Ann. Probab. 50 (4), 1478-1537, (July 2022) DOI: 10.1214/22-AOP1566
Giambattista Giacomin, Hubert Lacoin
KEYWORDS: Lattice free field, disordered pinning model, localization transition, Critical behavior, disorder relevance, multiscale analysis, 60K35, 60K37, 82B27, 82B44
We continue the study, initiated in (J. Eur. Math. Soc. (JEMS) 20 (2018) 199–257), of the localization transition of a lattice free field , , 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
where , and is an i.i.d. centered field. A transition takes place when the average pinning potential h goes past a threshold : from a delocalized phase , where the field is macroscopically repelled by the substrate, to a localized one 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 , that is . Here, we obtain the sharp critical behavior of the free energy approaching criticality:
Moreover, we give a precise description of the trajectories of the field in the same regime: to leading order as the absolute value of the field is except on a vanishing fraction of sites ( is the single site variance of the free field).