Logarithmic terms in discrete heat kernel expansions in the quadrant

TL;DR

This paper develops a new method to compute asymptotic expansions of lattice walk counts in the quarter plane, revealing for the first time the presence of logarithmic terms in such asymptotics.

Contribution

It introduces a novel approach using elliptic functions and Jacobi transformation to derive complete asymptotic expansions, including logarithmic terms, for lattice walks in the quadrant.

Findings

Logarithmic terms appear in the asymptotics of certain lattice walk models.

The method applies to both algebraic and infinite group models.

Coefficients relate to polyharmonic functions.

Abstract

In the context of lattice walk enumeration in cones, we consider the number of walks in the quarter plane with fixed starting and ending points, prescribed step-set and given length. After renormalization, this number may be interpreted as a discrete heat kernel in the quadrant. We propose a new method to compute complete asymptotic expansions of these numbers of walks as their length tends to infinity, based on two main ingredients: explicit expressions for the underlying generating functions in terms of elliptic Jacobi theta functions along with a duality known as Jacobi transformation. This duality allows us to pass from a classical Taylor expansion of the series to an expansion at the critical point of the model. We work through two examples. First, we present our approach on the well-known Kreweras model, which is algebraic, and show how to obtain a complete asymptotic expansion in this case. We then consider a more generic (so-called infinite group) model, and find the associated complete asymptotic expansion. In this second case, we prove the existence of logarithmic terms in the asymptotic expansion, and we relate the coefficients appearing in the expansion to polyharmonic functions. To our knowledge, this is the first time that logarithmic terms have been observed in the asymptotics of a class of lattice walks confined to a quadrant.