Sparse matrices: convergence of the characteristic polynomial seen from infinity

Simon Coste

Published 2021-06-01Version 1

We prove that the reverse characteristic polynomial $\det(I_n - zA_n)$ of a random $n \times n$ matrix $A_n$ with iid $\mathrm{Bernoulli}(d/n)$ entries converges in distribution towards the random infinite product $\prod_{\ell = 1}^\infty(1-z^\ell)^{Y_\ell}$ where $Y_\ell$ are independent $\mathrm{Poisson}(d^\ell/\ell)$ random variables. We show that this random function is a Poisson analog of the Gaussian holomorphic chaos and give some of its properties. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erd\H{o}s-R\'enyi digraphs.