Reinforced Loop Soup via Wilson’s Algorithm

The goal of this note is twofold: first, we explain the relation between the isomorphism theorems in the context of the vertex reinforced jump process, discovered in [7, 8], and the standard isomorphism theorems for Markovian jump processes; second, we introduce the vertex reinforced counterpart of the standard Poissonian loop soup developed by Le Jan [30]. To achieve these goals, we propose a variant of Wilson’s algorithm that incorporates reinforcement. We establish isomorphism theorems for the erased loops and the random walk generated by this algorithm; in particular, we provide a concrete construction of the reinforced loop soup via a random process with a reinforcement mechanism.