Comparison theorems for the extreme eigenvalues of a random symmetric matrix

Summary · qwen2.5:32b

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 matching Gaussian random matrix, strengthening previous results and providing corollaries for minimum eigenvalues and spectral norms. This methodology enhances existing bounds on eigenvalues in various fields including spectral graph theory and quantum information theory, and provides the first complete proof for the injectivity properties of sparse random dimension reduction maps conjectured by Nelson & Nguyen in 2013.

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Learn about comparison theorems for extreme eigenvalues of random symmetric matrices and their applications in various fields such as spectral graph theory, quantum information theory, high-dimensional statistics, and numerical linear algebra.

Excerpt

arXiv:2603.04365v3 Announce Type: replace Abstract: 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 whose statistics match the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm; the proof strategy also extends to matrix martingale sequences. 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.