Asymptotics of determinants with a rotation-invariant weight and discontinuities along circles

We study the moment generating function of disk counting statistics a two-dimensional determinantal point process which generalizes complex Ginibre process. This involves an n×n determinant whose weight is supported on whole plane, rotation-invariant, and has discontinuities along circles centered at 0. These can be thought as analogue jump-type Fisher-Hartwig singularities. In this paper, we obtain large n asymptotics for determinant, up to including term order n?12. allow any finite number in bulk, one discontinuity edge, bounded away from bulk. As application, all cumulants n?12, both bulk edge. improves best known results process, general values our parameters these are completely new. Our proof makes novel use uniform incomplete gamma function.