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 $ϕ=(ϕ(x))_{x \in Z^d}$, $d\ge 3$, 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 $$ \sum_{x\in Z^d }(βω_x+h)δ_x,$$ where $δ_x=1_{[-1,1]}(ϕ(x))$, and $(ω_x)_{x\in Z^d}$ is an IID centered field. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c(β)$: from a delocalized phase $h h_c(β)$ 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 $ω=ω_x$, that is $-h_c(β)=λ(β):=\log E[e^{βω}]$. Here we obtain the sharp critical behavior of the free energy approaching criticality: $$\lim_{u\searrow 0} \frac{ F(β,h_c(β)+u)}{u^2}= \frac{1}{2\, \textrm{Var}\left(e^{βω-λ(β)}\right)}.$$ Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is $\sqrt{2σ_d^2\vert\log(h-h_c(β))\vert}$ to leading order when $h\searrow h_c(β)$ except on a vanishing fraction of sites ($σ_d^2$ is the single site variance of the free field).