## 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 $\mathit{\varphi}={(\mathit{\varphi}(\mathit{x}))}_{\mathit{x}\in {\mathbb{Z}}^{\mathit{d}}}$, $\mathit{d}\ge 3$, 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

$$\sum _{\mathit{x}\in {\mathbb{Z}}^{\mathit{d}}}(\mathit{\beta}{\mathit{\omega}}_{\mathit{x}}+\mathit{h}){\mathit{\delta}}_{\mathit{x}},$$

where ${\mathit{\delta}}_{\mathit{x}}={\mathbf{1}}_{[-1,1]}(\mathit{\varphi}(\mathit{x}))$, and ${({\mathit{\omega}}_{\mathit{x}})}_{\mathit{x}\in {\mathbb{Z}}^{\mathit{d}}}$ is an i.i.d. centered field. A transition takes place when the average pinning potential *h* goes past a threshold ${\mathit{h}}_{\mathit{c}}(\mathit{\beta})$: from a delocalized phase $\mathit{h}<{\mathit{h}}_{\mathit{c}}(\mathit{\beta})$, where the field is macroscopically repelled by the substrate, to a localized one $\mathit{h}>{\mathit{h}}_{\mathit{c}}(\mathit{\beta})$ 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 $\mathit{\omega}={\mathit{\omega}}_{\mathit{x}}$, that is $-{\mathit{h}}_{\mathit{c}}(\mathit{\beta})=\mathit{\lambda}(\mathit{\beta}):=log\mathbb{E}[{\mathit{e}}^{\mathit{\beta}\mathit{\omega}}]$. Here, we obtain the sharp critical behavior of the free energy approaching criticality:

$$\underset{\mathit{u}\searrow 0}{lim}\frac{\mathrm{d}(\mathit{\beta},{\mathit{h}}_{\mathit{c}}(\mathit{\beta})+\mathit{u})}{{\mathit{u}}^{2}}=\frac{1}{2Var({\mathit{e}}^{\mathit{\beta}\mathit{\omega}-\mathit{\lambda}(\mathit{\beta})})}.$$

Moreover, we give a precise description of the trajectories of the field in the same regime: to leading order as $\mathit{h}\searrow {\mathit{h}}_{\mathit{c}}(\mathit{\beta})$ the absolute value of the field is $\sqrt{2{\mathit{\sigma}}_{\mathit{d}}^{2}|log(\mathit{h}-{\mathit{h}}_{\mathit{c}}(\mathit{\beta}))|}$ except on a vanishing fraction of sites (${\mathit{\sigma}}_{\mathit{d}}^{2}$ 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

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

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