# Eigenvalue Unfolding in Spacing Distributions of Random Matrices

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This Demonstration shows the effects of the unfolding procedure adopted in random matrix theory for rescaling eigenvalues of random Hermitian matrices before computing the nearest-neighbor spacing distribution of the matrix eigenvalues [1]. The purpose of unfolding is to separate large-scale variations in the eigenvalue spacings from local fluctuations. This can be achieved only if the local average varies on scales larger than the single-spacing scale. To achieve this goal, the average staircase function needs to be computed for the average density of eigenvalues . Then this staircase function is evaluated at each of the corresponding eigenvalues, hence . The obtained sequence of unfolded eigenvalues exhibits average spacing equal to one and uniform density distribution. It is therefore straightforward to verify that the nearest-neighbor spacing distribution follows the Wigner surmise function for the Gaussian orthogonal ensemble (GOE) distribution.

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Contributed by: Jessica Alfonsi (June 2021)

(Padova, Italy)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: default setting with matrix size

Snapshot 2: same as Snapshot 1 but with unfolded eigenvalues

Snapshot 3: unfolded eigenvalues for matrix size

References

[1] T. Timberlake, "Random Numbers and Random Matrices: Quantum Chaos Meets Number Theory," *American Journal of Physics*, 74(6), 2006 pp. 547–553. doi:10.1119/1.2198883.

[2] T. Timberlake. "Random Numbers and Random Matrices." (May 27, 2021) sites.berry.edu/ttimberlake/research/random-numbers-and-random-matrices.

## Permanent Citation