The shape of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>d SOS surface above a wall

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

doi:10.1016/j.crma.2012.07.006

Cited by 15 publications

(17 citation statements)

References 17 publications

(25 reference statements)

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“…-Interfacial adsorption at the interface between two equilibrium phases [13,17]. -Geometry of the top-most layer of the 2 + 1-dimensional SOS model above a wall [7]. -Island of activity in kinetically constrained models [5].…”

Section: Physical Motivationsmentioning

confidence: 99%

An Invariance Principle to Ferrari–Spohn Diffusions

Ioffe

Shlosman

Velenik

2015

Commun. Math. Phys.

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Abstract:We prove an invariance principle for a class of tilted 1+1-dimensional SOS models or, equivalently, for a class of tilted random walk bridges in Z + . The limiting objects are stationary reversible ergodic diffusions with drifts given by the logarithmic derivatives of the ground states of associated singular SturmLiouville operators. In the case of a linear area tilt, we recover the Ferrari-Spohn diffusion with log-Airy drift, which was derived in [12] in the context of Brownian motions conditioned to stay above circular and parabolic barriers.

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Section: Physical Motivationsmentioning

confidence: 99%

An Invariance Principle to Ferrari–Spohn Diffusions

Ioffe

Shlosman

Velenik

2015

Commun. Math. Phys.

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“…As a direct corollary of Theorem 3 it was deduced in [15] that the following upper bound on the fluctuations of all level lines Γ…”

Section: Introductionmentioning

confidence: 98%

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Caputo¹,

Lubetzky²,

Martinelli³

et al. 2016

J. Eur. Math. Soc.

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ABSTRACT. Consider the classical (2 + 1)-dimensional Solid-On-Solid model above a hard wall on an L × L box of Z 2 . The model describes a crystal surface by assigning a non-negative integer height ηx to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is proportional to exp(−βH(η)), where β is the inverse-temperature and H(η) sums the absolute values of height differences between neighboring sites.We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability the height of almost all sites is concentrated on two levels, H(L) = ⌊(1/4β) log L⌋ and H(L) − 1. Moreover, for most values of L the height is concentrated on the single value H(L). Next, we study the ensemble of level lines corresponding to the heights (H(L), H(L)−1, . . .). We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes L k , the rescaled macroscopic level line at height H(L k ) − n has a limiting shape if the fractional parts of (1/4β) log L k converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square Q and its boundary has a flat component on the boundary of Q. Finally, the highest macroscopic level line has L 1/3+o(1) k fluctuations along the flat part of the boundary of its limiting shape.

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“…A rigorous study of this phenomenon in the (2 + 1)D Solid-On-Solid (SOS) model-a low temperature approximation of the 3D Ising model-dates back to Bricmont, El Mellouki and Fröhlich [1] in 1986, where it was shown that, in the presence of a hard wall at height 0, the typical height of a site in the bulk is propelled to order log L. Thereafter, a detailed description of the shape of this random surface was obtained by Caputo et al [4][5][6], showing that it typically becomes rigid at a height which is one of two consecutive (explicit) integers, through a sequence of nested level lines each encompassing a (1−ε)-fraction of the sites (analogous behavior was later established [16] for the more general family of |∇φ| p -random surface model, where the SOS model is the case p = 1). The level lines near the center sides of the box behave as random walks-a ubiquitous feature of interfaces in low temperature spin systems-albeit with cube-root fluctuations, as their laws are tilted by the entropic repulsion effect.…”

Section: Introductionmentioning

confidence: 99%

On the limiting law of line ensembles of Brownian polymers with geometric area tilts

Dembo¹,

Lubetzky²,

Zeitouni³

2022

Preprint

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We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the level lines of the (2 + 1)d SOS model above a hard wall, was studied in two works from 2019 by Caputo, Ioffe and Wachtel. In those works, the tightness of the law of the top k paths, for any fixed k, was established under either zero or free boundary conditions, which in the former setting implied the existence of a limit via a monotonicity argument. Here we address the open problem of a limit under free boundary conditions: we prove that as the interval length, followed by the number of paths, go to ∞, the top k paths converge to the same limit as in the free boundary case, as conjectured by Caputo, Ioffe and Wachtel.

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The shape of the (2+1)d SOS surface above a wall

Cited by 15 publications

References 17 publications

An Invariance Principle to Ferrari–Spohn Diffusions

An Invariance Principle to Ferrari–Spohn Diffusions

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

On the limiting law of line ensembles of Brownian polymers with geometric area tilts

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