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).


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


Zbl 1430.60084
Full Text: DOI arXiv


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