Spectral gap and cutoff phenomenon for the Gibbs sampler of $\nabla \phi$ interfaces with convex potential

Abstract

We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on ${\mathbb{R}}^N$ describing $\nabla \phi$ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by ${\mathrm{gap}}_N=1-\cos (\pi /N)$, and that for all $\epsilon \in (0,1)$, its $\epsilon$-mixing time satisfies $T_N(\epsilon )\sim \frac{\log N}{2{\mathrm{gap}}_N}$ as $N\to \infty$, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential.

38 pages