We analyze the spectral distribution of the adjacency matrix and the graph Laplacian for a wide variety of random trees. Using soft arguments which seem to be applicable in a wide variety of settings, we show that the empirical spectral distribution for a number of random tree models, converges to a constant (model dependent) distribution. We also analyze the kernel of the spectrum and prove as...

Let B n = (1/N)T 1/2 n is a Hermitian square root of the nonnegative definite Hermitian matrix T n. It is shown in Bai and Silverstein (1998) that, under certain conditions on the eigenvalues of T n , with probability one no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large n. For these n the interval corresponds t...

Let X = (Xi,j)m×n,m ≥ n, be a complex Gaussian random matrix with mean zero and variance 1 n , let S = XX be a sample covariance matrix. In this paper we are mainly interested in the limiting behavior of eigenvalues when m n → γ ≥ 1 as n → ∞. Under certain conditions on k, we prove the central limit theorem holds true for the k-th largest eigenvalues λ(k) as k tends to infinity as n → ∞. The pr...