論文

Stochastic large-scale interacting systems can be studied via the observables, i.e. functions on the underlying configuration space. In our previous article (Bannai et al., 2024), we introduced the concept of uniform functions, which is a suitable class of functions on configuration spaces underlying stochastic systems on infinite graphs. An important consequence is the successful characterization of conserved quantities without introducing the notion of stationary distributions. In this article, we further develop the theory of uniform functions and construct the theory independently of any choice of a base state. Furthermore, we generalize the notion of interactions given in Bannai et al. (2024) to accommodate the case where there are multiple possible state transitions on adjacent vertices. We then prove that if the interaction is exchangeable, then any uniform function which gives a global conserved quantity can be expressed as a sum of local conserved quantities of the interaction. Contrary to our previous article, we do not need to assume that the interaction is irreducibly quantified. This shows that our theory of uniform functions on configuration spaces over infinite graphs with transition structure given by an exchangeable interaction is a natural framework to study general stochastic large-scale interacting systems. While some of the ideas in this article are based on Bannai et al. (2024), the current article is logically independent and self-contained.

Statistical mechanics explains the properties of macroscopic phenomena based on the movements of microscopic particles such as atoms and molecules. Movements of microscopic particles can be represented by large-scale interacting systems. In this article, we systematically study combinatorial objects which we call interactions, given as symmetric directed graphs representing the possible transitions of states on adjacent sites of large-scale interacting systems. Such interactions underlie various standard stochastic processes such as the exclusion processes, generalized exclusion processes, multi-species exclusion processes, lattice gas with energy processes, and the multi-lane exclusion processes. We introduce the notion of equivalences of interactions using their space of conserved quantities. This allows for the classification of interactions reflecting the expected macroscopic properties. In particular, we prove that when the set of local states consists of two, three or four elements, then the number of equivalence classes of separable interactions are respectively one, two and five. We also define the wedge sums and box products of interactions, which give systematic methods for constructing new interactions from existing ones. Furthermore, we prove that the irreducibly quantified condition for interactions, which implicitly plays an important role in the theory of hydrodynamic limits, is preserved by wedge sums and box products. Our results provide a systematic method to construct and classify interactions, offering abundant examples suitable for considering hydrodynamic limits.

The purpose of this article is to newly define the p-adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special values of p-adic L-functions interpolating nonpositive values of Hecke L-functions of the totally real field in terms of special values of these p-adic polylogarithms.

It is known that the special values at nonpositive integers of a Dirichlet L-function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L- functions of totally real fields. Hecke L-functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393—417], we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shin- tani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.References

In this article, we construct the Hodge realization of the polylogarithm class in the equivariant Deligne–Beilinson cohomology of a certain algebraic torus associated to a totally real field. We then prove that the de Rham realization of this polylogarithm gives the Shintani generating class, a cohomology class generating the values of the Lerch zeta functions of the totally real field at nonpositive integers. Inspired by this result, we give a conjecture concerning the specialization of this polylogarithm class at torsion points, and discuss its relation to the Beilinson conjecture for Hecke characters of totally real fields.

In this article, we investigate the topological structure of large-scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large-scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan's decomposition theorem for a general class of large-scale interacting systems.