9 research outputs found
Limits for circular Jacobi beta-ensembles
Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the
circular Jacobi -ensemble, which is a generalization of the Dyson
circular -ensemble but equipped with an additional parameter , and
further studied its limiting spectral measure. We calculate the scaling limits
for expected products of characteristic polynomials of circular Jacobi
-ensembles. For the fixed constant , the resulting limit near the
spectrum singularity is proven to be a new multivariate function. When , the scaling limits in the bulk and at the soft edge agree with those of
the Hermite (Gaussian), Laguerre (Chiral) and Jacobi -ensembles proved
in the joint work with P Desrosiers "Asymptotics for products of characteristic
polynomials in classical beta-ensembles", Constr. Approx. 39 (2014),
arXiv:1112.1119v3. As corollaries, for even the scaling limits of point
correlation functions for the ensemble are given. Besides, a transition from
the spectrum singularity to the soft edge limit is observed as goes to
infinity. The positivity of two special multivariate hypergeometric functions,
which appear as one factor of the joint eigenvalue densities for spiked
Jacobi/Wishart -ensembles and Gaussian -ensembles with source,
will also be shown.Comment: 26 page
Expanding the Fourier transform of the scaled circular Jacobi ensemble density
The family of circular Jacobi 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 ensemble are used
for this purpose. These include linear differential equations satisfied by the
scaled spectral density for and , and the loop equation
hierarchy. The polynomials in the variable which occur in the
expansion coefficents are found to have special properties analogous to those
known for the structure function of the circular ensemble, specifically
in relation to the zeros lying on the unit circle 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.Comment: 30 page