Universality

2018 ◽  
Author(s):  
Mark Adler
Keyword(s):  
Random Matrices ◽  
Singular Points ◽  
Unitary Matrix ◽  
Edge Points ◽  
Universality Classes ◽  
Correlation Kernels ◽  
Orthogonal And Symplectic Ensembles ◽  
Matrix Ensembles ◽  
Basic Characteristics

This article deals with the universality of eigenvalue spacings, one of the basic characteristics of random matrices. It first discusses the heuristic meaning of universality before describing the standard universality classes (sine, Airy, Bessel) and their appearance in unitary, orthogonal, and symplectic ensembles. It then examines unitary matrix ensembles in more detail and shows that universality in these ensembles comes down to the convergence of the properly scaled eigenvalue correlation kernels. It also analyses the Riemann–Hilbert method, along with certain non-standard universality classes that arise at singular points in the limiting spectrum. Finally, it considers the limiting kernels for each of the three types of singular points, namely interior singular points, singular edge points, and exterior singular points.

scholarly journals Modelling equilibration of local many-body quantum systems by random graph ensembles

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 273 ◽  
Author(s):  
Daniel Nickelsen ◽  
Michael Kastner
Keyword(s):  
Hilbert Space ◽  
Random Matrices ◽  
Network Theory ◽  
Maximum Flow ◽  
Quantum Systems ◽  
Complex Matrix ◽  
Many Body ◽  
Random Matrix Ensembles ◽  
Matrix Ensembles ◽  
Exact Diagonalisation

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.

scholarly journals Dynamical correlation functions for products of random matrices

2015 ◽  
Vol 04 (04) ◽  
pp. 1550020 ◽  
Author(s):  
Eugene Strahov
Keyword(s):  
Random Matrices ◽  
Discrete Time ◽  
Random Matrix ◽  
Correlation Functions ◽  
Singular Values ◽  
Matrix Product ◽  
Dynamical Correlation ◽  
Products Of Random Matrices ◽  
Special Cases ◽  
Correlation Kernels

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.

scholarly journals Gap probabilities for edge intervals in finite Gaussian and Jacobi unitary matrix ensembles

Nonlinearity ◽  
2000 ◽  
Vol 13 (5) ◽  
pp. 1439-1464 ◽  
Author(s):  
N S Witte ◽  
P J Forrester ◽  
Christopher M Cosgrove
Keyword(s):  
Unitary Matrix ◽  
Matrix Ensembles ◽  
Gap Probabilities

scholarly journals Singular values and evenness symmetry in random matrix theory

2016 ◽  
Vol 28 (5) ◽  
pp. 873-891 ◽  
Author(s):  
Folkmar Bornemann ◽  
Peter J. Forrester
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Functional Form ◽  
Matrix Theory ◽  
Stereographic Projection ◽  
Singular Values ◽  
Unitary Symmetry ◽  
Matrix Ensembles ◽  
Gap Probabilities ◽  
Orthogonal Ensembles

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.

scholarly journals Correlation kernels for sums and products of random matrices

2015 ◽  
Vol 04 (04) ◽  
pp. 1550017 ◽  
Author(s):  
Tom Claeys ◽  
Arno B. J. Kuijlaars ◽  
Dong Wang
Keyword(s):  
Random Matrix ◽  
Singular Values ◽  
Contour Integral ◽  
Unitary Matrix ◽  
Unitary Matrices ◽  
Correlation Kernel ◽  
Integral Formulas ◽  
Sum Formula ◽  
Correlation Kernels ◽  
Double Contour

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.

A Study of Image Weak-Edge Detection Combination of BP Neural Networks

2011 ◽  
Vol 225-226 ◽  
pp. 21-25
Author(s):  
Jing Bing Yang ◽  
Hui Ding ◽  
Shu Dong Zhang
Keyword(s):  
Neural Networks ◽  
Edge Detection ◽  
Training Sample ◽  
Detection Methods ◽  
Bp Neural Networks ◽  
Edge Points ◽  
Image Edge ◽  
Basic Characteristics ◽  
Edge Detection Method ◽  
Weak Edge

This paper proposes an image weak-edge detection method based on the combination of edge features and BP neural networks. Through analyzing the basic characteristics of the image edge points, we construct 8 groups 3-D feature vectors as the training sample set, combining with the learning function based on gradient descent momentum and the Levenberg-Marquardt training function, to train the BP neural network, further complete the image edge detection. Finally, compared with the traditional edge detection methods, the experimental results show that this method can detect the weak-edge and corner-edge much better.

scholarly journals Barrier billiard and random matrices

2021 ◽  
Author(s):  
Eugene B Bogomolny
Keyword(s):  
Quantum Mechanics ◽  
Barrier Height ◽  
Random Matrices ◽  
Low Complexity ◽  
Integrable Models ◽  
Independent Interest ◽  
Unitary Matrix ◽  
Quantum Properties ◽  
Spectral Statistics ◽  
Hopf Method

Abstract The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener-Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well described by the semi-Poisson distributions.

Transitions between universality classes of random matrices

1990 ◽  
Vol 65 (19) ◽  
pp. 2325-2328 ◽  
Author(s):  
Georg Lenz ◽  
Fritz Haake
Keyword(s):  
Random Matrices ◽  
Universality Classes
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