Dynamics Of (2 + 1)-dimensional sos surfaces above: A wall: Slow mixing induced by entropic repulsion1

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 × L box of Z² with zero boundary conditions, at large inverse-temperature β. It was shown by Bricmont, El Mellouki and Fröhlich [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 = (1/β) logL. 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 = (1/4β) logL + 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 ∈ (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_MIX ≥ e^cL. We also provide the matching upper bound T_MIX ≤ 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.