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arXiv · 2412.01086

Spectral radii of sparse non-Hermitian random matrices

Hyungwon Han
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Abstract

We provide estimates for the spectral radii of an $n\times n$ sparse non-Hermitian random matrix $Z$ with general entries in the regime $p=d/n$ where $0<d<1$ is fixed. Utilizing the structural results of (Łuczak, '90), we show that the spectral radius $ρ(Z)$ is $0$ with probability converging to some nonzero value, and satisfies the inequality $(ϕ(n))^{-1}\leq ρ(Z)\leq ϕ(n)$ in the asymptotic sense for any function $ϕ$ satisfying $\lim_{n\to\infty}ϕ(n)=\infty$ with the remaining probability.

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Hyungwon Han. 2024-12-02. Spectral radii of sparse non-Hermitian random matrices. https://arxiv.org/abs/2412.01086

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