Characteristic Polynomials of Sparse Non-Hermitian Random Matrices

Abstract

We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of sparse non-Hermitian random matrices \(X_n\) whose entries have the form \(x_{jk}=d_{jk}w_{jk}\) with iid complex standard Gaussian \(w_{jk}\) and normalised iid Bernoulli(p) \(d_{jk}\). It is shown that, as \(p\rightarrow \infty \), the local asymptotic behavior of the second correlation function of characteristic polynomials near \(z_0\in \mathbb {C}\) coincides with those for Ginibre ensemble: it converges to a determinant with Ginibre kernel in the bulk \(|z_0|<1\), and it is factorized if \(|z_0|>1\). For the finite \(p>0\), the behavior is different and exhibits the transition between different regimes depending on values of p and \(|z_0|^2\).