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Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. (English) Zbl 1331.60185

Authors’ abstract: Edge-reinforced random walk (ERRW), introduced by D. Coppersmith and P. Diaconis in 1986 [unpublished], is a random process which takes values in the vertex set of a graph \(G\) and is more likely to cross edges it has visited before.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 B. Davis and S. Volkov [Probab. Theory Relat. Fields 123, No. 2, 281–300 (2002; Zbl 1009.60027)], 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 M. R. Zirnbauer [Commun. Math. Phys. 141, No. 3, 503–522 (1991; Zbl 0746.58014)]. This enables us to deduce that VRJP and ERRW are positive recurrent on any graph of bounded degree for large reinforcement, and that the VRJP is transient on \(\mathbb{Z} ^{d}\) \(,d\geq 3\), for small reinforcement, using results of M. Disertori and T. Spencer [Commun. Math. Phys. 300, No. 3, 659–671 (2010; Zbl 1203.82017)] and M. Disertori et al. [ibid. 300, No. 2, 435–486 (2010; Zbl 1203.82018)].

MSC:

60K37 Processes in random environments
60G50 Sums of independent random variables; random walks
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J75 Jump processes (MSC2010)
81T25 Quantum field theory on lattices
81T60 Supersymmetric field theories in quantum mechanics
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References:

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