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Gaussian unitary ensemble

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1: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7(i) Definition
Jacobi Form
Confluent Form
Integral Representation
2: 32.14 Combinatorics
The distribution function F ( s ) given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of n × n Hermitian matrices; see Tracy and Widom (1994). …
3: Bibliography F
  • B. D. Fried and S. D. Conte (1961) The Plasma Dispersion Function: The Hilbert Transform of the Gaussian. Academic Press, London-New York.
  • Y. V. Fyodorov (2005) Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.
  • 4: 4.44 Other Applications
    For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). For an application of the Lambert W -function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). …
    5: 26.21 Tables
    It also contains a table of Gaussian polynomials up to [ 12 6 ] q . …
    6: 35.10 Methods of Computation
    See Yan (1992) for the F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on 𝐎 ( m ) applied to a generalization of the integral (35.5.8). …
    7: 3.2 Linear Algebra
    §3.2(i) Gaussian Elimination
    Iterative Refinement
    §3.2(ii) Gaussian Elimination for a Tridiagonal Matrix
    8: 35.9 Applications
    §35.9 Applications
    9: 7.1 Special Notation
    The notations P ( z ) , Q ( z ) , and Φ ( z ) are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions.
    10: 26.9 Integer Partitions: Restricted Number and Part Size
    26.9.4 [ m n ] q = j = 1 n 1 q m n + j 1 q j , n 0 ,
    is the Gaussian polynomial (or q -binomial coefficient); see also §§17.2(i)17.2(ii). …
    26.9.5 n = 0 p k ( n ) q n = j = 1 k 1 1 q j = 1 + m = 1 [ k + m 1 m ] q q m ,