Entropic repulsion for massless fields

[摘要] We consider the anharmonic crystal, or lattice massless field, with 0-boundary conditions outside D-N = ND boolean AND Z(d), D subset of or equal to R-d and N a large natural number, that is the finite volume Gibbs measure P-N on {phi is an element of R-Zd : phi(x) = 0 for every x is not an element of D-N} with Hamiltonian Sigma(x similar to y) V(phi(x) - phi(y)), V a strictly convex even function. We establish various bounds on P-N(Omega(+)(D-N)), where Omega(+)(D-N) = {phi: phi(x) greater than or equal to 0 for all x is an element of D-N}. Then we extract from these bounds the asymptotics (N --> infinity) of P-N(.\Omega(+)(D-N)): roughly speaking we show that the field is repelled by a hard-wall to a height of O(root log N) in d greater than or equal to 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 P-N(.\Omega(+)(D-N)) 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-Sjostrand representation). (C) 2000 Elsevier Science B.V. All rights reserved.

[发布日期] 2000-10-01

[关键词] Gibbs measures;massless fields;entropic repulsion;random surfaces;random walk in random environment