The geometry of random walk isomorphism theorems

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