173 research outputs found

    Deviations from the Circular Law

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    Consider Ginibre's ensemble of N×NN \times N non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance 1N\frac{1}{N}. As N↑∞N \uparrow \infty the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane. In this note we describe fluctuations about this {\em Circular Law}. First we obtain finite NN formulas for the covariance of certain linear statistics of the eigenvalues. Asymptotics of these objects coupled with a theorem of Costin and Lebowitz then result in central limit theorems for a variety of these statistics

    Small deviations for beta ensembles

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    We establish various small deviation inequalities for the extremal (soft edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both settings, upper bounds on the variance of the largest eigenvalue of the anticipated order follow immediately

    Beta ensembles, stochastic Airy spectrum, and a diffusion

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    We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schroedinger operator -d^2/dx^2 + x + (2/beta^{1/2}) b_x' restricted to the positive half-line, where b_x' is white noise. In doing so we extend the definition of the Tracy-Widom(beta) distributions to all beta>0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.Comment: Revised content, new results. In particular, Theorems 1.3 and 5.1 are ne
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