2020/09/04 by Vadim Gorin, Victor Kleptsyn, Gorin, Vadim +1 · 5 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #BETA (programming language) #Beta function (physics) #Computer science #Determinantal point process #Eigenvalues and eigenvectors #FOS: Mathematics #FOS: Physical sciences #Gaussian #Geometry #Hermite polynomials #Integrable system #Materials science #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Matrix (chemical analysis) #Orthogonal polynomials #Partition function (quantum field theory) #Physics #Probability (math.PR) #Pure mathematics #Quantum mechanics #Quaternion #Random Matrices and Applications #Random matrix #Real line #Relationship between string theory and quantum field theory #Scaling #Scaling limit #Symplectic geometry #math-ph #math.MP #math.PR
paper · pdf · open access · doi:10.48550/arxiv.2009.02006
published in arXiv (Cornell University) (Cornell University) · 57 pages. v3: clarifications and simulations added; to appear in Journal of European Mathematical Society
openalex publication_date 2020/09/04 · arxiv created 2021/12/28 · arxiv updated 2021/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a theory of multilevel distributions of eigenvalues which complements the Dyson's threefold β=1,2,4 approach corresponding to real/complex/quaternion matrices by β=∞ point. Our central objects are G∞E ensemble, which is a counterpart of classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and Airy∞ line ensemble, which is a collection of continuous curves serving as a scaling limit for largest eigenvalues at β=∞. We develop two points of views on these objects. Probabilistic one treats them as partition functions of certain additive polymers collecting white noise. Integrable point of view expresses their distributions through the so-called associated Hermite polynomials and integrals of Airy function. We also outline universal appearances of our ensembles as scaling limits.