Central limit theorem for linear eigenvalue statistics of diluted random matrices
We discuss the linear eigenvalue statistics of large random graph in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for the test functions with two derivatives the fluctuations of linear eigenvalue statistics converges in distribution to the Gaussian random variable with zero mean and the variance which coincides with "non gaussian" part of the Wigner ensemble variance.
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.