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However, in the parabolic barrier crossing, the velocity increases rapidly (exponentially), therefore, the contribution from the memory kernel integral is dominated by the most recent term only. This implies that in the long time scale, we should expect an effective description, in which the system feels only the insta... |
[EQUATION] where [MATH] is a renormalized friction coefficient. Our argument suggests that it is this renormalization that is behind the universal exponential decay in the long time limit of transition path time distribution. This asymptotic behavior sets already in at [MATH] as seen numerically (Fig. ). |
5.2 Comparison with experiments: possible implications Differently from the Kramers’ time, which is characterised by an exponential dependence on the barrier height [MATH] , the average TPT in the overdamped limit scales logarithmically [MATH] , where [MATH] is the inverse temperature. We have shown here that the logar... |
We believe that our results are helpful in interpreting experimental results. Indeed the barrier height as determined from a comparison between experiments and a model for diffusion in a parabolic potential without memory terms gave values that were much lower than those determined by other means. Our calculations have... |
{acknowledgement} Discussions with M. Caraglio and M. Laleman are gratefully acknowledged. Appendix A: Power law memory kernel The solution of the generalized Langevin equation ( ) with power law memory kernel is obtained by performing its Laplace transform |
[EQUATION] where [MATH] indicates the Laplace transform of the function [MATH] . To obtain the previous equation we used the convolution theorem (the Laplace transform of a convolution product is the product of the Laplace transforms) and the fact that the Laplace transform of a derivative is |
[EQUATION] (in our case the initial condition is [MATH] ). Solving ( 33 ) we get [EQUATION] The Laplace transform of the power law kernel ( ) is |
[EQUATION] therefore Eq. ( 35 ) takes the form [EQUATION] To perform the inverse transform we use the following relation [EQUATION] |
where [MATH] is known as Mittag-Leffler function 32 . To handle the two terms in the left hand side of ( 37 ) one can use ( 38 ) with [MATH] |
and [MATH] . For this purpose it is convenient to introduce the functions [EQUATION] where [MATH] is a characteristic rate of the process. Inverting ( 37 ) we get |
[EQUATION] Averaging over noise we get the average position, or equivalently the deterministic solution [EQUATION] while the variance ( ) is |
[EQUATION] (the details of the calculation of this integral are given in Appendix ). We finally combine the above results to find |
[MATH] , see ( [EQUATION] where [MATH] is the barrier height. This proves Eq. ( 15 of the main text. The Mittag-Leffler function behaves asymptotically as 32 |
[EQUATION] which implies [EQUATION] Hence [EQUATION] For small arguments, the Mittag-Leffler function behaves as 32 [EQUATION] hence |
[EQUATION] This implies that [MATH] diverges for small [MATH] [EQUATION] 6.1 The average transition path time To calculate the average TPT we follow the calculation outlined in 18 |
[EQUATION] where we have made the change of variables [MATH] and used the definition of the error function. The integral in the numerator can not be performed exactly. We can get an approximation for [MATH] |
where the integrals are determined by the large [MATH] -limit of [MATH] From ( 39 ), ( 44 ) and ( 45 ) one gets for [MATH] large |
[EQUATION] which can be inverted to [EQUATION] If we insert ( 54 ) into ( 52 ) and make also here an expansion for large [MATH] we finally get |
[EQUATION] where [MATH] is the Euler-Mascheroni constant. 6.2 The most likely transition path time Another interesting quantity we can infer from the results is |
[MATH] the most likely value of the TPT. This is obtained by solving the equation [EQUATION] which from ( ) implies [MATH] (where the dot indicates the time derivative), or using ( 15 ): |
[EQUATION] Using the asymptotic [MATH] expansion ( 46 ) one has [EQUATION] and to leading order in [MATH] the solution of ( 57 ) becomes |
[EQUATION] Appendix B: Integral ( 43 To compute the integral ( 43 ) we start from the double Laplace transform of the function [MATH] We have |
[EQUATION] To get rid of the absolute value we split the integral in two domains so to obtain [EQUATION] The integrals can be easily computed using a change of variables and the property ( 38 ). The above expression can be rearranged as follows |
[EQUATION] The double inverse Laplace transform of the first term is easy as this term is the product of a function of [MATH] and a function of [MATH] . One has two independent inverse Laplace transform and from ( 38 one sees that this generates [MATH] |
The second term in ( 62 ) is a product of two fractions. In the first one, one recognises the double Laplace transform of [MATH] . For the second one we use |
[EQUATION] Invoking the convolution theorem of double Laplace transforms, the second term of ( 62 ) is therefore the double Laplace transform of the convolution [EQUATION] Putting everything together we have [EQUATION] from which ( 43 ) follows by putting [MATH] |
# Source: arxiv 1806.11159 # Title: A1-invariants in Galois cohomology and a claim of Morel # Sections: all # Downloaded: 2026-03-03T05:20:01.499226+00:00 |
[MATH] -invariants in Galois cohomology and a claim of Morel Abstract We establish variants of the results in for invariants taking values in a strictly homotopy invariant sheaf. As an application, we prove the folklore result of Morel that [MATH] |
[MATH] -invariants of Étale Algebras This entire section is basically a minor variation of arguments from . Related results were also obtained by Morel (unpublished) and Hirsch , Theorem 2.3.12] |
Throughout, we fix a field [MATH] . Let [MATH] denote the category of smooth [MATH] -schemes and [MATH] the category of presheaves on [MATH] . As usual, if [MATH] and [MATH] is an essentially smooth [MATH] -scheme, then |
[MATH] makes sense. Definition We call [MATH] homotopy invariant if [MATH] for all [MATH] Definition We call [MATH] unramified if [MATH] , for all connected [MATH] the canonical map [MATH] is injective, and moreover |
[EQUATION] (intersection in [MATH] ). Remark 1 Suppose that [MATH] is perfect and [MATH] is a strictly homotopy invariant sheaf of abelian groups. Then [MATH] is unramified , Lemma 6.4.4] |
In this section we are interested in the presheaf [MATH] which assigns to [MATH] the set of isomorphism classes of étale [MATH] -schemes, everywhere of rank [MATH] . (This is neither homotopy invariant nor unramified, of course.) |
Definition A rank [MATH] versal étale scheme is an étale morphism [MATH] , everywhere of rank [MATH] , such that if [MATH] is any rank [MATH] étale algebra over a finitely generated field extension [MATH] , then [MATH] is obtained from [MATH] by pullback along a morphism [MATH] |
We will make good use of the following result. Theorem 2 , Proposition 24.6(2)) There exists a smooth [MATH] -scheme [MATH] , an irreducible divisor [MATH] and a finite étale morphism [MATH] of rank [MATH] , such that [MATH] is versal. |
Moreover, let [MATH] be the generic point of [MATH] [MATH] the Henselization of [MATH] in the generic point of [MATH] and [MATH] the generic point of |
[MATH] . The finite étale [MATH] -scheme [MATH] splits as a disjoint union [MATH] with [MATH] of rank two (unless [MATH] ). Proof. |
Let us review the construction of [MATH] . Let [MATH] be the evident map. Let [MATH] be the branch locus of [MATH] ; the reference proves that this is an irreducible divisor. Since étale algebras over fields are simple, it is clear that [MATH] is versal. |
Let [MATH] be the completion of [MATH] and write [MATH] for the generic point of [MATH] . The reference proves that [MATH] , with [MATH] of rank two and [MATH] |
unramified . What this means is that there exists a finite étale morphism [MATH] with [MATH] , beginning of Section 11, Proposition 24.2(3) and Definition 24.3] . Since [MATH] is a morphism of henselian local rings inducing an isomorphism on residue fields, it follows that there exists a finite étale morphism [MATH] wi... |
, Tag 04GK] . Lemma (2) below then furnishes us with a retraction [MATH] . Thus [MATH] splits as [MATH] , and [MATH] must have the same degree as [MATH] , i.e. 2. This concludes the proof. |
We used the following result, which is surely well-known. Lemma 3 Let [MATH] be a henselian DVR with completion [MATH] , fraction field [MATH] and completed fraction field [MATH] |
1. Let [MATH] be an étale algebra with completion [MATH] . If [MATH] satisfies a separable polynomial with coefficients in [MATH] , then [MATH] |
2. Let [MATH] be étale [MATH] -algebras. Then [MATH] Proof. (1) We may assume that [MATH] is a field. The normalization [MATH] of [MATH] in [MATH] is finite , Tag 032L] , and hence [MATH] , being a domain, is local henselian |
, Tag 04GH(1)] . We may thus replace [MATH] by [MATH] and assume that [MATH] . Let [MATH] be a uniformizer of [MATH] ; then [MATH] It follows that [MATH] for [MATH] sufficiently large and still satisfies a separable polynomial; hence we may assume that [MATH] . This reduced statement is a well-known characterisation of... |
(2) We may assume that [MATH] , where [MATH] is a separable polynomial. Then [MATH] is the set of elements [MATH] with [MATH] . By (1), such [MATH] lie in [MATH] . It follows that [MATH] is surjective. Injectivity is clear since [MATH] |
etc. are all injective. Lemma 4 Let [MATH] be the localisation of a smooth scheme in a point of codimension one. Write [MATH] for the Henselization, [MATH] for the generic point and |
[MATH] for the generic point of [MATH] . If [MATH] is a Nisnevich sheaf, then the following diagram is cartesian: [EQUATION] Proof. |
Let [MATH] be an étale neighbourhood of the closed point, and [MATH] the generic point of [MATH] . Then [EQUATION] is a distinguished Nisnevich square; hence applying [MATH] yields a cartesian square. Since [MATH] is obtained as the filtered inverse limit of the [MATH] and filtered colimits (of sets) commute with finit... |
Recall that an étale algebra [MATH] is called multiquadratic if it is a (finite) product of copies of [MATH] and quadratic separable extensions of [MATH] |
Corollary 5 Let [MATH] be a homotopy invariant, unramified Nisnevich sheaf of sets and [MATH] any morphism (of presheaves of sets). Assume there exists [MATH] such that for any field [MATH] and any multiquadratic étale algebra [MATH] we have [MATH] . Then for any [MATH] and any [MATH] we have [MATH] |
Proof. This is essentially the same as the proof of , Theorem 24.4] Since [MATH] is unramified, it suffices to prove the claim when [MATH] is the spectrum of a field. We proceed by induction on [MATH] . If [MATH] there is nothing to do. |
Suppose now that [MATH] is a field, [MATH] and [MATH] with [MATH] multiquadratic (but non-zero). Define an invariant [MATH] of [MATH] over [MATH] via |
[MATH] . By assumption [MATH] if [MATH] is multiquadratic, hence [MATH] by induction. We conclude that [MATH] Now let [MATH] be the versal morphism from Theorem . We consider [MATH] , where [MATH] . I claim that if [MATH] is a point of codimension one, then [MATH] is in the image of [MATH] . If [MATH] this is clear, be... |
Since [MATH] is unramified, it follows that [MATH] lies in the image of [MATH] . But [MATH] , and so there exists [MATH] such that [MATH] . Since [MATH] is versal this implies that [MATH] for any [MATH] and any [MATH] . But [MATH] , so [MATH] . This concludes the induction step. |
Application: computing [MATH] Now let [MATH] be a perfect field of characteristic not two. We write [MATH] for the category of strictly homotopy invariant Nisnevich sheaves (of abelian groups) on [MATH] [MATH] for the category of presheaves of monoids, with morphisms the morphisms of monoids. We have obvious forgetful ... |
[EQUATION] We write [MATH] , and [MATH] . Then the functors [MATH] and [MATH] have (potentially partially defined) left adjoints denoted |
[MATH] and [MATH] . We make use of the following result of Morel. Theorem 6 , Theorem 3.46) The morphism of presheaves of sets [MATH] [MATH] exhibits [MATH] as [MATH] |
Let [MATH] denote the presheaf of monoids [MATH] ; the monoidal operation is given by disjoint union of étale schemes. For an étale algebra [MATH] , denote by [MATH] the class of its trace form. We shall now prove the result advertised in the heading, in the following guise. |
Proposition 7 Let [MATH] be a perfect field of characteristic not two. The morphism of presheaves of monoids [MATH] [MATH] exhibits [MATH] as [MATH] |
Proof. Let [MATH] be any morphism, where [MATH] is arbitrary. For [MATH] let [MATH] . Since we are in characteristic not two, [MATH] is étale. Define [MATH] by mapping [MATH] to |
[EQUATION] (The reason for this formula is that [MATH] , and hence [MATH] .) By Theorem this induces [MATH] . I claim that the following diagram commutes: |
[EQUATION] By Corollary it suffices to show this for multi-quadratic algebras over fields [MATH] . But we are dealing with morphisms of monoids into unramified sheaves, so it suffices to show this for [MATH] and [MATH] , where [MATH] is a finitely generated field extension. Now |
[EQUATION] since [MATH] and [MATH] are isomorphic. Finally [EQUATION] (note that [MATH] ). Hence to prove the claim we need to show that [MATH] . Consider [MATH] . Then, applying Theorem again, we get a commutative diagram |
[EQUATION] Since [MATH] (indeed [MATH] ) and [MATH] , this proves the claim. Consequently we have proved that any morphism [MATH] factors through [MATH] . Since the image of [MATH] generates [MATH] (as an unramified sheaf of abelian groups), this factorization is unique. This concludes the proof. |
# Source: arxiv 1806.11282 # Title: Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs # Sections: all # Downloaded: 2026-03-03T01:44:29.609918+00:00 |
Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs Abstract We study the problem of approximating the Ising model partition function with complex parameters on bounded degree graphs. We establish a deterministic polynomial-time approximation scheme for the partition function when the inte... |
Introduction The Ising model partition function plays an important role in combinatorics and statistical physics. In this paper we study the problem of approximating the Ising model partition function in the complex parameter regime on bounded degree graphs. This work is motivated by the close relationship to quantum c... |
We establish a deterministic polynomial-time approximation scheme for the Ising model partition function on bounded degree graphs when the interactions and external fields are absolutely bounded close to zero (Corollary . This provides a lower bound on when the interactions and external fields cause approximations of t... |
Barvinok’s approach considers the Taylor expansion of the logarithm of a polynomial about an easy to evaluate point. Suppose that we can show that the complex zeros of the polynomial lie in the exterior of a closed disc centred at this point, then it follows that a truncated Taylor expansion provides an additive approx... |
To construct an algorithm from this approach we need to be able to compute the coefficients of the truncated Taylor expansion. Barvinok showed that computing these coefficients can be reduced to computing the leading coefficients of the polynomial itself. However, to achieve the accuracy required for an approximation s... |
Patel and Regts Patel and Regts ( 2017 showed that, for several classes of graph polynomials on bounded degree graphs, the leading coefficients can be computed in polynomial time. Their approach is based on expressing the coefficients as linear combinations of connected induced subgraph counts of size logarithmic in th... |
Barvinok and Soberón Barvinok and Soberón ( 2017 established a deterministic quasi-polynomial time algorithm for approximating the multivariate graph homomorphism partition function on bounded degree graphs when the matrix entries are absolutely bounded close to one. In the case that all matrix entries are exactly equa... |
In order to establish a polynomial-time approximation scheme for the Ising model partition function, we provide an approximation-preserving polynomial-time reduction to a restricted version of the multivariate graph homomorphism partition function (Proposition . We extend the results of Barvinok and Soberón Barvinok an... |
Previous work by Liu, Sinclair, and Srivastava Liu et al. 2019 studied the problem of approximating the ferromagnetic Ising model partition function based on the location of complex zeros. They gave a deterministic polynomial-time approximation scheme for the Ising model partition function in the ferromagnetic regime f... |
Further work has considered the problem of approximating the Ising model partition function on bounded degree graphs based on the decay of correlations property. Sinclair, Srivastava, and Thurley Sinclair et al. 2014 established a deterministic polynomial-time approximation scheme for the anti-ferromagnetic Ising model... |
Our final result is a polynomial-time algorithm for approximating certain output probability amplitudes of quantum circuits (Corollary . This algorithm is based on the observation that complex-valued Ising model partition functions arise in the output probability amplitudes of quantum circuits De las Cuevas et al. 2011... |
This paper is structured as follows. In Section II , we introduce the multivariate graph homomorphism partition function and establish a deterministic polynomial-time algorithm for approximating a restricted version of this partition function on bounded degree graphs when the matrix entries are absolutely bounded close... |
II Graph Homomorphism Partition Functions graph homomorphism between two graphs [MATH] and [MATH] is an adjacency-preserving map between the vertex sets, i.e., a map [MATH] such that [MATH] implies [MATH] . Graph homomorphisms generalise the notion of graph colouring Hell and Nešetřil ( 2004 ; for example, a graph homo... |
Hell and Nešetřil Hell and Nešetřil ( 1990 proved that the problem of deciding if a homomorphism between two graphs [MATH] and [MATH] exists is NP -complete . Dyer and Greenhill Dyer and Greenhill ( 2000 showed that the corresponding counting problem is #P -hard , unless the graph has some special structure; otherwise ... |
Definition 1 (Graph homomorphism partition function) Let [MATH] be a graph and let [MATH] be a [MATH] symmetric matrix. Then the graph homomorphism partition function is defined by |
[EQUATION] The graph homomorphism partition function evaluates to many important combinatorial quantities, including counting the number of graph homomorphisms, proper colourings, and independent sets Barvinok ( 2016b |
The complexity of computing graph homomorphism partition functions has been widely studied. Dyer and Greenhill Dyer and Greenhill ( 2000 showed that computing [MATH] when [MATH] is a fixed symmetric binary matrix is either in or #P -hard . Moreover, they showed that these hardness results hold even for graphs of maximu... |
The graph homomorphism partition function can be generalised by assigning a [MATH] symmetric matrix to each edge. The multivariate graph homomorphism partition function is defined as follows. |
Definition 2 (Multivariate graph homomorphism partition function) Let [MATH] be a graph with the [MATH] symmetric matrices [MATH] assigned to its edges. Then the multivariate graph homomorphism partition function is defined by |
[EQUATION] When the matrices are all equal, it is clear that the multivariate and standard graph homomorphism partition functions are equivalent. |
For convenience, let us define the polydisc consisting of all sets of [MATH] symmetric matrices with matrix entries absolutely bounded close to one. |
Definition 3 [MATH] For a graph [MATH] [MATH] , and [MATH] , we define [MATH] to be the closed polydisc consisting of all sets of [MATH] symmetric matrices [MATH] , such that [MATH] for all [MATH] and all [MATH] |
Barvinok and Soberón Barvinok and Soberón ( 2017 gave a quasi-polynomial time algorithm for approximating [MATH] when [MATH] is a graph of maximum degree at most [MATH] and [MATH] lies in the interior of the closed polydisc [MATH] . Here, [MATH] is an absolute constant. The absolute constants come from Barvinok’s monog... |
Definition 4 [MATH] For [MATH] , we define the absolute constant [MATH] by [EQUATION] Remark A simple numerical search gives [MATH] [MATH] [MATH] , and [MATH] . In general, we have [MATH] |
We shall consider a restricted version of the multivariate graph homomorphism partition function, in which the sum is restricted to map a subset of vertices to a fixed index. |
Definition 5 (Restricted multivariate graph homomorphism partition function) Let [MATH] be a graph with the [MATH] symmetric matrices [MATH] assigned to its edges. Further let [MATH] be a subset of [MATH] and let [MATH] be an integer. Then the restricted multivariate graph homomorphism partition function is defined by |
[EQUATION] The advantage of considering the restricted multivariate graph homomorphism partition function is that, when reduced from the Ising partition function, it will allows us to implement an external magnetic field. This reduction is described in detail in Appendix |
We extend the results of Barvinok and Soberón Barvinok and Soberón ( 2017 and Patel and Regts Patel and Regts ( 2017 to give a deterministic polynomial-time approximation scheme for the restricted multivariate graph homomorphism partition function. We have the following theorem. |
Theorem 1 (restate=[name=restatement]GraphHomomorphismPartitionFunctionBoundedDegreeGraphs) Fix [MATH] and [MATH] . There is a deterministic polynomial-time approximation scheme for the restricted multivariate graph homomorphism partition function [MATH] for all graphs [MATH] of maximum degree at most [MATH] and all [M... |
We prove Theorem in Appendix . Our proof requires a result of Barvinok (Barvinok, 2016b , Theorem 7.1.4) , which states that [MATH] does not vanish on graphs of maximum degree at most [MATH] when [MATH] lies in the interior of the closed polydisc [MATH] |
Lemma 2 (Barvinok Barvinok ( 2016b Fix [MATH] . For any graph [MATH] of degree at most [MATH] and any [MATH] in the closed polydisc [MATH] , the restricted multivariate graph homomorphism partition function does not vanish, i.e., [MATH] for all [MATH] and all [MATH] |
Our proof also requires the following lemma, which states that we can efficiently compute the constant term and inverse power sums of the roots of [MATH] |
Lemma 3 (restate=[name=restatement]GraphHomomorphismPartitionFunctionComputeConstantTermAndInversePowerSums) Fix [MATH] [MATH] , and [MATH] . Let [MATH] be a graph of maximum degree at most [MATH] with the [MATH] symmetric matrices [MATH] assigned to its edges. Further let [MATH] be the roots of the polynomial [MATH] .... |
We prove Lemma in Appendix . For convenience, let us define the closed disc [MATH] of radius [MATH] centred at the origin. Definition 6 |
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