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The disordered lattice free field pinning model approaching criticality. (English) Zbl 1503.60146

Summary: We continue the study, initiated in [the authors, J. Eur. Math. Soc. (JEMS) 20, No. 1, 199–257 (2018; Zbl 1430.60084)], of the localization transition of a lattice free field \(\phi ={(\phi (x))_{x\in{\mathbb{Z}^d}}}\), \(d\ge 3\), 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 \[ \sum \limits_{x\in{\mathbb{Z}^d}}(\beta{\omega_x}+h){\delta_x}, \] where \({\delta_x}={\mathbf{1}_{[-1,1]}}(\phi (x))\), and \({({\omega_x})_{x\in{\mathbb{Z}^d}}}\) is an i.i.d. centered field. A transition takes place when the average pinning potential \(h\) goes past a threshold \({h_c}(\beta )\): from a delocalized phase \(h<{h_c}(\beta)\), where the field is macroscopically repelled by the substrate, to a localized one \(h>{h_c}(\beta)\) where the field sticks to the substrate. In [loc. cit.], the critical value of \(h\) is identified and it coincides, up to the sign, with the log-Laplace transform of \(\omega ={\omega_x} \), that is \(-{h_c}(\beta )=\lambda (\beta ):=\log \mathbb{E}[{e^{\beta \omega }}]\). Here, we obtain the sharp critical behavior of the free energy approaching criticality: \[ \underset{u\searrow 0}{\lim }\frac{\text{d}(\beta ,{h_c}(\beta )+u)}{{u^2}}=\frac{1}{2\operatorname{Var}({e^{\beta \omega -\lambda (\beta )}})}. \] Moreover, we give a precise description of the trajectories of the field in the same regime: to leading order as \(h\searrow{h_c}(\beta )\) the absolute value of the field is \(\sqrt{2{\sigma_d^2}|\log (h-{h_c}(\beta ))|}\) except on a vanishing fraction of sites \(({\sigma_d^2}\) is the single site variance of the free field).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
82B27 Critical phenomena in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

Citations:

Zbl 1430.60084
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References:

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