Dynamics of $(2+1)$-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion

TL;DR

This paper analyzes the Glauber dynamics of the 2+1D SOS surface above a wall, revealing that entropic repulsion causes slow, exponential mixing times due to metastable transitions between height levels.

Contribution

It precisely determines the equilibrium height caused by entropic repulsion and establishes exponential lower and upper bounds for the mixing time of the dynamics.

Findings

Surface height scales as (1/4β) log L with high probability.

Mixing time is exponentially large in L, specifically between e^{cL} and e^{c'L}.

Removing the floor constraint eliminates the exponential slow mixing behavior.

Abstract

We study the Glauber dynamics for the $(2+1)\mathrm{D}$ Solid-On-Solid model above a hard wall and below a far away ceiling, on an $L\times L$ box of $\mathbb{Z}^2$ with zero boundary conditions, at large inverse-temperature $\beta$. It was shown by Bricmont, El Mellouki and Fr\"{o}hlich [J. Stat. Phys. 42 (1986) 743-798] that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height $H\asymp(1/\beta)\log L$. As an essential step in understanding the effect of entropic repulsion on the Glauber dynamics we determine the equilibrium height $H$ to within an additive constant: $H=(1/4\beta)\log L+O(1)$. We then show that starting from zero initial conditions the surface rises to its final height $H$ through a sequence of metastable transitions between consecutive levels. The time for a transition from height $h=aH$, $a\in(0,1)$, to height $h+1$ is roughly $\exp(cL^a)$ for some constant $c>0$. In particular, the mixing time of the dynamics is exponentially large in $L$, that is, $T_{\mathrm{MIX}}\geq e^{cL}$. We also provide the matching upper bound $T_{\mathrm{MIX}}\leq e^{c'L}$, requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques. Finally, to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in $L$.