Selberg integral theory and Muttalib-Borodin ensembles
PJ Forrester, JR Ipsen
ADVANCES IN APPLIED MATHEMATICS | ACADEMIC PRESS INC ELSEVIER SCIENCE | Published : 2018
We study Muttalib–Borodin ensembles — particular eigenvalue PDFs on the half-line — with classical weights, i.e. Laguerre, Jacobi or Jacobi prime. We show how the theory of the Selberg integral, involving also Jack and Schur polynomials, naturally leads to a multi-parameter generalisation of these particular Muttalib–Borodin ensembles, and also to the explicit form of underlying biorthogonal polynomials of a single variable. A suitable generalisation of the original definition of the Muttalib–Borodin ensemble allows for negative eigenvalues. In the cases of generalised Gaussian, symmetric Jacobi and Cauchy weights, we show that the problem of computing the normalisations and the biorthogonal..View full abstract
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Awarded by Australian Research Council
This work was supported by the Australian Research Council, through grant DP170102028, and through the Centre of Excellence for Mathematical and Statistical Frontiers.