Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach

TL;DR

This paper establishes near-optimal mixing time bounds for certain stochastic surface models in Z^3 and SOS, using a mean curvature approach and coupling techniques, with implications for spectral gap estimates.

Contribution

It introduces a novel mean curvature-based method to bound mixing times of monotone surfaces and SOS interfaces, unifying approaches for different height fluctuation scales.

Findings

Achieved O(L^2 polylog(L)) mixing time bounds for both models.

Demonstrated the effectiveness of mean curvature heuristics in stochastic surface dynamics.

Provided lower bounds on spectral gaps, indicating near-diffusive relaxation times.

Abstract

We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z^3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D.B. Wilson in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (logarithmic for monotone surfaces and L^{1/2} for the SOS model). Finally, combining techniques from kinetically constrained spin systems together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L^2 up to logarithmic corrections for the spectral gap of the SOS model with horizontal size L and unbounded heights.