On walks avoiding a quadrant

Abstract : Two-dimensional (random) walks in cones are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures. While walks restricted to the first quadrant have been studied a lot, the case of planar, non-convex cones---equivalent to the three-quarter plane after a linear transform---has been approached only recently. In this article we develop an analytic approach to the case of walks in three quadrants. The advantage of this method is to provide uniform treatment in the study of models corresponding to different step sets. After splitting the three quadrants in two symmetric convex cones, the method is composed of three main steps: write a system of functional equations satisfied by the counting generating function, which may be simplified into one single equation under symmetry conditions; transform the functional equation into a boundary value problem; and finally solve this problem, using a concept of anti-Tutte's invariant. The result is a contour-integral expression for the generating function. Such systems of functional equations also appear in queueing theory with the famous Join-the-Shortest-Queue model, which is still an open problem in the non-symmetric case.
Type de document :
Pré-publication, Document de travail
27 pages, 14 figures. 2018
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01848287
Contributeur : Amélie Trotignon <>
Soumis le : mardi 24 juillet 2018 - 14:43:13
Dernière modification le : mercredi 1 août 2018 - 01:13:34

Identifiants

  • HAL Id : hal-01848287, version 1
  • ARXIV : 1807.08610

Collections

Citation

Kilian Raschel, Amélie Trotignon. On walks avoiding a quadrant. 27 pages, 14 figures. 2018. 〈hal-01848287〉

Partager

Métriques

Consultations de la notice

40

Téléchargements de fichiers

13