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Decimation of the Dyson–Ising ferromagnet

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  • van Enter, Aernout
  • Le Ny, Arnaud

Abstract

We study the decimation to a sublattice of half the sites of the one-dimensional Dyson–Ising ferromagnet with slowly decaying long-range pair potentials of the form 1|i−j|α, deep in the phase transition region (1<α≤2 and low temperature). We prove non-Gibbsianness of the decimated measures at low enough temperatures by exhibiting a point of essential discontinuity for the (finite-volume) conditional probabilities of decimated Gibbs measures. This result complements previous work proving conservation of Gibbsianness for fastly decaying potentials (α>2) and provides an example of a “standard” non-Gibbsian result in one dimension, in the vein of similar results in higher dimensions for short-range models. We also discuss how these measures could fit within a generalized (almost vs. weak) Gibbsian framework. Moreover we comment on the possibility of similar results for some other transformations.

Suggested Citation

  • van Enter, Aernout & Le Ny, Arnaud, 2017. "Decimation of the Dyson–Ising ferromagnet," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3776-3791.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:11:p:3776-3791
    DOI: 10.1016/j.spa.2017.03.007
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    References listed on IDEAS

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    1. Maes, C. & Redig, F. & Moffaert, A. Van & Leuven, K. U., 1999. "Almost Gibbsian versus weakly Gibbsian measures," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 1-15, January.
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