July 2022 The disordered lattice free field pinning model approaching criticality
Giambattista Giacomin, Hubert Lacoin
Author Affiliations +
Ann. Probab. 50(4): 1478-1537 (July 2022). DOI: 10.1214/22-AOP1566


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))xZd, d3, 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 δx=1[1,1](ϕ(x)), and (ωx)xZd 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:


Moreover, we give a precise description of the trajectories of the field in the same regime: to leading order as hhc(β) the absolute value of the field is 2σd2|log(hhc(β))| except on a vanishing fraction of sites (σd2 is the single site variance of the free field).

Funding Statement

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.


Download Citation

Giambattista Giacomin. Hubert Lacoin. "The disordered lattice free field pinning model approaching criticality." Ann. Probab. 50 (4) 1478 - 1537, July 2022. https://doi.org/10.1214/22-AOP1566


Received: 1 July 2020; Revised: 1 October 2021; Published: July 2022
First available in Project Euclid: 11 May 2022

MathSciNet: MR4420425
zbMATH: 1503.60146
Digital Object Identifier: 10.1214/22-AOP1566

Primary: 60K35 , 60K37 , 82B27 , 82B44

Keywords: Critical behavior , disorder relevance , disordered pinning model , Lattice free field , localization transition , multiscale analysis

Rights: Copyright © 2022 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.50 • No. 4 • July 2022
Back to Top