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arXiv · 2503.04503

Reinforced Loop Soup via Wilson's Algorithm

Yinshan Chang·Yichao Huang·Dang-Zheng Liu·Xiaolin Zeng
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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.

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Yinshan Chang, Yichao Huang, Dang-Zheng Liu, Xiaolin Zeng. 2025-03-06. Reinforced Loop Soup via Wilson's Algorithm. https://arxiv.org/abs/2503.04503

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