Abstract We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is two-parameter generalization of complex Ginibre process. In this paper, we prove that probability no points lie on any number annuli centered at 0 satisfies large n asymptotics form $$\begin{aligned} \exp \Bigg ( C_{1} n^{2} + C_{2} \log C_{3} C_{4} \sqrt{n} C_{5}\log C_{6} ...