We study the Ginibre ensemble of $N\times N$ complex random matrices and compute exactly, for any finite $N$, the full distribution as well as all the cumulants of the number $N_r$ of eigenvalues within a disk of radius $r$ centered at the origin. In the limit of large $N$, when the average density of eigenvalues becomes uniform over the unit disk, we show that for $0<r<1$ the fluctuations of $N_r$ around its mean value $\langle N_r\rangle \approx N{r}^2$ display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order $O\left(N^{1/4}\right)$, (ii) an intermediate regime where $N_r-\langle N_r\rangle =O\left(\sqrt{N}\right)$, and (iii) a large deviation regime where $N_r-\langle N_r\rangle =O\left(N\right)$. This intermediate behavior (ii) had been overlooked in previous studies and we show here that it ensures a smooth matching between the typical and the large deviation regimes. In addition, we demonstrate that this intermediate regime controls all the (centered) cumulants of $N_r$, which are all of order $O\left(\sqrt{N}\right)$. We show that the intermediate deviation function that describes these intermediate fluctuations can be computed explicitly and we demonstrate that it is universal, i.e., it holds for a large class of complex random matrices. Our analytical results are corroborated by precise “importance sampling” Monte Carlo simulations.

• Received 6 April 2019
• Corrected 12 October 2020

DOI:https://doi.org/10.1103/PhysRevE.100.012137