Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy

Lacoin, Hubert

doi:10.5802/jep.110

Cited by 7 publications

(5 citation statements)

References 33 publications

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“…Comparison with another interface repulsion phenomenon. The disorder-induced repulsion phenomenon highlighted in the present paper bears some analogy with the entropic repulsion phenomenon observed in the SOS model constrained to remain positive and recently studied in detail by one of the authors [25,26]. The introduction of disorder has in fact effects that are very similar to those induced by the imposing a positivity constraint to the SOS model: the phase transition is smoothened, it vanishes like of (h−h c ) ν with ν ≥ 2 approaching the critical point (as opposed to linearly for the model without constraint), and the interface is repelled to a distance from level zero that diverges in this limit.…”

Section: Localization Strategy and Sketch Of Proofssupporting

confidence: 75%

The disordered lattice free field pinning model approaching criticality

Giacomin¹,

Lacoin²

2019

Preprint

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We continue the study, initiated in [21], of the localization transition of a lattice free field φ = (φ(x)) x∈Z d , d ≥ 3, in presence of a quenched disordered substrate. The presence of the substrate affects the interface at the spatial sites in which the interface height is close to zero. This corresponds to the Hamiltonian Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is 2σ 2 d | log(h − hc(β))| to leading order when h hc(β) except on a vanishing fraction of sites (σ 2 d is the single site variance of the free field).

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Section: Localization Strategy and Sketch Of Proofssupporting

confidence: 75%

The disordered lattice free field pinning model approaching criticality

Giacomin¹,

Lacoin²

2019

Preprint

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“…(A similar infinite sequence of critical points along which the interface height diverges occurs in the wetting problem, where in lieu of the bulk external field λ the surface is tilted by λ ř v 1 φv"0 , rewarding only sites that are pinned to height 0. A detailed understanding of this was developed by Lacoin [30,31] confirming the predicted phenomena [13]. See also [1,17] and the excellent survey by Ioffe and Velenik [27] for more on the layering phenomena associated with wetting/pinning.)…”

Section: Introductionmentioning

confidence: 69%

Concentration inequalities for polynomials of contracting Ising models

Gheissari¹,

Lubetzky²,

Peres³

2018

Electron. Commun. Probab.

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We study the concentration of a degree-d polynomial of the N spins of a general Ising model, in the regime where single-site Glauber dynamics is contracting. For d = 1, Gaussian concentration was shown by Marton (1996) and Samson (2000) as a special case of concentration for convex Lipschitz functions, and extended to a variety of related settings by e.g., Chazottes et al. (2007) and Kontorovich and Ramanan (2008). For d = 2, exponential concentration was shown by Marton (2003) on lattices. We treat a general fixed degree d with O(1) coefficients, and show that the polynomial has variance O(N d ) and, after rescaling it by N −d/2 , its tail probabilities decay as exp(−c r 2/d ) for deviations of r ≥ C log N .

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“…Let us describe shortly here what we believe should occur, when β is sufficiently large A step towards this result was performed in [3]: analyticity and the existence of the infinite volume measure at level n were proved to hold in the interval (J n+2+ε , J n+3−ε ) for β ≥ β n , where β n diverges when n goes to infinity. A scheduled sequel to the present paper aims to bring more of this conjecture on rigorous ground [25].…”

Section: About Layering Transitionsmentioning

confidence: 99%

“…Between these layering transition, the free energy should be analytic in both parameters h and β. This aspect of the problem is to be developed in a second paper [25].…”

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confidence: 99%

Wetting and Layering for Solid-on-Solid I: Identification of the Wetting Point and Critical Behavior

Lacoin

2018

Commun. Math. Phys.

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We provide a complete description of the low temperature wetting transition for the two dimensional Solid-On-Solid model. More precisely we study the integer-valued field (φ(x)) x∈Z 2 , associated associated to the energy functionalIt is known since the pioneering work Chalker [14] of that for every β, there exists hw(β) > 0 delimiting a transition between a delocalized phase (h < hw(β)) where the proportion of points at level zero vanishes, and a localized phase (h > hw(β)) where this proportion is positive. We prove in the present paper that for β sufficiently large we have hw(β) = log e 4β e 4β − 1 .Furthermore we provide a sharp asymptotic for the free energy at the vicinity of the critical point: we show that close to hw(β), the free energy is approximately piecewise affine and that the points of discontinuity for the derivative of the affine approximation forms a geometric sequence accumulating on the right of hw(β). This asymptotic behavior provides a strong evidence for the conjectured existence of countably many "layering transitions" at the vicinity of the critical point, corresponding to jumps for the typical height of the field.

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Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy

Abstract: We consider the Solid-On-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field (φ(x)) x∈Z 2 , and the energy functional

Cited by 7 publications

References 33 publications

The disordered lattice free field pinning model approaching criticality

The disordered lattice free field pinning model approaching criticality

Concentration inequalities for polynomials of contracting Ising models

Wetting and Layering for Solid-on-Solid I: Identification of the Wetting Point and Critical Behavior

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