Title: Reinforced Loop Soup via Wilson's Algorithm Show Chinese title

Title: 通过威尔逊算法的强化环汤

Abstract: The goal of this note is twofold: first, we explain the relation between the isomorphism theorems in the context of vertex reinforced jump process discovered in [BHS19, BHS21] and the standard Markovian isomorphism theorems for Markovian jump processes; second, we introduce the vertex reinforced counterpart of the standard Poissonian loop soup developed by Le Jan [LJ10]. To this end, we propose an algorithm that can be viewed as a variant of Wilson's algorithm with reinforcement. We establish the isomorphism theorems for the erased loops and the random walk from this algorithm, and in particular provide a concrete construction of the reinforced loop soup via a random process with a reinforcement mechanism. Show Chinese abstract

Abstract: 本注释的目标有两个：首先，我们解释在[ BHS19, BHS21 ]中发现的顶点强化跳跃过程中的同构定理与标准马尔可夫同构定理之间的关系；其次，我们引入由Le Jan [LJ10]开发的标准泊松环汤的顶点强化对应物。为此，我们提出一种算法，可以视为带有强化的Wilson算法的一种变体。我们从该算法中建立了擦除环和随机游走的同构定理，并特别通过具有强化机制的随机过程提供了强化环汤的具体构造。