French Japanese Conference on Probability & Interactions

Outliers of Perturbations of Banded Toeplitz Matrices

7 mars 2024, 11:20

50m

Centre de conférences Marilyn et James Simons (Le Bois-Marie)

Orateur

Prof. Mireille Capitaine (IMT)

Description

Let $T_n(\mathbf{a})$ be a $n\times n$ Toeplitz matrix with symbol $\mathbf{a}:{\mathbb{S}}^1\to \mathbb{C}$ given by the Laurent polynomial $\mathbf{a}(\lambda )=\sum _{k=-r}^s{a}_k{\lambda }^k$. We consider the matrix
$$M_n=T_n(\mathbf{a})+\sigma \frac{X_n}{\sqrt{n}},$$ where $\sigma >0$ and $X_n$ is some noise matrix whose entries are centered i.i.d. random variables of unit variance. When $n$ goes to infinity, the empirical spectral distribution of $M_n$ converges towards a probability measure ${\beta }_{\sigma }$ on $\mathbb{C}$. The objective of this talk is to describe, when $n$ is large, the eigenvalues of $M_n$ in closed regions of $\mathbb{C}\mathrm{\setminus }\text{support}({\beta }_{\sigma })$ which we will call the outlier eigenvalues.
This is a joint work with Charles Bordenave and Fran\c{c}ois Chapon.

Documents de présentation

slidesCapitaine7mars2024.pdf

Vidéo