On the spectrum of random graphs
On the spectrum of random graphs
Understanding the distribution of the spectrum as the dimensionÌýgrows is one of the main problems in random matrix theory. This includes,Ìýamong others, the study of the limiting spectral distribution and theÌýbehavior at the boundary of the support of the limiting measure. It is knownÌýthat the empirical spectral distribution of a square random matrix (resp.Ìýsymmetric) with i.i.d. centered entries with unit variance converges to theÌýcircular law (resp. semi-circular) as the dimension grows.ÌýIn this talk, we are interested in the stability of these results and theÌýbehavior of the spectrum when the i.i.d assumption is relaxed. Random graphsÌýprovide models encapsulating sparsity and dependence.ÌýThe talk will investigate: 1-The limiting spectral distribution of randomÌýregular graphs, 2-The behavior of the extreme eigenvalues/singular valuesÌýand the spectral gap of random graphs.