[Paper Review] Selection of measure and a Large Deviation Principle for the general XY model

Artur O. Lopes, Jairo K. Mengue|arXiv (Cornell University)|Jun 15, 2011

Mathematical Dynamics and Fractals22 references3 citations

TL;DR

This paper establishes a Large Deviation Principle (LDP) for the general one-dimensional XY model in thermodynamic formalism, showing that as the inverse temperature $ c \to \infty $, Gibbs measures $ \mu_c $ selected by the Ruelle operator converge weak* to maximizing measures for a H"older potential $ f $. The key result identifies the deviation function as $ R_+^\infty = \sum_{j=0}^\infty R_+(\sigma^j) $, where $ R_+ = \beta(f) + V \circ \sigma - V - f $, and $ V $ is a calibrated subaction, under the assumption of a unique maximizing measure.

ABSTRACT

We consider $(M,d)$ a connected and compact manifold and we denote by $X$ the Bernoulli space $M^{\mathbb{N}}$. 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 are going to prove that any measure selected by $μ_{c}$, as $c o +\infty$, is a maximizing measure for $f$. We also show, when the maximizing probability measure is unique, that it is true a Large Deviation Principle, with the deviation function $R_{+}^{\infty}=\sum_{j=0}^\infty R_{+} (σ^f)$, where $R_{+}:= β(f) + V\circσ- V - f$, and, $V$ is any calibrated subaction.

Motivation & Objective

• To establish a Large Deviation Principle (LDP) for the general one-dimensional XY model in the zero-temperature limit.
• To characterize the asymptotic behavior of Gibbs measures $ \mu_c $ as $ c \to \infty $, corresponding to infinite inverse temperature.
• To identify the deviation function in the LDP using a calibrated subaction $ V $ and the function $ R_+ = \beta(f) + V \circ \sigma - V - f $.
• To prove that any weak* limit of $ \mu_c $ as $ c \to \infty $ is a maximizing measure for the H"older potential $ f $.
• To show that when the maximizing measure is unique, the LDP holds with the specified deviation function.

Proposed method

• Use of the Ruelle operator $ L_{cf} $ with eigenfunction $ h_c $, eigenmeasure $ \nu_c $, and Gibbs measure $ \mu_c = h_c \nu_c $.
• Definition of the normalized function $ g_c = cf + \log h_c - \log(h_c \circ \sigma) - \log \beta_c $ for the Ruelle operator $ L_{cf} $.
• Introduction of the deviation function $ R_+^\infty = \sum_{j=0}^\infty R_+(\sigma^j) $, where $ R_+ = \beta(f) + V \circ \sigma - V - f $, with $ V $ a calibrated subaction.
• Application of double limit arguments: $ \lim_{c,n \to \infty} \frac{1}{c} \log \int_A e^{c R_-^k} \, dm $, to analyze large deviation rates.
• Use of empirical measures $ \nu_n $ defined by $ \phi \mapsto \frac{1}{n} \sum_{j=0}^{n-1} \phi(\sigma^j(z)) $, and their convergence to $ \mu_\infty $ under $ R_-^\infty(z) > -\infty $.
• Leverage properties of calibrated subactions and Hölder continuity to control the variation of $ R_- $, and derive bounds on the logarithmic moment generating functions.

Experimental results

Research questions

• RQ1 Does the Gibbs measure $ \mu_c $ for the general XY model converge to a maximizing measure as $ c \to \infty $?
• RQ2 What is the explicit form of the deviation function in the Large Deviation Principle for the zero-temperature limit of the XY model?
• RQ3 How does the calibrated subaction $ V $ contribute to the structure of the deviation function $ R_+^\infty $?
• RQ4 Under what conditions does the LDP hold with the given deviation function?
• RQ5 Can the asymptotic behavior of $ \mu_c $ be characterized via the iterated Ruelle operator and preimage structure of the shift?

Key findings

• Any weak* limit of a subsequence of $ \mu_c $ as $ c \to \infty $ is a maximizing measure for the potential $ f $.
• When the maximizing measure $ \mu_\infty $ is unique, a Large Deviation Principle holds with deviation function $ R_+^\infty = \sum_{j=0}^\infty R_+(\sigma^j) $, where $ R_+ = \beta(f) + V \circ \sigma - V - f $.
• The LDP is established via a double limit argument: $ \lim_{c,n \to \infty} \frac{1}{c} \log \int_A e^{c R_-^k} \, dm $, leading to the lower bound $ \sup_{z \in A} R_-^\infty(z) \geq \limsup_{k \to \infty} \sup_{z: \sigma^k(z) = y_0} R_-^k(z) $.
• The support of the maximizing measure $ \mu_\infty $ is contained in the set of accumulation points of sequences $ \{y_n\} $ with $ R_-(y_n) = 0 $, and such sequences are shown to accumulate at points in $ \text{supp}(\mu_\infty) $.
• If $ R_-^\infty(z) > -\infty $, then the empirical measures $ \nu_n $ associated to $ z $ converge weak* to $ \mu_\infty $.
• The result $ \liminf_{c,n \to \infty} \frac{1}{c} \log (L_{g_c}^n \chi_A)(x) \geq \sup_{z \in A} R_-^\infty(z) = -\inf_{z \in A} R_+^\infty(z) $ confirms the LDP with the proposed deviation function.