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23
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
Partition functions for matrix models and isomonodromic tau functions
, 2002
"... We derive the explicit relationship between the partition function for (generalized) onematrix models with polynomial potentials and the isomonodromic tau function for the 2 × 2 polynomial differential system satisfied by the associated orthogonal polynomials. ..."
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Cited by 33 (14 self)
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We derive the explicit relationship between the partition function for (generalized) onematrix models with polynomial potentials and the isomonodromic tau function for the 2 × 2 polynomial differential system satisfied by the associated orthogonal polynomials.
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.
Distribution of the first particle in discrete orthogonal . . .
, 2002
"... We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q) logderivative of the weight function is rational. In a number of classical special cases the recurrence proce ..."
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Cited by 21 (7 self)
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We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q) logderivative of the weight function is rational. In a number of classical special cases the recurrence procedure is equivalent to the difference and qPainlevé equations of [10], [17]. Our approach is based on the formalism of discrete integrable operators and discrete Riemann–Hilbert problems developed in [3], [4].
τFUNCTION EVALUATION OF GAP PROBABILITIES IN ORTHOGONAL AND SYMPLECTIC MATRIX ENSEMBLES
, 2002
"... It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact τfunctions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or deriva ..."
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Cited by 12 (4 self)
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It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact τfunctions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise τfunctions for certain Painlevé systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two τfunctions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two τfunctions gives the gap probability in the corresponding unitary symmetry case, while one of those τfunctions is the gap probability in the corresponding orthogonal symmetry case. 1
REPRESENTATION THEORY AND RANDOM POINT PROCESSES
, 2004
"... On a particular example we describe how to state and to solve the problem of harmonic analysis for groups with infinite–dimensional dual space. The representation theory for such groups differs in many respects from the conventional theory. We emphasize a remarkable connection with random point pr ..."
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Cited by 11 (8 self)
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On a particular example we describe how to state and to solve the problem of harmonic analysis for groups with infinite–dimensional dual space. The representation theory for such groups differs in many respects from the conventional theory. We emphasize a remarkable connection with random point processes that arise in random matrix theory. The paper is an extended version of the second author’s talk at the Congress.
Analytic theory of difference equations with rational and elliptic coefficients and the . . .
, 2004
"... ..."
How instanton combinatorics solves Painleve ́ VI, V and III’s
"... Abstract. We elaborate on a recently conjectured relation of Painleve ́ transcendents and 2D CFT. General solutions of Painleve ́ VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGTrelated to instanton partition functions in N = 2 supersymmetric gauge theo ..."
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Cited by 8 (2 self)
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Abstract. We elaborate on a recently conjectured relation of Painleve ́ transcendents and 2D CFT. General solutions of Painleve ́ VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGTrelated to instanton partition functions in N = 2 supersymmetric gauge theories with Nf = 0, 1, 2, 3, 4. Resulting combinatorial series representations of Painleve ́ functions provide an efficient tool for their numerical computation at finite values of the argument. The series involve sums over bipartitions which in the simplest cases coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the GUE, and allorder conformal perturbation theory expansions of correlation functions in the sineGordon field theory at the freefermion point. 1.
On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices
 J. NONLINEAR MATH. PHYS.
, 2002
"... It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space ..."
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Cited by 7 (1 self)
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It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic τfunctions.