Phase transitions for continuous-spin Ising ferromagnets

Abstract

We study the comparison of continuous-spin ferromagnetic Ising models which differ only in their a priori single-spin weighting measures, and characterize the relationship of two even weighting measures ν′, ν on R such that the spin expectations of any ferromagnet with single-spin weighting measure ν′ are less than those of the same ferromagnet with single-spin measure ν. Combining these comparison results with an extension of Bortz and Griffiths' variant of the Peierls argument, we prove that any (nontrivial) continuous-spin ferromagnetic Ising model of dimension at least 2 with translation-invariant pair interaction is spontaneously magnetized at low temperature. Thus, phase transitions are generic in ferromagnetic Ising models of dimension at least 2.