Expanding the Fourier Transform of the Scaled Circular Jacobi β Ensemble Density

Abstract

The family of circular Jacobi β ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding N→ ∞ bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobi β ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for β= 2 and β= 4 , and the loop equation hierarchy. The polynomials in the variable u= 2 / β which occur in the expansion coefficents are found to have special properties analogous to those known for the structur..