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

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

.. _sphx_glr_api_gallery_rmt_paper_replications_goe_gue_crossover_demo.py:


The Pandey-Mehta GOE-GUE Crossover
==================================

The Pandey-Mehta crossover ensemble interpolates continuously between
time-reversal-symmetric (GOE, :math:`\beta=1`) and time-reversal-broken
(GUE, :math:`\beta=2`) level statistics via a single real parameter
:math:`\lambda`:

.. math::

    H(\lambda) = A + i\lambda B,

where :math:`A` is an independent real symmetric (GOE-type) matrix and
:math:`B` is an independent real *antisymmetric* matrix, so that
:math:`i\lambda B` is Hermitian and :math:`H(\lambda)` is Hermitian for
any real :math:`\lambda`. At :math:`\lambda=0`, :math:`H = A` recovers
GOE exactly; as :math:`\lambda` grows, the antisymmetric imaginary part
breaks time-reversal symmetry, and the raw eigenvalue spread grows as
:math:`\sqrt{n(1+\lambda^2)}`.

This example reproduces the Pandey-Mehta crossover ensemble
:math:`H(\lambda) = A + i\lambda B`: as the time-reversal-breaking
parameter :math:`\lambda` grows from 0, the consecutive-spacing-ratio
statistic (which needs no unfolding) moves continuously away from the
GOE surmise and toward the GUE surmise -- the first quantitative
demonstration that "how broken" a symmetry is can be read directly off
spectral statistics.

Reference: A. Pandey, M. L. Mehta, Commun. Math. Phys. 87 (1982) 449.

Run:
    python examples/paper_replications/goe_gue_crossover_demo.py

.. GENERATED FROM PYTHON SOURCE LINES 35-97



.. image-sg:: /api/gallery/rmt/paper_replications/images/sphx_glr_goe_gue_crossover_demo_001.png
   :alt: Pandey-Mehta crossover: level statistics interpolate continuously between GOE and GUE as time-reversal symmetry is broken, lambda=0.0, lambda=0.05, lambda=1.0, Crossover at n=400: GOE-like to GUE-like
   :srcset: /api/gallery/rmt/paper_replications/images/sphx_glr_goe_gue_crossover_demo_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Saved goe_gue_crossover_replication.png






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.stats import kstest

    import physicskit.rmt as rmt

    N = 400
    N_SAMPLES = 20
    SEED = 2026

    goe_surmise = rmt.stats.RatioSurmise(beta=1)
    gue_surmise = rmt.stats.RatioSurmise(beta=2)

    fig, axd = plt.subplot_mosaic(
        [["l1", "l2", "l3"], ["ks", "ks", "ks"]],
        figsize=(12, 8),
    )

    # --- Panels "l1"-"l3": ratio-statistic histograms at three lambda values ---
    r_grid = np.linspace(0, 1, 400)

    for lam, key in zip([0.0, 0.05, 1.0], ["l1", "l2", "l3"], strict=True):
        ens = rmt.ensembles.GOEGUECrossoverEnsemble(n=N, lam=lam, seed=SEED)
        spectrum = ens.sample(n_samples=N_SAMPLES)
        ratios = rmt.stats.ratio_statistics(spectrum)
        ax = axd[key]
        ax.plot(r_grid, goe_surmise.pdf(r_grid), "k-", lw=2, label="GOE surmise")
        ax.plot(r_grid, gue_surmise.pdf(r_grid), "k--", lw=2, label="GUE surmise")
        ax.hist(ratios, bins=50, density=True, alpha=0.5, color="steelblue", label=f"lambda={lam}")
        ax.set_xlabel("r (consecutive spacing ratio)")
        ax.set_ylabel("density P(r)")
        ax.set_title(f"lambda={lam}")
        ax.legend(fontsize=8)

    # --- Panel "ks": KS distance to GOE/GUE surmises vs. lambda ---
    lambdas = np.array([0.0, 0.01, 0.02, 0.05, 0.1, 0.3, 1.0])
    ks_goe = []
    ks_gue = []
    for lam in lambdas:
        ens = rmt.ensembles.GOEGUECrossoverEnsemble(n=N, lam=float(lam), seed=SEED + 1)
        spectrum = ens.sample(n_samples=N_SAMPLES)
        ratios = rmt.stats.ratio_statistics(spectrum)
        ks_goe.append(kstest(ratios, goe_surmise.cdf).statistic)
        ks_gue.append(kstest(ratios, gue_surmise.cdf).statistic)

    ax = axd["ks"]
    ax.plot(lambdas, ks_goe, "o-", color="steelblue", label="KS distance to GOE")
    ax.plot(lambdas, ks_gue, "s-", color="indianred", label="KS distance to GUE")
    ax.set_xscale("symlog", linthresh=0.01)
    ax.set_xlabel("lambda (symmetry-breaking parameter)")
    ax.set_ylabel("KS statistic")
    ax.set_title(f"Crossover at n={N}: GOE-like to GUE-like")
    ax.legend(fontsize=8)

    fig.suptitle(
        "Pandey-Mehta crossover: level statistics interpolate continuously between GOE and GUE as time-reversal symmetry is broken",
    )
    fig.tight_layout()
    out_path = "goe_gue_crossover_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 2.630 seconds)


.. _sphx_glr_download_api_gallery_rmt_paper_replications_goe_gue_crossover_demo.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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