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

Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli

Research output: Contribution to journalArticlepeer-review


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(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.

Original languageEnglish (US)
Pages (from-to)1516-1589
Number of pages74
JournalAnnals of Probability
Issue number4
StatePublished - Jul 2014


  • Glauber dynamics
  • Mixing times
  • Random surface models
  • SOS model

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


Dive into the research topics of 'Dynamics Of (2 + 1)-dimensional sos surfaces above: A wall: Slow mixing induced by entropic repulsion1'. Together they form a unique fingerprint.

Cite this