Large deviation principle for empirical measures of once-reinforced random walks on finite graphs

Abstract

A $\hat{I}´$ once-reinforced random walk ($\hat{I}´$-ORRW) on connected graph is a self-interacting random walk which moves to its neighbors at each step according to the weights of the edges at that time, where the weights are $1$ on edges that have not been traversed and $\hat{I}´$ otherwise. In this paper, we prove a large deviation principle for empirical measures of $\hat{I}´$-ORRWs on finite connected graphs using a modified weak convergence approach. The rate function of the large deviation principle exhibits a phase transition at the $\hat{I}´=1$.