(with Reza Gheissari). Arxiv

The Gaussian Free Field (GFF) is a canonical random surface in probability theory generalizing Brownian motion to higher dimensions, 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. More precisely, on a box of side-length \(n\) in \(\mathbb{ℤ}^d\), when started out of equilibrium, we establish that for every \(d\ge 1\), this Markov process exhibits cutoff at time \(\frac{d}{\pi^2}n^2\log n\) with an \(O(n^2)\)-sized window. 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. Our proof relies crucially on uncovering an "exact" representation of the DGFF surface evolution in terms of backwards random walk trajectories with spatially dependent jump times.