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 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 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 Scaling limit of the one-dimensional XXZ Heisenberg chain with easy axis anisotropy

1999 ◽  
Vol 562 (3) ◽  
pp. 497-523 ◽  
Author(s):  
Tamás Hauer ◽  
Attila Rákos ◽  
Ferenc Woynarovich
Keyword(s):  
Easy Axis ◽  
Scaling Limit ◽  
Heisenberg Chain ◽  
One Dimensional ◽  
The One

scholarly journals NON-PERTURBATIVE SOLUTION OF THE SUPER-VIRASORO CONSTRAINTS

1993 ◽  
Vol 08 (13) ◽  
pp. 1205-1214 ◽  
Author(s):  
K. BECKER ◽  
M. BECKER
Keyword(s):  
Partition Function ◽  
Matrix Model ◽  
Scaling Limit ◽  
Virasoro Constraints ◽  
Perturbative Solution ◽  
The One ◽  
Genus Expansion ◽  
Double Scaling Limit

We present the solution of the discrete super-Virasoro constraints to all orders of the genus expansion. Integrating over the fermionic variables we get a representation of the partition function in terms of the one-matrix model. We also obtain the non-perturbative solution of the super-Virasoro constraints in the double scaling limit but do not find agreement between our flows and the known supersymmetric extensions of KdV.

The non-uniform stationary measure for discrete-time quantum walks in one dimension

2015 ◽  
Vol 15 (11&12) ◽  
pp. 1060-1075
Author(s):  
Norio Konno ◽  
Masato Takei
Keyword(s):  
Eigenvalue Problem ◽  
Discrete Time ◽  
Quantum Walks ◽  
Stationary Measure ◽  
Unitary Matrix ◽  
Uniform Measure ◽  
Simple Argument ◽  
Stationary Measures ◽  
Time Quantum ◽  
The One

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 $\times$ 2 unitary matrix $U$. In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when $U$ is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.

Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble

2021 ◽  
pp. 2250020
Author(s):  
Peter J. Forrester
Keyword(s):  
Statistical Physics ◽  
Scaling Limit ◽  
Pair Potential ◽  
Exact Expression ◽  
Correlation Kernel ◽  
Local Scaling ◽  
Gaussian Unitary Ensemble ◽  
Matrix Ensemble ◽  
Many Body Systems ◽  
Global And Local

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little [Formula: see text]-Jacobi polynomial. From their large [Formula: see text] form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.

scholarly journals AN OPERATOR FORMALISM FOR UNITARY MATRIX MODELS

1991 ◽  
Vol 06 (29) ◽  
pp. 2727-2739 ◽  
Author(s):  
K. N. ANAGNOSTOPOULOS ◽  
M. J. BOWICK ◽  
N. ISHIBASHI
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Pseudodifferential Operators ◽  
Scaling Limit ◽  
Unitary Matrix ◽  
String Equation ◽  
Operator Formalism ◽  
Multicritical Point ◽  
Compact Formulation ◽  
Double Scaling Limit

We analyze the double scaling limit of unitary matrix models in terms of trigonometric orthogonal polynomials on the circle. In particular we find a compact formulation of the string equation at the kth multicritical point in terms of pseudodifferential operators and a corresponding action principle. We also relate this approach to the mKdV hierarchy which appears in the analysis in terms of conventional orthogonal polynomials on the circle.

EXCITATIONS AND INTERACTIONS IN d=1 STRING THEORY

1991 ◽  
Vol 06 (11) ◽  
pp. 1961-1984 ◽  
Author(s):  
ANIRVAN M. SENGUPTA ◽  
SPENTA R. WADIA
Keyword(s):  
Matrix Model ◽  
Periodic Wave ◽  
Density Fluctuations ◽  
Dirac Fermion ◽  
Unitary Matrix ◽  
Additional Parameter ◽  
Continuum Approach ◽  
The Matrix ◽  
The One ◽  
The Continuum

We discuss the singlet sector of the d=1 matrix model in terms of a Dirac fermion formalism. The leading order two- and three-point functions of the density fluctuations are obtained by this method. This allows us to construct the effective action to that order and hence provide the equation of motion. This equation is compared with the one obtained from the continuum approach. We also compare continuum results for correlation functions with the matrix model ones and discuss the nature of gravitational dressing for this regularization. Finally, we address the question of boundary conditions within the framework of the d=1 unitary matrix model, considered as a regularized version of the Hermitian model, and study the implications of a generalized action with an additional parameter (analogous to the θ parameter) which give rise to quasi-periodic wave functions.

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