2023/06/01 by Peter J. Forrester, Forrester, Peter J., Bo-Jian Shen +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Chemistry · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Protein Interaction Studies and Fluorescence Analysis
paper · pdf · doi:10.48550/arxiv.2306.00525
openalex publication_date 2023/06/01 · openalex created_date 2023/06/04 · openalex updated_date 2026/07/28
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 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 structure function of the circular β 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.