# Ensemble Statistics for Periodic and Random Level Systems

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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).

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Contributed by: Jessica Alfonsi (August 2022)

(Padova, Italy)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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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