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Leandro Cioletti

Publications and source records attributed to Leandro Cioletti.

12 recordsLinked to original sources
arXiv2024

Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples

In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalize the Perron-Frobenius-Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined $ζ$-function at its critical parameter. We then generalize Sharp's construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the $ζ$-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.

math.DS↗
arXiv2019

Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures. One of the main results of the article is to show that a probability is DLR-Gibbs (associated to a continuous translation invariant specification), if and only if, is an eigenprobability for the transpose of the Ruelle operator. Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present

math.DS↗
arXiv2017

Correlation Inequalities and Monotonicity Properties of the Ruelle Operator

Let $X = \{1,-1\}^\mathbb{N}$ be the symbolic space endowed with the product order. A Borel probability measure $μ$ over $X$ is said to satisfy the FKG inequality if for any pair of continuous increasing functions $f$ and $g$ we have $μ(fg)-μ(f)μ(g)\geq 0$. In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive specifications our proof has advantage of to be easily adapted for suitable subshifts. We review (and provide proofs in our setting) some classical results about the long-range Ising model on the lattice $\mathbb{N}$ and use them to deduce some monotonicity properties of the associated Ruelle operator and their relations with phase transitions. As is widely known, for some continuous potentials does not exists a positive continuous eigenfunction associated to the spectral radius of the Ruelle operator acting on $C(X)$. Here we employed some ideas related to the involution kernel in order to solve the main eigenvalue problem in a suitable sense - for a class of potentials having low regularity. From this we obtain an explicit tight upper bound for the main eigenvalue (consequently for the pressure) of the Ruelle operator associated to Ising models with $1/r^{2+\varepsilon}$ interaction energy. Extensions of the Ruelle operator to suitable Hilbert Spaces are considered and a theorem solving to the main eigenvalue problem (in a weak sense) is obtained by using the Lions-Lax-Milgram theorem.

math.DS↗
arXiv2017

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 $\liminf h_i> 0$, where ${\bf h} = (h_i)_{i \in \Z^d}$ is the external magnetic field.

math-ph↗
arXiv2017

Spectral Properties of the Ruelle Operator on the Walters Class over Compact Spaces

Recently the Ruelle-Perron-Fröbenius theorem was proved for Hölder potentials defined on the symbolic space $Ω=M^{\mathbb{N}}$, where (the alphabet) $M$ is any compact metric space. In this paper, we extend this theorem to the Walters space $W(Ω)$, in similar general alphabets. We also describe in detail an abstract procedure to obtain the Fréchet-analyticity of the Ruelle operator under quite general conditions and we apply this result to prove the analytic dependence of this operator on both Walters and Hölder spaces. The analyticity of the pressure functional on Hölder spaces is established. An exponential decay of the correlations is shown when the Ruelle operator has the spectral gap property. A new (and natural) family of Walters potentials (on a finite alphabet derived from the Ising model) not having an exponential decay of the correlations is presented. Because of the lack of exponential decay, for such potentials we have the absence of the spectral gap for the Ruelle operator. The key idea to prove the lack of exponential decay of the correlations are the Griffiths-Kelly-Sherman inequalities.

math.DS↗
arXiv2017

Interactions, Specifications, DLR probabilities and the Ruelle Operator in the One-Dimensional Lattice

In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these "in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures). Among other things we extended the classical notion of a Gibbsian specification on $\mathbb{N}$ in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims that the construction of the conformal Measures in Dynamical Systems for Walters potentials, using the Ruelle operator, can be formulated in terms of Specification. We also describe the Ising model, with $1/r^{2+\varepsilon}$ interaction energy, in the Thermodynamic Formalism setting and prove that its associated potential is in Walters space - we present an explicit expression. We also provide an alternative way for obtaining the uniqueness of the DLR-Gibbs measures.

math.DS↗
arXiv2015

Graphical Representations for Ising and Potts Models in General External Fields

This work is concerned with the theory of Graphical Representation for the Ising and Potts Models over general lattices with non-translation invariant external field. We explicitly describe in terms of the Random Cluster Representation the distribution function and, consequently, the expected value of a single spin for the Ising and $q$-states Potts Models with general external fields. We also consider the Gibbs States for the Edwards-Sokal Representation of the Potts Model with non-translation invariant magnetic field and prove a version of the FKG Inequality for the so called General Random Cluster Model (GRC Model) with free and wired boundary conditions in the non-translation invariant case. Adding the amenability hypothesis on the lattice, we obtain the uniqueness of the infinite connected component and the quasilocality of the Gibbs Measures for the GRC Model with such general magnetic fields. As a final application of the theory developed, we show the uniqueness of the Gibbs Measures for the Ferromagnetic Ising Model with a positive power law decay magnetic field, as conjectured in [8].

math.PR↗
arXiv2014

Equivalence of optimal $L^1$-inequalities on Riemannian Manifolds

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \geq 2$. This paper concerns to the validity of the optimal Riemannian $L^1$-Entropy inequality \[ {\bf Ent}_{dv_g}(u) \leq n \log \left(A_{opt} \|D u\|_{BV(M)} + B_{opt}\right) \] for all $u \in BV(M)$ with $\|u\|_{L^1(M)} = 1$ and existence of extremal functions. In particular, we prove that this optimal inequality is equivalent a optimal $L^1$-Sobolev inequality obtained by Druet [6].

math.AP↗
arXiv2014

Diffusive-Ballistic Transition in Random Polymers with Drifts and Repulsive Long-Range Interactions

This paper leads with a random polymer model in $\Z^2$ having long-range self-repulsive interactions. By comparison with a long range one-dimensional ferromagnetic Ising model we shown that the polymer models we considered here undergo a phase transition in terms of the inverse temperature $β$. In the second part of this work we shown, using the Lee-Yang Circle Theorem, that our random polymer model with drifts satisfies the, Wu Liming [7], $C^2$ regularity condition. As consequence we obtain a Central Limit Theorem for the model.

math.PR↗
arXiv2013

On nonlinear Schrödinger equations with random potentials: existence and probabilistic properties

In this paper we are concerned with nonlinear Schrödinger equations with random potentials. Our class includes continuum and discrete potentials. Conditions on the potential $V_ω$ are found for existence of solutions almost sure $ω$. We study probabilistic properties like central limit theorem and law of larger numbers for the obtained solutions by independent ensembles. We also give estimates on the expected value for the $L^{\infty}$-norm of the solution showing how it depends on the size of the potential.

math.AP↗
arXiv2013

Multihomogeneous Normed Algebras and Polynomial Identities

In this paper we consider PI-algebras $A$ over $\R$ or $\C$. It is well known that in general such algebras are not normed algebras. In fact, there is a nilpontent commutative algebra which is not a normed algebra, see [1]. Here we address the question of whether it is possible to find a normed PI-algebra $B$ with the same polynomial identities as $A$, and moreover, whether there is some Banach PI-algebra with this property. Our main theorem provides an affirmative answer for this question and moreover we also show the existence of a Banach Algebra with the same polynomial identities as $A$. As a byproduct we prove that if $A$ is a normed PI-algebra and its completion is nil, then $A$ is nilpotent. By introducing the concept of multihomogeneous norm we obtain as an application of our main results that if $\FX$ is multihomogeneus normed algebra and $A$ is a PI-algebra such that the completion of the quotient space $\FX/Id(A)$ is nil, then $A$ is nilpotent. Both applications are extensions of the study initiated in [4].

math.RA↗