TRACY-WIDOM LIMIT FOR FREE SUM OF RANDOM MATRICES

• 成果类型: Article
• 署名作者: Ji, Hong chang; Park, Jaewhi
• 署名单位: Institute of Science & Technology - Austria; Korea Advanced Institute of Science & Technology (KAIST)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1705
• 发表日期: 2025
• 页码: 239-298
• 关键词: spectral statistics; LARGEST EIGENVALUE; UNIVERSALITY; THEOREM; edge

摘要

We consider fluctuations of the largest eigenvalues of the random matrix model A + UBU* where A and B are N x 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.