.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/rmt/paper_replications/wigner_semicircle_demo.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_rmt_paper_replications_wigner_semicircle_demo.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_rmt_paper_replications_wigner_semicircle_demo.py:


Wigner's Semicircle Law
=======================

GOE (:math:`\beta=1`), GUE (:math:`\beta=2`), and GSE (:math:`\beta=4`)
are the Gaussian random matrix ensembles built from real symmetric,
complex Hermitian, and quaternionic self-dual Hermitian matrices
respectively, with independent Gaussian entries. After rescaling
eigenvalues by :math:`\sqrt{n\beta}` (:math:`n` the matrix size), the
empirical spectral density of all three ensembles converges, as
:math:`n \to \infty`, to Wigner's semicircle law on
:math:`[-R, R]` (:math:`R=2` here):

.. math::

    p(x) = \frac{2}{\pi R^2}\sqrt{R^2 - x^2}, \qquad |x| \leq R.

This example reproduces that law -- the same limiting curve for all
three values of :math:`\beta` -- by histogramming Monte Carlo GOE/GUE/GSE
spectra and overlaying the theoretical semicircle.

Reference:
E. Wigner, Ann. Math. 62 (1955) 548; Ann. Math. 67 (1958) 325.

Run:
    python examples/paper_replications/wigner_semicircle_demo.py

.. GENERATED FROM PYTHON SOURCE LINES 28-68



.. image-sg:: /api/gallery/rmt/paper_replications/images/sphx_glr_wigner_semicircle_demo_001.png
   :alt: Wigner semicircle law -- N=800, 40 independent samples per ensemble, GOE (beta=1) KS=0.0007, GUE (beta=2) KS=0.0006, GSE (beta=4) KS=0.0006
   :srcset: /api/gallery/rmt/paper_replications/images/sphx_glr_wigner_semicircle_demo_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Saved wigner_semicircle_replication.png






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import physicskit.rmt as rmt

    ENSEMBLES = [
        ("GOE (beta=1)", rmt.ensembles.GOE),
        ("GUE (beta=2)", rmt.ensembles.GUE),
        ("GSE (beta=4)", rmt.ensembles.GSE),
    ]

    N = 800
    N_SAMPLES = 40
    SEED = 2026

    fig, axes = plt.subplots(1, 3, figsize=(13, 4), sharey=True)
    x = np.linspace(-2.2, 2.2, 500)
    benchmark = rmt.validation.WignerSemicircle()

    for ax, (label, cls) in zip(axes, ENSEMBLES, strict=True):
        ensemble = cls(n=N, seed=SEED)
        spectrum = ensemble.sample(n_samples=N_SAMPLES)
        result = benchmark.validate(spectrum, seed=SEED)

        centers, counts = rmt.stats.empirical_density(spectrum, bins=80)
        ax.bar(centers, counts, width=centers[1] - centers[0], alpha=0.6, label="empirical ESD", color="steelblue")
        ax.plot(x, benchmark.theoretical_pdf(x), "k-", lw=2, label="Wigner semicircle")
        ax.set_title(f"{label}\nKS={result.ks_statistic:.4f}")
        ax.set_xlabel(r"$\lambda / \sqrt{N\beta}$")
        ax.legend(fontsize=8)

    axes[0].set_ylabel("density")
    fig.suptitle(
        f"Wigner semicircle law -- N={N}, {N_SAMPLES} independent samples per ensemble",
    )
    fig.tight_layout()
    out_path = "wigner_semicircle_replication.png"
    fig.savefig(out_path, dpi=150)
    print(f"Saved {out_path}")


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 1.073 seconds)


.. _sphx_glr_download_api_gallery_rmt_paper_replications_wigner_semicircle_demo.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: wigner_semicircle_demo.ipynb <wigner_semicircle_demo.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: wigner_semicircle_demo.py <wigner_semicircle_demo.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: wigner_semicircle_demo.zip <wigner_semicircle_demo.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_