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 journalArticle

Abstract

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
Volume42
Issue number4
DOIs
StatePublished - 2014

Fingerprint

Entropic Repulsion
Glauber Dynamics
Mixing Time
Zero
Ceiling
Solid Model
Consecutive
Initial conditions
Upper bound
Statistics
Boundary conditions
Temperature

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Dynamics Of (2 + 1)-dimensional sos surfaces above : A wall: Slow mixing induced by entropic repulsion1. / Caputo, Pietro; Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan; Toninelli, Fabio Lucio.

In: Annals of Probability, Vol. 42, No. 4, 2014, p. 1516-1589.

Research output: Contribution to journalArticle

Caputo, Pietro ; Lubetzky, Eyal ; Martinelli, Fabio ; Sly, Allan ; Toninelli, Fabio Lucio. / Dynamics Of (2 + 1)-dimensional sos surfaces above : A wall: Slow mixing induced by entropic repulsion1. In: Annals of Probability. 2014 ; Vol. 42, No. 4. pp. 1516-1589.
@article{33bad007473b41be9e7181e2dc047971,
title = "Dynamics Of (2 + 1)-dimensional sos surfaces above: A wall: Slow mixing induced by entropic repulsion1",
abstract = "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 Fr{\"o}hlich [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.",
keywords = "Glauber dynamics, Mixing times, Random surface models, SOS model",
author = "Pietro Caputo and Eyal Lubetzky and Fabio Martinelli and Allan Sly and Toninelli, {Fabio Lucio}",
year = "2014",
doi = "10.1214/13-AOP836",
language = "English (US)",
volume = "42",
pages = "1516--1589",
journal = "Annals of Probability",
issn = "0091-1798",
publisher = "Institute of Mathematical Statistics",
number = "4",

}

TY - JOUR

T1 - Dynamics Of (2 + 1)-dimensional sos surfaces above

T2 - A wall: Slow mixing induced by entropic repulsion1

AU - Caputo, Pietro

AU - Lubetzky, Eyal

AU - Martinelli, Fabio

AU - Sly, Allan

AU - Toninelli, Fabio Lucio

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Glauber dynamics

KW - Mixing times

KW - Random surface models

KW - SOS model

UR - http://www.scopus.com/inward/record.url?scp=84903896859&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903896859&partnerID=8YFLogxK

U2 - 10.1214/13-AOP836

DO - 10.1214/13-AOP836

M3 - Article

AN - SCOPUS:84903896859

VL - 42

SP - 1516

EP - 1589

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -