Examples#
Runnable, self-contained scripts reproducing the key breakthroughs behind
physicskit.rmt, organized around Dyson’s threefold way: each script
reproduces a classic random matrix theory result to numerical precision
against its published closed-form law or reference (see
Breakthroughs in Random Matrix Theory for the full chronology).
Each script in this gallery is self-contained and can be run directly with
python examples/rmt/<section>/<script>.py. Every script also carries an
RST module docstring as its title/description and uses # %% markers to
split narrative text from code, which is exactly what Sphinx-Gallery renders
into the pages below – the script is the source of truth for what you see,
not a copy of it.
Sections#
edge_statistics – the Tracy-Widom soft-edge laws for the largest eigenvalue of the GOE, GUE, and GSE.
gaussian_ensembles – bulk spectra of Hermitian random matrices: Wigner’s semicircle law, the Two-Body Random Ensemble’s Gaussian (not semicircular) many-body density of states, Anderson localization in power-law banded matrices, the Altland-Zirnbauer Bogoliubov-de Gennes classes’ eigenvalue symmetry, and the Dumitriu-Edelman tridiagonal model that makes continuum-beta simulation possible at all.
non_hermitian – complex spectra: Ginibre’s circular law, the Feinberg-Zee single ring theorem, and Bender-Boettcher PT-symmetry breaking in pseudo-Hermitian ensembles.
spacings – local and long-range spectral correlations: the Wigner surmise, Dyson’s circular ensembles, the Keating-Snaith moments of the CUE characteristic polynomial, the Poisson (Berry-Tabor) integrable baseline, the Pandey-Mehta GOE-GUE crossover, the Bohigas-Giannoni-Schmit spectral rigidity and universality, and the Wigner-Dyson many-body level statistics of Kitaev’s SYK model.
wishart – covariance-type ensembles: Wishart’s original sample-covariance construction, the Marchenko-Pastur law, the Wachter law for Jacobi (MANOVA) ensembles, and Page’s conjecture for the average entanglement entropy of a random quantum state.
Edge statistics#
Fluctuations of the extreme eigenvalues: the Tracy-Widom soft-edge laws for the largest eigenvalue of the GOE, GUE, and GSE.
Gaussian ensembles#
Bulk spectra of Hermitian random matrices: Wigner’s semicircle law, the Two-Body Random Ensemble’s Gaussian (not semicircular) many-body density of states, Anderson localization in power-law banded matrices, the Altland-Zirnbauer Bogoliubov-de Gennes classes’ \(\pm E\) eigenvalue symmetry, and the Dumitriu-Edelman tridiagonal model for continuum \(\beta\).
The Altland-Zirnbauer Tenfold Way: BdG Eigenvalue Symmetry
Anderson Localization in Power-Law Banded Matrices
Embedded Ensembles: the Two-Body Random Ensemble Is Gaussian, not Semicircle
Non-Hermitian ensembles#
Complex spectra: Ginibre’s circular law, the Feinberg-Zee single ring theorem, and Bender-Boettcher PT-symmetry breaking in pseudo-Hermitian ensembles.
Bender-Boettcher PT-Symmetry Breaking in Pseudo-Hermitian Ensembles
Spacings and spectral correlations#
Level repulsion and its consequences: the Wigner surmise, Dyson’s circular ensembles and the sine kernel, the Keating-Snaith moments of the CUE characteristic polynomial, the Poisson (Berry-Tabor) integrable baseline, the Pandey-Mehta GOE-GUE crossover, Bohigas-Giannoni-Schmit spectral rigidity and universality, and the Wigner-Dyson many-body level statistics of Kitaev’s SYK model.
The Berry-Tabor Conjecture: Poisson Statistics for Integrable Spectra
The Bohigas-Giannoni-Schmit Conjecture: Spectral Rigidity and Universality
Dyson’s Threefold Way: the Circular Ensembles COE, CUE, CSE
Kitaev’s SYK Model: Majorana Fermions, Wigner-Dyson Statistics
The Riemann Zeta Connection: Keating-Snaith Moments of the CUE Characteristic Polynomial
Wishart and Jacobi ensembles#
Covariance-type ensembles: Wishart’s original sample-covariance construction, the Marchenko-Pastur law, the Wachter law for Jacobi (MANOVA) ensembles, and Page’s conjecture for the average entanglement entropy of a random quantum state.
Page’s Conjecture: the Average Entanglement Entropy of a Random State
