Solid-On-Solid interfaces with disordered pinning

Abstract

We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The interface is modeled by the graph of a function φ : Z 2 → Z and the disorder is given by a fixed realization of a field of IID centered random variables (ωx) x∈Z 2 . The Hamiltonian of the system depends on three parameters α, β > 0 and h ∈ R which determine respectively the intensity of nearest neighbor interaction the amplitude of disorder and the mean value of the interaction with the substrate, and is given by the expressionWe focus on the large-β/rigid phase phase of the Solid-On-Solid (SOS) model. In that regime, we provide a sharp description of the phase transition in h from a localized phase to a delocalized one corresponding respectivelly to a positive and vanishing fraction of points with φ(x) = 0. We prove that the critical value for h corresponds to that of the annealed model and is given by hc(α) = − log E[e αω ], and that near the critical point, the free energy displays the following critical behaviorVar [e αω ] E [e αω ] 2 . The positive constant θ1(β) > 0 is defined by the asymptotic probability of spikes for the infinite volume SOS with 0 boundary condition θ1(β) := limn→∞ e 4βn P β (φ(0) = n). This particular form of asymptotic behavior is the signature of an accumulation of layering transitions: At the level of heuristics the solution nu to the variational problem appearing in the asymptotics correspond to the typical distance to the defect plane at which φ localizes. The angular points appearing in the asymptotics correspond to transitional value for this integer observable which tends to infinity as u → 0+.