Sums of GUE matrices and concentration of hives from correlation decay of eigengaps

• 成果类型: Article
• 署名作者: Narayanan, Hariharan; Sheffield, Scott; Tao, Terence
• 署名单位: Tata Institute of Fundamental Research (TIFR); Massachusetts Institute of Technology (MIT); University of California System; University of California Los Angeles
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01250-4
• 发表日期: 2024
• 页码: 1121-1165
• 关键词: honeycomb model; horns problem; PROOF

摘要

Associated to two given sequences of eigenvalues lambda(1) >= ... >=lambda(n) and mu(1) >= ... = mu(n) is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as n -> infinity. Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand-Tsetlin polytopes; known results by Klartag on theKLSconjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni-Erdos-Schroder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process.