Numerical Simulation of GUE Two-Point Correlation and Cluster Functions
- PDF / 7,518,787 Bytes
- 8 Pages / 595.224 x 790.955 pts Page_size
- 78 Downloads / 210 Views
NUCLEAR PHYSICS
Numerical Simulation of GUE Two-Point Correlation and Cluster Functions Adam James Sargeant1 Received: 27 August 2020 / Accepted: 15 September 2020 © Sociedade Brasileira de F´ısica 2020
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. Keywords Random matrix theory · Gaussian unitary ensemble · Correlation functions · Cluster functions
1 Introduction Random matrices were introduced by Wishart in multivariate statistics in the 1920s [1] and Wigner in the study of neutron resonances in the 1950s [2]. Since then random matrix theory has found further applications in nuclear physics [3–8] as well as in quantum and wave chaos [9, 10], quantum chromodynamics [11], mesoscopic physics [12, 13], quantum gravity [14], numerical computation [15], number theory [16–18] and complex systems [19]. The main objects of analytical studies of correlations of the eigenvalues of random matrices are typically the correlation functions themselves, in particular the two-point correlation function, while the main objects of numerical studies are typically derivative correlation measures such as spacing distributions and the number variance, which are more convenient numerically and are simpler to interpret visually [10, 20, 21]. There is however insight to be gained from numerical simulations of the correlation functions Adam James Sargeant
[email protected] 1
Departamento de Ciˆencias Exatas e Aplicadas, Instituto de Ciˆencias Exatas e Aplicadas, Universidade Federal de Ouro Preto, Rua Trinta e Seis, 115, Loanda, Jo˜ao Monlevade, Minas Gerais, 35931-008, Brazil
themselves and in this paper we numerically calculate the GUE two-point correlation and cluster functions directly from their definitions in terms of delta functions and compare the numerical calculations with some known analytical formulas. The paper is organised as follows: In Section 2 the GUE is defined and the level density is discussed. In Section 3 the correlation and cluster functions are defined and some known analytical results are listed. In Section 4 numerical simulations of the correlation and cluster functions are presented and compared with the analytical formulas listed in Section 3. In Se
Data Loading...
