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.

Download (ext.)

Journal Article | Published | English

Scopus indexed
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.
All files available under the following license(s):
Copyright Statement:
This Item is protected by copyright and/or related rights. [...]

Link(s) to Main File(s)
Access Level
OA Open Access

Export

Marked Publications

Open Data ISTA Research Explorer

Sources

arXiv 2110.05147

Search this title in

Google Scholar