arXiv subjects

Eduardo A. Silva

Publications and source records attributed to Eduardo A. Silva.

3 recordsLinked to original sources

arXiv2023

Some Probabilistic Properties of General Topological Markov Chains

In this paper, we utilize the framework of Markov processes to attain a more probabilistic perspective on the theory of transfer operators. In doing so, we establish a functional central limit theorem (FLCT) for an $O(N)$ model associated with Dyson potential on the one-dimensional lattice. We also proof a FLCT on a non-compact alphabet setting, for a model associated with a Dyson type potential on the one-dimensional lattice. A Breimann ergodic theorem for equilibrium measures arising from the Ruelle Operator Formalism is proved. Furthermore, we obtain a qualitative criterion (strong transitivity) to determine when the conformal measure has full support. As an application, we show how to connect the Ruelle operator framework with the perspective of the Hopf theory of Markov Processes.

math.DS↗

arXiv2019

Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator Approach

In this paper we use a recent version of the Ruelle-Perron-Frobenius Theorem to compute, in terms of the maximal eigendata of the Ruelle operator, the pressure derivative of translation invariant spin systems taking values on a general compact metric space. On this setting the absence of metastable states for continuous potentials on one-dimensional one-sided lattice is proved. We apply our results, to show that the pressure of an essentially one-dimensional Heisenberg-type model, on the lattice $\mathbb{N}\times \mathbb{Z}$, is Fréchet differentiable, on a suitable Banach space. Additionally, exponential decay of the two-point function, for this model, is obtained for any positive temperature.

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↗