Random Walks in the Quarter-Plane: Advances in Explicit Criterions for the Finiteness of the Associated Group in the Genus 1 Case

TL;DR

This paper provides a concrete criterion, based on determinant cancellation, to determine when the group associated with a genus 1 random walk in the quarter-plane is finite, advancing theoretical understanding of such stochastic processes.

Contribution

It introduces a specific, polynomial-based criterion for finiteness of the walk's group in the genus 1 case, extending previous theoretical frameworks.

Findings

Finiteness criterion expressed as determinant cancellation

Criterion depends polynomially on walk coefficients

Applicable to algebraic curves of genus 1

Abstract

In the book [FIM], original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps, the general solution being obtained via reduction to boundary value problems. Among other things, an important quantity, the so-called group of the walk, allows to deduce theoretical features about the nature of the solutions. In particular, when the \emph{order} of the group is finite, necessary and sufficient conditions have been given in [FIM] for the solution to be rational or algebraic. In this paper, when the underlying algebraic curve is of genus $1$, we propose a concrete criterion ensuring the finiteness of the group. It turns out that this criterion can be expressed as the cancellation of a determinant of a matrix of order 3 or 4, which depends in a polynomial way on the coefficients of the walk.