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![The leftmost figure shows the eigenvalues, denoted by the small circles, of a single 500×500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$500 \times 500$$\end{document} iid random matrix 1500X500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\sqrt{500}}X_{500}$$\end{document} with Gaussian entries. The rightmost figure shows the eigenvalues, denoted by small circles, of a product of four independent 500×500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$500\times 500$$\end{document} random matrices, each scaled by 1500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\sqrt{500}}$$\end{document}, where the entries in each random matrix are independent iid Gaussian random variables](publication/332543759/figure/fig1/AS:961874577350667@1606340178585/The-leftmost-figure-shows-the-eigenvalues-denoted-by-the-small-circles-of-a-single.png)
The leftmost figure shows the eigenvalues, denoted by the small circles, of a single 500×500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$500 \times 500$$\end{document} iid random matrix 1500X500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\sqrt{500}}X_{500}$$\end{document} with Gaussian entries. The rightmost figure shows the eigenvalues, denoted by small circles, of a product of four independent 500×500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$500\times 500$$\end{document} random matrices, each scaled by 1500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\sqrt{500}}$$\end{document}, where the entries in each random matrix are independent iid Gaussian random variables
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Consider the product \(X = X_{1}\cdots X_{m}\) of m independent \(n\times n\) iid random matrices. When m is fixed and the dimension n tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of X for analytic test functions. We show that the limiting variance is universal in the sense that it does not depend on m (th...
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Citations
... The fluctuation formula (3.72) with β = 2 has been shown to remain valid in the case of the eigenvalues of products of i.i.d. complex random matrices in [116], albeit for a class of test function contained strictly inside the eigenvalue support; see too [50]. ...
... By asymptotic analysis of the Stieltjes transform, Vasilchuk [55] proved a central limit theorem for linear statistics of eigenvalues of a certain product of unitary matrices. A recent, general result of Coston-O'Rourke [19] showed Gaussian fluctuations for the eigenvalues of products of Wigner matrices. ...
We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index \(\beta = 1,2,4\) respectively) where time corresponds to the number of terms in the product. More generally, we consider the \(\beta \)-Jacobi product process obtained by extrapolating to arbitrary \(\beta > 0\). For fixed time (i.e. number of factors is constant), we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When \(\beta = 2\), our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann–Burda–Kieburg and Liu–Wang–Wang across time. In the arbitrary \(\beta > 0\) case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural \(\beta \)-generalization of the interpolating process.
... As mentioned above, a functional CLT for Wigner matrices was proved in [8,15], the former covering also mesoscopic regimes. For non-Hermitian matrices, [9,17,18] presented tracial CLT on macroscopic scale for analytic test functions, and later in [7] it was generalized to H 2 test functions both in macroscopic and partially mesoscopic scales. ...
For large dimensional non-Hermitian random matrices $X$ with real or complex independent, identically distributed, centered entries, we consider the fluctuations of $f(X)$ as a matrix where $f$ is an analytic function around the spectrum of $X$. We prove that for a generic bounded square matrix $A$, the quantity $\mathrm{Tr}f(X)A$ exhibits Gaussian fluctuations as the matrix size grows to infinity, which consists of two independent modes corresponding to the tracial and traceless parts of $A$. We find a new formula for the variance of the traceless part that involves the Frobenius norm of $A$ and the $L^{2}$-norm of $f$ on the boundary of the limiting spectrum.
... Polynomial test functions via the alternative moment method were considered by Nourdin and Peccati in [68]. The analytic method of [72] was recently extended by Coston and O'Rourke [34] to fluctuations of linear statistics for products of i.i.d. matrices. ...
We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.
... is the m-th power of the uniform distribution µ ∞ = µ 1 ∞ on the complex disc, see also [33]. The Gaussian case has been treated in [10,2], more general models can be found in [25,17,7,1,22], for the convergence of the singular values see [5] and furthermore for local results we refer to [31,24,32,18,12]. ...
... The techniques based upon the first four moment matching [59,44] are insensitive to the symmetry class, hence these results are obtained in parallel for both real and complex ensembles. Beyond this method, however, most results on CLT for non-Hermitian matrices were restricted to the complex case [25,33,47,51,52,54], see the introduction of [22] for a detailed history, as well as for references to the analogous CLT problem for Hermitian ensembles and log-gases. The special role that the real axis plays in the spectrum of the real case substantially complicates even the explicit formulas for the Ginibre ensemble both for the density [26] as well as for the k-point correlation functions [37,12,42]. ...
... In the case when the entries of the n×n matrix X n are iid random variables with mean zero, unit variance, and which satisfy some additional moment and regularity S. O'Rourke has been supported in part by NSF grants ECCS-1610003 and DMS-1810500. 1 requirements, Rider and Silverstein [25] showed that S n [f ](X n / √ n) converges in distribution to a Gaussian random variable as n → ∞ for test functions f analytic in a neighborhood of the disk {z ∈ C : |z| ≤ 4}. This result has been extended and generalized in subsequent works; see, for example, [7,8,16,18,19,22,26]. ...
... We conclude this section by specializing Theorems 1.5 and 1.6 to a few specific examples. The first of these is an immediate consequence of Theorem 2.2 from [8] and our main results above. Corollary 1.8. ...
For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues $\lambda_1, \ldots, \lambda_n$, the seminal work of Rider and Silverstein asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n f(\lambda_i)$ converge to a Gaussian distribution for sufficiently nice test functions $f$. We study the fluctuations of $\sum_{i=1}^{n-K} f(\lambda_i)$, where $K$ randomly chosen eigenvalues have been removed from the sum. In this case, we identify the limiting distribution and show that it need not be Gaussian. Our results hold for the case when $K$ is fixed as well as the case when $K$ tends to infinity with $n$. The proof utilizes the predicted locations of the eigenvalues introduced by E. Meckes and M. Meckes. As a consequence of our methods, we obtain a rate of convergence for the empirical spectral distribution of $X_n$ to the circular law in Wasserstein distance, which may be of independent interest.
... A similar result for polynomial test functions via moment method was given by Nourdin and Peccati in [48]. The analytic method of [49] was recently extended by Coston and O'Rourke [26] to fluctuations of linear statistics for products of i.i.d. matrices. ...
We consider large non-Hermitian random matrices $X$ with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having $2+\epsilon$ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [Rider, Silverstein 2006], or the distribution of the matrix elements needed to be Gaussian [Rider, Vir\'ag 2007], or at least match the Gaussian up to the first four moments [Tao, Vu 2016; Kopel 2015]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of $X$ with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian Motions.
... By asymptotic analysis of the Stieltjes transform, Vasilchuk [45] proved a central limit theorem for linear statistics of eigenvalues of a certain product of unitary matrices. A recent, general result of Coston-O'Rourke [17] showed Gaussian fluctuations for the eigenvalues of products of Wigner matrices. ...
We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index $\beta = 1,2,4$ respectively) where time corresponds to the number of terms in the product. More generally, we consider the $\beta$-Jacobi product process obtained by extrapolating to arbitrary $\beta > 0$. When the time scaling is preserved, we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When $\beta = 2$, our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann-Burda-Kieburg and Liu-Wang-Wang across time. In the arbitrary $\beta > 0$ case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural $\beta$-generalization of the interpolating process.
... CLT for polynomial f and real valued M in [21]. More recently, CLT for products of random matrices were found in [10,16]; and words of random matrices were found in [11]. ...
We show that the fluctuations of the linear eigenvalue statistics of a non-Hermitian random band matrix of bandwidth $b_{n}$ with a continuous variance profile $w_{\nu}(x)$ converges to a $N(0,\sigma_{f}^{2}(\nu))$, where $\nu=\lim_{n\to\infty}(2b_{n}/n)$. We obtain an explicit formula for $\sigma_{f}^{2}(\nu)$, which depends on the test function, and $w_{\nu}$. When $\nu=1$, the formula is consistent with Rider, and Silverstein (2006). We also compute an explicit formula for $\sigma_{f}^{2}(0)$. We show that $\sigma_{f}^{2}(\nu)\to \sigma_{f}^{2}(0)$ as $\nu\downarrow 0$.
