3D positive lattice walks and spherical triangles

Abstract

In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [30], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard decomposition, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision.Date: December 25, 2019. 1991 Mathematics Subject Classification. 05A15.

The asymptotics of reflectable weighted walks in arbitrary dimension

“…We show how to prove this property in the concluding remarks. The excursion enumeration formulas of Denisov and Wachtel [6] agree with ours for the known 2D and 3D cases [4,2]. The 1D results of Banderier and Flajolet [1] also agree with our formulas.…”

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

“…In other words, in dimension 3, one has to understand the principal eigenvalue λ 1 of spherical triangles. This connection between three-dimensional positive lattice walks and spherical triangles has been studied in [5], see also [21] in relation with a Brownian pursuit problem.…”

“…Dimension 2. Regarding the combinatorial model of walks in the quarter plane, the domain (5) simply becomes an arc of circle, see Figure 1 for a few examples. More precisely, if the walk is driftless and has identity covariance matrix, then L is just the identity and ( 5) is a quarter of circle.…”

“…We now explore the case of dimension 3. First, the domain (5) to consider is the trace on the sphere S 2 of LR 3 + , which by construction is a spherical triangle, see Figure 2 for a few examples. In other words, in dimension 3, one has to understand the principal eigenvalue λ 1 of spherical triangles.…”

Walks with Small Steps in the 4D-Orthant

“…They introduced the notions of dimension of a lattice walk model and Hadamard decomposition of a step set, which allow to reduce some of the problem to walks in lattices of lower dimension, and they used these new concepts as well as the classical orbit sum method for proving D-finiteness in certain cases. Bacher, Kauers, and Yatchak [1] have extended this work to step sets of arbitrary size, Du, Hou, and Wang provided non-D-finiteness results for many cases [14], and most recently, Bogosel, Perrollaz, Raschel, and Trotignon [3] have systematically explored the asymptotic behaviour of counting sequences for walks in the octant and observed a striking relation between the nature of the generating function and the angles of certain triangles on the sphere. Despite all this progress, there are still many open questions related to walks in the octant.…”