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Ci-dessous, les différences entre deux révisions de la page.
Les deux révisions précédentes Révision précédente Prochaine révision | Révision précédente Prochaine révisionLes deux révisions suivantes | ||
mega:seminaire [2020/02/20 18:54] – [Calendrier 2019-2020] Guillaume BARRAQUAND | mega:seminaire [2020/03/06 19:43] – [Prochaine séance] Guillaume BARRAQUAND | ||
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Ligne 13: | Ligne 13: | ||
* **Calendrier.** https:// | * **Calendrier.** https:// | ||
===== Prochaine séance ===== | ===== Prochaine séance ===== | ||
- | Vendredi **7 février**, salle 421 le matin, amphi Hermite l' | + | Vendredi **13 mars**, amphi Darboux |
- | * 10h30-12h00: | + | * 10h30-12h00: |
- | Dans ce mini-cours, je présenterai le modèle | + | Un processus ponctuel est dit rigide (ou number-rigide) si pour tout compact fixé, la donnée de la configuration à l' |
- | * 14h00-15h00: | + | * 14h00-15h00: |
- | Dans cet exposé, | + | Dans mon exposé, à partir |
- | * 15h30-16h30: | + | * 15h30-16h30: |
- | On étudie un système d’équations linéaires $x= 1+Ax$ pour lequel la matrice carrée $A$ est aléatoire à entrées i.i.d. On montre que asymptotiquement, quand les dimensions de la matrice $A$ tendent vers l’infini, l’obtention de solutions positives $x$ est liée à la normalisation de la matrice. Cette question est motivée par l’étude de réseaux trophiques en écologie mathématique. Travail en collaboration avec Pierre Bizeul. | + | In this talk, I would like to advertise an equality between two objects from very different areas of mathematical physics. This bridges the Gaussian Multiplicative Chaos, which plays an important role in certain conformal field theories, and a reference model in random matrices. The main tool is an explicit description in terms of coefficients known as |
+ | - canonical moments in statistics | ||
+ | - Verblunsky coefficients in the literature for orthogonal polynomials | ||
+ | - non-linear Fourier coefficients in harmonic analysis | ||
+ | On the one hand, in 1985, J.P Kahane introduced a random measure called the Gaussian Multiplicative Chaos (GMC), now an important object to in the study of turbulence. Morally, this is the measure whose Radon-Nikodym derivative w.r.t to Lebesgue is the exponential of a log correlated Gaussian field. In the cases of interest, this Gaussian field is a Schwartz distribution but not a function. As such, the construction of GMC needs to be done with care. In particular, in 2D, the GFF (Gaussian Free Field) is a random Schwartz distribution because of the logarithmic singularity of the Green kernel in 2D. Here we are interested in the 1D case on the circle. | ||
+ | On the other hand, it is known since Verblunsky (1930s) that a probability measure on the circle is entirely determined by the coefficients appearing in the recurrence of orthogonal polynomials. Furthermore, Killip and Nenciu (2000s) have given a realization of the CBE, an important model in random matrices, thanks to random orthogonal polynomials of the circle. | ||
+ | The goal is to give a precise theorem whose loose form is CBE = GMC. Although it was known that random matrices exhibit log-correlated features, such an exact correspondence is quite a surprise. | ||
===== Calendrier 2019-2020 ===== | ===== Calendrier 2019-2020 ===== | ||
* **Organisateurs 2019-2020.** | * **Organisateurs 2019-2020.** |