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We show that in the point process limit of the bulk eigenvalues of $beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $lambda$ is given by [bigl( kappa_{beta}+o(1)bigr)lambda^{gamma_{beta}}expbiggl(-{bet a}{64}lambda^2+biggl({beta}{8}-{1}{4}biggr)lambdabiggr)] as $lambdatoinfty$, where [gamma_{beta}={1}{4}biggl({beta}{2}+{2}{beta}-3biggr)] and $kappa_{beta}$ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157--165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron--Martin--Girsanov transformation in stochastic calculus.
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This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries. The proof relies on the Erd{H o}s-Schlein-Yau dynamic approach. We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle. This observable also provides a simple and unified proof of universality in the bulk and at the edge, which is quantitative. To illustrate this, we give the first explicit rate of convergence to the Tracy-Widom distribution for generalized Wigner matrices.
This is a brief survey of classical and recent results about the typical behavior of eigenvalues of large random matrices, written for mathematicians and others who study and use matrices but may not be accustomed to thinking about randomness.
We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2ntimes 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k}=lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,0}= -frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ where $zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{ngeq 1}$, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k_n}=-frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ provided $lim_{nrightarrow infty} left(n^{-1/2}log(n)right) k_{n}=0$.
We consider the real eigenvalues of an $(N times N)$ real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $tau_Nin [0,1]$. In the almost-Hermitian regime where $1-tau_N=Theta(N^{-1})$, we obtain the large-$N$ expansion of the mean and the variance of the number of the real eigenvalues. Furthermore, we derive the limiting empirical distributions of the real eigenvalues, which interpolate the Wigner semicircle law and the uniform distribution, the restriction of the elliptic law on the real axis. Our proofs are based on the skew-orthogonal polynomial representation of the correlation kernel due to Forrester and Nagao.
We study the statistics of the largest eigenvalues of $p times p$ sample covariance matrices $Sigma_{p,n} = M_{p,n}M_{p,n}^{*}$ when the entries of the $p times n$ matrix $M_{p,n}$ are sparse and have a distribution with tail $t^{-alpha}$, $alpha>0$. On average the number of nonzero entries of $M_{p,n}$ is of order $n^{mu+1}$, $0 leq mu leq 1$. We prove that in the large $n$ limit, the largest eigenvalues are Poissonian if $alpha<2(1+mu^{{-1}})$ and converge to a constant in the case $alpha>2(1+mu^{{-1}})$. We also extend the results of Benaych-Georges and Peche [7] in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.
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