.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/stochastic_processes/plot_02_polya_random_walks.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_probability_stochastic_processes_plot_02_polya_random_walks.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_probability_stochastic_processes_plot_02_polya_random_walks.py:


Pólya's theorem: a drunk man finds his way home, a drunk bird may not
======================================================================

A simple random walk returns to its starting point with probability 1
in one and two dimensions but only about 34% of the time in three. The
simulated fraction of walks that have returned by step :math:`n` keeps
climbing toward 1 in 1D (slowly, like :math:`1 - 1/\sqrt{\pi n/2}`), crawls
upward in 2D, and levels off near 0.3405 in 3D.

.. GENERATED FROM PYTHON SOURCE LINES 13-17

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.probability import random_walk_return_fraction, return_probability_1d, simple_random_walk








.. GENERATED FROM PYTHON SOURCE LINES 18-20

A 2D walk
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-29

.. code-block:: Python


    walk = simple_random_walk(n_walks=1, n_steps=5000, dim=2, seed=3)[0]
    fig, ax = plt.subplots()
    ax.plot(*walk.T, lw=0.6)
    ax.plot(0, 0, "ro", label="origin")
    ax.set_aspect("equal")
    ax.legend()
    ax.set_title("5000 steps of a 2D simple random walk")




.. image-sg:: /api/gallery/probability/stochastic_processes/images/sphx_glr_plot_02_polya_random_walks_001.png
   :alt: 5000 steps of a 2D simple random walk
   :srcset: /api/gallery/probability/stochastic_processes/images/sphx_glr_plot_02_polya_random_walks_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, '5000 steps of a 2D simple random walk')



.. GENERATED FROM PYTHON SOURCE LINES 30-32

Return fractions by dimension
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-45

.. code-block:: Python


    horizons = [10, 30, 100, 300, 1000]
    fig, ax = plt.subplots()
    for dim in (1, 2, 3):
        fracs = [random_walk_return_fraction(dim, n, n_walks=4000, seed=dim) for n in horizons]
        print(f"d={dim}: " + ", ".join(f"{f:.3f}" for f in fracs))
        ax.semilogx(horizons, fracs, "o-", label=f"d = {dim} (simulated)")
    ax.semilogx(horizons, [return_probability_1d(n) for n in horizons], "k:", label="d = 1 exact")
    ax.axhline(0.3405, color="0.5", ls="--", label="d = 3 limit 0.3405")
    ax.set_xlabel("steps")
    ax.set_ylabel("fraction returned to origin")
    ax.legend()
    ax.set_title("Pólya (1921): recurrence in d ≤ 2, transience in d = 3")



.. image-sg:: /api/gallery/probability/stochastic_processes/images/sphx_glr_plot_02_polya_random_walks_002.png
   :alt: Pólya (1921): recurrence in d ≤ 2, transience in d = 3
   :srcset: /api/gallery/probability/stochastic_processes/images/sphx_glr_plot_02_polya_random_walks_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    d=1: 0.748, 0.846, 0.912, 0.951, 0.972
    d=2: 0.433, 0.520, 0.595, 0.645, 0.692
    d=3: 0.253, 0.293, 0.317, 0.331, 0.337

    Text(0.5, 1.0, 'Pólya (1921): recurrence in d ≤ 2, transience in d = 3')




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

   **Total running time of the script:** (0 minutes 0.327 seconds)


.. _sphx_glr_download_api_gallery_probability_stochastic_processes_plot_02_polya_random_walks.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_polya_random_walks.ipynb <plot_02_polya_random_walks.ipynb>`

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

      :download:`Download Python source code: plot_02_polya_random_walks.py <plot_02_polya_random_walks.py>`

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

      :download:`Download zipped: plot_02_polya_random_walks.zip <plot_02_polya_random_walks.zip>`


.. only:: html

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

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