Decimation of the Dyson-Ising Ferromagnet

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 interactions of the form $\frac{1}{{|i-j|}^{\alpha}}$, in the phase transition region (1< $\alpha \leq$ 2, and low temperature). We prove non-Gibbsianness of the decimated measure at low enough temperatures by exhibiting a point of essential discontinuity for the finite-volume conditional probabilities of decimated Gibbs measures. Thus 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 resuts 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.