The disordered lattice free field pinning model approaching criticality

Abstract

We continue the study, initiated in [Giacomin and Lacoin, JEMS 2018], of the localization transition of a lattice free field $\varphi =(\varphi (x){)}_{x\in Z^d}$, $d\ge 3$, 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 $$\sum _{x\in Z^d}(\beta {\omega }_x+h){\delta }_x,$$ where ${\delta }_x=1_{[-1,1]}(\varphi (x))$, and $({\omega }_x{)}_{x\in Z^d}$ is an IID centered field. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c(\beta )$: from a delocalized phase $h<h_c(\beta )$, where the field is macroscopically repelled by the substrate, to a localized one $h>h_c(\beta )$ where the field sticks to the substrate. In [Giacomin and Lacoin, JEMS 2018] the critical value of $h$ is identified and it coincides, up to the sign, with the $\log$-Laplace transform of $\omega ={\omega }_x$, that is $-h_c(\beta )=\lambda (\beta ):=\log E[e^{\beta \omega }]$. Here we obtain the sharp critical behavior of the free energy approaching criticality: $$\underset{u\searrow 0}{\lim }\frac{F(\beta ,h_c(\beta )+u)}{u^2}=\frac{1}{2\text{Var}\left(e^{\beta \omega -\lambda (\beta )}\right)}.$$ Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is $\sqrt{2{\sigma }_d^2|\log (h-h_c(\beta ))|}$ to leading order when $h\searrow h_c(\beta )$ except on a vanishing fraction of sites (${\sigma }_d^2$ is the single site variance of the free field).

62 pages