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).
G.G. also acknowledges support from Grant ANR-15-CE40-0020. H.L. acknowledges support from a productivity Grant from CNPq and a Jovem Cientísta do Nosso Estado grant from FAPERJ.
This work has been performed in part while G.G. was visiting IMPA with the support of the Franco-Brazilian network in mathematics.
"The disordered lattice free field pinning model approaching criticality." Ann. Probab. 50 (4) 1478 - 1537, July 2022. https://doi.org/10.1214/22-AOP1566