论文标题
上限的最大偏差
Upper bounds for the maximum deviation of the Pearcey process
论文作者
论文摘要
Pearcey过程是随机矩阵理论中的通用点过程，取决于\ Mathbb {r} $中的参数$ρ\。令$ n（x）$为随机变量，它计算此过程中的点数，该变量属于间隔$ [-x，x] $。在本说明中，我们建立以下全局刚度上限：\ begin {align*} \ lim_ {s \ to \ infty} \ mathbb p \ left（\ sup_ {x> s} \ s} \ left | \ left | \ frac {n（x） - \ big（\ big） \ frac {3 \ sqrt {3}} {4π} x^{\ frac {4} {3}} - \ frac {\ sqrt {\ sqrt {3}ρ} {2π} \ frac {4 \ sqrt {2}} {3π} +ε\ right）= 1，\ end end {align*}，其中$ε> 0 $是任意的。我们还获得了相似的上限，以实现点的最大偏差，并获得单个波动的中心限制定理。证明是简短的，将DAI，Xu和Zhang的最新结果与Charlier和Claeys的另一个结果相结合。
The Pearcey process is a universal point process in random matrix theory and depends on a parameter $ρ\in \mathbb{R}$. Let $N(x)$ be the random variable that counts the number of points in this process that fall in the interval $[-x,x]$. In this note, we establish the following global rigidity upper bound: \begin{align*} \lim_{s \to \infty}\mathbb P\left(\sup_{x> s}\left|\frac{N(x)-\big( \frac{3\sqrt{3}}{4π}x^{\frac{4}{3}}-\frac{\sqrt{3}ρ}{2π}x^{\frac{2}{3}} \big)}{\log x}\right| \leq \frac{4\sqrt{2}}{3π} + ε\right) = 1, \end{align*} where $ε> 0$ is arbitrary. We also obtain a similar upper bound for the maximum deviation of the points, and a central limit theorem for the individual fluctuations. The proof is short and combines a recent result of Dai, Xu and Zhang with another result of Charlier and Claeys.
