School of Mathematics and Statistics - Research Publications
Permanent URI for this collection
Search Results
1 - 10 of 10
-
ItemCounting formulas associated with some random matrix averages(ACADEMIC PRESS INC ELSEVIER SCIENCE, 2006-08)
-
ItemThe importance of the Selberg integral(AMER MATHEMATICAL SOC, 2008)
-
ItemPieri-type formulas for the non-symmetric Jack polynomials(BIRKHAUSER VERLAG AG, 2004)
-
Item
-
ItemPainleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons(SPRINGER, 2003)The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlev\'e VI transcendent in $\sigma$-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlev\'e V and VI. We discuss how our results can be used to compute the ground state occupations.
-
Item
-
ItemHermite and Laguerre β-ensembles:: Asymptotic corrections to the Eigenvalue density(ELSEVIER SCIENCE BV, 2006-05-29)
-
Item
-
ItemApplication of the τ-function theory of Painleve equations to random matrices:: PVI, the JUE, CyUE, cJUE and scaled limits(CAMBRIDGE UNIV PRESS, 2004-06)Abstract Okamoto has obtained a sequence ofτ-functions for the PVIsystem 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 re-expressed 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 parameterN, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVItheory. We show that the Hamiltonian also satisfies an equation related to the discrete PVequation, 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 PVtranscendent inσform. 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) (parameteraa non-negative integer) and Laguerre symplectic ensemble (LSE) (parameteraan even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to theτ-function evaluation of the largest eigenvalue in the finite LOE and LSE with parametera =0; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.
-
ItemInterrelationships between orthogonal, unitary and symplectic matrix ensembles(Cambridge University Press, 2001)