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

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

.. _sphx_glr_api_gallery_statphys_percolation_plot_percolation_threshold.py:


Percolation: a purely geometric phase transition
====================================================

Percolation theory shows that a sharp, universal phase transition does not
require energy, temperature, or even dynamics -- pure geometry suffices. As
the occupation probability :math:`p` of a random lattice increases past a
critical threshold :math:`p_c`, an infinite spanning cluster suddenly
appears. This example uses Hoshen-Kopelman cluster labeling to estimate the
spanning probability curve :math:`P_{\text{span}}(p)` for site percolation
on the square lattice (:math:`p_c \approx 0.593`), and visualizes the
fractal, self-similar spanning cluster right at criticality.

.. GENERATED FROM PYTHON SOURCE LINES 14-24

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.statphys.chapters.percolation import Percolation2D
    from physicskit.statphys.visualizers.lattice_render import (
        plot_cluster_size_distribution,
        plot_percolation_clusters,
    )








.. GENERATED FROM PYTHON SOURCE LINES 25-27

The spanning probability curve sharpens with system size
--------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-40

.. code-block:: Python

    plt.figure(figsize=(6, 4))
    p_values = np.linspace(0.4, 0.8, 25)
    for L in [16, 32, 64]:
        perc = Percolation2D(L=L, mode="site", seed=0)
        p, P_span = perc.spanning_probability(p_values, n_trials=80)
        plt.plot(p, P_span, marker="o", ms=3, label=f"L = {L}")
    plt.axvline(Percolation2D.P_C_SITE, color="k", linestyle="--", linewidth=1, label="$p_c$")
    plt.xlabel("occupation probability $p$")
    plt.ylabel("$P_{\\mathrm{span}}(p)$")
    plt.title("Site percolation: sharpening transition with system size")
    plt.legend()
    plt.tight_layout()




.. image-sg:: /api/gallery/statphys/percolation/images/sphx_glr_plot_percolation_threshold_001.png
   :alt: Site percolation: sharpening transition with system size
   :srcset: /api/gallery/statphys/percolation/images/sphx_glr_plot_percolation_threshold_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 41-43

The spanning cluster at criticality is a fractal
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-51

.. code-block:: Python

    perc = Percolation2D(L=128, p=Percolation2D.P_C_SITE, mode="site", seed=1)
    d_f = perc.fractal_dimension(n_trials=10)
    print(f"Estimated fractal dimension at p_c: {d_f:.3f} (theory: 91/48 = {91 / 48:.3f}; finite-size bias is significant at L=128)")

    fig, ax = plt.subplots(figsize=(6, 6))
    plot_percolation_clusters(perc, ax=ax)
    plt.tight_layout()




.. image-sg:: /api/gallery/statphys/percolation/images/sphx_glr_plot_percolation_threshold_002.png
   :alt: site percolation, p = 0.593
   :srcset: /api/gallery/statphys/percolation/images/sphx_glr_plot_percolation_threshold_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Estimated fractal dimension at p_c: 2.258 (theory: 91/48 = 1.896; finite-size bias is significant at L=128)




.. GENERATED FROM PYTHON SOURCE LINES 52-59

Cluster sizes follow a power law at criticality
-----------------------------------------------------
Away from p_c, the cluster size distribution has a well-defined
characteristic scale and falls off exponentially; exactly at p_c, that
scale diverges and n_s ~ s^{-tau} instead, with the Fisher exponent
tau = 187/91 for 2D percolation -- the geometric analogue of a diverging
susceptibility at a thermal critical point.

.. GENERATED FROM PYTHON SOURCE LINES 59-70

.. code-block:: Python

    all_sizes = []
    for _trial in range(20):
        perc.generate(Percolation2D.P_C_SITE)
        all_sizes.append(perc.cluster_size_distribution())
    all_sizes = np.concatenate(all_sizes)

    fig, ax = plt.subplots(figsize=(6, 4.5))
    plot_cluster_size_distribution(all_sizes, ax=ax)
    ax.set_title("Percolation cluster size distribution at $p_c$")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/statphys/percolation/images/sphx_glr_plot_percolation_threshold_003.png
   :alt: Percolation cluster size distribution at $p_c$
   :srcset: /api/gallery/statphys/percolation/images/sphx_glr_plot_percolation_threshold_003.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 5.112 seconds)


.. _sphx_glr_download_api_gallery_statphys_percolation_plot_percolation_threshold.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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