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Entropic repulsion for massless fields

Author

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  • Deuschel, Jean-Dominique
  • Giacomin, Giambattista

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).

Suggested Citation

  • Deuschel, Jean-Dominique & Giacomin, Giambattista, 2000. "Entropic repulsion for massless fields," Stochastic Processes and their Applications, Elsevier, vol. 89(2), pages 333-354, October.
    • Handle: RePEc:eee:spapps:v:89:y:2000:i:2:p:333-354
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    Cited by:

    1. Funaki, Tadahisa & Olla, Stefano, 2001. "Fluctuations for [backward difference][phi] interface model on a wall," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 1-27, July.
    2. Deuschel, Jean-Dominique & Nishikawa, Takao, 2007. "The dynamic of entropic repulsion," Stochastic Processes and their Applications, Elsevier, vol. 117(5), pages 575-595, May.

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