Abstract: The Gaussian free field (GFF) is expected to be the universal scaling limit of a host of random surface models arising in lattice statistical physics: these include the height functions of the dimer model, the solid-on-solid model, and interfaces in the low-temperature Ising model. In that context, one is often interested in the time evolution of such random surfaces under a local update Markov process called the Glauber dynamics. More precisely, one is interested in the mixing time, or time for this dynamics to converge to equilibrium when started far out of equilibrium.
In the first part of this talk, we overview what is known for the mixing times of statistical physics models of random curves and random surfaces. While the one-dimensional setting has seen spectacular progress, models in higher dimensions have mostly resisted an analogous analysis. In this direction, we will describe a new result identifying the exact location of mixing and the cutoff phenomenon for the Glauber dynamics of the discrete Gaussian free field (DGFF), the natural pre-limiting object of the GFF, in every dimension. In the second part of the talk, we sketch the proof of this result. The proof goes through an exact representation for the DGFF dynamics in terms of random walk trajectories with space-dependent jump times, and quenched heat kernel bounds for these random walks.