Mixing profile for Glauber dynamics of the discrete Gaussian Free Field starting from super-harmonic functions

Abstract

We study the convergence rate of the heat-bath Glauber dynamics for the Discrete Gaussian Free Field on arbitrary connected finite graphs. We show that, when starting from super-harmonic initial conditions, the evolution enjoys a strong form of monotonicity. This allows us to get a sharp mixing profile as the size of the graphs diverges. More precisely, we show that mixing occurs at time $\frac{1}{2λ}\log(\mathcal{E})$ with window $\mathcal{O}(1/λ)$, where $λ$ is the spectral gap of the graph Laplacian and $\mathcal{E}$ is the energy of the super-harmonic initial condition. This result holds for arbitrary graphs that do not exhibit extreme connectivity properties (one way or the other). In particular, it holds for finite boxes of the grid $\mathbb{Z}^d$, in dimension $d\geq 3$.