Ensemble Statistics for Periodic and Random Level Systems

This Demonstration elaborates upon a very popular paper on random matrix theory [1]. This introduces a few statistical results on periodic and random level systems obtained from sampling values from different distributions: uniform, Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE). Results from the pre-computed zeros of the Riemann zeta function are also shown in relation to the statistical measurements known for the GUE (see [2–4] and the Wolfram Documentation).
In this Demonstration, the distribution of gaps between consecutive levels and the distribution of the two-point correlation function are represented by histograms; the latter measures the number of pairs of levels separated by any given distance . The computed data points in the related histograms are also plotted against some relevant analytical functions for comparison. Random systems with level values sampled from the uniform distribution in range closely follow an exponential gap distribution (see [5]), whereas jiggled level systems obtained from eigenvalues of GOE and GUE distributed matrices follow the corresponding Wigner surmise functions for their gap distribution (see [6] and references therein), which resemble Gaussian functions [1]. The corresponding two-point correlation function for the jiggled system from the GUE distribution closely follows the analytical function known from Montgomery's conjecture for the pair correlation function (see Thumbnail) which also fits very well the interpair distance distribution for the Riemann zeta zeros [2–4]. A very similar result can be found for the GOE system. All coding details can be obtained by inspecting the source code.
After setting the level system, you can plot the distribution for either the gap or the two-point correlation function. You can also tick the checkbox "show pictorial level system" to picture the variety of level systems considered in RMT-based computations. The setter "number of levels" lets you increase the numbers of levels to improve agreement between statistical data points and the corresponding fitting functions.

SNAPSHOTS

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DETAILS

Snapshot 1: periodic systems with uniform level spacing exhibit a single point in their histogram gap distribution at
Snapshot 2: same system as in Snapshot 1 with histogram distribution of the two-point correlation function; points are found at the corresponding unit distances
Snapshot 3: random level systems with levels sampled from uniform distribution function follow an exponential law in their histogram gap distribution
Snapshot 4: same system as in Snapshot 3 with histogram distribution of the two-point correlation function for interpair normalized distances following the Wigner surmise for GUE ensemble in a very crude approximation
Snapshot 5: jiggled systems with random level spacings as obtained from rescaled eigenvalues of the GOE distribution follow the GOE Wigner surmise in their histogram gap distribution
Snapshot 6: same system as in Snapshot 5 with histogram distribution of the two-point correlation function for interpair rescaled distances following the Montgomery conjecture in a very crude approximation
Snapshot 7: jiggled systems with random level spacings as obtained from rescaled eigenvalues of the GUE distribution follow the GUE Wigner surmise in their histogram gap distribution
Snapshot 8: computed zeros of the Riemann zeta function follow the GUE Wigner surmise in their histogram gap distribution as in Snapshot 7
Snapshot 9: computed zeros of the Riemann zeta function in their histogram distribution of the two-point correlation function for interpair rescaled distances follow very closely the analytical function from Montgomery's conjecture as it occurs with two-point correlation function of rescaled eigenvalues of the GUE distribution in the Thumbnail
References
[1] B. Hayes, "The Spectrum of Riemannium," American Scientist, 91(4), 2003 pp. 296–300. www.americanscientist.org/article/the-spectrum-of-riemannium.
[2] A. Odlyzko, "On the Distribution of Spacings between Zeros of the Zeta Function," Mathematics of Computation, 48(177), 1987 pp. 273–308. doi:10.2307/2007890. Data archives of computed values: www.dtc.umn.edu/~odlyZko/zeta_tables/index.html.
[3] Wikipedia. "Montgomery's Pair Correlation Conjecture." (May 11, 2022) en.wikipedia.org/wiki/Montgomery%27s_pair_correlation_conjecture.
[4] "Calculating the Density of Nearest Neighbours." (May 11, 2022) mathematica.stackexchange.com/questions/49172/calculating-the-density-of-nearest-neighbours.
[5] J. Alfonsi. "Spacing Distribution of Random Numbers" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/SpacingDistributionOfRandomNumbers.
[6] J. Alfonsi. "Eigenvalue Unfolding in Spacing Distributions of Random Matrices" demonstrations.wolfram.com/EigenvalueUnfoldingInSpacingDistributionsOfRandomMatrices.
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