Entropic repulsion for massless fields

We consider the anharmonic crystal, or lattice massless field, with 0-boundary conditions outside DN=ND∩Zd, D⊆Rd and N a large natural number, that is the finite volume Gibbs measure PN on {ϕ∈RZd:ϕx=0 for every x∉DN} with Hamiltonian ∑x∼yV(ϕx−ϕy), V a strictly convex even function. We establish various bounds on PN(Ω+(DN)), where Ω+(DN)={ϕ:ϕx⩾0 for all x∈DN}. Then we extract from these bounds the asymptotics (N→∞) of PN(·|Ω+(DN)): roughly speaking we show that the field is repelled by a hard-wall to a height of O(log N) in d⩾3 and of O(log N) in d=2. If we interpret ϕx as the height at x of an interface in a (d+1)-dimensional space, our results on the conditioned measure PN(·|Ω+(DN)) clarify some aspects of the effect of a hard-wall on an interface. Besides classical techniques, like the FKG inequalities and the Brascamp–Lieb inequalities for log-concave measures, we exploit a representation of the random field in term of a random walk in dynamical random environment (Helffer–Sjöstrand representation).