Sparse matrices: convergence of the characteristic polynomial seen from infinity

We prove that the reverse characteristic polynomial $\det (I_n-z{A}_n)$ of a random $n\times n$ matrix $A_n$ with iid $\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}(d\mathrm{/}n)$ entries converges in distribution towards the random infinite product $\prod_{\ell = 1}^\infty(1-z^\ell)^{Y_\ell}$where$Y_\ell$areindependent$$\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}$(d^\ell/\ell)$ 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 Erd\H{o}s-R'enyi digraphs: for every $d>1$, the greatest eigenvalue of $A_n$ is close to $d$ and the second greatest is smaller than $\sqrt{d}$, a Ramanujan-like property for irregular digraphs. For $d<1$, the only non-zero eigenvalues of $A_n$ 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 $\mathrm{\infty }$ with $n$, slower than $n^{o(1)}$. 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, as in the dense case studied in a recent paper \cite{bordenave2020convergence}. In the semi-sparse regime, the empirical spectral distribution of $A_n\mathrm{/}\sqrt{d_n}$ converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.