.. _sphx_glr_api_gallery_rmt_non_hermitian:

Non-Hermitian ensembles
-----------------------

Complex spectra: Ginibre's circular law, the Feinberg-Zee single ring
theorem, and Bender-Boettcher PT-symmetry breaking in pseudo-Hermitian
ensembles.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Bender and Boettcher observed that certain non-Hermitian Hamiltonians with PT (parity-time) symmetry can nonetheless have entirely real spectra. This ensemble realizes the equivalent condition of pseudo-Hermiticity, P H P^{-1} = H^\dagger for a fixed Hermitian involution P = \mathrm{diag}(I_p, -I_q) (n=p+q). Writing H in the matching 2\times 2 block form">

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  .. image:: /api/gallery/rmt/non_hermitian/images/thumb/sphx_glr_plot_bender_boettcher_pt_symmetry_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/non_hermitian/plot_bender_boettcher_pt_symmetry`

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      <div class="sphx-glr-thumbnail-title">Bender-Boettcher PT-Symmetry Breaking in Pseudo-Hermitian Ensembles</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="For a bi-unitarily-invariant non-Hermitian matrix M = U \, \mathrm{diag}(s_1, \dots, s_n)\, V, built from independent Haar-random unitary matrices U, V and singular values s_i, the single ring theorem states that as n \to \infty the eigenvalues fill the annulus r_{\text{in}} \leq |z| \leq r_{\text{out}}, with radii fixed by only the first two moments of the limiting singular-value distribution:">

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  .. image:: /api/gallery/rmt/non_hermitian/images/thumb/sphx_glr_plot_feinberg_zee_single_ring_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/non_hermitian/plot_feinberg_zee_single_ring`

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      <div class="sphx-glr-thumbnail-title">The Feinberg-Zee Single Ring Theorem</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The Ginibre ensembles GinOE (real), GinUE (complex), and GinSE (quaternion) drop the Hermitian/unitary symmetry constraint entirely: their n \times n matrices have i.i.d. Gaussian entries with no further structure, so eigenvalues are genuinely complex, spread over a two-dimensional region rather than confined to the real line or a circle. After rescaling by \sqrt{n} (Ginibre&#x27;s exact result, not just asymptotic), the eigenvalues fill the unit disk uniformly as n \to \infty -- the circular law. Because a uniform density on a disk is awkward to validate directly, this example instead checks the radial marginal: integrating out the angle, the magnitude r = |\lambda| has the simple closed-form density">

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  .. image:: /api/gallery/rmt/non_hermitian/images/thumb/sphx_glr_plot_ginibre_circular_law_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/non_hermitian/plot_ginibre_circular_law`

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      <div class="sphx-glr-thumbnail-title">Ginibre's Circular Law</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/rmt/non_hermitian/plot_bender_boettcher_pt_symmetry
   /api/gallery/rmt/non_hermitian/plot_feinberg_zee_single_ring
   /api/gallery/rmt/non_hermitian/plot_ginibre_circular_law