Numerical Simulation of GUE Two-Point Correlation and Cluster Functions

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.