Results 1  10
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15
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
Interpretations of some parameter dependent generalizations of classical matrix ensembles
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Increasing subsequences and the hardtosoftedge transition in matrix ensembles
 J. Phys. A
"... Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the ..."
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Cited by 23 (4 self)
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Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) ≤ l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to BaikDeiftJohansson and BaikRains. 1
Determinantal Correlations for Classical Projection Processes
, 801
"... Recent applications in queuing theory and statistical mechanics have isolated the process formed by the eigenvalues of successive minors of the GUE. Analogous eigenvalue processes, formed in general from the eigenvalues of nested sequences of matrices resulting from random corank 1 projections of cl ..."
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Cited by 16 (3 self)
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Recent applications in queuing theory and statistical mechanics have isolated the process formed by the eigenvalues of successive minors of the GUE. Analogous eigenvalue processes, formed in general from the eigenvalues of nested sequences of matrices resulting from random corank 1 projections of classical random matrix ensembles, are identified for the LUE and JUE. The correlations for all these processes can be computed in a unified way. The resulting expressions can then be analyzed in various scaling limits. At the soft edge, with the rank of the minors differing by an amount proportional to N 2/3, the scaled correlations coincide with those known from the soft edge scaling of the Dyson Brownian motion model. 1 1
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
Symmetrized models of last passage percolation and nonintersecting lattice paths
, 2007
"... It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l). We present a theory of such results based on nonintersecting lattice paths, and integration te ..."
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Cited by 8 (0 self)
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It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l). We present a theory of such results based on nonintersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.
Correlation functions for random involutions
 INTERNAT. MATH. RES. NOTICES (2006), ARTICLEID
, 2005
"... Our interest is in the scaled joint distribution associated with kincreasing subsequences for random involutions with a prescribed number of fixed points. We proceed by specifying in terms of correlation functions the same distribution for a Poissonized model in which both the number of symbols in ..."
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Cited by 7 (0 self)
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Our interest is in the scaled joint distribution associated with kincreasing subsequences for random involutions with a prescribed number of fixed points. We proceed by specifying in terms of correlation functions the same distribution for a Poissonized model in which both the number of symbols in the involution, and the number of fixed points, are random variables. From this, a dePoissonization argument yields the scaled correlations and distribution function for the random involutions. These are found to coincide with the same quantities known in random matrix theory from the study of ensembles interpolating between the orthogonal and symplectic universality classes at the soft edge, the interpolation being due to a rank 1 perturbation.
The antisymmetric GUE minor process
, 804
"... Abstract. Our study is initiated by a multicomponent particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species j consists of ⌊j/2 ⌋ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for thi ..."
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Cited by 7 (2 self)
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Abstract. Our study is initiated by a multicomponent particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species j consists of ⌊j/2 ⌋ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for this particle system is obtained, and is shown in a suitable scaling limit to coincide with the joint eigenvalue PDF for the process formed by the successive minors of antisymmetric GUE matrices, which in turn we compute from first principles. The correlations for this process are determinantal and we give an explicit formula for the corresponding correlation kernel in terms of Hermite polynomials. Scaling limits of the latter are computed, giving rise to the Airy kernel, extended Airy kernel and bead kernel at the soft edge and in the bulk, as well as a new kernel at the hard edge. 1.
Operators Associated with Soft and Hard Spectral Edges from Unitary Ensembles
, 2006
"... Using Hankel operators and shiftinvariant subspaces on Hilbert space, this paper develops the theory of the operators associated with soft and hard edges of eigenvalue distributions of random matrices. Tracy and Widom introduced a projection operator W to describe the soft edge of the spectrum of ..."
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Cited by 4 (1 self)
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Using Hankel operators and shiftinvariant subspaces on Hilbert space, this paper develops the theory of the operators associated with soft and hard edges of eigenvalue distributions of random matrices. Tracy and Widom introduced a projection operator W to describe the soft edge of the spectrum of the Gaussian unitary ensemble. The subspace WL2 is simply invariant under the translation semigroup eitD (t ≥ 0) and invariant under the Schrödinger semigroup eit(D2 +x) 2 (t ≥ 0); these properties characterize WL via Beurling’s theorem. The Jacobi ensemble of random matrices has positive eigenvalues which tend to accumulate near to the hard edge at zero. This paper identifies a pair of unitary groups that satisfy the von Neumann–Weyl anticommutation relations and leave invariant certain subspaces of L2 (0, ∞) which are invariant for operators with Jacobi kernels. Such Tracy–Widom operators are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill’s equation give a new family of Tracy–Widom type operators.
CORRELATION KERNELS FOR DISCRETE SYMPLECTIC AND ORTHOGONAL ENSEMBLES
, 712
"... Abstract. In [41] H. Widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials. We obtain similar results for discrete ensembles with rational discrete logarithmic derivative, and compute explicitly correlatio ..."
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Cited by 4 (4 self)
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Abstract. In [41] H. Widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials. We obtain similar results for discrete ensembles with rational discrete logarithmic derivative, and compute explicitly correlation kernels associated to the classical Meixner and Charlier weights. Contents