Vertexreinforced random walk on arbitrary graphs
Abstract
VertexReinforced Random Walk (VRRW), defined by Pemantle (1988a), is a random process in a continuously changing environment which is more likely to visit states it has visited before. We consider VRRW on arbitrary graphs and show that on almost all of them, VRRW visits only finitely many vertices with a positive probability. We conjecture that on all graphs of bounded degree, this happens a.s., and provide a proof only for trees of this type. We distinguish between several different patterns of localization and explicitly describe the longrun structure of VRRW, which depends on whether a graph contains triangles or not. While the results of this paper generalize those obtained by Pemantle and Volkov (1998) for Z,ideas of proofs are different and typically based on a large deviation principle rather than a martingale approach.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 1999
 arXiv:
 arXiv:math/9907196
 Bibcode:
 1999math......7196V
 Keywords:

 Mathematics  Probability