On the spectrum of random graphs

Pierre Youssef, Paris VII
Fine Hall 214

Understanding the distribution of the spectrum as the dimension grows is one of the main problems in random matrix theory. This includes, among others, the study of the limiting spectral distribution and the behavior at the boundary of the support of the limiting measure. It is known that the empirical spectral distribution of a square random matrix (resp. symmetric) with i.i.d. centered entries with unit variance converges to the circular law (resp. semi-circular) as the dimension grows. In this talk, we are interested in the stability of these results and the behavior of the spectrum when the i.i.d assumption is relaxed. Random graphs provide models encapsulating sparsity and dependence. The talk will investigate: 1-The limiting spectral distribution of random regular graphs, 2-The behavior of the extreme eigenvalues/singular values and the spectral gap of random graphs.