Enhanced Interface Repulsion from Quenched Hard–wall Randomness

We consider the harmonic crystal, or massless free field, φ = {φx}x∈Zd , d ≥ 3, that is the centered Gaussian field with covariance given by the Green function of the simple random walk on Z. Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition φx to be larger than σx, σ = {σx}x∈Zd is an IID field (which is also independent of φ), for every x in a large region DN = ND ∩ Z , with N a positive integer and D ⊂ R a rather general bounded subset of R. We are mostly motivated by results for given typical realizations of σ (quenched set–up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d + 1)–dimensional space, constrained not to go below a inhomogeneous substrate that acts as a hard wall. This substrate is mostly flat, but presents some rare anomalous spikes. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ0 is heavier than Gaussian, while essentially no effect is observed if the tail is sub–Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, φ and σ, leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large Deviation type asymptotics as N ր ∞ of the probability that φ lies above σ in DN . We will consider the annealed case too. It turns out that quenched and annealed asymptotics coincide and this fact plays a role in the proofs and concurs to building an understanding of the phenomenon. 2000 Mathematics Subject Classification: 82B24, 60K35, 60G15