Upper bound on the decay of correlations in the plane rotator model with long-range random interaction

Abstract

We give an upper bound on the decay of correlation function for the plane rotator model with Hamiltonian

$$ - \frac{1}{2}\mathop \sum \limits_{xy} \frac{{J_{xy} \cos (\theta _x - \theta _y )}}{{\| {x - y} \|^{({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2} + \varepsilon )^d } }}$$

in dimensiond=1 andd=2 when (J _xy are independent random variables with mean zero.