Cluster Expansions and Applications to the Ising Model

Abstract: Cluster expansions are a key tool used to derive series expansions for many quantities of interest. We will derive the cluster expansion and apply it to the Ising model in two regimes.  We then show an explicit series expansion for the pressure for the Ising model in a strong magnetic field. Lastly, we show exponential decay of correlations for the Ising model in the very low temperature regime.