Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.
We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n by n square matrix. The maximum absolute values of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the limiting distribution of the spectral radius. Our results reduce to those in Jiang and Qi [26] when the rectangular mat…
In this paper, we consider $m$ independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the $m$ rectangular matrices is an $n$ by $n$ square matrix. We study the limiting empirical spectral distributions of the product where the dimension of the product matrix goes to infinity, and $m$ may change with the dimension of the product matrix and diverge. We give a complete description for the limiting distribution of the empirical spectral distributions for the product matrix and illustrate…
We consider large non-Hermitian real or complex random matrices $X$ with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of $X$ are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy-Widom distribution at the spectral edges of the Wigner ensemble.
We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called $L^2$-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher-Hartwig singularities. Using Riemann-Hilbert methods, we prove a rather general Fisher-Hartwig formula for one-cut regular unitary invariant ensembles.
We derive estimates for the largest and smallest singular values of sparse rectangular $N\times n$ random matrices, assuming $\lim_{N,n\to\infty}\frac nN=y\in(0,1)$. We consider a model with sparsity parameter $p_N$ such that $Np_N\sim \log^{α}N$ for some $α>1$, and assume that the moments of the matrix elements satisfy the condition $\mathbf E|X_{jk}|^{4+δ}\le C<\infty$. We assume also that the entries of matrices we consider are truncated at the level $(Np_N)^{\frac12-\varkappa}$ with $\varkappa:=\fracδ{2(4+δ)}$.
R-transforms for non-Hermitian matrices: a spherical integral approach
Econophysics Lab, Institut Louis Bachelier（经济物理实验室，路易·巴舍利耶研究所）；LadHyX, UMR CNRS 7646, Ecole Polytechnique, Institut Polytechnique de Paris（拉迪希克斯实验室，法国国家科学研究中心联合研究单位第7646号，巴黎综合理工学院，巴黎理工学院）；Capital Fund Management（资本基金管理公司）
A recent approach to the Beck-Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models: matrices with Bernoulli or Poisson entries. For Poisson matrices, we further characterize the discrepancy for any rectangular aspect ratio. These results give sharp answers to questions of Hoberg and Rothvoss (SODA 2019) and Franks and Saks (Random Structures Algorithms 2020). Our main tool is a conditional second moment method combined with Stein's method of exch…
R-transforms for non-Hermitian block random matrices: a spherical integral approach
Institut Louis Bachelier（路易·巴舍利耶研究所）；Econophysics Lab, Institut Louis Bachelier（经济物理学实验室，路易·巴舍利耶研究所）；LadHyX, UMR CNRS 7646, École polytechnique, Institut Polytechnique de Paris（拉迪克斯实验室，法国国家科学研究中心联合研究单位7646，巴黎综合理工学院，巴黎理工学院）
We compare the Ornstein-Uhlenbeck process for the Gaussian Unitary Ensemble to its non-hermitian counterpart - for the complex Ginibre ensemble. We exploit the mathematical framework based on the generalized Green's functions, which involves a new, hidden complex variable, in comparison to the standard treatment of the resolvents. This new variable turns out to be crucial to understand the pattern of the evolution of non-hermitian systems. The new feature is the emergence of the coupling between the flow of eigenvalues and that of left/right eigenvectors. We analyze local and global equilibria…
In this manuscript we consider denoising of large rectangular matrices: given a noisy observation of a signal matrix, what is the best way of recovering the signal matrix itself? For Gaussian noise and rotationally-invariant signal priors, we completely characterize the optimal denoiser and its performance in the high-dimensional limit, in which the size of the signal matrix goes to infinity with fixed aspects ratio, and under the Bayes optimal setting, that is when the statistician knows how the signal and the observations were generated. Our results generalise previous works that considered …
In this short note we address a gaussian property of normal vectors in random non-Hermitian matrices. The approach uses a simple geometric and comparison technique.
We investigate the product of $n$ complex non-Hermitian, independent random matrices, each of size $N\times N$ in the class of elliptic matrices, with independent identically distributed entries. The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process.
The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of quantitative invertibility in the non-Hermitian setting, and review some applications of this line of research.
We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular law for the empirical spectral distribution of these matrices when the 4th moments of their entries are controlled.
We study the Brown measure of certain non-hermitian operators arising from Voiculescu's free probability theory. Usually those operators appear as the limit in *-moments of certain ensembles of non-hermitian random matrices, and the Brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices. A prominent class for our operators is given by polynomials in *-free variables. Other explicit examples include R-diagonal elements and elliptic elements, for which the Brown measure was already known, and a new class of triangular-elliptic elements. Our me…
We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le ΛN$ for some constant $Λ\ge 1$. Let $X$ be an $M\times n$ random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let $T$ be an $N\times M$ deterministic matrix with comparable singular values $c\le s_{N}(T) \le s_{1}(T) \le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\times n$ deterministic matrix with $\|A\|=O(\sqrt{N})$. Then we prove that for any $ε>0$, the sm…
We consider products of independent square random non-Hermitian matrices. More precisely, let $n\geq 2$ and let $X_1,\ldots,X_n$ be independent $N\times N$ random matrices with independent centered entries with variance $N^{-1}$. It was shown by Götze and Tikhomirov and by Soshnikov and O'Rourke that the limit of the empirical spectral distribution of the product $X_1\cdots X_n$ is supported in the unit disk. We prove that if the entries of the matrices $X_1,\ldots,X_n$ satisfy uniform subexponential decay condition, then the spectral radius of $X_1\cdots X_n$ converges to 1 almost surely as $…
