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Preprints, Working Papers, ... Year : 2021

Sparse matrices: convergence of the characteristic polynomial seen from infinity

Simon Coste

Abstract

We prove that the reverse characteristic polynomial $\det(I_n - zA_n)$ of a random $n \times n$ matrix $A_n$ with iid $\mathrm{Bernoulli}(d/n)$ entries converges in distribution towards the random infinite product $\prod_{\ell = 1}^\infty(1-z^\ell)^{Y_\ell}$ where $Y_\ell$ are independent $\mathrm{Poisson}(d^\ell/\ell)$ random variables. We show that this random function is a Poisson analog of the Gaussian holomorphic chaos and give some of its properties. 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.

Dates and versions

hal-03247769 , version 1 (03-06-2021)

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Simon Coste. Sparse matrices: convergence of the characteristic polynomial seen from infinity. 2021. ⟨hal-03247769⟩
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