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160
Application of the τfunction theory of Painlevé equations to random matrices
 PV, PIII, the LUE, JUE and CUE
, 2002
"... Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidim ..."
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Cited by 75 (20 self)
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Okamoto has obtained a sequence of τfunctions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be reexpressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a nonnegative
The discrete and continuous Painlevé VI hierarchy and the Garnier systems
 Glasgow Math. J. 43A
"... We present a general scheme to derive higherorder members of the Painlevé VI (PVI) hierarchy of ODE’s as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations ..."
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Cited by 47 (10 self)
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We present a general scheme to derive higherorder members of the Painlevé VI (PVI) hierarchy of ODE’s as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.
Essays on the theory of elliptic hypergeometric functions
, 2008
"... We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An ..."
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Cited by 44 (10 self)
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We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.
Hypergeometric solutions to the qPainlevé equations
 Internat. Math. Res. Notices
"... Hypergeometric solutions to seven qPainlevé equations in Sakai’s classification are constructed. Geometry of plane curves is used to reduce the qPainlevé equations to the threeterm recurrence relations for qhypergeometric functions. 1 ..."
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Cited by 38 (5 self)
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Hypergeometric solutions to seven qPainlevé equations in Sakai’s classification are constructed. Geometry of plane curves is used to reduce the qPainlevé equations to the threeterm recurrence relations for qhypergeometric functions. 1
MODULI OF STABLE PARABOLIC CONNECTIONS, RIEMANNHILBERT CORRESPONDENCE AND GEOMETRY OF PAINLEVÉ EQUATION OF TYPE VI, Part I
, 2003
"... In this paper, we will give a complete geometric background for the geometry of Painlevé V I and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space M α n (t, λ, L) of stable parabolic connection on P1 with logarithmic poles at D(t) = t1 + · · · + tn a ..."
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Cited by 35 (10 self)
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In this paper, we will give a complete geometric background for the geometry of Painlevé V I and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space M α n (t, λ, L) of stable parabolic connection on P1 with logarithmic poles at D(t) = t1 + · · · + tn as well as its natural compactification. Moreover the moduli space R(Pn,t)a of Jordan equivalence classes of SL2(C)representations of the fundamental group π1(P 1 \D(t), ∗) are defined as the categorical quotient. We define the RiemannHilbert correspondence RH: M α n (t, λ, L) − → R(Pn,t)a and prove that RH is a bimeromorphic proper surjective analytic map. Painlevé and Garnier equations can be derived from the isomonodromic flows and Painlevé property of these equations are easily derived from the properties of RH. We also prove that the smooth parts of both moduli spaces have natural symplectic structures and RH is a symplectic resolution of singularities of R(Pn,t)a, from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Bäcklund transformations, special solutions of these equations.
A study on the fourth qPainlevé equation
 J. Phys. A: Math. Gen
"... A qdifference analogue of the fourth Painlevé equation is proposed. Its symmetry structure and some particular solutions are investigated. ..."
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Cited by 34 (8 self)
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A qdifference analogue of the fourth Painlevé equation is proposed. Its symmetry structure and some particular solutions are investigated.
Discrete gap probabilities and discrete Painlevé equations
 DUKE MATH J
, 2003
"... We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1kernel to {s, s + 1,...}, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm ..."
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Cited by 29 (6 self)
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We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1kernel to {s, s + 1,...}, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a Poissonized Plancherel measure and a zmeasure, or as normalized Toeplitz determinants with symbols eη(ζ +ζ −1) and (1 + ξζ)
Isomonodromy transformations of linear systems of difference equations
"... Abstract. We introduce and study “isomonodromy ” transformations of the matrix linear difference equation Y (z + 1) = A(z)Y (z) with polynomial (or rational) A(z). Our main result is a construction of an isomonodromy action of Z m(n+1)−1 on the space of coefficients A(z) (here m is the size of matr ..."
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Cited by 26 (2 self)
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Abstract. We introduce and study “isomonodromy ” transformations of the matrix linear difference equation Y (z + 1) = A(z)Y (z) with polynomial (or rational) A(z). Our main result is a construction of an isomonodromy action of Z m(n+1)−1 on the space of coefficients A(z) (here m is the size of matrices and n is the degree of A(z)). The (birational) action of certain rank n subgroups can be described by difference analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. We also show that both the classical Schlesinger equations and the Schlesinger transformations known in the isomonodromy theory, can be obtained as limits of our action in two different limit regimes. Similarly to the continuous case, for m = n = 2 the difference Schlesinger equations and their qanalogs yield discrete Painlevé equations; examples include dPII, dPIV, dPV, and qPVI.
Recurrences for elliptic hypergeometric integrals
, 2006
"... In [15], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generali ..."
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Cited by 22 (7 self)
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In [15], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (qhypergeometric) integral identities as limits from the elliptic level.