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The Eigenvalue Spacing of IID Random Matrices and Related Least Singular Value Results


This thesis studies the spacing between eigenvalues of random matrices with independent and identically distributed (iid) entries. Tail estimates on the minimum distance between any pair of eigenvalues are proven. In particular, we establish that the spectrum of an iid random matrix is simple with high probability. A key technical result is a new least singular value tail estimate for shifted matrices of the form $A_n-zI_n$, where $A_n$ is an iid random matrix with real entries and $z$ is a complex scalar.

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