| Les deux révisions précédentes Révision précédente Prochaine révision | Révision précédente Prochaine révisionLes deux révisions suivantes |
| mega:start [2018/01/09 14:41] – [Exposés à venir 2017-2018] male | mega:start [2018/02/15 14:45] – [Exposés 2017-2018] male |
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| * Vendredi **12 janvier** | * Vendredi **12 janvier** |
| * 10h30-12h00: mini cours par **[[http://www.camillemale.com|Camille Male]]** sur les méthodes non commutatives en matrices aléatoires | * 10h30-12h00: mini cours par **[[http://www.camillemale.com|Camille Male]]** sur les méthodes non commutatives en matrices aléatoires |
| * 14h30-15h45: **[[https://sites.google.com/site/torbenkruegermath/|Torben Krüger]]** //Random matrices with slow correlation decay \\ // The resolvent of a large dimensional self-adjoint random matrix approximately satisfies the matrix Dyson equation (MDE) up to a random error. We show that for random matrices with arbitrary expectation and slow decay of correlation among its entries this error matrix converges to zero both in an isotropic and averaged sense with optimal rates of convergence as the dimension tends to infinity. This result requires a delicate cancellation (self-energy renormalization) which is seen through a diagrammatic cumulant expansion that automatically exploits the cancellation to all orders. Furthermore, we provide a comprehensive isotropic stability analysis of the MDE down to the length scale of the eigenvalue spacing. This analysis is then used to show convergence of the resolvent to the non-random solution of the MDE and to prove that the local eigenvalue statistics are universal, i.e. they do not depend on the distribution of the entries of the random matrix under consideration (Wigner-Dyson-Mehta spectral universality). Joint work with Oskari Ajanki & Laszlo Erdös & Dominik Schröder. | * 14h30-15h45: **[[https://sites.google.com/site/torbenkruegermath/|Torben Krüger]]** //Random matrices with slow correlation decay \\ // |
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| * 15h45-17h00: **[[http://www.iecl.univ-lorraine.fr/~Jeremie.Unterberger/|Jérémie Unterberger]]** //TBA// | * 15h45-17h00: **[[http://www.iecl.univ-lorraine.fr/~Jeremie.Unterberger/|Jérémie Unterberger]]** //Global fluctuations for 1D log-gas dynamics\\ // |
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| * Vendredi **9 février** | * Vendredi **9 février** |
| * 10h30-12h00: mini cours par **[[http://romaincouillet.hebfree.org|Romain Couillet]]** | * 10h30-12h00: mini cours par **[[http://romaincouillet.hebfree.org|Romain Couillet]]** //matrices aléatoires et l'apprentissage machine \\ // |
| * 14h30-15h45: **[[https://perso.univ-rennes1.fr/nizar.demni/Sitenizar/Accueil.html|Nizar Demni]]** // \\ // | * 14h00-15h00: **[[https://perso.univ-rennes1.fr/nizar.demni/Sitenizar/Accueil.html|Nizar Demni]]** //Etats quantiques Browniens et polynome de Jacobi dans le simplexe \\ // |
| * 15h45-17h00: | * 15h30-16h30: **[[https://www.lpsm.paris//pageperso/boutil/|Cédric Boutillier]]** //Discrete differential geometry and integrable models on isoradial graphs \\ // |
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| * Vendredi **16 mars** | * Vendredi **16 mars** |
| * 10h30-12h00: mini cours par **[[http://www.proba.jussieu.fr/pageperso/levy/|Thierry Lévy]]** | * 10h30-12h00: mini cours par **[[http://www.proba.jussieu.fr/pageperso/levy/|Thierry Lévy]]** |
| * 14h30-15h45: **[[http://www.math.ku.dk/~mikosch/|Thomas Mikosch]]** // \\ // | * 14h00-15h00: **[[http://www.math.ku.dk/~mikosch/|Thomas Mikosch]]** //The largest eigenvalues of the sample covariance matrix in the heavy-tail case\\ //Heavy tails of a time series are typically modeled by power law tails with a positive tail index $\alpha$. We refer to such time series as regularly varying with index $\alpha$. Regular variation of a time series translates into power law tail behavior of the partial sums of the time series above high threshold. This was observed early on in work by A.V. Nagaev (1969) and S.V. Nagaev (1979) who considered sums of iid regularly varying random variables. These results are referred to as heavy-tail or Nagaev-type large deviations. The goal of this lecture is to argue that heavy-tail large deviations are useful tools when dealing with the eigenvalues of the sample covariance matrix of dimension $p\times n$ when $p\to\infty$ as $n\to\infty$ in those cases when one can identify the dominating entries in this matrix. These are the diagonal entries in the iid and some other cases. A similar argument allows one to identify the dominating entries if the time series has a linear dependence structure with regularly varying noise. These techniques are an alternative approach to earlier results by Soshnikov (2004,2006), Auffinger, Ben Arous, Peche (2009), Belinschi, Dembo, Guionnet (2009). They also allow one to deal with certain classes of matrices with dependent heavy-tailed entries. This is joint work with Richard A. Davis (Columbia) and Johannes Heiny (Aarhus). |
| * 15h45-17h00: **[[http://umr-math.univ-mlv.fr/membres/tian.peng|Peng Tian]]** // \\ // | |
| | * 15h30-16h30: **[[http://umr-math.univ-mlv.fr/membres/tian.peng|Peng Tian]]** //Large Random Matrices of Long Memory Stationary Processes: Asymptotics and fluctuations of the largest eigenvalue \\ //Given $n$ i.i.d. samples $(\boldsymbol{\vec x}_1, \cdots, \boldsymbol{\vec x}_n)$ of a $N$-dimensional long memory stationary process, it has recently been proved that the limiting spectral distribution of the sample covariance matrix, $$\frac 1n \sum_{i=1}^n \boldsymbol{\vec x}_i \boldsymbol{\vec x}^*_i$$ has an unbounded support as $N,n\to \infty$ and $\frac Nn\to c\in (0,\infty)$. As a consequence, its largest eigenvalue $$\lambda_{\max} \left( \frac 1n \sum_{i=1}^n \boldsymbol{\vec x}_i \boldsymbol{\vec x}^*_i\right)$$ tends to $+\infty$. In this talk, we will describe its asymptotics and fluctuations, tightly related to the features of the underlying population covariance matrix, which is of a Toeplitz nature. This is a joint work with Florence Merlevède and Jamal Najim. |
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| * Vendredi **6 avril** | * Vendredi **6 avril** |
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| * Vendredi **11 mai** | * Vendredi **11 mai** |
| | * 10h30-12h00: mini cours par **[[http://google.com/search?q=Maxime+Février+Maths|Maxime Février]]** |
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| * Vendredi **8 juin** | * Vendredi **8 juin** |