The disordered lattice free field pinning model approaching criticality

We continue the study, initiated in (J. Eur. Math. Soc. (JEMS) 20 (2018) 199–257), of localization transition a lattice free field ϕ=(ϕ(x))x∈Zd, d≥3, presence quenched disordered substrate. The substrate affects interface at spatial sites which height is close to zero. This corresponds Hamiltonian ∑x∈Zd(βωx+h)δx, where δx= 1[−1,1](ϕ(x)), and (ωx)x∈Zd an i.i.d. centered field. A takes place when average pinning potential h goes past threshold hc(β): from delocalized phase hhc(β) sticks In critical value identified it coincides, up sign, with log-Laplace transform ω=ωx, that −hc(β)=λ(β):=logE[eβω]. Here, we obtain sharp behavior energy approaching criticality: limu↘0 d(β,hc(β)+u) u2=1 2Var(eβω−λ(β)). Moreover, give precise description trajectories same regime: leading order as h↘hc(β) absolute 2σd2|log(h−hc(β))| except on vanishing fraction (σd2 single site variance field).