Mixing Time for the Solid-on-Solid Model

TL;DR

This paper provides an upper bound on the mixing time of the Glauber dynamics for the solid-on-solid model, revealing insights into contour evolution and introducing novel analytical techniques with broader applicability.

Contribution

It establishes a tight upper bound on the mixing time of the model's Markov chain and introduces new analytical methods for studying contour dynamics.

Findings

Mixing time is bounded by O(n^{3.5}) factors logarithmic in n.

The bound is tight within a factor of O(sqrt{n}).

New analytical techniques are developed for contour evolution analysis.

Abstract

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.