Universal objects of the infinite beta random matrix theory
Copyright policy )Universal objects of the infinite beta random matrix theory
We develop a theory of multilevel distributions of eigenvalues which complements Dyson’s threefold \beta=1,2,4 approach corresponding to real/complex/quaternion matrices by \beta=\infty point. Our central objects are the G \infty E ensemble, which is a counterpart of the classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and the 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 view on these objects. The probabilistic one treats them as partition functions of certain additive polymers collecting white noise. The integrable point of view expresses their distributions through the so-called associated Hermite polynomials and integrals of the Airy function. We also outline universal appearances of our ensembles as scaling limits.
- University of California, Berkeley United States
- University of Rennes 1 France
- UNIVERSITY OF WISCONSIN-MADISON United States
- Mathematics Research Institute of Rennes France
- University of Rennes 2 France
Composite material, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Eigenvalues, singular values, and eigenvectors, Random matrices (algebraic aspects), Matrix (chemical analysis), [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 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, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Relationship between string theory and quantum field theory, freezing, Quantum mechanics, random matrices, Quantum, 510, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Beta function (physics), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], beta ensembles, FOS: Mathematics, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Airy process, Mathematical Physics, Eigenvalues and eigenvectors, BETA (programming language), Algebra over a field, Physics, Probability (math.PR), Quantum gravity, Pure mathematics, 500, Random matrix, Mathematical Physics (math-ph), Computer science, Materials science, Programming language, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Random matrices (probabilistic aspects), Combinatorics, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Mathematics, Mathematics - Probability
Composite material, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Eigenvalues, singular values, and eigenvectors, Random matrices (algebraic aspects), Matrix (chemical analysis), [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], 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, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Relationship between string theory and quantum field theory, freezing, Quantum mechanics, random matrices, Quantum, 510, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Beta function (physics), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], beta ensembles, FOS: Mathematics, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Airy process, Mathematical Physics, Eigenvalues and eigenvectors, BETA (programming language), Algebra over a field, Physics, Probability (math.PR), Quantum gravity, Pure mathematics, 500, Random matrix, Mathematical Physics (math-ph), Computer science, Materials science, Programming language, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Random matrices (probabilistic aspects), Combinatorics, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Mathematics, Mathematics - Probability
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