Reinforced Loop Soup via Wilson's Algorithm

doi:10.1007/s00023-026-01710-2

Reinforced Loop Soup via Wilson's Algorithm

2026-01-01

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DOI：10.1007/s00023-026-01710-2

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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.

Keywords:

RANDOM-WALK REPRESENTATION

CLASSICAL SPIN SYSTEMS

JUMP PROCESS

MARKOV-PROCESSES

LOCALIZATION

ISOMORPHISM

THEOREMS

TIME

Reinforced Loop Soup via Wilson's Algorithm

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