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 [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 factorization, 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.
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01764327
Contributeur : Amélie Trotignon <>
Soumis le : mardi 17 avril 2018 - 12:51:52
Dernière modification le : jeudi 10 mai 2018 - 02:04:02

Fichiers

3DWalks_SphericalTriangles.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-01764327, version 1
  • ARXIV : 1804.06245

Citation

B Bogosel, V Perrollaz, K. Raschel, A Trotignon. 3D POSITIVE LATTICE WALKS AND SPHERICAL TRIANGLES. 2018. 〈hal-01764327〉

Partager

Métriques

Consultations de la notice

127

Téléchargements de fichiers

19