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arXiv 1104.5360math.PRmath.CV
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关于重尾系数随机多项式的复根的分布

On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients

Friedrich Götze, Dmitry Zaporozhets

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8 pages
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摘要：

考虑具有i.i.d.的随机多项式$G_n(z)=ξ_nz^n+...+ξ_1z+ξ_0$。复值系数假设$\log(1+\log(1+|ξ_0|))$的分布具有缓慢变化的尾部。则$G_n$的复根的分布在概率上集中于两个中心圆，如$n\to\infty$，并且在辐角上是均匀的，如$n\to\infty$。圆的半径为$|ξ_0/ξ_τ|^{1/τ}$和$|ξ_τ/ξ_n|^{1/(n-τ)}$，其中$ξ_τ$表示具有最大模量的系数。

英文摘要：

Consider a random polynomial $G_n(z)=ξ_nz^n+...+ξ_1z+ξ_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\log(1+\log(1+|ξ_0|))$ has a slowly varying tail. Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\to\infty$, to two centered circles and is uniform in the argument as $n\to\infty$. The radii of the circles are $|ξ_0/ξ_τ|^{1/τ}$ and $|ξ_τ/ξ_n|^{1/(n-τ)}$, where $ξ_τ$ denotes the coefficient with the maximum modulus.

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