Moments of characteristic polynomials in certain random neural networks

Abstract

Abstract We consider large neural networks in cognitive neuropsychology whose synaptic connectivity matrices are randomly chosen from correlated Gaussian random matrices. We focus on the moments of characteristic polynomials and prove that the limiting even and odd moments at the edge are given by the largest eigenvalue distribution in the Gaussian Symplectic Ensemble (GSE) and in the induced GSE ensemble, respectively. Our results show that there exists a duality relation between the real Ginibre ensemble and the GSE ensemble via the moment of characteristic polynomials and the largest eigenvalue.