Phase coexistence of gradient Gibbs states

Abstract

We consider the (scalar) gradient fields $\eta =({\eta }_b)$--with $b$ denoting the nearest-neighbor edges in ${\mathbb{Z}}^2$--that are distributed according to the Gibbs measure proportional to $\texte^{-βH(η)}ν(\textdη)$. Here $H={\sum }_{b}V({\eta }_b)$ is the Hamiltonian, $V$ is a symmetric potential, $\beta >0$ is the inverse temperature, and $\nu$ is the Lebesgue measure on the linear space defined by imposing the loop condition ${\eta }_{b_1}+{\eta }_{b_2}={\eta }_{b_3}+{\eta }_{b_4}$ for each plaquette $(b_1,b_2,b_3,b_4)$ in ${\mathbb{Z}}^2$. For convex $V$, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their tilt. We describe a mechanism by which the gradient Gibbs measures with non-convex $V$ undergo a structural, order-disorder phase transition at some intermediate value of inverse temperature $\beta$. At the transition point, there are at least two distinct gradient measures with zero tilt, i.e., $E{\eta }_b=0$.

3 figs, PTRF style files included

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