Three-dimensional extension of lattice Yang–Mills Langevin convergence

Extend the convergence of lattice Yang–Mills Langevin dynamics from two dimensions to three dimensions, including the non-abelian setting beyond the abelian U(1) case studied here.

Background

The paper places its contribution in the context of previously established scaling-limit results for two-dimensional lattice gauge theories. In particular, the convergence of the invariant measures and Langevin dynamics of a broad class of discrete lattice gauge theories to their continuum limits had been proved in two dimensions.

The three-dimensional problem is substantially more singular and analytically difficult. The present work addresses only the abelian U(1) theory and proves convergence, after DeTurck gauge fixing, to the one-form stochastic heat equation. Thus, the broader extension of the two-dimensional convergence theory to three dimensions—especially for non-abelian lattice Yang–Mills models—remains unresolved.