Large gap asymptotics on annuli in the random normal matrix model

Abstract

AbstractWe 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 $$\begin{aligned} \exp \Bigg ( C_{1} n^{2} + C_{2} n \log n + C_{3} n + C_{4} \sqrt{n} + C_{5}\log n + C_{6} + {\mathcal {F}}_{n} + \mathcal {O}\Big ( n^{-\frac{1}{12}}\Big )\Bigg ), \end{aligned}$$ exp ( C 1 n 2 + C 2 n log n + C 3 n + C 4 n + C 5 log n + C 6 + F n + O ( n - 1 12 ) ) , where n is the number of points of the process. We determine the constants $$C_{1},\ldots ,C_{6}$$ C 1 , … , C 6 explicitly, as well as the oscillatory term $${\mathcal {F}}_{n}$$ F n 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 $$C_{1},\ldots ,C_{4}$$ C 1 , … , C 4 were previously known, (ii) when the hole region is an unbounded annulus, only $$C_{1},C_{2},C_{3}$$ C 1 , C 2 , C 3 were previously known, and (iii) when the hole region is a regular annulus in the bulk, only $$C_{1}$$ C 1 was previously known. For general values of our parameters, even $$C_{1}$$ C 1 is new. A main discovery of this work is that $${\mathcal {F}}_{n}$$ F n 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.