Author

# Edoardo Vivo

Other affiliations: University of Parma

Bio: Edoardo Vivo is an academic researcher from Charles III University of Madrid. The author has contributed to research in topics: Eigenvalues and eigenvectors & Random matrix. The author has an hindex of 8, co-authored 10 publications receiving 213 citations. Previous affiliations of Edoardo Vivo include University of Parma.

##### Papers

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TL;DR: In this article, the joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential are analyzed analytically for finite-size matrices.

Abstract: We calculate analytically, for finite-size matrices, joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential. We focus on the ratios of two spacings between three consecutive real eigenvalues, as well as certain generalizations such as the overlapping ratios. The resulting formulas are further analyzed in detail in two specific cases: the β-Hermite and the β-Laguerre cases, for which we offer explicit calculations for small N. The analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles, and with the level statistics of a quantum many-body lattice model and zeros of the Riemann zeta function.

63 citations

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TL;DR: In this article, the joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential are analyzed analytically for finite-size matrices.

Abstract: We calculate analytically, for finite-size matrices, joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential. We focus on the ratios of two spacings between three consecutive real eigenvalues, as well as certain generalizations such as the overlapping ratios. The resulting formulas are further analyzed in detail in two specific cases: the beta-Hermite and the beta-Laguerre cases, for which we offer explicit calculations for small N. The analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles, and with the level statistics of a quantum many-body lattice model and zeros of the Riemann zeta function.

51 citations

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TL;DR: In this article, it was shown that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads (N1,N 2) are analytically known from the Jacobi ensemble of random matrix theory.

Abstract: We point out that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads (N1 ,N 2) are analytically known from the Jacobi ensemble of random matrix theory. Using this result and a simple linear statistic, we give an exact and non-perturbative expression for moments of the form � λ m � for m> −|N1 − N2 |− 1 and β = 2, thus improving the existing results in the literature. Secondly, we offer an independent derivation of the average density and higher order correlation function for β = 2,4 which does not make use of the orthogonal polynomials technique. This result may be relevant for an efficient numerical implementation avoiding determinants. PACS numbers: 73.23.−b, 73.50.Td, 05.45.Mt, 73.63.Kv

44 citations

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TL;DR: In this article, the Jacobi ensemble of random matrix theory was used to derive the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads.

Abstract: We point out that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads $(N_1,N_2)$ are analytically known from the Jacobi ensemble of Random Matrix Theory. Using this result and a simple linear statistic, we give an exact and non-perturbative expression for moments of the form $ $ for $m>-|N_1-N_2|-1$ and $\beta=2$, thus improving the existing results in the literature. Secondly, we offer an independent derivation of the average density and higher order correlation functions for $\beta=2,4$ which does not make use of the orthogonal polynomials technique. This result may be relevant for an efficient numerical implementation avoiding determinants.

34 citations

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TL;DR: In this paper, an experimental assessment of surface kinetic roughening properties that are anisotropic in space was performed for two specific instances of silicon surfaces irradiated by ion-beam sputtering under diverse conditions.

Abstract: We report an experimental assessment of surface kinetic roughening properties that are anisotropic in space. Working for two specific instances of silicon surfaces irradiated by ion-beam sputtering under diverse conditions (with and without concurrent metallic impurity codeposition), we verify the predictions and consistency of a recently proposed scaling Ansatz for surface observables like the two-dimensional (2D) height power spectral density (PSD). In contrast with other formulations, this ansatz is naturally tailored to the study of two-dimensional surfaces, and allows us to readily explore the implications of anisotropic scaling for other observables, such as real-space correlation functions and PSD functions for 1D profiles of the surface. Our results confirm that there are indeed actual experimental systems whose kinetic roughening is strongly anisotropic, as consistently described by this scaling analysis. In the light of our work, some types of experimental measurements are seen to be more affected by issues like finite space resolution effects, etc. that may hinder a clear-cut assessment of strongly anisotropic scaling in the present and other practical contexts.

21 citations

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TL;DR: The eigenstate thermalization hypothesis (ETH) as discussed by the authors is a natural extension of quantum chaos and random matrix theory (RMT) that allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath.

Abstract: This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the i...

1,536 citations

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TL;DR: The eigenstate thermalization hypothesis (ETH) as mentioned in this paper is a natural extension of quantum chaos and random matrix theory (RMT) and it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath.

Abstract: This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.

985 citations