A Cluster Expansion and the Decay of Correlations of the 1D Long-Range Ising Model at Low Temperatures

In this work, a convergent low-temperature cluster expansion of the one-dimensional long-range ferromagnetic Ising model with polynomial decay $\alpha \in (1,2]$ is developed; that is, $J(r)=r^{-\alpha }$. As an application, the $n$-point correlations are studied and the two-point correlation is shown to be algebraic with rate of decay exactly $\alpha$.