Circular Law Theorem for Random Markov Matrices
Résumé
Let $(X_{i,j})$ be an infinite array of i.i.d. non negative real random variables with unit mean, finite positive variance $\sigma^2$, and finite fourth moment. Let $M$ be the $n\times n$ random Markov matrix with i.i.d. rows defined by $M_{i,j}=X_{i,j}/(X_{i,1}+\cdots+X_{i,n})$. It belongs to the Dirichlet Markov Ensemble when $X_{1,1}$ follows an exponential law. We show that with probability one, the empirical spectral distribution $\frac{1}{n}(\delta_{\lambda_1}+\cdots+\delta_{\lambda_n})$ of $\sqrt{n}M$ converges weakly as $n\to\infty$ to the uniform law on the disc $\{z\in\mathbb{C};|z|\leq \sigma\}$ and moreover, the spectral gap of $M$ is of order $n^{-1/2}$ for large enough $n$. There is for now a gap in the proof of Theorem 1.3 (convergence to the circular law).
Origine : Fichiers produits par l'(les) auteur(s)