.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/rmt/paper_replications/circular_law_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_circular_law_demo.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_rmt_paper_replications_circular_law_demo.py:


Ginibre's Circular Law
======================

The Ginibre ensembles GinOE (real), GinUE (complex), and GinSE
(quaternion) drop the Hermitian/unitary symmetry constraint entirely:
their :math:`n \times n` matrices have i.i.d. Gaussian entries with no
further structure, so eigenvalues are genuinely complex, spread over a
two-dimensional region rather than confined to the real line or a
circle. After rescaling by :math:`\sqrt{n}` (Ginibre's exact result,
not just asymptotic), the eigenvalues fill the unit disk uniformly as
:math:`n \to \infty` -- the circular law. Because a uniform density on
a disk is awkward to validate directly, this example instead checks the
radial marginal: integrating out the angle, the magnitude
:math:`r = |\lambda|` has the simple closed-form density

.. math::

    f(r) = 2r, \qquad 0 \leq r \leq 1

(equivalently :math:`F(r) = r^2`), against which the empirical
eigenvalue radii from GinOE/GinUE/GinSE are compared via a
Kolmogorov-Smirnov test. GinOE and GinSE eigenvalues are not confined
to the disk's generic bulk in exactly the same way as GinUE (GinOE
mixes real eigenvalues with complex-conjugate pairs; GinSE's
:math:`2n` eigenvalues come in genuinely distinct conjugate pairs, not
Kramers-degenerate ones), but all three share the same limiting radial
law. This example reproduces Ginibre's circular law for GinOE, GinUE,
GinSE: eigenvalue scatter in the complex plane against the exact
unit-disk boundary, plus the radial density check used for quantitative
validation.

Reference: J. Ginibre, J. Math. Phys. 6 (1965) 440.

Run:
    python examples/paper_replications/circular_law_demo.py

.. GENERATED FROM PYTHON SOURCE LINES 38-85



.. image-sg:: /api/gallery/rmt/paper_replications/images/sphx_glr_circular_law_demo_001.png
   :alt: Ginibre's circular law -- N=400, GinOE (real), GinUE (complex), GinSE (quaternion), radial density KS=0.0216, radial density KS=0.0202, radial density KS=0.0157
   :srcset: /api/gallery/rmt/paper_replications/images/sphx_glr_circular_law_demo_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Saved circular_law_replication.png






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import physicskit.rmt as rmt

    N = 400
    SEED = 2026

    fig, axes = plt.subplots(2, 3, figsize=(13, 8))
    theta_circle = np.linspace(0, 2 * np.pi, 300)
    r_grid = np.linspace(0, 1.2, 300)

    for col, (label, cls) in enumerate(
        [("GinOE (real)", rmt.ensembles.GinOE), ("GinUE (complex)", rmt.ensembles.GinUE), ("GinSE (quaternion)", rmt.ensembles.GinSE)]
    ):
        ensemble = cls(n=N, seed=SEED)
        spectrum = ensemble.sample(n_samples=1)
        eigs = spectrum.rescaled[0]

        ax = axes[0, col]
        ax.scatter(eigs.real, eigs.imag, s=4, alpha=0.6, color="darkorange")
        ax.plot(np.cos(theta_circle), np.sin(theta_circle), "k-", lw=1.5)
        ax.set_title(label)
        ax.set_aspect("equal")
        ax.set_xlabel("Re")

        # radial density check (pooled over more samples for a clean curve)
        spectrum_many = ensemble.sample(n_samples=20)
        radii = np.abs(spectrum_many.rescaled.ravel())
        benchmark = rmt.validation.CircularLaw()
        result = benchmark.validate(spectrum_many, seed=SEED)

        ax2 = axes[1, col]
        ax2.hist(radii, bins=50, density=True, alpha=0.5, color="steelblue", label="empirical")
        ax2.plot(r_grid, benchmark.theoretical_pdf(r_grid), "k-", lw=2, label="f(r)=2r")
        ax2.set_title(f"radial density\nKS={result.ks_statistic:.4f}")
        ax2.set_xlabel("|eigenvalue|")
        ax2.legend(fontsize=8)

    axes[0, 0].set_ylabel("Im")
    axes[1, 0].set_ylabel("density")
    fig.suptitle(f"Ginibre's circular law -- N={N}")
    fig.tight_layout()
    out_path = "circular_law_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 25.336 seconds)


.. _sphx_glr_download_api_gallery_rmt_paper_replications_circular_law_demo.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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