Entropic repulsion to the middle layer
We consider $\nabla\varphi$ height functions with even and convex interaction energy $W$ on the lattice $\mathbb{Z}^d$, which are restricted to take values in the set $\{-S, \ldots, S\}$ for some integer $S \ge 1$. We study the effect of entropic repulsion, which tends to push the spin values to the middle layer. We prove that the model has a unique Gibbs measure, with exponential decay of correlations, in the cases: (a) Dimension $d=2$ at all temperatures. (b) Dimensions $d\ge3$ at all temperatures, for a wide class of $W$ with non-increasing second derivative, including the family $W(x)=|x|^p$ for $p\in[1,2]$. (c) Dimensions $d\ge 3$ at both low and high temperatures, $T\in (0,\frac{W(1)d}{4(\ln d+\ln 8)})\cup(d W(2S),\infty]$, with the normalization $W(0)=0$. At low temperatures, our proof provides an alternative to Pirogov--Sinai methods. Conversely, we exhibit a class of even and convex interaction energies $W$ which, in high dimensions and suitable temperature regimes, have multiple Gibbs measures. Though uniqueness may fail, we show that the magnetization of every Gibbs measure lies in $(-\frac{1}{2},\frac{1}{2})$. This implies the delocalization of the model restricted to take values in $\{0,1,\ldots\}$ (i.e., conditioned to lie above a floor) for all dimensions, any even and convex $W$, and all temperatures. Our methods extend to additional setups: We prove that height functions taking values in the real interval $[-1,1]$ always have a unique Gibbs measure, a result previously proved only for the quadratic interaction. For height functions taking values in $\{-S+\frac{1}{2}, \ldots, S-\frac{1}{2}\}$, $S \ge1 $ integer, we prove that the magnetization of every Gibbs measure lies in $(-1,1)$. The special case $W(x)=x^2$ of our results addresses questions left open in the work of Bricmont--El Mellouki--Fr\"ohlich (1986).