Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE

Forrester, Peter J.; Witte, N. S.

doi:10.1007/s002200100422

Cited by 142 publications

(223 citation statements)

References 31 publications

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“…It is known that the simplest correlation functions, i.e. the moments of the characteristic polynomials, can be described in terms of non-linear differential equations (see works of Forrester and White [23,24], and also the paper by Kanzieper [41], and by Splittorff and Verbaarschot [48]). As for more complicated correlation functions a description in terms of differential equations is unknown.…”

Section: Summary and Discussionmentioning

confidence: 99%

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Strahov

Fyodorov

2003

Commun. Math. Phys.

144

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We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of β = 2 symmetry class.

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Section: Summary and Discussionmentioning

confidence: 99%

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Strahov

Fyodorov

2003

Commun. Math. Phys.

144

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“…It was realized by one of the present authors [10,11,12,13,14,15] that theory initiated in [8] could be further developed and used to express the average of (9) with respect to (14) and its limiting forms as the Laguerre -(2λ) ensemble and the Gaussian -(2λ) ensemble, as m -dimensional integrals. Because the role of n and m is effectively interchanged, these integration identities have been referred to as duality formulas [16,17,18]. One of their uses, as we will demonstrate in the case of the average of (11) with respect to (10), is in the computation of the large n asymptotics.…”

Section: Duality Formulas For Multiple Integrals and Asymptotic Amentioning

confidence: 99%

Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

Forrester

Frankel

Garoni

2003

Journal of Mathematical Physics

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The density matrix for the impenetrable Bose gas in Dirichlet and Neumann boundary conditions can be written in terms of n l=1 | cos φ1 − cos θ l || cos φ2 − cos θ l | , where the average is with respect to the eigenvalue probability density function for random unitary matrices from the classical groups Sp(n) and O + (2n) respectively. In the large n limit log-gas considerations imply that the average factorizes into the product of averages of the form n l=1 | cos φ − cos θ l | . By changing variables this average in turn is a special case of the function of t obtained by averaging n l=1 |t − x l | 2q over the Jacobi unitary ensemble from random matrix theory. The latter task is accomplished by a duality formula from the theory of Selberg correlation integrals, and the large n asymptotic form is obtained. The corresponding large n asymptotic form of the density matrix is used, via the exact solution of a particular integral equation, to compute the asymptotic form of the low lying effective single particle states and their occupations, which are proportional to √ N .

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“…For the integer powers this can be seen from the original Kajiwara-Ohta determinant formula for the rational solutions of PIV ( [16], c.f. [22]) and it was later explored by Forrester and Witte [13].…”

Section: Introductionmentioning

confidence: 99%

Painlevé IV and degenerate Gaussian unitary ensembles

Chen

Feigin

2006

J. Phys. A: Math. Gen.

123

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We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t , this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painlevé IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable. † ychen@ic.ac.uk

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Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE

Cited by 142 publications

References 31 publications

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

Painlevé IV and degenerate Gaussian unitary ensembles

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