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

TL;DR

This paper proves that the Gibbs sampler for certain convex potential models exhibits a universal spectral gap and cutoff phenomenon, with mixing times scaling logarithmically with system size, independent of potential specifics.

Contribution

It establishes a universal spectral gap and cutoff behavior for the Gibbs sampler of $ abla ext{varphi}$ interfaces with convex potentials under minimal assumptions.

Findings

Spectral gap is always given by 1 - cos(π/N).

Mixing time scales as (log N)/(2 * spectral gap).

Cutoff phenomenon is rigorously established.

Abstract

We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on $\mathbb{R}^N$ describing $\nabla\varphi$ 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.