On $H^{2|2}$ Isomorphism theorems and reinforced loop soup

TL;DR

This paper establishes new isomorphism theorems connecting supersymmetric hyperbolic fields, reinforced loop soups, and Markov processes, enhancing understanding of these probabilistic models through novel proofs and formulas.

Contribution

It introduces supersymmetric hyperbolic isomorphism theorems and a BFS-Dynkin's isomorphism for reinforced loop soup, linking various probabilistic fields and processes.

Findings

Supersymmetric hyperbolic isomorphism theorems relate VRJP and $H^{2|2}$ fields.

A BFS-Dynkin's isomorphism theorem for reinforced loop soup is proven.

Alternative proof of BFS-Dynkin's isomorphism for VRJP via Feynman-Kac.

Abstract

We show that supersymmetric (susy) hyperbolic isomorphism theorems that relate Vertex Reinforced Jump Processes and $H^{2|2}$ field, introduced in [2] and [3], are annealed version of isomorphism theorems relating Markov processes and Gaussian free field, with the help of a Bayes formula that relates susy hyperbolic field to susy free field. On the other hand, we also prove a BFS-Dynkin's isomorphism theorem for reinforced loop soup. Moreover, we provide yet another proof of BFS-Dynkin's isomorphism for VRJP a la Feynman-Kac.