Disorder and wetting transition: The pinned harmonic crystal in dimension three or larger

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 $N\to\infty$.