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

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.