DYNAMICS OF VERTEX-REINFORCED RANDOM WALKS

• 成果类型: Article
• 署名作者: Benaim, Michel; Tarres, Pierre
• 署名单位: University of Neuchatel; Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP609
• 发表日期: 2011
• 页码: 2178-2223
• 关键词: NUMBER; graphs

摘要

We generalize a result from Volkov [Ann. Probab. 29 (2001) 66-91] and prove that, on a large class of locally finite connected graphs of bounded degree (G, similar to)and symmetric reinforcement matrices a = (a(i), (j))(i),(j is an element of G), the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete d-partite subgraph with possible loops plus its outer boundary. We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph G satisfies the property that its support is a complete d-partite subgraph of G with possible loops, for some d >= 1. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory. Next we generalize the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117-136] and Bena m [Ann. Probab. 25 (1997) 361-392] relating the asymptotic behavior of the VRRW to replicator dynamics. This enables us to conclude that, given any neighborhood of a strictly stable equilibrium with support S, the following event occurs with positive probability: the walk localizes on S boolean OR partial derivative S (where partial derivative S is the outer boundary of S) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighborhood.