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Cutoff for the Glauber dynamics of the lattice free field

Shirshendu Ganguly and Reza Gheissari

Vol. 4 (2023), No. 2, 433–475
DOI: 10.2140/pmp.2023.4.433
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 2, when started out of equilibrium, the Glauber dynamics for the DGFF exhibit cutoff at time (2π2)n2 log n.

Keywords
Gaussian free field, random surface, Glauber dynamics, mixing time, cutoff
Mathematical Subject Classification
Primary: 60G15, 60J25, 82C41
Milestones
Received: 26 June 2022
Revised: 17 November 2022
Accepted: 2 January 2023
Published: 31 May 2023
Authors
Shirshendu Ganguly
Department of Statistics
University of California, Berkeley
Berkeley, CA
United States
Reza Gheissari
Departments of Mathematics
Northwestern University
Evanston, IL
United States
Department of Statistics and EECS
University of California, Berkeley
Berkeley, CA
United States