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Journal articles on the topic 'Random unitary matrix'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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1

Pluhař, Z., and H. A. Weidenmüller. "Ergodicity of Unitary Random-Matrix Ensembles." Annals of Physics 282, no. 2 (2000): 247–69. http://dx.doi.org/10.1006/aphy.2000.6029.

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2

Bornemann, Folkmar, and Peter J. Forrester. "Singular values and evenness symmetry in random matrix theory." Forum Mathematicum 28, no. 5 (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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3

Freilikher, V., E. Kanzieper, and I. Yurkevich. "Unitary random-matrix ensemble with governable level confinement." Physical Review E 53, no. 3 (1996): 2200–2209. http://dx.doi.org/10.1103/physreve.53.2200.

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4

Pluhař, Z., and H. A. Weidenmüller. "Ergodicity of Random-Matrix Theories: The Unitary Case." Physical Review Letters 84, no. 13 (2000): 2833–36. http://dx.doi.org/10.1103/physrevlett.84.2833.

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5

Claeys, Tom, Arno B. J. Kuijlaars, and Dong Wang. "Correlation kernels for sums and products of random matrices." Random Matrices: Theory and Applications 04, no. 04 (2015): 1550017. http://dx.doi.org/10.1142/s2010326315500173.

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Let [Formula: see text] be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of [Formula: see text] and [Formula: see text], where [Formula: see text] is a complex Ginibre matrix and [Formula: see text] is a truncated unitary matrix. We also consider the product of [Formula: see text] and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum [Formula: see text] where [Formula: see text] is a GUE matrix and [Formula: see text] is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of [Formula: see text] follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.

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6

Hughes, C. P., J. P. Keating, and Neil O'Connell. "On the Characteristic Polynomial¶ of a Random Unitary Matrix." Communications in Mathematical Physics 220, no. 2 (2001): 429–51. http://dx.doi.org/10.1007/s002200100453.

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7

Borodin, Alexei, Vadim Gorin, and Eugene Strahov. "Product Matrix Processes as Limits of Random Plane Partitions." International Mathematics Research Notices 2020, no. 20 (2019): 6713–68. http://dx.doi.org/10.1093/imrn/rny297.

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AbstractWe consider a random process with discrete time formed by squared singular values of products of truncations of Haar-distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives determinantal formulas for (dynamical) correlation functions and a contour integral representation for the correlation kernel. The relation with the Schur processes implies that the continuous limit of marginals for q-distributed plane partitions coincides with the joint law of squared singular values for products of truncations of Haar-distributed random unitary matrices. We provide structural reasons for this coincidence that may also extend to other classes of random matrices.

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8

Grimm, Uwe. "Level-spacing distributions of the Gaussian unitary random matrix ensemble." physica status solidi (b) 241, no. 9 (2004): 2139–47. http://dx.doi.org/10.1002/pssb.200404784.

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9

Bertini, Bruno, Pavel Kos, and Tomaž Prosen. "Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits." Communications in Mathematical Physics 387, no. 1 (2021): 597–620. http://dx.doi.org/10.1007/s00220-021-04139-2.

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10

Nechita, Ion. "On the Separability of Unitarily Invariant Random Quantum States: The Unbalanced Regime." Advances in Mathematical Physics 2018 (2018): 1–13. http://dx.doi.org/10.1155/2018/7105074.

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We study entanglement-related properties of random quantum states which are unitarily invariant, in the sense that their distribution is left unchanged by conjugation with arbitrary unitary operators. In the large matrix size limit, the distribution of these random quantum states is characterized by their limiting spectrum, a compactly supported probability distribution. We prove several results characterizing entanglement and the PPT property of random bipartite unitarily invariant quantum states in terms of the limiting spectral distribution, in the unbalanced asymptotical regime where one of the two subsystems is fixed, while the other one grows in size.

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11

Johansson, Kurt. "The longest increasing subsequence in a random permutation and a unitary random matrix model." Mathematical Research Letters 5, no. 1 (1998): 68–82. http://dx.doi.org/10.4310/mrl.1998.v5.n1.a6.

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12

Witte, N. S., P. J. Forrester, and Christopher M. Cosgrove. "Integrability, random matrices and Painlevé transcendents." ANZIAM Journal 44, no. 1 (2002): 41–50. http://dx.doi.org/10.1017/s1446181100007896.

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AbstractThe probability that an interval I is free of eigenvalues in a matrix ensemble with unitary symmetry is given by a Fredholm determinant. When the weight function in the matrix ensemble is a classical weight function, and the interval I includes an endpoint of the support, Tracy and Widom have given a formalism which gives coupled differential equations for the required probability and some auxiliary quantities. We summarize and extend earlier work by expressing the probability and some of the auxiliary quantities in terms of Painlevé transcendents.

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13

Hiary, G. A., and M. O. Rubinstein. "Uniform asymptotics of the coefficients of unitary moment polynomials." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2128 (2010): 1073–100. http://dx.doi.org/10.1098/rspa.2010.0430.

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Keating and Snaith showed that the 2 k th absolute moment of the characteristic polynomial of a random unitary matrix evaluated on the unit circle is given by a polynomial of degree k 2 . In this article, uniform asymptotics for the coefficients of that polynomial are derived, and a maximal coefficient is located. Some of the asymptotics are given in an explicit form. Numerical data to support these calculations are presented. Some apparent connections between the random matrix theory and the Riemann zeta function are discussed.

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14

Kieburg, Mario, and Holger Kösters. "Exact relation between singular value and eigenvalue statistics." Random Matrices: Theory and Applications 05, no. 04 (2016): 1650015. http://dx.doi.org/10.1142/s2010326316500155.

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We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that each of these joint densities determines the other one. Moreover, we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore, we show how to generalize the relation between the singular value and eigenvalue statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.

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15

Meckes, Elizabeth, and Kathryn Stewart. "Eigenvalue rigidity for truncations of random unitary matrices." Random Matrices: Theory and Applications 10, no. 01 (2020): 2150015. http://dx.doi.org/10.1142/s2010326321500155.

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We consider the empirical eigenvalue distribution of an [Formula: see text] principal submatrix of an [Formula: see text] random unitary matrix distributed according to Haar measure. For [Formula: see text] and [Formula: see text] large with [Formula: see text], the empirical spectral measure is well approximated by a deterministic measure [Formula: see text] supported on the unit disc. In earlier work, we showed that for fixed [Formula: see text] and [Formula: see text], the bounded-Lipschitz distance [Formula: see text] between the empirical spectral measure and the corresponding [Formula: see text] is typically of order [Formula: see text] or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.

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16

Dong, Zhishan, Tiefeng Jiang, and Danning Li. "Circular law and arc law for truncation of random unitary matrix." Journal of Mathematical Physics 53, no. 1 (2012): 013301. http://dx.doi.org/10.1063/1.3672885.

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17

Bourgade, P., C. P. Hughes, A. Nikeghbali, and M. Yor. "The characteristic polynomial of a random unitary matrix: A probabilistic approach." Duke Mathematical Journal 145, no. 1 (2008): 45–69. http://dx.doi.org/10.1215/00127094-2008-046.

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18

Fyodorov, Yan V., and Jonathan P. Keating. "Freezing transitions and extreme values: random matrix theory, and disordered landscapes." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, no. 2007 (2014): 20120503. http://dx.doi.org/10.1098/rsta.2012.0503.

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We argue that the freezing transition scenario , previously conjectured to occur in the statistical mechanics of 1/ f -noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p N ( θ ) of large N × N random unitary (circular unitary ensemble) matrices U N ; i.e. the extreme value statistics of p N ( θ ) when . In addition, we argue that it leads to multi-fractal-like behaviour in the total length μ N ( x ) of the intervals in which | p N ( θ )|> N x , x >0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ ( s ) over stretches of the critical line of given constant length and present the results of numerical computations of the large values of ). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

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19

Olver, Sheehan, Raj Rao Nadakuditi, and Thomas Trogdon. "Sampling unitary ensembles." Random Matrices: Theory and Applications 04, no. 01 (2015): 1550002. http://dx.doi.org/10.1142/s2010326315500021.

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We develop a computationally efficient algorithm for sampling from a broad class of unitary random matrix ensembles that includes but goes well beyond the straightforward to sample Gaussian unitary ensemble (GUE). The algorithm exploits the fact that the eigenvalues of unitary ensembles (UEs) can be represented as a determinantal point process whose kernel is given in terms of orthogonal polynomials. Consequently, our algorithm can be used to sample from UEs for which the associated orthogonal polynomials can be numerically computed efficiently. By facilitating high accuracy sampling of non-classical UEs, the algorithm can aid in the experimentation-based formulation or refutation of universality conjectures involving eigenvalue statistics that might presently be unamenable to theoretical analysis. Examples of such experiments are included.

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20

Van Peski, Roger. "Spectral distributions of periodic random matrix ensembles." Random Matrices: Theory and Applications 10, no. 01 (2019): 2150011. http://dx.doi.org/10.1142/s2010326321500118.

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Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as [Formula: see text]-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an [Formula: see text] Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu–Kopp–Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a ‘complex version’ of a [Formula: see text] submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.

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21

Claeys, Tom, Arno Kuijlaars, and Maarten Vanlessen. "Multi-critical unitary random matrix ensembles and the general Painlevé II equation." Annals of Mathematics 168, no. 2 (2008): 601–41. http://dx.doi.org/10.4007/annals.2008.168.601.

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22

Kieburg, M., and A. Monteleone. "Local tail statistics of heavy-tailed random matrix ensembles with unitary invariance." Journal of Physics A: Mathematical and Theoretical 54, no. 32 (2021): 325201. http://dx.doi.org/10.1088/1751-8121/ac0d6c.

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23

Diaconis, Persi, and Mehrdad Shahshahani. "On the eigenvalues of random matrices." Journal of Applied Probability 31, A (1994): 49–62. http://dx.doi.org/10.1017/s0021900200106989.

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Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.

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24

Blower, Gordon, and Andrew McCafferty. "Discrete Tracy–Widom operators." Proceedings of the Edinburgh Mathematical Society 52, no. 3 (2009): 545–59. http://dx.doi.org/10.1017/s001309150700140x.

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AbstractIntegrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.

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25

Strahov, Eugene. "Dynamical correlation functions for products of random matrices." Random Matrices: Theory and Applications 04, no. 04 (2015): 1550020. http://dx.doi.org/10.1142/s2010326315500203.

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We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case, we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlation functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard–Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.

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26

Kanazawa, Takuya. "Unitary matrix integral for two-color QCD and the GSE-GUE crossover in random matrix theory." Physics Letters B 819 (August 2021): 136416. http://dx.doi.org/10.1016/j.physletb.2021.136416.

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27

Itoi, Chigak. "Universal wide correlators in non-gaussian orthogonal, unitary and symplectic random matrix ensembles." Nuclear Physics B 493, no. 3 (1997): 651–59. http://dx.doi.org/10.1016/s0550-3213(97)00158-2.

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28

Skvortsov, M. A., D. M. Basko, and V. E. Kravtsov. "Energy absorption in time-dependent unitary random matrix ensembles: Dynamic versus anderson localization." Journal of Experimental and Theoretical Physics Letters 80, no. 1 (2004): 54–60. http://dx.doi.org/10.1134/1.1800215.

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29

Kieburg, Mario, Johan Grönqvist, and Thomas Guhr. "Arbitrary rotation invariant random matrix ensembles and supersymmetry: orthogonal and unitary-symplectic case." Journal of Physics A: Mathematical and Theoretical 42, no. 27 (2009): 275205. http://dx.doi.org/10.1088/1751-8113/42/27/275205.

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30

Marchal, O. "Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices." Random Matrices: Theory and Applications 04, no. 03 (2015): 1550011. http://dx.doi.org/10.1142/s2010326315500112.

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The purpose of this paper is to study the eigenvalues [Formula: see text] of Ut where U is a large N×N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first-orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

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31

Forrester, Peter J., and Anthony Mays. "Finite-size corrections in random matrix theory and Odlyzko’s dataset for the Riemann zeros." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2182 (2015): 20150436. http://dx.doi.org/10.1098/rspa.2015.0436.

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Odlyzko has computed a dataset listing more than 10 9 successive Riemann zeros, starting from a zero number to beyond 10 23 . This dataset relates to random matrix theory as, according to the Montgomery–Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and co-workers, have used N × N random unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natural process to apply to the dataset is to minimize it by deleting each member independently with some specified probability, and we proceed to compute empirical two-point correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterize the order 1/ N 2 correction term to the spacing distribution for random unitary matrices in terms of a second-order differential equation with coefficients that are Painlevé transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. In comparison to the Riemann zero data accurate agreement is exhibited.

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32

KAVALOV, AL R., R. L. MKRTCHYAN, and L. A. ZURABYAN. "RANDOM MATRICES WITH DISCRETE SPECTRUM AND FINITE TODA CHAINS." Modern Physics Letters A 06, no. 39 (1991): 3627–33. http://dx.doi.org/10.1142/s0217732391004188.

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Restricting the eigenvalues of matrices in random matrix models produces different models (Hermitian, unitary, (anti)symmetric, Penner's, etc.). We consider the model in which the eigenvalues receive values from some discrete finite set of points, establish the connection of such a model with a finite Toda chain and study the details of this connection. We derive also the string equation, which in the limit, when eigenvalues become dense on a real axis, tends to the usual string equation.

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33

Neuschel, Thorsten. "Spectral densities of singular values of products of Gaussian and truncated unitary random matrices." Random Matrices: Theory and Applications 09, no. 04 (2019): 2050014. http://dx.doi.org/10.1142/s2010326320500148.

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We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case, we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian, we derive elementary expressions for the limiting densities using suitable parameterizations for the spectral variable. Moreover, in all cases we study the behavior of the densities at the boundary of the spectrum.

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34

JAIN, SANJAY. "UNIVERSAL CORRELATIONS IN RANDOM MATRICES: QUANTUM CHAOS, THE 1/r2 INTEGRABLE MODEL, AND QUANTUM GRAVITY." Modern Physics Letters A 11, no. 15 (1996): 1201–19. http://dx.doi.org/10.1142/s0217732396001223.

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Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-D integrable model with the 1/r2 interaction (the Calogero-Sutherland-Moser system) and 2-D quantum gravity. We review the connection of RMT with these areas. We also discuss the method of loop equations for determining correlation functions in RMT, and smoothed global eigenvalue correlators in the two-matrix model for Gaussian orthogonal, unitary and symplectic ensembles.

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35

Farrell, Brendan, and Raj Rao Nadakuditi. "Local spectrum of truncations of Kronecker products of Haar distributed unitary matrices." Random Matrices: Theory and Applications 04, no. 01 (2015): 1550001. http://dx.doi.org/10.1142/s201032631550001x.

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We address the local spectral behavior of the random matrix [Formula: see text] where U is a Haar distributed unitary matrix of size n × n, the factor k is at most c0 lg n for a small constant c0 > 0, and Π1, Π2 are arbitrary projections on [Formula: see text] of ranks proportional to nk. We prove that in this setting the k-fold Kronecker product behaves similarly to the well-studied case when k = 1.

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36

Malacarne, Sara, and Sergey Neshveyev. "Martin boundaries of the duals of free unitary quantum groups." Compositio Mathematica 155, no. 6 (2019): 1171–93. http://dx.doi.org/10.1112/s0010437x19007322.

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Given a free unitary quantum group $G=A_{u}(F)$, with $F$ not a unitary $2\times 2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-${\hat{G}}$-invariant, irreducible, finite-range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.

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37

Akemann, G., and M. Bender. "Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles." Journal of Mathematical Physics 51, no. 10 (2010): 103524. http://dx.doi.org/10.1063/1.3496899.

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38

Itoi, Chigak, Hisamitsu Mukaida, and Yoshinori Sakamoto. "Replica method for wide correlators in Gaussian orthogonal, unitary and symplectic random matrix ensembles." Journal of Physics A: Mathematical and General 30, no. 16 (1997): 5709–25. http://dx.doi.org/10.1088/0305-4470/30/16/014.

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39

Killip, Rowan, and Rostyslav Kozhan. "Matrix Models and Eigenvalue Statistics for Truncations of Classical Ensembles of Random Unitary Matrices." Communications in Mathematical Physics 349, no. 3 (2016): 991–1027. http://dx.doi.org/10.1007/s00220-016-2658-z.

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40

ABUL-MAGD, A. Y., and M. ABDEL-MAGEED. "KAPPA-DEFORMED RANDOM-MATRIX THEORY BASED ON KANIADAKIS STATISTICS." Modern Physics Letters B 26, no. 10 (2012): 1250059. http://dx.doi.org/10.1142/s0217984912500595.

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We present a possible extension of the random-matrix theory, which is widely used to describe spectral fluctuations of chaotic systems. By considering the Kaniadakis non-Gaussian statistics, characterized by the index κ (Boltzmann–Gibbs entropy is recovered in the limit κ → 0), we propose the non-Gaussian deformations (κ ≠ 0) of the conventional orthogonal and unitary ensembles of random matrices. The joint eigenvalue distributions for the κ-deformed ensembles are derived by applying the principle maximum entropy to Kaniadakis entropy. The resulting distribution functions are base invariant as they depend on the matrix elements in a trace form. Using these expressions, we introduce a new generalized form of the Wigner surmise valid for nearly-chaotic mixed systems, where a basis-independent description is still expected to hold. We motivate the necessity of such generalization by the need to describe the transition of the spacing distribution from chaos to order, at least in the initial stage. We show several examples about the use of the generalized Wigner surmise to the analysis of the results of a number of previous experiments and numerical experiments. Our results suggest the entropic index κ as a measure for deviation from the state of chaos. We also introduce a κ-deformed Porter–Thomas distribution of transition intensities, which fits the experimental data for mixed systems better than the commonly-used gamma-distribution.

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41

Benaych-Georges, Florent, Guillaume Cébron, and Jean Rochet. "Fluctuation of Matrix Entries and Application to Outliers of Elliptic Matrices." Canadian Journal of Mathematics 70, no. 1 (2018): 3–25. http://dx.doi.org/10.4153/cjm-2017-024-8.

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AbstractFor any family of N ⨯ N randommatrices that is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type Tr(AkM), where the matrix M is deterministic (such random variables include, for example, the normalized matrix entries of Ak). A consequence is the asymptotic independence of the projection of the matrices Ak onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Formof the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other.These phenomena have already been observed with random matrices from the Single Ring Theorem.

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42

HAAGERUP, UFFE, and STEEN THORBJØRNSEN. "ASYMPTOTIC EXPANSIONS FOR THE GAUSSIAN UNITARY ENSEMBLE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 15, no. 01 (2012): 1250003. http://dx.doi.org/10.1142/s0219025712500038.

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Let g : ℝ → ℂ be a C∞-function with all derivatives bounded and let tr n denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value 𝔼{ tr n(g(Xn))} for a rather general class of random matrices Xn, including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a [Formula: see text] random matrix Xn that [Formula: see text] where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients αj(g), j ∈ ℕ, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov { Tr n[f(Xn)], Tr n[g(Xn)]}, where f is a function of the same kind as g, and Tr n = n tr n. Special focus is drawn to the case where [Formula: see text] and [Formula: see text] for λ, μ in ℂ\ℝ. In this case the mean and covariance considered above correspond to, respectively, the one- and two-dimensional Cauchy (or Stieltjes) transform of the [Formula: see text].

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43

Hiai, Fumio, Dénes Petz, and Yoshimichi Ueda. "A Free Logarithmic Sobolev Inequality on the Circle." Canadian Mathematical Bulletin 49, no. 3 (2006): 389–406. http://dx.doi.org/10.4153/cmb-2006-039-7.

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AbstractFree analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

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44

Smolyarenko, I. E., and B. D. Simons. "Parametric spectral statistics in unitary random matrix ensembles: from distribution functions to intra-level correlations." Journal of Physics A: Mathematical and General 36, no. 12 (2003): 3551–67. http://dx.doi.org/10.1088/0305-4470/36/12/339.

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45

Pastur, L., and M. Shcherbina. "Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles." Journal of Statistical Physics 86, no. 1-2 (1997): 109–47. http://dx.doi.org/10.1007/bf02180200.

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46

Xu, Shuai-Xia, and Dan Dai. "Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System." Communications in Mathematical Physics 365, no. 2 (2018): 515–67. http://dx.doi.org/10.1007/s00220-018-3257-y.

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47

Meckes, E. S., and M. W. Meckes. "A Sharp Rate of Convergence for the Empirical Spectral Measure of a Random Unitary Matrix." Journal of Mathematical Sciences 238, no. 4 (2019): 530–36. http://dx.doi.org/10.1007/s10958-019-04255-4.

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48

Warrier, Latha S. "Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm." Journal of Computational Methods in Physics 2013 (September 11, 2013): 1–5. http://dx.doi.org/10.1155/2013/235624.

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The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector Va. The eigenstate is a basis vector in the orthonormal eigenspace. Finding another eigenvalue, using a random approximate eigenvector, may require many trials as the trial may repeatedly result in the eigenvalue measured earlier. We present a method involving orthogonalization of the eigenstate obtained in a trial. It is used as the Va for the next trial. Because of the orthogonal construction, Abrams-Lloyd algorithm will not repeat the eigenvalue measured earlier. Thus, all the eigenvalues are obtained in sequence without repetitions. An operator that anticommutes with a unitary operator orthogonalizes the eigenvectors of the unitary. We implemented the method on the programming language model of quantum computation and tested it on a unitary matrix representing the time evolution operator of a small spin chain. All the eigenvalues of the operator were obtained sequentially. Another use of the first eigenvector from Abrams-Lloyd algorithm is preparing a state that is the uniform superposition of all the eigenvectors. This is possible by nonorthogonalizing the first eigenvector in all dimensions and then applying the Abrams-Lloyd algorithm steps stopping short of the last measurement.

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49

Sabri, H., Sh S. Hashemi, B. R. Maleki, and M. A. Jafarizadeh. "Generalization of Brody distribution for statistical investigation." Random Matrices: Theory and Applications 03, no. 04 (2014): 1450017. http://dx.doi.org/10.1142/s2010326314500178.

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In this paper, Brody distribution is generalized to explore the Poisson, Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble limits of Random Matrix Theory in the nearest neighbor spacing statistic framework. Parameters of new distribution are extracted via Maximum Likelihood Estimation technique for different sequences. This general distribution suggests more exact results in comparison with the results of other estimation methods and distribution functions.

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Kota, V. K. B., and Manan Vyas. "Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles." Annals of Physics 359 (August 2015): 252–89. http://dx.doi.org/10.1016/j.aop.2015.04.029.

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