Norm Convergence Rate for Multivariate Quadratic Polynomials of Wigner Matrices
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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2023
- DOI:
- 10.48550/arXiv.2308.16778
- arXiv:
- arXiv:2308.16778
- Bibcode:
- 2023arXiv230816778F
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics;
- Mathematics - Functional Analysis;
- Mathematics - Operator Algebras;
- 60B20;
- 15B52
- E-Print:
- 38 pages