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391
Rational surfaces associated with affine root systems and geometry of the Painlevé equations
, 1999
"... We present a geometric approach to the theory of Painlev'e equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anticanonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsyste ..."
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Cited by 162 (6 self)
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We present a geometric approach to the theory of Painlev'e equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anticanonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsystems of E (1) 8 inside the Picard lattice of X. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlev'e equations, and that the above action constitutes their group of symmetries by Backlund transformations. The six Painlev'e differential equations appear as degenerate cases of this construction. In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure.
Fredholm Determinants, Differential Equations and Matrix Models
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1994
"... Orthogonal polynomial random matrix models of N x N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (φ(x)φ(y) — ψ(x)φ(y))/x — y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the und ..."
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Cited by 148 (20 self)
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Orthogonal polynomial random matrix models of N x N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (φ(x)φ(y) — ψ(x)φ(y))/x — y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is the union of intervals J = [J ™ =1 (βiju Λ 2J) The emphasis is on the determinants thought of as functions of the endpoints a k. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as φ and φ satisfy a certain type of differentiation formula. The (φ, φ) pairs for the sine, Airy, and Bessel kernels satisfy such relations, as do the pairs which arise in the finite N Hermite, Laguerre and Jacobi ensembles and in matrix models of 2D quantum gravity. Therefore we shall be able to write down the systems of PDE's for these ensembles as special cases of the general system. An analysis of these equations will lead to explicit representations in terms of Painleve transcendents for the distribution functions of the largest and smallest eigenvalues in the finite N Hermite and Laguerre ensembles, and for the distribution functions of the largest and smallest singular values of rectangular matrices (of arbitrary dimensions) whose entries are independent identically distributed complex Gaussian variables. There is also an exponential variant of the kernel in which the denominator is replaced by e bx — e by, where b is an arbitrary complex number. We shall find an analogous system of differential equations in this setting. If b = i then we can interpret our operator as acting on (a subset of) the unit circle in the complex plane. As an application of this we shall write down a system of PDE's for Dyson's circular ensemble of N x N unitary matrices, and then an ODE if J is an arc of the circle.
Double scaling limit in the random matrix model: the RiemannHilbert approach
"... Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1. ..."
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Cited by 86 (11 self)
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Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.
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 78 (23 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
A qanalog of the sixth Painlevé equation
, 1995
"... A qdifference analog of the sixth Painlevé equation is presented. It arises as the condition for preserving the connection matrix of linear qdifference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit and special solutions ..."
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Cited by 73 (3 self)
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A qdifference analog of the sixth Painlevé equation is presented. It arises as the condition for preserving the connection matrix of linear qdifference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit and special solutions in terms of qhypergeometric functions are also discussed.
On the distributions of the lengths of the longest monotone subsequences in random words
"... We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k → ∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant rep ..."
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Cited by 67 (8 self)
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We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k → ∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k×k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N → ∞ limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k × k hermitian matrices of trace zero. I.
Monodromy of certain PainlevéVI transcendents and reflection groups
 Invent. Math
"... Abstract. We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β = γ = 0, δ = 1 and α arbitrary. We 2 introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analyt ..."
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Cited by 60 (11 self)
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Abstract. We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β = γ = 0, δ = 1 and α arbitrary. We 2 introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painlevé VI equation.
The Asymptotics of Monotone Subsequences of Involutions
, 2001
"... We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the ..."
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Cited by 60 (6 self)
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We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the number of fixed points, (1) the TracyWidom distributions for the laxgest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of the authors in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the RiemannHilbert analysis for the orthogonal polynomials by Delft, Johansson and the first author in [3].
Geometry and analytic theory of Frobenius manifolds
, 1998
"... Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifol ..."
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Cited by 56 (3 self)
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Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories.