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29
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
Exact scaling functions for onedimensional stationary KPZ growth
, 2004
"... With deep appreciation dedicated to Giovanni JonaLasinio at the occasion of his 70th birthday. We determine the stationary twopoint correlation function of the onedimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a ..."
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Cited by 58 (11 self)
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With deep appreciation dedicated to Giovanni JonaLasinio at the occasion of his 70th birthday. We determine the stationary twopoint correlation function of the onedimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a RiemannHilbert problem related to the Painlevé II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore [1] obtained from the mode coupling approximation. 1
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.
Moduli spaces of dconnections and difference Painlevé equations
 DUKE MATH. J
, 2004
"... We show that difference Painlevé equations can be interpreted as isomorphisms of moduli spaces of dconnections on P1 with given singularity structure. We also derive a new difference equation that lifts to an isomorphism between A (1)∗ 2 –surfaces in Sakai’s classification [25]. It is the most gene ..."
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Cited by 19 (1 self)
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We show that difference Painlevé equations can be interpreted as isomorphisms of moduli spaces of dconnections on P1 with given singularity structure. We also derive a new difference equation that lifts to an isomorphism between A (1)∗ 2 –surfaces in Sakai’s classification [25]. It is the most general difference Painlevé equation known so far, and it degenerates to both difference Painlevé V and classical (differential) Painlevé VI equations.
Discrete Painlevé equations and random matrix averages
, 2003
"... The τfunction theory of Painlevé systems is used to derive recurrences in the rank n of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlevé equations. The random matrix averages include cases which can be interpreted as eigenvalue ..."
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Cited by 9 (2 self)
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The τfunction theory of Painlevé systems is used to derive recurrences in the rank n of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlevé equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The recurrences are illustrated by computing the value of a sequence of these distributions as n varies, and demonstrating convergence to the value of the appropriate limiting distribution.
Growth models, random matrices and Painlevé transcendents
 Nonlinearity
"... The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0,0) to (1,1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of t ..."
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Cited by 8 (2 self)
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The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0,0) to (1,1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of the longest increasing subsequence in a random permutation. The cumulative distribution of the longest path length can be written in terms of an average over the unitary group. Versions of the Hammersley process in which the points are constrained to have certain symmetries of the square allow similar formulas. The derivation of these formulas is reviewed. Generalizing the original model to have point sources along two boundaries of the square, and appropriately scaling the parameters gives a model in the KPZ universality class. Following works of Baik and Rains, and Prähofer and Spohn, we review the calculation of the scaled cumulative distribution, in which a particular Painlevé II transcendent plays a prominent role. 1
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.
Isomonodromic deformation theory and the nexttodiagonal correlations of the anisotropic square lattice Ising model
 J. Phys. A: Math. Theor
"... Abstract. In 1980 Jimbo and Miwa evaluated the diagonal twopoint correlation function of the square lattice Ising model as a τfunction of the sixth Painlevé system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomo ..."
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Cited by 7 (0 self)
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Abstract. In 1980 Jimbo and Miwa evaluated the diagonal twopoint correlation function of the square lattice Ising model as a τfunction of the sixth Painlevé system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomonodromy theory was constructed based on biorthogonal polynomials on the unit circle with regular semiclassical weights, for which the diagonal Ising correlations arise as the leading coefficient of the polynomials specialised appropriately. Here we demonstrate that the nexttodiagonal correlations of the anisotropic Ising model are evaluated as one of the elements of this isomonodromic system or essentially as the CauchyHilbert transform of one of the biorthogonal polynomials. For the square lattice Ising model on the infinite lattice an unpublished result of Onsager (see [13]) gives that the diagonal spinspin correlation 〈σ0,0σN,N 〉 has the Toeplitz determinant form (1) 〈σ0,0σN,N 〉 = det(ai−j(k))1≤i,j≤N,