Abstract
We consider the least singular value of M=R∗XT+U∗YV, where R, T, U, V are independent Haar-distributed unitary matrices and X, Y are deterministic diagonal matrices. Under weak conditions on X and Y, we show that the limiting distribution of the least singular value of M, suitably rescaled, is the same as the limiting distribution for the least singular value of a matrix of i.i.d. Gaussian random variables. Our proof is based on the dynamical method used by Che and Landon to study the local spectral statistics of sums of Hermitian matrices.
Highlights
We consider the least singular value of M = R∗XT + U ∗Y V, where R, T, U, V are independent Haar-distributed unitary matrices and X, Y are deterministic diagonal matrices
In this work we prove universality of the least singular value for the matrix
Based on resolvent estimates and a precise analysis of the short-time behavior of Dyson Brownian motion [28, 39, 44, 58, 59], it has succeeded in its original goal of establishing the universality of local spectral statistics for Wigner matrices [29, 41, 45,46,47,48,49,50, 61], and has since been applied to investigate universality for numerous other random matrix models
Summary
The problem of effectively bounding the least singular value of a random matrix with independent entries has received tremendous attention from mathematicians and computer scientists [17, 35, 36, 62, 66,67,68,69,70, 72, 74,75,76,77]. Based on resolvent estimates and a precise analysis of the short-time behavior of Dyson Brownian motion [28, 39, 44, 58, 59], it has succeeded in its original goal of establishing the universality of local spectral statistics for Wigner matrices [29, 41, 45,46,47,48,49,50, 61], and has since been applied to investigate universality for numerous other random matrix models. We comment on an interesting difference between the real and complex cases which does not arise in the Hermitian model
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