Dynamics of Vertex - Reinforced Random Walks Michel

We generalize a result from Volkov (2001,[21]) and prove that, on an arbitrary graph of bounded degree (G, ∼) and for any symmetric reinforcement matrix a = (a i,j) i∼j , the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete d-partite subgraph 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 for some d 2. 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. relating the asymptotic behaviour of the VRRW to replicator dynamics. This enables us to conclude that, given any neighbourhood of a strictly stable equilibrium with support S, the following event occurs with positive probability: the walk localizes on S ∪ ∂S, (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 neighbourhood. Let (Ω, F, P) be a probability space. Let (G, ∼) be a locally finite symmetric graph, and let V (G) be its vertex set which we sometimes also denote by G for