The disordered lattice free field pinning model approaching criticality

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).