Mathematics Colloquia and Seminars

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Description

ABSTRACT . We will consider models of (2+1)-dimensional random surfaces at low tem- perature. The surface is described by a discrete height η x ∈ Z attached to each vertex x of a L × L box Λ in Z 2 . Outside the box the height is fixed to be zero. The probability of a surface configuration {η x } x∈Λ is proportional to exp(−βH(η)), where β 1 is p the inverse-temperature and H(η) = x,y |∇ x,y η| , where p ≥ 1 and the sum runs over nearest neighbor vertices. When p = 1 the model is the Solid-On-Solid model, for p = 2 is the Discrete Gaussian Model and for p = +∞ it is the restricted SOS model with ∇ x,y η ∈ {−1, 0, +1}. We will study the effect of the entropic repulsion produced by the constraint η x ≥ 0 ∀x ∈ Λ. We will show that, w.h.p. and apart small local fluctuations, the surface concentrates at a well defined height H(L) which is determined by the local large deviations without the constraint. In the discrete Gaussian case H(L) differs from the value predicted by Bricmont, El Meloukki and Fr ̈ ohlich (’86). For p = 1 we are also able to determine the limiting shape of the level curves and prove that their fluctuation exponent is 1/3. In this case we can also compute the asymptotic as L → ∞ of the probability that the unconstrained surface is non-negative. The results have been obtained in collaboration with P. Caputo, E. Lubetzky, A. Sly and F.L. Toninelli.