Edge-reinforced branching random walk on a triangle

Abstract

The edge-reinforced random walk (ERRW) is a random process on the vertices of a graph that is more likely to cross the edges it has visited in the past. Depending on the strength of the reinforcement, the ERRW of a single particle can either exhibit localisation (eventually moving back and forth across a single edge) or remain transient. We consider a model where a single ERRW is replaced by that of an exponentially growing number of random particles, and we study its localisation properties on the triangle. Using the dynamical systems approach we analyse the frequencies with which the edges are traversed and prove their almost sure convergence. We discuss the scenarios when those frequencies become negligible for one or two edges (dominance). We also discuss the situation when an edge stops being traversed entirely (monopoly).