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Numerical Simulation of GUE Two-Point Correlation and Cluster Functions

  • Nuclear Physics
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Abstract

Numerical simulations of the two-point eigenvalue correlation and cluster functions of the Gaussian unitary ensemble (GUE) are carried out directly from their definitions in terms of deltas functions. The simulations are compared with analytical results which follow from three analytical formulas for the two-point GUE cluster function: (i) Wigner’s exact formula in terms of Hermite polynomials, (ii) Brezin and Zee’s approximate formula which is valid for points with small enough separations and (iii) French, Mello and Pandey’s approximate formula which is valid on average for points with large enough separations. It is found that the oscillations present in formulas (i) and (ii) are reproduced by the numerical simulations if the width of the function used to represent the delta function is small enough and that the non-oscillating behaviour of formula (iii) is approached as the width is increased.

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Acknowledgments

I would like to acknowledge happy collaborations with Mahir Saleh Hussein and Mauricio Porto Pato which involved random matrices in nuclear physics and were the precursor to this work.

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  1. Departamento de Ciências Exatas e Aplicadas, Instituto de Ciências Exatas e Aplicadas, Universidade Federal de Ouro Preto, Rua Trinta e Seis, 115, Loanda, João Monlevade, Minas Gerais, 35931-008, Brazil

    Adam James Sargeant

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Appendix

Appendix

In this appendix we offer some considerations on the coding of the ensemble average in (14) for the two-point correlation function. Let us denote the ith eigenvalue of the kth realisation of the ensemble by Eik and define \(N\times k_{\max \limits }\) matrices X and Y whose elements are δ(Ex − Eik) and δ(Ey − Eik) respectively. Then the two-point correlation function R of energies Ex and Ey may be conveniently coded in Octave or MATLAB by

$$ \begin{array}{@{}rcl@{}} &&\mathtt{D = X * Y^{\prime}} \end{array} $$
(A.1)
$$ \begin{array}{@{}rcl@{}} &&\mathtt{R = (sum(D(:)) - trace(D))/kmax} \end{array} $$
(A.2)

where the N × N matrix D is the outer product of X and Y' (the transpose of Y) and sum(D(:)) is the grand sum of its elements.

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Sargeant, A.J. Numerical Simulation of GUE Two-Point Correlation and Cluster Functions. Braz J Phys 51, 308–315 (2021). https://doi.org/10.1007/s13538-020-00802-6

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