On the absence of breakdown of symmetry for the plane rotator model with long-range random interaction

Abstract

We study the plane rotator model with hamiltonian

$$ - \frac{1}{2}\sum\limits_{x \ne y} {J_{xy} \frac{{\cos (\theta _x - \theta _y )}}{{\left| {\left. {x - y} \right|} \right.^{3 + \in } }}}$$

whereJ _xy for different pair (x, y) are independent symmetric random variables. It is proved that for almost allJ, all the Gibbs statesP(J) are rotation invariant.