# Phase coexistence of gradient Gibbs measures

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A gradient Gibbs measure is the projection to the gradient variables $\eta_b=\phi_y-\phi_x$

of the Gibbs measure of the form

$$

P(\textd\phi)=Z^{-1}\exp\Bigl\{-\beta\sum_{\langle x,y\rangle}V(\phi_y-\phi_x)\Bigr\}\textd\phi,

$$

where $V$ is a potential, $\beta$ is the inverse temperature and $\textd\phi$ is the product

Lebesgue measure. The simplest example is the (lattice) Gaussian free field

$V(\eta)=\frac12\kappa\eta^2$. A well known result of Funaki and Spohn (and Sheffield)

asserts that, for any uniformly-convex $V$, the possible infinite-volume measures of this type are

characterized by the \emph{tilt}, which is a vector $u\in\R^d$ such that

$E(\eta_b)=u\cdot b$ for any (oriented) edge $b$. I will discuss a simple example

for which this result fails once $V$ is sufficiently non-convex thus showing that

the conditions of Funaki-Spohn's theory are generally optimal. The underlying

mechanism is an order-disorder phase transition known, e.g., from the context

of the $q$-state Potts model with sufficiently large $q$. Based on joint work

with Roman Koteck\'y.