The Cauchy Two-Matrix Model

Bertola, Marco; Gekhtman, Michael; Szmigielski, Jacek

doi:10.1007/s00220-009-0739-y

Cited by 64 publications

(176 citation statements)

References 38 publications

(192 reference statements)

Supporting

Mentioning

176

Contrasting

Order By: Relevance

“…These polynomials were related to matrix models with external source [3,19,20,63] and nonintersecting paths [27,28,54]. See also [13,14] for a 3 3 matrix-valued RH problem describing a Cauchy two-matrix model. The steepest-descent analysis of the larger size RH problems revealed several new features that are not present in the analysis of the 2 2 RH problem (1.10).…”

Section: Two-matrix Models and Biorthogonal Polynomialsmentioning

confidence: 99%

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Duits

Kuijlaars

2008

Comm Pure Appl Math

133

View full text Add to dashboard Cite

The eigenvalue statistics of a pair .M 1 ; M 2 / of n n Hermitian matrices taken randomly with respect to the measurecan be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest-descent analysis of a 4 4 matrix-valued RiemannHilbert problem characterizing one of the families of biorthogonal polynomials in the special case W .y/ D y 4 =4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M 1 (when averaged over M 2 ) in the global and local regime as n ! 1 in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.

show abstract

Section: Two-matrix Models and Biorthogonal Polynomialsmentioning

confidence: 99%

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Duits

Kuijlaars

2008

Comm Pure Appl Math

133

View full text Add to dashboard Cite

show abstract

“…(A4) and (A5). Using Cramer's rule, the biorthogonal polynomials, as well as the norms, can be expressed in terms of the bimoments as [34,35],…”

Section: Two-matrix Model and Biorthogonal Polynomialsmentioning

confidence: 99%

Products of rectangular random matrices: Singular values and progressive scattering

2013

View full text Add to dashboard Cite

We discuss the product of M rectangular random matrices with independent Gaussian entries, which have several applications, including wireless telecommunication and econophysics. For complex matrices an explicit expression for the joint probability density function is obtained using the Harish-Chandra-Itzykson-Zuber integration formula. Explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two-matrix model and the method of biorthogonal polynomials. This generalizes the classical result for the so-called Wishart-Laguerre Gaussian unitary ensemble (or chiral unitary ensemble) at M=1, and previous results for the product of square matrices. The correlation functions are given by a determinantal point process, where the kernel can be expressed in terms of Meijer G-functions. We compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices. The location of the end points of support for the latter are analyzed in detail for general M. Finally, we consider the so-called ergodic mutual information, which gives an upper bound for the spectral efficiency of a MIMO communication channel with multifold scattering.

show abstract

“…(5.11) It is interesting that these kernels appeared earlier in another random matrix model, namely in the Cauchy two-matrix model with linear potentials, see [10,12].…”

Section: Special Case M =mentioning

confidence: 99%

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Kuijlaars

Zhang

2014

Commun. Math. Phys.

126

261

View full text Add to dashboard Cite

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

show abstract

The Cauchy Two-Matrix Model

Cited by 64 publications

References 38 publications

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Products of rectangular random matrices: Singular values and progressive scattering

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Contact Info

Product

Resources

About