The study of random matrices finds many important applications in physics, mathematics, biology, statistics and finance, as random matrix ensembles offer simple but nontrivial models of strongly correlated systems. The derivation of the joint probability distribution of eigenvalues (JPDE) is one of the most important successes of random matrix theory, since spectral observables defined in terms of the eigenvalues, including the spectral density and correlation functions, follow from the JPDE. For non-Hermitian random matrices with Gaussian distributed elements, Ginibre deduced the JPDE for matrices with complex and real quaternion entries, a special case which can be mapped in the Boltzmann distribution characterizing an electrostatic system of interacting charges. This electrostatic analogy underlies Dyson’s Coulomb fluid approach, in which the spectral observables follow from the partition function of an analogous physical system. The situation is considerable more difficult in the Ginibre ensemble with real matrix elements owing to the existence of a finite fraction of real eigenvalues.

Perhaps the most fundamental spectral observable is the so-called number statistics or full counting statistics giving the distribution of the number N_{D} of eigenvalues contained in a certain domain D. The study of the fluctuations of N_{D} is a rich mathematical problem in itself, arising in a wide variety of applications including the study of the ground state of non-interacting fermions in a harmonic trap, the number of stable directions around the stationary points of disordered energy landscapes, the number of relevant fluctuation modes in principal component analysis, the localized or extended nature of eigenstates in disordered quantum systems, and the stability of large interacting biological systems such as neural networks and ecosystems.

Thanks to the well-developed machinery of the Coulomb fluid method, a complete picture of the typical and rare fluctuations of N_{D} has emerged for Gaussian Hermitian random matrices with complex, real, and real quaternion entries. For non-Hermitian random matrices, the question of how many eigenvalues lie outside a disk in the complex plane has been addressed in the case of the real Ginibre ensemble. However, the number statistics has been fully studied only for the complex Ginibre ensemble for which the Coulomb fluid method is readily applied. Such studies are restricted to circular domains D. Ironically, non-Hermitian random matrices with real entries are very relevant for applications, especially in the study of high-dimensional non-equilibrium systems, yet there is no generic analytic method for this case. Hence, the fluctuations of N_{D} remain poorly characterized.

In a recent paper, LML External Fellow Isaac Pérez Castillo and Fernando Metz, working with Edgar Guzmán-González, Antonio Tonatiúh Ramos Sánchez, introduce an analytic approach to determine the fluctuations of the number N_{D} of eigenvalues inside a domain D ⊂ C of arbitrary shape. They show that the study of the number statistics can be formulated for arbitrary ensembles of infinitely large non-Hermitian random matrices, with real or complex elements, without relying on the analytic knowledge of the JPDE. Among other results, the researchers derive explicit results for the statistics of N_{D} in the case of symmetric adjacency matrices of random graphs with asymmetric couplings, for which an analytic expression for the JPDE is not available. The main outcome is a set of effective equations, valid for infinitely large random matrices, which determine all the cumulants and the large deviation function controlling, respectively, the typical and rare fluctuations of N_{D}. The exactness of this theoretical approach is fully supported by numerical results obtained from the direct diagonalization of finite random matrices.

The paper is available at https://journals.aps.org/pre/abstract/10.1103/PhysRevE.103.062108

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