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Register a new userSpectrum of deformed random matrices and free probability
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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.
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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.
We investigate eigenvectors of rank-one deformations of random matrices $boldsymbol B = boldsymbol A + theta boldsymbol {uu}^*$ in which $boldsymbol A in mathbb R^{N times N}$ is a Wigner real symmetric random matrix, $theta in mathbb R^+$, and $boldsymbol u$ is uniformly distributed on the unit sphere. It is well known that for $theta > 1$ the eigenvector associated with the largest eigenvalue of $boldsymbol B$ closely estimates $boldsymbol u$ asymptotically, while for $theta < 1$ the eigenvectors of $boldsymbol B$ are uninformative about $boldsymbol u$. We examine $mathcal O(frac{1}{N})$ correlation of eigenvectors with $boldsymbol u$ before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law. This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of second kind. These polynomials form an orthogonal set with respect to the semicircle weighting function. This law is an increasing function in the support of semicircle law for eigenvalues $(-2: ,+2)$. Therefore, most of energy of the unknown deforming vector is concentrated in a $cN$-dimensional ($c<1$) known subspace of $boldsymbol B$. We use a combinatorial approach to prove the result.
Let $O(2n+ell)$ be the group of orthogonal matrices of size $left(2n+ellright)times left(2n+ellright)$ equipped with the probability distribution given by normalized Haar measure. We study the probability begin{equation*} p_{2n}^{left(ellright)} = mathbb{P}left[M_{2n} , mbox{has no real eigenvalues}right], end{equation*} where $M_{2n}$ is the $2ntimes 2n$ left top minor of a $(2n+ell)times(2n+ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $ntimes n$ that depends on the truncation parameter $ell$. For $ell=1$ the matrix coincides with the Hilbert matrix and we prove begin{equation*} p_{2n}^{left(1right)} sim n^{-3/8}, mbox{ when }n to infty. end{equation*} We also discuss connections of the above to the persistence probability for random Kac polynomials.
We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random band matrices and the problem of their Anderson transition. We finally describe a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.
We consider $N$ by $N$ deformed Wigner random matrices of the form $X_N=H_N+A_N$, where $H_N$ is a real symmetric or complex Hermitian Wigner matrix and $A_N$ is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of $X_N$ for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on the characteristic function method in [47], local laws for the Green function of $X_N$ in [3, 46, 51] and analytic subordination properties of the free additive convolution [24, 41]. We also prove the analogous results for high-dimensional sample covariance matrices.
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