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

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 .. 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/.)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=(1/4.)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(cLa) for some constant c>0. In particular, the mixing time of the dynamics is exponentially large in L, that is, TMIX.ecL. We also provide the matching upper bound TMIX.ec.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.