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Relevant bibliographies by topics / Random matrices ensembles

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Academic literature on the topic 'Random matrices ensembles'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Random matrices ensembles"

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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Random matrices ensembles"

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Reynolds, Alexi Kirsty. "β-ensembles of random matrices and Jacobi-type matrices." Thesis, University of Bristol, 2017. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730894.

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Andersson, Kasper. "A Review of Gaussian Random Matrices." Thesis, Linköpings universitet, Matematisk statistik, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-171649.

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While many university students get introduced to the concept of statistics early in their education, random matrix theory (RMT) usually first arises (if at all) in graduate level classes. This thesis serves as a friendly introduction to RMT, which is the study of matrices with entries following some probability distribution. Fundamental results, such as Gaussian and Wishart ensembles, are introduced and a discussion of how their corresponding eigenvalues are distributed is presented. Two well-studied applications, namely neural networks and PCA, are discussed where we present how RMT can be applied
Medan många stöter på statistik och sannolikhetslära tidigt under sina universitetsstudier så är det sällan slumpmatristeori (RMT) dyker upp förän på forskarnivå. RMT handlar om att studera matriser där elementen följer någon sannolikhetsfördelning och den här uppsatsen presenterar den mest grundläggande teorin för slumpmatriser. Vi introducerar Gaussian ensembles, Wishart ensembles samt fördelningarna för dem tillhörande egenvärdena. Avslutningsvis så introducerar vi hur slumpmatriser kan användas i neruonnät och i PCA.

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Holcomb, Diane, and Benedek Valkó. "Overcrowding asymptotics for the Sine(beta) process." INST MATHEMATICAL STATISTICS, 2017. http://hdl.handle.net/10150/625509.

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We give overcrowding estimates for the Sine(beta) process, the bulk point process limit of the Gaussian beta-ensemble. We show that the probability of having exactly n points in a fixed interval is given by e(-beta/2n2) log(n)+O(n(2)) as n -> infinity. We also identify the next order term in the exponent if the size of the interval goes to zero.

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Rochet, Jean. "Isolated eigenvalues of non Hermitian random matrices." Thesis, Sorbonne Paris Cité, 2016. http://www.theses.fr/2016USPCB030/document.

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Dans cette thèse, il est question de spiked models pour des matrices aléatoires nonhermitiennes. Plus précisément, on considère des matrices de type A+P, tel que le rang de P reste borné indépendamment de la taille de la matrice qui tend vers l’infini, et tel que A est une matrice aléatoire non-hermitienne. Tout d’abord, on prouve que dans le cas où la matrice P possède des valeurs propres hors du bulk, quelques valeurs propres de A+P (appelées outliers) apparaissent loin de celui-ci. Ensuite, on regarde les fluctuations des outliers de A autour de leurs limites et on montre que celles-ci ont la même distribution que les valeurs propres d’une certaine matrice aléatoire de taille finie. Ce genre de phénomène avait déjà été observé pour des modèles hermitiens. De manière inattendue, on montre que les vitesses de convergence des outliers varient (en fonction de la Réduite de Jordan de P) et que des corrélations peuvent apparaître entre des outliers situés à une distance macroscopique l’un de l’autre. Le premier modèle de matrices non-hermitiennes que l’on considère provient du théorème du Single Ring que l’on doit à Guionnet, Krishnapur et Zeitouni. Un autre modèle étudié est celui des matrices dites “presque” hermitiennes : c’est-à-dire lorsque A est hermitienne mais P ne l’est pas. Enfin, on regarde aussi les outliers pour des matrices Elliptiques Gaussiennes. Cette thèse traite aussi de la convergence en loi de variables aléatoires du type Tr( f (A)M) où A est une matrice du théorème du Single Ring et f est une fonction holomorphe sur un voisinage du bulk et la norme de Frobenius de M est de l’ordre de √N. En corollaire de ce résultat, on obtient des théorèmes type “Centrale Limite” pour les statistiques linéaires de A (pour des fonctions tests holomorphes) et des projections de rang finies de la matrice A (comme par exemple les entrées de la matrice)
This thesis is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type A+P, where the rank of P stays bounded as the dimension goes to infinity and where the matrix A is a non Hermitian random matrix. We first prove that if P has some eigenvalues outside the bulk, then A+P has some eigenvalues (called outliers) away from the bulk. Then, we study the fluctuations of the outliers of A around their limit and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of P) and that some correlations can appear between outliers at a macroscopic distance from each other. The first non Hermitian model studied comes from the Single Ring Theorem due to Guionnet, Krishnapur and Zeitouni. Then we investigated spiked models for nearly Hermitian random matrices : where A is Hermitian but P isn’t. At last, we studied the outliers of Gaussian Elliptic random matrices. This thesis also investigates the convergence in distribution of random variables of the type Tr( f (A)M) where A is a matrix from the Single Ring Theorem and f is analytic on a neighborhood of the bulk and the Frobenius norm of M has order √N. As corollaries, we obtain central limit theorems for linear spectral statistics of A (for analytic test functions) and for finite rank projections of f (A) (like matrix entries)

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Zhou, Da Sheng. "Eigenvalues statistics for restricted trace ensembles." Thesis, University of Macau, 2010. http://umaclib3.umac.mo/record=b2182958.

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6

Veneziani, Alexei Magalhães. "Ensembles de matrizes aleatórias normais: projeção, comportamento assintótico e universalidade dos autovalores." Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-20052008-101058/.

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Uma matriz `A IND.N´ de ordem N ´e normal se e somente se comuta com sua adjunta. Nesta tese investigamos a estatística dos autovalores (no plano complexo) de ensembles de matrizes aleatórias normais quando a ordem N destas tende a infinito. A função distribuição de probabilidade no espaço das matrizes normais atribui, como na mecânica estatística, um peso de Boltzmann `e POT.-NF(`A IND.N´)´ a cada realização `A IND.N´ destas matrizes, onde F é uma função a valores reais invariante por transformações unitárias. Realizando uma mudança de variáveis (das variáveis de entrada para as variáveis espectrais), escrevemos a distribuição marginal conjunta dos autovalores `{`z IND.i´} POT.N´ `IND.i=1´, bem como a função de n-pontos correspondente a vários ensembles, como o determinante de um núcleo integral associado. A partir deste formalismo bem estabelecido na literatura, apresentaremos nesta tese dois tipos de resultados: Primeiramente, explorando a semelhança da distribuição conjunta dos autovalores a um problema variacional sobre as medidas de equilíbrio eletrostático de cargas sujeitas a um potencial externo V : C ? R (escolhendo F(`A IND.N´) = ```sigma´ POT.N´ IND.i´=1 V (`z IND.i´)), podemos aplicar a teoria de potenciais logarítmicos para obter a única medida de equilíbrio coincidente com a função de 1-ponto destes ensembles. Com base nesta teoria, propomos nesta tese um método de interpolação analítica capaz de projetar a medida de equilíbrio dos ensembles normais em medidas de equilíbrio dos ensembles hermitianos e unitários correspondentes. Ilustramos o procedimento com várias aplicações. O segundo tipo de resultados utiliza o método de ponto de sela ao nícleo integral da família de ensembles de matrizes normais com potenciais `V IND.`alfa´´ (z) = `|z| POT.`alfa´´ , z `PERTENCE A´ C e `alfa´ `PERTENCE A´ ]0,`INFINITO´[. Analogamente ao que foi demonstrado em ensembles hermitianos por Deift, estabelecemos por intermédio desta expansão um conceito similar de universalidade para esta família, fazendo uso de mapas conformes e a teoria de espaços de Segal-Bargmann. Sobre o sentido de universalidade definido por G. Oas, mostramos que a afirmação de universalidade neste sentido por este autor é incorreta quando a cauda desta probabilidade é levada em conta.
A matrix `A IND.N´ of order N is normal if and only if it commutes with its adjoint. In the present thesis we investigate the eigenvalues statistics (in the complex plane) of ensembles of normal random matrices when their order N tends to infinite. The probability distribution function in the space of normal matrices attributes, as in statistical mechanics, a Boltzmann weight `e POT.-NF(`A IND.N´)´ at each matrix realization `A IND.N´, where F is a real-valued function invariant by unitary transformations. By performing a change of variables (from entry variables to spectral variables) we write the marginal joint distribution of eigenvalues {`z IND.i´} POT.N´ `IND.i=1´, as well as the n-points functions corresponding to several ensembles, as the determinant of an associated integral kernel. From this formalism well-established in the literature, we shall present in this thesis two types of results: Firstly, exploiting the similarity of joint distribution of eigenvalues to a variational problem on electrostatic equilibrium measures of charges subjected to an external potential V : C - > R (by choosing F(`A IND.N´) = ```sigma´ POT.N´ IND.i´=1 V (`z IND.i´)), we can apply the theory of logarithmic potentials to obtain the unique equilibrium measure coinciding with the 1-point function of these ensembles. Based on this theory, we propose in this thesis a method of analytical interpolation capable of projecting the equilibrium measure of normal ensembles in equilibrium measures of corresponding Hermitian and unitary ensembles. We give several applications of this procedure. The second type of results utilizes the saddle point method applied to integral kernel of a family of normal matrix ensembles with potentials `V IND.`alfa´´ (z) = `|z| POT.`alfa´´ , z `PERTENCE A´ C e `alfa´ `PERTENCE A´ ]0,`INFINITO´[. Similarly to what has been shown in hermitian ensembles by Deift, we established by mean of this expansion a similar concept of universality for this family, making use of conformal maps and theory of Segal-Bargmann space. Concerning the universality defined by G. Oas, we show that the universality claimed by this author is incorrect when the tail of this probability is taking into account.

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Silva, Roberto da. "Distribuição de autovalores de matrizes aleatórias." Universidade de São Paulo, 2000. http://www.teses.usp.br/teses/disponiveis/43/43133/tde-11062002-103116/.

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Em uma detalhada revisão nós obtemos a lei do semi-círculo para a densidade de estados no ensemble gaussiano de Wigner. Também falamos sobre a analogia eletrostática de Dyson, enxergando os autovalores como cargas que se repelem no círculo unitário, mostrando que nesse caso a densidade de estados é uniforme. Em um contexto mais geral nós obtemos a lei do semicírculo, provando o teorema de Glivenko-Cantelli para variáveis fortemente correlacionadas usando um método combinatorial de contagem de trajetos, o que nos dá subsídios para falar em estabilidade da lei do semi-círculo. Também, nesta dissertação nós estudamos as funções de correlação nos ensembles gaussiano e circular, mostrando que sob um adequado reescalamento elas são idênticas. Outros ensembles nesta dissertação foram investigados usando o Método de Gram para o caso em que os autovalores são limitados em um intervalo. Computamos a densidade de estados para cada um desses ensembles. Mais precisamente no ensemble de Chebychev, os resultados foram obtidos analiticamente e nesse ensemble além da densidade de estados, também traçamos grácos da função de correlação truncada.
In a detailed review we obtain a semi-circle law for the density of states in theWigners Gaussian Ensemble. Also we talk about Dysons Analogy, seeing the eigenvalues like charges that repulse themselves in the unitary circle, showing that this case the density of states is uniform. In a more general context we obtain the semi-circle law, proving the Glivenko-Cantelli Theorem to strongly correlated variables, using a combinatorial method of Paths' Counting. Thus we are showing the stability of the semi-circle Law. Also, in this dissertation we study the correlation functions in the Gaussian and Circular ensembles showing that using the Gram's Method in the case that eigenvalues are limited in a interval. In these ensembles we computed the density of states. More precisely, in a Chebychev ensemble the results were obtained analytically. In this ensemble, we also obtain graphics of the truncated correlation function.

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Santos, Gabriel Marinello de Souza. "Matrizes aleatórias no ensemble." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-06112014-154150/.

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O estudo de matrizes aleatórias na física tradicionalmente ocorre no contexto dos modelos de Wigner e na estatística por modelos de Wishart, que se conectam através do threefold way de Dyson para matrizes aleatórias reais, complexas e de quaternios indexadas respectivamente pelo índice B = 1; 2; 4 de Dyson. Estudos recentes mostraram o caminho para que estes modelos fossem generalizados para valores reais de B, permitindo o estudo de ensembles com índice arbitrário. Neste trabalho, estudamos as propriedades estatísticas destes sistemas e exploramos a física subjacente nos modelos de Wigner e Wishart e investigamos, através de cálculos numéricos, os efeitos de localização nos modelos de geral. Também introduzimos quebras na simetria desta nova forma e estudamos numericamente os resultados da estatística dos sistemas perturbados.
The study of random matrices in physics has traditionally occurred in the context of Wigner models and in statistics by Wishart models, which are connected through Dyson\'s threefold way for real, complex and quaternion random matrices index by the Dyson _ = 1; 2; 4 index, respectively. Recent studies have shown the way by which these models are generalized for real values of _, allowing for the study the ensembles with arbitrary index. In this work, we study the statistical properties of these systems and explore the underlying physics in Wigner\'s and Wishart\'s models through and investigate through numerical calculations the e_ects of localization in general _ models. We also introduce symmetry breaks in this new form and study numerically the results of the statistics of the disturbed systems.

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Matić, Rada. "Estimation Problems Related to Random Matrix Ensembles." Doctoral thesis, 2006. http://hdl.handle.net/11858/00-1735-0000-0006-B406-B.

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Wang, Dong. "Spiked models in Wishart ensemble /." 2008.

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Books on the topic "Random matrices ensembles"

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1973-, Gioev Dimitri, ed. Random matrix theory: Invariant ensembles and universality. Providence, R.I: American Mathematical Society, 2009.

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Ferrari, Patrik L., and Herbert Spohn. Random matrices and Laplacian growth. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.39.

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This article reviews the theory of random matrices with eigenvalues distributed in the complex plane and more general ‘beta ensembles’ (logarithmic gases in 2D). It first considers two ensembles of random matrices with complex eigenvalues: ensemble C of general complex matrices and ensemble N of normal matrices. In particular, it describes the Dyson gas picture for ensembles of matrices with general complex eigenvalues distributed on the plane. It then presents some general exact relations for correlation functions valid for any values of N and β before analysing the distribution and correlations of the eigenvalues in the large N limit. Using the technique of boundary value problems in two dimensions and elements of the potential theory, the article demonstrates that the finite-time blow-up (a cusp–like singularity) of the Laplacian growth with zero surface tension is a critical point of the normal and complex matrix models.

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Kravtsov, Vladimir. Heavy-tailed random matrices. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.13.

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This article considers non-Gaussian random matrices consisting of random variables with heavy-tailed probability distributions. In probability theory heavy tails of distributions describe rare but violent events which usually have a dominant influence on the statistics. Furthermore, they completely change the universal properties of eigenvalues and eigenvectors of random matrices. This article focuses on the universal macroscopic properties of Wigner matrices belonging to the Lévy basin of attraction, matrices representing stable free random variables, and a class of heavy-tailed matrices obtained by parametric deformations of standard ensembles. It first examines the properties of heavy-tailed symmetric matrices known as Wigner–Lévy matrices before discussing free random variables and free Lévy matrices as well as heavy-tailed deformations. In particular, it describes random matrix ensembles obtained from standard ensembles by a reweighting of the probability measure. It also analyses several matrix models belonging to heavy-tailed random matrices and presents methods for integrating them.

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Majumdar, Satya N. Random growth models. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.38.

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This article discusses the connection between a particular class of growth processes and random matrices. It first provides an overview of growth model, focusing on the TASEP (totally asymmetric simple exclusion process) with parallel updating, before explaining how random matrices appear. It then describes multi-matrix models and line ensembles, noting that for curved initial data the spatial statistics for large time t is identical to the family of largest eigenvalues in a Gaussian Unitary Ensemble (GUE multi-matrix model. It also considers the link between the line ensemble and Brownian motion, and whether this persists on Gaussian Orthogonal Ensemble (GOE) matrices by comparing the line ensembles at fixed position for the flat polynuclear growth model (PNG) and at fixed time for GOE Brownian motions. Finally, it examines (directed) last passage percolation and random tiling in relation to growth models.

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Speicher, Roland. Random banded and sparse matrices. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.23.

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This article discusses some mathematical results and conjectures about random band matrix ensembles (RBM) and sparse matrix ensembles. Spectral problems of RBM and sparse matrices can be expressed in terms of supersymmetric (SUSY) statistical mechanics that provides a dual representation for disordered quantum systems. This representation offers important insights into nonperturbative aspects of the spectrum and eigenfunctions of RBM. The article first presents the definition of RBM ensembles before considering the density of states, the behaviour of eigenvectors, and eigenvalue statistics for RBM and sparse random matrices. In particular, it highlights the relations with random Schrödinger (RS) and the role of the dimension of the lattice. It also describes the connection between RBM and statistical mechanics, the spectral theory of large random sparse matrices, conjectures and theorems about eigenvectors and local spacing statistics, and the RS operator on the Cayley tree or Bethe lattice.

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Eynard, Bertrand. Random matrices and loop equations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0007.

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This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.

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Kota, V. K. B. Embedded Random Matrix Ensembles in Quantum Physics. Springer, 2014.

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Kuijlaars, Arno. Supersymmetry. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.7.

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This article examines conceptual and structural issues related to supersymmetry. It first provides an overview of generating functions before discussing supermathematics, with a focus on Grassmann or anticommuting variables, vectors and matrices, groups and symmetric spaces, and derivatives and integrals. It then considers various applications of supersymmetry to random matrices, such as the representation of the ensemble average and the Hubbard–Stratonovich transformation, along with its generalization and superbosonization. It also describes matrix δ functions and an alternative representation as well as important and technically challenging problems that supersymmetry addresses beyond the invariant and factorizing ensembles. The article concludes with an analysis of the supersymmetric non-linear σ model, Brownian motion in superspace, circular ensembles and the Colour-Flavour-Transformation.

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Bohigas, Oriol, and Hans Weidenmuller. History – an overview. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.2.

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This article discusses the first four decades of the history of random matrix theory (RMT), that is, until about 1990. It first considers Niels Bohr's formulation of the concept of the compound nucleus, which is at the root of the use of random matrices in physics, before analysing the development of the theory of spectral fluctuations. In particular, it examines the Wishart ensemble; Dyson's classification leading to the three canonical ensembles — Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), and Gaussian Symplectic Ensemble (GSE); and the breaking of a symmetry or an invariance. It also describes how random matrix models emerged from quantum physics, more specifically from a statistical approach to the strongly interacting many-body system of the atomic nucleus. The article concludes with an overview of data on nuclear resonances, many-body theory, chaos, number theory, scattering theory, replica trick and supersymmetry, disordered solids, and interacting fermions and field theory.

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Akemann, Gernot, Jinho Baik, and Philippe Di Francesco, eds. The Oxford Handbook of Random Matrix Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.001.0001.

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This handbook showcases the major aspects and modern applications of random matrix theory (RMT). It examines the mathematical properties and applications of random matrices and some of the reasons why RMT has been very successful and continues to enjoy great interest among physicists, mathematicians and other scientists. It also discusses methods of solving RMT, basic properties and fundamental objects in RMT, and different models and symmetry classes in RMT. Topics include the use of classical orthogonal polynomials (OP) and skew-OP to solve exactly RMT ensembles with unitary, and orthogonal or symplectic invariance respectively, all at finite matrix size; the supersymmetric and replica methods; determinantal point processes; Painlevé transcendents; the fundamental property of RMT known as universality; RNA folding; two-dimensional quantum gravity; string theory; and the mathematical concept of free random variables. In addition to applications to mathematics and physics, the book considers broader applications to other sciences, including economics, engineering, biology, and complex networks.

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Book chapters on the topic "Random matrices ensembles"

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Livan, Giacomo, Marcel Novaes, and Pierpaolo Vivo. "Classical Ensembles: Wishart-Laguerre." In Introduction to Random Matrices, 89–95. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-70885-0_13.

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Pastur, Leonid, and Mariya Shcherbina. "Wigner ensembles." In Eigenvalue Distribution of Large Random Matrices, 525–82. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/surv/171/18.

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Pastur, Leonid, and Mariya Shcherbina. "Gaussian ensembles: Semicircle law." In Eigenvalue Distribution of Large Random Matrices, 35–67. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/surv/171/02.

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Pastur, Leonid, and Mariya Shcherbina. "Wishart and Laguerre ensembles." In Eigenvalue Distribution of Large Random Matrices, 177–210. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/surv/171/07.

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Pastur, Leonid, and Mariya Shcherbina. "Classical compact groups ensembles: Global regime." In Eigenvalue Distribution of Large Random Matrices, 211–48. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/surv/171/08.

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Pastur, Leonid, and Mariya Shcherbina. "Classical compact group ensembles: Further results." In Eigenvalue Distribution of Large Random Matrices, 249–74. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/surv/171/09.

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Pastur, Leonid, and Mariya Shcherbina. "Gaussian ensembles: Joint eigenvalue distribution and related results." In Eigenvalue Distribution of Large Random Matrices, 101–27. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/surv/171/04.

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Pastur, Leonid, and Mariya Shcherbina. "Gaussian ensembles: Central Limit Theorem for linear eigenvalue statistics." In Eigenvalue Distribution of Large Random Matrices, 69–100. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/surv/171/03.

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Li, Na, Martin Crane, Heather J. Ruskin, and Cathal Gurrin. "Random Matrix Ensembles of Time Correlation Matrices to Analyze Visual Lifelogs." In MultiMedia Modeling, 400–411. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-04114-8_34.

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Pastur, Leonid, and Mariya Shcherbina. "Gaussian unitary ensemble." In Eigenvalue Distribution of Large Random Matrices, 129–58. Providence, Rhode Island: American Mathematical Society, 2011. http://dx.doi.org/10.1090/surv/171/05.

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Conference papers on the topic "Random matrices ensembles"

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STOLZ, MICHAEL. "ENSEMBLES OF HERMITIAN RANDOM MATRICES ASSOCIATED TO SYMMETRIC SPACES." In Proceedings of the Fourth German–Japanese Symposium. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812832825_0019.

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Benet, Luis, Saúl Hernández-Quiroz, Thomas H. Seligman, Kurt B. Wolf, Luis Benet, Juan Mauricio Torres, and Peter O. Hess. "Fidelity decay of the two-level bosonic embedded ensembles of random matrices." In SYMMETRIES IN NATURE: SYMPOSIUM IN MEMORIAM MARCOS MOSHINSKY. AIP, 2010. http://dx.doi.org/10.1063/1.3537867.

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Weaver, Richard L. "On Eigenmode Statistics and Power Variances in Randomly Shaped Membranes." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0165.

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Abstract The eigenmodes and eigenfrequencies of finite differenced membranes with rough boundaries are obtained numerically. Such systems are approximations to reverberation rooms. The statistics of the eigenfrequencies are found to be identical to those of the Gaussian Orthogonal Ensemble (GOE) of random matrices. The Wigner conjecture that the modes may be understood to be uncorrelated random superpositions of plane waves with no significant inter-modal correlations is also corroborated. The consequences of these eigen-statistics upon the statistics of responses are discussed.

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El Gamal, Aly, Navid Naderializadeh, and A. Salman Avestimehr. "When does an ensemble of matrices with randomly scaled rows lose rank?" In 2015 IEEE International Symposium on Information Theory (ISIT). IEEE, 2015. http://dx.doi.org/10.1109/isit.2015.7282706.

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