Decimation of the Dyson–Ising ferromagnet

Aernout van Enter and Arnaud Le Ny

Stochastic Processes and their Applications, 2017, vol. 127, issue 11, 3776-3791

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.

Keywords: Long-range Ising models; Hidden phase transitions; Generalized Gibbs measures (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1016/j.spa.2017.03.007

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