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

We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on ${\mathbb{R}}^N$ describing $\mathrm{\nabla }\phi$ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by ${\mathrm{g}\mathrm{a}\mathrm{p}}_N=1-\cos (\pi \mathrm{/}N)$, and that for all $ϵ\in (0,1)$, its $ϵ$-mixing time satisfies $T_N(\epsilon)\sim \frac{\log N}{2$\mathrm{g}\mathrm{a}\mathrm{p}$_N}$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.