# Dynamics of vertex-reinforced random walks - Mathematics > Probability

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Abstract: We generalize a result from Volkov Ann. Probab. 29 2001 66-91 and provethat, on a large class of locally finite connected graphs of bounded degree$G,\sim$ and symmetric reinforcement matrices $a=a {i,j} {i,j\in G}$, thevertex-reinforced random walk VRRW eventually localizes with positiveprobability on subsets which consist of a complete $d$-partite subgraph withpossible loops plus its outer boundary. We first show that, in general, anystable equilibrium of a linear symmetric replicator dynamics with positivepayoffs on a graph $G$ satisfies the property that its support is a complete$d$-partite subgraph of $G$ with possible loops, for some $d\ge1$. This resultis used here for the study of VRRWs, but also applies to other contexts such asevolutionary models in population genetics and game theory. Next we generalizethe result of Pemantle Probab. Theory Related Fields 92 1992 117-136 andBena\-{{\i}}m Ann. Probab. 25 1997 361-392 relating the asymptoticbehavior of the VRRW to replicator dynamics. This enables us to conclude that,given any neighborhood of a strictly stable equilibrium with support $S$, thefollowing event occurs with positive probability: the walk localizes on$S\cup\partial S$ where $\partial S$ is the outer boundary of $S$ and thedensity of occupation of the VRRW converges, with polynomial rate, to astrictly stable equilibrium in this neighborhood.

Author: ** Michel Benaïm, Pierre Tarrès**

Source: https://arxiv.org/