Vol. 2, No. 3, 2021

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Covariant Symanzik identities

Adrien Kassel and Thierry Lévy

Vol. 2 (2021), No. 3, 419–475
Abstract

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik’s foundational work on the subject.

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Keywords
discrete potential theory, Laplacian on vector bundles, Gaussian free vector field, random walks, covariant Feynman–Kac formula, Poissonian ensembles of Markovian loops, local times, isomorphism theorems, discrete gauge theory, holonomy
Mathematical Subject Classification
Primary: 60J55, 60J57, 81T25, 82B20
Milestones
Received: 12 May 2020
Revised: 20 December 2020
Accepted: 16 March 2021
Published: 15 October 2021
Authors
Adrien Kassel
Unité de Mathématiques Pures et Appliquées
CNRS - ENS Lyon
Lyon
France
Thierry Lévy
Laboratoire de Probabilités, Statistique et Modélisation
Sorbonne Université
Paris
France