The disordered lattice free field pinning model approaching criticality

Abstract

We continue the study, initiated in (J. Eur. Math. Soc. (JEMS) 20 (2018) 199–257), of the localization transition of a lattice free field ϕ=(ϕ(x))x∈Zd, d≥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 ∑x∈Zd(βωx+h)δx, where δx= 1[−1,1](ϕ(x)), and (ωx)x∈Zd is an i.i.d. centered field. A transition takes place when the average pinning potential h goes past a threshold hc(β): from a delocalized phase h<hc(β), where the field is macroscopically repelled by the substrate, to a localized one h>hc(β) where the field sticks to the substrate. In (J. Eur. Math. Soc. (JEMS) 20 (2018) 199–257), the critical value of h is identified and it coincides, up to the sign, with the log-Laplace transform of ω=ωx, that is −hc(β)=λ(β):=logE[eβω]. Here, we obtain the sharp critical behavior of the free energy approaching criticality: limu↘0 d(β,hc(β)+u) u2=1 2Var(eβω−λ(β)). Moreover, we give a precise description of the trajectories of the field in the same regime: to leading order as h↘hc(β) the absolute value of the field is 2σd2|log(h−hc(β))| except on a vanishing fraction of sites (σd2 is the single site variance of the free field).