Enumeration of walks in multidimensional orthants and reflection groups

TL;DR

This paper investigates the algebraic and spectral properties of multidimensional orthant walks, establishing connections between their associated groups, reflection groups, and asymptotic enumeration, advancing understanding of walk models and their classifications.

Contribution

It introduces a novel link between the transformation groups of walk models and reflection groups, and derives new spectral asymptotics for polyhedral domains related to these walks.

Findings

The associated group of walk models can be characterized as infinite under certain conditions.

New spectral results for eigenvalues of polyhedral nodal domains are established.

Asymptotic enumeration formulas for walk excursions are derived from spectral properties.

Abstract

We consider (random) walks in a multidimensional orthant. Using the idea of universality in probability theory, one can associate a unique polyhedral domain to any given walk model. We use this connection to prove two sets of new results. First, we are interested in a group of transformations naturally associated with any small step model; as it turns out, this group is central to the classification of walk models. We show a strong connection between this group and the reflection group through the walls of the polyhedral domain. As a consequence, we can derive various conditions for the combinatorial group to be infinite. Secondly, we consider the asymptotics of the number of excursions, whose critical exponent is known to be computable in terms of the eigenvalue of the above polyhedral domain. We prove new results from spectral theory on the eigenvalues of polyhedral nodal domains. We believe that these results are interesting in their own right; they can also be used to find new exact asymptotic results for walk models corresponding to these nodal polyhedral domains.