Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach

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 2 polylog(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 [17]. 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. Wilson [18] 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 (log L for monotone surfaces and √ L for the SOS model). Finally, combining techniques from kinetically constrained spin systems [2] together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L 2 polylog(L) for the spectral gap of the SOS model with horizontal size L and unbounded heights.2000 Mathematics Subject Classification: 60K35, 82C20

Hydrodynamic Limit Equation for a Lozenge Tiling Glauber Dynamics

“…In terms of monotone stacking of cubes, the dynamics corresponds to adding/removing a cube to/from a column, with transition rate 1, provided the cube stacking remains monotone after the update. Recently it was proven that, if we restrict the single-flip dynamics to domains of diameter L, under certain restrictions on the domain shape the mixing time is of order O(L 2+o (1) ) as L → ∞ [3,11]. These results support the idea that the correct time-scale to observe a hydrodynamic limit should be diffusive (i.e.…”

“…This will be called the "single-flip dynamics" in the following. As discussed for instance in [2,3], the single-flip dynamics coincides with the zero-temperature Glauber dynamics of +/− spin interfaces of the three-dimensional Ising model with zero magnetic field, where spins flip one by one. In terms of monotone stacking of cubes, the dynamics corresponds to adding/removing a cube to/from a column, with transition rate 1, provided the cube stacking remains monotone after the update.…”

Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation

“…Thick edges are dimers of a perfect matching of H L (the matching can be extended to a 4-periodic perfect matching of the infinite lattice H). Dimers are interlaced: for instance, type-3 (i.e., horizontal) dimers b, b are interlaced with type-3 dimers b (1) , b (2) . (Note that dimer b is the same (by periodicity) as dimer b ; that is why it is drawn as dashed).…”

“…Proof of Proposition 5.3. Claim (1). In this case λ = π ρ , the measure obtained as the L → ∞ limit of the uniform measure on Ω ρ (L) with ρ (L) → ρ.…”

(2 + 1)-Dimensional Interface Dynamics: Mixing Time, Hydrodynamic Limit and Anisotropic KPZ Growth

“…In this situation, Funaki and Spohn [16] proved convergence of the height profile to 6 One can express µ also via a variational principle, see [40]. 7 For models in dimension d ≤ 2 the law of the interface does not have a limit as → 0, since the variance of hx diverges as → 0. However, the law of the gradients of h does have a limit and the transition rates c n x (h) are actually functions of the gradients of h only, by translation invariance in the vertical direction.…”