Wetting Transition of the Two-Dimensional Solid-on-Solid Model with Quenched Disorder
Abstract
We study the wetting transition for a two-dimensional Solid-On-Solid (SOS) interface above a hard wall in the presence of quenched disorder. The interface is represented by a nonnegative integer-valued height function ϕ, with Hamiltonian ℋω(ϕ)=β∑x∼ y|ϕ(x)−ϕ(y)|−∑x(h+αωx−λ(α))1{ϕ(x)=0}, where (ωx)x∈ℤ2 is an i.i.d. centered field, α≥0 is the disorder strength, and λ(α)=log𝔼[eαω0]. This normalization is chosen so that the annealed model coincides with the corresponding homogeneous wetting model. As the wall attraction h increases, the interface undergoes a transition from a delocalized phase, in which contacts with the wall have vanishing density, to a localized phase with a positive density of contacts. For sufficiently large β, the homogeneous wetting point and its sharp near-critical free-energy behavior are known. We prove that, for every fixed α≥0, the quenched critical point coincides with the homogeneous wetting point, hc(β,α)=hw(β)=−log(1−e−4β). Moreover, writing u=h−hw(β), we show that the quenched excess free energy satisfies F̅β(α,u)=F̅homo(β,u)+o(u3) as u↓0, where F̅homo(β,u) denotes the leading homogeneous wetting asymptotic and satisfies F̅homo(β,u)asymp u3. Thus, at the level of both the critical point and the leading near-critical free-energy asymptotics, the quenched model has the same behavior as the homogeneous SOS wetting model. This contrasts with the disordered SOS pinning model, in which quenched disorder leaves the critical point unchanged but modifies the leading critical behavior of the free energy.
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BibTeX
@misc{choi2026wetting,
title = {{Wetting Transition of the Two-Dimensional Solid-on-Solid Model with Quenched Disorder}},
author = {Seokun Choi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01157},
eprint = {2610.01157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01157},
}
RIS
TY - UNPB
TI - Wetting Transition of the Two-Dimensional Solid-on-Solid Model with Quenched Disorder
AU - Seokun Choi
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01157
UR - https://arxiv.org/abs/2610.01157
AB - We study the wetting transition for a two-dimensional Solid-On-Solid (SOS) interface above a hard wall in the presence of quenched disorder. The interface is represented by a nonnegative integer-valued height function ϕ, with Hamiltonian ℋ^ω(ϕ)=β∑_x∼ y|ϕ(x)−ϕ(y)|−∑_x(h+αω_x−λ(α))1_{ϕ(x)=0}, where (ω_x)_x∈ℤ^2 is an i.i.d. centered field, α≥0 is the disorder strength, and λ(α)=log𝔼[e^αω_0]. This normalization is chosen so that the annealed model coincides with the corresponding homogeneous wetting model. As the wall attraction h increases, the interface undergoes a transition from a delocalized phase, in which contacts with the wall have vanishing density, to a localized phase with a positive density of contacts. For sufficiently large β, the homogeneous wetting point and its sharp near-critical free-energy behavior are known. We prove that, for every fixed α≥0, the quenched critical point coincides with the homogeneous wetting point, h_c(β,α)=h_w(β)=−log(1−e^−4β). Moreover, writing u=h−h_w(β), we show that the quenched excess free energy satisfies F̅_β(α,u)=F̅_homo(β,u)+o(u^3) as u↓0, where F̅_homo(β,u) denotes the leading homogeneous wetting asymptotic and satisfies F̅_homo(β,u)asymp u^3. Thus, at the level of both the critical point and the leading near-critical free-energy asymptotics, the quenched model has the same behavior as the homogeneous SOS wetting model. This contrasts with the disordered SOS pinning model, in which quenched disorder leaves the critical point unchanged but modifies the leading critical behavior of the free energy.
ER -