Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy

TL;DR

This paper analyzes the Solid-On-Solid model with a wall, demonstrating the existence of layering transitions, Gibbs states, and the regularity of the free energy, revealing a detailed phase structure for large interaction parameter.

Contribution

It provides a rigorous analysis of layering transitions and Gibbs states in the SOS model interacting with a wall, including the differentiability of free energy and coexistence of states.

Findings

Existence of a sequence of critical parameters for layering transitions.

Differentiability of free energy except at specific transition points.

Unique Gibbs states localized at different heights within certain parameter intervals.

Abstract

We consider the Solid-On-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field $(\phi(x))_{x\in \mathbb Z^2}$, and the energy functional $$V(\phi)=\beta \sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_{x}\left( h{\bf 1}_{\{\phi(x)=0\}}-\infty{\bf 1}_{\{\phi(x)<0\}} \right).$$ 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 state 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 $n-1$. These results combined with those obtained in [23] provide a complete description of the wetting and layering transition for SOS.