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Phys Rev E. 2019 Jul;100(1-1):012137. doi: 10.1103/PhysRevE.100.012137.

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

Author information

1
LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.
2
Departamento de Física Cuántica y Fotónica, Instituto de Física, UNAM, P.O. Box 20-364, 01000 Mexico Distrito Federal, Mexico.
3
London Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom.
4
International Centre for Theoretical Sciences, TIFR, Bangalore 560089, India.

Abstract

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 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 〈N_{r}〉≈Nr^{2} display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order O(N^{1/4}), (ii) an intermediate regime where N_{r}-〈N_{r}〉=O(sqrt[N]), and (iii) a large deviation regime where N_{r}-〈N_{r}〉=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 N_{r}, 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.

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