On the second correlation function of the characteristic polynomials of sparse Non-Hermitian matrices

Ph. Dr. I. Afanasiev

2023- 2024

We consider the asymptotic local behavior of the second correlation function of the characteristic polynomials of sparse non-Hermitian random matrices X_n whose entries have the form x_jk=d_jkw_jk with iid complex standard Gaussian w_jk and normalised iid Bernoulli(p) d_jk. It is shown that, as p®∞, the local asymptotic behavior of the second correlation function of characteristic polynomials near z₀ÎC coincides with those for Ginibre ensemble: it converges to a determinant with Ginibre kernel in the bulk |z₀|<1, and it is factorized if |z₀|>1. For the finite p>0, the behavior is different and exhibits the transition between three different regimes depending on values of p and |z₀|².