Cauchy–Laguerre Two-Matrix Model and the Meijer-G Random Point Field

Bertola, Marco; Gekhtman, Michael; Szmigielski, Jacek

doi:10.1007/s00220-013-1833-8

Cited by 69 publications

(122 citation statements)

References 21 publications

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“…We derive a double contour integral representation of K n , which allows us to find its scaling limit at the origin (hard edge). The limiting kernels generalize the classical Bessel kernel, and if M = 2, it coincides with the limiting kernels in the Cauchy-Laguerre two-matrix model recently studied by Bertola, Gekhtman and Szmigielski in [12]. Universality suggests that the new limiting kernels should apply to more general situations for the products of independent complex random matrices, thus, representing a new universality class.…”

Section: Biorthogonal Functions and The Correlation Kernelsupporting

confidence: 55%

“…We conclude this paper by giving the integrable form of the limiting kernels derived in Theorem 5.3. Our argument follows [12,Section 5], where this was shown for the case M = 2.…”

Section: Integrable Form Of the Limiting Kernelsmentioning

confidence: 91%

“…(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%

See 2 more Smart Citations

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

Kuijlaars

Zhang

2014

Commun. Math. Phys.

126

264

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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.

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Section: Biorthogonal Functions and The Correlation Kernelsupporting

confidence: 55%

“…We conclude this paper by giving the integrable form of the limiting kernels derived in Theorem 5.3. Our argument follows [12,Section 5], where this was shown for the case M = 2.…”

Section: Integrable Form Of the Limiting Kernelsmentioning

confidence: 91%

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

Kuijlaars

Zhang

2014

Commun. Math. Phys.

126

264

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“…In this setting, CBOPs can be uniquely determined based on the orthogonal relation (2.2) [9,27]. Denote τ k = det(I i,j ) i,j=0,··· ,k−1 .…”

Section: Cauchy Biorthogonal Polynomialsmentioning

confidence: 99%

“…from which a four-term recurrence relation and related Riemann-Hilbert problem can be characterized [9].…”

Section: Cauchy Biorthogonal Polynomialsmentioning

confidence: 99%

The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

2018

J Nonlinear Sci

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This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hierarchy with the help of orthogonal polynomials theory and Toda-type equations. Starting from the symmetric reduction of Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the τ -function of the CKP hiearchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems.

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Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits

Akemann

Strahov

2018

Ann. Henri Poincaré

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Product matrix processes are multi-level point processes formed by the singular values of random matrix products. In this paper we study such processes where the products of up to m complex random matrices are no longer independent, by introducing a coupling term and potentials for each product. We show that such a process still forms a multi-level determinantal point processes, and give formulae for the relevant correlation functions in terms of the corresponding kernels.For a special choice of potential, leading to a Gaussian coupling between the mth matrix and the product of all previous m − 1 matrices, we derive a contour integral representation for the correlation kernels suitable for an asymptotic analysis of large matrix size n. Here, the correlations between the first m − 1 levels equal that of the product of m − 1 independent matrices, whereas all correlations with the mth level are modified. In the hard edge scaling limit at the origin of the spectra of all products we find three different asymptotic regimes. The first regime corresponding to weak coupling agrees with the multi-level process for the product of m independent complex Gaussian matrices for all levels, including the m-th. This process was introduced by one of the authors and can be understood as a multi-level extension of the Meijer G-kernel introduced by Kuijlaars and Zhang. In the second asymptotic regime at strong coupling the point process on level m collapses onto level m − 1, thus leading to the process of m − 1 independent matrices. Finally, in an intermediate regime where the coupling is proportional to n 1 2 , we obtain a family of parameter dependent kernels, interpolating between the limiting processes in the weak and strong coupling regime. These findings generalise previous results of the authors and their coworkers for m = 2. 16 3. Hard edge scaling limits of the multi-level determinantal processes 17 4. The interpolating multi-level determinantal process 20 5. An integration formula for coupled matrices 21 6. Proof of Theorem 2.1 25

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Cauchy–Laguerre Two-Matrix Model and the Meijer-G Random Point Field

Cited by 69 publications

References 21 publications

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

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

The Cauchy Two-Matrix Model, C-Toda Lattice and CKP Hierarchy

Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits

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