Leandro Cioletti

ON THE GENERAL ONE-DIMENSIONAL XY MODEL: POSITIVE AND ZERO TEMPERATURE, SELECTION AND NON-SELECTION

We consider (M, d) a connected and compact manifold and we denote by the Bernoulli space Mℤ. The analogous problem on the half-line ℕ is also considered. Let be an observable. Given a temperature T, we analyze the main properties of the Gibbs state . In order to do our analysis, we consider the Ruelle operator associated to , and we get in this procedure the main eigenfunction . Later, we analyze selection problems when the temperature goes to zero: (a) existence, or not, of the limit , a question about selection of subactions, and, (b) existence, or not, of the limit , a question about selection of measures. The existence of subactions and other properties of Ergodic Optimization are also considered. The case where the potential depends just on the coordinates (x0, x1) is carefully analyzed. We show, in this case, and under suitable hypotheses, a Large Deviation Principle, when T → 0, graph properties, etc. Finally, we will present in detail a result due to van Enter and Ruszel, where the authors show, for a particular example of potential A, that the selection of measure in this case, does not happen.

Phase Transition in Ferromagnetic Ising Models with Non-uniform External Magnetic Fields

In this article we study the phase transition phenomenon for the Ising model under the action of a non-uniform external magnetic field. We show that the Ising model on the hypercubic lattice with a summable magnetic field has a first-order phase transition and, for any positive (resp. negative) and bounded magnetic field, the model does not present the phase transition phenomenon whenever lim inf hi>0, where

Phase Transitions in One-Dimensional Translation Invariant Systems: A Ruelle Operator Approach

\mathbf{h}=(h_{i})_{i\in \mathbb{Z}^{d}}

Phase Transitions in Ferromagnetic Ising Models with Spatially Dependent Magnetic Fields

is the external magnetic field.

Spectral properties of the Ruelle operator for product-type potentials on shift spaces: SPECTRAL PROPERTIES OF THE RUELLE OPERATOR

We consider a family of potentials

Graphical Representations for Ising and Potts Models in General External Fields

f

Equivalence of optimal L1-inequalities on Riemannian manifolds

f, derived from the Hofbauer potentials, on the symbolic space