Department Seminars & Colloquia




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This talk is about the spectra of non-Hermitian random matrix models. Their asymptotic analysis reveals remarkable high dimensional phenomena, which are, among other aspects, related to entropy maximization and free probability theory. We  will present some of the general phenomena and methodologies, and we will focus mostly on the circular law phenomenon. We will present several works, notably in collaboration with Charles Bordenave and Pietro Caputo. We will also present briefly some few open problems.

Host: 폴정     English     2019-04-09 08:54:19

The real Ginibre ensemble consists of square real matrices whose entries are i.i.d. standard normal random variables. In the large dimension limit, the empirical distribution converges to the circular law in a disk. This is same as the complex Ginibre ensemble. However, unlike the complex version, there is a positive probability that there are real eigenvalues. The law of the largest real eigenvalue was studied by Rider, Sinclair and Poplavskyi, Tribe, Zaboronski. Building on their work, we will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system which is related to the nonlinear Schr¨odinger equation. The results of this talk are based on a recent joint with Thomas Bothner (King's College London).

Host: 폴정     English     2019-04-26 10:42:34

Describing the eigenvalue distribution of the sum of two general Hermitian matrices is basic question going back to Weyl. If the matrices have high dimensionality and are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of the sum is given by the free additive convolution of the respective spectral distributions. This result was obtained by Voiculescu on the macroscopic scale. In this talk, I show that it holds on the microscopic scale all the way down to the eigenvalue spacing in the bulk and at the regular edges. This shows a remarkable rigidity phenomenon for the eigenvalues. Joint work with Z.G. Bao and L. Erdos. 

Host: 폴정     English     2019-04-24 10:50:49