SEMINARIO Probabilita'

Hubert Lacoin (IMPA, Rio de Janeiro, Brasil)

Titolo: Wetting and Layering for Solid-and-Solid

Martedi' 10 Aprile 2018 ORE 14:30

Dipartimento di Matematica e Fisica Universita' degli Studi Roma Tre Largo San Leonardo Murialdo,1 - Pal.C - AULA 311

Abstract Solid-on-Solid (SOS) was introduced in the early 50s as a simplified model for lattice interfaces. It is believed to display the same low temperature behavior as three-dimensiona systems with phase coexistence while being considerably easier to analyze. The objective of this talk is to present the result we recently obtained for SOS interacting with a solid substrate, which is the problem associated with the following energy functional $$ V(phi)=eta sum_{xsim y}|phi(x)-phi(y)|-sum_{x}left(h{f 1}_{{phi(x)=0}}-infty{f 1}_{{phi(x)<0}} ight), $$ for $(phi(x))_{xin mathbb Z^2}$ (the graph of $phi$ representing the interface). We prove that for eta sufficiently large, there exists a decreasing sequence $(h^*_n(eta))_{nge 0}$, satisfying $lim_{n oinfty}h^*_n(eta)=h_w(eta)$, and such that: (A) The free energy associated with the system is infinitely differentiable on $mathbb R setminus left({h^*_n}_{nge 1}cup h_w( eta) ight)$, and not differentiable on ${h^*_n}_{nge 1}$. (B) For each nge 0 within the interval $(h^*_{n+1},h^*_n)$ (with the convention $h^*_0=infty$), there exists a unique translation invariant Gibbs state which is localized around height $n$, while at a point of non-differentiability, at least two ergodic Gibbs states coexist. The respective typical heights of these two Gibbs states are $n-1$ and $n$. The value $h^*_n$ corresponds thus to a first order layering transition from level $n$ to level $n-1$.