Work

Norm convergence rate for multivariate quadratic polynomials of Wigner matrices

Year: 2024

Type: article

Abstract: 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.

Source: Journal of Functional Analysis

DOI: 10.1016/j.jfa.2024.110647

Authors: Jacob Fronk, Torben Krüger, Yuriy Nemish

Institution: University of Copenhagen

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,670

Funder: Villum Fonden