scholarly journals A note on the limiting mean distribution of singular values for products of two Wishart random matrices

2013 ◽  
Vol 54 (8) ◽  
pp. 083303 ◽  
Author(s):  
Lun Zhang
Keyword(s):  
Random Matrices ◽  
Singular Values

scholarly journals On the Asymptotic Distribution of Singular Values of Powers of Random Matrices

2014 ◽  
Vol 199 (2) ◽  
pp. 68-87
Author(s):  
N. V. Alexeev ◽  
F. Götze ◽  
A. N. Tikhomirov
Keyword(s):  
Asymptotic Distribution ◽  
Random Matrices ◽  
Singular Values

scholarly journals Singular Values of Products of Ginibre Random Matrices

2016 ◽  
Vol 138 (2) ◽  
pp. 135-184 ◽  
Author(s):  
N. S. Witte ◽  
P. J. Forrester
Keyword(s):  
Random Matrices ◽  
Singular Values

scholarly journals On the interval of fluctuation of the singular values of random matrices

2017 ◽  
Vol 19 (5) ◽  
pp. 1469-1505 ◽  
Author(s):  
Olivier Guédon ◽  
Alexander Litvak ◽  
Alain Pajor ◽  
Nicole Tomczak-Jaegermann
Keyword(s):  
Random Matrices ◽  
Singular Values

scholarly journals Complex random matrices have no real eigenvalues

2018 ◽  
Vol 07 (01) ◽  
pp. 1750014 ◽  
Author(s):  
Kyle Luh
Keyword(s):  
Random Matrices ◽  
Random Variables ◽  
Principal Component ◽  
Random Variable ◽  
Singular Values ◽  
Singular Value ◽  
Small Ball ◽  
Circular Law ◽  
Least Singular Value ◽  
Tail Bound

Let [Formula: see text] where [Formula: see text] are iid copies of a mean zero, variance one, subgaussian random variable. Let [Formula: see text] be an [Formula: see text] random matrix with entries that are iid copies of [Formula: see text]. We prove that there exists a [Formula: see text] such that the probability that [Formula: see text] has any real eigenvalues is less than [Formula: see text] where [Formula: see text] only depends on the subgaussian moment of [Formula: see text]. The bound is optimal up to the value of the constant [Formula: see text]. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form [Formula: see text] where [Formula: see text] is a deterministic complex matrix with the condition that [Formula: see text] for some constant [Formula: see text] depending on the subgaussian moment of [Formula: see text]. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.[Formula: see text]N.S.[Formula: see text] 46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.[Formula: see text] 169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62(12) (2009) 1707–1739].

scholarly journals Dynamical correlation functions for products of random matrices

2015 ◽  
Vol 04 (04) ◽  
pp. 1550020 ◽  
Author(s):  
Eugene Strahov
Keyword(s):  
Random Matrices ◽  
Discrete Time ◽  
Random Matrix ◽  
Correlation Functions ◽  
Singular Values ◽  
Matrix Product ◽  
Dynamical Correlation ◽  
Products Of Random Matrices ◽  
Special Cases ◽  
Correlation Kernels

We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case, we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlation functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard–Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.

Sign in / Sign up

Export Citation Format

Share Document