Entropic repulsion for massless fields

Jean-Dominique Deuschel and Giambattista Giacomin

Stochastic Processes and their Applications, 2000, vol. 89, issue 2, 333-354

Abstract: We consider the anharmonic crystal, or lattice massless field, with 0-boundary conditions outside and N a large natural number, that is the finite volume Gibbs measure on for every x[negated set membership]DN} with Hamiltonian [summation operator]x~yV([phi]x-[phi]y), V a strictly convex even function. We establish various bounds on , where [Omega]+(DN)={[phi]:[phi]x[greater-or-equal, slanted]0 for all x[set membership, variant]DN}. Then we extract from these bounds the asymptotics (N-->[infinity]) of : roughly speaking we show that the field is repelled by a hard-wall to a height of in d[greater-or-equal, slanted]3 and of O(log N) in d=2. If we interpret [phi]x as the height at x of an interface in a (d+1)-dimensional space, our results on the conditioned measure 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).

Keywords: Gibbs; measures; Massless; fields; Entropic; repulsion; Random; surfaces; Random; walk; in; random; environment (search for similar items in EconPapers)
Date: 2000
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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