Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model

Ercolani, Nicholas M.; McLaughlin, Kenneth D.T-R

doi:10.1016/s0167-2789(01)00173-7

Cited by 48 publications

(106 citation statements)

References 31 publications

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“…4) In this work, we use essentially the same definition of orthogonality, but extend it to the case of polynomials V 1 and V 2 with arbitrary (possibly complex) coefficients, with the contours of integration no longer restricted to the real axis, but taken as curves in the complex plane starting and ending at ∞, chosen so that the integrals are convergent. The orthogonality relations determine the two families uniquely, if they exist [13,3].…”

Section: Introductionmentioning

confidence: 99%

“…(See [12,22] and references therein for further background on 2-matrix models, and [9,14,15,16,13] for other more recent developments.) The study of the large N limit of matrix integrals is of considerable interest in physics, since many physical systems having a large number of strongly correlated degrees of freedom (quantum chaos, mesoscopic conductors,…”

Section: Introductionmentioning

confidence: 99%

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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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…With the two-matrix model (1.1), we associate two sequences of monic polynomials (p k,n ) ∞ k=0 and (q k,n ) ∞ k=0 , where deg p k,n = deg q k,n = k, called biorthogonal polynomials, defined by the property Existence and uniqueness of these polynomials was proved by Ercolani and McLaughlin in [9]. They also showed that the zeros of these polynomials are real and simple.…”

Section: Two-matrix Model and Biorthogonal Polynomialsmentioning

confidence: 99%

“…It will be one of our results that the zeros interlace, see Theorem 2.1. It is well-known that the eigenvalue correlations of the matrices M 1 and M 2 from (1.1) are determinantal with correlation kernels that can be expressed in terms of the biorthogonal polynomials and their transforms, see [2,3,9,10,14]. As an example of these relations we have that E [det(xI n − M 1 )] = p n,n (x), E [det(yI n − M 2 )] = q n,n (y), which show that the diagonal biorthogonal polynomials p n,n and q n,n can be considered as 'typical' characteristic polynomials for M 1 and M 2 , respectively.…”

Section: Two-matrix Model and Biorthogonal Polynomialsmentioning

confidence: 99%

A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

2011

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We consider the two sequences of biorthogonal polynomials (p k,n )and (q k,n ) ∞ k=0 related to the Hermitian two-matrix model with potentials V (x) = x 2 /2 and W (y) = y 4 /4+ty 2 . From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p n,n as n → ∞. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behavior for a certain negative value of t.We also prove a general result about the interlacing of zeros of biorthogonal polynomials.

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“…Эрколани и Л. Маклолин [83] доказали, что эти полиномы однозначно опре-делены условием (6.3), а нули у них вещественные и простые. Совсем недавно было показано, что их нули перемежаются [84].…”

Section: двухматричная модельunclassified

Аппроксимации Эрмита - Паде И Ансамбли Совместно Ортогональных Многочленов

Аптекарев¹,

Aptekarev²,

Койэлаарс³

et al. 2011

Uspekhi Mat. Nauk

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Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model

Cited by 48 publications

References 31 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

A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

Аппроксимации Эрмита - Паде И Ансамбли Совместно Ортогональных Многочленов

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