Date
2015
Lien vers un document non conservé dans cette base
https://arxiv.org/abs/1111.3991v4
Indexation documentaire
Sciences connexes (physique, astrophysique)
Subject
Self-interacting random walk; reinforcement; random walk in random environment; sigma models; supersymmetry; de Finetti theorem
Nom de la revue
Journal of the European Mathematical Society
Volume
17
Numéro
9
Date de publication
2015
Pages article
2353-2378
Nom de l'éditeur
Springer
Auteur
Sabot, Christophe
Tarres, Pierre
Type
Article accepté pour publication ou publié
Résumé en anglais
Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 [8], is a random process which takes values in the vertex set of a graph G and is more likely to cross edges it has visited before. We show that it can be represented in terms of a vertex-reinforced jump process (VRJP) with independent gamma conductances; the VRJP was conceived by Werner and first studied by Davis and Volkov [10, 11], and is a continuous-time process favouring sites with more local time. We calculate, for any finite graph G, the limiting measure of the centred occupation time measure of VRJP, and interpret it as a supersymmetric hyperbolic sigma model in quantum field theory, introduced by Zirnbauer in 1991 [35].This enables us to deduce that VRJP and ERRWare positive recurrent on any graph of bounded degree for large reinforcement, and that the VRJP is transient on Zd,d≥3, for small reinforcement, using results of Disertori and Spencer [15] and Disertori, Spencer and Zirnbauer [16].