
- 'GPU.bib'
- 'dmk.bib'
title: 'A Massive Data Parallel Computational Framework for Petascale/Exascale Hybrid Computer Systems'
---
hybrid system ,stencil computations ,CFD ,computational framework ,large scale scientific application
Introduction {#introduction .unnumbered}
============
Heterogeneous systems are becoming more common on High Performance Computing (HPC) systems. Even using tools like CUDA [@cuda40] and OpenCL [@opencl11] it is a non-trivial task to obtain optimal performance on the GPU. Approaches to simplifying this task include Merge [@Linderman:2008:MPM:1353536.1346318] (a library based framework for heterogeneous multi-core systems), Zippy [@CGF:CGF1131] (a framework for parallel execution of codes on multiple GPU’s), BSGP [@Hou:2008:BBG:1360612.1360618] (a new programming language for general purpose computation on the GPU) and CUDA-lite [@springerlink:10.1007/978-3-540-89740-8_1] (an enhancement to CUDA that transforms code based on annotations). In addition, efforts are underway to improve compiler tools for automatic parallelization and optimization of affine loop nests for GPU’s [@Baskaran:2008:CFO:1375527.1375562] and for automatic translation of OpenMP parallelized codes to CUDA [@Lee:2009:OGC:1594835.1504194].
In this paper we present an alternative approach: a new computational framework for the development of massively data parallel scientific codes applications suitable for use on such petascale/exascale hybrid systems built upon the highly scalable Cactus framework [@CS_Goodale02a; @CS_Cactusweb] As the first non-trivial demonstration of its usefulness, we successfully developed a new 3D CFD code that achieves improved performance.
Cactus Computational Framework
==============================
The Cactus framework [@CS_Goodale02a; @CS_Cactusweb] was designed and developed to enhance programming productivity in large-scale science collaborations. The design of Cactus allows scientists and engineers to develop independent components for Cactus without worrying about portability issues on computing systems. The common infrastructure provided by Cactus also enables developing scientific codes that work across different disciplines. This approach emphasizes code re-usability, and greatly simplifies constructing sound interfaces and well-tested and well-supported software. As the name *Cactus* suggests, the Cactus framework consists of a central core called *flesh*, which provides infrastructure and interfaces for modular components called *thorns*.
Building upon the flesh, thorns can provide implementations for computational concepts such as parallelization, mesh refinement, I/O, check-pointing, web servers, and so forth. The Cactus Computational Toolkit (CCTK) is a collection of thorns which provide basic computational capabilities. Application thorns make use of the CCTK via the Cactus flesh API. Cactus is well suited for domain discretizations via regular, block-structured grids as are common e.g. for higher order finite differences. The *Carpet* AMR library [@CS_Schnetter-etal-03b; @CS_carpet_web] implements the recursive block-structured AMR algorithm by Berger and Oliger [@CS_Berger84], and provides support for multi-block (or multi-patch) domain discretizations. A set of explicit time integration schemes such as Runge-Kutta methods are provided by a Method of Lines time integrator. Overall, the Cactus framework hides the detailed implementations of Carpet and other utility thorns from application developers.
MPI-Based Data Parallelism in Cactus
------------------------------------
The Cactus framework adopts the idea of data parallelism in its design and implementation. In Cactus, the computational grid is decomposed into multiple components that are distributed between processes, and the same set of operations are applied to each. The communication component of Cactus uses the Message Passing Interface (MPI) to exchange data between processes. In Cactus, it is the task of a special *driver* component to set up storage for variables, partition the grid between MPI processes, and manage inter-process communication. Unlike physical boundaries where the boundary data can be set or calculated from boundary conditions, data at inter-process boundaries need to be copied from other processes where the neighboring grid components are located. This is implemented via a *ghost region* at the inter-process boundaries that is automatically set up by the driver. The necessary size of a ghost region depends on the numerical algorithms used and can be selected as parameters at run-time.
Parallelization on CPU-GPU Hybrid Systems
-----------------------------------------
The data parallelism in Cactus matches well with the features of CUDA [@cudasite] and OpenCL [@openclsite] in supporting programming on hybrid computer systems. On the computational framework level, there is not much difference between CUDA and OpenCL when targeting NVIDIA Fermi-class GPU’s. In this work we only focus on the parallelization on the CUDA architecture, and will present a computational framework based on OpenCL in a later publication.
Based upon the CUDA architecture, we build an MPI-CUDA based computational framework in Cactus \[fig:mpicuda\]. It enables a simple, semi-automatic, yet efficient implementation and execution of CUDA-enabled applications. Auto-tuning enables efficient data distribution between nodes, effectively hiding additional cost introduced by GPU-host and host-host interconnections.The computational overhead in such a generic framework is greatly reduced by overlapping data transfers and computation with the asynchronous data transfers and concurrent copy and execution supported in CUDA. With the help from such a computational framework, application developers can then spend more time optimizing the numerical kernel itself, implementing more efficient algorithms in these kernels, and (most importantly) advancing the science content in their code.
This system has been tested and benchmarked on a 3D CFD implementation (see section \[CFD\]) based on a finite difference discretization of Navier-Stokes equations.
![The Cactus computational framework manages the domain decomposition and communication among the split domains via MPI. Computations are performed primarily on GPU’s. The data transfers between CPU’s as well as to and from the GPU’s are concurrent with the computation.[]{data-label="fig:mpicuda"}](figs/mpi_cuda.pdf){width="95.00000%"}
GPGPU Programming in Cactus
===========================
Achieving efficient execution on a GPU often requires careful analysis of the application followed by extensive testing and tuning. For many important problems, such as linear algebra routines, this work has been done and packaged into libraries for convenient use by others [@volkov08].
Iterative grid techniques are widely used, and seem like a good fit for the high floating point density of GPU’s. But because each investigator may run a different grid kernel a simple library routine would not achieve wide use. GPU implementations of iterative grid algorithms must deal with the problem of ghost zone exchange made more tedious by GPU memory access constraints, among other factors. On conventional cluster systems iterative grid application programmers do not need to consider such issues when using a framework like Cactus. Cactus manages data communication between a cluster’s nodes, including ghost zone exchange, so that application code need only operate on that data. The problems related to ghost zone interchange between CUDA blocks is similar in many ways to ghost zone interchange between processors in a cluster CPU configuration.
In this work the Cactus framework has been extended to cover GPU execution via an architecture neutral programming abstraction to highly optimize finite difference operations in a multithreaded computing environment (see list \[list:kerneldef\]).
![The workflow chart of a CaCUDA-based application. The upper box shows the generation of the CaCUDA kernel headers at the code compilation stage. The lower box shows how the variables are evolved to the next time step. []{data-label="fig:cacudaworkflow"}](figs/cacudaworkflow){width="65.00000%"}
CaCUDA Kernel Abstraction
=========================
The task of simplifying the generation of CUDA code for a finite differencing code is not a straightforward one. Shared arrays with appropriate stencil sizes have to be carefully managed, and data needed by the stencil has to be streamed in while calculations proceed. It is possible to abstract away much of the difficult work into boiler plate code, but doing so requires some extra machinery. We design and implement a programming abstraction in the Cactus framework to enable automatic generation from a set of highly optimized templates to simplify code construction. The workflow chart of a typical CaCUDA-based application can be found in figure \[fig:cacudaworkflow\].
There are three major components in our CaCUDA Kernel abstraction.
1. *CaCUDA Kernel Descriptor* is used to declare the variables that will be needed in the GPGPU computation, and identify a few relevant properties.
2. *CaCUDA Templates* are a set of templates which are highly optimized for particular types of computational tasks and optimization strategies.
3. *CaCUDA Code Generator* is used to parse the descriptors and automatically generate CUDA-based macros. The code generator is based on Piraha[@Brandt10a], which implements a type of parsing expression grammar[@Ford2004]. Due to the page limit, we do not list the templates and the sample code generated by CaCUDA. More about the CaCUDA project can be found at the CaCUDA project site [@cacudacode].
<!-- -->
CCTK_CUDA_KERNEL UPDATE_VELOCITY
TYPE=3DBLOCK
STENCIL="1,1,1,1,1,1"
TILE="16,16,16"
{
CCTK_CUDA_KERNEL_VARIABLE CACHED=YES INTENT=SEPARATEINOUT
{
vx, vy, vz
} "VELOCITY"
CCTK_CUDA_KERNEL_VARIABLE CACHED=YES INTENT=IN
{
p
} "PRESSURE"
CCTK_CUDA_KERNEL_PARAMETER
{
density
} "DENSITY"
}
The above kernel abstraction can be integrated in a straightforward manner as a thorn (module), *CaCUDA* in Cactus without touching the flesh (core infrastructure). The kernel descriptor in this abstraction is similar in both format and functionality to those Cactus Configuration Language (CCL) files, which are used to declare global data structures, runtime parameters, and the way various C or Fortran subroutines interact through the schedule tree. The abstractions already used by Cactus are: param.ccl, configuration.ccl, schedule.ccl, and interface.ccl. To this set we add an additional declarative file called cacuda.ccl. The Cactus Framework already has a mechanism, implemented through the configuration.ccl file, by which discovery of additional preprocessing code can be enabled prior to the compilation of C/Fortran code.
CFD Implementation {#CFD}
==================
Computational Fluid Dynamics (CFD) is one of the branches of fluid mechanics which uses numerical methods and algorithms to solve and analyze fluid flows. It is successfully used in various fields of science and engineering such as weather forecasting, aerodynamic optimization of body shapes (e.g. planes, cars, ships), gas reservoir uncertainty analysis. Unfortunately accurate CFD simulations need great computational power. It is very important to adapt existing algorithms to new hybrid architectures and execute them in a massively parallel manner.
Background and Governing Equations
----------------------------------
The CFD numerical method is governed by Navier-Stokes incompressible equations which are derived from Newton’s second law (conservation of momentum) and conservation of mass (incompressibility). The Navier-Stokes equations are
$$\begin{aligned}
\label{navier_eqs}
\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \mathbf{\nabla}) \mathbf{u} &= -\nabla \phi + \nu \mathbf{\nabla}^2 \mathbf{u} + \mathbf{f}
\\
\nabla \cdot \mathbf{u} &= 0
$$
where $\mathbf{u}$ is the velocity field, $\nu$ is the the kinematic viscosity, $\mathbf{f}$ is the body force, $\phi$ is the modified pressure (pressure over density). The presented equations need to be further discretized in order to perform proper simulation. In this process we have followed [@VOF_Hirt79] and [@NASA_VOF2D_Torrey]. The equations are discretized using a finite-difference method. The computational domain is distributed onto a regular rectangular and staggered grid. The computations are performed in the stencil pattern. This implies that calculations are performed in close neighborhood of each grid’s cell.
Code Validation and Verification
--------------------------------
A homogeneous distribution of computations for the lid-driven cavity problem with a Reynolds number of 100 was used to benchmark the overall performance of the framework and verify the numerical implementation. We show the quantitative comparison of midsection centerline velocity with those by Ghia etc. [@GhiaLDC] in figure \[fig:comparison\].
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The figure to the left shows the quantitative comparison of midsection centerline velocity with those by Ghia etc. [@GhiaLDC]. The one to the right shows the contours of the X component of the velocity field along Y axis.[]{data-label="fig:comparison"}](figs/ldc_vx_y.png "fig:"){width="50.00000%"} ![The figure to the left shows the quantitative comparison of midsection centerline velocity with those by Ghia etc. [@GhiaLDC]. The one to the right shows the contours of the X component of the velocity field along Y axis.[]{data-label="fig:comparison"}](figs/ldc_vx_contour.png "fig:"){width="50.00000%"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
While these results come from a terascale machine, there is no logical barrier to continued scaling, and we plan to continue scaling studies as resources become available.
Performance and Scalability
---------------------------
We carried out performance and scaling tests on a 6 node GPU cluster at Cyfronet. Each node had 2 Tesla M2050 GPU’s, two Intel Xeon X5670 processors running at 2.93GHz, and an infiniband interconnect. The CFD code was measured for both the standalone code and the CaCUDA framework-based code. The performance results of one node were 43.5 and 58 GFlop/s for the standalone simulation and the simulation implemented within CaCUDA respectively. The scalability results of the 3D CFD code that makes use of the CaCUDA framework as well as the standalone version are shown in figure \[fig:cacudacfdbench\].
![This plot compares the speed-up of the CFD code built with CaCUDA to the standalone, handwritten implementation. Speed-ups are computed relative to the performance of the standalone code on a single node using a single GPU.[]{data-label="fig:cacudacfdbench"}](figs/cacudacfd_new){width="90.00000%"}
Conclusions
===========
In this paper an implementation of a new generic capability for computing on hybrid CPU/GPU architectures in the Cactus computational framework has been presented. The capability to handle the data exchange between GPU and CPU address space and deploying the computations in the hybrid environment was implemented as a new thorn “CaCUDA”. Moreover the application remarkably facilitates the implementation process by generating the templates of all declared kernel functions. Due to the flexibility and extensibility of the Cactus framework no changes to the Cactus flesh were necessary, guaranteeing that existing features and user implemented thorns are not affected by this addition.
As a test case application of these new framework’s features an incompressible CFD code has been implemented to test the overall performance and scalability. The results proving its usability have been presented.
Our current effort is focused on minimizing the costs of the data exchange between GPU and CPU and optimizing the boundary exchange. Further integration in this area may improve performance and scalability.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the Cybertools (http://cybertools .loni.org) project (NSF award 701491), the NG-CHC project (NSF award 1010640) through the Louisiana Board of Regents , and the NSF award PIF-0904015 (CIGR). This work used the computer resources provided by LSU/LONI. This research was supported in part by PL-Grid Infrastructure. This work is also supported by the UCoMS project under award number MNiSW(Polish Ministry of Science and Higher Education) Nr 469 1 N - USA/2009 in close collaboration with U.S. research institutions involved in the U.S. Department of Energy (DOE) funded grant under award number DE-FG02-04ER46136 and the Board of Regents, State of Louisiana, under contract no. DOE/LEQSF(2004-07). The authors want to thank our colleagues at both CCT and PSNC for great ideas and discussions. The authors want to thank Soon-Heum Ko from the National Supercomputing Centre at Linköping in Sweden for helping validating the CFD code.
|
---
abstract: '[We consider numerical methods for thermodynamic sampling, *i.e.* computing sequences of points distributed according to the Gibbs-Boltzmann distribution, using Langevin dynamics and overdamped Langevin dynamics (Brownian dynamics). A wide variety of numerical methods for Langevin dynamics may be constructed based on splitting the stochastic differential equations into various component parts, each of which may be propagated exactly in the sense of distributions. Each such method may be viewed as generating samples according to an associated invariant measure that differs from the exact canonical invariant measure by a stepsize-dependent perturbation. We provide error estimates à la Talay-Tubaro on the invariant distribution for small stepsize, and compare the sampling bias obtained for various choices of splitting method. We further investigate the overdamped limit and apply the methods in the context of driven systems where the goal is sampling with respect to a nonequilibrium steady state. Our analyses are illustrated by numerical experiments.]{} [Langevin dynamics; Stochastic differential equations; Numerical discretization; Canonical sampling; Molecular dynamics; Talay-Tubaro expansion; Nonequilibium.]{}'
author:
- |
[Benedict Leimkuhler[^1], Charles Matthews[^2] ]{}\
University of Edinburgh, School of Mathematics,\
James Clerk Maxwell Building, Edinburgh, EH9 3JZ, UK\
[and]{}\
[Gabriel Stoltz]{}[^3]\
Université Paris-Est, CERMICS (ENPC),\
INRIA, F-77455 Marne-la-Vallée, FRANCE
bibliography:
- 'ref.bib'
title: The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics
---
Introduction {#sec:introduction}
============
A fundamental purpose of molecular simulation is the computation of macroscopic quantities, typically through averages of functions of the variables of the system with respect to a given probability measure $\mu$ which defines the macroscopic state of the system. We consider systems described by a separable Hamiltonian $$\label{ham}
H(q,p) = V(q) + \frac12 p^T M^{-1}p,$$ where $q = (q_1,\dots,q_N)$ and $p=(p_1,\dots,p_N)$ respectively are the vectors of positions and momenta of $N$ particles in dimension $d$, $V$ is a potential energy function and $M$ is a positive definite mass matrix, typically a diagonal matrix. The Hamiltonian (\[ham\]) represents a fully classical molecular dynamics model. For instance, a fluid of $N$ argon atoms is well described by pairwise interactions among the nuclei, where the potential $V(q) = \sum_{1 {\leqslant}i < j {\leqslant}N} v(|q_i-q_j|).$ The distance based potential $v(r)$ may be fitted to Buckingham or Lennard-Jones forms (for instance, see [@frenksmit] or [@allentild]). These short-ranged potentials model van der Waals type interactions including both Pauli repulsion (the inability of the populated electron shells to interpenetrate) and dispersion due to temporary dipoles forming in the charge clouds surrounding the nuclei. In more complicated molecular systems, other potential energy functions are used to capture local covalent bond structure and Coulombic interactions due to charges on the atoms. Coarse-grained classical models may amalgamate several degrees of freedom, as for example when a molecule is replaced by a rigid body description. Classical molecular dynamics models are now a standard and widespread tool in almost every field of science and engineering. For example, see [@ne] for some applications in engineering, [@dd] for a discussion of the use of molecular dynamics in drug discovery and see also the motivation provided in classical textbooks on molecular simulation such as [@allentild; @frenksmit; @Schlick; @Tuckerman].
In the most common setting, the probability measure $\mu$ with respect to which averages are computed corresponds to the canonical ensemble. Its distribution is defined by the Boltzmann-Gibbs density, which models the configurations of a conservative system in contact with a heat bath at fixed temperature ${\mathrm{T}}$: $$\label{eq:canonical_measure}
\mu({\mathrm{d}q} \,{\mathrm{d}p}) = Z^{-1} {\mathrm{e}}^{-\beta H(q,p)} \, {\mathrm{d}q} \,{\mathrm{d}p},$$ where $\beta^{-1} = k_B{\mathrm{T}}$ with $k_B$ Boltzmann’s constant and $Z$ is a normalization constant ensuring that the integral of $\mu$ over the entirety of phase space is unity.
Molecular dynamics can be used for the study of a wide range of thermodynamic and structural properties. Typically, observables are chosen which capture the features of interest and numerical studies are aimed at computing the averages of these observables accurately. For instance, the average pressure in a three-dimensional fluid such as liquid argon is obtained by computing $\mathcal{P} = \mathbb{E}_\mu(\psi)$, the expectation of an observable $\psi$ with respect to the canonical measure $\mu$, where the pressure observable $\psi$ is defined as $$\psi(q,p) = \frac{1}{3 \mathcal{V}} \left( p^T M^{-1} p - \sum_{i=1}^N q_i \cdot \nabla_{q_i} V(q)
\right),$$ $\mathcal{V}$ being the physical volume of the box occupied by the fluid. By studying the variation in pressure with changes in a thermodynamic parameter (temperature or density), one may obtain part of the phase diagram of the material. Other observables may be used to model the determination of molecular form (shape and size) or structural rearrangement under different ambient conditions. It is for instance increasingly common to use molecular dynamics in biology to reveal allosteric mechanisms related to protein function or drug binding; in such cases the observable may measure the distance between two particular groups of atoms or their relative alignment; see [@dd] for examples and further references contained therein.
Numerically, the high-dimensional averages with respect to $\mu$ are often approximated as ergodic averages along discrete stochastic paths (Markov chains) constructed through numerical solution of certain stochastic differential equations (SDEs). There are two principal sources of approximation error in the computation of average properties such as $\mathbb{E}_\mu(\psi)$: (i) systematic bias (or [*perfect sampling bias*]{}) related to the use of a discretization method for the SDEs (and usually proportional to a power of the integration stepsize ${{\Delta t}}$), and (ii) statistical errors, due to the finite lengths of the sampling paths involved and the underlying variance of the random variables; see the presentation in Section 2.3.1 of [@LRS10]. In this article we are concerned with the systematic bias, specifically the systematic bias in long-term simulation, i.e. with respect to the invariant (or nonequilibrium steady-state) distribution.
One of the most popular choices of SDE system for sampling purposes is Langevin dynamics, which is given by: $$\label{langevin}
\left\{ \begin{aligned}
{\mathrm{d}q}_t & = M^{-1} p_t \, {\mathrm{d}t}, \\
{\mathrm{d}p}_t & = -\nabla V(q_t) \, {\mathrm{d}t} - \gamma M^{-1} p_t \, {\mathrm{d}t} + \sqrt{\frac{2\gamma}{\beta}} \, {\mathrm{d}W}_t,
\end{aligned} \right.$$ where ${\mathrm{d}W}_t$ is a standard $dN$-dimensional Wiener process. The friction intensity $\gamma>0$ is a free parameter which may be adjusted to enhance sampling efficiency. Under suitable conditions, the dynamics is ergodic for the Boltzmann-Gibbs distribution (see for instance [@Talay02; @MSH01; @CLS07] and references therein).
We will also be interested in nonequilibrium situations where a given system is subject to nonconservative driving and dissipative perturbations. In this case, the averages may be taken with respect to a stationary distribution which has no simple functional form. The simulation of nonequilibrium systems in their steady-states is one popular way to compute transport coefficients such as the thermal conductivity or the shear viscosity, as the linear response of an appropriate average property (see for instance [@EM08; @Tuckerman]). We discuss a specific example in Section \[sec:noneq\_systems\]: the computation of the mobility of a particle, which measures the tendency of the particle to flow in the direction of an external forcing. The mobility is related to the self-diffusion through Einstein’s relation (see below).
The aim of this work is to provide a numerical analysis of the perfect sampling bias in Langevin dynamics arising from numerical schemes obtained by a splitting strategy, building on studies such as [@Talay02] or [@BO10], and clarifying the sampling properties of recently proposed schemes (see [@SkeelIzaguirre; @Melchionna; @BussiParinello; @ParisDudesCharlieKnowsTheReference; @LM12]). Of particular interest is the behavior of methods in the overdamped limit $\gamma\to+\infty$ and variations of Langevin dynamics incorporating nonequilibrium forcings such as the addition of non-gradient forces (in which case the invariant measure is unknown). The idea behind splitting schemes for stochastic differential equations is to decompose the generator of the dynamics into a sum of generators associated with dynamics which are analytically integrable, or at least very simple to integrate. We refer to the individual splitting terms of the dynamics as “elementary dynamics” in the sequel. One example in the context of Langevin dynamics is the splitting scheme based on a symplectic integration of the Hamiltonian part of the dynamics combined with an exact treatment of the fluctuation-dissipation part. Such methods are more convenient to implement in molecular simulation codes than the implicit schemes proposed in [@Talay02] or [@MSH01], and are also efficient in practice (see [@LM13]). Some essential elements of the numerical analysis on the accuracy of such splitting schemes have been provided in [@BO10].
We focus in this article on the case where the position space is compact (e.g. a torus) since this is most relevant from the point-of-view of applications in condensed matter physics and biology, where periodic boundary conditions are typically used. This assumption simplifies the treatment of the Fokker-Planck operator associated to Langevin dynamics, and, with additional smoothness assumptions on the potential energy function, ensures regularity properties, discrete spectrum and spectral gap. In particular is the unique invariant probability measure of the Langevin process. We assume for simplicity that the positions belong to the torus $\mathcal{M}=(L\mathbb{T})^{dN}$ where $L>0$ denotes the size of the simulation cell, and denote by ${\mathcal{E}}= \mathcal{M} \times \mathbb{R}^{dN}$ the state space of the system, *i.e.* the set of all admissible configurations $(q,p)$.
Let us emphasize that we expect our results to hold for unbounded position spaces, under appropriate assumptions on the potential energy function. Our proofs may however require non-trivial modifications, using in particular the tools and the results from [@MSH01; @Talay02; @BO10; @Kopec]. Generalizations to other dynamics similar to Langevin dynamics such as Generalized Langevin Dynamics (see [@Mori; @Z73]), Dissipative Particle Dynamics (see [@HK92; @EW95]) or Nosé-Hoover-Langevin dynamics (see [@SaDeCh07; @LeNoTh09]) are also possible, although a rigorous extension would require substantial work in view of the estimates needed involving the generator of the dynamics for instance (see the discussion in Remark \[rmk:structure\_proof\]).
In practice, since Langevin dynamics is discretized, averages computed along a single trajectory converge to averages with respect to a measure $\mu_{\gamma,{{\Delta t}}}$, which is an approximation to $\mu$ in the sense that there exists a function $f_{\alpha,\gamma}$ for which $$\label{eq:error_estimate_intro}
\int_{\mathcal{E}}\psi(q,p) \, \mu_{\gamma,{{\Delta t}}}({\mathrm{d}q} \, {\mathrm{d}p}) = \int_{\mathcal{E}}\psi(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) + {{\Delta t}}^\alpha \int_{\mathcal{E}}\psi(q,p) f_{\alpha,\gamma}(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) + \mathrm{O}({{\Delta t}}^{\alpha+1}),$$ see Section \[sec:error\_estimates\_finite\_friction\] for precise statements. Of course, the momenta are usually trivial to sample since they are distributed according to a Gaussian measure. The primary issue is therefore to sample positions according to the marginal of the canonical measure: $$\label{eq:marginal_mu}
\overline{\mu}({\mathrm{d}q}) = \widetilde{Z}^{-1} {\mathrm{e}}^{-\beta V(q)} \, {\mathrm{d}q}.$$ Denoting by $\overline{\mu}_{\gamma,{{\Delta t}}}({\mathrm{d}q})$ the marginal of the invariant measure for the numerical scheme in the position variables, and by $$\label{eq:pi}
(\pi \varphi)(q) = \int_{{\mathbb{R}}^{dN}} \varphi(q,p) \, \kappa({\mathrm{d}p}), \qquad \kappa({\mathrm{d}p}) = \left(\frac{2\pi}{\beta}\right)^{-dN/2} \sqrt{\mathrm{det}(M)} \exp\left(-\frac{\beta p^T M^{-1} p}{2}\right) {\mathrm{d}p},$$ the partial average of a function $\varphi$ with respect to the momentum variable, the error estimate becomes, for observables which depend only on the position variable, $$\int_\mathcal{M} \psi(q) \, \overline{\mu}_{\gamma,{{\Delta t}}}({\mathrm{d}q}) = \int_\mathcal{M} \psi(q) \, \overline{\mu}({\mathrm{d}q}) + {{\Delta t}}^\alpha \int_\mathcal{M} \psi(q) (\pi f_{\alpha,\gamma})(q) \, \overline{\mu}({\mathrm{d}q}) + \mathrm{O}({{\Delta t}}^{\alpha+1}).$$
Let us conclude this introduction by noting that alternative sampling strategies are available: the bias in the invariant measure sampled by discretization of Langevin dynamics could in principle be eliminated by employing a Metropolis-Hastings procedure (see [@MRRTT53; @Hastings70] and the discussion in Section 2.2 of [@LRS10]). Another advantage of superimposing a Metropolis-Hastings procedure upon a discretization of Langevin dynamics is that it stabilizes the numerical scheme even for forces $-\nabla V$ which are not globally Lipschitz. The numerical analysis of Langevin-based Metropolis integrators has been performed in [@BV09] and [@BV12], where strong error estimates are provided. On the other hand, it is not always possible or desirable to use a Metropolis correction. First, the average acceptance probability in the Metropolis step for Langevin-like dynamics in general decreases exponentially with the dimension of the system for a *fixed* timestep (see for instance [@KP91]). In fact, the timestep should be reduced as some inverse power of the system size in order to maintain a constant acceptance rate (see the recent works on Metropolization of Hamiltonian dynamics by [@BPRSS13], following the strategy pioneered in [@RGG97; @RR98]). There are ways to limit the decrease of the ratio, by either changing the dynamics or the measure used to compute the Metropolis ratio (see for instance [@IH04] in the context of Hamiltonian dynamics), or by evolving only parts of the system (see [@BV12]). The latter strategy may however complicate the implementation of parallel algorithms for the simulation of very large systems, especially if long-range potentials are used (as acknowledged in Remark 2.5 of [@BV12]). This may be a reason why Metropolis corrections are not often implemented in popular molecular dynamics packages such as NAMD. Second, the variance of the computed averages may increase since rejections occur, and the numerical trajectory is therefore more correlated in general than for rejection-free dynamics. Lastly, the Metropolis procedure requires that the invariant measure of the system be known. This is the case for equilibrium systems, but no longer is the case for nonequilibrium systems subjected to external forcings such as a temperature gradient or a non-gradient force (this is the framework considered in Section \[sec:noneq\_systems\] of this article, see for instance the dynamics ).
Summary of the results and organization of the paper {#summary-of-the-results-and-organization-of-the-paper .unnumbered}
----------------------------------------------------
We focus in this article on first- and second-order splitting schemes, relying on Lie-Trotter decompositions of the evolution. This restriction is motivated both by pedagogical purposes and by the dominant role in applications played by second-order splitting schemes. Let us however emphasize that most of our results could, in principle, be extended to higher-order decompositions.
Results corresponding to discretizations of the equilibrium Langevin dynamics and computation of static average properties are gathered in Section \[sec:equilibrium\], while nonequilibrium systems and the computation of transport properties are discussed in Section \[sec:noneq\_systems\] (relying on the computation of the mobility or autodiffusion coefficient as an illustration). The proofs of all our results can be found in Section \[sec:proofs\].
Let us now highlight some of our contributions.
- In the equilibrium setting, we rigorously ground in Section \[sec:error\_estimates\_finite\_friction\] the results presented in [@LM12] giving the leading order correction to the invariant measure with respect to ${{\Delta t}}$ for general splitting schemes, via a Talay-Tubaro expansion (see [@TT90]). We carefully study all possible splitting schemes, taking advantage of what we call the “TU lemma” (Lemma \[lem:TU\]) to relate invariant measures of various splitting schemes where the elementary dynamics are integrated in different orders. From a technical viewpoint, our proofs are a variation on the standard way of establishing similar results since we use the specific structure of splitting schemes to conveniently write evolution operators as compositions of the semigroups of the elementary dynamics (working at the level of generators, as in [@DF12]; see also [@MST10] for a related approach based on solution of appropriate Poisson equations). The structure of the proof is highlighted in Section \[sec:proof\_thm:error\_first\_order\_schemes\], see Remark \[rmk:structure\_proof\].
- We show in Section \[sec:num\_estimation\_correction\] how the leading order correction to equilibrium averages can be estimated on-the-fly by approximating a time-integrated correlation function. This can be seen as a practical way of numerically solving a Poisson equation (a standard way of proceeding when studying linear response of nonequilibrium systems) and is an alternative to Romberg extrapolation to eliminate the leading order correction as done in [@TT90].
- We carefully study the overdamped regime $\gamma \to +\infty$ in Section \[sec:ovd\_limit\], making use in particular of uniform resolvent estimates obtained in Theorem \[lem:bounds\_CL\_gamma\] thanks to a uniform hypocoercivity property;
- We provide error estimates for the computation of transport coefficients, by assessing the bias arising in the numerical discretization of either (i) the computation of integrated time-correlation functions expressing transport coefficients via Green-Kubo formulae; or (ii) ergodic averages of steady-state nonequilibrium dynamics where the equilibrium evolution is perturbed by a non-gradient force and the transport coefficient is extracted from the linear response of some quantity of interest (see Section \[sec:noneq\_systems\]). The latter approach is illustrated by the study of the mobility, which measures the response in the average velocity arising from a constant external force exerted on the system. We also study the consistency of the numerical estimations in the overdamped limit.
Some numerical simulations are provided to illustrate the most important results (see Section \[sec:numerics\] and \[sec:error\_transport\]).
Error estimates for the invariant measure for equilibrium dynamics {#sec:equilibrium}
==================================================================
We start by giving some properties of Langevin dynamics in Section \[sec:ppties\_Lang\] (most results are well-known, except for the material on the overdamped limit $\gamma \to +\infty$ presented in Section \[sec:ovd\_lim\_Lang\]). The numerical schemes we consider are then described in Section \[sec:splitting\_schemes\], their ergodic properties being discussed in Section \[sec:ergo\_num\_scheme\]. Error estimates for the invariant measure are provided in Section \[sec:error\_estimates\_finite\_friction\]. We then show in Section \[sec:num\_estimation\_correction\] how to estimate the leading order correction term through an appropriate integrated correlation function. An important side result of this section is the development error estimates for Green-Kubo type formulas. Finally, we study the errors on the invariant measures in the overdamped limit in Section \[sec:ovd\_limit\]. Let us emphasize that we will make use of the following assumption throughout this work:\
[Assumption]{} 1: The potential $V$ belongs to $C^\infty(\mathcal{M},\mathbb{R})$.\
The above assumption is quite restrictive since typical potentials used in molecular simulation, such as the Lennard-Jones potential, have singularities. Although ergodicity for Langevin dynamics with singular potentials has been recently proved in [@CG10], there are still many issues with singular potentials, including the existence and uniqueness of an invariant measure for numerical schemes (see [@MSH01]), and the derivation of appropriate bounds or estimates on the resolvent of the generator of Langevin dynamics (all the results presented in Section \[sec:ergodicity\] below are obtained under the assumption of smooth potentials). Since the latter estimates are fundamental for our work, we have to restrict ourselves to smooth potentials. Of course, from a more practical viewpoint, it could also be argued that the potential energy function could be smoothed out by appropriate high energy truncations and regularizations, and that such regularizations should not affect too much the average properties of the system since high energy states are quite unlikely under the canonical measure.
Functional analysis setting and notation {#functional-analysis-setting-and-notation .unnumbered}
----------------------------------------
The reference Hilbert space for our analysis is the Hilbert space $L^2(\mu)$. As in [@Talay02] for instance, we will consider errors in the average of smooth functions whose derivatives grow at most polynomially (the space $\mathcal{S}$ defined below). In fact, since the position space is compact, only the growth in the momentum variable has to be controlled.
The polynomial growth of a function can be characterized by the Lyapunov functions: $${\mathcal{K}}_s(q,p) = 1 + |p|^{2s},$$ for $s \in \mathbb{N}^* = \{1,2,3,\ldots \}$. This allows us to define the following Banach spaces of functions of polynomial growth $$L^\infty_{{\mathcal{K}}_s} = \left \{ \psi \textrm{ measurable } \, \left| \, \frac{\psi}{{\mathcal{K}}_s} \in L^\infty({\mathcal{E}}) \right. \right\},$$ endowed with the norms $$\| \psi \|_{L^\infty_{{\mathcal{K}}_s}} = \left\| \frac{\psi}{{\mathcal{K}}_s}\right\|_{L^\infty}.$$ To characterize the growth of the derivatives, we introduce the spaces $W^{m,\infty}_{{\mathcal{K}}_s}$ defined as $$W^{m,\infty}_{{\mathcal{K}}_s} = \Big\{ f \in L^\infty_{{\mathcal{K}}_s} \ \Big| \ \forall r \in \mathbb{N}^{2dN}, \ |r| {\leqslant}m, \ \partial^r f \in L^\infty_{{\mathcal{K}}_s} \Big\},$$ where $|r| = r_1+r_2+\dots+r_{2dN}$, and $\partial^r$ stands for $\partial_{q_1}^{r_1} \dots \partial_{q_{dN}}^{r_{dN}} \partial_{p_1}^{r_{dN+1}} \dots \partial_{p_{dN}}^{r_{2dN}}$.
The set $\mathcal{S}$ of smooth functions is the set of functions $f \in L^2(\mu)$ such that, for any $m {\geqslant}0$, there exists $s {\geqslant}0$ (depending on $f$ and $m$) so that $f \in W^{m,\infty}_{{\mathcal{K}}_s}$. The subset $\widetilde{\mathcal{S}} \subset \mathcal{S}$ is composed of the functions with average zero with respect to $\mu$: $$\widetilde{\mathcal{S}} = \left\{ f \in \mathcal{S} \ \left| \ \int_{\mathcal{E}}f\,d\mu = 0 \right.\right\}.$$
Some of our results will be stated in the weighted Sobolev spaces $H^m(\mu)$ defined as $$H^m(\mu) = \left\{ f \in L^2(\mu) \ \left| \ \forall r \in \mathbb{N}^{2dN}, \ |r| {\leqslant}m, \ \partial^r f \in L^2(\mu) \right.\right\},$$ endowed with the norm $$\| f \|^2_{H^m(\mu)} = \| u \|^2_{L^2(\mu)} + \sum_{\substack{r \in \mathbb{N}^{2dN} \cr 1{\leqslant}|r| {\leqslant}m}} \| \partial^r f \|^2_{L^2(\mu)}.$$ Note that $W^{m,\infty}_{{\mathcal{K}}_s} \subset H^m(\mu)$ since the function ${\mathcal{K}}_s$ is in $L^2(\mu)$. We will also occasionally need the Sobolev spaces $H^m(\kappa)$ of functions of the $p$ variable only whose derivatives up to order $m$ are square-integrable with respect to the probability measure $\kappa(dp)$.
Unless stated otherwise, all the operators appearing below are by default considered as operators defined on the core $\mathcal{S}$, with range contained in $\mathcal{S}$. Some results are stated on extensions of the operators under consideration to (sub)spaces of $H^1(\mu)$ or $L^\infty_{{\mathcal{K}}_s}$. With some abuse of notation, we will denote the extension of operators by the same letter. The appropriate domain of the operators should always be clear from the context. When an operator $T$ is defined on the core $\mathcal{S}$, we denote by $T^*$ its formal adjoint, which is the operator defined on $\mathcal{S}$ such that, for all $(f,g)\in\mathcal{S}^2$, $$\langle f, Tg\rangle_{L^2(\mu)} = \int_{\mathcal{E}} f(q,p) \, (Tg)(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) = \int_{\mathcal{E}} (T^*f)(q,p) \, g(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) = \langle T^*f, g\rangle_{L^2(\mu)}.$$ When $T$ is a differential operator with smooth coefficient (which will be the case in many situations here), the action of the formal adjoint is found using integration by parts.
Properties of equilibrium Langevin dynamics {#sec:ppties_Lang}
-------------------------------------------
Langevin dynamics can be seen as Hamiltonian dynamics perturbed by an Ornstein-Uhlenbeck process in the momenta with friction coefficient $\gamma > 0$: $$\label{eq:Langevin}
\left\{ \begin{aligned}
{\mathrm{d}q}_t & = M^{-1} p_t \, {\mathrm{d}t}, \\
{\mathrm{d}p}_t & = -\nabla V(q_t) \, {\mathrm{d}t} - \gamma M^{-1} p_t \, {\mathrm{d}t} + \sqrt{\frac{2\gamma}{\beta}} \, {\mathrm{d}W}_t,
\end{aligned} \right.$$ where $W_t$ is a $dN$-dimensional standard Brownian motion and $M$ is the mass matrix of the system. We assume that the mass matrix is diagonal: $M = \mathrm{diag}(m_1 \mathrm{I}_d,\dots,m_N \mathrm{I}_d)$, so that momenta are Gaussian random vectors under the canonical measure, with unit covariance, and hence the components of $p$ are very easy to sample. Note that we formulate here the dynamics using friction forces proportional to the velocity of the particles.
The existence and uniqueness of strong solutions is guaranteed when the position space is compact since the kinetic energy function $1+|p|^2$ is a Lyapunov function, see for instance Theorem 5.9 in [@rey-bellet]. We will sometimes denote by $(q_{\gamma,t},p_{\gamma,t})$ the solution of this equation to emphasize the dependence on the friction coefficient.
In order to describe more conveniently splitting schemes, it is useful to introduce the elementary dynamics with generators (defined on the core $\mathcal{S}$) $$\label{eq:def_ABC}
A = M^{-1} p \cdot \nabla_q,
\qquad
B = -\nabla V(q) \cdot \nabla_p,
\qquad
C = -M^{-1} p \cdot \nabla_p + \frac1\beta \Delta_p.$$ The generator $\mathcal{L}_\gamma$ for equilibrium Langevin dynamics , defined on the core $\mathcal{S}$, is the sum of the generators of the elementary dynamics: $$\mathcal{L}_\gamma = A + B + \gamma C,$$ where $\mathcal{L}_0 = A+B$ is the generator associated with the Hamiltonian part of the dynamics. The invariance of the canonical measure $\mu$ defined in for Langevin dynamics can be rewritten in terms of the generator $\mathcal{L}_\gamma$: for any test function $\varphi \in \mathcal{S}$, $$\label{eq:inv_mu_by_L}
\int_\mathcal{E} \mathcal{L}_\gamma \varphi \, {\mathrm{d}\mu} = 0.$$ In fact, the operators $A+B$ and $C$ separately preserve $\mu$. Recall also that, thanks to the compact embedding of $$H^1(\kappa) \cap \mathrm{Ker}(\pi) = \left\{ f \in H^1(\kappa) \, \left| \int_{\mathbb{R}^{dN}} f(p) \, \kappa({\mathrm{d}p}) = 0 \right. \right \}$$ in $L^2(\kappa) \cap \mathrm{Ker}(\pi)$, it is easy to show that the operator $C^{-1}$ is compact and positive definite on $L^2(\kappa) \cap \mathrm{Ker}(\pi)$. It is also easy to check that $$(A+B)^* = -(A+B), \qquad C^* = C,$$ where, we recall, the adjoints are formally defined as operators on $\mathcal{S}$ through integration by parts. Note that the formal adjoint $$\label{eq:reversibility_Langevin}
\mathcal{L}^*_\gamma = -(A + B) + \gamma C$$ defined on $\mathcal{S}$ has an action quite similar to the action of the generator $\mathcal{L}_\gamma$ defined on $\mathcal{S}$. Functional estimates valid for (extensions of) $\mathcal{L}_\gamma$ will therefore also hold for (extensions of) the formal adjoint of this operator. The equality expresses the reversibility up to momentum reversal of Langevin dynamics with respect to the invariant measure $\mu$ (see the discussion in Section 2.2.3 of [@LRS10]). In particular, introducing the bounded, unitary operator on $L^2(\mu)$ $$\label{eq:op_R}
(\mathcal{R}\varphi)(q,p) = \varphi(q,-p),$$ can be reformulated $\mathcal{R} \mathcal{L}_\gamma \mathcal{R} = \mathcal{L}_\gamma^*$.
### Ergodicity results {#sec:ergodicity}
The ergodicity of Langevin dynamics for $\gamma > 0$, understood either as the almost sure convergence of time averages along a realization of the dynamics, or the long-time convergence of the law of the process to $\mu$, is well established, see for instance [@MSH01; @Talay02; @CLS07] and references therein. These references rely on the use of Lyapunov functions, following strategies of proofs pioneered in the Markov Chain community (see [@MeynTweedie]), although alternative proofs relying on analytical tools exist (see [@rey-bellet; @HM11]). In any case, the evolution semigroup can be given a meaning in a weighted $L^\infty$ space, and the measure $\mu$ is the unique invariant measure of the dynamics. This property can be translated as $\mathrm{Ker}({\mathcal{L}_\gamma}) = \mathbb{C}\mathbf{1}$.
An alternative way to prove the long-time convergence of the law of the process is to use subelliptic or hypocoercive estimates as studied in [@Talay02; @EH03; @HN04; @Villani; @HP08]. An important result of hypocoercivity in this case is that there exist $K_\gamma, \lambda_\gamma > 0$ such that the semigroup ${\mathrm{e}}^{t {\mathcal{L}_\gamma}}$, defined on the core $\widetilde{\mathcal{S}}$, can be extended to a bounded operator on an appropriate subspace of $H^1(\mu)$: $$\label{eq:semigroup_estimates_H1mu}
\| {\mathrm{e}}^{t {\mathcal{L}_\gamma}} \|_{\mathcal{B}({\mathcal{H}}^1)} {\leqslant}K_\gamma {\mathrm{e}}^{-\lambda_\gamma t},$$ where the subspace $${\mathcal{H}}^1 = H^1(\mu) \backslash \mathrm{Ker}({\mathcal{L}_\gamma}) = \left\{ u \in H^1(\mu) \ \left| \ \int_{\mathcal{E}}u \, {\mathrm{d}\mu} = 0 \right. \right\}$$ of the Hilbert space $H^1(\mu)$ is endowed with the norm $\| u \|^2_{H^1(\mu)} = \| u \|_{L^2(\mu)}^2 + \| \nabla_p u \|_{L^2(\mu)}^2 + \| \nabla_q u\|_{L^2(\mu)}^2$, and $\| \cdot \|_{\mathcal{B}({\mathcal{H}}^1)}$ is the operator norm on ${\mathcal{H}}^1$. A similar bound holds for ${\mathrm{e}}^{t {\mathcal{L}_\gamma}^*}$. In particular, the operators $\mathcal{L}_\gamma$ and ${\mathcal{L}_\gamma}^*$ are invertible on ${\mathcal{H}}^1$, and $$\label{eq:stability_H1}
\left\| {\mathcal{L}_\gamma}^{-1} \right\|_{\mathcal{B}({\mathcal{H}}^1)} {\leqslant}\frac{K_\gamma}{\lambda_\gamma}.$$ Note also that the same bound holds for $(\mathcal{L}_\gamma^*)^{-1}$.
For unbounded position spaces, the potential $V$ has to satisfy some assumptions for to hold (such as a Poincaré inequality for ${\mathrm{e}}^{-\beta V}$), but these assumptions are trivially satisfied when the position space is compact, as is the case here. An important issue is the dependence on $\gamma$ of the constants $K_\gamma,\lambda_\gamma$, or at least the dependence on $\gamma$ of the resolvent norm $\left\| {\mathcal{L}_\gamma}^{-1} \right\|_{\mathcal{B}({\mathcal{H}}^1)}$. This is made precise in the results presented below in Section \[sec:ham\_limit\] and \[sec:ovd\_lim\_Lang\].
Before presenting these asymptotic estimates, let us first recall that a careful analysis of the proof presented in [@Talay02], as provided by [@Kopec], allows to prove the following result.
\[thm:stability\_S\] The space $\widetilde{\mathcal{S}}$ is stable under ${\mathcal{L}_\gamma}^{-1}$ and $({\mathcal{L}_\gamma}^*)^{-1}$.
This result is of fundamental importance in our proofs. It allows to state that, if the operators $T_1,\dots,T_M$ are well defined operators from $\widetilde{\mathcal{S}}$ to $\widetilde{\mathcal{S}}$, then the operator ${\mathcal{L}_\gamma}^{-1} T_M {\mathcal{L}_\gamma}^{-1} \dots {\mathcal{L}_\gamma}^{-1} T_1 {\mathcal{L}_\gamma}^{-1}$ also is a well defined operator from $\widetilde{\mathcal{S}}$ to $\widetilde{\mathcal{S}}$.
### Hamiltonian limit $\gamma \to 0$ {#sec:ham_limit}
When $\gamma = 0$, Langevin dynamics reduces to the Hamiltonian dynamics, whose generator $A+B$ has a kernel much larger than $\mathrm{Ker}({\mathcal{L}_\gamma}) = \mathbb{C}\mathbf{1}$. It is therefore expected that the operator norm of ${\mathcal{L}_\gamma}^{-1}$ diverges as $\gamma \to 0$. The rate of divergence is made precise in the following theorem, summarizing the results from Theorem 1.6 and Proposition 6.3 of [@HP08].
\[thm:Ham\_limit\_Lgam\] Denote by $\| \cdot \|_{\mathcal{B}({\mathcal{H}}^0)}$ the operator norm on the subspace $$\label{eq:def_H0}
{\mathcal{H}}^0 = \left\{ u \in L^2(\mu) \ \left| \ \int_{\mathcal{E}}u \, {\mathrm{d}\mu} = 0 \right. \right\}$$ of the Hilbert space $L^2(\mu)$. There exists two constants $c_-, c_+ > 0$ such that, for any $0 < \gamma {\leqslant}1$, $$\frac{c_-}{\gamma} {\leqslant}\left\| \mathcal{L}_\gamma^{-1} \right\|_{\mathcal{B}(\mathcal{H}^0)} {\leqslant}\frac{c_+}{\gamma}.$$
We state the result with the upper bound $\gamma {\leqslant}1$, but it holds in fact for $0 < \gamma {\leqslant}\gamma_{\rm max}$ for any finite value $\gamma_{\rm max}>0$. Note also that the same bound holds for $(\mathcal{L}_\gamma^*)^{-1}$.
### Overdamped limit $\gamma \to +\infty$ {#sec:ovd_lim_Lang}
The overdamped limit can be obtained by either letting the friction go to infinity in together with an appropriate rescaling of time; or by letting masses go to 0. When discussing overdamped limits in this article, we will always set the mass matrix $M$ to identity and consider the limit $\gamma \to +\infty$. Since we restrict our attention to the invariant measure of the system, the time rescaling is not relevant.
Let us describe more precisely the convergence result. It is shown in Section 2.2.4 of [@LRS10] for instance that the solutions of observed over long times, namely $(q_{\gamma,\gamma s},p_{\gamma,\gamma s})_{s {\geqslant}0}$, converge pathwise on finite time intervals $s \in [0,t]$ to the solutions of overdamped Langevin dynamics $$\label{eq:overdamped}
{\mathrm{d}Q}_t = -\nabla V(Q_t) \, {\mathrm{d}t} + \sqrt{\frac2\beta} \, {\mathrm{d}W}_t,$$ with the same initial condition $Q_0 = q_{\gamma,0}$. The process is ergodic on the compact position space $\mathcal{M}$, with unique invariant probability measure $\overline{\mu}({\mathrm{d}q})$ defined in . Its generator $${\mathcal{L}_{\rm ovd}}= -\nabla V(q) \cdot \nabla_q + \frac1\beta \Delta_q,$$ defined on the core $\mathcal{S} \cap \mathrm{Ker}(\pi) = C^\infty(\mathcal{M})$, is an elliptic operator which is symmetric on $L^2(\overline{\mu})$, with compact resolvent (see for instance the discussion and the references in Section 2.3.2 of [@LRS10]). It is easy to see that the inverse operator ${\mathcal{L}_{\rm ovd}}^{-1}$ can be extended to a bounded operator from $$\widetilde{H}^m(\overline{\mu}) = \left\{ \varphi \in H^m(\overline{\mu}) \, \left| \int_\mathcal{M} \varphi \, {\mathrm{d}\overline{\mu}} = 0 \right.\right\}$$ to $\widetilde{H}^{m+2}(\overline{\mu})$. Let us finally mention that the set of $C^\infty(\mathcal{M})$ functions with average zero with respect to $\overline{\mu}$ is of course stable with respect to ${\mathcal{L}_{\rm ovd}}^{-1}$.
The following result gives bounds on the resolvent of the Langevin generator in the overdamped regime, and in fact quantifies the difference between the resolvent ${\mathcal{L}_\gamma}^{-1}$ and the resolvent ${\mathcal{L}_{\rm ovd}}^{-1}$ appropriately rescaled by a factor $\gamma$.
\[lem:bounds\_CL\_gamma\] There exist two constants $c_-, c_+ > 0$ such that, for any $\gamma {\geqslant}1$, $$\label{eq:crude_bounds_Lgamma_ovd}
c_- \gamma {\leqslant}\| {\mathcal{L}}_\gamma^{-1} \|_{\mathcal{B}({\mathcal{H}}^1)} {\leqslant}c_+ \gamma.$$ More precisely, there exists a constant $K > 0$ such that, for any $\gamma {\geqslant}1$, $$\label{eq:divergent_behavior_Lgamma}
\begin{aligned}
\left\| {\mathcal{L}}_\gamma^{-1} - \gamma {\mathcal{L}_{\rm ovd}}^{-1} \pi - p^T \nabla_q{\mathcal{L}_{\rm ovd}}^{-1}\pi + {\mathcal{L}_{\rm ovd}}^{-1} \pi (A+B) C^{-1} ({\mathrm{Id}}- \pi) \right \|_{\mathcal{B}({\mathcal{H}}^1)} {\leqslant}\frac{K}{\gamma}, \\
\left\| \left({\mathcal{L}}_\gamma^*\right)^{-1} - \gamma {\mathcal{L}_{\rm ovd}}^{-1} \pi + p^T \nabla_q{\mathcal{L}_{\rm ovd}}^{-1}\pi - {\mathcal{L}_{\rm ovd}}^{-1} \pi (A+B) C^{-1} ({\mathrm{Id}}- \pi) \right \|_{\mathcal{B}({\mathcal{H}}^1)} {\leqslant}\frac{K}{\gamma},
\end{aligned}$$ where the operator $\pi$ is defined in , and $(C^{-1}\psi)(q,p)$ is understood as applying the operator $C^{-1}$ to the function $\psi(q,\cdot) \in L^2(\kappa)$ for all values of $q \in \mathcal{M}$.
Note that the function ${\mathcal{L}_{\rm ovd}}^{-1} \pi f$ is well defined since, as $f$ belongs to ${\mathcal{H}}^1$, the function $\pi f$ has a vanishing average with respect to $\overline{\mu}$. The fact that ${\mathcal{L}_{\rm ovd}}^{-1} \pi (A+B) C^{-1} ({\mathrm{Id}}- \pi)$ is bounded on ${\mathcal{H}}^1$ is discussed in the proof of Theorem \[lem:bounds\_CL\_gamma\]. An important ingredient in the proof is the following estimate, which we call uniform hypocoercivity estimate.
\[lem:bounded\_resolvent\_perp\] Consider the following subspace of $\mathcal{H}^1$: $${\mathcal{H}}_\perp^1 = \left\{ u \in {\mathcal{H}}^1 \ \left| \ \overline{u}(q) = \int_{{\mathbb{R}}^{dN}} u(q,p) \, \kappa({\mathrm{d}p}) = 0 \right. \right\}.$$ There exists a constant $K > 0$ such that, for any $\gamma {\geqslant}1$, $$\forall f \in {\mathcal{H}}^1_\perp, \qquad \| {\mathcal{L}}_\gamma^{-1} f \|_{H^1(\mu)} {\leqslant}K \| f \|_{H^1(\mu)}.$$
The proofs of Theorem \[lem:bounds\_CL\_gamma\] and Lemma \[lem:bounded\_resolvent\_perp\] are provided in Section \[sec:proof\_L\_gamma\_large\].
Splitting schemes for equilibrium Langevin dynamics {#sec:splitting_schemes}
---------------------------------------------------
We present in this section the splitting schemes to be examined in this article. These schemes can be described by evolution operators $P_{{\Delta t}}$ defined on the core $\mathcal{S}$ (but which can be extended to bounded operators on $L^\infty(\mathcal{E})$), and which are such that the Markov chain $(q^n,p^n)$ generated by the discretization satisfies $$P_{{\Delta t}}\psi(q,p) = \mathbb{E}\Big( \psi\left(q^{n+1},p^{n+1}\right)\Big| (q^n,p^n) = (q,p)\Big).$$ We also briefly give some ergodicity results obtained by minor extensions or variations of existing results in the literature (see in particular [@MSH01; @Talay02; @BO10; @BH13]). Since these ergodicity issues are by now a rather standard and well-understood matter, especially for compact position spaces, we provide only elements of proofs in Section \[sec:proof\_ergodicity\_MC\].
### First-order splitting schemes {#sec:splitting_schemes_1st}
First-order schemes are obtained by a Lie-Trotter splitting of the elementary evolutions generated by $A,B,\gamma C$. The motivation for this splitting is that all elementary evolutions are analytically integrable (see the expressions of the associated semigroups in ). There are 6 possible schemes, whose evolution operators (defined on the core $\mathcal{S}$) are of the general form $$P^{Z,Y,X}_{{\Delta t}}= {\mathrm{e}}^{{{\Delta t}}Z} {\mathrm{e}}^{{{\Delta t}}Y} {\mathrm{e}}^{{{\Delta t}}X},$$ with all possible permutations $(Z,Y,X)$ of $(A, B, \gamma C)$. For instance, the numerical scheme associated with $P_{{\Delta t}}^{B,A,\gamma C}$ is $$\label{eq:Langevin_splitting}
\left\{ \begin{aligned}
\widetilde{p}^{n+1} & = p^n - {{\Delta t}}\, \nabla V(q^{n}), \\
q^{n+1} & = q^n + {{\Delta t}}\, M^{-1} \widetilde{p}^{n+1}, \\
p^{n+1} & = \alpha_{{\Delta t}}\widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha^2_{{\Delta t}}}{\beta}M} \, G^n,
\end{aligned} \right.$$ where $\alpha_{{\Delta t}}= \exp(-\gamma M^{-1} {{\Delta t}})$, and $(G^n)$ are independent and identically distributed Gaussian random vectors with identity covariance. The simulation of the dynamics with generator $C$ is very simple for diagonal mass matrix $M$ since $\alpha_{{\Delta t}}$ is a diagonal matrix. Note that the order of the operations performed on the configuration of the system is the inverse of the order of the operations mentioned in the superscript of the evolution operator $P_{{\Delta t}}^{B,A,\gamma C}$ when read from right to left. This inversion is known as *Vertauschungssatz* (see for instance the discussion in Section III.5.1 of [@HairerLubichWanner06]). It arises from the fact that the numerical method modifies the distribution of the variables, whereas the evolution operator encodes the evolution of observables (determined by the adjoint of the operator encoding the evolution of the distribution).
The iterations of the three schemes associated with $P^{\gamma C,B,A}_{{\Delta t}},P^{B,A,\gamma C}_{{\Delta t}},P^{A,\gamma C,B}_{{\Delta t}}$ share a common sequence of update operations, as for $P^{\gamma C,A,B}_{{\Delta t}},P^{A,B,\gamma C}_{{\Delta t}},P^{B,\gamma C,A}_{{\Delta t}}$. More precisely, we mean that equalities of the following form hold: $$\label{eq:similar_evolution_operators}
\left(P^{A,B,\gamma C}_{{\Delta t}}\right)^n = T_{ {{\Delta t}}} \left(P^{\gamma C,A,B}_{{\Delta t}}\right)^{n-1} U_{\gamma,{{\Delta t}}},
\qquad
U_{\gamma,{{\Delta t}}} = {\mathrm{e}}^{\gamma {{\Delta t}}C},
\qquad
T_{ {{\Delta t}}} = {\mathrm{e}}^{{{\Delta t}}A} {\mathrm{e}}^{{{\Delta t}}B}.$$ It is therefore not surprising that the invariant measures of the schemes with operators composed in the same order have very similar properties, as made precise in Theorem \[thm:error\_first\_order\_schemes\], relying on Lemma \[lem:TU\].
### Second-order schemes {#sec:splitting_schemes_2nd}
Second-order schemes are obtained by a Strang splitting of the elementary evolutions generated by $A,B,\gamma C$. There are also 6 possible schemes, which are of the general form $$P^{Z,Y,X,Y,Z}_{{\Delta t}}= {\mathrm{e}}^{{{\Delta t}}Z/2} {\mathrm{e}}^{{{\Delta t}}Y/2} {\mathrm{e}}^{{{\Delta t}}X} {\mathrm{e}}^{{{\Delta t}}Y/2} {\mathrm{e}}^{{{\Delta t}}Z/2},$$ with the same possible orderings as for first-order schemes. Again, these schemes can be classified into three groups depending on the ordering of the operators once the elementary one-step evolution is iterated: (i) $P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}, P_{{\Delta t}}^{A,B,\gamma C,B,A}$, (ii) $P_{{\Delta t}}^{\gamma C,A,B,A,\gamma C}, P_{{\Delta t}}^{B,A,\gamma C,A,B}$, and (iii) $P_{{\Delta t}}^{B,\gamma C,A,\gamma C,B}, P_{{\Delta t}}^{A,\gamma C,B,\gamma C,A}$. We discard the latter category since the invariant measures of the associated numerical schemes are not consistent with $\overline{\mu}$ in the overdamped limit (see Section \[sec:ovd\_limit\]).
### Geometric Langevin Algorithms {#sec:splitting_schemes_GLA}
In fact, as already proved in [@BO10] (see also Corollary \[cor:error\_GLA\] below), second order accuracy of the invariant measure can be obtained by resorting to a first-order splitting between the Hamiltonian and the Ornstein-Uhlenbeck parts, and discretizing the Hamiltonian part with a second-order scheme. This corresponds to the following evolution operators of Geometric Langevin Algorithm (GLA) type: $$\label{eq:GLA_schemes}
\begin{aligned}
P_{{\Delta t}}^{\gamma C,A,B,A} = {\mathrm{e}}^{\gamma {{\Delta t}}C} {\mathrm{e}}^{{{\Delta t}}A /2} {\mathrm{e}}^{{{\Delta t}}B} {\mathrm{e}}^{{{\Delta t}}A/2}, & \qquad P_{{\Delta t}}^{\gamma C,B,A,B} = {\mathrm{e}}^{\gamma {{\Delta t}}C} {\mathrm{e}}^{{{\Delta t}}B /2} {\mathrm{e}}^{{{\Delta t}}A} {\mathrm{e}}^{{{\Delta t}}B/2}, \\
P_{{\Delta t}}^{A,B,A,\gamma C} = {\mathrm{e}}^{{{\Delta t}}A /2} {\mathrm{e}}^{{{\Delta t}}B} {\mathrm{e}}^{{{\Delta t}}A/2} {\mathrm{e}}^{\gamma {{\Delta t}}C}, & \qquad P_{{\Delta t}}^{B,A,B,\gamma C} = {\mathrm{e}}^{{{\Delta t}}B /2} {\mathrm{e}}^{{{\Delta t}}A} {\mathrm{e}}^{{{\Delta t}}B/2} {\mathrm{e}}^{\gamma {{\Delta t}}C}.
\end{aligned}$$
Ergodicity results for splitting schemes {#sec:ergo_num_scheme}
----------------------------------------
Let us now give some technical results on the ergodic behavior of the splitting schemes presented in Section \[sec:splitting\_schemes\].In this section we denote the evolution operator by $P_{{\Delta t}}$ (supressing the dependence on the friction parameter $\gamma$ although the constants appearing in the results below a priori depend on this parameter). Ergodicity results for a fixed value of ${{\Delta t}}$ are obtained with techniques similar to the ones presented in [@MeynTweedie], by mimicking the proofs presented for certain discretization schemes of the Langevin equation in [@MSH01; @Talay02; @BO10]. A more subtle point is to obtain rates of convergence which are uniform in the timestep ${{\Delta t}}$, as done in [@BH13] for a class of Metropolis-Hastings schemes based on a discretization of overdamped Langevin dynamics in unbounded spaces as the proposal. We are able here to prove such results by relying on the fact that the position space $\mathcal{M}$ is compact.
The proof is based on two preliminary results, namely a uniform drift inequality or Lyapunov condition and a uniform minorization condition (see Section \[sec:proof\_ergodicity\_MC\] for the proofs). The term uniform refers to estimates which are independent of the timestep ${{\Delta t}}$. To obtain such estimates, we have to consider evolutions over fixed times $T \simeq n{{\Delta t}}$, which amounts to iterating the elementary evolution $P_{{\Delta t}}$ over $\lceil T/{{\Delta t}}\rceil$ timesteps (where $\lceil x \rceil$ denotes the smallest integer larger than $x$).
\[lem:Lyapunov\] For any $s^* \in \mathbb{N}^*$, there exist ${{\Delta t}}^* > 0$ and $C_a,C_b >0$ such that, for any $1 {\leqslant}s {\leqslant}s^*$ and $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$, $$\label{eq:moment_estimates}
P_{{\Delta t}}{\mathcal{K}}_s {\leqslant}{\mathrm{e}}^{-C_a {{\Delta t}}} {\mathcal{K}}_s + C_b {{\Delta t}}.$$ In particular, for any $T > 0$, $$\label{eq:moment_estimates_uniform}
P_{{\Delta t}}^{\lceil T/{{\Delta t}}\rceil} {\mathcal{K}}_s {\leqslant}\exp(-C_a T) {\mathcal{K}}_s + \frac{C_b {{\Delta t}}}{1 - {\mathrm{e}}^{-C_a {{\Delta t}}}}.$$
\[lem:minorization\] Consider $T > 0$ sufficiently large, and fix any $p_{\rm max} > 0$. There exist ${{\Delta t}}^*, \alpha > 0$ and a probability measure $\nu$ such that, for any bounded, measurable non-negative function $f$, and any $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$, $$\inf_{|p| {\leqslant}p_{\rm max}} \left( P_{{\Delta t}}^{\lceil T/{{\Delta t}}\rceil}f \right)(q,p) {\geqslant}\alpha \int_{\mathcal{E}}f(q,p) \, \nu({\mathrm{d}q} \, {\mathrm{d}p}).$$
Lemma \[lem:minorization\] ensures that Assumption 2 in [@HM11] holds for any choice of Lyapunov function ${\mathcal{K}}_s$ ($s {\geqslant}1$), provided $p_{\rm max}$ is chosen to be sufficiently large. The uniform minorization condition can formally be rewritten as $$\forall (q_0,p_0) \in \mathcal{M} \times B(0,p_{\rm max}),
\qquad
P_{{\Delta t}}\Big((q_0,p_0),{\mathrm{d}q}\,{\mathrm{d}p}\Big) {\geqslant}\alpha \nu({\mathrm{d}q}\,{\mathrm{d}p}).$$ We present a direct proof of Lemma \[lem:minorization\] in Section \[sec:proof\_ergodicity\_MC\]. Extending this result to unbounded position spaces is much more difficult in general, see for instance the recent works [@KV06; @KV13] and [@BH13] where non-degeneracy of the noise is assumed.
Let us now precisely state the ergodicity result.
\[prop:ergodicity\_MC\] Fix $s^* {\geqslant}1$. For any $0 < \gamma < +\infty$, there exists ${{\Delta t}}^*>0$ such that, for any $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$, the Markov chain associated with $P_{{\Delta t}}$ has a unique invariant probability measure $\mu_{\gamma,{{\Delta t}}}$, which admits a density with respect to the Lebesgue measure ${\mathrm{d}q} \, {\mathrm{d}p}$, and has finite moments: There exists $R>0$ such that, for any $1 {\leqslant}s {\leqslant}s^*$, $$\label{eq:moment_estimate}
\int_{\mathcal{E}}{\mathcal{K}}_s \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} {\leqslant}R < +\infty,$$ uniformly in the timestep ${{\Delta t}}$. There also exist $\lambda, K > 0$ (depending on $s^*$ and $\gamma$ but not on ${{\Delta t}}$) such that, and for all functions $f \in L^\infty_{{\mathcal{K}}_s}$, the following holds for almost all $(q,p) \in {\mathcal{E}}$: $$\label{eq:ergodicity_num}
\forall n \in \mathbb{N}, \qquad \left| \left(P_{{\Delta t}}^n f\right)(q,p) - \int_{\mathcal{E}}f {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} \right| {\leqslant}K \, {\mathcal{K}}_s(q,p) \, {\mathrm{e}}^{-\lambda n {{\Delta t}}} \, \| f \|_{L^\infty_{{\mathcal{K}}_s}}.$$
Let us again emphasize that, compared to the results of [@MSH01; @Talay02; @BO10], the only new estimate is the uniform-in-${{\Delta t}}$ decay rate in as obtained in [@BH13] for Metropolis schemes. These uniform estimates follow from an application of the results of [@HM11] to the sampled chain $P_{{\Delta t}}^{\lceil T/{{\Delta t}}\rceil}$ (see Section \[sec:proof\_ergodicity\_MC\] for more detail). Recall also that the convergence rates we obtain of course depend on the friction parameter $\gamma$.
An interesting consequence of the above estimates is that we are able to obtain uniform control of the resolvent of the operator ${\mathrm{Id}}-P_{{\Delta t}}$ extended to appropriate Banach spaces. Such a bound will prove useful to control approximation errors in Green-Kubo type formulas (see Section \[sec:num\_estimation\_correction\]). Note indeed that the estimate implies the operator bound $$\left \| P_{{\Delta t}}^n \right\|_{\mathcal{B}(L^\infty_{{\mathcal{K}}_s,{{\Delta t}}})} {\leqslant}K \, {\mathrm{e}}^{-\lambda n {{\Delta t}}},$$ on the Banach space $$L^\infty_{{\mathcal{K}}_s,{{\Delta t}}} = \left\{ \psi \in L^\infty_{{\mathcal{K}}_s} \, \left| \, \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} = 0 \right. \right\}.$$ The Banach space $L^\infty_{{\mathcal{K}}_s,{{\Delta t}}}$ depends both on ${{\Delta t}}$ and $\gamma$ through $\mu_{\gamma,{{\Delta t}}}$, although the dependence on $\gamma$ is not explicitly written. This proves that the series $$\sum_{n=0}^{+\infty} P_{{\Delta t}}^n$$ is well defined as a bounded operator on $L^\infty_{{\mathcal{K}}_s,{{\Delta t}}}$, and is in fact equal to $({\mathrm{Id}}-P_{{\Delta t}})^{-1}$ since $$({\mathrm{Id}}-P_{{\Delta t}}) \sum_{n=0}^{+\infty} P_{{\Delta t}}^n = {\mathrm{Id}}.$$ We also have the bound $$\left\| \left({\mathrm{Id}}- P_{{\Delta t}}\right)^{-1} \right \|_{\mathcal{B}(L^\infty_{{\mathcal{K}}_s,{{\Delta t}}})} {\leqslant}\sum_{n=0}^{+\infty} \left \| P_{{\Delta t}}^n \right\|_{\mathcal{B}(L^\infty_{{\mathcal{K}}_s,{{\Delta t}}})} {\leqslant}\frac{K}{1-{\mathrm{e}}^{-\lambda {{\Delta t}}}} {\leqslant}\frac{2K}{\lambda {{\Delta t}}}$$ provided ${{\Delta t}}$ is sufficiently small. Let us summarize this result as follows.
\[corr:resolvent\_estimates\_I\_Pdt\] For any $s^* \in \mathbb{N}^*$, there exist ${{\Delta t}}^* > 0$ and $R > 0$ such that, for all $0 {\leqslant}s {\leqslant}s^*$, a uniform resolvent bound holds: for any $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$, $$\label{eq:unif_bound_Linf_Lis}
\left\| \left(\frac{{\mathrm{Id}}- P_{{\Delta t}}}{{{\Delta t}}}\right)^{-1} \right \|_{\mathcal{B}(L^\infty_{{\mathcal{K}}_s,{{\Delta t}}})} {\leqslant}R.$$
Error estimates for finite frictions {#sec:error_estimates_finite_friction}
------------------------------------
In this section we study the error of the average of sufficiently smooth functions, which allows us to characterize the corrections to the invariant measure. In Theorems \[thm:error\_first\_order\_schemes\] and \[thm:error\_second\_order\_schemes\], below, we characterize all the first- and second-order splittings; the technique of proof allows us to provide a rigorous study of the error estimates in the overdamped regime (see Section \[sec:ovd\_limit\]) and for nonequilibrium systems (see Section \[sec:noneq\_systems\]).
If only the order of magnitude of the correction is of interest, and not the expression of the correction in itself, no regularity result with regard to the derivatives is required (see [@BO10]), in contrast to situations where such corrections are explicitly considered, as in [@Talay02] for instance.
### Relating invariant measures of two numerical schemes
We classified in Section \[sec:splitting\_schemes\] the numerical schemes according to the order of appearance of the elementary operators. More precisely, we considered schemes to be similar when the global ordering of the operators is the same but the operations are started and ended differently, as in above (see also below for an abstract definition). This choice of classification is motivated by the following lemma which demonstrates how we may straightforwardly obtain the expression of the invariant measure of one scheme when the expression for another one is given.
We state the result in an abstract fashion for two schemes $P_{{\Delta t}}= U_{{\Delta t}}T_{{\Delta t}}$ and $Q_{{\Delta t}}= T_{{\Delta t}}U_{{\Delta t}}$ (which implies the condition below). See for a concrete example.
\[lem:TU\] Consider two numerical schemes with associated evolution operators $P_{{\Delta t}},Q_{{\Delta t}}$ bounded on $L^\infty(\mathcal{E})$, for which there exist bounded operators $U_{{\Delta t}},T_{{\Delta t}}$ on $L^\infty(\mathcal{E})$ such that, for all $n {\geqslant}1$, $$\label{eq:relation_Q_P}
Q_{{\Delta t}}^n = T_{{\Delta t}}P_{{\Delta t}}^{n-1} U_{{\Delta t}}.$$ We also assume that both schemes are ergodic with associated invariant measures denoted respectively by $\mu_{P,{{\Delta t}}}$, $\mu_{Q,{{\Delta t}}}$: For almost all $(q,p) \in {\mathcal{E}}$ and $f \in L^\infty(\mathcal{E})$, $$\label{eq:ergoditicy_TU}
\lim_{n \to +\infty} P_{{\Delta t}}^n f(q,p) = \int_{\mathcal{E}}f \, {\mathrm{d}\mu}_{P,{{\Delta t}}},
\qquad
\lim_{n \to +\infty} Q_{{\Delta t}}^n f(q,p) = \int_{\mathcal{E}}f \, {\mathrm{d}\mu}_{Q,{{\Delta t}}}.$$ Then, for all $\varphi \in L^\infty(\mathcal{E})$, $$\label{eq:TU_relation}
\int_{\mathcal{E}}\varphi \, {\mathrm{d}\mu}_{Q,{{\Delta t}}} = \int_{\mathcal{E}}\left(U_{{\Delta t}}\varphi \right) {\mathrm{d}\mu}_{P,{{\Delta t}}}.$$
Ergodicity results such as are implied by conditions such as .
The proof of this result relies on the simple observation that, for a given initial measure $\rho$ with a smooth density with respect to the Lebesgue measure, the ergodicity assumption ensures that, for a bounded measurable function $\varphi$, $$\int_{\mathcal{E}}\varphi \, {\mathrm{d}\mu}_{Q,{{\Delta t}}} = \lim_{n \to +\infty} \int_{\mathcal{E}}Q_{{\Delta t}}^n \varphi \, {\mathrm{d}\rho} = \lim_{n \to +\infty} \int_{\mathcal{E}}T_{{\Delta t}}P_{{\Delta t}}^{n-1} \left(U_{{\Delta t}}\varphi \right) \, {\mathrm{d}\rho}.$$ Now, we use the ergodicity property with $f$ replaced by $U_{{\Delta t}}\varphi$ to obtain the following convergence for almost all $(q,p) \in \mathcal{E}$: $$\lim_{n \to +\infty} P_{{\Delta t}}^{n-1} \left(U_{{\Delta t}}\varphi \right)(q,p) = \int_{\mathcal{E}}U_{{\Delta t}}\varphi \, {\mathrm{d}\mu}_{P,{{\Delta t}}} = a_{{\Delta t}}.$$ Since $T_{{\Delta t}}$ preserves constant functions, there holds $$\int_\mathcal{E} T_{{\Delta t}}(a_{{\Delta t}}\mathbf{1}) \, {\mathrm{d}\rho} = a_{{\Delta t}}\int_\mathcal{E} \mathbf{1} \, {\mathrm{d}\rho} = a_{{\Delta t}},$$ which finally gives .
Let us now show how we will use Lemma \[lem:TU\] in the sequel. Assume that a weak error estimate holds on the invariant measure $\mu_{P,{{\Delta t}}}$: there exist $\alpha {\geqslant}1$ and a function $f_\alpha \in \mathcal{S}$ such that $$\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{P,{{\Delta t}}} = \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu} + \Delta t^\alpha \int_{\mathcal{E}}\psi \, f_\alpha \, {\mathrm{d}\mu} + \Delta t^{\alpha + 1} r_{\psi,\alpha,{{\Delta t}}},$$ with $|r_{\psi,\alpha,{{\Delta t}}}| {\leqslant}K$ for ${{\Delta t}}$ sufficiently small. Combining this equality and , the following expansion is obtained for $\mu_{Q,{{\Delta t}}}$: $$\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{Q,{{\Delta t}}} = \int_{\mathcal{E}}\left(U_{{\Delta t}}\psi \right) {\mathrm{d}\mu}_{P,{{\Delta t}}}
= \int_{\mathcal{E}}\left(U_{{\Delta t}}\psi \right) {\mathrm{d}\mu} + \Delta t^\alpha \int_{\mathcal{E}}\left(U_{{\Delta t}}\psi \right) f_\alpha \, {\mathrm{d}\mu} + \Delta t^{\alpha + 1} r_{U_{{\Delta t}}\psi,\alpha,{{\Delta t}}}.$$ In general, for an evolution operator $U_{{\Delta t}}$ preserving the measure $\mu$ at order $\delta {\geqslant}1$, we can write $$U_{{\Delta t}}= {\mathrm{Id}}+ \Delta t \, \mathcal{A}_1 + \dots + \Delta t^{\delta-1} \mathcal{A}_{\delta-1} + \Delta t^{\delta} \, S_\delta + \Delta t^{\delta + 1} \, R_{\delta, {{\Delta t}}},$$ where all the operators on the right hand side are defined on the core $\mathcal{S}$, and the operators $\mathcal{A}_k$ preserve the measure $\mu$: $$\forall \varphi \in \mathcal{S}, \qquad \int_\mathcal{E} \mathcal{A}_k \varphi \, d\mu = 0,$$ while the operator $S_{\delta}$ does not. Typically, $\mathcal{A}_k$ is a composition of the operators $A+B$ and $C$. In addition, for a given function $\varphi \in \mathcal{S}$, the remainder $R_{\delta, {{\Delta t}}} \varphi$ is uniformly bounded for ${{\Delta t}}$ sufficiently small. Three cases should then be distinguished:
(i) When $\delta {\geqslant}\alpha+1$, the weak error in the invariant measure $\mu_{Q,{{\Delta t}}}$ is of the same order as for $\mu_{P,{{\Delta t}}}$ since $$\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_Q = \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu} + \Delta t^\alpha \int_{\mathcal{E}}\psi \, f_\alpha \, {\mathrm{d}\mu} + \Delta t^{\alpha + 1} \widetilde{r}_{\psi,\alpha,\delta,{{\Delta t}}}.$$
(ii) For $\delta {\leqslant}\alpha-1$, the weak error in the invariant measure $\mu_Q$ arises at dominant order from the operator $U_{{\Delta t}}$: $$\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_Q = \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu} + \Delta t^\delta \int_{\mathcal{E}}\psi \, \left(S_\delta^*\mathbf{1}\right) \, {\mathrm{d}\mu} + \Delta t^{\delta + 1} \widetilde{r}_{\psi,\alpha,\delta,{{\Delta t}}}.$$
(iii) The interesting case corresponds to $\alpha = \delta$. In this situation, $$\label{eq:interesting_TU_situation}
\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_Q = \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu} + \Delta t^\alpha \int_{\mathcal{E}}\psi \, \left(f_\alpha+S_\alpha^*\mathbf{1}\right) \, {\mathrm{d}\mu} + \Delta t^{\alpha + 1} \widetilde{r}_{\psi,\alpha,\delta,{{\Delta t}}}.$$ An increase in the order of the error on the invariant measure is obtained when the leading order correction vanishes for all admissible observables $\psi$, that is, if and only if $f_\alpha+S_\alpha^*\mathbf{1} = 0$.
### First-order schemes
The following result characterizes at leading order the invariant measure of the schemes based on a first-order splitting (see Section \[sec:splitting\_schemes\_1st\]). We first study the error estimates in the invariant measure of the schemes $P_{{\Delta t}}^{\gamma C, B, A}$, $P_{{\Delta t}}^{\gamma C, A,B}$ (which can be interpreted as GLA schemes with a symplectic Euler discretization of the Hamiltonian part, see [@BO10]), and then deduce error estimates for the four remaining schemes introduced in Section \[sec:splitting\_schemes\_1st\] by making use of Lemma \[lem:TU\]. The proof can be read in Section \[sec:proof\_thm:error\_first\_order\_schemes\].
\[thm:error\_first\_order\_schemes\] Consider any of the first order splittings presented in Section \[sec:splitting\_schemes\_1st\], and denote by $\mu_{\gamma,{{\Delta t}}}({\mathrm{d}q} \, {\mathrm{d}p})$ its invariant measure. Then there exists a function $f_{1,\gamma} \in \widetilde{\mathcal{S}}$ such that, for any function $\psi \in \mathcal{S}$, $$\label{eq:error_first_order_schemes}
\int_{\mathcal{E}}\psi(q,p) \, \mu_{\gamma,{{\Delta t}}}({\mathrm{d}q} \, {\mathrm{d}p}) = \int_{\mathcal{E}}\psi(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) + {{\Delta t}}\int_{\mathcal{E}}\psi(q,p) f_{1,\gamma}(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) + {{\Delta t}}^2 r_{\psi,\gamma,{{\Delta t}}},$$ where the remainder $r_{\psi,\gamma,{{\Delta t}}}$ is uniformly bounded for ${{\Delta t}}$ sufficiently small. The expressions of the correction functions $f_{1,\gamma}$ depend on the numerical scheme at hand. They are defined as $$\label{eq:correction_first_order_schemes}
\begin{aligned}
{\mathcal{L}_\gamma}^* f_{1}^{\gamma C,B,A} & = -\frac12 (A+B)g, \qquad g(q,p) = \beta p^T M^{-1} \nabla V(q),\\
f_{1}^{\gamma C,A,B} & = f_{1}^{A,B,\gamma C} = - f_{1}^{B,A,\gamma C} = - f_{1}^{\gamma C,B,A},\\
f_{1}^{A,\gamma C,B} & = - f_{1}^{B,\gamma C,A} = f_{1}^{\gamma C,B,A} - g.\\
\end{aligned}$$
It would in fact possible to obtain bounds on the the remainder $r_{\psi,\gamma,{{\Delta t}}}$ with respect to $\psi$, thanks to functional inequalities given in Appendix A of [@Kopec].
The equations could be analytically solved if, instead of the fluctuation/dissipation operator $C$, we were using the mass-weighted differential operator as in [@LM12]: $$C_M = -p^T \nabla_p + \frac1\beta M : \nabla_p^2.$$ The corresponding generator $\mathcal{L}_{\gamma,M} = A+B + \gamma C_M$ defined on the core $\mathcal{S}$ is associated with Langevin dynamics where the friction force is proportional to the momenta rather than velocities. A simple computation shows that $$-\frac12 (A+B)g = \mathcal{L}_{\gamma,M}^* \left(\frac{\beta}{2} V - g\right).$$ The condition would be replaced by $\mathcal{L}_{\gamma,M}^* f_1^{\gamma C,B,A} = -(A+B)g/2$, so that $f_1^{\gamma C,B,A} = \beta V/2 - g + c$ where $c$ is a constant ensuring that $f_1^{\gamma C,B,A}$ has a vanishing average with respect to $\mu$.
### Hamiltonian limit of the correction term
For first order splitting schemes, the limit of the leading order correction term in can be studied in the limit when $\gamma \to 0$. Not surprisingly, it turns out that the leading order correction is the first term in the expansion of the modified Hamiltonian of the symplectic Euler method in powers of ${{\Delta t}}$. In contrast to the more complete proof we are able to present for the overdamped limit (see Section \[sec:ovd\_limit\]), we were not able to study the behavior of the remainder terms $r_{\psi,\gamma,{{\Delta t}}}$ in . There is a technical obstruction to controlling these remainders from the way we prove our results since the limiting operator $\mathcal{L}_0 = A+B$ is not invertible. Let us also mention that studying the corresponding Hamiltonian limit for second order schemes turns out to be a much more difficult question (see Remark \[rmk:Hamiltonian\_limit\_second\_order\]).
\[prop:Ham\_limit\_correction\] There exists a constant $K > 0$ such that, for all $0 < \gamma {\leqslant}1$, $$\left\| f_1^{\gamma C, B,A} - \frac\beta2 p^T M^{-1} \nabla V \right\|_{L^2(\mu)} {\leqslant}K \gamma,$$ with similar estimates for $f_1^{B, \gamma C, A}$ and $f_1^{B,A,\gamma C}$; and $$\left\| f_1^{\gamma C, A,B} + \frac\beta2 p^T M^{-1} \nabla V \right\|_{L^2(\mu)} {\leqslant}K \gamma,$$ with similar estimates for $f_{1}^{A,\gamma C,B}$ and $f_{1}^{A,B,\gamma C}$.
The proof of this result is provided in Section \[sec:proof\_Ham\_limit\].
### Second-order schemes {#second-order-schemes}
The following result characterizes at leading order the invariant measure of the schemes based on a second-order splitting (see Section \[sec:splitting\_schemes\_2nd\]).
\[thm:error\_second\_order\_schemes\] Consider any of the second order splittings presented in Section \[sec:splitting\_schemes\_2nd\], and denote by $\mu_{\gamma,{{\Delta t}}}({\mathrm{d}q} \, {\mathrm{d}p})$ its invariant measure. Then there exists a function $f_{2,\gamma} \in \widetilde{\mathcal{S}}$ such that, for any function $\psi \in \mathcal{S}$, $$\label{eq:error_second_order_schemes}
\int_{\mathcal{E}}\psi(q,p) \, \mu_{\gamma,{{\Delta t}}}({\mathrm{d}q} \, {\mathrm{d}p}) = \int_{\mathcal{E}}\psi(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) + {{\Delta t}}^2 \int_{\mathcal{E}}\psi(q,p) f_{2,\gamma}(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) + {{\Delta t}}^4 r_{\psi,\gamma,{{\Delta t}}},$$ where the remainder $r_{\psi,\gamma,{{\Delta t}}}$ is uniformly bounded for ${{\Delta t}}$ sufficiently small. The expressions of the correction functions $f_{2,\gamma}$ depend on the numerical scheme at hand. They are defined as $$\label{eq:correction_second_order_schemes}
\begin{aligned}
{\mathcal{L}_\gamma}^* f_{2}^{\gamma C,B,A,B,\gamma C} & = \frac{1}{12} (A+B) \left[\left(A+\frac{B}{2}\right)g\right], \qquad g(q,p) = \beta p^T M^{-1} \nabla V(q),\\
{\mathcal{L}_\gamma}^* f_{2}^{\gamma C,A,B,A,\gamma C} & = -\frac{1}{12} (A+B) \left[\left(B+\frac{A}{2}\right)g\right], \\
f_{2}^{A,B,\gamma C,B,A} & = f_{2}^{\gamma C,B,A,B,\gamma C} + \frac18 (A+B)g, \\
f_{2}^{B,A,\gamma C,A,B} & = f_{2}^{\gamma C,A,B,A,\gamma C} - \frac18 (A+B)g. \\
\end{aligned}$$
It can be checked that the expressions of $f_{2}^{B,A,\gamma C,A,B}$ and $f_{2}^{A,B,\gamma C,B,A}$ agree with the ones presented in [@LM12]. Let us emphasize that no ${{\Delta t}}^3$ correction term appears in after the ${{\Delta t}}^2$ term. In fact, a more careful treatment would allow us to write an error expansion in terms of higher orders of ${{\Delta t}}$, with only even powers of ${{\Delta t}}$ appearing.
The proof of this result is given in Section \[sec:proof\_thm:error\_second\_order\_schemes\]. We use as reference schemes for the proofs the schemes $P_{{\Delta t}}^{\gamma C,A,B,A,\gamma C}$, $P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}$. These schemes indeed turn out to be particularly convenient to study the overdamped limit.
The results from Theorem \[thm:error\_second\_order\_schemes\] allow us to obtain error estimates for the so-called Geometric Langevin Algorithms (GLA) introduced in [@BO10]. Recall the somewhat surprising result that the error in the invariant measure of the GLA schemes is of order $\Delta t^p$ for a discretization of order $p$ of the Hamiltonian part, even though the weak and strong orders of the scheme are only one. The following result complements the estimate given in [@BO10] by making precise the leading order corrections to the invariant measure of the numerical scheme with respect to the canonical measure (see the proof in Section \[sec:proof\_cor:error\_GLA\]).
\[cor:error\_GLA\] Consider one of the GLA schemes defined in , and denote by $\mu_{\gamma,{{\Delta t}}}({\mathrm{d}q} \, {\mathrm{d}p})$ its invariant measure. Then there exist functions $f_{2,\gamma}, f_{3,\gamma} \in \widetilde{\mathcal{S}}$ such that, for any function $\psi \in \mathcal{S}$, $$\label{eq:error_GLA_general}
\begin{aligned}
\int_{\mathcal{E}}\psi(q,p) \, \mu_{\gamma,{{\Delta t}}}({\mathrm{d}q} \, {\mathrm{d}p}) & = \int_{\mathcal{E}}\psi(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) + {{\Delta t}}^2 \int_{\mathcal{E}}\psi(q,p) f_{2,\gamma}(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) \\
& \quad + {{\Delta t}}^3 \int_{\mathcal{E}}\psi(q,p) f_{3,\gamma}(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) + {{\Delta t}}^4 r_{\psi,\gamma,{{\Delta t}}},
\end{aligned}$$ where the remainder $r_{\psi,\gamma,{{\Delta t}}}$ is uniformly bounded for ${{\Delta t}}$ sufficiently small. The expressions of the correction functions $f_{2,\gamma}$ and $f_{3,\gamma}$ are $$\begin{aligned}
f_2^{\gamma C, A,B,A} = f_2^{\gamma C, A,B,A, \gamma C}, & \qquad f_3^{\gamma C, A,B,A} = -\frac{\gamma}{2} Cf_2^{\gamma C, A,B,A}, \\
f_2^{\gamma C, B,A,B} = f_2^{\gamma C, B,A,B \gamma C}, & \qquad f_3^{\gamma C, B,A,B} = -\frac{\gamma}{2} Cf_2^{\gamma C, B,A,B}.
\end{aligned}
$$
Note that the leading order term of the error is the same as for the corresponding second order splitting schemes. The next order correction (of order ${{\Delta t}}^3$) vanishes for functions $\psi$ depending only on the position variable $q$.
\[rmk:Hamiltonian\_limit\_second\_order\] Proving a result similar to Proposition \[prop:Ham\_limit\_correction\] for second order splitting schemes or GLA schemes turns out to be much more difficult, although we formally expect that the limit of $f_{2,\gamma}$ as $\gamma \to 0$ is the first order correction of the modified Hamiltonian constructed by backward analysis. From , it should indeed be the case that $f_{2}^{\gamma C,B,A,B,\gamma C}$ converges to $$f_2^{B,A,B} = -\frac{1}{12} \left(A+ \frac{B}{2}\right)g.$$ Moreover, as we already mentioned before Proposition \[prop:Ham\_limit\_correction\], we are not able to uniformly control remainder terms in the error expansion as $\gamma\to 0$.
Numerical estimation of the correction term {#sec:num_estimation_correction}
-------------------------------------------
The results of Section \[sec:error\_estimates\_finite\_friction\] show that the leading order correction terms for the average of an observable $\psi \in \mathcal{S}$ can be written as $$\label{eq:leading_correction}
\int_{\mathcal{E}}\psi(q,p) f_{\gamma}(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}),$$ where the function $f_\gamma \in \widetilde{\mathcal{S}}$ is the solution of a Poisson equation $$\label{eq:Poisson_leading_correction}
{\mathcal{L}_\gamma}^* f_{\gamma} = g_\gamma,$$ the function $g_\gamma \in \widetilde{\mathcal{S}}$ depending on the numerical scheme at hand (the fact that $f_\gamma \in \widetilde{\mathcal{S}}$ is a consequence of Theorem \[thm:stability\_S\]). It is in general impossible to analytically solve , and very difficult to numerically approximate its solution since it is a high-dimensional partial differential equation. It is however possible to rewrite as an integrated correlation function, a quantity which is amenable to numerical approximation. This is a standard way of computing transport coefficients based on Green-Kubo formulae, see the summary provided in Section \[sec:def\_transport\_coeff\]. It provides here a way to compute the first order correction in the perfect sampling bias with a single simulation (as an alternative to Romberg extrapolation, which requires at least two simulations at different timesteps, see [@TT90]).
### Error estimates
The approach we follow is based on the following operator identity (which makes sense in ${\mathcal{H}}^1$ for instance, in view of ) $${\mathcal{L}_\gamma}^{-1} = -\int_0^{+\infty} {\mathrm{e}}^{t {\mathcal{L}_\gamma}} \, {\mathrm{d}t}.$$ Since $$\int_{\mathcal{E}}\left( {\mathrm{e}}^{t {\mathcal{L}_\gamma}} \psi \right) \, g_\gamma \, {\mathrm{d}\mu} = \mathbb{E} \Big( \psi(q_t,p_t) g_\gamma(q_0,p_0) \Big),$$ where the expectation is taken over all initial conditions $(q_0,p_0)$ distributed according to $\mu$ and over all realizations of equilibrium Langevin dynamics , the leading order correction term can be rewritten as $$\label{eq:rewriting_correction}
\int_{\mathcal{E}}\psi(q,p) f_{\gamma}(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}) = -\int_0^{+\infty} \mathbb{E} \Big( \psi(q_t,p_t) g_\gamma(q_0,p_0) \Big) {\mathrm{d}t}.$$ The following result (proved in Section \[sec:proof\_approx\_GK\_formula\]) shows how to approximate quantities such as up to errors $\mathrm{O}({{\Delta t}}^\alpha)$, when the invariant measure of the numerical scheme is correct to terms of order $\mathrm{O}({{\Delta t}}^\alpha)$ (as discussed in Section \[sec:error\_estimates\_finite\_friction\]). The fundamental ingredient is the replacement of the observable $\psi$ by some modified observable, in the spirit of backward analysis. Let us emphasize that we do not require the numerical scheme to be of weak or strong order $p$ in itself. For instance, GLA schemes are only first order correct on trajectories (as proved in [@BO10]), but nonetheless may have invariant measures which are very close to $\mu$. To somewhat simplify the notation and state our result in a more general fashion since it can be used in other contexts than Langevin dynamics (see [@FHS14] for an application to Metropolis-Hastings schemes), we do not denote explicitly all the dependencies on $\gamma$ although the reader should keep them in mind.
\[thm:approx\_GK\_formula\] Consider a numerical method with an invariant measure $\mu_{{\Delta t}}$ having bounded moments at all orders (i.e. is satisfied) and such that, for $\psi \in \mathcal{S}$, $$\label{eq:asssumption_order_method}
\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{{\Delta t}}= \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu} + {{\Delta t}}^\alpha r_{\psi,{{\Delta t}}},$$ where the remainder $r_{\psi,{{\Delta t}}}$ is uniformly bounded for ${{\Delta t}}$ small enough. Suppose in addition that its evolution operator $P_{{\Delta t}}$ is such that, for any $\psi \in \mathcal{S}$, $$\label{eq:expansion_I_Pdt}
-\frac{{\mathrm{Id}}-P_{{\Delta t}}}{{{\Delta t}}} \psi = {\mathcal{L}}_\gamma\psi + {{\Delta t}}S_1\psi + \dots + {{\Delta t}}^{\alpha-1} S_{\alpha-1}\psi + {{\Delta t}}^\alpha \widetilde{R}_{\alpha,{{\Delta t}}}\psi,$$ where $S_1\psi,\dots,S_{\alpha-1}\psi,\widetilde{R}_{\alpha,{{\Delta t}}}\psi \in \mathcal{S}$ and there exists $s>0$ such that the remainder $\widetilde{R}_{\alpha,{{\Delta t}}}\psi$ is uniformly bounded in $L^\infty_{{\mathcal{K}}_s}$ for ${{\Delta t}}$ sufficiently small. Assume finally that $P_{{\Delta t}}$ satisfies the uniform ergodicity condition (hence holds). Then, the integrated correlation of two observables $\psi, \varphi \in \widetilde{\mathcal{S}}$ can be approximated by a Riemann sum up to an error of order ${{\Delta t}}^\alpha$: $$\label{eq:correction_GK_general}
\int_0^{+\infty} \mathbb{E} \Big( \psi(q_t,p_t) \varphi(q_0,p_0) \Big) {\mathrm{d}t} = {{\Delta t}}\sum_{n=0}^{+\infty} \mathbb{E}_{{\Delta t}}\left(\widetilde{\psi}_{{{\Delta t}},\alpha}\left(q^{n},p^{n}\right)\varphi\left(q^0,p^0\right)\right) + {{\Delta t}}^\alpha r^{\psi,\varphi}_{{\Delta t}},$$ where $r_{{\Delta t}}^{\psi,\varphi}$ is uniformly bounded for ${{\Delta t}}$ sufficiently small, the expectation $\mathbb{E}_{{\Delta t}}$ is over all initial conditions $(q_0,p_0)$ distributed according to $\mu_{{\Delta t}}$ and over all realizations of the Markov chain induced by $P_{{\Delta t}}$, and the modified observable $\widetilde{\psi}_{{{\Delta t}},\alpha} \in \mathcal{S}$ reads $$\widetilde{\psi}_{{{\Delta t}},\alpha} = \psi_{{{\Delta t}},\alpha} - \int_{\mathcal{E}}\psi_{{{\Delta t}},\alpha} \, {\mathrm{d}\mu}_{{\Delta t}}, \qquad
\psi_{{{\Delta t}},\alpha} = \left({\mathrm{Id}}+ {{\Delta t}}\,S_1 {\mathcal{L}_\gamma}^{-1} + \dots + {{\Delta t}}^{\alpha-1} S_{\alpha-1}{\mathcal{L}_\gamma}^{-1} \right)\psi.
$$
The assumptions of this theorem are satisfied for the splitting schemes considered in this article (see the comment after for the boundedness of the remainder $\widetilde{R}_{\alpha,{{\Delta t}}}\psi$).
In the particular case $\alpha=2$, which is in fact the most relevant one from a practical viewpoint, it is possible not to modify the observable $\psi$ when the discrete generator is correct at order $2$ (see below for a precise statement), upon considering a time discretization of the integral which leads to errors of order ${{\Delta t}}^2$, for instance a trapezoidal rule. The following result is obtained by an appropriate application of Theorem \[thm:approx\_GK\_formula\] (see Section \[sec:proof\_approx\_GK\_formula\] for the proof).
\[cor:GK\_trapezoidal\] Consider a numerical scheme satisfying the assumptions of Theorem \[thm:approx\_GK\_formula\], and whose discrete generator is in addition correct at order 2: for any $\psi \in \mathcal{S}$, $$\label{eq:expansion_I_Pdt_order_2}
-\frac{{\mathrm{Id}}-P_{{\Delta t}}}{{{\Delta t}}}\psi = {\mathcal{L}}_\gamma\psi + \frac{{\Delta t}}2 {\mathcal{L}}_\gamma^2\psi + {{\Delta t}}^2 \widetilde{R}_{{{\Delta t}}}\psi.$$ Then, for two observables $\varphi,\psi \in \widetilde{\mathcal{S}}$, $$\label{eq:correction_GK_second}
\begin{aligned}
& \int_0^{+\infty} \mathbb{E} \Big( \psi(q_t,p_t) \varphi(q_0,p_0) \Big) {\mathrm{d}t} \\
& \qquad \qquad = \frac{{\Delta t}}2 \mathbb{E}_{{\Delta t}}\Big(\psi_{{{\Delta t}},0}\left(q^0,p^0\right) \varphi\left(q^0,p^0\right)\Big) + {{\Delta t}}\sum_{n=1}^{+\infty} \mathbb{E}_{{\Delta t}}\Big(\psi_{{{\Delta t}},0}\left(q^{n},p^{n}\right)\varphi\left(q^0,p^0\right)\Big) + {{\Delta t}}^2 r^{\psi,\varphi}_{{\Delta t}},
\end{aligned}$$ where $r_{{\Delta t}}^{\psi,\varphi}$ is bounded for ${{\Delta t}}$ sufficiently small and $$\psi_{{{\Delta t}},0} = \psi - \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{{\Delta t}}.$$
### Numerical approximation
There are two principal ways to estimate the expectations in or , using either several independent realizations of the nonequilibrium dynamics or a single, long trajectory, see for instance the discussion in Section 13.4 of [@Tuckerman]. When $K$ independent realizations $(q^{n,k},p^{n,k})$ are generated for $N_{\rm iter}$ timesteps each, starting from initial conditions distributed according to $\mu_{{\Delta t}}$, the expectation in may be approximated using empirical averages of the correlation functions as $$\frac{{{\Delta t}}}{K} \sum_{k=1}^K \sum_{n=0}^{N_{\rm iter}} \left[ \psi_{{{\Delta t}},\alpha}\left(q^{n,k},p^{n,k}\right) - \Psi_{{{\Delta t}},\alpha}^{K,N_{\rm iter}} \right]\varphi\left(q^{0,k},p^{0,k}\right),$$ where $\alpha = 1$ and $\psi_{{{\Delta t}},1} = \psi$ for first order splittings; while $\alpha = 2$ and $\psi_{{{\Delta t}},2} = (1+{{\Delta t}}{\mathcal{L}}_\gamma/2)\psi$ for second order ones since $S_1 = {\mathcal{L}}_\gamma^2/2$ for the schemes presented in Section \[sec:splitting\_schemes\_2nd\] (see for instance ). The empirical average $\Psi_{{{\Delta t}},p}^{M,N_{\rm iter}}$ reads $$\Psi_{{{\Delta t}},\alpha}^{M,N_{\rm iter}} = \frac{1}{K (1+N_{\rm iter})} \sum_{k=1}^K \sum_{n=0}^{N_{\rm iter}}
\psi_{{{\Delta t}},\alpha}\left(q^{n,k},p^{n,k}\right).$$ This formula highlights the other errors arising from the discretization: (i) a statistical error related to the finiteness of $K$ and to the fact that initial conditions are obtained in practice by subsampling a single, long trajectory; (ii) a truncation error related to the finiteness of $N_{\rm iter}$.
### Numerical illustration {#sec:numerics}
We illustrate the convergence results and for a simple two-dimensional system. We denote $q=(x,y) \in \mathcal{M} = (2\pi \mathbb{T})^2$, and consider the potential $$V(q) = 2 \cos(2x) + \cos(y).$$ The inverse temperature is fixed to $\beta = 1$ and we consider a trivial mass matrix $M = \mathrm{Id}$ with unit friction $\gamma=1$. Trajectory data is taken from $10^3$ independent runs of fixed time interval $2\times 10^8$, with the aim to compute the integral of the velocity autocorrelation function, which corresponds to $\psi(q,p) = \varphi(q,p) = M^{-1}p$ in . Using the second order $P_{{\Delta t}}^{\gamma C, B,A,B, \gamma C}$ scheme, applying the appropriate correction function gives the predicted order ${{\Delta t}}^2$ result, while the standard Riemann approximation has errors of order ${{\Delta t}}$. In the numerical results in Figure \[fig:corr\] the corrected approximation gives marginally better results than the trapezoidal rule (though of the same order) due to additional higher order terms being included.
Let us now numerically confirm the error estimates --. More precisely, we show that, provided the leading correction term is estimated by discretizing using and subtracted from the estimated result, canonical averages are estimated up to errors of order ${{\Delta t}}^4$ for second order splittings instead of ${{\Delta t}}^2$ without the correction. We use the same trajectory data as above to approximate the canonical average of the total system energy $H$. We test the effectiveness of the correction both in practice and principle, by computing the observed average and correction term in the same simulation in the former case, while computing a more accurate correction term independently in the latter case (using a smaller timestep ${{\Delta t}}=0.1$). The results are shown in the right panel of Figure \[fig:corr\].
![\[fig:corr\] Left: The error in the value of the integrated velocity autocorrelation function is compared at a number of timesteps when computed using a Riemann sum or the correction term provided in . The result from computing the integral using the trapezoidal rule is also shown. Right: The error in the computed average of total energy is plotted, with the correction term computed using the same stepsize demonstrating the practical application of the method. We can test the validity of in principle by computing the correction more accurately at a smaller timestep in a separate simulation, this result is labelled as the ‘exact correction’. All results are computed using the scheme associated with $P_{{\Delta t}}^{\gamma C, B,A,B, \gamma C}$ with $\beta=\gamma = 1$. ](corr.eps){width="\textwidth"}
Overdamped limit {#sec:ovd_limit}
----------------
We study in this section the overdamped limit $\gamma \to +\infty$, assuming that the mass matrix is $M = {\mathrm{Id}}$. We first study the consistency of the invariant measures of limiting numerical schemes in Section \[sec:ovd\_schemes\], before stating precise convergence results for second order splitting schemes in Section \[sec:rigorous\_estimates\_ovd\].Ultimately, we relate in Section \[sec:ovd\_limit\_correction\] the overdamped limit of the correction terms obtained for finite $\gamma$ to the correction obtained by directly studying the overdamped limit.
### Overdamped limits of splitting schemes {#sec:ovd_schemes}
The only part of the numerical schemes where the friction enters is the Ornstein-Uhlenbeck process on momenta. The limit $\gamma \to +\infty$ for ${{\Delta t}}> 0$ fixed amounts to resampling momenta according to the Gaussian distribution $\kappa({\mathrm{d}p})$ at all timesteps. For instance, the numerical scheme associated with the evolution operator $P_{{\Delta t}}^{\gamma C, B,A,B,\gamma C}$ reduces to $$q^{n+1} = q^n - \frac{{{\Delta t}}^2}{2} \nabla V(q^n) + \frac{{{\Delta t}}}{\sqrt{\beta}} \, G^n,
$$ where $(G^n)$ are independent and identically distributed Gaussian random vectors with identity covariance. This is indeed a consistent discretization of the overdamped process with an effective timestep $h = {{\Delta t}}^2/2$, and the invariant measure of this numerical scheme is close to $\overline{\mu}$. Other schemes may have non-trivial large friction limits and invariant measures close to $\overline{\mu}$. This is the case for the scheme associated with the evolution operator $P_{{\Delta t}}^{B,A,\gamma C,A,B}$, for which the limiting discrete dynamics reads (see [@LM12]) $$\begin{aligned}
q^1 &= q^0- \frac{{{\Delta t}}^2}{4} \nabla V(q^0) + \frac{{{\Delta t}}}{2\sqrt{\beta}} (G^0+G^{1}),\\
q^{n+1} &= q^n - \frac{{{\Delta t}}^2}{2} \nabla V(q^n) + \frac{{{\Delta t}}}{2\sqrt{\beta}} (G^n+G^{n+1}), \quad \textrm{for } n>0.\end{aligned}$$ Note that $(q^n)$ is not a Markov chain due to the correlations in the random noises.
On the other hand, the limits of the invariant measures associated with certain schemes are not consistent with the canonical measure $\overline{\mu}$. This is the case for the first-order schemes, as well as the second order splittings listed in item (iii) in Section \[sec:splitting\_schemes\_2nd\]. For instance, the limit of the scheme associated with $P_{{\Delta t}}^{\gamma C,A,B}$ reads $$q^{n+1} = q^n + \frac{{{\Delta t}}}{\sqrt{\beta}} \, G^n.$$ The invariant measure of this Markov chain is the uniform measure on $\mathcal{M}$, and is therefore very different from the invariant measure $\overline{\mu}$ of the continuous dynamics (it amounts to setting $V = 0$). As another example, consider the limit of the scheme associated with $P_{{\Delta t}}^{\gamma C,B,A}$: $$q^{n+1} = q^n - {{\Delta t}}^2 \nabla V(q^n) + \frac{{{\Delta t}}}{\sqrt{\beta}} \, G^n.$$ This is the Euler-Maruyama discretization of with an effective timestep $h={{\Delta t}}^2$ but an inverse temperature $2\beta$ rather than $\beta$.
### Rigorous error estimates {#sec:rigorous_estimates_ovd}
The following result quantifies the errors of the invariant measure of second order splitting schemes of Langevin dynamics, for large values of $\gamma$. We restrict ourselves to the second order splittings where the Ornstein-Uhlenbeck part is either at the ends or in the middle (categories (i) and (ii) in Section \[sec:splitting\_schemes\_2nd\]). From a technical viewpoint, we are able here to bound remainder terms uniformly in $\gamma$ by relying on the properties of the limiting operator ${\mathcal{L}_{\rm ovd}}^{-1}$. The result we obtain is the following (see Section \[sec:proof\_thm:ovd\_limit\] for the proof).
\[thm:ovd\_limit\] Consider any of the second order splittings presented in Section \[sec:splitting\_schemes\_2nd\], denote by $\mu_{\gamma,{{\Delta t}}}({\mathrm{d}q} \, {\mathrm{d}p})$ its invariant measure, and by $\overline{\mu}_{\gamma,{{\Delta t}}}({\mathrm{d}q})$ its marginal in the position variable. Then there exists a function $f_{2,\infty} = f_{2,\infty}(q) \in C^\infty(\mathcal{M})$, with average zero with respect to $\overline{\mu}$, such that, for any smooth $\psi = \psi(q) \in C^\infty(\mathcal{M})$ and $\gamma {\geqslant}1$, $$\int_\mathcal{M} \psi(q) \, \overline{\mu}_{\gamma,{{\Delta t}}}({\mathrm{d}q}) = \int_\mathcal{M} \psi \, {\mathrm{d}\overline{\mu}} + {{\Delta t}}^2 \int_\mathcal{M} \psi \, f_{2,\infty}\, {\mathrm{d}\overline{\mu}} + r_{\psi,\gamma,{{\Delta t}}},$$ where the remainder is of order ${{\Delta t}}^4$ up to terms exponentially small in $\gamma {{\Delta t}}$. More precisely, there exist constants $a,b {\geqslant}0$ and ${c}> 0$ (all depending on $\psi$) such that $$\left|r_{\psi,\gamma,{{\Delta t}}}\right| {\leqslant}a {{\Delta t}}^4 + b \, {\mathrm{e}}^{-{c}\gamma {{\Delta t}}}.$$ The expression of $f_{2,\infty}$ depends on the numerical scheme at hand: $$\label{eq:correction_overdamped}
\begin{aligned}
f^{\gamma C,B,A,B,\gamma C}_{2,\infty}(q) & = \frac18 \left( -2\Delta V + \beta |\nabla V|^2 + a_{\beta,V}\right), \qquad a_{\beta,V} = \int_\mathcal{M} \Delta V \, {\mathrm{d}\overline{\mu}} = \beta \int_\mathcal{M} \left|\nabla V\right|^2 \, {\mathrm{d}\overline{\mu}},\\
f_{2,\infty}^{A,B,\gamma C, B,A}(q) & = -\frac18 \left(\Delta V - a_{\beta,V}\right), \\
f^{\gamma C,A,B,A,\gamma C}_{2,\infty}(q) & = \frac18 \left( \Delta V - \beta |\nabla V|^2\right), \\
f^{B,A,\gamma C,A,B}_{2,\infty}(q) & = 0.
\end{aligned}$$
The real number $a_{\beta,V}$ ensures that all functions $f_{2,\infty}$ are of average zero with respect to $\overline{\mu}$. Two comments are in order. Note first that the result is stated for observables which depend only on the position variable $q$ since the limiting case $\gamma \to +\infty$ corresponds to a dynamics on the positions only. There is anyway no restriction in stating the result using such observables since, as already discussed in the introduction, the error on the marginal in the position variables is the relevant error, momenta being trivial to sample exactly under the canonical measure. Secondly, let us emphasize that the ${{\Delta t}}^2$ correction term vanishes for the method associated with $P_{{\Delta t}}^{B,A,\gamma C,A,B}$ (as already noted in [@LM12]). This means that the corresponding discretization of overdamped Langevin dynamics (formally obtained by setting $\gamma = +\infty$) has an invariant measure which is correct at second-order in the effective timestep $h = {{\Delta t}}^2/2$.
### Overdamped limit of the correction terms {#sec:ovd_limit_correction}
In order to relate the convergence result from Theorem \[thm:ovd\_limit\] to the error estimates from Theorem \[thm:error\_second\_order\_schemes\], we prove that the limits of the correction functions $f_{2,\gamma}$ as $\gamma \to +\infty$ agree with the functions defined in (see Section \[sec:proof\_prop:ovd\_limit\_correction\] for the proof). This can be seen as a statement regarding the permutation of the limits $\gamma \to +\infty$ and ${{\Delta t}}\to 0$ for the leading correction term, namely, for a smooth function $\psi = \psi(q) \in C^\infty(\mathcal{M})$, $$\begin{aligned}
\lim_{{{\Delta t}}\to 0} \lim_{\gamma \to +\infty} \frac{1}{{{\Delta t}}^2} \left( \int_\mathcal{M} \psi \, {\mathrm{d}\overline{\mu}}_{\gamma,{{\Delta t}}} - \int_\mathcal{M} \psi \, {\mathrm{d}\overline{\mu}}\right) & = \lim_{\gamma \to +\infty} \lim_{{{\Delta t}}\to 0} \frac{1}{{{\Delta t}}^2} \left( \int_\mathcal{M} \psi \, {\mathrm{d}\overline{\mu}}_{\gamma,{{\Delta t}}} - \int_\mathcal{M} \psi \, {\mathrm{d}\overline{\mu}}\right) \\
& = \lim_{\gamma \to +\infty} \int_\mathcal{M} \psi \left(\pi f_{2,\gamma}\right) {\mathrm{d}\overline{\mu}}\\
& = \int_\mathcal{M} \psi \, f_{2,\infty} \, {\mathrm{d}\overline{\mu}}.
\end{aligned}$$ The precise result is the following:
\[prop:ovd\_limit\_correction\] There exists a constant $K > 0$ such that, for all $\gamma {\geqslant}1$, $$\begin{aligned}
\left\| f_2^{\gamma C, B,A,B,\gamma C} - \frac18 \left( -2 \Delta V + \beta |\nabla V|^2 + a_{\beta,V} \right)\right\|_{H^1(\mu)} & {\leqslant}\frac{K}{\gamma}, \\
\left\| f_2^{A,B,\gamma C, B,A} - \frac18 \left(-2\Delta V + \beta p^T (\nabla^2 V)p + a_{\beta,V} \right)\right\|_{H^1(\mu)} & {\leqslant}\frac{K}{\gamma}, \\
\left\| f_2^{\gamma C, A,B,A,\gamma C} - \frac18 \left( \Delta V - \beta |\nabla V|^2\right) \right\|_{H^1(\mu)} & {\leqslant}\frac{K}{\gamma}, \\
\left\| f_2^{B,A,\gamma C,A,B} - \frac18 \left( \Delta V - \beta p^T (\nabla^2 V)p \right) \right\|_{H^1(\mu)} & {\leqslant}\frac{K}{\gamma}, \\
\end{aligned}$$ where the constant $a_{\beta,V}$ is defined in .
Note that, as expected, the averages with respect to $\kappa({\mathrm{d}p})$ of the above limiting functions coincide with the functions $f_{2,\infty}$ given in , that is, $\pi f_{2,\gamma} = f_{2,\infty} + \mathrm{O}(\gamma^{-1})$.
Let us also mention that the overdamped limit of the correction function $f_{1,\gamma}$ for first order splittings is not well defined. This is not surprising since the invariant measures of the corresponding numerical schemes are not consistent with $\overline{\mu}$, as discussed in Section \[sec:ovd\_schemes\]. For instance, combining and the expressions of the correction functions , we see that there exists a constant $K>0$ such that $$\label{eq:divergence_f1}
\left\| f_{1}^{\gamma C,B,A} + \frac{\gamma \beta}{2} {\mathcal{L}_{\rm ovd}}^{-1} \mathcal{L}_{\mathrm{ovd},M} V \right\|_{H^1(\mu)} {\leqslant}K,$$ where the operator $$\mathcal{L}_{\mathrm{ovd},M} = -M^{-1} \nabla V \cdot \nabla_q + \frac1\beta M : \nabla^2,$$ defined on $\mathcal{S}$, is the generator of the overdamped Langevin dynamics with non-trivial mass matrix: $${\mathrm{d}q}_t = -M^{-1} \nabla V(q_t)\,{\mathrm{d}t} + \sqrt{\frac{2}{\beta}} M^{-1/2} \, {\mathrm{d}W}_t.$$ Note that, when $M = {\mathrm{Id}}$, the solution can in fact be analytically computed as $f_{1}^{\gamma C,B,A} = -\beta ( \gamma V + p^T \nabla V)/2$. In any case, shows that $f_{1}^{\gamma C,B,A}$ diverges as $\gamma \to +\infty$.
Nonequilibrium dynamics and the computation of transport coefficients {#sec:noneq_systems}
=====================================================================
We discuss in this section the numerical estimation of transport properties such as the thermal conductivity, the shear stress, etc. (see [@EM08; @Tuckerman] for general physical presentations of the computation of transport coefficients, and Section 3.1 of [@HDR] for a mathematically oriented introduction).
We consider the prototypical case of the estimation of the autodiffusion coefficient. In this situation, it is relevant to consider a nonequilibrium perturbation of standard equilibrium Langevin dynamics, where some external forcing arising from a constant force $F \in \mathbb{R}^{dN}$ is imposed on the system: $$\label{eq:noneq_Langevin}
\left\{ \begin{aligned}
{\mathrm{d}q}_t & = M^{-1} p_t \, {\mathrm{d}t}, \\
{\mathrm{d}p}_t & = \Big( -\nabla V(q_t) + \eta F \Big) {\mathrm{d}t} - \gamma M^{-1} p_t \, {\mathrm{d}t} + \sqrt{\frac{2\gamma}{\beta}} \, {\mathrm{d}W}_t.
\end{aligned} \right.$$ We denote by $${\widetilde{\mathcal{L}}}= F \cdot \nabla_p$$ the generator of the perturbation (considered as an operator on $L^2(\mu)$, with core $\mathcal{S}$). Note that the constant force $F$ does not derive from the gradient of a smooth function defined on $\mathcal{M}$. (It would indeed seem that this force derives from $-F^T q$, but this potential is not periodic.) Therefore, the expression of the invariant measure is unknown, but can be nonetheless obtained as an expansion in powers of $\eta$ when the magnitude of the forcing is sufficiently small (see Section \[sec:def\_transport\_coeff\]). The effect of the force is to create a non-zero average velocity in the direction of $F$. The magnitude of the average velocity is a property of the system under consideration. For small forcings, it is linear in $\eta$, with a constant of proportionality called the *mobility* (see the definition below), related to the autodiffusion coefficient through .
As shown in [@JPS14], it is possible to consider more general forcing terms $F(q)$ which do not derive from the gradient of a periodic function. A popular example is provided by shearing forces where the particles experience a force in some direction, whose intensity depends on the coordinates of the system in another direction.
We will also be interested in the overdamped limit of the nonequilibrium dynamics , which reads $$\label{eq:noneq_ovd_Langevin}
{\mathrm{d}q}_t = \Big( -\nabla V(q_t) + \eta F \Big) {\mathrm{d}t} + \sqrt{\frac{2}{\beta}} \, {\mathrm{d}W}_t.$$ The generator of this dynamics is ${\mathcal{L}_{\rm ovd}}+ \eta {\widetilde{\mathcal{L}}}_{\rm ovd}$ with ${\widetilde{\mathcal{L}}}_{\rm ovd} = F \cdot \nabla_q$ (all operators being defined on the core $\mathcal{S}$). In this case the physically relevant response turns out to be the average force $-F\cdot \nabla V$ exerted in the direction $F$.
Definition of transport coefficients {#sec:def_transport_coeff}
------------------------------------
Following the strategy advertised in [@rey-bellet] (using the kinetic energy as a Lyapunov function), it is easy to show that the dynamics has a unique invariant probability measure $\mu_{\gamma,\eta}({\mathrm{d}q}\,{\mathrm{d}p})$ with a smooth density with respect to the Lebesgue measure for any value of $\eta \in \mathbb{R}$. The mobility $\nu_{F,\gamma}$ is defined as the linear response of the velocity in the direction $F$ as the magnitude of the forcing goes to 0: $$\label{eq:def_nu_NEMD}
\nu_{F,\gamma} = \lim_{\eta \to 0} \frac{1}{\eta} \int_{\mathcal{E}}F^T M^{-1} p \, \mu_{\gamma,\eta}({\mathrm{d}q} \, {\mathrm{d}p}).$$ From linear response theory (see for example the presentation in [@HDR Section 3.1], and the short summary provided in Section \[sec:proof\_LRT\]), it can be shown that $$\label{eq:def_nu_LRT}
\nu_{F,\gamma} = \int_{\mathcal{E}}F^T M^{-1} p \, f_{0,1,\gamma}(q,p) \, \mu({\mathrm{d}q} \, {\mathrm{d}p}), \qquad {\mathcal{L}_\gamma}^* f_{0,1,\gamma} = -{\widetilde{\mathcal{L}}}^* \mathbf{1} = -\beta F^T M^{-1}p.$$ The mobility can therefore be rewritten as the integrated autocorrelation function of the velocity in the direction $F$: $$\label{eq:def_nu_GK}
\nu_{F,\gamma} = \beta \int_0^{+\infty} \mathbb{E}\Big[\big(F^T M^{-1}p_t\big)\big(F^T M^{-1}p_0\big)\Big] {\mathrm{d}t},$$ where the expectation is over all initial conditions $(q_0,p_0)$ distributed according to $\mu$ and over all realizations of the equilibrium Langevin dynamics . From this relation, it is easily seen that the mobility in the direction $F$ is related to the autodiffusion coefficient $$\label{eq:def_nu_Einstein}
D_{F,\gamma} = \lim_{t \to +\infty} \frac{\mathbb{E}\Big[\big(F\cdot(q_t-q_0)\big)^2\Big]}{2t}$$ as $$\nu_{F,\gamma} = \beta D_{F,\gamma}.$$ In practice, the two most popular ways of estimating a transport coefficient rely on the Green-Kubo formula and the linear response of nonequilibrium dynamics in their steady-states . Since the error estimates for Green-Kubo type formulas have already been discussed in Theorem \[thm:approx\_GK\_formula\], we will restrict ourselves in the sequel to the analysis of the numerical errors introduced by nonequilibrium methods.
### Overdamped limit {#overdamped-limit}
The overdamped limit of the mobility $\nu_{F,\gamma}$ is studied in [@HP08], where the authors consider the autodiffusion coefficient $D_{F,\gamma}$. First, it is easily shown that the overdamped dynamics admits a unique invariant probability measure, which we denote by $\overline{\mu}_\eta({\mathrm{d}q})$. The mobility for the overdamped dynamics is defined from the linear response of the projected force $-F \cdot \nabla V$ as $$\label{eq:ovd_mobility}
\overline{\nu}_F = \lim_{\eta \to 0} \frac1\eta \int_\mathcal{M} -F^T \nabla V(q) \, \overline{\mu}_\eta({\mathrm{d}q})
= \beta \int_\mathcal{M} F^T \nabla V(q) {\mathcal{L}_{\rm ovd}}^{-1} \left(F^T \nabla V(q)\right) \overline{\mu}({\mathrm{d}q}).$$ The derivation of this formula is very similar to that leading to . The following result summarizes the limiting behavior of the mobility as the friction increases (recall that we set mass matrices to identity when studying overdamped limits).
\[lem:ovd\_mobility\] There exists $K > 0$ such that, for any $\gamma {\geqslant}1$, $$\left|\gamma \nu_{F,\gamma} - \overline{\nu}_F - |F|^2 \right| {\leqslant}\frac{K}{\gamma}.$$
This result is already contained in [@HP08], but we nonetheless provide a short alternative proof in Section \[sec:proof\_lem:ovd\_mobility\] (see Remark \[rmk:ovd\_Einstein\] for a more precise comparison of the results). It shows that, in the overdamped regime $\gamma \to +\infty$, $$\label{eq:computatio_nu_F_gamma}
\nu_{F,\gamma} = \frac{|F|^2 + \overline{\nu}_F}{\gamma} + \mathrm{O}\left(\frac{1}{\gamma^2}\right),$$ which suggests to estimate $\nu_{F,\gamma}$ using the linear response of $F^T \nabla V$ for large frictions since this quantity is expected to be a good approximation of $\overline{\nu}_F$ – instead of relying on the standard linear response result , for which the response is of order $1/\gamma$ and is hence difficult to reliably estimate. Error estimates on the numerical approximation are deduced from below.
Numerical schemes for the nonequilibrium Langevin dynamics
----------------------------------------------------------
We present in this section numerical schemes approximating solutions of . These schemes reduce to the schemes presented in Section \[sec:splitting\_schemes\] when $\eta = 0$. Since the aim is to decompose the evolution generated by ${\mathcal{L}_\gamma}+ \eta {\widetilde{\mathcal{L}}}$ into analytically integrable parts, there are two principal options: either replace $B$ by $$B_\eta = B+\eta{\widetilde{\mathcal{L}}},$$ or replace $C$ by $C+\eta{\widetilde{\mathcal{L}}}$. However, the schemes built on the latter option do not perform correctly in the overdamped limit since their invariant measures are not consistent with the invariant measures of nonequilibrium overdamped Langevin dynamics . More precisely, consider for instance the first order scheme generated by $P_{{{\Delta t}}}^{A,B,\gamma C + \eta {\widetilde{\mathcal{L}}}} = {\mathrm{e}}^{\Delta t \, A} {\mathrm{e}}^{\Delta t \, B} {\mathrm{e}}^{\Delta t (\gamma C + \eta {\widetilde{\mathcal{L}}})}$ in the case when $M = {\mathrm{Id}}$: $$\left\{ \begin{aligned}
q^{n+1} & = q^n + {{\Delta t}}\, p^n, \\
\widetilde{p}^{n+1} & = p^n - {{\Delta t}}\, \nabla V(q^{n+1}), \\
p^{n+1} & = \alpha_{{\Delta t}}\widetilde{p}^{n+1} + \frac{1-\alpha_{{\Delta t}}}{\gamma} \, \eta F + \sqrt{\frac{1-\alpha^2_{{\Delta t}}}{\beta}} \, G^n,
\end{aligned} \right.$$ where $\alpha_{{\Delta t}}$ is defined after , and $(G^n)$ is a sequence of independent and identically distributed Gaussian random vectors with identity covariance. As $\gamma \to +\infty$, a standard Euler-Maruyama discretization of the equilibrium overdamped Langevin dynamics (*i.e.* $\eta = 0$) is obtained, whereas we would like to obtain a consistent discretization of nonequilibrium overdamped Langevin dynamics . We therefore instead consider schemes obtained by replacing $B$ with $B + \eta {\widetilde{\mathcal{L}}}$, such as the first order splitting $$P_{{{\Delta t}}}^{A,B+ \eta {\widetilde{\mathcal{L}}},\gamma C} = {\mathrm{e}}^{\Delta t \, A} {\mathrm{e}}^{\Delta t (B+ \eta {\widetilde{\mathcal{L}}})} {\mathrm{e}}^{\gamma \Delta t \, C},$$ or the second order splitting $$P_{{{\Delta t}}}^{\gamma C,B+ \eta {\widetilde{\mathcal{L}}},A,B+ \eta {\widetilde{\mathcal{L}}},C} = {\mathrm{e}}^{\gamma \Delta t \, C/2} {\mathrm{e}}^{\Delta t (B+ \eta {\widetilde{\mathcal{L}}})/2} {\mathrm{e}}^{\Delta t \, A} {\mathrm{e}}^{\Delta t (B+ \eta {\widetilde{\mathcal{L}}})/2} {\mathrm{e}}^{\gamma \Delta t \, C/2}.$$ The numerical scheme associated with the first order splitting scheme $P_{{{\Delta t}}}^{A,B+ \eta {\widetilde{\mathcal{L}}},\gamma C}$ $$\left\{ \begin{aligned}
q^{n+1} & = q^n + {{\Delta t}}\, p^n, \\
\widetilde{p}^{n+1} & = p^n + {{\Delta t}}\Big(-\nabla V(q^{n+1}) + \eta F \Big), \\
p^{n+1} & = \alpha_{{\Delta t}}\widetilde{p}^{n+1} + \sqrt{\frac{1-\alpha^2_{{\Delta t}}}{\beta}} \, G^n,
\end{aligned} \right.
$$ indeed is, in the limit as $\gamma \to +\infty$, a consistent discretization of the nonequilibrium Langevin dynamics , and its invariant measure turns out to converge to the invariant measure of in the limit ${{\Delta t}}\to 0$.
Following the method of proof of Proposition \[prop:ergodicity\_MC\], it can be shown that there exists a unique invariant measure $\mu_{\gamma,\eta,{{\Delta t}}}$ for the corresponding Markov chain. The crucial point is that the gradient structure of the force term is never used explicitly in the proofs since we rely solely on the boundedness of the force, so that we are able to obtain convergence results and moment estimates that are independent of the magnitude $\eta$ of the forcing term provided $\eta$ is in a bounded subset of $\mathbb{R}$. We denote below by $P_{\gamma,\eta,{{\Delta t}}}$ the evolution operator associated with the numerical schemes.
\[prop:ergodicity\_MC\_noneq\] Fix $s^* {\geqslant}1$ and $\eta^* > 0$. For any $0 < \gamma < +\infty$, there exists ${{\Delta t}}^*$ such that, for any $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$ and $\eta \in [-\eta^*,\eta^*]$, the Markov chain associated with $P_{\gamma,\eta,{{\Delta t}}}$ has a unique invariant probability measure $\mu_{\gamma,\eta,{{\Delta t}}}$, which admits a density with respect to the Lebesgue measure ${\mathrm{d}q} \, {\mathrm{d}p}$, and has finite moments: There exists $R > 0$ such that, for any $1 {\leqslant}s {\leqslant}s^*$, $$\int_{\mathcal{E}}{\mathcal{K}}_s \, {\mathrm{d}\mu}_{\gamma,\eta,{{\Delta t}}} {\leqslant}R < +\infty,$$ uniformly in the timestep ${{\Delta t}}$ and the forcing magnitude $\eta$. There also exist $\lambda, K > 0$ (depending on $s^*$, $\gamma$ and $\eta^*$ but not on ${{\Delta t}}$) such that, for all functions $f \in L^\infty_{{\mathcal{K}}_s}$, the following holds for almost all $(q,p) \in {\mathcal{E}}$: $$\forall n \in \mathbb{N}, \qquad \left| \left(P_{\gamma,\eta,{{\Delta t}}}^n f\right)(q,p) - \int_{\mathcal{E}}f {\mathrm{d}\mu}_{\gamma,\eta,{{\Delta t}}} \right| {\leqslant}K \, {\mathcal{K}}_s(q,p) \, {\mathrm{e}}^{-\lambda n{{\Delta t}}} \, \| f \|_{L^\infty_{{\mathcal{K}}_s}}.$$
Let us emphasize that we do not have any control on the convergence rate $\lambda$ in terms of $\eta^*$, and it could well be that $\lambda$ goes to 0 as $\eta^*$ increases.
Error estimates on transport coefficients from nonequilibrium methods {#sec:error_transport}
---------------------------------------------------------------------
The following result provides error estimates for the invariant measure of the first order or second order splittings schemes of Section \[sec:splitting\_schemes\_2nd\] when $B$ is replaced by $B_\eta$.
\[thm:error\_noneq\] Denote by $p$ the order of the splitting scheme, by $f_{\alpha,0,\gamma}$ the leading order correction function in the case $\eta = 0$ as given by Theorem \[thm:error\_first\_order\_schemes\] for $\alpha=1$ and by Theorem \[thm:error\_second\_order\_schemes\] for $\alpha=2$. Then, there exists a function $f_{\alpha,1,\gamma} \in \widetilde{\mathcal{S}}$ such that, for any smooth function $\psi \in \mathcal{S}$, there exist ${{\Delta t}}^*,\eta^* > 0$ and a constant $K > 0$ for which, for all $\eta \in [-\eta^*,\eta^*]$ and $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$, $$\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{\gamma,\eta,{{\Delta t}}} = \int_{\mathcal{E}}\psi \Big(1+ \eta f_{0,1,\gamma} + {{\Delta t}}^\alpha f_{\alpha,0,\gamma} + \eta {{\Delta t}}^\alpha f_{\alpha,1,\gamma} \Big) {\mathrm{d}\mu} + r_{\psi,\gamma,\eta,{{\Delta t}}},$$ where $f_{0,1,\gamma}$ is defined in , and $$\left|r_{\psi,\gamma,\eta,{{\Delta t}}}\right| {\leqslant}K(\eta^2 + {{\Delta t}}^{\alpha+1}),
\qquad
\left|r_{\psi,\gamma,\eta,{{\Delta t}}} - r_{\psi,\gamma,0,{{\Delta t}}}\right| {\leqslant}K \eta (\eta + {{\Delta t}}^{\alpha+1}).$$
The proof of this result can be found in Section \[sec:proof\_noneq\]. Note that the remainder term now collects higher order terms both as powers of the timestep and the nonequilibrium parameter $\eta$. The estimates we obtain on the remainder are however compatible with taking the linear response limit, as made precise by the following error estimate on the transport coefficient (which is an immediate consequence of Theorem \[thm:error\_noneq\]). In order to state the result, we introduce the reference linear response for an observable $\psi$ $$\mathscr{D}_{\psi,\gamma,0} = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{\gamma,\eta} - \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{\gamma} \right),$$ and its numerical approximation $$\mathscr{D}_{\psi,\gamma,{{\Delta t}}} = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{\gamma,\eta,{{\Delta t}}} - \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} \right).$$ It is often the case that $\psi$ has a vanishing average with respect to $\mu$, as is the case for the function $F^T M^{-1}p$ in . In general, it however has a non-zero average with respect to the invariant measure $\mu_{\gamma,{{\Delta t}}}$ of the numerical scheme associated with a discretization of the equilibrium dynamics.
\[cor:noneq\] There exist ${{\Delta t}}^*,\eta^* > 0$ and a constant $K > 0$ such that, for all $\eta \in [-\eta^*,\eta^*]$ and $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$, $$\mathscr{D}_{\psi,\gamma,{{\Delta t}}} = \mathscr{D}_{\psi,\gamma,0} + {{\Delta t}}^\alpha \int_{\mathcal{E}}\psi \, f_{\alpha,1,\gamma} \, {\mathrm{d}\mu} + {{\Delta t}}^{\alpha+1} r_{\psi,\gamma,{{\Delta t}}},$$ where $r_{\psi,\gamma,{{\Delta t}}}$ is uniformly bounded.
In particular, we obtain the following estimate on the numerically computed mobility: $$\begin{aligned}
\nu_{F,\gamma,{{\Delta t}}} & = \lim_{\eta \to 0} \frac{1}{\eta} \left(\int_{\mathcal{E}}F^T M^{-1} p \, \mu_{\gamma,\eta,{{\Delta t}}}({\mathrm{d}q}\,{\mathrm{d}p}) - \int_{\mathcal{E}}F^T M^{-1} p \, \mu_{\gamma,0,{{\Delta t}}}({\mathrm{d}q}\,{\mathrm{d}p}) \right) \label{eq:LR_num_mob} \\
& = \nu_{F,\gamma} + {{\Delta t}}^\alpha \int_{\mathcal{E}}F^T M^{-1} p \, f_{\alpha,1,\gamma} \, {\mathrm{d}\mu} + {{\Delta t}}^{\alpha+1} r_{\gamma,{{\Delta t}}}, \label{eq:prediction_num}\end{aligned}$$ where the reference mobility $\nu_{F,\gamma}$ is defined in .
### Numerical illustration {#numerical-illustration}
We consider the same system as in Section \[sec:numerics\], with an external force $F=(1,0)$ and $K+1$ forcing strengths $\eta_k = (k-1) \Delta \eta$ uniformly spaced in the interval $[0,\eta_{\rm max}]$ with $\eta_{\rm max} = 0.5$ (so that $\Delta \eta = \eta_{\rm max}/K$). We fix the friction to $\gamma = 1$ and the inverse temperature to $\beta = 1$. We use a coupling strategy to reduce the statistical noise in the computation of the linear response . The $K+1$ replicas of the system are started at the same position $q= (0,0)$, with the same velocity (sampled according to the canonical measure $\mu$). Each replica experiences the force $-\nabla V + \eta_k F$ (Note that the first replica experiences the reference force $-\nabla V$ corresponding to a discretization of the equilibrium dynamics). Most importantly, the same Gaussian random numbers $G^n$ are used for all replicas to discretize the Brownian motion. Although not carefully documented here, this coupling strategy tremendously decreases the statistical error in the computed linear response. Such a coupling strategy was already proposed for exclusion processes in [@GoodmanKin]. However, our experience shows that it fails for higher dimensional systems with more complex potentials (such as Lennard-Jones fluids).
For a given value of the timestep ${{\Delta t}}$, we denote by $(q^{k,n},p^{k,n})_{n {\geqslant}0}$ the discrete trajectory of the $k$th replica. The linear response in the projected average velocity $\delta v_{\eta_k}$ is approximated over $N_{\rm iter}$ integration steps as $$\begin{aligned}
\delta v_{\eta_k} & = \int_{\mathcal{E}}F^T M^{-1} p \, \mu_{{{\Delta t}},\eta_k}({\mathrm{d}q}\,{\mathrm{d}p}) - \int_{\mathcal{E}}F^T M^{-1} p \, \mu_{{{\Delta t}},0}({\mathrm{d}q}\,{\mathrm{d}p}) \\
& \simeq \frac{1}{N_{\rm iter}} \sum_{n=1}^{N_{\rm iter}}F^T M^{-1} \Big(p^{k,n}-p^{1,n}\Big) = {\widehat{v}_{\eta_k}}^{\,\,N_{\rm iter}}.
\end{aligned}$$ We then estimate the mobility by a linear fit on the first $K' = 10$ values of $\widehat{v}_{\eta_k}^{\,\,N_{\rm iter}}$ considered as a function of $\eta_k$ (see Figure \[fig:noneq\], left). The value $\nu_{F,\gamma,{{\Delta t}}}$ is the estimated slope in the fit. The behavior of the mobility $\nu_{F,\gamma,{{\Delta t}}}$ as a function of the timestep is presented in Figure \[fig:noneq\] (right) for the numerical schemes associated with the first order splitting $P_{{\Delta t}}^{A,B_\eta,\gamma C}$ and the second order splitting $P_{{\Delta t}}^{\gamma C, B_\eta,A,B_\eta, \gamma C}$. We used $N_{\rm iter} = 4 \times 10^{11}$ for the first order scheme, and $N_{\rm iter} = 2.5 \times 10^{11}$ for the second order one. The statistical error is very small and error bars are therefore not reported. The computed mobilities can be fitted for small ${{\Delta t}}$ as $$\nu_{F,\gamma,{{\Delta t}}} \simeq 0.0740 + 0.0817{{\Delta t}}$$ for the first-order splitting and $$\nu_{F,\gamma,{{\Delta t}}} \simeq 0.0741 + 0.197{{\Delta t}}^2$$ for the second order splitting scheme, in agreement with the theoretical prediction .
![\[fig:noneq\] Left: Linear response of the average velocity $\delta v_\eta$ as a function of $\eta$ ($K = 50$) for the scheme associated with $P_{{\Delta t}}^{\gamma C, B_\eta,A,B_\eta, \gamma C}$ and ${{\Delta t}}= 0.01, \gamma = 1$. A linear fit on the first ten values gives $\delta v_\eta \simeq 0.07416 \eta$, so that $\nu_{F,\gamma,{{\Delta t}}} = 0.07416$ in this case. Right: Scaling of the mobility $\nu_{F,\gamma,{{\Delta t}}}$ for the first order scheme $P_{{\Delta t}}^{A,B_\eta,\gamma C}$ and the second order scheme $P_{{\Delta t}}^{\gamma C, B_\eta,A,B_\eta, \gamma C}$ (with $\gamma = 1$). The fits respectively give $\nu_{F,\gamma,{{\Delta t}}} \simeq 0.0740 + 0.0817{{\Delta t}}$ and $\nu_{F,\gamma,{{\Delta t}}} \simeq 0.0741 + 0.197{{\Delta t}}^2$.](LR.eps "fig:"){width="7cm"} ![\[fig:noneq\] Left: Linear response of the average velocity $\delta v_\eta$ as a function of $\eta$ ($K = 50$) for the scheme associated with $P_{{\Delta t}}^{\gamma C, B_\eta,A,B_\eta, \gamma C}$ and ${{\Delta t}}= 0.01, \gamma = 1$. A linear fit on the first ten values gives $\delta v_\eta \simeq 0.07416 \eta$, so that $\nu_{F,\gamma,{{\Delta t}}} = 0.07416$ in this case. Right: Scaling of the mobility $\nu_{F,\gamma,{{\Delta t}}}$ for the first order scheme $P_{{\Delta t}}^{A,B_\eta,\gamma C}$ and the second order scheme $P_{{\Delta t}}^{\gamma C, B_\eta,A,B_\eta, \gamma C}$ (with $\gamma = 1$). The fits respectively give $\nu_{F,\gamma,{{\Delta t}}} \simeq 0.0740 + 0.0817{{\Delta t}}$ and $\nu_{F,\gamma,{{\Delta t}}} \simeq 0.0741 + 0.197{{\Delta t}}^2$.](noneq.eps "fig:"){width="7cm"}
Error estimates in the overdamped limit
---------------------------------------
We now study the numerical errors arising in the simulation of nonequilibrium systems in the large friction limit. We restrict ourselves to the second order splittings where the Ornstein-Uhlenbeck part is either at the ends or in the middle (categories (i) and (ii) in Section \[sec:splitting\_schemes\_2nd\]). To state the result, we introduce the first order correction to the invariant measure in terms of the magnitude of the nonequilibrium forcing, namely (recall ${\widetilde{\mathcal{L}}}_{\rm ovd} = F \cdot \nabla_q$) $${\mathcal{L}_{\rm ovd}}^* f_{0,1,\infty} = -{\widetilde{\mathcal{L}}}_{\rm ovd}^* \mathbf{1} = -\beta F^T \nabla V.$$ A simple computation based on shows that the functions $f_{0,1,\gamma}$ defined in converge in $H^1(\mu)$ to $f_{0,1,\infty}$ (recall that we assume $M = {\mathrm{Id}}$ in the overdamped regime).
\[thm:error\_estimate\_noneq\_ovd\] Denote by $\overline{\mu}_{\gamma,\eta,{{\Delta t}}}({\mathrm{d}q})$ the marginal of the invariant measure $\mu_{\gamma,\eta,{{\Delta t}}}$ of an admissible second order splitting scheme in the position variable, and by $f_{2,0,\infty}$ the leading order correction function in the case $\eta = 0$ as given by Theorem \[thm:ovd\_limit\]. Then, there exists a function $f_{2,1,\infty} \in \widetilde{\mathcal{S}}$ such that, for any $\psi \equiv \psi(q) \in C^\infty(\mathcal{M})$, there exist ${{\Delta t}}^*,\eta^* > 0$ and constants $K,{c}> 0$ such that, for all $\eta \in [-\eta^*,\eta^*]$, $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$ and $\gamma {\geqslant}1$, $$\int_\mathcal{M} \psi(q) \, \overline{\mu}_{\gamma,\eta,{{\Delta t}}}({\mathrm{d}q}) = \int_\mathcal{M} \psi(q) \Big(1+ \eta f_{0,1,\infty}(q) + {{\Delta t}}^2 f_{2,0,\infty}(q) + \eta {{\Delta t}}^2 f_{2,1,\infty} \Big) \overline{\mu}({\mathrm{d}q}) + r_{\psi,\gamma,\eta,{{\Delta t}}},$$ with $$\left|r_{\psi,\gamma,\eta,{{\Delta t}}}\right| {\leqslant}K\left(\eta^2 + {{\Delta t}}^{3} + {\mathrm{e}}^{-{c}\gamma {{\Delta t}}}\right),
\qquad
\left|r_{\psi,\gamma,\eta,{{\Delta t}}} - r_{\psi,\gamma,0,{{\Delta t}}}\right| {\leqslant}K \eta (\eta + {{\Delta t}}^{3} + {\mathrm{e}}^{-{c}\gamma {{\Delta t}}}).$$
The proof is presented in Section \[sec:proof\_thm:error\_estimate\_noneq\_ovd\]. This result allows us to estimate the error in the computation of the transport coefficient $\nu_{F,\gamma}$ based on and Lemma \[lem:ovd\_mobility\]. Indeed, studying the linear response of the observable $-F^T \nabla V$ and defining $$\overline{\nu}_{F,\gamma,{{\Delta t}}} = -\lim_{\eta \to 0} \frac1\eta \left(\int_\mathcal{M} F^T \nabla V(q) \, \overline{\mu}_{\gamma,\eta,{{\Delta t}}}({\mathrm{d}q}) - \int_\mathcal{M} F^T\nabla V(q) \, \overline{\mu}_{\gamma,{{\Delta t}}}({\mathrm{d}q}) \right),$$ there holds $$\overline{\nu}_F = \overline{\nu}_{F,\gamma,{{\Delta t}}} - {{\Delta t}}^2 \int_\mathcal{M} F^T \nabla V(q) f_{2,1,\infty}(q) \, \overline{\mu}({\mathrm{d}q}) + r_{\psi,\gamma,{{\Delta t}}},$$ with $|r_{\psi,\gamma,{{\Delta t}}}| {\leqslant}a({{\Delta t}}^3 + {\mathrm{e}}^{-{c}\gamma {{\Delta t}}})$ for some $a > 0$. Therefore, in view of , $$\label{eq:estimate_overline_nu_F}
\nu_{F,\gamma} = \frac{|F|^2 + \overline{\nu}_F}{\gamma} + \mathrm{O}\left(\frac{1}{\gamma^2}\right) = \frac{|F|^2 + \overline{\nu}_{F,\gamma,{{\Delta t}}}}{\gamma} + \mathrm{O}\left(\frac{1}{\gamma^2},\frac{{{\Delta t}}^2}{\gamma},\frac{{\mathrm{e}}^{- {c}\gamma {{\Delta t}}}}{\gamma}\right).$$ In the latter expression, $\overline{\nu}_{F,\gamma,{{\Delta t}}}$ can be numerically estimated, in a manner similar to that presented at the end of Section \[sec:error\_transport\].
Proofs of the results {#sec:proofs}
=====================
Unless otherwise stated, the default norm $\| f \|$ and scalar product $\langle f,g \rangle$ are the ones associated with the Hilbert space $L^2(\mu)$. Recall that, unless otherwise mentioned, all operators are defined on $\mathcal{S}$, and that formal adjoint operators are by default considered on $L^2(\mu)$. Recall also that $$\label{eq:decomposition_C}
C = -\frac1\beta \nabla_p^* \nabla_p = -\frac1\beta \sum_{i=1}^N \sum_{\alpha = 1}^d \partial_{p_{i,\alpha}}^* \partial_{p_{i,\alpha}},$$ with $p_i = (p_{i,1},\dots,p_{i,d})$ since $\partial_{p_{i,\alpha}}^* = -\partial_{p_{i,\alpha}} + \beta p_{i,\alpha}$.
Large friction behavior of $\mathcal{L}_\gamma^{-1}$ {#sec:proof_L_gamma_large}
----------------------------------------------------
The proof of Lemma \[lem:bounded\_resolvent\_perp\] follows the same lines as the proof of uniform hypocoercive estimates in the corrected version of Theorem 3 in [@JS12] (see the erratum [@JS12erratum] or the updated preprint version [@JS12preprint]). We provide a simplified version of it for completeness.
We show that the operator $\mathcal{L}_\gamma$ is uniformly hypocoercive for $\gamma {\geqslant}1$. The aim is to obtain bounds on the inverse $\mathcal{L}^{-1}_\gamma$ extended to ${\mathcal{H}}^1_\perp$. To this end, we decompose ${\mathcal{L}_\gamma}$ for $\gamma {\geqslant}1$ as $${\mathcal{L}_\gamma}= \mathcal{L}_1 + (\gamma-1) C.$$ The proof of Theorem 6.2 in [@HP08] shows that there exists $\widetilde{\alpha} > 0$ such that, for all $u \in \mathcal{S}$, $$-\left\langle\left\langle u, \mathcal{L}_1 u \right\rangle\right\rangle
{\geqslant}\widetilde{\alpha} \left\langle\left\langle u, u \right\rangle\right\rangle,$$ where the norm induced by $\left\langle\left\langle \cdot, \cdot\right\rangle\right\rangle$ is equivalent to the $H^1(\mu)$ norm. More precisely, $\left\langle\left\langle \cdot, \cdot\right\rangle\right\rangle$ is the bilinear form defined by $$\left\langle\left\langle u, v\right\rangle\right\rangle
= a\left\langle u, v\right\rangle
+ b \left\langle \nabla_p u,\nabla_pv\right\rangle
- \langle \nabla_p u, \nabla_q v\rangle
- \langle \nabla_q u, \nabla_p v\rangle
+ b\langle \nabla_q u,\nabla_q v\rangle,$$ with appropriate coefficients $a \gg b \gg 1$. It follows that there exists $\alpha > 0$ independent of $\gamma$ such that $$\label{eq:eq_for_coercivity_Lgam}
\alpha \left\| u \right\|_{H^1(\mu)}^2 - (\gamma-1)
\left\langle\left\langle u, C u \right\rangle\right\rangle
{\leqslant}-\left\langle\left\langle u, {\mathcal{L}_\gamma}u \right\rangle\right\rangle.$$ Let us now show that $$\label{eq:estimate_positiviy_A}
\forall u \in {\mathcal{H}}^1_\perp \cap \mathcal{S}, \qquad -\left\langle\left\langle u, C u \right\rangle\right\rangle
{\geqslant}0.$$ Using the rewriting of the operator $C$, and the commutation relations $[\partial_{p_{i,\alpha}}, \partial_{p_{j,\alpha'}}^*] = \beta \delta_{\alpha,\alpha'}\delta_{ij}$, a simple computation shows $$\begin{aligned}
\left\langle\left\langle u, \left(\partial_{p_{i,\alpha}}\right)^*\partial_{p_{i,\alpha}} u \right\rangle\right\rangle
& = (a+\beta b) \| \partial_{p_{i,\alpha}} u \|^2 + b \|\nabla_p \partial_{p_{i,\alpha}} u\|^2 \nonumber \\
& \quad + b \|\nabla_q \partial_{p_{i,\alpha}} u\|^2 - 2\langle \nabla_q \partial_{p_{i,\alpha}} u, \nabla_p \partial_{p_{i,\alpha}} u\rangle - \beta \langle \partial_{q_{i,\alpha}} u, \partial_{p_{i,\alpha}} u\rangle \nonumber \\
& {\geqslant}\left(a+\beta \left(b-\frac12\right)\right) \| \partial_{p_{i,\alpha}} u \|^2 + (b-1) \|\nabla_p \partial_{p_{i,\alpha}} u\|^2 \label{eq:ineg_pi*pi} \\
& \qquad + (b-1) \|\nabla_q \partial_{p_{i,\alpha}} u\|^2 - \frac{\beta}{2} \| \partial_{q_{i,\alpha}} u \|^2. \nonumber\end{aligned}$$ Now, since the Gaussian measure $\kappa({\mathrm{d}p})$ satisfies a Poincaré inequality, there exists a constant $A > 0$ such that, for all $i = 1,\dots,N$ and $\alpha = 1,\dots,d$, $$ \| \partial_{q_{i,\alpha}}u \|^2 {\leqslant}A \| \nabla_p \partial_{q_{i,\alpha}} u \|^2.
$$ Note indeed that $\partial_{q_{i,\alpha}}u$ has a vanishing average with respect to the Gaussian measure $\kappa({\mathrm{d}p})$ because $$\int_{{\mathbb{R}}^{dN}} \partial_{q_{i,\alpha}} u(q,p) \, \kappa({\mathrm{d}p}) = \partial_{q_{i,\alpha}} \overline{u}(q) = 0$$ for functions $u \in {\mathcal{H}}^1_\perp$. Therefore, $$\sum_{i=1}^N \sum_{\alpha=1}^d \| \partial_{q_{i,\alpha}} u \|^2 {\leqslant}A \sum_{i,j=1}^N \sum_{\alpha,\alpha' = 1}^d \| \partial_{p_{j,\alpha'}} \partial_{q_{i,\alpha}} u \|^2 = A \sum_{j=1}^N \sum_{\alpha' = 1}^d \| \nabla_{q} \partial_{p_{j,\alpha'}}\|^2.$$ Summing on $i \in \{ 1,\dots,N\}$ and $\alpha \in \{1,\dots,d\}$, the quantity is seen to be non-negative for an appropriate choice of constants $a \gg b \gg 1$.
From , we then deduce that there exists a constant $K>0$ such that, for any $\gamma {\geqslant}1$ and for any $u \in {\mathcal{H}}^1_\perp \cap \mathcal{S}$, it holds $\left\| u \right\|_{H^1(\mu)} {\leqslant}K \| {\mathcal{L}_\gamma}u \|_{H^1(\mu)}$. Taking inverses and passing to the limit in ${\mathcal{H}}^1_\perp$ gives $$\forall \gamma {\geqslant}1, \quad \forall u \in {\mathcal{H}}^1_\perp, \qquad \left\| {\mathcal{L}_\gamma}^{-1} u \right\|_{H^1(\mu)} {\leqslant}K \| u \|_{H^1(\mu)},$$ which is the desired result.
We are now in position to give the proof of Theorem \[lem:bounds\_CL\_gamma\].
We write the proof for ${\mathcal{L}_\gamma}^{-1}$. The estimates for $({\mathcal{L}_\gamma}^*)^{-1}$ are obtained by using ${\mathcal{L}_\gamma}^* = \mathcal{R} {\mathcal{L}_\gamma}\mathcal{R}$ (the momentum reversal operator being defined in ), and the fact that $\mathcal{R} C \mathcal{R} = C$, $\mathcal{R} {\mathcal{L}_{\rm ovd}}\mathcal{R} = {\mathcal{L}_{\rm ovd}}$ and $\mathcal{R} (A+B) \mathcal{R} = -(A+B)$.
The lower bound in could be obtained directly provided $V$ is not constant, by considering the special case $${\mathcal{L}}_\gamma \Big( p^T \nabla V + \gamma (V-v)\Big) = p^T M^{-1} \left(\nabla^2 V\right)p - |\nabla V|^2,$$ where $v$ is a constant chosen such that $p^T \nabla V + \gamma (V-v)$ has a vanishing average with respect to $\mu$. This example is also useful to motivate the fact that, in general, solutions of the Poisson equation ${\mathcal{L}}_\gamma u_\gamma = f$ have divergent parts of order $\gamma$ as $\gamma \to +\infty$.
Let us now turn to the refined upper and lower bounds , which we prove using techniques from asymptotic analysis. Consider $f \in {\mathcal{H}}^1$, and $u_\gamma \in {\mathcal{H}}^1$ the unique solution of the following Poisson equation ${\mathcal{L}}_\gamma u_\gamma = f$. The above example suggests the following expansion in inverse powers of $\gamma$: $$\label{eq:ansatz_u_pi}
u_\gamma = \gamma u^{-1} + u^0 + \frac1\gamma u^1 + \dots$$ To rigorously prove this expansion, we first proceed formally, taking as an ansatz, plugging it into ${\mathcal{L}}_\gamma u = f$ and identifying terms according to powers of $\gamma$. This leads to $$\begin{aligned}
C u^{-1} & = 0, \\
(A+B)u^{-1} + Cu^0 & = 0, \\
(A+B)u^0 + Cu^1 & = f.\end{aligned}$$ The first equality implies that $u^{-1} = u^{-1}(q)$ since $C$ satisfies a Poincaré inequality on $L^2(\kappa)$ (where $\kappa$ is defined in ). The second then reduces to $Cu^0 = -M^{-1} p \cdot \nabla_q u^{-1}$, from which we deduce $u^0(q,p) = p^T\nabla u^{-1}(q) + \widetilde{u}^0(q)$. Inserting this expression in the third equality gives $$Cu^1 = f - p^T M^{-1} \left(\nabla^2 u^{-1}\right) p - p^T M^{-1} \nabla \widetilde{u}^0 + (\nabla V)^T \nabla u^{-1}.$$ The solvability condition for this equation is that the right-hand side has a vanishing average with respect to $\kappa$, *i.e.* belongs to the kernel of $\pi$. This condition reads $$\frac1\beta \Delta u^{-1} - (\nabla V)^T \nabla u^{-1} = \pi f,$$ so that $u^{-1} = {\mathcal{L}_{\rm ovd}}^{-1} \pi f$ (which is well defined since $\pi f$ has a vanishing average with respect to $\overline{\mu}$). Note that the function $u^{-1}$ is in $H^{n+2}(\overline{\mu})$ when $f \in H^n(\mu)$ (by elliptic regularity, using also the fact that ${\mathrm{e}}^{-\beta V(q)}$ is a smooth function bounded from above and below on $\mathcal{M}$), so that $p^T M^{-1} (\nabla^2 u^{-1}) p$ belongs to $L^2(\mu)$. The equation determining $u^1$ then reduces to $$Cu^1 = (f-\pi f) - p^T M^{-1} \nabla \widetilde{u}^0 - p^T M^{-1} \left(\nabla^2 u^{-1}\right) p + \frac1\beta\Delta u^{-1}.$$ Since $C(p^T A p) = -p^T M^{-1} (A + A^T)p + 2\beta^{-1} \mathrm{Tr}(A)$, we can choose $$u^1(q,p) = \left[C^{-1}(f -\pi f)\right](q,p) + \frac12 p^T (\nabla^2 u^{-1}(q)) p + p^T \nabla_q \widetilde{u}^0(q).$$ Coming back to , we see that the proposed approximate solution is such that $$\label{eq:difference_with_exact_sol}
{\mathcal{L}}_\gamma\left(u_\gamma - \gamma u^{-1} - u^0 - \frac1\gamma u^1\right) = -\frac1\gamma (A+B)u^1.$$ We now choose $\widetilde{u}^0$ such that $(A+B)u^1$ belongs to ${\mathcal{H}}^1_\perp$, which amounts to $$\pi(A+B)p^T \nabla_q \widetilde{u}^0 = {\mathcal{L}_{\rm ovd}}\widetilde{u}^0 = -\pi(A+B)C^{-1}(f -\pi f).$$ It is easily checked that $\widetilde{u}^0 = -{\mathcal{L}_{\rm ovd}}^{-1}\pi(A+B)C^{-1}(f -\pi f)$ is a well defined element in ${\mathcal{H}}^1$ for $f \in H^1(\mu)$: first, $C^{-1}(f -\pi f) \in {\mathcal{H}}^1$, so $(A+B)C^{-1}(f -\pi f) \in L^2(\mu)$. Finally, the image under ${\mathcal{L}_{\rm ovd}}^{-1}\pi$ of any function in $L^2(\mu)$ is a function of average zero with respect to $\overline{\mu}$, depending only on the position variable $q$ and belonging to $H^2(\overline{\mu})$; hence to ${\mathcal{H}}^1$.
Combining and Lemma \[lem:bounded\_resolvent\_perp\], we see that there exists a constant $R > 0$, such that, for all $\gamma {\geqslant}1$, it holds $\| u_\gamma - \gamma u^{-1} - u^0 \|_{H^1(\mu)} {\leqslant}R \| f \|_{H^1(\mu)}/\gamma$ for the above choices of functions $u^{-1}, u^0$. This gives .
Ergodicity results for numerical schemes {#sec:proof_ergodicity_MC}
----------------------------------------
We write the proof for the scheme associated with the evolution operator $P_{{\Delta t}}^{B,A,\gamma C}$, starting by the case $s=1$, before turning to the general case $s {\geqslant}2$. The proofs for other schemes are very similar, and we therefore skip them.
The numerical scheme corresponding to $P_{{\Delta t}}^{B,A,\gamma C}$ is . We introduce $m \in (0,+\infty)$ such that $m {\leqslant}M {\leqslant}m^{-1}$ (in the sense of symmetric matrices). A simple computation shows that $$\begin{aligned}
\mathbb{E}\left[\left.\left(p^{n+1}\right)^2\,\right|\,\mathcal{F}_n\right] & = \left(p^n- {{\Delta t}}\nabla V(q^n)\right)^T \alpha_{{\Delta t}}^2 \left(p^n- {{\Delta t}}\nabla V(q^n)\right) + \frac1\beta {\mathrm{Tr}}\left[\left(1-\alpha_{{\Delta t}}^2\right)M^2\right] \\
& {\leqslant}{\mathrm{e}}^{-2m\gamma{{\Delta t}}} \left(p^n\right)^2 + 2 {{\Delta t}}\left\|\nabla V\right\|_{L^\infty} \left|p^n\right| + {{\Delta t}}^2 \left\|\nabla V\right\|_{L^\infty}^2 + \frac{1-{\mathrm{e}}^{-2\gamma{{\Delta t}}/m}}{\beta m^2} \\
& {\leqslant}\left({\mathrm{e}}^{-2m\gamma{{\Delta t}}} + \varepsilon {{\Delta t}}\right) \left(p^n\right)^2 + {{\Delta t}}\left( \frac1\varepsilon + {{\Delta t}}\right) \left\|\nabla V\right\|_{L^\infty}^2 + \frac{1-{\mathrm{e}}^{-2\gamma{{\Delta t}}/m}}{\beta m^2}.
\end{aligned}$$ We choose for instance $\varepsilon = m\gamma$, in which case $$0 {\leqslant}{\mathrm{e}}^{-2m\gamma{{\Delta t}}} + \varepsilon {{\Delta t}}{\leqslant}\exp\left( -C_a {{\Delta t}}\right), \qquad C_a = \frac{m\gamma}{2},$$ for ${{\Delta t}}$ sufficiently small, and $$0 {\leqslant}{{\Delta t}}\left( \frac1\varepsilon + {{\Delta t}}\right) \left\|\nabla V\right\|_{L^\infty}^2 + \frac{1-{\mathrm{e}}^{-2\gamma{{\Delta t}}/m}}{\beta m^2} {\leqslant}\widetilde{C}_b {{\Delta t}},
\qquad
\widetilde{C}_b = \frac{2}{m\gamma} \left\|\nabla V\right\|_{L^\infty}^2 + \frac{4\gamma}{\beta m^3},$$ for ${{\Delta t}}$ sufficiently small. Finally, since ${\mathcal{K}}_2(q,p) = 1+|p|^2$, $$\mathbb{E}\left[\left. {\mathcal{K}}_2\left(q^{n+1},p^{n+1}\right)\,\right|\,\mathcal{F}_n\right] {\leqslant}{\mathrm{e}}^{-C_a{{\Delta t}}} {\mathcal{K}}_2\left(q^n,p^n\right) + 1-{\mathrm{e}}^{-C_a{{\Delta t}}} + \widetilde{C}_b {{\Delta t}}{\leqslant}{\mathrm{e}}^{-C_a{{\Delta t}}} {\mathcal{K}}_2\left(q^n,p^n\right) + C_b {{\Delta t}},$$ for ${{\Delta t}}$ sufficiently small. This gives . To obtain , we iterate the bound : $$\begin{aligned}
P_{{\Delta t}}^n {\mathcal{K}}_s & {\leqslant}{\mathrm{e}}^{-C_a \,n{{\Delta t}}} {\mathcal{K}}_s + C_b {{\Delta t}}\left(1 + {\mathrm{e}}^{-C_a{{\Delta t}}} + \dots + {\mathrm{e}}^{-C_a\, (n-1){{\Delta t}}}\right) {\leqslant}{\mathrm{e}}^{-C_a \, n{{\Delta t}}} {\mathcal{K}}_s + \frac{C_b {{\Delta t}}}{1-{\mathrm{e}}^{-C_a{{\Delta t}}}}. \end{aligned}$$ The computations are similar for a general power $s {\geqslant}2$. We write $p^{n+1} = \alpha_{{\Delta t}}p^n + \delta_{{\Delta t}}$ with $\delta_{{\Delta t}}= - \alpha_{{\Delta t}}{{\Delta t}}\nabla V(q^n) + \sqrt{\beta^{-1}(1-\alpha_{{\Delta t}}^2)M} \, G^n$. Note that $\delta_{{\Delta t}}$ is of order ${{\Delta t}}^{1/2}$ because of the random term. We work componentwise, using the assumption that $M$ is diagonal, so that, denoting by $m_i$ the mass of the $i$th degree of freedom, $$\begin{aligned}
\left(p_i^{n+1}\right)^{2s} & = \left({\mathrm{e}}^{-\gamma {{\Delta t}}/m_i} p_i^n + \delta_{i,{{\Delta t}}} \right)^{2s} \\
& = {\mathrm{e}}^{-2s \gamma {{\Delta t}}/m_i} \left(p_i^n\right)^{2s} + 2s \, {\mathrm{e}}^{-(2s-1)\gamma {{\Delta t}}/m_i} \left(p_i^n\right)^{2s-1} \delta_{i,{{\Delta t}}} \\
& \ \ \ + s(2s-1) {\mathrm{e}}^{-2(s-1)\gamma {{\Delta t}}/m_i} \left(p_i^n\right)^{2(s-1)} \delta_{i,{{\Delta t}}}^2 + \dots
\end{aligned}$$ Taking expectations, $$\begin{aligned}
& \mathbb{E}\left[ \left. \left(p_i^{n+1}\right)^{2s} \, \right| \, \mathcal{F}_n \right]
= {\mathrm{e}}^{-2s \gamma {{\Delta t}}/m_i} \left(p_i^n\right)^{2s} - 2s \, {{\Delta t}}\, {\mathrm{e}}^{-2s \gamma {{\Delta t}}/m_i} \left(p_i^n\right)^{2s-1} \partial_{q_i}V(q^n) \\
& \ \ \ + s(2s-1) {\mathrm{e}}^{-2(s-1)\gamma {{\Delta t}}/m_i} \left(p_i^n\right)^{2(s-1)} \left({{\Delta t}}^2 {\mathrm{e}}^{-2 \gamma {{\Delta t}}/m_i} \partial_{q_i}V(q^n) + \frac{(1-{\mathrm{e}}^{-2 \gamma {{\Delta t}}/m_i})m_i}{\beta} \right) \\
& \ \ \ + {{\Delta t}}^2 r_{s,{{\Delta t}},i}(q^n) \left(1+\left(p^n\right)^{2s-3}\right),
\end{aligned}$$ where the remainder $r_{s,{{\Delta t}}}(q^n)$ is uniformly bounded as ${{\Delta t}}\to 0$. Distinguishing between $|p_i| {\geqslant}1/\varepsilon$ and $|p_i| {\leqslant}1/\varepsilon$, we have $$|p_i|^{2s -m} {\leqslant}\varepsilon^m (p_i)^{2s} + \frac{1}{\varepsilon^{2s-m}},$$ from which we obtain $$\mathbb{E}\left[ \left. \left(p_i^{n+1}\right)^{2s} \, \right| \, \mathcal{F}_n \right] {\leqslant}\widehat{a}_{{{\Delta t}},\varepsilon,i} \left(p_i^n\right)^{2s} + \widehat{b}_{{{\Delta t}},\varepsilon,i},$$ with $$\begin{aligned}
\widehat{a}_{{{\Delta t}},\varepsilon,i} & = {\mathrm{e}}^{-2s \gamma {{\Delta t}}/m_i} + 2s\varepsilon {{\Delta t}}\| \partial_{q_i}V \|_{L^\infty} \\
& \ \ \ + s(2s-1) \varepsilon^2 \left({{\Delta t}}^2 \| \partial_{q_i}V \|_{L^\infty} + \frac{(1-{\mathrm{e}}^{-2 \gamma {{\Delta t}}/m_i})m_i}{\beta}\right) + \varepsilon^3 {{\Delta t}}^2 \| r_{s,{{\Delta t}},i} \|_{L^\infty} ,
\end{aligned}$$ and $$\begin{aligned}
\widehat{b}_{{{\Delta t}},\varepsilon,i} & = \frac{2s}{\varepsilon} {{\Delta t}}\| \partial_{q_i}V \|_{L^\infty} \\
& \ \ \ + \frac{s(2s-1)}{\varepsilon^2} \left({{\Delta t}}^2 \| \partial_{q_i}V \|_{L^\infty} + \frac{(1-{\mathrm{e}}^{-2 \gamma {{\Delta t}}/m_i})m_i}{\beta}\right) + {{\Delta t}}^2\left(1 + \frac{1}{\varepsilon^3}\right) \| r_{s,{{\Delta t}},i} \|_{L^\infty}.
\end{aligned}$$ The proof is then concluded as in the case $s=1$ by choosing $\varepsilon$ sufficiently small (independently of ${{\Delta t}}$).
It is sufficient to prove the result for indicator functions of Borel sets $A = A_q \times A_p \subset {\mathcal{E}}$, where $A_q \subset \mathcal{M}$ while $A_p \subset \mathbb{R}^{dN}$ (see [@Rudin]). We therefore aim at proving $$\mathbb{P}\left( \left(q^n,p^n\right) \in A \, \left| \, \left|p^0\right| {\leqslant}p_{\rm max} \right.\right) {\geqslant}\alpha \, \nu(A),$$ for a well chosen probability measure $\nu$ and a constant $\alpha > 0$. The idea of the proof is to explicitly rewrite $q^n$ and $p^n$ as perturbations of the reference evolution corresponding to $\nabla V = 0$ and $(q^0,p^0) = (0,0)$. Since we consider smooth potentials and the position space is compact, the perturbation can be uniformly controlled when the initial momenta are within a compact set.
We write the proof for the scheme associated with the evolution operator $P_{{\Delta t}}^{B,A,\gamma C}$, as in the proof of Lemma \[lem:Lyapunov\]. A simple computation shows that, for $n {\geqslant}1$, $$q^n = q^0 + {{\Delta t}}M^{-1} \left(p^{n-1} + \dots + p^0 \right) - {{\Delta t}}^2 M^{-1} \Big( \nabla V(q^{n-1}) + \dots + \nabla V(q^0) \Big),$$ and $$\begin{aligned}
p^n = \alpha_{{\Delta t}}^n \, p^0 & - {{\Delta t}}\, \alpha_{{\Delta t}}\left( \nabla V(q^{n-1}) + \alpha_{{\Delta t}}\nabla V(q^{n-2}) + \dots + \alpha_{{\Delta t}}^{n-1} \nabla V(q^0) \right) \\
& + \sqrt{\frac{1-\alpha_{{\Delta t}}^2}{\beta} M}\left(G^{n-1} + \alpha_{{\Delta t}}G^{n-2} + \dots + \alpha_{{\Delta t}}^{n-1} G^0 \right).
\end{aligned}$$ Denote by $\mathcal{G}^n$ the centered Gaussian random variable $$\mathcal{G}^n = \sqrt{\frac{1-\alpha_{{\Delta t}}^2}{\beta} M}\left(G^{n-1} + \alpha_{{\Delta t}}G^{n-2} + \dots + \alpha_{{\Delta t}}^{n-1} G^0 \right).$$ Introduce also $$\begin{aligned}
F^n & = -\alpha_{{\Delta t}}\left( \nabla V(q^{n-1}) + \alpha_{{\Delta t}}\nabla V(q^{n-2}) + \dots + \alpha_{{\Delta t}}^{n-1} \nabla V(q^0) \right), \\
\mathscr{P}^n & = \alpha_{{\Delta t}}^n \, p^0 + {{\Delta t}}\, F^n, \\
\mathscr{Q}^n & = q^0 + {{\Delta t}}M^{-1} \left({{\Delta t}}\sum_{m=0}^{n-1} F^m + \frac{1-\alpha_{{\Delta t}}^n}{1-\alpha_{{\Delta t}}} p^0 \right) - {{\Delta t}}^2 M^{-1} \Big( \nabla V(q^{n-1}) + \dots + \nabla V(q^0) \Big).\\
\end{aligned}$$ With this notation, $$p^n = \mathscr{P}^n + \mathcal{G}^n, \qquad q^n = \mathscr{Q}^n + \widetilde{\mathcal{G}}^n,$$ where $$\begin{aligned}
\widetilde{\mathcal{G}}^n & = {{\Delta t}}M^{-1} \sum_{m=1}^{n-1} \mathcal{G}^m \\
& = {{\Delta t}}\sqrt{\frac{1-\alpha_{{\Delta t}}^2}{\beta} M^{-1}}\Big(G^{n-2} + (1+\alpha_{{\Delta t}}) G^{n-3} + \dots + (1+\alpha_{{\Delta t}}+\dots+\alpha_{{\Delta t}}^{n-2}) G^0 \Big)
\end{aligned}$$ is a centered Gaussian random variable. Now, $$\label{eq:first_ineq_minorization}
\mathbb{P}\left( \left(q^n,p^n\right) \in A \, \left| \, \left|p^0\right| {\leqslant}p_{\rm max} \right.\right)
= \mathbb{P}\left(\left. \left(\widetilde{\mathcal{G}}^n,\mathcal{G}^n\right) \in \left(A_q -\mathscr{Q}^n\right) \times \left( A_p - \mathscr{P}^n\right) \right| \, \left|p^0\right| {\leqslant}p_{\rm max} \right).$$ In fact, we consider in the sequel that the random variable $\widetilde{\mathcal{G}}^n$ has values in $\mathbb{R}^{dN}$ rather than $\mathcal{M}$ and understand $A_q -\mathscr{Q}^n$ as a subset of $\mathbb{R}^{dN}$ rather than $\mathcal{M}$. This amounts to neglecting the possible periodic images, and henceforth reduces the probability on the right-hand side of the above inequality. This is however not a problem since we seek a lower bound.
Note that ${{\Delta t}}\, F^n$ is uniformly bounded: using $0 {\leqslant}\alpha_{{\Delta t}}{\leqslant}\exp(-\gamma m {{\Delta t}})$ in the sense of symmetric, positive matrices (with $m {\leqslant}M {\leqslant}m^{-1}$), $$\left| {{\Delta t}}\, F^n \right| {\leqslant}\| \nabla V \|_{L^\infty} \, \frac{{{\Delta t}}}{1-\exp(-\gamma m {{\Delta t}})} {\leqslant}\frac{2 }{m\gamma} \, \| \nabla V \|_{L^\infty}$$ provided ${{\Delta t}}$ is sufficiently small. Therefore, there exists a constant $R > 0$ (depending on $p_{\rm max}$) and ${{\Delta t}}^*>0$ such that, for all timesteps $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$ and corresponding integration steps $0 {\leqslant}n {\leqslant}T/{{\Delta t}}$, $$\label{eq:second_ineq_minorization}
\left| \mathscr{Q}^n \right|{\leqslant}R, \qquad \left| \mathscr{P}^n \right|{\leqslant}R.$$ A lengthy but straightforward computation shows that the variance of the centered Gaussian vector $\left(\widetilde{\mathcal{G}}^n,\mathcal{G}^n\right)$ is $$\mathscr{V}^n = \mathbb{E}\left[ \left(\widetilde{\mathcal{G}}^n,\mathcal{G}^n\right)^T \left(\widetilde{\mathcal{G}}^n,\mathcal{G}^n\right) \right] = \begin{pmatrix} \mathscr{V}^n_{qq} & \mathscr{V}^n_{qp} \\ \mathscr{V}^n_{qp} & \mathscr{V}^n_{pp} \end{pmatrix}$$ with $$\left\{ \begin{aligned}
\mathscr{V}^n_{qq} & = \frac{{{\Delta t}}\, (1-\alpha_{{\Delta t}}^2)}{(1-\alpha_{{\Delta t}})^2} M^{-1}\left( (n-1){{\Delta t}}- \frac{2{{\Delta t}}\, \alpha_{{\Delta t}}}{1-\alpha_{{\Delta t}}} (1-\alpha_{{\Delta t}}^{n-1}) + \frac{{{\Delta t}}\, \alpha_{{\Delta t}}^2}{1-\alpha_{{\Delta t}}^2} \left(1-\alpha_{{\Delta t}}^{2(n-1)}\right)\right),\\
\mathscr{V}^n_{qp} & = \frac{{{\Delta t}}\, \alpha_{{\Delta t}}}{\beta (1-\alpha_{{\Delta t}})} \Big(1 - \alpha_{{\Delta t}}^{n-1} (1+\alpha_{{\Delta t}}) + \alpha_{{\Delta t}}^{2n-1} \Big),\\
\mathscr{V}^n_{pp} & = \frac{M}{\beta}(1-\alpha_{{\Delta t}}^{2n}).
\end{aligned} \right.$$ To check that this expression is appropriate, we note that it converges as ${{\Delta t}}\to 0$ with $n{{\Delta t}}\to T$ to the variance of the limiting continuous process $${\mathrm{d}q}_t = M^{-1} p_t \, {\mathrm{d}t}, \qquad {\mathrm{d}p}_t = -\gamma M^{-1} p_t \, {\mathrm{d}t} + \sqrt{\frac{2\gamma}{\beta}} \, {\mathrm{d}W}_t,$$ starting from $(q_0,p_0) = (0,0)$, which reads $$\mathscr{V} = \begin{pmatrix} \mathscr{V}_{qq} & \mathscr{V}_{qp} \\ \mathscr{V}_{qp} & \mathscr{V}_{pp} \end{pmatrix},$$ with $$\left\{
\begin{aligned}
\mathscr{V}_{qq} & = \frac{1}{\beta \gamma}\left(2T - \frac{M}{\gamma}\left(3 - 4 \, \alpha_T + \alpha_T^2\right)\right),\\
\mathscr{V}_{qp} & = \frac{M}{\beta \gamma}\left(1-\alpha_T\right)^2, \\
\mathscr{V}_{pp} & = \frac{M}{\beta} \left(1-\alpha_T^2\right).
\end{aligned}
\right.$$ Upon reducing ${{\Delta t}}^* > 0$, it holds $\mathscr{V}/2 {\leqslant}\mathscr{V}^{\lceil T/{{\Delta t}}\rceil} {\leqslant}2\mathscr{V}$ for $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$. In particular, $\mathscr{V}^{\lceil T/{{\Delta t}}\rceil}$ is invertible for $T$ sufficiently large. For a set $E_q \times E_p \subset \mathbb{R}^{2dN}$, it then holds that $$\begin{aligned}
\mathbb{P}\left( \left(\widetilde{\mathcal{G}}^{\lceil T/{{\Delta t}}\rceil},\mathcal{G}^{\lceil T/{{\Delta t}}\rceil}\right) \in E \right) &=
(2\pi)^{-dN} \mathrm{det}\left(\mathscr{V}^{\lceil T/{{\Delta t}}\rceil}\right)^{-1/2} \int_{E_q \times E_p} \exp\left( -\frac12 x^T \left(\mathscr{V}^{\lceil T/{{\Delta t}}\rceil}\right)^{-1} x \right) \, {\mathrm{d}x} \nonumber \\
& {\geqslant}\pi^{-dN} 2^{-3dN/2} \mathrm{det}\left(\mathscr{V}\right)^{-1/2} \int_{E_q \times E_p} \exp\left( -x^T \mathscr{V}^{-1} x \right) \, {\mathrm{d}x}. \label{eq:third_ineq_minorization}\end{aligned}$$ The result follows by combining -- and introducing the probability measure $$\nu(A_q \times A_p) = Z_R^{-1} \inf_{|\mathscr{Q}|, |\mathscr{P}| {\leqslant}R} \int_{(A_q - \mathscr{Q}) \times (A_p-\mathscr{P})} \exp\left( -x^T \mathscr{V}^{-1} x \right) \, {\mathrm{d}x},$$ as well as $\alpha = Z_R \pi^{-dN} 2^{-3dN/2} \mathrm{det}\left(\mathscr{V}\right)^{-1/2}$.
We only prove and since the other results are standard. To obtain the bound , we first note that, by the results of [@HM11], there exists $\widetilde{\lambda} > 0$ such that, for any function $f \in L^\infty_{{\mathcal{K}}_s,{{\Delta t}}}$ and $0 < {{\Delta t}}{\leqslant}{{\Delta t}}^*$ (the critical timestep being given by Lemmas \[lem:Lyapunov\] and \[lem:minorization\]), the following holds for almost all $(q,p) \in \mathcal{E}$: $$\forall m \in \mathbb{N}, \qquad \left| \left( \left[P_{{\Delta t}}^{\lceil T/{{\Delta t}}\rceil}\right]^m f\right)(q,p) \right| {\leqslant}K \, {\mathcal{K}}_s(q,p) \, {\mathrm{e}}^{-\widetilde{\lambda} m} \, \| f \|_{L^\infty_{{\mathcal{K}}_s}}.$$ For a general index $n \in \mathbb{N}$, we write $$n = m_n \left\lceil \frac{T}{{{\Delta t}}} \right\rceil + \widetilde{n}, \qquad 0 {\leqslant}\widetilde{n} {\leqslant}\left\lceil \frac{T}{{{\Delta t}}} \right\rceil - 1,$$ so that, using the contractivity property $|P_{{\Delta t}}f(q,p) | {\leqslant}|f(q,p)|$, $$\left| P_{{\Delta t}}^n f(q,p) \right| {\leqslant}K \, {\mathcal{K}}_s(q,p) \, {\mathrm{e}}^{-\widetilde{\lambda} m_n} \, \| f \|_{L^\infty_{{\mathcal{K}}_s}}.$$ Introducing $\lambda = \widetilde{\lambda}/T$, the argument of the exponent reads $$\widetilde{\lambda} m_n = \lambda (n-\widetilde{n}){{\Delta t}}\, \frac{T}{{{\Delta t}}}\left\lceil \frac{T}{{{\Delta t}}} \right\rceil^{-1} {\geqslant}\frac{\lambda n {{\Delta t}}}{2} - \lambda T,$$ when ${{\Delta t}}$ is sufficiently small. This gives .
The moment estimate is obtained by averaging with respect to the invariant measure: $$\int_{\mathcal{E}}\left(P_{{\Delta t}}{\mathcal{K}}_s\right) {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} {\leqslant}{\mathrm{e}}^{-C_a {{\Delta t}}} \int_{\mathcal{E}}{\mathcal{K}}_s \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} + C_b {{\Delta t}}.$$ Since $\mu_{\gamma,{{\Delta t}}}$ is invariant, $$\int_{\mathcal{E}}\left(P_{{\Delta t}}{\mathcal{K}}_s\right) {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} = \int_{\mathcal{E}}{\mathcal{K}}_s \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}},$$ so that $$\left(1-{\mathrm{e}}^{-C_a {{\Delta t}}}\right) \int_{\mathcal{E}}{\mathcal{K}}_s \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} {\leqslant}C_b {{\Delta t}},$$ which gives the desired result with $R=2C_b/C_a$ for instance, provided ${{\Delta t}}$ is sufficiently small.
Some useful results {#sec:useful}
-------------------
### Expansion of the evolution operator {#sec:expansion_evolution}
We give in this section an expression for the evolution operator $$P_t = {\mathrm{e}}^{t A_M} \dots {\mathrm{e}}^{t A_1},$$ which can easily be compared to the evolution operator ${\mathrm{e}}^{t (A_1+\dots+A_M)}$. We assume that the generators $A_i$ of all elementary dynamics are well defined operators on a core $X$, with image in $X$ (typically, $X = \mathcal{S}$ or a subset of this space such as $\widetilde{\mathcal{S}}$). We also assume that the elementary evolution semigroups ${\mathrm{e}}^{t A_i}$, as well as $P_t$, are well defined on $X$ with values in $X$. These semigroups may be extended to bounded operators on an appropriate Banach space using the Hille-Yosida theorem for instance (see [@Pazy]). All the operator equalities stated in this section have to be considered in the strong sense, namely $T_1 = T_2$ means $T_1 \varphi = T_2 \varphi$ for all $\varphi \in X$.
It is easy to check that the operators $A,B,C$ defined in map $\mathcal{S}$ to itself. It is in fact possible to analytically write down the action of the associated semigroups: $$\label{eq:analytic_expressions_semigroups}
\left\{ \begin{aligned}
\left({\mathrm{e}^{t A}}\varphi\right)(q,p) & = \varphi\Big(q+tM^{-1}p,p\Big), \\
\left({\mathrm{e}^{t B}}\varphi\right)(q,p) & = \varphi\Big(q,p-t\nabla V(q)\Big), \\
\left({\mathrm{e}^{t C}}\varphi\right)(q,p) & = \int_{{\mathbb{R}}^{dN}} \varphi\left(q,{\mathrm{e}}^{-\gamma M^{-1} t}p + \left(\frac{1-{\mathrm{e}}^{-2\gamma M^{-1} t}}{\beta}M\right)^{1/2} x\right) \frac{{\mathrm{e}}^{-|x|^2/2}}{(2\pi)^{dN/2}} \, {\mathrm{d}x}.
\end{aligned} \right.$$
Coming back to the general case, the key building block for the subsequent numerical analysis is the following equality: $$P_t = P_0 + t \left. \frac{dP_t}{dt} \right|_{t=0} + \frac{t^2}{2} \left. \frac{d^2 P_t}{dt^2} \right|_{t=0} + \dots + \frac{t^n}{n!} \left. \frac{d^n P_t}{dt^n} \right|_{t=0} + \frac{t^{n+1}}{n!} \int_0^1 (1-\theta)^n \left. \frac{d^{n+1} P_s}{ds^{n+1}} \right|_{s = \theta t} \, {\mathrm{d}\theta}.$$ Now, $$\begin{aligned}
\frac{dP_t}{dt} & = A_M {\mathrm{e}}^{t A_M} \dots {\mathrm{e}}^{t A_1} + {\mathrm{e}}^{t A_M} A_{M-1} {\mathrm{e}}^{t A_{M-1}} \dots {\mathrm{e}}^{t A_1} + \dots
+ {\mathrm{e}}^{t A_M} \dots {\mathrm{e}}^{t A_1} A_1 \\
& = {\mathcal{T}}\left[ (A_1+\dots+A_M) P_t \right]
\end{aligned}$$ where ${\mathcal{T}}$ is a notation indicating that the operators with the smallest indices (or their associated semigroups) are farthest to the right. In fact, simple computations show that $$\frac{d^n P_t}{dt^n} = {\mathcal{T}}\Big[ (A_1+\dots+A_M)^n P_t \Big].$$ Therefore, the following equality holds when applied to functions $\varphi \in X$: $$\label{eq:P_t_expansion}
\begin{aligned}
P_t \varphi & = \varphi + t(A_1+\dots+A_M)\varphi + \frac{t^2}{2}{\mathcal{T}}\Big[ (A_1+\dots+A_M)^2 \Big]\varphi +\dots + \frac{t^n}{n!}{\mathcal{T}}\Big[ (A_1+\dots+A_M)^n \Big]\varphi \\
& \ \ + \frac{t^{n+1}}{n!} \int_0^1 (1-\theta)^n {\mathcal{T}}\Big[ (A_1+\dots+A_M)^{n+1} P_{\theta t}\Big]\varphi\, {\mathrm{d}\theta}.
\end{aligned}$$
### Baker-Campbell-Hausdorff (BCH) formula {#sec:BCH}
It is important to rewrite the various terms in the right-hand side of in a form more amenable to analytical computations. More precisely, it is convenient to write the following equality in terms of operators defined on $X$: $${\mathcal{T}}\Big[ (A_1+\dots+A_M)^n \Big] = (A_1+\dots+A_M)^n + S_n,$$ where the operator $S_n$ involves commutators $[A_i,A_j]$, which can also be defined as operators on $X$ with values in $X$. In fact, the algebraic expressions of the operators $S_n$ can be formally obtained from the BCH formula for first order splittings (see for instance [@HairerLubichWanner06 Section III.4.2]): for $M=3$, $${\mathrm{e}}^{{{\Delta t}}A_3} {\mathrm{e}}^{{{\Delta t}}A_2} {\mathrm{e}}^{{{\Delta t}}A_1} = {\mathrm{e}}^{{{\Delta t}}\mathcal{A}},
\qquad \mathcal{A} = A_1+A_2+A_3 + \frac{{{\Delta t}}}{2} \Big( [A_3,A_1+A_2] + [A_2,A_1]\Big) + \dots,$$ and from the symmetric BCH formula for second order involving 3 operators (obtained by composition of the standard BCH formula involving 2 operators): $$\label{eq:BCH}
{\mathrm{e}}^{{{\Delta t}}A_1/2} {\mathrm{e}}^{{{\Delta t}}A_2/2} {\mathrm{e}}^{{{\Delta t}}A_3} {\mathrm{e}}^{{{\Delta t}}A_2/2} {\mathrm{e}}^{{{\Delta t}}A_1/2}
= {\mathrm{e}}^{{{\Delta t}}\mathcal{A}},$$ with $$\begin{aligned}
\mathcal{A} = A_1+A_2+A_3 + \frac{{{\Delta t}}^2}{12} \left( [A_3,[A_3,A_2]] + [A_2+A_3,[A_2+A_3,A_1]] \phantom{\frac12} \right.&\\
\left. - \frac12 [A_2,[A_2,A_3]] - \frac12 [A_1,[A_1,A_2+A_3]]\right) + \dots&
\end{aligned}
$$ where we do not write down the expressions of the higher order terms ${{\Delta t}}^{2n}$ (for $n {\geqslant}2$). Let us insist that these formulas are only formal (since the operators appearing the argument of the exponential on the right-hand side involve more and more derivatives), but nonetheless allow us to find the algebraic expressions of $S_n$ upon formally expanding the exponential as $${\mathrm{e}}^{{{\Delta t}}\mathcal{A}} = {\mathrm{Id}}+ {{\Delta t}}\mathcal{A} + \frac{{{\Delta t}}^2}{2} \mathcal{A}^2 + \dots$$ and identifying terms with the same powers of ${{\Delta t}}$ in .
### Approximate inverse operators {#sec:approx_inv}
Consider an operator $A$ defined on some core $X$ (typically some subspace of $\mathcal{S}$), and whose inverse is also defined on $X$ in the following sense: for any $g \in X$, there exist $f \in X$ such that $Af=g$. We denote by $A^{-1}g$ the element $f \in X$. At this stage, we do not assume any boundedness in an appropriate operator norm for $A^{-1}$ or some extension of it. We next consider a perturbation ${{\Delta t}}^\alpha B$ for some exponent $\alpha {\geqslant}1$, where $B$ is also defined on $X$ and has values in $X$. In the typical situations encountered in this article, $B$ is not bounded with respect to $A$ in an appropriate operator norm since it may involve higher order derivatives than $A$ does. It is therefore impossible in general to properly define the inverse of $A + {{\Delta t}}^\alpha B$.
However, it is possible to introduce an approximate inverse, which we define as an operator $Q_{{{\Delta t}},n}$ from $X$ to $X$ such that there exists an operator $\widetilde{Q}_{{{\Delta t}},n}$ from $X$ to $X$ for which the following equality holds for any function $f\in X$: $$\label{eq:pseudo_inverse_general_definition}
(A + {{\Delta t}}^\alpha B)Q_{{{\Delta t}},n}f = f + {{\Delta t}}^{(n+1)\alpha} \widetilde{Q}_{{{\Delta t}},n}f.$$ To this end we simply truncate the formal series expansion of the inverse of the operator $A + {{\Delta t}}^\alpha \, B = A({\mathrm{Id}}+ {{\Delta t}}^\alpha \, A^{-1} B)$, which formally reads $A^{-1} - {{\Delta t}}^\alpha \, A^{-1}B A^{-1} + {{\Delta t}}^{2\alpha} \, A^{-1}B A^{-1}B A^{-1} + \dots$. For instance, $Q_{{{\Delta t}},1} = A^{-1} - {{\Delta t}}^\alpha \, A^{-1}B A^{-1}$ and $Q_{{{\Delta t}},2} = A^{-1} - {{\Delta t}}^\alpha \, A^{-1}B A^{-1} + {{\Delta t}}^{2\alpha} \, A^{-1}B A^{-1}B A^{-1}$ indeed are operators from $X$ to $X$ satisfying , respectively with $n=1$ and $n=2$.
Proof of Theorem \[thm:error\_first\_order\_schemes\] {#sec:proof_thm:error_first_order_schemes}
-----------------------------------------------------
We write the proof for the scheme associated with $P_{{\Delta t}}^{\gamma C, B, A} = {\mathrm{e}}^{\gamma {{\Delta t}}C} {\mathrm{e}}^{{{\Delta t}}B} {\mathrm{e}}^{{{\Delta t}}A}$, the proof for the scheme $P_{{\Delta t}}^{\gamma C, A, B}$ following the same lines. The results for the other schemes are then obtained with the TU lemma (Lemma \[lem:TU\]). Without loss of generality, we perform the proof for a function $\psi$ with average zero with respect to $\mu$ (recovering the general case by adding a constant to $\psi$ in the final expression).
#### Proof of .
First, note that, by definition of the invariant measure $\mu_{\gamma,{{\Delta t}}}$, it holds that, for any $\varphi \in \mathcal{S}$, $$\label{eq:avg_0_mu_dt}
\int_{\mathcal{E}}\left(\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}}\right) \varphi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} = 0.$$ The next step is to choose the correction function $f_{1,\gamma}$. Using the results of Section \[sec:useful\], a simple computation shows that $$\label{eq:P_dt_eq_order1}
P_{{\Delta t}}^{\gamma C, B, A} = {\mathrm{Id}}+ {{\Delta t}}{\mathcal{L}_\gamma}+ \frac{{{\Delta t}}^2}{2} \left({\mathcal{L}_\gamma}^2 + S_1\right) + {{\Delta t}}^3 R_{1,{{\Delta t}}}, \qquad S_1 = [C,A+B] + [B,A],$$ where the subscript index 1 refers to the order of the splitting, and where all operators are understood as operators on $\mathcal{S}$. More precisely, $$R_{1,{{\Delta t}}} = \frac12 \int_0^1 (1-\theta)^2 \mathcal{R}_{\theta \Delta t} \, {\mathrm{d}\theta},$$ where $\mathcal{R}_s$ is a finite linear combination of terms of the form $C^\gamma {\mathrm{e}^{s C}}B^\beta {\mathrm{e}^{s B}}A^\alpha {\mathrm{e}^{s A}}$ with $\alpha,\beta,\gamma {\geqslant}0$ and $\alpha + \beta + \gamma = 3$. In any case, $R_{1,{{\Delta t}}}$ is a differential operator involving at most 6 derivatives, and with smooth coefficients of at most polynomial growth. It is easily seen that $R_{1,{{\Delta t}}}\psi$ is uniformly bounded in some space $L^\infty_{{\mathcal{K}}_s}$ (with $s$ chosen sufficiently large) for ${{\Delta t}}$ small enough when $\psi \in \mathcal{S}$. Therefore, for any $\varphi \in \mathcal{S}$ and $f_{1,\gamma} \in \widetilde{\mathcal{S}}$, $$\begin{aligned}
& \int_{\mathcal{E}}\left[\left(\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}}\right) \varphi\right] (1+{{\Delta t}}f_{1,\gamma}) \, {\mathrm{d}\mu} \\
& \qquad = -\int_{\mathcal{E}}\left[\left({\mathcal{L}_\gamma}+ \frac{{{\Delta t}}}{2} \left({\mathcal{L}_\gamma}^2 + S_1\right) + {{\Delta t}}^2 R_{1,{{\Delta t}}}\right)\varphi\right] (1+{{\Delta t}}f_{1,\gamma}) \, {\mathrm{d}\mu} \\
& \qquad = -{{\Delta t}}\int_{\mathcal{E}}\left( \frac12 S_1 \varphi + ({\mathcal{L}_\gamma}\varphi) f_{1,\gamma} \right)d\mu
- {{\Delta t}}^2 \int_{\mathcal{E}}\left( \left[\frac{1}{2} \left({\mathcal{L}_\gamma}^2 + S_1\right)\varphi\right] f_{1,\gamma} + (R_{1,{{\Delta t}}}\varphi) (1+{{\Delta t}}f_{1,\gamma}) \right) {\mathrm{d}\mu}.
\end{aligned}$$ The dominant term on the right-hand side can be written, using integration by parts, $$\int_{\mathcal{E}}\left(\frac12 S_1 \varphi + ({\mathcal{L}_\gamma}\varphi) f_{1,\gamma} \right) {\mathrm{d}\mu} = \int_{\mathcal{E}}\varphi \left[ \frac12 S_1^* \mathbf{1} + {\mathcal{L}_\gamma}^* f_{1,\gamma}\right] {\mathrm{d}\mu}.$$ In view of , we choose the correction function in order to eliminate the dominant term: $$\label{eq:choice_f1}
{\mathcal{L}_\gamma}^* f_{1,\gamma} = -\frac12 S_1^* \mathbf{1}.$$ Relying on Theorem \[thm:stability\_S\] and , the function $f_{1,\gamma}$ is a well defined element from $\widetilde{\mathcal{S}}$ since the right-hand side of belongs to $\widetilde{\mathcal{S}}$. A direct computation using integration by parts indeed shows that $S_1^* \mathbf{1} \in \mathcal{S}$ (see below). The centering condition follows from the fact that $\mathbf{1} \in \mathrm{Ker}(S_1)$: indeed, $$\int_{\mathcal{E}}S_1^* \mathbf{1} \, {\mathrm{d}\mu} = \int_{\mathcal{E}}S_1 \mathbf{1} \, {\mathrm{d}\mu} = 0.$$ With the choice , $$\label{eq:first_order_corrected}
\begin{aligned}
& \int_{\mathcal{E}}\left[\left(\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}}\right) \varphi\right] (1+{{\Delta t}}f_{1,\gamma}) \, {\mathrm{d}\mu} \\
& \qquad = - {{\Delta t}}^2 \int_{\mathcal{E}}\left( \left[\frac{1}{2} \left({\mathcal{L}_\gamma}^2 + S_1\right)\varphi\right] f_{1,\gamma} + (R_{1,{{\Delta t}}}\varphi) (1+{{\Delta t}}f_{1,\gamma}) \right) {\mathrm{d}\mu}.
\end{aligned}$$ We would like, at this stage, to replace the observable $\varphi$ appearing on the left hand side by the function $$\left(\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}}\right)^{-1} \psi.$$ However, we do not have any control on the derivatives of this function (Corollary \[corr:resolvent\_estimates\_I\_Pdt\] allows to control the norm of the function, not of its derivatives), whereas such a control is required to bound the remainder terms. In order to use an approximate inverse operator involving iterated powers of $\mathcal{L}_\gamma^{-1}$ (see Section \[sec:approx\_inv\]), we first need to make sure that all operators are defined on $\widetilde{\mathcal{S}}$, with values in $\widetilde{\mathcal{S}}$. This is the case for $\mathcal{L}_\gamma$ and its inverse, but not for the other operators appearing in , which have values in $\mathcal{S}$. We therefore project out averages with respect to $\mu$. Define to this end the projector $$\label{eq:def_Pi}
\Pi^\perp f = f - \int_\mathcal{E} f \, {\mathrm{d}\mu},$$ which maps $\mathcal{S}$ to $\widetilde{\mathcal{S}}$. Then, for a function $\varphi \in \widetilde{\mathcal{S}}$ (for which $\Pi^\perp \varphi = \varphi$), can be rewritten as $$\begin{aligned}
& \int_{\mathcal{E}}\left[ \Pi^\perp \frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}} \Pi^\perp \varphi\right] (1+{{\Delta t}}f_{1,\gamma}) \, {\mathrm{d}\mu} \\
& \qquad = \frac{1}{{{\Delta t}}} \int_{\mathcal{E}}P_{{\Delta t}}^{\gamma C,B,A} \varphi \, {\mathrm{d}\mu} - {{\Delta t}}^2 \int_{\mathcal{E}}\left( \left[\frac{1}{2} \left({\mathcal{L}_\gamma}^2 + S_1\right)\varphi\right] f_{1,\gamma} + (R_{1,{{\Delta t}}}\varphi) (1+{{\Delta t}}f_{1,\gamma}) \right) {\mathrm{d}\mu},
\end{aligned}$$ where we have used the fact that $f_{1,\gamma}$ is of average zero with respect to $\mu$. On the other hand, may be rewritten $$\int_{\mathcal{E}}\Pi^\perp \frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}} \Pi^\perp \varphi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} = \frac{1}{{{\Delta t}}} \int_{\mathcal{E}}P_{{\Delta t}}^{\gamma C,B,A} \varphi \, {\mathrm{d}\mu}.$$ Therefore, $$\label{eq:first_order_corrected_bis}
\begin{aligned}
& \int_{\mathcal{E}}\left[ \Pi^\perp \frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}} \Pi^\perp \varphi\right] (1+{{\Delta t}}f_{1,\gamma}) \, {\mathrm{d}\mu} - \int_{\mathcal{E}}\Pi^\perp \frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}} \Pi^\perp \varphi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} \\
& \qquad \qquad = - {{\Delta t}}^2 \int_{\mathcal{E}}\left( \left[\frac{1}{2} \left({\mathcal{L}_\gamma}^2 + S_1\right)\varphi\right] f_{1,\gamma} + (R_{1,{{\Delta t}}}\varphi) (1+{{\Delta t}}f_{1,\gamma}) \right) {\mathrm{d}\mu}.
\end{aligned}$$ As a consequence of the presence of the projection $\Pi^\perp$, all of the operators in are restricted to the range of $\Pi^\perp$, *i.e.* the following equality holds on $\widetilde{\mathcal{S}}$: $$-\Pi^\perp \frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C, B, A}}{{{\Delta t}}} \Pi^\perp = {\mathcal{L}_\gamma}+ \frac{{{\Delta t}}}{2} \left({\mathcal{L}_\gamma}^2 + \Pi^\perp S_1 \Pi^\perp \right) + {{\Delta t}}^2 \Pi^\perp R_{1,{{\Delta t}}} \Pi^\perp.$$ We therefore introduce the operator $$Q_{1,{{\Delta t}}} = -{\mathcal{L}_\gamma}^{-1} + \frac{{\Delta t}}2 (\Pi^\perp + {\mathcal{L}_\gamma}^{-1} \Pi^\perp S_1 \Pi^\perp {\mathcal{L}_\gamma}^{-1}),$$ which is a well defined operator from $\widetilde{\mathcal{S}}$ to $\widetilde{\mathcal{S}}$ such that $$\left(\Pi^\perp \frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A}}{{{\Delta t}}} \Pi^\perp \right) Q_{1,{{\Delta t}}} = \Pi^\perp + {{\Delta t}}^2 Z_{1,{{\Delta t}}},$$ where $Z_{1,{{\Delta t}}}$ maps $\mathcal{S}$ to $\mathcal{S}$. We finally replace $\varphi$ by $Q_{1,{{\Delta t}}}\psi$ in . This gives (recall that $\Pi^\perp \psi = \psi$ by assumption) $$\int_{\mathcal{E}}\psi \, (1+{{\Delta t}}f_{1,\gamma}) \, {\mathrm{d}\mu} - \int_\mathcal{E} \psi \, {\mathrm{d}\mu_{{\Delta t}}} = {{\Delta t}}^2 \int_{\mathcal{E}}\left[\left(\widetilde{R}_{1,{{\Delta t}}} \psi\right) f_{1,\gamma} + \widehat{R}_{1,{{\Delta t}}} \psi\right] {\mathrm{d}\mu},
$$ where the functions $\widetilde{R}_{1,{{\Delta t}}} \psi, \widehat{R}_{1,{{\Delta t}}} \psi$ belong to $\mathcal{S}$ when $\psi$ does. The integral on the right-hand side is uniformly bounded for small ${{\Delta t}}$ (using the fact that the functions appearing in the integral are in $\mathcal{S}$ and relying on Proposition \[prop:ergodicity\_MC\]). This gives for the splitting scheme $P_{{\Delta t}}^{\gamma C, B, A}$.
#### Proof of .
The function $f_1^{\gamma C,B,A} \in \widetilde{\mathcal{S}}$ (denoted by $f_{1,\gamma}$ above) is uniquely determined by the equation $${\mathcal{L}_\gamma}^* f_1^{\gamma C,B,A} = -\frac12 S_1^* \mathbf{1} = - \frac12 \Big( [C,A+B] + [B,A] \Big)^* \mathbf{1}, \qquad \int_\mathcal{E} f_1^{\gamma C,B,A} \, {\mathrm{d}\mu} = 0,$$ where we have used $[{\mathcal{L}_\gamma}^2]^* \mathbf{1} = 0$ to simplify the right-hand side. Now, $[C,A+B]^* = [C,A+B]$ since $C^* = C$ and $(A+B)^* = -(A+B)$. Therefore, $[C,A+B]^* \mathbf{1} = 0$. In addition, $$[B,A]^* \mathbf{1} = - (A+B)^*g = (A+B)g,$$ since $A^* = -A + g$ and $B^* = -B -g$. Therefore, $$\label{eq:S1_star}
S_1^* \mathbf{1} = (A+B)g.$$ This gives the first expression in .
To obtain the expressions of $f_1^{A,\gamma C,B}$ and $f_1^{B,A,\gamma C}$, we use the TU lemma, where the operators $U_{{\Delta t}}$ respectively read ${\mathrm{e}}^{\gamma {{\Delta t}}C}{\mathrm{e}}^{{{\Delta t}}B} = {\mathrm{Id}}+ {{\Delta t}}(B+\gamma C) + {{\Delta t}}^2 R_{{\Delta t}}$ and ${\mathrm{e}}^{\gamma {{\Delta t}}C}$ (which preserves $\mu$). We actually are in a situation similar to : $$f_1^{B,A,\gamma C} = f_1^{\gamma C,B,A}, \qquad f_1^{A,\gamma C,B} = f_1^{\gamma C,B,A} + B^* \mathbf{1}.$$ The expressions for the first order corrections when the operators $A$ and $B$ are exchanged are obtained by noting that the sign of $S_1^* \mathbf{1}$ is changed and that $f_1^{B,\gamma C,A} = f_1^{\gamma C,A,B} + A^* \mathbf{1}$.
\[rmk:structure\_proof\] Let us highlight the structure of the proof, in order to make clear which technical extensions are required in order to state error estimates for other dynamics:
(i) first, an expansion of the evolution operator $P_{{\Delta t}}$ in powers of ${{\Delta t}}$ has to be written out. This step is usually quite simple although sometimes algebraically involved. The expansion of $P_{{\Delta t}}$ is the same as the one used to prove weak error estimates;
(ii) second, good control on the resolvent has to be established, such as the stability result provided by Theorem \[thm:stability\_S\]. This step may already be quite complicated since it involves proving that $\mu$ is the unique invariant measure, and that the resolvent can be inverted for functions with average zero with respect to $\mu$. A typical way to do so is to establish decay properties of the semigroup. Such decay estimates may be hard to obtain for degenerate noises;
(iii) the existence of an invariant measure $\mu_{{{\Delta t}}}$ for the numerical scheme has to be demonstrated (uniqueness is not required), typically by finding a Lyapunov function. Again, this may be difficult if the dynamics is highly degenerate.
Once the above steps have been performed, the correction function can be identified as the solution of a Poisson equation, by comparing the average of $({\mathrm{Id}}-P_{{\Delta t}})\varphi$ under $\mu$ and $\mu_{{\Delta t}}$. The remainder of the proof allows one to state error estimates for any smooth function (and not just functions in the range of ${\mathrm{Id}}- P_{{\Delta t}}$) using appropriate pseudo-inverses.
Proof of Proposition \[prop:Ham\_limit\_correction\] {#sec:proof_Ham_limit}
----------------------------------------------------
We use a very standard strategy: first, we propose an ansatz for the correction term $f_{1,\gamma}$ as $$f_{1,\gamma} = f_1^0 + \gamma f_1^1 + \gamma^2 f_1^2 + \dots,$$ then identify the two leading order terms in this expression, and finally use the resolvent estimate of Theorem \[thm:Ham\_limit\_Lgam\] to conclude. Note that our ansatz is not obvious since the estimate of Theorem \[thm:Ham\_limit\_Lgam\] shows that, in general, a leading order correction term of order $1/\gamma$ should be considered. It turns out however that, due to the specific structure of the right-hand side of (namely the fact that the right-hand is at leading order in $\gamma$ the image under the Hamiltonian operator of some function), such a divergent leading order term is not necessary.
Consider for instance the case when $f_{1,\gamma}$ is $f_1^{\gamma C, B, A}$. This function solves $$\Big[-(A+B)+\gamma C\Big]f_1^{\gamma C, B, A} = -\frac12 (A+B)g, \qquad \int_\mathcal{E} f_1^{\gamma C, B, A} \, {\mathrm{d}\mu} = 0,$$ so that we consider the ansatz $f_1^{\gamma C, B, A} = g/2 + \gamma f_1^1 + \dots$. Identifying terms with same powers of $\gamma$, we see that the correction term $f_1^1$ should satisfy $$(A+B)f_1^1 = \frac12 Cg = \frac\beta2 p^T M^{-2}\nabla V.$$ Possible solutions are defined up to elements of the kernel of $A+B$ (which contains function of the form $\varphi \circ H$). One possible choice is to set $f_1^1 = \beta p^T M^{-2} p/4 + c_1^1$, where the constant $c_1^1$ is chosen in order for $f_1^1$ to have a vanishing average with respect to $\mu$. Then, $${\mathcal{L}_\gamma}^*\left(f_1^{\gamma C, B, A} -\frac{g}{2}-\gamma f_1^1\right) = \gamma^2 Cf_1^1.$$ In view of Theorem \[thm:Ham\_limit\_Lgam\], this implies that there exists a constant $K > 0$ such that $$\left\| f_1^{\gamma C, B, A} - \frac{g}{2}-\gamma f_1^1 \right\|_{L^2(\mu)} {\leqslant}K\gamma,$$ for $\gamma {\leqslant}1$, which gives the desired estimate on $f_1^{\gamma C, B, A}$. Similar computations give the estimate on $f_1^{\gamma C, A, B}$, while the estimates on the remaining functions are obtained from .
Proof of Theorem \[thm:error\_second\_order\_schemes\] {#sec:proof_thm:error_second_order_schemes}
------------------------------------------------------
The proof follows the same lines as the proof for the first order splitting schemes (see Section \[sec:proof\_thm:error\_first\_order\_schemes\]). We present only the required modifications. We write the proof for $P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}$ since the correction term has a much simpler right-hand side than $P_{{\Delta t}}^{A,B,\gamma C,B,A}$.
#### Proof of .
Expanding up to terms of order ${{\Delta t}}^5$ the formal expression of $P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}$ given by the BCH expansion , we obtain the following equality (as operators on $\mathcal{S}$) $$P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C} = {\mathrm{Id}}+ {{\Delta t}}({\mathcal{L}_\gamma}+ {{\Delta t}}^2 S_2) + \frac{{{\Delta t}}^2}{2}\Big({\mathcal{L}_\gamma}^2 + {{\Delta t}}^2 ({\mathcal{L}_\gamma}S_2 + S_2 {\mathcal{L}_\gamma}) \Big) + \frac{{{\Delta t}}^3}{6} {\mathcal{L}_\gamma}^3 + \frac{{{\Delta t}}^4}{24} {\mathcal{L}_\gamma}^4 + {{\Delta t}}^5 R_{2,{{\Delta t}}},
$$ where $$R_{2,{{\Delta t}}} = \frac{1}{24} \int_0^1 (1-\theta)^4 \mathcal{R}_{\theta \Delta t} \, {\mathrm{d}\theta},$$ $\mathcal{R}_s$ being a finite linear combination of terms of the form $C^\gamma {\mathrm{e}^{s C}}B^\beta {\mathrm{e}^{s B}}A^\alpha {\mathrm{e}^{s A}}$ with $\alpha,\beta,\gamma {\geqslant}0$ and $\alpha + \beta + \gamma = 5$; and $$\label{eq:def_S2}
S_2 = \frac{1}{12} \left( S_{2,0} + \gamma S_{2,1} + \gamma^2 S_{2,2}\right),$$ with $$\left\{ \begin{aligned}
S_{2,0} & = [A,[A,B]] - \frac12 [B,[B,A]], \\
S_{2,1} & = [A+B,[A+B,C]], \\
S_{2,2} & = -\frac12 [C,[C,A+B]].
\end{aligned} \right.$$ Therefore, $$\label{eq:I_Pdt_eq_order2}
\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}}{{{\Delta t}}} = -{\mathcal{L}_\gamma}- \frac{{\Delta t}}2 {\mathcal{L}_\gamma}^2 - {{\Delta t}}^2 \left( \frac16 {\mathcal{L}_\gamma}^3 + S_2\right) - \frac{{{\Delta t}}^3}{2} \left( \frac{1}{12} {\mathcal{L}_\gamma}^4 + S_2 {\mathcal{L}_\gamma}+ {\mathcal{L}_\gamma}S_2 \right) - {{\Delta t}}^4 R_{2,{{\Delta t}}}.$$ We choose $f_2^{\gamma C,B,A,B,\gamma C} \in \widetilde{\mathcal{S}}$ as the unique solution of the Poisson equation ${\mathcal{L}_\gamma}^* f_2^{\gamma C,B,A,B,\gamma C} = -S_2^* \mathbf{1}$ (which is indeed well posed since the right hand side has a vanishing average with respect to $\mu$ since it is in the image of $S_2$, and is regular as shown by below). Then, for a function $\varphi \in \mathcal{S}$, $$\begin{aligned}
& \int_{\mathcal{E}}\left(\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}}{{{\Delta t}}}\right)\varphi \, \left(1+{{\Delta t}}^2 f_2^{\gamma C,B,A,B,\gamma C} \right) {\mathrm{d}\mu} = \\
& \qquad \qquad -\frac{{{\Delta t}}^3}{2} \int_{\mathcal{E}}S_2 {\mathcal{L}_\gamma}\varphi + \left({\mathcal{L}_\gamma}^2 \varphi \right) f_2^{\gamma C,B,A,B,\gamma C} \, {\mathrm{d}\mu} - {{\Delta t}}^4 \int_{\mathcal{E}}\left[ \widetilde{R}_{2,{{\Delta t}}} \varphi + \widehat{R}_{2,{{\Delta t}}} \varphi f_2^{\gamma C,B,A,B,\gamma C} \right] {\mathrm{d}\mu},
\end{aligned}$$ where many terms cancel by the invariance of $\mu$ by $\left({\mathcal{L}_\gamma}^\alpha\right)^*$ (for integer powers $\alpha$). The leading order term on the right-hand side in fact vanishes since it can be rewritten as $$\int_{\mathcal{E}}S_2 {\mathcal{L}_\gamma}\varphi + {\mathcal{L}_\gamma}^2 \varphi \, f_2^{\gamma C,B,A,B,\gamma C} \, {\mathrm{d}\mu} = \int_{\mathcal{E}}{\mathcal{L}_\gamma}\varphi \left( S_2^* \mathbf{1} + {\mathcal{L}_\gamma}^*f_2^{\gamma C,B,A,B,\gamma C} \right) {\mathrm{d}\mu} = 0.$$ Therefore, $$\int_{\mathcal{E}}\left(\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}}{{{\Delta t}}}\right)\varphi \, \left(1+{{\Delta t}}^2 f_2^{\gamma C,B,A,B,\gamma C} \right) {\mathrm{d}\mu} = - {{\Delta t}}^4 \int_{\mathcal{E}}\left[ \widetilde{R}_{2,{{\Delta t}}} \varphi + \widehat{R}_{2,{{\Delta t}}} \varphi f_2^{\gamma C,B,A,B,\gamma C} \right] {\mathrm{d}\mu}.$$ We then restrict the above equality to functions $\varphi \in \widetilde{\mathcal{S}}$, project out the average with respect to $\mu$ of the first factor in the integral on the left using the projector $\Pi^\perp$ introduced in , and finally replace $\varphi$ by $Q_{2,{{\Delta t}}} \psi$ where $Q_{2,{{\Delta t}}}$ is an approximate inverse satisfying $$\Pi^\perp \frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}}{{{\Delta t}}} \Pi^\perp Q_{2,{{\Delta t}}} = \Pi^\perp + {{\Delta t}}^4 Z_{{\Delta t}}.$$ The proof is concluded as in Section \[sec:proof\_thm:error\_first\_order\_schemes\].
#### Proof of .
To evaluate the expression $S_2^* \mathbf{1}$, we need to compute the actions of the formal adjoints of the various commutators. Using $C\mathbf{1} = (A+B)\mathbf{1} = 0$ and $$C^* = C, \qquad A^* = -A + g, \qquad B^* = -B-g,$$ straightforward computations show that $S_{2,2}^* \mathbf{1} = S_{2,1}^*\mathbf{1} = 0$. In addition, since $$A\left(g^2\right) = 2g Ag, \qquad B\left(g^2\right) = 2g \, Bg,$$ it follows that $$\begin{aligned}
\big([A,[A,B]]\big)^*\mathbf{1} & = \left(A^2 B-2ABA + BA^2 \right)^*\mathbf{1} = \left( B^* A^* - 2 A^* B^* - (A^*)^2 \right)g \\
& = \Big( (B+g)(A-g) - 2 (A-g)(B+g) - (A-g)^2 \Big)g \\
& = (BA-2AB-A^2)g = -(A+B)Ag,
\end{aligned}$$ where we have used $AB g = BA g$ (as can be checked by a direct computation). A similar computation shows that $\big([B,[B,A]]\big)^*\mathbf{1} = (-AB+2BA+B^2)g = (A+B)Bg = ABg$ (since $B^2g = 0$ by a direct verification). Finally, $$\label{eq:S2_star}
S_{2}^*\mathbf{1} = -\frac{1}{12} (A+B)\left(A + \frac{B}{2}\right)g.$$ To obtain the expression of $f_{2}^{A,B,\gamma C,B,A}$, we use the TU lemma with the operator $$U_{{\Delta t}}= {\mathrm{e}}^{\gamma{{\Delta t}}C/2} {\mathrm{e}}^{{{\Delta t}}B/2} {\mathrm{e}}^{{{\Delta t}}A/2}.$$ A simple computation shows that $$U_{{\Delta t}}^* \mathbf{1} = \mathbf{1} + \frac{{{\Delta t}}^2}{8} (A+B)g + {{\Delta t}}^3 R_{{\Delta t}}^* \mathbf{1}.$$ In fact, it can be shown that the ${{\Delta t}}^3$ term does not pollute the remainder since the next order correction in the invariant measure has to be of order ${{\Delta t}}^4$ (see ). The expressions for $f_{2}^{\gamma C,A,B,A,\gamma C}$ and $f_{2}^{B,A,\gamma C,A,B}$ are obtained in a similar manner.
Proof of Corollary \[cor:error\_GLA\] {#sec:proof_cor:error_GLA}
-------------------------------------
The proof relies on the results of Theorem \[thm:error\_second\_order\_schemes\] and the TU lemma (Lemma \[lem:TU\]). More precisely, the error estimate is established by following the same lines of proof as for second order splitting schemes, except that the contributions of order ${{\Delta t}}^3$ do not vanish. We then use the TU lemma by considering the GLA evolution as the reference, and express the invariant measure of second order splitting schemes in terms of the invariant measure of the GLA scheme. For instance, consider $P_{{\Delta t}}^{\gamma C,B,A,B}$ and $P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}$, in which case $U_{{\Delta t}}= {\mathrm{e}}^{\gamma {{\Delta t}}C/2}$. Then, $$\begin{aligned}
&\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{{\Delta t}}^{\gamma C,B,A,B,\gamma C} = \int_{\mathcal{E}}\left( U_{{\Delta t}}\psi \right) {\mathrm{d}\mu}_{{\Delta t}}^{\gamma C,B,A,B} \\
& = \int_{\mathcal{E}}U_{{\Delta t}}\psi \, {\mathrm{d}\mu} + {{\Delta t}}^2 \int_{\mathcal{E}}(U_{{\Delta t}}\psi) f_2^{\gamma C,B,A,B} \, {\mathrm{d}\mu}+ {{\Delta t}}^3 \int_{\mathcal{E}}(U_{{\Delta t}}\psi) f_3^{\gamma C,B,A,B} \, {\mathrm{d}\mu} + {{\Delta t}}^4 r_{\psi,\gamma,{{\Delta t}}} \\
& = \int_{\mathcal{E}}\psi \, {\mathrm{d}\mu} + {{\Delta t}}^2 \int_{\mathcal{E}}\psi \, f_2^{\gamma C,B,A,B} \, {\mathrm{d}\mu}+ {{\Delta t}}^3 \int_{\mathcal{E}}\psi \left(f_3^{\gamma C,B,A,B}+\frac{\gamma}{2}Cf_2^{\gamma C,B,A,B}\right) {\mathrm{d}\mu} + {{\Delta t}}^4 \widetilde{r}_{\psi,\gamma,{{\Delta t}}},
\end{aligned}$$ where we have used the invariance of $\mu$ by $U_{{\Delta t}}$. The comparison with - gives the desired result.
Approximation of integrated correlation functions {#sec:proof_approx_GK_formula}
-------------------------------------------------
The proof makes use of the projection operator defined on $\mathcal{S}$ as (compare ) $$\Pi^\perp_{{\Delta t}}\varphi = \varphi - \int_\mathcal{E} \varphi \, {\mathrm{d}\mu}_{{\Delta t}}.$$ The range of $\Pi^\perp_{{\Delta t}}$ is contained in the set of functions with average zero with respect to the invariant measure $\mu_{{\Delta t}}$ of the numerical scheme. We first introduce the invariant measure for the numerical scheme, using the fact that $-{\mathcal{L}}_\gamma^{-1} \psi$ has zero average with respect to $\mu$: $$\begin{aligned}
\int_{\mathcal{E}}\left(-{\mathcal{L}}_\gamma^{-1} \psi\right) \varphi \, {\mathrm{d}\mu} & = \int_{\mathcal{E}}\left(-{\mathcal{L}}_\gamma^{-1} \psi\right) \Pi^\perp_{{\Delta t}}\varphi \, {\mathrm{d}\mu} \nonumber \\
& = \int_{\mathcal{E}}\left(-{\mathcal{L}}_\gamma^{-1} \psi\right) \Pi^\perp_{{\Delta t}}\varphi \, {\mathrm{d}\mu}_{{\Delta t}}+ {{\Delta t}}^\alpha r^{\psi,\varphi}_{{\Delta t}}, \nonumber \\
& = \int_{\mathcal{E}}\Pi^\perp_{{\Delta t}}\left(-{\mathcal{L}}_\gamma^{-1} \psi\right) \Pi^\perp_{{\Delta t}}\varphi \, {\mathrm{d}\mu}_{{\Delta t}}+ {{\Delta t}}^\alpha r^{\psi,\varphi}_{{\Delta t}}, \label{eq:introduce_mu_dt_in_correlation}\end{aligned}$$ where $r^{\psi,\varphi}_{{\Delta t}}$ is uniformly bounded for ${{\Delta t}}$ sufficiently small by . In addition, by , $$\begin{aligned}
-\Pi^\perp_{{\Delta t}}{\mathcal{L}}_\gamma^{-1} \psi & = -\Pi^\perp_{{\Delta t}}\left({{\Delta t}}\sum_{n=0}^{+\infty} P_{{\Delta t}}^n \right) \Pi^\perp_{{\Delta t}}\left(\frac{{\mathrm{Id}}- P_{{\Delta t}}}{{{\Delta t}}}\right) {\mathcal{L}_\gamma}^{-1} \psi \\
& = {{\Delta t}}\left(\sum_{n=0}^{+\infty} \left[ \Pi^\perp_{{\Delta t}}P_{{\Delta t}}\Pi^\perp_{{\Delta t}}\right]^n \right) \left({\mathcal{L}}_\gamma + {{\Delta t}}S_1 + \dots + {{\Delta t}}^{\alpha-1} S_{\alpha-1} + {{\Delta t}}^\alpha \widetilde{R}_{\alpha,{{\Delta t}}}\right) {\mathcal{L}_\gamma}^{-1} \psi, \\
& = {{\Delta t}}\sum_{n=0}^{+\infty} \left[ \Pi^\perp_{{\Delta t}}P_{{\Delta t}}\Pi^\perp_{{\Delta t}}\right]^n \widetilde{\psi}_{{{\Delta t}},\alpha} + {{\Delta t}}^\alpha \left(\frac{{\mathrm{Id}}- P_{{\Delta t}}}{{{\Delta t}}}\right)^{-1} \Pi^\perp_{{\Delta t}}\widetilde{R}_{\alpha,{{\Delta t}}} {\mathcal{L}_\gamma}^{-1} \psi.
\end{aligned}$$ Note that the sum on the right hand side is well defined in view of the decay estimates . Plugging the above equality in leads to $$\begin{aligned}
\int_{\mathcal{E}}\left(-{\mathcal{L}}_\gamma^{-1} \psi\right) \varphi \, {\mathrm{d}\mu} & = {{\Delta t}}\int_\mathcal{E} \sum_{n=0}^{+\infty} \left( \Pi_{{\Delta t}}^\perp P_{{\Delta t}}^n \widetilde{\psi}_{{{\Delta t}},\alpha} \right) \left( \Pi^\perp_{{\Delta t}}\varphi \right) {\mathrm{d}\mu}_{{\Delta t}}\\
& \qquad + {{\Delta t}}^\alpha \int_\mathcal{E} \left( \left(\frac{{\mathrm{Id}}- P_{{\Delta t}}}{{{\Delta t}}} \right)^{-1} \Pi^\perp_{{\Delta t}}\widetilde{R}_{\alpha,{{\Delta t}}} {\mathcal{L}_\gamma}^{-1} \psi\right)\Pi^\perp_{{\Delta t}}\varphi \, {\mathrm{d}\mu}_{{\Delta t}}+ {{\Delta t}}^\alpha r^{\psi,\varphi}_{{\Delta t}}.
\end{aligned}$$ The second term on the right hand side is uniformly bounded in view of the moment estimates on $\mu_{{\Delta t}}$, the resolvent bounds provided by Corollary \[corr:resolvent\_estimates\_I\_Pdt\] and the uniform boundedness of the remainder $\widetilde{R}_{\alpha,{{\Delta t}}}f$ for a given, smooth function $f$. Since $$\int_\mathcal{E} \sum_{n=0}^{+\infty} \left( \Pi_{{\Delta t}}^\perp P_{{\Delta t}}^n \widetilde{\psi}_{{{\Delta t}},\alpha} \right)\left( \Pi^\perp_{{\Delta t}}\varphi \right) {\mathrm{d}\mu}_{{\Delta t}}= \int_\mathcal{E} \sum_{n=0}^{+\infty} \left(P_{{\Delta t}}^n \widetilde{\psi}_{{{\Delta t}},\alpha} \right) \varphi \, {\mathrm{d}\mu}_{{\Delta t}}= \sum_{n=0}^{+\infty} \mathbb{E}_{{\Delta t}}\left(\widetilde{\psi}_{{{\Delta t}},\alpha}\left(q^{n},p^{n}\right)\varphi\left(q^0,p^0\right)\right),$$ finally follows.
The idea is to start from and to appropriately rewrite the first order correction term. We use to this end with $\psi$ replaced by its first order correction $(\psi_{{{\Delta t}},2} - \psi)/{{\Delta t}}= S_1{\mathcal{L}}_\gamma^{-1}\psi$, and discard terms of order ${{\Delta t}}^2$: $$\int_0^{+\infty} \mathbb{E} \Big( S_1{\mathcal{L}}_\gamma^{-1}\psi(q_t,p_t) \varphi(q_0,p_0) \Big) {\mathrm{d}t} = {{\Delta t}}\sum_{n=0}^{+\infty} \mathbb{E}_{{\Delta t}}\Big(S_1{\mathcal{L}}_\gamma^{-1}\psi\left(q^{n+1},p^{n+1}\right)\varphi_{{{\Delta t}},0}\left(q^0,p^0\right)\Big) + {{\Delta t}}\,r^{\psi,\varphi}_{{\Delta t}},$$ where $r^{\psi,\varphi}_{{\Delta t}}$ is uniformly bounded for ${{\Delta t}}$ sufficiently small and $\varphi_{{{\Delta t}},0} = \Pi_{{\Delta t}}^\perp \varphi$. On the other hand, using $S_1 = \mathcal{L}_\gamma^2/2$, $$\int_0^{+\infty} \mathbb{E} \Big( S_1{\mathcal{L}}_\gamma^{-1}\psi(q_t,p_t) \varphi(q_0,p_0) \Big) {\mathrm{d}t} = -\int_{\mathcal{E}}{\mathcal{L}}_\gamma^{-1}S_1{\mathcal{L}}_\gamma^{-1} \psi \, \varphi \, {\mathrm{d}\mu} = -\frac12 \int_{\mathcal{E}}\psi \varphi \, {\mathrm{d}\mu},$$ so that, $$\begin{aligned}
& {{\Delta t}}\sum_{n=0}^{+\infty} \mathbb{E}_{{\Delta t}}\Big( \left(S_1{\mathcal{L}}_\gamma^{-1}\psi\right)_{{{\Delta t}},0}\left(q^{n+1},p^{n+1}\right)\varphi\left(q^0,p^0\right)\Big) \\
& \qquad = {{\Delta t}}\sum_{n=0}^{+\infty} \mathbb{E}_{{\Delta t}}\Big(S_1{\mathcal{L}}_\gamma^{-1}\psi\left(q^{n+1},p^{n+1}\right)\Pi_{{\Delta t}}^\perp\varphi\left(q^0,p^0\right)\Big) \\
& \qquad = \int_0^{+\infty} \mathbb{E} \Big( S_1{\mathcal{L}}_\gamma^{-1}\psi(q_t,p_t) \Pi_{{\Delta t}}^\perp\varphi(q_0,p_0) \Big) {\mathrm{d}t} - {{\Delta t}}\,r^{\psi,\varphi}_{{\Delta t}}\\
& \qquad = -\frac12 \int_{\mathcal{E}}\psi \, \Pi_{{\Delta t}}^\perp\varphi \, {\mathrm{d}\mu} - {{\Delta t}}\,r^{\psi,\varphi}_{{\Delta t}}= -\frac12 \int_{\mathcal{E}}\psi_{{{\Delta t}},0} \varphi \, {\mathrm{d}\mu} - {{\Delta t}}\,r^{\psi,\varphi}_{{\Delta t}}\\
& \qquad = -\frac12 \mathbb{E}_{{\Delta t}}( \psi_{{{\Delta t}},0} \varphi) + {{\Delta t}}\, \widetilde{r}^{\psi,\varphi}_{{\Delta t}}.
\end{aligned}$$ This gives .
Proof of Theorem \[thm:ovd\_limit\] {#sec:proof_thm:ovd_limit}
-----------------------------------
We write the proof for the evolution operator $P_{{\Delta t}}^{\gamma C, A,B,A,\gamma C}$ first, and mention then how to extend the result to $P_{{\Delta t}}^{B,A,\gamma C,A,B}$ using the TU lemma. The proofs for $P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}$ and $P_{{\Delta t}}^{A,B,\gamma C,B,A}$ are very similar, so we only briefly mention the required modifications. By default, all operators appearing in this section are defined on the core $\mathcal{S}$.
#### Reduction to a limiting operator up to exponentially small terms.
Let us introduce the evolution operator corresponding to the standard position Verlet scheme: $P_{{\rm ham},{{\Delta t}}} = {\mathrm{e}}^{{{\Delta t}}A/2} {\mathrm{e}}^{{{\Delta t}}B} {\mathrm{e}}^{{{\Delta t}}A/2}$, so that $P_{{\Delta t}}^{\gamma C, A,B,A\gamma C} = {\mathrm{e}}^{\gamma {{\Delta t}}C/2} P_{{\rm ham},{{\Delta t}}} \rm\, e^{\gamma {{\Delta t}}C/2}$. On the other hand, we have the following convergence result, whose proof is postponed to the end of the section.
\[lem:cv\_etC\] Fix $s^* \in \mathbb{N}^*$. Then, there exist $K,\alpha > 0$ such that, for any $1 {\leqslant}s {\leqslant}s^*$ and any $t {\geqslant}0$, $$\left\| {\mathrm{e}}^{\gamma t C} - \pi \right\|_{\mathcal{B}(L^\infty_{{\mathcal{K}}_s})} {\leqslant}K {\mathrm{e}}^{-\alpha \gamma t}.$$
This suggests to consider the limiting operator $P_{\infty,{{\Delta t}}} = \pi P_{{\rm ham},{{\Delta t}}} \pi$ and write $$\label{eq:distance_to_limiting_operator}
P_{{\Delta t}}^{\gamma C, A,B,A,\gamma C} - P_{\infty,{{\Delta t}}} = \Big( {\mathrm{e}}^{\gamma {{\Delta t}}C/2} - \pi\Big) P_{{\rm ham},{{\Delta t}}} \pi + {\mathrm{e}}^{\gamma {{\Delta t}}C/2} P_{{\rm ham},{{\Delta t}}} \Big( {\mathrm{e}}^{\gamma {{\Delta t}}C/2} - \pi \Big).$$ For a given smooth function $\varphi \in \mathcal{S}$ which depends only on the position variable $q$, $$\label{eq:ovd_diff_first_term}
\int_{\mathcal{E}}\left({\mathrm{Id}}- P_{{\Delta t}}^{\gamma C, A,B,A,\gamma C}\right)\varphi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} = 0 =
\int_{\mathcal{E}}({\mathrm{Id}}- P_{\infty,{{\Delta t}}} )\varphi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} + r_{\varphi,\gamma,{{\Delta t}}}^1,$$ with the remainder $$r_{\varphi,\gamma,{{\Delta t}}}^1 = \int_{\mathcal{E}}\left( P_{\infty,{{\Delta t}}}-P_{{\Delta t}}^{\gamma C, A,B,A,\gamma C}\right)\varphi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}}.$$ On the other hand, $$\label{eq:ovd_diff_second_term}
\int_{\mathcal{E}}\left[\left({\mathrm{Id}}- P_{{\Delta t}}^{\gamma C, B,A,B,\gamma C}\right)\varphi \right] (1 + {{\Delta t}}^2 f_{2,\infty}) \, {\mathrm{d}\mu} = \int_{\mathcal{E}}\left[({\mathrm{Id}}- P_{\infty,{{\Delta t}}})\varphi \right] (1 + {{\Delta t}}^2 f_{2,\infty}) \, {\mathrm{d}\mu} + r_{\varphi,\gamma,{{\Delta t}}}^2,$$ with the remainder $$r_{\varphi,\gamma,{{\Delta t}}}^2 = \int_{\mathcal{E}}\left[\left(P_{\infty,{{\Delta t}}} - P_{{\Delta t}}^{\gamma C, B,A,B,\gamma C}\right)\varphi \right] (1 + {{\Delta t}}^2 f_{2,\infty}) \, {\mathrm{d}\mu}.$$ The idea is that the remainders $r_{\varphi,\gamma,{{\Delta t}}}^1$ and $r_{\varphi,\gamma,{{\Delta t}}}^2$ are exponentially small when the function $\varphi$ is sufficiently smooth (see below for a more precise discussion, once $\varphi$ has been replaced by $Q_{{\Delta t}}\psi$ with $Q_{{\Delta t}}$ an appropriate approximate inverse). Therefore, the leading order terms in the error estimate are obtained by considering the limiting operator $P_{\infty,{{\Delta t}}}$ only.
#### Error estimates for the limiting operator $P_{\infty,{{\Delta t}}}$.
We now study the error estimates associated with $P_{\infty,{{\Delta t}}}$, following the strategy used in Section \[sec:proof\_thm:error\_first\_order\_schemes\]. We first use the results of Section \[sec:expansion\_evolution\] with $M=3$, $A_1 = A_3 = A/2$ and $A_2 = B$ to expand $P_{{\rm ham},{{\Delta t}}}$, so that $$\label{eq:dvpmt_P_infty}
P_{\infty,{{\Delta t}}} = \pi + {{\Delta t}}\pi(A+B)\pi + \frac{{{\Delta t}}^2}{2} \pi (A+B)^2\pi + \frac{{{\Delta t}}^3}{6} \pi S_3 \pi + \frac{{{\Delta t}}^4}{24} \pi S_4 \pi + \frac{{{\Delta t}}^5}{120} \pi S_5 \pi + {{\Delta t}}^6 \pi R_{{\Delta t}}\pi,$$ with $S_i = {\mathcal{T}}[(A_1+A_2+A_3)^i]$. To give more precise expressions of the operators appearing on the right-hand side of the above equality, we use the following facts: $$\label{eq:rules_ovd_1}
\forall n \in \mathbb{N}, \qquad B^n \pi = 0, \qquad \pi A^{2n+1} \pi = 0,$$ and $$\label{eq:rules_ovd_2}
\forall n {\geqslant}m+1, \qquad B^n A^m \pi = 0.$$ In addition, $$\pi A^2 \pi = \frac1\beta \Delta_q \pi,
\qquad
BA\pi = - \nabla V \cdot \nabla_q \pi.$$ Using these rules in leads to $$\label{eq:rules_A+B_carre}
\pi(A+B)\pi = 0, \qquad \pi(A+B)^2\pi = \pi(A^2+BA)\pi = {\mathcal{L}_{\rm ovd}}\pi.$$ The operator $S_3$ is a combination of terms of the form $A^a B^b A^c$ with $a+b+c=3$ and $a,b,c \in \mathbb{N}$. In view of -, only the terms with $c {\geqslant}1$ and $b {\leqslant}c$ have to be considered, so that only $BA^2$ and $ABA$ remain. A simple computation shows that $BA^2\pi \varphi$ and $ABA\pi\varphi$ are functions linear in $p$, so that $\pi BA^2 \pi = \pi ABA\pi\varphi = 0$. Finally, $\pi S_3 \pi = 0$. A similar reasoning shows that $\pi S_5 \pi = 0$ and that many terms appearing in the expression of $S_4$ also disappear.
Plugging the above results in and introducing $h={{\Delta t}}^2/2$, $$P_{\infty,{{\Delta t}}} = \pi + h \pi {\mathcal{L}_{\rm ovd}}\pi + \frac{h^2}{6}\pi \left(A^4 + \frac32 A^2 BA + \frac32 ABA^2 + \frac32 B^2 A^2 + \frac12 BA^3 \right)\pi+ h^3 R_{\infty,{{\Delta t}}}.$$ Using $$\label{eq:rules_A4_BA3_etc}
\begin{aligned}
\pi A^4 \pi \varphi & = \frac{3}{\beta^2} \Delta_q^2 \pi \varphi = 3 \left(\pi A^2 \pi\right)^2 \varphi, \\
\pi B A^3 \pi \varphi & = -\frac3\beta \nabla V \cdot \nabla_q \Big(\Delta_q \pi \varphi \Big) = 3 \pi BA \pi A^2 \pi \varphi, \\
\pi B^2 A^2 \pi \varphi & = 2 (\nabla V)^T \Big(\nabla_q^2 \pi\varphi\Big) \nabla V, \\
\pi ABA^2 \pi \varphi & = -\frac2\beta \left(\nabla^2 V : \nabla^2 \varphi + \nabla V \cdot \nabla (\Delta \varphi)\right), \\
\pi A^2BA \pi \varphi & = -\frac1\beta \left(2\nabla^2 V : \nabla^2 \varphi + \nabla V \cdot \nabla (\Delta \varphi) + \nabla (\Delta V) \cdot \nabla \varphi \right) = \pi A^2 \pi BA \pi \varphi, \\
\end{aligned}$$ it follows $$\begin{aligned}
& \left(A^4 + \frac32 A^2 BA + \frac32 ABA^2 + \frac32 B^2 A^2 + \frac12 BA^3 \right)\pi \varphi \\
& \qquad = \frac{3}{\beta^2} \Delta_q^2\varphi - \frac{6}{\beta} \nabla^2 V : \nabla^2 \varphi - \frac{6}{\beta}\nabla V \cdot \nabla (\Delta \varphi) - \frac{3}{2\beta} \nabla (\Delta V) \cdot \nabla \varphi + 3 (\nabla V)^T (\nabla^2 \varphi)\nabla V.
\end{aligned}$$ A straightforward computation shows that $$\mathcal{L}_{\rm ovd}^2 \varphi = \frac{1}{\beta^2} \Delta_q^2 \varphi - \frac{2}{\beta} \nabla^2 V : \nabla^2 \varphi - \frac{2}{\beta}\nabla V \cdot \nabla (\Delta \varphi) - \frac{1}{\beta} \nabla (\Delta V) \cdot \nabla \varphi + (\nabla V)^T (\nabla^2 \varphi)\nabla V + (\nabla V)^T (\nabla^2 V) \nabla \varphi.$$ Therefore, $$\pi\left(A^4 + \frac32 A^2 BA + \frac32 ABA^2 + \frac32 B^2 A^2 + \frac12 BA^3 \right)\pi = 3 \left(\mathcal{L}_{\rm ovd}^2 + D\right)\pi,$$ with $$\label{eq:def_D}
D \varphi = \frac{1}{2\beta} \nabla (\Delta V) \cdot \nabla \varphi - (\nabla V)^T (\nabla^2 V) \nabla \varphi.$$ In conclusion, $$\label{eq:final_expansion_dt_inf}
P_{\infty,{{\Delta t}}} = \pi + h \mathcal{L}_{\rm ovd} + \frac{h^2}{2} \left(\mathcal{L}_{\rm ovd}^2 + D\right)\pi + h^3 R_{\infty,{{\Delta t}}}.$$ Let us emphasize that this operator acts on functions of $q$ (we define it on $\mathcal{S} \cap \mathrm{Ker}(\pi) = C^\infty(\mathcal{M})$), that $\pi$ is the identity operator for functions which are independent of $p$, and note that for any $\phi\in C^\infty(\mathcal{M})$, $$\label{eq:expansion_P_inf_dt}
\frac{\pi-P_{\infty,{{\Delta t}}}}{h} \phi = -{\mathcal{L}_{\rm ovd}}\phi - \frac{h}{2} \left({\mathcal{L}_{\rm ovd}}^2 +D \right)\phi -h^2 R_{{\Delta t}}\phi.$$ In fact, proceeding as in Section \[sec:proof\_thm:error\_first\_order\_schemes\], we project out averages with respect to $\overline{\mu}({\mathrm{d}q})$ in order to properly define approximate inverses. Introduce to this end the projector $$\overline{\Pi}^\perp \phi = \phi - \int_\mathcal{M} \phi(q) \, \overline{\mu}({\mathrm{d}q})$$ defined on the core $C^\infty(\mathcal{M})$. The equality then implies the following equality on $C^\infty(\mathcal{M}) \cap \mathrm{Ran}(\overline{\Pi}^\perp)$: $$\overline{\Pi}^\perp \frac{\pi-P_{\infty,{{\Delta t}}}}{h} \overline{\Pi}^\perp = -{\mathcal{L}_{\rm ovd}}- \frac{h}{2} \left({\mathcal{L}_{\rm ovd}}^2 + \overline{\Pi}^\perp D \overline{\Pi}^\perp \right) -h^2 \overline{\Pi}^\perp R_{{\Delta t}}\overline{\Pi}^\perp.$$ An approximate inverse of the operator appearing on the left hand side of the above equality is thus $$Q_h = -{\mathcal{L}_{\rm ovd}}^{-1} + \frac{h}{2} \left( \overline{\Pi}^\perp + {\mathcal{L}_{\rm ovd}}^{-1} \overline{\Pi}^\perp D \overline{\Pi}^\perp {\mathcal{L}_{\rm ovd}}^{-1} \right).$$ Denote by $\overline{\mu}_{\infty,{{\Delta t}}}({\mathrm{d}q})$ the invariant measure of the Markov chain generated by the limiting method $P_{\infty,{{\Delta t}}}$. Proceeding as in Section \[sec:proof\_thm:error\_first\_order\_schemes\] by first identifying the leading order correction $f_{2,\infty}$, projecting out averages with respect to $\overline{\mu}({\mathrm{d}q})$ using $\overline{\Pi}^\perp$, and replacing $\overline{\Pi}^\perp \varphi$ by $Q_h \psi$, the equality allows us to obtain $$\label{eq:error_estimate_limiting_operator}
\int_\mathcal{M} \psi(q) \, \overline{\mu}_{\infty,{{\Delta t}}}({\mathrm{d}q}) = \int_\mathcal{M} \psi(q) \, \overline{\mu}({\mathrm{d}q}) + {{\Delta t}}^2 \int_\mathcal{M} \psi(q) f_{2,\infty}(q) \, \overline{\mu}({\mathrm{d}q}) + {{\Delta t}}^4 \overline{r}_{{{\Delta t}},\psi},$$ where $f_{2,\infty}$ is the unique solution of $$\label{eq:def_f2infty}
\mathcal{L}_{\rm ovd} f_{2,\infty} = -\frac14 D^* \mathbf{1}.$$ A more explicit expression can be obtained by noting that $$D\varphi = \frac12 \nabla \left(\frac1\beta \Delta V - |\nabla V|^2\right) \cdot \nabla \varphi,$$ so that (recalling $\mathcal{L}_{\rm ovd} = - \beta^{-1} \nabla^* \nabla = -\beta^{-1} \sum_{i=1}^{dN} \partial_{q_i}^* \partial_{q_i}$ where the formal adjoints are taken on $L^2(\overline{\mu})$) $$\begin{aligned}
\int_\mathcal{M} \varphi \left(D^*\mathbf{1}\right) \, {\mathrm{d}\overline{\mu}} & = \int_\mathcal{M} D\varphi \, {\mathrm{d}\overline{\mu}} = \frac12 \int_\mathcal{M} \varphi \nabla^* \nabla \left(\frac1\beta \Delta V - |\nabla V|^2\right) \, {\mathrm{d}\overline{\mu}} \\
& = -\frac{1}{2} \int_\mathcal{M} \varphi \, \mathcal{L}_{\rm ovd}\left(\Delta V - \beta |\nabla V|^2\right) \, {\mathrm{d}\overline{\mu}}.
\end{aligned}$$ Since $f_{2,\infty}$ should have a vanishing average with respect to $\mu$, this proves that $$\label{eq:f_2_infty}
f_{2,\infty}(q) = \frac18 \left( \Delta V - \beta |\nabla V|^2\right) + a,$$ where the constant $a$ is adjusted to account for the constraint of vanishing average. A simple computation shows that it is equal to the constant $a_{\beta,V}$ defined in .
In fact, it is possible for the scheme considered here to precisely determine the leading order correction for numerical averages by noting that $$\label{eq:magic_identity_Laplacian}
\frac{1}{\beta} \int_\mathcal{M} \Delta \varphi \, {\mathrm{d}\overline{\mu}} = -\int_\mathcal{M} \varphi \left( \Delta V - \beta |\nabla V|^2\right) \, {\mathrm{d}\overline{\mu}},$$ so that finally $$\int_\mathcal{M} \psi(q) \, \overline{\mu}_{\infty,{{\Delta t}}}({\mathrm{d}q}) = \int_\mathcal{M} \psi(q) \, \overline{\mu}({\mathrm{d}q}) - \frac{{{\Delta t}}^2}{8\beta} \int_\mathcal{M} \Delta \psi(q) \, \overline{\mu}({\mathrm{d}q}) + {{\Delta t}}^4 r_{{{\Delta t}},\psi}.$$
#### Conclusion of the proof.
We now come back to - and replace $\overline{\Pi}^\perp \varphi$ by $Q_h \psi$: $$\label{eq:final_error_estimate_ovd}
\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}} = \int_{\mathcal{E}}\psi (1 + {{\Delta t}}^2 f_{2,\infty}) \, {\mathrm{d}\mu} + r^1_{\psi,\gamma,{{\Delta t}}} + r^2_{\psi,\gamma,{{\Delta t}}} + {{\Delta t}}^4 \overline{r}_{{{\Delta t}},\psi},$$ where $\overline{r}_{{{\Delta t}},\psi}$ is the same as in , while $$\begin{aligned}
r^1_{\psi,\gamma,{{\Delta t}}} = \int_{\mathcal{E}}\left( P_{\infty,{{\Delta t}}}-P_{{\Delta t}}^{\gamma C, A,B,A,\gamma C}\right)Q_h\psi \, {\mathrm{d}\mu}_{\gamma,{{\Delta t}}}, \\
r^2_{\psi,\gamma,{{\Delta t}}} = \int_{\mathcal{E}}\left[\left(P_{\infty,{{\Delta t}}} - P_{{\Delta t}}^{\gamma C, B,A,B,\gamma C}\right)Q_h \psi \right] (1 + {{\Delta t}}^2 f_{2,\infty}) \, {\mathrm{d}\mu}. \\
\end{aligned}$$ We then integrate with respect to momenta in , and bound the remainders by $K {\mathrm{e}}^{-{c}\gamma {{\Delta t}}}$ in view of the decomposition and Lemma \[lem:cv\_etC\] (the operators $P_{{\rm ham},{{\Delta t}}}$ and ${\mathrm{e}}^{\gamma {{\Delta t}}C/2}$ being bounded on $L^\infty_{{\mathcal{K}}_s}$ uniformly in ${{\Delta t}}$).
#### Proof of for $f^{B,A,\gamma C,A,B}_{2,\infty}$
We set $$U_{\gamma,{{\Delta t}}} = {\mathrm{e}}^{\gamma{{\Delta t}}C/2} {\mathrm{e}}^{{{\Delta t}}A/2} {\mathrm{e}}^{{{\Delta t}}B/2},
\qquad
T_{\gamma,{{\Delta t}}} = {\mathrm{e}}^{{{\Delta t}}B/2} {\mathrm{e}}^{{{\Delta t}}A/2} {\mathrm{e}}^{\gamma{{\Delta t}}C/2},$$ so that $P_{{\Delta t}}^{B,A,\gamma C,A,B} = T_{\gamma,{{\Delta t}}}U_{\gamma,{{\Delta t}}}$ while $P_{{\Delta t}}^{\gamma C,A,B,A,\gamma C} = U_{\gamma,{{\Delta t}}} T_{\gamma,{{\Delta t}}}$. By the TU lemma, $$\begin{aligned}
\int_{\mathcal{E}}\psi \, {\mathrm{d}\mu}_{{{\Delta t}}}^{B,A,\gamma C,A,B} & = \int_{\mathcal{E}}\left(U_{\gamma,{{\Delta t}}}\psi\right)d\mu_{{\Delta t}}^{\gamma C,A,B,A,\gamma C} \nonumber \\
& = \int_{\mathcal{E}}\left(U_{\infty,{{\Delta t}}}\psi\right)d\mu_{{\Delta t}}^{\gamma C,A,B,A,\gamma C} + \int_{\mathcal{E}}\left(U_{\gamma,{{\Delta t}}}-U_{\infty,{{\Delta t}}}\right)\psi \, {\mathrm{d}\mu}_{{\Delta t}}^{\gamma C,A,B,A,\gamma C}, \label{eqn_rhsbacab}\end{aligned}$$ where we have introduced $U_{\infty,{{\Delta t}}} = \pi {\mathrm{e}}^{{{\Delta t}}A/2} {\mathrm{e}}^{{{\Delta t}}B/2}$. The second term on the right hand side can be bounded by $K {\mathrm{e}}^{-{c}\gamma {{\Delta t}}}$ in view of Lemma \[lem:cv\_etC\] and the moment estimate . For the first term in the right-hand side of , we use and the following expansion (using the rules -): for $\psi \in \mathcal{S}$, $$U_{\infty,{{\Delta t}}} \psi = U_{\infty,{{\Delta t}}} \pi \psi = \psi + \frac{{{\Delta t}}^2}{8} \pi A^2 \pi \psi + {{\Delta t}}^4 \widetilde{r}_{\psi,{{\Delta t}}} = \psi + \frac{{{\Delta t}}^2}{8\beta} \Delta \psi + {{\Delta t}}^4 \widetilde{r}_{\psi,{{\Delta t}}},$$ where the remainder $\widetilde{r}_{\psi,{{\Delta t}}}$ is uniformly bounded for ${{\Delta t}}$ sufficiently small. Therefore, $$\int_{\mathcal{E}}\left(U_{\infty,{{\Delta t}}}\psi\right)d\mu_{{\Delta t}}^{\gamma C,A,B,A,\gamma C} = \int_{\mathcal{E}}\psi (1 + {{\Delta t}}^2 f_{2,\infty}) \, {\mathrm{d}\mu} + \frac{{{\Delta t}}^2}{8\beta} \int_{\mathcal{E}}\Delta \psi \, {\mathrm{d}\mu} + \widehat{r}_{\psi,\gamma,{{\Delta t}}},$$ where $f_{2,\infty}$ is given in . The remainder $\widehat{r}_{\psi,\gamma,{{\Delta t}}}$ is the sum of terms of order ${{\Delta t}}^4$ and others which can be bounded by $K {\mathrm{e}}^{-{c}\gamma {{\Delta t}}}$. We conclude by resorting to to compute the formal adjoint of the operator $\Delta_q$ on $L^2(\mu)$.
#### Proof of for $f^{\gamma C,B,A,B,\gamma C}_{2,\infty}$ and $f^{A,B,\gamma C,B,A}_{2,\infty}$.
We mimic the above proof for the evolution operator $P_{{\Delta t}}^{\gamma C,B,A,B,\gamma C}$. The equality still holds, but the operator $S_4$ now reads $$S_4 = A^4 + 2 BA^2 + \frac32 B^2A^2,$$ so that $$D\varphi = \frac2\beta \nabla^2 V : \nabla^2\varphi + \frac1\beta \nabla(\Delta V)\cdot \nabla\varphi - \nabla V^T (\nabla^2 V) \nabla\varphi.$$ A simple computation shows that $$\int_\mathcal{M} D\varphi \, {\mathrm{d}\overline{\mu}} = -\frac1\beta \int_\mathcal{M} \nabla\left(\Delta V - \frac\beta2 |\nabla V|^2 \right) \cdot \nabla \varphi \, {\mathrm{d}\overline{\mu}} = \int_\mathcal{M} {\mathcal{L}_{\rm ovd}}\left(\Delta V - \frac\beta2 |\nabla V|^2 \right) \varphi \, {\mathrm{d}\overline{\mu}},$$ so that, in view of , $$f^{\gamma C,B,A,B,\gamma C}_{2,\infty} = -\frac14 \left( \Delta V - \frac\beta2 |\nabla V|^2 -\frac{a_{\beta,V}}{2} \right).$$ The expression of $f^{A,B,\gamma C,B,A}_{2,\infty}$ is obtained via the TU lemma, introducing the limiting operator $$U_{\infty,{{\Delta t}}}\pi = \pi {\mathrm{e}}^{{{\Delta t}}B/2} {\mathrm{e}}^{{{\Delta t}}A/2}\pi = \pi + \frac{{{\Delta t}}^2}{8} \pi (A^2 + 2BA)\pi + {{\Delta t}}^4 R_{{\Delta t}},$$ so that $$f^{A,B,\gamma C,B,A}_{2,\infty} = f^{\gamma C,B,A,B,\gamma C}_{2,\infty} + \frac18 \Big( \pi (A^2 + 2BA)\pi \Big)^* \mathbf{1} = f^{\gamma C,B,A,B,\gamma C}_{2,\infty} + \frac18 \Big( \pi BA \pi \Big)^* \mathbf{1} = -\frac18 \left( \Delta V - a_{\beta,V} \right).$$
Let us conclude this section with the proof of Lemma \[lem:cv\_etC\].
The conclusion follows for instance by an application of Theorem 8.7 in [@rey-bellet], considering as a reference dynamics the Ornstein-Uhlenbeck process $${\mathrm{d}p}_t = -M^{-1} p_t \, {\mathrm{d}t} + \sqrt{\frac{2\gamma}{\beta}} \, {\mathrm{d}W}_t$$ with generator $C$ defined on functions of $\mathcal{S}$ which are independent of $q$ (recall that the unique invariant probability measure of this process is $\kappa({\mathrm{d}p})$). To apply the theorem, we need to show that ${\mathcal{K}}_s$ is a Lyapunov function for any $s {\geqslant}1$. We compute $$C {\mathcal{K}}_s = \left(-2s p^T p + \frac{2s(dN+2s-2)}{\beta} \right)|p|^{2(s-1)}
{\leqslant}- {\mathcal{K}}_s + b_s$$ for an appropriate constant $b_s {\geqslant}0$. This shows the existence of constants $R_s, \alpha_s$ such that $$\left| \left({\mathrm{e}}^{t C}f\right)(p) - \int_{\mathbb{R}^{dN}} f(p) \, \kappa({\mathrm{d}p}) \right| {\leqslant}R_s {\mathrm{e}}^{-\alpha_s t} \| f\|_{L^\infty_{{\mathcal{K}}_s}({\mathrm{d}p})} {\mathcal{K}}_s(p),$$ where the notation $L^\infty_{{\mathcal{K}}_s}({\mathrm{d}p})$ emphasizes that the supremum is taken over a function of the momentum variable only. The desired result now follows by applying the above bound to the function $\psi(q,\cdot)$ for any element $\psi \in L^\infty_{{\mathcal{K}}_s}$, and taking the supremum over $q$.
Proof of Proposition \[prop:ovd\_limit\_correction\] {#sec:proof_prop:ovd_limit_correction}
----------------------------------------------------
Recall that we set $M = {\mathrm{Id}}$ for overdamped limits. We consider first $f_2^{\gamma C, B,A,B, \gamma C}$, which satisfies . Let us first compute the right-hand side. Since $$\left[\left(A+\frac12 B\right)g\right] = \beta \left(p^T (\nabla^2 V) p - \frac12 |\nabla V|^2 \right),$$ a simple computation shows that $$\widetilde{g} = \frac{1}{12} (A+B) \left[\left(A+\frac12 B\right)g\right] = \frac{\beta}{12} \Big[ (\nabla^3 V) : (p \otimes p \otimes p) - 3p^T (\nabla^2 V) \nabla V \Big].$$ Note that the above function has average zero with respect to $\kappa$. We then apply Theorem \[lem:bounds\_CL\_gamma\] to obtain $$\left\| f_2^{\gamma C, B,A,B, \gamma C} - {\mathcal{L}_{\rm ovd}}^{-1} \pi (A+B) C^{-1} \widetilde{g} \right\|_{H^1(\mu)} {\leqslant}\frac{K}{\gamma}.$$ Since $$C \Big[ (\nabla^3 V) : (p \otimes p \otimes p) \Big] = -3 (\nabla^3 V) : (p \otimes p \otimes p) + \frac{6}{\beta} p^T \nabla \left(\Delta V\right),$$ it is easily checked that $$\begin{aligned}
C^{-1} \widetilde{g} & = -\frac{\beta}{36} (\nabla^3 V) : (p \otimes p \otimes p) - \frac{1}{6} p^T\nabla (\Delta V) + \frac{\beta}{4} p^T (\nabla^2 V) \nabla V \\
& = -\frac{\beta}{36} A^3 \pi V + A\pi \left(- \frac16 (\Delta V) + \frac{\beta}{8} |\nabla V|^2\right).
\end{aligned}$$ To compute $\pi (A+B) C^{-1} \widetilde{g}$, we rely on and and obtain $$\begin{aligned}
\pi (A+B) C^{-1} \widetilde{g} & = -\frac{1}{12} \left(\frac1\beta \Delta^2 V - \nabla V \cdot \nabla (\Delta V) \right) + {\mathcal{L}_{\rm ovd}}\left(- \frac16 (\Delta V) + \frac{\beta}{8} |\nabla V|^2\right) \\
& = {\mathcal{L}_{\rm ovd}}\left(-\frac14 \Delta V + \frac{\beta}{8} |\nabla V|^2\right).
\end{aligned}$$ This allows us to conclude that the limit of $f_2^{\gamma C, B,A,B, \gamma C}$ is the argument of the operator ${\mathcal{L}_{\rm ovd}}$ in the previous line, up to an additive constant chosen to ensure that $f_2^{\gamma C, B,A,B, \gamma C}$ has a vanishing average with respect to $\mu$ (which turns out to be $a_{\beta,V}/8$). We deduce the limit for $f_2^{A,B, \gamma C,B,A}$ with since $(A+B)g = p^T(\nabla^2 V)p-|\nabla V|^2$.
The expressions for the limits of $f_2^{\gamma C, A,B,A \gamma C}$ and $f_2^{B,A, \gamma C,A,B}$ are obtained in a similar fashion.
Linear response theory {#sec:proof_LRT}
----------------------
### Definition of the mobility in
We briefly sketch the discussion in [@HDR Section 3.1] (see in particular Theorem 3.1 in this reference). Hypoellipticity arguments show that the measure $\mu_{\gamma,\eta}$ has a smooth density with respect to the Lebesgue measure. It moreover formally satisfies the Fokker-Planck equation $$\label{eq:FP_xi}
\left({\mathcal{L}_\gamma}+\eta {\widetilde{\mathcal{L}}}\right)^* h_{\gamma,\eta} = 0, \qquad \mu_{\gamma,\eta}({\mathrm{d}q} \, {\mathrm{d}p}) = h_{\gamma,\eta}(q,p) \mu({\mathrm{d}q}\,{\mathrm{d}p}), \qquad \int_{\mathcal{E}}{\mathrm{d}\mu}_{\gamma,\eta} = 1.$$ This equation can be given a rigorous meaning when $\eta$ is sufficiently small. We rely on the following result (proved at the end of this section), which is itself based on the fact that $\left({\mathcal{L}_\gamma}^*\right)^{-1}$ can be extended to a bounded operator on ${\mathcal{H}}^0$ (see Theorem \[thm:Ham\_limit\_Lgam\] and the comment after it).
\[lem:relatively\_bounded\_perturbation\] The operator $({\mathcal{L}_\gamma}^*)^{-1}{\widetilde{\mathcal{L}}}^*$, considered as an operator on the Hilbert space $\mathcal{H}^0 = L^2(\mu) \cap \{ \mathbf{1} \}^\perp$ introduced in , is bounded.
Denoting by $r$ the spectral radius of $({\mathcal{L}_\gamma}^*)^{-1}{\widetilde{\mathcal{L}}}^* \in \mathcal{B}(\mathcal{H}^0)$, it is easily checked that $\left({\mathcal{L}_\gamma}+\eta {\widetilde{\mathcal{L}}}\right)^*$ is invertible for $|\eta| < r^{-1}$ with $$\left[ \left({\mathcal{L}_\gamma}+\eta {\widetilde{\mathcal{L}}}\right)^* \right]^{-1} = \left(\sum_{n=0}^{+\infty} (-\eta)^n \left[\left({\mathcal{L}_\gamma}^*\right)^{-1}{\widetilde{\mathcal{L}}}^*\right]^n\right)\left({\mathcal{L}_\gamma}^*\right)^{-1}.$$ Therefore, a straightforward computation shows that $$\label{eq:f_gamma_eta}
h_{\gamma,\eta}(q,p) = 1 + \sum_{n=1}^{+\infty} (-\eta)^n \left[\left({\mathcal{L}_\gamma}^*\right)^{-1}{\widetilde{\mathcal{L}}}^*\right]^n \mathbf{1}$$ is an admissible solution of , and it is in fact the only one in view of the uniqueness of the invariant probability measure (since $h_{\gamma,\eta}$ can be shown to be nonnegative). Note that the normalization of the measure $h_{\gamma,\eta} {\mathrm{d}\mu}$ does not depend on $\eta$. Finally, $$\int_{\mathcal{E}}F^T M^{-1}p \, \mu_{\gamma,\eta}({\mathrm{d}q} \, {\mathrm{d}p}) = -\eta \int_{\mathcal{E}}F^T M^{-1}p \left[\left({\mathcal{L}_\gamma}^*\right)^{-1}{\widetilde{\mathcal{L}}}^*\mathbf{1} \right] \mu({\mathrm{d}q} \, {\mathrm{d}p}) + \eta^2 r_{\eta,\gamma},$$ with $r_{\eta,\gamma}$ uniformly bounded as $\eta \to 0$. This gives .
Note first that the image of ${\widetilde{\mathcal{L}}}^*$ is contained in ${\mathcal{H}}^0$ since, for any $u \in \mathcal{S}$, $$\int_{\mathcal{E}}{\widetilde{\mathcal{L}}}^*u \, {\mathrm{d}\mu} = \int_{\mathcal{E}}u \left({\widetilde{\mathcal{L}}}\mathbf{1}\right) {\mathrm{d}\mu} = 0.$$ It is therefore possible to give a meaning to the operator $({\mathcal{L}_\gamma}^*)^{-1}{\widetilde{\mathcal{L}}}^*$ as an operator on $\widetilde{\mathcal{S}}$. We then check that the perturbation ${\widetilde{\mathcal{L}}}$ is ${\mathcal{L}_\gamma}$-bounded (with relative bound 0, in fact): for $u \in \widetilde{\mathcal{S}}$, $$\left\| {\widetilde{\mathcal{L}}}u \right\|^2_{L^2(\mu)} {\leqslant}|F|^2 \| \nabla_p u\|^2_{L^2(\mu)} = -\beta |F|^2 \langle u, {\mathcal{L}_\gamma}u\rangle_{L^2(\mu)} {\leqslant}\beta |F|^2 \| u \|_{L^2(\mu)} \left\| {\mathcal{L}_\gamma}u \right\|_{L^2(\mu)},$$ so that, for $u \in {\mathcal{H}}^0$ (recall that ${\mathcal{L}_\gamma}^{-1}u$ is well defined in this case), $$\left\| {\widetilde{\mathcal{L}}}{\mathcal{L}_\gamma}^{-1} u \right\|^2_{L^2(\mu)} {\leqslant}\beta |F|^2 \| u \|_{L^2(\mu)}\left\| {\mathcal{L}_\gamma}^{-1} u \right\|_{L^2(\mu)} {\leqslant}\beta |F|^2 \left\| {\mathcal{L}_\gamma}^{-1} \right\|_{\mathcal{B}({\mathcal{H}}^0)} \| u \|^2_{L^2(\mu)}.$$ This proves that ${\widetilde{\mathcal{L}}}{\mathcal{L}_\gamma}^{-1}$ is bounded, hence its adjoint is bounded as well.
### Proof of Lemma \[lem:ovd\_mobility\] {#sec:proof_lem:ovd_mobility}
Recall that we set mass matrices to identity when considering overdamped limits. Since $${\mathcal{L}_\gamma}\left(F^T p\right) = - \gamma F^T p - F^T \nabla V,$$ it follows (using first to compute the linear response and then to obtain the asymptotic behavior of ${\mathcal{L}_\gamma}^{-1}(F^T\nabla V)$ as $\gamma \to +\infty$) $$\begin{aligned}
\gamma \nu_{F,\gamma} & = \lim_{\eta \to 0} \frac{\gamma}{\eta} \int_{\mathcal{E}}F^T p \, \mu_{\gamma,\eta}({\mathrm{d}q}\,{\mathrm{d}p}) = \lim_{\eta \to 0} \frac1\eta \int_{\mathcal{E}}\left[-F^T \nabla V(q)-{\mathcal{L}_\gamma}\left(F^T p\right)\right] \mu_{\gamma,\eta}({\mathrm{d}q}\,{\mathrm{d}p}) \\
& = \beta \int_{\mathcal{E}}F^T p \, {\mathcal{L}_\gamma}^{-1} \left[F^T \nabla V(q)+{\mathcal{L}_\gamma}\left(F^T p\right)\right] \mu({\mathrm{d}q}\,{\mathrm{d}p}) \\
& = |F|^2 + \beta \int_{\mathcal{E}}\left(F^T p\right) \left[p^T \nabla_q {\mathcal{L}_{\rm ovd}}^{-1}\left(F^T \nabla V\right)\right] \mu({\mathrm{d}q}\,{\mathrm{d}p}) + \frac1\gamma r_\gamma \\
& = |F|^2 + \int_\mathcal{M} \left(F^T \nabla_q^* \mathbf{1} \right) {\mathcal{L}_{\rm ovd}}^{-1}\left(F^T \nabla V\right) \, \overline{\mu}({\mathrm{d}q}) + \frac1\gamma r_\gamma \\
& = |F|^2 + \beta \int_\mathcal{M} \left(F^T \nabla V \right) {\mathcal{L}_{\rm ovd}}^{-1}\left(F^T \nabla V\right) \, \overline{\mu}({\mathrm{d}q}) + \frac1\gamma r_\gamma \\
& = |F|^2 + \overline{\nu}_F + \frac1\gamma r_\gamma,
\end{aligned}$$ where $r_\gamma$ is uniformly bounded for $\gamma {\geqslant}1$. This gives the desired result.
\[rmk:ovd\_Einstein\] The article [@HP08] in fact studies the limiting behavior of the autodiffusion coefficient, as computed from : $$\beta \overline{D}_F = \int_\mathcal{M} \left| F + \nabla_q {\mathcal{L}_{\rm ovd}}^{-1} \left(F\cdot \nabla V\right) \right|^2 {\mathrm{d}\overline{\mu}}.$$ Using ${\mathcal{L}_{\rm ovd}}= -\beta^{-1} \nabla_q^* \nabla_q$, a simple computation shows $$\begin{aligned}
\beta \overline{D}_F
& = |F|^2 + 2 \int_\mathcal{M}F^T \nabla_q {\mathcal{L}_{\rm ovd}}^{-1} \left(F\cdot \nabla V\right) \, {\mathrm{d}\overline{\mu}}
+ \int_\mathcal{M} \left|\nabla_q {\mathcal{L}_{\rm ovd}}^{-1} \left(F\cdot \nabla V\right)\right|^2 {\mathrm{d}\overline{\mu}} \\
& = |F|^2 + 2 \int_\mathcal{M} \left(F^T \nabla_q^*\mathbf{1} \right) {\mathcal{L}_{\rm ovd}}^{-1} \left(F\cdot \nabla V\right) \, {\mathrm{d}\overline{\mu}} + \int_\mathcal{M} \nabla_q^* \nabla_q {\mathcal{L}_{\rm ovd}}^{-1} \left(F\cdot \nabla V\right)\, {\mathcal{L}_{\rm ovd}}^{-1} \left(F\cdot \nabla V\right)\, {\mathrm{d}\overline{\mu}} \\
& = |F|^2 + \beta \int_\mathcal{M} \left(F^T \nabla V \right) {\mathcal{L}_{\rm ovd}}^{-1} \left(F\cdot \nabla V\right) \, {\mathrm{d}\overline{\mu}},
\end{aligned}$$ so that $\beta \overline{D}_F = |F|^2 + \overline{\nu}_F$.
Proof of Theorem \[thm:error\_noneq\] {#sec:proof_noneq}
-------------------------------------
The proof again is along the lines of the proof written in Section \[sec:proof\_thm:error\_first\_order\_schemes\], and we are therefore very brief, mentioning only the most important modifications.
#### Case $\alpha=1$.
Let us first consider the first order scheme $P_{{{\Delta t}}}^{\gamma C,B+ \eta {\widetilde{\mathcal{L}}},A}$. Using the notation introduced in Section \[sec:expansion\_evolution\], and recalling the definition $B_\eta = B + \eta {\widetilde{\mathcal{L}}}$, we write $$\label{eq:expansion_P_dt_eta}
P_{{{\Delta t}}}^{\gamma C,B + \eta{\widetilde{\mathcal{L}}},A} = {\mathrm{Id}}+ {{\Delta t}}\left(A+B_\eta+\gamma C\right)+ \frac{{{\Delta t}}^2}{2} {\mathcal{T}}\left[\Big( A+B_\eta+\gamma C \Big)^2\right] + \frac{{{\Delta t}}^3}{2} R_{\eta,{{\Delta t}}},$$ with $$R_{\eta,{{\Delta t}}} = \int_0^1 (1-\theta)^2 \, {\mathcal{T}}\left[(A+B_\eta+\gamma C)P_{\theta {{\Delta t}}}^{\gamma C,B + \eta{\widetilde{\mathcal{L}}},A}\right]^3 {\mathrm{d}\theta}.$$ All the operators appearing in the expressions above are defined on the core $\mathcal{S}$, and have values in $\mathcal{S}$. Since $${\mathrm{e}}^{\theta {{\Delta t}}B_\eta} - {\mathrm{e}}^{\theta {{\Delta t}}B} = \eta \int_0^1 {\mathrm{e}}^{\theta s B_\eta} \, {\widetilde{\mathcal{L}}}\, {\mathrm{e}}^{\theta (1-s) B} \, ds,$$ it is easy to see that the operator $R_{\eta,{{\Delta t}}}$ can be rewritten as the sum of two contributions: $R_{\eta,{{\Delta t}}} = R_{0,{{\Delta t}}} + \eta \widetilde{R}_{\eta,{{\Delta t}}}$, where, for $\psi \in \mathcal{S}$, the smooth function $\widetilde{R}_{\eta,{{\Delta t}}} \psi$ can be uniformly controlled in $\eta$ for $|\eta| {\leqslant}1$. Finally, the evolution operator can be rewritten as $$\label{eq:Pdt_expansion_finite_gamma_eta}
P_{{{\Delta t}}}^{\gamma C,B + \eta{\widetilde{\mathcal{L}}},A} = {\mathrm{Id}}+ {{\Delta t}}\left({\mathcal{L}_\gamma}+ \eta {\widetilde{\mathcal{L}}}\right) + \frac{{{\Delta t}}^2}{2} \left({\mathcal{L}_\gamma}^2 + S_1 + \eta D_1 \right) + {{\Delta t}}^2 \mathscr{R}_{\eta,{{\Delta t}}},$$ where $S_1$ is defined in (which corresponds to the case $\eta = 0$), $D_1 = (2\gamma C+B){\widetilde{\mathcal{L}}}+ {\widetilde{\mathcal{L}}}(2A+B)$, and $$\mathscr{R}_{\eta,{{\Delta t}}} = \frac{{{\Delta t}}}{2} R_{0,{{\Delta t}}} + \frac{\eta {{\Delta t}}}{2} \widetilde{R}_{\eta,{{\Delta t}}} + \frac{\eta^2}{2} {\widetilde{\mathcal{L}}}^2.$$ We then compute, for $\varphi \in \mathcal{S}$ and $f_{1,1,\gamma} \in \widetilde{\mathcal{S}}$ to be chosen later, $$\begin{aligned}
& \int_{\mathcal{E}}\left[ \left(\frac{{\mathrm{Id}}-P_{{{\Delta t}}}^{\gamma C,B + \eta{\widetilde{\mathcal{L}}},A}}{{{\Delta t}}}\right)\varphi \right] \left(1 + {{\Delta t}}f_{1,0,\gamma} + \eta f_{0,1,\gamma} + \eta {{\Delta t}}f_{1,1,\gamma} \right) \, {\mathrm{d}\mu} \\
& = -\int_{\mathcal{E}}\left[\left({\mathcal{L}_\gamma}+ \eta {\widetilde{\mathcal{L}}}+ \frac{{{\Delta t}}}{2} \left({\mathcal{L}_\gamma}^2 + S_1 + \eta D_1 \right) + {{\Delta t}}\mathscr{R}_{\eta,{{\Delta t}}} \right)\varphi \right] \left(1 + {{\Delta t}}f_{1,0,\gamma} + \eta f_{0,1,\gamma} + \eta {{\Delta t}}f_{1,1,\gamma} \right) \, {\mathrm{d}\mu} \\
& = -\eta \int_{\mathcal{E}}\left[ {\widetilde{\mathcal{L}}}\varphi + ({\mathcal{L}_\gamma}\varphi) f_{0,1,\gamma} \right]{\mathrm{d}\mu} - {{\Delta t}}\int_{\mathcal{E}}\left[ \frac12 S_1 \varphi + ({\mathcal{L}_\gamma}\varphi) f_{1,0,\gamma} \right]{\mathrm{d}\mu} \\
& \ \ \ - \eta{{\Delta t}}\int_{\mathcal{E}}\left[ \left({\widetilde{\mathcal{L}}}\varphi\right) f_{1,0,\gamma} + \frac12\left({\mathcal{L}_\gamma}^2 + S_1\right)\varphi \, f_{0,1,\gamma} + ({\mathcal{L}_\gamma}\varphi) f_{1,1,\gamma} + \frac12 D_1\varphi \right]{\mathrm{d}\mu} \\
& \ \ \ - \eta^2 \int_{\mathcal{E}}\left({\widetilde{\mathcal{L}}}\varphi\right) (f_{0,1,\gamma} + {{\Delta t}}f_{1,1,\gamma})\, {\mathrm{d}\mu} - \frac{{{\Delta t}}^2}{2} \int_{\mathcal{E}}\left[\left({\mathcal{L}_\gamma}^2 + S_1 + \eta D_1 \right)\varphi \right] (f_{1,0,\gamma} + \eta f_{1,1,\gamma}) \, {\mathrm{d}\mu} \\
& \ \ \ - {{\Delta t}}\int_{\mathcal{E}}\mathscr{R}_{\eta,{{\Delta t}}}\varphi \left(1 + {{\Delta t}}f_{1,0,\gamma} + \eta f_{0,1,\gamma} + \eta {{\Delta t}}f_{1,1,\gamma} \right) \, {\mathrm{d}\mu}.
\end{aligned}$$ The first two terms in the last expression vanish by definition of $f_{0,1,\gamma}$ and $f_{1,0,\gamma}$, while the third one vanishes when the function $f_{1,1,\gamma}$ is defined by the Poisson equation $$\label{eq:def_f_1_1_gamma}
{\mathcal{L}_\gamma}^* f_{1,1,\gamma} = - {\widetilde{\mathcal{L}}}^* f_{1,0,\gamma} - \frac12\left({\mathcal{L}_\gamma}^2 + S_1\right)^* f_{0,1,\gamma} - \frac12 D_1^* \mathbf{1}.$$ It is easy to check that the right-hand side of this equation has a vanishing average with respect to $\mu$ (integrating with respect to $\mu$ and letting the adjoints of the operators act on $\mathbf{1}$). We then project using $\Pi^\perp$ and introduce the approximate inverse, defined on $\widetilde{\mathcal{S}}$ as $$\begin{aligned}
Q_{\eta,{{\Delta t}}} & = -{\mathcal{L}_\gamma}^{-1} + \eta {\mathcal{L}_\gamma}^{-1} \Pi^\perp{\widetilde{\mathcal{L}}}\Pi^\perp {\mathcal{L}_\gamma}^{-1} + \frac{{{\Delta t}}}{2} \left[\Pi^\perp + {\mathcal{L}_\gamma}^{-1}\Pi^\perp\left(S_1 + \eta D_1\right)\Pi^\perp {\mathcal{L}_\gamma}^{-1} \right] \\
& \ \ - \frac{\eta{{\Delta t}}}{2} {\mathcal{L}_\gamma}^{-1} \Pi^\perp{\widetilde{\mathcal{L}}}\Pi^\perp {\mathcal{L}_\gamma}^{-1}\left({\mathcal{L}_\gamma}^2 + \Pi^\perp S_1\Pi^\perp + \eta \Pi^\perp D_1\Pi^\perp \right) {\mathcal{L}_\gamma}^{-1} \\
& \ \ - \frac{\eta{{\Delta t}}}{2} {\mathcal{L}_\gamma}^{-1} \left({\mathcal{L}_\gamma}^2 + \Pi^\perp S_1\Pi^\perp + \eta \Pi^\perp D_1\Pi^\perp \right) {\mathcal{L}_\gamma}^{-1} \Pi^\perp {\widetilde{\mathcal{L}}}\Pi^\perp {\mathcal{L}_\gamma}^{-1},
\end{aligned}$$ obtained by truncating the formal series expansion of the inverse operator by discarding terms associated with $\eta^2$ or ${{\Delta t}}^2$. The approximate inverse is such that $$\Pi^\perp \left(\frac{{\mathrm{Id}}-P_{{{\Delta t}}}^{\gamma C,B + \eta{\widetilde{\mathcal{L}}},A}}{{{\Delta t}}}\right) \Pi^\perp Q_{\eta,{{\Delta t}}} = \Pi^\perp + \eta^2 \mathcal{R}^1_{\eta,{{\Delta t}}} + {{\Delta t}}^2 \mathcal{R}^2_{\eta,{{\Delta t}}},$$ with $\mathcal{R}^2_{\eta,{{\Delta t}}} = \mathcal{R}^2_{0,{{\Delta t}}} + \eta \widetilde{\mathcal{R}}^2_{\eta,{{\Delta t}}}$. We then replace $\Pi^\perp \varphi$ by $Q_{\eta,{{\Delta t}}} \psi$ and conclude as in Section \[sec:proof\_thm:error\_first\_order\_schemes\].
#### Case $\alpha=2$.
The result for the second order splitting is obtained by appropriate modifications of the proof written above for $p=1$, similar to the ones introduced in Section \[sec:proof\_thm:error\_second\_order\_schemes\]. We will therefore mention only the most important point, which is the following. Replacing $B$ by $B_\eta$ in the expansion , we see that $$\begin{aligned}
\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B_\eta,A,B_\eta,\gamma C}}{{{\Delta t}}} & = -{\mathcal{L}_\gamma}- \eta {\widetilde{\mathcal{L}}}- \frac{{\Delta t}}2 ({\mathcal{L}_\gamma}+\eta {\widetilde{\mathcal{L}}})^2 - {{\Delta t}}^2 \left( \frac16 ({\mathcal{L}_\gamma}+\eta{\widetilde{\mathcal{L}}})^3 + S_{2} + \eta \widetilde{S}_{2,\eta} \right) - {{\Delta t}}^3 R_{\eta,{{\Delta t}}} \\
& = -{\mathcal{L}_\gamma}- \eta {\widetilde{\mathcal{L}}}- \frac{{\Delta t}}2 {\mathcal{L}_\gamma}^2 - \frac{\eta{{\Delta t}}}{2} \left({\mathcal{L}_\gamma}{\widetilde{\mathcal{L}}}+ {\widetilde{\mathcal{L}}}{\mathcal{L}_\gamma}\right) - \frac{\eta^2{{\Delta t}}}{2} {\widetilde{\mathcal{L}}}^2 - {{\Delta t}}^2 \left( \frac16 {\mathcal{L}_\gamma}^3 + S_{2} \right) \\
& \ \ \ - \eta {{\Delta t}}^2 \left(\frac16 \left({\mathcal{L}_\gamma}^2 {\widetilde{\mathcal{L}}}+ {\mathcal{L}_\gamma}{\widetilde{\mathcal{L}}}{\mathcal{L}_\gamma}+ {\widetilde{\mathcal{L}}}{\mathcal{L}_\gamma}^2 \right) + \widetilde{S}_{2,0}\right) + \mathscr{R}_{\eta,{{\Delta t}}},
\end{aligned}$$ where $\mathscr{R}_{\eta,{{\Delta t}}}$ regroups operators of order ${{\Delta t}}^{3+\alpha} \eta^{\alpha'}$ or ${{\Delta t}}^{2+\alpha} \eta^{2+\alpha'}$ for $\alpha,\alpha' {\geqslant}0$, the operator $S_2$ is defined in and $\widetilde{S}_{2,\eta}$ satisfies $$\begin{aligned}
12 \, \widetilde{S}_{2,\eta} & = \left[A,\left[A,{\widetilde{\mathcal{L}}}\right]\right] - \frac12 \left[B,\left[{\widetilde{\mathcal{L}}},A\right]\right]- \frac12 \left[{\widetilde{\mathcal{L}}},\left[B,A\right]\right] + \gamma \left[{\widetilde{\mathcal{L}}},\left[A+B,C\right]\right] + \gamma \left[A+B,\left[{\widetilde{\mathcal{L}}},C\right]\right] \\
& \ \ \ - \frac{\gamma^2}{2} \left[C,\left[C,{\widetilde{\mathcal{L}}}\right]\right] + \eta \left( \gamma \left[{\widetilde{\mathcal{L}}},\left[{\widetilde{\mathcal{L}}},C\right]\right] - \frac12 \left[{\widetilde{\mathcal{L}}},\left[{\widetilde{\mathcal{L}}},A\right]\right] \right).
\end{aligned}$$ We next compute the dominant terms in $$\int_{\mathcal{E}}\left[\left(\frac{{\mathrm{Id}}-P_{{\Delta t}}^{\gamma C,B_\eta,A,B_\eta,\gamma C}}{{{\Delta t}}}\right)\varphi \right] \left(1 + {{\Delta t}}^2 f_{2,0,\gamma} + \eta f_{0,1,\gamma} + \eta {{\Delta t}}^2 f_{2,1,\gamma} \right) \, {\mathrm{d}\mu}.$$ We consider only contributions of the form $\eta^\alpha {{\Delta t}}^{\alpha'}$ with $\alpha = 0,1$ and $0 {\leqslant}\alpha' {\leqslant}2$. The contributions in ${{\Delta t}},{{\Delta t}}^2$ are the same as in the case $\eta = 0$ and therefore vanish. The contribution in $\eta$ vanishes in view of the choice of $f_{0,1,\gamma}$. For the same reason, the contribution in $\eta{{\Delta t}}$ vanishes as well: $$- \frac{\eta{{\Delta t}}}{2} \int_{\mathcal{E}}\left[ \left({\mathcal{L}_\gamma}{\widetilde{\mathcal{L}}}+ {\widetilde{\mathcal{L}}}{\mathcal{L}_\gamma}\right)\varphi + \left({\mathcal{L}_\gamma}^2\varphi\right) f_{0,1,\gamma} \right] \, {\mathrm{d}\mu} = - \frac{\eta{{\Delta t}}}{2} \int_{\mathcal{E}}\left( {\mathcal{L}_\gamma}\varphi\right) \left({\widetilde{\mathcal{L}}}^*\mathbf{1} + {\mathcal{L}_\gamma}^* f_{0,1,\gamma}\right) {\mathrm{d}\mu} = 0.$$ The contribution in $\eta{{\Delta t}}^2$ is proportional to $$\int_{\mathcal{E}}\left[ \left(\frac{{\mathcal{L}_\gamma}^2 {\widetilde{\mathcal{L}}}+ {\mathcal{L}_\gamma}{\widetilde{\mathcal{L}}}{\mathcal{L}_\gamma}+ {\widetilde{\mathcal{L}}}{\mathcal{L}_\gamma}^2}{6} + \widetilde{S}_{2,0}\right)\varphi + \left({\widetilde{\mathcal{L}}}\varphi\right) f_{2,0,\gamma} + \left[\left( \frac{{\mathcal{L}_\gamma}^3}{6} + S_{2} \right)\varphi\right] f_{0,1,\gamma} + \left({\mathcal{L}_\gamma}\varphi\right) f_{2,1,\gamma} \right] {\mathrm{d}\mu}.$$ The requirement that this expression vanishes for all functions $\varphi \in \mathcal{S}$ characterizes the function $f_{2,1,\gamma}$ (the discussion on the solvability of this equation following the same lines as the discussion on the solvability of ). The proof is then concluded as in the case $p=1$.
Proof of Theorem \[thm:error\_estimate\_noneq\_ovd\] {#sec:proof_thm:error_estimate_noneq_ovd}
----------------------------------------------------
The proof of this result is obtained by modifying the proof of Theorem \[thm:ovd\_limit\] presented in Section \[sec:proof\_thm:ovd\_limit\] by taking into account the nonequilibrium perturbation, as done in the proof of Theorem \[thm:error\_noneq\] presented in Section \[sec:proof\_noneq\]. We will therefore be very brief and only mention the most important modifications.
We write the proof for the scheme associated with the evolution operator $P_{{\Delta t}}^{\gamma C,A,B_\eta,A,\gamma C}$ for instance (since this is the case explicitly treated in Section \[sec:proof\_thm:ovd\_limit\] for $\eta = 0$). First, arguing as in Section \[sec:proof\_thm:ovd\_limit\], we see that it is possible to replace $P_{{\Delta t}}^{\gamma C,A,B_\eta,A,\gamma C}$ by $$\pi P_{{\rm ham},{{\Delta t}},\eta} \pi = \pi {\mathrm{e}}^{{{\Delta t}}A/2} {\mathrm{e}}^{{{\Delta t}}B_\eta} {\mathrm{e}}^{{{\Delta t}}A/2} \pi$$ up to error terms in the invariant measure which are exponentially small in $\gamma {{\Delta t}}$. Note that $B_\eta = (F-\nabla V)\cdot \nabla_p$, so that the rules - are still valid. Therefore, introducing again $h = {{\Delta t}}^2/2$, $$\begin{aligned}
& \pi P_{{\rm ham},{{\Delta t}},\eta} \pi \\
& = \pi + \frac{{{\Delta t}}^2}{2} \pi (A+B_\eta)^2 \pi + \frac{{{\Delta t}}^4}{24}\pi \left(A^4 + \frac32 A^2 B_\eta A + \frac32 A B_\eta A^2 + \frac32 B_\eta^2 A^2 + \frac12 B_\eta A^3\right)\pi + {{\Delta t}}^6 R_{{{\Delta t}},\eta} \\
& = \pi + h \pi \left({\mathcal{L}_{\rm ovd}}+ \eta\left[{\widetilde{\mathcal{L}}}(A+B) + (A+B){\widetilde{\mathcal{L}}}\right] + \eta^2 {\widetilde{\mathcal{L}}}^2 \right) \pi + \frac{h^2}{2} \left({\mathcal{L}_{\rm ovd}}^2 + D + \eta \widetilde{D}_1 + \eta^2 \widetilde{D}_2\right) \pi \\
& \ \ \ + {{\Delta t}}^6 R_{{{\Delta t}},\eta},
\end{aligned}$$ where $D$ is defined in , and the expressions of the operators $\widetilde{D}_i$ ($i=1,2$) are obtained by expanding the various terms $A^aB_\eta^b A^c$ in powers of $\eta$. Keeping only the dominant terms, we arrive at $$\pi P_{{\rm ham},{{\Delta t}},\eta} \pi = \pi + h {\mathcal{L}_{\rm ovd}}\pi + \frac{h^2}{2}\left({\mathcal{L}_{\rm ovd}}^2 + D\right) + \eta h \pi \left[{\widetilde{\mathcal{L}}}(A+B) + (A+B){\widetilde{\mathcal{L}}}\right]\pi + \frac{\eta h^2}{2} \widetilde{D}_1 + \mathscr{R}_{{{\Delta t}},\eta}.$$ Since $$\pi \left({\widetilde{\mathcal{L}}}(A+B) + (A+B){\widetilde{\mathcal{L}}}\right) \pi = \pi {\widetilde{\mathcal{L}}}A \pi = {\widetilde{\mathcal{L}}}_{\rm ovd},$$ we conclude $$\pi P_{{\rm ham},{{\Delta t}},\eta} \pi = \pi + h \left( {\mathcal{L}_{\rm ovd}}+ \eta {\widetilde{\mathcal{L}}}_{\rm ovd}\right) \pi + \frac{h^2}{2}\left({\mathcal{L}_{\rm ovd}}^2 + D + \eta \widetilde{D}_1 \right) + \mathscr{R}_{{{\Delta t}},\eta}.$$ This relation is the analogue of in the overdamped limit, and the remainder of the proof is carried on following the strategy presented in Section \[sec:proof\_thm:ovd\_limit\].
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Francis Nier for several fruitful discussions on the properties of Langevin-type generators. Ben Leimkuhler and Charles Matthews acknowledge the support of the Engineering and Physical Sciences Research Council (UK) and grant EP/G036136/1. G. Stoltz was partially supported by the project DYMHOM (De la dynamique moléculaire, via l’homogénéisation, aux modèles macroscopiques de poroélasticité et électrocinétique) from the program NEEDS (Projet fédérateur Milieux Poreux MIPOR). G. Stoltz also benefited from the scientific environment of the Laboratoire International Associé between the Centre National de la Recherche Scientifique and the University of Illinois at Urbana-Champaign.
[^1]: Email: b.leimkuhler@ed.ac.uk
[^2]: Email: c.matthews@ed.ac.uk
[^3]: Corresponding Author. Email: stoltz@cermics.enpc.fr
|
---
abstract: |
The time dependent pion emission rate and the mean life time of the pion source in the center of mass frame of the two-jet events have been determined for $e^+-e^- \to hadrons$ at $\sqrt{s} = 20 - 50$ GeV by the Dynamical String Model. It was established that the time needed for the creation of the pion source is $5$ fm/c, whereas its life time is $\tau_0 \approx 7 - 13$ fm/c for energies $\sqrt{s}
= 20 - 50$ GeV, respectively.
---
-3 cm -2 cm 23 cm
\
$^a$B. Iványi, $^a$Zs. Schram, $^a$K. Sailer and $^b$W. Greiner\
$^a$Department for Theoretical Physics,\
Kossuth Lajos University,\
H-4010 Debrecen, Pf. 5, Hungary\
$^b$Institute for Theoretical Physics,\
Johann Wolfgang Goethe University,\
D-60054 Frankfurt am Main 11, Postfach 111932, Germany\
Introduction {#introduction .unnumbered}
============
The purpose of this work is to obtain the reliable time evolution of the hadronization process for two-jet events $e^+ e^- \to hadrons$ at the energies $\sqrt{s} \approx 20-50$ GeV in the framework of the Dynamical String Model. This is achieved by making use of the particular sensitivity of the simulated Bose-Einstein correlation to the decay constant of the hadronic string. We show that it is possible to arrive at an agreeable quantitative description of both the Bose-Einstein correlation and the single particle data by an appropriate choice of the string decay constant, being close to its value predicted in Ref.[@nyolcb]. Having settled the time scale of the hadronization process in this way, we determine the time dependent emission rate and the mean life time of the pion source.
The Dynamical String Model was developed in order to describe high energy hadronic processes[@elso]. In the model the hadrons are represented by classical open strings and their free motion is governed by the Nambu-Goto action. Two types of string interactions are introduced. The excited hadronic strings decay due to the quark-antiquark pair creation in the chromoelectric field of the flux tube represented by the string. The collision of hadronic strings is modelled by the arm-exchange mechanism called rearrangement. It is assumed that the strings move freely between two interactions. In addition to, hadronic strings have a finite transverse size ensuring the right order of magnitude of the hadronic cross sections.
The parameters of the model are as follows. The string tension $\kappa \approx 1$ GeV$/$fm is determined from the slope of the leading Regge trajectory of hadron resonances. The radius $R$ of hadronic strings is fitted to the proton-proton total cross section. The third parameter of the model, the decay constant of the hadronic strings can be expressed in terms of the string radius and the string tension making use of the analogy of the hadronic string with the chromoelectric flux tube [@masodik]. Recently, there have been given theoretical arguments in favour of this analogy [@harmadik]. Nevertheless, such an analogy does not lead to an unambiguous relationship between the string tension $\kappa$ and the chromoelectric field strength $\cal E$ as discussed in [@negyedik]. Changing this relationship modifies the value of the decay constant for a given radius $R$. The limitations for the above mentioned relationship and for the possible radius values, and their optimal choice ($e{\cal E} = 1.5 \kappa$ , $R = 0.5$ fm) were given in [@negyedik] comparing simulated and experimental single-particle distributions for elementary hadronic processes.
In this work we introduce the discrete final state resonances in the Dynamical String Model and investigate their effect on the single-particle distributions for $2$-jet events $e^+ e^- \to hadrons$ at $\sqrt{s} =
$ 20 - 50 GeV. Then we redetermine the string radius via the decay constant obtained by comparing the results of numerical simulations with experimental data for the Bose-Einstein correlation of identical pions. Finally we present the time dependence of the pion emission.
Final state resonances {#final-state-resonances .unnumbered}
======================
At the very beginning of any kind of high-energy hadronic processes highly excited hadrons are produced. So it is plausible to describe these processes in terms of classical strings possessing a continuous mass spectrum. However, the experimentally observed particles are low energy hadrons which definitely have a discrete mass spectrum. Therefore the Dynamical String Model dealing originally only with strings belonging to the continuous mass spectrum, has been improved by introducing the discrete final state hadron resonances below the mass threshold $1$ GeV. These final hadron states are described by strings in the rotating rod mode. Their properties (rest mass, decay width, the average momentum of their decay products) are chosen according to the phenomenology[@otodik]. Thus all the mesons from $\pi$ to $a_0$ are included except of those containing strange (anti)quarks.
The treatment of decays needs a special care when the parent particles belong to the continuous mass spectrum and the daughters to the discrete one. It is assumed that the probability of choosing any species of hadron resonances is proportional to the degree of degeneracy of this state and the inverse of its rest mass squared [@hatodik].
The improved model described above was tested using the same fit parameters as for the test of its original version in [@negyedik]. We concluded that the single-particle distributions are in quantitative agreement with the experimental data for the $2$-jet events $e^+ e^- \to hadrons$ at $\sqrt{s} \approx 20-50$ GeV[@hetedik]. The effect of the discrete hadron resonances turned out to be significant only for the transverse momentum distribution (Fig. 1). The width of the distribution increased due to the decay of discrete hadron resonances.
Bose-Einstein correlations {#bose-einstein-correlations .unnumbered}
==========================
The Bose-Einstein correlation has also been investigated in the framework of the Dynamical String Model to extract information about the detailed phase space structure of the simulated events $e^+ e^- \to hadrons$. We show that the correlation function is rather sensitive to the value of the decay constant. Thus it provides a better tool to fix the parameters of the model than the single-particle distributions.
The Bose-Einstein correlation of identical bosons is the consequence of the symmetrization of their wave function. It is reflected on the enhancement of the number of like sign pion pairs at low momentum differences in hadronic processes.
The correlation function is defined by $$\begin{aligned}
\label{domi}
C ( {\bf p}_1 , {\bf p}_2 ) =
\frac{P ( {\bf p}_1 , {\bf p}_2 )}{P ( {\bf p}_1 ) P ( {\bf p}_2 )}\end{aligned}$$ where $P ( {\bf p}_1 , {\bf p}_2 )$ is the two-particle momentum distribution and $P ( {\bf p}_i )$ is the single-particle momentum distribution. The product in the denominator of (\[domi\]) can be replaced by $P_0 (
{\bf p}_1, {\bf p}_2 )$ which is the two-particle distribution in the lack of quantum interference effects. If the source function $g ( x , {\bf p} )$ of the emitted particles is known, the correlation function in the plane wave approximation will read[@nyolcadik] $$\begin{aligned}
\label{corr}
C ( {\bf p}_1, {\bf p}_2 ) =
\frac{ \int d^4 x d^4 x' g ( x , {\bf p}_1 ) g ( x' , {\bf p}_2 )
[ 1 + \cos ( p_1 - p_2 )^{\mu} ( x - x' )_{\mu} ] }
{ \int d^4 x d^4 x' g ( x , {\bf p}_1 ) g ( x' , {\bf p}_2 ) }\end{aligned}$$ (with the Lorentz index $\mu$). For a homogeneous source, if the momentum of the emitted particle is independent of its space-time coordinate $x$ , the source function is a product $g ( x , {\bf p} ) = \rho ( x ) g ( {\bf p} )$. Then the correlation function is given by $$\begin{aligned}
\label{furier}
C ( {\bf p}_1, {\bf p}_2 ) =
1 + | \tilde{\rho} ( {\bf p}_1 - {\bf p}_2 ) |^2\end{aligned}$$ where the $\tilde{\rho}({\bf p})$ is the Fourier transform of $\rho ( {\bf x} )$, and the width of the correlation function is proportional to the inverse of the size of the source in space.
The source function of the produced pions has been obtained by numerical simulation in the Dynamical String Model. The test for the homogeneity of the source function shows that only the direction of the momentum of the produced particle is independent of its space-time coordinate. The magnitude of the momentum increases as the spatial distance increases between the creation point of the particle and the point of the initial quark-antiquark pair creation. This can be expected from the picture of inside-outside cascade of the hadronization in the string model, where the low momentum states are populated first in time and consequently in space[@nyolca].
Making use of the simulated source function the correlation function has been determined according to Eq. (\[corr\]). Similarly to the presentation of the TPC data[@kilencedik], the correlation function was calculated as the function of the following single variables: $q=|(p_1-p_2)^{\mu}(p_1-p_2)_{\mu}|^{1/2}$, $q_T$ (the component of momentum difference ${\bf p}_1-{\bf p}_2$ perpendicular to the momentum sum ${\bf p}_1+{\bf p}_2$) and $q_0$ (the energy difference of the pion pair). In the latter two cases the restrictions $q_0 < 0.2$ GeV and $q_T < 0.2$ GeV/c were applied, respectively. Furthermore the same cuts were used as in the experiment to get an appropriate sample of events, namely only the pions with momentum $0.15$ GeV/c $< |{\bf p}| <
1.45$ GeV/c were chosen.
The calculation was carried out at first for the parameters (taken from [@negyedik]) giving the best fit to the single-particle distributions and practically no Bose-Einstein effect was found (Fig. 2). The width of the fitted Gaussian function gave for the radius of the pion source an order of magnitude larger value than the experimental source size in space-time. This was basicly due to the rather large decay constant slowing down the pion emission tremendously.
At this point one can ask whether the input parameters can be fixed to reproduce the experimental data for the Bose-Einstein correlation function and the single-particle distributions simultaneously. Hence the correlation function is more sensitive to the variation of the decay constant than the single-particle spectra, we tried to fit the correlation function by an appropriate choice of the decay constant and then to find out the string radius and the proper relationship between the string tension $\kappa$ and the chromoelectric field strength ${\cal E}$. We found a qualitatively good description of the Bose-Einstein correlation for $e{\cal E} = 2 \kappa$ and $R = 0.6$ fm (Figs. 3-5). The ‘new’ relationship and radius value are still in the range allowed by the single-particle spectra [@negyedik]. The value of the decay constant is now $\Lambda=1.25$ fm$^{-2}$. This implies that the average ’life time’ of the expanding quark-antiquark pair in the center of mass system is $0.9$ fm/c, which is comparable to the ’life time’ ($1.2 \pm 0.1$ fm/c) of the string between two coloured nucleons extracted from the rapidity distribution of protons in the collisions of proton on proton[@nyolcb].
By adjusting the decay constant the qualitative agreement is only achieved for the $q_0$ dependence of the correlation function (Fig. 5). This agreement guarantees that the time scale of the hadronization process is correct in our model.
The single-particle distributions like the transverse momentum distributions, the Feynman $x_F$ distributions and the rapidity distributions are only slightly modified by the new decay constant (Figs. 6-8). The calculated average charged particle multiplicities show a softer energy dependence than the experimental data (Fig. 9).
It is the main advantage of the Dynamical String Model that it follows up the space-time evolution of the hadronization process directly. So having fixed the time scale of the process as described above, we determined the time dependence of the pion emission in the center of mass system of the colliding $e^+e^-$ pair (Fig. 10). It can be seen that the pion emission rate has a maximum at about $5$ fm/c independently of the c.m. energy.
For times $t>5$ fm/c the pion emission rate decreases exponentially $dN/dt \propto e^{-t/T_0}$, where $T_0$ is the mean life time of the pion source in the c.m. system of the 2-jet events (Table 1). These values of $T_0$ cannot be compared directly to the mean life time $\tau_0=(0.62 \pm 0.25)$ fm/c of the pion source extracted from the dependence of the Bose-Einstein correlation function on $q_0$ and $q_T$[@kilencedik], because $\tau_0$ measures the life time of the pion source in its rest frame. On the other hand from the ratio of the mean life time in the laboratory and to that in the rest frame the speed of the pion source can be estimated, $\gamma=(1-v^2/c^2)$ and $\gamma=T_0/\tau_0$. Considering the very simplified picture that the pion emission takes place nearby the leading (anti)quark a kind of effective mass of the pion source can be estimated using the speed of the source (Table 2). This estimation leads to the rest mass $\sqrt{s}/2\gamma \approx 1.0$ GeV of the pion source.
The maximum of the pion emission rate is found at around $2$ fm/c proper time (Fig. 11). The slope of the decay of the pion source is determined by the characteristic proper time $2.2$ fm/c.
Conclusions {#conclusions .unnumbered}
===========
It was established that the Dynamical String Model improved by introducing the discrete final state hadron resonances provides one of better quality single-particle spectra for the process $e^+ e^- \to hadrons$ at $\sqrt{s} \approx 20 - 50$ GeV than the original version of the model. As the consequence of the decay of the discrete resonances the transverse momentum distributions are now in quantitative agreement with the experimental data for small transverse momenta up to $p_T < 2$ GeV/c.
The simulations in the framework of the Dynamical String Model have shown that the Bose-Einstein correlation for the input parameters representing the best fit to the single-particle distributions (obtained in [@negyedik]) is in contradiction with the TPC experiment. It has also been established that the simulated Bose-Einstein correlation is rather sensitive to the value of the decay constant. Making use of the ambiguities of the hadronic string - flux tube analogy, it was possible to choose the string radius $R = 0.6$ fm and achieve a decay constant providing a rather good agreement of both the simulated Bose-Einstein correlation and the single particle distributions with the experimental data. The appropriate string decay constant implies the life time $0.9$ fm/c of the hadronic string which is close to the value obtained in Ref.[@nyolcb].
Working with this decay constant that allows to recover the observed $q_0$ dependence of the Bose-Einstein correlation, the time scale of the hadronization process was fixed in a reliable way. Then the pion emission rate and the mean life time of the pion source were determined. It was established that the pion source builds up nearly in the first $5$ fm/c in the laboratory frame ($2$ fm/c in proper time), independently of the c.m. energy. Its mean life time is linearly rising with the c.m. energy. Comparing the mean life time detemined by the pion emission rate to the mean life time extracted from the Bose-Einstein correlation an effective rest mass ($1.0$ GeV) of the pion source nearby the leading (anti)quark can be found.
Acknowledgement {#acknowledgement .unnumbered}
===============
One of the authors (B.I.) wishes to express his thanks to J. Salo and F. Kun for helping in the data analysis and B. Müller for drawing his attention to Ref.[@nyolcb]. This work was supported by the EC project ERB-CIPA-CT92-4023, proposal 3293 and by the Hungarian Research Fund OTKA 2192/91.
c.m. energy (GeV) $22$ $35$ $50$
------------------- ----------------- ---------------- -----------------
$T_0$ (fm/c) $7.35 \pm 0.13$ $10.1 \pm 0.1$ $13.2 \pm 0.15$
: $T_0$ parameters of the exponential decay of the pion emission rate (see text) for different c.m. energies.
c.m. energy (GeV) $22$ $35$ $50$
------------------- ----------------- ----------------- -----------------
m (GeV) $0.93 \pm 0.02$ $1.07 \pm 0.01$ $1.17 \pm 0.01$
: The “effective mass” of the pion source (see text) for different c.m. energies.
=1.0
=1.0
=1.0
=1.0
=1.0
=1.0
=1.0
=1.0
=1.0
=1.0
=1.0
[99]{} K. Sailer, W. Greiner and B. Müller, [*in:*]{} “Quark-Gluon Plasma” R. C. Hwa, ed.,World Scientific, Singapore (1990) p. 299 Th. Schönfeld, A. Schäfer, B. Müller, K. Sailer, J. Reinhardt and W. Greiner, [*Phys. Lett.*]{} B247:5 (1990) M. G. Olsson, S. Veseli, and K. Williams, preprint MADPH-95-885, hep-ph/9505205 K. Sailer, Th. Schönfeld, Zs. Schram, A. Schäfer and W. Greiner, [*Int. J. Mod. Phys.*]{} A6:4395 (1991) Particle Data Group, [*Phys. Rev.*]{} D45:II.6 (1992) X. Artru and G. Mennessier, [*Nucl. Phys.*]{} B70:93 (1974) B. Iványi, Zs. Schram, K. Sailer and W. Greiner, [*in:*]{} Hot and Dense Nuclear Matter, Proc. NASI (Bodrum, 1993), ed. by W. Greiner (Plenum press), New York p. 751 B. Lörstad, [*Int. J. Mod. Phys.*]{} A4:2861 (1989) B. Andersson, G. Gustafson, G.Ingelman and T. Sjöstrand [*Phys. Rep.*]{} 97:31 (1983) W. Busza, T. Dreyer, M. Erdmann, [*Z. Phys.*]{} C48:167 (1990) H. Aihara at al., (TPC) [*Phys. Rev.* ]{} D31:996 (1985) P. Mättig, [*Phys. Rep.* ]{} 177:141 (1989)
|
---
abstract: 'We report electrical measurements showing the degradation processes of LaMnO$_{3-y}$ (LaMnO) in LaMnO/normal metal interface in both point contact and planar-type junctions. Immediately after the preparation of the interface, the degradation process was followed by measuring the evolution of the junction resistance versus time. This process is characterized by the appearance of a second maximum in the resistance vs. temperature (R-T) dependence at temperatures lower than the Curie temperature T$_c$, at which the metal-insulator transition occurs in the bulk. These effects are explained in terms of the formation of a depleted interface layer in LaMnO caused by an out-diffusion of oxygen from the manganite surface to the normal metal. This assumption is confirmed by XPS measurement. Similar results on LaSrMnO$_{3-y}$ interfaces are also obtained.'
address:
- 'Institute of Electrical Engineering, Slovak Academy of Sciences, Dubravska cesta 9, 84239 Bratislava, Slovak Republic'
- 'Instituto de Ciencia de Materiales de Sevilla, Centro de Investigaciones Científicas “Isla de la Cartuja”, C/ Americo Vespucio, s/n, 41092 Isla de la Cartuja, Sevilla, Spain'
- 'Institute of Electrical Engineering, Slovak Academy of Sciences, Dubravska cesta 9, 84239 Bratislava, Slovak Republic '
- 'University of Nice Sophia Antipolis, UMR CNRS 6622, Parc Valrose, 06108 Nice Cedex 2, France'
author:
- 'A.Plecenik and K. Fröhlich'
- 'J.P. Espinós and J.P.Holgado'
- 'A. Halabica and M. Pripko'
- 'A. Gilabert'
title: 'Degradation of LaMnO$_{3-y}$ surface layer in LaMnO$_{3-y}$/ metal interface'
---
\#1[[$\backslash$\#1]{}]{}
pacs: [ 73.40.Cg, 68.35.Fx, 79.60.Jv]{}
The magnetoresistive rare earth perovskites La$_{1-x}$A$_x$MnO$_{3-y}$ (LaAMnO, A=Ca, Sr, Pb), which include LaMnO, have been in the research spotlight for a decade. Optimally doped LaAMnO exhibits a paramagnetic - insulating behavior above the Curie temperature T$_c$ and a ferromagnetic - metallic behavior below Tc. In the recent years, the giant magnetoresistance phenomenon (MR) has been intensively studied in these materials. However till low-field, MR effects in granular films and tunneling of spin-polarized electrons through an insulating barrier in LaAMnO/Insulator/LaAMnO magnetic tunnel junctions (MTJ) gave promise to their technological applications like magnetic field sensor and nonvolatile magnetic random access memory. One can expect a near 100% value of tunneling magnetoresistance (TMR) in tunnel junctions based on half metallic LaAMnO ferromagnet. TMR depends on the quality of the magnetic state of the LaAMnO surface, interface quality, surface roughness of the electrodes, etc. Anomalous tunneling behavior observed in MTJ could be explained by extrinsic factors playing a dominant role in the MTJ interfaces. This assumption has been made by several authors. Park et al. [@Park98] showed that even in a fully oxidized LaSrMnO sample, the magnetic properties at the surface boundary are significantly different from those of the bulk. In addition, several experiments indicated the creation of an oxygen depleted layer of LaAMnO interfaces. Mieville et al. [@Mieville98] studied the interface resistance and transport across conducting ferromagnetic oxide/metal interfaces. They measured a very high resistance of LaSrMnO/Al(Nb) junctions, which could be explained by the existence of a degraded surface layer of ferromagnetic oxide due to a loss of oxygen. Also de Teresa et al. [@Teresa00] observed a dependence of TMR on the type of tunneling barrier. The present work was stimulated by the above mentioned results, but mainly by the results published in our previous paper [@Gilabert01]. LaMnO/Al$_2$O$_3$/Nb junctions were fabricated for the study of the influence of illumination on the electrical properties of MTJ. The Al$_2$O$_3$ insulating barrier was prepared using well-known thin films Nb technology [@Gurwitch83]: An aluminium thin film was sputtered on LaMnO, and then oxidized in an oxygen atmosphere. Finally a semi-transparent Nb upper electrode was sputtered on the top. Using the same technique with the same preparation parameters, the width of the barrier created on the LaMnO was about two times higher than that for Nb/Al$_2$O$_3$/Nb junctions.
In this paper we study the LaMnO surface properties in contact with Al, In, Au and Pb metals in order to explain the physical processes in the interface which give rise to an unusual behavior of MTJ. The LaMnO/normal metal point contact exhibits a change of resistance immediately after its preparation. In the planar junctions (more stable than point contact) the appearance of a second maximum in the R-T dependence was observed several hours after junction preparation. These effects are explained by an out-diffusion of oxygen from LaMnO, and they were confirmed by XPS spectroscopy.
200 nm thick LaMnO epitaxial thin films with T$_c$=270 K were deposited on single crystalline SrTiO$_3$ substrates by low pressure liquid source metal-organic chemical vapour deposition (MOCVD) [@Frohlich95]. LaMnO/metal point contacts were realized using a holder which exerted a constant pressure of the tip onto the sample. Bulk Au, Al, or Pb were used for the upper electrode. The shape of the sharp point was prepared by mechanical sharpening. After the electrical measurements, a 200x200 $\mu m^2$ geometric area was evaluated for the Pb tip.
For the preparation of planar junctions, a 40 $\mu$m - wide and 260 $\mu$m - long LaMnO$_3$ base electrode was formed by wet etching through resist mask. After stripping the photoresist, the area for deposition of next layers by lift-off technique was defined in positive photoresist. Shortly before the deposition of the barrier and the upper electrodes, the surface of LaMnO$_3$ was etched to eliminate a contaminated and degraded upper layer. A 100 nm thick Al or In layer was deposited by thermal evaporation. Junctions with an area of 40$\times$40 $\mu$m$^2$ were finalized after the removal of the photoresist.\
The R-T and resistance vs. time (R-t) characteristics were measured by a computer controlled four point method .
XPS spectra were obtained in an ESCALAB 210 spectrometer that consisted of two separate independently-pumped chambers. Pressures in the range of 6x10$^{-11}$ mbar and 10$^{-8}$ mbar were obtained in the analysis and preparation chambers, respectively. Al has been evaporated in the analysis chamber from a resistively heated filament made of Al wire wrapped around a thick tungsten wire. The power was maintained to give a reproducible evaporation rate of 1 monolayer of Al per minute. Control evaporation carried out onto a clean Au foil showed that under our experimental conditions, only metallic Al is evaporated, maintaining this oxidation state throughout all the acquisition time. A hemispherical electron energy analyser working in the pass energy constant mode at a value of 50 eV was used. Unmonochromatized Al Ka radiation was used as the excitation source. Spectra were energy-calibrated by taking the La3d5/2 peak at 834.6 eV (BE). The spectra were acquired at 90 (normal) and 20 (grazing) degrees with respect to sample surface.
In Fig 1, the LaMnO/metal point contact resistance dependence versus time shows a significant increase at several hundred seconds after the preparation of the contact. These changes were observed even if noble metal (Au) was used as the upper electrode material. Similar results on high-T$_c$ superconductors (HTS) were described in our previous work [@Grajcar92]. The change of the point contact resistance was explained within an oxygen out-diffusion model.\
The total point contact resistance with a tip made of nonreactive metal (e.g. Au) can be expressed as $R =
R_M + R_{LaMnO} + R_T$, where $R_T$ is the tunneling resistance and $R_M$ and $R_{LaMnO}$ the contact resistances with metal and LaMnO, respectively. We suppose that $R_M$ and $R_T$ are constant in time for stable nonreactive metals, and the change of resistance $R$ is then given by the change of $R_{LaMnO}$. The creation of additional barriers from a LaMnO oxygen depleted layer as well as from the oxide of the upper metallic electrode (created from reactive metal like Al, Pb) creates a complex interface. But in all cases, the Curie temperature $T_c$ in the LaMnO oxygen depleted region must be changed also.
In the next measurements the planar junctions were used because they were more stable than the point contact ones. Because Al is a very reactive metal with a very short time of interaction, indium as an upper electrode was used for the following measurement. The R-T characteristic measured on a LaMnO/In planar junction at different times ( 3, 5 and 23 hours) after preparation of the contacts is shown in Fig.2. We observed the classical maximum correlated with the I-M transition of the bulk material at $T_c$ around 275 K of the bulk material. This maximum did not change very much with time. It means that the oxygen content in the bulk material does not change. After a long time (23 hours) a second maximum appears around T’$_c$= 185 K in the R-T characteristics. We know that in LaAMnO systems the Tc decreases with a decrease of the oxygen content, and it disappears for deoxygenated samples that exhibit semiconducting properties. This new maximum is related to the oxygen depleted interface, and it is broader due to a distribution of T’$_c$. This value of T’$_c$= 185 K corresponds to an average oxygen content at the interface of y around 0.1.[@Leon-Guevara97]
To confirm the occurrence of out-diffusion processes in the LaMnO surface layer, XPS measurements of LaMnO/Al interfaces were studied. Fig.3 showns spectra of the O1s, Mn2p, Al2p and La3d regions measured on clean (original) LaMnO surface (curves a) at the normal acquisition angle, after the deposition of around 1 monolayer of Al (curves b), at the normal adquisition angle, around 2 monolayers of Al, at the normal adquisition angle (curves c), and curves d were measured as curves c at a grazing adquisition angle. One can see the following features in the XPS spectra after the deposition of Al:\
- two peaks correlated with Al and Al$_2$O$_3$,\
- two peaks in the O1s spectra correlated with oxygen bonding in LaMnO$_3$ at 529.9 eV and Al$_2$O$_3$ at 531.6 eV,\
- shift of the peaks in the Mn2p spectra due to the decreasing of Mn valency.
The deposition of the first monolayer of Al on the LaMnO surface provokes the oxidation of Al to Al$^{3+}$, whose peak arises at about 75 eV (BE). This behaviour was tested by a simultaneous deposition of Al on Au. The detection of metallic Al during the whole experiment (more than 1 hour) exclude the oxidation of Al due to the residual O$_2$ partial preasure in the analytic chamber. In additional, when a second amount of Al was deposited (curve c in Fig.3a), most of it remained metallic (peak at 72.6 eV BE). Finally, when this situation is examined at the grazing adquisition angle (curve d in Fig.3a), the relative intensity of the peak from Al(0), in comparison with that from Al$^{+3}$, increases substantially, indicating that metallic Al is located on the top of the Al$_2$O$_3$.
The oxidation of Al in the LaMnO$_3$/Al interface accompanying the appearence of two peaks in the O1s spectra is correlated with a change of binding energy due to the creation of Al$_2$O$_3$. The smooth second peak was measured also on clean LaMnO surface (curve a in Fig.3c) as well as after cleaning treatment. The origin of the second peak can be explained by the contamination of LaMnO by CO$_3^=$ species within the preparation procedure of the LaMnO thin film. Anyway, the intensity of the peak at the energy of 520.6 eV increase after the deposition of Al (curves b and c in Fig.3c) due to the arising of O$^=$ ions in Al$_2$O$_3$. The measurement at the grazing angle shows that Al$_2$O$_3$ is created on the LaMnO surface (the global intensity of Al2p signal grows and the Mn and La signals decrease). These facts aslo indicate the existence of metallic Al on the top of Al$_2$O$_3$.
A significant chemical shift of Mn2p peak from 642.5 eV before Al evaporation to 641.4 eV after Al evaporation strongly indicates a decrease of Mn valency to +3 state with a subsequent change of the Mn$^{3+}$/Mn$^{4+}$ ratio, typical for oxygen-deficient LaAMnO.
From XPS measurements described above, we can confirm the loss of oxygen from the LaMnO in the LaMnO/Al interface and the creation of a LaMnO/Al$_2$O$_3$/Al junction with a subsequent change in the Mn$^{3+}$/Mn$^{4+}$ ratio.
In conclusion, we presented R-T and R-t measurements on LaMnO/metal point contact and planar junctions. The time evolution of point contact junction resistance as well an arising of a second peak on the R-T characteristics were observed. These effects were explained in terms of the formation of a depleted interface layer in LaMnO$_{3-x}$ caused by the out-diffusion of oxygen from the manganite surface to the normal metal. This assumption was confirmed by XPS measurements . The similar results (XPS as well as time evolution of point contact resistance) on LaSrMnO magnetoresistive thin films were also obtained.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported by the Slovak Grant Agency for Science (Grants No.2/7199/20 and 1/7072/20) and in part by the European Commission (project GRT-CT-2000-05001 MULTIMETOX)
J.-H.Park, E.Vescovo. H.-J.Kim, C.Kwon, R.Ramesh and T.Venkatesan [**81**]{}, Phys.Rev.Lett. [**81**]{}, 1953 (1998). L.Mieville, D.Worledge, T.H.Geballe, R.Contreras and K.Char, Appl.Phys.Lett. [**73**]{}, 1736 (1998). J.M. de Teresa, A.Barthélémy, J.P.Contour and A.Fert, J.Magn.Magn.Mater. [**211**]{}, 160 (2000). A.Gilabert, A.Plecenik, K.Fröhlich, Š.Gaži, M.Pripko, Ž.Mozolová, D.Machajdík, Š.Beňačka and M.G.Medici, Appl.Phys.Lett. [**78**]{}, 1712 (2001). M.Gurwitch, M.A.Washington and H.A.Huggins, Appl.Phys.Lett. [**42**]{}, 472 (1983). K. Fröhlich, J. Šouc, D. Machajdík, A.P. Kobzev, F. Weiss, J. P. Senateur, K.H. Dahmen, Journ. de Physique [**IV C5**]{}, 533 (1995). M.Grajcar, A.Plecenik, Š.Beňačka, Ju.Revenko, V.M.Svistunov, Physica C [**218**]{}, 82 (1993). A.M.De Leon-Guevara; P.Berthet; J.Berthon; F.Millot; R.Revcolevschi; A.Anane; C.Dupas; K.Le Dang; J.P.Renard; P.Veillet Phys.Rev. B [**56**]{}, 6031(1997)
|
---
abstract: |
The refined inertia of a square real matrix $A$ is the ordered $4$-tuple $(n_+, n_-, n_z, 2n_p)$, where $n_+$ (resp., $n_-$) is the number of eigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is the number of zero eigenvalues of $A$, and $2n_p$ is the number of nonzero pure imaginary eigenvalues of $A$. For $n \geq 3$, the set of refined inertias $\mathbb{H}_n=\{(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\}$ is important for the onset of Hopf bifurcation in dynamical systems. We say that an $n\times n$ sign pattern ${\cal A}$ requires $\mathbb{H}_n$ if $\mathbb{H}_n=\{\operatorname{ri}(B) | B \in Q({\cal A})\}$. Bodine et al. conjectured that no $n\times n$ irreducible sign pattern that requires $\mathbb{H}_n$ exists for $n$ sufficiently large, possibly $n\ge 8$. However, for each $n \geq 4$, we identify three $n\times n$ irreducible sign patterns that require $\mathbb{H}_n$, which resolves this conjecture.
15B35, 15A18, 05C50
Eigenvalues; Refined inertia; Sign pattern.
author:
- |
Wei Gao[^1], Zhongshan Li, Lihua Zhang\
Dept of Math and Stat, Georgia State University, Atlanta, GA 30302-4110, USA
title: 'Sign patterns that require $\mathbb{H}_n$ exist for each $n\geq 4$ '
---
-1.6cm -1.9cm
Introduction
============
The refined inertia of a square real matrix $B$, denoted $\operatorname{ri}(B)$, is the ordered $4$-tuple $(n_+(B),$ $ n_-(B), n_z(B), 2n_p(B))$, where $n_+(B)$ (resp., $n_-(B)$) is the number of eigenvalues of $B$ with positive (resp., negative) real part, $n_z(B)$ is the number of zero eigenvalues of $B$, and $2n_p(B)$ is the number of pure imaginary eigenvalues of $B$. Refined inertias were introduced in [@Deaett10], and have been the focus of study in several recent papers such as [@Bodine12; @Gao14; @Garnett13; @Garnett14; @Olesky13; @Yu12].
A [*sign pattern*]{} ([*matrix*]{}) is a matrix whose entries are from the set $\{+, -, 0\}$. For a real matrix $B$, sgn($B$) is the sign pattern matrix obtained by replacing each positive (resp., negative, zero) entry of $B$ by $+$ (resp., $-, 0$). For an $n\times n$ sign pattern matrix $\cal A$, the qualitative class of $\cal A$, denoted $Q(\cal A)$, is defined as $Q({\cal A})=\{B\in M_n({\mathbb{R}}) \mid \hbox{sgn}(B)={\cal A}\}$.
A [*permutation sign pattern*]{} is a square sign pattern with entries from the set $\{0, +\}$, where the entry $+$ occurs precisely once in each row and in each column. A [*signature sign pattern*]{} is a square diagonal sign pattern all of whose diagonal entries are nonzero. Let ${\cal A}_1$ and ${\cal A}_2$ be two square sign patterns of the same order. Sign pattern ${\cal A}_1$ is said to be [*permutationally similar*]{} to ${\cal A}_2$ if there exists a permutation sign pattern ${\cal P}$ such that ${\cal A}_2 ={\cal P}^{T} {\cal A}_1 {\cal P}$. Sign pattern ${\cal A}_1$ is said to be [*signature similar*]{} to ${\cal A}_2$ if there exists a signature sign pattern ${\cal D}$ such that ${\cal A}_2 ={\cal D}{\cal A}_1 {\cal D}$. Two sign patterns are said to be *equivalent* if one can be obtained from the other by transposition, signature similarity, permutation similarity, or any combination of these.
Let $n\ge 3$ and let $\mathbb{H}_n=\{(0, n, 0, 0),\ (0, n-2, 0, 2),\ (2, n-2, 0, 0)\}$. As pointed out by Bodine et al. [@Bodine12], $\mathbb{H}_n$ is an important set of refined inertias which can signal the onset of periodic solutions by Hopf bifurcation in dynamical systems. We say that an $n\times n$ sign pattern ${\cal A}$ [*requires*]{} $\mathbb{H}_n$ if $\mathbb{H}_n=\{\operatorname{ri}(B) | B \in Q({\cal A})\}$, and ${\cal A}$ [*allows*]{} $\mathbb{H}_n$ if $\mathbb{H}_n\subseteq \{\operatorname{ri}(B) | B \in Q({\cal A})\}$. In [@Bodine12], the authors made the following conjecture.
\[conj\]No $n\times n$ irreducible sign pattern that requires $\mathbb{H}_n$ exists for $n$ sufficiently large, possibly $n\ge 8$.
In this paper, for each $n\geq 4$, we identify three $n\times n$ irreducible sign patterns that each require $\mathbb{H}_n$, which negatively resolves the preceding conjecture.
Let ${\cal A}_1$, ${\cal A}_2$, ${\cal A}_3$ be sign patterns of order $n \geq 4$ defined by $${\cal A}_1=\left[\begin{array}{cccccc}
+&+&+&\cdots&+&+\\
-&0&\\
-&&-&\\
\vdots&&&\ddots\\
-&&&&-&\\
-&&&&&-
\end{array}\right],\ \ \
{\cal A}_2= \left[\begin{array}{cccccc}
-&+&+&\cdots&+&+\\
-&0&\\
+&&-&\\
\vdots&&&\ddots\\
+&&&&-&\\
+&&&&&-
\end{array}\right],$$ $${\cal A}_3=\left[\begin{array}{cccccc}
-&+&+&\cdots&+&+\\
+&0&\\
+&&-&\\
\vdots&&&\ddots\\
+&&&&-&\\
-&&&&&+
\end{array}\right],$$ where all the off-diagonal entries except those on the first row or first column are zeros. We will show that for $i=1,2,3$, the ${\cal A}_i$ require $\mathbb{H}_n$ for each $n\ge 4$.
Throughout what follows, we let $B$ denote a real matrix of order $n \geq 4$ of the form $$B=\left[\begin{array}{cccccccc}
a_1&1&1&\cdots&1&1\\
a_2&0&&\\
a_3&&-b_1&\\
\vdots&&&\ddots\\
a_{n-1}&&&&-b_{n-3}\\
a_{n}&&&&&-b_{n-2}
\end{array}\right],\eqno{(1.1)}$$ where $b_j>0$ for $j=1,2,\dots, n-3$, and suitable real values for $a_1, a_2, \dots, a_n$ and $b_{n-2}$ are taken so that $B\in Q({\cal A}_i)$ for some $i \in \{ 1,2,3 \}$.
The sign patterns ${\cal A}_i$ allow $\mathbb{H}_n$ for $i=1,2,3$
=================================================================
Note that the $4\times 4$ sign patterns ${\cal A}_1$, ${\cal A}_2$ and ${\cal A}_3$ given in Section 1 are equivalent to ${\cal S}_5$, ${\cal S}_2$ and ${\cal S}_4$ defined in [@Garnett13 p.624], respectively. Thus the next lemma follows from Theorems 2.5, 2.6, and 2.8 in [@Garnett13].
\[lemma:H4\]The $4\times 4$ sign patterns ${\cal A}_i$ require $\mathbb{H}_4$ for $i=1,2,3$.
We now show that for $i=1,2,3$, the $n\times n$ sign patterns ${\cal A}_i$ allow $\mathbb{H}_n$ for each $n\ge 5$.
\[allow:Hn\]For $i=1,2,3$, the sign patterns ${\cal A}_i$ allow $\mathbb{H}_n$ for each $n\ge 5$, .
Choose any $i \in \{1, 2, 3\}$. By Lemma \[lemma:H4\], for each of the refined inertias $(0,4,0,0)$, $(0,2,0,2)$ and $(2,2,0,0)$, there exist suitable values of $a_1,a_2,a_3,a_4,b_1,b_2$ such that $$B_{4\times 4}=\left[\begin{array}{cccc}
a_1&1&1&1\\
a_2&0&\\
a_3&&-b_1&\\
a_4&&&-b_2
\end{array}\right]\in Q({\cal A}_i)$$ has this refined inertia.
For $n\ge 5$, consider the $n\times n$ matrix $$B=\left[\begin{array}{ccccccc}
a_1&1&1&1&\cdots&1&1\\
a_2&0&\\
\frac{a_3}{n-3}&&-b_1&\\
\frac{a_3}{n-3}&&&-b_1&\\
\vdots&&&&\ddots\\
\frac{a_3}{n-3}&&&&&-b_1&\\
a_4&&&&&&-b_2
\end{array}\right].$$ Then $B\in Q({\cal A}_i)$, and $$|\lambda I-B|=\left|\begin{array}{ccccccc}
\lambda-a_1 &-1&-1&-1&\cdots&-1&-1\\
-a_2&\lambda &\\
-\frac{a_3}{n-3}&&\lambda +b_1&\\
-\frac{a_3}{n-3}&&&\lambda +b_1&\\
\vdots&&&&\ddots\\
-\frac{a_3}{n-3}&&&&&\lambda +b_1&\\
-a_4&&&&&&\lambda +b_2
\end{array}\right|$$ $$=\left|\begin{array}{ccccccc}
\lambda-a_1 &-1&-1&-1&\cdots&-1&-1\\
-a_2&\lambda &\\
-a_3&&\lambda +b_1&\lambda +b_1&\cdots&\lambda +b_1\\
-\frac{a_3}{n-3}&&&\lambda +b_1&\\
\vdots&&&&\ddots\\
-\frac{a_3}{n-3}&&&&&\lambda +b_1&\\
-a_4&&&&&&\lambda +b_2
\end{array}\right|$$ $$=\left|\begin{array}{ccccccc}
\lambda-a_1 &-1&-1&0&\cdots&0&-1\\
-a_2&\lambda &\\
-a_3&&\lambda +b_1\\
-\frac{a_3}{n-3}&&&\lambda +b_1&\\
\vdots&&&&\ddots\\
-\frac{a_3}{n-3}&&&&&\lambda +b_1&\\
-a_4&&&&&&\lambda +b_2
\end{array}\right|$$ $$=(\lambda +b_1)^{n-4}\left|\begin{array}{ccccccc}
\lambda-a_1 &-1&-1&-1\\
-a_2&\lambda &\\
-a_3&&\lambda +b_1\\
-a_4&&&\lambda +b_2
\end{array}\right|
=(\lambda +b_1)^{n-4}|\lambda I-B_{4\times 4}|.$$ So the multiset of the eigenvalues of $B$ is given by $\sigma(B)=\{-b_1,\dots,-b_1\}\cup \sigma(B_{4\times 4})$, in which each set is interpreted as a multiset. It follows that $n_-(B)=n_-(B_{4\times 4})+(n-4)$, $n_+(B)=n_+(B_{4\times 4})$, $n_z(B)=n_z(B_{4\times 4})$, and $2n_p(B)=2n_p(B_{4\times 4})$. Thus the $n\times n$ sign pattern ${\cal A}_i$ allows $\mathbb{H}_n=\{(0,n,0,0),\ (0,n-2,0,2),\ (2,n-2,0,0)\}$.
The main result
===============
In this section, we establish that for $i=1,2,3$, the $n\times n$ sign patterns ${\cal A}_i$ require $\mathbb{H}_n$ for each $n\ge 5$. As in the introduction, throughout this section we let $B$ be a real matrix in the form $(1.1)$ of order $n \geq 5$.
First, we consider the case that all the $b_j$ are distinct for $j=1, 2, \dots,n-2$ and show the following result.
\[thm:distint\_b\_i\]Let $n\ge 5$ and let $B$ have the form $(1.1)$. If all the $b_j$ are distinct for $j=1, 2, \dots,n-2$, then $\operatorname{ri}(B) \in \mathbb{H}_n$.
Since all the $b_j$ are distinct for $j=1, 2, \dots,n-2$, we may assume, applying a permutation similarity if necessary, that $B$ is subjected to $b_1>b_2>\dots>b_{n-2}$ in Theorem \[thm:distint\_b\_i\]. To prove Theorem \[thm:distint\_b\_i\], we need the following lemmas. We also assume that $b_1>b_2>\dots>b_{n-2}$ in Lemmas \[lemma3.2\]–\[lemma-even-odd\].
\[lemma3.2\]For $j=1, 2, \dots, n-2$, $$|b_jI+B|=-a_{j+2}b_j\prod_{\substack{m=1 \\ m\ne j}}^{n-2}(b_j-b_m).$$
Note that row $j+2$ as well as column $j+2$ of $b_jI+B$ has exactly one nonzero entry, namely the first entry, which may be used to zero out all other entries in the first row or the first column without affecting the determinant. Hence, $$\begin{aligned}
|b_jI+B| & =\left| \begin{array}{ccccccccc}
0&0&0&\cdots&0&1&0&\cdots&0\\
0&b_j&\\
0&&b_j-b_1&\\
\vdots&&&\ddots\\
0&&&&b_j-b_{j-1}&\\
a_{j+2}&&&&&0\\
0&&&&&&b_j-b_{j+1}\\
\vdots&&&&&&&\ddots\\
0&&&&&&&&b_j-b_{n-2}
\end{array}\right| \notag \\
& =-a_{j+2}b_j\prod_{\substack{m=1\\ m\ne j}}^{n-2}(b_j-b_m). \hspace{7.5cm} \Box \notag\end{aligned}$$
In view of Lemma \[lemma3.2\], the following two results are straightforward.
\[lemma3.4\]Suppose $B \in Q({\cal A}_i)$ for $i \in \{1,2,3\}$. Then $$\hbox{\em sgn}(\det(b_jI+B)) = \begin{cases}
(-)^{j+1} &\text{for $j=1,2,\dots,n-2$ if $i =1$};\\
(-)^{j} &\text{for $j=1,2,\dots,n-2$ if $i =2$};\\
(-)^{j} &\text{for $j=1,2,\dots,n-3$ if $i =3$}.
\end{cases}$$
\[lemma3.3\]The eigenvalues of $B$ do not include $-b_j$ for any $j \in \{1,2,\dots,n-2\}$.
\[remark3.5\]Suppose $B \in Q({\cal A}_i)$ for $i \in \{1,2,3\}$. Then $n_{-}(B)\geq n-4 \geq 1$. Furthermore, if $i \neq 3$, $n_{-}(B) \geq n-3$.
Observe that by Lemma \[lemma3.4\], the real function $p(t) = \det (tI -B)$ takes on nonzero values of opposite signs at $-b_j$ and $-b_{j+1}$, for $j=1, 2, \dots, n-4$. Thus, by the Intermediate Value Theorem, $p(t)$ has at least one real zero in each open interval $(-b_j,-b_{j+1})$. It follows that the matrix $B$ has at least one real eigenvalue in $(-b_j,-b_{j+1})$, for $j=1, 2, \dots, n-4$. Thus $n_{-}(B)\geq n-4 \geq 1$. Furthermore, if $i \in\{1, 2\}$, then by Lemma \[lemma3.4\], $B$ has at least one real eigenvalue in $(-b_j,-b_{j+1})$, for $j=1, 2, \dots, n-3$, so we have $n_{-}(B) \geq n-3$.
\[remark3.6\] $\operatorname{sgn}(\det(B))=(-)^n$. Furthermore, $n_z(B)=0$, and $n_-(B)$ and $n$ have the same parity.
Expanding the determinant along the second column reveals that $\hbox{sgn}(\det(B))=(-)^n$. Consequently, $n_z(B)=0$. It follows that $\operatorname{sgn}(\det(B))=(-)^n = (-)^{n_-(B)}$. Hence, $n_-(B)$ and $n$ have the same parity.
For any $r \in {\mathbb{R}}$, define $\Delta(r)$ to be the number of eigenvalues $\lambda$ of $B$ in the closed left half-plane with $\hbox{Re}(\lambda) \le -r$. It is clear that $$n_-(B)\ge\Delta(b_{n-3})=\Delta(b_1)
+\sum_{j=1}^{n-4}[\Delta(b_{j+1})-\Delta(b_j)]. \eqno{(3.1)}$$
\[lemma3.6\]For $j=1,2,\dots, n-3$, $n_-(b_jI+B)$ and $\Delta(b_j)$ have the same parity.
Note that $\lambda$ is an eigenvalue of $B$ if and only if $b_j+\lambda$ is an eigenvalue of $b_jI+B$, that the non-real eigenvalues of $b_jI+B$ occur in conjugate pairs, and that $-b_1, -b_2, \dots, -b_{n-3}$ are not eigenvalues of $B$ by Lemma \[lemma3.3\]. We see that for $j=1,2,\dots, n-3$,
- $n_-(b_jI+B)=$ the number of eigenvalues $\lambda$ of $B$ satisfying $\hbox{Re}(\lambda)< -b_j$;
- $\Delta(b_j)=$ the number of eigenvalues $\lambda$ of $B$ satisfying $\hbox{Re}(\lambda)\le -b_j$;
- the number of eigenvalues $\lambda$ of $B$ satisfying $\hbox{Re}(\lambda)= -b_j$ is even.
So $n_-(b_jI+B)$ and $\Delta(b_j)$ have the same parity.
\[lemma-even-odd\]Let $k=n_-(B)$. Then $k\ge n-2$.
If $i=1$ or $i=2$, then by Lemma \[remark3.5\], we have $k=n_-(B)\geq n-3$. By Lemma \[remark3.6\], $k$ and $n$ have the same parity. It follows that $k \geq n-2$, as desired. Hence, assume $i = 3$.
We claim that for every $j \le n-3$, the parity of $j$ and $\Delta(b_{j})$ are the same. Otherwise, if there exists an even index $j$ $ \le n-3 $ such that $\Delta(b_{j})$ is odd, by Lemmas \[lemma3.4\] and \[lemma3.6\], we have that $\det(b_{j}I+B)>0$ and $n_-(b_{j}I+B)$ is odd, which is a contradiction; if there exists an odd index $j$ $ \le n-3 $ such that $\Delta(b_{j})$ is even, by Lemmas \[lemma3.4\] and \[lemma3.6\], we have that $\det(b_{j}I+B)<0$ and $n_-(b_{j}I+B)$ is even, which is a contradiction.
Thus $\Delta(b_{1})$ is odd, and $\Delta(b_{j+1})-\Delta(b_{j}) > 0 $ is odd for $1\le j\le n-4$. So by (3.1), $$\begin{aligned}
k& \ge\Delta(b_{n-3})=\Delta(b_1)
+\sum_{j=1}^{n-4}[\Delta(b_{j+1})-\Delta(b_j)]
\ge n-3. \notag\end{aligned}$$ By Lemma \[remark3.6\], $k$ and $n$ have the same parity. It follows that $k \geq n-2$.
We now complete the proof of Theorem \[thm:distint\_b\_i\].
[**Proof of Theorem 3.1**]{} By Lemmas \[remark3.6\] and \[lemma-even-odd\], we have $n_z(B)=0$ and $n_-(B)=n-2$ or $n_-(B)=n$. It follows that $\operatorname{ri}(B)\in \mathbb{H}_n$.
We are now ready to establish the main result.
\[thm:main\]For $i=1,2,3$, the $n\times n$ sign patterns ${\cal A}_i$ require $\mathbb{H}_n$ for each $n\ge 4$.
Fix any $i \in \{ 1, 2,3 \}$. We proceed by induction on the order $n$ of ${\cal A}_i$.
By Lemma \[lemma:H4\], the result holds for $n=4$.
Suppose that the $(n-1)\times (n-1)$ sign pattern ${\cal A}_i$ requires $\mathbb{H}_{n-1}$ for some $n\ge 5$. We prove that the $n\times n$ sign pattern ${\cal A}_i$ requires $\mathbb{H}_n$. By Theorem \[allow:Hn\], ${\cal A}_i$ allows $\mathbb{H}_n$. Thus we only need to prove that $\operatorname{ri}(B) \in \mathbb{H}_n$ for every $B \in Q({\cal A}_i)$.
For any $B \in Q({\cal A}_i)$, by performing a diagonal similarity on $B$ if necessary, we may assume that $B$ has the form (1.1). If all the $b_j$ are distinct for $j=1, 2, \dots,n-2$, then by Theorem \[thm:distint\_b\_i\] $\operatorname{ri}(B) \in \mathbb{H}_n$.
Now suppose that two of the $b_j$ are the same for $j=1, 2, \dots,n-2$. Note that in the case of $B\in Q({\cal A}_3)$, $b_{n-2}$ is different from each $b_j$ with $j\leq n-3$ as $b_j>0>b_{n-2}$. By performing a permutational similarity if necessary, without loss of generality, we may assume that $b_1=b_2$. Then $$|\lambda I-B|=\left|\begin{array}{ccccccc}
\lambda-a_1&-1&-1&-1&-1&\cdots&-1\\
-a_2&\lambda&\\
-a_3&&\lambda+b_1&\\
-a_4&&&\lambda+b_1&\\
-a_5&&&&\lambda+b_3\\
\vdots&&&&&\ddots\\
-a_n&&&&&&\lambda+b_{n-2}
\end{array}\right|$$ $$=\left|\begin{array}{ccccccc}
\lambda-a_1&-1&-1&-1&-1&\cdots&-1\\
-a_2&\lambda&\\
-a_3&&\lambda+b_1&\\
-a_3-a_4&&\lambda+b_1&\lambda+b_1&\\
-a_5&&&&\lambda+b_3\\
\vdots&&&&&\ddots\\
-a_n&&&&&&\lambda+b_{n-2}
\end{array}\right|$$ $$=\left|\begin{array}{ccccccc}
\lambda-a_1&-1&0&-1&-1&\cdots&-1\\
-a_2&\lambda&\\
-a_3&&\lambda+b_1&\\
-a_3-a_4&&&\lambda+b_1&\\
-a_5&&&&\lambda+b_3\\
\vdots&&&&&\ddots\\
-a_n&&&&&&\lambda+b_{n-2}
\end{array}\right|$$ $$=(\lambda+b_1)\left|\begin{array}{cccccc}
\lambda-a_1&-1&-1&-1&\cdots&-1\\
-a_2&\lambda&\\
-a_3-a_4&&\lambda+b_1&\\
-a_5&&&\lambda+b_3\\
\vdots&&&&\ddots\\
-a_n&&&&&\lambda+b_{n-2}
\end{array}\right|.$$ Take the $(n-1)\times (n-1)$ matrix $$B_1=\left[\begin{array}{cccccc}
a_1&1&1&1&\cdots&1\\
a_2&0&\\
(a_3+a_4)&&-b_1\\
a_5&&&-b_3\\
\vdots&&&&\ddots\\
a_n&&&&&-b_{n-2}
\end{array}\right].$$ Then $$\sigma(B)=\{-b_1\}\cup \sigma(B_1),$$ Note that $B_1 \in Q({\cal A}_i)$ has order $n-1$. By the induction hypothesis, ${\cal A}_i$ of order $n-1$ requires $\mathbb{H}_{n-1}=\{(0,n-1,0,0),\ (0,n-3,0,2),\ (2,n-3,0,0)\}$. Thus $\operatorname{ri}(B)$ is one of $(0, n ,0,0)$, $(0,n-2,0,2)$ and $(2,n-2,0,0)$. It follows that $\operatorname{ri}(B)\in \mathbb{H}_n$.
This completes the proof.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors express their sincere thanks to the anonymous referee for valuable suggestions which greatly improved the exposition of the paper.
[11]{}\[preferences\]
[^1]: Corresponding author. E-mail: wgao2@gsu.edu
|
---
abstract: 'We investigate spectral properties of the tensor products of two completely positive and trace preserving linear maps (also known as quantum channels) acting on matrix algebras. This leads to an important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels the multiplicative domain of their tensor product splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of maximal unital proper \*-subalgebras (MUPSA) of a matrix algebra we provide a non-trivial upper bound of the recently-introduced multiplicative index of a unital channel. This bound gives a criteria on when a channel cannot be factored into a product of two different channels. We construct examples of channels which cannot be realized as a tensor product of two channels in any way. With these techniques and results, we found some applications in quantum information theory.'
author:
- Sam Jaques and Mizanur Rahaman
bibliography:
- 'jaques-rahaman.bib'
title: Spectral Properties of Tensor Products of Channels
---
[^1]
Introduction
============
If we have a linear map acting on a matrix algebra that can be expressed as a tensor product of matrix algebras, and the map itself can be expressed as a tensor product of two other linear maps, there may be few similarities between the constituent maps and the larger linear map they produce. If we restrict ourselves to special classes of linear maps and special domains of matrix algebras, then the tensor product adds no extra complexity. Our goal in this paper is to use the multiplicative domain to characterize some of these properties for trace-preserving, completely positive maps on matrix algebras. These maps are also known as quantum channels, which we refer to as channels.
The multiplicative domain of a linear map ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ is the set of all matrices $x\in {\mathcal{M}}_d$ such that, for all $y\in {\mathcal{M}}_d$, ${{\mathcal{E}}}(xy)={{\mathcal{E}}}(x){{\mathcal{E}}}(y)$ and ${{\mathcal{E}}}(yx)={{\mathcal{E}}}(y){{\mathcal{E}}}(x)$. When the linear map is completely positive, this specific set has received much attention in operator theory and operator algebras ([@choi1], [@paulsen]-chapter 4, [@stormer]-section 2.1). In this context, it characterizes certain distinguishability measures. A completely positive linear map acts like a homomorphism on the multiplicative domain, and hence studying this domain can reveal structure and properties of the linear map.
In quantum information theory ([@johnston], [@choi2], [@kribs-spekkens]), the multiplicative domain contains the unitarily correctable codes and noiseless subsystems. Studying the multiplicative domain of tensor products sheds light on error correction in bipartite systems.
It turns out that we can capture most of the spectral properties of the tensor product of channels simply by investigating the multiplicative behavior. Note that the spectral properties of a channel acting on one copy of a quantum system have been well explored ([@wolf], [@ergodic], [@evans-krohn], [@inverse-eigenvalue]) for various purposes, mainly in an effort to understand the dynamics of a system evolving through quantum measurements. In quantum dynamical systems, the ergodicity of a channel [@ergodic] and its decoherence-free subspaces [@veronica] are important spectral properties . When the underlying domain is a bipartite system, the spectral properties of product channels can be hard to analyze, but we can use the multiplicative domain as a tool to understand them.
As in previous work on quantum error correction (e.g., [@johnston][@kribs-spekkens]), we restrict our focus to unital channels because the multiplicative domain has less structure in non-unital channels. In particular, the multiplicative domain of a unital channel can be described using the Kraus operators. Using this, we can characterize certain channels and derive facts beyond the multiplicative structure.
The paper is organized as follows: firstly, in Section \[sec:mutiplicatve\_domain\_of\_products\], we show that the multiplicative domain of a tensor product of unital channels “splits" nicely with the tensor product. We use this to prove that the peripheral spectra of two unital channels will precisely determine whether the fixed points of their tensor product will also split or not. This analysis provides the necessary and sufficient condition on when the tensor product of two ergodic (or primitive) channels is again ergodic (or primitive). Here we recapture some of the results obtained by [@Luczak], [@Watanabe] in a very different way based on the analysis of multiplicative domain.
Since [@miza] showed that repeated applications of a finite-dimensional channel produces a chain in the lattice of unital \*-subalgebras of ${\mathcal{M}}_d$, we characterize such algebras in Section \[sec:MUPSAS\]. This provides an easy way to enumerate the lattice of unital \*-subalgebras of ${\mathcal{M}}_d$, as well as providing a limit on the length of chains in the lattice that is linear in the dimension. This finding can be of independent interest because it provides a finer analysis of the structure of unital \*-subalgebras in ${\mathcal{M}}_d$. In turn, this allows us to use the multiplicative index, introduced in [@miza], to show that certain channels cannot be product channels. We give examples of channels with large multiplicative indices in Sections \[sec:ETB\_channels\] and \[sec:Schur\_channels\], thus showing that these cannot be product channels.
Next, in Section \[Sec:strictly-contractive\], we consider channels which are strictly contractive with respect to some distinguishability measures that frequently arise in information theory. Using our results in the previous sections we prove that the tensor product of two strictly contractive channels with respect to certain distinguishability measures is again strictly contractive provided the measures allow recovery maps. We make use of the reversibility and monotonicity properties of these measures under channels, which is a wide topic of current research ([@Jencova1], [@Jencova2], [@f-div1], [@f-div2]).
As a final application, we show that unitary-correctable quantum codes (UCC) gain nothing through tensor products.
Background and Notation
-----------------------
Throughout this paper we will use the following notation:
- ${{\mathcal{E}}},\Psi$ will refer to quantum channels, that is, completely positive, trace-preserving linear operators from $B({\mathcal{H}})$ to $B({\mathcal{H}})$ for some finite dimensional Hilbert space ${\mathcal{H}}$. In this paper we identify $B({\mathcal{H}})$ with ${\mathcal{M}}_d$, the $d\times d$ complex matrices. It is well known that a quantum channel ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow{\mathcal{M}}_d$ is always represented by a set of (non-unique) Kraus operators $\{a_{j}\}_{j=1}^{n}$ in ${\mathcal{M}}_d$ such that for all $x\in {\mathcal{M}}_d$, we have $${{\mathcal{E}}}(x)=\displaystyle\sum_{j=1}^{n}a_{j}xa_{j}^*,$$ where $\displaystyle\sum_{j=1}^{n}a_{j}^*a_{j}=1$. Here $1$ represents the identity matrix in ${\mathcal{M}}_d$. In the dual picture ${{\mathcal{E}}}$ is realized as a unital completely positive map and denoted by ${{\mathcal{E}}}^*$ acting on ${\mathcal{M}}_d$ again and satisfying the relation $${{\text{Tr}}}({{\mathcal{E}}}(x)y)={{\text{Tr}}}(x{{\mathcal{E}}}^*(y)),$$ for every $x,y\in {\mathcal{M}}_d$. This linear map ${{\mathcal{E}}}^*$ is the adjoint of ${{\mathcal{E}}}$ with respect to the Hilbert-Schmidt inner product which is defined as $\langle a,b\rangle_{HS}={\text{Tr}}(ab^*)$, for all $a,b\in {\mathcal{M}}_d$.
An important note is that many papers work in the dual framework, where a quantum channel is necessarily unital but may not be trace-preserving. Hence, these papers refer to these maps as unital channels, or UCP maps. In our work, where all channels are trace-preserving, unitality is an extra condition that limits our results to a particular subset of quantum channels.
- Lowercase letters from the end of the Latin alphabets, $x,y,z$, will refer to matrices in ${\mathcal{M}}_d$. The letters $p,q$ will refer to projections in ${\mathcal{M}}_d$.
- Greek letters, $\varphi,\zeta$ will refer to either vectors in ${\mathbb{C}}^d$ or partitions of $\{1,\cdots,n\}$ for $n\in{\mathbb{N}}$. We will use $[n]$ to denote the set $\{1,\cdots,n\}$.
- Stylized letters from the beginning of the Latin alphabet, ${\mathscr{A}},{\mathscr{B}},\mathscr{C}$, will refer to sub-algebras of ${\mathcal{M}}_d$. For a set $S$, the algebra generated by $S$ will be denoted $\text{alg}(S)$ and the \*-algebra generated by $S$ will be denoted $\text{alg}^*(S)$.
- For a quantum channel ${{\mathcal{E}}}$, ${\mathcal{M}}_{{\mathcal{E}}}$ denotes the multiplicative domain and also $\rm{Fix}_{{{\mathcal{E}}}}$ denotes the set of fixed points of ${{\mathcal{E}}}$, that is, $$\rm{Fix}_{{{\mathcal{E}}}}=\{a\in {\mathcal{M}}_d \ | \ {{\mathcal{E}}}(a)=a\}.$$
There are a number of useful characterizations of the multiplicative domain we will use extensively.
\[choi\] For a unital completely positive map ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$, the multiplicative domain ${\mathcal{M}}_{{\mathcal{E}}}$ is a C$^*$-subalgebra of ${\mathcal{M}}_d$ and moreover, it is equal to the following set: $${\mathcal{M}}_{{\mathcal{E}}}=\{x\in {\mathcal{M}}_d \ | \ {{\mathcal{E}}}(x^*x)={{\mathcal{E}}}(x)^*{{\mathcal{E}}}(x), \ {{\mathcal{E}}}(xx^*)={{\mathcal{E}}}(x){{\mathcal{E}}}(x^*)\}.$$
The following theorem is also useful in describing the multiplicative domain of a unital channel.
\[kribs-spekkens\] For a unital channel ${{\mathcal{E}}}$, we have the relation $${\mathcal{M}}_{{\mathcal{E}}}=\rm{Fix}_{{{\mathcal{E}}}^*\circ{{\mathcal{E}}}}.$$ Here ${{\mathcal{E}}}^*$ is the adjoint of ${{\mathcal{E}}}$
The next theorem connects the fixed points set and the Kraus operators of a channel.
\[cummutant-fixed pnt\] Let ${{\mathcal{E}}}$ be a unital channel represented as ${{\mathcal{E}}}(x)=\displaystyle\sum_{j=1}^{n}a_{j}xa_{j}^*$. Then the fixed point set $\rm{Fix}_{{\mathcal{E}}}$ is an algebra and it equals to the commutant of the \*-algebra (${\mathscr{A}}$) generated by $\{a_1,\cdots, a_{n}\}$. That is $$\rm{Fix}_{{\mathcal{E}}}={\mathscr{A}}',$$ where ${\mathscr{A}}'$ represents the commutant of the algebra ${\mathscr{A}}$.
It follows that ${\mathcal{M}}_{{\mathcal{E}}}$ is a \*-closed subalgebra of ${\mathcal{M}}_d$ containing the fixed points of ${{\mathcal{E}}}$. As with all finite \*-algebras, it is generated by a set of projections. For any projection $p\in {\mathcal{M}}_{{\mathcal{E}}}$, ${{\mathcal{E}}}(p)$ is a projection of the same rank, and $1-p$ is also in ${\mathcal{M}}_{{\mathcal{E}}}$. We say that ${\mathcal{M}}_{{\mathcal{E}}}$ is *trivial* if ${\mathcal{M}}_{{\mathcal{E}}}={\mathbb{C}}1$; if ${\mathcal{M}}_{{\mathcal{E}}}$ is non-trivial, then it must contain at least two orthogonal projections.
For any unital channel ${{\mathcal{E}}}$ and any $k\in{\mathbb{N}}$, ${\mathcal{M}}_{{{\mathcal{E}}}^{k+1}}\subseteq {\mathcal{M}}_{{{\mathcal{E}}}^k}$ [@miza], and hence there is some $N\in{\mathbb{N}}$ such that for any $n\geq N$, ${\mathcal{M}}_{{{\mathcal{E}}}^n}={\mathcal{M}}_{{{\mathcal{E}}}^N}$. Following [@miza], we denote this algebra ${\mathcal{M}}_{{{\mathcal{E}}}^\infty}$ and refer to it as the *stabilized multiplicative domain* of ${{\mathcal{E}}}$.
The *multiplicative index* of a unital quantum channel ${{\mathcal{E}}}$ is the minimum $n\in{\mathbb{N}}$ such that ${\mathcal{M}}_{{{\mathcal{E}}}^n}={\mathcal{M}}_{{{\mathcal{E}}}^\infty}$.
We denote the multiplicative index of ${{\mathcal{E}}}$ by $\kappa({{\mathcal{E}}})$. Another useful result is Lemma 2.2 from [@miza]:
\[lem:md\_composition\] If ${{\mathcal{E}}}_1,{{\mathcal{E}}}_2$ are two unital quantum channels, then $${\mathcal{M}}_{{{\mathcal{E}}}_2\circ{{\mathcal{E}}}_1}=\{x\in {\mathcal{M}}_{{{\mathcal{E}}}_1} \ | \ {{\mathcal{E}}}_1(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_2}\}.$$
The Multiplicative Domain of Product Channels {#sec:mutiplicatve_domain_of_products}
=============================================
Splitting problem for subalgebras in tensor product
---------------------------------------------------
The splitting problem for a von Neumann subalgebra (or a C$^*$-subalgebra) of a tensor product of algebras has remained one of the most important problems in operator algebra. One of the early results that drew a lot of attention on this problem is due to L. Ge and R. Kadison:
[(Ge-Kadison, 1996, [@Ge-Kadison])]{} Let $\mathcal{M}, \mathcal{N}$ be two von Neumann algebras and assume that $\mathcal{M}$ is a factor. If $\mathcal{A}\subseteq \mathcal{M}\bar{\otimes}\mathcal{N}$ is a subalgebra that contains $\mathcal{M}\otimes \mathbb{C}1$, then $$\mathcal{A}=\mathcal{M}\otimes \mathcal{B},$$ for some von Neumann subalgebra $\mathcal{B}$ of $\mathcal{N}$.
There have been a lot of improvements and new research into the splitting problem. See [@splitting1], [@splitting2],[@splitting3],[@splitting4] for more information on this area.
Here we examine the multiplicative domain of the tensor product of two channels ${{\mathcal{E}}}_1,{{\mathcal{E}}}_2$ acting on $\mathcal{M}_{d}$ and $\mathcal{M}_{d'}$ separately. Since the multiplicative domain $\mathcal{M}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}$ is a C$^*$-subalgebra of $\mathcal{M}_{d}\otimes\mathcal{M}_{d'}$, it is natural to ask whether this subalgebra splits into tensor product of two subalgebras. We show that for unital channels, the multiplicative domain is unchanged by the tensor product. To prove this claim we need the following lemma. For our purposes, if ${\mathscr{A}}_1$ and ${\mathscr{A}}_2$ are two algebras and $S\subseteq{\mathscr{A}}_1$ and $R\subseteq{\mathscr{A}}_2$ are two sets, possibly without any algebraic structure themselves, then $S\otimes R$ is defined as $\{s\otimes r | s\in S,r\in R\}$.
\[lem:proj\_tensor\] Let $S\subseteq {\mathcal{M}}_d$ and $R\subseteq {\mathcal{M}}_{d'}$. If, for every $s\in S$, there is a projection $p\in {\rm{span}}(S)$ such that $ps=s$, and for every $r\in R$ there is a projection $q\in {\rm{span}}(R)$ such that $rq=r$, then ${\rm{alg}}(S)\otimes {\rm{alg}}(R)={\rm{alg}}(S\otimes R)$. If there also exist such $p,q$ for for all $s\in S^*=\{s^*|s\in S\}$ and all $r\in R^*$, then ${\rm{alg}}^*(S)\otimes{\rm{alg}}^*(R)={\rm{alg}}^*(S\otimes R)$.
Note that for any two sets $S$ and $R$, $S\subseteq {\rm{alg}}(S)$ and $R\subseteq {\rm{alg}}(R)$, so $S\otimes R\subseteq {\rm{alg}}(S)\otimes {\rm{alg}}(R)$, so ${\rm{alg}}(S\otimes R)\subseteq {\rm{alg}}(S)\otimes {\rm{alg}}(R)$.
For the reverse inclusion, free products of elements of $S$ will span ${\rm{alg}}(S)$, and free products of elements of $R$ will span ${\rm{alg}}(R)$. The tensor product of spanning sets is a spanning set of the tensor product, so elements of the form $s_1\cdots s_n\otimes r_1\cdots r_m$, with $s_i\in S$ and $r_j\in R$, will span ${\rm{alg}}(S)\otimes{\rm{alg}}(R)$. We take an arbitrary element of this form, $s_1\cdots s_n\otimes r_1\cdots r_m$, and then take $p\in {\rm{span}}(S)$ such that $ps_1=s_1$ and $q\in {\rm{span}}(R)$ such that $r_mq=r_m$. Then: $$\begin{aligned}
s_1\cdots s_n\otimes r_1\cdots r_m=&ps_1\cdots s_n\otimes r_1\cdots r_mq\\
=&(p\otimes r_1\cdots r_m)(s_1\cdots s_n\otimes q)\\
=&(p\otimes r_1)(p\otimes r_2)\cdots (p\otimes r_m)(s_1\otimes q)(s_2\otimes q)\cdots (s_n\otimes q)
\end{aligned}$$ Since $p\in {\rm{span}}(S)$, then there are elements $\{s'_j\}_{j=1}^n$ in $S$ such that $p=\sum_{j=1}^na_js'_j$. But for any $r_i$, $s'_j\otimes r_i\in S\otimes R$, so the sum $\sum a_j(s'_j\otimes r_i)=p\otimes r_i$ is in ${\rm{alg}}(S\otimes R)$. Similarly, $s_i\otimes q\in {\rm{alg}}(S\otimes R)$. Thus, the product above is also in ${\rm{alg}}(S\otimes R)$, and thus all the basis elements of ${\rm{alg}}(S)\otimes {\rm{alg}}(R)$ are in ${\rm{alg}}(S\otimes R)$, so ${\rm{alg}}(S)\otimes {\rm{alg}}(R)\subseteq alg(S\otimes R)$.
For the \*-algebras, a very similar logic holds. Free products of the form $s_1\cdots s_n\otimes r_1\cdots r_m$, $s_i\in S\cup S^*$ and $r_i\in R\cup R^*$, will span ${\rm{alg}}^*(S)\otimes{\rm{alg}}^*(R)$. Take and arbitrary element of this form and let $p$ and $q$ be projections defined as before, i.e., $ps_1=s_1$ and $r_mq=r_m$. Suppose $r_i$ is in $R^*$, so $r_i=\tilde{r}_i^*$, with $\tilde{r}_i\in R$. Projections are self-adjoint, so $p\otimes r_i=p^*\otimes \tilde{r}_i^*=(p\otimes \tilde{r}_i)^*$ is in ${\rm{alg}}^*(S\otimes R)$. Similarly, if $s_i=\tilde{s}_i^*$ is in $S^*$, then $s_i\otimes q=(\tilde{s}_i\otimes q)^*\in {\rm{alg}}^*(S\otimes R)$. Thus the same decomposition can be done as the one above: $$s_1\cdots s_n\otimes r_1\cdots r_m=(s_1\otimes q)\cdots (s_n\otimes q)(p\otimes r_1)\cdots (p\otimes r_m)$$ And since all of the terms on the right-hand side are in ${\rm{alg}}^*(S\otimes R)$, then ${\rm{alg}}^*(S)\otimes{\rm{alg}}^*(R)={\rm{alg}}^*(S\otimes R)$.
\[thm:md\_tensor\] For any two unital quantum channels ${{\mathcal{E}}}_1,{{\mathcal{E}}}_2$, $${\mathcal{M}}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}={\mathcal{M}}_{{{\mathcal{E}}}_1}\otimes {\mathcal{M}}_{{{\mathcal{E}}}_2}.$$
Let ${{\mathcal{E}}}_1(x)=\displaystyle\sum_{i=1}^ma_ixa_i^*$ and ${{\mathcal{E}}}_2(x)=\displaystyle\sum_{i=1}^nb_ixb_i^*$ be the Kraus decomposition of ${{\mathcal{E}}}_1$ and ${{\mathcal{E}}}_2$ respectively. Trace preservation implies $1=\displaystyle\sum_{i=1}^m a_i^*a_i=\displaystyle\sum_{j=1}^{n}b_{j}^*b_{j}$.
The Kraus operators of ${{\mathcal{E}}}_1^*\circ{{\mathcal{E}}}_1$ are $\{a_i^*a_j\}$ for any $i,j$. Define $S=\{a_i^*a_j|1\leq i,j\leq m\}$. Similarly, let $R=\{b_i^*b_j|1\leq i,j\leq n\}$ be the set Kraus operators of ${{\mathcal{E}}}_2^*\circ{{\mathcal{E}}}_2$. Then the Kraus operators of ${{\mathcal{E}}}_1^*\circ{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2^*\circ{{\mathcal{E}}}_2 (=({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)^*\circ({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2))$ are $\{a_i^*a_j\otimes b_k^*b_l:1\leq i,j\leq m \ \text{and} \ 1\leq k,l\leq n\}$, or $S\otimes R$. Since $1\in {\rm{span}}(S)$ and $1\in {\rm{span}}(R)$, we have the necessary projections to use Lemma \[lem:proj\_tensor\]. Hence we have that $${\rm{alg}}^*(S\otimes R)={\rm{alg}}^*(S)\otimes {\rm{alg}}^*(R).$$ Now the finite dimensional \*-algebras are von Neumann algebras and by the commutant-tensor product theorem for von Neumann algebras ( [@kadison-ringroseII], Theorem 11.2.16) we have that $${\rm{alg}}^*(S\otimes R)'={{\rm{alg}}^*(S)}'\otimes {{\rm{alg}}^*(R)}'.$$ Then by Theorem \[cummutant-fixed pnt\] ${\rm{alg}}^*(S\otimes R)'=\rm{Fix}_{({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)^*\circ({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)}$ and ${\rm{alg}}^*(S)'=\rm{Fix}_{{{\mathcal{E}}}_1^*\circ{{\mathcal{E}}}_1}$ and ${\rm{alg}}^*(R)'=\rm{Fix}_{{{\mathcal{E}}}_2^*\circ{{\mathcal{E}}}_2}$, thus $$\rm{Fix}_{({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)^*\circ({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)}=\rm{Fix}_{{{\mathcal{E}}}_1^*\circ{{\mathcal{E}}}_1}\otimes \rm{Fix}_{{{\mathcal{E}}}_2^*\circ{{\mathcal{E}}}_2}.$$ Now invoking Theorem \[kribs-spekkens\] and noting that ${\mathcal{M}}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}=\rm{Fix}_{({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)^*\circ({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)}$ we immediately obtain $${\mathcal{M}}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}={\mathcal{M}}_{{{\mathcal{E}}}_1}\otimes {\mathcal{M}}_{{{\mathcal{E}}}_2}.$$
Since the multiplicative domain behaves well with the tensor product, it leads to a simple form for the multiplicative index:
\[prop:kappa\_tensor\_bound\] Given two unital channels ${{\mathcal{E}}}_1:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d,{{\mathcal{E}}}_2:{\mathcal{M}}_{d'}\rightarrow {\mathcal{M}}_{d'}$, then $\kappa({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)=\max\{\kappa({{\mathcal{E}}}_1),\kappa({{\mathcal{E}}}_2)\}$ (where $\kappa$ is the multiplicative index).
If $k\geq \max\{\kappa({{\mathcal{E}}}_1),\kappa({{\mathcal{E}}}_2)\}(=:\kappa_{\max})$, then: $${\mathcal{M}}_{({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)^k}={\mathcal{M}}_{{{\mathcal{E}}}_1^k\otimes{{\mathcal{E}}}_2^k}={\mathcal{M}}_{{{\mathcal{E}}}_1^k}\otimes {\mathcal{M}}_{{{\mathcal{E}}}_2^k}={\mathcal{M}}_{{{\mathcal{E}}}_1^\infty}\otimes {\mathcal{M}}_{{{\mathcal{E}}}_2^\infty}.$$ That is, the multiplicative domain is constant after $\kappa_{\max}$, so $\kappa({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)\leq\kappa_{\max}$. Then suppose $k<\kappa_{\max}$ (and, without loss of generality, suppose $\kappa({{\mathcal{E}}}_2)=\kappa_{\max}$). By a similar logic: $${\mathcal{M}}_{({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)^k}={\mathcal{M}}_{{{\mathcal{E}}}_1^k}\otimes {\mathcal{M}}_{{{\mathcal{E}}}_2^k}\subsetneq {\mathcal{M}}_{{{\mathcal{E}}}_1^{k+1}}\otimes {\mathcal{M}}_{{{\mathcal{E}}}_2^{k+1}}={\mathcal{M}}_{({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)^{k+1}}.$$ Since the multiplicative domain is still strictly decreasing with $k$, then $\kappa({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)>k$ and the result follows.
The above proposition implies the following corollary:
\[splitting stabilising algebra\] For unital channels ${{\mathcal{E}}}_1,{{\mathcal{E}}}_2$ we have $${\mathcal{M}}_{{({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)}^{\infty}}={\mathcal{M}}_{{{\mathcal{E}}}_1^\infty}\otimes{\mathcal{M}}_{{{\mathcal{E}}}_2^\infty}.$$
Fixed Points of Product Channels {#sec:fixed_points_of_products}
--------------------------------
For a unital channel ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$, the fixed point set $\rm{Fix}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}$ is a subalgebra of ${\mathcal{M}}_d\otimes\mathcal{M}_{d'}$ and unlike the multiplicative domain case, this subalgebra does not split nicely. However, using Theorem \[thm:md\_tensor\], we can provide an exact description of this algebra and characterize when this subalgebra splits and recapture the result of [@Luczak]. Our results are specific cases of [@Watanabe] and [@Luczak], but through a vastly different approach. The spectrum of the tensor product of two channels is known to be the set product of the two spectra, but this theorem characterizes the eigen operators as only the obvious choices. In what follows $\mathbb{T}$ represents the unit circle in the complex plane. Note that (see [@wolf]) for any quantum channel ${{\mathcal{E}}}$, all the eigenvalues lie in the closed unit disc of the complex plane. We define the spectrum ($\rm{Spec}_{{{\mathcal{E}}}}$) of ${{\mathcal{E}}}$ as follows $$\rm{Spec}_{{{\mathcal{E}}}}=\{\lambda\in \mathbb{C} \ | \ (\lambda 1-{{\mathcal{E}}}) \ \text{is \ not \ \ invertible \ on} \ {\mathcal{M}}_d \},$$ where $1$ is the identity operator on ${\mathcal{M}_{d}}$. The set $\rm{Spec}_{{{\mathcal{E}}}}\cap \mathbb{T}$ is called the *peripheral eigenvalues* and the corresponding eigenoperators are called *peripheral eigenvectors*.
\[thm:fixed\_point\_products\] Let ${{\mathcal{E}}}_1:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d,{{\mathcal{E}}}_2:{\mathcal{M}}_{d'}\rightarrow {\mathcal{M}}_{d'}$ be two unital quantum channels. Then for any $\lambda\in \mathbb{T}$: $${\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}z \in {\mathcal{M}}_d\otimes {\mathcal{M}}_{d'} \midsetr {{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2(z)=\lambda z\rightset = {\rm{span}}{\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}x_i\otimes y_i \midsetr {{\mathcal{E}}}_1(x_i)=\mu_1 x_i,{{\mathcal{E}}}_2(y_i)=\frac{\lambda}{\mu_1} y_i\rightset.$$
Let $\lambda\in \mathbb{T}$ . For the left inclusion, suppose there are two numbers $\mu_1,\mu_2$ such that ${{\mathcal{E}}}_1(x)=\mu_1 x$ and ${{\mathcal{E}}}_2(y)=\mu_2 y$ for matrices $x,y$ and $\lambda=\mu_1\mu_2$. Then $${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2(x\otimes y)=(\mu_1 x)\otimes (\mu_2 y)=\lambda (x\otimes y).$$ For the right inclusion, let $z$ be a matrix such that ${{\mathcal{E}}}_1\otimes {{\mathcal{E}}}_2(z)=\lambda z$. By Theorem 2.5 from [@miza], we know that the peripheral eigenvectors of a channel are precisely the stabilized multiplicative domain. Thus: $$z\in {\mathcal{M}}_{({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)^\infty}={\mathcal{M}}_{{{\mathcal{E}}}_1^\infty}\otimes {\mathcal{M}}_{{{\mathcal{E}}}_2^\infty}.$$ We can then represent $z$ as $z=\sum_{i=1}^mx_i\otimes y_i$, where $x_i\in {\mathcal{M}}_{{{\mathcal{E}}}_1^\infty}$ and $y_i\in {\mathcal{M}}_{{{\mathcal{E}}}_2^\infty}$. By the same theorem, we know that $x_i$ is a linear combination of peripheral eigenvectors of ${{\mathcal{E}}}_1$. Thus we can further decompose $z$ as $$z=\sum_{i=1}^{m'}x_i'\otimes y_i$$ where the $\{x_i'\}$ are linearly independent and ${{\mathcal{E}}}_1(x_i')=\mu_ix_i'$ with $\mu_i\in\mathbb{T}$. This gives us: $${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2(z)=\sum_{i=1}^{m'}\mu_i x_i'\otimes {{\mathcal{E}}}_2(y_i).$$ But by choice of $z$, ${{\mathcal{E}}}_1\otimes {{\mathcal{E}}}_2(z)=\lambda z = \lambda \sum_{i=1}^{m'}x_i'\otimes y_i$. By the linear independence of $\{x_i'\}$, we have that $\lambda y_i=\mu_i{{\mathcal{E}}}_2(y_i)$, i.e., ${{\mathcal{E}}}_2(y_i)=\frac{\lambda}{\mu_i}y_i$. This holds for all $i$, giving the required inclusion.
Using the above theorem we obtain the following corollary which first appeared in [@Luczak], Corollary 13 in a more general context. However our method of obtaining this result is significantly different from [@Luczak].
\[cor:fixed\_point\_split\] For two unital channels ${{\mathcal{E}}}_1$ and ${{\mathcal{E}}}_2$ with spectra $\rm{Spec}_{{{\mathcal{E}}}_1}$ and $\rm{Spec}_{{{\mathcal{E}}}_2}$ respectively, the fixed point algebra splits if and only if the intersection of the peripheral spectra is trivial. That is, $$\rm{Fix}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}=\rm{Fix}_{{{\mathcal{E}}}_1}\otimes
\rm{Fix}_{{{\mathcal{E}}}_2},$$ if and only if $\rm{Spec}_{{{\mathcal{E}}}_1}\cap \rm{Spec}_{{{\mathcal{E}}}_2}\cap\mathbb{T}
=\{1\}$.
The fixed points are the special case of peripheral eigen-operators where $\lambda=1$. Using Theorem \[thm:fixed\_point\_products\], we have that the fixed points are given by $$\rm{Fix}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}=\text{span}{\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}x_i\otimes y_i\midsetr {{\mathcal{E}}}_1(x_i)=\mu x_i,{{\mathcal{E}}}_2(y_i)=\overline{\mu}y_i,\vert\mu\vert=1\rightset.$$ This set will equal $\rm{Fix}_{{{\mathcal{E}}}_1}\otimes\rm{Fix}_{{{\mathcal{E}}}_2}$ if and only if there is no $\mu\in\rm{Spec}_{{{\mathcal{E}}}_1}\cap\mathbb{T}$ with $\overline{\mu}\in\rm{Spec}_{{{\mathcal{E}}}_2}\cap\mathbb{T}$. Since the spectrum of a quantum channel is closed under conjugation, this means $\mu$ would need to be in both spectra. Thus, the spectrum will split if and only if the intersection of the spectra is trivial.
Theorem \[thm:fixed\_point\_products\] is particularly helpful to analyze the ergodicity or irreducibility of tensor product of quantum channels. We provide the definition of such channels below:
A channel ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ is called irreducible if there is no non-trivial projection $p\in {\mathcal{M}}_d$ such that ${{\mathcal{E}}}(p)\leq \lambda p$, for $\lambda>0$.
An irreducible channel ${{\mathcal{E}}}$ is called primitive if the set of peripheral eigenvalues contains only 1, that is if $\rm{Spec}_{{\mathcal{E}}}\cap \mathbb{T}=\{1\}$.
We note down some properties of irreducible positive linear maps:
[[(see [@evans-krohn])]{}]{}\[thm:evans-krohn\] Let ${{\mathcal{E}}}$ be a positive linear map on ${\mathcal{M}_{d}}$ and let $r$ be its spectral radius. Then
1. There is a non zero positive element $x \in {\mathcal{M}}_d$ such that ${{\mathcal{E}}}(x)=rx$
2. If ${{\mathcal{E}}}$ is irreducible and if a positive $y \in {{\mathcal{M}_{d}}}$ is an eigenvector of ${{\mathcal{E}}}$ corresponding to some eigenvalue $s$ of ${{\mathcal{E}}}$, then $s = r$ and $y$ is a positive scalar multiple of $x$.
3. If ${{\mathcal{E}}}$ is unital, irreducible and satisfies the Schwarz inequality for positive linear maps then
- $r=1$ and $\rm{Fix}_{{{\mathcal{E}}}}=\mathbb{C}1$.
- Every peripheral eigenvalue $\lambda \in \rm{Spec_{{{\mathcal{E}}}}\cap \mathbb{T}} $ is simple and the corresponding eigenspace is spanned by a unitary $u_{\lambda}$ which satisfies ${{\mathcal{E}}}(u_{\lambda}x)=\lambda u_{\lambda}{{\mathcal{E}}}(x)$, for all $x\in {\mathcal{M}_{d}}$.
- The set $\Gamma=\rm{Spec_{{{\mathcal{E}}}}\cap\mathbb{T}}$ is a cyclic subgroup of the group $\mathbb{T}$ and the corresponding eigenvectors form a cyclic group which is isomorphic to $\Gamma$ under the isomorphism $\lambda \rightarrow u_{\lambda}$.
Often irreducible channels are called *ergodic channels*. Ergodic/irreducible positive maps have been a great topic of interest (see [@evans-krohn], [@farenick], [@ergodic]). The study of such maps enriched the analysis of non-commutative Perron-Frobenius theory. Although ergodicity of a quantum dynamical system (discrete or continuous) has received much attention, the same analysis in the tensor product framework has been talked about less except [@Watanabe] and [@Luczak]. Here we present necessary and sufficient conditions for a channel to be irreducible and primitive in the tensor product system. By the aid of Theorem \[thm:fixed\_point\_products\] we recapture Theorem 5.3 in [@Watanabe].
Let ${{\mathcal{E}}}_1$ be an irreducible unital quantum channel with $n$ peripheral eigenvalues $\Gamma_n$. Then:
1. The product ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_1$ is irreducible if and only if ${{\mathcal{E}}}_1$ is also primitive, in which case ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_1$ is also primitive.
2. For any primitive unital channel ${{\mathcal{E}}}_2$, ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ is irreducible.
3. If ${{\mathcal{E}}}_2$ is irreducible with $m$ peripheral eigenvalues $\Gamma_m$, then ${{\mathcal{E}}}_1\otimes {{\mathcal{E}}}_2$ is irreducible if and only if $gcd(n,m)=1$.
\(1) For ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_1$ to be irreducible, its fixed points would need to be ${\mathbb{C}}(1\otimes 1)$, meaning the fixed points would have to split. By Corollary \[cor:fixed\_point\_split\], this would occur if and and only if the peripheral spectrum of ${{\mathcal{E}}}_1$ is trivial, meaning ${{\mathcal{E}}}_1$ is primitive. Since the spectrum of a quantum channel is contained in the unit disc, in this case the peripheral spectrum of ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_1$ will still be trivial and thus it will be primitive.
\(2) Since ${{\mathcal{E}}}_2$ is primitive, its only eigenvalue is 1 with eigenvector $1$. Thus the fixed points of ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ will split, and since both fixed point algebras are trivial, the product will also be trivial.
For item (3), if $gcd(n,m)=1$, then the two cyclic groups $\Gamma_n, \Gamma_m$ intersect trivially and hence by Corollary \[cor:fixed\_point\_split\] we get $\rm{Fix}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}=\rm{Fix}_{{{\mathcal{E}}}_1}\otimes\rm{Fix}
_{{{\mathcal{E}}}_2}=\mathbb{C}1\otimes\mathbb{C}1=\mathbb{C}(1\otimes 1)$. Conversely, if ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ is irreducible, then $\rm{Fix}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}=\mathbb{C}1$. From Theorem \[thm:evans-krohn\] we know that the peripheral spectrum of ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ is a cyclic subgroup of some order $N$. Since $\rm{Fix}_{{{\mathcal{E}}}_1}=\mathbb{C}1=\rm{Fix_{{{\mathcal{E}}}_2}}$, it is evident that this can only happen if the fixed point algebra splits. By Corollary \[cor:fixed\_point\_split\] again we conclude that $\Gamma_n\cap\Gamma_m=\{1\}$; that is, $gcd(n,m)=1$.
Theorem \[thm:fixed\_point\_products\] gives structure to the eigenspaces of these eigenvalues. For some intuition on this, a channel acts like an automorphism on its stabilized multiplicative domain, so in some sense it is “normal” on this subalgebra. The eigenspaces of the tensor product of two normal matrices will simply be the products of the original eigenspaces, and here something similar holds for the “normal part” of the channel.
Restrictions on the Multiplicative Index {#sec:restrictions_on_kappa}
========================================
Maximal Unital Proper \*-Subalgebras (MUPSAs) {#sec:MUPSAS}
---------------------------------------------
Proposition \[prop:kappa\_tensor\_bound\] restricts which channels can be product channels, since the multiplicative index must be the same as the multiplicative index of one of the channels in the product. Our goal is thus to restrict the possible values of the multiplicative index. An obvious bound is the dimension of the matrix algebra, $d^2$, but in fact we can do much better by looking at chains of maximal unital proper \*-subalgebras, defined in the obvious way as follows:
An algebra ${\mathscr{A}}$ is a maximal unital proper \*-subalgebra (for convenience, a “MUPSA") of a C\*-algebra ${\mathscr{B}}$ if ${\mathscr{A}}$ is unital proper \*-subalgebra of ${\mathscr{B}}$ (meaning ${\mathscr{A}}\neq {\mathscr{B}}$, $1\in{\mathscr{A}}$, and ${\mathscr{A}}^*={\mathscr{A}}$) such that if $\tilde{{\mathscr{A}}}$ is another unital proper \*-algebra with ${\mathscr{A}}\subseteq \tilde{{\mathscr{A}}}\subseteq{\mathscr{B}}$, then either $\tilde{{\mathscr{A}}}={\mathscr{B}}$ or $\tilde{{\mathscr{A}}}={\mathscr{A}}$.
While there are many possible forms of a subalgebra of ${\mathcal{M}}_d$, restricting to MUPSAs allows us to precisely characterize their structure, up to isomorphism. We use the Wedderburn decomposition extensively. For a matrix algebra ${\mathscr{A}}$, one can always decompose it as $${\mathscr{A}}\cong \bigoplus_{r=1}^m {\mathcal{M}}_{n_r}\otimes 1_{k_r}.$$ This is the Wedderburn decomposition.
\[lem:MdMUPSA\] If ${\mathscr{A}}$ is a MUPSA of ${\mathcal{M}}_d$, then (up to isomorphism) ${\mathscr{A}}={\mathcal{M}}_{d-r}\oplus {\mathcal{M}}_r$, where $1\leq r\leq d-1$.
Let ${\mathscr{A}}$ be a \*-subalgebra of ${\mathcal{M}}_d$. Then let $${\mathscr{A}}=\bigoplus_{r=1}^m{\mathcal{M}}_{n_r}\otimes 1_{k_r}$$ be the Wedderburn decomposition of ${\mathscr{A}}$. If $m\geq 3$, then the following subalgebra $$\tilde{{\mathscr{A}}}={\mathcal{M}}_{n_1}\otimes 1_{k_1}\oplus {\mathcal{M}}_{\sum_{r=2}^mn_rk_r}$$ will strictly contain ${\mathscr{A}}$, but be strictly contained in ${\mathcal{M}}_d$, contradicting the maximality of ${\mathscr{A}}$. If $m=1$, then ${\mathscr{A}}={\mathcal{M}}_{d/p}\otimes 1_p$ (for some number $p$ dividing $d$). Then ${\mathscr{A}}\subsetneq {\mathcal{M}}_{d/p}\oplus {\mathcal{M}}_{d(p-1)/p}$, contradicting maximality of ${\mathscr{A}}$. Thus $m=2$, and ${\mathscr{A}}={\mathcal{M}}_{n_1}\otimes 1_{k_1}\oplus {\mathcal{M}}_{n_2}\otimes 1_{k_2}$. If $k_1>1$, then ${\mathscr{A}}$ is a proper subalgebra of $$\underbrace{{\mathcal{M}}_{n_1}\oplus\cdots\oplus {\mathcal{M}}_{n_1}}_\text{$p$ times} \oplus {\mathcal{M}}_{n_2}\otimes 1_{k_2}$$ which in turn is a proper subalgebra of ${\mathcal{M}}_d$, again contradicting maximality. The same argument applies to $k_2$, and thus $${\mathscr{A}}={\mathcal{M}}_{n_1}\oplus {\mathcal{M}}_{n_2}.$$ Since ${\mathscr{A}}$ is unital, $n_1+n_2=d$, so we can write $n_1=d-r$ and $n_2=r$, for $0\leq r\leq d$. If $r=0$ or $r=d$, then ${\mathscr{A}}={\mathcal{M}}_d$, so $1\leq r\leq d-1$.
\[thm:any\_MUPSA\] Let ${\mathscr{B}}$ be a unital \*-subalgebra of ${\mathcal{M}}_d$ with Wedderburn decomposition ${\mathscr{B}}=\bigoplus_{r=1}^m{\mathcal{M}}_{n_r}\otimes 1_{k_r}$. If ${\mathscr{A}}$ is a MUPSA of ${\mathscr{B}}$, then, up to unitary equivalence, ${\mathscr{A}}$ has one of the following forms:
1. $${\mathscr{A}}=\left({\mathcal{M}}_{n_j-s}\otimes 1_{k_j}\right)\oplus\left({\mathcal{M}}_{s}\otimes 1_{k_j}\right)\oplus\bigoplus_{r=1,r\neq j}^m{\mathcal{M}}_{n_r}\otimes 1_{k_r}$$ for some $1\leq j\leq m$ and some $s$ such that $1\leq s\leq n_j-1$, or
2. $${\mathscr{A}}=\left({\mathcal{M}}_{n_j}\otimes 1_{k_j+k_i}\right)\oplus\bigoplus_{r=1,r\neq i,j}^m{\mathcal{M}}_{n_r}\otimes 1_{k_r}$$ for some $i,j$ such that $1\leq i,j\leq m$ and $n_j=n_i$.
Before the proof, we recall a result of Bratteli’s from [@bratteli2] that will be very useful.
\[prop:bratteli\] Let ${\mathscr{A}}\cong\oplus_{r=1}^\ell {\mathcal{M}}_{a_r}$ and ${\mathscr{B}}\cong\oplus_{r=1}^{m}{\mathcal{M}}_{n_r}$ as algebraic isomorphisms, with ${\mathscr{A}}\subseteq{\mathscr{B}}$. Then there exist integers $p_{rs}\in{\mathbb{N}}\cup\{0\}$ for $r\in\{1,\cdots,m\}$ and $s\in\{1,\cdots,\ell\}$ such that we can identify ${\mathscr{A}}$ with $$\bigoplus_{r=1}^{m}\left(\bigoplus_{s=1}^\ell {\mathcal{M}}_{a_s}\otimes 1_{p_{rs}}\right),$$ with the convention that, for any two matrix algebras ${\mathcal{M}}_{n_1}$ and ${\mathcal{M}}_{n_2}$, ${\mathcal{M}}_{n_1}\oplus ({\mathcal{M}}_{n_2}\otimes 1_0)={\mathcal{M}}_{n_1}$.
This is an informal statement of the proposition, but it says that every block ${\mathcal{M}}_{a_r}$ in ${\mathscr{A}}$ is embedded into zero or more blocks of ${\mathscr{B}}$. Note that the equivalences ignore the tensor factors in the usual Wedderburn decomposition, since these affect only the norms, not the algebraic structure. Hence, to prove Theorem \[thm:any\_MUPSA\] we will first use Bratteli’s result for the algebraic structure, then recover the norms.
\[lem:bratteli\_MUPSA\] Let ${\mathscr{A}}$ and ${\mathscr{B}}$ be matrix algebras such that ${\mathscr{A}}$ is a MUPSA of ${\mathscr{B}}$, with $${\mathscr{B}}=\bigoplus_{r=1}^m{\mathcal{M}}_{n_r},\, {\mathscr{A}}\cong\bigoplus_{r=1}^\ell{\mathcal{M}}_{a_r},$$ and the embedding of ${{\mathscr{A}}}$ into ${{\mathscr{B}}}$ has the form $$\bigoplus_{r=1}^{m}\left(\bigoplus_{s=1}^\ell {\mathcal{M}}_{a_s}\otimes 1_{p_{rs}}\right).$$ Then, up to a permutation of the blocks of ${\mathscr{A}}$, either:
1. The number of blocks in ${\mathscr{A}}$ is $m+1$, and there is an index $j\in[m]$, such that:
- For all $r\neq j$, ${\mathcal{M}}_{a_r}={\mathcal{M}}_{n_r}$ and $p_{rs}=\delta_{rs}$.
- There is some $t$ with $1\leq t\leq n_j-1$ such that ${\mathcal{M}}_{a_j}={\mathcal{M}}_t$, ${\mathcal{M}}_{a_{m+1}}={\mathcal{M}}_{n_j-t}$, $p_{js}=\delta_{js}+\delta_{(m+1)s}$.
2. The number of blocks in ${\mathscr{A}}$ is $m-1$, and there are indices $i,j\in[m]$, $j<i$, such that:
- For all $r< i$, ${\mathcal{M}}_{a_r}={\mathcal{M}}_{n_r}$ and $p_{rs}=\delta_{rs}$.
- For all $r> i$, ${\mathcal{M}}_{a_{r-1}}={\mathcal{M}}_{n_{r}}$ and $p_{rs}=\delta_{(r-1)s}$.
- ${\mathcal{M}}_{n_i}={\mathcal{M}}_{n_j}$ and $p_{is}=\delta_{sj}$.
This lemma states that, with one or two exceptions, every block of ${\mathscr{A}}$ maps surjectively into a block of ${\mathscr{B}}$. For the remaining block(s), either there are two blocks of ${\mathscr{A}}$ that map into one block of ${\mathscr{B}}$, or there is one block of ${\mathscr{A}}$ that maps to two blocks of ${\mathscr{B}}$.
Note that we assume ${\mathscr{B}}$ is equal to the structure without tensor products, but we can only assume ${\mathscr{A}}$ is isomorphic to such a structure. The decomposition of ${\mathscr{A}}$ given in the statement of Lemma \[lem:bratteli\_MUPSA\] ignores the dimension, and the embedding into ${\mathscr{B}}$ may not be isometric. Indeed, if case 2 holds, then one block of ${\mathscr{A}}$ will contain a tensor product with $1_2$.
For all $r\in[m]$, define ${\mathscr{A}}_{n_r}$ as the $r$th block of the embedding of ${\mathscr{A}}$, i.e.: $${\mathscr{A}}_{n_r}=\bigoplus_{s=1}^\ell {\mathcal{M}}_{i_s}\otimes 1_{p_{rs}}\subseteq {\mathcal{M}}_{n_r}.\label{eq:mupsa1}$$ With this notation, we have that $${\mathscr{A}}_{\mathscr{B}}\subseteq\bigoplus_{r=1}^m {\mathscr{A}}_{n_r}\subseteq {{\mathscr{B}}}$$ where ${{\mathscr{A}}}_{{\mathscr{B}}}$ is the image of ${{\mathscr{A}}}$ of the embedding into ${{\mathscr{B}}}$. Note that ${\mathscr{A}}_{{\mathscr{B}}}$ must also be a MUPSA of ${\mathscr{B}}$.
For each $r$, ${\mathscr{A}}_{n_r}$ may be a proper subalgebra of ${\mathcal{M}}_{n_r}$ or not. Suppose there is some $j$ where it is a proper subalgbera. Then we can take the subalgebra $\tilde{{\mathscr{A}}}$ defined by $$\tilde{{\mathscr{A}}}={\mathscr{A}}_{n_{j}}\oplus\bigoplus_{r\neq j}{\mathcal{M}}_{n_r}$$ and this will be a proper subalgebra of ${{\mathscr{B}}}$ and it will contain ${{\mathscr{A}}}_{{\mathscr{B}}}$. Since ${\mathscr{A}}_{{\mathscr{B}}}$ is also a MUPSA, $\tilde{{\mathscr{A}}}={\mathscr{A}}_{{\mathscr{B}}}$. Thus, ${\mathscr{A}}$ must have the form of $\tilde{{\mathscr{A}}}$, so ${\mathscr{A}}$ can have at most one $j$ such that ${\mathscr{A}}_{n_{j}}$ is a proper subalgebra of ${\mathcal{M}}_{n_{j}}$.
In this case, we can argue that ${\mathscr{A}}_{n_{j}}$ must itself be a MUPSA of ${\mathcal{M}}_{n_{j}}$, or ${{\mathscr{A}}}$ would not be maximal - we could take a MUPSA as the $j$th block instead. By Lemma \[lem:MdMUPSA\], ${\mathscr{A}}_{n_{j}}$ must have the form ${\mathcal{M}}_{n_j-t}\oplus {\mathcal{M}}_t$ for some $t$ with $1\leq t\leq n_j-1$. This proves Part (1).
The other possible situation is where ${\mathscr{A}}_{n_r}={\mathcal{M}}_{n_r}$ for all $r$. This means that in the notation of Equation \[eq:mupsa1\], there can only be one block of ${\mathscr{A}}$ in each block of ${\mathscr{B}}$, so for each $r$, there is a unique $s_0(r)$ such that $p_{rs_0(r)}=1$, and $p_{rs}=0$ for all $s\neq s_0(r)$. This means that the embedding of ${\mathscr{A}}$ into ${\mathscr{B}}$ looks like $$\bigoplus_{s=1}^\ell {\mathcal{M}}_{a_s}\mapsto\bigoplus_{r=1}^{m}\left({\mathcal{M}}_{a_{s_0(r)}}\right).$$ The direct sum on the left is not all of ${\mathscr{A}}$, it is only isomorphic to ${\mathscr{A}}$. A block on the left might appear twice in the embedding if there is some $i\neq j$ such that $s_0(i)=s_0(j)$. This is how, even though each block is surjectively covered by the embedding, ${\mathscr{A}}$ can still be a proper subalgebra of ${\mathscr{B}}$, since ${\mathscr{B}}$ has more freedom between blocks.
If $\ell=m$, then each block of ${\mathscr{A}}$ embeds surjectively into each block of ${\mathscr{B}}$, implying the contradictory statement that ${\mathscr{A}}={\mathscr{B}}$. Thus $\ell<m$. This means there must be some $i$ and $j$ such that $s_0(i)=s_0(j)$. That is, some block of ${\mathscr{A}}$ maps to two blocks in ${\mathscr{B}}$. We define an algebra $\tilde{{\mathscr{A}}}$ with ${\mathscr{A}}\subseteq\tilde{{\mathscr{A}}}\subseteq{\mathscr{B}}$ such that $$\tilde{{\mathscr{A}}}=\bigoplus_{s\neq i,s\leq m}{\mathcal{M}}_{n_s}\mapsto \bigoplus_{r=1}^m\left(({\mathcal{M}}_{n_r}\otimes 1_{1-\delta_{ri}})\oplus ({\mathcal{M}}_{n_j}\otimes 1_{\delta_{ri}})\right).$$ That is, $\tilde{{\mathscr{A}}}$ is just all of the blocks of ${\mathscr{B}}$ except the $i$th block; to embed it into ${\mathscr{B}}$, we use the identity on all blocks, and send a copy of the $j$th block to the $i$th block of ${\mathscr{B}}$. Since we required that each block ${\mathscr{A}}_{n_j}={\mathcal{M}}_{n_j}$, then ${\mathcal{M}}_{n_j}={\mathcal{M}}_{n_i}$. Clearly, $\tilde{{\mathscr{A}}}$ is a proper subalgebra of ${\mathscr{B}}$, and by this construction, $\tilde{{\mathscr{A}}}$ must contain ${\mathscr{A}}$. Hence $\tilde{{\mathscr{A}}}={\mathscr{A}}$.
Thus a MUPSA must have the form of $\tilde{{\mathscr{A}}}$ for some blocks $i$ and $j$, hence $\ell=m-1$ and in all other blocks, ${\mathscr{A}}$ and ${\mathscr{B}}$ are equal. This proves Part (2).
Given $${\mathscr{B}}=\bigoplus_{r=1}^m{\mathcal{M}}_{n_r}\otimes 1_{k_r}$$ we can define a new algebra $\tilde{{\mathscr{B}}}$ as $$\tilde{{\mathscr{B}}}=\bigoplus_{r=1}^m{\mathcal{M}}_{n_r}.$$ This will be \*-isomorphic, but not isometric, to ${\mathscr{B}}$. The natural isomorphism $\phi:\tilde{{\mathscr{B}}}\rightarrow {\mathscr{B}}$ can be defined as $$\phi(x_1,\cdots,x_m)=(x_1\otimes 1_{k_1},\cdots,x_m\otimes 1_{k_m}).$$ Then we can let $\tilde{{\mathscr{A}}}=\phi^{-1}({\mathscr{A}})\subseteq\tilde{{\mathscr{B}}}$. In fact, $\tilde{{\mathscr{A}}}$ will be a MUPSA of $\tilde{{\mathscr{B}}}$, since any subalgebra of $\tilde{{\mathscr{B}}}$ can map to a subalgebra of ${\mathscr{B}}$.
Then, ignoring tensor products, we can write $$\tilde{{\mathscr{A}}}\cong\bigoplus_{r=1}^\ell{\mathcal{M}}_{a_r}$$ and apply Lemma \[lem:bratteli\_MUPSA\] and consider the two cases.
In the first case, $\ell=m+1$, and the decomposition of $\tilde{{\mathscr{A}}}$ looks like $$\tilde{{\mathscr{A}}}\mapsto {\mathcal{M}}_{t}\oplus{\mathcal{M}}_{n_j-t}\oplus\bigoplus_{r\neq j;r\leq m}{\mathcal{M}}_{n_r}.$$ By dimension counting, this must actually equal $\tilde{{\mathscr{A}}}$, so $$\tilde{{\mathscr{A}}}= {\mathcal{M}}_{t}\oplus{\mathcal{M}}_{n_j-t}\oplus\bigoplus_{r\neq j;r\leq m}{\mathcal{M}}_{n_r}.$$ Then we can write $${\mathscr{A}}=\phi(\tilde{{\mathscr{A}}})=({\mathcal{M}}_t\otimes 1_{k_j})\oplus({\mathcal{M}}_{n_j-t}\otimes 1_{k_j})\oplus\bigoplus_{r\neq j;r\leq m}{\mathcal{M}}_{n_r}\otimes 1_{k_r},$$ thus proving Part (1).
In the second case, $\ell=m-1$ and the embedding of $\tilde{{\mathscr{A}}}$ into $\tilde{{\mathscr{B}}}$ is $$\tilde{{\mathscr{A}}}\mapsto ({\mathcal{M}}_{n_j}\otimes 1_2)\oplus\bigoplus_{r\neq i,j;r\leq m}{\mathcal{M}}_{n_r}.$$ Here we’ve used the fact that ${\mathcal{M}}_{n_j}$ maps into two blocks and replaced these two blocks with a tensor product with $1_2$. Once again, by dimension counting, this is not just an embedding, it is the actual structure of $\tilde{{\mathscr{A}}}$. Hence, we can write ${\mathscr{A}}$ as $\phi(\tilde{{\mathscr{A}}})$. To handle the $j$th block, note that any element of $\tilde{{\mathscr{A}}}$ has the same elements in the $i$ and $j$ components, so it looks like $(x_j,x_j)$. When we apply $\phi$ to these components, they become $(x_j\otimes 1_{k_i},x_j\otimes 1_{k_j})=x_j\otimes 1_{k_i+k_j}$. Thus, $${\mathscr{A}}=\phi(\tilde{{\mathscr{A}}})=({\mathcal{M}}_{n_j}\otimes 1_{k_i+k_j})\oplus\bigoplus_{r\neq i,j;r\leq m}{\mathcal{M}}_{n_r}\otimes 1_{k_r}.$$ This proves part (2).
This characterization of MUPSAs also characterizes the lattice of proper \*-algebras of ${\mathcal{M}}_d$. For example, if $d=4$, then the MUPSAs form the lattice shown in Figure \[fig:lattice\_diagram\]. Note that in the figure, the length of the longest chain of subalgebras, including ${\mathcal{M}}_4$, is 7. The next lemma generalizes this.
(M4) at (2,7) [${\mathcal{M}}_4$]{}; (M3M1) at (0,6) [${\mathcal{M}}_3\oplus {\mathcal{M}}_1$]{}; (M2M2) at (4,6) [${\mathcal{M}}_2\oplus {\mathcal{M}}_2$]{}; (M2M1M1) at (0,5) [${\mathcal{M}}_2\oplus {\mathcal{M}}_1\oplus {\mathcal{M}}_1$]{};‘ (M1M1M1M1) at (0,4) [${\mathcal{M}}_1\oplus {\mathcal{M}}_1\oplus {\mathcal{M}}_1\oplus {\mathcal{M}}_1$]{}; (M1x2M1M1) at (0,3) [${\mathcal{M}}_1\otimes 1_2\oplus {\mathcal{M}}_1\oplus {\mathcal{M}}_1$]{}; (M2x2) at (4,4) [${\mathcal{M}}_2\otimes 1_2$]{}; (M1x3M1) at (0,2) [${\mathcal{M}}_1\otimes 1_3\oplus {\mathcal{M}}_1$]{}; (M1x2M1x2) at (4,2) [${\mathcal{M}}_1\otimes 1_2\oplus {\mathcal{M}}_1\otimes 1_2$]{}; (M1x4) at (2,1) [${\mathcal{M}}_1\otimes 1_4$]{}; (M4) edge (M3M1) edge (M2M2) (M3M1) edge (M2M1M1) (M2M2) edge (M2x2) edge (M2M1M1) (M2x2) edge (M1x2M1x2) (M1x2M1x2) edge (M1x4) (M2M1M1) edge (M1M1M1M1) (M1M1M1M1) edge (M1x2M1M1) (M1x2M1M1) edge (M1x3M1) edge (M1x2M1x2) (M1x3M1) edge (M1x4);
\[fig:lattice\_diagram\]
\[lem:chain\_length\] Let $\{{\mathscr{A}}_1,\cdots,{\mathscr{A}}_n={\mathbb{C}}1\}$ be a descending chain of unital subalgebras of ${\mathcal{M}}_d$ and let ${\mathscr{A}}_1=\bigoplus_{r=1}^m{\mathcal{M}}_{n_r}\otimes 1_{k_r}$. Then the length of the chain is at most $\sum_{r=1}^m(2n_r-1)$.
If ${\mathscr{A}}_{i+1}$ is not a MUPSA of ${\mathscr{A}}_i$, then there must be a chain of MUPSAs going from ${\mathscr{A}}_i$ to ${\mathscr{A}}_{i+1}$, and this will only increase the length of the chain. So, without loss of generality, assume that each algebra is a MUPSA of the previous one.
For ${\mathscr{A}}_i=\bigoplus_{r=1}^{m_i}{\mathcal{M}}_{n_{i_r}}\otimes 1_{k_{i_r}}$, define $\chi({\mathscr{A}}_i)=\sum_{r=1}^{m_i}(2n_{i_r}-1)$. We will use induction on $\chi({\mathscr{A}}_1)$ to show that the length of the chain is at most $\chi({\mathscr{A}}_1)$. Since $2n_r-1\geq 1$ for all $n_r$, if $\chi({\mathscr{A}}_1)=1$, then ${\mathscr{A}}_1={\mathcal{M}}_1\otimes 1_d={\mathbb{C}}1$. Then the length of the chain is just 1, which equals $\chi({\mathscr{A}}_1)$.
Suppose the hypothesis holds for all chains starting with algebras ${\mathscr{A}}_1$ such that $\chi({\mathscr{A}}_1)<y$ for some $y$. Suppose we have a chain with ${\mathscr{A}}_1=\bigoplus_{r=1}^m{\mathcal{M}}_{n_r}\otimes 1_{k_r}$ such that $\chi({\mathscr{A}}_1)=y$. Then the next algebra in the chain, ${\mathscr{A}}_2$, is a MUPSA of ${\mathscr{A}}_1$, and by Lemma \[thm:any\_MUPSA\], it has two possible forms:
1. ${\mathscr{A}}_2=({\mathcal{M}}_{n_j-s}\otimes 1_{k_j})\oplus ( {\mathcal{M}}_s\otimes 1_{k_j})\oplus\bigoplus_{r=1,r\neq j}^m {\mathcal{M}}_{n_r}\otimes 1_{k_r}$. In this case: $$\begin{aligned}
\chi({\mathscr{A}}_2)=&\sum_{r=1,\neq j}(2n_r-1)+(2(n_j-s)-1)+(2s-1)\\
=&\sum_{r=1,\neq j}(2n_r-1)+(2n_j-1)-1\\
=&\sum_{r=1}^m(2n_r-1)-1\\
=&\chi({\mathscr{A}}_1)-1,
\end{aligned}$$ which is less than $y$, so we can apply induction and declare that the length of the chain $\{{\mathscr{A}}_2,\cdots, {\mathscr{A}}_n\}$ is at most $\chi({\mathscr{A}}_2)$; adding 1 when we add ${\mathscr{A}}_1$ takes the maximum length to $\chi({\mathscr{A}}_2)+1=\chi({\mathscr{A}}_1)$.
2. ${\mathscr{A}}_2=({\mathcal{M}}_{n_j}\otimes 1_{k_j+k_i})\oplus\bigoplus_{r=1,r\neq j,i}^m {\mathcal{M}}_{n_r}\otimes 1_{k_r}$. Then, noting that $2n_i-1\geq 1$, that $\chi({\mathscr{A}}_2)$ is $$\begin{aligned}
\sum_{r=1,r\neq i,j}^m (2n_r-1) + (2n_j-1) \leq& \sum_{r=1,r\neq i,j}^m (2n_r-1)+(2n_j-1)+(2n_i-1)-1\\
=&\sum_{r=1}^m(2n_r-1)-1\\
=&\chi({\mathscr{A}}_1)-1
\end{aligned}$$ Again, we apply induction and add the remaining algebra to show that the length of the chain is at most $\chi({\mathscr{A}}_1)$.
Since the multiplicative domains of powers of a unital channel give a chain of unital \*-subalgebras, Lemma \[lem:chain\_length\] gives a bound on the multiplicative index:
\[thm:kappa\_bound\] Let ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ be a unital quantum channel for $d\geq 2$. Then the multiplicative index $\kappa({{\mathcal{E}}})\leq 2d-2$.
For a channel ${{\mathcal{E}}}$, $\{{\mathcal{M}}_{{\mathcal{E}}},{\mathcal{M}}_{{{\mathcal{E}}}^2},\cdots,{\mathcal{M}}_{{{\mathcal{E}}}^\kappa}\}$ is a descending chain of subalgebras, so $\kappa$ must be less than the maximum length of such a chain. As a chain of subalgebras this has a maximum length of $2d-1$ by Lemma \[lem:chain\_length\], which could only be achieved if ${\mathcal{M}}_{{\mathcal{E}}}={\mathcal{M}_{d}}$. However, if ${\mathcal{M}}_{{\mathcal{E}}}={\mathcal{M}_{d}}$, then the channel must be unitary and $\kappa({{\mathcal{E}}})=1\leq 2(d-1)$. If ${\mathcal{M}}_{{\mathcal{E}}}\neq {\mathcal{M}_{d}}$, then the length of the chain does not achieve the maximum and must be at most $2(d-1)$.
The proof of Theorem \[thm:kappa\_bound\] uses only Lemma \[lem:chain\_length\] and the fact that ${\mathcal{M}}_{{\mathcal{E}}}$ is \*-closed and unital but does not use the structure of the channel itself. It’s possible that the structure of ${\mathcal{M}}_{{\mathcal{E}}}$ gives a tighter bound for the multiplicative index than $2(d-1)$. So far the the largest multiplicative index we have constructed is $d$. Sections \[sec:ETB\_channels\] and \[sec:Schur\_channels\] illustrate our examples. Note that in order to get a non-trivial value for $\kappa$, one must choose ${\mathcal{M}}_{{\mathcal{E}}}$ to be a proper subalgebra of ${\mathcal{M}}_d$ and not ${\mathcal{M}}_d$ itself. So the value obtained for the length of the longest chain in Figure \[fig:lattice\_diagram\] is one more than the possible maximum value of $\kappa$ determined by the Theorem \[thm:kappa\_bound\].
Example: Entanglement Breaking Channels {#sec:ETB_channels}
---------------------------------------
There are many equivalent definitions of an entanglement breaking channel (see [@entng-brkng], [@stormer2008]), but for our purposes it is most convenient to define it in terms of Kraus operators. A channel ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ is an entanglement breaking channel if it can be written as: $${{\mathcal{E}}}(x)=\sum_{i=1}^n\varphi_i\zeta_i^* x\zeta_i\varphi_i^*,$$ where $\zeta_i,\varphi_i\in{\mathbb{C}}^d$. We note down a useful result below which can be deduced from $\cite{stormer2008}$:
[[(St[ø]{}rmer]{}, [@stormer2008])]{} Let ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow{\mathcal{M}}_d$ be a unital entanglement breaking channel. Then the multiplicative domain of ${{\mathcal{E}}}$ is an abelian C$^*$-algebra.
Using the above theorem we get an upper bound for the multiplicative index of unital entanglement breaking channels:
\[prop:etb\_bound\] If ${{\mathcal{E}}}$ is a unital entanglement breaking channel, then $\kappa({{\mathcal{E}}})\leq d$.
Consider the chain $\{{\mathcal{M}}_{{\mathcal{E}}},{\mathcal{M}}_{{{\mathcal{E}}}^2},\cdots,{\mathcal{M}}_{{{\mathcal{E}}}^\kappa}\}$. Since ${\mathcal{M}}_{{\mathcal{E}}}$ is abelian, it is a subalgebra of a maximal abelian subalgebra, which has the form ${\mathcal{M}}_1\oplus\cdots\oplus {\mathcal{M}}_1$ for ${\mathcal{M}}_d$. Following Lemma \[lem:chain\_length\], the length of the chain is at most $\sum_{i=1}^d (2(1)-1)=d$, and thus that is the maximum possible value of $\kappa({{\mathcal{E}}})$.
The vectors $\{\zeta_i\}$ that form the Kraus operators of an entanglement breaking channel need not be linearly independent, but if the channel is unital and the vectors are linearly independent, then they are orthonormal. These are the only channels we consider, since in this case the multiplicative domain is easily calculated.
\[prop:etb\_md\_struct\] Let ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ be a unital quantum channel such that ${{\mathcal{E}}}(x)=\sum_{i=1}^d\varphi_i\zeta_i^*x\zeta_i\varphi_i^*$, and let $\{\varphi_i\}_{i=1}^d$ and $\{\zeta_i\}_{i=1}^d$ be orthonormal. Then ${\mathcal{M}}_{{\mathcal{E}}}={\rm{span}}\{\zeta_i\zeta_i^*|1\leq i\leq d\}$.
We will use Theorem \[kribs-spekkens\], which states that ${\mathcal{M}}_{{\mathcal{E}}}=\rm{Fix}_{{{\mathcal{E}}}^*\circ{{\mathcal{E}}}}$. First we compute the Kraus operators of ${{\mathcal{E}}}^*\circ{{\mathcal{E}}}$, which are all the products of the form $\zeta_i\varphi_i^*\varphi_j\zeta_j^*=\delta_{ij}\zeta_i\zeta_j^*$. Hence the set of Kraus operators is $\{\zeta_i\zeta_i^*\}_{i=1}^d$. This spans a maximal abelian subalgebra, and hence its span is its own commutant. Theorem \[cummutant-fixed pnt\] states that the fixed points of a unital channel are the commutant of its Kraus operators, so these will also span the multiplicative domain.
The next lemma characterizes the projections in ${\mathcal{M}}_{{{\mathcal{E}}}^k}$ for higher powers of $k$. Since the multiplicative domain is spanned by its projections, this characterizes the multiplicative domain.
\[prop:uebc\_md\_struc\] If ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ is a unital entanglement breaking channel given by ${{\mathcal{E}}}(x)=\sum_i\varphi_i\zeta_i^*x\zeta_i\varphi_i^*$, then for $1\leq r\leq d$ and any projection $p$, $p\in {\mathcal{M}}_{{{\mathcal{E}}}^r}$ if and only if there exists subsets $K_1,\cdots,K_r$ of $[1,\cdots, d]$ such that $\vert K_i\vert=\rm{rank}(p)$ for all $i$, $\sum_{k\in K_1}\zeta_k\zeta_k^*=p$, and $\sum_{k\in K_i}\varphi_k\varphi_k^*=\sum_{k\in K_{i+1}}\zeta_k\zeta_k^*$ for all $1\leq i\leq r-1$.
To prove one direction, suppose the sets $K_1,\cdots,K_r$ exist. It’s clear that for any $K\subseteq[d]$: $$\label{eq:sum-proj.}
{{\mathcal{E}}}\left(\sum_{k\in K}\zeta_k\zeta_k^*\right)=\sum_{i=1}^n\sum_{k\in K}\varphi_i\zeta_i^*\zeta_k\zeta_k^*\zeta_i\varphi_i^*=\sum_{k\in K}\varphi_k\varphi_k^*.$$ However, we know that $\sum_{k\in K_i}\varphi_k\varphi_k^*=\sum_{k\in K_{i+1}}\zeta_k\zeta_k^*$. Thus, from the Equation \[eq:sum-proj.\], ${{\mathcal{E}}}\left(\sum_{k\in K_i}\zeta_k\zeta_k^*\right)=\sum_{k\in K_{i+1}}\zeta_k\zeta_k^*$, which is still a projection in the multiplicative domain. Since we can repeat this process $r-1$ times, then ${{\mathcal{E}}}^{r-1}(p)$ is still in ${\mathcal{M}}_{{\mathcal{E}}}$; thus $p\in {\mathcal{M}}_{{{\mathcal{E}}}^r}$.
For the converse, we will use induction to show that the sets $K_1,\cdots, K_r$ exist. For $r=1$, this reduces to $p=\sum_{k\in K}\zeta_k\zeta_k^*$, which is proven by Proposition \[prop:etb\_md\_struct\]. Then suppose the sets exist for all numbers up to $r-1$ and that $p\in {\mathcal{M}}_{{{\mathcal{E}}}^r}$ is a projection. Then $p$ must also be in ${\mathcal{M}}_{{{\mathcal{E}}}^{r-1}}$, meaning there exist sets $K_1,\cdots, K_{r-1}$ such that $\sum_{k\in K_i}\varphi_k\varphi_k^*=\sum_{k\in K_{i+1}}\zeta_k\zeta_k^*$, for $i\leq r-2$. Now since $p\in {\mathcal{M}}_{{{\mathcal{E}}}^r}$, ${{\mathcal{E}}}^{r-1}(p)\in {\mathcal{M}}_{{\mathcal{E}}}$. By the logic in the proof of the previous direction, we have that ${{\mathcal{E}}}^{r-1}(p)=\sum_{k\in K_{r-1}}\varphi_k\varphi_k^*$. If this is in the multiplicative domain, it must be a projection of the form $\sum_{k\in K}\zeta_k\zeta_k^*$ for some $K\subseteq[d]$. Thus, $\sum_{k\in K_{r-1}}\varphi_k\varphi_k^*=\sum_{k\in K}\zeta_k\zeta_k^*$; set $K_r=K$ and the conclusion holds.
This requirement, where the sums of two different sets of rank-one projections add up to the same projection, is a central part of this construction. Thus, we define it as follows:
A set of vectors $\{\varphi_i\}_{i=1}^n$ in ${\mathbb{C}}^d$ is **non-comparable to** another set $\{\zeta_i\}_{i=1}^m$ if there are no two proper subsets $K_1\subsetneq[n-1],K_2\subsetneq[m-1]$ such that $\sum_{k\in K_1}\varphi_k\varphi_k^*=\sum_{k\in K_2}\zeta_k\zeta_k^*$.
An operator $X:{\mathbb{C}}^d\rightarrow {\mathbb{C}}^d$ is said to be non-comparable with respect to a basis $\{\varphi_i\}_{i=1}^d$ if $\{X\varphi_i\}_{i=1}^d$ is non-comparable to $\{\varphi_i\}_{i=1}^d$.
As an example, the basis: $${\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}\frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\\0\\0\end{pmatrix},\frac{1}{\sqrt{2}}\begin{pmatrix}1\\-1\\0\\0\end{pmatrix},
\begin{pmatrix}0\\0\\1\\0\end{pmatrix},\begin{pmatrix}0\\0\\0\\1\end{pmatrix}\rightset$$ is comparable to the basis: $${\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}\begin{pmatrix}\tfrac{1}{2}\\\tfrac{\sqrt{3}}{2}\\0\\0\end{pmatrix},\begin{pmatrix}\tfrac{\sqrt{3}}{2}\\-\tfrac{1}{2}\\0\\0\end{pmatrix},
\frac{1}{\sqrt{2}}\begin{pmatrix}0\\0\\1\\1\end{pmatrix},\frac{1}{\sqrt{2}}\begin{pmatrix}0\\0\\-1\\1\end{pmatrix}\rightset$$ even though all of the vectors are distinct. In the notation of Proposition \[prop:uebc\_md\_struc\], $K_1=K_2=\{1,2\}$.
As an example of a non-comparable operator, the $d\times d$ normalized discrete Fourier transform matrix $\Delta$, where $\Delta_{ij}=\frac{1}{\sqrt{d}}\omega^{(i-1)(j-1)}$ for some primitive $d$th root of unity $\omega$, is non-comparable to the canonical basis. To see this, the projections $(e_1e_1^*, \cdots, e_de_{d}^*)$ constructed from the canonical basis $(e_1,,\cdots,e_d)$ span the diagonal matrices. However, the projections generated by $\Delta e_i e_i^*\Delta^*$ all have constant diagonals. Thus, the only linear combination of these projections that is diagonal is the identity.
Using this definition, we can construct entanglement breaking channels on ${\mathcal{M}}_d$ with any multiplicative index up to $d$.
\[thm:etb\_kappa\_d\] For any integer $r$ between $1$ and $d$, there is a unital entanglement-breaking channel ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ such that $\kappa({{\mathcal{E}}})=r$.
Let $\{\zeta_i\}_{i=1}^d$ be any orthonormal basis. For $r=1$, set ${{\mathcal{E}}}(x)=\sum_{i=1}^d\zeta_i\zeta_i^*x\zeta_i\zeta_i^*$. It’s clear that the fixed points $(\rm{Fix}_{{\mathcal{E}}})$ of this channel are simply ${\rm{span}}\{\zeta_i\zeta_i^*|1\leq i\leq d\}$, which is also ${\mathcal{M}}_{{\mathcal{E}}}$. Since $\rm{Fix}_{{\mathcal{E}}}\subseteq{\mathcal{M}}_{{{\mathcal{E}}}^\infty}\subseteq{\mathcal{M}}_{{\mathcal{E}}}$ in general, we have ${\mathcal{M}}_{{\mathcal{E}}}={\mathcal{M}}_{{{\mathcal{E}}}^\infty}$ and $\kappa({{\mathcal{E}}})=1$.
For $r\geq 2$, let $\{\zeta_i\}_{i=1}^d$ be any orthonormal basis of ${\mathbb{C}}^d$. Let $U$ be a unitary matrix in ${\mathcal{M}}_{d-r+2}$ that is non-comparable to $\{\zeta_i\}_{i=1}^{d-r+2}$. Define $\{\varphi_i\}$ such that $\varphi_i=U\zeta_i,i\leq d-r+2$, and $\varphi_i=\zeta_i$ otherwise. Then define $$\begin{aligned}
{{\mathcal{E}}}(x)=&\sum_{i=1}^d \varphi_i\zeta_{i-1}^*x \zeta_{i-1}\varphi_i^*\\
=&\sum_{i=2}^{d-r+1}\varphi_i\zeta_{i-1}^*x\zeta_{i-1}\varphi_i^* + \sum_{i=d-r+2}^{d-1}\zeta_i\zeta_{i-1}^*x\zeta_{i-1}\zeta_i^* + \varphi_1\zeta_d^*x\zeta_d\varphi_1^*.\end{aligned}$$
(where $\zeta_0=\zeta_d$). In the middle sum, we have replaced $\varphi_i=\zeta_i$, since that was how we constructed $\varphi_i$. From Proposition \[prop:etb\_md\_struct\], we know that the multiplicative domain of ${{\mathcal{E}}}$ is the span of $\{\zeta_i\zeta_i^*\}_{i=1}^d$. Since $\varphi_i=\zeta_i$ for $i>d-r+2$, we also have that ${\rm{span}}\{\varphi_i\varphi_i^*\}_{i=d-r+3}^d$ is in the multiplicative domain. Further, since the first $d-r+2$ vectors $\varphi_i$ are non-comparable to $\zeta_i$, we have that for any set $K\subseteq [d-r+2]$ with $\vert K\vert<d-r+2$, $\sum_{k\in K}\varphi_k\varphi_k^*$ is not in the multiplicative domain.
The map ${{\mathcal{E}}}$ maps the first $d-r+1$ projections in ${\mathcal{M}}_{{\mathcal{E}}}$ outside of the multiplicative domain, and then “pushes" the remaining projections to $\zeta_d\zeta_d^*$, whose image is also outside of the multiplicative domain.
To prove this, take the rank-1 projection $\zeta_{d-r+2}\zeta_{d-r+2}^*$. Note that, by orthogonality, for $d-r+2\leq j\leq d-1$, ${{\mathcal{E}}}(\zeta_j\zeta_j^*)=\zeta_{j+1}\zeta_{j+1}^*$, which is also in ${\mathcal{M}}_{{\mathcal{E}}}$. So applying ${{\mathcal{E}}}$ $m$ times gives ${{\mathcal{E}}}^m(\zeta_j\zeta_j^*)=\zeta_{j+m}\zeta_{j+m}^*$ (for $j\leq d-m$). However, suppose $j=d$. Then ${{\mathcal{E}}}(\zeta_d\zeta_d^*)=\varphi_1\varphi_1^*\notin {\mathcal{M}}_{{\mathcal{E}}}$. Thus, for any $j$ with $d-r+2\leq j$, ${{\mathcal{E}}}^{d-j}(\zeta_j\zeta_j^*)=\zeta_d\zeta_d^*\in {\mathcal{M}}_{{\mathcal{E}}}$, but ${{\mathcal{E}}}^{d-j+1}(\zeta_j\zeta_j^*)\notin {\mathcal{M}}_{{\mathcal{E}}}$. Thus, $\zeta_j\zeta_j^*$ is in ${\mathcal{M}}_{{{\mathcal{E}}}^{d-j+1}}$ but not ${\mathcal{M}}_{{{\mathcal{E}}}^{d-j+2}}$; thus, the multiplicative index must be at least $d-j+2$. Since this pattern works for $j\geq d-r+2$, we get that the multiplicative index must be at least $r$.
To show that this is also an upper bound, let $K$ be an arbitrary subset of $[d]$. We need to show that if $p_K=\sum_{k\in K}\zeta_k\zeta_k^*$ is in ${\mathcal{M}}_{{{\mathcal{E}}}^r}$, then $K=[d]$ and $p_K=1$. The idea is as follows: If there is any rank-1 projection missing from $p_K$, then this creates a “hole". If the hole is in the first $d-r+2$ projections, then the image is not in the multiplicative domain. However, the action of ${{\mathcal{E}}}$ will move the hole until it is, eventually, in the first $d-r+2$ projections.
Suppose that $j\notin K$ for some $j\in [d]$ (the hole). If $j=d$ or $j<d-r+2$, then ${{\mathcal{E}}}(p_K)$ is a projection onto a subspace that is (partially) spanned by a strict subset of $\{\varphi_i\varphi_i^*\}_{1\leq i\leq d-r+2}$, which is, by construction, non-comparable to $\{\zeta_i\zeta_i\}$. Thus, ${{\mathcal{E}}}(p_K)$ cannot be written as a sum of $\zeta_j\zeta_j^*$, and thus is it is not in the multiplicative domain. Now consider that if $j\notin K$ for $p_K$ but ${{\mathcal{E}}}(p_K)=p_{K'}$ for some $K'$, then $j+1\notin K'$ (since the only way to get $\zeta_{j+1}\zeta_{j+1}^*$ would be if $\zeta_j\zeta_j^*$ was in the decomposition of $P_K$, which it is not). Thus, if $j\geq d$, then $P_K\notin {\mathcal{M}}_{{{\mathcal{E}}}^2}$. The pattern continues and thus $p_K\notin {\mathcal{M}}_{{{\mathcal{E}}}^m}$ if it is missing any $j\geq d-m+2$. Since we require $P_k\in {\mathcal{M}}_{{{\mathcal{E}}}^r}$, we must have that $j\leq d-r+2$. From before, any missing $j$ must be more than $d-r+2$. Thus, to be in ${\mathcal{M}}_{{{\mathcal{E}}}^r}$ any missing $j$ must satisfy both $j\leq d-r+2$ and $j>d-r+2$, a contradiction. Thus, there is no $j\notin K$; $K=[d]$ and $p_K=1$.
As an example in ${\mathcal{M}}_3$, to get a multiplicative index of 3, we can take the canonical basis $\{e_1,e_2,e_3\}$ and the basis $\{f_1,f_2,f_3\}$ formed by applying the $2\times 2$ discrete Fourier transform matrix to $e_1$ and $e_2$. Let $\omega$ be a primitive 3rd root of unity. Using Theorem \[thm:etb\_kappa\_d\], we can constuct a channel ${{\mathcal{E}}}$ with Kraus operators of: $$K_1=f_1e_3^*=\tfrac{1}{\sqrt{2}}\begin{pmatrix}0&0&1\\0&0&1\\0&0&0\end{pmatrix},K_2=f_2e_1^*=\tfrac{1}{\sqrt{2}}\begin{pmatrix}1&0&0\\-1&0&0\\0&0&0\end{pmatrix},K_3=f_3e_2^*=\begin{pmatrix}0&0&0\\0&0&0\\0&1&0\end{pmatrix}.$$ By Proposition \[prop:etb\_md\_struct\], the multiplicative domain of ${{\mathcal{E}}}$ is all the diagonal matrices, so we take a diagonal matrix $D=\begin{pmatrix}a&0&0\\0&b&0\\0&0&c\end{pmatrix}$. Then $${{\mathcal{E}}}(D)=\frac{1}{2}\begin{pmatrix}a+c&-a+c&0\\-a+c&a+c&0\\0&0&2b\end{pmatrix}.$$ For this to be in ${\mathcal{M}}_{{\mathcal{E}}}$, and thus $D\in {\mathcal{M}}_{{{\mathcal{E}}}^2}$, the off-diagonal terms must be 0, so $a=c$. If we set $D'$ to be $\begin{pmatrix}a&0&0\\0&b&0\\0&0&a\end{pmatrix}$, then $${{\mathcal{E}}}^2(D')=\frac{1}{2}\begin{pmatrix}a+b&-a+b&0\\-a+b&a+b&0\\0&0&2a\end{pmatrix}$$ and for $D'$ to be in ${\mathcal{M}}_{{{\mathcal{E}}}^3}$, $a$ must equal $b$ and thus $D'$ is a multiple of the identity. Since ${\mathcal{M}}_{{\mathcal{E}}}\subsetneq {\mathcal{M}}_{{{\mathcal{E}}}^2}\subsetneq {\mathcal{M}}_{{{\mathcal{E}}}^3}={\mathbb{C}}1$, the multiplicative index of ${{\mathcal{E}}}$ is 3.
Example: Schur Channels {#sec:Schur_channels}
-----------------------
Given two matrices $a,b\in {\mathcal{M}}_d$, the Schur product $a\bullet b$ is defined as the matrix $(a\bullet b)_{ij}=a_{ij}b_{ij}$. A positive semidefinite matrix $b$ whose diagonals are all 1 induces a quantum channel $\mathcal{T}_b$ whose action is given by $\mathcal{T}_b(x)=b\bullet x$. Such channels are necessarily unital. We will call these channels Schur channels.
We will let $b=\begin{pmatrix} J_{d-1}&0\\0&1\end{pmatrix}$, where $J_{d-1}$ is the $d-1\times d-1$ matrix of all ones and let $u$ be the permutation matrix corresponding to $(1 2 3\cdots d)$ in the symmetric group on $[d]$.
We can construct a channel ${{\mathcal{E}}}={\mathcal}{U}\circ\mathcal{T}_b$, and this will have multiplicative index $d-1$. Here $\mathcal{U}$ denotes the unitary channel $\mathcal{U}(\cdot)=u(\cdot)u^*$.To see this, let $p=\tfrac{1}{2}(E_{11}+E_{22}+E_{12}+E_{21})$. This is a projection, and if $k\leq d-2$, $${{\mathcal{E}}}^k(p)=\tfrac{1}{2}(E_{k+1,k+1}+E_{k+2,k+2}+E_{k+1,k+2}+E_{k+2,k+1}).$$ However, $d-1$ applications of ${{\mathcal{E}}}$ to $p$ gives $${{\mathcal{E}}}^{d-1}(p)={{\mathcal{E}}}(\tfrac{1}{2}(E_{d-1,d-1}+E_{dd}+E_{d-1,d}+E_{d,d-1})=\tfrac{1}{2}(E_{11}+E_{dd}),$$ which is no longer a projection. Thus $p\in {\mathcal{M}}_{{{\mathcal{E}}}^{d-2}}$ but not ${\mathcal{M}}_{{{\mathcal{E}}}^{d-1}}$, so $\kappa({{\mathcal{E}}})\geq d-1$. Since ${\mathcal{M}}_{{{\mathcal{E}}}}\cong {\mathcal{M}}_{d-1}\oplus {\mathcal{M}}_1$ and ${\mathcal{M}}_{{{\mathcal{E}}}^\infty}$ must contain every diagonal matrix, then we can use the same logic as Lemma \[lem:chain\_length\] to see that the chain of multiplicative domains can have length at most $d-1$. Thus, $\kappa({{\mathcal{E}}})=d-1$.
Product Channels {#sec:product_channels}
----------------
Looking at Proposition \[prop:kappa\_tensor\_bound\], the bound on the multiplicative index is linear in the dimension, but cannot increase for product channels. This creates a gap between the possible multiplicative indices of an arbitrary channel and a product channel.
\[thm:index\_factor\] Let ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ be a unital quantum channel. If ${{\mathcal{E}}}=\otimes_{i=1}^n {{\mathcal{E}}}_i$ where ${{\mathcal{E}}}_i:{\mathcal{M}}_{d_i}\rightarrow {\mathcal{M}}_{d_i}$, then $\kappa({{\mathcal{E}}})\leq \max_i\{ 2(d_i-1)\}$. Specifically, if $\kappa({{\mathcal{E}}})\geq d-1$, then ${{\mathcal{E}}}$ cannot be factored in any way.
Suppose ${{\mathcal{E}}}=\otimes_{i=1}^n{{\mathcal{E}}}_i$, where ${{\mathcal{E}}}_i:{\mathcal{M}}_{d_i}\rightarrow {\mathcal{M}}_{d_i}$. Then by Proposition \[prop:kappa\_tensor\_bound\] $$\kappa({{\mathcal{E}}})=\max_i\{\kappa({{\mathcal{E}}}_i)\}\leq \max_i\{2(d_i-1)\}.$$
If $n=2$, then $d_1=d/s$ and $d_2=s$ for some $s\vert d$. Since $2\leq s\leq d/2$, then $\max_i\{2(d_i-1)\}\leq 2(d/2-1)=d-2$. Any other factorization will have even smaller components; thus, $\kappa({{\mathcal{E}}})\leq d-2$ for any channel that can be factored. By contrapositive, if $\kappa({{\mathcal{E}}})\geq d-1$, ${{\mathcal{E}}}$ cannot be factorized.
Our examples in Sections \[sec:ETB\_channels\] and \[sec:Schur\_channels\] show that it is possible for a channel to be in this gap, and thus we know that neither example can be written as a product of two channels. We summarize this into the following corollary.
\[cor-not-factorable\] For any $d>2$, the channels on ${\mathcal{M}}_d$ described in Example \[sec:Schur\_channels\] and in Theorem \[thm:etb\_kappa\_d\] (with $r=d-1$ or $r=d$) cannot be factored into tensor product of channels acting on smaller subsystems of ${\mathcal{M}}_d$.
Note that for a unitary $u\in {\mathcal{M}}_d\otimes\mathcal{M}_{d'}$ which is not of the form $u_1\otimes u_2$, where $u_1$ is a unitary in ${\mathcal{M}}_d$ and $u_2$ is a unitary in $\mathcal{M}_{d'}$ respectively, the unitary channel on ${\mathcal{M}}_d\otimes\mathcal{M}_{d'}$ defined by $$Ad_{u}(x)=uxu^*, \ \forall x\in {\mathcal{M}}_d\otimes\mathcal{M}_{d'},$$ is a channel that cannot be factored into product channels (See Example 6.17 in [@math-language-qit]). The Swap operator is one such unitary, where the Swap operator ($W$) is defined on the product of two Hilbert spaces by $W(\xi\otimes\eta)=\eta\otimes \xi$. Although for unitary conjugation maps, there are ways (see [@Busch]) to decide if they can be factored into two unitary channels or not, for arbitrary channels, it’s a hard problem to decide. As the eigenvalues of product channels are all possible product of the constituent channels, analysing the set of eigenvalues is one such criteria. Even when the eigenvalues match up, Theorem \[thm:index\_factor\] provides a new criteria to decide if a channel can be factored into product channels and Corollary \[cor-not-factorable\] demonstrates the use of this particular criteria.
Separable Channels {#separable_channels}
------------------
While a channel may not be a product of two channels, it could be a convex combination of product channels, known as a separable channel.
A channel ${{\mathcal{E}}}:{\mathcal{M}}_d\otimes {\mathcal{M}}_c\rightarrow {\mathcal{M}}_d\otimes {\mathcal{M}}_c$ is called *separable* if $${{\mathcal{E}}}=\sum_{i=1}^n\lambda_i{{\mathcal{E}}}_{i_1}\otimes {{\mathcal{E}}}_{i_2},$$ where ${{\mathcal{E}}}_{i_1}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ and ${{\mathcal{E}}}_{i_2}:{\mathcal{M}}_c\rightarrow {\mathcal{M}}_c$ are quantum channels, $\lambda_i\geq 0$, and $\sum_{i=1}^n\lambda_i=1$.
There has been much interest in the separability of states ([@math-language-qit], [@Watrous-book], [@Bengtsson]), but less so in the separability of channels ([@Watrous-book] Chapter 6). Note that the term “separable channel" is not universal. Some authors prefer to use separable channels just to refer to the sum of product channels; however, we use “separable channels" to mean convex combination of product channels. Unfortunately, our technique fails to provide answers on separability, as convex combinations can increase the multiplicative index of a channel. We will use the following lemma to examine this:
\[lem:convex\_mult\] Let ${{\mathcal{E}}}=\lambda{{\mathcal{E}}}_1+(1-\lambda){{\mathcal{E}}}_2$. Then $${\mathcal{M}}_{{{\mathcal{E}}}^k}={\mathcal{M}}_{{{\mathcal{E}}}_1^k}\cap {\mathcal{M}}_{{{\mathcal{E}}}_2^k}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1^n(x)={{\mathcal{E}}}_2^n(x),1\leq n\leq k\}.$$
We will prove this by induction. The result is already established for $k=1$ by Choi ([@choi1], Theorem 3.3) where it was proved that with ${{\mathcal{E}}},{{\mathcal{E}}}_1,{{\mathcal{E}}}_2$ as above we have $${\mathcal{M}}_{{\mathcal{E}}}={\mathcal{M}}_{{{\mathcal{E}}}_1}\cap{\mathcal{M}}_{{{\mathcal{E}}}_2}\cap\{x\in {\mathcal{M}}_d \ | \ {{\mathcal{E}}}_1(x)={{\mathcal{E}}}_2(x)\}.$$
so let $k\in{\mathbb{N}}$ and assume the inductive hypothesis. Then $${{\mathcal{E}}}^k=\lambda{{\mathcal{E}}}_1{{\mathcal{E}}}^{k-1}+(1-\lambda){{\mathcal{E}}}_2{{\mathcal{E}}}^{k-1}.$$ Again by using Choi’s theorem, this makes: $${\mathcal{M}}_{{{\mathcal{E}}}^k}={\mathcal{M}}_{{{\mathcal{E}}}_1{{\mathcal{E}}}^{k-1}}\cap {\mathcal{M}}_{{{\mathcal{E}}}_2{{\mathcal{E}}}^{k-1}}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1{{\mathcal{E}}}^{k-1}(x)={{\mathcal{E}}}_2{{\mathcal{E}}}^{k-1}(x)\}.$$ Note that by Lemma \[lem:md\_composition\] the first two sets are subsets of ${\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}$, so we can assume any $x$ in the last set is in ${\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}$. Thus, we can rewrite it as: $${\mathcal{M}}_{{{\mathcal{E}}}^k}={\mathcal{M}}_{{{\mathcal{E}}}_1{{\mathcal{E}}}^{k-1}}\cap {\mathcal{M}}_{{{\mathcal{E}}}_2{{\mathcal{E}}}^{k-1}}\cap \{x\in {\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}|{{\mathcal{E}}}_1{{\mathcal{E}}}^{k-1}(x)={{\mathcal{E}}}_2{{\mathcal{E}}}^{k-1}(x)\}.$$ We can write ${{\mathcal{E}}}^{k-1}=\sum_{i=1}^{2^{k-1}}c_i{{\mathcal{E}}}_{i_1}\cdots{{\mathcal{E}}}_{i_{k-1}}$ (where $i_j\in\{1,2\}$ and $\sum_ic_i=1$). If $x\in {\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}$ then by the inductive hypothesis this means ${{\mathcal{E}}}_1^n(x)={{\mathcal{E}}}_2^n(x)$, for all $n\leq k-1$. Thus, $$\begin{aligned}
{{\mathcal{E}}}_{i_1}\cdots{{\mathcal{E}}}_{i_{k-1}}(x)=&{{\mathcal{E}}}_{i_1}\cdots{{\mathcal{E}}}_{i_{k-2}}{{\mathcal{E}}}_{i_{k-2}}(x)\\
=&{{\mathcal{E}}}_{i_1}\cdots{{\mathcal{E}}}_{i_{k-3}}{{\mathcal{E}}}_{i_{k-3}}^2(x)\\
\cdots&\\
=&{{\mathcal{E}}}_{i_1}{{\mathcal{E}}}_{i_1}^{k-2}(x)\\
=&{{\mathcal{E}}}_1^{k-1}(x)={{\mathcal{E}}}_2^{k-1}(x)
\end{aligned}$$ Thus, ${{\mathcal{E}}}^{k-1}(x)=\sum_{i=1}^{2^{k-1}}c_i{{\mathcal{E}}}_1^{k-1}(x)={{\mathcal{E}}}_1^{k-1}(x)={{\mathcal{E}}}_2^{k-1}(x)$. Hence: $$\begin{aligned}
{\mathcal{M}}_{{{\mathcal{E}}}^k}=&{\mathcal{M}}_{{{\mathcal{E}}}_1{{\mathcal{E}}}^{k-1}}\cap {\mathcal{M}}_{{{\mathcal{E}}}_2{{\mathcal{E}}}^{k-1}}\cap \{x\in {\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}|{{\mathcal{E}}}_1{{\mathcal{E}}}_1^{k-1}(x)={{\mathcal{E}}}_2{{\mathcal{E}}}_2^{k-1}(x)\}\\
=&{\mathcal{M}}_{{{\mathcal{E}}}_1{{\mathcal{E}}}^{k-1}}\cap {\mathcal{M}}_{{{\mathcal{E}}}_2{{\mathcal{E}}}^{k-1}}\cap {\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1^k(x)={{\mathcal{E}}}_2^k(x)\}
\end{aligned}$$ For the first term, Lemma \[lem:md\_composition\] gives $${\mathcal{M}}_{{{\mathcal{E}}}_1{{\mathcal{E}}}^{k-1}}=\{x\in {\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}|{{\mathcal{E}}}_1(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_ 1}\}={\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_1}\}.$$ Similarly for ${\mathcal{M}}_{{{\mathcal{E}}}_2{{\mathcal{E}}}^{k-1}}$. Combining these results and using the inductive hypothesis for the structure of ${\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}$: $$\begin{aligned}
{\mathcal{M}}_{{{\mathcal{E}}}^k}=&\{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_1}\}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_2(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_2}\}\\
&\cap {\mathcal{M}}_{{{\mathcal{E}}}^{k-1}}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1^k(x)={{\mathcal{E}}}_2^k(x)\}\\
=&\{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_1}\}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_2(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_2}\}\cap {\mathcal{M}}_{{{\mathcal{E}}}_1^{k-1}}\cap {\mathcal{M}}_{{{\mathcal{E}}}_2^{k-1}}\\
&\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1^n(x)={{\mathcal{E}}}_2^n(x), 1\leq n\leq k-1\}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1^k(x)={{\mathcal{E}}}_2^k(x)\}\\
=&{\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}x\in {\mathcal{M}}_{{{\mathcal{E}}}_1^{k-1}}\midsetl{{\mathcal{E}}}_1(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_1}\rightset\cap {\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}x\in {\mathcal{M}}_{{{\mathcal{E}}}_2^{k-1}}\midsetl{{\mathcal{E}}}_2(x)\in {\mathcal{M}}_{{{\mathcal{E}}}_2}\rightset\\
&\cap\{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1^n(x)={{\mathcal{E}}}_2^n(x), 1\leq n\leq k\}\\
=&{\mathcal{M}}_{{{\mathcal{E}}}_1^k}\cap {\mathcal{M}}_{{{\mathcal{E}}}_2^k}\cap \{x\in {\mathcal{M}}_d|{{\mathcal{E}}}_1^n(x)={{\mathcal{E}}}_2^n(x), 1\leq n\leq k\}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:MUCP_convex}
\end{aligned}$$
The problem that allows $\kappa({{\mathcal{E}}})>\max\{\kappa({{\mathcal{E}}}_1),\kappa({{\mathcal{E}}}_1)\}$ is the final set: It might be that the two channels stabilize quickly, but they have different actions on their stabilized multiplicative domains. As an example, consider the two channels ${{\mathcal{E}}}_1,{{\mathcal{E}}}_2:{\mathcal{M}}_3\rightarrow {\mathcal{M}}_3$ given by: $${{\mathcal{E}}}_1(x)=\frac{1}{3}\begin{pmatrix}1&1&1\\\omega&1&\omega^2\\\omega^2&1&\omega\end{pmatrix}x\begin{pmatrix}1&\omega&\omega^2\\1&1&1\\1&\omega^2&\omega\end{pmatrix},$$ $${{\mathcal{E}}}_2(x)=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}x\begin{pmatrix}0&0&1\\1&0&0\\0&0&1\end{pmatrix},$$ where $\omega=e^{i2\pi/3}$. Both of these channels have multiplicative index 1, since they are unitary and ${\mathcal{M}}_{{\mathcal{E}}}={\mathcal{M}}_{{{\mathcal{E}}}^\infty}={\mathcal{M}}_3$. Let ${{\mathcal{E}}}=\tfrac{1}{2}{{\mathcal{E}}}_1+\tfrac{1}{2}{{\mathcal{E}}}_2$. We calculated that $${\mathcal{M}}_{{\mathcal{E}}}={\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}\begin{pmatrix} a&b&-c\\b&2b+a-c&b\\-c&b&a\end{pmatrix}\midsetl a,b,c\in{\mathbb{C}}\rightset,\, {\mathcal{M}}_{{{\mathcal{E}}}^2}={\mathbb{C}}1$$ and thus $\kappa({{\mathcal{E}}})=2$, greater than the multiplicative index of either channel in the convex combination.
Extending this idea, we let ${{\mathcal{E}}}:{\mathcal{M}}_9\rightarrow {\mathcal{M}}_9$ be defined as $${{\mathcal{E}}}=\tfrac{1}{2}{{\mathcal{E}}}_1\otimes {{\mathcal{E}}}_1 + \tfrac{1}{2}{{\mathcal{E}}}_2\otimes {{\mathcal{E}}}_2.$$ This is a separable channel, and both product channels in the convex combination have a multiplicative index of 1 (by Theorem \[thm:kappa\_bound\]) but using numerical methods, we found that $\kappa({{\mathcal{E}}})=2$. Thus, we do not have a direct analog of Theorem \[thm:index\_factor\] for separable channels.
However, the size of the multiplicative domain of a convex combination of channels is limited by the size of the multiplicative domain of each channel in the convex combination; thus, for the multiplicative index to be higher than any of the constituent channels, the last set in Equation \[eq:MUCP\_convex\] must be the limiting factor. As a specific case of this, we have the following proposition. For the proof, recall that for three algebras ${\mathscr{A}},{\mathscr{B}},\mathscr{C}$, if $1\in {\mathscr{B}}$, then: $$(1\otimes {\mathscr{A}})\cap ({\mathscr{B}}\otimes\mathscr{C})=1\otimes ({\mathscr{A}}\cap\mathscr{C}).$$
Let $\{{{\mathcal{E}}}_1,\cdots,{{\mathcal{E}}}_n\}$ be unital quantum channels on ${\mathcal{M}}_d$ such that ${\mathcal{M}}_{{{\mathcal{E}}}_i^\infty}={\mathbb{C}}1$ for some $i$ and $\{\Psi_1,\cdots,\Psi_n\}$ be unital channels on ${\mathcal{M}}_c$. Then if $${{\mathcal{E}}}=\sum_{i=1}^n\lambda_i{{\mathcal{E}}}_i\otimes\Psi_i$$ where $\lambda_i\geq 0,\sum_i\lambda_i=1$, then $\kappa({{\mathcal{E}}})\leq \max\{2d-2,2c-2\}$.
Let $k\geq\max\{2d-2,2c-2\}$. Using Lemma \[lem:convex\_mult\] and Theorem \[thm:md\_tensor\], we have that: $$\begin{aligned}
{\mathcal{M}}_{{{\mathcal{E}}}^k}=&\bigcap_{i=1}^n {\mathcal{M}}_{{{\mathcal{E}}}_i^k\otimes \Psi_i^k} \cap {\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}x\in {\mathcal{M}}_{dc}\midsetr {{\mathcal{E}}}_i^k\otimes\Psi_i^k(x)={{\mathcal{E}}}_j^k\otimes \Psi_j^k(x),1\leq i,j\leq n\leq k\rightset\\
=&\bigcap_{i=1}^n {\mathcal{M}}_{{{\mathcal{E}}}_i^\infty}\otimes {\mathcal{M}}_{\Psi_i^\infty} \cap {\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}x\in {\mathcal{M}}_{dc}\midsetr {{\mathcal{E}}}_i^k\otimes\Psi_i^k(x)={{\mathcal{E}}}_j^k\otimes \Psi_j^k(x),1\leq i,j\leq n\leq k\rightset
\end{aligned}$$ Since ${\mathcal{M}}_{{{\mathcal{E}}}_i^\infty}={\mathbb{C}}1$ for some $i$, $$\bigcap_{i=1}^n {\mathcal{M}}_{{{\mathcal{E}}}_i^\infty}\otimes {\mathcal{M}}_{\Psi_i^\infty}={\mathbb{C}}1\otimes \bigcap_{i=1}^n {\mathcal{M}}_{\Psi_i^\infty}.$$ This gives us: $$\begin{aligned}
{\mathcal{M}}_{{{\mathcal{E}}}^k}=&{\mathbb{C}}1\otimes \bigcap_{i=1}^n {\mathcal{M}}_{\Psi_i^\infty}\cap{\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}x\in {\mathcal{M}}_{dc}\midsetr {{\mathcal{E}}}_i^k\otimes\Psi_i^k(x)={{\mathcal{E}}}_j^k\otimes \Psi_j^k(x),1\leq i,j\leq n\leq k\rightset\\
=&{\mathbb{C}}1\otimes \bigcap_{i=1}^n {\mathcal{M}}_{\Psi_i^\infty}\cap{\left\{\left.}
\newcommand{\midsetl}{\right\vert\left.}
\newcommand{\midsetr}{\right.\left\vert}
\newcommand{\rightset}{\right.\right\}}1\otimes x\midsetr x\in {\mathcal{M}}_c, \Psi_i^k( x)= \Psi_j^k(x),1\leq i,j\leq n\leq k\rightset\\
=&{\mathbb{C}}1\otimes {\mathcal{M}}_{\Psi^k}
\end{aligned}$$ where $\Psi=\sum_{i=1}^n\lambda_i\Psi_i$. Since $\Psi:{\mathcal{M}}_c\rightarrow {\mathcal{M}}_c$, then by Theorem \[thm:kappa\_bound\], $\kappa(\Psi)\leq 2c-2\leq k$, so ${\mathcal{M}}_{\Psi^k}={\mathcal{M}}_{\Psi^\infty}$. Thus: $${\mathcal{M}}_{{{\mathcal{E}}}^k}={\mathbb{C}}1\otimes {\mathcal{M}}_{\Psi^\infty}$$ for all $k\geq\max\{2d-2,2c-2\}$; hence the multiplicative domain is stable, so $\kappa({{\mathcal{E}}})\leq\max\{2d-2,2c-2\}$.
The contrapositive states that if ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ is separable and $\kappa({{\mathcal{E}}})\geq d-1$, then none of the channels in the decomposition of ${{\mathcal{E}}}$ can have a trivial stabilized multiplicative domain.
Products of Strictly Contractive Channels {#Sec:strictly-contractive}
=========================================
Let $\mathfrak{d}$ be any distinguishability measure which is monotone with respect to a quantum channel. More precisely let ${{\mathcal{E}}}:{\mathcal{M}_{d}}\rightarrow{\mathcal{M}_{d}}$ be a channel such that the following holds $$\mathfrak{d}({{\mathcal{E}}}(\rho),{{\mathcal{E}}}(\sigma))\leq\mathfrak{d}(\rho,\sigma),$$ for all density matrices $\rho,\sigma$. A wide class of examples are the quantum f-divergence ([@f-div1], [@f-div2]), quantum relative entropy ([@Watrous-book], Theorem 5.38), quantum R[é]{}nyi entropy ([@Frank-monotonicity]), fidelity ([@Watrous-book], Theorem 3.30), and trace metric ([@math-language-qit], Proposition 4.37). A channel that does not preserve the measure for any pair of distinct density matrices is called a **strictly contractive channel**. Formally we define:
Given a distinguishability measure $\mathfrak{d}$, a channel ${{\mathcal{E}}}:{\mathcal{M}}_d\rightarrow{\mathcal{M}}_d$ is called a strictly contractive channel with respect to $\mathfrak{d}$ if for every pair of distinct density operators $\rho,\sigma$ in ${\mathcal{M}}_d$, we have $$\mathfrak{d}({{{\mathcal{E}}}(\rho),{{\mathcal{E}}}(\sigma)})<\mathfrak{d}(\rho,\sigma)$$
Note that in ([@Raginsky] and [@Doug-Miza]) the authors studied maps with respect to the trace metric and the Bures metric respectively and showed that these maps are dense in the set of all quantum channels. Since such channels are found in abundance, we study contractive maps with respect to any distinguishability measure. For these maps we have the following observation:
\[strictly-contractive\] For unital strictly contractive channels ${\mathcal{E}}$ as defined above, the multiplicative domain ${\mathcal{M}_{\mathcal{E}}}$ is trivial, that is, ${\mathcal{M}_{\mathcal{E}}}=\mathbb{C}1$.
Suppose ${\mathcal{M}_{\mathcal{E}}}$ is not trivial. Because $1$ is always in ${\mathcal{M}_{\mathcal{E}}}$, we assume that there exists at least one non-trivial element in ${\mathcal{M}_{\mathcal{E}}}$. Since for unital channels we have the relation ${\mathcal{M}_{\mathcal{E}}}=\rm{Fix}_{{\mathcal{E}}^*\circ{\mathcal{E}}}$, using the monotonicity property of $\mathfrak{d}$ under ${\mathcal{E}}$ we have that for any $\rho,\sigma\in {\mathcal{M}_{\mathcal{E}}}$, $$\mathfrak{d}(\rho,\sigma)=\mathfrak{d}({\mathcal{E}}^*\circ{\mathcal{E}}(\rho),{\mathcal{E}}^*\circ{\mathcal{E}}(\sigma))\leq\mathfrak{d}({\mathcal{E}}(\rho),{\mathcal{E}}(\sigma))<\mathfrak{d}(\rho,\sigma).$$ So we arrive at a contradiction and hence ${\mathcal{M}_{\mathcal{E}}}=\mathbb{C}1$.
As pointed out in ([@Raginsky]), it is a hard problem to decide whether the tensor product of strictly contractive channels is strictly contractive or not. We provide an answer in affirmative if the measure allows the recovery map.
Among all the distinguishability measures, there are certain measures that allow reversibility of a channel. Given a distinguishable measure $\mu$, a channel ${{\mathcal{E}}}$ is called reversible on a certain subset $\mathcal{S}$ of density operators if there exists a recovery map $\mathcal{R}$ such that $\mathcal{R}\circ{\mathcal{E}}(x)=x$, for all $x\in \mathcal{S}$. This property is often referred to as sufficiency ([@Jencova1], [@Jencova2]). There are certain measures which allow reversible maps if the measure of a pair of density matrices is preserved by the channel (for example Renyi entropy, quantum entropy etc. ([@f-div2], [@Jencova1], [@Jencova2])). However there are some measures (trace, fidelity ([@Mosonyi], [@Jencova1])) that do not allow the recovery maps even if the channel preserves these measure for some density operators. We denote $\mathfrak{D}_{R}$ to be the measures that allow unital recovery maps if the channel preserves the measure of two density operators. For these measures we have the following theorem:
Let $\mathfrak{d}\in \mathfrak{D}_{R}$. With respect to the measure $\mathfrak{d}$, two unital channels ${{\mathcal{E}}}_1,{{\mathcal{E}}}_2$ are strictly contractive if and only if ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ is also strictly contractive.
Suppose both ${{\mathcal{E}}}_1$ and ${{\mathcal{E}}}_2$ are strictly contractive but ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ is not. Then there exists a pair of distnct $\rho,\sigma\in {\mathcal{M}_{d}}\otimes{\mathcal{M}_{d}}$ such that $$\mathfrak{d}(\rho,\sigma)=\mathfrak{d}({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2(\rho),{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2(\sigma)).$$ Now since $\mathfrak{d}$ allows a recovery map, we get a (unital) channel $\mathcal{R}$ such that $$\mathcal{R}\circ({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)(\rho)=\rho \ \text{and} \ \mathcal{R}\circ({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)(\sigma)=\sigma.$$ Now this means ([@miza], Proposition 5.2) we get $\rho,\sigma\in \mathcal{M}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}$. Since we get from Theorem \[thm:md\_tensor\] that $\mathcal{M}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}={\mathcal{M}}_{{{\mathcal{E}}}_1}\otimes\mathcal{M}_
{{{\mathcal{E}}}_2}$ and we know by Lemma \[strictly-contractive\] that ${\mathcal{M}}_{{{\mathcal{E}}}_1}=\mathbb{C}1=\mathcal{M}_{{{\mathcal{E}}}_2}$, we obtain $\mathcal{M}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}=\mathbb{C}1$, contradicting the existence of the pair $\rho,\sigma\in {\mathcal{M}_{d}}\otimes{\mathcal{M}_{d}}$. Hence ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ is strictly contractive.
For the converse, if, without loss of generality, ${{\mathcal{E}}}_1$ is not strictly contractive, then it admits a recovery map $\mathcal{R}_1$ such that there are two linearly independent matrices $\rho,\sigma$ with $\mathcal{R}_1\circ{{\mathcal{E}}}_1(\rho)=\rho$ and similarly for $\sigma$. But then the map $\mathcal{R}_1\otimes 1$ will reverse the action of ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ on the matrices $\rho\otimes 1$ and $\sigma\otimes 1$. This would imply that ${{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2$ cannot be strictly contractive.
Applications {#sec:error_correction}
============
In the context of quantum information theory, the multiplicative domain of a channel was studied in connection to the scheme of error correction ([@choi2], [@johnston]). Specifically, Theorem 11 in [@choi2] states that the multiplicative domain is precisely the algebra over the largest unitarily correctable code (UCC). Now suppose we take two channels ${{\mathcal{E}}}_1:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ and ${{\mathcal{E}}}_2:{\mathcal{M}}_c\rightarrow {\mathcal{M}}_c$ and consider the correctable codes of ${{\mathcal{E}}}_1\otimes {{\mathcal{E}}}_2$. Clearly, if ${\mathcal}{C}_1$ and ${\mathcal}{C}_2$ are unitarily correctable codes for ${{\mathcal{E}}}_1$ and ${{\mathcal{E}}}_2$, respectively, then ${\mathcal}{C}_1\otimes{\mathcal}{C}_2$ is a correctable code for ${{\mathcal{E}}}_1\otimes {{\mathcal{E}}}_2$; however, we would like to have a larger code. Is there a code ${\mathcal}{C}$ that cannot be decomposed into ${\mathcal}{C}={\mathcal}{C}_1\otimes{\mathcal}{C}_2$? Using Theorem \[thm:md\_tensor\], we can show that for unital channels, there are no larger unitarily correctable codes.
Let ${{\mathcal{E}}}_1:{\mathcal{M}}_d\rightarrow {\mathcal{M}}_d$ and ${{\mathcal{E}}}_2:{\mathcal{M}}_c\rightarrow {\mathcal{M}}_c$ be two unital quantum channels. Then $UCC({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)$ is precisely $UCC({{\mathcal{E}}}_1)\otimes UCC({{\mathcal{E}}}_2)$.
Using Theorem 11 from [@choi2], $UCC({{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2)={\mathcal{M}}_{{{\mathcal{E}}}_1\otimes{{\mathcal{E}}}_2}$, and using Theorem \[thm:md\_tensor\], this is equal to ${\mathcal{M}}_{{{\mathcal{E}}}_1}\otimes {\mathcal{M}}_{{{\mathcal{E}}}_2}=UCC({{\mathcal{E}}}_1)\otimes UCC({{\mathcal{E}}}_2)$.
For separable channels, the situation does not improve either. If ${{\mathcal{E}}}=\sum_{i=1}^n\lambda_i{{\mathcal{E}}}_i\otimes\Psi_i$, then by Choi’s theorem ([@choi1], Theorem 3.3), ${\mathcal{M}}_{{\mathcal{E}}}\subseteq {\mathcal{M}}_{{{\mathcal{E}}}_i}\otimes {\mathcal{M}}_{\Psi_i}$ for all $i$. Thus, the largest unitarily correctable code of a separable unital channel is at most as large as the tensor product of the largest unitarily correctable codes for any of its constituent product channels.
As a natural converse question, if we start with an error correcting code for a channel ${{\mathcal{E}}}:{\mathcal{M}}_d\otimes {\mathcal{M}}_c\rightarrow {\mathcal{M}}_d\otimes {\mathcal{M}}_c$, can we correct the code with a unital product channel ${\mathcal}{R}_1\otimes{\mathcal}{R}_2$? First, a lemma:
\[lem:ucc\_kappa\] If ${{\mathcal{E}}}$ is a unital channel, ${\mathcal}{C}$ is the largest $UCC$ (i.e., the multiplicative domain), and ${\mathcal}{R}$ corrects ${\mathcal}{C}$, then $\kappa({\mathcal}{R})\geq\kappa({{\mathcal{E}}})$.
Let $k\leq\kappa({{\mathcal{E}}})$. As a first result, ${\mathcal}{R}^k$ inverts ${{\mathcal{E}}}^k$ on ${\mathcal{M}}_{{{\mathcal{E}}}^k}$. To see this, let $x$ be in ${\mathcal{M}}_{{{\mathcal{E}}}^k}$. Then ${{\mathcal{E}}}^k(x)$ is in the image ${{\mathcal{E}}}({\mathcal{M}}_{{\mathcal{E}}})$. Since ${\mathcal}{R}\circ{{\mathcal{E}}}$ is the identity on ${\mathcal{M}}_{{\mathcal{E}}}$, then ${\mathcal}{R}{{\mathcal{E}}}^k(x)={{\mathcal{E}}}^{k-1}(x)$. Repeating this argument with each ${\mathcal}{R}$ gives ${\mathcal}{R}^k{{\mathcal{E}}}^k(x)=x$.
If $p$ is any projection in ${\mathcal{M}}_{{{\mathcal{E}}}^k}$ of rank $r$, then ${{\mathcal{E}}}^k(p)$ is also a projection of rank $r$. Since ${\mathcal}{R}^k({{\mathcal{E}}}^k(p))=p$, then ${\mathcal}{R}^k$ takes a rank-$r$ projection to a rank-$r$ projection - in other words, ${{\mathcal{E}}}^k(p)$ is in ${\mathcal{M}}_{ {\mathcal}{R}^k}$.
By Theorem 2.5 from [@miza], there is a basis of ${\mathcal{M}}_{{\mathcal}{R}^\infty}$ consisting of peripheral eigenoperators of ${\mathcal}{R}$. Let $x$ be a such a basis element for some eigenvalue $\lambda$. Suppose $x$ is the image ${{\mathcal{E}}}^k(y)$ for some $k<\kappa({{\mathcal{E}}})$ and some $y\in {\mathcal{M}}_{{{\mathcal{E}}}}\setminus {\mathcal{M}}_{{{\mathcal{E}}}^\infty}$. Then we would have that ${\mathcal}{R}^k(x)={\mathcal}{R}^k({{\mathcal{E}}}^k(y))=y$, but we also have that ${\mathcal}{R}^k(x)=\lambda^kx$. Thus ${{\mathcal{E}}}^k(y)=\lambda^{-k}y$ meaning $y\in {\mathcal{M}}_{{{\mathcal{E}}}^\infty}$, a contradiction. Thus, $x$ is not in the image of ${{\mathcal{E}}}^k({\mathcal{M}}_{{\mathcal{E}}}\setminus {\mathcal{M}}_{{{\mathcal{E}}}^\infty})$. This means that if we take $x'\in {\mathcal{M}}_{{{\mathcal{E}}}^{\kappa-1}}\setminus {\mathcal{M}}_{{{\mathcal{E}}}^\infty}$, then ${{\mathcal{E}}}^{\kappa-1}(x')$ is in ${\mathcal{M}}_{{\mathcal}{R}^{\kappa-1}}$ but is not in ${\mathcal{M}}_{{\mathcal}{R}^\infty}$. Thus, $\kappa({\mathcal}{R})\geq\kappa({{\mathcal{E}}})$.
As an immediate corollary, we have an extension of Theorem \[thm:index\_factor\] for unital error corrections:
If ${\mathcal}{C}$ is the largest $UCC$ for a unital channel ${{\mathcal{E}}}:{\mathcal{M}}_d\otimes {\mathcal{M}}_c\rightarrow {\mathcal{M}}_d\otimes {\mathcal{M}}_c$ and $\kappa({{\mathcal{E}}})\geq \max\{2(d-1),2(c-1)\}$, then ${\mathcal}{C}$ cannot be corrected by any product channel ${\mathcal}{R}_1\otimes{\mathcal}{R}_2$. So if the recovery operator is chosen to be a unitary, then it must be a global unitary channel.
Let ${\mathcal}{R}$ be any channel that corrects ${\mathcal}{C}$. From Lemma \[lem:ucc\_kappa\], $\kappa({\mathcal}{R})\geq\max\{2(d-1),2(c-1)\}$, and so from Theorem \[thm:index\_factor\], ${\mathcal}{R}$ cannot be written as a product channel.
Note that ${{\mathcal{E}}}^*$ will also correct ${\mathcal}{C}$ for ${{\mathcal{E}}}$, and ${{\mathcal{E}}}$ will correct ${{\mathcal{E}}}({\mathcal}{C})$ for ${{\mathcal{E}}}^*$, so we have a final corollary:
For a unital channel ${{\mathcal{E}}}$, $\kappa({{\mathcal{E}}})=\kappa({{\mathcal{E}}}^*)$.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was initiated at the University of Regina under the supervision of Dr. D. Farenick and supported in part by an NSERC Discovery Grant (Farenick) and a University of Regina Graduate Student Research Fellowship.
The first author acknowledges the NSERC Canadian Graduate Scholarship and funding from the Department of Combinatorics and Optimization at the University of Waterloo, and the second author acknowledges the Post-Doctoral Fellowship of the Department of Pure Mathematics at the University of Waterloo.
We also want to thank our referees, whose suggestions made the proofs of Theorem \[thm:any\_MUPSA\] and Proposition \[prop:etb\_md\_struct\] much shorter.
[^1]: [*Key words:* Quantum channel; Multiplicative domain; Spectral property; Tensor product; Fixed points]{}
|
---
abstract: 'Using a quantumlike description for light propagation in nonhomogeneous optical fibers, quantum information processing can be implemented by optical means. Quantum-like bits (qulbits) are associated to light modes in the optical fiber and quantum gates to segments of the fiber providing an unitary transformation of the mode structure along a space direction. Simulation of nonlinear quantum effects is also discussed.'
author:
- 'M. A. Man’ko[^1] [^2] , V. I. Man’ko and R. Vilela Mendes[^3]'
title: 'Quantum computation by quantum-like systems'
---
PACS: 03.67.Lx
Introduction
============
The quantum computer idea [@Ben; @Feyn; @Deut] uses the possibility to code numerical information by vectors in Hilbert space. In the simplest case, a two-dimensional Hilbert space is said to code a [*qubit*]{}. A physical realization of a qubit might be a spin 1/2 particle or a two-level atom. Calculations are carried out by unitary time-evolution transforming an input state vector into a final state vector in Hilbert space. Several characteristic features distinguish the quantum computation paradigm from classical computation. First, because of the superposition principle, the qubit space contains all the complex linear combinations, rather than just two states as in a classical bit. Second, in a quantum space of $n$ qubits (of dimensionality $2^{n}$) there are both factorized and entangled states, the latter having no correspondence in the space of $n$ classical bits. Third, quantum evolution operates simultaneously in the $2^{n}$ states, implying intrinsic exponential parallelism of quantum computation. To extract the result from a quantum computation one has to observe the system, that is, to project it in one of the exponentially many states, thus losing most of the exponential amount of information generated by the unitary evolution. However, it is possible to take advantage of the exponentiality of quantum computation using the interference mechanism characteristic of quantum mechanics. In short, it is the combined effect of [*entanglement*]{}, [*exponential parallelism*]{} and [*interference*]{} that may allow the full potential of quantum computation to be realized [@Aharonov].
According to the Church thesis, a classical Turing machine can simulate the computation of any computable function. Therefore a classical computer can simulate any computation of a quantum computer. The problem is how long the simulation will take to run. In particular it is know that some problems that are computed in exponential time by classical computers might be solved in polynomial time by quantum computers. Therefore, to take full advantage of the quantum computation algorithms, one requires physical devices obeying quantum laws.
There are many classical systems that physically implement some of the features of quantum computation[@Spree; @FP; @CerfPR; @Kwiat]. For example electromagnetic waves may be linearly superimposed and interfere. Nevertheless, given the qualitative differences between classical and quantum mechanics, no classical system, [*where computations correspond to evolution in real time*]{}, may ever implement simultaneously all the features of quantum computation. Otherwise we would have proved the physical equivalence of classical and quantum mechanics.
Notice however that quantum computation [*is not*]{} quantum mechanics. Quantum computation is a mathematical algorithm that uses all the mathematical features of quantum mechanics. In particular it is irrelevant for the algorithm how the Hilbert space is physically implemented and whether the unitary evolution is taking place along a real time direction or along some other coordinate. It is here that quantum-like systems may play a role. Quantum-like systems are classical systems which obey equations formally identical to the Schrödinger equation, but where the role of time is played by a space coordinate. Therefore, insofar as they obey equations mathematically identical to those of quantum mechanics, they may implement all the quantum computation operations, provided the unitary evolution is interpreted not as evolution in time, but as evolution along a space coordinate. Because of this exchange of the role of the coordinates, there is no contradiction with the non-equivalence of classical and quantum mechanics.
In Sect.2 we give an overview of quantum-like systems and then, concentrating on fiber optics phenomena, we discuss how quantum information may be coded on the fiber and the kind of non-homogeneities and interactions needed to implement a set of universal quantum gates. Here we have concentrated on harmonic light modes. An alternative scheme might be developed based on soliton propagation on the fibers.
The fact that the unitary evolution needed for a quantum gate is obtained by setting up a nonuniform refraction index profile along the fiber, means that the unitary evolution becomes permanently coded in the hardware. This may be physically more convenient than to control a sequence of operations each time the gate is operated.
Abrams and Lloyd[@Abrams] have suggested that a stronger computation model would be obtained with non-linear terms in the quantum evolution. So far, all experimental evidence favors strict linearity of quantum mechanics. However, this is not so in quantum-like systems where non-linear effects may easily be introduced. Actual implementations of non-linear quantum computation might therefore use quantum-like systems.
Quantum-like systems
====================
More than half a century ago, Fock and Leontovich have shown that paraxial beams of electromagnetic radiation in the parabolic approximation can be described by a Schrödinger-like equation [@Leon; @Fock-Leon]. The role of time in this equation is played by the spatial (longitudinal) coordinate of the light beam, the role of Planck’s constant is played by the light wavelength, and the role of potential energy by the index of refraction of the medium. Thus, the paraxial beam of light, a purely classical object, is a quantum-like system obeying equations formally identical to those of quantum mechanics.
Given the Helmholz equation for a component of the electric field, obtained for a fixed frequency, neglecting media dispersion and polarization $$\frac{\partial ^{2}E}{\partial ^{2}x}+\frac{\partial ^{2}E}{\partial z^{2}}%
+k^{2}n^{2}(x,z)\,E=0 \label{5}$$ (we use the planar configuration, $\lambda =2\pi /k$ is the wavelength in vacuum, and $z$ the longitudinal coordinate).
Introduce the complex function $\psi (x,z)$, which is the slowly varying amplitude of the electric field in $$E(x,z)=n_{0}^{-1/2}(z)\,\psi (x,z)\,\exp \left[ ik\int_{0}^{z}n_{0}(\xi
)~d\xi \right] \label{4}$$ The ansatz (\[4\]) reduces the Helmholz equation (\[5\]) to a Schrödinger-like equation $$i\lambda \,\frac{\partial \psi (x,z)}{\partial z}=-\frac{\lambda ^{2}}{4\pi
n_{0}(z)}\,\frac{\partial ^{2}\psi (x,z)}{\partial x^{2}}+U(x,z)\,\psi (x,z)
\label{3}$$ $U(x,z)$ being an effective potential related to the index of refraction of the medium, $n(x,z)$$$U(x,z)=\frac{\pi }{n_{0}(z)}\,\left[ n_{0}^{2}(z)-n^{2}(x,z)\right] \,$$ and $n_{0}(z)=n(0,z)$ the index of refraction at the beam axis. In deriving this equation, second-order $z-$derivatives of $\psi $ and derivatives of the function $n_{0}(z)$ were neglected. This is justified for slow variation of the index of refraction along the beam axis over distances of order of one wavelength $$\frac{\lambda }{n_{0}^{2}(z)}\left| \frac{dn_{0}(z)}{dz}\right| \ll 1\,.$$
The Fock–Leontovich approximation is a basis for the description of light-beam propagation in optical fibers [@Gloge-Marcuse; @Marcuse; @Arnaud; @MankoLeeSpr] leading to $$\begin{aligned}
i\lambda \,\frac{\partial \psi (x,y,z)}{\partial z} &=&-\frac{\lambda ^{2}}{%
4\pi n_{0}(z)}\left( \frac{\partial ^{2}\psi (x,y,z)}{\partial x^{2}}+\frac{%
\partial ^{2}\psi (x,y,z)}{\partial y^{2}}\right) \label{6} \\
&&+\frac{\pi }{n_{0}(z)}\left[ n_{0}^{2}(z)-n^{2}(x,y,z)\right] \psi (x,y,z)
\nonumber\end{aligned}$$ which is Schrödinger-like for a wave function depending on the transversal coordinates $x$ and $y$. The longitudinal $z$ coordinate plays the role of time in the Schrödinger equation. The unitary $z$-evolution of the electromagnetic complex amplitude is described by the evolution operator $\hat{U}(z)$ $$\hat{U}(z,z_{0})\psi (x,y,z_{0})=\psi (x,y,z),$$ associated to the Hamiltonian $$\hat{H}(z)=\left( \frac{\hat{p}_{x}^{2}}{2}+\frac{\hat{p}_{y}^{2}}{2}\right)
\frac{1}{n_{0}(z)}+U(x,y,z). \label{7}$$ with $\hat{p}_{x}=-i\lambda \,\frac{\partial }{\partial x}\,,\hat{p}%
_{y}=-i\lambda \,\frac{\partial }{\partial y}\,$ and a potential function $$U(x,y,z)=\frac{\pi }{n_{0}(z)}\left[ n_{0}^{2}(z)-n^{2}(x,y,z)\right] .$$
Other quantum-like systems are reviewed in [@Erici; @Caserta]. An important example is sound-wave propagation in acoustic waveguides [@Brekhovskikhbook]. Acoustic waves in the paraxial approximation are well described by a Schrödinger-like equation. Charged-particle beams were also recently discussed as quantumlike systems [@FedMie; @FedManPRE]. Light beams inside diode lasers have been treated as quantumlike systems as well, due to the waveguide structure of their active region [@Gevork; @Rita]. As remarked in [@Bielefeld] the variety of quantumlike classical systems provides a wide range of possibilities for perfect simulation of quantum computation operations by classical means.
Modes and gates in optical fibers
=================================
In an optical fiber, the light modes, being solutions of the Schrödinger-like equation (\[6\]), have all the properties of quantum-mechanical wave functions including the entanglement phenomenon.
Light modes in the fiber are used to code quantum-like bits ([*qulbits*]{}). How many qulbits may live in one optical fiber? In the simplest case, which is considered here, light modes with a fixed frequency are considered. But, in the same optical fiber, light of different frequencies may be used. The Helmholtz equation holds for each frequency, with an index of refraction profile that may be different for different frequencies. Hence, in the same optical fiber, one may store many qulbits simply by exploiting the propagation of light beams with different frequencies. Interaction of photons with different frequencies may provide useful computation effects.
The Fock–Leontovich approximation is obtained from the Helmholtz equation for the components of the electric or magnetic field. This equation is a scalar approximation which neglects the tensorial structure of the dielectric constant and the polarization. Also neglected are time and space dispersion, related to the nonlocal linear response of the medium to electromagnetic perturbations. Taking into account these effects would provide an even richer framework to accommodate qulbits in the fiber.
We consider now the scalar fixed frequency situation described by Eq. (\[6\]). When the index of refraction profile has the form of an inverse well, the optical fiber traps discrete modes $\psi _{n_{1}n_{2}}(x,y,z)$, $n_{1}$ and $n_{2}$ being integer labels.
With light modes on a fiber and the unitary $z-$evolution associated to Eq.(\[6\]) one may perform quantum computation over continuous variables in a way similar to the one proposed for time evolution in Ref.[@Lloyd1]. All the physical interactions needed for the construction of polynomial Hamiltonians are available by the choice of the appropriate refraction profile and by Kerr interactions. Also, as explained below, by restricting oneself to finite-dimensional subspaces of excitations, one may perform the same operations as in quantum computation with discrete variables.
There are several ways to code information by light modes in a fiber which may be useful for quantum information processing, in particular those associated to different choices of basis in self-focusing potentials. The important self-focusing case is associated to quadratic potentials of the form $$U(x,y,z)=a(z)x^{2}+b(z)y^{2}+d(z)xy+e(z)x+f(x)y+l(z), \label{3.1}$$ and in this case, an explicit solution may be obtained for the four $z-$dependent integrals of motion[@MankoLeeSpr] $$\left(
\begin{array}{l}
\hat{x}_{0}(z) \\
\hat{y}_{0}(z)
\end{array}
\right) =\hat{U}(z,z_{0})\left(
\begin{array}{l}
\hat{x} \\
\hat{y}
\end{array}
\right) \hat{U}^{-1}(z,z_{0}),\qquad \left(
\begin{array}{l}
\hat{p}_{x0}(z) \\
\hat{p}_{y0}(z)
\end{array}
\right) =\hat{U}(z,z_{0})\left(
\begin{array}{l}
\hat{p}_{x} \\
\hat{p}_{y}
\end{array}
\right) \hat{U}^{-1}(z,z_{0}) \label{3.2}$$ Defining the boson integrals of motion $$\left(
\begin{array}{l}
a(z) \\
a^{\dagger }(z)
\end{array}
\right) =\frac{1}{\sqrt{2}}\left(
\begin{array}{l}
\hat{x}_{0}(z)+i\hat{p}_{x0}(z) \\
\hat{x}_{0}(z)-i\hat{p}_{x0}(z)
\end{array}
\right) ,\left(
\begin{array}{l}
b(z) \\
b^{\dagger }(z)
\end{array}
\right) =\frac{1}{\sqrt{2}}\left(
\begin{array}{l}
\hat{y}_{0}(z)+i\hat{p}_{y0}(z) \\
\hat{y}_{0}(z)-i\hat{p}_{y0}(z)
\end{array}
\right) \label{3.3}$$ several basis may be constructed. The discrete Fock-state modes are solutions to the eigenvalue equation $$\begin{aligned}
a^{\dagger }(z)a(z) &\mid &n_{1},n_{2},z\rangle =n_{1}\mid
n_{1},n_{2},z\rangle , \label{S9} \\
b^{\dagger }(z)b(z) &\mid &n_{1},n_{2},z\rangle =n_{2}\mid
n_{1},n_{2},z\rangle ,\qquad n_{1},n_{2}=0,1,2,\ldots \label{S10}\end{aligned}$$ The Fock-state modes are obtained from the fundamental mode $\mid
0,0,z\rangle $ by $$\mid n_{1},n_{2},z\rangle =\frac{a_{1}^{\dagger n_{1}}(z)b^{\dagger n_{2}}(z)%
}{\sqrt{n_{1}!n_{2}!}}\mid 0,0,z\rangle . \label{S24}$$
A spin-like description of the Fock-state modes is related to $SU(2)$-subgroup of the Weyl-symplectic group in two dimensions. This is related to the Jordan–Schwinger map $$J_{+}(z)=a^{\dagger }(z)b(z);\quad J_{-}(z)=b^{\dagger }(z)a(z);\quad
J_{3}(z)=\frac{1}{2}\left( a^{\dagger }(z)a(z)-b^{\dagger }(z)b(z)\right)
\label{3.4}$$ Irreducible representation spaces for this subgroup are spanned by states $%
\left\{ \mid n_{1},n_{2},z\rangle ;n_{1}+n_{2}=N\right\} $ for each fixed $N$.
On the other hand, coherent modes in the optical fiber are labelled by two complex numbers $\alpha $ and $\beta $, $$\mid \alpha ,\beta ,z\rangle =\exp \left[ \alpha a^{\dagger }(z)-\alpha
^{*}a(z)\right] \exp \left[ \beta b^{\dagger }(z)-\beta ^{*}b(z)\right] \mid
0,0,z\rangle , \label{S26}$$ Using this basis, the self-focusing fiber could be considered a Gaussian channel for numerical information. For arbitrary choices of the complex numbers $\alpha $ and $\beta $ this is an overcomplete set. However choosing the numbers on the von Neumann lattice[@Boon] $$\alpha _{m_{1},n_{1}}=\frac{1}{\sqrt{2}}\left( n_{1}+i2\pi m_{1}\right)
;\quad \beta _{m_{2},n_{2}}=\frac{1}{\sqrt{2}}\left( n_{2}+i2\pi m_{2}\right)
\label{3.5}$$ and excluding two pairs (for example $m_{1}=n_{1}=m_{2}=n_{2}=0$) one obtains a discrete [*complete*]{} set of coherent modes, which might provide a basis for the coding of a large amount of quantum-like information in the fiber.
We now analyze the question of what unitary transformations may be obtained by evolution of the light modes along the fiber. The simplest possibility is by a change of the index of refraction profile. From Eq.(\[6\]) we may write a path integral representation for the evolution of the light mode along the fiber $$\psi (x,y,z)=G\left( x,x_{0},y,y_{0},z\right) \psi (x_{0},y_{0},z)
\label{N1}$$ with $$G\left( x,x_{0},y,y_{0},z\right) =\int_{\left( x_{0},y_{0},0\right)
}^{\left( x,y,z\right) }d^{2}\overrightarrow{\xi }\exp \left\{ \frac{i}{%
\lambda }\int_{0}^{z}d\tau \left[ \pi n_{0}(z)\stackrel{\bullet }{\xi }%
^{2}(z)-U(\overrightarrow{\xi },z)\right] \right\} \label{N2}$$ $\overrightarrow{\xi }$ being a two-dimensional vector on the fiber sections and $$U(\overrightarrow{\xi },z)=\frac{\pi }{n_{0}\left( z\right) }\left[
n_{0}^{2}(z)-n^{2}(\overrightarrow{\xi },z)\right] \label{N3}$$ One sees from Eq.(\[N2\]) that adjusting the index of refraction profile changes not only the potential but also the coefficient of the kinetic term. Let us consider a self-focusing quadratic potential and define, as before, $%
a^{\dagger },a$ and $b^{\dagger },b$ to be creation and annihilation operators for harmonic modes along the $x$ and $y-$directions respectively. For simplicity we drop the $z$ argument in the operators. Then, changing the index of refraction profile gives us direct access to the generators $$a^{\dagger }a+b^{\dagger }b;a^{\dagger }+a;\left( a^{\dagger }+a\right)
^{2};b^{\dagger }+b;\left( b^{\dagger }+b\right) ^{2};\left( a^{\dagger
}+a\right) \left( b^{\dagger }+b\right) \label{N4}$$ By a simple reasoning using the Baker-Campbell-Hausdorff formula one concludes that by a non-uniform change of the index of refraction one may obtain in Eq.(\[N2\]) all the operations of the Weyl-symplectic group in two dimensions. This group is not compact. Therefore the unitary representations of the full group are not finite-dimensional. This might be explored for information manipulation schemes were an unbounded number of states is manipulated. However for operations on a finite number of qulbits, it is the compact subgroups that are important. In particular a useful subgroup is the $SU\left( 2\right) $ group described before in Eq.(\[3.4\]). Finite-dimensional irreducible spaces for this subgroup are $$a^{\dagger n}b^{\dagger m}|0>,\quad n+m=2k \label{N6}$$ for $k=0,\frac{1}{2},1,\frac{3}{2},2,...$ with dimension $2k+1$. In this finite-dimensional spaces all unitary operations may be implemented on the fiber by changing the index of refraction profile.
To perform universal quantum computation it is necessary, at least, to have arbitrary unitary transformations on a single qulbit and a CNOT operation on two qulbits[@Barenco]. According to the discussion above the $\left|
0\right] ,\left| 1\right] $ qulbit states may be coded, for example, as follows $$\left| 1\right] =a^{\dagger }|0>,\quad \left| 0\right] =b^{\dagger }|0>
\label{N7}$$ This being a $k=\frac{1}{2}$ two-dimensional representation, all unitary transformations may be performed in this space by the $SU\left( 2\right) $ subgroup.
For the CNOT operation we may code the $\left| 1\right] -$state of the control bit as the application of two energy quanta along the $x-$ direction and the $\left| 0\right] -$state of the control bit as the application of two energy quanta along the $y-$direction. For the target bit we use the same coding as in (\[N7\]). Therefore $$\left| 11\right] =a^{\dagger 3}|0>,\left| 01\right] =a^{\dagger 2}b^{\dagger
}|0>,\left| 10\right] =a^{\dagger }b^{\dagger 2}|0>,\left| 00\right]
=b^{\dagger 3}|0> \label{N8}$$ the first label in $\left| \alpha \beta \right] $ being the label of the target qulbit and the second the label of the control qulbit. The subspace spanned by (\[N8\]) is a four-dimensional $SU(2)-$irreducible subspace. Therefore all unitary transformations may be implemented in this subspace and, in particular, the CNOT operation.
We have therefore proved that, with this coding, the self-focusing fiber is capable of universal quantum computation. With higher order potentials many other possibilities would be available. For example, an interacting term of the form $\eta a^{\dagger }ab^{\dagger }b$ appears in the Kerr Hamiltonian. This allows to perform phase operations on $a^{\dagger }$ modes gated by the $b^{\dagger }$ excitations. For example by coding the target qulbit as $$\left| 1\right] =\frac{1}{\sqrt{2}}\left( 1-a^{\dagger }\right) |0>,\quad
\left| 0\right] =\frac{1}{\sqrt{2}}\left( 1+a^{\dagger }\right) |0>$$ and the control qulbit as $$\left| 1\right] =b^{\dagger }|0>,\quad \left| 0\right] =|0>$$ the operator $\exp \left( i\pi a^{\dagger }ab^{\dagger }b\right) $ implements the CNOT gate.
The self-focusing fiber is a versatile medium to code quantum-like information and this is the reason we have emphasized the existence of several light mode basis. In Eq.(\[N7\]) above, a qulbit is coded using modes in the $x$ and $y$ directions. We might as well have used $x-$modes only and coded the qulbit using the quadratures $\hat{x}$ and $\hat{p}_{x}$. Then, light propagation with the symmetric self-focusing Hamiltonian corresponding to the operator $$U=e^{i\left[ (\hat{p}_{x}^{2}/2+(\hat{x}^{2}/2)\right] \pi /4}$$ yields $$\left(
\begin{array}{c}
\hat{p}_{x} \\
\hat{x}
\end{array}
\right) \rightarrow \frac{1}{\sqrt{2}}\left(
\begin{array}{c}
\hat{p}_{x}+\hat{x} \\
\hat{p}_{x}-\hat{x}
\end{array}
\right) ,$$ a Hadamard gate transformation. Observation of such transformation can be done measuring the position and direction of rays in the optical fiber for Gaussian wave packets.
Conclusions
===========
1\) Classical systems that are quantum-like, in the sense that their evolution along a space direction is described by a Schrödinger equation, possess a high potential for information processing including quantum computation. A promising system of this type consists of a light beam propagating along an optical fiber. It should be noticed however that similar possibilities exist with other systems, for example acoustic waves propagating along an acoustic waveguide. Practical implementation problems to be addressed are the choice of the optical fiber and the definition of a coding standard for the qulbits. The variety of materials used and a fairly well developed optical fiber technology give us hope that the model Hamiltonians needed for the operations of quantum computing may be physically implemented in this medium.
2\) The fact that the unitary evolution of the quantum-like systems is associated to a space dimension, means that the unitary transformation is implemented in the hardware, rather than requiring a precise sequence of temporal operations. Also and this might be useful for mass production purposes, once a non-uniform refraction index profile is set up on the material, many different gates may be obtained simply by cutting fiber segments of different lengths.
3\) Finally we should also point out the easy possibility to imitate nonlinear quantum mechanics by means of the nonlinear media response to the electromagnetic radiation. This might provide, as suggested in [@Abrams], even more powerful quantum computation algorithms.
[99]{} P. Benioff; J. Stat. Phys. [**22**]{}, 563 (1980); [**29**]{}, 515 (1982).
R. P. Feynman; Int. J. Theor. Phys. [**21**]{}, 467 (1982); Found. of Phys. 16, 507 (1986).
D. Deutsch; Proc. Roy. Soc. (London) A [**400**]{}, 97 (1985).
D. Aharonov, “Quantum computation,” quant-ph/9812037, Annual Reviews of Computational Physics, Vol. 6, ed. Dietrich Stauffer, World Scientific, 1998.
R. J. Spreeuw; quant-ph/0009066 v2.
R. J. Spreeuw; Found. Phys. [**28**]{}, 361 (1998).
N. J. Cerf, C. Adami, and P. G. Kwiat; Phys. Rev. A [**57**]{}, R1477 (1998).
P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt and H. G. White; J. Mod. Opt. [**47**]{}, 257 (2000).
D. Abrams and S. Lloyd; Phys. Rev. Lett. [**83**]{}, 5162 (1999).
M. A. Leontovich; Izv. Akad. Nauk SSSR, Ser. Fiz., [**8**]{}, 16 (1944).
V. A. Fock and M. A. Leontovich; Zh. Éksp. Teor. Fiz., [**16**]{}, 557 (1946).
D. Gloge and D. Marcuse; J. Opt. Soc. Am. [**59**]{}, 1629 (1969).
D. Marcuse; [*Light Transmission Optics*]{}, Van Nostrand, New York 1972.
J. A. Arnaud, [*Beam and Fiber Optics*]{}, Academic, New York 1976.
V. I. Man’ko, in [*Lie Methods in Optics,*]{} J. Sanchez Mondragon and K.-B. Wolf (Eds.), Lecture Notes in Physics 250, pp. 193-207, Springer, Berlin 1986.
R. Fedele and P. K. Shukla (eds.); [*Quantum-like Models and Coherent Effects*]{}, World Scientific, Singapore 1995.
S. De Martino, S. De Nicola, S. De Siena, R. Fedele, and G. Miele (eds.); [*New Perspectives in Physics of Mesoscopic Systems: Quantum-like Descriptions and Macroscopical Coherence Phenomena*]{}, World Scientific, Singapore 1997.
L. M. Brekhovskikh; [*Waves in slab media*]{}, Nauka, Moscow 1973.
R. Fedele and G. Miele; Nuovo Cim. D [**13**]{}, 1527 (1991).
R. Fedele and V. I. Man’ko; Phys. Rev. E [**58**]{}, 992 (1998); R. Fedele, M. A. Man’ko, and V. I. Man’ko, J. Russ. Laser Res. (Kluwer Academic/Plenum Publishers), [**21**]{}, 1 (2000); J. Opt. Soc. Am. A [**17**]{}, 2506 (2000).
M. A. Man’ko and G. T. Mikaelyan; Sov. J. Quantum Electron., [**16**]{}, 895 (1986); “Modes and mode conversions in active semiconductor waveguides,” in: [*The Nonlinear Optics of Semiconductor Lasers*]{}, [*Proceedings of the P.N. Lebedev Physical Institute*]{}, Nauka, Moscow (1986), Vol. 166, p. 126 \[Nova Science, New York (1987), Vol. 166, p. 170\].
M. A. Man’ko; “Some aspects of nonlinear optics of semiconductor lasers,” in: M. Bertolotti (ed.), [*ECOOSA-90 Quantum Optics*]{}, IOP Publ. Ltd. (1991), p. 247.
M. A. Man’ko; “Noncommutative tomography of analytic signal and entaglement in the probability representation of quantum mechanics,” Talk at the Second Bielefeld Workshop on Quantum Information and Complexity (12-14 October 2000), Eprint No. 2000/006, \[http://www.uni-bielefeld.de/ZIF/complexity/publications.html\].
S. Lloyd and S. L. Braunstein; Phys. Rev. Lett. [**82**]{}, 1784 (1999).
V. V. Dodonov and V. I. Man’ko; [*Invariants and Evolution of Nonstationary Quantum Systems*]{}, [*Proceedings of the Lebedev Physical Institute*]{}, Vol. 183, Nova Science, New York 1989.
M. Boon and J. Zak; Phys. Rev. B [**18**]{}, 6744 (1978).
A. Barenco, C. H. Bennet, R. Cleave, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin and H. Weinfurter; Phys. Rev. A52, 3457 (1995)
[^1]: P. N. Lebedev Physical Institute, Leninskii Prospect 53, Moscow 117924, Russia, e-mail: mmanko@sci.lebedev.ru
[^2]: Zentrum für interdisziplinäre Forschung, Universität Bielefeld, Wellenberg 1, 33615 Bielefeld, Germany
[^3]: Grupo de Física Matemática, Complexo Interdisciplinar, Universidade de Lisboa, Av. Gama Pinto, 2 - P1699 Lisboa Codex, Portugal, e-mail: vilela@cii.fc.ul.pt
|
---
abstract: 'The existence of Alexander–equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by A. Degtyarev. However, up to now, no explicit example of such a pair was available (only the existence was known). In this paper, we construct the first concrete example.'
address:
- 'Department of Mathematical Sciences, University of Aarhus, Building 1530, Ny Munkegade, DK–8000 Aarhus C, Denmark'
- 'Department of Mathematics, Tokyo University of Science, 26 Wakamiya–cho, Shinjuku–ku, Tokyo 162–8601, Japan'
author:
- Christophe Eyral and Mutsuo Oka
title: 'Alexander–equivalent Zariski pairs of irreducible sextics'
---
[^1]
Introduction
============
Let $\mathcal{M}(\Sigma,d)$ be the moduli space of reduced curves of degree $d$ in $\mathbb{CP}^2$ with a prescribed configuration of singularities $\Sigma$.[^2] A pair of curves $(C,C')$ in $\mathcal{M}(\Sigma,d)$ is said to be a *Zariski pair* if it satisfies the following two conditions (cf. [@A]):
1. \[c1\] $C$ and $C'$ have the same *combinatoric*, that is, there exist regular neighbourhoods $T(C)$ and $T(C')$ of $C$ and $C'$ respectively such that the pairs $(T(C),C)$ and $(T(C'),C')$ are homeomorphic;
2. \[c2\] the pairs $(\mathbb{CP}^2,C)$ and $(\mathbb{CP}^2,C')$ are not homeomorphic.
It is easy to check that if both $C$ and $C'$ are *irreducible* then the first condition is always satisfied. The first Zariski pair appears in the works by O. Zariski [@Z1; @Z2; @Z3] (see also [@A] and [@O3]). The members of the pair are irreducible curves of degree $6$, which is the smallest degree for which Zariski pairs exist.
The existence of a Zariski pair in a moduli space gives an information about its connected components. Indeed, if $\mathcal{M}(\Sigma,d)$ has a Zariski pair $(C,C')$, then $C$ and $C'$ necessarily belong to different connected components (cf. [@Z4; @Z5; @LR]) — in particular, if $\mathcal{M}(\Sigma,d)$ has a Zariski pair, then it is not connected. (The converse statement is not clear. Two curves coming from two different connected components may have the same embedded topology.)
Now to check whether a given a pair of curves $C,C' \in \mathcal{M}(\Sigma,d)$ with the same combinatoric is a Zariski pair, one can first try to calculate the generic Alexander polynomials of the curves. If these polynomials are different, then the curves do not have the same embedded topology, and $(C,C')$ is a Zariski pair. However, it may happen that these polynomials are the same although $(C,C')$ is a Zariski pair. In this case the pair is said to be *Alexander–equivalent*. The first example of such a pair was given by E. Artal Bartolo and J. Carmona Ruber [@AC] for *reducible* curves of degree 7. The first examples dealing with *irreducible* curves are due to the second author [@O1] (curves of degree $12$) and [@O2] (curves of degree $8$). In [@D1], A. Degtyarev proved that Alexander–equivalent Zariski pairs also appear on irreducible curves of degree $6$. However he did not give any explicit example (only the existence is proved). The aim of the present paper is to construct a concrete example of such a pair.
Statement of the result
=======================
Let $(X\colon Y\colon Z)$ be homogeneous coordinates on $\mathbb{CP}^2$ and $(x,y)$ the affine coordinates defined by $x:=X/Z$ and $y:=Y/Z$ on $\mathbb{CP}^2 \setminus \{Z=0\}$, as usual. We consider the projective curves $C$ and $C'$ in $\mathbb{CP}^2$ defined by the affine equations $f(x,y)=0$ and $f'(x,y)=0$ respectively, where $$\begin{aligned}
f(x,y) & := & {\frac {369}{364}}\,{y}^{6}+{y}^{5}x-{\frac {197}{91}}\,{y}^{5}+{
\frac {207}{182}}\,{y}^{4}{x}^{2}-{\frac {185}{91}}\,{y}^{4}x+{\frac {
235}{182}}\,{y}^{4}+\\
& & {\frac {87}{182}}\,{y}^{3}{x}^{3}-{\frac {255}{182
}}\,{y}^{3}{x}^{2}+{\frac {97}{91}}\,{y}^{3}x-{\frac {1}{7}}\,{y}^{3}+{\frac {101
}{364}}\,{y}^{2}{x}^{4}-{\frac {47}{91}}\,{y}^{2}{x}^{3}+\\
& & {\frac {7}{26
}}\,{y}^{2}{x}^{2}-{\frac {3}{91}}\,{y}^{2}x+{\frac {1}{364}}\,{y}^{2}
+{\frac {5}{182}}\,y{x}^{5}-{\frac {11}{182}}\,y{x}^{4}+{\frac {1}{26}}\,y{x}^{3}-\\
& & {\frac {1}{182}}\,y{x}^{2}+{\frac {1}{364}}\,{x}^{6}-{\frac {1}{182}}
\,{x}^{5}+{\frac {1}{364}}\,{x}^{4},\\
&&\\
f'(x,y) & := & -{\frac {4}{3}}\,{y}^{6}+ \left( -{\frac {8}{9}} \, {x}^{2}+4\,x+1 \right) {y}^{4}+ \left( -{\frac {4}{9}}\,{x}^{4}+{\frac {26}{9}} \,{x}^{3}-{\frac {14}{3}}\,{x}^{2}-2\,x \right) {y}^{2}\\
& & +{\frac {1}{9}}\,{x}^{6}+{\frac {2}{9}}\,{x}^{5}-{\frac {17}{9}}\,{x}^{4}+2
\,{x}^{3}+{x}^{2}.\end{aligned}$$ Both $C$ and $C'$ are *irreducible sextics* with the set of singularities $\textbf{A}_9\oplus 2\textbf{A}_4$. (We recall that a point $P$ in a curve $D$ is said to be an $\textbf{A}_n$–singularity ($n\geq 1$) if the germ $(D,P)$ is topologically equivalent to the germ at the origin of the curve defined by $x^2+y^{n+1}=0$.) For the curve $C$, the $\textbf{A}_4$–singularities are located at $(1,0)$ and $(0,1)$ while the $\textbf{A}_9$–singularity is at $(0,0)$. For $C'$, the $\textbf{A}_4$–singularities are at $(1,\pm 1)$ and the $\textbf{A}_9$–singularity is at $(0,0)$. The *real* plane sections of $C$ and $C'$ are shown in Figure \[figa92a4rps\] and \[figa92a4narps\] respectively. (In the figures we do not respect the numerical scale.) The curve $C'$ is symmetric with respect to the $x$–axis. The curve $C$ has no particular symmetry. Notice that after the analytic change of coordinates $$(x,y) \mapsto \left(x-\frac{1}{3}\, y^4+y^2,y\right),$$ the equation of $C'$ near the origin takes the form $$x^2+\frac{4}{27}\, y^{10}+\mbox{higher terms} = 0.$$ As the leading term $x^2+(4/27)\, y^{10}$ is positive on $\mathbb{R}^2\setminus \{(0,0)\}$, the origin is an isolated point of the real plane section of $C'$ (cf. Figure \[figa92a4narps\]). Finally, since $C$ and $C'$ are not of torus type (cf. [@OP] and [@P]), their generic Alexander polynomials are trivial (cf. [@D1]).
![\[figa92a4rps\]$\{(x,y)\in\mathbb{R}^2\ ;\ f(x,y)=0\}$](figa92a4rps.eps){width="8cm" height="5cm"}
![\[figa92a4narps\]$\{(x,y)\in\mathbb{R}^2\ ;\ f'(x,y)=0\}$](figa92a4narps.eps){width="8cm" height="5cm"}
\[maintheorem\] The fundamental group $\pi_1(\mathbb{CP}^2\setminus C)$ is isomorphic to $\mathbb{Z}/6\mathbb{Z}$ while $\pi_1(\mathbb{CP}^2\setminus C')$ is isomorphic to $\mathbb{D}_{10}\times (\mathbb{Z}/3\mathbb{Z})$, where $\mathbb{D}_{10}$ is the dihedral group of order $10$. In particular $(C,C')$ is an Alexander–equivalent Zariski pair and the moduli space of irreducible sextics with the set of singularities $\textbf{\emph{A}}_9\oplus 2\textbf{\emph{A}}_4$ has at least two connected components.[^3]
Theorem \[maintheorem\] is proved in sections \[abelian\] and \[nonabelian\] below.
The curve $C'$ is an example of so–called *$\mathbb{D}_{10}$–sextics*. (A $\mathbb{D}_{10}$–sextic is a non–torus irreducible sextic with simple singularities and whose fundamental group[^4] factors to the dihedral group $\mathbb{D}_{10}$.) The existence of $\mathbb{D}_{10}$–sextics was first proved, purely arithmetically, by A. Degtyarev [@D1] who showed that there exist exactly 8 equisingular deformation families of such curves, one family for each of the following sets of singularities:
$4\textbf{A}_4,\ 4\textbf{A}_4\oplus\textbf{A}_1,\
4\textbf{A}_4\oplus2\textbf{A}_1,\ 4\textbf{A}_4\oplus\textbf{A}_2,$ $\textbf{A}_9\oplus2\textbf{A}_4,\
\textbf{A}_9\oplus2\textbf{A}_4\oplus\textbf{A}_1,\
\textbf{A}_9\oplus2\textbf{A}_4\oplus\textbf{A}_2,\
2\textbf{A}_9.$
First explicit examples and fundamental groups of $\mathbb{D}_{10}$–sextics were given in [@D2] (see also [@EO] for the sets of singularities $4\textbf{A}_4$ and $4\textbf{A}_4\oplus\textbf{A}_1$).
Furthermore, Degtyarev also observed in [@D1] that the configurations $$4\textbf{A}_4,\ 4\textbf{A}_4\oplus\textbf{A}_1,\
\textbf{A}_9\oplus2\textbf{A}_4,\
\textbf{A}_9\oplus2\textbf{A}_4\oplus\textbf{A}_1,\
2\textbf{A}_9$$ can be realized not only by $\mathbb{D}_{10}$–sextics but also by irreducible non–torus sextics $D$ for which the group $\pi_1(\mathbb{CP}^2\setminus D)$ does *not* factors to $\mathbb{D}_{10}$. (In particular, since the Alexander polynomial of an irreducible non–torus sextic with simple singularities is always trivial, these 5 sets of singularities give rise to Alexander–equivalent Zariski pairs of irreducible sextics.) However, Degtyarev did not give any explicit equation of such a curve $D$ (only the existence is proved) and did not calculate the group $\pi_1(\mathbb{CP}^2\setminus D)$. The curve $C$ in Theorem \[maintheorem\] is the first explicit example of such a curve — i.e., a curve whose fundamental group does not admit a dihedral quotient although its set of singularities can be realized by a $\mathbb{D}_{10}$–sextic as well. (In particular, the pair $(C,C')$ in Theorem \[maintheorem\] is also the first concrete example of an Alexander–equivalent Zariski pair dealing with irreducible sextics.)
Fundamental group of $\mathbb{CP}^2\setminus C$ {#abelian}
===============================================
In this section, we show that $\pi_1(\mathbb{CP}^2\setminus C)\simeq \mathbb{Z}/6\mathbb{Z}$. In fact, it suffices to prove that $\pi_1(\mathbb{CP}^2\setminus C)$ is abelian. Indeed, by Hurewicz’s theorem, if $\pi_1(\mathbb{CP}^2\setminus C)$ is abelian, then it is isomorphic to first integral homology group $H_1(\mathbb{CP}^2\setminus C)$ and it is well known that $H_1(\mathbb{CP}^2\setminus C)\simeq \mathbb{Z}/6\mathbb{Z}$.
To show that $\pi_1(\mathbb{CP}^2\setminus C)$ is abelian, we use Zariski–van Kampen’s theorem with the pencil given by the vertical lines $L_{\eta}\colon x=\eta$, $\eta\in\mathbb{C}$ (cf. [@Z1] and [@vK]). We always take the point $(0\colon 1\colon 0)$ as base point for our fundamental groups. This point is nothing but the axis of the pencil, which is also the point at infinity of the lines $L_{\eta}$. Note that it does not belong to the curve.
The discriminant $\Delta_y(f)$ of $f$ as a polynomial in $y$ is the polynomial in $x$ given by $$\begin{aligned}
\Delta_y(f)(x) = a_0\, {x}^{15}\, \, (x-1)^{7}\, (858898351\,{x
}^{8}-1278576626\,{x}^{7}-359900737\,{x}^{6}+\\
1017975356\,{x}^{5}-56181608\,{x}^{4}- 170653568\,{x}^{3}+ 2388080\,{x}^{2}+2072000\,x-96000),\end{aligned}$$ where $a_0\in\mathbb{Q}\setminus\{0\}$. This polynomial has exactly 10 distinct complex roots:
$\eta_1 \approx - 0.7408$, $\eta_2 \approx - 0.3914$, $\eta_3 \approx - 0.1309$, $\eta_4 = 0$,
$\eta_5 \approx 0.0598$, $\eta_6 \approx 0.0778$, $\eta_7 \approx 0.6274$,
$\eta_8 \approx 0.9933 - i\, 0.1446$, $\eta_9 =\bar\eta_8\approx 0.9933 + i\, 0.1446$, $\eta_{10} = 1$.
The singular lines of the pencil are the lines $L_{\eta_j}$ ($1\leq j\leq 10$) corresponding to these 10 roots. The lines $L_{\eta_4}$ and $L_{\eta_{10}}$ intersect the curve at its singular points. All the other singular lines are tangent to $C$. See Figure \[figa92a4rps\].
![\[figa92a4eta4\]Generators at $x=\eta_4+\varepsilon$](figa92a4eta4.eps){width="8cm" height="5cm"}
We consider the generic line $L_{\eta_4+\varepsilon}$ and choose generators $\xi_1,\ldots,\xi_6$ of the fundamental group $\pi_1(L_{\eta_4+\varepsilon}\setminus C)$ as in Figure \[figa92a4eta4\], where $\varepsilon>0$ is small enough. The $\xi_k$’s ($1\leq k\leq 6$) are *lassos oriented counter–clockwise* around the 6 intersection points of the line $L_{\eta_4+\varepsilon}$ with the curve — i.e., the 6 complex roots of the equation $f(\eta_4+\varepsilon,y)=0$. (In the figures, a lasso is represented by a path ending with a bullet.) The Zariski–van Kampen theorem says that $$\pi_1(\mathbb{CP}^2\setminus C) \simeq
\pi_1(L_{\eta_4+\varepsilon}\setminus C) \big/ G,$$ where $G$ is the normal subgroup of $\pi_1(L_{\eta_4+\varepsilon}
\setminus C)$ generated by the monodromy relations associated with the singular lines of the pencil. To find these relations, we fix a ‘standard’ system of generators $\sigma_1,\ldots,\sigma_{10}$ for the fundamental group $\pi_1(\mathbb{C}\setminus \{\eta_1,
\ldots, \eta_{10}\})$ as follows. Each $\sigma_j$ is a lasso (oriented counter–clockwise) around $\eta_j$ with base point $\eta_4+\varepsilon$. For $j\not= 8,9$, the tail of $\sigma_j$ is a union of real segments and half–circles around the exceptional parameters $\eta_l$ ($l\not=j$) located in the real axis between the base point $\eta_4+\varepsilon$ and $\eta_j$. Its head is the circle $\mathbb{S}_\varepsilon(\eta_j)$ with centre $\eta_j$ and radius $\varepsilon$. The lasso $\sigma_8$ corresponding to the non–real root $\eta_8$ is given by $\zeta\theta\zeta^{-1}$, where $\theta$ is the loop obtained by moving $x$ once on the circle $\mathbb{S}_\varepsilon(\eta_8)$, starting at $\Re(\eta_8)+i\, (\Im(\eta_8)+\varepsilon)$, while $\zeta$ is the path obtained when $x$ moves on the real axis from $\eta_4+\varepsilon$ to $\eta_5-\varepsilon$, makes half–turn on the circle $\mathbb{S}_\varepsilon(\eta_5)$, from $\eta_5-\varepsilon$ to $\eta_5+\varepsilon$, moves on the real axis from $\eta_5+\varepsilon$ to $\eta_6-\varepsilon$, makes half–turn on the circle $\mathbb{S}_\varepsilon(\eta_6)$, from $\eta_6-\varepsilon$ to $\eta_6+\varepsilon$, moves on the real axis from $\eta_6+\varepsilon$ to $\eta_7-\varepsilon$, makes half–turn on the circle $\mathbb{S}_\varepsilon(\eta_7)$, from $\eta_7-\varepsilon$ to $\eta_7+\varepsilon$, moves on the real axis from $\eta_7+\varepsilon$ to $\Re(\eta_8)$, and finally moves in a straight line from $\Re(\eta_8)$ to $\Re(\eta_8)+i\, (\Im(\eta_8)+\varepsilon)$. (Here $\Re(\eta_8)$ and $\Im(\eta_8)$ denote the real and the imaginary parts of $\eta_8$ respectively.) The lasso $\sigma_9$ is defined similarly from a loop $\theta$ and a path $\zeta$ meeting at $\Re(\eta_9)+i\, (\Im(\eta_9)-\varepsilon)$. The monodromy relations around the singular line $L_{\eta_j}$ are obtained by moving the generic ‘fibre’ $F\simeq L_{\eta_4+\varepsilon}
\setminus C$ isotopically ‘above’ the loop $\sigma_j$, and by identifying each $\xi_k$ ($1\leq k\leq 6$) with its image by the terminal homeomorphism of this isotopy. (For details see [@Z1] and [@vK].)
The remaining of the proof is to determine these relations.
![\[figa92a4eta5\]Generators at $x=\eta_5-\varepsilon$](figa92a4eta5.eps){width="8cm" height="5cm"}
Monodromy relations at $x=\eta_5$ {#monodromy-relations-at-xeta_5 .unnumbered}
---------------------------------
When $x$ moves on the real axis from $\eta_4+\varepsilon$ to $\eta_5-\varepsilon$, the 6 complex roots of the equation (in $y$) $f(x,y)=0$ (and, consequently, the 6 generators $\xi_1,\ldots\xi_6$) are deformed as in Figure \[figa92a4eta5\]. The singular line $L_{\eta_5}$ is tangent to the curve at the simple point $P\approx(\eta_5,0.0095)$, and the intersection multiplicity $I(L_{\eta_5},C;P)$ of this line with the curve at this point is 2. Therefore, by the implicit functions theorem, the germ $(C,P)$ is given by $$x-\eta_5=b_0\, (y-0.0095)^2+\mbox{higher terms},$$ where $b_0\not=0$. It follows that when $x$ runs once counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_5)$, starting at $\eta_5-\varepsilon$, the variable $y$ makes half–turn counter–clockwise on the dotted circle around $0.0095$ (cf. Figure \[figa92a4eta5\]). The monodromy relation at $x=\eta_5$ is then given by $$\label{rela92a4eta5}
\xi_5=\xi_4.$$
Monodromy relations at $x=\eta_6$ {#monodromy-relations-at-xeta_6 .unnumbered}
---------------------------------
At $x=\eta_5-\varepsilon$, the generators are as in Figure \[figa92a4eta5\]. In Figure \[figa92a4eta6\], we show how the $\xi_k$’s are deformed when $x$ first makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_5)$, from $\eta_5-\varepsilon$ to $\eta_5+\varepsilon$, and then moves on the real axis from $\eta_5+\varepsilon$ to $\eta_6-\varepsilon$. The singular line $L_{\eta_6}$ is also tangent to $C$ at a simple point $P'$ and $I(L_{\eta_6},C;P')=2$. Therefore, as above, the monodromy relation we are looking for is simply given by $$\label{rela92a4eta6}
\xi_3=\xi_4^{-1}\xi_6\xi_4.$$
![\[figa92a4eta6\]Generators at $x=\eta_6-\varepsilon$](figa92a4eta6.eps){width="8cm" height="5cm"}
Monodromy relations at $x=\eta_4$ {#monodromy-relations-at-xeta_4 .unnumbered}
---------------------------------
The singular line $L_{\eta_4}$ passes through the singular points $(0,0)$ and $(0,1)$ which are singularities of type $\textbf{A}_9$ and $\textbf{A}_4$ repectively. At $(0,0)$, the curve has two branches $K_-$ and $K_+$ given by $$K_\pm:\quad y=x^2+5\, x^3+51\, x^4+(503\pm 32 \sqrt{6})\, x^5 +\mbox{higher terms.}$$ It follows that when $x$ runs once counter–clockwise on the circle $\mathbb{S}_\varepsilon (\eta_4)$, starting at $\eta_4+\varepsilon$, the generators $\xi_6$ and $\xi_5=\xi_4$ make $5$ turns counter–clockwise on the corresponding dotted circle (cf. Figure \[figa92a4eta4\]). The monodromy relation around $L_{\eta_4}$ that comes from the singular point $(0,0)$ is then given by $$\label{rela92a4eta400}
\xi_4=(\xi_6\xi_4)^5\cdot\xi_4\cdot(\xi_6\xi_4)^{-5}.$$ At $(0,1)$, a Puiseux parametrization of $C$ is given by $$x=t^2,\quad y=1-\frac{1}{2}\, t^4+\frac{1}{10}\,i\, \sqrt{5}\, t^5+\mbox{higher terms.}$$ Hence, when $x$ goes once counter–clockwise on the circle $\mathbb{S}_\varepsilon (\eta_4)$, starting at $\eta_4+\varepsilon$, the generators $\xi_1$ and $\xi_2$ make $(5/2)$–turn counter–clockwise on the corresponding dotted circle (cf. Figure \[figa92a4eta4\]). The monodromy relation around $L_{\eta_4}$ that comes from the singular point $(0,1)$ is then given by $$\label{rela92a4eta401}
\xi_1=(\xi_2\xi_1)^2\cdot\xi_2\cdot(\xi_2\xi_1)^{-2}.$$
Monodromy relations at $x=\eta_3$ {#monodromy-relations-at-xeta_3 .unnumbered}
---------------------------------
At $x=\eta_4+\varepsilon$, the generators are as in Figure \[figa92a4eta4\]. Now, when $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon (\eta_4)$, from $\eta_4+\varepsilon$ to from $\eta_4-\varepsilon$, and then moves on the real axis from $\eta_4-\varepsilon$ to $\eta_3+\varepsilon$, the $\xi_k$’s are deformed as in Figure \[figa92a4eta3\], where $$\gamma:=(\xi_6\xi_4)^{-2}\cdot\xi_4^{-1}\xi_6\xi_4\cdot(\xi_6\xi_4)^2.$$ The singular line $L_{\eta_3}$ is tangent to the curve at a simple point, with intersection multiplicity 2, and the monodromy relation we are looking for is given by $$\label{rela92a4eta3}
\xi_4=\gamma.$$
![\[figa92a4eta3\]Generators at $x=\eta_3+\varepsilon$](figa92a4eta3.eps){width="8cm" height="5cm"}
![\[figa92a4eta2\]Generators at $x=\eta_2+\varepsilon$](figa92a4eta2.eps){width="8cm" height="5cm"}
Monodromy relations at $x=\eta_2$ {#monodromy-relations-at-xeta_2 .unnumbered}
---------------------------------
When $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_3)$, from $\eta_3+\varepsilon$ to $\eta_3-\varepsilon$, then moves on the real axis from $\eta_3-\varepsilon$ to $\eta_2+\varepsilon$, the $\xi_k$’s are deformed as in Figure \[figa92a4eta2\], where $$\begin{aligned}
& & \alpha:=\bigl((\xi_6\xi_4)^{-2}\cdot\xi_4\cdot(\xi_6\xi_4)^2\bigr)
\cdot\xi_4\cdot \bigl((\xi_6\xi_4)^{-2} \cdot\xi_4 \cdot (\xi_6\xi_4)^2\bigr)^{-1},\\
& & \beta:=\xi_4^{-1} \cdot (\xi_6\xi_4)^{-2}\cdot \xi_4 \cdot(\xi_6\xi_4)^2 \cdot \xi_4.\end{aligned}$$ The monodromy relation at $x=\eta_2$ is also an usual multiplicity 2 tangent relation: $$\label{rela92a4eta2}
\xi_3=\xi_1^{-1}\xi_2\xi_1.$$
![\[figa92a4eta1\]Generators at $x=\eta_1+\varepsilon$](figa92a4eta1.eps){width="8cm" height="5cm"}
Monodromy relations at $x=\eta_1$ {#monodromy-relations-at-xeta_1 .unnumbered}
---------------------------------
In Figure \[figa92a4eta1\], we show how the $\xi_k$’s are deformed when $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_2)$, from $\eta_2+\varepsilon$ to $\eta_2-\varepsilon$, then moves on the real axis from $\eta_2-\varepsilon$ to $\eta_1+\varepsilon$. The monodromy relation around $L_{\eta_1}$ is a multiplicity 2 tangent relation given by $$\label{rela92a4eta1}
\xi_1=\beta.$$ Equivalently $(\xi_6\xi_4)^2 \cdot \xi_4 = \xi_4^{-1}(\xi_6\xi_4)^2 \cdot \xi_4 \xi_1$. Since $(\xi_6\xi_4)^2 \cdot \xi_4 = \xi_4^{-1}(\xi_6\xi_4)^2\cdot\xi_6\xi_4$ (by (\[rela92a4eta3\])), it follows that $$\label{rela92a4s1}
\xi_4\xi_1=\xi_6\xi_4.$$ Combined with (\[rela92a4eta6\]), this gives $$\label{rela92a4s2}
\xi_3=\xi_1.$$ Combined with (\[rela92a4eta2\]), this in turn implies $$\label{rela92a4s3}
\xi_2=\xi_1.$$
Monodromy relations at $x=\eta_7$ {#monodromy-relations-at-xeta_7 .unnumbered}
---------------------------------
We recall that, at $x=\eta_6-\varepsilon$, the generators are as in Figure \[figa92a4eta6\]. When $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_6)$, from $\eta_6-\varepsilon$ to $\eta_6+\varepsilon$, the $\xi_k$’s are deformed as shown in Figure \[figa92a4eta6bis\], where $$\delta:=(\xi_6\xi_4\xi_6^{-1}\xi_4)\cdot\xi_1\cdot (\xi_6\xi_4\xi_6^{-1}\xi_4)^{-1}.$$
![\[figa92a4eta6bis\]Generators at $x=\eta_6+\varepsilon$](figa92a4eta6bis.eps){width="8cm" height="5cm"}
\[lemmaa92a4\] When $x$ moves on the real axis from $\eta_6+\varepsilon$ to $\eta_7-\varepsilon$, the $\xi_k$’s are deformed as Figure \[figa92a4eta7\].
Lemma \[lemmaa92a4\] is not obvious and will be proved at the end of this section. Before, let us complete the calculation of $\pi_1(\mathbb{CP}^2\setminus C)$.
![\[figa92a4eta7\]Generators at $x=\eta_7-\varepsilon$](figa92a4eta7.eps){width="8cm" height="5cm"}
The monodromy relation at $x=\eta_7$ is a multiplicity $2$ tangent relation given by $$\delta=\xi_1.$$ Notice that, by (\[rela92a4s1\]), this relation can also be written as $$\label{rela92a4eta7s}
\xi_4\xi_1\xi_4\xi_1 = \xi_1\xi_4\xi_1\xi_4.$$ Moreover, still using (\[rela92a4s1\]), we can write the relation (\[rela92a4eta400\]) as $$\xi_4=(\xi_4\xi_1)^5\cdot \xi_4\cdot (\xi_4\xi_1)^{-5}.$$ The latter, combined with (\[rela92a4eta7s\]), implies $$\xi_4\xi_1=\xi_1\xi_4.$$ This already shows that the fundamental group $\pi_1(\mathbb{CP}^2\setminus C)$ is abelian. (We do not need to consider the monodromy relations around $L_{\eta_8}$, $L_{\eta_9}$ and $L_{\eta_{10}}$.)
To complete the calculation it remains to prove Lemma \[lemmaa92a4\].
We consider the polynomial $$h(x,u,v):=f(x,u+i\, v)$$ for $x$, $u$ and $v$ real. We denote by $f_e(x,u,v)$ and $f_o(x,u,v)$ the real and the imaginary part of $h(x,u,v)$ respectively. They have degree $6$ and $5$ respectively in $v$. Suppose there exists a number $x_0\in [\eta_6+\varepsilon,\eta_7-\varepsilon]$ such that $4$ complex solutions of the equation $f(x_0,y)=0$ are on the same vertical line $u=u_0$ in the complex plane $(\mathbb{C},y=u+i\, v)$. This implies that the equations $$f_e(x_0,u_0,v)=f_o(x_0,u_0,v)=0$$ have $4$ common real solutions $v_1$, $v_2$, $v_3$ and $v_4$. These solutions are non–zero since the equation $\Delta_y(f)(x)=0$ does not have any solution in $[\eta_6+\varepsilon,\eta_7-\varepsilon]$. Therefore, the equations $$f_e(x_0,u_0,v)=f_{oo}(x_0,u_0,v)=0,$$ where $f_{oo}(x,u,v)=f_o(x,u,v)/v$ (notice that $v$ divides $f_o(x,u,v)$, and $f_{oo}(x,u,v)$ is then a polynomial), also have $v_1$, $v_2$, $v_3$ and $v_4$ as common solutions. As $f_{oo}$ has degree $4$ in $v$, it follows that $f_{oo}(x_0,u_0,v)$ divides $f_e(x_0,u_0,v)$. Therefore the remainder $R(x,u,v)$ of $f_e$ by $f_{oo}$, as a polynomial in $v$, must be identically zero for $u=u_0$ and $x=x_0$ (of course, $R$ is written as $R=R'/R''$ where $R'$ is a polynomial in $x$, $u$ and $v$ while $R''$ is a polynomial depending just on $x$ and $u$). By an easy computation, we see that $R=(R_2'/R_2'')\, v^2 + (R_0'/R_0'')$, where $R_2'$, $R_2''$, $R_0'$ and $R_0''$ are polynomials in $x$ and $u$. Thus, $(x_0,u_0)$ is a common real solution of the equations $$\label{R}
R_2'(x,u)=R_0'(x,u)=0.$$ This implies that $x_0$ is a root of the resultant $\mbox{Res}_u(R_2',R_0')$ of $R_2'$ and $R_0'$ as polynomials in $u$. Note that the condition $\mbox{Res}_u(R_2',R_0')(x_0)=0$ is necessary to have a real partner $u_0$ such that $R_2'(x_0,u_0)=R_0'(x_0,u_0)=0$ but it is not sufficient since the possible partner $u_0$ might be non–real. There are $5$ real solutions $x_{01}, \ldots, x_{05}$ of the equation $\mbox{Res}_u(R_2',R_0')(x)=0$ in the interval $[\eta_6+\varepsilon,\eta_7-\varepsilon]$. Each of them gives a real number, say $u_{0j}$ ($1\leq j\leq 5$), such that $(x_{0j},u_{0j})$ is a solution of (\[R\]). Now, we have to check if these $5$ solutions give $4$ real roots of the polynomial $v\mapsto f_{oo}(x_0,u_0,v)$. Only the solution $(x_0,u_0):\approx(0.1205,0.0075)$ satisfies this requirement. Therefore we can have one (and only one) overcrossing of $4$ complex roots. To check if it is the case, we look at the solutions $y$ of the equation, in $y$, $f(x,y)=0$ for values of $x$ near $x_0$. `Maple` actually gives an overcrossing.
Fundamental group of $\mathbb{CP}^2\setminus C'$ {#nonabelian}
================================================
In this section, we prove that $\pi_1(\mathbb{CP}^2\setminus C') \simeq \mathbb{D}_{10}\times (\mathbb{Z}/3\mathbb{Z})$. As above, we use Zariski–van Kampen’s theorem with the pencil given by the vertical lines $L_{\eta}\colon x=\eta$, $\eta\in\mathbb{C}$.
The discriminant $\Delta_y(f)$ of $f$ as a polynomial in $y$ is the following polynomial in $x$: $$\Delta_y(f)(x) = a_0\,{x}^{12} \left( {x}^{4}+2\,{x}^{3}-17\,{x}^{2} +18\,x+9 \right) \left( 25\,{x}^{2}-15\,x-9 \right)^{2} \left( x-1 \right)^{10},$$ where $a_0\in\mathbb{Q}\setminus \{0\}$. This polynomial has exactly 8 distinct complex roots:
$\eta_1 \approx -5.5758$, $\eta_2 \approx -0.3708$, $\eta_3 \approx - 0.3677$,
$\eta_4 = 0$, $\eta_5 \approx 0.9708$, $\eta_6 = 1$,
$\eta_7 \approx 1.9718 - i\, 0.7077$, $\eta_8 = \bar \eta_7 \approx 1.9718 + i\, 0.7077$.
The singular lines of the pencil are the lines $L_{\eta_j}$ ($1\leq j\leq 8$) corresponding to these 8 roots. The lines $L_{\eta_4}$ and $L_{\eta_6}$ intersect the curve at its singular points, while all the other singular lines are tangent to the curve. See Figure \[figa92a4narps\].
Here we start with the generic line $L_{\eta_4-\varepsilon}$ and we choose generators $\xi_1,\ldots,\xi_6$ of $\pi_1(L_{\eta_4-\varepsilon}\setminus C')$ as in Figure \[figa92a4naeta4\], where $\varepsilon>0$ is small enough. (To determine the position of the complex roots of the equation $f(\eta_4-\varepsilon,y)=0$ one may use (\[b0\]) and (\[pp0\]) below.)
![\[figa92a4naeta4\]Generators at $x=\eta_4-\varepsilon$](figa92a4naeta4.eps){width="8cm" height="5cm"}
As above, to find the monodromy relations around the singular lines $L_{\eta_j}$ ($1\leq j\leq 8$) of the pencil, we fix a ‘standard’ system of generators $\sigma_1, \ldots,\sigma_8$ of the fundamental group $\pi_1(\mathbb{C}\setminus \{\eta_1,\ldots, \eta_8\})$, where each $\sigma_j$ is a lasso (oriented counter–clockwise) around $\eta_j$ and with base point $\eta_4-\varepsilon$. For $j\not= 7,8$, the tail of $\sigma_j$ is a union of real segments and half–circles around the exceptional parameters $\eta_l$ ($l\not=j$) located in the real axis between the base point $\eta_4-\varepsilon$ and $\eta_j$, while its head is the circle $\mathbb{S}_\varepsilon(\eta_j)$. The lasso $\sigma_7$ corresponding to the non–real root $\eta_7$ has the form $\zeta\theta\zeta^{-1}$, where $\theta$ is the loop obtained by moving $x$ once on the circle $\mathbb{S}_\varepsilon(\eta_7)$, starting at $\Re(\eta_7)+i\, (\Im(\eta_7)+\varepsilon)$, and $\zeta$ the path obtained when $x$ makes half–turn on the circle $\mathbb{S}_\varepsilon(\eta_4)$, from $\eta_4-\varepsilon$ to $\eta_4+\varepsilon$, moves on the real axis from $\eta_4+\varepsilon$ to $\eta_5-\varepsilon$, makes half–turn on the circle $\mathbb{S}_\varepsilon(\eta_5)$, from $\eta_5-\varepsilon$ to $\eta_5+\varepsilon$, moves on the real axis from $\eta_5+\varepsilon$ to $\eta_6-\varepsilon$, makes half–turn on the circle $\mathbb{S}_\varepsilon(\eta_6)$, from $\eta_6-\varepsilon$ to $\eta_6+\varepsilon$, moves on the real axis from $\eta_6+\varepsilon$ to $\Re(\eta_7)$, and finally moves in a straight line from $\Re(\eta_7)$ to $\Re(\eta_7)+i\, (\Im(\eta_7)+\varepsilon)$. The lasso $\sigma_8$ is defined similarly from a loop $\theta$ and a path $\zeta$ meeting at $\Re(\eta_8)+i\, (\Im(\eta_8)-\varepsilon)$.
The monodromy relations are now given as follows.
Monodromy relations at $x=\eta_3$ {#monodromy-relations-at-xeta_3-1 .unnumbered}
---------------------------------
When $x$ moves on the real axis from $\eta_4-\varepsilon$ to $\eta_3+\varepsilon$, the $\xi_k$’s are deformed as in Figure \[figa92a4naeta3\]. The singular line $L_{\eta_3}$ is tangent to $C'$ at one simple point, with intersection multiplicity 2, so the monodromy relation around this line is simply given by $$\label{rela92a4naeta3}
\xi_4=\xi_3.$$
![\[figa92a4naeta3\]Generators at $x=\eta_3+\varepsilon$](figa92a4naeta3.eps){width="6cm" height="6cm"}
Monodromy relations at $x=\eta_2$ {#monodromy-relations-at-xeta_2-1 .unnumbered}
---------------------------------
Now, when $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_3)$, from $\eta_3+\varepsilon$ to $\eta_3-\varepsilon$, and then moves on the real axis from $\eta_3-\varepsilon$ to $\eta_2+\varepsilon$, the $\xi_k$’s are deformed as in Figure \[figa92a4naeta2\]. The singular line $L_{\eta_2}$ is tangent to the curve at two simple points, in both case with intersection multiplicity 2. The monodromy relations around this line are given by $$\begin{aligned}
\label{rela92a4naeta21} && \xi_5=\xi_3,\\
\label{rela92a4naeta22} && \xi_3=\xi_2.\end{aligned}$$
![\[figa92a4naeta2\]Generators at $x=\eta_2+\varepsilon$](figa92a4naeta2){width="6cm" height="6cm"}
![\[figa92a4naeta1\]Generators at $x=\eta_1+\varepsilon$](figa92a4naeta1){width="8cm" height="5cm"}
Monodromy relations at $x=\eta_1$ {#monodromy-relations-at-xeta_1-1 .unnumbered}
---------------------------------
In Figure \[figa92a4naeta1\], we show how the $\xi_k$’s are deformed when $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_2)$, from $\eta_2+\varepsilon$ to $\eta_2-\varepsilon$, and then moves on the real axis from $\eta_2-\varepsilon$ to $\eta_1+\varepsilon$. The monodromy relation around $L_{\eta_1}$ is a multiplicity 2 tangent relation: $$\label{rela92a4naeta1}
\xi_6=\xi_2^2\xi_1\xi_2^{-2}.$$
Notice that the relations (\[rela92a4naeta3\])–(\[rela92a4naeta1\]) show that the *vanishing relation at infinity* can be written as $$\label{rela92a4nainfini}
(\xi_2\xi_2\xi_1)^2=e,$$ where $e$ is the unit element.
Monodromy relations at $x=\eta_4$ {#monodromy-relations-at-xeta_4-1 .unnumbered}
---------------------------------
At $x=\eta_4-\varepsilon$, the generators are shown in Figure \[figa92a4naeta4\]. By (\[rela92a4naeta3\]), (\[rela92a4naeta21\]) and (\[rela92a4naeta22\]), Figure \[figa92a4naeta4\] is the same as Figure \[figa92a4naeta4bis\]. The singular line $L_{\eta_4}$ passes through the origin which a type $\textbf{A}_9$ singular point of the curve. At this point the curve has two branches $K_+$ and $K_-$ given by $$\label{b0}
K_{\pm}:\quad x=y^2-{\frac{1}{3}}\, y^4\pm{\frac{2}{9}}\, i\, \sqrt{3}\, y^5+\, \mbox{higher terms.}$$ An easy computation shows that Puiseux parametrizations of these branches are: $$\label{pp0}
K_{\pm}:\quad x=t^2,\quad y=t+{\frac{1}{6}}\, t^3\mp{\frac{\sqrt{3}}{9}}\, i\, t^4+\, \mbox{higher terms.}$$ The monodromy relations at $x=\eta_4$ are then given by $$\begin{aligned}
\label{rela92a4naeta41} & & \xi_2\xi_1\xi_2^{-1} = (\xi_6\xi_2)\cdot \xi_6\xi_2\xi_6^{-1}\cdot (\xi_6\xi_2)^{-1},\\
&\label{rela92a4naeta42} & \xi_2 = (\xi_6\xi_2)^2\cdot\xi_6\cdot (\xi_6\xi_2)^{-2},\\
&\label{rela92a4naeta43} & \xi_2=(\xi_6\xi_2^3\xi_1)\cdot \xi_2\xi_1\xi_2^{-1}\cdot (\xi_6\xi_2^3\xi_1)^{-1},\\
&\label{rela92a4naeta44} & \xi_6=(\xi_6\xi_2^3\xi_1)\cdot (\xi_2\xi_1)\cdot\xi_2\cdot(\xi_2\xi_1)^{-1}\cdot (\xi_6\xi_2^3\xi_1)^{-1}.\end{aligned}$$ Note that, by (\[rela92a4naeta1\]), all of them are equivalent to $$\begin{aligned}
\label{rela92a4nas1}
\xi_1\xi_2\xi_1\xi_2\xi_1 = \xi_2\xi_1\xi_2\xi_1\xi_2.\end{aligned}$$
![\[figa92a4naeta4bis\]Generators at $x=\eta_4-\varepsilon$](figa92a4naeta4bis){width="8cm" height="5cm"}
Monodromy relations at $x=\eta_5$ {#monodromy-relations-at-xeta_5-1 .unnumbered}
---------------------------------
At $x=\eta_4-\varepsilon$, the generators are as in Figure \[figa92a4naeta4bis\]. Now, when $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_4)$, from $\eta_4-\varepsilon$ to $\eta_4+\varepsilon$, then moves on the real axis from $\eta_4+\varepsilon$ to $\eta_5-\varepsilon$, the $\xi_k$’s are deformed as in Figure \[figa92a4naeta5\], where $$\begin{aligned}
&& \alpha:=(\xi_2^{-1}\xi_6\xi_2)^{-1}\cdot \xi_6\xi_2\xi_6^{-1}\cdot (\xi_2^{-1}\xi_6\xi_2), \\
&& \beta:=(\xi_1\xi_2^{-1})^{-1}\cdot (\xi_2\xi_1)\cdot \xi_2\cdot (\xi_2\xi_1)^{-1}
\cdot (\xi_1\xi_2^{-1}).\end{aligned}$$ The singular line $L_{\eta_1}$ is tangent to the curve at two simple points, in both cases with intersection multiplicity 2. The monodromy relations at $x=\eta_5$ are then given as follows: $$\begin{aligned}
\label{rela92a4naeta51} & & (\xi_1\xi_2^{-1})^{-1}\cdot \xi_2\cdot (\xi_1\xi_2^{-1}) = (\xi_2\xi_1)^{-1}\cdot \xi_1\cdot (\xi_2\xi_1),\\
&\label{rela92a4naeta52} & (\xi_6\xi_2\xi_6^{-1})\cdot \xi_2\cdot (\xi_6\xi_2\xi_6^{-1})^{-1} = \xi_2^{-1}\xi_6\xi_2.\end{aligned}$$ By (\[rela92a4naeta1\]) and (\[rela92a4nas1\]), both of them are equivalent to $$\begin{aligned}
\label{rela92a4nas2}
\xi_2\xi_1\xi_1 = \xi_1\xi_1\xi_2.\end{aligned}$$
![\[figa92a4naeta5\]Generators at $x=\eta_5-\varepsilon$](figa92a4naeta5){width="8cm" height="5cm"}
Monodromy relations at $x=\eta_6$ {#monodromy-relations-at-xeta_6-1 .unnumbered}
---------------------------------
In Figure \[figa92a4naeta6\], we show how the generators are deformed when $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_5)$, from $\eta_5-\varepsilon$ to $\eta_5+\varepsilon$, and then moves on the real axis from $\eta_5+\varepsilon$ to $\eta_6-\varepsilon$. The singular line $L_{\eta_6}$ passes through the points $(1,1)$ and $(1,-1)$ which are both singularities of type $\textbf{A}_4$. Puiseux parametrizations of the curve at these points are given by: $$\begin{aligned}
(1,1):\quad & x=1+t^2,\ & y=1+t^4+3\, i\, \sqrt{3}\, t^5 + \mbox{higher terms},\\
(1,-1):\quad & x=1+t^2,\ & y=-1-t^4-3\, i\, \sqrt{3}\, t^5 + \mbox{higher terms}.\end{aligned}$$ The monodromy relations at $x=\eta_6$ are then given by $$\begin{aligned}
\label{rela92a4naeta61} & & \xi_2\cdot \xi_1\xi_2\xi_1^{-1}\cdot \xi_2\cdot \xi_1\xi_2\xi_1^{-1}\cdot \xi_2 = \xi_1\xi_2\xi_1^{-1}\cdot \xi_2\cdot \xi_1\xi_2\xi_1^{-1}\cdot \xi_2\cdot\xi_1\xi_2\xi_1^{-1},\\
&\label{rela92a4naeta62} & \alpha = (\xi_2^{-1}\xi_6\xi_2\alpha)^2 \cdot \xi_2^{-1}\xi_6\xi_2\cdot (\xi_2^{-1}\xi_6\xi_2\alpha)^{-2}.\end{aligned}$$ Notice that (\[rela92a4naeta61\]) is automatically satisfied, while (\[rela92a4naeta62\]) can be written as $$\begin{aligned}
\label{rela92a4nas3}
(\xi_2\xi_1)^4 = \xi_1\xi_2\end{aligned}$$ or, equivalently, as $$\begin{aligned}
\label{rela92a4nas33}
(\xi_2\xi_1\xi_1)^2 = e.\end{aligned}$$ Indeed, using (\[rela92a4nas2\]) under the form $\xi_1^{-1}\xi_2\xi_1 = \xi_1\xi_2\xi_1^{-1}$, the relation (\[rela92a4naeta61\]) can be written as $$\begin{aligned}
\xi_2 \cdot (\xi_1^{-1}\xi_2\xi_1 \cdot \xi_2 \cdot \xi_1)\xi_2\xi_1^{-1} \cdot \xi_2 = \xi_1^{-1}\xi_2\xi_1 \cdot \xi_2 \cdot (\xi_1^{-1}\xi_2\xi_1 \cdot \xi_2 \cdot \xi_1)\xi_2\xi_1^{-1}, \end{aligned}$$ which is nothing but $\xi_1\xi_2\cdot \xi_2\xi_1\xi_2 = \xi_2\xi_1\cdot \xi_2\xi_2\xi_1$ by (\[rela92a4nas1\]). By (\[rela92a4nainfini\]) this equality is always satisfied. The relation (\[rela92a4naeta62\]) is written as $$\begin{aligned}
\xi_6\xi_2\xi_6^{-1}\cdot (\xi_2^{-1}\xi_6\xi_2\cdot \xi_6\xi_2\xi_6^{-1})^2 =
(\xi_2^{-1}\xi_6\xi_2\cdot \xi_6\xi_2\xi_6^{-1})^2 \cdot \xi_2^{-1}\xi_6\xi_2,\end{aligned}$$ which is the same as $$\begin{aligned}
\xi_6\xi_2\xi_6^{-1}\cdot (\xi_6\xi_2\xi_6^{-1}\cdot \xi_2)^2 =
(\xi_6\xi_2\xi_6^{-1}\cdot \xi_2)^2 \cdot \xi_2^{-1}\xi_6\xi_2\end{aligned}$$ by (\[rela92a4naeta52\]). Equivalently $\xi_6\xi_2\xi_6^{-1}\cdot \xi_2 \cdot \xi_6\xi_2\xi_6^{-1} = \xi_2\xi_6\xi_2$. By (\[rela92a4naeta1\]), $\xi_6$ can be eliminated so $$\begin{aligned}
\label{rela92a4nas5}
\xi_1\xi_2\cdot (\xi_2\xi_1)^{-1}\cdot \xi_2\xi_2\xi_1 = \xi_2\xi_1\xi_2\xi_1\xi_2^{-1}.\end{aligned}$$ By (\[rela92a4nainfini\]) this is the same as $\xi_1\xi_2 = (\xi_2\xi_1)^4$, while (\[rela92a4nainfini\]) and (\[rela92a4nas1\]) show that (\[rela92a4nas5\]) is also the same as $(\xi_2\xi_1\xi_1)^2=e$.
![\[figa92a4naeta6\]Generators at $x=\eta_6-\varepsilon$](figa92a4naeta6){width="8cm" height="5cm"}
It remains to find the monodromy relations around the singular lines $L_{\eta_7}$ and $L_{\eta_8}$ corresponding to the non–real roots $\eta_7$ and $\eta_8$ of the discriminant $\Delta_y(f)(x)$.
Monodromy relations at $x=\eta_7$ and $x=\eta_8$ {#monodromy-relations-at-xeta_7-and-xeta_8 .unnumbered}
------------------------------------------------
Figure \[figa92a4naeta6\] shows the generators at $x=\eta_6-\varepsilon$. In Figure \[figa92a4naeta7\] (respectively Figure \[figa92a4naeta8\]), we show how the $\xi_k$’s are deformed when $x$ makes half–turn counter–clockwise on the circle $\mathbb{S}_\varepsilon(\eta_6)$, from $\eta_6-\varepsilon$ to $\eta_6+\varepsilon$, then moves on the real axis from $\eta_6+\varepsilon$ to $\Re(\eta_7)$, and finally moves straight along the line $(\Re(\eta_7),\eta_7)$ from $\Re(\eta_7)$ to $\Re(\eta_7)+i\, (\Im(\eta_7)+\varepsilon)$ (respectively along the line $(\Re(\eta_8),\eta_8)$ from $\Re(\eta_8)$ to $\Re(\eta_8)+i\, (\Im(\eta_8)-\varepsilon)$), where $$\gamma:=(\xi_2^{-1}\xi_6\xi_2\alpha)^{-1} \cdot \alpha \cdot (\xi_2^{-1}\xi_6\xi_2\alpha).$$ (In these figures we concentrate only on the generators which may give *a priori* some relations.) The monodromy relations at $x=\eta_7$ and $x=\eta_8$ are multiplicity 2 tangent relations given by $$\label{rela92a4naeta7} \gamma=\xi_2,$$ and $$\label{rela92a4naeta8} (\alpha \gamma)^{-1} \cdot \xi_2^{-1}\xi_6\xi_2 \cdot (\alpha \gamma)=\xi_1^{-1}\xi_2\xi_1,$$ respectively. In fact, these relations are automatically satisfied. Indeed, the relation (\[rela92a4naeta7\]) is written as $$\xi_6\xi_2\xi_6^{-1} \cdot (\xi_2^{-1}\xi_6\xi_2\cdot \xi_6\xi_2\xi_6^{-1}) \cdot \xi_2^{-1}\xi_6 = (\xi_2^{-1}\xi_6\xi_2\cdot \xi_6\xi_2\xi_6^{-1}) \cdot \xi_2^{-1}\xi_6\xi_2.$$ But, by (\[rela92a4naeta52\]), we know that $\xi_2^{-1}\xi_6\xi_2\cdot \xi_6\xi_2\xi_6^{-1} = \xi_6\xi_2\xi_6^{-1} \cdot \xi_2$, so the relation above is always satisfied. Now, using (\[rela92a4naeta7\]), the relation (\[rela92a4naeta8\]) is written as $$(\xi_2^{-1}\xi_6\xi_2\cdot \xi_6\xi_2\xi_6^{-1}) \cdot \xi_2^{-1}\xi_6\xi_2 \cdot \xi_2 = \xi_6\xi_2\xi_6^{-1} \cdot \xi_2^{-1}\xi_6\xi_2 \cdot \xi_2 \cdot \xi_1^{-1}\xi_2\xi_1,$$ which is equivalent to $\xi_2\xi_6\xi_2\xi_2 = \xi_6\xi_2\xi_2\xi_1^{-1}\xi_2\xi_1$ by (\[rela92a4naeta52\]). The latter is always satisfied, by (\[rela92a4naeta1\]).
![\[figa92a4naeta7\]Generators at $x=\Re(\eta_7)+i\, (\Im(\eta_7)+\varepsilon)$](figa92a4naeta7){width="8cm" height="5cm"}
![\[figa92a4naeta8\]Generators at $x=\Re(\eta_8)+i\, (\Im(\eta_8)-\varepsilon)$](figa92a4naeta8){width="8cm" height="5cm"}
Altogether, we have proved that the fundamental group $\pi_1(\mathbb{CP}^2\setminus C')$ is presented by the generators $\xi_1$ and $\xi_2$ and the relations (\[rela92a4nainfini\]), (\[rela92a4nas1\]), (\[rela92a4nas2\]) and (\[rela92a4nas3\]). The relation (\[rela92a4nainfini\]) can be written as $$\label{rela92a4nas4}
(\xi_2\xi_1\xi_2)^2=e.$$ This shows that (\[rela92a4nas1\]) is equivalent to (\[rela92a4nas3\]). The relation (\[rela92a4nas2\]) is automatically satisfied. Indeed, by (\[rela92a4nas3\]), it is equivalent to $$(\xi_2\xi_1\xi_1)^2=(\xi_1\xi_2\xi_1)^2.$$ But both sides are equal to $e$, by (\[rela92a4nas3\]) under the form (\[rela92a4nas33\]). Hence, $\pi_1(\mathbb{CP}^2\setminus C')$ is simply presented by the generators $\xi_1$ and $\xi_2$ and the relations (\[rela92a4nas3\]) and (\[rela92a4nas4\]). After the change $a:=\xi_2\xi_1\xi_2$ and $b:=\xi_2\xi_1$, the presentation is also given by $$\pi_1(\mathbb{CP}^2\setminus C') \simeq \bigl\langle\, a,\, b\, \mid a^2=e,\, aba=b^4\, \bigr\rangle.$$
The generator $b$ satisfies the following two properties:
1. $b^{15}=e$;
2. $b^5$ is in the centre of $\pi_1(\mathbb{CP}^2\setminus C')$.
Since $a^2=e$, the relation $aba=b^4$ gives $b^{16}=ab^4a=b$, that is, $b^{15}=e$ as desired. To show that $b^5$ is in the centre of $\pi_1(\mathbb{CP}^2\setminus C')$ we write: $$\begin{aligned}
b^5ab^{-5}a^{-1} = b\cdot b^4\cdot ab^{-5}a^{-1}
= b\cdot aba\cdot ab^{-5}a^{-1}=\\
ba\cdot b^{-4}\cdot a^{-1}
= ba\cdot a^{-1}b^{-1}a^{-1}\cdot a^{-1}
= e.\end{aligned}$$
It follows from the lemma that $\pi_1(\mathbb{CP}^2\setminus C')$ is also presented as: $$\begin{aligned}
\pi_1(\mathbb{CP}^2\setminus C') & \simeq
& \bigl\langle
a,\, b\mid a^2=e,\ aba=b^4,\, b^{15}=e,\, b^5a=ab^5
\bigr\rangle \\
& \simeq & \bigl\langle
a,\, b,\, c,\, d\mid a^2=b^{15}=e,\, aba=b^4,\, b^5a=ab^5,\, c=b^6, \\
&& \qquad d=b^5,\, da=ad,\, db=bd,\, dc=cd
\bigr\rangle \\
& \simeq & \bigl\langle
a,\, b,\, c,\, d\mid a^2=b^{15}=e,\, aba=b^4,\, c=b^6,\, d=b^5,\\
&& \qquad b=cd^{-1},\, da=ad,\, db=bd,\, dc=cd
\bigr\rangle \\
& \simeq & \bigl\langle
a,\, c,\, d\mid a^2=c^5=d^3=e,\, acd^{-1}a=c^4d^{-1},\, da=ad,\, dc=cd
\bigr\rangle \\
& \simeq & \bigl\langle
a,\, c,\, d\mid a^2=c^5=d^3=e,\, aca=c^4,\, da=ad,\, dc=cd
\bigr\rangle \\
& \simeq & \mathbb{D}_{10}\times (\mathbb{Z}/3\mathbb{Z}).\end{aligned}$$
[10]{}
E. Artal Bartolo, *Sur les couples de Zariski*, J. Algebraic Geom. **3** (1994) 223–247.
E. Artal Bartolo and J. Carmona Ruber, *Zariski pairs, fundamental groups and Alexander polynomials*, J. Math. Soc. Japan **50** (1998) 521–543.
A. Degtyarev, *Oka’s conjecture on irreducible plane sextics*, J. London Math. Soc. (to appear).
A. Degtyarev, *On irreducible sextics with non–abelian fundamental group*, Proceedings of Niigata–Toyama Conferences 2007, Adv. Stud. Pure Math. (to appear).
C. Eyral and M. Oka, *A proof of a conjecture of Degtyarev on non–torus plane sextics*, Proceedings of Niigata–Toyama Conferences 2007, Adv. Stud. Pure Math. (to appear).
Lê D. T. and C. P. Ramanujam, *The invariance of Milnor number implies the invariance of the topological type*, Amer. J. Math. **98** (1976) 67–78.
M. Oka, *Symmetric plane curves with nodes and cusps*, J. Math. Soc. Japan **44** (3) (1992) 375–414.
M. Oka, *Flex curves and their applications*, Geometriae Dedicata **75** (1999) 67–100.
M. Oka, *A new Alexander–equivalent Zariski pair*, Acta Math. Vietnam. **27** (3) (2002) 349–357.
M. Oka and D. T. Pho, *Classification of sextics of torus type*, Tokyo J. Math. **25** (2) (2002) 399–433.
D. T. Pho, *Classification of singularities on torus curves of type $(2,3)$*, Kodai Math. J. **24** (2001) 259–284.
E. R. van Kampen, *On the fundamental group of an algebraic curve*, Amer. J. Math. **55** (1933) 255–260.
O. Zariski, *On the problem of existence of algebraic functions of two variables possessing a given branch curve*, Amer. J. Math. **51** (1929) 305–328.
O. Zariski, *On the irregularity of cyclic multiple planes*, Ann. of Math. **32** (1931) 445–489.
O. Zariski, *The topological discriminant group of a Riemann surface of genus $p$*, Amer. J. Math. **59** (1937) 335–358.
O. Zariski, *Studies in equisingularity II. Equisingularity in codimension 1 (and characteristic zero)*, Amer. J. Math. **87** (1965) 972–1006.
O. Zariski, *Contribution to the problem of equisingularity*, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Ed. Cremonese, Roma (1970), 261–343.
[^1]:
[^2]: By such a moduli space we mean the quotient space $\mathcal{C}(\Sigma,d)/\text{PGL}(3,\mathbb{C})$ of the space $\mathcal{C}(\Sigma,d)$ of reduced plane curves with degree $d$ and set of singularities $\Sigma$ by the ‘standard’ group action of $\text{PGL}(3,\mathbb{C})$.
[^3]: Though the existence of the structure of an algebraic variety on a moduli space is not always obvious, the moduli space we consider here has such a structure. The last assertion in the theorem then implies this moduli space has at least two irreducible components as well.
[^4]: We always mean the fundamental group of the complement of the curve.
|
---
abstract: 'Let $G$ be a graph of hyperbolic groups with 2-ended edge groups. We show that $G$ is hierarchically hyperbolic if and only if $G$ has no distorted infinite cyclic subgroup. More precisely, we show that $G$ is hierarchically hyperbolic if and only if $G$ does not contain certain quotients of Baumslag–Solitar groups. As a consequence, we obtain several new results about this class, such as quadratic isoperimetric inequality and finite asymptotic dimension.'
author:
- 'Bruno Robbio, Davide Spriano'
bibliography:
- 'Bibliography.bib'
title: 'Hierarchical hyperbolicity of hyperbolic-2-decomposable groups'
---
Introduction
============
When trying to understand a complicated group, a strategy that often proves to be successful is to reduce it to several, more manageable, groups. This can be achieved in several ways, and for this paper we will consider groups that split as graphs of groups with 2-ended edge groups. For the sake of brevity, if $P$ is a property of a group, we say that a group is *$P$-2-decomposable* if it splits as a graph of groups with 2-ended edge groups and vertex groups satisfying property $P$.
Considering groups of this form is not a novelty in geometric group theory. An important example is the class of $\mathbb{Z}$-2-decomposable groups, also known as *generalized Baumslag–Solitar groups* (GBS groups). Although we will not dive deeply in the theory of GBS groups from a traditional viewpoint, it is worth noting that this class has been extensively studied and shown to be an extremely rich object to analyse from multiple points of view. To name a few, GBS groups have been studied in relation with JSJ decompositions ([@ForesterGBS]), quasi-isometries ([@RipsSubgroupsSmallCancellation]), automorphisms ([@LevittAutGBS]) and cohomological dimension ([@krophollerGBS]). For a general overview of results on GBS groups we refer to the survey by Robinson ([@robinsonGBS]).
This paper focuses on hyperbolic-2-decomposable groups satisfying a technical condition called *balancedness*. A group $G$ is said to be balanced if for every $g\in G$ of infinite order, whenever $hg^ih^{-1}=g^j$ for some $h\in G$ it follows that $|i|=|j|$. The notion of balancedness played an important role in the theory of graphs of groups. In [@wise2000subgroup], the author shows that a free-2-decomposable group is subgroup separable if and only if it is balanced. In [@ShepherdWoodhouse:QIrigidity], the authors extend Wise’s result to (virtually-free)-2-decomposable groups, obtaining quasi-isometrical rigidity for certain balanced groups. In [@button2015balanced] the author studies the relation between possible acylindrical actions of (torsion-free)-2-decomposable groups in connection with balancedness of such groups.
In our main result (Theorem \[thmi: HHG iff no BS subgroups\] below) we show that every balanced hyperbolic-2-decomposable group is a *hierarchically hyperbolic group*. This theorem can be thought of as a hierarchically hyperbolic group version of two corollaries of the Bestvina–Feighn combination theorem that give necessary and sufficient conditions for a hyperbolic-2-decomposable group to be hyperbolic ([@BestvinaFeighnCombination Corollary Section 7] and [@BestvinaFeighnCombinationAddendum Corollary 2.3]).
Before discussing the theory of hierarchically hyperbolic groups, let us point out some consequences of hierarchical hyperbolicity.
\[theorem: main theorem applications\] Let $G$ be a hyperbolic-2-decomposable group. If $G$ is balanced, then
1. $G$ has finite asymptotic dimension ([@HHSAsdim2015 Theorem A]);
2. $G$ is coarse median in the sense of Bowditch ([@bowditch2013coarse Definition 2.1]) and, therefore, satisfies a quadratic Dehn function ([@HHSII Corollary 7.5]);
3. every top-dimensional quasi-flat in $G$ is at bounded distance from a union of standard orthants [@HHSFlats Theorem A];
4. if $G$ is virtually torsion-free then either $G$ has uniform exponential growth or there exists a space $E$ such that $G$ is quasi-isometric to $\mathbb{Z}\times E$ ([@abbott2019exponentialgrowth Theorem 1.1]);
5. $G$ satisfies the Morse local-to-global property, in particular, given stable subgroups $H$, $K$ of $G$, there are sufficient conditions to ensure $\langle H, K \rangle \cong H\ast_{H \cap K}K$ ([@russell2019local Theorem G and Theorem 4.20]);
6. a finite family of subgroups $\{H_i\}$ is hyperbolically embedded in $G$ if and only if it is a family of almost-malnormal strongly quasiconvex subgroups ([@russellsprianotran:convexity Theorem 8.1]);
7. there is a linear bound on the length of the shortest conjugator for any pair of conjugate Morse elements in $G$ ([@abbott2018conjugator Theorem A]);
8. $G$ does not contain distorted cyclic subgroups ([@HHSBoundaries Theorem 7.1] and [@durham2018corrigendum Theorem 3.1]).
To the best of our knowledge, all the items of Theorem \[theorem: main theorem applications\] were previously unknown for hyperbolic-2-decomposable groups.
Hierarchically hyperbolic groups (HHGs) were introduced by Behrstock, Hagen and Sisto in [@HHSI] and [@HHSII] and have been widely studied since. The key intuition that led to the definition is that several aspects of the machinery developed by Masur and Minsky to study the mapping class groups in [@MasurMinskyI; @MasurMinskyII] could be generalized to a much larger class of groups. Examples of hierarchically hyperbolic groups are most (conjecturally all) cubical groups ([@HagenSusse]), in particular all right-angled Artin and Coxeter groups; groups hyperbolic relative to peripheral hierarchically hyperbolic groups ([@HHSII]); many 3-manifold groups ([@HHSII]); certain quotients of mapping class groups ([@behrstock2020combinatorial; @HHSAsdim2015]) and many more. Given the number of interesting examples in the class of hierarchically hyperbolic groups, it is perhaps surprising that being a hierarchically hyperbolic group has several deep consequences, as Corollary \[theorem: main theorem applications\] suggests. As the full definition of hierarchically hyperbolic group is quite technical, we postpone it until Section \[subsection: basics of hhs\]. For the time being, it is enough to know that a hierarchically hyperbolic group structure $(G,\mathfrak{S})$ on $G$ consists of a collection of $\delta$–hyperbolic spaces $\{\mathcal{C}V\mid V\in\mathfrak{S}\}$, and projections $\pi_V$ from $\mathrm{Cay}(G)$ onto the various hyperbolic spaces $\mathcal{C}V$, satisfying a number of axioms.
A property of hierarchically hyperbolic groups that plays a key role in this paper is the fact that they do not have distorted cyclic subgroups. As a first application, this property provides a source of examples of groups that are not hierarchically hyperbolic. The standard example of groups with distorted cyclic subgroups are Baumslag–Solitar groups $BS(m,n)=\langle a,t\mid ta^nt^{-1}=a^m\rangle$ with $\vert m \vert \neq \vert n \vert$, often called *non-Euclidean* Baumslag–Solitar subgroups. In particular, if a group has a non-Euclidean Baumslag–Solitar subgroup, then the group cannot be hierarchically hyperbolic. When considering hyperbolic-2-decomposable groups, this is a clear obstruction to hierarchical hyperbolicity. The main result of the paper is that, up to some issues with torsion, this is essentially the only obstruction. More precisely, we say that a *non-Euclidean almost Baumslag–Solitar group* is a group generated by two infinite order elements $a,b$ such that we have $ba^{m}b^{-1} = a^{n}$ for some $\vert m \vert \neq \vert n \vert$. Equivalently, a non-Euclidean almost Baumslag–Solitar group is a quotient of a non-Euclidean Baumslag–Solitar group where the image of the element $a$ has infinite order (see Definition \[def: almost bs groups\] and successive remarks).
Then we have the following theorem, which is a shortened version of Corollary \[corollary: main result version 2\].
\[thmi: HHG iff no BS subgroups\] Let $G$ be a hyperbolic-2-decomposable group. The following are equivalent.
1. $G$ admits a hierarchically hyperbolic group structure.
2. $G$ does not contain a distorted infinite cyclic subgroup.
3. $G$ does not contain a non-Euclidean almost Baumslag–Solitar group.
Moreover, if $G$ is virtually torsion-free, condition (3) can be replaced by
1. $G$ does not contain a non-Euclidean Baumslag–Solitar group.
We believe that Item (3’) of Theorem \[thmi: HHG iff no BS subgroups\] should be true even without the assumption of $G$ being virtually torsion-free. See the Questions section for further discussion.
Theorem \[thmi: HHG iff no BS subgroups\] is a combination theorem for hierarchically hyperbolic groups: if edge and vertex groups of a graph of groups satisfy certain conditions, then the (fundamental group of the) graph of groups is hierarchically hyperbolic. Combination theorems for hierarchically hyperbolic groups are not new, and indeed Theorem \[thmi: HHG iff no BS subgroups\] relies on a combination theorem for hierarchically hyperbolic groups ([@berlai2018refined Theorem C]). However, there are important differences with many other results of the same form. Firstly, Theorem \[thmi: HHG iff no BS subgroups\] can be used as a black-box. The statement of the theorem is elementary and does not require familiarity with hierarchically hyperbolic groups to be understood. Secondly, Theorem \[thmi: HHG iff no BS subgroups\] imposes no condition of geometric nature on the edge embeddings, such as, say, the images of the edge groups in the vertex groups to form an *almost-malnormal* collection (Definition \[def: almost malnormal collection\]).
Unlike Theorem \[thmi: HHG iff no BS subgroups\], several important results in the literature require almost-malnormality of edge groups. For instance, the groundbreaking work of Haglund–Wise and Hsu–Wise ([@HaglundWise:ACombination; @HsuWise:Cubulating]), which is crucial in Agol’s proof of the virtual Haken conjecture ([@AgolHaken]), provides a combination theorem for virtually compact special groups where one of the key condition is the almost-malnormality of the edge groups in the vertex groups.
**Detecting almost Baumslag–Solitar subgroups:** In general, checking whether a given graph of groups contains an almost Baumslag–Solitar subgroup may be challenging. For this reason, we introduce the notion of *balanced edges*. An edge $e$ of a graph of groups ${\mathcal{G}}$ is a balanced edge if for every infinite order element $g \in G_e$ and $h \in \pi_1({\mathcal{G}}-e)$ $$\text{ if }hg^ih^{-1}=g^j \text{ then }|i|=|j|.$$
We then have the following criterion to detect almost Baumslag–Solitar subgroups.
\[thm: non-euclidean BS iff unbalanced edge intro section\] Let ${\mathcal{G}}$ be a graph of groups where none of the vertex groups contain distorted cyclic subgroups. Then $\pi_1({\mathcal{G}})$ contains a non-Euclidean almost Baumslag–Solitar subgroup if and only if ${\mathcal{G}}$ has an unbalanced edge.
The proof of this result can be found in Theorem \[thm: non-euclidean BS iff unbalanced edge\].
The absence of unbalanced edges is a significantly weaker condition than almost-malnormality of edge groups. Indeed, whenever the underlying graph of ${\mathcal{G}}$ is a tree, all the edges will be automatically balanced (Remark \[remark: edges in tree are balanced\]), even if the edge groups do not form almost-malnormal collections. In particular, we conclude that if $G = H_1 \ast_C H_2$ where $H_i$ are hyperbolic and $C$ is 2-ended, then $G$ does not contain non-Euclidean almost Baumslag–Solitar subgroups. As a consequence, we have the following.
\[corol: corollary to mainT intro version\] Let $G = H_1 \ast_C H_2$ where $H_i$ are hyperbolic and $C$ is 2-ended. Then $G$ is a hierarchically hyperbolic group.
This result is proved in Corollary \[corol: corollary to mainT\].
Let us discuss briefly the main aspects of the proof of Theorem \[thmi: HHG iff no BS subgroups\].
**Idea of the proofs:** Ultimately, our goal is to show that the combination theorem for hierarchically hyperbolic groups ([@berlai2018refined], see \[comb\_thm\_ver2\]) can be applied to hyperbolic-2-decomposable groups with no non-Euclidean almost Baumslag–Solitar subgroups. However, verifying the hypotheses of the combination theorem present several challenges. Firstly, we need to construct hierarchically hyperbolic group structures on all vertex and edge groups, and then verify that such structures satisfy the hypotheses of the combination theorem. The key hypothesis to check here is that the embedding of the edge groups into the vertex groups are *glueing hieromorphisms*. Avoiding the technicalities (discussed in Definition \[def: glueing\_hieromorphism\]), given hierarchically hyperbolic groups $(G,\mathfrak{S}),(G',\mathfrak{S}')$ a necessary condition to the existence of a glueing hieromorphism is that $\mathfrak{S}\subseteq\mathfrak{S}'$. That is to say, the set of hyperbolic spaces forming the hierarchical structure of $G$ is a subset of the hierarchical structure of $G'$. This is one of the key difficulties in equipping the edge and vertex groups with hierarchically hyperbolic structure: the structure of every edge needs to be a subset of the structure of both the vertices it is incident to. However, since we do not require almost-malnormality, the structures of two edge groups incident to the same vertex group may interact with each other, which in turn influences the structures of the other vertices adjacent to such edges and so on.
To overcome this issue, we find hierarchically hyperbolic structures on the edge groups that are compatible whenever the edge groups are not almost-malnormal. This is done by fixing a reference group, in this case the dihedral group, and pulling back the same hierarchical structure to all the various edge groups. The key notion used is the one of *linearly parametrizable graph of groups*. A graph of groups ${\mathcal{G}}$ is linearly parametrizable if there is a homomorphism $\Phi \colon \pi_1({\mathcal{G}}) \to \dihed$ such that $\Phi\vert_{G_u}\colon G_u \to \dihed$ is a quasi-isometry for each vertex or edge group $G_u$. Similarly a group $G$ is linearly parametrizable if $G = \pi_1({\mathcal{G}})$ for some linearly parametrizable graph of groups ${\mathcal{G}}$. Note that, by definition, a linearly parametrizable group is (2-ended)-2-decomposable. Perhaps the reader will not be surprised that balancedness is the key for the converse to hold. This result is a consequence of Theorem \[thm: balanced edges and BS version 2\]
Let $G$ be a group. Then $G$ is linearly parametrizable if and only if $G$ is (2-ended)-2-decomposable and balanced.
As hinted before, linearly parametrizable graph of groups satisfy the hypotheses of the combination theorem for hierarchically hyperbolic groups, yielding a version of Theorem \[thmi: HHG iff no BS subgroups\] that holds for (2-ended)-2-decomposable groups.
To extend such a result to hyperbolic-2-decomposable groups, we introduce the concept of *conjugacy graph* (Definition \[definition: conjugacy graph\]). The conjugacy graph is a graph of groups associated to each commensurability class of edge groups. If the group $\pi_1({\mathcal{G}})$ is balanced, then all the conjugacy graphs are linearly parametrized. Then, using a relative hyperbolicity argument, we construct hierarchically hyperbolic structures on the vertex groups that are compatible with the various conjugacy graphs (Theorem \[theorem: readaptation of relative HHG\]).
Questions
---------
**The non virtually torsion-free case:** our results are stated differently for the case of virtually torsion-free groups. The main problem being that we could not determine in the class of hyperbolic-2-decomposable groups whether all non-Euclidean almost Baumslag–Solitar groups contain a Baumslag–Solitar subgroup.
Does every non-Euclidean almost Baumslag–Solitar subgroup of a hyperbolic-2-decomposable group contain a non-Euclidean Baumslag–Solitar subgroup?
We stress that this question has a positive answer for certain torsion-free groups. In [@levitt2015quotients Proposition 7.5] the author shows that the question has a positive answer for GBS groups. In [@button2015balanced Proposition 9.6] the author extends the result to (torsion-free hyperbolic)-2-decomposable groups. However, the results appearing in those papers rely heavily on the absence of torsion. As we will see in Section \[section: graph of infinite virtually cyclic groups\], it is enough to assume that $G$ is virtually torsion-free. It is perhaps also worth noting that a graph of virtually torsion-free groups may not have a virtually torsion-free fundamental group, even when the edge groups are assumed to be of finite index in its neighbouring vertex groups. This is illustrated, for instance, in the examples appearing in these [mathoverflow replies[^1].](https://mathoverflow.net/questions/330632/is-an-hnn-extension-of-a-virtually-torsion-free-group-virtually-torsion-free)
**Generalization to HHG-2-decomposable** In our proofs, hyperbolicity of the edge groups is used only in Theorem \[theorem: readaptation of relative HHG\] and Lemma \[remark: elementarizer\]. Thus we expect that finding appropriate replacements for the two results above will yield a sufficient condition for a (hierarchically hyperbolic)-2-decomposable group to be hierarchically hyperbolic. However, the question becomes harder when asking for a full characterization. As remarked before, all hierarchically hyperbolic groups are balanced, hence balancedness is surely a necessary condition in Question 2.
Under which conditions a (hierarchically hyperbolic)-2-decomposable group is hierarchically hyperbolic?
A possible strategy to answer this question would be to extend the tools developed in Section \[section: graph of word hyperbolic groups\] to the class of hierarchically hyperbolic groups. That is to say, provide conditions guaranteeing that the hierarchically hyperbolic structure of edge groups can be included in the one of the vertex group.
However, we don’t think this strategy would work in the general case. For instance, consider $\mathbb{Z}^2$-2-decomposable groups (also known as *tubular groups*). If one vertex has three incoming edges, defining pairwise linearly independent lines, there is no straightforward way of defining a hierarchically hyperbolic group structure on $\mathbb{Z}^2$ that contains each edge group.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The first author would like to thank Montserrat Casals-Ruiz, Mark Hagen and Ilya Kazachkov for numerous and attentive discussions on the work presented in this paper and also for reading and commenting very early versions of it. The second author would like to thank Alessandro Sisto for numerous helpful discussions and suggestions. We thank Daniel Woodhouse and Matteo Pintonello for several early inputs on this paper. We would also like to thank Jason Behrstock and Harry Petyt for numerous comments that helped to improve the exposition of the paper.
The first author was supported by the University of the Basque Country through the grant PIF17/241. He also acknowledges the support of the ERC grant PCG-336983, Basque Government Grant IT974-16, and Ministry of Economy, Industry and Competitiveness of the Spanish Government Grant PID2019-107444GA-I00.
The second author was partially supported by the Swiss National Science Foundation (grant \# 182186).
Graph of groups and balanced groups {#section: graph of groups and balanced groups}
===================================
Graph of groups
---------------
We start by recalling the definition of a graph of groups. As usually with graph of groups, we will consider oriented edges. Many of the results of this subsection are probably known to experts. We include them here for the reader’s convenience and to uniformize notation.
A *graph* $\Gamma$ consists of sets $V(\Gamma)$, $E(\Gamma)$ and maps $$\begin{aligned}
E(\Gamma) &\to V(\Gamma) \times V(\Gamma); & E(\Gamma) &\to E(\Gamma) \\
e &\mapsto (e^{+}, e^{-}) & e & \mapsto \bar{e}\end{aligned}$$ satisfying $\bar{\bar{e}} = e$, $\bar{e}\neq e$ and $\bar{e}^{-} = e^{+}$.
The elements of $V(\Gamma)$ are called *vertices*, the ones of $E(\Gamma)$ are called *edges*, the vertex $e^{-}$ is the *source* of $e$, $e^{+}$ is the *target* and $\bar{e}$ is the *reverse edge*. A graph $\Gamma$ is *finite* if both $V(\Gamma), E(\Gamma)$ are finite sets. A *subgraph* of $\Gamma$ is a graph $\Gamma'$ such that $V(\Gamma') \subseteq V(\Gamma)$ and $E(\Gamma') \subseteq E(\Gamma)$. Given a graph $\Gamma$, it is standard to associate to it a $\Delta$–complex $\vert \Gamma \vert$. We say that $\Gamma$ is *connected* if $\vert \Gamma \vert$ is. We say that a graph $\Gamma$ is a *tree* if $\vert \Gamma \vert$ is simply connected. We say that a subgraph $T$ of $\Gamma$ is a *spanning tree* if $V(T) = V(\Gamma)$ and $T$ is a tree.
A *graph of group* ${\mathcal{G}}$ consists of a finite graph $\Gamma$, a collection of groups $\{G_v \mid v \in V(\Gamma)\}$, $\{G_e \mid e \in E(\Gamma)\}$ and injective homorphisms $\phi_{e^\pm}:G_e \to G_{e^{\pm}}$ such that
1. $G_{e} = G_{\bar{e}}$;
2. $\phi_{e^{+}} = \phi_{\bar{e}^{-}}$.
We will often use the notation $V({\mathcal{G}})$ to denote $V(\Gamma)$ and similarly for $E({\mathcal{G}})$.
Let ${\mathcal{G}}= (\Gamma, \{G_v\}, \{G_e\}, \{\phi_{e^{\pm}}\})$ be a graph of groups. We define the group $F{\mathcal{G}}$ as: $$F{\mathcal{G}} = \left(\bigast_{v \in V(\Gamma)}G_v\right) \ast \left(\bigast_{e \in E(\Gamma)} \langle t_{e} \rangle\right).$$ Let $T$ be a spanning tree of $\Gamma$. Then the *fundamental group* of ${\mathcal{G}}$ with respect to $T$, denoted by $\pi_1({\mathcal{G}},T)$, is the group obtained adding the following relations to $F{\mathcal{G}}$:
1. $t_e = t_{\bar{e}}^{-1}$;
2. $t_e = 1$ if $e \in E(T)$;
3. $t_e \phi_{e^{+}}(x)t_e^{-1} = \phi_{e^{-}}(x)$ for all $x \in G_e$.
\[rmk: pi\_1 of G does not depend on the spannig tree\] The group $\pi_1({\mathcal{G}}, T)$ does not depend on the choice of the spanning tree, meaning that for different spanning trees $T, T'$ there is an isomorphism $\pi_1({\mathcal{G}},T )\to \pi_1({\mathcal{G}},T')$. For this reason, we will often denote $\pi_1({\mathcal{G}},T)$ simply by $\pi_1({\mathcal{G}})$ (see, for instance [@SerreTrees Proposition 20]).
Unless otherwise specified, we will represent the elements of $\pi_1({\mathcal{G}})$ in the alphabet $\bigcup_{v \in V({\mathcal{G}})} G_v \cup \bigcup_{e \in E({\mathcal{G}})}\langle t_{e}\rangle$. That is, we write each element $g \in \pi_1({\mathcal{G}})$ as $g = x_0 x_1 \dots x_k$ where either $x_i \in G_v$ for some $v$, or $x_i = t_{e}^{m}$ for some $e\in E({\mathcal{G}})$. Moreover, we will assume that if $1 \neq x_i \in G_v$, then $x_{i+1} \not\in G_v$, and similarly if $1 \neq x_i \in \langle t_e \rangle$, then $t_{i+1} \not \in \langle t_e \rangle$. Note that this is not a restrictive assumption as if $x_i, x_{i+1} \in G_v$, then we replace them by the element $x'= x_{i}x_{i+1} \in G_v$, and similarly for $\langle t_e \rangle$. Finally, we will assume that if $x_i$ has the form $t_e^{\epsilon}$, then $\epsilon \geq 0$. Indeed, otherwise substitute $t_e^{\epsilon}$ with $t_{\bar{e}}^{-\epsilon}$.
For many purposes it is convenient to choose a way to write elements of $\pi_1({\mathcal{G}})$ that takes the geometry of the graph in account.
A word $w$ is written in *path form* if $$w = g_0t_{e_1}^{\epsilon_1}g_1\ldots t^{\epsilon_n}_{e_n}g_n,$$ where $\varepsilon_i=\pm 1$ and we require $g_i \in G_{e_{i+1}^-}$ and $g_i \in G_{e_i^+}$, whenever defined, and $g_0, g_n \in G_v$ for some $v$. As a consequence, $e_1, \dots, e_n$ form a closed path in $\Gamma$. We say that the path form is *based* at $v$.
\[rmk: writing elements in path form\] Let $u$ be any word in the alphabet $\bigcup G_v \cup \{t_{e}\}_{e \in E({\mathcal{G}})}$. It is always possible to replace $u$ with some $p$ written in path form such that $u$ and $p$ represent the same element of $\pi_1({\mathcal{G}},T)$. Moreover, the loop of edges associated can be based at any vertex of ${\mathcal{G}}$. Indeed, suppose that the beginning of $u$ is of the form $g_0 g_1$, with $g_0 \in G_v$, $g_1 \in G_w$. Choose a path $e_1, \dots, e_m$ in $T$ between $v$ and $w$. This is always possible since $T$ is a spanning tree. Then replace the beginning of $u$ with $g_0 t_{e_1} 1 t_{e_{2}} \dots t_{e_n}g_1$, where $1$ represents the trivial element. The case where one (or both) of $g_0, g_1$ were stable letters is analogous. Since we added only stable letters corresponding to edges in the spanning tree, we did not change the group element represented. Proceeding in this way we obtain a word $p'$ written in path form that represents the same element of $u$. Suppose that the loop associated to $p'$ is based at some vertex $v$, and we want to have a word based at some other vertex $w$. Again, by considering a path $e_1, \dots , e_m$ connecting $v$ and $w$ in the spanning tree $T$, we can conjugate $p'$ by $t_{e_1} 1 t_{e_2} \dots t_{e_n}$ to obtain the desired word $p$.
In particular, every element $g\in \pi_1({\mathcal{G}})$ can be written in path form.
Following the above argument, an element $\omega$ can always be written in path form $\omega=g_0t_{e_1}^{\varepsilon_1}\ldots t_{e_n}^{\varepsilon_n}g_n$ where $\varepsilon_i=1$ for every $i$. We adopt this notation when considering an element written in path form in a graph of groups where the underlying graph has more than one edge. We believe that this renders the proofs of results more readable. However, if we are working with either a simple HNN extension or free product with amalgamation then we allow $\varepsilon$ to be $-1$.
\[thm: normal form thm\] Let ${\mathcal{G}}$ be a graph of groups and let $g = g_0 t_{e_1} \dots t_{e_n}g_n$ be written in path form. Then if $g = 1$ in $\pi_1({\mathcal{G}})$, there is $i$ such that $e_{i} = \bar{e}_{i+1}$ and $g_i \in \phi_{{e_i}^{+}}(G_{e_i})$.
This is a well known result. For a detailed proof see [@Bogopolski Theorem 16.10].
Note, when the underlying graph of ${\mathcal{G}}$ is a tree, we have $t_{e_i}= 1$ in $\pi_1({\mathcal{G}})$ for all edges $e_i$. Hence, we can greatly simplify path forms to simply be $g_0 \dots g_n$ where $g_i \in G_v$ for some vertex $v$ and $g_{i \pm 1} \not \in G_v$, and modify the normal form theorem accordingly.
Let ${\mathcal{G}}$ be a graph of groups. A path word $g = g_0 t_{e_1} \dots t_{e_n}g_n$ is written in *reduced form* if for each $i$ such that $e_i = \bar{e}_{i+1}$ it follows that $g_i \not\in \phi_{t_{e_i^+}}(G_{e_i})$.
\[cor: existence reduced form\] For every $g \in \pi_1({\mathcal{G}},T)$ and $v \in V({\mathcal{G}})$ it is possible to write $g$ in a reduced form based at the vertex $v$.
We recall now the notion of commensurable subgroups. Note that this should not be confused with the weaker condition of *abstract commensurability:* two groups $A, B$ are abstractly commensurable if they contain isomorphic finite index subgroups.
\[def: commensurable subgroups\] Let $G$ be a group and $A, B \leq G$ be subgroups. We say that $A$ and $B$ are *commensurable* if there exists $g \in G$ such that $gAg^{-1} \cap B$ has finite index in $B$ and $A \cap g^{-1} B g$ has finite index in $A$.
Moreover, we say that two elements $a,b\in G$ are non-commensurable if $\langle a\rangle$ and $\langle b\rangle$ are non-commensurable in $G$.
When $A, B \leq G$ are infinite virtually cyclic, being commensurable amounts to the existence of some $g \in G$ such that $\lvert A^g\cap B\rvert = \infty$. Conversely, if $|A^g\cap B|<\infty$ for every $g\in G$ then $A$ and $B$ are not commensurable in $G$.
A handy application of the normal form Theorem is the following.
\[lemma: conjugacy path in G\] Let ${\mathcal{G}}$ be a graph of groups, let $v, w \in V({\mathcal{G}})$ and $x \in G_v -\{1\}$, $y \in G_w-\{1\}$. Then $x,y$ are conjugate in $\pi_1({\mathcal{G}},T)$ if and only if there is a sequence of edges $e_1, \dots, e_n$ between $v$ and $w$ and elements $g_i$ satisfying $g_i \in G_{e_{i}^{+}}, g_i \in G_{e^{-}_{i+1}}$, whenever defined, such that: $$(g_0 t_{e_1} g_{1} \dots t_{e_n}g_n) x (g_0 t_{e_1} g_{1} \dots t_{e_n}g_n)^{-1} = y.$$ Moreover, for each $g_i$ we have $ \phi_{e_{i+1}^{-}}(G_{e_{i+1}}) \cap g_i\phi_{e_{i}^{+}}(G_{e_i})g_{i}^{-1} \neq \{1\}$.
One implication is clear, we need to show the other. Suppose $x,y$ are conjugate and let $h \in \pi_1({\mathcal{G}})$ be such that $hxh^{-1}=y$. By Corollary \[cor: existence reduced form\], there is a reduced path word $u= u_0 t_{e_1} u_{1} \dots t_{e_m}u_m$ based at the vertex $v$ that represents $h$. Choose a shortest path $f_1, \dots, f_s$ of $T$ that connects $w$ and $v$ and let $p = t_{f_1} 1 t_{f_2} \dots t_{f_s}$. Then we have $(pu)x(pu)^{-1} = y$, where both sides of the equations are path words based at $w$. If we multiply by $y^{-1}$, we have that $(pu)x(pu)^{-1}y^{-1} = 1$, where both sides of the equation are path words. Spelling it out we have: $$\left[ \left(t_{f_1} 1 t_{f_2} \dots t_{f_s}\right)\left( u_0 t_{e_1} u_{1} \dots t_{e_m}u_m \right)\right]x\left[ \left( u_m^{-1} t^{-1}_{e_m} \dots u^{-1}_{1} t^{-1}_{e_a}u^{-1}_0 \right)\left(t^{-1}_{f_1} 1 t^{-1}_{f_2} \dots t^{-1}_{f_s}\right)\right]y^{-1} = 1.$$
By the normal form Theorem (Theorem \[thm: normal form thm\]), in the left hand side of the equation there is a subword of the form $t_{e} g t_{\bar{e}}$, with $g \in \phi_{e^{+}}(G_e)$. Our goal is to perform reductions to assume that every such occurrence contains the $x$. So, suppose this is not the case. Without loss of generality the subword must appear in $\left[ \left(t_{f_1} 1 t_{f_2} \dots t_{f_s}\right)\left( u_0 t_{e_1} u_{1} \dots t_{e_m}u_m \right)\right]$. Since $u$ was assumed to be reduced and $f_1, \dots, f_s$ is a shortest path, the subword must be $t_{f_s}u_0t_{e_1}$, where $u_0 = \phi_{f_s^{+}}(z)$ for some $z \in G_{f_s}$. Then replace $t_{f_s}u_0t_{e_1}$ by $\phi_{f^{-}_s}(z)$, and perform the symmetric change on the other side of the $x$. Note that this process reduces the length of the path $f_1, \dots, f_s$ by one. In particular, it has to terminate.
So, assume that no reduction can be performed in $pu = \left[ \left(t_{f_1} 1 t_{f_2} \dots t_{f_s}\right)\left( u_0 t_{e_1} u_{1} \dots t_{e_m}u_m \right)\right]$. If $pu = h_0 \in G_w$, and hence $x,y\in G_w$ are conjugate in $G_w$, we are done. So suppose this is not the case. We need to have that $u_m x u_m^{-1} = \phi_{e_m^{+}}(z)$ for some $z \in \G_{e_m}$. Substitute $t_{e_m} u_m x u_m^{-1}t_{e_m}^{-1}$ with $\phi_{e_m^{-}}(z)$. By proceeding as above, we obtain the claim for each $u_i$.
Whenever we are working with a graph of groups, it is often the case that we are interested in studying a subgraph of groups. To that end, we adopt the following notation.
\[notation: subgraph notation\] Let $\mathcal{G}$ be a graph of groups and $\Gamma$ its underlying graph. If $\Lambda\subseteq\Gamma$ is a connected subgraph, then we can define the subgraph of groups $\mathcal{G}|_{\Lambda}$, where the underlying graph is $\Lambda$, every vertex and edge in $\Lambda$ has the same associated groups as in $\mathcal{G}$.
We call $\mathcal{G}|_{\Lambda}$ the *subgraph of groups spanned by $\Lambda$*.
Let ${\mathcal{G}}$ be a graph of groups and let $\Lambda \subseteq \Gamma$ be a subgraph. Let $T'\subseteq\Lambda$ be a spanning tree of $\Lambda$ defined as $T'=T\cap\Lambda$. Then, there exists a group injection $\pi_1({\mathcal{G}}|_{\Lambda},T')\hookrightarrow \pi_1({\mathcal{G}},T)$.
We define the function $\iota:\pi_1({\mathcal{G}}|_{\Lambda},T')\to\pi_1({\mathcal{G}},T)$ on generators such that $\iota|_{G_v}$ is the identity for every $v\in\Lambda$. Let $g\in\pi_1({\mathcal{G}},T)$ be written in reduced form as $g=g_0t_{e_1}^{\varepsilon_1}\ldots t_{e_n}^{\varepsilon_n}g_n$. By the normal form theorem, $\iota(g)=1$ if and only if there exists $i$ such that $e_i=\overline{e_{i+1}}$ and $g_i\in\phi_{e_i
^+}(G_{e_i})$. By definition, this is equivalent to $g=1$ in $\pi_1({\mathcal{G}},T')$.
Balanced groups
---------------
As mentioned in the introduction, a fundamental notion throughout the paper is the notion of balanced group.
Let $G$ be a group and $g\in G$. We say that $g$ is balanced either if $g$ has finite order, or if whenever $g^n$ is conjugate to $g^m$, it must follow $|n|=|m|$. We say that a group $G$ is *balanced* if every element is balanced.
\[lemma: finite index and balance\] Let $G$ be a group and assume that there exists a balanced subgroup $H$ of $G$ of finite index. Then, $G$ is balanced.
We are now going to study how balanced groups behave under amalgamated products and HNN extension over virtually cyclic groups. A key property of virtually cyclic groups that will be used throughout the paper is that if $a, b$ are infinite order elements of a virtually cyclic group, then there are $N, M$ such that $a^N = b^M$.
\[lemma: amalgam and balance\] Let $C$ be a virtually cyclic group and $G = A\ast_C B$. Then $G$ is balanced if and only if $A, B$ are.
One implication is clear. To show the converse, let $g\in G$ be an infinite order element and let $h\in G$ be such that $hg^nh^{-1}=g^m$ for $|n|\neq|m|$. If $g$ is acts hyperbolically on the Bass-Serre tree $T$ corresponding to $G$, then the translation length $\ell_G(g)$ is positive. Moreover, $\ell_G(g^n)=|n|\ell_G(g)$ and $\ell_G(hgh^{-1})=\ell_G(g)$. Thus, if $hg^nh^{-1}=g^m$ then $|n|=|m|$, which is a contradiction. Thus, we can assume that $g$ acts elliptically on $T$.
Therefore, there exists $x$ such that $xgx^{-1}$ belongs in $A$ or $B$. Assume without loss of generality that $xgx^{-1}\in A$. We have $$\label{eq: first amalgamated product of balanced} (xhx^{-1})(xgx^{-1})^n(xhx^{-1})^{-1}=(xgx^{-1})^m.$$ If we write $a = (xgx^{-1})\in A$ and $k = xhx^{-1}$, Equation becomes $ka^{n}k^{-1} = a^m$. Write $k$ in normal form $k_0 \cdots k_s$, where $k_i \in A-1$ or $B-1$. We have $$(k_0\cdots k_s)a^{Tn}(k_0 \cdots k_s)^{-1} a^{-Tm} = 1.$$ There are now two cases. First, assume that no powers of $a$ can be conjugated into $C$, for instance, this happens whenever $\vert C \vert \leq \infty$. Then by the normal form theorem, $s=0$ $k_0 \in A$ and hence $A$ was not balanced.
So suppose that there is some power $a^{\epsilon}$ of $a$ that can be conjugated into $C$. Up to conjugating $a$ and $k$ and taking powers of $a$, we can assume that $a \in C$ and $k a^{n} k^{-1} = a^{m}$ holds. Again, consider the normal form $k = k_0 \dots k_s$. We will proceed by induction on $s$.
*Case $s = 0$*. In this case we have $k_0 a^{n} k_0^{-1} = a^m$. Since $a \in C$, if $k_0 \in A$ (resp. $B$), we have that $A$ (resp. $B$) is unbalanced.
*Induction step*. Suppose that the claim holds for $k$ with normal-form length $s-1$. We will show that it holds for length $s$. Consider the equation $ka^{n} k^{-1} = a^{m}$ and assume that $k$ has normal-form length $s$. Observe that for each $T$ the equation $ka^{Tn}k^{-1} = a^{Tm}$ still holds. We will show that, for $T$ large enough, we can write $ka^{Tn}k^{-1} = a^{Tm}$ as $k'c^{n'}(k')^{-1} = c^{m'}$ with $c \in C$, $\vert n' \vert \neq \vert m'\vert$ and $k'$ with normal-form length at most $s-1$. Then we are done by induction hypothesis.
We have $$(k_0\cdots k_s)a^{n}(k_0 \cdots k_s)^{-1} = a^{m}.$$ By the normal form theorem, $ b = k_sa^{n}k_s^{-1} \in C$. Since $C$ is 2-ended, there is $c \in C$ and $P_1, P_2, P_3, P_4$ such that $a^{P_1} = c^{P_2}$ and $b^{P_3} = c^{P_4}$. Let $K = k_0\cdots k_{s-1}$. Then we have $$\label{eq: second amalgamated product of balanced}
K k_sa^{P_1 P_3n}k_s^{-1} K^{-1} = a^{P_1P_3m}$$ Let’s focus on the left-hand side only, conjugating it by $K$. We have $$k_s c^{P_2P_3 n} k_s^{-1} = k_s a^{P_1P_3 n} k_s^{-1}= b^{P_1P_3} = c^{P_1P_4}.$$ Since $k_s$ belongs to either $A$ or $B$, all the elements of the above series of equations are in one between $A,B$, say $A$. Since $A$ is balanced, we need to have $\vert P_2 P_3 n \vert = \vert P_1 P_4 \vert$. Thus, up to possibly substituting $n$ with $-n$, we can write the left-hand-side of Equation as $K c^{P_2 P_3 n} K^{-1}$. Now, applying the equality $a^{P_1} = c^{P_2}$ to the right-hand-side of Equation , we have $$Kc^{P_1P_4}K^{-1}=K c^{P_2 P_3 n} K^{-1} = c^{P_2 P_3 m}.$$ We are now done by induction hypothesis.
By applying repeatedly the previous lemma, we obtain the following corollary.
\[corollary: tree product and balance\] If $G$ is a balanced-2-decomposable group such that the underlying graph is a tree, then $G$ is balanced.
It is straightforward to check that HNN extensions of balanced groups are not balanced in general: simply consider $BS(2,3)$ as the HNN extension $\langle a,t\mid ta^2t^{-1}=a^3\rangle\cong\langle a\rangle\ast_{ta^2t^{-1}=a^3}$.
To finish this subsection we include results that give sufficient conditions for an HNN extension over a balanced group to be balanced. We stress that these results are modified versions of [@button2015balanced Proposition 6.3] and [@button2015balanced Theorem 6.4]. They have been modified as to allow torsion.
\[prop: balance and absence of conjugation\] Let $H$ be a balanced group, $A, B \leq H$ be virtually cyclic subgroups and $\phi\colon A \to B$ be a isomorphism. Let $G = H \ast_{\phi}$. Then,
1. If $g\in H$ but no power of $g$ is conjugate in $H$ into $A\cup B$ then $g$ is still balanced in $G$.
2. If $A$ and $B$ are non-commensurable in $H$, then $G$ is also a balanced group.
Suppose $g$ was not balanced in $G$. Hence there is $h\in G-H$ such that $hg^{p} h^{-1} = g^{q}$ for some $\vert p \vert \neq \vert q \vert$. Since $h \in G-H$, we can write $ h = h_1t^{\varepsilon_1}\ldots h_{r-1}t^{\varepsilon_r}h_r$ in reduced form. By assumption $h_r g h_r^{-1}$ does not belong to $A$ nor $B$, and hence $h g^{q} h^{-1}$ cannot represent an element of $H$. Thus, $h \in H$ and since $H$ is balanced $\vert q \vert = \vert p \vert$.
For the second item, we only need to check the balancedeness of elliptic elements in $G$, since a translation length argument similar to that of Lemma \[lemma: amalgam and balance\] rules out unbalancedeness of hyperbolic elements. Thus, if $G$ is unbalanced, by the first item there must exist an unbalanced infinite order element $h\in H$ such that some power of $h$ can be conjugated into $A\cup B$. Therefore, we can assume without loss of generality that $h\in A\cup B$. Assume that $h=a\in A$. Since $a$ is unbalanced, there is some $g\in G$ such that $ga^ig^{-1}=a^j$ with $|i|\neq |j|$. Let $g=h_1t^{\varepsilon_1}\ldots h_rt^{\varepsilon_r}$ be the reduced form expression in $G$. Since $gh^ig^{-1}=h^j$ has normal form length $1$, there must exist some possible reduction in $(h_1t^{\varepsilon_1}\ldots h_rt^{\varepsilon_r})h^i(h_1t^{\varepsilon_1}\ldots h_rt^{\varepsilon_r})^{-1}$. There are two possible ways that this could happen: either $\varepsilon_r=1$ and $h_rh^ih_r^{-1}\in A$ or $\varepsilon_r=-1$ and $h_rh^ih_r^{-1}\in B$. If the latter occurs, then the proof is complete, as $h_rh^ih_r^{-1}$ is an infinite order element in $A^{h_r}\cap B$. Assume now that the former case occurs. Since $A$ is a 2-ended balanced group, there must exist $k$ such that $h_ra^{ik}h_r^{-1}=a^{\pm ik}$. Therefore, $t^{\varepsilon_r}h_ra^{ik}h_r^{-1}t^{-\varepsilon_r}=ta^{\pm ik}t^{-1}=b^{\pm ik}$. Again, as before, we have two possibilities: either $h_{r-1}b^{\pm ik}h_{r-1}^{-1}$ belongs in $B$ and $\varepsilon_{r-1}=-1$ or $h_{r-1}b^{\pm ik}h_{r-1}^{-1}$ belongs in $A$ and $\varepsilon_{r-1}=1$. If the latter occurs, the proof is complete. If the former occurs, since $B$ is a 2-ended balanced group, then $h_{r-1}b^{\pm ikk'}h_{r-1}^{-1}=b^{\pm ikk'}$ for some $k'$. We can continue performing reductions in the expression of $ga^ig^{-1}$ and at each step we have the same dichotomy where either the proof is complete or we can continue reducing. Note that at some point of the reduction we obtain $h_i$ such that $A
^{h_i}\cap B$ or $A\cap B^{h_i}$ is infinite. Indeed, otherwise for some $K\neq 0$ the equality $ga^{Ki}g^{-1}=a^{Kj}$ would hold for $|Ki|=|Kj|$, contradicting the assumption.
\[corollary: HNN and balance\] Let $G$ be an HNN extension of the balanced group $H$ with stable letter $t$ and 2-ended associated subgroups $A$ and $B$ of $H$. Let $a\in A,b\in B$ be infinite order elements such that $tat^{-1}=b$. Moreover, suppose that there is $h\in H$ conjugating a power of $a$ to a power of $b$, so that $ha^ih^{-1}=b^j$. Then $G$ is balanced if and only if for every pair of elements $a,b$ as above we have $|i|=|j|$.
One implication is clear, we now show that $G$ is balanced provided that for every $h\in H$ such that $ha^ih^{-1}=b^j$ for some $i,j$ it follows that $|i|=|j|$.
Assume that $G$ is an unbalanced group. Therefore, by the second assertion in the previous proposition, there must exist some $h'\in H$ such that $A\cap h'Bh'^{-1}$ is infinite. Since HNN extensions are defined up to conjugation of the corresponding embedding maps, by conjugating by $h'$ we can assume that $A\cap B$ is infinite in $H$. By the first assertion in the previous proposition, the only elements that can be unbalanced are those $h\in H$ that can be conjugate in $H$ into $A\cup B$. Thus, we can assume without loss of generality that the unbalanced elements in $G$ belong in $A\cup B$. Therefore, if $G$ is unbalanced, we can assume that for some $a\in A$ there is some $g\in G$ such that $ga^ng^{-1}=a^m$ for some $|n|\neq|m|$. We will induct on the length of the reduced form of $g$ to show that $ga^ng^{-1}=a^m$ implies $|n|=|m|$, obtaining a contradiction.
Let $g=h_0t^{\varepsilon_1}h_1\ldots t^{\varepsilon_r}h_r$ be the reduced expression of $g$. Let us say that $r$ denotes the reduced form length of $g$. Assume that $r=0$. That is to say, $g\in H$. Since $H$ is balanced, we have $\vert n \vert = \vert m \vert$.
Assume now that the claim holds for elements of reduced form length $r-1$, and let $g$ of reduced form length $r$ be such that $ga^ng^{-1}=a^m$. We denote by $b\in B$ the element such that $tat^{-1}=b$. Note that if the equation $ga^ng^{-1}=a^m$ holds in $G$, then for every $T$ we have that $ga^{Tn}g^{-1}=a^{Tm}$ for every $T>0$. Since the element $ga^ng^{-1}=a^m$ belongs in $H$, by the normal form theorem, $ga^ng^{-1}$ must admit some reduction in its reduced form. There are two ways that this reduction can occur: either $\varepsilon_r=1$ and $h_ra^nh_r^{-1}$ belongs in $A$ or $\varepsilon_1=-1$ and $h_ra
^nh_r^{-1}$ belongs in $B$. Note that in the former case, since $A$ is 2-ended and balanced, there must exist some $k$ such that $h_ra^{kn}h_r^{-1}=a^{\pm kn}$. Therefore, $t_r^{\varepsilon}h_ra^{kn}h_r^{-1}t_r^{-\varepsilon_r}=b^{\pm kn}$. In the latter case we have that $h_ra^nh_r^{-1}=b'\in B$. Since $B$ is a 2-ended group, there must exist $l_1,l_2$ such that $(b')^{l_1}=b^{l_2}$. Thus, $h_ra^{nl_1}h_r^{-1}=(b')^{l_1}=b^{l_2}$. By assumption, we must have that $|nl_1|=|l_2|$. Therefore, in the latter case we have that $t^{-1}h_ra^{nl_1}h_r^{-1}t=t^{-1}b^{\pm l_2}t=a^{\pm l_2}=a^{\pm nl_1}$. In both cases, we use the induction step to conclude $|kn|=|km|$ or $|l_1n|=|l_1m|$ respectively. In particular, since $k \neq 0 \neq l_1$, we conclude $\vert n \vert = \vert m \vert$.
Hierarchically hyperbolic groups {#subsection: basics of hhs}
================================
\[HHS\_definition\] A $q$–quasigeodesic metric space $(\mathcal{X},d_\mathcal{X})$ is *hierarchically hyperbolic* if there exist $\delta\geqslant 0$, an index set $\mathfrak{S}$, and a set $\{\mathcal{C}W\mid W\in\mathfrak{S}\}$ of $\delta$–hyperbolic spaces $(\mathcal{C}U,d_U)$, such that the following conditions are satisfied:
1. [**(Projections)**]{}\[axiom1\] There is a set $\{\pi_W\colon\mathcal{X}\to 2^{\mathcal{C}W}\mid W\in\mathfrak{S}\}$ of projections that send points in $\mathcal{X}$ to sets of diameter bounded by some $\xi\geqslant 0$ in the hyperbolic spaces $\mathcal{C}W\in\mathfrak{S}$. Moreover, there exists $K$ so that all $W\in\mathfrak{S}$, the coarse map $\pi_{W}$ is $(K,K)$–coarsely lipschitz and $\pi_W(\mathcal{X})$[^2] is $K$–quasiconvex in $\mathcal{C}W$.
2. [**(Nesting)**]{} The index set $\mathfrak{S}$ is equipped with a partial order $\sqsubseteq$ called *nesting*, and either $\mathfrak{S}$ is empty or it contains a unique $\sqsubseteq$–maximal element. When $V\sqsubseteq W$, $V$ is nested into $W$. For each $W\in\mathfrak{S}$, $W\sqsubseteq W$, and with $\mathfrak{S}_W$ we denote the set of all $V\in\mathfrak{S}$ that are nested in $W$. For all $V,W\in\mathfrak{S}$ such that $V\sqsubsetneq W$ there is a subset $\rho_W^V\subseteq \mathcal{C}W$ with diameter at most $\xi$, and a map $\rho_V^W\colon \mathcal{C}W\to 2^{\mathcal{C}V}$.
3. \[A3\] [**(Orthogonality)**]{} The set $\mathfrak{S}$ has a symmetric and antireflexive relation $\perp$ called *orthogonality*. Whenever $V\sqsubseteq W$ and $W\perp U$, then $V\perp U$ as well. For each $Z\in \mathfrak{S}$ and each $U\in\mathfrak{S}_Z$ for which $\{V\in\mathfrak{S}_Z\mid V\perp U\}\neq\emptyset$, there exists $\cont_\perp^ZU\in \mathfrak{S}_Z\setminus\{Z\}$ such that whenever $V\perp U$ and $V\sqsubseteq Z$, then $V\sqsubseteq \cont_\perp^ZU$.
4. [**(Transversality and Consistency)**]{}\[HHS\_definition\_4\] If $V,W\in\mathfrak{S}$ are not orthogonal and neither is nested into the other, then they are transverse: $V\pitchfork W$. There exists $\kappa_0\geqslant 0$ such that if $V\pitchfork W$, then there are sets $\rho_W^V\subseteq \mathcal{C}W$ and $\rho_V^W\subseteq \mathcal{C}V$, each of diameter at most $\xi$, satisfying $$\label{consistent_transversal}
\min\bigl\{d_W(\pi_W(x),\rho_W^V),d_V(\pi_V(x),\rho_V^W)\bigr\}\leqslant \kappa_0,\qquad \forall\ x\in\mathcal{X}.$$ Moreover, for $V\sqsubseteq W$ and for all $x\in\mathcal{X}$ we have that $$\label{consistent_nesting}
\min\bigl\{d_W(\pi_W(x),\rho_W^V),\diam_{\mathcal{C}V}(\pi_V(x)\cup\rho_V^W(\pi_W(x)))\bigr\}\leqslant \kappa_0.$$ In the case of $V\sqsubseteq W$, we have that $d_U(\rho^V_U,\rho^W_U)\leqslant \kappa_0$ whenever $U\in\mathfrak{S}$ is such that either $W\sqsubsetneq U$, or $W\pitchfork U$ and $U\not\perp V$.
5. [**(Finite complexity)**]{} There is a natural number $n\geqslant0$, the complexity of $\mathcal{X}$ with respect to $\mathfrak{S}$, such that any set of pairwise $\sqsubseteq$–comparable elements of $\mathfrak{S}$ has cardinality at most $n$.
6. [**(Large links)**]{} There exist $\lambda\geqslant 1$ and $E\geqslant \max\{\xi,\kappa_0\}$ such that, given any $W\in\mathfrak{S}$ and $x,x'\in\mathcal{X}$, there exists $\{T_i\}_{i=1,\dots,\lfloor N\rfloor}\subset \mathfrak{S}_W\setminus\{W\}$ such that for all $T\in\mathfrak{S}_W\setminus\{W\}$ either $T\in\mathfrak{S}_{T_i}$ for some $i$, or $d_T(\pi_T(x),\pi_T(x'))< E$, where $N=\lambda d_W(\pi_W(x),\pi_W(x'))+\lambda$. Moreover, $d_W(\pi_W(x),\rho_W^{T_i})\leqslant N$ for all $i$.
7. [**(Bounded geodesic image)**]{} For all $W\in\mathfrak{S}$, all $V\in\mathfrak{S}_W \setminus \{W\}$ and all geodesics $\gamma$ of $\mathcal{C}W$, either $\diam_{\mathcal{C}V}(\rho_V^W(\gamma))\leqslant E$ or $\gamma\cap \mathcal{N}_E(\rho_W^V)\neq \emptyset$.
8. [**(Partial realization)**]{} There is a constant $\alpha$ satisfying: let $\{V_j\}$ be a family of pairwise orthogonal elements of $\mathfrak{S}$, ad let $p_j\in\pi_{V_j}(\mathcal{X})\subseteq \mathcal{C}V_j$. Then there exists $x\in\mathcal{X}$ such that
- $d_{V_j}\bigl(\pi_{V_j}(x),p_j\bigr)\leqslant \alpha$ for all $j$;
- for all $j$ and all $V\in\mathfrak S$ such that $V\pitchfork V_j$ or $V_j\sqsubseteq V$ we have $d_V(\pi_V(x),\rho^{V_j}_V)\leq\alpha$.
9. [**(Uniqueness)**]{} For each $\kappa\geqslant 0$ there exists $\theta_u=\theta_u(\kappa)$ such that if $x,y\in\mathcal{X}$ and $d(x,y)\geqslant \theta_u$, then there exists $V\in\mathfrak{S}$ such that $d_V(x,y)\geqslant\kappa$.
The inequalities of the fourth axiom are called *consistency inequalities*.
\[rmk: HHS are QI invariant\] Being a hierarchically hyperbolic space is a quasi-isometric invariant property. That is, if $(\mathcal{X},\mathfrak{S})$ hierarchically hyperbolic and $q \colon {\mathcal{X}} \to {\mathcal{Y}}$ is a quasi-isometry, then $\mathcal{Y}$ is a hierarchically hyperbolic space, and the hierarchical structure coincides with the one of ${\mathcal{X}}$. Indeed, if $\bar{q}$ is a quasi-inverse of $q$, define all the hyperbolic spaces as the one of ${\mathcal{X}}$ and the projections as $\pi_U\circ \bar{q}$. Then checking the axioms is a tedious (but straightforward) work. We will see in the following sections that this is not the case for hierarchically hyperbolic groups (Definition \[def:hhg\]). The issue boils down to the fact that the quasi-inverse $\bar{q}$ might not be equivariant.
\[def:HHS\_hieromorphism\] Let $(\mathcal{X},\mathfrak{S})$ and $(\mathcal{X}',\mathfrak{S}')$ be hierarchically hyperbolic spaces. A *hieromorphism* is a triple $\phi=\bigl(\phi,\phi^\diamondsuit,\{\phi^\ast_U\}_{U\in\mathfrak{S}}\bigr)$, where $\phi\colon \mathcal{X}\to\mathcal{X}'$ is a map, $\phi^\diamondsuit\colon \mathfrak{S}\to\mathfrak{S}'$ is an injective map that preserves nesting, transversality and orthogonality, and, for every $U\in\mathfrak{S}$, the maps $\phi^*_U\colon \mathcal{C}U\to\mathcal{C}\phi^\diamondsuit(U)$ are quasi-isometric embeddings with uniform constants.
Moreover, the following two diagrams coarsely commute (again with uniform constants), for all $U,V\in\mathfrak{S}$ such that $U\sqsubseteq V$ or $U\pitchfork V$: $$\label{coarsely.commuting.diagrams}
\xymatrix{
\mathcal{X}\ar[r]^{\phi}\ar[d]_{\pi_U} & \mathcal{X}'\ar[d]^{\pi_{\phi^\diamondsuit(U)}}\\
\mathcal{C}U\ar[r]_{\phi^*_U}&\mathcal{C}\phi^\diamondsuit(U)
}\qquad\qquad\qquad
\xymatrix{
\mathcal{C}U\ar[rr]^{\phi^*_U}\ar[d]_{\rho^U_V} && \mathcal{C}\phi^\diamondsuit(U)\ar[d]^{\rho^{\phi^\diamondsuit(U)}_{\phi^\diamondsuit(V)}}\\
\mathcal{C}V\ar[rr]_{\phi^*_V}&&\mathcal{C}\phi^\diamondsuit(V)
}$$
\[def:fullness\_definition\] A hieromorphism $\phi\colon(\mathcal{X},\mathfrak{S})\to(\mathcal{X}',\mathfrak{S}')$ is *full* if:
1. there exists $\xi$ such that the maps $\phi^*_U\colon\mathcal{C}U\to\mathcal{C}\phi^\diamondsuit(U)$ are $(\xi,\xi)$–quasi-isometries, for all $U\in\mathfrak{S}$;
2. if $S$ denotes the $\sqsubseteq$–maximal element of $\mathfrak{S}$, then for all $U'\in\mathfrak{S}'$ nested into $\phi^\diamondsuit(S)$ there exists $U\in\mathfrak{S}$ such that $U'=\phi^\diamondsuit(U)$.
\[def:hhg\] We say that a group $G$ is *hierarchically hyperbolic* if it acts on a hierarchically hyperbolic space $(\mathcal{X},\mathfrak{S})$ satisfying the following conditions:
1. The action of $G$ on $\mathcal{X}$ is proper and cobounded;
2. $G$ acts on ${\mathcal{X}}$ by uniform hieromorphisms (i.e the constants involved in Defintion \[def:HHS\_hieromorphism\] are uniform for every $g\in G$), and the action on ${\mathfrak{S}}$ has finitely many orbits.
By definition, if $(G, {\mathfrak{S}})$ is a hierarchically hyperbolic group and $g \in G$, multiplication by $g$ coarsely satisfies the two diagrams of Equation . However, it is always possible to modify the structure to obtain commutativity on the nose, as described in [@durham2018corrigendum Section 2.1]. When considering hierarchically hyperbolic groups, we will always assume such equivariance on the nose.
Convexity
---------
In this paper, we will make use of two notions of convexity. The first one, called hierarchical quasiconvexity, heavily relies on the hierarchical structure. For instance, it is not quasi-isometric invariant. For a more precise account, we refer to [@russellsprianotran:convexity].
A subset $Y$ of an HHS $(X,{\mathfrak{S}})$ is hierarchically quasiconvex if there is a function $k:[0,\infty)\to\mathbb{R}$ such that every $\pi_U(Y)$ is $k(0)$–quasiconvex, and any point $x\in X$ with $d_U(x,Y)\leq r$ for all $U\in{\mathfrak{S}}$ satisfies $d_X(x,Y)\leq k(r)$.
Although we will not use this fact, we recall that one of the main motivations of hierarchical quasiconvexity is that hierarchically quasiconvex subsets of an HHS are HHSs themselves [@HHSII Proposition 5.6].
To detect hierarchical quasiconvexity sometimes it is convenient to check a stronger property.
A subset $Y$ of a quasigeodesic space $X$ is said to be *strongly quasiconvex* if there is a function $M:[1,\infty)\to \mathbb{R}$ such that every $\lambda$–quasigeodesic in $X$ with endpoints in $Y$ stays $M(\lambda)$–close to $Y$.
\[thm: RST strong quasiconvex\] Let $(G, {\mathfrak{S}})$ be a hierarchically hyperbolic group and $Y\subseteq G$ be a subset. Then if $Y$ is strongly quasiconvex, it is hierarchically quasiconvex, where the constants determine each other.
A special case of strongly quasiconvex subsets is given by peripheral subgroups of relatively hyperbolic groups.
\[lem: peripherals are strongly QC\] Let $P$ be a peripheral subgroup in the relatively hyperbolic group $G$. Then $P$ is strongly quasiconvex.
In the case of hyperbolic spaces, relative hyperbolicity and strong quasi-convexity are intimately related.
\[def: almost malnormal collection\] We say that a collection of subgroups $\{H_i\}_{i=1}^n$ of $G$ is almost-malnormal if $H_i\cap gH_jg^{-1}$ is finite unless $i=j$ and $g\in H_i$.
\[thm:bowditch\] Let $G$ be a hyperbolic group and $\{H_i\}_{i=1}^n$ be a finite family of subgroups of $G$. Then $G$ is hyperbolic relative to $\{H_i\}$ if and only if $\{H_i\}$ is an almost-malnormal family of strongly quasiconvex subgroups.
To finish we recall one last useful property of peripheral subgroups.
\[lem: cpproj is C. Lipschitz\] Let $P$ be a peripheral subgroup of the relatively hyperbolic group $G$. Then the closest point projection on $P$ is coarsely Lipschitz.
Combination theorem {#subsec: combination theorems}
-------------------
\[def: glueing\_hieromorphism\] Let $(H, {\mathfrak{S}}_1)$ and $(G, {\mathfrak{S}}_2)$ be hierarchically hyperbolic groups. A *glueing hieromorphism* between $H$ and $G$ is a group homomorphism $\phi \colon H \to G$ that can be realized as a full hieromorphism $(\phi, \phi^\diamondsuit, \phi^\ast_U)$ such that the image $\phi(H)$ is hierarchically quasi-convex in $G$ and the maps $\phi^\ast_U \colon {{\mathcal C}}U \to {{\mathcal C}}\phi^\diamondsuit U$ are isometries for each $U \in {\mathfrak{S}}_1$. If the map $\phi \colon H \to G$ is injective, we say that the glueing hieromorphism is injective.
\[intersectionproperty\_definition\] A hierarchically hyperbolic space $(\mathcal{X},\mathfrak{S})$ has the *intersection property* if the index set admits an operation $\wedge\colon (\mathfrak{S}\cup\{\emptyset\})\times(\mathfrak{S}\cup\{\emptyset\})\to\mathfrak{S}\cup\{\emptyset\}$ satisfying the following properties for all $U,V,W\in\mathfrak{S}$:
1. $V\wedge \emptyset=\emptyset \wedge V=\emptyset$;
2. $U\wedge V=V\wedge U$;
3. $(U\wedge V)\wedge W=U\wedge (V\wedge W)$;
4. $U\wedge V\sqsubseteq U$ and $U\wedge V\sqsubseteq V$ whenever $U\wedge V\in\mathfrak{S}$;
5. if $W\sqsubseteq U$ and $W\sqsubseteq V$, then $W\sqsubseteq U\wedge V$.
A hierarchically hyperbolic space $(\mathcal{X},\mathfrak{S})$ is said to have clean containers if $U\perp \cont_\perp^Z U$ for all $U,Z\in\mathfrak{S}$, as originally defined in [@ABD Definition 3.4].
We also recall a combination theorem for hierarchically hyperbolic groups.
\[comb\_thm\_ver2\] Let ${\mathcal{G}} = \bigl(\Gamma,\{G_v\}_{v\in V},\{G_e\}_{e\in E},\{\phi_{e^\pm}\colon G_e\to G_{e^\pm}\}_{ e\in E}\bigr)$ be a finite graph of hierarchically hyperbolic groups. Suppose that:
1. each $\phi_{e^\pm}$ is a $(K,K)$–coarsely Lipschitz map and a glueing hieromorphism;
2. each vertex group has the intersection property and clean containers.
Then $\pi_1({\mathcal{G}})$ is a hierarchically hyperbolic group with the intersection property and clean containers.
Not all fundamental groups of a graph of groups have a hierarchical hyperbolic structure, as the following remark shows.
\[remark: absence of BS in HHG\] If $G$ is a hierarchically hyperbolic group, then $G$ cannot have a subgroup isomorphic to $BS(n,m)=\langle a,t\mid ta^nt^{-1}=a^m\rangle$, with $\vert n \vert \neq \vert m \vert$. Indeed, suppose there is an embedding $\iota: BS(n,m)\hookrightarrow G$. We have that $\iota(a)$ is an infinite order element of $G$. By [@HHSBoundaries Theorem 7.1] and [@durham2018corrigendum Theorem 3.1], $\iota(a)$ is undistorted, which is a contradiction.
More generally, if a group $G$ has a hierarchical hyperbolic structure, then it cannot be unbalanced, as it cannot contain infinite undistorted cyclic subgroups. The rest of the paper is dedicated to investigate if the converse also holds for (fundamental group of) graph of groups. More precisely, we show that the converse holds for the class of hyperbolic-2-decomposable groups.
Hierarchical hyperbolicity of (2-ended)-2-decomposable groups {#section: graph of infinite virtually cyclic groups}
=============================================================
In this section, we focus on (2-ended)-2-decomposable groups. That is to say, graphs of groups where every vertex and edge group is 2-ended. We begin the section by recalling some useful results on 2-ended groups.
Two-ended groups
----------------
In this subsection, we recall basic results and remarks on the structure of two-ended groups. An important result of these type of groups is known as the structure theorem for infinite virtually cyclic groups. Throughout the paper, we will make use of this fact on many occasions.
\[lemma:basic\_fact\_virt\_cyclic\] If $G$ is an infinite virtually cyclic group, then either
1. $G$ admits a surjection with finite kernel onto the infinite cyclic group $\mathbb{Z}$, or
2. $G$ admits a surjection with finite kernel onto the infinite dihedral group $\mathbb{D}_{\infty}$
We recall that the *infinite dihedral group* is the group defined by the presentation $\mathbb{D}_\infty = \langle r, s \mid srs = r^{-1}, s^2 \rangle$. Note that every element of $\mathbb{D}_\infty$ can be written as $s^\epsilon r^k$, for $\epsilon\in \{0,1\}$ and $k \in \mathbb{Z}$. Moreover, every element of the form $sr^k$ has order 2, and an element of the form $r^k$ has infinite order precisely when $k \neq 0$. Using those observations, we have the following lemma.
\[lem: Kernel to dihedral is always the same\] Let $G$ be a virtually cyclic group. Let $\Phi_1$, $\Phi_2 \colon G \to \mathbb{D}_\infty$ be homomorphisms with finite kernel and finite index image. Then $\Ker (\Phi_1) = \Ker (\Phi_2)$.
As before, $\mathbb{D}_\infty = \langle a, b \mid bab = a^{-1}, b^2 \rangle$. Suppose that there is $g \in G$ such that $g \in \Ker(\Phi_1)$ and $g \not \in \Ker(\Phi_2)$. Since $g \in \Ker(\Phi_1)$, we conclude that $g$ has finite order, otherwise $\lvert \Ker(\Phi_1)\rvert = \infty$. Since $\Phi_2(G)$ has finite index in $\mathbb{D}_\infty$ there exists $c \in G$ such that $\Phi_2(c)$ has infinite order. In particular there exist $k_1 \in \mathbb{Z}, k_2 \in \mathbb{Z}-\{0\}$ such that $\Phi_2(g) = ba^{k_1}$ and $\Phi_2(c) = a^{k_2}$, and so $\Phi_2(gc) = ba^{k_1+k_2}$. Again, $gc$ has to have finite order to not contradict $\lvert \Ker(\Phi_2)\rvert < \infty$ . However, since $g \in \Ker(\Phi_1)$ we have that $\Phi_1(gc) = \Phi_1(c)$, and so $gc$ cannot have finite order. From this we conclude $\Ker(\Phi_1) \subseteq \Ker(\Phi_2)$. The symmetric argument yields the claim.
\[rmk: virtually cyclic surjection dichotomy\] Note that an infinite virtually cyclic group $G$ cannot surject onto both $\mathbb{Z}$ and $\mathbb{D}_{\infty}$ with finite kernel. Indeed, assume that two surjective homomorphisms $\Phi:G\to\mathbb{Z}$ and $\Phi':G\to\mathbb{D}_{\infty}$ exist. Since $\mathbb{Z}$ embeds into $\mathbb{D}_{\infty}$ with finite index image, we can regard $\Phi$ as a homomorphism from $G$ to $\mathbb{D}_{\infty}$ with finite kernel and finite index image. Let $s\in\mathbb{D}_{\infty}$ be the generator of order two and let $g\in G$ be an element such that $\Phi'(g)=s$. Since $s^2=1$, we have that $g^2\in\Ker(\Phi')$; by Lemma \[lem: Kernel to dihedral is always the same\] we have that $g^2\in\Ker(\Phi)$. Since $\mathbb{Z}$ is torsion-free, $\Phi(g)^2=1$ if and only if $\Phi(g)=1$. Since $\Ker(\Phi)=\Ker(\Phi')$, it follows that $\Phi'(g)=1$, which is a contradiction.
Pulling back hierarchical structures
------------------------------------
Recall that GBS groups are (infinite cyclic)-2-decomposable groups.
\[definition: GBS groups\] We say that a group $G$ is a Generalized Baumslag–Solitar group if there exists a finite graph of infinite cyclic groups ${\mathcal{G}}$ for which $G\cong\pi_1({\mathcal{G}})$.
\[lemma: subgroups of graph of 2 ended groups\] Let $G$ be a (2-ended)-2-decomposable group and let $H\leq G$. If $H$ is torsion-free, then $H$ is either a GBS group or a free group.
Let $G_v$ be a vertex group in $\mathcal{G}$. Since $H$ is torsion-free, there are two possibilities: either $H\cap G_v$ is trivial or it is infinite cyclic. Since every edge group has finite index in its neighbouring vertex groups, if $H\cap G_v$ is trivial, then $H\cap G_w$ is trivial for every other vertex $w$. Then $H$ acts on the Bass-Serre tree corresponding to $\mathcal{G}$ with trivial stabilizers. This is equivalent to $H$ being a free group.
If $H\cap G_v$ is non trivial, then it is of finite index in $G_v$, since $G_v$ is two-ended. Therefore, since the Bass-Serre tree of $\mathcal{G}$ is locally finite, the group $H$ acts with infinite cyclic stabilizers on a locally finite tree. That is to say, $H$ splits as a finite graph of groups with infinite cyclic vertex groups and the result follows.
Let $G,H$ be finitely generated groups and let $S_G,S_H$ be generating sets of $G$ and $H$ respectively. We say that a group homomorphism $f:H\to G$ is a *quasi-isometric homomorphism* if $f:(G,d_{S_G})\to (H,d_{S_H})$ is a quasi-isometry.
Recall that a group homomorphism $f:G\to H$ yields a quasi-isometry for some (hence, any) generating sets $S_H,S_G$ if and only if $|\Ker(f)|<\infty$ and $|H:\Im(f)|<\infty$.
As we have seen in Remark \[rmk: HHS are QI invariant\], the hierarchically hyperbolic structure on geodesic metric spaces can be pushed out and pulled back via quasi-isometries. For hierarchically hyperbolic groups, however, this is not true, as group actions are in general not equivariant with respect to any quasi-isometry. The next lemma describes how to pull back hierarchically hyperbolic group structures on a group $H$ via quasi-isometric homomorphisms. Recall the definition of glueing hieromorphism (Definition \[def: glueing\_hieromorphism\]).
\[lemma: pulling back hierarchical structures\] Let $(G, {\mathfrak{S}}_G)$ be a hierarchically hyperbolic group and let $f \colon H \to G$ be a quasi-isometric homomorphism. Then $H$ can be endowed with a hierarchically hyperbolic structure ${\mathfrak{S}}_H$ defined as follows.
1. The set ${\mathfrak{S}}_H$ coincides with ${\mathfrak{S}}_G$, and the associated hyperbolic spaces also coincide.
2. The projections $\pi^H_U \colon H \to {{\mathcal C}}U$ are defined as the composition $\pi^G_U \circ f$, where $\pi^G_U \colon G \to {{\mathcal C}}U$ is the projection associated to $(G, {\mathfrak{S}}_G)$.
3. The relations between the elements of ${\mathfrak{S}}_H$ are unchanged, and so are the maps $\rho_V^U$.
Moreover, $f$ is a glueing hieromorphism between $H$ and $G$.
Since $f$ has finite kernel and finite index image, it is clear that $f$ induces a quasi-isometry. Thus $(H, {\mathfrak{S}}_H)$ is a hierarchically hyperbolic space. In order to show that it is a hierarchically hyperbolic group, we now show that the structure induced above is $H$–equivariant. Since $G$ acts on ${\mathfrak{S}}_G$, we obtain that $H$ acts on ${\mathfrak{S}}$ as well via $f$. Since $f(H)$ has finite index in $G$, we obtain that the action has finitely many orbits. We now show that every $h\in H$ and $U\in\mathfrak{S}_H$ there exists an isometry $h_U:\mathcal{C}U\to\mathcal{C}hU$ such that the following diagram commutes $$\label{coarsely.commuting.diagrams.hhg}
\xymatrix{
H\ar[r]^{h}\ar[d]_{\pi_U} & H\ar[d]^{\pi_{hU}}\\
\mathcal{C}U\ar[r]_{h_U}&\mathcal{C}hU
}$$ Indeed, if we define $h_U$ as the isometry induced by $f(h)$ on $\mathcal{C}U$ we obtain that $h_U\circ\pi^H_U(h')=f(h)^*_U\circ\pi_U^G\circ f(h')=\pi_{hU}^G(f(h)\cdot f(h'))=\pi_{hU}^H(h\cdot h')$ for every $h'\in H$.
If $f \colon H \to G$ is as in Lemma \[lemma: pulling back hierarchical structures\], we say that ${\mathfrak{S}}_H$ is the *pullback* of the hierarchical structure on $G$ and denote it by $f^\ast ({\mathfrak{S}}_G)$.
From the above we immediately obtain the following lemma:
\[lem: functioriality of pullback\] Let $(G, {\mathfrak{S}})$ be a hierarchically hyperbolic group and let $H, K$ be groups such that there exist quasi-isometric homomorphisms $f_1 \colon K \to H$ and $f_2 \colon H \to G$. Let $f = f_2 \circ f_1$. Then $f^\ast {\mathfrak{S}} = f^\ast_1 \left(f^\ast_2 {\mathfrak{S}}\right)$, and the map $f$ is a glueing homomorphism.
Linearly parametrizable graph of groups
---------------------------------------
\[def: linearly parametrized\] Let $\mathcal{G}$ be a graph of groups. We say that $\mathcal{G}$ is *linearly parametrized* if there is a map $\Phi \colon \pi_1(\mathcal{G}) \to \mathbb{D}_\infty$ such that for each vertex or edge group $G$, the restriction $\Phi|_G$ has finite kernel and finite-index image (i.e $\Phi|_G$ is a quasi-isometric homomorphism).
\[thm: inducing hhg structure\] Let $\mathcal{G}$ be a linearly parametrized graph of groups and let $G = \pi_1 (\mathcal{G})$. Then, $G$ admits a hierarchically hyperbolic group structure.
Let $\Phi \colon G \to \mathbb{D}_{\infty}$ be the map witnessing the linear parametrization of $G$. Equip $\mathbb{D}_\infty$ with the trivial hierarchically hyperbolic group structure $(\mathbb{D}_\infty, {\mathfrak{T}})$, where ${\mathfrak{T}}$ contains a single element $T$ and ${{\mathcal C}}T$ coincides with a Cayley graph for $\mathbb{D}_\infty$. Endow every vertex $G_v$ with the pullback structure $(G_v, {\Phi |_{G_v}}^\ast ({\mathfrak{T}}))$, and endow analogously the edge groups. We claim that this turns ${\mathcal{G}}$ into a graph of groups that satisfies the hypothesis of Theorem \[comb\_thm\_ver2\]. Since the HHG structure on each vertex group consists of a single element, it satisfies the intersection property and clean containers. Let $e$ be an edge, $v$ a vertex incident to $e$, and let $\varphi \colon G_e \to G_v$ be an injective homomorphism. Since both $G_e$ and $G_v$ are infinite virtually cyclic, we have that $\varphi$ is a quasi-isometric homomorphism. Thus, by Lemma \[lem: functioriality of pullback\], it induces a glueing hieromorphism. Since $e$ and $v$ were generic, the result follows.
Thus, from now on we will focus on determining which graphs of 2-ended groups can be linearly parametrized. We begin by showing which amalgams and HNN extensions of linearly parametrizable groups can be linearly parametrized.
\[lemma: amalgam of linearly parametrized\] Let $\mathcal{G}_1$ and ${\mathcal{G}}_2$ be linearly parametrized graphs of groups, and let $\mathcal{G}$ be a graph of groups obtained connecting ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ with an edge such that the corresponding edge group is 2-ended. Then, ${\mathcal{G}}$ is linearly parametrized.
Let $e$ be the added edge and let $G_e$ be the associated group. We want to show that there are maps $\Phi_1 \colon \pi_1({\mathcal{G}}_1) \to \mathbb{D}_\infty$ and $\Phi_2 \colon \pi_1({\mathcal{G}}_2) \to \mathbb{D}_\infty$ that agree on $G_e$ such that their restriction to vertex/edges subgroups has finite kernel and finite index image. Then the universal property of the amalgamated product yields the desired map $\Phi \colon \pi_1 ({\mathcal{G}}) \to \mathbb{D}_\infty$.
Let $\Phi_1 \colon \pi_1({\mathcal{G}}_1) \to \mathbb{D}_\infty$ be the function parametrizing ${\mathcal{G}}_1$, and let $\Phi_2$ be the one for ${\mathcal{G}}_2$. Consider the two restrictions $\Phi_i|_{G_e}$, for $i\in \{1,2\}$. Since $G_e$ is an infinite group by assumption, its image has finite index in the vertex groups adjacent to it. In particular, the restrictions $\Phi_i|_{G_e}$ have finite kernel and finite index image. By Lemma \[lem: Kernel to dihedral is always the same\], we conclude $\Ker (\Phi_1|_{G_e}) = \Ker (\Phi_2|_{G_e})$. We concentrate now on the images $\Phi_i(G_e)$ which, by the previous argument, are isomorphic. An infinite index subgroup of the dihedral group has to have the form $\langle s^k \rangle$ or $\langle s^k, rs^l\rangle$, for some $k, l \in \mathbb{Z}-\{0\}$. Suppose that the subgroups $\Phi_i(G_e)$ have the form $\langle s^{k_i}, ra^{l_i}\rangle $ respectively (the case where they are both cyclic is analogous). Note that the map $\rho_l \colon \mathbb{D}_\infty \to \mathbb{D}_\infty $ which sends $s \to s$ and $r \to rs^l$ is an isomorphism. Thus, up to postcomposing $\Phi_i$ with $\rho_{-l_i}$ we can assume that the images $\Phi_i(G_e)$ have the form $\langle s^{k_i}, r\rangle $ respectively.
Let $\tau_{k} \colon \mathbb{D}_\infty \to \mathbb{D}_\infty$ be the map that sends $s \to s^k$ and $r \to r$. Note that $\tau_k$ is an injection with finite index image, thus postcomposing with $\tau_k$ does not alter the fact that a map has finite kernel and finite index image. It is now straightforward to verify that the maps $\Phi_1 : = \tau_{k_2} \circ\Phi_1$ and $\Phi_2 := \tau_{k_1} \circ \Phi_2$ satisfy the desired requirements.
A result of this type in HNN extensions does not hold in general, as the following example shows:
If $H=\langle a\rangle$ is an infinite cyclic group, then it can be linearly parametrized via the map $\Phi:H\to\mathbb{D}_{\infty}$ that sends $a\mapsto r$. Let us construct an HNN extension over $H$ by adding a stable letter $t$ that conjugates $a^2$ to $a^3$. That is to say, $G=H\ast_{ta^2t^{-1}=a^3}$.
Assume that $\Phi$ can be extended to $\widehat{\Phi}:G\to \mathbb{D}_{\infty}$ that linearly parametrizes $G$. As a consequence we obtain that the relation $\widehat{\Phi}(t)\widehat{\Phi}(a)^2\widehat{\Phi}(t)^{-1}=\widehat{\Phi}(a)^3$ holds in $\mathbb{D}_{\infty}$. As virtually cyclic groups are balanced, $\widehat{\Phi}(t)$ must be trivial. Since $\widehat{\Phi}(a)=\Phi(a)=r$, we obtain as a consequence that $r
^2=r^3$ in $\mathbb{D}_{\infty}$, which is a contradiction. Thus, $\Phi$ cannot be extended to a linear parametrization of $G$.
To determine which HNN extensions of linearly parametrizable groups can be linearly parametrized, we introduce the notion of balanced edge.
\[def: balanced edges\] Let ${\mathcal{G}}$ be a graph of groups and $e$ be an edge of ${\mathcal{G}}$. We say that $e$ is *balanced* if the following holds. Let ${\mathcal{H}} = {\mathcal{G}}- e$, and let $\phi_+, \phi_- \colon G_e \to \pi_1({\mathcal{H}})$ be the morphisms associated to $e$. Then for every infinite order element $a \in G_e$, if there exists $h \in \pi_1({\mathcal{H}})$ such that $$\label{equation: unbalanced edge}
h\phi_{+}(a)^{i}h^{-1}= \phi_{-}(a)^j,$$ it follows that $\vert i \vert = \vert j\vert$.
\[remark: edges in tree are balanced\] Note that if an edge $e$ in a graph of groups $\mathcal{G}$ is unbalanced then $\pi_1(\mathcal{G})$ is unbalanced. Moreover, by Corollary \[corollary: tree product and balance\] we have that unbalanced edges can never exist in a graph of groups where the underlying graph is a tree.
\[lem: parametrizing HNN\] Let ${\mathcal{H}}$ be a linearly parametrized graph of groups and let ${\mathcal{G}}$ be obtained from ${\mathcal{H}}$ by adding an edge $e$ with infinite associated edge group. Then ${\mathcal{G}}$ is linearly parametrized if and only if $e$ is balanced.
Let $A,B$ be the images of the edge group, and let $\psi \colon A \to B$ be the induced isomorphism. Let $\Phi\colon H = \pi_1({\mathcal{H}}) \to \dihed$ be the map that linearly parametrizes $H$. As usual, we use the presentation $\mathbb{D}_{\infty}=\langle r,s\mid srs^{-1}=r^{-1},s^2=1\rangle$. We start by showing that the second condition implies the first.
Consider the subgroups $\Phi(A), \Phi(B)\leq \mathbb{D}_\infty$. Note that every infinite order element of $A$ has to be sent to $r^n$ for some $n \in \mathbb{Z}- \{0\}$. Indeed, those are the only infinite order elements of $\mathbb{D}_{\infty}$, and since $\Phi|_{A}$ has finite kernel, infinite order elements cannot be mapped to torsion ones. A similar argument applies for $B$. Thus, $\Phi(A) \cap \langle r \rangle$ has finite index in $\langle r\rangle$.
Let $\lvert n\rvert$ and $\lvert m\rvert$ be the index of $\langle\Phi(A)\rangle\cap\langle r\rangle$ in $\langle r\rangle$ and of $\langle\Phi(B)\rangle\cap\langle r\rangle$ in $\langle r\rangle$ respectively. We now show that $\lvert n\rvert=\lvert m\rvert$. Let $a \in A$ be such that $\Phi(a)$ generates $\Phi(A) \cap \langle r\rangle$. Observe that there exists $h \in H$ and $i > 0$ such that $ha^ih^{-1}=\psi(a)^j$, for some $j >0$. Indeed, since ${\mathcal{H}}$ is linearly parametrized, all its vertices and edges groups are infinite virtually cyclic, and the underlying graph is connected. Thus, $G_v$ and $G_w$ are commensurable. By assumption, we need to have $\lvert i\rvert= \lvert j\rvert$. Thus, $ha^i h^{-1}=\psi(a)^i$ and, therefore, $\Phi(a)^i=\Phi(\psi(a))^{\pm j}$. By mutiplicativity of index of subgroups we obtain $\lvert\langle\Phi(a)\rangle:\langle r\rangle\rvert=\lvert\langle\Phi(\psi(a))\rangle:\langle r\rangle\rvert$. This shows that $|n| \leq |m|$. The symmetric argument obtained choosing $b \in B$ such that $\Phi(b)$ generates $\Phi(B) \cap \langle r \rangle$ and considering $\psi^{-1}(b)$ provides the other inequality. Thus $|n| = |m|$.
Define a map $\psi':\Phi(A)\to\Phi(B)$ as $\psi'(\Phi(x))=\Phi(\psi(x))$. By Lemma \[lem: Kernel to dihedral is always the same\], $\mathrm{Ker}(\Phi)\vert_{A} = \mathrm{Ker}(\Phi)\vert_{B}$. Thus, $\psi'$ is a well defined, injective homomorphism. Since $\psi$ is surjective, so is $\psi'$, showing that $\psi'$ is an isomorphism. Since $\Phi(a)$ generates $\Phi(A) \cap \langle r \rangle$ and $\psi'(\Phi(a))$ generates $\Phi(B) \cap \langle r \rangle$, we have $\Phi(a) = r^m$, $\Phi(\phi(a)) = r^n$ with $\vert m \vert = \vert n \vert$.
In particular, $\Phi$ extends to a homomorphism $\Phi':G\to(\mathbb{D}_{\infty})\ast_{\psi'}$. Consider the presentation $(\mathbb{D}_{\infty})\ast_{\psi'} = \langle s,r,t\mid srs^{-1}=r^{-1},s^2=1, t\psi'(\Phi(x))t^{-1}=\Phi(x)\quad \forall x\in A\rangle$. Let $\rho:\mathbb{D}_{\infty}\ast_{\psi'}\to\mathbb{D}_{\infty}$ be defined as $\rho(s)=s,\rho(r)=r$ and $\rho(t)=s^{\lvert n-m\rvert/2\lvert n\rvert}$. Then the map $\widetilde{\Phi}=\rho\circ\Phi':G\to\mathbb{D}_{\infty}$ linearly parametrizes $G$.
To show that the first condition implies the second one, we argue by contradiction. Consider the presentation $G = \langle H, t \vert tgt^{-1} = \psi (g), \forall g \in A \rangle$ and assume that for some $h\in H$ and infinite order $a \in A$ we have $ha^ih^{-1}=\psi(a)^j$ with $\lvert i\rvert\neq\lvert j\rvert$. Therefore, $ta^it^{-1}=a^j$. Applying $\widetilde{\Phi}$ we have $\widetilde{\Phi}(e)\widetilde{\Phi}(a)^i\widetilde{\Phi}(e)^{-1}=\widetilde{\Phi}(a)^j$. However, since $\mathbb{D}_{\infty}$ is virtually cyclic, by Lemma \[lemma: finite index and balance\] it follows that $\lvert i\rvert$ must be equal to $\lvert j\rvert$, which is a contradiction.
Combining the above two lemmas we obtain the following.
\[coro: linearly parametr iff all edges balanced\] Let ${\mathcal{G}}$ be a graph of groups with 2-ended vertex and edge groups. Then ${\mathcal{G}}$ is linearly parametrizable if and only if all edges are balanced.
Assume that ${\mathcal{G}}$ is linearly parametrizable by a map $\Phi$ and let $e\in E({\mathcal{G}})$. If $e$ belongs in a spanning tree of ${\mathcal{G}}$ then $e$ is a balanced edge by Remark \[remark: edges in tree are balanced\]. Assume now that $e$ does not belong in a spanning tree. Note first that the subgraph of groups ${\mathcal{G}}-e$ of ${\mathcal{G}}$ is also linearly parametrizable, as we can use the restricted map $\widetilde{\Phi}=\Phi|_{\pi_1({\mathcal{G}}-e)}$ as linear parametrization. If $e$ is unbalanced, then by Lemma \[lem: parametrizing HNN\] we obtain that $\widetilde{\Phi}$ cannot be extended to $\pi_1({\mathcal{G}}-e)\ast_{t_e}\cong\pi_1({\mathcal{G}})$, which is a contradiction. Thus, every edge $e$ must be balanced.
To show the converse, let $T$ be a spanning tree in ${\mathcal{G}}$. Since every vertex group is 2-ended, we can repeatedly apply Lemma \[lemma: amalgam of linearly parametrized\] to show that the subgraph of groups ${\mathcal{G}}|_T$ is linearly parametrizable. If every edge in ${\mathcal{G}}$ is balanced, then we can add one by one the remaining edges in ${\mathcal{G}}$ to $T$ and apply Lemma \[lem: parametrizing HNN\] at each step to obtain the result.
Characterizations of hierarchical hyperbolicity
-----------------------------------------------
With the following lemma, we establish a relation between those graphs of groups that can be linearly parametrized and those which have balanced fundamental group.
\[Lem: unbalanced group iff unbalanced edge\] Let ${\mathcal{G}}$ be a graph of groups with balanced vertex groups. Then $\pi_1({\mathcal{G}})$ is unbalanced if and only if it contains an unbalanced edge.
By definition, if ${\mathcal{G}}$ contains an unbalanced edge then $\pi_1({\mathcal{G}})$ is unbalanced. Assume now that $\pi_1({\mathcal{G}})$ is unbalanced. Let $T$ be a spanning tree of the underlying graph $\Gamma$ of ${\mathcal{G}}$. Start adding edges in $\Gamma\setminus T$ to $T$ until we obtain a subgraph $\Lambda$ of $\Gamma$ such that $\pi_1({\mathcal{G}}|_{\Lambda})$ is unbalanced and $\pi_1({\mathcal{G}}|_{\Lambda - e})$ is balanced. Split $\pi_1({\mathcal{G}}|_{\Lambda})$ as $\pi_1({\mathcal{G}}|_{\Lambda-e})\ast_{t_e}$, and let $A, B \in \pi_1({\mathcal{G}}|_{\Lambda-e})$ be the subgroups associated to the HNN extension. By Corollary \[corollary: HNN and balance\], there is an infinite order element $a \in A$ and $h \in \pi_1({\mathcal{G}}|_{\Lambda-e})$ such that $$ha^{p}h^{-1} = ta^{q}t^{-1},$$ for $\vert p \vert \neq \vert q \vert$, showing that $e$ is an unbalanced edge.
The final ingredient for the proof of the main theorem of this section is the so-called almost Baumslag–Solitar group, which we now introduce.
\[def: almost bs groups\]\[**Almost Baumslag–Solitar**\] A group $G$ is called an *almost Baumslag–Solitar group* if it can be generated by two infinite order elements $a,s$ and the relation $sa^is^{-1}=a^j$ holds in $G$ for $i,j\neq 0$. An almost Baumslag–Solitar subgroup is *non-Euclidean* if $\vert i \vert \neq \vert j \vert$.
\[remark: almost BS groups\] Almost Baumslag–Solitar groups can look very different from traditional Baumslag–Solitar groups. For instance, any group with presentation $\langle a, b\mid ba^nb^{-1}=a^m, R\rangle$ where $R$ is a non-trivial relator on $\{a,b\}$ that does not forces $a$ nor $b$ to be of finite order is an almost Baumslag–Solitar group. Note, moreover, that an almost Baumslag–Solitar group can be obtained as a quotient of some Baumslag–Solitar group, but such quotient is not, in general, an isomorphism.
A common theme throughout the rest of the paper will be finding almost Baumslag–Solitar subgroups. Typically, we will find elements $a, s \in G$ such that $a$ has infinite order and the relation $sa^is^{-1}=a^j$ holds. When $|i|\neq |j|$, we can immediately conclude that $\langle a, s \rangle$ is a non-Euclidean Baumslag–Solitar subgroup. Indeed, if $s$ was of finite order then there would exist some $k>0$ such that $s^k=1$. Therefore, as $sa^is^{-1}=a^j$, it follows that $s^ka^{i^k}s^{-k}=a^{j^k}$ contradicting that $a$ is of infinite order.
An interesting question to ask is under which conditions does an almost Baumslag–Solitar group contain $BS(m,n)$ for some $m,n$. In [@levitt2015quotients Proposition 7.5] it is shown that if a non-Euclidean almost Baumslag–Solitar group $G$ can be embedded into a GBS group, then $G$ will contain some $BS(m,n)$ for $|m|\neq |n|$. In [@button2015balanced Corollary 9.6] it is shown that if a non-Euclidean almost Baumslag–Solitar group $G$ can be embedded into the fundamental group of a graph of torsion-free balanced groups with cyclic edge subgroups then $G$ will contain some $BS(m,n)$ for $|m|\neq|n|$. Following the same spirit, in Corollary \[corollary: main result\] we show equivalent conditions under which a non-Euclidean almost Baumslag–Solitar group contains some $BS(m,n)$ for $|m|\neq|n|$.
\[cor: unbalanced implies almost BS\] Let ${\mathcal{G}}$ be a graph of groups containing an unbalanced edge. Then
1. $\pi_1({\mathcal{G}})$ contains a non-Euclidean almost Baumslag–Solitar subgroup;
2. if $\pi_1({\mathcal{G}})$ is virtually torsion-free then $\pi_1({\mathcal{G}})$ must contain a non-Euclidean Baumslag–Solitar subgroup.
By definition of balanced edges (Definition \[def: balanced edges\]), if $e$ is unbalanced and $\phi_{\pm}$ are the monomorphisms associated to the edge $e$, then there exists an infinite order element $a'\in G_e$ and $h\in\pi_1(\mathcal{G}-e)$ such that $h\phi_+(a')^ih^{-1}=\phi_-(a')^j$ for some $|i|\neq |j|$. Let $a$ denote $\phi_+(a')$ and $s$ denote $t_eh$ for short. By assumption, $a$ has infinite order, and so $s \neq 1$. Then $\langle a, s \rangle$ is a non-Euclidean almost Baumslag–Solitar group.
If, in addition, $\pi_1(\mathcal{G})$ is virtually torsion-free then there exists $N>1$ such that $a
^N$ and $s^N$ belongs in a torsion-free subgroup of $\pi_1(\mathcal{G})$. Note that $$\begin{aligned}
s^N a^{N \cdot i^N}s^{-N} & = s^{N-1} (s (a^i)^{N \cdot i^{N-1}} s^{-1}) s^{-(N-1)} = \\
& = s^{N-1} ( (a^j)^{N \cdot i^{N-1}} ) s^{-(N-1)}= \\
&= s^{N-2}(s (a^i)^{JN \cdot i^{N-2}} s^{-1}) s^{-(N-2)} = \\
&= \cdots = a^{N \cdot j^{N}}\end{aligned}$$ Therefore, the relation $s^N(a^{Ni^N})s^{-N}=a^{Nj^N}$ is satisfied in a torsion-free subgroup $Q$ of $\pi_1(\mathcal{G})$. By Lemma \[lemma: subgroups of graph of 2 ended groups\], $Q$ is a generalized Baumslag–Solitar group. Since $Ni^N/Nj^N=(i/j)^N\neq\pm 1$, by [@levitt2015quotients Proposition 7.5] the subgroup $\langle a^N,s^N\rangle$ contains some non-Euclidean Baumslag–Solitar group.
Combining Lemma \[Lem: unbalanced group iff unbalanced edge\] with Corollary \[cor: unbalanced implies almost BS\] we obtain Theorem \[thm: non-euclidean BS iff unbalanced edge intro section\] from the introduction:
\[thm: non-euclidean BS iff unbalanced edge\] Let ${\mathcal{G}}$ be a graph of groups where none of the vertex groups contain distorted cyclic subgroups. Then $\pi_1({\mathcal{G}})$ contains a non-Euclidean almost Baumslag–Solitar subgroups if and only if ${\mathcal{G}}$ has an unbalanced edge.
If $G=\pi_1({\mathcal{G}})$ contains a non-Euclidean almost Baumslag–Solitar subgroup then it is unbalanced. By Lemma \[Lem: unbalanced group iff unbalanced edge\] we obtain that ${\mathcal{G}}$ must contain some unbalanced edge. Corollary \[cor: unbalanced implies almost BS\] shows the converse.
We are now ready to prove the main result of this section.
\[thm: balanced edges and BS\] Let $\mathcal{G}$ be a graph of groups, where all vertex and edge groups are two-ended. Assume moreover that $\pi_1(\mathcal{G})$ is virtually torsion-free. Then the following are equivalent.
1. $\pi_1({\mathcal{G}})$ admits a hierarchically hyperbolic group structure.
2. ${\mathcal{G}}$ is linearly parametrizable.
3. $\pi_1({\mathcal{G}})$ is balanced.
4. $\pi_1({\mathcal{G}})$ does not contain $\mathrm{BS}(m,n)$ with $\vert m \vert \neq \vert n\vert$.
5. $\pi_1({\mathcal{G}})$ does not contain a distorted infinite cyclic subgroup.
By Corollary \[coro: linearly parametr iff all edges balanced\] we have that $\pi_1({\mathcal{G}})$ is linearly parametrizable if and only if every edge $e$ in ${\mathcal{G}}$ is balanced. Moreover, by Lemma \[Lem: unbalanced group iff unbalanced edge\] we have that every edge in ${\mathcal{G}}$ is balanced if and only if $\pi_1({\mathcal{G}})$ is balanced.
Assume that $\pi_1({\mathcal{G}})$ is unbalanced. Therefore, by Lemma \[Lem: unbalanced group iff unbalanced edge\] there is an edge $e$, an infinite order element $a \in G_e$ and an element $h \in \pi_1({\mathcal{G}}- e)$ such that $$h\phi_{+}(a)^{i}h^{-1}= \phi_{-}(a)^j,$$ with $\vert i \vert \neq \vert j \vert$. Let $x= \phi_+(a)$ and $y= \phi_{-}(a)$. Since $e$ is unbalanced, there is a spanning tree that does not contain $e$. In particular, we can assume there is a stable letter $t$ associated to the edge $e$ such that $tyt^{-1}= x$. We claim that $\langle x \rangle$ is distorted. Note that $x$ is of infinite order. To simply notation, we will write $ A \approx^{r} B$ if $\vert A - B \vert \leq r$. We have: $$\begin{aligned}
d\left(1, x^{N\cdot i }\right) \approx^{2\vert h \vert} d\left(1, hx^{N \cdot i} h^{-1}\right) = d\left(1, y^{N \cdot j}\right) \approx^{2\vert t \vert} d\left(1, x^{N \cdot j}\right).\end{aligned}$$ This is to say, for each $N$ we have $\left\vert d\left(1, x^{N \cdot i}\right) - d\left(1, x^{N \cdot j}\right) \right\vert \leq 2\left( \vert h \vert + \vert t \vert\right)$. Since $\vert i \vert \neq \vert j \vert$, it is now a standard argument to show that $\langle x \rangle $ is distorted. Indeed, restating the argument before for a general exponent $M$ we have $d\left(x^M, x^{\left\lfloor \frac{\vert j \vert}{\vert i \vert} M\right\rfloor}\right) \leq \vert h \vert + \vert t \vert + i$. Assuming that $\vert i \vert > \vert j \vert$, we can iterate the inequality above to obtain that $d(1, X^M)$ is comparable to $\log_{\frac{\vert j \vert}{\vert i \vert}}(M) \cdot (\vert h \vert + \vert t \vert + i)$. That is to say, $d(1, X^M)$ grows logarithmically, showing that the map $n \mapsto x^{n}$ cannot be a quasi-isometric embedding.
Assume that $\pi_1({\mathcal{G}})$ is unbalanced. Therefore, by Lemma \[Lem: unbalanced group iff unbalanced edge\], ${\mathcal{G}}$ must contain an unbalanced edge. The second item of Corollary \[cor: unbalanced implies almost BS\] concludes the proof.
Follows from [@HHSBoundaries Theorem 7.1] and [@durham2018corrigendum Theorem 3.1].
Follows from Theorem \[thm: inducing hhg structure\].
Since non-Euclidean Baumslag–Solitar groups contain distorted cyclic subgroups if $G$ contains some non-Euclidean Baumslag–Solitar subgroup we obtain the result.
\[thm: balanced edges and BS version 2\] Let $\mathcal{G}$ be a graph of groups, where all vertex and edge groups are two-ended. Then the following are equivalent.
1. $\pi_1({\mathcal{G}})$ admits a hierarchically hyperbolic group structure.
2. ${\mathcal{G}}$ is linearly parametrized.
3. $\pi_1({\mathcal{G}})$ is balanced.
4. $\pi_1({\mathcal{G}})$ does not contain a non-Euclidean almost Baumslag–Solitar subgroup.
5. $\pi_1({\mathcal{G}})$ does not contain a distorted infinite cyclic subgroup.
Assume that $\pi_1({\mathcal{G}})$ is unbalanced. Therefore, by Lemma \[Lem: unbalanced group iff unbalanced edge\], ${\mathcal{G}}$ must contain an unbalanced edge. The first item of Corollary \[cor: unbalanced implies almost BS\] shows the implication $4\Rightarrow 3$. The rest of the implications are the same as in Theorem \[thm: balanced edges and BS\].
Hierarchical hyperbolicity of hyperbolic-2-decomposable groups {#section: graph of word hyperbolic groups}
==============================================================
The goal of this section is to extend Theorems \[thm: balanced edges and BS\] and \[thm: balanced edges and BS version 2\] to (hyperbolic)-2-decomposable groups. In this case, we cannot use linear parametrization to obtain hierarchical hyperbolicity. To solve this issue, the first step is to consider (2-ended)-2-decomposable graphs of groups associated to the various edge groups, called *conjugacy graphs*. Intuitively, a conjugacy graph records if an edge group is involved in the presence of Baumslag–Solitar groups. Indeed, it turns out that if all conjugacy graphs are lineraly parametrized, then the group does not contain non-Euclidean Baumslag–Solitar subgroups. Firstly we will prove a combination theorem for relatively hyperbolic groups (Theorem \[theorem: readaptation of relative HHG\]) that allows us to obtain structures on the vertex groups compatible with the ones on the edge groups.
We begin by showing the following lemma. This allows us, without loss of generality, to restrict our attention to graphs of hyperbolic groups with infinite virtually cyclic edge groups.
Let ${\mathcal{G}}$ be a graph of groups such that $\pi_1({\mathcal{G}})$ is infinite and ${\mathcal{G}}$ has hyperbolic vertex groups and virtually cyclic edge groups. Then there exists a finite graph of groups ${\mathcal{G}}'$ with infinite hyperbolic vertex groups and 2-ended edge groups such that $\pi_1 ({\mathcal{G}}')=\pi_1(\mathcal{G})$.
Given a graph of groups ${\mathcal{H}}$ let $F({\mathcal{H}})$ be the set of edges with finite associated edge group, that is $\{e \in E({\mathcal{H}}) \mid \vert G_e \vert \leq \infty\}$. Let ${\mathcal{G}}_0= {\mathcal{G}}$. We will produce a sequence of graph of groups ${\mathcal{G}}_i$ such that $\pi_1({\mathcal{G}}_i) \cong \pi_1({\mathcal{G}})$, ${\mathcal{G}}_i$ has hyperbolic vertex groups and virtually cyclic edge groups and $\vert F({\mathcal{G}}_i)\vert < \vert F({\mathcal{G}}_{i-1})\vert$. Since the graph of groups is finite, eventually we will find ${\mathcal{G}}_n$ such that $F({\mathcal{G}}_n) = \emptyset$. In particular, if ${\mathcal{G}}_n$ has at least one edge, then the associated edge group is infinite. Hence, the vertex groups needs to be infinite and we are done. If there are no edges, then there is a single vertex labelled by $\pi_1({\mathcal{G}})$, which is hyperbolic by construction. Since, by assumption $\pi_1({\mathcal{G}})$ is infinite, we are done.
Suppose ${\mathcal{G}}_{i}$ is defined. Firstly, suppose that there is $e \in F({\mathcal{G}}_{i})$ such that there exists a spanning tree $T_e$ of ${\mathcal{G}}_{i}$ containing $e$ (recall that $\pi_1({\mathcal{G}})$ does not depend on the choice of spanning tree, as pointed out in Remark \[rmk: pi\_1 of G does not depend on the spannig tree\]). Then the subgroup $G_{e^+}\ast_{G_e}G_{e^-}$ is hyperbolic by Theorem [@BestvinaFeighnCombination Corollary Section 7]. Then let ${\mathcal{G}}_{i+1}$ be defined from ${\mathcal{G}}_{i}$ by replacing the edge $e$ and the incident vertices by a single vertex with associated group $G_{e^+}\ast_{G_e}G_{e^-}$, and leaving the other edge maps unchanged. By doing this, we still have hyperbolic vertex groups and virtually cyclic edge groups.
So, suppose that no element of $F({\mathcal{G}}_i)$ can be included in a spanning tree. This is to say that all elements of $F({\mathcal{G}}_i)$ are loops. Let $e \in F({\mathcal{G}}_i)$, and let $v$ be the vertex incident to it. Then by [@BestvinaFeighnAddendum Corollary 2.3], the HNN extesion $G_v \ast_{G_e}$ is hyperbolic. Then we define ${\mathcal{G}}_{i+1}$ as the graph of groups obtained from ${\mathcal{G}}_i$ by removing the edge $e$ and changing the vertex group of $v$ to $G_v \ast_{G_e}$.
From now on, whenever we state a result on a graph of hyperbolic groups ${\mathcal{G}}$ we will always assume that the associated edge groups $G_e$ are virtually cyclic and infinite. In other words, from now on we assume that the groups considered are hyperbolic-2-decomposable.
Given a vertex group $G_v$, one of the main challenges that we have to face in this setting is the fact that the incoming edge groups do not necessarily form an almost-malnormal collection in $G_v$ (Definition \[def: almost malnormal collection\]). As a consequence, these edge groups may not be geometrically separated so as to include them in the hierarchical hyperbolic structure of $G_v$. The following theorem solves this problem, and it is pivotal in the proof of the main theorem in this section. We also stress that it is a consequence of [@HHSII Theorem 9.1].
\[theorem: readaptation of relative HHG\] Let $G$ be a group hyperbolic relative to a family of hierarchically hyperbolic groups $\{(H_i, {\mathfrak{S}}_i)\}_{i=1}^n$. Suppose that there is a finite family of subgroups $\{K_\alpha\}_{\alpha\in\Lambda}$ and homomorphisms $\phi_\alpha: K_\alpha \to G$ such that for each $\alpha$ there exists $i$ and $g \in G$ such that $\phi_\alpha (K_\alpha)$ has finite index in $H_i^g$. Finally, suppose that each group $K_\alpha$ is equipped with a hierarchically hyperbolic structure ${\mathfrak{K}}_\alpha$ such that $\phi_\alpha^{g^{-1}} \colon (K_\alpha,{\mathfrak{K}}_\alpha) \to (H_i,\mathfrak{S}_i)$ is a glueing hieromorphism.
Then there is a hierarchically hyperbolic structure $(G, {\mathfrak{S}})$ on $G$ such that $\phi_\alpha$ is a glueing hieromorphism for every $\alpha$. Moreover, if all $(H_i, {\mathfrak{S}}_i)$ satisfy the intersection property, so does $(G, {\mathfrak{S}})$, and similarly for clean containers.
This theorem is an adaptation of [@HHSII Theorem 9.1], in which the authors provide an explicit hierarchical hyperbolic structure on $G$ and prove that it satisfies the hierarchical hyperbolic axioms. We will follow almost verbatim the part of the proof that describes such a structure on $G$, but we will not verify the axioms since this already appears in [@HHSII Theorem 9.1]. We will conclude the proof by showing that the maps $\phi_\alpha$ can be realized as glueing hieromorphisms.
**The structure:** For each $i=1, \ldots,n$ and each left coset of $H_i$ in $G$, fix a representative $gH_i$. Let $g\mathfrak{S}_i$ be a copy of $\mathfrak{S}_i$ with its associated hyperbolic spaces and projections in such a way that there is a hieromorphism $H_i\to gH_i$ equivariant with respect to the conjugation isomorphism $H_i\to H_i^g$. Let $\widehat{G}$ be the hyperbolic space obtained by coning-off $G$ with respect to the peripherals $\{H_i\}$, and let $\mathfrak{S}=\{\widehat{G}\}\cup\bigsqcup_{g\in g}\bigsqcup_i\mathfrak{S}_{gH_i}$. The relation of nesting, orthogonality or transversality between hyperbolic spaces belonging to the same copy $\mathfrak{S}_{gH_i}$ are the same as in $\mathfrak{S}_{H_i}$. Further, if $U,V$ belong in two different copies of different cosets, then we impose transversality between them. Finally, for every $U\in \mathfrak{S}_{gH_i}$ we declare that $U$ is nested into $\widehat{G}$.
The projections are defined as follows: $\pi_{\widehat{G}} \colon G \to \widehat{G}$ is the inclusion, which is coarsely surjective and hence has quasiconvex image. For each $U \in {\mathfrak{S}}_{gH_i}$, let ${\mathfrak{g}_{_}\left(gH_i\right)} \colon G \to gH_i$ be the closest-point projection onto $gH_i$ and let $\pi^G_U = \pi^{H_i}_U \circ {\mathfrak{g}_{_}\left(gH_i\right)}$, to extend the domain of $\pi_U$ from $gH_i$ to $G$. Since each $\pi^{H_i}_U$ was coarsely Lipschitz on ${{\mathcal C}}U$ with quasiconvex image, and the closest-point projection in $G$ is uniformly coarsely Lipschitz (Lemma \[lem: cpproj is C. Lipschitz\]), the projection $\pi^{G}_U$ is uniformly coarsely Lipschitz and has quasiconvex image. For each $U,V \in {\mathfrak{S}}_{gH_i}$, the various $\rho_U^V$ and $\rho_V^U$ are already defined. If $U \in {\mathfrak{S}}_{gH_i}$ and $V \in {\mathfrak{S}}_{g'H_j}$, then $\rho_V^U = \pi_V({\mathfrak{g}_{_}\left(g'H_j\right)}(gH_i))$. Finally, for $ U \neq \widehat{G}$, we define $\rho^U_{\widehat{G}}$ to be the cone-point over the unique $gH_i$ with $U \in {\mathfrak{S}}_{gH_i}$, and $\rho_U^{\widehat{G}} \colon \widehat{G} \to {{\mathcal C}}U$ is defined as follows: for $x \in G$, let $\rho_U^{\widehat{G}}(x) = \pi^G_U(x)$. If $x \in \widehat{G}$ is a cone point over $g'H_j \neq gH_i$, let $\rho_U^{\widehat{G}}(x) = \rho_U^{S_{g'H_j}}$, where $S_{g'H_j}$ is the $\nest$–maximal element of ${\mathfrak{S}}_{g'H_j}$. The cone-point over $gH_i$ may be sent anywhere in ${{\mathcal C}}U$.
By [@HHSII Theorem 9.1], the construction above endows $(G, {\mathfrak{S}})$ with a hierarchically hyperbolic group structure.
**Hieromorphisms:** Fix $\alpha$. By assumption there exists $i$ and $g \in G$ such that $\phi_\alpha(K_\alpha) \subseteq H_i^{g}$. Moreover, $\Phi_\alpha = \phi_\alpha^{g^{-1}}\colon (K_\alpha, {\mathfrak{K}}_\alpha) \to (H_i, {\mathfrak{S}}_i)$ is a glueing hieromorphism. Our goal is to show that $\phi\colon (K_\alpha, {\mathfrak{K}}_\alpha) \to (G, {\mathfrak{S}})$ can be equipped with a glueing hieromorpism structure.
To simplify notation we will drop the $\alpha$ and $i$ subscript and denote $(K, {\mathfrak{K}}) = (K_\alpha, {\mathfrak{K}}_\alpha)$, $\phi = \phi_\alpha$, $(H, {\mathfrak{S}}_H ) = (H_i, {\mathfrak{S}}_i)$ and so on.
For every $V \in {\mathfrak{K}}$, define $\phi^\diamondsuit (V) = g \Phi^\diamondsuit(V)$ and $\phi_V^\ast = g^\ast \circ \Phi^\ast_V$, where $g^\ast$ is the isometry associated to the multiplication $g \in G$. By assumption, the maps $\Phi^\ast_V \colon {{\mathcal C}}V \to {{\mathcal C}}\Phi^\diamondsuit V$ are isometries, and for each $U \in {\mathfrak{S}}_H$, the space ${{\mathcal C}}_H U$ and the space ${{\mathcal C}}_G gU$ are isometric. Thus, the maps $\phi_V^\ast$ are isometries.
We need to show that the following two diagrams coarsely commute. $$\xymatrix{
K\ar[r]^{\phi}\ar[d]_{\pi^K_V} & G\ar[d]^{\pi^G_{\phi^\diamondsuit(V)}}\\
\mathcal{C}V\ar[r]_{\phi^*_U}&\mathcal{C}\phi^\diamondsuit(V)
}\qquad\qquad\qquad
\xymatrix{
\mathcal{C}V\ar[rr]^{\phi^*_V}\ar[d]_{\rho^V_U} && \mathcal{C}\phi^\diamondsuit(V)\ar[d]^{\rho^{\phi^\diamondsuit(V)}_{\phi^\diamondsuit(U)}}\\
\mathcal{C}U\ar[rr]_{\phi^*_U}&&\mathcal{C}\phi^\diamondsuit(U)
}$$ This is a matter of unwinding the definitions. We will check the first one, the second is analogous. So, let $x\in K$. Recall that $\phi(x) = g\Phi(x)g^{-1} \in gH_ig^{-1}$. Then $$\label{eq: first eq Theorem on rely hyp HHS}
\begin{split}
\pi^{G}_{\phi^\diamondsuit(V)}(\phi(x)) = g^\ast\circ \pi^{H_i}_{\Phi^\diamondsuit(V)} \circ g^{-1}
= g^\ast \circ \pi_{\Phi^\diamondsuit (V)}^{H_i}({\mathfrak{g}_{_}\left(gH_i\right)}(\Phi(x) g^{-1})).
\end{split}$$ Note that $d(\Phi(x)g^{-1}, gH_i) \leq \vert g \vert $. Since all the maps are coarsely Lipschitz, there is a uniform bound between $\pi_{\Phi^\diamondsuit (V)}^{H_i}({\mathfrak{g}_{_}\left(gH_i\right)}(\Phi(x) g^{-1}))$ and $\pi_{\Phi^\diamondsuit (V)}^{H_i}(\Phi(x))$. That is, up to a uniformly bounded error, we can write Equation \[eq: first eq Theorem on rely hyp HHS\] as $$\label{eq: second eq Theorem on rely hyp HHS}
\pi^{G}_{\phi^\diamondsuit(V)}(\phi(x)) = g^\ast \left(\pi_{\Phi^\diamondsuit (V)}^{H_i}(\Phi(x))\right).$$ On the other hand, we have $$\label{eq: third eq Theorem on rely hyp HHS}
\phi^\ast_V \circ \pi_V^K(x) = g^\ast \left(\Phi^\ast_U \circ \pi_V^K(x) \right).$$ Since $g^\ast$ is an isometry, Equations and give the result. Note that the constant of the coarse commutativity depends on $g$. However, since there are only finitely many pairs $(K_\alpha, H_i)$, we obtain uniformity. Hence, the map $\phi$ can be equipped with a hieromorphism structure. By construction, the maps $\phi^\ast_U$ are isometries, and the hieromorphism is full. To see that it has hierarchically quasiconvex image, observe that its image is at finite Hausdorff distance from a peripheral subgroup, hence it is strongly quasiconvex (Lemma \[lem: peripherals are strongly QC\]). Then it is hierarchically quasiconvex by Theorem \[thm: RST strong quasiconvex\]. [@russellsprianotran:convexity Thorem 6.3].
**Intersection property and clean containers:** We start by checking clean containers, that is checking that for each $U \nest T \in {\mathfrak{S}}$ we have $U \bot \cont_\perp^T U$. If $U = \widehat{G}$ there is nothing to check. Hence, assume $U \in g{\mathfrak{S}}_i$ and let $gS_i$ be the $\nest$–maximal element of $g{\mathfrak{S}}_i$. Recall that the relations on ${\mathfrak{S}}$ are defined such that if $ U,V \in {\mathfrak{S}}-\{\widehat{G}\}$ are not transverse, then there is $i \in \{1, \dots, n\}$ and $g \in G$ such that $U,V \in g{\mathfrak{S}}_i$. In particular, $U \bot V$ implies $U,V \in g{\mathfrak{S}}_i$. Hence, $\cont_\perp^{\widehat{G}} U = \cont_\perp^{gS_i} U$. Moreover, if $U \nest T$ and $T \neq \widehat{G}$, it follows $T \in g{\mathfrak{S}}_i$. Since we assumed that $(H_i, {\mathfrak{S}}_i)$ has clean containers, we have $U \bot \cont_\perp^T U$ for all $T \in g{\mathfrak{S}}_i$, completing the proof.
Consider now the intersection property. By hypothesis, for each $g{\mathfrak{S}}_i$ the map $\wedge^{gH_i}$ is defined. Then define $\wedge \colon ({\mathfrak{S}} \cup \{\emptyset\}) \times ({\mathfrak{S}} \cup \{\emptyset\}) \to ({\mathfrak{S}} \cup \{\emptyset\})$ by considering the symmetric closure of the following: $$U \wedge V = \begin{cases} U & \text{ if } V = \widehat{G}\\
U \wedge^{gH_i} V & \text{ if } U,V \in g{\mathfrak{S}}_i \text{ for some } i, g\\
\emptyset & \text{ otherwise.}\end{cases}$$ The only property to verify that does not follow directly is that if $U \in g{\mathfrak{S}}_i$ and $V \in g'{\mathfrak{S}}_j$ with $g{\mathfrak{S}}_i \neq g'{\mathfrak{S}}_j$, then there is no $W$ nested in both $U,V$. But if such a $W$ existed, then it needs to belong to both $g{\mathfrak{S}}_i$ and $g'{\mathfrak{S}}_j$, a contradiction.
Commensurability and conjugacy graph
------------------------------------
In this subsection we extend the results obtained in Section \[section: graph of infinite virtually cyclic groups\] to the general setting. The key object that will allow us to do this is the conjugacy graph (Definition \[definition: conjugacy graph\]). This is a graph of groups that, combined with Theorem \[theorem: readaptation of relative HHG\], provides vertex groups with a hierarchical hyperbolic structure realizing edge maps as glueing hieromorphisms.
As the vertex groups in the graphs of groups considered are not 2-ended, the whole graph of groups cannot be linearly parametrized. Moreover, the edge groups do not necessarily embed into vertex groups in an almost-malnormal way. To overcome those problems, we will consider the elementary closure of subgroups. A systematic study of elementary closures of WPD subgroups (which include cyclic subgroups of hyperbolic groups as a special case) is carried out in [@DGO], where the authors show such subgroups need to be hyperbolically embedded in the ambient group. For the sake of self-containment, we recall some useful properties of the elementary closure.
Let $G$ be a group and let $H$ be a subgroup of $G$. We define the *elementary closure* of $H$ in $G$ as the subgroup $$E_G(H)=\{g\in G\mid \dhaus{(gH,H)}<\infty\}.$$
\[lem:elementary closure is maximal for commensurability\] Let $H, K$ be subgroups of $G$ such that $H \cap K$ has finite index in both $H$ and $K$, then $K \leq E_G(H)$.
Let $k \in K$ and $h \in H$. Our goal is to uniformly bound $d(kh, H)$. Since $H\cap K$ has finite index in $H$, there is $k_0 \in H \cap K$ at uniformly bounded distance from $h$. Note that $kk_0\in K$. Since $H\cap K$ has finite index in $K$, there is $h_0 \in H \cap K$ at uniformly bounded distance from $k k_0$. By triangular inequality, we get a uniform bound on $d(kh, h_0)$.
Note that, in general, $H$ will not have finite index in $E_G(H)$. A simple example of this is given by considering the subgroup $\langle a \rangle $ in $\langle a \rangle \oplus \langle b \rangle \cong \mathbb{Z}^2$. Indeed, in this case we would have $E_{\mathbb{Z}^2}(\langle a \rangle) = \mathbb{Z}^2$. This is not the case, however, for 2-ended subgroups of hyperbolic groups.
\[remark: elementarizer\] Let $G$ be a hyperbolic group and $H$ be a 2-ended subgroup. Then $E_G(H)$ is 2-ended.
In particular, observe that $E_G(H)$ has to be the maximal cyclic subgroup containing $H$. This yields the following useful lemma.
\[remark: commensurability and malnormality\] Let $H_1, \dots, H_n$ be 2-ended subgroups of a hyperbolic group $G$. Then
1. $H_i$ and $H_j$ are commensurable in $G$ if and only if $E_G(H_i)$ and $E_G(H_j)$ are conjugate to each other.
2. $\{E_G(H_1),\ldots,E_G(H_n)\}$ is an almost-malnormal collection if and only if $H_i$ and $H_j$ are non-commensurable for every $i\neq j$;
Since $H_i$ has finite index in $E_G(H_i)$, we have that $E_G(H_i)$ and $E_G(H_j)$ are commensurable if and only if $H_i$ and $H_j$ are. In particular, this shows one implication. Suppose that $E_G(H_i)$ and $E_G(H_j)$ are commensurable. Up to conjugating one of them we have that $gE_G(H_i)g^{-1} \cap E_G(H_j)$ has infinite index in both $gE_G(H_i)g^{-1}$, and $ E_G(H_j)$. By Lemma \[lem:elementary closure is maximal for commensurability\] we have $gE_G(H_i)g^{-1} \leq E_G(E_G(H_j)) = E_G(H_j)$ and, by symmetry, $E_G(H_j) \leq gE_G(H_i)g^{-1}$. Hence, $E_G(H_i)$ and $E_G(H_j)$ are conjugate.
For the second item, observe that if $E_G(H_i)$ and $E_G(H_j)$ are not commensurable, since they are 2-ended groups it must follow $\vert E_G(H_i) \cap gE_G(H_j) g^{-1} \vert \leq \infty$ for all $g \in G$. Hence they are almost-malnormal.
We now introduce the conjugacy graph associated to an edge group.
Let $G$ be a group and let $\mathcal{P}$ be a collection of 2-ended subgroups of $G$. We denote by $\approx$ the equivalence relation on $\mathcal{P}$ induced by commensurability. That is to say, $P_1\approx P_2$ whenever $P_1, P_2$ are commensurable (as in Definition \[def: commensurable subgroups\]). For each $P\in\mathcal{P}$ we use $\llbracket P\rrbracket$ to denote its commensurability class.
Let ${\mathcal{G}}$ be a graph of groups with 2-ended edge groups.
Consider the multiset $$U=\{\phi_{e^+}(G_e), \phi_{e^-}(G_e)\mid e \in E(\Gamma)\}$$ of all the images of edge groups into vertex groups counted with repetitions.
Let $\sim_0$ be the relation on $U$ defined by imposing $H_1 \sim_0 H_2$ whenever either there exists $e$ such that $H_1 = \phi_{e^+}(G_e)$ and $H_2 = \phi_{e^-}(G_e)$, or $H_1, H_2\in G_v$ for some $v$ and $H_1\approx H_2$ in $G_v$. Extend $\sim_0$ to an equivalence relation $\sim$ on $U$ by taking the transitive closure of $\sim_0$.
For a vertex group $H$, we denote by $[H]$ its equivalence class with respect to $\sim$.
\[definition: conjugacy graph\] Let ${\mathcal{G}}$ be a graph of groups with 2-ended edge groups and let $[H]$ be the equivalence class of an edge group in ${\mathcal{G}}$. We define the *conjugacy graph* associated to $[H]$ as the graph of groups $\Delta_{[H]}$ defined as follows.
For each vertex group $G_v \in {\mathcal{G}}$, let $[H]_v = \{H' \in [H]\mid H'\leq G_v\}$.
[**Vertices:** ]{}For each vertex $v$ of the original graph ${\mathcal{G}}$ and commensurability class $\llbracket K \rrbracket$ of $[H]_v$, add one vertex $v_K$ to $\Delta_{[H]}$. Choose once and for all a representative $K\in \llbracket K \rrbracket$ and define $E_{G_v}(K)$ to be the vertex group associated to $v_K$.
[**Edges:** ]{}For each edge $e \in \Gamma$ such that $\phi_{e^+}(G_e) \in [H]$, add an edge between $\llbracket \phi_{e^+}(G_e) \rrbracket$ and $\llbracket \phi_{e^-}(G_e)\rrbracket$, with associated edge group $G_e$. To define the edge maps, let $K$ be the chosen representative of $\llbracket \phi_{e^+}(G_e) \rrbracket$. Then there is $h \in G_{e^+}$ such that $\phi_{e^+}(G_e)^h \subseteq E_{G_{e^+}}(K)$. If $\phi_{e^+} \colon G_e \to G_{e^+}$ was the edge map of ${\mathcal{G}}$, let the attaching map of $\Delta_{[H]}$ be defined as $\phi_{e^+}^h \colon G_e \to E_{G_{e^+}}(K)$. Note that, by Remark \[remark: commensurability and malnormality\], this map is well defined.
\[remark: remark on conjugacy graph\] In this paper, we consider only graphs of groups with 2-ended edge groups. In particular, by Lemma \[remark: elementarizer\] the vertex groups of the conjugacy graphs are 2-ended. As the edge groups of the conjugacy graphs are the same as the original edge groups, the conjugacy graphs have 2-ended vertex and edge groups.
Let $\mathbb{F}_2=\langle a,b\rangle$ be the free group of rank $2$ and consider the group $G$ to be $\pi_1(\mathcal{G})=\mathbb{F}_2\ast_{ta^3t^{-1}=ba^2b^{-1}}$. By construction, the splitting of $G$ has one vertex $v$ with associated vertex group $G_v=\mathbb{F}_2$ and one edge $e$ with associated cyclic edge group $G_e$. We now construct the conjugacy graph $\Delta_{[G_e]}$ associated to $[G_e]$. Note first that the images of the single edge group are commensurable in the vertex group, as $b\langle a^3\rangle b^{-1}\cap\langle ba^2b^{-1}\rangle$ is infinite. Thus, there is a single conjugacy class of $[G_e]$ in $\mathbb{F}_2$ and, therefore, a single vertex in $\Delta_{[H]}$. The associated vertex group of $\Delta_{[H]}$ is $bE_{\mathbb{F}_2}(a^2)b^{-1}=b\langle a\rangle b^{-1}$. There is also a single edge group in $\Delta_{[H]}$ with associated edge group equal to the one in $\mathcal{G}$. The associated attaching maps are $\phi_{e^+}$ and $\phi_{e^-}^b$. The conjugacy graph associated to $[G_e]$ results in the group $\langle a\rangle\ast_{ta^2t^{-1}=a^3}$.
In the following two lemmas, we describe how is the linear parametrization in a graph of 2-ended groups extended to the general setting using the conjugacy graph.
\[lemma: conjugacy path and conjugacy graph\] Let $G\cong\pi_1({\mathcal{G}})$ be a graph of hyperbolic groups with 2-ended edge subgroups and let $e$ be an edge in the underlying graph of ${\mathcal{G}}$. If $\Delta_{[G_e]}$ denotes the conjugacy graph associated to $[G_e]$, then $e$ is unbalanced in $\mathcal{G}$ if and only if $\pi_1(\Delta_{[G_e]})$ is unbalanced.
Assume first that $\mathcal{G}$ contains an unbalanced edge $e$. Therefore, there exists an infinite order element $a\in G_e$ and $h\in\pi_1({\mathcal{G}}-e)$ such that $h\phi_{e^+}(a)^ih^{-1}=\phi_{e^-}(a)^j$ for some $|i|\neq |j|$. By Lemma \[lemma: conjugacy path in G\] there is a path $e_1,\ldots, e_k$ in the graph of ${\mathcal{G}}-e$ with $A_{e(1)} = G_\alpha, B_{e(k)} = G_\beta$ such that $B_{e_j}^{h_j}\cap A_{e_{j+1}}$ is non-trivial for every $j=1,\ldots,k-1$ (i.e $E_{G_{e_j
^+}}(B_{e_j})^{h_j}=E_{G_{e_j^+}}(A_{e_{j+1}})$) and elements $h_0 \in G_\alpha$ and $h_i \in G_{b(e_i)}$ satisfying $$\label{equation: conjugacy path 3}
(t_{e_k}h_k\cdots h_1h_0) \phi_{e^+}(a)^i (t_{e_k}h_k\cdots h_1h_0)^{-1}=\phi_{e^-}(a)^j,$$ for some $|i|\neq |j|$.
This means that the conjugacy graph $\Delta_{[G_e]}$ splits as $\pi_1(\Delta_{[G_e]}-e)\ast_{t_e}$. Recall that by definition the attaching maps in $\Delta_{[G_e]}$ are defined as conjugates $\phi_{e'^+}^{h_{e'}}$ in $G_{e'^+}$ of the attaching maps $\phi_{e'^+}$ in $\mathcal{G}$. Therefore, since $\phi_{e^+}(g),\phi_{e^-}(g')$ are conjugate in $\pi_1(\mathcal{G})$, following Equation we obtain that $\phi_{e^+}(g)^i=\phi_{e^-}(g)^j$ in $\pi_1(\Delta_{[G_e]}-e)$ where $|i|\neq |j|$.
Assume now that, $\pi_1(\Delta_{[G_e]})$ is unbalanced. We can apply Lemma \[lemma: conjugacy path in G\] to obtain, $$\label{equation second implication unbalanced CG implies unbalanced}(h_kt^{\epsilon_k}_{e_k}\cdots h_1t_{e_1}^{\epsilon_1}h_0)a^p(h_kt^{\epsilon_k}_{e_k}\cdots h_1t_{e_1}^{\epsilon_1}h_0)^{-1}=a^q,$$ for some $|p|\neq |q|$. Here, $a$ is of infinite order, the various elements $h_i$ and $a$ belong to vertex groups and at least one $\epsilon_i$ is non zero. Our goal is to modify the above equation to obtain an analogous one that holds in $\pi_1({\mathcal{G}})$. Let $H_0$ be the vertex group of $\Delta_{[G_e]}$ that contains $a$ and let $H_1$ be the other vertex group adjacent to $e_1$ in $\Delta_{[G_e]}$ (possibly, $H_0 = H_1$). Let $x \in H_1$ be such that $(t_{e_1}^{\epsilon_1}h_0)a^p(t_{e_1}^{\epsilon_1}h_0)^{-1} = x$ in $\pi_1(\Delta_{[G_e]})$. By definition of conjugacy graphs, there are vertex groups $G_0, G_1$ of ${\mathcal{G}}$ such that $H_i \leq G_i$. Since the attaching maps in the conjugacy graph are defined as a conjugates of the attaching maps of ${\mathcal{G}}$, there exist $k_i \in G_i$ such that the following holds in $\pi_1({\mathcal{G}})$: $$(k_1t_{e_1}^{\epsilon_1}h_0k_0)a^p(k_1t_{e_1}^{\epsilon_1}h_0k_0)^{-1} = x$$ Let $y_1 = (k_1t_{e_1}^{\epsilon_1}h_0k_0)$. Proceeding in this way, we find an element $y_k = y$ of $\pi_1({\mathcal{G}}-e)$ such that $$ya^{p}y^{-1} = a^q$$ with $\vert p \vert \neq \vert q \vert$, showing that $e$ is unbalanced in ${\mathcal{G}}$.
\[lemma: 2 implies 1\] Let ${\mathcal{G}}$ be a graph of groups with hyperbolic vertices and 2-ended edge subgroups. Suppose, moreover, that for each edge $e$ the conjugacy graph $\Delta_{[G_e]}$ is linearly parametrizable. Then $\pi_1({\mathcal{G}})$ admits a hierarchically hyperbolic group structure.
For each vertex $v\in V({\mathcal{G}})$ let $\{e_i\}$ be the set of incoming edges and let $E(G_{e_i^{+}})$ be the elementary closure of the images of the edge groups in $G_v$. Choose representatives $\{ E_i\}$ of the commensurability classes $\{\llbracket E(G_{e_i^{+}})\rrbracket\}$. Note that, by Remark \[remark: commensurability and malnormality\], $\{ E_i\}$ forms an almost-malnormal collection of subgroups. In particular, $G_v$ is hyperbolic relative to $\{E_i\}$ by Theorem \[thm:bowditch\].
By assumption, the conjugacy graph $\Delta_{[G_e]}$ associated to $[G_e]$ is linearly parametrizable for every $e$. That is to say, for every edge $e$ there exists $\Phi_{[G_e]}:\pi_1(\Delta_{[G_e]})\to\mathbb{D}_{\infty}^{(e)}$ such that $\Phi_{[G_e]}|_{G_x}:G_x\to\mathbb{D}_{\infty}^{(e)}$ is a quasi-isometry, where $G_x$ is either a vertex or edge group of $\Delta_{[G_e]}$. We endow the various groups $G_x$ with the hierarchical hyperbolic structure $(G_x,\{\mathbb{D}_{\infty}^{(e)}\})$ as described in Lemma \[lemma: pulling back hierarchical structures\]. In particular, this allows to equip with a hierarchically hyperbolic group structure every edge group of ${\mathcal{G}}$ and every group $E_i \leq G_v$ as before. Note that this is well defined. Indeed, suppose that $e,f$ are edges incoming in $v$ and $E(\phi_{e^{+}}(G_e)), E(\phi_{f^{+}}(G_f))$ are conjugate. Then $e\sim f$ and hence $E(\phi_{e^{+}}(G_e))$ and $E(\phi_{f^{+}}(G_f))$ are identified in the conjugacy graph. Thus the hierarchically hyperbolic structure of the representative $E$ does not depend on choices. Finally, note that since the trivial hierarchically hyperbolic structure on $\dihed$ satisfies the intersection property and clean containers, so do all the hierarchically hyperbolic structures considered thus far.
Note that we are now in the hypotheses of Theorem \[theorem: readaptation of relative HHG\], allowing us to equip every vertex group with a hierarchically hyperbolic structure $(G_v,\mathfrak{S}_v)$ that turn the edge maps into glueing hieromorphisms $(G_e,\mathfrak{S}_e)\hookrightarrow (G_v,\mathfrak{S}_v)$. Moreover $(G_v, {\mathfrak{S}}_v)$ satisfy the intersection property and clean containers. Applying Theorem \[comb\_thm\_ver2\] we obtain that $\pi_1({\mathcal{G}})$ is a hierarchically hyperbolic group.
We now show the proof of the main results of the section and the paper.
\[corollary: main result\] Let ${\mathcal{G}}$ be a graph of groups with hyperbolic vertices and 2-ended edge subgroups. Assume that $G = \pi_1({\mathcal{G}})$ is virtually torsion-free. The following are equivalent:
1. $G$ is a hierarchically hyperbolic group;
2. the conjugacy graph associated to every equivalence class of edges is linearly parametrizable;
3. $G$ does not contain $BS(m,n)$ for $\lvert n\lvert\neq \lvert m\lvert$;
4. $G$ is balanced;
5. $G$ does not contain an infinite distorted cyclic subgroup.
Follows from [@HHSBoundaries Theorem 7.1] and [@durham2018corrigendum Theorem 3.1].
If $G$ is non-balanced, then by Corollary \[Lem: unbalanced group iff unbalanced edge\], ${\mathcal{G}}$ contains an unbalanced edge and hence a non-Euclidean Baumslag–Solitar subgroup. Since these subgroups contain an infinite distorted subgroup we obtain the implication.
By definition, a balanced group cannot contain a non-Euclidean Baumslag–Solitar subgroup.
Assume that $\Delta_{[G_e]}$ is not linearly parametrizable for some edge $e$. Theorem \[thm: balanced edges and BS\] implies that there exists an edge $e\in\Gamma\setminus T$ which is unbalanced in $\Delta_{[G_e]}$. Moreover, Lemma \[lemma: conjugacy path and conjugacy graph\] ensures that there exists an unbalanced edge in $\mathcal{G}$. By lemma \[Lem: unbalanced group iff unbalanced edge\] we obtain that $G$ must contain some non-Euclidean Baumslag–Solitar group.
Follows from Lemma \[lemma: 2 implies 1\]
\[corollary: main result version 2\] Let ${\mathcal{G}}$ be a graph of groups with hyperbolic vertices and 2-ended edge subgroups. The following are equivalent:
1. $G$ is a hierarchically hyperbolic group;
2. the conjugacy graph associated to every equivalence class of edges is linearly parametrizable;
3. $G$ does not contain a non-Euclidean almost Baumslag–Solitar group;
4. $G$ is balanced;
5. $G$ does not contain an infinite distorted cyclic subgroup.
The implications are the same as in Corollary \[corollary: main result\], except for $4\Rightarrow 3$ and $3\Rightarrow 2$, which we now show.
By definition, a balanced group cannot contain a non-Euclidean almost Baumslag–Solitar group.
Assume that $\Delta_{[G_e]}$ is not linearly parametrizable for some edge $e$. Since $\Delta_{[G_e]}$ is a graph of 2-ended groups (Remark \[remark: remark on conjugacy graph\]), Theorem \[thm: balanced edges and BS version 2\] implies that $\pi_1(\Delta_{[G_e]})$ is unbalanced. Therefore, Lemma \[lemma: conjugacy path and conjugacy graph\] ensures that there exists an unbalanced edge in $\mathcal{G}$. By Corollary \[cor: unbalanced implies almost BS\] we obtain that $G$ must contain some non-Euclidean almost Baumslag–Solitar group.
As a consequence of this we obtain the following corollary that was included in the introduction:
\[corol: corollary to mainT\] Let $G = H_1 \ast_C H_2$ where $H_i$ are hyperbolic and $C$ is 2-ended. Then $G$ is a hierarchically hyperbolic group.
It follows from Lemma \[lemma: amalgam and balance\] that $G$ is balanced. From the previous Corollary, we obtain the result.
[^1]: https://mathoverflow.net/questions/330632/is-an-hnn-extension-of-a-virtually-torsion-free-group-virtually-torsion-free
[^2]: If $A\subseteq \mathcal{X}$, by $\pi_U(A)$ we mean $\bigcup_{a\in A}\pi_U(a)$.
|
---
abstract: 'We investigate the mechanism that leads to systematic deviations in cluster Monte Carlo simulations when correlated pseudo-random numbers are used. We present a simple model, which enables an analysis of the effects due to correlations in several types of pseudo-random-number sequences. This model provides qualitative understanding of the bias mechanism in a class of cluster Monte Carlo algorithms.'
address:
- ' Landau Institute for Theoretical Physics, 117940 GSP-1 Moscow V-334, Russia'
- |
Department of Applied Physics, Delft University of Technology,\
Lorentzweg 1, 2628 CJ Delft, The Netherlands
author:
- 'L.N. Shchur'
- 'J.R. Heringa'
- 'H.W.J. Blöte'
title: ' Simulation of a directed random-walk model: the effect of pseudo-random-number correlations'
---
,
and
random-number generator, directed random walk
Introduction {#sec_intro}
============
The Monte Carlo method to obtain statistical averages for a model is widely used when exact calculations are not available. The error, expected from a standard statistical analysis, is proportional to the inverse square root of the number of randomly chosen samples. One may find biased results if the samples are not representative. Upper bounds to the error are not known in general. Standard statistical analysis of simulation data suffices to obtain the errors of a random statistical nature. Statistically independent random numbers do not introduce a bias in Monte Carlo simulations. However, it is not possible to achieve independence with a deterministic recipe, as is commonly used in Monte Carlo calculations[@HN]. Although one generates seemingly irregular numbers the underlying production rule induces relations or correlations between the pseudo-random numbers. The production rule itself already constitutes a correlation between the generated pseudo-random numbers. The correlations may cause a bias in the simulation results that is difficult to assess.
Substitution of the production rule in itself leads to further correlations involving more random numbers. By repeated substitutions one finds a hierarchy of correlations. Such hierarchies were studied by Compagner[@AC1; @AC2]. Due to the multitude of possible correlations it is not feasible to study all of them. One has to analyse which correlations are detrimental to the intended use of the pseudo-random numbers. Compagner notes, that correlations between few numbers or closely spaced numbers are most important. The effect of such correlations has actually been observed in Metropolis type Monte Carlo simulations of Ising models [@DISP; @Wilding].
In this work, we investigate the consequences of correlations between pseudorandom numbers in an example of a simple random-walk model. An analysis of the effect of the correlations present when a given generator is used may guide us in the choice of an appropriate production rule. In Section \[sec\_model\] we introduce our stochastic walk model. In Section \[sec\_rng\] we discuss the class of generators we used. Deviations caused by the correlations are described in Section \[sec\_core\] for shift register generators and in Section \[sec\_fib\] for the lagged Fibonacci random number generator. In the last Section we give a discussion and conclusion.
The walk model {#sec_model}
==============
The Ising simulations using cluster updates with the Wolff [@Wolff] method seem to be very sensitive to deficiencies in a number of pseudo-random-number generators [@FLW; @SB]. The crucial element of such simulations is the formation of clusters. These are formed by joining spins which are connected by ‘active’ bonds. In ferromagnetic models bonds can only be active, if the spins on either side of the bond are in the same state. The probability, that a bond is active, still depends on the coupling between the spins.
In order to study the random-number generator induced bias we replace the Wolff cluster formation process by a simpler one. First we place the Ising spins on a one-dimensional lattice. Second the cluster is grown only in one direction (or, equivalently, the cluster formation always begins at an open boundary of the spin chain). Third the cluster is grown in a configuration of parallel spins only. This third simplification is less far reaching than it may seem. It does not modify the Ising statistics of the system. The probability that Ising spins, coupled with a strength $K$, are parallel is $\left(1+e^{-2K}\right)^{-1}$. If they are parallel the random-cluster model yields a probability $(1-e^{-2K})$ that they are connected by an active bond, i.e. that the two spins belong to the same cluster. Thus, in Wolff simulations of the one-dimensional Ising model the probability that each next spin is included in the cluster, equals $\mu=\tanh(K)$: a constant depending on $K$ only, just as in our simplified model.
We may interpret the cluster formation in the simplified model as a directed one-dimensional random walk. A walk starts on one site (see Fig. \[walk\]). At discrete times $n$ the walker steps to the neighbouring site in a fixed direction (to the right in Fig. \[walk\]) with probability $\mu$. Thus the walker cannot recur to a site once visited. If the step is not made, a new walk is initialized. The decision, whether to start a new walk or not, does (ideally) not depend on previous decisions. The probability to start a new walk at a certain time equals $\nu = 1 - \mu$, independent of the length of the walk. The probability $P(n)$ that this process generates a walk covering precisely $n$ sites, satisfies $$P(n)=\mu^{n-1}(1-\mu)
\label{eq_prob}$$ and the expectation value of the number of sites visited is $$\langle n \rangle=\frac{1}{1-\mu}$$ One can formulate this random walk process in terms of the following two algorithms. We distinguish two possible ways to generate the decision whether to step to the next site or to initialize a new walk . The algorithms differ by the underlined statements.
==Algorithm P\
\
Initialize walk statistics $\{n_L\}$\
\
generate a random number $r$\
(with $0\leq r <1$)\
\
[**for**]{} walk := 1 [**to**]{} total number [**do**]{}\
{ walk length $L$:= 1\
\
{ $L:=L +1$\
generate a random number $r$ }\
\
$n_L:=n_L+1$ }\
[**enddo**]{}\
\
output $\{n_L\}$\
==\
\
Initialize walk statistics $\{n_L\}$\
\
generate a random number $r$\
(with $0\leq r <1$)\
\
[**for**]{} walk := 1 [**to**]{} total number [**do**]{}\
{ walk length $L$:= 1\
$\underline{1-\mu \leq r}$ [**do**]{}\
{ $L:=L +1$\
generate a random number $r$ }\
\
$n_L:=n_L+1$ }\
[**enddo**]{}\
\
output $\{n_L\}$\
Using perfect random numbers, independent and uniformly distributed between 0 and 1, both algorithms would yield consistent and unbiased averages. However, in real simulations, the random numbers are correlated, as a consequence of the deterministic production rule. This causes the distribution of $n$-tuples of consecutive pseudo-random numbers to be non-uniform in the unit hyper-cube especially for larger dimensionality $n$. A certain degree of non-uniformity is acceptable for the numbers used in the directed random walk algorithm. It is sufficient, that $n$-tuples with components larger or smaller than the quantity used for the decision ($\mu$ in Algorithm $P$ and $1-\mu$ in Algorithm $N$) are present in the right proportions. But even this weaker requirement is not satisfied in practice. This affects the expectation values of the simulation results. These deviations are discussed in Sections \[sec\_core\] and \[sec\_fib\].
Random generators {#sec_rng}
=================
In a numerical Monte Carlo calculation as described in Section \[sec\_model\] one has to decide between stepping with probability $\mu$ and initialization of a new walk with probability $\nu \equiv 1-\mu$. One needs a random sequence of two decision outputs, where one occurs with the step probability $\mu$ and the other with the initialization probability $1-\mu$. To this purpose we use pseudo-random numbers $r$ with $0\leq r <1$ and compare them to the relevant probability.
We generate the pseudo-random numbers with a number of rules. One is the Generalized Feedback Shift Register (GFSR) method [@LP; @KS]. A sequence of pseudo-random integers $r(i)$, represented by their binary expansion (usually 32 bits), is generated by the rule $r(i)=r(i-p)\oplus r(i-q_1)\oplus \cdots \oplus r(i-q_j)$, where the exclusive-or operation $\oplus$ is applied bit-wise and the feedback positions (lags) are ordered according to $p>q_1>q_2>\cdots >q_j$. If $j=1$, we denote the feedback lag by $q$. The leading bits (as well as all other bits) separately form a sequence, that is generated by a feedback shift register. The maximum length equals $2^p -1$. Production rules that generate maximum-length bit-sequences, are characterized by primitive polynomials [@Gol]. Tables of primitive polynomials are listed in [@rule]. The most widely used random number generator based on the GFSR method is R250 [@KS] with $p=250$, $q=103$ and $r(i)=r(i-250)\oplus r(i-103)$.
Where primitive trinomials are known, one can obtain other primitive polynomials by decimation procedures [@Gol]. For example, using every third number from a sequence generated with a rule derived from a trinomial is equivalent to using a rule specified by a pentanomial, which adds two terms to the primitive trinomial. For a rule based on primitive polynomials of degree $p$ only sequences up to $p$ successive bits or numbers are independent. In order to prevent problems one should avoid unwanted initializations. For instance, if the $k$-th bit of each initialized integer is zero, the same will hold for the subsequently generated pseudo-random integers.
Furthermore we shall consider the lagged Fibonacci method: $r(i)=r(i-p)+r(i-q)$, where the addition is understood modulo $2^l$, where $l$ is the number of bits in a computer word. The maximum sequence length[@Dai; @Marsfib] is $2^{l-1} ( 2^p - 1)$.
Examples of the correlation effect: Shift Registers. {#sec_core}
====================================================
An important factor contributing to the bias for a given random-number generator is associated with the fact that, whenever a new walk is initialized, the preceding decision was [*not*]{} to visit another site. The last random number $r_P(0)$ used in the previous walk in Algorithm $P$ thus obeys $r_P(0)\geq \mu$ (or obeys $r_N(0)<1-\mu=\nu$ in Algorithm $N$). If the new walk visits another site, the next random number $r_P(1) < \mu$ (or $r_N(1) \geq \nu$). If the computer words consist of $l$ bits, only probabilities that are a multiple of $2^{-l}$ are faithfully represented. We suppose that the first $p$ numbers of a generalized shift-register sequence may be considered as independent (as if produced by an ideal random number generator), i.e. the decisions with respect to the first $p-1$ steps occur with probability $\mu$ independent of the history. The probability of a walk with less than $p$ steps is that given by Eq. (\[eq\_prob\]).
However, deviations will occur for step number $p$. The $p$-th number in the sequence depends on the bits in $r(0)$ and those in the integer $r(p-q)$ on the feedback position. This causes the initialization probability at this point to differ from the value $\nu$ it should have. We denote the actual probability by $\nu^*$. This probability will depend on the chosen random generator (GFSR or lagged Fibonacci, etc.). In this Section we will focus attention on shift-register random-number generators. The case of the lagged Fibonacci recipe is treated in the next Section.
First, we analyze some simple examples. If $\mu=\frac{1}{2}$ just one random bit $b$ is needed to generate the decision, where both outcomes occur with equal probability. The random bit $b$ denotes the leading bit of the pseudo-random number $r$. The completion of the preceding walk implies, except for the first walk, that $b(0)=1$ when Algorithm $P$ is used. In the case that a decision is to be made on step $p$, the completion of the preceding steps means that $b(i)=0$ for $0<i<p$. Thus, irrespective of the number $j+1$ of feedback positions $p,q_1,q_2,...,q_j$, $b(p)=1$ so that the walk ends, and a new one is initialized. The actual initialization probability is denoted $\nu^*(p)$ and satisfies $\nu^*_P(p)=1$ instead of the desired $\nu = 1- \mu $. The maximum walk length is $p$ and occurs with a probability of twice that expected from Eq. (\[eq\_prob\]).
In the case that a decision has to be made on step $p$ of Algorithm $N$, $b(0)=0$ and $b(i)=1$ for $0<i<p$. If the number of additional feedback bits is odd, which is the case for maximum-length sequences, $b(p)=1$ and the walker always proceeds to site $p$. The actual initialization probability thus satisfies $\nu^*_N(p)=0$. However, one has $b(p+1) = 0$, so that $\nu^*_N(p+1)=1$. Thus all walks with a length exceeding $p-1$ have a length $p+1$. Length $p$ occurs with probability 0, length $p+1$ with probability $P^*(p+1)={\left(\frac{1}{2}\right)}^{p-1}$.
These results can be generalized for all $\mu = 2^{-m}$ and positive integer $m$. The walk lengths will be smaller than $p+1$ in Algorithm $P$, as the leading $m$ bits of $r(p)$ will be the same as those of $r(0)$. For Algorithm $N$ the leading $m$ bits of $r(p+1)$ have to be zero. Lengths larger than $p+1$ do not occur.
If $\mu > \frac{1}{2}$, much longer walks may occur, if the GFSR-generator is used. As an example of this regime we take $\mu = \frac{5}{8}$. One needs three bits to make a decision. The possible values are represented by the integers $\tilde{r} \equiv \lfloor 8 r\rfloor$ where the brackets denote the integer part.
In the case of Algorithm $P$ a decision on step $p$ requires $\tilde{r}(0) \geq 5$ and $\tilde{r}(i) <5 $ for $0<i<p$. We assume that all admissible numbers for $\tilde{r}(i)$ with $i<p$ occur with equal probability, which strictly holds only for the first walk. For sequences generated with a production rule derived from a primitive trinomial, $\tilde{r}(p)$ equals 4, 5, 6 and 7 with probability $\frac{1}{5}$, 1, 2 and 3 with probability $\frac{1}{15}$ and 0 with probability 0. This non-uniform distribution of $\tilde{r}$ leads to a probability $\nu^*_P(p)=\frac{3}{5}$ for the walk to end at step $p$. This is different from the desired probability $\nu=\frac{3}{8}$. The probability of walks of length $p$ thus equals $P^*(p)={\left(\frac{5}{8}\right)}^{p-1}\frac{3}{5}$.
For $\tilde{r}(i)$ with $$p<i<\left\{\begin{array}{lll}
p+q& \mathrm{if}&q<\frac{p}{2}\\
2p-q& \mathrm{if}&q>\frac{p}{2}\\
\end{array}
\right.
\label{ineq}$$ the numbers 4, 5, 6 and 7 occur with probability $\frac{2}{25}$, the numbers 1, 2 and 3 with probability $\frac{4}{25}$ and the number 0 with probability $\frac{1}{5}$. Then $\nu^*_P(i) = \frac{6}{25}$. Chains of length $p+1$ occur with probability $P^*(p+1)={\left(\frac{5}{8}\right)}^{p-1}\frac{2}{5}\cdot\frac{6}{25}$ provided $q\neq 1,p-1$. Similar non-uniformities of the distribution lead to $\nu^*_P(p+q)=\frac{2}{5}$ or $\nu^*_P(2p-q)=\frac{2}{5}$.
For a production rule derived from a primitive pentanomial with lags $p>q_1>q_2>q_3$ the actual initialization probability $\nu^*_P(p)=\frac{57}{125}$ and $\nu^*_P(i)=\frac{204}{625}$ for $i$ larger than $p$ and less than the minimum of $p+q_3$, $p+q_2-q_3$, $p+q_1-q_2$ and $2p-q_1$. The deviations are less than for trinomials.
Whenever a decision on step $p$ is made in the case of Algorithm $N$, the pseudo-random numbers satisfy $\tilde{r}(0) \leq 2$ and $\tilde{r}(i)>2$ for $0<i<p$. Assuming that $\tilde{r}(i)$ with $i<p$ occurs with equal probability for all possibilities consistent with a walk length larger than $p-1$, $\tilde{r}(p)$ equals 4, 5, 6 and 7 with probability $\frac{1}{5}$, 1, 2 and 3 with probability $\frac{1}{15}$ and 0 with probability 0 in the case of a primitive trinomial. So $\nu^*_N(p)=\frac{2}{15}$. The probability of walks of length $p$ thus equals $P^*(p)={\left(\frac{5}{8}\right)}^{p-1}\frac{2}{15}$. For $\tilde{r}(i)$ with $i$ obeying inequality (\[ineq\]) the values 4, 5, 6 and 7 occur with probability $\frac{2}{25}$, the values 1, 2 and 3 with probability $\frac{4}{25}$ and the value 0 with probability $\frac{1}{5}$. Thus $\nu^*_N(i)=\frac{13}{25}$. Chains of length $p+1$ occur with probability $P^*(p+1)={\left(\frac{5}{8}\right)}^{p-1}\frac{13}{15}\cdot\frac{13}{25}$, if $q\neq 1,p-1$. In a similar way $\nu^*_N(p+q)=\frac{37}{65}$ or $\nu^*_N(2p-q)=\frac{37}{65}$.
In the examples a resonance in $\nu^*(i)$ versus $i$ is found, when the value of $r(0)$ affects the value of $r(i)$. The first one encountered is a reflection of the production rule itself; for trinomials a relation between lags 0, $q$ and $p$. The next one is a four-point correlation resulting from the interference of the production rule and a shifted version of the production rule. The correlation between lags 0, $q$ and $p$ and that between lags $q$, $2q$ and $p+q$ leads to a correlation between lags 0, $2q$, $p$ and $p+q$. Similarly the correlation between lags 0, $q$ and $p$ and that between $p-q$, $p$ and $2p-q$ leads to a correlation between 0, $p-q$, $q$ and $2p-q$. The deviation of the initialization probability has a different sign for Algorithm $P$ and Algorithm $N$, if the new correlation was between an odd number of lags. It has the same sign for the four-point correlation.
If a walk is longer than $p$ in the case of Algorithm $P$ for a general real step probability $\mu > \frac{1}{2}$, then $r_P(0)\geq \mu$ and $r_P(i) < \mu$ for $0<i<p$. As a consequence of these inequalities, the production rule generates $r(p)$ with a non-uniform probability distribution. For a GFSR-rule derived from a trinomial $r(p)=r(0)\oplus r(p-q)$. The exclusive-or operation with a fixed operand is a permutation of the numbers smaller than 1. If both $r(0)$ and $r(p-q)$ are larger than $\mu$ the result is less than $\frac{1}{2}$. This implies that all possibilities for $r(p)> \mu$ are realized for the reduced set of numbers $r(p-q)<\mu$. So the probability that the walk ends, when it has visited $p$ sites, is not equal to $\nu=1-\mu$, but $\nu^*_P(p)=\frac{\nu}{\mu}$. By similar arguments for $i$ obeying (\[ineq\]) $\nu^*_P(i)=(2\mu-1)\nu / {\mu^2}$.
For Algorithm $N$ the number $r_N(0) <\nu$ and $r_N(i)\geq \nu$. If $\nu = 2^{-m}$ with $m$ a positive integer, then $r_N(0)$ has $m$ leading bits equal to zero, so $r_N(p)$ has the leading $m$ bits the same as $r_N(p-q)$ and $\nu^*_N(p)=0$. A similar result is found for all lengths for which a three-point correlation involving $r_N(0)$ is induced by the production rule. In particular in the case of a rule derived from a primitive trinomial this holds for all lengths $2^k p$ with $k$ a non-negative integer, because a decimation by 2 leads to a sequence generated by the same rule[@Gol]. For general $\nu$ a more complicated argument leads to $$\nu^*_N(p)=\frac{2(2^{n_b-l+1}-\nu)(\nu-2^{n_b-l})}{\nu\mu}$$ and for $i$ obeying (\[ineq\]) $$\nu^*_N(i)=\frac{\nu}{\mu}-
\frac{2(2^{n_b-l+1}-\nu)(\nu-2^{n_b-l})}{\nu\mu}$$ with $$n_b=\lceil \log_2(\nu 2^l +1) \rceil-1.$$ where $\lceil x \rceil$ is the smallest integer $\ge x$.
The deviation of the actual initialization probability $\nu^*$ from the ideal value $\nu$ causes the probability of a walk to visit $n$ sites $P^*(n)$ to deviate from the value for uncorrelated numbers (\[eq\_prob\]). We define $$\delta P(n)\equiv \frac{P^*(n)}{P(n)}-1,\;
P^*(n)=\prod_{i=1}^{n-1}(1-\nu^*(i))\; \nu^*(n).
\label{dev_prob}$$ For $i<p$ we expect no deviation, so $\delta P = 0$. Substitution in (\[dev\_prob\]) of the values found for $\nu^*$ leads to $\delta P_P(p)=\frac{\nu}{\mu}$ for Algorithm $P$. With $i$ obeying inequality (\[ineq\]) $$\delta P_P(i)=\frac{(2\mu-1)^2}{\mu^4}
\left(\frac{3\mu^2-3\mu+1}{\mu^3}\right)^{i-p-1}-1$$
To give an impression of the deviations for lengths beyond the first new low-order correlation we use numerical calculations. The results of computer simulations of $10^9$ walks are presented in Fig. \[fig\_P\] for the GFSR with $(p,q)=(89,38)$, Algorithm $P$ and $\mu=31/32$. We mention two reasons to choose these particular values of $p$ and $q$. First, simulations with the desired accuracy typically need more than $10^{12}$ random numbers. Therefore, the length $p$ of the shift register should be greater than $\log_2 10^{12} \approx 40$. Second, we do not want to choose $p$ much larger than necessary because the observability of the bias decreases with $p$.
The error bars were computed using $100$ bins of $10^7$ walks each. The predicted resonances are easily seen for the walk lengths $p$, $p+q$, $2p-q$, $3p-2q$, etc., which are linear combinations of the feedback positions in the production rule. The result for walk length $2p$ is not shown in the Figure because of its large deviation of $\delta P(2p)=0.109(3)$. The value of the calculated deviation $\delta P(p)=0.0328(7)$ is in good agreement with the value $\frac{\nu}{\mu}\approx0.0323$.
The results of the calculations with the same parameters using Algorithm $N$ are shown in Fig. \[fig\_N\]. The deviations at lengths equal to $2^kp$ ($k$=0,1,2...) are equal to $-1$ and are not shown in the Figure. The deviation $\delta P(p+1) = 0.0651(8)$ is in agreement with the value $(1-\mu^2)/{\mu^2}\approx0.0656$. A comparison of Fig. \[fig\_P\] and Fig. \[fig\_N\] shows, that the main resonances have different signs for the two algorithms. This is in qualitative agreement with the results of cluster simulations of the two-dimensional Ising model by Shchur and Blöte [@SB] who indeed found deviations in thermodynamical quantities with opposite signs for comparison strategies $P$ and $N$.
In order to give an example of the deviations of random walk statistics for a more complicated production rule we use the decimated sequence with ($p$,$q_1$,$q_2$,$q_3$)=(89,72,55,38), Algorithm $P$ and $\mu=\frac{31}{32}$ in a random walk simulation. Because the feedback positions for this sequence are equally spaced, the four numbers with lags 106, 38, 17 and 0 are correlated. This four-point correlation may be more important in applications than the five-point correlation implied by the production rule. This is the case for the numerical results of the random walk statistics (Fig. \[fig\_dec\]). The deviation at the shift-register length $\delta P(p)=1/{31^3}$ is small compared to the error bars. The deviations for the two four-point correlations at walk lengths 106 and 212 are of a similar magnitude as the deviations for four-point correlations for the trinomial (Fig. \[fig\_P\]). The number of four-point correlations is smaller than for the trinomial and no three-point correlations are found. Deviations tend to be weaker for higher order correlations. This agrees with results of Wolff simulations of Ising models[@SB], where the deviations were smaller for decimated sequences.
Correlation effects due to the lagged Fibonacci recipe. {#sec_fib}
=======================================================
For lagged Fibonacci generators the relation between the leading bits is less simple. The distribution of numbers is not completely uniform for lagged Fibonacci sequences. Deviations of uniformity have been analysed in [@Pinch]. We assume that all sequences $r(0)\ldots r(p-1)$ are equally probable. If the walk has a certain length the previous numbers $r(i)$ obey the appropriate inequality.
We first consider $\mu=\frac{1}{2}$ again. For Algorithm $P$, $r(0) \geq \frac{1}{2}$ and $r(i) < \frac{1}{2}$ for $0<i<p$, in the case of a decision on step $p$. These inequalities cause the production rule to generate $r(p)$ with a non-uniform probability distribution. The distribution can be calculated as the convolution of the distributions for the feedback lags. For numbers of infinite precision it has the symmetry property $P(r)=P(1-r)$ in the case of a rule derived from a primitive trinomial. The finite word length $l$ leads to corrections of order $2^{-l}$. We neglect these corrections because they are small compared to the statistical errors in our simulations.
The actual initialization probability conserves its ideal value $\nu^*_P(p) = \frac{1}{2}$ by the symmetry property. The same symmetry holds for the distribution of $r(i)$ with $i$ obeying inequality (\[ineq\]). Thus $\nu^*_P(i) = \frac{1}{2}$. For the next random number the probability distribution does no longer obey $P(r)=P(1-r)$. The actual initialization probability $\nu^*_P(p+q)=\frac{1}{3}$ or $\nu_P^*(2p-q)=\frac{1}{3}$. The same results are obtained for algorithm $N$.
As in the case of the GFSR rule it is possible to calculate the values of the actual initialization probability for arbitrary $\mu$. For a lagged Fibonacci rule derived from a trinomial the actual initialization probability becomes $\nu^*_P(p)=\frac{\nu}{2 \mu}$. For $i$ obeying inequality (\[ineq\]) the actual probability $\nu^*_P(i)=(3\mu-1)\nu / {2\mu^2}$. For strategy $N$ we get the same actual probabilities. The expressions for $\nu^*$ grow progressively more complicated for larger walk lengths. We therefore refrain from showing them.
Numerically calculated deviations of the resulting probability $P^*(n)$ of a walk of length $n$ are shown in Fig. \[fig\_F\] for the lagged Fibonacci rule with feedback positions $(p,q)=(89,38)$. Error bars were computed using the same parameters as in the previous Section. Because of the number of bits $l=30$ used in the random numbers the finite-word-length corrections are quite small. As those differences are small compared to the statistical errors, the results for Algorithms $P$ and $N$ are equal within error bars for all walk lengths, unlike in the case of shift registers. The value of the deviation at walk length $p$ (not shown in the Figure) is equal to $\delta P(p)=-0.4841(4)$ in good agreement with the value of $\frac{1}{2\mu}-1\approx -0.4839$. The deviations have the same order of magnitude for both recipes of random number generation we considered in this paper. This does not support the claim that the lagged Fibonacci method performs better than the GFSR-method[@Marsfib].
Conclusions {#sec_dis}
===========
Correlation between random numbers can influence the results of a Monte Carlo calculation of a simple random walk model. No deviations occur in the distribution of walk lengths shorter than the magnitude of the largest feedback position. The walk length statistics is affected for larger lengths. In some cases the difference in results using comparison strategy $N$ and $P$ gives an indication of the magnitude of the bias. The magnitude of the deviations tends to be larger for generators derived from primitive trinomials than for primitive pentanomials. Similar effects occur in the cluster-size distribution in a cluster simulation of the Ising model[@LNS].
The magnitude of the deviations $\nu - \nu^*$ depends only on the comparison strategy, and on the value $\mu $. It does not depend on the particular values of feedback positions. In the case of other feedback positions, for example R250 the Kirkpatrick-Stoll Random Number Generator, one should accordingly rescale the horizontal axes of Figures 2-4 by $250/89$. However, continuation of the walks to larger $n$ leads to increased scatter so that the effects become less prominent.
These results are relevant for the Wolff cluster simulation of spin models in more than one dimension. In such simulations, each spin in the cluster may have more than one neighbour that has to be included in the cluster. If so, the addresses of these neighbouring spins are temporarily stored, e.g. in a memory called ‘stack’( for a hardware implementation see [@cspc]). For each entry in the stack, it has to be checked whether further neighbours have to be added to the cluster, and thus whether further additions to the stack memory have to be made. Spin addresses that have thus been processed, are removed from the stack. The number of addresses in the stack is a fluctuating variable, and the cluster is completed when the stack is empty. Typically the stack memory contains more than one spin address, in which case one random number is not sufficient to end the cluster formation process. However, it is obvious that the completion of a cluster is strongly correlated with the values of the preceding pseudo-random numbers.
The bias in the random numbers preceding the construction of a new cluster will lead to a significant correlation between $n-1$ pseudo-random numbers in the case of an $n$-point production rule, although the correlation is weaker than the correlation between $n$ numbers imposed by the production rule. For a 3-point rule this 2-point correlation combined with the bias in the random numbers allowing a further growth of the cluster causes a bias for the $p$-th pseudo-random number used in the construction of a cluster. It is known[@SB] that the bias in the simulation results (for three-point production rules) becomes largest, when $p$ is equal to the average cluster size.
For higher $n$ the bias-producing mechanism is less simple and it seems plausible that the bias in the $p$-th pseudo-random number will be weaker. Thus one expects that the deviation in the statistics of cluster sizes will be stronger, when less pseudo-random numbers are involved in the production rule. Indeed for the case of shift-registers with a $n$-point production rule, the bias in the thermodynamics in a Wolff simulation increases strongly with decreasing $n$ [@SB], where rules with higher values of $n$ can be generated either by decimation or by combining the numbers generated by two or more rules with an exclusive-or.
We acknowledge productive discussions with A. Compagner, W. Selke, D. Stauffer and A.L. Talapov. L.N.S. is grateful to the Computational Physics Group at TU-Delft, where this work was initiated, for their kind hospitality. This work is partially supported by grants RFBR 93-02-2018, NWO 07-13-210, INTAS-93-211.
[10]{} H. Niederreiter, Random number generation and quasi-Monte Carlo methods, SIAM, Philadelphia, 1992. A. Compagner, Phys. Rev. E52 (1995) 5634. A. Compagner, J. Stat. Phys. 63 (1991) 883. A. Hoogland, J. Spaa, B. Selman and A. Compagner, J. Comp. Phys. 51 (1983) 250; A. Hoogland, A. Compagner and H.W.J. Blöte, [*The Delft Ising System Processor*]{}, Chapter 7 of [*Architecture and Performance of Specialized Computers*]{}, ed. B. Alder, in the series [*Computational Techniques*]{} (Academic, New York, 1988). F. Schmid and N.B. Wilding, Int. J. Mod. Phys. C6 (1995) 781 U. Wolff, Phys. Rev. Lett. 62 (1989) 361. A.M. Ferrenberg, D.P. Landau and Y.J. Wong, Phys. Rev. Lett. 69 (1992) 3382; W. Selke, A.L. Talapov and L.N. Shchur, JETP Lett. 58 (1993) 665; P.D. Coddington, Int. J. Mod. Phys. C5 (1994) 547; I. Vattulainen, T. Ala-Nissila and K. Kankaala, Phys. Rev. Lett. 73 (1994) 2513; I. Vattulainen, T. Ala-Nissila and K. Kankaala, Phys. Rev. E52 (1995) 3205, P.D. Coddington, Int. J. Mod. Phys. C7 (1996) 295. L.N. Shchur and H.W.J. Blöte, to be published. T.G. Lewis and W.H. Payne, J. Assoc. Comput. Mach. 20 (1973) 456-468 S. Kirkpatrick and E.P. Stoll, J. Comput. Phys. 40 (1981) 517. S.W. Golomb, Shift Register Sequences, (Holden-Day, Inc., San Francisco, 1967). J.R. Heringa, H.W.J. Blöte and A. Compagner, Int. J. Mod. Phys. C3 (1992) 561; M. Živković, Math. Comp. 62 (1994) 385. Z.-D. Dai, J. Cryptology 5 (1992) 193; M. Ward, Trans. Amer. Math. Soc. 35 (1933) 600. G. Marsaglia and I.-H. Tsay, Lin. Alg. Appl. 67 (1985) 147. R.G.E. Pinch, in: C. Boyd, ed., Proceedings 5th IMA Conference on Coding and Cryptography 1995, Springer Lecture Notes in Computer Science 1025 (1995) 188. L.N. Shchur (unpublished). A.L. Talapov, V.B. Andreichenko, Vl.S. Dotsenko and L.N. Shchur, Int. J. Mod. Phys. C4 (1993) 787.
=
=
=
=
=
|
---
abstract: 'We study a skew product with a curve of neutral points. We show that there exists a unique absolutely continuous invariant probability measure, and that the Birkhoff averages of a sufficiently smooth observable converge to a normal law or a stable law, depending on the average of the observable along the neutral curve.'
author:
- 'S. Gouëzel'
- 'Sébastien Gouëzel [^1]'
bibliography:
- 'biblio.bib'
date: November 2003
title: 'Statistical properties of a skew product with a curve of neutral points [^2]'
---
Introduction
============
Let $T:M\to M$ be a map on a compact space. While uniformly hyperbolic or uniformly expanding dynamics are well understood, problems arise when there are neutral fixed points (where the differential of $T$ has an eigenvalue equal to $1$). The one-dimensional case has been thoroughly studied, particularly when $T$ has only one neutral fixed point (see [@liverani_saussol_vaienti] and references therein). The normal form at the fixed point dictates the asymptotics of the dynamics, and in particular the speed of mixing, and the convergence of Birkhoff sums to limit laws ([@gouezel:stable]).
In this article, we study the same type of phenomenon, but in higher dimension. Contrary to [@hu:almost_hyperbolic], [@pollicott_yuri:indifferent] (where the case of isolated fixed points is considered), our models admit a whole invariant neutral curve. We show that the one-dimensional results remain essentially true.
More precisely, define a map $T_\alpha$ on $[0,1]$ by $$T_\alpha(x)=\left\{ \begin{array}{cl}
x(1+2^\alpha x^\alpha) &\text{if }0{\leqslant}x{\leqslant}1/2
\\
2x-1 &\text{if }1/2<x{\leqslant}1
\end{array}\right.$$ It has a neutral fixed point at $0$, behaving like $x(1+x^\alpha)$. To mix different such behaviors, we consider a skew product, similar to the Alves-Viana map ([@viana:multidim_attr]) but where the unimodal maps are replaced by $T_\alpha$. Let $\alpha:S^1 \to (0,1)$ be a map with minimum $\operatorname{{\alpha_{\text{min}}}}$ and maximum $\operatorname{{\alpha_{\text{max}}}}$. Assume that
1. $\alpha$ is $C^2$.
2. $0<\operatorname{{\alpha_{\text{min}}}}<\operatorname{{\alpha_{\text{max}}}}<1$.
3. $\alpha$ takes the value $\operatorname{{\alpha_{\text{min}}}}$ at a unique point $x_0$, with $\alpha''(x_0)>0$.
4. $\operatorname{{\alpha_{\text{max}}}}< \frac{3}{2}\operatorname{{\alpha_{\text{min}}}}$ (which implies $\operatorname{{\alpha_{\text{max}}}}<\operatorname{{\alpha_{\text{min}}}}+1/2$).
These conditions are for example satisfied by $\alpha(\omega)=\operatorname{{\alpha_{\text{min}}}}+{\varepsilon}(1+\sin(2\pi \omega))$ where $\operatorname{{\alpha_{\text{min}}}}\in (0,1)$ and ${\varepsilon}$ is small enough.
We define a map $T$ on $S^1 \times [0,1]$ by $$\label{definit_T}
T(\omega,x)= (F(\omega), T_{\alpha(\omega)}(x))$$ where $F(\omega)=4\omega$.
In the following, we will generalize to this skew product the one-dimensional results on the maps $T_\alpha$. First of all, in Section \[section\_invariante\], we prove that there exists a unique absolutely continuous invariant probability measure $m$, whose density $h$ is in fact Lipschitz on every compact subset of $S^1\times (0,1]$ (Theorem \[existence\_mesure\_invariante\]). In Section \[limit\_Markov\], we prove limit theorems for abstract Markov maps (using a method essentially due to [@melbourne_torok] and recalled in Appendix \[appendice:loi\_stable\], and estimates of [@aaronson_denker] and [@gouezel:stable]). Finally, in Sections \[section\_estimee\_Xn\] and \[section:limite\], we study the limit laws of Birkhoff sums for the skew product $T$, and we obtain the convergence to a normal law or a stable law, depending on the value of $\operatorname{{\alpha_{\text{min}}}}$. We obtain the following theorem (see Theorem \[enonce\_theoreme\_limite\] for more details).
Set $$A=\frac{1}{4\left( \operatorname{{\alpha_{\text{min}}}}^{3/2}
\sqrt{\frac{\pi}{2\alpha''(x_0)}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}}\int_{S^1\times \{1/2\}}
h\operatorname{dLeb},$$ where $h$ is the density of the absolutely continuous invariant probability measure.
Let $f$ be a Lipschitz function on $S^1\times [0,1]$, with $\int
f{\, {\rm d}}m=0$. Write $c=\int_{S^1\times\{0\}} f \operatorname{dLeb}$ and $S_n
f=\sum_{k=0}^{n-1} f\circ T^k$. Then
- If $\operatorname{{\alpha_{\text{min}}}}<1/2$, there exists $\sigma^2 {\geqslant}0$ such that $\frac{1}{\sqrt{n}} S_n f \to {\mathcal{N}}(0,\sigma^2)$.
- If $\operatorname{{\alpha_{\text{min}}}}=1/2$ and $c\not=0$, then $\frac{S_n f}{\sqrt{ \frac{c^2 A }{4} n (\ln n)^2}} \to
{\mathcal{N}}(0,1)$.
- If $1/2<\operatorname{{\alpha_{\text{min}}}}<1$ and $c\not =0$, then $\frac{S_n f}{n^{\operatorname{{\alpha_{\text{min}}}}}\sqrt{\operatorname{{\alpha_{\text{min}}}}\ln n}} \to Z$, where the random variable $Z$ has an explicit stable distribution.
- If $1/2{\leqslant}\operatorname{{\alpha_{\text{min}}}}<1$ and $c=0$, then there exists $\sigma^2 {\geqslant}0$ such that $\frac{1}{\sqrt{n}} S_n f \to
{\mathcal{N}}(0,\sigma^2)$.
An interesting feature of this example is that its study involves sophisticated mixing properties of $F$, particularly a multiple decorrelation property, proved in Appendix \[appendice:pene\] using [@pene:averaging].
Theorems of [@gouezel:stable] could be used instead of the method of [@melbourne_torok] to get the limit laws. However, this elementary method is interesting in its own right, and can be generalized more easily to other settings than the results of [@gouezel:stable] (in particular to the case of more neutral fixed points).
The specific form of $F$ is of no importance at all, the results remain true when $F$ is $C^2$ with $|F'| {\geqslant}4$ (for example $F(\omega)=d\omega$ with $d {\geqslant}4$). In the same way, the only important properties of the maps $T_\alpha$ are their normal form close to $0$ and the fact that they are Markov. Finally, the hypothesis $\alpha''(x_0)\not=0$ is only useful for limit theorems, and can be replaced by: $\exists m,
\alpha^{(m)}(x_0)\not=0$ (but the normalizing factors have to be modified accordingly). For the sake of simplicity, we will restrict ourselves in what follows to the aforementioned case.
In this article, $a(n) \sim b(n)$ means that $a(n)/b(n) \to 1$ when $n\to \infty$. The integral with respect to a probability measure will sometimes be denoted by $E(\cdot)$. Finally, $\lfloor
x \rfloor$ will denote the integer part of $x$.
Invariant measure {#section_invariante}
=================
An important property of the map $T$, that will be used thoroughly in what follows, is that it is Markov: there exists a partition of the space such that every element of this partition is mapped by $T$ on a union of elements of this partition. In fact, we will consider $T_Y$ (the induced map on $Y=S^1 \times (1/2,1]$), which is also Markov, and expanding, contrary to $T$. We will apply to $T_Y$ classical results on expanding Markov maps (also called *Gibbs-Markov* maps), which we recall in the next paragraph.
Markov maps and invariant measures
----------------------------------
Let $(Y,{\mathcal{B}},m_Y)$ be a standard probability space, endowed with a bounded metric $d$. A non-singular map $T_Y$ defined on $Y$ is said to be a *Markov map* if there exists a finite or countable partition $\alpha$ of $Y$ such that $\forall a\in
\alpha$, $m_Y(a)>0$, $T_Y(a)$ is a union (mod $0$) of sets of $\alpha$, and $T_Y:a \to T_Y(a)$ is invertible. In this case, $\alpha$ is a *Markov partition* for $T_Y$.
A Markov map $T_Y$ (with a Markov partition $\alpha$) is a *Gibbs-Markov map* ([@aaronson:book]) if
1. $T_Y$ has the big image property: $\inf_{a\in
\alpha} m_Y(T_Y(a))>0$.
2. \[enumere\_expansion\] There exists $\lambda>1$ such that $\forall a\in \alpha, \forall
x,y\in a, d(T_Yx, T_Y y){\geqslant}\lambda d(x,y)$.
3. \[enumere\_distortion\] Let $g$ be the inverse of the jacobian of $T_Y$, i.e. on a set $a\in \alpha$, $g(x)=\frac{{\, {\rm d}}m_Y}{{\, {\rm d}}\left(m_Y \circ
(T_Y)_{|a}\right)} (x)$. Then there exists $C>0$ such that for all $a\in \alpha$, for almost all $x,y\in a$, $$\left|1-\frac{g(x)}{g(y)} \right| {\leqslant}C d(T_Yx,T_Yy).$$
This definition is slightly more general than the definition of [@aaronson:book]: the distance $d=d_\tau$ considered there is given by $d_\tau(x,y)=\tau^{s(x,y)}$ where $\tau<1$ and $s(x,y)$ is the separation time of $x$ and $y$, i.e. $$\label{definit_separation}
s(x,y)=\inf\{ n\in {\mathbb{N}}{\ |\ }\nexists a\in \alpha, T^n x\in a, T^n y\in
a\}.$$
The proof of [@aaronson:book Theorem 4.7.4] still works in our context, and gives:
\[existe\_mesure\_invariante\] Let $T_Y$ be a transitive Gibbs-Markov map ($\forall a,b \in
\alpha, \exists n\in {\mathbb{N}}, m_Y(T_Y^n a \cap b)>0$) such that $\operatorname{Card}(\alpha_*)<\infty$, where $\alpha_*$ is the partition generated by the images $T_Y(a)$ for $a\in \alpha$. Then $T_Y$ is ergodic, and there exists a unique absolutely continuous (with respect to $m_Y$) invariant probability measure, denoted by $\mu_Y$.
Moreover, $\mu_Y=h m_Y$ where the density $h$ is bounded and bounded away from $0$, and Lipschitz on every set of $\alpha_*$.
Preliminary estimates
---------------------
To apply Theorem \[existe\_mesure\_invariante\], we will construct a Markov partition, and control the distortion of the inverse branches of $T_Y$.
We will write $T_\omega^n=T_{\alpha(F^{n-1}\omega)}
\circ\cdots\circ T_{\alpha(\omega)}$, whence $T^n(\omega,x)=(F^n\omega, T_\omega^n(x))$. Write also $d((\omega_1,x_1),(\omega_2,x_2))= |\omega_1-\omega_2|+|x_1-x_2|$. A point of $S^1 \times [0,1]$ will be denoted by ${\textbf{x}}=(\omega,x)$. Finally, set $\operatorname{{d_{\text{vert}}}}((\omega_1,x_1),(\omega_2,x_2))=|x_2-x_1|$.
Define $X_0(\omega)=1$, $X_1(\omega)=1/2$, and for $n{\geqslant}2$, $X_n(\omega)$ is the preimage in $[0,1/2]$ of $X_{n-1}(F\omega)$ by $T_{\alpha(\omega)}$. These $X_n$ will be useful in the construction of a Markov partition for $T$ (paragraph \[construit\_markov\]).
\[estime\_croissance\_grossiere\_Xn\] There exists $C>0$ such that $\forall n\in {\mathbb{N}}^*, \forall \omega\in
S^1$, $$\frac{1}{Cn^{1/\operatorname{{\alpha_{\text{min}}}}}} {\leqslant}X_n(\omega) {\leqslant}\frac{C}{n^{1/\operatorname{{\alpha_{\text{max}}}}}}.$$
Write $Z_1=1/2$ and $T(Z_{n+1})=Z_n$ where $T(x)=x(1+2^{\operatorname{{\alpha_{\text{max}}}}}
x^{\operatorname{{\alpha_{\text{min}}}}})$. We easily check inductively that $Z_n{\leqslant}X_n(\omega)$ for every $\omega$, since $T(x){\geqslant}T_{\alpha(\omega)}(x)$ for every $\omega$. It is thus sufficient to estimate $Z_n$ to get the minoration. As $T(x){\geqslant}x$, the sequence $Z_n$ is decreasing, and nonnegative, whence it tends to a fixed point of $T$, necessarily $0$.
We have $$\begin{aligned}
\frac{1}{Z_n^{\operatorname{{\alpha_{\text{min}}}}}}&
=\frac{1}{Z_{n+1}^{\operatorname{{\alpha_{\text{min}}}}}}\left(1+2^{\operatorname{{\alpha_{\text{max}}}}} Z_{n+1}^{\operatorname{{\alpha_{\text{min}}}}}\right)^{-\operatorname{{\alpha_{\text{min}}}}}
=\frac{1}{Z_{n+1}^{\operatorname{{\alpha_{\text{min}}}}}}\left(1-\operatorname{{\alpha_{\text{min}}}}2^{\operatorname{{\alpha_{\text{max}}}}} Z_{n+1}^{\operatorname{{\alpha_{\text{min}}}}}
+o(Z_{n+1}^{\operatorname{{\alpha_{\text{min}}}}})\right)
\\&
=\frac{1}{Z_{n+1}^{\operatorname{{\alpha_{\text{min}}}}}}-\operatorname{{\alpha_{\text{min}}}}2^{\operatorname{{\alpha_{\text{max}}}}}+o(1).
\end{aligned}$$ A summation gives $\frac{1}{Z_m^{\operatorname{{\alpha_{\text{min}}}}}}\sim m \operatorname{{\alpha_{\text{min}}}}2^{\operatorname{{\alpha_{\text{max}}}}}$, whence $Z_m \sim C/m^{1/\operatorname{{\alpha_{\text{min}}}}}$, which concludes the minoration.
The majoration is similar, using a sequence $Z'_n$ with $Z'_n {\geqslant}X_n(\omega)$.
*We fix once and for all a large enough constant $D$.* The following definition is analogous to a definition of Viana ([@viana:multidim_attr]).
Let $\psi: K\to [0,1]$, where $K$ is a subinterval of $S^1$. We say that the graph of $\psi$ is an admissible curve if $\psi$ is $C^1$ with $|\psi'|{\leqslant}D$.
Let $\psi$ be an admissible curve, defined on $K$ with $|K| <1/4$, and included in $K\times [0,1/2]$ or $K\times (1/2,1]$. Then the image of $\psi$ by $T$ is still an admissible curve.
Let $(u,v)$ be a tangent vector at $(\omega,x)$ with $|v|{\leqslant}D
|u|$, we have to check that its image $(u',v')$ by $DT(\omega,x)$ still satisfies $|v'|{\leqslant}D |u'|$.
Assume first that $x{\leqslant}1/2$, whence $u'=4u$ and $v'=(1+(2x)^{\alpha(\omega)} (\alpha(\omega)+1))v+ x
\ln(2x)\alpha'(\omega)(2x)^{\alpha(\omega)}u$. As $\alpha(\omega){\leqslant}\operatorname{{\alpha_{\text{max}}}}{\leqslant}1$, we get $|v'|{\leqslant}3|v|+C|u|$ for a constant $C$ (which depends only on ${\left\| \alpha' \right\|}_\infty$). Thus, $$\frac{|v'|}{|u'|} {\leqslant}\frac{3}{4} \frac{|v|}{|u|} +\frac{C}{4}.$$ This will give $|v'|/|u'| {\leqslant}D$ if $\frac{3}{4}D +\frac{C}{4}
{\leqslant}D$, which is true if $D$ is large enough.
Assume then that $x>1/2$. Then $u'=4u$ and $v'=2v$, and there is nothing to prove.
\[controle\_rapport\_horz\_vert\] Let $(\omega_1,x_1)$ and $(\omega_2,x_2)$ be two points in $S^1\times [0,1/2]$ with $|x_1-x_2|{\leqslant}D|\omega_1-\omega_2|$ and $|\omega_1-\omega_1|{\leqslant}\frac{1}{8}$. Then their images satisfy $|x'_1-x'_2|{\leqslant}D|\omega'_1-\omega'_2|$.
Use a segment between the two points: it is an admissible curve, whence its image is still admissible.
The Markov partition {#construit_markov}
--------------------
Set $Y=S^1 \times (1/2,1]$. For ${\textbf{x}}\in Y$, set ${\varphi}_Y({\textbf{x}})=\inf\{n>0 {\ |\ }T^n({\textbf{x}})\in Y\}$: this is the first return time to $Y$, everywhere finite. The map $T_Y({\textbf{x}}):=T^{{\varphi}_Y({\textbf{x}})}({\textbf{x}})$ is the map induced by $T$ on $Y$. We will show that $T_Y$ is a Gibbs-Markov map, by constructing an appropriate Markov partition.
If $I$ is an interval of $S^1$, we will abusively write $I\times
[X_{n+1},X_n]$ for $\{(\omega,x){\ |\ }\omega \in I, x\in
[X_{n+1}(\omega), X_n(\omega)]\}$.
Set $I_n(\omega)=[X_{n+1}(\omega),X_n(\omega)]$ (or $\{\omega\}
\times [X_{n+1}(\omega),X_n(\omega)]$, depending on the context). By definition of $X_n$, $T$ maps $\{\omega\}\times I_n(\omega)$ bijectively on $\{F\omega\} \times I_{n-1}(F\omega)$. Thus, the interval $I_n(\omega)$ returns to $[1/2,1]$ in exactly $n$ steps.
Let $Y_n(\omega)$ be the preimage in $[1/2,1]$ of $X_{n-1}(F\omega)$ under $T_{\alpha(\omega)}$. Thus, the interval $J_n(\omega)= [Y_{n+1}(\omega),Y_n(\omega)]$ returns to $[1/2,1]$ in $n$ steps.
*We fix once and for all $0<{{\varepsilon}_0}<\frac{1}{8}$, small enough so that $D{{\varepsilon}_0}$ is less than the length of every interval $I_1(\omega)$*. (This condition will be useful in distortion estimates).
*Let $q$ be large enough so that $\frac{1}{4^q}<{{\varepsilon}_0}$, and consider $A_{s,n}=\left[
\frac{s}{4^{q+n}},\frac{s+1}{4^{q+n}}\right] \times J_n$, for $n\in {\mathbb{N}}^*$ and $0{\leqslant}s{\leqslant}4^{q+n}-1$*: this set is mapped by $T^n$ on $\left[\frac{s}{4^q},\frac{s+1}{4^q}\right] \times
[1/2,1]$. Let $K_0,\ldots,K_{4^q-1}$ be the sets $\left[\frac{i}{4^q},\frac{i+1}{4^q}\right] \times [1/2,1]$. Then the map $T_Y$ is an isomorphism between each $A_{s,n}$ and some $K_i$. Consequently, the map $T_Y$ is Markov for the partition $\{A_{s,n}\}$, and it has the big image property.
To apply Theorem \[existe\_mesure\_invariante\], we need expansion (for in the definition of Gibbs-Markov maps) and distortion control (for ). The expansion is given by the next proposition, and the distortion is estimated in the following paragraph.
On the intervals $[X_3(\omega),X_1(\omega)]$, the derivative of $T_{\alpha(\omega)}$ is greater than $1$, whence greater than a constant $2>\lambda>1$, independent of $\omega$.
For $(\omega_1,x_1)$ and $(\omega_2,x_2)\in S^1\times [0,1]$, set $$\label{definit_d'}
d'((\omega_1,x_1),(\omega_2,x_2))=a|x_1-x_2|+|\omega_1-\omega_2|$$ where $a=\frac{1-\lambda/4}{D}$.
\[dilate\_markov\] On each $A_{s,n}$, the map $T^n$ is expanding by at least $\lambda$ for the distance $d'$.
For $n=1$ (the points return directly to $S^1\times [1/2,1]$), everything is linear, and the result is clear. Assume $n{\geqslant}2$.
Take $(\omega_1,x_1)$ and $(\omega_2,x_2)\in A_{s,n}$, with for example $x_2{\geqslant}x_1$. The points $(\omega_1,x_1)$ and $(\omega_2,x_1)$ return to $S^1\times [1/2,1]$ after at least $n$ iterations (by hypothesis for the first point, and the second point is under $(\omega_2,x_2)$). We can use Corollary \[controle\_rapport\_horz\_vert\] $n-1$ times, and get that in vertical distance, $\operatorname{{d_{\text{vert}}}}( T^n(\omega_1,x_1),T^n(\omega_2,x_1))
{\leqslant}D|F^n\omega_1-F^n\omega_2|$. In particular, $T_{\omega_2}^n(x_1) {\geqslant}T_{\omega_1}^n(x_1) - D {\varepsilon}_0 {\geqslant}1/2-D {\varepsilon}_0$. Thus, by definition of ${\varepsilon}_0$, $T^n(\omega_2,x_1) \in I_i(F^n \omega_2)$ for $i=0$ or $1$, whence $T^{n-1}(\omega_2,x_1) \in [X_3(F^{n-1} \omega_2),
X_1(F^{n-1}\omega_2)]$. Note that $T^{n-1}(\omega_2,x_2)$ belongs to the same interval (in fact, $T^{n-1}_{\omega_2}(x_2) \in
[X_2(F^{n-1}\omega_2), X_1(F^{n-1}\omega_2)]$). Moreover, the $T_\alpha$ are expanding, whence $\operatorname{{d_{\text{vert}}}}(T^{n-1}(\omega_2,x_1),
T^{n-1}(\omega_2,x_2)) {\geqslant}|x_1-x_2|$. We apply once more $T$, which expands at least by $\lambda$ on $[X_3(F^{n-1}\omega_2),
X_1(F^{n-1} \omega_2)]$ by definition of $\lambda$, and get $\operatorname{{d_{\text{vert}}}}(T^n(\omega_2,x_1),T^n(\omega_2,x_2)) {\geqslant}\lambda|x_1-x_2|$.
Finally, $$\begin{aligned}
d'(T^n(\omega_1,x_1),T^n(\omega_2,x_2))&
=a
\operatorname{{d_{\text{vert}}}}(T^n(\omega_1,x_1),T^n(\omega_2,x_2))+|F^n\omega_1-F^n\omega_2|
\\&
{\geqslant}a \operatorname{{d_{\text{vert}}}}(T^n(\omega_2,x_1),T^n(\omega_2,x_2))-a \operatorname{{d_{\text{vert}}}}(
T^n(\omega_1,x_1),T^n(\omega_2,x_1))
\\& \hphantom{=\ }+|F^n\omega_1-F^n\omega_2|
\\&
{\geqslant}a \lambda|x_1-x_2| -aD
|F^n\omega_1-F^n\omega_2|+|F^n\omega_1-F^n\omega_2|.
\end{aligned}$$ The proposition will be proved if $(1-aD)|F^n\omega_1-F^n\omega_2|
{\geqslant}\lambda |\omega_1-\omega_2|$. But $$(1-aD)|F^n\omega_1-F^n\omega_2|
=(1-aD)4^n |\omega_1-\omega_2|
{\geqslant}(1-aD)4 |\omega_1-\omega_2|
=\lambda |\omega_1-\omega_2|.$$
Distortion bounds
-----------------
\[lemme\_distortion\_deplac\_horz\] There exists a constant $E>0$ such that $$\begin{split}
\forall n>0, \forall \omega_1,\omega_2&\in S^1 \text{ with
}|\omega_1-\omega_2|{\leqslant}\frac{{{\varepsilon}_0}}{4^n}, \forall x_1 \in
J_n(\omega_1) \text{ with }T_{\omega_2}^{n-1} x_1{\leqslant}1/2,\\&
\left| \ln (T_{\omega_1}^n)'(x_1)-\ln (T_{\omega_2}^n)'(x_1)\right|
{\leqslant}E |F^n \omega_1 -F^n \omega_2|.
\end{split}$$
We use Corollary \[controle\_rapport\_horz\_vert\] $n-1$ times and get for $0{\leqslant}k{\leqslant}n$ that $|T_{\omega_1}^k x_1 -T_{\omega_2}^k
x_1|{\leqslant}D|F^k\omega_1-F^k\omega_2|$.
In particular, for $k=n$, $|T_{\omega_1}^n x_1|{\geqslant}1/2$, whence $|T_{\omega_2}^n x_1|{\geqslant}1/2-D{{\varepsilon}_0}$. Consequently, $T^n(\omega_2,x_1)\in I_i(F^n\omega_2)$ for some $i\in \{0,1\}$, by definition of ${{\varepsilon}_0}$. An inverse induction gives $T^k(\omega_2,x_1) \in I_{n-k+i}(F^k\omega_2)$.
For $x{\leqslant}1/2$ and $\omega\in S^1$, write $G(\omega,x)=\ln
T_{\alpha(\omega)}'(x)=
\ln\left(1+(\alpha(\omega)+1)(2x)^{\alpha(\omega)}\right)$. Then $$\frac{\partial G}{\partial x}(\omega,x)
=\frac{(\alpha(\omega)+1)\alpha(\omega)2^{\alpha(\omega)}
x^{\alpha(\omega)-1}}{1+(\alpha(\omega)+1)(2x)^{\alpha(\omega)}}
{\leqslant}C x^{\operatorname{{\alpha_{\text{min}}}}-1}$$ and $$\left|\frac{\partial G}{\partial \omega}(\omega,x)\right|
=\left|\frac{\alpha'(\omega)(2x)^{\alpha(\omega)}
+(\alpha(\omega)+1)\alpha'(\omega)\ln(2x)(2x)^{\alpha(\omega)}}
{1+(\alpha(\omega)+1)(2x)^{\alpha(\omega)}} \right|
{\leqslant}C.$$
Lemma \[estime\_croissance\_grossiere\_Xn\], and the fact that $T^k(\omega_1,x_1)\in I_{n-k}(F^k\omega_1)$ and $T^k(\omega_2,x_1)
\in I_{n-k+i}(F^k\omega_2)$ with $i{\leqslant}1$, give that the second coordinates of $T^k(\omega_1,x_1)$ and $T^k(\omega_2,x_1)$ are ${\geqslant}\frac{1}{C (n-k+1)^{1/\operatorname{{\alpha_{\text{min}}}}}}$. On the set of points $(\omega,x)$ with $x{\geqslant}\frac{1}{C (n-k+1)^{1/\operatorname{{\alpha_{\text{min}}}}}}$, the estimates on the partial derivatives of $G$ show that this function is $C (n-k+1)^{1/\operatorname{{\alpha_{\text{min}}}}-1}$-Lipschitz, whence $$\begin{aligned}
|G(T^k(\omega_1,x_1))-G(T^k(\omega_2,x_1))| &
{\leqslant}C (n-k+1)^{1/\operatorname{{\alpha_{\text{min}}}}-1} d((T^k(\omega_1,x_1),T^k(\omega_2,x_1))
\\&
{\leqslant}C (n-k+1)^{1/\operatorname{{\alpha_{\text{min}}}}-1} (1+D)|F^k \omega_1-F^k \omega_2|
\\&
{\leqslant}C (n-k+1)^{1/\operatorname{{\alpha_{\text{min}}}}-1} (1+D) 4^k |\omega_1-\omega_2|.
\end{aligned}$$ Finally, $$\begin{aligned}
\left| \ln (T_{\omega_1}^n)'(x_1)-\ln (T_{\omega_2}^n)'(x_1)\right|&
{\leqslant}\sum_{k=0}^{n-1}|G(T^k(\omega_1,x_1))-G(T^k(\omega_2,x_1))|
\\&
{\leqslant}C 4^n |\omega_1-\omega_2| \sum_{k=0}^{n-1} (n-k+1)^{1/\operatorname{{\alpha_{\text{min}}}}-1} 4^{k-n}
\\&
{\leqslant}C |F^n\omega_1-F^n\omega_2| \sum_{l=1}^\infty (l+1)^{1/\operatorname{{\alpha_{\text{min}}}}-1} 4^{-l}.
\end{aligned}$$ The last sum is finite, which concludes the proof.
For $n{\geqslant}2$, write $J_n^+(\omega)=[Y_{n+2}(\omega),Y_n(\omega)]$. Thus, if $n{\geqslant}1$, $J_{n+1}^+(\omega)$ is the preimage of $I_n^+(F\omega)$, defined by $I_n^+(F\omega)=[X_{n+2}(F\omega),X_{n}(F\omega)]$. These intervals will appear naturally in distortion controls, since we have seen in the proof of Lemma \[lemme\_distortion\_deplac\_horz\] that, if we move away horizontally from a point of $J_n(\omega_1)$, we find a point of $J_{n+i}(\omega_2)$ for $i\in
\{0,1\}$, i.e. in $J_n^+(\omega_2)$.
\[distortion\_verticale\_bornee\] There exists a constant $C$ such that $$\forall n{\geqslant}0, \forall \omega\in S^1, \forall x,y\in
J_n^+(\omega),\ \left|\ln (T_\omega^n)'(x)-\ln(T_\omega^n)'(y)
\right| {\leqslant}C |T_\omega^n(x)-T_\omega^n(y)|.$$
Recall that the Schwarzian derivative of an increasing diffeomorphism $g$ of class $C^3$ is $Sg(x)=\frac{g'''(x)}{g'(x)}-\frac{3}{2}\left(\frac{g''(x)}{g'(x)}
\right)^2$. The composition of two functions with nonpositive Schwarzian derivative still has a nonpositive Schwarzian derivative.
For $\tau>0$, the Koebe principle ([@demelo_vanstrien Theorem IV.1.2]) states that, if $Sg{\leqslant}0$, and $J\subset J'$ are two intervals such that $g(J')$ contains a $\tau$-scaled neighborhood of $g(J)$ (i.e. the intervals on the left and on the right of $g(J)$ in $g(J')$ have length at least $\tau |g(J)|$), then there exists a constant $K(\tau)$ such that $$\forall x,y\in J, \left|\ln g'(x) - \ln g'(y)\right| {\leqslant}K(\tau) \frac{|x-y|}{|J|}.$$ This implies that the distortion of $g$ is bounded on $J$, whence it is possible to replace the bound on the right with $K'(\tau)
\frac{|g(x)-g(y)|}{|g(J)|}$.
In our case, if $0<\alpha<1$, the left branch of $T_\alpha$ has nonpositive Schwarzian derivative, since $T_\alpha'''<0$ and $T_\alpha'>0$. Let in particular $g$ be the composition of the left branches of $T_{\alpha(F^{n-1}\omega)},\ldots,
T_{\alpha(F\omega)}$, and of the right branch of $T_{\alpha(\omega)}$. Then, on $J_n^+$, we have $T_\omega^n=g$, and $g$ has nonpositive Schwarzian derivative.
We want to see that $\left|\ln (T_\omega^n)'(x) - \ln
(T_\omega^n)'(y)\right|{\leqslant}C|T_\omega^n(x)-T_\omega^n(y)|$. For this, we apply the Koebe principle to $J=J_n^+$ and $J'=[1/2+\delta,2]$ for $\delta$ very small. Then $g(J)=[X_2,1]$ while $g(J')$ contains $[\delta',2]$ for $\delta'>0$, arbitrarily small if $\delta$ is small enough. As the $X_{2}$ are uniformly bounded away from $0$, there exists $\tau>0$ (independent of $\omega$ and $n$) such that $g(J')$ contains a $\tau$-scaled neighborhood of $g(J)$. The Koebe principle then gives the desired result.
\[prop\_distortion\_bornee\] There exists a constant $C$ such that, for every $A_{s,n}$, for every $(\omega_1,x_1)$ and $(\omega_2,x_2)\in A_{s,n}$, $$\left| \frac{\det DT^n(\omega_1,x_1)}{\det DT^n(\omega_2,x_2)}-1
\right|{\leqslant}C d(T^n(\omega_1,x_1),T^n(\omega_2,x_2)).$$
The matrix $DT^n(\omega,x)$ is upper triangular, with $4^n$ in the upper left corner. Thus, we have to show that $$\left| \ln(T_{\omega_1}^n)'(x_1)-\ln(T^n_{\omega_2})'(x_2)
\right|{\leqslant}C d(T^n(\omega_1,x_1),T^n(\omega_2,x_2)).$$ Assume for example $x_2{\geqslant}x_1$, which implies that $T_{\omega_2}^k(x_1){\leqslant}1/2$ for $k=0,\ldots,n-1$. Lemma \[lemme\_distortion\_deplac\_horz\] can be applied to $x_1$, $\omega_1$ and $\omega_2$, and gives in particular that $x_1\in
J_n^+(\omega_2)$.
Write $$\begin{aligned}
\left| \ln(T_{\omega_2}^n)'(x_2)-\ln(T^n_{\omega_1})'(x_1)
\right|
&{\leqslant}\left| \ln(T_{\omega_2}^n)'(x_2)-\ln(T^n_{\omega_2})'(x_1)
\right|
+ \left| \ln(T_{\omega_2}^n)'(x_1)-\ln(T^n_{\omega_1})'(x_1)
\right|
\\&
{\leqslant}C
d(T^n(\omega_2,x_2)),T^n(\omega_2,x_1))
+ E|F^n\omega_2-F^n \omega_1|
\end{aligned}$$ by Lemmas \[lemme\_distortion\_deplac\_horz\] and \[distortion\_verticale\_bornee\]. For the first term, $$\begin{aligned}
d(T^n(\omega_2,x_2),T^n(\omega_2,x_1))
&{\leqslant}d(T^n(\omega_2,x_2),T^n(\omega_1,x_1))
+d(T^n(\omega_1,x_1),T^n(\omega_2,x_1))
\\&
{\leqslant}d(T^n(\omega_2,x_2),T^n(\omega_1,x_1))
+ (D+1) |F^n \omega_1-F^n \omega_2|
\end{aligned}$$ using admissible curves.
As $|F^n \omega_1-F^n \omega_2|{\leqslant}d(T^n(\omega_1,x_1),T^n(\omega_2,x_2))$, we get the conclusion.
Construction of the invariant measure
-------------------------------------
The previous estimates and Theorem \[existe\_mesure\_invariante\] easily give that $T_Y$ admits an invariant measure, with Lipschitz density. Inducing gives an invariant measure for $T$, whose density is Lipschitz on each set $S^1\times (X_{n+1},X_n)$. However, this does not exclude discontinuities on $S^1\times X_n$, which is not surprising since $T$ itself has a discontinuity on $S^1 \times \{1/2\}$, which will then propagate to the other $X_n$, since the measure is invariant.
However, in the one-dimensional case, Liverani, Saussol and Vaienti ([@liverani_saussol_vaienti]) have proved that the density is really continuous everywhere, since they constructed it as an element of a cone of continuous functions. This fact remains true here:
\[existence\_mesure\_invariante\] The map $T$ admits a unique absolutely continuous invariant probability measure ${\, {\rm d}}m$. Moreover, this measure is ergodic. Finally, the density $h=\frac{{\, {\rm d}}m}{\operatorname{dLeb}}$ is Lipschitz on every compact subset of $S^1 \times (0,1]$.
Consider the map $T_Y$ induced by $T$ on $Y=S^1\times (1/2,1]$. It is Markov for the partition $\alpha=\{A_{s,n}\}$, and transitive for this partition since $T_Y^2(a)=Y$ for all $a\in \alpha$. Moreover, it is expanding for $d'$ on each set of the partition (Proposition \[dilate\_markov\]) and its distortion is Lipschitz (Proposition \[prop\_distortion\_bornee\], and $d$ equivalent to $d'$).
Theorem \[existe\_mesure\_invariante\] shows that $T_Y$ admits a unique absolutely continuous invariant probability measure ${\, {\rm d}}m_Y=h \operatorname{dLeb}$, which is ergodic. Moreover, the density $h$ is Lipschitz (for the distance $d'$, whence for the usual one) on each element of the partition $\alpha_*$ generated by the sets $T_Y(a)$, i.e. on the sets $K_i$.
To construct an invariant measure for the initial map $T$, we use the classical induction process ([@aaronson:book Section 1.1.5]): let ${\varphi}_Y$ be the return time to $Y$ under $T$, then $\mu=\sum_{n=0}^\infty T_*^n(m_Y | {\varphi}_Y>n)$ is invariant. To check that the new measure has finite mass, we have to see that $\sum m_Y({\varphi}_Y>n) <\infty$. As ${\, {\rm d}}m_Y$ and $\operatorname{dLeb}$ are equivalent, we check it for $\operatorname{dLeb}$. We have $$\operatorname{Leb}({\varphi}_Y>n)=\operatorname{Leb}(S^1 \times [1/2,Y_{n+1}])=\frac{1}{2}
\operatorname{Leb}(S^1\times [0,X_n])
{\leqslant}\frac{1}{2} \frac{C}{n^{1/\operatorname{{\alpha_{\text{max}}}}}},$$ using Lemma \[estime\_croissance\_grossiere\_Xn\]. As $\operatorname{{\alpha_{\text{max}}}}<1$, this is summable.
We know that $h$ is Lipschitz on the sets $[\frac{s}{4^q},\frac{s+1}{4^q}]\times [1/2,1]$, we have to prove the continuity on $\{s/4^q\}\times [1/2,1]$, which is not hard: these numbers $s/4^q$ are artificial, since they depend on the arbitrary choice of a Markov partition on $S^1$. We can do the same construction using other sets than the $A_{s,n}$. For example, set $A'_{s,n}=\left[ \frac{1}{3}+\frac{s}{4^{q+n}},
\frac{1}{3}+\frac{s+1}{4^{q+n}}\right]\times J_n$, and $K'_i=\left[\frac{1}{3}+\frac{i}{4^q},
\frac{1}{3}+\frac{i+1}{4^q}\right]$. Since $1/3$ is a fixed point of $F$, the map $T_Y$ is Markov for the partition $\{A'_{s,n}\}$, and each of these sets is mapped on a set $K'_i$. Thus, the same arguments as above apply, and prove that $h$ is Lipschitz on each set $K'_i$. Since the boundaries of the sets $K_i$ and $K'_i$ are different, this shows that $h$ is in fact Lipschitz on $S^1\times
[1/2,1]$.
We show now that $h$ is Lipschitz on $S^1 \times [X_2,1]$. Note that it is slightly incorrect to say that $h$ is Lipschitz, since $h$ is defined only almost everywhere. Nevertheless, if we prove that $|h({\textbf{x}})-h({\textbf{y}})|{\leqslant}Cd({\textbf{x}},{\textbf{y}})$ for almost all ${\textbf{x}}$ and ${\textbf{y}}$, then there will exist a unique version of $h$ which is really Lipschitz. Thus, all the equalities we will write until the end of this proof will be true only almost everywhere.
Let $A_{s,n}^+ = \left[\frac{s}{4^{q+n}},
\frac{s+1}{4^{q+n}}\right]\times J_n^+$: $T^n$ is a diffeomorphism between $A_{s,n}^+$ and $K_i^+ =
\left[\frac{i}{4^q},\frac{i+1}{4^q}\right]\times [X_2,1]$. We fix some $K^+=K_i^+=I\times [X_2,1]$, and we show that $h$ is Lipschitz on $K^+$. Let $U_1,U_2,\ldots$ be the inverse branches of $T^{n_1},T^{n_2},\ldots$ whose images all coincide with $K^+$. Let $T_Y$ be the map induced by $T$ on $Y=S^1\times [1/2,1]$. Then $h\operatorname{dLeb}_{|Y}$ is invariant under $T_Y$, which means that, for each ${\textbf{x}}\in I\times [1/2,1]$, $$h({\textbf{x}})=\sum JU_j({\textbf{x}}) h(U_j {\textbf{x}})$$ where $JU_j$ is the jacobian of $U_j$.
Let $Z=S^1\times [X_2,1]$, and $T_Z$ be the map induced by $T$ on $Z$. Since $h\operatorname{dLeb}_{|Z}$ is also invariant under $T_Z$, we have the same kind of equation as above. For ${\textbf{x}}\in I\times
[X_2,1/2]$, all its preimages under $T_Z$ are in $S^1 \times
[1/2,1]$, and the invariance gives that $$h({\textbf{x}})=\sum JU_j({\textbf{x}}) h(U_j {\textbf{x}}).$$
We have shown that, for every ${\textbf{x}}\in S^1 \times [X_2,1]$, $$h({\textbf{x}})=\sum JU_j({\textbf{x}}) h(U_j {\textbf{x}}).$$ This means that $h$ is invariant under some kind of transfer operator, even though it is not a real transfer operator since the images of the maps $U_j$ are not disjoint, and since they do not cover the space. In particular, the images of the $U_j$ are included in $S^1 \times [1/2,1]$, and we already know that $h$ is Lipschitz on this set.
The bounds of the previous paragraphs still apply to the distortion of the $U_j$, and their expansion. In particular, $\left|1-\frac{JU_j({\textbf{y}})}{JU_j({\textbf{x}})}\right|{\leqslant}Cd({\textbf{x}},{\textbf{y}})$ for a constant $C$ independent of $j$, and $|h(U_j {\textbf{x}})-h(U_j {\textbf{y}})| {\leqslant}C d(U_j {\textbf{x}},U_j {\textbf{y}}){\leqslant}D d({\textbf{x}},{\textbf{y}})$ (since $h$ is Lipschitz on the image of $U_j$). Thus, $$\begin{aligned}
|h({\textbf{x}})-h({\textbf{y}})|&
{\leqslant}\sum |JU_j({\textbf{x}})h(U_j {\textbf{x}})-JU_j({\textbf{y}})h(U_j {\textbf{y}})|
\\&
{\leqslant}\sum |JU_j({\textbf{x}})| \left|1-\frac{JU_j({\textbf{y}})}{JU_j({\textbf{x}})}\right|
|h(U_j
{\textbf{x}})| +\sum |JU_j({\textbf{y}})| |h(U_j {\textbf{x}})-h(U_j {\textbf{y}})|
\\&
{\leqslant}Cd({\textbf{x}},{\textbf{y}})\sum |JU_j({\textbf{x}})| + Dd({\textbf{x}},{\textbf{y}}) \sum |JU_j({\textbf{y}})|.
\end{aligned}$$ It remains to prove that $\sum |JU_j({\textbf{x}})|$ is bounded. The bound on distortion gives $JU_j({\textbf{x}}) \asymp \operatorname{Leb}(\operatorname{Im}U_j)$, whence $\sum JU_j({\textbf{x}}){\leqslant}C\sum \operatorname{Leb}(\operatorname{Im}U_j)$, which is finite since every point of $I\times[1/2,1]$ is in the image of at most two maps $U_j$.
We have proved that $h$ is Lipschitz on $S^1\times [X_2,1]$, except maybe on $\{\frac{s}{4^q}\}\times [X_2,1]$. As above, using another Markov partition, we exclude the possibility of discontinuities there. Thus, $h$ is Lipschitz on $S^1 \times
[X_2,1]$.
To prove that $h$ is Lipschitz on $S^1\times [X_k,1]$, we do exactly the same thing, except that we consider $[Y_{n+k},Y_n]$ instead of $J_n^+=[Y_{n+2},Y_n]$. As above, writing $U_1,U_2,\ldots$ for the inverse branches of $T^n$ defined on a set $[\frac{s}{4^{n+q}},\frac{s+1}{4^{n+q}}]\times [Y_{n+k},Y_n]$ and whose image is $K'=[\frac{i}{4^q},\frac{i+1}{4^q}] \times
[X_k,1]=I\times [X_k,1]$, we show that $h({\textbf{x}})=\sum JU_j({\textbf{x}})
h(U_j{\textbf{x}})$ for ${\textbf{x}}\in K'$. In fact, for ${\textbf{x}}\in I\times
[X_{l},X_{l-1}]$, we use the invariance of $h\operatorname{dLeb}$ under the map induced by $T$ on $S^1 \times [X_l,1]$. We conclude finally as above, using the fact that $h$ is Lipschitz on $S^1\times
[1/2,1]$, which contains the images of the $U_j$.
This concludes the proof, since every compact subset of $S^1\times(0,1]$ is contained in $S^1 \times [X_k,1]$ for large enough $k$.
Limit theorems for Markov maps {#limit_Markov}
==============================
We want to establish limit theorems for Birkhoff sums, of the form $\sum_{k=0}^{n-1} f(T^k x)$. We give in this section an abstract result, valid for a map that induces a Gibbs-Markov map on a subset of the space (which is the case of our skew product). Related limit theorems have been proved in [@gouezel:stable], but we will show here a slightly different result, which requires more control on the return time ${\varphi}$ but is more elementary, using Theorem \[thm\_probabiliste\_general\] proved in Appendix \[appendice:loi\_stable\] and inspired by results of Melbourne and Török ([@melbourne_torok]) for flows. An advantage of this new method is that, contrary to [@gouezel:stable], it can easily be extended to stable laws of index $1$.
If $Z_0,\ldots,Z_{n-1},\ldots$ are independent identically distributed random variables with zero mean, the sums $\frac{1}{B_n} \sum_{k=0}^{n-1} Z_k$ (where $B_n$ is a real sequence) converge to a nontrivial limit in essentially three cases: if $Z_k \in L^2$, there is convergence to a normal law for $B_n=\sqrt{n}$. There is also convergence to a normal law, but with a different normalization, if $P(|Z_k|>x)=x^{-2}l(x)$ with $L(x):=2\int_1^x \frac{l(u)}{u}{\, {\rm d}}u$ unbounded and slowly varying (i.e. $L:(0,\infty) \to (0,\infty)$ satisfies $\forall a
>0, \lim_{x\to \infty} L(ax)/L(x)=1$) – this is in particular true when $l$ itself is slowly varying. Finally, if $P(Z_k>x)=(c_1+o(1))x^{-p}L(x)$ and $P(Z_k<-x)=(c_2+o(1))x^{-p}L(x)$, where $L$ is slowly varying and $p\in (0,2)$, we have convergence (for a good choice of $B_n$) to a limit law called stable law. Moreover, these are the only cases where there is a convergence ([@feller:2]).
In the dynamical setting, we will prove the same kind of limit theorems, still with three possible cases: $L^2$, normal nonstandard, and stable. The normalizations will moreover be the same as in the probabilistic setting.
\[thm\_abstrait\_markov\] Let $T:X\to X$ be an ergodic transformation preserving a probability measure $m$. Assume that there exists a subset $Y$ of $X$ with $m(Y)>0$ such that the first return map $T_Y(x)=T^{{\varphi}(x)}(x)$ (where ${\varphi}(x)= \inf\{ n>0 {\ |\ }T^n(x) \in
Y\}$) is Gibbs-Markov for $m_{|Y}$, a partition $\alpha$ of $Y$ such that ${\varphi}$ is constant on each element of $\alpha$, and a distance $d$ on $Y$.
Let $f:X \to {\mathbb{R}}$ be an integrable map with $\int f=0$, such that $f_Y(y):=\sum_{n=0}^{{\varphi}(y)-1} f(T^n y)$ satisfies $$\label{condition_markov_abstrait}
\sum_{a\in \alpha} m(a) D f_Y(a) <\infty$$ where $$D f_Y(a)=\inf\{ C >0 {\ |\ }\forall x,y\in a, |f_Y(x)-f_Y(y)| {\leqslant}C
d(x,y)\}.$$
Set $M(y)=\max_{1{\leqslant}k {\leqslant}{\varphi}(y)} \left| \sum_{j=0}^{k-1}
f\circ T^j(y) \right|$.
Then:
- Assume that $f_Y \in L^2$ and $M \in L^2$. Assume moreover that ${\varphi}$ satisfies one of the following hypotheses:
- ${\varphi}\in L^2$.
- $m({\varphi}> x)=x^{-p} L(x)$ where $L$ is slowly varying and $p \in (1,2]$.
Then there exists $\sigma^2 {\geqslant}0$ such that $\frac{1}{\sqrt{n}}
S_n f \to {\mathcal{N}}(0,\sigma^2)$.
- Assume that $m(|f_Y| > x)=x^{-2} l(x)$, with $L(x):=2\int_1^x\frac{l(u)}{u}{\, {\rm d}}u$ unbounded and slowly varying. Assume moreover that $m(M
> x) {\leqslant}C x^{-2} l(x)$, and $m({\varphi}>x)=(c+o(1)) x^{-2}l(x)$. Let $B_n\to \infty$ satisfy $n L(B_n)=B_n^2$. Then $\frac{1}{B_n} S_n f \to {\mathcal{N}}(0,1)$.
- Assume that $m(f_Y>x)=(c_1+o(1))x^{-p}L(x)$ and $m(f_Y<-x)=(c_2+o(1))x^{-p} L(x)$ where $L$ is a slowly varying function, $p\in (1,2)$, and $c_1,c_2{\geqslant}0$ with $c_1+c_2>0$. Assume also that $m(M>x) {\leqslant}C x^{-p} L(x)$, and $m({\varphi}>x)=(c_3+o(1)) x^{-p}L(x)$. Let $B_n\to \infty$ satisfy $n
L(B_n)=B_n^p$. Then $\frac{1}{B_n} S_n f \to Z$ where the random variable $Z$ has a characteristic function given by $$E(e^{itZ})=e^{-c|t|^p \left( 1-i\beta \operatorname{sgn}(t) \tan
\left(\frac{p\pi}{2} \right) \right)}$$ with $c=(c_1+c_2) \Gamma(1-p) \cos \left(\frac{p\pi}{2} \right)$ and $\beta=\frac{c_1-c_2}{c_1+c_2}$.
Note that $M(y){\leqslant}\sum_{j=0}^{{\varphi}(y)-1} |f(T^j y)|=|f|_Y(y)$. Thus, if the integrability hypotheses of the theorem are satisfied by $|f|_Y$ (which will often be the case), they are automatically satisfied by $M$.
In the second case of the theorem, when $l$ itself is slowly varying, then $L$ is automatically slowly varying.
The second case of the theorem is not the most general possible result, since one may have convergence to a normal law even when the function $l$ is not slowly varying (what really matters is that $L$ is slowly varying). The theorem can be extended without problem to this more general setting, but the result becomes more complicated to state. In the applications, the statement given in Theorem \[thm\_abstrait\_markov\] will be sufficient.
The idea is to use Theorem \[thm\_probabiliste\_general\]: we have to check all its hypotheses. We will use the notations of this theorem, and in particular write $E_Y(u)=\frac{\int_Y u {\, {\rm d}}m}{m(Y)}$.
We first treat the third case (stable law), using the results of [@aaronson_denker] (and the generalizations of [@gouezel:stable]). Let $s(x,y)$ be the separation time of $x$ and $y$ defined in , $\tau=1/\lambda$ and $d_\tau=\tau^s$ the corresponding metric. Since every iteration of $T_Y$ expands by at least $\lambda$, we get $d(x,y){\leqslant}C d_\tau(x,y)$. In particular, we can assume without loss of generality that $d=d_\tau$, which is the setting of [@aaronson_denker] and [@gouezel:stable].
Let $P$ be the transfer operator associated to $T_Y$ (i.e.defined by $\int u\cdot v\circ T_Y = \int P(u)\cdot v$), and $P_t(u)=P(e^{itf_Y} u)$. Let ${\mathcal{L}}$ be the space of bounded Lipschitz functions (i.e. such that there exists $C$ such that, $\forall a\in \alpha, \forall x,y\in a, |g(x)-g(y)|{\leqslant}C
d(x,y)$). Theorem 5.1 of [@aaronson_denker] ensures that, for small enough $t$, $P_t$ acting on ${\mathcal{L}}$ has an eigenvalue $\lambda(t)=e^{-\frac{c}{m(Y)}|t|^p \left( 1-i\beta \operatorname{sgn}(t) \tan
\left(\frac{p\pi}{2} \right) \right)L(|t|^{-1})(1+o(1))}$, the remaining part of its spectrum being contained in a disk of radius ${\leqslant}1-\delta<1$. In fact, this theorem requires that $Df_Y(a)$ is bounded, but [@gouezel:stable Theorem 3.8] shows that it remains true under the weaker assumption $\sum m(a)D
f_Y(a)<\infty$.
The slow variation of $L$ easily implies that $\lambda\left(\frac{t}{B_n} \right)^{\lfloor n m(Y) \rfloor} \to
e^{-c|t|^p \left( 1-i\beta \operatorname{sgn}(t) \tan \left(\frac{p\pi}{2}
\right) \right)}$, whence, for $g\in {\mathcal{L}}$, $$\label{converge_dans_L}
E_Y\left(g e^{i\frac{t}{B_n}
S^Y_{\lfloor n m(Y) \rfloor} f_Y}\right) \to E_Y(g)E(e^{itZ})$$ where the random variable $Z$ is as in the statement of the theorem (see [@aaronson_denker] or [@gouezel:stable] for more details). We can not apply this result to $g={\varphi}$, since ${\varphi}$ is not bounded. However, ${\varphi}$ is Lipschitz and integrable, whence $P{\varphi}\in {\mathcal{L}}$ ([@aaronson_denker Proposition 1.4]). Equation applied to $P{\varphi}$ gives $E_Y\left({\varphi}e^{i\frac{t}{B_n} S^Y_{\lfloor n
m(Y) \rfloor} f_Y\circ T_Y }\right) \to E(e^{itZ})$, since $E_Y(P{\varphi})=E_Y({\varphi})=1$ by Kac’s Formula. Let $k(n)$ be a sequence such that $\lfloor k(n)m(Y) \rfloor = \lfloor n m(Y) \rfloor
-1$. Since $k(n)\sim n$, the same arguments give in fact that $E_Y\left({\varphi}e^{i\frac{t}{B_n} S^Y_{\lfloor k(n)
m(Y) \rfloor} f_Y\circ T_Y }\right) \to E(e^{itZ})$, i.e. $E_Y\left({\varphi}e^{i\frac{t}{B_n} (S^Y_{\lfloor n
m(Y) \rfloor} f_Y - f_Y)}\right) \to E(e^{itZ})$. The difference between this term and $E_Y\left({\varphi}e^{i\frac{t}{B_n} (S^Y_{\lfloor n
m(Y) \rfloor} f_Y }\right)$ is bounded by $E_Y\left(
{\varphi}\left|e^{-i\frac{t}{B_n}
f_Y}-1\right| \right)$, which tends to $0$ by dominated convergence. Thus, $$\label{eq_presque_bonne}
E_Y\left({\varphi}e^{i\frac{t}{B_n}
S^Y_{\lfloor n m(Y) \rfloor} f_Y} \right)
\to E(e^{itZ}).$$ This is . Finally, since $L$ is slowly varying, the equation $n L(B_n)=B_n^p$ implies that $\sup_{r{\leqslant}2n} \frac{B_r}{B_n}<\infty$, $\inf_{r{\geqslant}n} \frac{B_r}{B_n}>0$ (using for example [@feller:2 Corollary page 274]).
Let ${\varepsilon}>0$, we bound $m(M {\geqslant}{\varepsilon}B_n)$. $$m(M {\geqslant}{\varepsilon}B_n)
{\leqslant}C ({\varepsilon}B_n)^{-p} L( {\varepsilon}B_n)
=C {\varepsilon}^{-p} B_n^{-p} L(B_n)
\frac{L({\varepsilon}B_n)}{L(B_n)}.$$ But $B_n^{-p}L(B_n)=\frac{1}{n}$ by definition of $B_n$, and $\frac{L({\varepsilon}B_n)}{L(B_n)}$ tends to $1$ since $L$ is slowly varying. Thus, $m(M {\geqslant}{\varepsilon}B_n) {\leqslant}\frac{D}{n}$, which proves .
Hypothesis \[hypothese\_3\] of Theorem \[thm\_probabiliste\_general\] is satisfied for $b=1$, according to the Birkhoff Theorem applied to ${\varphi}-E_Y({\varphi})$ (and because $T_Y$ is ergodic, which is a consequence of the ergodicity of $T$). Finally, the hypothesis on the distribution of ${\varphi}$ ensures, once again by [@aaronson_denker], that $\frac{S_{\lfloor
nm(Y)\rfloor}^Y
{\varphi}-n m(Y)E_Y({\varphi})}{B_n}$ converges in distribution. Thus, is satisfied. We can use Theorem \[thm\_probabiliste\_general\], and get that $\frac{S_n f}{B_n}
\to Z$.
The proof of the second case of Theorem \[thm\_abstrait\_markov\] is exactly the same, using [@aaronson_denker:central] instead of [@aaronson_denker] to show the convergence in distribution of $\frac{S_{\lfloor nm(Y)\rfloor} ^Y f_Y}{B_n}$ and $\frac{S_{\lfloor
nm(Y) \rfloor} ^Y
{\varphi}-n m(Y) E_Y({\varphi})}{B_n}$.
In the first case ($f_Y\in L^2$), the proof is again identical when ${\varphi}\in L^2$, with $B_n=\sqrt{n}$, using [@guivarch-hardy] (or the remarks of [@aaronson_denker:central]). However, when $m({\varphi}>x)=x^{-p}
L(x)$, we check in a different way the hypotheses \[hypothese\_3\] and \[hypothese\_4\] of Theorem \[thm\_probabiliste\_general\]. [@aaronson_denker] ensures that, if $B'_n$ is given by $$\label{definit_bn}
nL(B'_n)=(B'_n)^p,$$ then $\frac{S_n^Y {\varphi}-n E_Y({\varphi})}{B'_n}$ converges in distribution. Moreover, [@gouezel:stable Lemma 3.4] proves that $Pf_Y \in {\mathcal{L}}$, and has a vanishing integral. As $P$ has a spectral gap on ${\mathcal{L}}$, $P^n f_Y \to 0$ exponentially fast. In particular, $\int f_Y\circ T_Y^n \cdot f_Y=\int (P^n f)\cdot
f=O((1-\delta)^n)$ for some $0<\delta<1$. Thus, as $f_Y\in L^2$, [@vitesse_birkhoff Theorem 16] gives that, for every $b>1/2$, $\frac{1}{N^b}\sum_{k=0}^{N-1}f_Y(T_Y^k) \to 0$ almost everywhere when $N\to \infty$. In the natural extension, $\int f_Y\circ
T_Y^{-n} \cdot f_Y=\int f_Y \cdot f_Y\circ T_Y^n$ decays also exponentially fast, whence the same argument gives that $\frac{1}{|N|^b}\sum_{k=0}^{N-1}f_Y(T_Y^k) \to 0$ when $N\to
-\infty$. Thus, Hypothesis \[hypothese\_3\] of Theorem \[thm\_probabiliste\_general\] is satisfied for any $b>1/2$. Let $\kappa>0$ be very small. As $L$ is slowly varying, $L(B'_n)=O((B'_n)^\kappa)$, whence Equation gives $B'_n=O(n^{1/(p-\kappa)})$. Thus, if $b<\frac{p}{2}$, we have $B'_n=O(B_n^{1/b})$, which implies .
Asymptotic behavior of $X_n$ {#section_estimee_Xn}
============================
We return to the study of the skew product . To prove limit theorems using Theorem \[thm\_abstrait\_markov\], we will need to estimate $m({\varphi}_Y>n)$, which is directly related to the speed of convergence of $X_n$ to $0$. This section will be devoted to the proof of the following theorem:
\[estimee\_Xn\_L1\] We have $$\left(\frac{n}{\sqrt{\ln n}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} X_n
\to \frac{1}{\left(2^{\operatorname{{\alpha_{\text{min}}}}} \operatorname{{\alpha_{\text{min}}}}^{3/2}
\sqrt{\frac{\pi}{2\alpha''(x_0)}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}}$$ almost everywhere and in $L^1$.
\[lemme\_asympt\_Xn\] We have $$E(e^{-(\alpha-\operatorname{{\alpha_{\text{min}}}})w})\sim \sqrt{\frac{\pi}{2\alpha''(x_0)}}
\frac{1}{\sqrt{w}} \text{
when }w\to\infty.$$
Write $\beta=\alpha-\operatorname{{\alpha_{\text{min}}}}$, and $f(b)=\operatorname{Leb}\{ \omega {\ |\ }\beta(\omega) \in [0,b)\}$. In a neighborhood of $x_0$ (the unique point where $\alpha$ takes its minimal value $\operatorname{{\alpha_{\text{min}}}}$), $\alpha$ behaves like the parabola $\operatorname{{\alpha_{\text{min}}}}+\frac{\alpha''(x_0)}{2}(x-x_0)^2$, whence $f(b) \sim \sqrt{\frac{2}{\alpha''(x_0)}} \sqrt{b}$ when $b
\to 0$.
Writing $P_\beta$ for the distribution of $\beta$, an integration by parts gives $$\begin{aligned}
E\left(e^{-(\alpha-\operatorname{{\alpha_{\text{min}}}})w}\right)&=\int_0^\infty e^{-b w} {\, {\rm d}}P_\beta(b)
=w \int_0^\infty e^{-b w} f(b) {\, {\rm d}}b
=\int_0^\infty e^{-u} f(u/w) {\, {\rm d}}u
\\ &
=\frac{1}{\sqrt{w}} \int_0^\infty e^{-u}
\left(\sqrt{w}f(u/w)\right) {\, {\rm d}}u.
\end{aligned}$$ But $e^{-u} \left(\sqrt{w}f(u/w)\right) \to e^{-u}
\sqrt{\frac{2}{\alpha''(x_0)}}\sqrt{u}$ when $w\to \infty$. There exists a constant $E$ such that $f(u){\leqslant}E \sqrt{u}$ (this is clear in a neighborhood of $0$, and elsewhere since $f$ is bounded), whence $e^{-u} \left(\sqrt{w}f(u/w)\right) {\leqslant}E
e^{-u}\sqrt{u}$ integrable. By dominated convergence, $$\int_0^\infty e^{-u}
\left(\sqrt{w}f(u/w)\right) {\, {\rm d}}u
\to \sqrt{\frac{2}{\alpha''(x_0)}} \int_0^\infty e^{-u} \sqrt{u}{\, {\rm d}}u
=\sqrt{\frac{2}{\alpha''(x_0)}} \frac{\sqrt{\pi}}{2}.$$
As in Proposition \[estime\_croissance\_grossiere\_Xn\], we write $$\frac{1}{X_n(F\omega)^{\operatorname{{\alpha_{\text{min}}}}}}=\frac{1}{X_{n+1}(\omega)^{\operatorname{{\alpha_{\text{min}}}}}}-\operatorname{{\alpha_{\text{min}}}}2^{\operatorname{{\alpha_{\text{min}}}}} (2X_{n+1}(\omega))^{\alpha(\omega) -\operatorname{{\alpha_{\text{min}}}}}
+O(X_{n+1}(\omega)^{2\alpha(\omega) -\operatorname{{\alpha_{\text{min}}}}}).$$ Proposition \[estime\_croissance\_grossiere\_Xn\] gives $$X_{n+1}(\omega)^{2\alpha(\omega)-\operatorname{{\alpha_{\text{min}}}}} {\leqslant}X_{n+1}(\omega)^{\operatorname{{\alpha_{\text{min}}}}}
{\leqslant}\frac{C}{(n+1)^{\operatorname{{\alpha_{\text{min}}}}/\operatorname{{\alpha_{\text{max}}}}}}
{\leqslant}\frac{C}{\sqrt{n+1}}$$ as $\operatorname{{\alpha_{\text{min}}}}/\operatorname{{\alpha_{\text{max}}}}{\geqslant}1/2$ by hypothesis. Thus, $$\frac{1}{X_{n+1}(\omega)^{\operatorname{{\alpha_{\text{min}}}}}}-\frac{1}{X_n(F\omega)^{\operatorname{{\alpha_{\text{min}}}}}}=2^{\operatorname{{\alpha_{\text{min}}}}} \operatorname{{\alpha_{\text{min}}}}(2X_{n+1}(\omega))^{\alpha-\operatorname{{\alpha_{\text{min}}}}} +O(1/\sqrt{n}).$$ Summing from $1$ to $n$, we get a constant $P$ (independent of $\omega$) such that $$\begin{aligned}
\label{minore_1/X_n}
\frac{1}{X_n(\omega)^{\operatorname{{\alpha_{\text{min}}}}}}{\geqslant}2^{\operatorname{{\alpha_{\text{min}}}}} \operatorname{{\alpha_{\text{min}}}}\left[\sum_{k=1}^{n} (2X_k(F^{n-k} \omega))^{\alpha(F^{n-k}\omega)-\operatorname{{\alpha_{\text{min}}}}}
-P\sqrt{n} \right]
\\
\label{majore_1/X_n}
\frac{1}{X_n(\omega)^{\operatorname{{\alpha_{\text{min}}}}}}{\leqslant}2^{\operatorname{{\alpha_{\text{min}}}}} \operatorname{{\alpha_{\text{min}}}}\left[\sum_{k=1}^n (2X_k(F^{n-k}\omega))^{\alpha(F^{n-k}\omega)-\operatorname{{\alpha_{\text{min}}}}}
+P\sqrt{n} \right]
\end{aligned}$$
Equation and Proposition \[estime\_croissance\_grossiere\_Xn\] imply that $$\label{definit_An}
\frac{\sqrt{\ln n}}{n} \frac{1}{2^{\operatorname{{\alpha_{\text{min}}}}} \operatorname{{\alpha_{\text{min}}}}X_n(\omega)^{\operatorname{{\alpha_{\text{min}}}}}}
{\geqslant}\frac{\sqrt{\ln n}}{n}\sum_{k=1}^n \left(\frac{2C^{-1}}{k^{1/\operatorname{{\alpha_{\text{min}}}}}}
\right)^{\alpha(F^{n-k} \omega)-\operatorname{{\alpha_{\text{min}}}}}
-P\sqrt{\frac{\ln n}{n}}=: A_n(\omega).$$
We first study the convergence of $A_n$. The functions $\alpha$ and $\alpha \circ F^{n-k}$ have the same distribution since $F$ preserve Lebesgue measure. Thus, by Lemma \[lemme\_asympt\_Xn\], $$E\left(\left(\frac{2C^{-1}}{k^{1/\operatorname{{\alpha_{\text{min}}}}}}\right)^{\alpha\circ F^{n-k}-\operatorname{{\alpha_{\text{min}}}}}
\right)
\sim \sqrt{\frac{\pi}{2\alpha''(x_0)}}
\frac{1}{\sqrt{\ln (k^{1/\operatorname{{\alpha_{\text{min}}}}})-\ln(2C^{-1})}}
\sim \sqrt{\frac{\pi \operatorname{{\alpha_{\text{min}}}}}{2\alpha''(x_0)}}\frac{1}{\sqrt{\ln k}}.$$ Summing, we get that $$E(A_n)\to C_1:=\sqrt{\frac{\pi \operatorname{{\alpha_{\text{min}}}}}{2\alpha''(x_0)}},$$ since $\sum_{k=2}^n \frac{1}{\sqrt{\ln k}} \sim \frac{n}{\sqrt{\ln
n}}$.
We will need $L^p$ estimates, for $p{\geqslant}1$. To get them, we use a result of Françoise Pène, recalled in Appendix \[appendice:pene\]. Let us denote by ${\left\| g \right\|}$ the Lipschitz norm of a function $g:S^1 \to {\mathbb{R}}$.
We define $f_k(\omega)=\left(\frac{2C^{-1}}{k^{1/\operatorname{{\alpha_{\text{min}}}}}}
\right)^{\alpha(\omega)-\operatorname{{\alpha_{\text{min}}}}}$, and $g_k=f_k-E(f_k)$. Thus, $A_n=\frac{\sqrt{\ln n}}{n}\sum_{k=1}^n f_k \circ
F^{n-k}-P\sqrt{\frac{\ln n}{n}}$. As $g'_k=\ln
\left(\frac{2C^{-1}}{k^{1/\operatorname{{\alpha_{\text{min}}}}}} \right)\alpha' f_k$, there exists a constant $L$ such that, for $k{\leqslant}n$, ${\left\| g_k \right\|} {\leqslant}L \ln
n$. As a consequence, Theorem \[thm\_borne\_Lp\_pene\] applied to $g_k/(L \ln n)$ gives $${\left\| A_n-E(A_n) \right\|}_p
= \frac{\sqrt{\ln n}}{n} L\ln n
{\left\| \sum_{k=1}^{n} g_k \circ F^{n-k} / (L \ln n) \right\|}_p
{\leqslant}\frac{\sqrt{\ln n}}{n} L\ln n K_p \sqrt{n},$$ i.e. $${\left\| A_n-E(A_n) \right\|}_p {\leqslant}L_p \sqrt{\frac{\ln^3 n}{n}}.$$
This implies in particular that $A_n$ converges almost everywhere to $C_1$. Namely, if $\delta>0$, $$\operatorname{Leb}\{|A_n-E(A_n)|>\delta\}
{\leqslant}\int \frac{|A_n-E(A_n)|^4}{\delta^4}
{\leqslant}\frac{L_4}{\delta^4} \left(\frac{\ln^3 n}{n}\right)^{4/2}$$ which is summable, and $E(A_n)\to C_1$.
We have $$\begin{aligned}
A_n(\omega)&
{\geqslant}\frac{\sqrt{\ln n}}{n} \left[\sum_{k=1}^n \left( \frac{2C^{-1}}
{k^{1/\operatorname{{\alpha_{\text{min}}}}}}\right)^{\operatorname{{\alpha_{\text{max}}}}-\operatorname{{\alpha_{\text{min}}}}} - P\sqrt{n} \right]
{\geqslant}\frac{\sqrt{\ln n}}{n} \bigl[ K n^{2-\operatorname{{\alpha_{\text{max}}}}/\operatorname{{\alpha_{\text{min}}}}} - P
\sqrt{n}\bigr]
\\&
{\geqslant}K' \frac{\sqrt{\ln n}}{n} n^{2-\operatorname{{\alpha_{\text{max}}}}/\operatorname{{\alpha_{\text{min}}}}}
\end{aligned}$$ since $\operatorname{{\alpha_{\text{max}}}}/\operatorname{{\alpha_{\text{min}}}}<3/2$. Thus, $${\left\| \frac{1}{A_n} \right\|}_\infty {\leqslant}K'' \frac{n^{\operatorname{{\alpha_{\text{max}}}}/\operatorname{{\alpha_{\text{min}}}}-1}}{\sqrt{\ln
n}}.$$ Note that $E(A_n)$ tends to $C_1 \not=0$, whence $\frac{1}{E(A_n)}$ is bounded. Thus, $$\begin{aligned}
{\left\| \frac{1}{A_n} - \frac{1}{E(A_n)} \right\|}_p &
{\leqslant}{\left\| \frac{1}{A_n} \right\|}_\infty \frac{1}{E(A_n)}
{\left\| A_n-E(A_n) \right\|}_p
{\leqslant}K''' \frac{n^{\operatorname{{\alpha_{\text{max}}}}/\operatorname{{\alpha_{\text{min}}}}-1}}{\sqrt{\ln
n}} L_p \sqrt{\frac{\ln^3 n}{n}}
\\&
= M_p \frac{\ln n}{n^\kappa}
\end{aligned}$$ where $\kappa=\frac{3}{2}-\frac{\operatorname{{\alpha_{\text{max}}}}}{\operatorname{{\alpha_{\text{min}}}}}>0$. In particular, $\frac{1}{A_n}$ tends to $\frac{1}{C_1}$ in every $L^p$. Equation shows that $$\label{un_petit_label}
\left(\frac{n}{\sqrt{\ln n}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} X_n {\leqslant}\frac{1}
{(2^{\operatorname{{\alpha_{\text{min}}}}}\operatorname{{\alpha_{\text{min}}}}A_n )^{1/\operatorname{{\alpha_{\text{min}}}}}}.$$ The right hand side tends to $$C_2:=\frac{1}{\left(2^{\operatorname{{\alpha_{\text{min}}}}}\operatorname{{\alpha_{\text{min}}}}^{3/2}
\sqrt{\frac{\pi}{2\alpha''(x_0)}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}}$$ in every $L^p$, and in particular in $L^1$. Thus, $$\label{eq_esperance}
\varlimsup E\left(\left( \frac{n}{\sqrt{\ln n}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} X_n
\right) {\leqslant}C_2.$$ Moreover, $A_n$ converges almost everywhere to $C_1$, whence yields that, almost everywhere, $$\label{eq_limsup}
\varlimsup \left( \frac{n}{\sqrt{\ln n}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}
X_n(\omega)
{\leqslant}C_2.$$
Set $Q=\sup_n \left(\frac{1}{E(A_n)}\right)+1$, we estimate $\operatorname{Leb}\left\{ \frac{1}{A_n} {\geqslant}Q\right\}$. If $p{\geqslant}1$, $$\operatorname{Leb}\left\{ \frac{1}{A_n} {\geqslant}Q\right\}
{\leqslant}\operatorname{Leb}\left\{ \left|\frac{1}{A_n} -\frac{1}{E(A_n)} \right|
{\geqslant}1 \right\}
{\leqslant}E\left( \left|\frac{1}{A_n} -\frac{1}{E(A_n)} \right| ^p
\right)
{\leqslant}\left( M_p \frac{\ln n}{n^\kappa} \right)^p.$$ In particular, choosing $p$ large enough gives $$\operatorname{Leb}\left\{ \frac{1}{A_n} {\geqslant}Q\right\}{\leqslant}\frac{M}{n^5}.$$ Setting $Q'=\frac{Q}{2^{\operatorname{{\alpha_{\text{min}}}}} \operatorname{{\alpha_{\text{min}}}}}$, thus yields that $$\operatorname{Leb}\left\{ X_n {\geqslant}\left(\frac{Q'\sqrt{\ln n}}{n}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}
\right\} {\leqslant}\frac{M}{n^5}.$$ Consequently, $U_n:=\left\{ \omega {\ |\ }\exists \sqrt{n} {\leqslant}k{\leqslant}n \text{ with } X_k(F^{n-k} \omega) {\geqslant}\left(\frac{Q'\sqrt{\ln
k}}{k}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} \right\}$ has a measure at most $\sum_{\sqrt{n}}^n \frac{M}{k^5} {\leqslant}\frac{M'}{n^2}$ (since $\operatorname{Leb}$ is invariant under $F^{n-k}$). Finally, Borel-Cantelli ensures that there is a full measure subset of $S^1$ on which $\omega \not \in U_n$ for large enough $n$.
Set $$A'_n(\omega)=\frac{\sqrt{\ln n}}{n} \left[\sum_{k=1}^n
\left(\frac{2(Q'\sqrt{\ln
k})^{1/\operatorname{{\alpha_{\text{min}}}}}}{k^{1/\operatorname{{\alpha_{\text{min}}}}}}\right)^{\alpha(F^{n-k} \omega) -\operatorname{{\alpha_{\text{min}}}}}
+ (P+1)\sqrt{n}\right].$$ As for $A_n$, we show that $A'_n \to C_1$ in every $L^p$ and almost everywhere.
Let $\omega$ be such that $\omega \not \in U_n$ for large enough $n$, and $A'_n(\omega) \to C_1$ (these properties are true almost everywhere). Then, for large enough $n$, Equation and the fact that $X_k(F^{n-k} \omega) {\leqslant}\left(\frac{Q'\sqrt{\ln k}}{k}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}$ for $\sqrt{n}{\leqslant}k{\leqslant}n$, yield that $$\begin{aligned}
\frac{1}{2^{\operatorname{{\alpha_{\text{min}}}}} \operatorname{{\alpha_{\text{min}}}}X_n(\omega)^{\operatorname{{\alpha_{\text{min}}}}}}&
{\leqslant}\left[\sum_{k=1}^{\sqrt{n}} 1+
\sum_{k=\sqrt{n}}^n \left(\frac{2(Q'\sqrt{\ln
k})^{1/\operatorname{{\alpha_{\text{min}}}}}}{k^{1/\operatorname{{\alpha_{\text{min}}}}}}\right)^{\alpha(F^{n-k}\omega) -\operatorname{{\alpha_{\text{min}}}}}
+P \sqrt{n} \right]
\\&
{\leqslant}\frac{n}{\sqrt{\ln n}} A'_n(\omega)\sim \frac{n}{\sqrt{\ln n}} C_1.
\end{aligned}$$ Thus, $$\label{eq_liminf}
\varliminf \left( \frac{n}{\sqrt{\ln n}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}
X_n(\omega)
{\geqslant}C_2.$$
Equations and prove that $\left( \frac{n}{\sqrt{\ln n}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} X_n$ tends almost everywhere to $C_2$. We get the convergence in $L^1$ from the inequality and the following elementary lemma.
Let $f_n$ be nonnegative functions on a probability space, with $f_n \to f$ almost everywhere, and $\varlimsup E(f_n) {\leqslant}E(f)<\infty$. Then $f_n \to f$ in $L^1$.
Limit theorems {#section:limite}
==============
Set $$\label{definit_A}
A=\frac{1}{4\left( \operatorname{{\alpha_{\text{min}}}}^{3/2}
\sqrt{\frac{\pi}{2\alpha''(x_0)}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}}\int_{S^1\times \{1/2\}}
h\operatorname{dLeb},$$ where $h$ is the density of $m$ with respect to $\operatorname{Leb}$.
In this section, we prove the following theorem:
\[enonce\_theoreme\_limite\] Let $f$ be a Hölder function on $S^1\times [0,1]$, with $\int f{\, {\rm d}}m=0$. Write $c=\int_{S^1\times\{0\}} f \operatorname{dLeb}$. Then
- If $\operatorname{{\alpha_{\text{min}}}}<1/2$, there exists $\sigma^2 {\geqslant}0$ such that $\frac{1}{\sqrt{n}} S_n f \to {\mathcal{N}}(0,\sigma^2)$.
- If $\operatorname{{\alpha_{\text{min}}}}=1/2$ and $c\not=0$, then $\frac{S_n f}{\sqrt{ \frac{c^2 A }{4} n (\ln n)^2}} \to
{\mathcal{N}}(0,1)$.
- If $1/2<\operatorname{{\alpha_{\text{min}}}}<1$ and $c\not =0$, then $\frac{S_n f}{n^{\operatorname{{\alpha_{\text{min}}}}}\sqrt{\operatorname{{\alpha_{\text{min}}}}\ln n}} \to Z$, where the random variable $Z$ has a characteristic function given by $$E(e^{itZ})
=e^{- A
|c|^{1/\operatorname{{\alpha_{\text{min}}}}} \Gamma(1-1/\operatorname{{\alpha_{\text{min}}}})\cos\left(\frac{\pi}{2\operatorname{{\alpha_{\text{min}}}}}\right) |t|^{1/\operatorname{{\alpha_{\text{min}}}}}
\left( 1-i\operatorname{sgn}(ct) \tan
\left(\frac{\pi}{2\operatorname{{\alpha_{\text{min}}}}} \right) \right)}$$
- If $1/2{\leqslant}\operatorname{{\alpha_{\text{min}}}}<1$ and $c=0$, assume also that there exists $\gamma
>0$ such that $|f(\omega,x)-f(\omega,0)|{\leqslant}Cx^\gamma$, with $\gamma>\operatorname{{\alpha_{\text{max}}}}\left(1-\frac{1}{2\operatorname{{\alpha_{\text{min}}}}}\right)$. Then there exists $\sigma^2 {\geqslant}0$ such that $\frac{1}{\sqrt{n}} S_n f \to
{\mathcal{N}}(0,\sigma^2)$.
The random variable $Z$ in the third case has a stable distribution of exponent $1/\operatorname{{\alpha_{\text{min}}}}$ and parameters $A|c|^{1/\operatorname{{\alpha_{\text{min}}}}}
\Gamma(1-1/\operatorname{{\alpha_{\text{min}}}})\cos\left(\frac{\pi}{2\operatorname{{\alpha_{\text{min}}}}}\right)$ and $\operatorname{sgn}(c)$.
To prove this theorem, we will use Theorem \[thm\_abstrait\_markov\]. For this, we need a control of $m({\varphi}_Y>n)$ which comes from the asymptotic behavior of $X_n$ proved in Theorem \[estimee\_Xn\_L1\]. It will also be necessary to estimate $m(f_Y>x)$, through the study of the integrability of $f_Y$ (Lemmas \[L2\_am\_petit\] and \[lemme\_dans\_Lp\]).
In the rest of this section, $f$ will be a Hölder function on $S^1\times [0,1]$, fixed once and for all. Recall that $f_Y(y)=\sum_{k=0}^{{\varphi}_Y(y)-1} f(T^k y)$, where ${\varphi}_Y$ is the first return time to $Y=S^1\times(1/2,1]$.
Estimates on measures
---------------------
\[controle\_mesure\_phi\] We have $$m({\varphi}_Y>n) \sim \left(\frac{\sqrt{\ln n}}{n}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} A$$ where $A$ is given by .
We have $$\begin{aligned}
m({\varphi}_Y> n)&
=\int_{S^1} \int_{1/2}^{Y_{n+1}(\omega)} h(\omega,u){\, {\rm d}}u {\, {\rm d}}\omega
=\int_{S^1} \int_0^{X_{n}(F\omega)/2} h(\omega,1/2+u){\, {\rm d}}u {\, {\rm d}}\omega
\\&
=\int_{S^1} \frac{X_{n}(F\omega)}{2} h(\omega,1/2){\, {\rm d}}\omega
+\int_{S^1} \int_0^{X_{n}(F\omega)/2} \bigl[h(\omega,1/2+u)-h(\omega,1/2)
\bigr] {\, {\rm d}}u{\, {\rm d}}\omega
\\&
=I+II.
\end{aligned}$$ As $\left(\frac{n}{\sqrt{\ln n}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} X_n(F\omega) \to
\frac{1}{\left(2^{\operatorname{{\alpha_{\text{min}}}}} \operatorname{{\alpha_{\text{min}}}}^{3/2}
\sqrt{\frac{\pi}{2\alpha''(x_0)}}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}}$ in $L^1$ and almost everywhere (Theorem \[estimee\_Xn\_L1\]) and $h(\omega,1/2)$ is bounded, we get that $I\sim
\left(\frac{\sqrt{\ln n}}{n}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} A$. Moreover, for large enough $n$, $|h(\omega,1/2+u)-h(\omega,1/2)|{\leqslant}{\varepsilon}$, whence $II=o\left(\frac{\sqrt{\ln n}}{n}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}$.
\[L2\_am\_petit\] If $\operatorname{{\alpha_{\text{min}}}}<1/2$, then $f_Y \in L^2(Y, {{\rm d}}m)$.
We have $$\begin{aligned}
\int f_Y^2 {\, {\rm d}}m &
{\leqslant}C\sum_n m({\varphi}_Y=n) n^2
=C \sum \bigl( m({\varphi}_Y>n-1)-m({\varphi}_Y>n)\bigr)n^2
\\&
{\leqslant}C\sum m({\varphi}_Y>n) n
\end{aligned}$$ which is summable since $m({\varphi}_Y>n)\sim A \left(\frac{\sqrt{\ln
n}}{n}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}$ with $1/\operatorname{{\alpha_{\text{min}}}}>2$.
\[lemme\_dans\_Lp\] Assume that $\int_{S^1\times \{0\}} f=0$. Let $\operatorname{{\alpha_{\text{max}}}}>\gamma>0$ be such that $|f(\omega,x)-f(\omega,0)| {\leqslant}C x^\gamma$. If $1<p<
\min\left( \frac{2}{\operatorname{{\alpha_{\text{min}}}}}, \frac{1}{\operatorname{{\alpha_{\text{min}}}}(1-\gamma/\operatorname{{\alpha_{\text{max}}}})} \right)$, then $f_Y \in L^p(Y,{{\rm d}}m)$.
As $h$ is bounded on $Y$, it is sufficient to prove that $f_Y \in
L^p(Y,\operatorname{dLeb})$.
Assume first that $f \equiv 0$ on $S^1 \times \{0\}$. Then, if ${\textbf{x}}=(\omega,x)$ satisfies ${\varphi}_Y({\textbf{x}})=n$, we have $f_Y({\textbf{x}})=\sum_0^{n-1} f(T^k {\textbf{x}})$. If $k {\geqslant}1$, $T_\omega^k(x){\leqslant}X_{n-k}(F^k \omega) {\leqslant}\frac{C}{(n-k)^{1/\operatorname{{\alpha_{\text{max}}}}}}$, whence $|f(T^k {\textbf{x}})|{\leqslant}\frac{C}{(n-k)^{\gamma/\operatorname{{\alpha_{\text{max}}}}}}$, and a summation yields that $|f_Y({\textbf{x}})|{\leqslant}C n^{1-\gamma/\operatorname{{\alpha_{\text{max}}}}}$. Thus, $$\begin{aligned}
\int |f_Y|^p &
{\leqslant}C \sum m({\varphi}_Y=n) n^{p(1-\gamma/\operatorname{{\alpha_{\text{max}}}})}
\\&
{\leqslant}C \sum m({\varphi}_Y>n) n^{p(1-\gamma/\operatorname{{\alpha_{\text{max}}}})-1}.
\end{aligned}$$ As $m({\varphi}_Y>n) \sim A \left(\frac{\sqrt{\ln n}}{n}
\right)^{1/\operatorname{{\alpha_{\text{min}}}}}$, this last series is summable as soon as $$-\frac{1}{\operatorname{{\alpha_{\text{min}}}}}+p\left(1-\frac{\gamma}{\operatorname{{\alpha_{\text{max}}}}}\right)-1 <-1,$$ which is the case by assumption on $p$.
Assume now that $f$ has a vanishing integral on $S^1$. Let $g(\omega,x)=f(\omega,0)$. The function $f-g$ vanishes on $S^1
\times \{0\}$, whence $f_Y-g_Y \in L^p$ according to the first part of this proof. Consequently, it is sufficient to prove that $g_Y \in L^p$. Write $\chi(\omega)=f(\omega,0)$ and $S_n
\chi(\omega)=\sum_{k=0}^{n-1} \chi(F^k \omega)$: then $g_Y(\omega,x)=S_{{\varphi}_Y(\omega,x)} \chi(\omega)$.
Let $M_n \chi(\omega)=\max_{k{\leqslant}n}|S_k \chi(\omega)|$. Let $\delta>0$, and $l=\frac{1+\delta}{\delta}$, so that $\frac{1}{l}+\frac{1}{1+\delta}=1$. We have $$\begin{aligned}
\int |g_Y|^p &
=\sum_{n=0}^{\infty} \int_{S^1}
\int_{1/2+X_{n}(F\omega)/2}^{1/2+X_{n-1}(F\omega)/2} \bigl|S_n
\chi(\omega)\bigr|^p {\, {\rm d}}u {\, {\rm d}}\omega
\\&
{\leqslant}\sum_{k=1}^{\infty} \int_{S^1}
\int_{1/2+X_{2^k}(F\omega)/2}^{1/2+X_{2^{k-1}}(F\omega)/2} |M_{2^k}
\chi(\omega)|^p {\, {\rm d}}u {\, {\rm d}}\omega
\\&
{\leqslant}\sum_{k=1}^\infty \int_{S^1} X_{2^{k-1}}(F\omega)
|M_{2^k}\chi(\omega)|^p {\, {\rm d}}\omega
{\leqslant}\sum_{k=1}^\infty {\left\| X_{2^{k-1}}\circ F \right\|}_{1+\delta}
{\left\| M_{2^k}\chi \right\|}_{lp}^p,
\end{aligned}$$ where the last inequality is Hölder inequality. If $\delta$ is small enough, $lp>2$, whence Corollary \[ineg\_max\] yields that ${\left\| M_{2^k}\chi \right\|}_{lp} {\leqslant}C k^{\frac{lp-1}{lp}}\sqrt{2^k}$. Moreover, $${\left\| X_{2^{k-1}}\circ F \right\|}_{1+\delta}
={\left\| X_{2^{k-1}} \right\|}_{1+\delta}
{\leqslant}\left( \int X_{2^{k-1}} \right)^{1/(1+\delta)}
\sim C \left( \frac{\sqrt{\ln(2^{k-1})}}{2^{k-1}}
\right)^{\frac{1}{(1+\delta) \operatorname{{\alpha_{\text{min}}}}}}$$ by Theorem \[estimee\_Xn\_L1\]. Thus, $\int|g_Y|^p<\infty$ if $\frac{1}{(1+\delta)\operatorname{{\alpha_{\text{min}}}}} > \frac{p}{2}$, and it is possible to choose $\delta$ such that this inequality is true, since $\frac{1}{\operatorname{{\alpha_{\text{min}}}}}>\frac{p}{2}$ by hypothesis.
Proof of Theorem \[enonce\_theoreme\_limite\]
---------------------------------------------
To apply Theorem \[thm\_abstrait\_markov\], we first check the condition . Let $\theta$ be the Hölder exponent of $f$. We will work with the distance $d_{
\lambda^{-\theta}}= \lambda^{-\theta s(x,y)}$. For this distance, $T_Y$ is a Gibbs-Markov map.
*Fact: if $f$ is $\theta$-Hölder on $S^1\times [0,1]$, then $$\sum m[A_{s,n}] D f_Y(A_{s,n}) <\infty.$$*
Recall that $D f_Y(A_{s,n})$ (defined in Theorem \[thm\_abstrait\_markov\]) is the best Lipschitz constant of $f_Y$ on $A_{s,n}$, here for the distance $d_{ \lambda^{-\theta}}$.
Take $(\omega_1,x_1)$ and $(\omega_2,x_2)\in A_{s,n}$ with for example $x_2{\geqslant}x_1$. This implies that $x_1\in J_n^+(\omega_2)$ and that, for $0{\leqslant}k{\leqslant}n$, $d(T^k(\omega_1,x_1),T^k(\omega_2,x_2)) {\leqslant}(1+D) |F^k \omega_1
-F^k \omega_2|$ (see the beginning of the proof of Proposition \[prop\_distortion\_bornee\]). Moreover, $d(T^k(\omega_2,x_1),T^k(\omega_2,x_2)) {\leqslant}d(T^n(\omega_2,x_1),T^n(\omega_2,x_2))$ (since, if $\omega$ is fixed, the map $T_{\alpha(\omega)}$ is expanding).
Thus, for $0{\leqslant}k{\leqslant}n$, $$\begin{aligned}
d(T^k(\omega_1,x_1),T^k(\omega_2,x_2))&
{\leqslant}d(T^k(\omega_1,x_1),T^k(\omega_2,x_1))
+d(T^k(\omega_2,x_1),T^k(\omega_2,x_2))
\\&
{\leqslant}(1+D)|F^k \omega_1 -F^k \omega_2|
+d(T^n(\omega_2,x_1),T^n(\omega_2,x_2))
\\&
{\leqslant}(1+D)|F^n \omega_1 -F^n \omega_2|
+d(T^n(\omega_1,x_1),T^n(\omega_2,x_1))
\\& \hphantom{=\ }
+d(T^n(\omega_1,x_1),T^n(\omega_2,x_2))
\\&
{\leqslant}(1+D)|F^n \omega_1 -F^n \omega_2|
+(1+D)|F^n \omega_1 -F^n \omega_2|
\\& \hphantom{=\ }
+d(T^n(\omega_1,x_1),T^n(\omega_2,x_2))
\\&
{\leqslant}(3+2D)d(T^n(\omega_1,x_1),T^n(\omega_2,x_2)).
\end{aligned}$$ We deduce that $$\begin{aligned}
|f_Y(\omega_1,x_1)-f_Y(\omega_2,x_2)|&
{\leqslant}\sum_{k=0}^{n-1}
|f(T^k(\omega_1,x_1))-f(T^k(\omega_2,x_2))|
\\&
{\leqslant}\sum_{k=0}^{n-1}C
d(T^k(\omega_1,x_1),T^k(\omega_2,x_2))^\theta
\\&
{\leqslant}C' n d(T^n(\omega_1,x_1),T^n(\omega_2,x_2))^\theta.
\end{aligned}$$ As $T_Y$ is expanding for the distance $d'$ (defined in , and equivalent to $d$), we get $$d(T^n(\omega_1,x_1),T^n(\omega_2,x_2)) {\leqslant}C
d_{\lambda^{-1}}(T^n(\omega_1,x_1),T^n(\omega_2,x_2))=C \lambda
d_{\lambda^{-1}}((\omega_1,x_1),(\omega_2,x_2)),$$ whence $d(T^n(\omega_1,x_1),T^n(\omega_2,x_2))^\theta {\leqslant}C
d_{\lambda^{-\theta}}((\omega_1,x_1),(\omega_2,x_2))$.
Thus, $Df_Y(A_{s,n}) {\leqslant}C n$, and $$\sum m(A_{s,n}) D f_Y(A_{s,n})
{\leqslant}C \sum m({\varphi}_Y=n) n
= C < +\infty,$$ by Kac’s Formula.
In the case $\operatorname{{\alpha_{\text{min}}}}<1/2$, Lemma \[L2\_am\_petit\] gives that $f_Y \in
L^2$. Moreover, $|f|_Y\in L^2$ for the same reason, and ${\varphi}\in
L^2$ (since ${\varphi}=g_Y$ for $g\equiv 1$, whence Lemma \[L2\_am\_petit\] applies also). We have already checked the condition , so we can apply (the first case of) Theorem \[thm\_abstrait\_markov\]. This yields the central limit theorem for $f$.
The second and third cases are analogous. Let us prove for example the third one, i.e. $1/2<\operatorname{{\alpha_{\text{min}}}}<1$ and $c\not=0$. Assume for example $c>0$. We estimate $m(f_Y>x)$.
*Fact: $m(f_Y>x) \sim \left(\frac{c \sqrt{\ln
x}}{x}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} A$ and $m(f_Y<-x)=o\left(\frac{\sqrt{\ln
x}}{x}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}$.*
We prove the estimate on $m(f_Y>x)$, the other one being similar.
Let $g\equiv c$ on $S^1 \times [0,1]$. Then $g_Y=nc$ on $[{\varphi}_Y=n]$, which implies that $m(g_Y>nc)=m({\varphi}_Y>n)\sim
\left(\frac{\sqrt{\ln n}}{n}\right)^{1/\operatorname{{\alpha_{\text{min}}}}} A$ by Lemma \[controle\_mesure\_phi\].
In the general case, consider $j=f-g$, and let us prove that $m(|j_Y|>x)=o\left(\frac{\sqrt{\ln x}}{x}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}$. As $f_Y=g_Y+j_Y$, it will give $$m(g_Y>x(1+{\varepsilon})) -m(|j_Y|>x{\varepsilon}) {\leqslant}m(f_Y>x) {\leqslant}m(g_Y>x(1-{\varepsilon}))+m(|j_Y|>x{\varepsilon}),$$ which gives the conclusion. Let $\gamma>0$ with $\gamma<\min(\theta,\operatorname{{\alpha_{\text{max}}}})$ (where $\theta$ is the Hölder coefficient of $f$). Lemma \[lemme\_dans\_Lp\] gives that $j_Y\in L^p$ if $p<\min\left(\frac{2}{\operatorname{{\alpha_{\text{min}}}}},\frac{1}{\operatorname{{\alpha_{\text{min}}}}(1-\gamma/\operatorname{{\alpha_{\text{max}}}})} \right)$. We can in particular choose $p>1/\operatorname{{\alpha_{\text{min}}}}$. Then $m(|j_Y|>x) {\leqslant}\int
\left(\frac{|j_Y|}{x}\right)^p =O(x^{-p})$, which concludes the proof of the fact.
The same fact holds for ${\varphi}_Y$ and $|f|_Y$, with the same proof, whence we are in the third case of Theorem \[thm\_abstrait\_markov\]. This gives the desired result.
Assume finally that $\frac{1}{2}{\leqslant}\operatorname{{\alpha_{\text{min}}}}<1$ and that $c=0$. Under the hypotheses of the theorem, we can apply Lemma \[lemme\_dans\_Lp\] with $p=2$, and get that $f_Y \in L^2$. The proof of this lemma shows in fact that the function $M$ (defined in Theorem \[thm\_abstrait\_markov\]) is also in $L^2$. Finally, Lemma \[controle\_mesure\_phi\] shows that $m[{\varphi}_Y>x] \sim
\left(\frac{\sqrt{\ln x}}{x}\right)^{1/\operatorname{{\alpha_{\text{min}}}}}A$. We have checked all the hypotheses of the first case of Theorem \[thm\_abstrait\_markov\].
Induced maps and limit theorems {#appendice:loi_stable}
===============================
The aim of this section is to prove very general results stating that, if a function satisfies a limit theorem for an induced map, it also satisfies one for the initial map. Similar theorems have been proved in [@gouezel:stable], by spectral methods. We will describe here a more elementary method, essentially due to Melbourne and Török for flows ([@melbourne_torok]).
If $Y$ is a subset of a probability space $(X,m)$, $T:X\to X$, and $T_Y$ is the induced map on $Y$, we will write $S_n^Y
g=\sum_{k=0}^{n-1} g\circ T_Y^k$: this is the Birkhoff sum of $g$, for the transformation $T_Y$. We will also write $E_Y(g)=\frac{\int_Y g}{m[Y]}$. Finally, for $t\in {\mathbb{R}}$, $\lfloor t
\rfloor$ denotes the integer part of $t$.
\[thm\_probabiliste\_general\] Let $T:X\to X$ be an ergodic endomorphism of a probability space $(X,m)$, and $f:X\to {\mathbb{R}}$ an integrable function with vanishing integral. Let $Y\subset X$ have positive measure. For $y\in Y$, write ${\varphi}(y)=\inf\{n>0 {\ |\ }T^n(y)\in Y\}$ and $f_Y(y)=\sum_{k=0}^{{\varphi}(y)-1} f(T^k y)$, and $M(y)=\max_{1{\leqslant}k{\leqslant}{\varphi}(y)} \left|\sum_{j=0}^{k-1} f(T^j y) \right|$.
We assume the following properties:
1. There exists a sequence $B_n\to +\infty$, with $\sup_{r {\leqslant}2n}
\frac{B_r}{B_n} < \infty$ and $\inf_{r {\geqslant}n} \frac{B_r}{B_n}>0$, such that $(f_Y, {\varphi})$ satisfies a mixing limit theorem for the normalization $B_n$: there exists a random variable $Z$ such that, for every $t\in {\mathbb{R}}$, $$\label{limite_mixing}
E_Y\left({\varphi}e^{it \frac{S^Y_{\lfloor n m(Y) \rfloor} f_Y}{B_n}}\right)
\to E_Y({\varphi}) E\left(e^{itZ}\right).$$
2. For every ${\varepsilon}>0$, there exists $C$ such that, for any $n\in
{\mathbb{N}}^*$, $$\label{majore_fY}
m\{y\in Y {\ |\ }M(y) {\geqslant}{\varepsilon}B_n \} {\leqslant}\frac{C}{n}.$$
3. \[hypothese\_3\] There exists $b>0$ such that, in the natural extension of $T_Y$, $\frac{1}{N^b} \sum_0^{N-1} f_Y(T_Y^k)$ tends almost everywhere to $0$ when $N \to \pm \infty$.
4. \[hypothese\_4\] For every ${\varepsilon}>0$, there exists $A>0$ and $N_0$ such that, for every $n{\geqslant}N_0$, $$\label{eq_hyp4}
m\left\{ y\in Y {\ |\ }\left| \frac{S_n^Y {\varphi}- n
E_Y({\varphi})}{B_n^{1/b}} \right| {\geqslant}A \right\} {\leqslant}{\varepsilon}.$$
Then the function $f$ satisfies also a limit theorem: $$E\left(e^{it \frac{S_n f}{B_n}}\right)
\to E(e^{it Z}),$$ i.e. $\frac{S_n f}{B_n}$ tends in distribution to $Z$.
The hypotheses of the theorem are tailor-made so that the following proof works, but they are in fact often satisfied in natural cases. Let us comment on these 4 hypotheses:
1. The convergence is very often satisfied when $f_Y$ satisfies a limit theorem. Namely, the martingale proofs or spectral proofs of limit theorems automatically give this kind of convergence.
2. If $Z_0,Z_1,\ldots$ are independent identically random variables such that $\frac{\sum_{0}^{n-1} Z_k}{B_n}$ converges in distribution to a nontrivial limit, then for all ${\varepsilon}>0$, there exists $C$ such that $P(|Z_0| {\geqslant}{\varepsilon}B_n) {\leqslant}\frac{C}{n}$: this is a consequence of the classification of the stable laws, see [@feller:2].
Here, we are not in the independent setting, and there is no such classification. However, the same kind of results holds very often: usually, it is not hard to check in practical cases that $m(|f_Y(x)| {\geqslant}{\varepsilon}B_n) {\leqslant}\frac{C}{n}$, since $f_Y$ satisfies a limit theorem by the first assumption. Set $|f|_Y(y)=\sum_{j=0}^{{\varphi}(y)-1} |f(T^j y)|$. As $|f|_Y$ and $f_Y$ have more or less the same distribution, $|f|_Y$ satisfies also often $$\label{majore_fY'}
\tag{\ref{majore_fY}'}
m\{ y\in Y {\ |\ }|f|_Y(y) {\geqslant}{\varepsilon}B_n \} {\leqslant}\frac{C}{n}.$$ Since $M {\leqslant}f_Y$, implies . Thus, it will often be sufficient to check . However, is sometimes strictly weaker than , because of cancellations, which is why we have stated the theorem with .
3. The natural extension is useful so that we can let $N$ tend to $-\infty$, and consider $T_Y^{-1}$ in the proof. Generally, Birkhoff’s Theorem yields that this assumption is satisfied for $b=1$. This is often sufficient. However, sometimes, it is important to have better estimates. It is then possible to use [@vitesse_birkhoff Theorem 16], for example: this theorem ensures that, if the correlations of $f_Y\in L^2$ decay at least as $O(1/n)$, then the hypothesis is satisfied for any $b>1/2$ (for $N \to -\infty$, use the fact that $\int f_Y \cdot f_Y \circ T_Y^n
= \int f_Y\circ T_Y^{-n} \cdot f_Y$, and apply the result to $T_Y^{-1}$).
4. The fourth assumption is weaker than $$\label{eq_hyp4'}
\tag{\ref{eq_hyp4}'}
\exists B'_n=O(B_n^{1/b}) \text{ such that }
\frac{S_n^Y {\varphi}- nE_Y({\varphi})}{B'_n}\text{ converges in
distribution.}$$ Moreover, ${\varphi}$ is often simpler than $f_Y$. Since $f_Y$ satisfies a limit theorem (this is more or less the first hypothesis), this is also often the case of ${\varphi}$, which implies . Thus, – and hence – are satisfied quite generally.
Without loss of generality, we can work in a tower, i.e. assume that $X=\{ (y,i) {\ |\ }y\in Y, i\in\{0,\ldots,{\varphi}(y)-1\} \}$ and that, for $i<{\varphi}(y)-1$, $T(y,i)=(y,i+1)$, while $T(y,{\varphi}(y)-1)=(T_Y(y),0)$. Namely, it is possible to build an extension of $X$ satisfying these properties, and it is equivalent to prove a limit theorem in $X$ or in this extension (see for example [@gouezel:stable Section 4.1]). Note that $E_Y({\varphi})=1/m(Y)$ by Kac’s Formula. Let $\pi$ be the projection from $X$ to $Y$, given by $\pi(y,i)=y$.
In this proof, we will write $S_t f(x)$, even when $t$ is not an integer, for $S_{\lfloor t \rfloor }f(x)$. In the same way, $T^t$ should be understood as $T^{\lfloor t \rfloor}$. We also extend $B_n$ to ${\mathbb{R}}_+$, setting $B_t:=B_{\lfloor t \rfloor}$.
As $T$ is ergodic, $T_Y$ is also ergodic ([@aaronson:book Proposition 1.5.2]). Birkhoff’s Theorem gives that $$\label{asymp_phin}
S_n^Y {\varphi}= \frac{n}{m(Y)} +o(n)$$ almost everywhere on $Y$. For $y\in Y$ and $N\in {\mathbb{N}}$, let $n(y,N)$ be the greatest integer $n$ such that $S_n^Y {\varphi}(y) < N$. If $y$ is such that $S_n^Y {\varphi}(y)=\frac{n}{m(Y)}+o(n)$ (which is true almost everywhere), then $n(y,N)$ is finite for every $N$, and $\frac{n(y,N)}{m(Y)} \sim N$, i.e.$$\label{renouvellement_basique}
\frac{n(y,N)}{N m(Y)} \to 1.$$
Since $\int_X e^{it (S_N^Y f_Y)\circ \pi }=\int_Y {\varphi}e^{it S_N^Y
f_Y}$, yields that $$\label{Thm_limite_SY}
\frac{(S^Y_{Nm(Y)} f_Y)\circ \pi}{B_N} \to Z$$ in distribution on $X$. The idea of the proof will be to see that $(S^Y_{Nm(Y)}f_Y)\circ \pi$ and $S_N f$ are close (this is not surprising, since one iteration of $T_Y$ corresponds roughly to $1/m(Y)$ iterations of $T$). This will give that $\frac{S_N
f}{B_N}$ tends to $Z$.
We write $$\begin{aligned}
S_N f(y,i)=
\left(S_N f(y,i)-S_N f(y,0) \right)
&+ \left(S_N f(y,0)-S^Y_{n(y,N)}f_Y(y) \right)
\\&
+\left( S^Y_{n(y,N)}f_Y(y) - S^Y_{Nm(Y)}f_Y(y) \right)
+S^Y_{ Nm(Y)}f_Y(y).
\end{aligned}$$ The last term, equal to $\bigl(S^Y_{N m(Y)} f_Y\bigr)\circ \pi$, satisfies a limit theorem by . To conclude the proof, we will see that the three other terms, divided by $B_N$, tend to $0$ in probability.
The second and third terms depend only on $y$. Thus, the following lemma will be useful to prove that they tend to $0$ on $X$:
\[proba\_sur\_Y\_donne\_proba\_sur\_X\] Let $f_n$ be a sequence of functions on $Y$, tending to $0$ in probability on $Y$. Then $f_n \circ \pi$ tends to $0$ in probability on $X$.
Take ${\varepsilon}>0$. As $f_n \to 0$ in probability, the measure of $E_n:=\{ y\in Y {\ |\ }|f_n(y)|{\geqslant}{\varepsilon}\}$ tends to $0$. As ${\varphi}\in L^1$, dominated convergence yields that $\int_{E_n} {\varphi}\to 0$, i.e. the measure of $\pi^{-1}(E_n)$ tends to $0$. But $\pi^{-1}(E_n)$ is exactly the set where $|f_n \circ \pi|{\geqslant}{\varepsilon}$.
*Fact: $\frac{1}{B_N} \left(S_N f(y,i)-S_N f(y,0) \right)$ tends to $0$ in probability on $X$.*
Set $V_N(y)=\sum_{i=0}^{{\varphi}(y)-1} |f \circ T^N(y,i)|$ on $Y$. Then ${\left\| V_N \right\|}_{L^1(Y)} ={\left\| f\circ T^N \right\|}_{L^1(X)}
={\left\| f \right\|}_{L^1(X)}$ since $T$ preserves the measure. Thus, $V_N/B_{N}$ tends to $0$ in $L^1(Y)$, and in probability. Lemma \[proba\_sur\_Y\_donne\_proba\_sur\_X\] yields that $\frac{1}{B_N}
V_N\circ \pi$ tends to $0$ in probability on $X$.
As $S_N f(y,i)-S_N f(y,0) =\sum_N^{N+i-1} f(T^k(y,0))
-\sum_0^{i-1} f(T^k(y,0))$, we get $|S_N f(y,i)-S_N f(y,0)| {\leqslant}V_N(y)+V_0(y)$. Thus, $\frac{1}{B_N} \left(S_N f(y,i)-S_N f(y,0)
\right)$ is bounded by a function going to $0$ in probability.
*Fact: $\frac{1}{B_N} \left(S_N f(y,0)-S_{n(y,N)}^Y
f_Y(y)\right)$ tends to $0$ in probability on $X$.*
By Lemma \[proba\_sur\_Y\_donne\_proba\_sur\_X\], it is sufficient to prove it on $Y$. We have $$\left|S_N f(y,0)-S_{n(y,N)}^Y f_Y(y)\right|
=
\left| \sum_{S_{n(y,N)}^Y {\varphi}(y)}^{N-1} f\circ T^k(y,0) \right|
{\leqslant}M\left(T_Y^{n(y,N)}y\right).$$ Let $a>0$ be very small, we show that $m\left\{ y {\ |\ }M\left(T_Y^{n(y,N)}y\right) {\geqslant}a B_N \right\} \to 0$.
Let ${\varepsilon}>0$. Let $C$ be such that $m( M(y) {\geqslant}a B_n) {\leqslant}\frac{C}{n}$, by . Set $\delta=\frac{{\varepsilon}}{2Cm(Y)}$. By , for large enough $N$, $$m\left\{\left|\frac{n(y,N)}{m(Y) N} - 1 \right|
{\geqslant}\delta\right\}
{\leqslant}{\varepsilon}.$$ When $\left|\frac{n(y,N)}{m(Y) N} - 1 \right| {\leqslant}\delta$, the fact that $M\left(T_Y^{n(y,N)}y\right) {\geqslant}aB_N$ implies that there exists $n \in [(1-\delta) m(Y)N, (1+\delta)m(Y)N]$ such that $M(T_Y^n y) {\geqslant}aB_N$. Thus, $$m\left\{ y {\ |\ }M\left(T_Y^{n(y,N)}y\right) {\geqslant}a B_N \right\}
{\leqslant}{\varepsilon}+
\sum_{n=(1-\delta) m(Y)N}^{(1+\delta) m(Y)N}
m\{M(T_Y^n y) {\geqslant}a B_N\}.$$
As $m$ is invariant by $T_Y$, we have $m\{M(T_Y^n y) {\geqslant}a B_N\}
=m\{ M {\geqslant}a B_N\} {\leqslant}\frac{C}{N}$. Thus, $$m\left\{ y {\ |\ }M\left(T_Y^{n(y,N)}y\right) {\geqslant}a B_N \right\}
{\leqslant}{\varepsilon}+
2\delta m(Y)N \frac{C}{N}
=2{\varepsilon}.$$
*Fact: $\frac{1}{B_N} \left(S^Y_{n(y,N)} f_Y- S^Y_{N
m(Y)}f_Y\right)$ tends to $0$ in probability on $X$ when $N \to
\infty$.*
By Lemma \[proba\_sur\_Y\_donne\_proba\_sur\_X\], it is sufficient to prove it on $Y$. Without loss of generality, we can use the natural extension and assume that $T_Y$ is invertible.
For $n<0$, write $S^Y_n f_Y=\sum_0^{|n|-1} f_Y\circ T_Y^{-j}$. Then, setting $\nu(y,N)=n(y,N)-N m(Y)$, $$\label{definit_nu}
S^Y_{n(y,N)} f_Y(y)- S^Y_{N m(Y)}f_Y(y)
=S^Y_{\nu(y,N)} f_Y \left(T^{N m(Y)}(y)\right).$$ If $A>0$ and $N\in {\mathbb{N}}$, as $E_Y({\varphi})=1/m(Y)$, we get $$\begin{gathered}
\{y{\ |\ }\nu(y,N) {\geqslant}A B_N^{1/b}\}
=\{ n(y,N) {\geqslant}A B_N^{1/b}+ N m(Y)\}
=\{ S^Y_{A B_N^{1/b}+Nm(Y)}{\varphi}< N\}
\\=
\left \{ \frac{S^Y_{A B_N^{1/b}+N m(Y)}{\varphi}- (A B_N^{1/b}+
Nm(Y))E_Y({\varphi})}{\left(B_{AB_N^{1/b}+Nm(Y)}\right)^{1/b}} <
-\frac{A}{m(Y)}\left(\frac{B_N}{B_{AB_N^{1/b}+Nm(Y)}}\right)^{1/b}
\right\}.
\end{gathered}$$ For some integer $k$, we have $N {\leqslant}2^k Nm(Y) {\leqslant}2^k
(AB_{Nm(Y)}^{1/b}+Nm(Y))$. The assumption $\sup_{r{\leqslant}2n}
\frac{B_r}{B_n} {\leqslant}C<\infty$ thus yields that $\frac{B_N}{B_{AB_{Nm(Y)}^{1/b}+Nm(Y)}} {\leqslant}C^k$. In particular, $$\{y{\ |\ }\nu(y,N) {\geqslant}A B_N^{1/b}\}
\subset \left \{ \frac{S^Y_{A B_N^{1/b}+N m(Y)}{\varphi}- (A B_N^{1/b}+
Nm(Y))E_Y({\varphi})}{\left(B_{AB_N^{1/b}+Nm(Y)}\right)^{1/b}} <
-\frac{AC^{k/b}}{m(Y)}
\right\}.$$ Consequently, if $A$ is large enough, Assumption \[hypothese\_4\] yields that $m\{y{\ |\ }\nu(y,N) {\geqslant}A B_N^{1/b}\} {\leqslant}{\varepsilon}$ for large enough $N$. We handle in the same way the set of points where $\nu(y,N) {\leqslant}-A B_N^{1/b}$, using the assumption $\inf_{r{\geqslant}n}\frac{B_r}{B_n}>0$.
We have thus proved: $$\label{convergence_nuy}
\forall {\varepsilon}>0, \exists A>0, \exists N_0>0, \forall N {\geqslant}N_0,\
m\{ y{\ |\ }|\nu(y,N)| {\geqslant}A B_N^{1/b} \} {\leqslant}{\varepsilon}.$$
Set $W_N(y)=\frac{1}{B_N} S_{\nu(y,N)} f_Y \left(T_Y^{ N
m(Y)}(y)\right)$, we will show that it tends to $0$ in distribution, which will conclude the proof, by . Take $a>0$, we show that $m(|W_N|>a) \to 0$ when $N\to \infty$.
Let ${\varepsilon}>0$. Assumption \[hypothese\_3\] ensures that there exists ${\widetilde}{Y}$ with $m({\widetilde}{Y}){\geqslant}m(Y)-{\varepsilon}$ and $N_1$ such that $\frac{1}{|N|^b}|S_N^Y f_Y| {\leqslant}{\varepsilon}$ on ${\widetilde}{Y}$, for every $|N|{\geqslant}N_1$. Define $Y'_N=\{y\in Y {\ |\ }|\nu(y,N)|< N_1\}$ and $Y''_N=\{y \in Y {\ |\ }|\nu(y,N)|{\geqslant}N_1\}$. We estimate first the contribution of $Y'_N$.
Set $\psi(y)=\sum_{-N_1}^{N_1-1} |f_Y\circ T_Y^j|$. Since $\psi$ is measurable, there exists a constant $C$ and a subset $Z$ of $Y$ with $m(Z){\geqslant}m(Y)-{\varepsilon}$ and $\psi {\leqslant}C$ on $Z$. Then, for $y\in Y'_N$, we have $|W_N(y)|{\leqslant}\frac{1}{B_N} \psi\left(T_Y^{N
m(Y)} y\right)$. Set $Z_N=Y'_N \cap T_Y^{- N m(Y)}(Z)$: it satisfies $m(Z_N) {\geqslant}m(Y'_N)-{\varepsilon}$. On $Z_N$, we have $|W_N|{\leqslant}\frac{C}{B_N}$, whence, for large enough $N$, $|W_N|<a$ on $Z_N$. Thus, for large enough $N$, $$m\left\{y \in Y'_N {\ |\ }|W_N(y)|{\geqslant}a\right\}
{\leqslant}m\left\{y\in Z_N {\ |\ }|W_N(y)|{\geqslant}a\right\}+{\varepsilon}={\varepsilon}.$$
We estimate then the contribution of $Y''_N$. Set ${\widetilde}{Y}''_N=Y''_N \cap T_Y^{-N m(Y)}({\widetilde}{Y})$, satisfying $m({\widetilde}{Y}''_N) {\geqslant}m(Y''_N)-{\varepsilon}$. Thus, $$m(|W_N|{\geqslant}a) {\leqslant}m\{y\in {\widetilde}{Y}''_N {\ |\ }|W_N(y)|{\geqslant}a\}+2{\varepsilon}.$$ On ${\widetilde}{Y}''_N$, $|\nu(y,N)|{\geqslant}N_1$, whence $\frac{1}{|\nu(y,N)|^b}\left|S^Y_{\nu(y,N)}f_Y \left(T_Y^{N
m(Y)}y\right)\right| {\leqslant}{\varepsilon}$. Thus, $|W_N(y)|{\leqslant}{\varepsilon}\frac{|\nu(y,N)|^b}{B_N}={\varepsilon}\left(\frac{|\nu(y,N)|}{B_N^{1/b}}\right)^b$. Consequently, $$m(|W_N|{\geqslant}a) {\leqslant}m\left (\frac{|\nu(y,N)|}{B_N^{1/b}} {\geqslant}\left(\frac{a}{{\varepsilon}}\right)^{1/b} \right) +2{\varepsilon}.$$ Thus, if ${\varepsilon}$ is small enough, and $N$ large enough, yields that $m(|W_N|{\geqslant}a) {\leqslant}3{\varepsilon}$.
The three facts we have just proved imply that $\frac{S_N
f(y,i)}{B_N}-\frac{S_{Nm(Y)}^Y f_Y(y)}{B_N} \to 0$ in distribution on $X$. As $\frac{S_{Nm(Y)}^Y f_Y(y)}{B_N} \to Z$ in distribution on $X$, by , this concludes the proof.
Multiple decorrelations and $L^p$-boundedness {#appendice:pene}
=============================================
The following theorem has been useful in this paper:
\[thm\_borne\_Lp\_pene\] Let $F: \omega \to 4\omega$ on the circle $S^1$. Then, for every $p \in [1,\infty)$, there exists a constant $K_p$ such that, for every $n\in {\mathbb{N}}$, for every $f_0,\ldots,f_{n-1}:S^1 \to {\mathbb{R}}$ bounded by $1$, of zero average and $1$-Lipschitz, $${\left\| \sum_{k=0}^{n-1} f_k \circ F^k \right\|}_p {\leqslant}K_p \sqrt{n}.$$
This result has essentially been proved by Françoise Pène, in a much broader context. Her proof depends on a property of multiple decorrelations, which is implied by the spectral gap of the transfer operator:
Let ${\left\| f \right\|}$ be the Lipschitz norm of the function $f$ on the circle $S^1$. Then, for every $m,m'\in {\mathbb{N}}$, there exist $C>0$ and $\delta<1$ such that, for every $N\in {\mathbb{N}}$, for every increasing sequences $(k_1,\ldots,k_m)$ and $(l_1,\ldots,l_{m'})$, for every Lipschitz functions $G_1,\ldots,G_m,H_1,\ldots, H_{m'}$, $$\label{decorr_mult}
\left|\operatorname{Cov}\left(\prod_{i=1}^m G_i \circ F^{k_i}, \prod_{j=1}^{m'}
H_j\circ F^{N+l_j}\right)\right|
{\leqslant}C \left(\prod_{i=1}^m {\left\| G_i \right\|}\right) \left(\prod_{j=1}^{m'}
{\left\| H_j \right\|} \right) \delta^{N-k_m}.$$
Here $\operatorname{Cov}(u,v)=\int uv -\int u \int v$.
Let ${\widehat}{F}$ be the transfer operator associated to $F$, and acting on Lipschitz functions. It is known that it admits a spectral gap and that its iterates are bounded, i.e. there exist constants $M>0$ and $\delta<1$ such that $\bigl\|{\widehat}{F}^n
f\bigr\|{\leqslant}M {\left\| f \right\|}$, and $\bigl\|{\widehat}{F}^n f\bigr\|{\leqslant}M\delta^n {\left\| f \right\|}$ if $\int f=0$.
We can assume that $N{\geqslant}k_m$ (otherwise, $\delta^{N-k_m}{\geqslant}1$, and the inequality becomes trivial). Then, writing ${\varphi}=\prod_{i=1}^m G_i \circ F^{k_i}$ and $\psi=\prod_{j=1}^{m'} H_j\circ F^{l_j}$, we get $$\begin{aligned}
\left|\operatorname{Cov}({\varphi},\psi\circ F^N)\right|
&
=\left|
\int \left({\varphi}-\int {\varphi}\right) \psi\circ F^N\right|
=\left|\int {\widehat}{F}^N \left({\varphi}-\int {\varphi}\right) \psi \right|
\\&
{\leqslant}{\left\| {\widehat}{F}^N \left({\varphi}-\int {\varphi}\right) \right\|} {\left\| \psi \right\|}_\infty.
\end{aligned}$$ But $$\begin{aligned}
{\widehat}{F}^N({\varphi})&
={\widehat}{F}^N \left(\prod G_i^{k_i}\right)
={\widehat}{F}^{N-k_m} (G_m {\widehat}{F}^{k_m-k_{m-1}}(G_{m-1}
{\widehat}{F}^{k_{m-1}-k_{m-2}}(\ldots {\widehat}{F}^{k_2-k_1} (G_1))\ldots)
\\&
=:{\widehat}{F}^{N-k_m}(\chi).
\end{aligned}$$ As the iterates of ${\widehat}{F}$ are bounded on Lipschitz functions, we get a bound on the Lipschitz norm of $\chi$: ${\left\| \chi \right\|}{\leqslant}M^{m-1} \prod {\left\| G_i \right\|}$. Moreover, $\int \chi=\int {\varphi}$, whence $$\begin{aligned}
{\left\| {\widehat}{F}^N \left({\varphi}-\int {\varphi}\right) \right\|}
&
={\left\| {\widehat}{F}^{N-k_m} \left(\chi -\int \chi\right) \right\|}
{\leqslant}M\delta^{N-k_m} {\left\| \chi-\int \chi \right\|}
\\&
{\leqslant}M\delta^{N-k_m} M^{m-1} \prod {\left\| G_i \right\|}.
\end{aligned}$$
When $p$ is an even integer, Theorem \[thm\_borne\_Lp\_pene\] is then a consequence of [@pene:averaging Lemma 2.3.4]. The Hölder inequality gives the general case.
\[remarque\_pene\] The same result holds for Hölder functions instead of Lipschitz functions, with the same proof.
We will also need the following result:
\[ineg\_max\_abstraite\] Let $T$ be a measure preserving transformation on a space $X$. Let $f:X\to {\mathbb{R}}$ and $p>2$ be such that $$\exists C>0, \forall n\in {\mathbb{N}}^*, {\left\| S_n f \right\|}_p {\leqslant}C \sqrt{n}.$$ Write $M_nf(x)=\sup_{1{\leqslant}k{\leqslant}n} |S_k f(x)|$. Then there exists a constant $K$ such that $$\forall n{\geqslant}2, {\left\| M_n f \right\|}_p {\leqslant}K (\ln n)^{\frac{p-1}{p}}
\sqrt{n}.$$
Let $n\in {\mathbb{N}}^*$. Let $k<2^n$, and write its binary decomposition $k=\sum_{j=0}^{n-1} {\varepsilon}_j 2^j$, with ${\varepsilon}_j \in
\{0,1\}$. Set $q_j=\sum_{l=j}^{n-1} {\varepsilon}_l 2^l$ (in particular, $q_0=k$ and $q_n=0$). Then $S_k f=\sum_{j=0}^{n-1}
(S_{q_j}f-S_{q_{j+1}}f)$. Consequently, the convexity inequality $(a_0+\ldots +a_{n-1})^p {\leqslant}n^{p-1} (a_0^p+\ldots+a_{n-1})^p$ gives that $$|S_k f|^p {\leqslant}n^{p-1} \sum_{j=0}^{n-1}
|S_{q_j}f-S_{q_{j+1}}f|^p.$$ Note that $q_{j+1}$ is of the form $\lambda 2^{j+1}$ with $0{\leqslant}\lambda {\leqslant}2^{n-j-1}-1$, and $q_j$ is equal to $q_{j+1}$ or $q_{j+1}+2^j$. Thus, $$|S_k f|^p {\leqslant}n^{p-1} \sum_{j=0}^{n-1} \left(
\sum_{\lambda=0}^{2^{n-j-1} -1} \left|S_{\lambda 2^{j+1} +2^j}f
-S_{\lambda 2^{j+1}}f \right|^p \right).$$ The right hand term is independent of $k$, and gives a bound on $|M_{2^n -1} f|^p$. Moreover, $$\int \left|S_{\lambda 2^{j+1} +2^j}f -S_{\lambda 2^{j+1}}f
\right|^p =\int \left|S_{2^j} f\right|^p {\leqslant}C^p \sqrt{2^j}^{p}.$$ Thus, we get $$\int |M_{2^n -1} f|^p {\leqslant}n^{p-1} \sum_{j=0}^{n-1} 2^{n-j-1} C^p 2^{pj/2}
{\leqslant}K n^{p-1} 2^n 2^{(\frac{p}{2}-1)n}
= K n^{p-1} \sqrt{2^n}^p.$$ For times of the form $2^n-1$, this is a bound of the form ${\left\| M_t \right\|}_p {\leqslant}K (\ln t)^{\frac{p-1}{p}} \sqrt{t}$. To get the same estimate for an arbitrary time $t$, it is sufficient to choose $n$ with $2^{n-1} {\leqslant}t<2^n$, and to note that $M_t {\leqslant}M_{2^n -1}$.
\[ineg\_max\] Let $F: \omega \to 4\omega$ on the circle $S^1$, let $\chi:S^1 \to
{\mathbb{R}}$ be a Hölder function with $0$ average, and $p>2$. Write $M_n
\chi(x)=\sup_{1{\leqslant}k{\leqslant}n} |S_k \chi(x)|$. Then there exists a constant $K$ such that $$\forall n{\geqslant}2, {\left\| M_n \chi \right\|}_p {\leqslant}K (\ln
n)^{\frac{p-1}{p}} \sqrt{n}.$$
Theorem \[thm\_borne\_Lp\_pene\] (or rather the remark following it, for the Hölder case) shows that ${\left\| S_n \chi \right\|}_p {\leqslant}C\sqrt{n}$. Consequently, Theorem \[ineg\_max\_abstraite\] gives the conclusion.
[^1]: Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm 75005 Paris (France). e-mail `Sebastien.Gouezel@ens.fr`
[^2]: *keywords*: intermittency, countable Markov shift, central limit theorem, stable laws. *2000 Mathematics Subject Classification:* 37A50, 37C40, 60F05
|
---
abstract: 'We report the discoveries of two, two-image gravitationally lensed quasars selected from the Sloan Digital Sky Survey: SDSS J0806+2006 at $z_s=1.540$ and SDSS J1353+1138 at $z_s=1.629$ with image separations of $\Delta{\theta}=1\farcs40$ and $\Delta{\theta}=1\farcs41$ respectively. Spectroscopic and optical/near-infrared imaging follow-up observations show that the quasar images have identical redshifts and possess extended objects between the images that are likely to be lens galaxies at $z_l \simeq 0.6$ in SDSS J0806+2006 and $z_l \simeq 0.3$ in SDSS J1353+1138. The field of SDSS J0806+2006 contains several nearby galaxies that may significantly perturb the system, and SDSS J1353+1138 has an extra component near its Einstein ring that is probably a foreground star. Simple mass models with reasonable parameters reproduce the quasar positions and fluxes of both systems.'
author:
- 'Naohisa Inada, Masamune Oguri, Robert H. Becker, Richard L. White, Michael D. Gregg, Paul L. Schechter, Yozo Kawano, Christopher S. Kochanek, Gordon T. Richards, Donald P. Schneider, J. C. Barentine, Howard J. Brewington, J. Brinkmann, Michael Harvanek, S. J. Kleinman, Jurek Krzesinski, Dan Long, Eric H. Neilsen, Jr., Atsuko Nitta, Stephanie A. Snedden, and Donald G. York'
title: 'SDSS J0806+2006 and SDSS J1353+1138: Two New Gravitationally Lensed Quasars from the Sloan Digital Sky Survey'
---
Introduction {#sec:intro}
============
Since the discovery of Q0957+561 [@walsh79], about 80 gravitationally lensed quasars have been discovered [@kochanek04]. Lensed quasars are not only intriguing phenomena but also have become indispensable astronomical tools, including probes of the cosmological parameters and the structure of galaxies [e.g., @refsdal64; @kochanek91]. In particular, the abundance of gravitational lenses in a well-defined source sample can be used to constrain dark energy [e.g., @turner90; @fukugita90; @chae02]. Unfortunately, the largest existing survey, the Cosmic Lens All-Sky Survey [CLASS; @myers03; @browne03], contains only 22 lensed radio sources (with a well-defined statistical sample of 13 lenses) discovered from ${\sim}10,000$ radio sources, which is still insufficient to place tight constraints on dark energy models. The Sloan Digital Sky Survey [SDSS; @york00] should lead to a significantly larger lens sample for attacking the dark energy problem. SDSS is expected to identify $10^5$ quasars spectroscopically [e.g., @schneider05] and $\approx10^6$ quasars photometrically [e.g., @richards04], which should lead to a sample of over $10^2$ lensed quasars given a standard lensing probability of $10^{-3}$ [@turner84]. Indeed, 12 new lensed quasars have been discovered from the SDSS quasars so far [@inada03a; @inada03b; @inada03c; @inada05; @johnston03; @morgan03; @pindor04; @pindor05; @oguri04; @oguri05; @burles05], in addition to recovering 5 previously known lensed quasars [@walsh79; @weymann80; @surdej87; @bade97; @oscoz97]. We can presently construct a well-defined statistical sample of 16 lensed SDSS quasars, but there remain many promising SDSS lensed quasar candidates for which the necessary follow-up observations are incomplete.
In this paper, we report on the discovery of two more gravitationally lensed quasars, SDSS J080623.70+200631.9 (SDSS J0806+2006) and SDSS J135306.35+113804.7 (SDSS J1353+1138). We present imaging and spectroscopic follow-up observations with the University of Hawaii 2.2-meter (UH88) telescope, the W. M. Keck Observatory’s Keck I and II telescopes, and the Magellan Consortium’s Landon Clay 6.5-m (LC6.5m) telescope [^1]. We model the systems to check that their geometries are consistent with the lensing hypothesis.
The structure of this paper is as follows. We describe our lens candidate selections from the SDSS data in §\[sec:sdss\]. The follow-up observations and mass modeling of SDSS J0806+2006 and SDSS J1353+1138 are presented in §\[sec:0806\] and §\[sec:1353\], respectively. We finally present a summary and give a conclusion in §\[sec:conc\]. Throughout the paper we assume a cosmological model with the matter density $\Omega_M=0.27$, cosmological constant $\Omega_\Lambda=0.73$, and Hubble constant $h=H_0/100{\rm km\,sec^{-1}Mpc^{-1}}=0.7$ [@spergel03].
Selecting Lensed Quasar Candidates from the SDSS {#sec:sdss}
================================================
The SDSS is conducting a photometric and spectroscopic survey of 10,000 square degrees of the sky approximately centered on the North Galactic Cap using the dedicated wide-field ($3^{\circ}$ field of view) 2.5-m telescope [@gunn05] at the Apache Point Observatory in New Mexico, USA. Photometric observations [@gunn98; @lupton99; @tucker05] are made in five optical filters [@fukugita96]. After automated data processing by the photometric pipeline [@lupton01; @lupton05], quasar and galaxy candidates are selected by the spectroscopic target selection algorithms [@eisenstein01; @richards02; @strauss02]. Spectra of these candidates are obtained according to the tiling algorithm of [@blanton03] using a multi-fiber spectrograph covering 3800[Å]{} to 9200[Å]{} at a resolution of R$\sim1800$. The data are very homogeneous, with an astrometric accuracy better than about $0\farcs1$ rms per coordinate [@pier03] and photometric zeropoint errors less than about 0.03 magnitude over the entire survey area [@hogg01; @smith02; @ivezic04]. The data have been released continuously to the public [@stoughton02; @abazajian03; @abazajian04; @abazajian05; @adelman05].
We selected the two objects, SDSS J0806+2006 and SDSS J1353+1138, as lensed quasar candidates from the SDSS spectroscopic quasar sample with the same algorithm used for the discovery of most SDSS lensed quasars [e.g., @inada03a]. Specifically, the algorithm uses the SDSS image parameters [dev\_L]{}, [exp\_L]{} and [star\_L]{} for the likelihood that a source can be modeled as a de Vaucouleurs profile, an exponential disk or a point source to quantify the structure of each quasar. Lensed quasars should be modeled poorly by all three profiles, so quasars with small values for all three likelihoods are good lens candidates [see @inada03a for more details].
The SDSS $i$-band images of SDSS J0806+2006 and SDSS J1353+1138 are shown in Figure \[fig:field08061353\]. Both systems are clearly resolved, making them excellent lens candidates since their spectra are clearly those of $z>1$ quasars. For SDSS J0806+2006, the total magnitudes (within ${\sim}2\farcs3$ aperture radius) of the system are $19.18\pm0.05$, $18.76\pm0.01$, $18.44\pm0.02$, $18.06\pm0.02$, and $17.92\pm0.05$, in $u$, $g$, $r$, $i$, and $z$, respectively. For SDSS J1353+1138, the total magnitudes ($ugriz$, within ${\sim}2\farcs1$ aperture radius) are $16.87\pm0.01$, $16.60\pm0.01$, $16.48\pm0.01$, $16.26\pm0.01$, and $16.20\pm0.02$, respectively. The redshifts of SDSS J0806+2006 and SDSS J1353+1138 measured from their SDSS spectra are $z_s=1.537\pm0.002$ and $z_s=1.623\pm0.003$, respectively. From the SDSS spectra, with a fiber aperture of $3\farcs0$, we know that the components of the candidates cannot have greatly dissimilar spectra. However, we cannot resolve the spectra of the individual components.
Thus, while the SDSS data are sufficient to identify these objects as lens candidates, additional observations are needed to confirm them as lensed quasars. Deeper and higher resolution images are needed to confirm the existence of multiple quasar images and to search for the lens galaxies, and spatially resolved spectra are needed to confirm that the quasar images have identical redshifts. We present this evidence for SDSS J0806+2006 in §3 and for SDSS J1353+1138 in §4.
SDSS J0806+2006 {#sec:0806}
===============
Imaging Observations {#sec:0806img}
--------------------
We obtained $V$, $R$ and $I$-band images of SDSS J0806+2006 using the 8k mosaic CCD camera at the UH88 telescope, on 2004 December 16. The pixel scale of the 8k mosaic CCD camera is 0235 ${\rm pixel^{-1}}$. The exposure time was 360 sec for each band. The typical seeing in the exposures was ${\sim}0\farcs7$, about a half of the typical seeing size of the SDSS (${\sim}1\farcs5$). Bias-subtracted and flat-field corrected images are shown in the upper panels of Figure \[fig:uh88\_0806\]. We also obtained an $H$-band image using the QUick InfraRed Camera (QUIRC) of the UH88 telescope on 2005 February 21 and a $K'$-band image using the Near InfraRed Camera [NIRC; @matthews94] of the Keck I telescope on 2005 April 24. The pixel scale of QUIRC is $0\farcs189$ ${\rm pixel^{-1}}$, and that of the NIRC is $0\farcs15$ ${\rm pixel^{-1}}$. The seeing was ${\sim}0\farcs7$ in the QUIRC observation, and it was ${\sim}0\farcs4$ in the NIRC observation. The total exposure time was 720 sec and 900 sec for $H$-band imaging and $K'$-band imaging, respectively. The $H$-band image is shown in Figure \[fig:uh88\_0806\], and the $K'$-band image is shown in Figure \[fig:keck\_0806\].
All the images ($VRIHK'$) clearly show two stellar components; we name these two stellar components A (eastern component) and B (western component), with A being the brighter component. To determine if there are any extended objects in between the stellar components, we subtracted point spread functions (PSFs) from the raw $VRIH$ images, adopting a nearby star as a template for the PSF. The results are shown in the lower panels of Figure \[fig:uh88\_0806\]; there is clearly residual flux between components A and B in all four images. This object (named G) is quite red, with colors of $V-R{\sim}1.1$ and $R-I{\sim}1.1$ that are similar to those of an early-type galaxy at $z_l{\sim}0.6$ [@fukugita95]. Such a redshift is consistent with the redshift of a absorption line system in the spectra of the quasars (see §\[sec:0806spc\]) and an estimate based on the Faber-Jackson relation (see §\[sec:0806model\]). Therefore, we conclude that component G is the lens galaxy. In the higher resolution NIRC $K'$-band image (left panel of Figure \[fig:keck\_0806\]), we can identify component G between components A and B even before PSF subtraction. We fit this image using GALFIT [@peng02] to find that G is well-modeled by a de Vaucouleurs profile of ellipticity $e=0.18$ and major axis position angle $\theta_e=51^\circ$.
The astrometry and photometry of components A, B and G are summarized in Table \[tab:0806\]. The optical images were calibrated using the standard star PG 0231+051 [@landolt92] and the $H$-band image was calibrated using the standard star FS 21 [@hawarden01]. We lack a photometric standard stars for the NIRC $K'$-band image, thus the data were used only for astrometry and the flux ratio constraints in the lens models. When we fit the NIRC $K'$-band image using a model consisting of the two quasar images and the lens galaxy, we find a quasar flux ratio of 0.53 as compared to the mean flux ratio of 0.67 (derived from fitting only the PSF models) for the optical bands that could be created by a modest amount of dust extinction or chromatic microlensing. We report the astrometry from using GALFIT to simultaneously fit components A, B and G in the NIRC $K'$-band image. The angular separation of components A and B is $1\farcs403\pm0\farcs012$.
Spectroscopic Observations {#sec:0806spc}
--------------------------
A spectroscopic observation of SDSS J0806+2006 was conducted with the Keck II telescope on 2005 April 12 in ${\sim}1\farcs0$ seeing. We used the echellette mode of the Echellette Spectrograph and Imager [ESI; @sutin97; @sheinis02] with the MIT-LL 2048$\times$4096 CCD camera. The spectral range was 3900[Å]{} to 11,000[Å]{} at a spectral resolution of $11.4~{\rm km\,s^{-1}pixel^{-1}}$ (R$\sim$27000). The exposure time was 900 sec. The $1\farcs0$-wide slit was oriented to observe components A and B simultaneously. The spectra of each component was extracted using the standard method (summing the fluxes in a window around the position of each component and subtracting the sky using neighboring windows on either side of the trace). We show the binned spectra in Figure \[fig:spec0806\]. Both spectra have the and emission lines at the same wavelength, with an estimated velocity difference of $20{\pm}70$ km s$^{-1}$ for the emission line. We summarize the redshifts calculated from these emission lines in Table \[tab:0806em\]. In addition, the spectral energy distributions (SEDs) are also very similar. The ratio of the spectra, which is shown in Figure \[fig:spec0806\], is almost constant (${\sim}0.7$) for a wide range of wavelengths, and it is consistent with the mean flux ratio of the optical images (0.67). Note, however, that the emission line flux ratios appear to differ slightly from that of the continuum, suggesting that there is some microlensing of the continuum by the stars in the lens galaxy.
In addition to the quasar emission lines, we found a strong absorption line system at $z=0.573$ in both spectra (see the inset of Figure \[fig:spec0806\]). We also found H and K ($\sim6200$ Å) and absorption lines ($\sim4100$ Å) at $z=0.573$ in the spectra. The fact that the redshift of this absorption line system is close to the estimated redshift (see §\[sec:0806img\]) of the probable lens galaxy (component G) and that absorption lines are frequently associated with galaxies [e.g., @bergeron91] suggests that the absorption is due to the lens galaxy. If this is the case, the lens redshift should be $z_l{\simeq}0.573$; we adopt this value for the mass models presented in the next subsection.
Mass Modeling {#sec:0806model}
-------------
To explore the lensing hypothesis further, we modeled the system using the two standard mass models: a Singular Isothermal Sphere with an external shear (SIS+shear) model, and a Singular Isothermal Ellipsoid (SIE) model. Both models have eight parameters: the Einstein radius $R_{\rm E}$, the shear $\gamma$ or ellipticity $e$ and its position angle ($\theta_\gamma$ or $\theta_e$), the position of the lens galaxy, and the position and flux of the source quasar. We have only eight constraints (the positions of A, B and G, and the fluxes of A and B), so we should be able to fit the data perfectly. We should, however, be able to do so with sensible values for the shear or ellipticity of the models. We used the component positions from Table \[tab:0806\] and the flux ratio of 0.53 derived from the GALFIT model of the NIRC $K'$-band image including the two stellar components and the lens galaxy.
We used [*lensmodel*]{} [@keeton01] to determine the model parameters, with the results summarized in Table \[table:model0806\]. The required ellipticities of $\gamma=0.03$ or $e=0.08$ are typical of other lensed systems and roughly consistent with that of the lens galaxy ($e=0.18$), but there is a significant misalignment between the position angle of the major axis of the models ($\theta_e{\simeq}-28^\circ$) and that of the lens galaxy ($\theta_e{=}51^\circ$). This might imply that the system is affected by a strong external perturbation [@keeton98]. Indeed, there are at least four galaxies within a 16$''$ radius of the lens (named G1–4, with the nearest galaxy only 4$''$ from the lens; see Table \[tab:0806gal\]), so the position angle of the models may be a compromise between that of the lens galaxy and the shear induced by G1. Finally, these models predict that the time delay between the images is ${\sim}32h^{-1}$days.
We can estimate the redshift of the lens galaxy based on the Faber-Jackson relation [@faber76]. From the Table 3 of @rusin03, we estimate that the lens magnitude should be $R{\sim}20.5$ for $\Delta{\theta}=1\farcs40$, $z_s{=}1.54$, and assuming $z_l=0.57$. While this is somewhat brighter than the observed $R$-band magnitude of galaxy G ($R=21.2$), it is roughly consistent given the $\pm0.6$ mag scatter [@rusin03]. The rough agreement of this estimate further supports for a lens galaxy redshift of $z_l{\simeq}0.573$.
SDSS J1353+1138 {#sec:1353}
===============
Imaging Observations {#sec:1353img}
--------------------
We obtained $V$, $R$, and $I$-band images of SDSS J1353+1138 using the 8K mosaic CCD camera of the UH88 telescope on 2004 May 25 in ${\sim}1\farcs0$ seeing. The exposure times were 120 sec for $V$ and 180 sec for $R$ and $I$, respectively. We also acquired an $H$-band image on the UH88 telescope using QUIRC on 2005 February 21. The exposure time was 720 sec, and the seeing was ${\sim}0\farcs7$. The images, shown in the upper panels of Figure \[fig:uh88\_1353\], clearly show two stellar components, which we named A (northern and brighter component) and B (southern component). Although it is not obvious in the raw $H$-band image of Figure \[fig:uh88\_1353\], an extended object can be seen between components A and B even before the PSF subtraction. We also obtained $g$ and $i$-band images using the Magellan Instant Camera (MagIC) at the LC6.5m telescope on 2005 April 15. The pixel scale of MagIC is $0\farcs069$ ${\rm pixel^{-1}}$. The seeing was ${\sim}0\farcs6$, and the exposure time was 360 sec for $g$-band and 480 sec for $i$-band.
After subtracting PSF models for the two stellar components, we clearly detect an extended red object positioned between them in all the residual images. These residual images are shown in Figure \[fig:uh88\_1353\] and \[fig:mag\_1353\] for the UH88 and Magellan data, respectively. We identify this extended object, which we denote as G, with the lens galaxy. We fit the $i$-band image using GALFIT to find that the lens galaxy is well-modeled with a de Vaucouleurs profile of ellipticity $e=0.50$ and major axis position angle $\theta_e{=}-62^\circ$. In addition to component G, there is an additional object which we named component C near image A. It is most easily seen in the $g$-band image after subtracting components A and B (see the middle panel of Figure \[fig:mag\_1353\]), but it can also be seen at the same position in the $i$-band image.
There are four possible interpretations of component C. First, it could be a chance superposition of a foreground star. It is well fit by the PSF (see the bottom panels of Figure \[fig:mag\_1353\]), and our lens models require only components A, B and G to reproduce the system with reasonable parameter values (see \[sec:1353model\]). Second, it could be an object related to the lens galaxy. However, lens models with significant extra mass at the position of component C generally fit the data very badly. Third, it could be a third image of the quasar. This seems unlikely since its colors are very different from components A and B (see Table \[tab:1353\]). Moreover, identifying it as a quasar image makes little sense as a lens geometry since it is in the wrong place to be a central, odd image (which should lie between components B and G) or to be part of a four-image system in which one of the A/B/C components is an unresolved image pair. Fourth, it could be emission from the host galaxy of the source quasar. Probably only the first possibility is permitted, although we cannot completely exclude the other possibilities.
We summarize astrometry and photometry of components A, B, C and G in Table \[tab:1353\]. The optical images were calibrated using the standard stars PG 1528+062 [@landolt92] and SA 107-351 [@smith02], and the $H$-band image was calibrated using the standard star FS 21 [@hawarden01]. The flux ratio of components A and B changes significantly with wavelength, from a mean flux ratio in the optical of $0.35$ to an $H$-band flux ratio of 0.54. The angular separation of components A and B is $1\farcs406\pm0\farcs007$.
Spectroscopic Observations {#sec:1353spc}
--------------------------
We used ESI on the Keck II telescope to obtain spectra of the two components of SDSS J1353+1138 on 2005 April 12. The instrumental set up was the same as in §\[sec:0806spc\] and we used an exposure time of 600 sec. The binned spectra are shown in Figure \[fig:spec1353\]. Both components have and emission lines at identical wavelengths, with a velocity difference of only $10{\pm}50$ km s$^{-1}$ for the emission line. We summarize the redshifts derived from the emission lines in Table \[tab:1353em\].
The flux ratio of the spectra is nearly constant (${\sim}0.4$) and consistent with that in the optical images ($\sim 0.35$). There is, however, a slight increase in the ratio redwards of about 5300 [Å]{} that may be due to contamination of the spectrum of component B by emission from the lens galaxy. Assuming this feature is due to the 4000[Å]{} break of the lens galaxy, it implies a lens redshift of $z_l \sim 0.3$. Since the colors of the lens galaxy of $V-R{\sim}0.3$, $R-I{\sim}1.0$, and $g-i{\sim}1.1$ are roughly consistent with those of an early-type galaxy at $z=0.1{\sim}0.3$ [@fukugita95], we adopt $z_l=0.3$ for mass modeling.
The quasar spectra also contain strong absorption line systems at $z=1.238$, $0.904$, and $0.637$, with their absorption lines. These are unlikely to be associated with the lens galaxy, given the large difference from the lens redshift estimated from the colors, and also given the fact that many (unlensed) quasars show absorption. We note that an absorption line due to a $z \simeq 0.3$ lens is undetectable in our spectra because it would lie beyond the atmospheric cutoff.
Mass Modeling {#sec:1353model}
-------------
We modeled SDSS J1353+1138 in the same way as in §\[sec:0806model\]. The positions of the quasars and lens galaxy are taken from Table \[table:model1353\] (neglecting component C). For the flux ratio of components A and B, we adopt $0.31$, which was derived from the GALFIT model of the MagIC $i$-band image. The results are summarized in Table \[table:model1353\]. The derived shear and ellipticity of $\gamma=0.05$ and $e=0.15$, are again typical for a lensed quasar system. The derived ellipticity (0.15) appears to be significantly smaller than the ellipticity of the observed light profile ($e{=}0.50$), but this difference is commonly seen in the lensed quasar systems [@keeton98]. However, in addition to the disagreement of the ellipticity, there is a significant misalignment between the position angle of the models ($\theta_e{\simeq}-34^\circ$) and that of the lens galaxy ($\theta_e{=}-62^\circ$). While this might be due to external perturbations, there are no nearby galaxies that can easily explain the misalignment. It could be due to component C, but we find that simple tests of lens models with mass located at the position of component C generally fail to fit the data. The time delay between A and B is predicted to be ${\sim}16h^{-1}$days, assuming the lens redshift of $z_l=0.3$.
As in §\[sec:0806model\] we also estimated the lens redshift based on the Faber-Jackson relation. Again we used Table 3 of @rusin03, $\Delta{\theta}=1\farcs41$, and $z_s{=}1.63$ to estimate that the lens magnitude should be $R{\sim}19.2$, assuming ${z_l}=0.3$. This estimate is in rough agreement with the observed $R$-band magnitude of galaxy G ($R=18.8$), again supporting for a lens redshift of $z_l\sim0.3$.
Summary {#sec:conc}
=======
We report the discovery of two doubly-imaged quasar lenses, SDSS J0806+2006 and SDSS 1353+1138. Both were selected from the SDSS spectroscopic quasar sample as lensed quasar candidates and confirmed in subsequent imaging and spectroscopic observations. SDSS J0806+2006 consists of two $z_s=1.540$ quasar images separated by $\Delta{\theta}=1\farcs40$ lensed by a galaxy at $z_l\simeq0.6$. The lens galaxy is closer to the fainter image as expected, and its redshift, as suggested by its magnitude, colors, and the presence of a absorption feature, is $z_l=0.573$. Several nearby galaxies may perturb this system and indicate that the lens galaxy is part of a small group. SDSS 1353+1138 consists of two $z_s=1.629$ quasar images separated by $\Delta{\theta}=1\farcs41$ with a lens galaxy at $z_l{\sim}0.3$. The redshift of the lens galaxy is estimated based on its magnitude, colors, and the spectral flux ratio between the two quasar images. There is an additional component, which we have labeled C, superposed on this system whose nature is presently unexplained. Observations using the [*Hubble Space Telescope*]{} are probably required to clarify its role in the lensed quasar system.
N. I. and M. O. are supported by JSPS through JSPS Research Fellowship for Young Scientists. A portion of this work was also performed under the auspices of the U.S. Department of Energy, National Nuclear Security Administration by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.
Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation.
Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Korean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
Abazajian, K., et al. 2003, , 126, 2081
Abazajian, K., et al. 2004, , 128, 502
Abazajian, K., et al. 2005, , 129, 1755
Adelman-McCarthy, J. K., et al. 2005, , in press (astro-ph/0507711)
Bade, N., Siebert, J., Lopez, S., Voges, W., & Reimers, D. 1997, A&A, 317, L13
Bergeron, J., & Boisse, P. 1991, A&A, 243, 344
Blanton, M. R., Lin, H., Lupton, R. H., Maley, F. M., Young, N., Zehavi, I., & Loveday, J. 2003, , 125, 2276
Browne, I. W. A., et al. 2003, , 341, 13
Burles, S., et al. 2005, , in preparation
Chae, K.-H., et al. 2002, Phys. Rev. Lett., 89, 151301
Eisenstein, D. J., et al. 2001, , 122, 2267
Faber, S. M. & Jackson, R. E. 1976, , 204, 668
Fukugita, M., Futamase, T., & Kasai, M. 1990, , 246, 24P
Fukugita, M., Shimasaku, K., & Ichikawa, T. 1995, PASP, 107, 945
Fukugita, M., Ichikawa, T., Gunn, J. E., Doi, M., Shimasaku, K., & Schneider, D. P. 1996, , 111, 1748
Gunn, J. E., et al. 1998, , 116, 3040
Gunn, J. E., et al. 2005, , submitted
Hawarden, T. G., Leggett, S. K., Letawsky, M. B., Ballantyne, D. R., & Casali, M. M. 2001, , 325, 563
Hogg, D. W., Finkbeiner, D. P., Schlegel, D. J., & Gunn, J. E. 2001, , 122, 2129
Inada, N., et al. 2003a, , 126, 666
Inada, N., et al. 2003b, , submitted
Inada, N., et al. 2003c, , 426, 810
Inada, N., et al. 2005, , 130, 1967
Ivezić, Ž., et al. 2004, AN, 325, 583
Johnston, D. E., et al. 2003, , 126, 2281
Keeton, C. R. 2001b, preprint (astro-ph/0102340)
Keeton, C. R., Kochanek, C. S., & Falco, E. E. 1998, , 509, 561
Kochanek, C. S. 1991, , 373, 354
Kochanek, C. S., Schneider, P., Wambsganss, J., 2004, Part 2 of Gravitational Lensing: Strong, Weak & Micro, Proceedings of the 33rd Saas-Fee Advanced Course, G. Meylan, P. Jetzer & P. North, eds. (Springer-Verlag: Berlin)
Landolt, A. U. 1992, , 104, 340
Lupton, R. H., Gunn, J. E., & Szalay, A. S. 1999, , 118, 1406
Lupton, R., Gunn, J. E., Ivezić, Z., Knapp, G. R., Kent, S., & Yasuda, N. 2001, in ASP Conf. Ser. 238, Astronomical Data Analysis Software and Systems X, ed. F. R. Harnden, Jr., F. A. Primini, and H. E. Payne (San Francisco: Astr. Soc. Pac.), p. 269 (astro-ph/0101420)
Lupton, R. 2005, , submitted
Matthews, K., & Soifer, B. T. 1994, ASSL Vol. 190: Astronomy with Arrays, The Next Generation, 239
Morgan, N. D., Snyder, J. A., & Reens, L. H. 2003, , 126, 2145
Myers, S. T., et al. 2003, , 341, 1
Oguri, M., et al. 2004, , 56, 399
Oguri, M., et al. 2005, , 622, 106
Oscoz, A., Serra-Ricart, M., Mediavilla, E., Buitrago, J., & Goicoechea, L. J. 1997, , 491, L7
Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, , 124, 266
Pier, J. R., Munn, J. A., Hindsley, R. B., Hennessy, G. S., Kent, S. M., Lupton, R. H., & Ivezić, Ź. 2003, , 125, 1559
Pindor, B., et al. 2004, , 127, 1318
Pindor, B., et al. 2005, , in press (astro-ph/0509296)
Refsdal, S. 1964, , 128, 307
Richards, G. T., et al. 2002, , 123, 2945
Richards, G. T., et al. 2004, , 155, 257
Rusin, D, et al. 2003, , 587, 143
Schneider, D. P., et al. 2005, , 130, 367
Sheinis, A. I., Bolte, M., Epps, H. W., Kibrick, R. I., Miller, J. S., Radovan, M. V., Bigelow, B. C., & Sutin, B. M. 2002, , 114, 851
Smith, J. A., et al. 2002, , 123, 2121
Spergel, D. N., et al. 2003, , 148, 175
Stoughton, C., et al. 2002, , 123, 485
Strauss, M. A., et al. 2002, , 124, 1810
Surdej, J., Swings, J.-P., Magain, P., Courvoisier, T. J.-L., & Borgeest, U. 1987, , 329, 695
Sutin, B. M. 1997, Proc. SIPE, 2871, 1116
Tucker, D. L., et al. 2005, , submitted
Turner, E. L., Ostriker, J. P., & Gott, J. R., III 1984, , 284, 1
Turner, E. L. 1990, , 365, L43
Walsh, D., Carswell, R. F., & Weymann, R. J. 1979, , 279, 381
Weymann, R. J., et al. 1980, , 285, 641
York, D. G., et al. 2000, , 120, 1579
[crrrrrr]{} A & 0.000[$\pm$]{}0.005 & 0.000[$\pm$]{}0.005 & 19.23[$\pm$]{}0.01 & 18.93[$\pm$]{}0.01 & 18.54[$\pm$]{}0.01 & 16.87[$\pm$]{}0.01\
B & $-$1.136[$\pm$]{}0.007 & $-$0.823[$\pm$]{}0.007 & 19.82[$\pm$]{}0.02 & 19.36[$\pm$]{}0.02 & 18.84[$\pm$]{}0.01 & 17.34[$\pm$]{}0.02\
G & $-$0.811[$\pm$]{}0.015 & $-$0.574[$\pm$]{}0.015 & 22.27[$\pm$]{}0.07 & 21.20[$\pm$]{}0.04 & 20.16[$\pm$]{}0.03 & 17.90[$\pm$]{}0.05\
[lccccccc]{} (1908.73) & 4847.18 & 42.4 & 1.5395$\pm$0.0005 & & 4847.99 & 43.8 & 1.5399$\pm$0.0007\
(2798.75) & 7110.25 & 48.2 & 1.5405$\pm$0.0004 & & 7109.44 & 52.0 & 1.5402$\pm$0.0006\
[cccccc]{} SIS+shear & $0.68$ & $0.03$ & $-28.0$ & $31.6$ & 4.63\
SIE & $0.69$ & $0.08$ & $-27.9$ & $32.5$ & 4.62\
[crrr]{} G1 & $-$3.766 & 0.965 & 0.11\
G2 & 5.957 & 2.451 & 0.03\
G3 & 3.480 & $-$8.784 & 0.09\
G4 & $-$13.770 & $-$7.970 & 0.07\
[crrrrrrrr]{} A & 0.000[$\pm$]{}0.003 & 0.000[$\pm$]{}0.004 & 17.08[$\pm$]{}0.01 & 16.74[$\pm$]{}0.01 & 16.44[$\pm$]{}0.01 & 15.16[$\pm$]{}0.01 & 16.91[$\pm$]{}0.01 & 16.77[$\pm$]{}0.01\
B & $-$0.267[$\pm$]{}0.003 & $-$1.380[$\pm$]{}0.004 & 18.63[$\pm$]{}0.02 & 17.63[$\pm$]{}0.01 & 17.43[$\pm$]{}0.01 & 15.83[$\pm$]{}0.02 & 18.50[$\pm$]{}0.01 & 17.88[$\pm$]{}0.01\
C & 0.107[$\pm$]{}0.044 & $-$0.358[$\pm$]{}0.044 & & & & & 20.96[$\pm$]{}0.06 & 21.67[$\pm$]{}0.10\
G & $-$0.255[$\pm$]{}0.008 & $-$1.041[$\pm$]{}0.008 & 19.06[$\pm$]{}0.03 & 18.80[$\pm$]{}0.02 & 17.80[$\pm$]{}0.01 & 16.16[$\pm$]{}0.02 & 19.84[$\pm$]{}0.01 & 18.77[$\pm$]{}0.01\
[lccccccc]{} (1908.73) & 5007.61 & 75.6 & 1.6235$\pm$0.0006 & & 5008.08 & 75.6 & 1.6238$\pm$0.0006\
(2798.75) & 7359.33 & 74.3 & 1.6295$\pm$0.0004 & & 7358.66 & 73.4 & 1.6293$\pm$0.0004\
[cccccc]{} SIS+shear & $0.71$ & $0.05$ & $-35.1$ & $16.2$ & 3.81\
SIE & $0.70$ & $0.15$ & $-34.3$ & $16.4$ & 3.75\
[^1]: The second telescope of the Magellan Project; a collaboration between the Observatories of the Carnegie Institution of Washington (OCIW), University of Arizona, Harvard University, University of Michigan, and Massachusetts Institute of Technology (MIT) to construct two 6.5 Meter optical telescopes in the southern hemisphere.
|
---
abstract: 'This paper shows that wirelessly powered backscatter communications is subject to a fundamental tradeoff between the harvested energy at the tag and the reliability of the backscatter communication, measured in terms of SNR at the reader. Assuming the RF transmit signal is a multisine waveform adaptive to the channel state information, we derive a systematic approach to optimize the transmit waveform weights (amplitudes and phases) in order to enlarge as much as possible the SNR-energy region. Performance evaluations confirm the significant benefits of using multiple frequency components in the adaptive transmit multisine waveform to exploit the nonlinearity of the rectifier and a frequency diversity gain.'
author:
- 'Bruno Clerckx, Zati Bayani Zawawi and Kaibin Huang [^1]'
title: 'Wirelessly Powered Backscatter Communications: Waveform Design and SNR-Energy Tradeoff'
---
Backscatter Communications, Waveform Design, SNR-Energy Tradeoff, Wireless Power Transfer
Introduction
============
The emergence of RFID technology in the last decade is the first sign of a serious interest for far-field wireless power transfer (WPT) and backscatter communications. RFID tags harvest energy from the transmit RF signal and rely on backscattering modulation to reflect and modulate the incoming RF signal for communication with an RFID reader. Since tags do not require oscillators to generate carrier signals, backscatter communications benefit from orders-of-magnitude lower power consumption than conventional radio communications [@Smith:2013]. Backscatter communication has recently received a renewed interest, in the context of the Internet-of-Things, with advances in backscatter communication theory and the development of sophisticated backscatter communication systems [@Boyer:2014; @Yang:2015; @Kellogg:2016; @Han:2017].
Backscatter communications commonly assume that the RF transmitter generates a sinusoidal continuous wave (CW). Significant progress has recently been made on the design of efficient signals for WPT [@Clerckx:2015; @Clerckx:2016b; @Huang:2016; @Zeng:2017]. In particular, multisine waveforms adaptive to the Channel State Information (CSI) have been shown particularly powerful in exploiting the rectifier nonlinearity and the frequency-selectivity of the channel so as to maximize the amount of harvested DC power [@Clerckx:2016b].
In this paper, we depart from this traditional CW transmission and leverage those recent progress in WPT signal design, and in particular the adaptive multisine wireless power waveform design, to show that wirelessly powered backscatter communications is subject to a fundamental tradeoff between the harvested energy at the tag and the SNR at the reader. Indeed, the SNR at the reader is a function of the backscatter channel (concatenation of the forward channel from transmitter to tag and backward channel from tag to reader) while the harvested energy at the tag is a function of the forward channel only. Due to the difference between those two channels, the optimal transmit waveform design for SNR and energy maximization are different. This suggests that adjusting the transmit waveform leads to a SNR-energy tradeoff.
Specifically, assuming that the CSI is perfectly available to the RF transmitter, we derive a systematic and optimal design of the transmit multisine waveform in order to enlarge as much as possible the SNR-energy region. Due to the non-linearity of the rectifier, the waveform design and the characterization of the region results from a non-convex posynomial maximization problem that can be solved iteratively using a successive convex approximation approach. Simulation results highlight that increasing the number of sinewaves in the transmit multisine waveform enlarges the SNR-energy region by exploiting the non-linearity of the rectifier and a frequency diversity gain.
*Notations:* Bold letters stand for vectors or matrices whereas a symbol not in bold font represents a scalar. $|.|$ and $\left\|.\right\|$ refer to the absolute value of a scalar and the 2-norm of a vector. $\mathcal{E}\left\{.\right\}$ refers to the averaging operator.
System Model {#system}
============
The overall system architecture is illustrated in Fig \[fig\_sim\_0\] (left).
Received Signal at the Tag
--------------------------
Consider a multisine signal (with $N$ sinewaves) transmitted by an RF transmitter at time $t$ over a single antenna $$\begin{aligned}
\label{WPT_waveform}
x(t)=\Re\left\{\sum_{n=0}^{N-1}w_{n}e^{j2\pi f_n t}\right\},\end{aligned}$$ with $w_{n}=s_{n}e^{j\phi_{n}}$ where $s_{n}$ and $\phi_{n}$ refer to the amplitude and phase of the $n^{th}$ sinewave at frequency $f_n$, respectively. We assume for simplicity that the frequencies are evenly spaced, i.e. $f_n=f_0+n\Delta_f$ with $\Delta_f$ the frequency spacing. The magnitudes and phases of the sinewaves can be collected into vectors $\mathbf{s}$ and $\mathbf{\Phi}$. The $n^{th}$ entry of $\mathbf{s}$ and $\mathbf{\Phi}$ are written as $s_{n}$ and $\phi_{n}$, respectively. The transmitter is subject to a transmit power constraint $\mathcal{E}\big\{\left|x\right|^2\big\}=\frac{1}{2}\left\|\mathbf{s}\right\|_F^2\leq P$.
The transmit waveform propagates through a multipath channel and is received at the single-antenna tag as $$\begin{aligned}
y(t)&=\sum_{n=0}^{N-1}s_{n}A_{n} \cos(2\pi f_n t+\psi_{n})\label{received_signal_ant_m}\\
&=\Re\left\{\sum_{n=0}^{N-1} h_n w_n e^{j 2\pi f_n t}\right\}\end{aligned}$$ where $h_{n}=A_{n}e^{j \bar{\psi}_{n}}$ is the forward channel frequency response at frequency $f_n$. The amplitude $A_{n}$ and the phase $\psi_{n}$ are such that $A_{n}e^{j \psi_{n}}=A_{n}e^{j \left(\phi_{n}+\bar{\psi}_{n}\right)}=e^{j \phi_{n}}h_{n}$.
![System architecture (left) and single diode rectifier at the tag (right).[]{data-label="fig_sim_0"}](system_architecture_rev1.eps){width="0.9\columnwidth"}
![System architecture (left) and single diode rectifier at the tag (right).[]{data-label="fig_sim_0"}](rectifier_rev1.eps){width="0.9\columnwidth"}
Tag’s Operation
---------------
We assume the tag only performs binary modulation. Binary 0 corresponds to a perfect impedance matching that completely absorbs the incoming signal (i.e. the reflection coefficient is 0). The signal absorbed by the tag during binary 0 operation is conveyed to a rectifier that converts the incoming RF signal into DC current. Binary 1 corresponds to a perfect impedance mismatch that competely reflects the incoming signal (i.e. the reflection coefficient is 1). The signal reflected during binary 1 operation is backscattered to a reader, whose objective is to decide upon the sequence of transmitted bits (0 or 1).
Rectenna Model and DC Current at the Tag
----------------------------------------
We will assume the same rectenna model as in [@Clerckx:2015; @Clerckx:2016b]. The rectenna is made of an antenna and a rectifier. The antenna model reflects the power transfer from the antenna to the rectifier through the matching network. A lossless antenna can be modelled as a voltage source $v_s(t)$ followed by a series resistance $R_{ant}$. Let $Z_{in} = R_{in} + j X_{in}$ denote the input impedance of the rectifier with the matching network. Assuming perfect matching during binary operation 0 ($R_{in} = R_{ant}$, $X_{in} = 0$), all the incoming RF power $P_{in,av}$ is transferred to the rectifier and absorbed by $R_{in}$, so that $P_{in,av} = \mathcal{E}\big\{\left|v_{in}(t)\right|^2\big\}/R_{in}$ with $v_{in}(t)=v_{s}(t)/2$ the input voltage to the rectifier as per Fig \[fig\_sim\_0\] (right). Since $P_{in,av} =\mathcal{E}\big\{\left|y(t)\right|^2\big\}$, $v_{in}(t)=y(t)\sqrt{R_{in}}=y(t)\sqrt{R_{ant}}$.
Consider a rectifier composed of a single diode followed by a low-pass filter with load ($R_L$). Denoting the voltage drop across the diode as $v_d(t)=v_{in}(t)-v_{out}(t)$ where $v_{out}(t)$ is the output voltage across the load resistor (see Fig \[fig\_sim\_0\]), a tractable behavioural diode model is obtained by Taylor series expansion of the diode characteristic equation $i_d(t)=i_s \big(e^{\frac{v_d(t)}{n v_t}}-1 \big)$ (with $i_s$ the reverse bias saturation current, $v_t$ the thermal voltage, $n$ the ideality factor equal to $1.05$) around a quiescent operating point $v_d=a$, namely $$\label{DiodeCurrent}
i_d(t)=\sum_{i=0}^{\infty }k_i' \left(v_d(t)-a\right)^i,$$ where $k_0'=i_s\big(e^{\frac{a}{n v_t}}-1\big)$ and $k_i'=i_s\frac{e^{\frac{a}{n v_t}}}{i!\left(n v_t\right)^i}$, $i=1,\ldots,\infty$.
Assume a steady-state response and an ideal low pass filter such that $v_{out}(t)$ is at constant DC level. Choosing $a=\mathcal{E} \left\{ v_d(t) \right\}=-v_{out}$, can be simplified as $i_d(t)=\sum_{i=0}^{\infty }k_i' v_{in}(t)^i=\sum_{i=0}^{\infty }k_i' R_{ant}^{i/2} y(t)^i$. Truncating the expansion to order 4, the DC component of $i_{d}(t)$ is the time average of the diode current, and is obtained as $i_{out}\approx k_0'+k_2' R_{ant}\mathcal{E}\left\{y(t)^2\right\}+k_4' R_{ant}^2\mathcal{E}\left\{y(t)^4\right\}$.
Backscatter Signal and SNR at the Reader
----------------------------------------
The backscatter signal received at the reader is given by $$\begin{aligned}
\label{back_rec_signal}
z(t)&=m\Re\left\{\sum_{n=0}^{N-1} h_{r,n} h_n w_n e^{j 2\pi f_n t}\right\}+n(t)\end{aligned}$$ where $m$ equals 0 or 1 for binary operation 0 and 1, respectively. The quantity $n(t)$ is the AWGN and $h_{r,n}=A_{r,n}e^{j \bar{\psi}_{r,n}}$ is the frequency response of the backward channel (from tag to reader) on frequency $n$.
After applying a product detector to each frequency and assuming ideal low pass filtering, the baseband signal on each frequency $n$ is given by $$z_n=h_{r,n} h_n w_n m +n_n$$ where $n_n \sim \mathcal{CN}(0,\sigma^2)$. The SNR after Maximum Ratio Combining (MRC) is finally given by $$\rho\left(\mathbf{s}\right)=\frac{\sum_{n=0}^{N-1}\left|h_{r,n} h_n w_n\right|^2}{\sigma^2}=\frac{\sum_{n=0}^{N-1} A_{r,n}^2 A_n^2 s_n^2}{\sigma^2}.$$
CSIT Assumption
---------------
We assume perfect CSIT, i.e. the forward $h_n$ and backscatter $h_n h_{r,n}$ channels are perfectly known $\forall n$ to the RF transmitter, so as to shape the transmit waveform dynamically as a function of the channel states to maximize $i_{out}$ and $\rho$. The backscatter channel $h_n h_{r,n}$ can be obtained at the RF transmitter by letting the reader send pilots, reaching the RF transmitter through backscattering. Backscatter and forward channels can then be estimated and obtained at the RF transmitter [@Yang:2015].
We also assume that the concatenated channel $h_{r,n} h_n w_n$ is perfectly known to the reader to perform MRC.
Waveform Optimization and SNR-Energy Region Characterization {#section_backscatter_waveform}
============================================================
Subject to a transmit power constraint $\frac{1}{2}\left\|\mathbf{s}\right\|^2\leq P$ and under the assumption of perfect CSIT, the maximization of the SNR suggests an adaptive single-sinewave strategy (ASS) that consists in transmitting all power on a single sinewave, namely the one corresponding to the strongest channel $\bar{n} = \arg \max_i A_i A_{i,r}$. On the other hand, the maximization of the harvested energy, namely $i_{out}$, is shown in [@Clerckx:2016b] to be equivalent to maximizing the quantity $$\label{diode_model_2}
z_{DC}\left(\mathbf{s},\mathbf{\Phi}\right)=k_2 R_{ant}\mathcal{E}\left\{y(t)^2\right\}+k_4 R_{ant}^2\mathcal{E}\left\{y(t)^4\right\}$$ where $k_i=\frac{i_s}{i!\left(n v_t\right)^i}$, $i=2,4$[^2]. The maximization of suggests allocating power over multiple sinewaves, and those with stronger frequency-domain channel gains are allocated more power, in order to exploit the non-linearity of the rectifier and the frequency diversity [@Clerckx:2016b]. Hence the design of efficient waveforms for backscatter communication is subject to a tradeoff between maximizing received SNR at the reader and maximizing harvested energy at the tag. Characterizing this SNR-energy tradeoff and the corresponding waveform design is the objective of this section.
$$\begin{aligned}
\label{z_DC}
z_{DC}(\mathbf{s},\mathbf{\Phi})&=\frac{k_{2}}{2}R_{ant}\left[\sum_{n=0}^{N-1} s_{n}^2A_{n}^2\right]+\frac{3k_{4}}{8}R_{ant}^2\left[\sum_{{\genfrac{}{}{0pt}{}{n_0,n_1,n_2,n_3}{n_0+n_1=n_2+n_3}}}\Bigg[\prod_{j=0}^3s_{n_j}A_{n_j}\Bigg]\cos(\psi_{n_0}+\psi_{n_1}-\psi_{n_2}-\psi_{n_3})\right].\end{aligned}$$
We can now define the achievable SNR-harvested energy (or more accurately SNR-DC current) region as $$\begin{gathered}
C_{SNR-I_{DC}}(P)\triangleq\Big\{(SNR,I_{DC}):SNR\leq \rho(\mathbf{s}),\Big. \\
\Big.I_{DC}\leq z_{DC}(\mathbf{s},\mathbf{\Phi}), \frac{1}{2}\left\|\mathbf{s}\right\|^2\leq P \Big\}.\end{gathered}$$ Optimal values $\mathbf{s}^{\star}$,$\mathbf{\Phi}^{\star}$ are to be found in order to enlarge as much as possible $C_{SNR-I_{DC}}$. The expression of $z_{DC}$ is provided in after plugging into .
We note that the phases of the waveform $\phi_n$ influences $z_{DC}$ but not the SNR. Hence we can choose the phases as in point-to-point WPT in [@Clerckx:2015], namely $\phi_{n}^{\star}=-\bar{\psi}_{n}$. This guarantees that all arguments of the cosine functions in $z_{DC}$ are equal to 0 in , which can simply be written as $$\begin{gathered}
\label{z_DC_opt_phase}
z_{DC}(\mathbf{s},\mathbf{\Phi}^{\star})=\frac{k_{2}}{2}R_{ant}\left[\sum_{n=0}^{N-1} s_{n}^2A_{n}^2\right]\\
+\frac{3k_{4}}{8}R_{ant}^2\sum_{{\genfrac{}{}{0pt}{}{n_0,n_1,n_2,n_3}{n_0+n_1=n_2+n_3}}}\prod_{j=0}^3s_{n_j}A_{n_j}.\end{gathered}$$ $\mathbf{\Phi}^{\star}$ is obtained by collecting $\phi_{n}^{\star}$ $\forall n$ into a vector.
Recall from [@Chiang:2007] that a monomial is defined as the function $g:\mathbb{R}_{++}^{N}\rightarrow\mathbb{R}:g(\mathbf{x})=c x_1^{a_1}x_2^{a_2}\ldots x_N^{a_N}$ where $c>0$ and $a_i\in\mathbb{R}$. A sum of $K$ monomials is called a posynomial and can be written as $f(\mathbf{x})=\sum_{k=1}^K g_k(\mathbf{x})$ with $g_k(\mathbf{x})=c_k x_1^{a_{1k}}x_2^{a_{2k}}\ldots x_N^{a_{Nk}}$ where $c_k>0$. As we can see from , $z_{DC}(\mathbf{s},\mathbf{\Phi}^{\star})$ is a posynomial.
In order to identify the achievable SNR-energy region, we formulate the optimization problem as an energy maximization problem subject to transmit power and SNR constraints $$\begin{aligned}
\label{back_opt_problem}
\max_{\mathbf{s}} \hspace{0.3cm}&z_{DC}(\mathbf{s},\mathbf{\Phi}^{\star})\\
\textnormal{subject to} \hspace{0.3cm} &\frac{1}{2}\left\|\mathbf{s}\right\|^2\leq P,\\
& \rho(\mathbf{s})\geq \overline{SNR}.\end{aligned}$$ It therefore consists in maximizing a posynomial subject to constraints. Unfortunately this problem is not a standard Geometric Program (GP) but it can be transformed to an equivalent problem by introducing an auxiliary variable $t_0$ $$\begin{aligned}
\label{back_opt_problem_eq}
\min_{\mathbf{s},t_0} \hspace{0.3cm} &1/t_0\\
\textnormal{subject to} \hspace{0.3cm} &\frac{1}{2}\left\|\mathbf{s}\right\|^2\leq P,\\
&t_0/z_{DC}(\mathbf{s},\mathbf{\Phi}^{\star})\leq1,\\&\frac{\overline{SNR}}{\rho(\mathbf{s})}\leq 1.\label{back_opt_problem_eq_4}\end{aligned}$$ This is known as a Reversed Geometric Program. A similar problem also appeared in the WPT waveform optimization [@Clerckx:2015; @Clerckx:2016b] and the rate-energy region characterization of Simultaneous Wireless Information and Power Transfer [@Clerckx:2016]. Note that $1/z_{DC}(\mathbf{s},\mathbf{\Phi}^{\star})$ and $1/\rho(\mathbf{s})$ are not posynomials, therefore preventing the use of standard GP tools. The idea is to replace the last two inequalities (in a conservative way) by making use of the arithmetic mean-geometric mean inequality.
Let $\left\{g_k(\mathbf{s},\mathbf{\Phi}^{\star})\right\}$ be the monomial terms in the posynomial $z_{DC}(\mathbf{s},\mathbf{\Phi}^{\star})=\sum_{k=1}^K g_k(\mathbf{s},\mathbf{\Phi}^{\star})$. Similarly we define $\left\{f_{n}(\mathbf{s})\right\}$ as the set of monomials of the posynomial $\rho(\mathbf{s})=\sum_{n=0}^{N-1}f_{n}(\mathbf{s})$ with $f_{n}(\mathbf{s})=s_n^2 A_n^2 A_{r,n}^2/\sigma^2$. For a given choice of $\left\{\gamma_k\right\}$ and $\left\{\beta_{n}\right\}$ with $\gamma_k,\beta_{n}\geq 0$ and $\sum_{k=1}^K \gamma_k=\sum_{i=1}^{I} \beta_{n}=1$, we perform single condensations and write the standard GP as $$\begin{aligned}
\label{standard_GP_back}
\min_{\mathbf{s},t_0} \hspace{0.3cm} &1/t_0\\
\textnormal{subject to} \hspace{0.3cm} &\frac{1}{2}\left\|\mathbf{s}\right\|^2\leq P,\\
&t_0\prod_{k=1}^K\left(\frac{g_k(\mathbf{s},\mathbf{\Phi}^{\star})}{\gamma_k}\right)^{-\gamma_k}\leq1,\\
&\overline{SNR}\prod_{n=0}^{N-1}\left(\frac{f_{n}(\mathbf{s})}{\beta_{n}}\right)^{-\beta_{n}}\leq 1.\label{standard_GP_back_3}\end{aligned}$$
It is important to note that the choice of $\left\{\gamma_k,\beta_{n}\right\}$ plays a great role in the tightness of the AM-GM inequality. An iterative procedure can be used where at each iteration the standard GP - is solved for an updated set of $\left\{\gamma_k,\beta_{n}\right\}$. Assuming a feasible set of magnitude $\mathbf{s}^{(i-1)}$ at iteration $i-1$, compute at iteration $i$ $\gamma_k=g_k(\mathbf{s}^{(i-1)},\mathbf{\Phi}^{\star})/z_{DC}(\mathbf{s}^{(i-1)},\mathbf{\Phi}^{\star})$ $k=1,\ldots,K$ and $\beta_{n}= f_{n}(\mathbf{s}^{(i-1)})/\rho(\mathbf{s}^{(i-1)})$, $n=0,\ldots,N-1$, and then solve problem - to obtain $\mathbf{s}^{(i)}$. Repeat the iterations till convergence. The whole optimization procedure is summarized in Algorithm \[Algthm\_OPT\_back\]. The successive approximation method used in the Algorithm \[Algthm\_OPT\_back\] is also known as a successive convex approximation. It cannot guarantee to converge to the global solution of the original problem, but yields a point fulfilling the KKT conditions [@Chiang:2007].
**Initialize**: $i\gets 0$, $\overline{SNR}$, $\mathbf{\Phi}^{\star}$, $\mathbf{s}$, $z_{DC}^{(0)}=0$ \[Algthm\_OPT\_WIPT\_step\_initialize\] $i\gets i+1$, $\ddot{\mathbf{s}}\gets \mathbf{s}$ $\gamma_k\gets g_k(\ddot{\mathbf{s}},\mathbf{\Phi}^{\star})/z_{DC}(\ddot{\mathbf{s}},\mathbf{\Phi}^{\star})$, $k=1,\ldots,K$ \[Algthm\_OPT\_back\_step\_gamma\] $\beta_{n}\gets f_{n}(\ddot{\mathbf{s}})/\rho(\ddot{\mathbf{s}})$, $n=0,\ldots,N-1$ \[Algthm\_OPT\_back\_step\_gamma\_2\] $\mathbf{s} \gets \arg \min \eqref{standard_GP_back}-\eqref{standard_GP_back_3}$ \[Algthm\_OPT\_back\_step\_OPT\] $z_{DC}^{(i)} \gets z_{DC}(\mathbf{s},\mathbf{\Phi}^{\star})$
Simulation Results {#simulations}
==================
We consider a centre frequency of 5.18GHz, 36dBm EIRP, 2dBi receive and transmit antenna gain at the tag and 2dBi receive antenna gain at the reader. The path loss between the transmitter and the tag and between the tag and the reader is 58dB for each link. A NLOS channel power delay profile is obtained from model B [@Medbo:1998b]. The channel taps each with an average power $\beta_l$ are independent, circularly symmetric complex random Gaussian distributed and normalized such that $\sum_l \beta_l=1$. This leads to an average receive power of -20dBm at the tag and -74dBm at the reader. The noise power $\sigma^2$ at the reader is fixed to -84dB. The simulation is run over a channel realization with a bandwidth $B=1,10$ MHz. The frequency responses of the forward and backward channels are illustrated in Fig \[fig\_sim\_1\]. The channel frequency response within the 1 MHz bandwisth is obtained by looking at Fig 1 between -0.5 MHz and 0.5 MHz. The frequency spacing of the multisine waveform is fixed as $\Delta_f=B/N$ and the $N$ sinewaves are centered around 5.18 GHz.
![Channel frequency responses over a 10 MHz bandwidth.[]{data-label="fig_sim_1"}](Freq_response_new.eps){width="0.9\columnwidth"}
![SNR-$I_{DC}$ trade-off with B=1MHz and B=10MHz.[]{data-label="fig_sim_2"}](snr_idc_1M_10M_a.eps){width="0.9\columnwidth"}
For the channel frequency responses of Fig \[fig\_sim\_1\], Algorithm \[Algthm\_OPT\_back\] is used, along with CVX [@CVX], to compute the optimal waveform and the corresponding SNR-$I_{DC}$ tradeoff, illustrated in Fig \[fig\_sim\_2\] for B=1MHz and B=10MHz. The extreme point on the x-axis (SNR maximization) is achieved using the ASS strategy. On the other hand, the maximum energy is in general achieved by allocating transmit power over multiple subcarriers (as a consequence of the non-linearity of the rectifier) [@Clerckx:2016b]. A *first* observation from Fig \[fig\_sim\_2\] is that SNR and $I_{DC}$ are indeed subject to a fundamental tradeoff, i.e. increasing one of them is likely to result in a decrease of the other one. Nevertheless, as the channel becomes more frequency flat or the bandwidth decreases, the SNR-$I_{DC}$ appears more rectangular. A *second* observation is that an increase in the number of frequency components $N$ of the multisine waveform results in an enlarged SNR-energy region. Indeed, by increasing $N$, the waveform exploits the nonlinearity of the rectifier and a frequency diversity gain, the latter being beneficial to both SNR and energy. A *third* observation is that the shape of the SNR-energy region highly depends on the channel realizations and bandwidth. In particular, for the specific channel realization of Fig \[fig\_sim\_1\], we note that the 1MHz bandwidth favours higher $I_{DC}$ while the 10 MHz bandwidth favours higher SNR. This can be explained as follows. Recall first that $I_{DC}$ is a function of the forward channel amplitudes $A_n$ $\forall n$, while the SNR is a function of the backscatter channel $A_n A_{r,n}$. From Fig \[fig\_sim\_1\], $A_n$ reaches its peak for frequencies between -1 MHz and 0.5 MHz. Since the multisine waveform with a power allocation over multiple frequency components helps increasing $I_{DC}$, allocating the $N$ frequencies uniformly within the 1MHz bandwidth leads to higher $I_{DC}$ than that obtained with a 10MHz bandwidth (which exhibits deep fades). On the other hand, $A_n A_{r,n}$ exhibits its largest gain around 2MHz, which is outside the 1MHz bandwidth. Since ASS maximizes the SNR, larger SNRs are obtained on the 10MHz channel.
Conclusions
===========
The paper derived a methodology to design adaptive transmit multisine waveforms for backscatter communications and characterize the fundamental tradeoff between conveying energy to the tag and enhancing the SNR of the backscatter communication link. Future interesting works consist in addressing the design of waveforms and the characterization of the SNR-energy region for more general setup including multiple antennas, multiple transmitters and multiple tags. The problem of CSI acquisition and its impact on the SNR-energy region is also of significant interest.
[1]{}
J.R. Smith, “Wirelessly Powered Sensor Networks and Computational RFID,” New York Springer 2013. C. Boyer and S. Roy, “Backscatter communication and RFID: Coding, energy, and MIMO analysis,” IEEE Trans. Commun., vol. 62, pp. 770-785, Mar. 2014. G. Yang, C.K. Ho and Y.L. Guan, “Multi-antenna Wireless Energy Transfer for Backscatter Communication Systems,” IEEE Journal on Sel. Areas in Comm., Vol. 33, No. 12, Dec 2015. B. Kellogg, V. Talla, S. Gollakota and J.R. Smith, “Passive Wi-Fi: Bringing Low Power to Wi-Fi Transmissions,” 13th USENIX Symp. on Networked Systems Design and Implementation, March 2016. K. Han, K. Huang, “Wirelessly Powered Backscatter Communication Networks: Modeling, Coverage and Capacity,” to be published in IEEE Trans. on Wireless Commun. B. Clerckx, E. Bayguzina, D. Yates, and P.D. Mitcheson, “Waveform Optimization for Wireless Power Transfer with Nonlinear Energy Harvester Modeling,” IEEE ISWCS 2015, August 2015, Brussels. B. Clerckx and E. Bayguzina, “Waveform Design for Wireless Power Transfer,” IEEE Trans on Sig Proc, Vol. 64, No. 23, Dec 2016. Y. Huang and B. Clerckx, “Waveform optimization for large-scale multi-antenna multi-sine wireless power transfer,” in Proc. IEEE SPAWC 2016. Y. Zeng, B. Clerckx and R. Zhang, “Communications and Signals Design for Wireless Power Transmission,” IEEE Trans. on Comm, 2017. B. Clerckx, “Waveform Optimization for SWIPT with Nonlinear Energy Harvester Modeling,” 20th International ITG Workshop on Smart Antennas, March 2016, Munich. M. Chiang, C. W. Tan, D. P. Palomar, D. O. Neill, and D. Julian, “Power control by geometric programming,” IEEE Trans. Wireless Commun., Vol. 6, No. 7, pp. 2640-2651, Jul. 2007. M. Grant, S. Boyd, and Y. Ye, “CVX: MATLAB software for disciplined convex programming \[Online\],” Available: http://cvxr.com/cvx/, 2015. J. Medbo, P. Schramm, “Channel Models for HIPERLAN/2 in Different Indoor Scenarios,” 3ERI085B, ETSI EP BRAN, March 1998.
[^1]: B. Clerckx and Z. Bayani Zawawi are with the EEE department at Imperial College London, London SW7 2AZ, UK (email: {b.clerckx,z.zawawimohd-zawawi13}@imperial.ac.uk). K. Huang is with the EEE department at University of Hong Kong (email: huangkb@eee.hku.hk). This work has been partially supported by the EPSRC of the UK under grant EP/P003885/1.
[^2]: Assuming $i_s=5 \mu A$, a diode ideality factor $n=1.05$ and $v_t=25.86 mV$, typical values are given by $k_2=0.0034$ and $k_4=0.3829$.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.