DISORDER AND WETTING TRANSITION: THE PINNED HARMONIC CRYSTAL IN DIMENSION THREE OR LARGER

• 成果类型: Article
• 署名作者: Giacomin, Giambattista; Lacoin, Hubert
• 署名单位: Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1312
• 发表日期: 2018
• 页码: 577-606
• 关键词: lattice free-field; entropic repulsion; gradient fields

摘要

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 is an element of 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 is an element of Zd) are i.i.d. centered and exponentially integrable random variables of unit variance and beta >= 0. 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 <= 0 and positive for h > 0. We show that, for beta not equal 0, the transition happens at h = h(c)(beta) := -log 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 SE arrow 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 -> infinity.