Random Walks on Groups and Monoids with a Markovian Harmonic Measure
Mairesse Jean, CNRS - Universite Paris 7
Abstract
We consider a transient nearest neighbor random walk on a group G
with finite set of generators S.
The pair (G,S) is assumed to admit a natural notion of
normal form words where only the last letter is
modified by multiplication by a generator.
The basic examples are the free products of a finitely generated free
group and a finite family of finite
groups, with natural generators.
We prove that the harmonic measure is Markovian of a particular
type. The transition matrix is entirely determined by the initial
distribution which is itself the unique solution of a finite set of
polynomial equations of degree two. This enables
to efficiently compute the drift, the entropy, the probability of ever
hitting an element, and the minimal
positive harmonic functions of the walk. The results extend to
monoids.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.