We prove that the reverse characteristic polynomial of a random matrix with iid entries converges in distribution towards the random infinite product
where are independent 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 , the greatest eigenvalue of is close to d and the second greatest is smaller than , a Ramanujan-like property for irregular digraphs. For , the only non-zero eigenvalues of 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 . 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 converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.
The author is supported by ERC NEMO, under the European Union’s Horizon 2020 research and innovation programme grant agreement number 788851.
The author thanks the three authors of  as well as Yizhe Zhu and Ludovic Stephan for various discussions on this method and comments on the paper.
"Sparse matrices: convergence of the characteristic polynomial seen from infinity." Electron. J. Probab. 28 1 - 40, 2023. https://doi.org/10.1214/22-EJP875