Dynamics of 2+1 dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion
Résumé
We study the Glauber dynamics for the (2+1)D Solid-On-Solid model above a hard wall and below a far away ceiling, on an $L \times L$ box of $Z^2$ with zero boundary conditions, at large inverse-temperature $\beta$. It was shown by Bricmont, El-Mellouki and Fröhlich (1986) 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 $\exp(c L^a)$ for some constant $c>0$. In particular, the mixing time of the dynamics is exponentially large in $L$, i.e., $T_{mix} \geq e^{c L}$. We also provide the matching upper bound $T_{mix} \leq e^{c' L}$, 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 $L$.