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 Fro ̈hlich [12] that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height H ≍ (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, i.e. 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.

Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, F.l. (2014). Dynamics of 2+1 dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion. ANNALS OF PROBABILITY, 42(4), 1516-1589 [10.1214/13-AOP836].

Dynamics of 2+1 dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion

CAPUTO, PIETRO;MARTINELLI, Fabio;
2014-01-01

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 Fro ̈hlich [12] that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height H ≍ (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, i.e. 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.
2014
Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, F.l. (2014). Dynamics of 2+1 dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion. ANNALS OF PROBABILITY, 42(4), 1516-1589 [10.1214/13-AOP836].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/138619
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