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November 2011 Dynamics of vertex-reinforced random walks
Michel Benaïm, Pierre Tarrès
Ann. Probab. 39(6): 2178-2223 (November 2011). DOI: 10.1214/10-AOP609

Abstract

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, ∼) and symmetric reinforcement matrices a = (ai,j)i,jG, 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 Benaïm [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 SS (where ∂ 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.

Citation

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Michel Benaïm. Pierre Tarrès. "Dynamics of vertex-reinforced random walks." Ann. Probab. 39 (6) 2178 - 2223, November 2011. https://doi.org/10.1214/10-AOP609

Information

Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1242.60044
MathSciNet: MR2932667
Digital Object Identifier: 10.1214/10-AOP609

Subjects:
Primary: 60G50
Secondary: 34F05 , 60J10 , 60K35

Keywords: entropy function , Martingales , random perturbations of dynamical systems , reinforced random walks , replicator dynamics

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 6 • November 2011
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