Vertex-reinforced random walk on arbitrary graphs
成果类型：
Article
署名作者：
Volkov, S
署名单位：
University of Bristol
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1008956322
发表日期：
2001
页码：
66-91
关键词：
摘要：
Vertex-reinforced random walk (VRRW), defined by Pemantle, 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 with probability 1, and provide a proof only for trees of this type. We distinguish between several different patterns of localization and explicitly describe the long-run 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 for Z(1), ideas of proofs are different and typically based on a large deviation principle rather than a martingale approach.