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

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))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

xZd(βωx+h)δx,

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:

limu0d(β,hc(β)+u)u2=12Var(eβωλ(β)).

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.

Acknowledgements

This work has been performed in part while G.G. was visiting IMPA with the support of the Franco-Brazilian network in mathematics.

Citation

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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

Information

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

Digital Object Identifier: 10.1214/22-AOP1566

Subjects:
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

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Vol.50 • No. 4 • July 2022
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