Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model
Sabot, Christophe; Tarres, Pierre (2015), Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model, Journal of the European Mathematical Society, 17, 9, p. 2353-2378. 10.4171/JEMS/559
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/1111.3991v4
Journal nameJournal of the European Mathematical Society
MetadataShow full item record
Abstract (EN)Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 , 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 .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  and Disertori, Spencer and Zirnbauer .
Subjects / KeywordsSelf-interacting random walk; reinforcement; random walk in random environment; sigma models; supersymmetry; de Finetti theorem
Showing items related by title and author.
ARFIMA Process : Tests and Applications at a White Noise Process, A Random Walk Process and the Stock Exchange Index CAC 40 Bourbonnais, Régis; Maftei, Magda Mara (2012-01) Article accepté pour publication ou publié