2023/08/31 by Jacob Fronk, Fronk, Jacob, Torben Krüger +3 · 1 citation
Mathematics · #15B52 #60B20 #Advanced Algebra and Geometry #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Graph theory and applications #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2308.16778
openalex publication_date 2023/08/31 · openalex created_date 2023/09/02 · openalex updated_date 2026/07/28
We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension N of the matrices grows to infinity, the operator norm of such polynomials q converges to a deterministic limit with a rate of convergence of N-2/3+o(1). Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviours.