.. _sphx_glr_api_gallery_rmt_spacings:

Spacings and spectral correlations
----------------------------------

Level repulsion and its consequences: the Wigner surmise, Dyson's circular
ensembles and the sine kernel, the Keating-Snaith moments of the CUE
characteristic polynomial, the Poisson (Berry-Tabor) integrable baseline,
the Pandey-Mehta GOE-GUE crossover, Bohigas-Giannoni-Schmit spectral
rigidity and universality, and the Wigner-Dyson many-body level statistics
of Kitaev's SYK model.


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

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    <div class="sphx-glr-thumbcontainer" tooltip="The Berry-Tabor conjecture holds that a classically integrable system&#x27;s quantum energy levels, having no level repulsion mechanism to correlate them, behave locally like an uncorrelated (Poisson) point process -- in sharp contrast to a chaotic system&#x27;s GOE/GUE/GSE-type statistics. PoissonEnsemble realizes this null model directly (independent, uniformly spaced levels with no repulsion built in). Two statistics distinguish the two regimes:">

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  .. image:: /api/gallery/rmt/spacings/images/thumb/sphx_glr_plot_berry_tabor_poisson_statistics_thumb.png
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  :doc:`/api/gallery/rmt/spacings/plot_berry_tabor_poisson_statistics`

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      <div class="sphx-glr-thumbnail-title">The Berry-Tabor Conjecture: Poisson Statistics for Integrable Spectra</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Beyond nearest-neighbor spacing, the number variance \Sigma^2(L) -- the variance of the eigenvalue count in a randomly placed window of (unfolded) length L -- probes longer-range spectral correlations. An uncorrelated (Poisson) sequence has \Sigma^2(L) = L; level repulsion makes the Gaussian ensembles&#x27; spectra far more rigid, with only logarithmic growth (Dyson-Mehta 1963):">

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  .. image:: /api/gallery/rmt/spacings/images/thumb/sphx_glr_plot_bohigas_giannoni_schmit_rigidity_thumb.png
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  :doc:`/api/gallery/rmt/spacings/plot_bohigas_giannoni_schmit_rigidity`

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      <div class="sphx-glr-thumbnail-title">The Bohigas-Giannoni-Schmit Conjecture: Spectral Rigidity and Universality</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Dyson&#x27;s circular ensembles are built from Haar-random unitary matrices rather than Gaussian Hermitian ones: CUE (\beta=2) is simply the eigenvalue phases of a Haar-random U(n) matrix; COE (\beta=1) uses the symmetric unitary matrix U = V^T V for V Haar-random in U(n); CSE (\beta=4) uses the self-dual unitary U = V^R V built from a Haar-random V \in U(2n) (its 2n eigenvalues occur in doubly degenerate pairs, of which n distinct phases are kept). Because Haar measure is exactly rotation-invariant, the eigenvalue phases \theta \in [0, 2\pi) are, after rescaling by n/(2\pi), exactly unfolded to unit mean spacing at any finite n -- no asymptotic unfolding is needed, unlike the Gaussian/Wishart ensembles.">

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  .. image:: /api/gallery/rmt/spacings/images/thumb/sphx_glr_plot_dyson_circular_ensembles_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/spacings/plot_dyson_circular_ensembles`

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      <div class="sphx-glr-thumbnail-title">Dyson's Threefold Way: the Circular Ensembles COE, CUE, CSE</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Unlike every classical ensemble in this package, the Sachdev-Ye-Kitaev (SYK) model is not built from i.i.d. matrix entries: it is an explicit many-body Hamiltonian acting on N Majorana fermions \gamma_1, \dots, \gamma_N (Hermitian operators satisfying the Clifford algebra \{\gamma_a, \gamma_b\} = 2\delta_{ab}),">

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  .. image:: /api/gallery/rmt/spacings/images/thumb/sphx_glr_plot_kitaev_syk_model_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/spacings/plot_kitaev_syk_model`

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      <div class="sphx-glr-thumbnail-title">Kitaev's SYK Model: Majorana Fermions, Wigner-Dyson Statistics</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The Pandey-Mehta crossover ensemble interpolates continuously between time-reversal-symmetric (GOE, \beta=1) and time-reversal-broken (GUE, \beta=2) level statistics via a single real parameter \lambda:">

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  .. image:: /api/gallery/rmt/spacings/images/thumb/sphx_glr_plot_pandey_mehta_goe_gue_crossover_thumb.png
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  :doc:`/api/gallery/rmt/spacings/plot_pandey_mehta_goe_gue_crossover`

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      <div class="sphx-glr-thumbnail-title">The Pandey-Mehta GOE-GUE Crossover</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="For U an n \times n Haar-random unitary matrix (CUE) and its characteristic polynomial evaluated on the unit circle, Z_n(\theta) = \det(I - U e^{-i\theta}), the Diaconis-Shahshahani / Keating-Snaith formula gives the exact moments (positive integer k; by Haar-measure rotation invariance the value does not depend on \theta):">

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  .. image:: /api/gallery/rmt/spacings/images/thumb/sphx_glr_plot_riemann_zeta_keating_snaith_moments_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/spacings/plot_riemann_zeta_keating_snaith_moments`

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      <div class="sphx-glr-thumbnail-title">The Riemann Zeta Connection: Keating-Snaith Moments of the CUE Characteristic Polynomial</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="For the Gaussian ensembles GOE (\beta=1), GUE (\beta=2), GSE (\beta=4), neighboring eigenvalues repel: small spacings are suppressed relative to a Poisson process. The (generalized) Wigner surmise, derived exactly from the eigenvalue spacing of a 2x2 beta-ensemble and normalized to unit mean spacing, models the nearest-neighbor spacing density (after unfolding to unit mean level density) as">

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  .. image:: /api/gallery/rmt/spacings/images/thumb/sphx_glr_plot_wigner_surmise_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/spacings/plot_wigner_surmise`

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      <div class="sphx-glr-thumbnail-title">The Wigner Surmise</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/rmt/spacings/plot_berry_tabor_poisson_statistics
   /api/gallery/rmt/spacings/plot_bohigas_giannoni_schmit_rigidity
   /api/gallery/rmt/spacings/plot_dyson_circular_ensembles
   /api/gallery/rmt/spacings/plot_kitaev_syk_model
   /api/gallery/rmt/spacings/plot_pandey_mehta_goe_gue_crossover
   /api/gallery/rmt/spacings/plot_riemann_zeta_keating_snaith_moments
   /api/gallery/rmt/spacings/plot_wigner_surmise