Spectral gap and cutoff phenomenon for the Gibbs sampler of ∇φ interfaces with convex potential

Abstract

We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on R describing ∇φ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by gapN = 1 − cos(π/N), and that for all ǫ ∈ (0, 1), its ǫ-mixing time satisfies TN (ǫ) ∼ logN 2 gapN as N → ∞, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential. MSC 2010 subject classifications: Primary 60J25; Secondary 37A25, 82C22.