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.

scholarly journals Multiplicative convolution of real asymmetric and real anti-symmetric matrices

2019 ◽  
Vol 10 (4) ◽  
pp. 467-492 ◽  
Author(s):  
Mario Kieburg ◽  
Peter J. Forrester ◽  
Jesper R. Ipsen
Keyword(s):  
Probability Distribution ◽  
Random Matrices ◽  
Singular Values ◽  
Symmetric Matrices ◽  
Orthogonal Matrices ◽  
Unitary Matrices ◽  
Correlation Kernel ◽  
Multiplicative Convolution ◽  
Special Case ◽  
Double Contour

AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each {X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.

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 Product Matrix Processes as Limits of Random Plane Partitions

2019 ◽  
Vol 2020 (20) ◽  
pp. 6713-6768
Author(s):  
Alexei Borodin ◽  
Vadim Gorin ◽  
Eugene Strahov
Keyword(s):  
Scaling Limit ◽  
Singular Values ◽  
Continuous Limit ◽  
Dynamical Correlation ◽  
Unitary Matrices ◽  
Correlation Kernel ◽  
Plane Partitions ◽  
Product Matrix ◽  
Schur Process ◽  
Random Unitary Matrices

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.

scholarly journals Local spectrum of truncations of Kronecker products of Haar distributed unitary matrices

2015 ◽  
Vol 04 (01) ◽  
pp. 1550001 ◽  
Author(s):  
Brendan Farrell ◽  
Raj Rao Nadakuditi
Keyword(s):  
Random Matrix ◽  
Kronecker Product ◽  
Unitary Matrix ◽  
Kronecker Products ◽  
Local Spectrum ◽  
Unitary Matrices ◽  
Spectral Behavior ◽  
Small Constant ◽  
Matrix Formula

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.

scholarly journals Singular values of products of random matrices and polynomial ensembles

2014 ◽  
Vol 03 (03) ◽  
pp. 1450011 ◽  
Author(s):  
Arno B. J. Kuijlaars ◽  
Dries Stivigny
Keyword(s):  
Scaling Limit ◽  
Singular Values ◽  
Unitary Matrix ◽  
Scaling Limits ◽  
Determinantal Point Process ◽  
Double Integral ◽  
Correlation Kernel ◽  
Products Of Random Matrices ◽  
Hard Edge ◽  
The One

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.

scholarly journals DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN
Keyword(s):  
General Scheme ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Additional Structure ◽  
Unitary Matrices ◽  
Decomposition Scheme ◽  
Single Qubit ◽  
Matlab Program

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

scholarly journals The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

10.37236/3709 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Jacob G. Martin ◽  
E. Rodney Canfield
Keyword(s):  
Random Matrix ◽  
Generating Functions ◽  
Numerical Algorithms ◽  
Singular Values ◽  
Gram Matrix ◽  
Efficient Computation ◽  
Underlying Distribution ◽  
Characteristic Roots ◽  
Speed Up ◽  
Theoretical Results

A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.

The Empirical Distribution of the Singular Values of a Random Hankel Matrix

2015 ◽  
Vol 14 (03) ◽  
pp. 1550027 ◽  
Author(s):  
Mansi Ghodsi ◽  
Nader Alharbi ◽  
Hossein Hassani
Keyword(s):  
Normal Distribution ◽  
Random Matrix ◽  
Empirical Distribution ◽  
Hankel Matrix ◽  
Singular Values ◽  
Shape Parameters ◽  
The Matrix ◽  
Non Parametric

The empirical distribution of the eigenvalues of the matrix HHT divided by its trace is considered, where H is a Hankel random matrix. The normal distribution with different parameters are considered and the effect of scale and shape parameters are evaluated. The correlation among eigenvalues are assessed using parametric and non-parametric association criteria.

scholarly journals Random matrix products: Universality and least singular values

2020 ◽  
Vol 48 (3) ◽  
pp. 1372-1410
Author(s):  
Phil Kopel ◽  
Sean O’Rourke ◽  
Van Vu
Keyword(s):  
Random Matrix ◽  
Singular Values ◽  
Random Matrix Products ◽  
Matrix Products
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