Fluctuations and Large Deviations for Extreme Eigenvalues of Deformed Random Matrices
This is joint work with F.~Benaych Georges and Alice Guionnet.
We consider a model of matrices with well-known spectrum (deterministic with converging spectral measure, Wigner, Wishart etc.) and add a random perturbation with finite rank and delocalized eigenvectors. We get so called spiked or deformed models which lately received quite a lot of attention. We investigate the asymptotic behavior of their extreme eigenvalues, in particular their fluctuations and large deviations properties.
We review the asymptotic behavior of a class of Toeplitz as well as related Hankel and Toeplitz + Hankel determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and a transition between them. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we mention results on the sine, Bessel, and confluent hypergeometric kernel determinants. The talk is based on the joint works with Tom Claeys, Percy Deift, Alexander Its, and Julia Vasilevska.