Dyson models: One-sided and two-sided aspects

Abstract: I will discuss Dyson models, long-range one-dimensional Ising spin systems with ferromagnetic pair interactions. I first will discuss how they provide an example for non-equivalence of one-sided and two-sided regularity properties. The switching between one-sided and two-sided descriptions lies at the base of Thermodynamic Formalism, which describes various Dynamical Systems as one-dimensional statistical mechanics models, usually with fast decaying interactions.. In particular at low temperatures there can be differences in continuity properties between a measure's one-sided and two-sided conditional probabilities. Gibbs measures for summable interactions can be characterised by the continuity of their two-sided (from both left and right) conditional probabilities, The one-sided analogue, where a spin depends continuously on the left only (the past), forms the class of g-measures. It was known before that a g-measure need not be a Gibbs measure. We show the opposite, that a Gibbs measure for a summable interaction, and in particular for a low-temperature Dyson interaction, can fail to be a g-measure.

In a somewhat different usage, one-sided versus two-sided systems describes models on a half-line as compared with the model on the whole line. One can consider the model on the whole line as a model on two half-lines with a coupling term between the half-lines. Properties of this coupling term, connecting either ordered or disordered half-lines, such as its boundedness, and integrability, explain the properties of the "metastates" -- the distribution over limit states- of the model with random boundary conditions at low temperatures. Also they explain the continuity, or the lack thereof, of the eigenfunctions of Ruelle transfer operators at high temperatures. Continuity of these eigenfunctions is a sufficient, but not necessary, condition for continuity of the one-sided conditional probabilities. We show that in certain circumstances the Radon-Nikodym density on the half-line, between coupled and decoupled Gibbs measures -- which is equal to the transfer operator eigenfunction--, can have stronger regularity properties than the analogous Radon-Nikodym density on the whole line.

The talk is based on joint works with Rodrigo Bissacot, Eric Endo, Roberto Fernández, Arnaud Le Ny, Mirmukhsin Makhmudov and Evgeny Verbitskiy

mathematical physics

Audience: researchers in the discipline

One world IAMP mathematical physics seminar

Series comments: In order to receive announcements, please send an email to IAMPseminars@gmail.com with “subscribe” in the subject line.