Decimation of the Dyson-Ising ferromagnet

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/vertical bar i-j vertical bar(alpha), deep in the phase transition region (1 < alpha <= 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 (alpha > 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. (C) 2017 Elsevier B.V. All rights reserved.