Inverse monodromy problem and Riemann-Hilbert factorization

Fokas, A. S.; Its, Alexander; Kapaev, Andrei; Novokshenov, V. Yu.

doi:10.1090/surv/128/03

Cited by 36 publications

(85 citation statements)

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“…Solving the equation, we have 15) where the constant α 1 is determined by comparing both sides at s = 0. For the chosen initial value r (0) = 1 α , the constant vanishes, thus we get Λ = 0, which is the third-order equation (2.10).…”

Section: Proof Of Proposition 2: Reduction To Piiimentioning

confidence: 99%

“…The vanishing lemma states that the null space is trivial, which implies that the singular integral equation (and thus Ψ 0 ) is solvable as a result of the Fredholm alternative theorem. More details can be found in [22,Proposition 2.4]; see also [10,12,15,17] for standard methods connecting RH problems with integral equations. Now we have the following solvability result:…”

Section: Solvability Of the Model Riemann-hilbert Problemmentioning

confidence: 99%

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Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Dai

Zhao

2014

Commun. Math. Phys.

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In this paper, we study the singularly perturbed Laguerre unitary ensemblewith V t (x) = x + t/x, x ∈ (0, +∞) and t > 0. Due to the effect of t/x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves ψ-functions associated with a special solution to a new thirdorder nonlinear differential equation, which is then shown equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.

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Section: Proof Of Proposition 2: Reduction To Piiimentioning

confidence: 99%

Section: Solvability Of the Model Riemann-hilbert Problemmentioning

confidence: 99%

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Dai

Zhao

2014

Commun. Math. Phys.

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“…Here S = −i/ √ 2π is the Stokes constant computed in [29,50], and both of the branches give the same result. Substituting (4.20), (4.21) and integrating q HM (−z) 2 twice in z, one finally obtains the asymptotics of the free energy F (1, −z) for z → ∞: …”

Section: Jhep09(2014)104mentioning

confidence: 78%

“…Properties of this solution to the Painlevé II equation (3.22), called the Hastings-McLeod solution q HM (s) [28], are extensively studied in the literature (see e.g. [29]). Thus we readily have the full nonperturbative free energy of the SUSY double-well matrix model in the form of (3.21) with ξ = 1.…”

Section: Free Energy and Instanton Summentioning

confidence: 99%

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

Nishigaki

Sugino

2014

J. High Energ. Phys.

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We present an analytic expression of the nonperturbative free energy of a double-well supersymmetric matrix model in its double scaling limit, which corresponds to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. To this end we draw upon the wisdom of random matrix theory developed by Tracy and Widom, which expresses the largest eigenvalue distribution of unitary ensembles in terms of a Painlevé II transcendent. Regularity of the result at any value of the string coupling constant shows that the third-order phase transition between a supersymmetry-preserving phase and a supersymmetry-broken phase, previously found at the planar level, becomes a smooth crossover in the double scaling limit. Accordingly, the supersymmetry is always broken spontaneously as its order parameter stays nonzero for the whole region of the coupling constant. Coincidence of the result with the unitary one-matrix model suggests that one-dimensional type 0 string theories partially correspond to the type IIA superstring theory. Our formulation naturally allows for introduction of an instanton chemical potential, and reveals the presence of a novel phase transition, possibly interpreted as condensation of instantons.

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“…In terms of definition (ii), we have a real τ (x) which is associated with a compactly supported real potential q(x) = −2 In random matrix theory, tau functions are introduced alongside integrable kernels that describe the distribution of eigenvalues of random matrices, especially the generalized unitary ensemble; see [14,26,31]. Let X be a n × n complex Hermitian matrix, let λ = (λ 1 ≤ λ 2 ≤ .…”

Section: Definition (I)mentioning

confidence: 99%

On tau functions for orthogonal polynomials and matrix models

Blower¹

2011

J. Phys. A: Math. Theor.

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Abstract. Let v be a real polynomial of even degree, and let ρ be the equilibrium probability measure for v with support S; so that, v(x) ≥ 2 log |x−y| ρ(dy)+C v for some constant C v with equality on S. Then S is the union of finitely many bounded intervals with endpoints δ j , and ρ is given by an algebraic weight w(x) on S. The system of orthogonal polynomials for w gives rise to the Magnus-Schlesinger differential equations. This paper identifies the τ function of this system with the Hankel determinant det[of ρ. The solutions of the Magnus-Schlesinger equations are realised by a linear system, which is used to compute the tau function in terms of a Gelfand-Levitan equation. The tau function is associated with a potential q and a scattering problem for the Schrödinger operator with potential q. For some algebro-geometric q, the paper solves the scattering problem in terms of linear systems. The theory extends naturally to elliptic curves and resolves the case where S has exactly two intervals.MSC (2000) classification: 60B20 (37K15)

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Inverse monodromy problem and Riemann-Hilbert factorization

Cited by 36 publications

References 0 publications

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

On tau functions for orthogonal polynomials and matrix models

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