Random walks on crystal lattices and multiple zeta functions

TL;DR

This paper introduces multidimensional zeta functions on crystal lattices to analyze random walks, enabling explicit construction of various multidimensional discrete distributions and extending classical lattice theories.

Contribution

It develops new multidimensional Euler and Shintani zeta functions directly on crystal lattices, expanding the analytical tools for random walks and discrete distributions in Euclidean spaces.

Findings

Defined multidimensional Euler and Shintani zeta functions on crystal lattices.

Constructed random walks with infinite and finite ranges using these zeta functions.

Provided explicit examples illustrating the application of the theory.

Abstract

Crystal lattices are known to be one of the generalizations of classical periodic lattices which can be embedded into some Euclidean spaces properly. As to make a wide range of multidimensional discrete distributions on Euclidean spaces more treatable, multidimensional Euler products and multidimensional Shintani zeta functions on crystal lattices are introduced. They are completely different from existing Ihara zeta functions on graphs in that our zeta functions are defined on crystal lattices directly. Via a concept of periodic realizations of crystal lattices, we make it possible to provide many kinds of multidimensional discrete distributions explicitly. In particular, random walks on crystal lattices whose range is infinite and such random walks whose range is finite are constructed by multidimensional Euler products and multidimensional Shintani zeta functions, respectively. We give some comprehensible examples as well.