Weighted Lattice Walks and Universality Classes

TL;DR

This paper analyzes weighted lattice walks in cones, deriving asymptotic formulas for the Gouyou-Beauchamps model and characterizing universality classes for general weighted walks, revealing complex behaviors including non-D-finite generating functions.

Contribution

It provides the first asymptotic analysis of the Gouyou-Beauchamps model and characterizes universality classes for weighted walks in cones, including combinatorial identities and non-D-finite cases.

Findings

Six asymptotic regimes identified for Gouyou-Beauchamps walks

Explicit asymptotic formulas parametrized by weights and starting points

Infinite models with non-D-finite generating functions

Abstract

In this work we consider two different aspects of weighted walks in cones. To begin we examine a particular weighted model, known as the Gouyou-Beauchamps model. Using the theory of analytic combinatorics in several variables we obtain the asymptotic expansion of the total number of Gouyou-Beauchamps walks confined to the quarter plane. Our formulas are parametrized by weights and starting point, and we identify six different asymptotic regimes (called universality classes) which arise according to the values of the weights. The weights allowed in this model satisfy natural algebraic identities permitting an expression of the weighted generating function in terms of the generating function of unweighted walks on the same steps. The second part of this article explains these identities combinatorially for walks in arbitrary cones and dimensions, and provides a characterization of universality classes for general weighted walks. Furthermore, we describe an infinite set of models with non-D-finite generating function.