WIAS Preprint No. 412, (1998)

The continuous spin random field model: Ferromagnetic ordering in d greater-than 3


  • Külske, Christof

2010 Mathematics Subject Classification

  • 82B44 82B26 82B28


  • Disordered Systems, Contour Models, Cluster Expansions, Renormalization Group, Random Field Model


We investigate the Gibbs-measures of ferromagnetically coupled continuous spins in double-well potentials subjected to a random field (our specific example being the Φ4 theory), showing ferromagnetic ordering in d ≥ 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 ≥ 3.

Appeared in

  • Rev. Math. Phys. 11, No. 10, (1999), pp. 1269-1314

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