2000/10/11 by Alexei Borodin, Grigori Olshanski · 1 citation
Mathematics · Physics and Astronomy · #Circular ensemble #Determinantal point process #Eigenvalues and eigenvectors #Ergodic theory #Geometry #Hermitian matrix #Invariant (physics) #Kernel (algebra) #Mathematical analysis #Mathematics #Measure (data warehouse) #Physics #Point process #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Sine #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #math.RT
paper · pdf · doi:10.1007/s002200100529
published as Comm. Math. Phys. 223 (2001), no. 1, 87--123 · 36 pages
arxiv created 2000/10/11 · openalex publication_date 2001/09/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into the well-known sine kernel which describes the local correlation in Circular and Gaussian Unitary Ensembles. Thus, the random point configuration of the sine process is interpreted as the random set of ``eigenvalues'' of infinite Hermitian matrices distributed according to the corresponding measure.