A stochastic analysis approach to lattice Yang--Mills

报告题目:  A stochastic analysis approach to lattice Yang--Mills

报告专家:朱蓉禅 教授(北京理工大学)




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.

报告人简介朱蓉禅,2007年本科毕业于四川大学数学基地班,2012年博士毕业于中科院数学与系统科学研究院和比勒菲尔德大学,现为北京理工大学教授。研究兴趣为随机偏微分方程,特别是奇异随机偏微分方程和量子场的交叉以及随机流体方程。2019年获批国家自然科学基金优秀青年基金,在CPAMCMPJFA AOPPTRFAAP等期刊上发表多篇论文。