Painlevé VI and the Unitary Jacobi Ensembles
Article first published online: 17 MAR 2010
© 2010 by the Massachusetts Institute of Technology
Studies in Applied Mathematics
Volume 125, Issue 1, pages 91–112, July 2010
How to Cite
Chen, Y. and Zhang, L. (2010), Painlevé VI and the Unitary Jacobi Ensembles. Studies in Applied Mathematics, 125: 91–112. doi: 10.1111/j.1467-9590.2010.00483.x
- Issue published online: 21 JUL 2010
- Article first published online: 17 MAR 2010
- (Received November 10, 2009)
The six Painlevé transcendants which originally appeared in the studies of ordinary differential equations have been found numerous applications in physical problems. The well-known examples among which include symmetry reduction of the Ernst equation which arises from stationary axial symmetric Einstein manifold and the spin-spin correlation functions of the two-dimensional Ising model in the work of McCoy, Tracy, and Wu.
The problem we study in this paper originates from random matrix theory, namely, the smallest eigenvalues distribution of the finite n Jacobi unitary ensembles which was first investigated by Tracy and Widom. This is equivalent to the computation of the probability that the spectrum is free of eigenvalues on the interval . Such ensembles also appears in multivariate statistics known as the double-Wishart distribution.
We consider a more general model where the Jacobi weight is perturbed by a discontinuous factor and study the associated finite Hankel determinant. It is shown that the logarithmic derivative of Hankel determinant satisfies a particular σ-form of Painlevé VI, which holds for the gap probability as well. We also compute exactly the leading term of the gap probability as .