### 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 Z^{2} 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(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 T_{MIX} ≤ 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1516-1589 |

Number of pages | 74 |

Journal | Annals of Probability |

Volume | 42 |

Issue number | 4 |

DOIs | |

State | Published - 2014 |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*42*(4), 1516-1589. https://doi.org/10.1214/13-AOP836

**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.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 42, no. 4, pp. 1516-1589. https://doi.org/10.1214/13-AOP836

}

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 -