Premium
Painlevé VI and the Unitary Jacobi Ensembles
Author(s) -
Chen Y.,
Zhang L.
Publication year - 2010
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2010.00483.x
Subject(s) - mathematics , eigenvalues and eigenvectors , wishart distribution , random matrix , jacobi operator , ising model , distribution (mathematics) , hankel matrix , mathematical physics , mathematical analysis , pure mathematics , quantum mechanics , orthogonal polynomials , jacobi polynomials , physics , multivariate statistics , statistics
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 .