Open Access
February 2018 Disorder and wetting transition: The pinned harmonic crystal in dimension three or larger
Giambattista Giacomin, Hubert Lacoin
Ann. Appl. Probab. 28(1): 577-606 (February 2018). DOI: 10.1214/17-AAP1312

Abstract

We consider the lattice Gaussian free field in $d+1$ dimensions, $d=3$ or larger, on a large box (linear size $N$) with boundary conditions zero. On this field, two potentials are acting: one, that models the presence of a wall, penalizes the field when it enters the lower half space and one, the pinning potential, rewards visits to the proximity of the wall. The wall can be soft, that is, the field has a finite penalty to enter the lower half-plane, or hard, when the penalty is infinite. In general, the pinning potential is disordered and it gives on average a reward $h\in\mathbb{R}$ (a negative reward is a penalty): the energetic contribution when the field at site $x$ visits the pinning region is $\beta\omega_{x}+h$, $\{\omega_{x}\}_{x\in\mathbb{Z}^{d}}$ are i.i.d. centered and exponentially integrable random variables of unit variance and $\beta\ge0$. In [J. Math. Phys. 41 (2000) 1211–1223], it is shown that, when $\beta=0$ (i.e., in the nondisordered model), a delocalization-localization transition happens at $h=0$, in particular the free energy of the system is zero for $h\le0$ and positive for $h>0$. We show that, for $\beta\neq0$, the transition happens at $h=h_{c}(\beta):=-\log\mathbb{E}\exp(\beta\omega_{x})$, and we find the precise asymptotic behavior of the logarithm of the free energy density of the system when $h\searrow h_{c}(\beta)$. In particular, we show that the transition is of infinite order in the sense that the free energy is smaller than any power of $h-h_{c}(\beta)$ in the neighborhood of the critical point and that disorder does not modify at all the nature of the transition. We also provide results on the behavior of the paths of the random field in the limit $N\to\infty$.

Citation

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Giambattista Giacomin. Hubert Lacoin. "Disorder and wetting transition: The pinned harmonic crystal in dimension three or larger." Ann. Appl. Probab. 28 (1) 577 - 606, February 2018. https://doi.org/10.1214/17-AAP1312

Information

Received: 1 July 2016; Revised: 1 March 2017; Published: February 2018
First available in Project Euclid: 3 March 2018

zbMATH: 06873691
MathSciNet: MR3770884
Digital Object Identifier: 10.1214/17-AAP1312

Subjects:
Primary: 60K35 , 60K37 , 82B27 , 82B44

Keywords: Critical behavior , Disorder irrelevance , disordered pinning model , Lattice Gaussian free field , localization transition

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.28 • No. 1 • February 2018
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