论文标题
随机多项式衍生物的半圆定律
A Semicircle Law for Derivatives of Random Polynomials
论文作者
论文摘要
令$ x_1，\ dots，x_n $ be $ n $独立且分布的随机变量，平均零，单位差异和所有剩余订单的有限矩。我们研究随机多项式$ p_n $，根为$ x_1，\ dots，x_n $。我们证明，对于$ \ ell \ in \ mathbb {n} $固定为$ n \ rightarrow \ infty $，$（n- \ ell） - $ p_n^{} $的$ th衍生物的行为就像hermite polynomial：compact，$ x $ compact，$ x $ compact，$ x $ \ frac {\ ell！} {n！} \ cdot p_n^{（n- \ ell）} \ left（\ frac {x} {x} {\ sqrt {n}}} \ right） Probabilists的Hermite多项式和$γ_n$是一个随机变量，将$ \ Mathcal {n}（0,1）$ Gaussian融合为$ n \ rightarrow \ rightarrow \ infty $。因此，当多次区分随机多项式时，存在普遍性现象：其余的根遵循Wigner半圆形分布。
Let $x_1, \dots, x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots, x_n$. We prove that for $\ell \in \mathbb{N}$ fixed as $n \rightarrow \infty$, the $(n-\ell)-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval,$${n^{\ell/2}} \frac{\ell!}{n!} \cdot p_n^{(n-\ell)}\left( \frac{x}{\sqrt{n}}\right) \rightarrow He_{\ell}(x + γ_n),$$ where $He_{\ell}$ is the $\ell-$th probabilists' Hermite polynomial and $γ_n$ is a random variable converging to the standard $\mathcal{N}(0,1)$ Gaussian as $n \rightarrow \infty$. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.
