Sparse matrices: convergence of the characteristic polynomial seen from infinity

We prove that the reverse characteristic polynomial det(In−zAn) of a random n×n matrix An with iid Bernoulli(d∕n) entries converges in distribution towards the random infinite product ∏ℓ=1∞(1−zℓ)Yℓ where Yℓ are independent Poisson(dℓ∕ℓ) random variables. We show that this random function is a Poisson analog of more classical Gaussian objects such as the Gaussian holomorphic chaos. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erdős-Rényi digraphs: for every d>1, the greatest eigenvalue of An is close to d and the second greatest is smaller than d, a Ramanujan-like property for irregular digraphs. For d<1, the only non-zero eigenvalues of An converge to a Poisson multipoint process on the unit circle. Our results also extend to the semi-sparse regime where d is allowed to grow to ∞ with n, slower than no(1). We show that the reverse characteristic polynomial converges towards a more classical object written in terms of the exponential of a log-correlated real Gaussian field. In the semi-sparse regime, the empirical spectral distribution of An∕ dn converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.