We study the moment generating function of the disk counting statistics of a two-dimensional determinantal point process which generalizes the complex Ginibre point process. This moment generating function involves an n × n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has discontinuities along circles centered at 0. These discontinuities can be thought of as a two-dimensional analogue of jump-type Fisher-Hartwig singularities. In this paper, we obtain large n asymptotics for this determinant, up to and including the term of order n − 1 2 . We allow for any number of discontinuities in the bulk, one discontinuity at the edge, and any number of discontinuities bounded away from the bulk. As an application, we obtain the large n asymptotics of all the cumulants of the disk counting function up to and including the term of order n − 1 2 , both in the bulk and at the edge. This improves on the best known results for the complex Ginibre point process, and for general values of our parameters these results are completely new. Our proof makes a novel use of the uniform asymptotics of the incomplete gamma function.

We consider the limiting process that arises at the hard edge of Muttalib-Borodin ensembles. This point process depends on θ > 0 and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the formwhere the constants ρ, a, and b have been derived explicitly via a differential identity in s and the analysis of a Riemann-Hilbert problem. Their method can be used to evaluate c (with more efforts), but does not allow for the evaluation of C. In this work, we obtain expressions for the constants c and C by employing a differential identity in θ. When θ is rational, we find that C can be expressed in terms of Barnes' G-function. We also show that the asymptotic formula can be extended to all orders in s.

We develop an inverse scattering transform formalism for the "good" Boussinesq equation on the line. Assuming that the solution exists, we show that it can be expressed in terms of the solution of a 3×3 matrix Riemann-Hilbert problem. The Riemann-Hilbert problem is formulated in terms of two reflection coefficients whose definitions involve only the initial data, and it has a form which makes it suitable for the evaluation of long-time asymptotics via Deift-Zhou steepest descent arguments.

Traceability, Risk management, Food safety, European regulation, I18, K32, Q18,