Spectral radii of sparse random matrices

We establish bounds on the spectral radii for a large class of
sparse random matrices, which includes the adjacency matrices of
inhomogeneous ErdH{o}s-R’enyi graphs. For the ErdH{o}s-R’enyi graph $G(n,d/n)$, our results imply that the smallest and second-largest
eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that $d gg log n$. This establishes a crossover in the behaviour of the extremal eigenvalues around $d sim log n$. Our results also apply to non-Hermitian sparse random matrices, corresponding to adjacency matrices of directed graphs. Joint work with Florent Benaych-Georges and Charles Bordenave.

• Speaker: Antti Knowles (Geneva)
• Tuesday 07 March 2017, 15:15–16:15
• Venue: MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.
• Series: Probability; organiser: Perla Sousi.