Universal objects of the infinite beta random matrix theory

We develop a theory of multilevel distributions of eigenvalues which complements the Dyson's threefold $\beta=1,2,4$ approach corresponding to real/complex/quaternion matrices by $\beta=\infty$ point. Our central objects are G$\infty$E ensemble, which is a counterpart of classical Gaussian Orth...

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Vydané v:Journal of the European Mathematical Society : JEMS
Hlavní autori: Gorin, Vadim, Kleptsyn, Victor
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: European Mathematical Society 2022
Predmet:
Dynamical Systems
Mathematical Physics
Mathematics
Probability
Quantum mechanics, Quantum gravity, Computer science, Relationship between string theory and quantum field theory, Composite material, Quantum, Programming language, Mathematics, Random matrix, BETA (programming language), Matrix (chemical analysis), Pure mathematics, Beta function (physics), Algebra over a field, Combinatorics, Eigenvalues and eigenvectors, Physics, Materials science
ISSN:1435-9855, 1435-9863
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Popis
Shrnutí:We develop a theory of multilevel distributions of eigenvalues which complements the Dyson's threefold $\beta=1,2,4$ approach corresponding to real/complex/quaternion matrices by $\beta=\infty$ point. Our central objects are G$\infty$E ensemble, which is a counterpart of classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and Airy$_{\infty}$ line ensemble, which is a collection of continuous curves serving as a scaling limit for largest eigenvalues at $\beta=\infty$. 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.
ISSN:1435-9855
1435-9863