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

Sabot, 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 -