.. _sphx_glr_api_gallery_rmt_paper_replications:

Paper replications
------------------

Classic random matrix theory results, organized around Dyson's threefold way
and its non-Hermitian/chiral/Bogoliubov-de Gennes extensions: Wishart's
original sample-covariance construction, the Wigner semicircle and surmise,
the Marchenko-Pastur and Wachter laws, the Tracy-Widom edge distributions,
Dyson's circular ensembles and the Keating-Snaith characteristic-polynomial
moments, spectral rigidity and universality, the Pandey-Mehta GOE-GUE
crossover, the Poisson (Berry-Tabor) integrable baseline, Ginibre's circular
law, the single ring theorem, PT-symmetry breaking, Anderson localization,
the Altland-Zirnbauer Bogoliubov-de Gennes symmetry classes, the Two-Body
Random Ensemble, Page's average-entanglement-entropy conjecture, the
Dumitriu-Edelman tridiagonal model, and Kitaev's SYK model.


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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="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/paper_replications/images/thumb/sphx_glr_bdg_classes_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/bdg_classes_demo`

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      <div class="sphx-glr-thumbnail-title">Altland-Zirnbauer Classes: BdG Eigenvalue Symmetry</div>
    </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/paper_replications/images/thumb/sphx_glr_circular_ensembles_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/circular_ensembles_demo`

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      <div class="sphx-glr-thumbnail-title">Dyson's Circular Ensembles: COE, CUE, CSE</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/paper_replications/images/thumb/sphx_glr_circular_law_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/circular_law_demo`

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      <div class="sphx-glr-thumbnail-title">Ginibre's Circular Law</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/paper_replications/images/thumb/sphx_glr_embedded_ensemble_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/embedded_ensemble_demo`

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      <div class="sphx-glr-thumbnail-title">The Two-Body Random Ensemble: Gaussian, not Semicircle</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/paper_replications/images/thumb/sphx_glr_goe_gue_crossover_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/goe_gue_crossover_demo`

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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/paper_replications/images/thumb/sphx_glr_keating_snaith_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/keating_snaith_demo`

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      <div class="sphx-glr-thumbnail-title">Keating-Snaith Moments of the CUE Characteristic Polynomial</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/paper_replications/images/thumb/sphx_glr_localization_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/localization_demo`

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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 Wishart (Laguerre) ensembles LOE (\beta=1), LUE (\beta=2), and LSE (\beta=4) are built from the sample covariance matrix W = X X^\dagger of an n \times m Gaussian data matrix X (n variables, m samples, m \geq n), with real, complex, or quaternionic entries respectively. The relevant control parameter is the aspect ratio \gamma = n/m \in (0, 1].">

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  .. image:: /api/gallery/rmt/paper_replications/images/thumb/sphx_glr_marchenko_pastur_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/marchenko_pastur_demo`

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      <div class="sphx-glr-thumbnail-title">The Marchenko-Pastur Law</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A Haar-random pure state on \mathbb{C}^n \otimes \mathbb{C}^k (n \leq k), traced out over the k-dimensional subsystem, leaves an n-dimensional reduced density matrix \rho = A A^\dagger / \mathrm{Tr}(A A^\dagger) built from an n \times k complex Ginibre matrix A (the &quot;induced measure&quot;; k=n is the flat Hilbert-Schmidt measure on density matrices). Page&#x27;s exact result gives the average von Neumann entropy S = -\sum_i \lambda_i \log \lambda_i (in nats, \lambda_i the eigenvalues of \rho) of this random reduced state, for any finite n, k:">

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  .. image:: /api/gallery/rmt/paper_replications/images/thumb/sphx_glr_page_curve_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/page_curve_demo`

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      <div class="sphx-glr-thumbnail-title">Page's Conjecture: the Average Entanglement Entropy of a Random State</div>
    </div>


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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/paper_replications/images/thumb/sphx_glr_poisson_berry_tabor_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/poisson_berry_tabor_demo`

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      <div class="sphx-glr-thumbnail-title">The Berry-Tabor Poisson Baseline</div>
    </div>


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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/paper_replications/images/thumb/sphx_glr_pt_symmetric_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/pt_symmetric_demo`

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      <div class="sphx-glr-thumbnail-title">PT-Symmetry Breaking in Pseudo-Hermitian Ensembles</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/paper_replications/images/thumb/sphx_glr_rigidity_universality_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/rigidity_universality_demo`

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      <div class="sphx-glr-thumbnail-title">Spectral Rigidity and Universality</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/paper_replications/images/thumb/sphx_glr_single_ring_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/single_ring_demo`

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      <div class="sphx-glr-thumbnail-title">The Single Ring Theorem</div>
    </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/paper_replications/images/thumb/sphx_glr_syk_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/syk_demo`

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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="For the Gaussian beta-ensembles GOE (\beta=1), GUE (\beta=2), and GSE (\beta=4), the bulk of the n-eigenvalue spectrum fills the semicircle [-2, 2], but the largest eigenvalue \lambda_{\max} fluctuates above the edge 2 on a much finer scale than the bulk spacing: rescaling">

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  .. image:: /api/gallery/rmt/paper_replications/images/thumb/sphx_glr_tracy_widom_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/tracy_widom_demo`

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      <div class="sphx-glr-thumbnail-title">Tracy-Widom Soft-Edge Laws</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/paper_replications/images/thumb/sphx_glr_tridiagonal_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/tridiagonal_demo`

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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="The Jacobi (MANOVA) ensemble is built from two independent Wishart-type matrices A and B, each n \times n and formed from m_1 and m_2 independent Gaussian &quot;samples&quot; of n variables respectively (m_1, m_2 \geq n). The double-Wishart ratio matrix">

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  .. image:: /api/gallery/rmt/paper_replications/images/thumb/sphx_glr_wachter_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/wachter_demo`

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      <div class="sphx-glr-thumbnail-title">The Wachter Law for Jacobi Ensembles</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/paper_replications/images/thumb/sphx_glr_wigner_semicircle_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/wigner_semicircle_demo`

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      <div class="sphx-glr-thumbnail-title">Wigner's Semicircle Law</div>
    </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/paper_replications/images/thumb/sphx_glr_wigner_surmise_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/wigner_surmise_demo`

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


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    <div class="sphx-glr-thumbcontainer" tooltip="For an m \times n data matrix X of m independent samples of n variables, the sample covariance (scatter) matrix">

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  .. image:: /api/gallery/rmt/paper_replications/images/thumb/sphx_glr_wishart_demo_thumb.png
    :alt:

  :doc:`/api/gallery/rmt/paper_replications/wishart_demo`

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      <div class="sphx-glr-thumbnail-title">Wishart's Sample Covariance Matrix</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/rmt/paper_replications/bdg_classes_demo
   /api/gallery/rmt/paper_replications/circular_ensembles_demo
   /api/gallery/rmt/paper_replications/circular_law_demo
   /api/gallery/rmt/paper_replications/embedded_ensemble_demo
   /api/gallery/rmt/paper_replications/goe_gue_crossover_demo
   /api/gallery/rmt/paper_replications/keating_snaith_demo
   /api/gallery/rmt/paper_replications/localization_demo
   /api/gallery/rmt/paper_replications/marchenko_pastur_demo
   /api/gallery/rmt/paper_replications/page_curve_demo
   /api/gallery/rmt/paper_replications/poisson_berry_tabor_demo
   /api/gallery/rmt/paper_replications/pt_symmetric_demo
   /api/gallery/rmt/paper_replications/rigidity_universality_demo
   /api/gallery/rmt/paper_replications/single_ring_demo
   /api/gallery/rmt/paper_replications/syk_demo
   /api/gallery/rmt/paper_replications/tracy_widom_demo
   /api/gallery/rmt/paper_replications/tridiagonal_demo
   /api/gallery/rmt/paper_replications/wachter_demo
   /api/gallery/rmt/paper_replications/wigner_semicircle_demo
   /api/gallery/rmt/paper_replications/wigner_surmise_demo
   /api/gallery/rmt/paper_replications/wishart_demo