Numerical Simulation of GUE Two-Point Correlation and Cluster Functions

TL;DR

This paper presents numerical simulations of GUE two-point correlation functions, comparing them with analytical formulas, and investigates how simulation parameters affect the reproduction of oscillatory and non-oscillatory behaviors.

Contribution

It demonstrates how numerical simulations can accurately reproduce analytical GUE correlation functions and explores the impact of delta function width on these results.

Findings

Simulations reproduce oscillations when delta function width is small.

Non-oscillating behavior emerges as delta function width increases.

Comparison validates analytical formulas against numerical results.

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.