Cluster Expansion and Decay of Correlations for Multidimensional Long-Range Ising Models

Abstract

We develop the cluster expansion for the multidimensional multiscaled contours defined by three of us. These contours are suitable for long-range Ising models with interaction $J_{xy}=J(|x-y|)=J/|x-y{|}^{\alpha }$, $J>0$, and $\alpha >d$. As an application of the convergence of the cluster expansion at low temperatures, we study the decay of the truncated two-point correlation functions, showing that the decay is algebraic with coefficient $\alpha$.

arXiv comment40 pages, 6 figures