# Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

Christophe Sabot; Pierre Tarrès

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 9, page 2353-2378
- ISSN: 1435-9855

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topSabot, Christophe, and Tarrès, Pierre. "Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model." Journal of the European Mathematical Society 017.9 (2015): 2353-2378. <http://eudml.org/doc/277733>.

@article{Sabot2015,

abstract = {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 $\mathbb \{Z\}^d , d \ge 3$, for small reinforcement, using results of Disertori and Spencer [15] and Disertori, Spencer and Zirnbauer [16].},

author = {Sabot, Christophe, Tarrès, Pierre},

journal = {Journal of the European Mathematical Society},

keywords = {self-interacting random walk; reinforcement; random walk in random environment; sigma models; supersymmetry; de Finetti theorem; edge-reinforced random walk; self-interaction; random environment; vertex-reinforced jump process; hyperbolic sigma model; supersymmetry; de Finetti's theorem},

language = {eng},

number = {9},

pages = {2353-2378},

publisher = {European Mathematical Society Publishing House},

title = {Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model},

url = {http://eudml.org/doc/277733},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Sabot, Christophe

AU - Tarrès, Pierre

TI - Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 9

SP - 2353

EP - 2378

AB - 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 $\mathbb {Z}^d , d \ge 3$, for small reinforcement, using results of Disertori and Spencer [15] and Disertori, Spencer and Zirnbauer [16].

LA - eng

KW - self-interacting random walk; reinforcement; random walk in random environment; sigma models; supersymmetry; de Finetti theorem; edge-reinforced random walk; self-interaction; random environment; vertex-reinforced jump process; hyperbolic sigma model; supersymmetry; de Finetti's theorem

UR - http://eudml.org/doc/277733

ER -

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