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SUMMARY: Edge reinforced random walks\, Vertex reinforced jump process\, a
 nd the SuSy hyperbolic sigma model. - Pierre Tarres (Oxford)
DTSTART:20120515T150000Z
DTEND:20120515T160000Z
UID:TALK37960@talks.cam.ac.uk
CONTACT:12974
DESCRIPTION:Edge-reinforced random walk (ERRW)\, introduced by Coppersmith
  and Diaconis\nin 1986\, is a random process which takes values in the ver
 tex set of a graph\nG\, and is more likely to cross edges it has visited\n
 before. We show that it can be represented in terms of a Vertex-reinforced
 \njump process (VRJP) with independent gamma conductances: the VRJP was\nc
 onceived by Werner and first studied by Davis and Volkov (2002\,2004)\, an
 d\nis a continuous-time process favouring sites with more local time.\n\nT
 hen we prove that the VRJP is a mixture of time-changed Markov\njump proce
 sses and calculate the mixing measure\, which we interpret as\na marginal 
 of the supersymmetric hyperbolic sigma model introduced\nby Disertori\, Sp
 encer and Zirnbauer (2010).\n\nThis enables us to deduce that VRJP and ERR
 W are positive recurrent on\ngraphs of bounded degree for large reinforcem
 ent\, and that VRJP is transient\nin dimension greater than or equal to 3 
 for small reinforcment\, using the\nprevious results of Disertori and Spen
 cer and Zirnbauer.\n\n(Joint work with Christophe Sabot)\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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