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

We study the Ginibre ensemble of N×N complex random matrices and compute exactly, for any finite N, the full distribution as well as all the cumulants of the number Nᵣ 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 Nᵣ around its mean value 〈Nᵣ〉≈Nr² display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order O(N1/4), (ii) an intermediate regime where Nᵣ-〈Nᵣ〉=O(sqrt[N]), and (iii) a large deviation regime where Nᵣ-〈Nᵣ〉=O(N). 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 Nᵣ, which are all of order O(sqrt[N]). 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.