On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review

On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review (2010)

Cached

Download Links

[www-m3.ma.tum.de]

by F. Bornemann

Citations:

2 - 0 self

BibTeX

@MISC{Bornemann10onthe,
    author = {F. Bornemann},
    title = {On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review},
    year = {2010}
}

Bookmark

OpenURL

Abstract

Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) β-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.

Citations

400	Orthogonal Polynomials - SZEGŐ - 1959
337	Asymptotics and special functions - Olver - 1974
253	Numerical integration - Davis, Rabinowitz - 1967
219	Orthogonal Polynomials and Random Matrices: a Riemann– Hilbert Approach - Deift - 1999
181	Level-spacing distributions and the Airy kernel - Tracy, Widom - 1994
121	Statistical theory of energy levels of complex systems - Dyson - 1962
101	H.: On orthogonal and symplectic matrix ensembles - Tracy, Widom - 1996
94	Exactly solvable field theories of closed strings - Brezin, Kazakov - 1990
81	Mathematics by Experiment, Plausible Reasoning - Borwein, Bailey - 2004
74	Fredholm determinants, differential equations and 23 models - Tracy, Widom
71	The spectrum edge of random matrix ensembles - Forrester - 1993
70	Correlation functions, cluster functions, and spacing distributions for random matrices - Tracy, Widom - 1998
67	On the distribution of spacings between the zeros of the zeta function - Odlyzko - 1987
66	Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica 1D - Jimbo, Miwa, et al. - 1980
55	H.: Scale invariance of the PNG droplet and the Airy process - Prähofer, Spohn - 2002
49	Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region,” Physical Review B13 - Wu, McCoy, et al. - 1976
47	A boundary value problem associated with the second Painleve transcendent and the Korteweg de Vries equation - Hastings, McLeod - 1980
40	Trace Ideals and Their - Simon - 1979
38	Barycentric lagrange interpolation - Berrut, Trefethen
36	An extension of Matlab to continuous functions and operators - Battles, Trefethen
26	The numerical stability of barycentric Lagrange interpolation - Higham
26	Matrix models of 2D gravity and isomonodromic - Moore - 1991
24	F: Scientific Computing with Ordinary Differential Equations - Deuflhard, Bornemann - 2002
23	Application of the τ-function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits - Forrester, Witte
22	Introduction to Random Matrices - Tracy, Widom - 1992
21	Integrable operators, in Differential Operators and Spectral Theory: M. Sh. Birman’s 70th anniversary collection - Deift - 1999
21	Random matrix theory - Edelman, Rao - 2005
19	Level spacing distributions and the Bessel kernel - Tracy, Widom - 1992
18	Traces and Determinants of Linear Operators - Gohberg, Goldberg, et al. - 2000
18	Lagrangian interpolation at the Chebyshev points xn,ν = cos(νπ/n), ν = 0(1)n; some unnoted advantages - Salzer - 1972
17	Universality of the distribution functions of random matrix theory, preprint - Tracy, Widom
14	Exact scaling functions for one-dimensional stationary KPZ growth - Praehofer, Spohn - 2004
13	Universality for mathematical and physical systems - Deift - 2007
13	la loi limite de l’espacement des valeurs propres d’une matrice aléatorie, Nucl - Gaudin, Sur - 1961
12	Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations - Dieng - 2005
12	Inter-relationships between orthogonal, unitary and symplectic matrix ensembles - Forrester, Rains
11	Matrix Algorithms. Vol. I: Basic Decompositions - Stewart - 1998
9	On the numerical evaluation of Fredholm determinants - Bornemann
9	Matrix kernels for the Gaussian orthogonal and symplectic ensembles - Tracy, Widom - 1994
9	On asymptotics for the Airy process - Widom - 2004
8	Asymptotics of Level-Spacing Distributions for Random Matrices - Basor, Tracy, et al. - 1992
8	Increasing subsequences and the hard-to-soft edge transition in matrix ensembles - Borodin, Forrester
7	Painlevé equations, nonlinear special functions - Clarkson - 2001
4	The chebop system for automatic solution of differential equations - Driscoll, Bornemann, et al.
3	M.: The Airy1 Process is not the Limit of the Largest Eigenvalue - Ferrari, Prähofer - 2008
3	Accuracy of the Semicircle Approximation for the Density of Eigenvalues of Random - Bronk - 1964
3	2006) Relationships between τ-functions and Fredholm determinant expressions for gap probabilities in random matrix theory - Desrosiers, Forrester
3	Numerical Methods for Eigenvalue Distributions of Random Matrices - Edelman, Persson
3	Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients - Stratton, Morse, et al. - 1956
2	2010) Asymptotic independence of the extreme eigenvalues of GUE - Bornemann