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

Fill an n × n matrix with independent complex Gaussians of variance 1/n. As n → ∞, the eigenvalues {z k } converge to a sum of an H 1 -noise on the unit disk and an independent H 1/2 -noise on the unit circle. More precisely, for C 1 functions of suitable growth, the distribution of n k=1 (f (z k ) − Ef (z k )) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the disk H 1 and the circle H 1/2 norms of f . As a consequence, with p n the characteristic polynomial, it is found that log |p n | − E log |p n | tends to the planar Gaussian free field conditioned to be harmonic outside the unit disk. Further, for polynomial test functions f , we prove that the limiting covariance structure is universal for a class of models including Haar distributed unitary matrices.

We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian sample-covariance structures ("beta ensembles") are described by the spectrum of a random diffusion generator. This generator may be mapped onto the "Stochastic Bessel Operator," introduced and studied by A. Edelman and B. Sutton in [6] where the corresponding convergence was first conjectured. Here, by a Riccati transformation, we also obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. All this pertains to the so-called hard edge of random matrix theory and sits in complement to the recent work [15] of the authors and B. Virág on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used below to prove there exists a transition between the soft and hard edge laws at all values of beta.

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

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Consider an ensemble of $N\times N$ non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite $(4+\epsilon)$ moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494--529] has shown the ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$ and letting $N\to \infty$, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type $X_N(f)=\sum_{k=1}^Nf(\lambda_k)$ where $\lambda_1,\lambda_2,...,\lambda_N$ denote the ensemble eigenvalues and the test function $f$ is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533--605], where the analogous result for random sample covariance matrices is established.Comment: Published at http://dx.doi.org/10.1214/009117906000000403 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as $n\rightarrow\infty$. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.Comment: Published in at http://dx.doi.org/10.1214/13-AAP958 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The current method allows researchers to obtain more detailed information on postural allocation compared with that in the use of a single activPAL on the thigh.