Universality of the least singular value for the sum of random matrices

Abstract

We consider the least singular value of $M=R^{\ast }XT+U^{\ast }YV$, 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.

arXiv commentrevisions

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Cited by in corpus (1)

• Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble