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12
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
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
2006, Relationships between τfunctions and Fredholm determinant expressions for gap probabilities in random matrix theory, Nonlinearity 19
"... Abstract. The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of τfunctions. Extending recent work relating to the soft edge, it is shown that these τfunctions, and their generalizations to contain a generating function lik ..."
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Abstract. The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of τfunctions. Extending recent work relating to the soft edge, it is shown that these τfunctions, and their generalizations to contain a generating function like parameter, can be expressed as Fredholm determinants. These same Fredholm determinants also occur in exact expressions for gap probabilities in scaled random matrix ensembles with unitary and symplectic symmetry. 1.
Gap probabilities for double intervals in hermitian random matrix ensembles as τfunctions  the Bessel kernel case
 In preparation
, 2001
"... The probability for the exclusion of eigenvalues from an interval (−x, x) symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter a (a generalisation of the sine kernel in the bulk scaling case), is consider ..."
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Cited by 4 (2 self)
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The probability for the exclusion of eigenvalues from an interval (−x, x) symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter a (a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a τfunction, in the sense of Okamoto, for the Painlevé system PIII. This then leads to a factorisation of the probability as the product of two τfunctions for the Painlevé ′. A previous study has given a formula of this type but system PIII involving PIII ′ systems with different parameters consequently implying an identity between products of τfunctions or equivalently sums of Hamiltonians. The probability Eβ(0; J; g(x); N) that a subset of the real line J is free of eigenvalues for an ensemble of N × N random matrices with eigenvalue probability density function proportional to (1) N∏ g(xl) l=1
Gap probabilities and applications to geometry and random topology
"... Abstract. We give an exact formula for the value of the derivative at zero of the gap probability fβ,n in finite Gaussian βensembles (β = 1, 2, 4). As n goes to infinity our computation provides: f ′β,n(0) ∼ − ..."
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Abstract. We give an exact formula for the value of the derivative at zero of the gap probability fβ,n in finite Gaussian βensembles (β = 1, 2, 4). As n goes to infinity our computation provides: f ′β,n(0) ∼ −
Random matrix theory and higher genus integrability: the quantum chiral Potts model
, 2002
"... Abstract We perform a random matrix theory (RMT) analysis of the quantum fourstate chiral Potts chain for different sizes of the chain up to size L = 8. Our analysis gives clear evidence of a Gaussian orthogonal ensemble (GOE) statistics, suggesting the existence of a generalized timereversal inva ..."
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Abstract We perform a random matrix theory (RMT) analysis of the quantum fourstate chiral Potts chain for different sizes of the chain up to size L = 8. Our analysis gives clear evidence of a Gaussian orthogonal ensemble (GOE) statistics, suggesting the existence of a generalized timereversal invariance. Furthermore, a change from the (generic) GOE distribution to a Poisson distribution occurs when the integrability conditions are met. The chiral Potts model is known to correspond to a (startriangle) integrability associated with curves of genus higher than zero or one. Therefore, the RMT analysis can also be seen as a detector of 'higher genus integrability'.
INTRINSIC VOLUMES OF SETS OF SINGULAR MATRICES AND APPLICATIONS REAL ALGEBRAIC GEOMETRY
"... Abstract. Let Σµ be the set of complex n × n matrices of Frobenius norm one and corank at least µ. We are interested in computing the intrinsic volumes of the two sets: Σµ ∩M(n,R) and Σµ ∩ Sym(n,R) (they are, respectively, the set of real and realsymmetric n×n matrices with Frobenius norm one and c ..."
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Abstract. Let Σµ be the set of complex n × n matrices of Frobenius norm one and corank at least µ. We are interested in computing the intrinsic volumes of the two sets: Σµ ∩M(n,R) and Σµ ∩ Sym(n,R) (they are, respectively, the set of real and realsymmetric n×n matrices with Frobenius norm one and corank at least µ). Using some Random Matrix Theory techniques, explicit formulas for their intrinsic volumes are obtained and an asymptotic analysis of these formulas is performed, obtaining: p(Σµ ∩M(n,R)) = Θ n
RANDOM MATRICES AND THE AVERAGE TOPOLOGY OF THE INTERSECTION OF TWO QUADRICS
"... Abstract. Let XR be the zero locus in RP n of one or two independently and Weyl distributed random real quadratic forms. Denoting by XC the complex part in CP n of XR and by b(XR) and b(XC) the sums of their Betti numbers, we prove that: (1) lim n→∞ ..."
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Abstract. Let XR be the zero locus in RP n of one or two independently and Weyl distributed random real quadratic forms. Denoting by XC the complex part in CP n of XR and by b(XR) and b(XC) the sums of their Betti numbers, we prove that: (1) lim n→∞