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

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

.. _sphx_glr_api_gallery_rmt_paper_replications_marchenko_pastur_demo.py:


The Marchenko-Pastur Law
========================

The Wishart (Laguerre) ensembles LOE (:math:`\beta=1`), LUE
(:math:`\beta=2`), and LSE (:math:`\beta=4`) are built from the sample
covariance matrix :math:`W = X X^\dagger` of an :math:`n \times m`
Gaussian data matrix :math:`X` (:math:`n` variables, :math:`m` samples,
:math:`m \geq n`), with real, complex, or quaternionic entries
respectively. The relevant control parameter is the aspect ratio
:math:`\gamma = n/m \in (0, 1]`.

As :math:`n, m \to \infty` with :math:`\gamma` fixed, the empirical
spectral density of :math:`W/m` converges to the Marchenko-Pastur law,
supported on :math:`[\lambda_-, \lambda_+]` with

.. math::

    \lambda_\pm = \sigma^2\left(1 \pm \sqrt{\gamma}\right)^2,

and density

.. math::

    p(x) = \frac{\sqrt{(\lambda_+ - x)(x - \lambda_-)}}
    {2\pi\,\gamma\,\sigma^2\, x}, \qquad \lambda_- < x < \lambda_+,

where :math:`\sigma^2` is the entry variance (here :math:`\sigma^2=1`).
This example reproduces that law for LOE, LUE, and LSE across several
values of :math:`\gamma`.

Reference:
V. A. Marchenko, L. A. Pastur, "Distribution of eigenvalues for some
sets of random matrices", Mat. Sb. 72 (1967) 507.

Run:
    python examples/paper_replications/marchenko_pastur_demo.py

.. GENERATED FROM PYTHON SOURCE LINES 39-81



.. image-sg:: /api/gallery/rmt/paper_replications/images/sphx_glr_marchenko_pastur_demo_001.png
   :alt: Marchenko-Pastur law across aspect ratios -- n=600, 30 samples per ensemble, gamma = n/m = 0.1, gamma = n/m = 0.4, gamma = n/m = 0.9
   :srcset: /api/gallery/rmt/paper_replications/images/sphx_glr_marchenko_pastur_demo_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Saved marchenko_pastur_replication.png






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import physicskit.rmt as rmt

    N = 600
    N_SAMPLES = 30
    SEED = 2026
    GAMMAS = [0.1, 0.4, 0.9]

    fig, axes = plt.subplots(1, len(GAMMAS), figsize=(13, 4), sharey=False)

    for ax, gamma in zip(axes, GAMMAS, strict=True):
        m = int(N / gamma)
        benchmark = rmt.validation.MarchenkoPastur(gamma=gamma)
        lo, hi = rmt.stats.mp_support(gamma)
        x = np.linspace(lo, hi, 400)

        for cls, color in [
            (rmt.ensembles.LOE, "steelblue"),
            (rmt.ensembles.LUE, "indianred"),
            (rmt.ensembles.LSE, "seagreen"),
        ]:
            ensemble = cls(n=N, m=m, seed=SEED)
            spectrum = ensemble.sample(n_samples=N_SAMPLES)
            centers, counts = rmt.stats.empirical_density(spectrum, bins=60)
            ax.plot(centers, counts, color=color, alpha=0.7, lw=1.5, label=cls.__name__)

        ax.plot(x, benchmark.theoretical_pdf(x), "k--", lw=2, label="Marchenko-Pastur")
        ax.set_title(f"gamma = n/m = {gamma}")
        ax.set_xlabel("eigenvalue")
        ax.legend(fontsize=8)

    axes[0].set_ylabel("density")
    fig.suptitle(
        f"Marchenko-Pastur law across aspect ratios -- n={N}, {N_SAMPLES} samples per ensemble",
    )
    fig.tight_layout()
    out_path = "marchenko_pastur_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 3.075 seconds)


.. _sphx_glr_download_api_gallery_rmt_paper_replications_marchenko_pastur_demo.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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