THE DISORDERED LATTICE FREE FIELD PINNING MODEL APPROACHING CRITICALITY

We continue the study, initiated in (J. Eur. Math. Soc. (JEMS) 20 (2018) 199-257), of the localization transition of a lattice free field phi = (phi(x))(x is an element of Zd), d >= 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 Sigma(x is an element of Zd) (beta omega(x) + h)delta(x), where delta(x) = 1([-1,1])(phi(x)), and (omega(x))(x is an element of Zd) 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 (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 omega = omega(x), that is - h(c)(beta) = lambda(beta) := logE[e(beta omega)]. Here, we obtain the sharp critical behavior of the free energy approaching criticality: lim(u SE arrow 0) d(beta, h(c)(beta) + u)/u(2) = 1/2 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 SE arrow h(c)(beta) the absolute value of the field is root 2 sigma(2)(d) vertical bar log(h - h(c)(beta))| except on a vanishing fraction of sites (sigma(2)(d) d is the single site variance of the free field).

Automated summary

This study continues the investigation of the localization transition of a lattice free field in the presence of a quenched disordered substrate. The critical behavior of the free energy and the trajectories of the field near criticality are precisely described in this research.