2021/06/01 by Coste, Simon · 1 citation
#05C80 #15B52 #30B20 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2106.00593
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-z^ℓ)Y_ℓ where Y_ℓ are independent Poisson(d^ℓ/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, as in the dense case studied in a recent paper \citebordenave2020convergence. 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.