Seminari del Dipartimento

Probabilita'

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 $$ V(\phi)=\beta \sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_{x}\left(h{\bf 1}_{\{\phi(x)=0\}}-\infty{\bf 1}_{\{\phi(x)<0\}} \right), $$ for $(\phi(x))_{x\in \mathbb Z^2}$ (the graph of $\phi$ representing the interface). We prove that for \beta sufficiently large, there exists a decreasing sequence $(h^*_n(\beta))_{n\ge 0}$, satisfying $\lim_{n\to\infty}h^*_n(\beta)=h_w(\beta)$, and such that: (A) The free energy associated with the system is infinitely differentiable on $\mathbb R \setminus \left(\{h^*_n\}_{n\ge 1}\cup h_w(\beta)\right)$, and not differentiable on $\{h^*_n\}_{n\ge 1}$. (B) For each n\ge 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 $n-1$.
 
org: CAPUTO Pietro