Geodesic
Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model
Journal of the European Mathematical Society, Tome 17 (2015) no. 9, pp. 2353-2378 Cet article a éte moissonné depuis la source EMS Press

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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].
DOI : 10.4171/jems/559
Classification : 60-XX, 81-XX
Keywords: Self-interacting random walk, reinforcement, random walk in random environment, sigma models, supersymmetry, de Finetti theorem 
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     author = {Christophe Sabot and Pierre Tarr\`es},
     title = {Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model},
     journal = {Journal of the European Mathematical Society},
     pages = {2353--2378},
     year = {2015},
     volume = {17},
     number = {9},
     doi = {10.4171/jems/559},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/559/}
}
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Christophe Sabot; Pierre Tarrès. Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. Journal of the European Mathematical Society, Tome 17 (2015) no. 9, pp. 2353-2378. doi: 10.4171/jems/559

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