Product Matrix Processes as Limits of Random Plane Partitions

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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Correlation kernels for sums and products of random matrices

Random Matrices Theory and Application ◽

10.1142/s2010326315500173 ◽

2015 ◽

Vol 04 (04) ◽

pp. 1550017 ◽

Cited By ~ 13

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.

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Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Advances in Pure and Applied Mathematics ◽

10.1515/apam-2018-0037 ◽

2019 ◽

Vol 10 (4) ◽

pp. 467-492 ◽

Cited By ~ 1

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.

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Singular values of products of random matrices and polynomial ensembles

Random Matrices Theory and Application ◽

10.1142/s2010326314500117 ◽

2014 ◽

Vol 03 (03) ◽

pp. 1450011 ◽

Cited By ~ 39

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.

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Efficiency of producing random unitary matrices with quantum circuits

Physical Review A ◽

10.1103/physreva.78.062329 ◽

2008 ◽

Vol 78 (6) ◽

Cited By ~ 19

Author(s):

Ludovic Arnaud ◽

Daniel Braun

Keyword(s):

Quantum Circuits ◽

Unitary Matrices ◽

Random Unitary Matrices

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

Random Matrices Theory and Application ◽

10.1142/s2010326322500204 ◽

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.

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