2019/04/03 by Bertrand Lacroix-A-Chez-Toine, Jeyson Andres Monroy Garzon, Christopher Sebastian Hidalgo Calva +4 · 1 citation
Physics and Astronomy · Mathematics · #cond-mat.stat-mech #cond-mat.dis-nn #math-ph #math.MP
paper · pdf · doi:10.1103/physreve.100.012137
published as Phys. Rev. E 100, 012137 (2019) · 10 pages, 3 Figures
arxiv created 2019/04/03 · arxiv updated 2019/07/30
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 ⟩ ≈ N r2 display three different regimes: (i) a typical Gaussian regime where the fluctuations are of order \cal O(N1/4), (ii) an intermediate regime where Nr - ⟨ Nr ⟩ = \cal O(√(N)), and (iii) a large deviation regime where Nr - ⟨ Nr ⟩ = \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 Nr, which are all of order \cal O(√(N)), and we compute them explicitly. Our analytical results are corroborated by precise "importance sampling" Monte Carlo simulations.