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.
Universality Behviour of Solutions of Hamiltonian PDEs in Critical Regimes
We study the solution of a class of Hamiltonian PDE near critical points and we show that the solution locally does not depend on the initial data and it is described by particular solutions of Painleve equations.