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

Abstract

The family of circular Jacobi beta 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 rightarrow infty bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobibeta ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for beta = 2 and beta = 4, and the loop equation hierarchy. The polynomials in the variable u=2/beta which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular beta ensemble, specifically in relation to the zeros lying on the unit circle |u|=1 and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.