Index distribution of the Ginibre ensemble

Abstract

Complex systems, and in particular random neural networks, are often described by randomly interacting dynamical systems with no specific symmetry. In that context, characterizing the number of relevant directions necessitates fine estimates on the Ginibre ensemble. In this fast track communication, we compute analytically the probability distribution of the number of eigenvalues NR with modulus greater than R (the index) of a large N × N random matrix in the real or complex Ginibre ensemble. We show that the fraction NR/N = p has a distribution scaling as exp ( − βN2ψR(p)) with β = 2 (respectively β = 1) for the complex (resp. real) Ginibre ensemble. For any p ∈ [0, 1], the equilibrium spectral densities as well as the rate function ψR(p) are explicitly derived. This function displays a third order phase transition at the critical (minimum) value , associated to a phase transition of the Coulomb gas. We deduce that, in the central regime, the fluctuations of the index NR around its typical value scale as N1/3.