# The asymptotics of reflectable weighted walks in arbitrary dimension

## TL;DR

This paper develops a general asymptotic enumeration formula for weighted lattice walks in the positive orthant, extending reflection principles to include weights and using singularity analysis of rational functions.

## Contribution

It introduces a novel asymptotic enumeration method for weighted walks in the orthant, generalizing previous unweighted models and applying singularity analysis techniques.

## Key findings

- Asymptotic formulas are derived for weighted walks ending anywhere in the orthant.
- The approach uses singularity analysis of multivariable rational functions.
- The results extend existing reflection principle methods to weighted models.

## Abstract

Gessel and Zeilberger generalized the reflection principle to handle walks confined to Weyl chambers, under some restrictions on the allowable steps. For those models that are invariant under the Weyl group action, they express the counting function for the walks with fixed starting and endpoint as a constant term in the Taylor series expansion of a rational function. Here, we focus on the simplest case, the Weyl groups $A_1^d$, which correspond to walks in the first orthant $\mathbb{N}^d$ taking steps from a subset of $\{\pm1, 0\}^d$ which is invariant under reflection across any axis. The principle novelty here is the incorporation of weights on the steps and the main result is a very general theorem giving asymptotic enumeration formulas for walks that end anywhere in the orthant. The formulas are determined by singularity analysis of multivariable rational functions, an approach that has already been successfully applied in numerous related cases.