# The geometry of random walk isomorphism theorems

## TL;DR

This paper extends classical random walk isomorphism theorems to hyperbolic and spherical geometries, involving non-Markovian processes and supersymmetric versions, with new proofs and applications in spin systems.

## Contribution

It generalizes isomorphism theorems to non-Euclidean geometries and non-Markovian processes, providing new proofs and applications in spin system analysis.

## Key findings

- Extended isomorphism theorems to hyperbolic and spherical geometries.
- Derived new proofs exploiting continuous symmetries.
- Provided applications including correlation decay and local time formulas.

## Abstract

The classical random walk isomorphism theorems relate the local times of a continuous-time random walk to the square of a Gaussian free field. A Gaussian free field is a spin system that takes values in Euclidean space, and this article generalises the classical isomorphism theorems to spin systems taking values in hyperbolic and spherical geometries. The corresponding random walks are no longer Markovian: they are the vertex-reinforced and vertex-diminished jump processes. We also investigate supersymmetric versions of these formulas. Our proofs are based on exploiting the continuous symmetries of the corresponding spin systems. The classical isomorphism theorems use the translation symmetry of Euclidean space, while in hyperbolic and spherical geometries the relevant symmetries are Lorentz boosts and rotations, respectively. These very short proofs are new even in the Euclidean case. Isomorphism theorems are useful tools, and to illustrate this we present several applications. These include simple proofs of exponential decay for spin system correlations, exact formulas for the resolvents of the joint processes of random walks together with their local times, and a new derivation of the Sabot--Tarr\`es formula for the limiting local time of the vertex-reinforced jump process.