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# Documents  60B20 | enregistrements trouvés : 56

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## On determinants of random matrices Zeitouni, Ofer | CIRM H

Post-edited

Research talks;Probability and Statistics

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## Mathematical research today and tomorrowviewpoints of seven fields medalistslectures given at the Institut d'Estudis CatalansJune Casacuberta, C. ; Castellet, M. | Springer-Verlag 1992

Congrès

ISBN 978-3-540-56011-1

Lecture notes in mathematics , 1525

Localisation : Collection 1er étage

géométrie diophantienne # géométrie non commutative # informatique théorique # modèle intégrable # noeud en mathématique et en physique # philosophie # équation différentielle non linéaire

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## Algebraic and geometric aspects of integrable systems and random matrices:Proceedings of the AMS special sessionBoston # january 6-7, 2012 Dzhamay, Anton ; Maruno, Kenichi ; Pierce, Virgil U. | American Mathematical Society 2013

Congrès

- xii; 345 p.
ISBN 978-0-8218-8747-9

Contemporary mathematics , 0593

Localisation : Collection 1er étage

courbe algébrique # équation de Painlevé # équation différentielle # système hamiltonnien # équation différentielle non-linéaire

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## Modern aspects of random matrix theory.Based on lectures delivered at the 2013 AMS short course on random matricesSan Diego # January 6-7, 2013 Vu, Van H. | American Mathematical Society 2014

Congrès

- viii; 174 p.
ISBN 978-0-8218-9471-2

Proceedings of symposia in applied mathematics , 0072

Localisation : Collection 1er étage

matrice aléatoire # théorie des nombres # algèbre linéaire # matrice de Wigner # probabilité libre

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## High dimensional probability VI:the Banff volume.Proceedings of the sixth high dimensional probability conference (HDP VI)Banff # october 9-14, 2011 Houdré, Christian ; Mason, David M. ; Rosinski, Jan ; Wellner, Jon A. | Birkhäuser 2013

Congrès

- xi; 373 p.
ISBN 978-3-0348-0489-9

Progress in probability , 0066

Localisation : Colloque 1er étage (BANF)

loi de probabilité # théorème limite # espace de dimension infinie # espace de Hilbert # espace de Banach # matrice aléatoire # statistique non paramétrique # processus empirique # concentration de la mesure # approximation forte # approximation faible # optimisation combinatoire # théorie des graphes aléatoires

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## Modern trends in constructive function theory:constructive functions 2014Conference in honor of Ed Saff's 70th birthdayNashville # May 26-30, 2014 Hardin, Douglas P. ; Lubinsky, Doron S. ; Simanek, Brian Z. | American Mathematical Society 2016

Congrès

- ix; 297 p.
ISBN 978-1-4704-2534-0

Contemporary mathematics , 0661

Localisation : Collection 1er étage

Edward B. Saff # théorie de l'approximation # analyse constructive

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## Probability and statistical physics in St. Petersburg.St. Petersburg school on probability and statistical physicsSt. Petersburg # June 18-29, 2012 Sidoravicius, V. ; Smirnov, S. | American Mathematical Society 2016

Congrès

- vi; 471 p.
ISBN 978-1-4704-2248-6

Proceedings of symposia in pure mathematics , 0091

Localisation : Collection 1er étage

probabilités # physique statistique # théorie ergodique # marche aléatoire # chaîne de Markov # modèle de Potts # mesure invariante # champ Gaussien

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## Non-linear partial differential equations, mathematical physics, and stochastic analysis:the Hedge Holden anniversary volume.Based on the presentations at the conference 'Non-linear PDEs, mathematical physics and stochastic analysis'Trondheim # July 4-7, 2016 Gesztesy, Fritz ; Hanche-Olsen, Harald ; Jakobsen, Espen R. ; Lyubarskii, Yurii ; Risebro, Nils Henrik ; Seip, Kristian | European Mathematical Society 2018

Congrès

- xvi; 486 p.
ISBN 978-3-03719-186-6

EMS series of congress reports

Localisation : Colloque 1er étage (TRON)

analyse infinitésimale # équation aux dérivées partielles # loi de conservation hyperbolique # analyse stochastique # théorie spectrale # évolution discrète # système complètement intégrable # matrice aléatoire # dynamique chaotique

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## Compressive sensing with time-frequency structured random matrices Rauhut, Holger | CIRM H

Multi angle

Special events;30 Years of Wavelets;Analysis and its Applications;Probability and Statistics;Numerical Analysis and Scientific Computing

One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This finding triggered an enormous research activity in recent years both in signal processing applications as well as their mathematical foundations. The present talk discusses connections of compressive sensing and time-frequency analysis (the sister of wavelet theory). In particular, we give on overview on recent results on compressive sensing with time-frequency structured random matrices.

Keywords: compressive sensing - time-frequency analysis - wavelets - sparsity - random matrices - $\ell_1$-minimization - radar - wireless communications
One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This ...

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## Free probability and random matrices Biane, Philippe | CIRM H

Multi angle

Research talks;Analysis and its Applications;Probability and Statistics

I will explain how free probability, which is a theory of independence for non-commutative random variables, can be applied to understand the spectra of various models of random matrices.

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## Correlation functions for some integrable systems with random initial data, theory and computation - Lecture 1 Grava, Tamara | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations;Probability and Statistics

We will investigate the form of spatio-temporal correlation functions for integrable models of systems of particles on the line. There are few analytical results for nonlinear systems, and so we start developing intuition from harmonic chains, where steepest descent analysis yields detailed asymptotic behaviour of the correlation functions in a variety of scaling limits. We will introduce integrable nonlinear lattices, explain the integrable solution procedure, as well as computational simulations to see dynamics of correlation functions in action. We will investigate the form of spatio-temporal correlation functions for integrable models of systems of particles on the line. There are few analytical results for nonlinear systems, and so we start developing intuition from harmonic chains, where steepest descent analysis yields detailed asymptotic behaviour of the correlation functions in a variety of scaling limits. We will introduce integrable nonlinear lattices, explain the integrable ...

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## Correlation functions for some integrable systems with random initial data, theory and computation - Lecture 2 McLaughlin, Kenneth D. T.-R. | CIRM H

Multi angle

Research School;Dynamical Systems and Ordinary Differential Equations;Probability and Statistics

We will investigate the form of spatio-temporal correlation functions for integrable models of systems of particles on the line. There are few analytical results for nonlinear systems, and so we start developing intuition from harmonic chains, where steepest descent analysis yields detailed asymptotic behaviour of the correlation functions in a variety of scaling limits. We will introduce integrable nonlinear lattices, explain the integrable solution procedure, as well as computational simulations to see dynamics of correlation functions in action. We will investigate the form of spatio-temporal correlation functions for integrable models of systems of particles on the line. There are few analytical results for nonlinear systems, and so we start developing intuition from harmonic chains, where steepest descent analysis yields detailed asymptotic behaviour of the correlation functions in a variety of scaling limits. We will introduce integrable nonlinear lattices, explain the integrable ...

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## Determinantal structure of eigenvector correlations in the complex Ginibre ensemble Akemann, Gernot | CIRM H

Multi angle

Exposés de recherche;Mathematical Physics;Probability and Statistics

We study the expectation of the matrix of overlaps of left and right eigenvectors in the complex Ginibre ensemble, conditioned on a fixed number of k complex eigenvalues.
The diagonal (k=1) and off-diagonal overlap (k=2) were introduced by Chalker and Mehlig. They provided exact expressions for finite matrix size N, in terms of a large determinant of size proportional to N. In the large-N limit these overlaps were determined on the global scale and heuristic arguments for the local scaling at the origin were given. The topic has seen a rapid development in the recent past. Our contribution is to derive exact determinantal expressions of size k x k in terms of a kernel, valid for finite N and arbitrary k.
It can be expressed as an operator acting on the complex eigenvalue correlation functions and allows us to determine all local correlations in the bulk close to the origin, and at the spectral edge. The methods we use are bi-orthogonal polynomials in the complex plane and the analyticity of the diagonal overlap for general k.
This is joint work with Roger Tribe, Athanasios Tsareas, and Oleg Zaboronski as appeared in arXiv:1903.09016 [math-ph]
We study the expectation of the matrix of overlaps of left and right eigenvectors in the complex Ginibre ensemble, conditioned on a fixed number of k complex eigenvalues.
The diagonal (k=1) and off-diagonal overlap (k=2) were introduced by Chalker and Mehlig. They provided exact expressions for finite matrix size N, in terms of a large determinant of size proportional to N. In the large-N limit these overlaps were determined on the global scale ...

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## Optimal global rigidity estimates in unitary invariant ensembles Claeys, Tom | CIRM H

Multi angle

Exposés de recherche;Mathematical Physics;Probability and Statistics

A fundamental question in random matrix theory is to understand how much the eigenvalues of a random matrix fluctuate.
I will address this question in the context of unitary invariant ensembles, by studying the global rigidity of the eigenvalues, or in other words the maximal deviation of an eigenvalue from its classical location.
Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types.
In addition to optimal rigidity estimates, our approach sheds light on the extreme values and on the fractal geometry of the eigenvalue counting function.
The talk will be based on joint work in progress with Benjamin Fahs, Gaultier Lambert, and Christian Webb.
A fundamental question in random matrix theory is to understand how much the eigenvalues of a random matrix fluctuate.
I will address this question in the context of unitary invariant ensembles, by studying the global rigidity of the eigenvalues, or in other words the maximal deviation of an eigenvalue from its classical location.
Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in ...

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## Random orthogonal polynomials: from matrices to point processes Holcomb, Diane | CIRM H

Multi angle

Exposés de recherche;Mathematical Physics;Probability and Statistics

For the commonly studied Hermitian random matrix models there exist tridiagonal matrix models with the same eigenvalue distribution and the same spectral measure $v_{n}$ at the vector $e_{1}$. These tridiagonal matrices give recurrence coefficients that can be used to build the family of random polynomials that are orthogonal with respect to νn. A similar bijection between spectral data and recurrence coefficients also holds for the Unitary ensembles. This time in stead of obtaining a tridiagonal matrix you obtain a sequence $\left \{ \alpha _{k} \right \}_{k=0}^{n-1}$ Szegö coefficients. The random orthogonal polynomials that are generated by this process may then be used to study properties of the original eigenvalue process.
These techniques may be used not just in the classical cases, but also in the more general case of $\beta$-ensembles. I will discuss various ways that orthogonal polynomials techniques may be applied including to show convergence of the Circular $\beta$-ensemble to $Sine_{\beta }$. I will finish by discussing a result on the maximum deviation of the counting function of Sineβ from it expected value. This is related to studying the phases of associated random orthogonal polynomials.
For the commonly studied Hermitian random matrix models there exist tridiagonal matrix models with the same eigenvalue distribution and the same spectral measure $v_{n}$ at the vector $e_{1}$. These tridiagonal matrices give recurrence coefficients that can be used to build the family of random polynomials that are orthogonal with respect to νn. A similar bijection between spectral data and recurrence coefficients also holds for the Unitary ...

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## Integrable systems and spectral curves Eynard, Bertrand | CIRM H

Multi angle

Exposés de recherche;Analysis and its Applications;Mathematical Physics;Probability and Statistics

Usually one defines a Tau function Tau(t_1,t_2,...) as a function of a family of times having to obey some equations, like Miwa-Jimbo equations, or Hirota equations.
Here we shall view times as local coordinates in the moduli-space of spectral curves, and define the Tau-function of a spectral curve Tau(S), in an intrinsic way, independent of a choice of coordinates. Deformations are tangent vectors, and the tangent space is isomorphic to the space of cycles (cf Goldman bracket), so that Hamiltonians can be represented by cycles.
All the integrable system formalism can then be represented geometrically in the space of cycles: the Poisson bracket is the intersection, the conserved quantities are periods, Miwa-Jimbo equations and Seiberg-Witten equations are a mere consequence of the definition, Hirota equation is a vanishing monodromy condition, and Virasoro-W constraint are automatically satisfied by our definition, showing that our Tau-function is also a conformal block. Our definition contains KdV, KP multicomponent KP, Hitchin systems, and probably all known classical integrable systems.
Usually one defines a Tau function Tau(t_1,t_2,...) as a function of a family of times having to obey some equations, like Miwa-Jimbo equations, or Hirota equations.
Here we shall view times as local coordinates in the moduli-space of spectral curves, and define the Tau-function of a spectral curve Tau(S), in an intrinsic way, independent of a choice of coordinates. Deformations are tangent vectors, and the tangent space is isomorphic to the ...

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## Universality in tiling models Van Moerbeke, Pierre | CIRM H

Multi angle

Exposés de recherche;Mathematical Physics;Probability and Statistics

We consider the domino tilings of a large class of Aztec rectangles. For an appropriate scaling limit, we show that, the disordered region consists of roughly two arctic circles connected with a finite number of paths. The statistics of these paths is governed by a kernel, also found in other models (universality). The kernel thus obtained is believed to be a master kernel, from which the kernels, associated with critical points, can all be derived. We consider the domino tilings of a large class of Aztec rectangles. For an appropriate scaling limit, we show that, the disordered region consists of roughly two arctic circles connected with a finite number of paths. The statistics of these paths is governed by a kernel, also found in other models (universality). The kernel thus obtained is believed to be a master kernel, from which the kernels, associated with critical points, can all be ...

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## Transfer matrix approach to 1d random band matrices Shcherbina , Tatyana | CIRM H

Multi angle

Exposés de recherche;Mathematical Physics;Probability and Statistics

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM. Joint project with Maria Shcherbina. Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method ...

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## Spherical Sherrington-Kirkpatrick model and random matrix Baik, Jinho | CIRM H

Multi angle

Exposés de recherche;Mathematical Physics;Probability and Statistics

The Spherical Sherrington-Kirkpatrick (SSK) model is defined by the Gibbs measure on a highdimensional sphere with a random Hamiltonian given by a symmetric quadratic function. The free energy at the zero temperature is the same as the largest eigenvalue of the random matrix associated with the quadratic function. Even for the finite temperature, there is a simple relationship between the free energy and the eigenvalues. We will discuss how one can study the fluctuations of the free energy using this relationship and results from random matrix theory. We will also discuss the distribution of the spin sampled from the Gibbs measure. The Spherical Sherrington-Kirkpatrick (SSK) model is defined by the Gibbs measure on a highdimensional sphere with a random Hamiltonian given by a symmetric quadratic function. The free energy at the zero temperature is the same as the largest eigenvalue of the random matrix associated with the quadratic function. Even for the finite temperature, there is a simple relationship between the free energy and the eigenvalues. We will discuss how one ...

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## Representations of classical Lie groups: two growth regimes Bufetov, Alexey | CIRM H

Multi angle

Exposés de recherche;Algebra;Combinatorics;Probability and Statistics

Asymptotic representation theory deals with representations of groups of growing size. For classical Lie groups there are two distinguished regimes of growth. One of them is related to representations of infinite-dimensional groups, and the other appears in combinatorial and probabilistic questions. In the talk I will discuss differences and similarities between these two settings.

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