@article{6511,
abstract = {Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {00911798},
journal = {Annals of Probability},
number = {3},
pages = {1270--1334},
publisher = {Project Euclid},
title = {{Local single ring theorem on optimal scale}},
doi = {10.1214/18-AOP1284},
volume = {47},
year = {2019},
}
@article{1528,
abstract = {We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.},
author = {Bao, Zhigang and Erdös, László},
issn = {01788051},
journal = {Probability Theory and Related Fields},
number = {3-4},
pages = {673 -- 776},
publisher = {Springer},
title = {{Delocalization for a class of random block band matrices}},
doi = {10.1007/s00440-015-0692-y},
volume = {167},
year = {2017},
}
@article{733,
abstract = {Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum.},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
journal = {Advances in Mathematics},
pages = {251 -- 291},
publisher = {Academic Press},
title = {{Convergence rate for spectral distribution of addition of random matrices}},
doi = {10.1016/j.aim.2017.08.028},
volume = {319},
year = {2017},
}
@article{1207,
abstract = {The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {00103616},
journal = {Communications in Mathematical Physics},
number = {3},
pages = {947 -- 990},
publisher = {Springer},
title = {{Local law of addition of random matrices on optimal scale}},
doi = {10.1007/s00220-016-2805-6},
volume = {349},
year = {2017},
}
@article{1434,
abstract = {We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3.},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
journal = {Journal of Functional Analysis},
number = {3},
pages = {672 -- 719},
publisher = {Academic Press},
title = {{Local stability of the free additive convolution}},
doi = {10.1016/j.jfa.2016.04.006},
volume = {271},
year = {2016},
}
@article{1504,
abstract = {Let Q = (Q1, . . . , Qn) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1, 2, . . . , n}. Let Z = (Z1, . . . , Zn), where Zj is the mean zero variance one random variable obtained by centralizing and normalizing Qj , j = 1, . . . , n. Assume that Xi , i = 1, . . . ,p are i.i.d. copies of 1/√ p Z and X = Xp,n is the p × n random matrix with Xi as its ith row. Then Sn = XX is called the p × n Spearman's rank correlation matrix which can be regarded as a high dimensional extension of the classical nonparametric statistic Spearman's rank correlation coefficient between two independent random variables. In this paper, we establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension, namely, p = p(n) and p/n→c ∈ (0,∞) as n→∞.We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni's cumulant method in [Ann. Statist. 36 (2008) 2553-2576] to bypass the so-called joint cumulant summability. In addition, we raise a two-step comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT, we then construct a distribution-free statistic to test complete independence for components of random vectors. Owing to the nonparametric property, we can use this test on generally distributed random variables including the heavy-tailed ones.},
author = {Bao, Zhigang and Lin, Liang and Pan, Guangming and Zhou, Wang},
journal = {Annals of Statistics},
number = {6},
pages = {2588 -- 2623},
publisher = {Institute of Mathematical Statistics},
title = {{Spectral statistics of large dimensional spearman s rank correlation matrix and its application}},
doi = {10.1214/15-AOS1353},
volume = {43},
year = {2015},
}
@article{1585,
abstract = {In this paper, we consider the fluctuation of mutual information statistics of a multiple input multiple output channel communication systems without assuming that the entries of the channel matrix have zero pseudovariance. To this end, we also establish a central limit theorem of the linear spectral statistics for sample covariance matrices under general moment conditions by removing the restrictions imposed on the second moment and fourth moment on the matrix entries in Bai and Silverstein (2004).},
author = {Bao, Zhigang and Pan, Guangming and Zhou, Wang},
journal = {IEEE Transactions on Information Theory},
number = {6},
pages = {3413 -- 3426},
publisher = {IEEE},
title = {{Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices}},
doi = {10.1109/TIT.2015.2421894},
volume = {61},
year = {2015},
}
@article{1505,
abstract = {This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form W N =Σ 1/2XX∗Σ 1/2 . Here, X = (xij )M,N is an M× N random matrix with independent entries xij , 1 ≤ i M,≤ 1 ≤ j ≤ N such that Exij = 0, E|xij |2 = 1/N . On dimensionality, we assume that M = M(N) and N/M → d ε (0, ∞) as N ∞→. For a class of general deterministic positive-definite M × M matrices Σ , under some additional assumptions on the distribution of xij 's, we show that the limiting behavior of the largest eigenvalue of W N is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd″os, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case (&Epsi = I ). Consequently, in the standard complex case (Ex2 ij = 0), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of W N converges weakly to the type 2 Tracy-Widom distribution TW2 . Moreover, in the real case, we show that whenΣ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit TW1 holds for the normalized largest eigenvalue of W N , which extends a result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal Σ and more generally distributed X . In summary, we establish the Tracy-Widom type universality for the largest eigenvalue of generally distributed sample covariance matrices under quite light assumptions on &Sigma . Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed.},
author = {Bao, Zhigang and Pan, Guangming and Zhou, Wang},
journal = {Annals of Statistics},
number = {1},
pages = {382 -- 421},
publisher = {Institute of Mathematical Statistics},
title = {{Universality for the largest eigenvalue of sample covariance matrices with general population}},
doi = {10.1214/14-AOS1281},
volume = {43},
year = {2015},
}
@article{1506,
abstract = {Consider the square random matrix An = (aij)n,n, where {aij:= a(n)ij , i, j = 1, . . . , n} is a collection of independent real random variables with means zero and variances one. Under the additional moment condition supn max1≤i,j ≤n Ea4ij <∞, we prove Girko's logarithmic law of det An in the sense that as n→∞ log | detAn| ? (1/2) log(n-1)! d/→√(1/2) log n N(0, 1).},
author = {Bao, Zhigang and Pan, Guangming and Zhou, Wang},
journal = {Bernoulli},
number = {3},
pages = {1600 -- 1628},
publisher = {Bernoulli Society for Mathematical Statistics and Probability},
title = {{The logarithmic law of random determinant}},
doi = {10.3150/14-BEJ615},
volume = {21},
year = {2015},
}