3D positive lattice walks and spherical triangles

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:172

Article

Bogosel, B.¹  Perrollaz, V²  Raschel, K.^2,3  Trotignon, A.^2,4 

[1] Ecole Polytech, CMAP, F-91120 Palaiseau, France

[2] Univ Tours, Inst Denis Poisson, Parc Grandmont, F-37200 Tours, France

[3] Univ Tours, CNRS, Parc Grandmont, F-37200 Tours, France

[4] Simon Fraser Univ, Dept Math, Burnaby, BC, Canada

关键词: Enumerative combinatorics;    Lattice paths in the octant;    Asymptotic analysis;    Spherical geometry;   

DOI  :  10.1016/j.jcta.2019.105189

来源: Elsevier

【 摘 要 】

In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], 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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free