DYNAMICS OF (2+1)-DIMENSIONAL SOS SURFACES ABOVE A WALL: SLOW MIXING INDUCED BY ENTROPIC REPULSION'

• 成果类型: Article
• 署名作者: Caputo, Pietro; Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan; Toninelli, Fabio Lucio
• 署名单位: Roma Tre University; Microsoft; University of California System; University of California Berkeley; Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP836
• 发表日期: 2014
• 页码: 1516-1589
• 关键词: kosterlitz-thouless transition; roughening transition; statistical-mechanics; ising-model; interface; tension; droplet

摘要

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 x L box of Z(2) with zero boundary conditions, at large inverse-temperature P. It was shown by Bricmont, El Mellouki and Frohlich [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 um 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 is an element of (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 Tivnix <= 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.