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

Abstract: $\delta$ once-reinforced random walks ($\delta$-ORRWs) are a type of self-interacting random walks defined on connected graphs with reinforcement factor $\delta>0$, moving to neighbours for each step. The transition probability is proposional to weights of edges, where the weights are $1$ on edges not traversed and $\delta$ otherwise. In this paper, we show a general version of large deviation principle for empirical measures of $\delta$-ORRWs on finite connected graphs by a modified weak convergence approach. We also obtain a phase transition of the rate function of large deviation principle with critical point $\delta_c=1$.