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
in
, when
started out of equilibrium, the Glauber dynamics for the DGFF exhibit cutoff at time
.
Keywords
Gaussian free field, random surface, Glauber dynamics,
mixing time, cutoff