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

Understand

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.