The isomonodromy approach to matric models in 2D quantum gravity

Fokas, A. S.; Its, Alexander; Kitaev, A. V.

doi:10.1007/bf02096594

Cited by 529 publications

(696 citation statements)

References 17 publications

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“…This provides the first main step towards a rigorous analysis of the N → ∞, N = O(1) limit of the partition function and the Fredholm kernels determining the spectral statistics of coupled random matrices (essential, e.g., to the question of universality in 2-matrix models). Such an analysis should follow similar lines to those previously successfully applied to ordinary orthogonal polynomials in the 1-matrix case [6,17,19,20,10,11]. The main difference in the 2-(or more) matrix case is that in the double-scaling limit the functional dependence of the free energy on the eigenvalue distributions is not as explicit as in the 1-matrix models [23,18].…”

Section: Summary and Comments On Large N Asymptotics In Multimatrix Mmentioning

confidence: 68%

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann?Hilbert Problem

Bertola

Eynard

Harnad

2003

Communications in Mathematical Physics

213

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We consider biorthogonal polynomials that arise in the study of a generalization of two-matrix Hermitian models with two polynomial potentials V 1 (x), V 2 (y) of any degree, with arbitrary complex coefficients.Finite consecutive subsequences of biorthogonal polynomials ("windows"), of lengths equal to the degrees of the potentials V 1 and V 2 , satisfy systems of ODE's with polynomial coefficients as well as PDE's (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.

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Section: Summary and Comments On Large N Asymptotics In Multimatrix Mmentioning

confidence: 68%

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann?Hilbert Problem

Bertola

Eynard

Harnad

2003

Communications in Mathematical Physics

213

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“…We prove our results by characterizing the orthogonal polynomials via the well-known 2 × 2 matrix valued Fokas-Its-Kitaev Riemann-Hilbert (RH) problem [17] and applying the Deift/Zhou steepest descent method [13] to analyze this RH problem asymptotically. This approach has been used many times before, see e.g.…”

Section: Outline Of the Rest Of The Papermentioning

confidence: 76%

“…For each fixed n, s, and t, we consider the Fokas-Its-Kitaev Riemann-Hilbert problem [17] characterizing the orthogonal polynomials p (n,s,t) k with respect to the weight functions e −nVs,t . We seek a 2 × 2 matrix-valued function Y (z) = Y (z; n, s, t) (we suppress the n, s, and t dependence for brevity) that satisfies the following conditions.…”

Section: Rh Problem For Orthogonal Polynomialsmentioning

confidence: 99%

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

Claeys

Vanlessen

2007

Commun. Math. Phys.

115

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We consider unitary random matrix ensembles Z −1 n,s,t e −n tr Vs,t(M) dM on the space of Hermitian n × n matrices M , where the confining potential V s,t is such that the limiting mean density of eigenvalues (as n → ∞ and s, t → 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P 2 I equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the wellknown connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P 2 I equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights e −nVs,t on R. The special solution of the P 2 I equation pops up in the n −2/7 -term of the asymptotics.

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“…An application of particular interest for us is the appearance of the first and second Painlevé equation in the theory of random matrices, see e.g. [21,18,3,6,7]. In random matrix ensembles with certain unitary invariant probability measures, the eigenvalues accumulate on a finite union of intervals when the dimension of the matrices grows [8,9,10].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit

Claeys

Грава

2009

Comm Pure Appl Math

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In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlevé II equation. We prove our results using the Riemann-Hilbert approach.

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The isomonodromy approach to matric models in 2D quantum gravity

Cited by 529 publications

References 17 publications

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann?Hilbert Problem

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann?Hilbert Problem

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit

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