.. _sphx_glr_api_gallery_rmt_gaussian_ensembles:

Gaussian ensembles
------------------

Bulk spectra of Hermitian random matrices: Wigner's semicircle law, the
Two-Body Random Ensemble's Gaussian (not semicircular) many-body density of
states, Anderson localization in power-law banded matrices, the
Altland-Zirnbauer Bogoliubov-de Gennes classes' :math:`\pm E` eigenvalue
symmetry, and the Dumitriu-Edelman tridiagonal model for continuum
:math:`\beta`.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Extending Dyson&#x27;s threefold way (GOE/GUE/GSE) with particle-hole (charge-conjugation) symmetry -- appropriate to Bogoliubov-de Gennes mean-field Hamiltonians of disordered/chaotic superconductors -- yields four more symmetry classes: D, C, CI, DIII. Each is a 2N \times 2N Hermitian matrix built from a block structure such as (class D, \beta=2)">

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  .. image:: /api/gallery/rmt/gaussian_ensembles/images/thumb/sphx_glr_plot_altland_zirnbauer_bdg_classes_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/gaussian_ensembles/plot_altland_zirnbauer_bdg_classes`

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      <div class="sphx-glr-thumbnail-title">The Altland-Zirnbauer Tenfold Way: BdG Eigenvalue Symmetry</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The power-law banded random matrix (PBRM) ensemble is a real symmetric n \times n matrix whose off-diagonal entry variance decays with distance from the diagonal as">

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  .. image:: /api/gallery/rmt/gaussian_ensembles/images/thumb/sphx_glr_plot_anderson_localization_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/gaussian_ensembles/plot_anderson_localization`

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      <div class="sphx-glr-thumbnail-title">Anderson Localization in Power-Law Banded Matrices</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The classical Gaussian ensembles GOE/GUE/GSE only exist at Dyson index \beta = 1, 2, 4, and are conventionally built as dense n \times n matrices costing O(n^3) to diagonalize. Dumitriu and Edelman (2002) showed that the tridiagonal matrix">

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  .. image:: /api/gallery/rmt/gaussian_ensembles/images/thumb/sphx_glr_plot_dumitriu_edelman_tridiagonal_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/gaussian_ensembles/plot_dumitriu_edelman_tridiagonal`

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      <div class="sphx-glr-thumbnail-title">The Dumitriu-Edelman Tridiagonal Model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Reproduces the celebrated French-Wong / Bohigas-Flores result: embedding a random two-body interaction into the many-particle Fock space (the Two-Body Random Ensemble, TBRE) produces correlated many-body matrix elements whose density of states approaches a GAUSSIAN, not Wigner&#x27;s semicircle, unlike an ordinary GOE Hamiltonian of the same dimension built from independent entries.">

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  .. image:: /api/gallery/rmt/gaussian_ensembles/images/thumb/sphx_glr_plot_embedded_two_body_ensemble_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/gaussian_ensembles/plot_embedded_two_body_ensemble`

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      <div class="sphx-glr-thumbnail-title">Embedded Ensembles: the Two-Body Random Ensemble Is Gaussian, not Semicircle</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="GOE (\beta=1), GUE (\beta=2), and GSE (\beta=4) are the Gaussian random matrix ensembles built from real symmetric, complex Hermitian, and quaternionic self-dual Hermitian matrices respectively, with independent Gaussian entries. After rescaling eigenvalues by \sqrt{n\beta} (n the matrix size), the empirical spectral density of all three ensembles converges, as n \to \infty, to Wigner&#x27;s semicircle law on [-R, R] (R=2 here):">

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  .. image:: /api/gallery/rmt/gaussian_ensembles/images/thumb/sphx_glr_plot_wigner_semicircle_law_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/gaussian_ensembles/plot_wigner_semicircle_law`

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      <div class="sphx-glr-thumbnail-title">Wigner's Semicircle Law</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/rmt/gaussian_ensembles/plot_altland_zirnbauer_bdg_classes
   /api/gallery/rmt/gaussian_ensembles/plot_anderson_localization
   /api/gallery/rmt/gaussian_ensembles/plot_dumitriu_edelman_tridiagonal
   /api/gallery/rmt/gaussian_ensembles/plot_embedded_two_body_ensemble
   /api/gallery/rmt/gaussian_ensembles/plot_wigner_semicircle_law