##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics

## Marek Biskup

#### UCLA

## Phase coexistence of gradient Gibbs measures

##### Abstract:

A gradient Gibbs measure is the projection to the gradient variables $\eta_b=\phi_y-\phi_x$ of the Gibbs measure of the form $$ P(d\phi)=Z^{-1}\exp\Bigl\{-\beta\sum_{\langle x,y\rangle}V(\phi_y-\phi_x)\Bigr\}d\phi, $$ where $V$ is a potential, $\beta$ is the inverse temperature and $d\phi$ is the product Lebesgue measure. The simplest example is the (lattice) Gaussian free field $V(\eta)={1 \over 2}\kappa\eta^2$. A well known result of Funaki and Spohn asserts that, for any uniformly-convex $V$, the possible infinite-volume measures of this type are characterized by the {\it tilt}, which is a vector $u\in{\bf R}^d$ 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.

Host: Jason Schweinsberg

### December 1, 2005

### 9:00 AM

### AP&M 6218

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