Tracy-Widom limit for free sum of random matrices
Ji HC, Park J. 2025. Tracy-Widom limit for free sum of random matrices. The Annals of Probability. 53(1), 239–298.
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Author
Ji, Hong ChangISTA;
Park, Jaewhi
Corresponding author has ISTA affiliation
Department
Abstract
We consider fluctuations of the largest eigenvalues of the random matrix model A + UBU∗ where A and B are N × N deterministic Hermitian (or symmetric) matrices and U is a Haar-distributed unitary (or orthogonal) matrix. We prove that the largest eigenvalue weakly converges to the GUE (or GOE) Tracy–Widom distribution, under mild assumptions on A and B to
guarantee that the density of states of the model decays as square root around
the upper edge. Our proof is based on the comparison of the Green function
along the Dyson Brownian motion starting from the matrix A + UBU∗ and
ending at time N−1/3+o(1). As a byproduct of our proof, we also prove an
optimal local law for the Dyson Brownian motion up to the constant time
scale.
Publishing Year
Date Published
2025-01-19
Journal Title
The Annals of Probability
Publisher
Institute of Mathematical Statistics
Acknowledgement
The work of H.C. Ji was partially supported by ERC Advanced Grant “RMTBeyond” No. 101020331. The work of J. Park was partially supported by National Research Foundation of Korea under grant number NRF-2019R1A5A1028324. The authors would like to thank Ji Oon Lee for helpful discussions.
Volume
53
Issue
1
Page
239 - 298
ISSN
IST-REx-ID
Cite this
Ji HC, Park J. Tracy-Widom limit for free sum of random matrices. The Annals of Probability. 2025;53(1):239-298. doi:10.1214/24-aop1705
Ji, H. C., & Park, J. (2025). Tracy-Widom limit for free sum of random matrices. The Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/24-aop1705
Ji, Hong Chang, and Jaewhi Park. “Tracy-Widom Limit for Free Sum of Random Matrices.” The Annals of Probability. Institute of Mathematical Statistics, 2025. https://doi.org/10.1214/24-aop1705.
H. C. Ji and J. Park, “Tracy-Widom limit for free sum of random matrices,” The Annals of Probability, vol. 53, no. 1. Institute of Mathematical Statistics, pp. 239–298, 2025.
Ji HC, Park J. 2025. Tracy-Widom limit for free sum of random matrices. The Annals of Probability. 53(1), 239–298.
Ji, Hong Chang, and Jaewhi Park. “Tracy-Widom Limit for Free Sum of Random Matrices.” The Annals of Probability, vol. 53, no. 1, Institute of Mathematical Statistics, 2025, pp. 239–98, doi:10.1214/24-aop1705.
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