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Journal articles on the topic 'Random matrices ensembles'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Akemann, Gernot, Eugene Strahov, and Tim R. Würfel. "Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles." Annales Henri Poincaré 21, no. 12 (October 14, 2020): 3973–4002. http://dx.doi.org/10.1007/s00023-020-00963-9.

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Abstract Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.

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2

Cheliotis, Dimitris. "Triangular random matrices and biorthogonal ensembles." Statistics & Probability Letters 134 (March 2018): 36–44. http://dx.doi.org/10.1016/j.spl.2017.10.010.

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3

Życzkowski, Karol, and Marek Kuś. "Interpolating ensembles of random unitary matrices." Physical Review E 53, no. 1 (January 1, 1996): 319–26. http://dx.doi.org/10.1103/physreve.53.319.

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4

Pozniak, Marcin, Karol Zyczkowski, and Marek Kus. "Composed ensembles of random unitary matrices." Journal of Physics A: Mathematical and General 31, no. 3 (January 23, 1998): 1059–71. http://dx.doi.org/10.1088/0305-4470/31/3/016.

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5

Kieburg, Mario. "Additive matrix convolutions of Pólya ensembles and polynomial ensembles." Random Matrices: Theory and Applications 09, no. 04 (November 8, 2019): 2150002. http://dx.doi.org/10.1142/s2010326321500027.

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Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work, we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose, we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results, we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example, we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore, we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.

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6

Kuijlaars, Arno B. J., and Dries Stivigny. "Singular values of products of random matrices and polynomial ensembles." Random Matrices: Theory and Applications 03, no. 03 (July 2014): 1450011. http://dx.doi.org/10.1142/s2010326314500117.

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Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. We also show that the squared singular values of the product of M - 1 complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and we derive a double integral representation for the correlation kernel associated with this ensemble. We use this to calculate the scaling limit at the hard edge, which turns out to be the same scaling limit as the one found by Kuijlaars and Zhang for the squared singular values of a product of M complex Ginibre matrices. Our final result is that these limiting kernels also appear as scaling limits for the biorthogonal ensembles of Borodin with parameter θ > 0, in case θ or 1/θ is an integer. This further supports the conjecture that these kernels have a universal character.

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7

Kirsch, Werner, and Thomas Kriecherbauer. "Random matrices with exchangeable entries." Reviews in Mathematical Physics 32, no. 07 (January 30, 2020): 2050022. http://dx.doi.org/10.1142/s0129055x20500221.

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We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general, the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding [Formula: see text]-operator norms. The key to our analysis is a generalization of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centered. Some of our results appear to be new even for such Wigner band matrices.

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8

Kieburg, Mario, and Holger Kösters. "Products of random matrices from polynomial ensembles." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 55, no. 1 (February 2019): 98–126. http://dx.doi.org/10.1214/17-aihp877.

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9

Ahmed, Zafar. "Gaussian-Random Ensembles of Pseudo-Hermitian Matrices." Czechoslovak Journal of Physics 54, no. 10 (October 2004): 1011–18. http://dx.doi.org/10.1023/b:cjop.0000043999.11289.a3.

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10

Nagao, Taro, and Peter J. Forrester. "Dynamical correlations for circular ensembles of random matrices." Nuclear Physics B 660, no. 3 (June 2003): 557–78. http://dx.doi.org/10.1016/s0550-3213(03)00292-x.

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11

Nagao, Taro, and Keith Slevin. "Laguerre ensembles of random matrices: Nonuniversal correlation functions." Journal of Mathematical Physics 34, no. 6 (June 1993): 2317–30. http://dx.doi.org/10.1063/1.530118.

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12

Shcherbina, M. "Edge Universality for Orthogonal Ensembles of Random Matrices." Journal of Statistical Physics 136, no. 1 (June 3, 2009): 35–50. http://dx.doi.org/10.1007/s10955-009-9766-5.

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13

Shcherbina, M. "On Universality for Orthogonal Ensembles of Random Matrices." Communications in Mathematical Physics 285, no. 3 (October 23, 2008): 957–74. http://dx.doi.org/10.1007/s00220-008-0648-5.

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14

Burkhardt, Paula, Peter Cohen, Jonathan DeWitt, Max Hlavacek, Steven J. Miller, Carsten Sprunger, Yen Nhi Truong Vu, Roger Van Peski, and Kevin Yang. "Random matrix ensembles with split limiting behavior." Random Matrices: Theory and Applications 07, no. 03 (June 19, 2018): 1850006. http://dx.doi.org/10.1142/s2010326318500065.

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We introduce a new family of [Formula: see text] random real symmetric matrix ensembles, the [Formula: see text]-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but [Formula: see text] eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as [Formula: see text]; the remaining [Formula: see text] are tightly constrained near [Formula: see text] and their distribution converges to the [Formula: see text] hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for complex and quaternionic analogues. We are able to isolate each regime separately through appropriate choices of weight functions for the eigenvalues and then an analysis of the resulting combinatorics.

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15

Sinclair, Christopher D., and Maxim L. Yattselev. "The reciprocal Mahler ensembles of random polynomials." Random Matrices: Theory and Applications 08, no. 04 (October 2019): 1950012. http://dx.doi.org/10.1142/s2010326319500126.

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We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree [Formula: see text] whose Mahler measure is bounded by a constant. After a change of variables, this reduces to a generalization of Ginibre’s complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval [Formula: see text] on the real axis in the complex plane. In the complex (real) case, the random roots form a determinantal (Pfaffian) point process, and in both cases, the empirical measure on roots converges weakly to the arcsine distribution supported on [Formula: see text]. Outside this region, the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from [Formula: see text]. These kernels as well as the scaling limits for the kernels in the bulk [Formula: see text] and at the endpoints [Formula: see text] are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.

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16

Nagao, Taro, and Peter J. Forrester. "Transitive ensembles of random matrices related to orthogonal polynomials." Nuclear Physics B 530, no. 3 (October 1998): 742–62. http://dx.doi.org/10.1016/s0550-3213(98)00501-x.

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17

Vivo, Pierpaolo, and Satya N. Majumdar. "On invariant 2×2 β -ensembles of random matrices." Physica A: Statistical Mechanics and its Applications 387, no. 19-20 (August 2008): 4839–55. http://dx.doi.org/10.1016/j.physa.2008.03.009.

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18

Prakash, Ravi, and Akhilesh Pandey. "Universal spectral correlations in ensembles of random normal matrices." EPL (Europhysics Letters) 110, no. 3 (May 1, 2015): 30001. http://dx.doi.org/10.1209/0295-5075/110/30001.

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19

Abul-Magd, A. Y., G. Akemann, and P. Vivo. "Superstatistical generalizations of Wishart–Laguerre ensembles of random matrices." Journal of Physics A: Mathematical and Theoretical 42, no. 17 (April 6, 2009): 175207. http://dx.doi.org/10.1088/1751-8113/42/17/175207.

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20

Kumar, Santosh, and Akhilesh Pandey. "Crossover ensembles of random matrices and skew-orthogonal polynomials." Annals of Physics 326, no. 8 (August 2011): 1877–915. http://dx.doi.org/10.1016/j.aop.2011.04.013.

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21

Miller, Steven J., Kirk Swanson, Kimsy Tor, and Karl Winsor. "Limiting spectral measures for random matrix ensembles with a polynomial link function." Random Matrices: Theory and Applications 04, no. 02 (April 2015): 1550004. http://dx.doi.org/10.1142/s2010326315500045.

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Consider the ensembles of real symmetric Toeplitz matrices and real symmetric Hankel matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work on real symmetric Toeplitz matrices shows that the spectral measures, or densities of normalized eigenvalues, converge almost surely to a universal near-Gaussian distribution, while previous work on real symmetric Hankel matrices shows that the spectral measures converge almost surely to a universal non-unimodal distribution. Real symmetric Toeplitz matrices are constant along the diagonals, while real symmetric Hankel matrices are constant along the skew diagonals. We generalize the Toeplitz and Hankel matrices to study matrices that are constant along some curve described by a real-valued bivariate polynomial (other authors refer to the dependencies among the matrix elements as arising from a link function). Using the Method of Moments and an analysis of the resulting Diophantine equations, we show that the spectral measures associated with linear bivariate polynomials converge in probability and almost surely to universal non-semicircular distributions. We prove that for certain choices these limiting distributions approach the semicircle in the limit of large values of the polynomial coefficients. We then prove that the spectral measures associated with the sum or difference of any two real-valued polynomials with different degrees converge in probability and almost surely to a universal semicircular distribution.

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22

Guhlich, Matthias, Jan Nagel, and Holger Dette. "Random block matrices generalizing the classical Jacobi and Laguerre ensembles." Journal of Multivariate Analysis 101, no. 8 (September 2010): 1884–97. http://dx.doi.org/10.1016/j.jmva.2010.03.013.

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23

Benet, L., and H. A. Weidenm ller. "Review of thek-body embedded ensembles of Gaussian random matrices." Journal of Physics A: Mathematical and General 36, no. 12 (March 13, 2003): 3569–93. http://dx.doi.org/10.1088/0305-4470/36/12/340.

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24

Akemann, Gernot, and Pierpaolo Vivo. "Power law deformation of Wishart–Laguerre ensembles of random matrices." Journal of Statistical Mechanics: Theory and Experiment 2008, no. 09 (September 1, 2008): P09002. http://dx.doi.org/10.1088/1742-5468/2008/09/p09002.

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25

Fyodorov, Y. V., B. A. Khoruzhenko, and A. Nock. "UniversalK-matrix distribution in β = 2 ensembles of random matrices." Journal of Physics A: Mathematical and Theoretical 46, no. 26 (June 7, 2013): 262001. http://dx.doi.org/10.1088/1751-8113/46/26/262001.

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26

Oliveira, Lucas H., and Marcel Novaes. "Immanants of blocks from random matrices in some unitary ensembles." Journal of Physics A: Mathematical and Theoretical 54, no. 29 (June 24, 2021): 295205. http://dx.doi.org/10.1088/1751-8121/ac0984.

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27

Mays, Anthony, and Anita Ponsaing. "An induced real quaternion spherical ensemble of random matrices." Random Matrices: Theory and Applications 06, no. 01 (January 2017): 1750001. http://dx.doi.org/10.1142/s2010326317500010.

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We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a [Formula: see text] complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to [Formula: see text] These matrices can also be constructed via a procedure called ‘inducing’, using a product of a Wishart matrix (with parameters [Formula: see text]) and a rectangular Ginibre matrix of size [Formula: see text]. The inducing procedure imposes a repulsion of eigenvalues from [Formula: see text] and [Formula: see text] in the complex plane with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of [Formula: see text], [Formula: see text] and [Formula: see text]. By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue [Formula: see text]-point correlation functions, and in particular the eigenvalue density (given by [Formula: see text]). We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of [Formula: see text] and [Formula: see text]. After a stereographic projection, the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behavior of the density near the real line based on analogous results in the [Formula: see text] and [Formula: see text] ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the [Formula: see text] induced spherical ensemble. This ensemble is a quaternionic analog of a model of a one-component charged plasma on a sphere, with soft wall boundary conditions.

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28

Nickelsen, Daniel, and Michael Kastner. "Modelling equilibration of local many-body quantum systems by random graph ensembles." Quantum 4 (May 28, 2020): 273. http://dx.doi.org/10.22331/q-2020-05-28-273.

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We introduce structured random matrix ensembles, constructed to model many-body quantum systems with local interactions. These ensembles are employed to study equilibration of isolated many-body quantum systems, showing that rather complex matrix structures, well beyond Wigner's full or banded random matrices, are required to faithfully model equilibration times. Viewing the random matrices as connectivities of graphs, we analyse the resulting network of classical oscillators in Hilbert space with tools from network theory. One of these tools, called the maximum flow value, is found to be an excellent proxy for equilibration times. Since maximum flow values are less expensive to compute, they give access to approximate equilibration times for system sizes beyond those accessible by exact diagonalisation.

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29

Webb, Zak. "The Clifford group forms a unitary 3-design." Quantum Information and Computation 16, no. 15&16 (November 2016): 1379–400. http://dx.doi.org/10.26421/qic16.15-16-8.

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Unitary k-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery. We also show that the Clifford group does not form a 4-design, thus characterizing how well random Clifford elements approximate Haar-random unitaries. Additionally, we show that the generalized Clifford group for qudits is not a 3-design unless the dimension of the qudit is a power of 2.

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30

FORRESTER, PETER J. "PROBABILITY DENSITIES AND DISTRIBUTIONS FOR SPIKED AND GENERAL VARIANCE WISHART β-ENSEMBLES." Random Matrices: Theory and Applications 02, no. 04 (October 2013): 1350011. http://dx.doi.org/10.1142/s2010326313500111.

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A Wishart matrix is said to be spiked when the underlying covariance matrix has a single eigenvalue b different from unity. As b increases through b = 2, a gap forms from the largest eigenvalue to the rest of the spectrum, and with b - 2 of order N-1/3 the scaled largest eigenvalues form a well-defined parameter dependent state. Recent works by Bloemendal and Virág [Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825], and Mo [Rank I real Wishart spiked model, Comm. Pure Appl. Math.65 (2012) 1528–1638], have quantified this parameter dependent state for real Wishart matrices from different viewpoints, and the former authors have done similarly for the spiked Wishart β-ensemble. The latter is defined in terms of certain random bidiagonal matrices. We use a recursive structure to give an alternative construction of the spiked and more generally the general variance Wishart β-ensemble, and we give the exact form of the joint eigenvalue PDF for the two matrices in the recurrence. In the case of real quaternion Wishart matrices (β = 4) the latter is recognized as having appeared in earlier studies on symmetrized last passage percolation, allowing the exact form of the scaled distribution of the largest eigenvalue to be given. This extends and simplifies earlier work of Wang, and is an alternative derivation to a result in [A. Bloemendal and B. Virág, Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825]. We also use the construction of the spiked Wishart β-ensemble from [A. Bloemendal and B. Virág, Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825] to give a simple derivation of the explicit form of the eigenvalue PDF.

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31

Forrester, Peter J. "Octonions in random matrix theory." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2200 (April 2017): 20160800. http://dx.doi.org/10.1098/rspa.2016.0800.

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The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random matrices by symmetry considerations. Only for N =2 is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for N =3. We then proceed to consider the matrix structure X † X , when X has random octonion entries. Analytic results are obtained from N =2, but are observed to break down in the 3×3 case.

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32

Götze, F., H. Kösters, and A. Tikhomirov. "Asymptotic spectra of matrix-valued functions of independent random matrices and free probability." Random Matrices: Theory and Applications 04, no. 02 (April 2015): 1550005. http://dx.doi.org/10.1142/s2010326315500057.

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We investigate the universality of singular value and eigenvalue distributions of matrix-valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution and prove universality under general conditions for singular value and eigenvalue distributions of products of independent matrices from spherical ensembles.

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33

Akemann, Gernot, Jesper R. Ipsen, and Eugene Strahov. "Permanental processes from products of complex and quaternionic induced Ginibre ensembles." Random Matrices: Theory and Applications 03, no. 04 (October 2014): 1450014. http://dx.doi.org/10.1142/s2010326314500142.

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We consider products of independent random matrices taken from the induced Ginibre ensemble with complex or quaternion elements. The joint densities for the complex eigenvalues of the product matrix can be written down exactly for a product of any fixed number of matrices and any finite matrix size. We show that the squared absolute values of the eigenvalues form a permanental process, generalizing the results of Kostlan and Rider for single matrices to products of complex and quaternionic matrices. Based on these findings, we can first write down exact results and asymptotic expansions for the so-called hole probabilities, that a disk centered at the origin is void of eigenvalues. Second, we compute the asymptotic expansion for the opposite problem, that a large fraction of complex eigenvalues occupies a disk of fixed radius centered at the origin; this is known as the overcrowding problem. While the expressions for finite matrix size depend on the parameters of the induced ensembles, the asymptotic results agree to leading order with previous results for products of square Ginibre matrices.

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34

Bornemann, Folkmar, and Peter J. Forrester. "Singular values and evenness symmetry in random matrix theory." Forum Mathematicum 28, no. 5 (September 1, 2016): 873–91. http://dx.doi.org/10.1515/forum-2015-0055.

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AbstractComplex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values – whereby only even, or odd, labels are observed – for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types – Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

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35

Delannay, R., and G. Le Caër. "Exact densities of states of fixed trace ensembles of random matrices." Journal of Physics A: Mathematical and General 33, no. 14 (March 31, 2000): 2611–30. http://dx.doi.org/10.1088/0305-4470/33/14/302.

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36

Kumar, Santosh, and Akhilesh Pandey. "Jacobi crossover ensembles of random matrices and statistics of transmission eigenvalues." Journal of Physics A: Mathematical and Theoretical 43, no. 8 (February 4, 2010): 085001. http://dx.doi.org/10.1088/1751-8113/43/8/085001.

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37

Cunden, Fabio Deelan, Francesco Mezzadri, and Pierpaolo Vivo. "A unified fluctuation formula for one-cutβ-ensembles of random matrices." Journal of Physics A: Mathematical and Theoretical 48, no. 31 (July 20, 2015): 315204. http://dx.doi.org/10.1088/1751-8113/48/31/315204.

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38

Akemann, Gernot, and Milan Cikovic. "Products of random matrices from fixed trace and induced Ginibre ensembles." Journal of Physics A: Mathematical and Theoretical 51, no. 18 (April 9, 2018): 184002. http://dx.doi.org/10.1088/1751-8121/aab8a9.

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39

Adamczak, Radosław, Rafał Latała, Alexander E. Litvak, Alain Pajor, and Nicole Tomczak-Jaegermann. "Geometry of log-concave ensembles of random matrices and approximate reconstruction." Comptes Rendus Mathematique 349, no. 13-14 (July 2011): 783–86. http://dx.doi.org/10.1016/j.crma.2011.06.025.

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40

Basor, Estelle L. "Distribution Functions for Random Variables for Ensembles of Positive Hermitian Matrices." Communications in Mathematical Physics 188, no. 2 (September 1, 1997): 327–50. http://dx.doi.org/10.1007/s002200050167.

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41

Kumar, Sachin, and Zafar Ahmed. "NEW SPECTRAL STATISTICS FOR ENSEMBLES OF 2 × 2 REAL SYMMETRIC RANDOM MATRICES." Acta Polytechnica 57, no. 6 (December 30, 2017): 418. http://dx.doi.org/10.14311/ap.2017.57.0418.

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We investigate spacing statistics for ensembles of various real random matrices where the matrix-elements have various Probability Distribution Function (PDF: <em>f(x)</em>) including Gaussian. For two modifications of 2 × 2 matrices with various PDFs, we derive the spacing distributions <em>p(s)</em> of adjacent energy eigenvalues. Nevertheless, they show the linear level repulsion near s = 0 as <em>αs</em> where <em>α</em> depends on the choice of the PDF. More interestingly when <em>f</em>(<em>x</em>) = <em>xe</em><sup>−x<sup>2</sup></sup> (<em>f</em>(0) = 0), we get cubic level repulsion near s = 0: <em>p(s)</em> ~ s<sup>3</sup>e<sup>−s<sup>2</sup></sup>.We also derive the distribution of eigenvalues <em>D</em>(ε) for these matrices.

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42

Veneziani, Alexei M., Tiago Pereira, and Domingos H. U. Marchetti. "Asymptotic integral kernel for ensembles of random normal matrices with radial potentials." Journal of Mathematical Physics 53, no. 2 (February 2012): 023303. http://dx.doi.org/10.1063/1.3688293.

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43

Pluhař, Z., and H. A. Weidenmüller. "Spectral fluctuation properties of constrained unitary ensembles of Gaussian-distributed random matrices." Journal of Physics A: Mathematical and Theoretical 42, no. 15 (March 24, 2009): 155203. http://dx.doi.org/10.1088/1751-8113/42/15/155203.

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44

Mehta, Madan Lal, and Jean-Marie Normand. "Moments of the characteristic polynomial in the three ensembles of random matrices." Journal of Physics A: Mathematical and General 34, no. 22 (May 24, 2001): 4627–39. http://dx.doi.org/10.1088/0305-4470/34/22/304.

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Benet, L., T. Rupp, and H. A. Weidenmüller. "Spectral Properties of the k-Body Embedded Gaussian Ensembles of Random Matrices." Annals of Physics 292, no. 1 (August 2001): 67–94. http://dx.doi.org/10.1006/aphy.2001.6156.

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Gorin, Vadim, and Adam W. Marcus. "Crystallization of Random Matrix Orbits." International Mathematics Research Notices 2020, no. 3 (April 3, 2018): 883–913. http://dx.doi.org/10.1093/imrn/rny052.

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Abstract Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta =1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of $\beta>0$ through associated special functions. We show that the $\beta \to \infty $ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta $ self-adjoint matrix with fixed eigenvalues is known as the $\beta $-corners process. We show that as $\beta \to \infty $ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.

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Eichelsbacher, Peter, and Lukas Knichel. "Fine asymptotics for models with Gamma type moments." Random Matrices: Theory and Applications 10, no. 01 (December 9, 2019): 2150007. http://dx.doi.org/10.1142/s2010326321500076.

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The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the so-called tenfold way, including the Bogoliubov–de Gennes and chiral ensembles from mesoscopic physics. We show that fixed-trace matrix ensembles can be analyzed as well. Finally, we add fine asymptotics for the [Formula: see text]-dimensional volume of the simplex with [Formula: see text] points in [Formula: see text] distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod-[Formula: see text] convergence to obtain extended limit theorems, Berry–Esseen bounds, precise moderate deviations, large and moderate deviation principles as well as local limit theorems. The work is especially based on the recent work of Dal Borgo et al. [Mod-Gaussian convergence for random determinants, Ann. Henri Poincaré (2018)].

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Hofmann-Credner, Katrin, and Michael Stolz. "Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices." Electronic Communications in Probability 13 (2008): 401–14. http://dx.doi.org/10.1214/ecp.v13-1395.

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Vasilchuk, Vladimir. "On the fluctuations of eigenvalues of multiplicative deformed unitary invariant ensembles." Random Matrices: Theory and Applications 05, no. 02 (April 2016): 1650007. http://dx.doi.org/10.1142/s2010326316500076.

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We consider the ensemble of [Formula: see text] random matrices [Formula: see text], where [Formula: see text] and [Formula: see text] are non-random, unitary, having the limiting Normalized Counting Measure (NCM) of eigenvalues, and [Formula: see text] is unitary, uniformly distributed over [Formula: see text]. We find the leading term of the covariance of traces of resolvent of [Formula: see text] and establish the Central Limit Theorem for sufficiently smooth linear eigenvalue statistics of [Formula: see text] as [Formula: see text].

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Desrosiers, Patrick, and Dang-Zheng Liu. "Selberg integrals, super-hypergeometric functions and applications to β-ensembles of random matrices." Random Matrices: Theory and Applications 04, no. 02 (April 2015): 1550007. http://dx.doi.org/10.1142/s2010326315500070.

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We study a new Selberg-type integral with n + m indeterminates, which turns out to be related to the deformed Calogero–Sutherland systems. We show that the integral satisfies a holonomic system of n + m non-symmetric linear partial differential equations. We also prove that a particular hypergeometric function defined in terms of super-Jack polynomials is the unique solution of the system. Some properties such as duality relations, integral formulas, Pfaff–Euler and Kummer transformations are also established. As a direct application, we evaluate the expectation value of ratios of characteristic polynomials in the classical β-ensembles of Random Matrix Theory.

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