Normality of smooth statistics for planar determinantal point processes

We consider smooth linear statistics of determinantal point processes on the complex plane, and their large scale asymptotics. We prove asymptotic normality in the finite variance case, where Soshnikov's theorem is not applicable. The setting is similar to that of Rider and Vir\'ag [Electron. J. Probab., 12, no. 45, 1238--1257, (2007)] for the complex plane, but replaces analyticity conditions by the assumption that the correlation kernel is reproducing. Our proof is a streamlined version of that of Ameur, Hedenmalm and Makarov [Duke Math J., 159, 31--81, (2011)] for eigenvalues of normal random matrices. In our case, the reproducing property is brought to bear to compensate for the lack of analyticity and radial symmetries.