Decimation of the Dyson-Ising Ferromagnet

Aernout van Enter (1) , Arnaud Le Ny (2)

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

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.

Mots clés: Long-range Ising models; hidden phase transitions; generalized Gibbs measures

Domaines: Probabilités [math.PR]

Aernout van Enter, Arnaud Le Ny. Decimation of the Dyson-Ising Ferromagnet. 2016. ⟨hal-01290716⟩