2012/05/31 by Pietro Caputo, Eyal Lubetzky, Fabio Martinelli +2 · 1 citation
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP
paper · pdf · doi:10.1214/13-aop836
published as Annals of Probability 2014, Vol. 42, No. 4, 1516-1589 · Published in at http://dx.doi.org/10.1214/13-AOP836 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv created 2014/07/24 · arxiv updated 2014/07/25
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\asymp(1/β)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=(1/4β)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∈(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.