Tracy-Widom limit for free sum of random matrices Ji, Hong Chang Park, Jaewhi 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 Statistics 2025 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/19039 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> eng info:eu-repo/semantics/altIdentifier/doi/10.1214/24-aop1705 info:eu-repo/semantics/altIdentifier/issn/0091-1798 info:eu-repo/semantics/altIdentifier/wos/001407834700007 info:eu-repo/semantics/altIdentifier/arxiv/2110.05147 info:eu-repo/semantics/openAccess