Asymptotics of decay of correlations for lattice spin fields at high temperatures. I. The Ising model

We find the asymptotic decrease of correlations <_{A +y},_B>,yZ^v+1, |y|, in the Ising model at high temperatures. For the case when monomials_A and_B both are odd, using the saddle-point method, we find the asymptotics of the correlations for any dimension . For even monomials_A,_B we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases: (1) =1 and the vectory has an arbitrary direction, (2)y is directed along a fixed axis and arbitrary . Here we use besides the saddle-point method, some arguments from scattering theory.