Random walks on crystal lattices and multiple zeta functions

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.