article published yes Hong Chang Ji author dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d Jaewhi Park author LaEr department Random matrices beyond Wigner-Dyson-Mehta project 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. Institute of Mathematical Statistics2025 eng 0091-1798 2110.05147 00140783470000710.1214/24-aop1705 531239 - 298 Ji, Hong Chang, and Jaewhi Park. “Tracy-Widom Limit for Free Sum of Random Matrices.” <i>The Annals of Probability</i>, vol. 53, no. 1, Institute of Mathematical Statistics, 2025, pp. 239–98, doi:<a href="https://doi.org/10.1214/24-aop1705">10.1214/24-aop1705</a>. Ji HC, Park J. Tracy-Widom limit for free sum of random matrices. <i>The Annals of Probability</i>. 2025;53(1):239-298. doi:<a href="https://doi.org/10.1214/24-aop1705">10.1214/24-aop1705</a> H. C. Ji and J. Park, “Tracy-Widom limit for free sum of random matrices,” <i>The Annals of Probability</i>, vol. 53, no. 1. Institute of Mathematical Statistics, pp. 239–298, 2025. Ji, H. C., &#38; Park, J. (2025). Tracy-Widom limit for free sum of random matrices. <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/24-aop1705">https://doi.org/10.1214/24-aop1705</a> Ji, Hong Chang, and Jaewhi Park. “Tracy-Widom Limit for Free Sum of Random Matrices.” <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2025. <a href="https://doi.org/10.1214/24-aop1705">https://doi.org/10.1214/24-aop1705</a>. Ji HC, Park J. 2025. Tracy-Widom limit for free sum of random matrices. The Annals of Probability. 53(1), 239–298. H.C. Ji, J. Park, The Annals of Probability 53 (2025) 239–298. 190392025-02-17T09:32:16Z2025-09-30T10:32:51Z