2016/11/30 by Sasha Sodin
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics #Conjecture #Critical point (mathematics) #Determinantal point process #Eigenvalues and eigenvectors #Interval (graph theory) #Mathematical analysis #Mathematics #Point process #Polynomial #Pure mathematics #Quantum mechanics #Random matrix #Riemann Xi function #Riemann hypothesis #Statistics #Z function #math.NT #math.PR
paper · pdf · doi:10.1093/qmath/hax033
minor revision; v3: incorporated referee suggestions and updated ref-s. To appear in Q. J. Math
openalex created_date 2016/12/08 · arxiv created 2017/06/07 · openalex publication_date 2017/06/22 · arxiv updated 2017/08/18 · openalex updated_date 2026/08/05
A one-parameter family of point processes describing the distribution of the critical points of the characteristic polynomial of large random Hermitian matrices on the scale of mean spacing is investigated. Conditionally on the Riemann hypothesis and the multiple correlation conjecture, we show that one of these limiting processes also describes the distribution of the critical points of the Riemann ξ-function on the critical line. We prove that each of these processes boasts stronger level repulsion than the sine process describing the limiting statistics of the eigenvalues: the probability to find k critical points in a short interval is comparable to the probability to find k + 1 eigenvalues there. We also prove a similar property for the critical points and zeros of the Riemann ξ-function, conditionally on the Riemann hypothesis, but not on the multiple correlation conjecture.