Author
Djalil Chafaï
Other affiliations: Paul Sabatier University, Institut de Mathématiques de Toulouse, Institut national de la recherche agronomique ...read more
Bio: Djalil Chafaï is an academic researcher from Paris Dauphine University. The author has contributed to research in topics: Random matrix & Circular law. The author has an hindex of 26, co-authored 105 publications receiving 2420 citations. Previous affiliations of Djalil Chafaï include Paul Sabatier University & Institut de Mathématiques de Toulouse.
Papers published on a yearly basis
Papers
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TL;DR: In this article, it was shown that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity.
Abstract: These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.
193 citations
TL;DR: In this article, the authors investigated the possibilities of choosing an optimal subset of data and found that the Jacobian method and the iterative method were the most suitable for the problem.
Abstract: Advanced infrared sounders will provide thousands of radiance data at every observation location. The number of individual pieces of information is not usable in an operational numerical weather-prediction context, and we have investigated the possibilities of choosing an optimal subset of data. These issues have been addressed in the context of optimal linear estimation theory, using simulated Infrared Atmospheric Sounding Interferometer data. Several methods have been tried to select a set of the most useful channels for each individual atmospheric profile. These are two methods based on the data resolution matrix, one method based on the Jacobian matrix, and one iterative method selecting sequentially the channels with largest information content. The Jacobian method and the iterative method were found to be the most suitable for the problem. The iterative method was demonstrated to always produce the best results, but at a larger cost than the Jacobian method. To test the robustness of the iterative method, a variant has been tried. It consists in building a mean channel selection aimed at optimizing the results over the whole database, and then applying to each profile this constant selection. Results show that this constant iterative method is very promising, with results of intermediate quality between the ones obtained for the optimal iterative method and the Jacobian method. The practical advantage of this method for operational purposes is that the same set of channels can be used for various atmospheric profiles.
149 citations
TL;DR: In this paper, it was shown that the empirical spectral distribution of a nxn random matrix with iid entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity, which is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices.
Abstract: These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with iid entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature
146 citations
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TL;DR: In this article, a short and self contained synthesis of Phi-Sobolev functional inequalities is provided, which generalise and unify various related and unrelated works involving what they call Phi- SOBE functional inequalities.
Abstract: Our aim is to provide a short and self contained synthesis which generalise and unify various related and unrelated works involving what we call Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies can be seen in particular as an inclusive interpolation between Poincare and Gross logarithmic Sobolev inequalities. In addition to the known material, extensions are provided and improvements are given for some aspects. Stability by tensor products, convolution, and bounded perturbations are addressed. We show that under simple convexity assumptions on Phi, such inequalities hold in a lot of situations, including hyper-contractive diffusions, uniformly strictly log-concave measures, Wiener measure (paths space of Brownian Motion on Riemannian Manifolds) and generic Poisson space (includes paths space of some pure jumps Levy processes and related infinitely divisible laws). Proofs are simple and relies essentially on convexity. We end up by a short parallel inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic Gases and Information Theories.
130 citations
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Journal Article•
28,685 citations
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TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.
7,116 citations
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
5,689 citations
TL;DR: Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944 as mentioned in this paper, and there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them.
Abstract: R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.
1,658 citations