## The Annals of Probability

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

Luiz Renato G. Fontes

#### 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.

#### Article information

Source
Ann. Probab., Volume 21, Number 3 (1993), 1394-1412.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176989123

Digital Object Identifier
doi:10.1214/aop/1176989123

Mathematical Reviews number (MathSciNet)
MR1235421

Zentralblatt MATH identifier
0785.60047

JSTOR
Fontes, Luiz Renato G. An Ordered Phase with Slow Decay of Correlations in Continuum $1/r^2$ Ising Models. Ann. Probab. 21 (1993), no. 3, 1394--1412. doi:10.1214/aop/1176989123. https://projecteuclid.org/euclid.aop/1176989123