A stochastic analysis approach to lattice Yang--Mills at strong coupling

TL;DR

This paper introduces a stochastic analysis framework for lattice Yang--Mills models at strong coupling, establishing ergodicity, convergence, and mass gap properties, with implications for large N limits and Wilson loop behavior.

Contribution

It develops a novel stochastic approach to analyze lattice Yang--Mills at strong coupling, proving ergodicity, uniqueness, and mass gap, and simplifying previous methods.

Findings

Unique invariant measure for Langevin dynamics

Exponential ergodicity under Wasserstein distance

Convergence of Wilson loops and existence of a mass gap

Abstract

We develop a new stochastic analysis approach to the lattice Yang--Mills model at strong coupling in any dimension $d>1$, with t' Hooft scaling $\beta N$ for the inverse coupling strength. We study their Langevin dynamics, ergodicity, functional inequalities, large $N$ limits, and mass gap. Assuming $|\beta| < \frac{N-2}{32(d-1)N}$ for the structure group $SO(N)$, or $|\beta| < \frac{1}{16(d-1)}$ for $SU(N)$, we prove the following results. The invariant measure for the corresponding Langevin dynamic is unique on the entire lattice, and the dynamic is exponentially ergodic under a Wasserstein distance. The finite volume Yang--Mills measures converge to this unique invariant measure in the infinite volume limit, for which Log-Sobolev and Poincar\'e inequalities hold. These functional inequalities imply that the suitably rescaled Wilson loops for the infinite volume measure has factorized correlations and converges in probability to deterministic limits in the large $N$ limit, and correlations of a large class of observables decay exponentially, namely the infinite volume measure has a strictly positive mass gap. Our method improves earlier results or simplifies the proofs, and provides some new perspectives to the study of lattice Yang--Mills model.