Ronkin/Zeta Correspondence

TL;DR

This paper establishes a novel connection between the Ronkin function and zeta functions for random and quantum walks, expanding the mathematical understanding of these functions in relation to quantum and classical stochastic processes.

Contribution

It introduces the first known relation linking the Ronkin function with zeta functions for quantum and random walks, including higher-dimensional cases.

Findings

Established a relation between Ronkin function and zeta functions for 1D random walks

Extended the relation to higher-dimensional random walks

Compared the quantum walk case with classical random walks

Abstract

The Ronkin function was defined by Ronkin in the consideration of the zeros of almost periodic function. Recently, this function has been used in various research fields in mathematics, physics and so on. Especially in mathematics, it has a closed connections with tropical geometry, amoebas, Newton polytopes and dimer models. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on Zeta Correspondence. The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Ronkin function and our zeta function for random walks and quantum walks. Firstly we consider this relation in the case of one-dimensional random walks. Afterwards we deal with higher-dimensional random walks. For comparison with the case of the quantum walk, we also treat the case of one-dimensional quantum walks. Our results bridge between the Ronkin function and the zeta function via quantum walks for the first time.