TY - JOUR

T1 - The geometry of random walk isomorphism theorems

AU - Bauerschmidt, Roland

AU - Helmuth, Tyler

AU - Swan, Andrew

N1 - Funding Information:
We thank Christophe Sabot for pointing out an error in an earlier version of this article. RB and TH would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Scaling limits, rough paths, quantum field theory” when work on this paper was undertaken; this work was supported by EPSRC grant no. EP/R014604/1. TH is supported by EPSRC grant no. EP/P003656/1.
Publisher Copyright:
© Association des Publications de l’Institut Henri Poincaré, 2021

PY - 2021/2

Y1 - 2021/2

N2 - 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–Tarrès formula for the limiting local time of the vertex-reinforced jump process.

AB - 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–Tarrès formula for the limiting local time of the vertex-reinforced jump process.

KW - Dynkin isomorphism

KW - Eisenbaum isomorphism

KW - Non-linear sigma models

KW - Ray–Knight identities

KW - Reinforced random walks

KW - Supersymmetry

KW - Vertex-reinforced jump process

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U2 - 10.1214/20-AIHP1083

DO - 10.1214/20-AIHP1083

M3 - Article

AN - SCOPUS:85104321593

SN - 0246-0203

VL - 57

SP - 408

EP - 454

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 1

ER -