Law of fractional logarithm for random matrices

2025/03/24 by Bao, Zhigang, Cipolloni, Giorgio, Erdős, László +2 · 2 citations
#60B20 #60G55 #82C10 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.18922

Abstract

We prove the Paquette-Zeitouni law of fractional logarithm (LFL) for the extreme eigenvalues [arXiv:1505.05627] in full generality, and thereby verify a conjecture from [arXiv:1505.05627]. Our result holds for any Wigner minor process and both symmetry classes, in particular for the GOE minor process, while [arXiv:1505.05627] and the recent full resolution of LFL by Baslingker et.~al.~[arXiv:2410.11836] cover only the GUE case which is determinantal. Lacking the possibility for a direct comparison with the Gaussian case, we develop a robust and natural method for both key parts of the proof. On one hand, we rely on a powerful martingale technique to describe precisely the strong correlation between the largest eigenvalue of an N× N Wigner matrix and its (N-k)× (N-k) minor if k≪ N^2/3. On the other hand, we use dynamical methods to show that this correlation is weak if k≫ N^2/3.

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• Numerical Radius of Non-Hermitian Random Matrices
• Bulk Universality for Sparse Complex non-Hermitian Random Matrices

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