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

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

.. _sphx_glr_api_gallery_rmt_paper_replications_tracy_widom_demo.py:


Tracy-Widom Soft-Edge Laws
==========================

For the Gaussian beta-ensembles GOE (:math:`\beta=1`), GUE
(:math:`\beta=2`), and GSE (:math:`\beta=4`), the bulk of the
:math:`n`-eigenvalue spectrum fills the semicircle :math:`[-2, 2]`, but
the largest eigenvalue :math:`\lambda_{\max}` fluctuates *above* the
edge :math:`2` on a much finer scale than the bulk spacing: rescaling

.. math::

    s = n^{2/3}(\lambda_{\max} - 2) \;\;(\beta=1,2), \qquad
    s = (2n)^{2/3}(\lambda_{\max} - 2) \;\;(\beta=4)

converges in distribution to the Tracy-Widom law :math:`F_\beta(s)`,
the universal "soft edge" statistic shared by a wide class of random
matrix and growth-process models.

All three laws are built from :math:`q(x)`, the Hastings-McLeod
solution of the Painleve II equation :math:`q''(x) = x q(x) + 2 q(x)^3`
with :math:`q(x) \sim \mathrm{Ai}(x)` as :math:`x \to +\infty`, via

.. math::

    F_2(s) = \exp\left(-\int_s^\infty (x-s)\, q(x)^2\, dx\right),

.. math::

    F_1(s) = \sqrt{F_2(s)}\,
    \exp\left(-\tfrac{1}{2}\int_s^\infty q(x)\, dx\right), \qquad
    F_4(s) = \sqrt{F_2(s)}\,
    \cosh\left(\tfrac{1}{2}\int_s^\infty q(x)\, dx\right).

This example draws :math:`\lambda_{\max}` from Monte Carlo GOE/GUE/GSE
samples, rescales it as above, and overlays the resulting histogram
against the corresponding :math:`F_1`, :math:`F_2`, or :math:`F_4`
density obtained by numerically integrating the Painleve II equation.

References:
C. A. Tracy, H. Widom, Commun. Math. Phys. 159 (1994) 151.
C. A. Tracy, H. Widom, Commun. Math. Phys. 177 (1996) 727.

Run:
    python examples/paper_replications/tracy_widom_demo.py

.. GENERATED FROM PYTHON SOURCE LINES 47-90



.. image-sg:: /api/gallery/rmt/paper_replications/images/sphx_glr_tracy_widom_demo_001.png
   :alt: Tracy-Widom soft edge -- N=500, 400 samples per ensemble, GOE (beta=1) KS=0.0414, GUE (beta=2) KS=0.0480, GSE (beta=4) KS=0.0486
   :srcset: /api/gallery/rmt/paper_replications/images/sphx_glr_tracy_widom_demo_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Saved tracy_widom_replication.png






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import physicskit.rmt as rmt

    N = 500
    N_SAMPLES = 400
    SEED = 2026

    fig, axes = plt.subplots(1, 3, figsize=(13, 4), sharey=True)
    s_grid = np.linspace(-5, 3, 400)

    for ax, (label, cls, beta) in zip(
        axes,
        [("GOE (beta=1)", rmt.ensembles.GOE, 1), ("GUE (beta=2)", rmt.ensembles.GUE, 2), ("GSE (beta=4)", rmt.ensembles.GSE, 4)],
        strict=True,
    ):
        ensemble = cls(n=N, seed=SEED)
        spectrum = ensemble.sample(n_samples=N_SAMPLES)
        benchmark = rmt.validation.TracyWidom(beta=beta)
        result = benchmark.validate(spectrum, seed=SEED)

        lam_max = rmt.stats.largest_eigenvalues(spectrum)
        scale = rmt.stats.tracy_widom_edge_scale(N, beta)
        edge_stat = scale * (lam_max - 2.0)

        ax.hist(edge_stat, bins=40, density=True, alpha=0.5, color="steelblue", label="empirical")
        theory_cdf = rmt.stats.tracy_widom_cdf(s_grid, beta)
        theory_pdf = np.gradient(theory_cdf, s_grid)
        tw_index = {1: 1, 2: 2, 4: 4}[beta]
        ax.plot(s_grid, theory_pdf, "k-", lw=2, label=f"Tracy-Widom F{tw_index}")
        ax.set_title(f"{label}\nKS={result.ks_statistic:.4f}")
        xlabel = r"$N^{2/3}(\lambda_{max} - 2)$" if beta != 4 else r"$(2N)^{2/3}(\lambda_{max} - 2)$"
        ax.set_xlabel(xlabel)
        ax.legend(fontsize=8)

    axes[0].set_ylabel("density")
    fig.suptitle(f"Tracy-Widom soft edge -- N={N}, {N_SAMPLES} samples per ensemble")
    fig.tight_layout()
    out_path = "tracy_widom_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 3.063 seconds)


.. _sphx_glr_download_api_gallery_rmt_paper_replications_tracy_widom_demo.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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