For shifted real or complex Ginibre matrices near the spectral edge, the least singular value has optimal tail probability of order x in the complex case and square-root x with a Gaussian imaginary-part damping in the real case.
Universality of the least singular value for the sum of random matrices
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abstract
We consider the least singular value of $M = R^* X T + U^* YV$, 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.
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2019 1verdicts
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Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble
For shifted real or complex Ginibre matrices near the spectral edge, the least singular value has optimal tail probability of order x in the complex case and square-root x with a Gaussian imaginary-part damping in the real case.