Asymptotics of correlation decrease for Gibbs spin fields

The asymptotics of decrease of the correlations $\langle F_0,F_x\rangle$ is considered in the case of Gibbs spin fields on a lattice $\mathbb Z^\nu$ of arbitrary dimension at high temperatures for a large class of functions $F_0$ ($F_x$ is the function $F_0$ “shifted” by the vector $x\in\mathbb Z^\nu$). The correlation in the case of shift of $F_0$ along the “time” axis is studied in most detail. In all the considered cases the leading (exponential) term of the asymptotic behavior is found, and also its pre-exponential factor.