Universality and critical behaviour in the chiral two-matrix model

Delvaux, Steven; Geudens, Dries; Zhang, Lun

doi:10.1088/0951-7715/26/8/2231

Cited by 7 publications

(7 citation statements)

References 46 publications

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“…In this case, Akemann et al derived the joint PDF of squared singular values of G and X and also explicit formulas for all spectral correlation functions, which opens up the possibility of asymptotic analysis for local statistics; see e.g. [1] and [20]. However, when L > N (at this stage Ω must be a rectangular matrix due to the existence of the trace operation in the exponent) or Ω is not scalar, to the best of our knowledge, there are no explicit formulas available for the joint PDF of squared singular values of G and X.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Liu

2017

Constr Approx

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Abstract. Consider the product GX of two rectangular complex random matrices coupled by a constant matrix Ω, where G can be thought to be a Gaussian matrix and X is a bi-invariant polynomial ensemble. We prove that the squared singular values form a biorthogonal ensemble in Borodin's sense, and further that for X being Gaussian the correlation kernel can be expressed as a double contour integral. When all but finitely many eigenvalues of ΩΩ * are equal, the corresponding correlation kernel is shown to admit a phase transition phenomenon at the hard edge in four different regimes as the coupling matrix changes. Specifically, the four limiting kernels in turn are the Meijer G-kernel for products of two independent Gaussian matrices, a new critical and interpolating kernel, the perturbed Bessel kernel and the finite coupled product kernel associated with GX. In the special case that X is also a Gaussian matrix and Ω is scalar, such a product has been recently investigated by Akemann and Strahov. We also propose a Jacobi-type product and prove the same transition.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Liu

2017

Constr Approx

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“…Following [10] we briefly mention the tacnode kernel in section 4.1. The implications of Theorem 2.8 for the Duits-Geudens critical kernel are discussed in section 4.A variation of the tacnode RH problem for the hard-edge tacnode and the chiral two-matrix model appears in [9,11]. It may be possible that explicit integral representations for the solution of these RH problems can be found as well.…”

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confidence: 99%

“…A variation of the tacnode RH problem for the hard-edge tacnode and the chiral two-matrix model appears in [9,11]. It may be possible that explicit integral representations for the solution of these RH problems can be found as well.…”

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confidence: 99%

The Tacnode Riemann–Hilbert Problem

Kuijlaars

2013

Constr Approx

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The tacnode Riemann-Hilbert problem is a 4 × 4 matrix valued RH problem that appears in the description of the local behavior of two touching groups of non-intersecting Brownian motions. The same RH problem was also found by Duits and Geudens to describe a new critical regime in the two-matrix model.Delvaux gave integral representations for some of the entries of the 4 × 4 matrix. We complement this work by presenting integral representations for all of the entries. As a consequence we give an explicit formula for the Duits-Geudens critical kernel * Recently, Delvaux [10] made the connection between the two sets of formulas by presenting integral representations for some of the entries of the solution of the tacnode RH problem. These entries are exactly the ones that play a role for the tacnode kernel in [12]. With these explicit formulas Delvaux could make the connection between the formulas in [12] and the ones by Ferrari and Vető [16] for the asymmetric tacnode. The paper [10] was inspired by the paper [6] by Baik, Liechty, and Schehr, where a connection between different sets of formulas for the maximal height and position of the Airy 2 process was made.The aim of this paper is to complement the work of [10] by providing integral representations for all the entries of the tacnode RH problem. Some of these remaining entries appear in the description of a critical kernel appearing in the two-matrix model as shown by Duits and Geudens [13]. We therefore find explicit integral formulas for the Duits-Geudens critical kernel.In section 2 we recall the tacnode RH problem with some of its properties, and in particular the connection with the Hastings-McLeod solution of Painlevé II. The main results of this paper are stated in Theorems 2.5 and 2.8 below. We compare the solution of the tacnode RH problem with the explicit solution of the usual 2 × 2 matrix-valued RH problem for the Hastings-McLeod solution in section 2.6.The proofs of the results are in section 3. A key role is played by Lemma 3.2 that describes solutions to a certain ODE system (3.2). The proof of this lemma follows along the lines of certain proofs in [10]. We give full details about the calculations in section 5. Following [10] we briefly mention the tacnode kernel in section 4.1. The implications of Theorem 2.8 for the Duits-Geudens critical kernel are discussed in section 4.A variation of the tacnode RH problem for the hard-edge tacnode and the chiral two-matrix model appears in [9,11]. It may be possible that explicit integral representations for the solution of these RH problems can be found as well. Other recent contributions [1,3,7,19] discuss further connections and properties of the tacnode process. Statement of results The tacnode RH problemThe tacnode RH problem asks for a 4 × 4 matrix-valued function M : C \ Γ M → C 4×4 Proposition 2.2 was proved in the case τ = 0 by Delvaux, Kuijlaars, and Zhang [12], and in the case r 1 = r 2 = 1, s 1 = s 2 with general τ by Duits and Geudens [13]. The proof in [13] extends to the general case as noted...

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“…(Chiral 2-matrix model). The RH problem 2.1 for M (ζ) was recently used to study a critical phenomenon in the chiral 2-matrix model [10]. This leads to a critical correlation kernel which is essentially different from the kernel K tacnode in (2.25) and which is a hard edge analogue of the Duits-Geudens kernel [14].…”

Section: )mentioning

confidence: 99%

Non-Intersecting Squared Bessel Paths at a Hard-Edge Tacnode

Delvaux

2013

Commun. Math. Phys.

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The squared Bessel process is a 1-dimensional diffusion process related to the squared norm of a higher dimensional Brownian motion. We study a model of n non-intersecting squared Bessel paths, with all paths starting at the same point a > 0 at time t = 0 and ending at the same point b > 0 at time t = 1. Our interest lies in the critical regime ab = 1/4, for which the paths are tangent to the hard edge at the origin at a critical time t * ∈ (0, 1). The critical behavior of the paths for n → ∞ is studied in a scaling limit with time t = t * + O(n −1/3 ) and temperature T = 1+O(n −2/3 ). This leads to a critical correlation kernel that is defined via a new Riemann-Hilbert problem of size 4 × 4. The Riemann-Hilbert problem gives rise to a new Lax pair representation for the Hastings-McLeod solution to the inhomogeneous Painlevé II equation q ′′ (x) = xq(x) + 2q 3 (x) − ν, where ν = α + 1/2 with α > −1 the parameter of the squared Bessel process. These results extend our recent work with Kuijlaars and Zhang [13] for the homogeneous case ν = 0.

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Universality and critical behaviour in the chiral two-matrix model

Cited by 7 publications

References 46 publications

Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

The Tacnode Riemann–Hilbert Problem

Non-Intersecting Squared Bessel Paths at a Hard-Edge Tacnode

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