Asymptotics of Weighted Reflectable Walks in $A_2$

TL;DR

This paper derives asymptotic formulas for weighted lattice walks in the $A_2$ Weyl chamber using analytic combinatorics, revealing universality classes and identifying new types of singularities in generating functions.

Contribution

It introduces new asymptotic results for weighted walks in $A_2$, including a novel singularity type not previously covered by ACSV theory.

Findings

Identified universality classes depending on walk weights.

Derived asymptotics for weighted walks in $A_2$.

Discovered a new singularity type in multivariate generating functions.

Abstract

Lattice walks are used to model various physical phenomena. In particular, walks within Weyl chambers connect directly to representation theory via the Littelmann path model. We derive asymptotics for centrally weighted lattice walks within the Weyl chamber corresponding to $A_2$ by using tools from analytic combinatorics in several variables (ACSV). We find universality classes depending on the weights of the walks, in line with prior results on the weighted Gouyou-Beauchamps model. Along the way, we identify a type of singularity within a multivariate rational generating function that is not yet covered by the theory of ACSV. We conjecture asymptotics for this type of singularity.