Universality for zeros of random analytic functions
Universality for zeros of random analytic functions
复制标题
随机解析函数零点的普遍性
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
D. Zaporozhets
中科院分区：
文献类型：
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作者：
Z. Kabluchko;D. Zaporozhets
设0；1；：独立同分布(I.I.D.)随机变量使得E log(1+j 0j)<；1.我们考虑形式为Gn(Z)=1Xk=0kfk；nZk；n logf(Tn)；n！U(T)为n！1，其中u(T)是某一函数，我们证明了该测度n弱收敛于某一确定性测度，该测度由u的勒让德{fichel变换来刻画.该极限测度是普适的，即它不依赖于k的分布.这一结果被应用于几个随机解析函数系综上，包括对应于三个二维常曲率几何的系综.作为另一个应用，我们证明了随机矩阵的循环定律的一个随机多项式模拟。
Let 0; 1;::: be independent identically distributed (i.i.d.) ran- dom variables such that E log(1 +j 0j) < 1. We consider random analytic functions of the form Gn(z) = 1 X k=0 kfk;nz k ; n logf(tn);n! u(t) as n!1, where u(t) is some function, we show that the measure n converges weakly to some deterministic measure which is characterized in terms of the Legendre{Fenchel transform of u. The limiting measure is universal, that is it does not depend on the distribution of the k's. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.