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Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the cases that the eigenvalue probability… (More)

- P. J. Forrester
- 2000

The scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplec-tic ensembles is evaluated in terms of a Painlevé V transcendent. This same Painlevé V transcendent is known from the work of Tracy and Widom, where it has been shown to specify the scaled distribution of the smallest eigenvalue in the Laguerre unitary ensemble. The… (More)

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… (More)

- T H Baker, P J Forrester
- 1997

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic N j=1 x j − x is computed exactly and shown to satisfy a central limit theorem as N → ∞. For the circular random matrix ensemble the p.d.f.'s for the linear statistics 1 2 N j=1 (θ j −π) and − N j=1 log 2| sin θ j /2| are calculated… (More)

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of˜E N (λ; a) := N l=1 χ (l) (−∞,λ] (λ − λ l) a , where χ (l) (−∞,λ] = 1 for λ l ∈ (−∞, λ] and χ (l) (−∞,λ] = 0 otherwise, and the… (More)

We consider Hermite and Laguerre β-ensembles of large N × N random matrices. For all β even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density which are based on special realizations of the generalized… (More)

- P J Forrester, N E Frankel
- 2004

Fisher-Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical… (More)

- T H Baker, P J Forrester
- 1997

The theory of non-symmetric Jack polynomials is developed independently of the theory of symmetric Jack polynomials, and this theory together with the relationship between the non-symmetric, symmetric and anti-symmetric Jack polynomials is used to deduce the corresponding results for the symmetric Jack polynomials.

- T H Baker, P J Forrester
- 1997

A q-analogue of the type A Dunkl operator and integral kernel We introduce the q-analogue of the type A Dunkl operators, which are a set of degree–lowering operators on the space of polynomials in n variables. This allows the construction of raising/lowering operators with a simple action on non-symmetric Macdonald polynomials. A bilinear series of… (More)

Our interest is in the scaled joint distribution associated with k-increasing 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… (More)