Mixing Time for the Solid-on-Solid Model

Understand

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.