Sparse matrices: convergence of the characteristic polynomial seen from infinity

Abstract

We prove that the reverse characteristic polynomial $det(I_n-z{A}_n)$ of a random $n\times n$ matrix $A_n$ with iid $\mathrm{Bernoulli}(d∕n)$ entries converges in distribution towards the random infinite product

$$\prod _{\ell =1}^{\mathrm{\infty }}{(1-z^{\ell })}^{Y_{\ell }}$$

where $Y_{\ell }$ are independent $\mathrm{Poisson}(d^{\ell }∕\ell )$ random variables. We show that this random function is a Poisson analog of 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 $A_n$ 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 ∞ 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. In the semi-sparse regime, the empirical spectral distribution of $A_n∕\sqrt{d_n}$ converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.