Appears in collection : French Japanese Conference on Probability and Interactions

Let $T_n({\bf a})$ be a $n\times n$ Toeplitz matrix with symbol ${\bf a}\colon \mathbb S^1 \to \mathbb{C}$ given by the Laurent polynomial ${\bf a}(\lambda) = \sum_{k=-r}^s a_k \lambda^k$. We consider the matrix $$ M_n = T_n({\bf a}) + \sigma \frac{X_n}{\sqrt{n}}, $$ 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 $\mathbb{C} \backslash \mbox{support}(\beta_\sigma)$ which we will call the outlier eigenvalues. This is a joint work with Charles Bordenave and Fran\c{c}ois Chapon.