Decimation of the Dyson-Ising FerromagnetReport as inadecuate

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1 Johann Bernoulli Institute of Mathematics and Computing Science 2 LAMA - Laboratoire d-Analyse et de Mathématiques Appliquées

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 , 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.

Keywords : Long-range Ising models hidden phase transitions generalized Gibbs measures

Author: Aernout Van Enter - Arnaud Le Ny -



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