Selection of measure and a large deviation principle for the general one-dimensionalXYmodel

Abstract

We consider (M, d) a connected and compact manifold and we denote by X the Bernoulli space M N . The shift acting on X is denoted by σ.We analyze the general XY model, as presented in a recent paper by A. T. Baraviera, L. M. Cioletti, A. O. Lopes, J. Mohr and R. R. Souza. Denote the Gibbs measure by µ c := h c ν c , where h c is the eigenfunction, and, ν c is the eigenmeasure of the Ruelle operator associated to cf . We will show that any measure selected by µ c , as c → +∞, is a maximizing measure for f . We also prove, when the maximizing probability measure is unique, that a certain Large Deviation Principle holds with the deviation function R ∞and V is any calibrated subaction. * arturoscar.lopes@gmail.com, Instituto de Matemática -UFRGS -Partially supported by DynEurBraz, CNPq, PRONEX -Sistemas Dinamicos, INCT, Convenio Brasil-Franca † jairokras@gmail.com, Instituto de Matemática -UFRGSLet f : X → R be a fixed Holder potential defined in the Bernoulli space X. We denote by m the Lebesgue probability on M. We suppose without loss of generality that the diameter of the manifold M is smaller than one. This distance induces another one, in the usual fashion, on M N [7].We are interested in the Gibbs state (for finite and zero temperature) associated to the potential f . This model is called the general XY model in [7]. We refer the reader to such work for a detailed explanation about the motivation for considering such kind of problems. We point out that in the literature in Physics what is called the XY model is the case when M = S 1 , and, the potential f depends just on a finite number of coordinates. In [7] and here the hypothesis are more general.Classical references in the XY model are [20], [35] and [39] where the spin can be in a circle. A nice reference for general results in Statistical Mechanics is [18].In order to define a transfer operator we need a probability a priori on M which we will denote by d m. In the case M = S 1 is usual to consider the Lebesgue measure dx (see [39]) as the a priori probability. In [32] it is consider the case of the one dimensional spin lattice with a general a priori probability and is presented results which are generalizations of the ones in [7].First we will recall some definitions and results from [7].Definition 1. Let C be the space of continuous functions from X = M N to R. We define the Ruelle operator L f acting on C, associated to the Holder potential f : M N → R, as the linear operator that takes w ∈ C, and send it to

Entropy and variational principle for one-dimensional lattice systems with a generala prioriprobability: positive and zero temperature

“…Now we return to study the Gibbs measures at zero temperature. In the case µ βA ⇀ µ ∞ , when β → ∞ (not just a subsequence), as we said before, we have selection of probability at temperature zero (see [40], [42], [41] for general positive results and [13] [14] [22] for negative results). The next result uses the variational principle proved in the previous section and the property that the entropy of an invariant probability is not positive.…”

“…In the case the maximizing probability for A is unique, we have selection of Gibbs probability at temperature zero. Questions related to the Large Deviation property on the XY model, when β → ∞, are considered in [42]. The existence of a calibrated subaction plays an important role in this kind of result.…”

On the General One-Dimensional Xy Model: Positive and Zero Temperature, Selection and Non-Selection

“…Some of the results presented here will be used in a future related paper [40]. We point out that the understanding of Statistical Mechanics via the Ruelle Operator (Transfer Operator) allows one to get eigen-functions, and, in the limit (in the logarithm scale), when temperature goes to zero, the subaction.…”

Large Deviations for Equilibrium Measures and Selection of Subaction

“…In [18,19] this result has been generalized. Given x ∈ X, n ∈ N and β > 0, consider the probability m x,β,n ∈ P (X) defined by w dm x,β,n = L n g β (w)(x).…”