An Ordered Phase with Slow Decay of Correlations in Continuum $1/r^2$ Ising Models

Abstract

For continuum $1/r^2$ Ising models, we prove that the critical value of the long range coupling constant (inverse temperature), above which an ordered phase occurs (for strong short range cutoff), is exactly 1. This leads to a proof of the existence of an ordered phase with slow decay of correlations. Our arguments involve comparisons between continuum and discrete Ising models, including (quenched and annealed) site diluted models, which may be of independent interest.