Comparison theorems for the extreme eigenvalues of a random symmetric matrix
This addresses eigenvalue estimation problems for researchers in random matrix theory and applied fields, providing incremental improvements with new applications.
The paper establishes a comparison theorem showing that the maximum eigenvalue of a sum of independent random symmetric matrices is dominated by that of a Gaussian random matrix with matching statistics, strengthening prior results. It applies this to improve eigenvalue bounds in areas like spectral graph theory and quantum information, including proving a conjecture about sparse random dimension reduction maps.
This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix that inherits its statistics from the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm. The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper improves on existing eigenvalue bounds for random matrices arising in spectral graph theory, quantum information theory, high-dimensional statistics, and numerical linear algebra. In particular, these techniques deliver the first complete proof that a sparse random dimension reduction map has the injectivity properties conjectured by Nelson & Nguyen in 2013.