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We consider the Gibbs sampler, or heat bath dynamics associated tolog-concave measures on $\mathbb{R}^N$ describing $\nabla\varphi$ interfaceswith convex potentials. Under minimal assumptions on the potential, we findthat 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 $T_N(\epsilon)\sim \frac{\logN}{2\mathrm{gap}_N}$ as $N\to\infty$, thus establishing the cutoff phenomenon.The results reveal a universal behavior in that they do not depend on thechoice of the potential.

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