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Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter
 In preparation
, 2002
"... A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is ..."
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A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously. 1
Painléve IV and degenerate Gaussian unitary ensembles
 J. Phys. A: Math. Gen
, 2006
"... We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a ..."
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We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painlevé IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable.
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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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
Introduction to the random matrix theory: Gaussian unitary ensemble and beyond
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Circular Jacobi ensembles and deformed Verblunsky coefficients
"... Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: c (n) δ,β ..."
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Cited by 9 (6 self)
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Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: c (n) δ,β
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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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.
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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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.
Painlevé expressions for LOE, LSE and interpolating ensembles
 Int. Math. Res. Not
"... We consider an ensemble which interpolates the Laguerre orthogonal ensemble and the Laguerre symplectic ensemble. This interpolating ensemble was introduced earlier by the author and Rains in connection with a last passage percolation model with a symmetry condition. In this paper, we obtain a Paine ..."
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We consider an ensemble which interpolates the Laguerre orthogonal ensemble and the Laguerre symplectic ensemble. This interpolating ensemble was introduced earlier by the author and Rains in connection with a last passage percolation model with a symmetry condition. In this paper, we obtain a Painelevé V expression for the distribution of the rightmost particle of the interpolating ensemble. Special cases of this result yield the Painlevé V expressions for the largest eigenvalues of Laguerre orthogonal ensemble and Laguerre symplectic ensemble of finite size. 1
Boundary conditions associated with the Painlevé III ′ and V evaluations of some random matrix averages, arXiv math.CA/0512142
"... Abstract. In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval (0, s) at the hard edge contains k eigenvalues, was evaluated in terms of a Painlevé V transcendent in σform. However the boundary condit ..."
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Abstract. In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval (0, s) at the hard edge contains k eigenvalues, was evaluated in terms of a Painlevé V transcendent in σform. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small s behaviour of the Painlevé V equation in σform known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlevé III ′ transcendent in σform. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function. 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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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,