The geometry of random walk isomorphism theorems

Roland Bauerschmidt, Tyler Helmuth, Andrew Swan

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)408-454
Number of pages47
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number1
StatePublished - Feb 2021


  • Dynkin isomorphism
  • Eisenbaum isomorphism
  • Non-linear sigma models
  • Ray–Knight identities
  • Reinforced random walks
  • Supersymmetry
  • Vertex-reinforced jump process

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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