A Stochastic Analysis Approach to Lattice Yang–Mills at Strong Coupling (Xiangchan Zhu)

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

　　Publication:

　　Communications in Mathematical Physics

　　DOI: 10.1007/s00220-022-04609-1

　　Author:

　　Hao Shen

　　Department of Mathematics, University of Wisconsin, Madison, USA.

　　Rongchan Zhu

　　Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

　　Beijing Key Laboratory on MCAACI, Beijing, China.

　　Xiangchan Zhu

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.

　　E-mail: zhuxiangchan@126.comRepublic of China