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Register a new userFree probability for purely discrete eigenvalues of random matrices
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In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescus sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.
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The aim of this paper is to show how free probability theory sheds light on spectral properties of deformed matricial models and provides a unified understanding of various asymptotic phenomena such as spectral measure description, localization and fluctuations of extremal eigenvalues, eigenvectors behaviour.
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 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.
Let $xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(xi)$ denote an $ntimes n$ random matrix with entries that are independent copies of $xi$. For $xi$ which is not uniform on its support, we show that begin{align*} mathbb{P}[M_{n}(xi)text{ is singular}] &= mathbb{P}[text{zero row or column}] + (1+o_n(1))mathbb{P}[text{two equal (up to sign) rows or columns}], end{align*} thereby confirming a folklore conjecture. As special cases, we obtain: (1) For $xi = text{Bernoulli}(p)$ with fixed $p in (0,1/2)$, [mathbb{P}[M_{n}(xi)text{ is singular}] = 2n(1-p)^{n} + (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n},] which determines the singularity probability to two asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. (2) For $xi = text{Bernoulli}(p)$ with fixed $p in (1/2,1)$, [mathbb{P}[M_{n}(xi)text{ is singular}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}.] Previously, only the much weaker upper bound of $(sqrt{p} + o_n(1))^{n}$ was known due to the work of Bourgain-Vu-Wood. For $xi$ which is uniform on its support: (1) We show that begin{align*} mathbb{P}[M_{n}(xi)text{ is singular}] &= (1+o_n(1))^{n}mathbb{P}[text{two rows or columns are equal}]. end{align*} (2) Perhaps more importantly, we provide a sharp analysis of the contribution of the `compressible part of the unit sphere to the lower tail of the smallest singular value of $M_{n}(xi)$.
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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