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

Abstract

The family of circular Jacobi $$\beta $$ 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 $$N \rightarrow \infty $$ N → ∞ bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobi$$\beta $$ β ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for $$\beta = 2$$ β = 2 and $$\beta = 4$$ β = 4 , and the loop equation hierarchy. The polynomials in the variable $$u=2/\beta $$ u = 2 / β which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular $$\beta $$ β ensemble, specifically in relation to the zeros lying on the unit circle $$|u|=1$$ | 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.