Large gap asymptotics on annuli in the random normal matrix model.

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form exp(C1n2+C2nlogn+C3n+C4n+C5logn+C6+Fn+O(n-112)),where n is the number of points of the process. We determine the constants C1,…,C6 explicitly, as well as the oscillatory term Fn which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only C1,…,C4 were previously known, (ii) when the hole region is an unbounded annulus, only C1,C2,C3 were previously known, and (iii) when the hole region is a regular annulus in the bulk, only C1 was previously known. For general values of our parameters, even C1 is new. A main discovery of this work is that Fn is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.