The continuous spin random field model: Ferromagnetic ordering in d>=3

Abstract

We investigate the Gibbs-measures of ferromagnetically coupled continuous spins in double-well potentials subjected to a random field (our specific example being the $ϕ^4$ theory), showing ferromagnetic ordering in $d\geq 3$ dimensions for weak disorder and large energy barriers. We map the random continuous spin distributions to distributions for an Ising-spin system by means of a single-site coarse-graining method described by local transition kernels. We derive a contour- representation for them with notably positive contour activities and prove their Gibbsianness. This representation is shown to allow for application of the discrete-spin renormalization group developed by Bricmont/Kupiainen implying the result in $d\geq 3$.