Lattice walks confined to an octant in dimension 3: (non-)rationality of the second critical exponent

TL;DR

This paper investigates the asymptotic behavior of lattice walks confined to a 3D octant, revealing that the associated heat kernel can exhibit both rational and non-rational critical exponents, with implications for the algebraic nature of generating functions.

Contribution

It demonstrates the existence of a 3D cone where the heat kernel's asymptotic expansion has mixed rational and non-rational exponents, using spectral and perturbation theory.

Findings

Existence of a 3D cone with mixed rational/non-rational exponents

Development of a new Hadamard formula for eigenvalue derivatives

Insights into the algebraic nature of walk enumeration asymptotics

Abstract

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three, there exists a cone such that the heat kernel (the continuous analogue of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles.