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

A $\delta$ once-reinforced random walk ($\delta$-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 $\delta$ otherwise. In this paper, we prove a large deviation principle for empirical measures of $\delta$-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 $\delta =1$.