Stochastic Foundations of Correlation Decay in Lattice Yang-Mills Theories

Abstract

In this thesis, we provide an accessible exposition of a recent paper [SZZ23] on lattice Yang--Mills theories over SO(N) or SU(N) at strong coupling. Viewing these lattice Yang-Mills theories as the invariant measures of the Markov semigroups associated to certain Langevin dynamics, we obtain a collection of functional inequalities (including Poincaré and log-Sobolev inequalities) via Bakry-Émery theory. Under a suitable choice of test function adapted to the specific application, these functional inequalities imply the existence of a mass gap as well as the factorization property for Wilson loop observables. We situate the results in their broader contexts, highlighting the connections between geometry and probability.