Upper bounds on correlation decay for one-dimensional long-range spin-glass models

Some recent results on one-dimensional spin-glass models with polynomially decreasing interactions are described.     

Sturmian Ground States in Classical Lattice–Gas Models

Journal of Statistical Physics - We construct for the first time examples of non-frustrated, two-body, infinite-range, one-dimensional classical lattice–gas models without periodic...     

Provable First-Order Transitions for Nonlinear Vector and Gauge Models with Continuous Symmetries

We consider various sufficiently nonlinear vector models of ferromagnets, of nematic liquid crystals and of nonlinear lattice gauge theories with continuous symmetries. We show, employing the method of Reflection Positivity and Chessboard Estimates, that they all exhibit first-order transitions in the temperature, when the nonlinearity parameter is large enough. The results hold in dimension 2 or more for the ferromagnetic models and the RP^N–1 liquid crystal models and in dimension 3 or more for the lattice gauge models. In the two-dimensional case our results clarify and solve a recent controversy about the possibility of such transitions. For lattice gauge models our methods provide the first proof of a first-order transition in a model with a continuous gauge symmetry.Acknowledgement We thank in particular E. Domany and A. Schwimmer who suggested to us to consider lattice gauge models, and also L. Chayes, D. v.d. Marel, A. Messager, K. Netocný, S. Romano and A. Sokal for stimulating discussions and/or correspondence. S.S. acknowledges the financial support of the RFFI grant 03-01-00444.     

Two Connections Between Random Systems and Non-Gibbsian Measures

In this contribution we discuss the role disordered (or random) systems have played in the study of non-Gibbsian measures. This role has two main aspects, the distinction between which has not always been fully clear: 1) From disordered systems: Disordered systems can be used as a tool; analogies with, as well as results and methods from the study of random systems can be employed to investigate non-Gibbsian properties of a variety of measures of physical and mathematical interest. 2) Of disordered systems: Non-Gibbsianness is a property of various (joint) measures describing quenched disordered systems. We discuss and review this distinction and a number of results related to these issues. Moreover, we discuss the mean-field version of the non-Gibbsian property, and present some ideas how a Kac limit approach might connect the finite-range and the mean-field non-Gibbsian properties.     

Ill-defined block-spin transformations at arbitrarily high temperatures

Examples are presented of block-spin transformations which map the Gibbs measures of the Ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non-Gibbsian measures. In this way we provide the first example of this kind of pathology for very high temperatures, and as a corollary also the first example of such a pathology happening at a critical point.     

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 $\frac{1}{{|i?j|}^{\alpha }}$, deep in the phase transition region ($1<\alpha \le 2$ 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.     

On Thermodynamic Limits of Entropy Densities

We give some sufficient conditions which guarantee that the entropy density in the thermodynamic limit is equal to the thermodynamic limit of the entropy densities of finite-volume (local) Gibbs states.     

(Almost) Gibbsian Description of the Sign Fields of SOS Fields

An example is presented of a measure on a lattice system which has a measure zero set of points (configurations) where some conditional probability can be discontinuous, but does not become a Gibbs measure under decimation (or other) transformations. We also discuss some related issues.