We consider the least singular value of , 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.
Z.C. is partially supported by NSF grant DMS-1607871. P.L. is partially supported by the NSF Graduate Research Fellowship Program under Grant DGE-1144152.
The authors thank Benjamin Landon for comments on a preliminary draft of this paper. They also thank the anonymous referees for their detailed comments, which substantially improved the paper.
"Universality of the least singular value for the sum of random matrices." Electron. J. Probab. 26 1 - 38, 2021. https://doi.org/10.1214/21-EJP603