Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble

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 ${\cal O}(N^{1/4})$, (ii) an intermediate regime where $N_r - \langle N_r \rangle = {\cal O}(\sqrt{N})$, and (iii) a large deviation regime where $N_r - \langle N_r \rangle = {\cal O}({N})$. This intermediate behaviour (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 (centred) cumulants of $N_r$, which are all of order ${\cal O}(\sqrt{N})$, and we compute them explicitly. Our analytical results are corroborated by precise "importance sampling" Monte Carlo simulations.