Hong-Chang Ji

We consider large-dimensional Hermitian or symmetric random matrices of the form [Formula: see text], where [Formula: see text] is a Wigner matrix and [Formula: see text] is a real diagonal matrix whose entries are independent of [Formula: see text]. For a large class of diagonal matrices [Formula: see text], we prove that the fluctuations of linear spectral statistics of [Formula: see text] for [Formula: see text] test function can be decomposed into that of [Formula: see text] and of [Formula:

Abstract We consider non‐Hermitian random matrices of the form , where is a general deterministic matrix and consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, that is, that the local density of eigenvalues is bounded by and (ii) that the expected condition number of any bulk eigenvalue is bounded by ; both results are optimal up to the factor . The latter result complements the very recent matching lower bound

High dimensional Haar random matrices are common objects in modern statistical learning theory. We consider the random matrix model $A^{1/2} UBU^* A^{1/2},$ where $A$ and $B$ are two $N \times N$ positive {definite} matrices satisfying some regularity conditions and $U$ is either an $N \times N$ Haar unitary or orthogonal random matrix. On the macroscopic scale, it is well-known that the empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution of t

Consider the random matrix model A1/2UBU∗A1/2, where A and B are two N×N deterministic matrices and U is either an N×N Haar unitary or orthogonal random matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991) 201–220), the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution of the limiting ESDs of A and B, denoted as μα⊠μβ, where μα and μβ are the limiting ESDs of A and B, respectively. In this paper, we study

For given two Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}_{+}=[0,\infty)$, we derive properties of the free multiplicative convolution $\mu\boxtimes\nu$ via its Cauchy-Stieltjes transform. In particular we prove that $\mu\boxtimes\nu$ always has no singular continuous part and, under certain conditions, that the density of its absolutely continuous part is bounded by $x^{-1}$. We also consider a special case in which $\mu$ and $\nu$ are compactly supported Jacobi measures on $(0,\i

We consider the Brown measure of the free circular Brownian motion, \boldsymbol a+\sqrt{t}\boldsymbol x , with an arbitrary initial condition \boldsymbol a , i.e. \boldsymbol a is a general non-normal operator and \boldsymbol x is a circular element * -free from \boldsymbol a . We prove that, under a mild assumption on \boldsymbol a , the density of the Brown measure has one of the following two types of behavior around each point on the boundary of its support – either (i) sharp cut, i.e. a jum

In this paper, we study the eigenvalues and eigenvectors of the spiked invariant multiplicative models when the randomness is from Haar matrices. We establish the limits of the outlier eigenvalues λ̂i and the generalized components (〈v,ûi〉 for any deterministic vector v) of the outlier eigenvectors ûi with optimal convergence rates. Moreover, we prove that the non-outlier eigenvalues stick with those of the unspiked matrices and the non-outlier eigenvectors are delocalized. The results also ho

We consider large-dimensional Hermitian random matrices of the form $W=M+\vartheta V$ where $M$ is a Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $M$. For a large class of diagonal matrices $V$, we prove that the fluctuations of linear spectral statistics of $W$ for analytic test function can be decomposed into that of $M$ and of $V$, and that each of those weakly converges to a Gaussian distribution. We also calculate the formulae fo

Abstract For general large non–Hermitian random matrices X and deterministic normal deformations A , we prove that the local eigenvalue statistics of $$A+X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> close to the critical edge points of its spectrum are universal. This concludes the proof of the third and last remaining typical universality class for non–Hermitian random matrices (for norm

We consider $N\times N$ non-Hermitian random matrices of the form $X+A$, where $A$ is a general deterministic matrix and $\sqrt{N}X$ consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, i.e. that the local density of eigenvalues is bounded by $N^{1+o(1)}$ and (ii) that the expected condition number of any bulk eigenvalue is bounded by $N^{1+o(1)}$; both results are optimal up to the factor $N^{o(1)}$. The latter

We consider fluctuations of the largest eigenvalues of the random matrix model A+UBU∗ where A and B are N×N deterministic Hermitian (or symmetric) matrices and U is a Haar-distributed unitary (or orthogonal) matrix. We prove that the largest eigenvalue weakly converges to the GUE (or GOE) Tracy–Widom distribution, under mild assumptions on A and B to guarantee that the density of states of the model decays as square root around the upper edge. Our proof is based on the comparison of the Green fu

Consider the random matrix model $A^{1/2} UBU^* A^{1/2},$ where $A$ and $B$ are two $N \times N$ deterministic matrices and $U$ is either an $N \times N$ Haar unitary or orthogonal random matrix. It is well-known that on the macroscopic scale, the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution of the limiting ESDs of $A$ and $B,$ denoted as $μ_α\boxtimes μ_β,$ where $μ_α$ and $μ_β$ are the limiting ESDs of $A$ and $B,$ respective

On étudie les fluctuations de f(X), où X est une matrice aléatoire non-hermitienne de grande taille à coefficients i.i.d. (réels ou complexes), et f une fonction analytique sur un domaine qui contient le spectre de X. On prouve que, pour une matrice carrée générique et bornée A, les fluctuations de la quantité trf(X)A sont asymptotiquement gaussiennes et comportent deux modes indépendants, correspondant aux composantes traciale et de trace nulle de A. Une nouvelle formule est établie pour la var

We consider fluctuations of the largest eigenvalues of the random matrix model $A+UBU^{*}$ where $A$ and $B$ are $N \times N$ deterministic Hermitian (or symmetric) matrices and $U$ is a Haar-distributed unitary (or orthogonal) matrix. We prove that the largest eigenvalue weakly converges to the Tracy-Widom distribution, under mild assumptions on $A$ and $B$ to guarantee that the density of states of the model decays as square root around the upper edge. Our proof is based on the comparison of t

Abstract Given two nondegenerate Borel probability measures $\mu$ and $\nu$ on ${\mathbb{R}}_{+}=[0,\infty )$, we prove that their free multiplicative convolution $\mu \boxtimes \nu$ has zero singular continuous part and its absolutely continuous part has a density bounded by $x^{-1}$. When $\mu$ and $\nu$ are compactly supported Jacobi measures on $(0,\infty )$ having power law behavior with exponents in $(-1,1)$, we prove that $\mu \boxtimes \nu$ is another Jacobi measure whose density has squ