Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces
Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces
复制标题
黎曼曲面上随机亚纯函数的零点和临界点配对
DOI：
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发表时间：
2013
期刊：
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通讯作者：
B. Hanin
中科院分区：
文献类型：
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作者：
B. Hanin
本文证明了一个复变函数的N次随机多项式p_N$的零点和临界点成对出现。更准确地说，如果$p_N$被条件化为对于C中的固定$xi，$p_N（xi）=0${0}，证明了在环$N^{-1-ep}<abs{z-xi}< N^{-1+ep}}$中存在唯一的临界点z，并且没有临界点以至少1-O（N^{-3/2+3ep}）的概率接近$xi$。我们还证明了一个类似的声明，在更一般的设置随机亚纯函数的封闭黎曼曲面。
We prove that zeros and critical points of a random polynomial $p_N$ of degree $N$ in one complex variable appear in pairs. More precisely, if $p_N$ is conditioned to have $p_N(xi)=0$ for a fixed $xi in Cackslashset{0},$ we prove that there is a unique critical point z in the annulus $N^{-1-ep}<abs{z-xi}< N^{-1+ep}}$ and no critical points closer to $xi$ with probability at least $1-O(N^{-3/2+3ep}).$ We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.