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- 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

15-06 ; 60-06 ; 00B25 ; 15B52 ; 60B20 ; 11C20 ; 05D40 ; 60H25 ; 62-07

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- 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

60-XX ; 62-XX ; 60-06 ; 60Exx ; 60Fxx ; 60Gxx ; 62Gxx ; 60B20 ; 15B52 ; 00B25

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- xv; 504 p.
ISBN 978-3-319-75995-1

Operator theory :
advances and applications
, 0268

Localisation : Collection 1er étage

théorie des opérateurs # théorie de la commande # analyse globale # matrice # variété # base

47-06 ; 15B05 ; 15B52 ; 42A16 ; 42C15 ; 47A10 ; 47B15 ; 47B35 ; 47F05 ; 47G30 ; 47L15 ; 58J40 ; 81U20 ; 93C55

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- 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

15B52 ; 35J10 ; 35L65 ; 35Q41 ; 35Q51 ; 35Q53 ; 37K10 ; 42B20 ; 46N20 ; 46N30 ; 46T12 ; 47B36 ; 47F05 ; 60H20 ; 68N30 ; 76S05 ; 33C45 ; 35A01 ; 35A02 ; 35L80 ; 37D45 ; 39A12 ; 47A10 ; 47N20 ; 47N30 ; 60B20

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Research talks;Analysis and its Applications;Probability and Statistics

We show that finite rank perturbations of certain random matrices fit in the framework of infinitesimal (type B) asymptotic freeness. This can be used to explain the appearance of free harmonic analysis (such as subordination functions appearing in additive free convolution) in computations of outlier eigenvalues in spectra of such matrices.

46L54 ; 15B52

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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.

15B52 ; 60B20 ; 46L53 ; 46L54

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Research School

I will give an overview of connections between Random Matrix Theory and Number Theory, in particular connections with the theory of the Riemann zeta-function and zeta functions defined in function fields. I will then discuss recent developments in which integrability plays an important role. These include the statistics of extreme values and connections with the theory of log-correlated Gaussian fields.

11M06 ; 15B52 ; 11Z05

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Research School

I will give an overview of connections between Random Matrix Theory and Number Theory, in particular connections with the theory of the Riemann zeta-function and zeta functions defined in function fields. I will then discuss recent developments in which integrability plays an important role. These include the statistics of extreme values and connections with the theory of log-correlated Gaussian fields.

11M06 ; 15B52 ; 11Z05

... Lire [+]

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Research School

I will give an overview of connections between Random Matrix Theory and Number Theory, in particular connections with the theory of the Riemann zeta-function and zeta functions defined in function fields. I will then discuss recent developments in which integrability plays an important role. These include the statistics of extreme values and connections with the theory of log-correlated Gaussian fields.

11M06 ; 15B52 ; 11Z05

... Lire [+]

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Research School

Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom’s original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.
Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of ...

60H25 ; 15B52

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Research School

Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom’s original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.
Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of ...

60H25 ; 15B52

... Lire [+]

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Research School

Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom’s original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.
Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of ...

60H25 ; 15B52

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research School

I will give an overview of connections between Random Matrix Theory and Number Theory, in particular connections with the theory of the Riemann zeta-function and zeta functions defined in function fields. I will then discuss recent developments in which integrability plays an important role. These include the statistics of extreme values and connections with the theory of log-correlated Gaussian fields.

11M06 ; 15B52 ; 11Z05

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research School

Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of certain Gaussian Hermitian matrices, these laws are now understood to govern the behavior of a wide range of nonlinear phenomena in mathematical physics (exclusion processes, random growth models, etc.)

My aim here will be to describe a relatively new approach to limit theorems for random matrices. Instead of focussing on some particular spectral statistic, one rather understands the large dimensional limit as a continuum limit, demonstrating that the matrices themselves converge to some random differential operators. This method is especially suited to the so-called beta ensembles, which generalize the classical Gaussian Unitary and Orthogonal Ensembles (GUE/GOE), and can be viewed in their own right as models of coulomb gases.

The first lecture will review the underlying analytic structure of the just mentioned classical ensembles (essential to, for example, Tracy and Widom’s original work), and then introduce the beta ensembles along with our main players: the stochastic Airy, Bessel, and Sine operators. These operators provide complete characterizations of the general edge and bulk statistics for the beta-ensembles and as such generalize all previously discovered limit theorems for say GUE/GOE. Lecture two will provide the rigorous framework for these operators, as well as an overview of the proofs of the implied operator convergence. The last lectures will be devoted to upshots and applications of these new characterizations of random matrix limits: tail estimates for general beta Tracy-Widom, a simple PDE description of ``the Baik-Ben Arous-Peche phase transition", approaches to universality, and so on.
Random matrix theory is an asymptotic spectral theory. For a given ensemble of $n$ by $n$ matrices, one aims to proves limit theorems for the eigenvalues as the dimension tends to infinity. One of the more remarkable aspects of the subject is that it has introduced important new points of concentration in the space of distributions. Take for example the Tracy-Widom laws. First discovered as the fluctuation limit for the spectral radius of ...

60H25 ; 15B52

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Exposés de recherche

For 20 years we have known that average values of characteristic polynomials of random unitary matrices provide a good model for moments of the Riemann zeta function. Now we consider mixed moments of characteristic polynomials and their derivatives, calculations which are motivated by questions on the distribution of zeros of the derivative of the Riemann zeta function.

15B52 ; 11M26 ; 11M06

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Exposés de recherche

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 ...

15B52 ; 60B20

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Exposés de recherche

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 ...

60B20 ; 15B52

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Exposés de recherche

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 ...

60B20 ; 15B52

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Exposés de recherche

An explicit relationship between certain cubic Hodge integrals on the Deligne-Mumford moduli space of stable algebraic curves and connected GUE correlators of even valencies, called the Hodge-GUE correspondence, was recently discovered. In this talk, we prove this correspondence by using the Virasoro constraints and by deriving the Dubrovin-Zhang loop equation. The talk is based on a series of joint work with Boris Dubrovin, Si-Qi Liu and Youjin Zhang. An explicit relationship between certain cubic Hodge integrals on the Deligne-Mumford moduli space of stable algebraic curves and connected GUE correlators of even valencies, called the Hodge-GUE correspondence, was recently discovered. In this talk, we prove this correspondence by using the Virasoro constraints and by deriving the Dubrovin-Zhang loop equation. The talk is based on a series of joint work with Boris Dubrovin, Si-Qi Liu and Youjin ...

53D45 ; 37K10 ; 15B52

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Exposés de recherche

We establish a new connection between moments of n×n random matrices $X_{n}$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s\in\mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order n→∞ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials. This is work in collaboration with Fabio Cunden, Neil O' Connell and Nick Simm. We establish a new connection between moments of n×n random matrices $X_{n}$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s\in\mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and ...

15B52 ; 05E05 ; 33C45

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