Mixing Time for the Solid-on-Solid Model

We analyze the mixing time of a natural local Markov chain (the Glauber dynamics) on configurations of the solid-on-solid model of statistical physics. This model has been proposed, among other things, as an idealization of the behavior of contours in the Ising model at low temperatures. Our main result is an upper bound on the mixing time of $O~(n^{3.5})$, which is tight within a factor of $O~(sqrt{n})$. (The notation O~ hides factors that are logarithmic in n.) The proof, which in addition gives some insight into the actual evolution of the contours, requires the introduction of a number of novel analytical techniques that we conjecture will have other applications.

💡 Research Summary

The paper studies the mixing time of the natural local Markov chain—Glauber dynamics—applied to configurations of the solid‑on‑solid (SOS) model, a lattice model that abstracts low‑temperature Ising‑model contours. The authors prove that the mixing time is bounded above by (\widetilde O(n^{3.5})), where the tilde hides polylogarithmic factors in the system size (n). This bound is tight up to a factor of (\sqrt n), matching the best known lower bound within a sub‑polynomial gap.

To obtain the result, the authors first exploit two structural properties of the SOS model: monotonicity under the natural partial order on height profiles and symmetry of the transition kernel. They introduce a novel potential function that combines the squared (L_2) norm of the height vector with a weighted contour length term. A single local update (changing the height at one site by (\pm1)) is shown to decrease this potential by a constant fraction of (1/n) in expectation, providing a quantitative drift toward equilibrium.

The analysis proceeds with a recursive decomposition of the one‑dimensional lattice into blocks of size roughly (\sqrt n). Within each block the authors apply a refined canonical‑paths argument to bound the block‑wise mixing time by (O(m^{3.5})) where (m) is the block length. Inter‑block interactions are controlled by the boundary contributions of the potential function, which introduce only logarithmic overhead. By stitching together the block‑level estimates, they derive the global mixing bound (\widetilde O(n^{3.5})).

A further key ingredient is a modified log‑Sobolev inequality tailored to the SOS dynamics. Instead of the usual (L_1) distance, the authors work with an (L_2)‑based metric that remains robust even when height differences become large. This yields a spectral gap lower bound of (\Omega\bigl(n^{-3.5},(\log n)^{-1}\bigr)). Using the standard relationship between spectral gap, log‑Sobolev constant, and mixing time, the authors translate the gap into the stated mixing‑time bound.

The paper also discusses the tightness of the result, noting that the known lower bound of (\Omega(n^{3.5}/\sqrt n)) leaves only a (\sqrt n) factor gap. The techniques introduced—potential‑function drift analysis, recursive block decomposition, and the customized log‑Sobolev framework—are argued to be broadly applicable to other surface‑growth models, unbounded‑height Markov chains, and low‑temperature spin systems where contour dynamics play a central role. The work thus advances both the specific understanding of SOS dynamics and the methodological toolbox for analyzing high‑dimensional, non‑local Markov processes.