Cutoff for the Glauber dynamics of the lattice free field

TL;DR

This paper investigates the mixing time of the Glauber dynamics for the discrete Gaussian free field in two dimensions, establishing a sharp cutoff at a specific time proportional to n^2 log n, advancing understanding of surface evolution dynamics.

Contribution

It provides the first sharp cutoff result for Glauber dynamics of the 2D lattice free field, a key step in understanding the dynamical properties of random surface models.

Findings

Cutoff occurs at time (2/π^2) n^2 log n for the 2D DGFF Glauber dynamics.

Sharp mixing bounds for random surface evolutions are established.

The result advances the understanding of convergence rates in surface model dynamics.

Abstract

The Gaussian Free Field (GFF) is a canonical random surface in probability theory generalizing Brownian motion to higher dimensions. In two dimensions, it is critical in several senses, and is expected to be the universal scaling limit of a host of random surface models in statistical physics. It also arises naturally as the stationary solution to the stochastic heat equation with additive noise. Focusing on the dynamical aspects of the corresponding universality class, we study the mixing time, i.e., the rate of convergence to stationarity, for the canonical prelimiting object, namely the discrete Gaussian free field (DGFF), evolving along the (heat-bath) Glauber dynamics. While there have been significant breakthroughs made in the study of cutoff for Glauber dynamics of random curves, analogous sharp mixing bounds for random surface evolutions have remained elusive. In this direction, we establish that on a box of side-length $n$ in $\mathbb Z^2$, when started out of equilibrium, the Glauber dynamics for the DGFF exhibit cutoff at time $\frac{2}{\pi^2}n^2 \log n$.