Janossy Densities II. Pfaffian Ensembles

We extend the main result of the companion paper math-ph/0212063 "Janossy Densities I. Determinantal Ensembles" (joint with Alexei Borodin) to the case of the pfaffian ensembles.Comment: Misprints are corrected. The paper is to appear in J.Stat.Phys, vo. 113, No. 3/4, pp.611-622, (2003

Central Limit Theorem for local linear statistics in classical compact groups and related combinatorial identities

We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn

A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices

Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X^* \* X (X^t \*X) converges to the Tracy-Widom law as $n, p$ (the dimensions of $X$) tend to $\infty$ in some ratio $n/p \to \gamma>0.$ We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy-Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner matrices allows to extend the results by Johansson and Johnstone to the case of $X$ with non-Gaussian entries, provided $n-p =O(p^{1/3}) .$ We also prove that \lambda_{max} \leq (n^{1/2}+p^{1/2})^2 +O(p^{1/2}\*\log(p)) (a.e.) for general $\gamma >0.$Comment: This is a preliminary version. Minor misprints are correcte

Gaussian limit for determinantal random point fields

We prove that under fairly general conditions properly rescaled determinantal random point field converges to a generalized Gaussian random process.Comment: This is the revised version accepted for publication in the Annals of Probability. The results of Theorems 1 and 2 are extended, minor misprints are correcte

Determinantal random point fields

The paper contains an exposition of recent as well as old enough results on determinantal random point fields. We start with some general theorems including the proofs of the necessary and sufficient condition for the existence of the determinantal random point field with Hermitian kernel and a criterion for the weak convergence of its distribution. In the second section we proceed with the examples of the determinantal random point fields from Quantum Mechanics, Statistical Mechanics, Random Matrix Theory, Probability Theory, Representation Theory and Ergodic Theory. In connection with the Theory of Renewal Processes we characterize all determinantal random point fields in R^1 and Z^1 with independent identically distributed spacings. In the third section we study the translation invariant determinantal random point fields and prove the mixing property of any multiplicity and the absolute continuity of the spectra. In the fourth (and the last) section we discuss the proofs of the Central Limit Theorem for the number of particles in the growing box and the Functional Central Limit Theorem for the empirical distribution function of spacings.Comment: To appear in the Russian Mathematical Surveys; small misprints are correcte

On the largest eigenvalue of a sparse random subgraph of the hypercube

We consider a sparse random subraph of the $n$-cube where each edge appears independently with small probability $p(n) =O(n^{-1+o(1)})$. In the most interesting regime when $p(n)$ is not exponentially small we prove that the largest eigenvalue of the graph is asymtotically equal to the square root of the maximum degree

Poisson Statistics for the Largest Eigenvalues of Wigner Random Matrices with Heavy Tails

We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics.Comment: I have found a couple of small mistakes in the auxiliary Lemmas 1 and 2 and made the necessary corrections. These changes do not affect the results of the pape

Statistics of Extreme Spacings in Determinantal Random Point Processes

We study translation-invariant determinantal random point fields on the real line. We prove, under quite general conditions, that the smallest nearest spacings between the particles in a large interval have Poisson statistics as the length of the interval goes to infinity.Comment: 16 page

Gaussian fluctuation for the number of particles in Airy, Bessel, sine and other determinantal random point fields

We prove the Central Limit Theorem for the number of eigenvalues near the spectrum edge for hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of Costin-Lebowitz Theorem we prove CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.Comment: The essential alterations are a slightly different formulation of the Costin-Lebowitz Theorem and the addition of Remark 4 in the section

On the lower bound of the spectral norm of symmetric random matrices with independent entries

We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by 2 \*\sigma - o(N^{-6/11+\epsilon}), where $\sigma^2$ is the variance of the matrix entries and $\epsilon$ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $\epsilon >0,$ one has \|A_N\| =2 \*\sigma + o(N^{-6/11+\epsilon}) with probability going to 1 as $N \to \infty.