Independence of linear spectral statistics and the point process at the edge of Wigner matrices
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Published:2024-08-07
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ISSN:2010-3263
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Container-title:Random Matrices: Theory and Applications
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language:en
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Short-container-title:Random Matrices: Theory Appl.
Author:
Banerjee Debapratim1ORCID
Affiliation:
1. Department of mathematics, Ashoka University, Plot no 2, Rajiv Gandhi Education City, Rai, Sonipat, Haryana 131029, India
Abstract
Let [Formula: see text] be a Wigner matrix of dimension [Formula: see text] with eigenvalues [Formula: see text] and [Formula: see text] be an analytic function on [Formula: see text] with polynomial growth. It is known that [Formula: see text] converges in distribution to a normal random variable with mean [Formula: see text] and a finite variance depending on [Formula: see text]. On the other hand, it is also known that [Formula: see text] converges in distribution to the GOE Tracy widom law. In this paper we prove that whenever the entries of the Wigner matrix are sub-Gaussian, [Formula: see text] is asymptotically independent of the point process at the edge of the spectrum. Hence, one gets that [Formula: see text] and [Formula: see text] are asymptotically independent. The main ingredient of the proof is based on a recent paper by Banerjee [A new combinatorial approach for tracy–widom law of wigner matrices, preprint (2022), arXiv:2201.00300]. The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.
Funder
Inspire faculty fellowship of Department of Science and Technology, Government of India
Publisher
World Scientific Pub Co Pte Ltd