旋转不变随机单形的薄壳理论
Thin-shell theory for rotationally invariant random simplices
摘要：
对于固定函数$G,H:[0,\infty)\to[0,\infty)$，考虑$\mathbb{R}^n$上的旋转不变概率密度的形式为\[ μ^n(ds) = \frac{1}{Z_n} G(\|s\|_2)\, e^{ - n H( \|s\|_2)} ds. \]我们证明了当$n$很大时，根据$μ^n$分布的随机向量$Y^n$的欧几里得范数$\|Y^n\|_2$满足高斯薄壳性质：$\|Y^n\|_2$的分布集中在一定值$s_0$附近，其波动近似为高斯型，其阶数为$1/\sqrt{n}$。我们应用这种薄壳性质的研究旋转不变的随机单形，单形的顶点组成的起源以及独立的随机向量$Y_1^n,\ldots,Y_p^n$分布根据$μ^n$。我们表明，对数体积的产生的单纯形表现出高度高斯行为，提供了一个推广和统一的设置Grote-Kabluchko-Thäle [极限定理随机单纯形在高维，ALEA，拉特。Am. J. Probab.数学统计16，141- 177（2019）]。最后，通过将随机单形的体积与随机行列式相关联，我们证明了如果$A^n$是$n \times n$随机矩阵，其条目是独立的标准高斯随机变量，则存在显式常数$c_0,c_1\in(0,\infty)$和绝对常数$C\in(0,\infty)$，使得\[\sup_{ s \in \mathbb{R}} \left| \mathbb{P} \left[ \frac{ \log \mathrm{det}(A^n) - \log(n-1)! - c_0 }{ \sqrt{ \frac{1}{2} \log n + c_1 }} < s \right] - \int_{-\infty}^s \frac{e^{ - u^2/2} du}{ \sqrt{ 2 π}} \right| < \frac{C}{\log^{3/2}n}, \]锐化Nguyen和Vu中的$1/\log^{1/3 + o(1)}n$界[Random matrices：Law of the determinant，Ann. Probab. 42（1）（2014），146- 167]。
英文摘要：
For fixed functions $G,H:[0,\infty)\to[0,\infty)$, consider the rotationally invariant probability density on $\mathbb{R}^n$ of the form \[ μ^n(ds) = \frac{1}{Z_n} G(\|s\|_2)\, e^{ - n H( \|s\|_2)} ds. \] We show that when $n$ is large, the Euclidean norm $\|Y^n\|_2$ of a random vector $Y^n$ distributed according to $μ^n$ satisfies a Gaussian thin-shell property: the distribution of $\|Y^n\|_2$ concentrates around a certain value $s_0$, and the fluctuations of $\|Y^n\|_2$ are approximately Gaussian with the order $1/\sqrt{n}$. We apply this thin shell property to the study of rotationally invariant random simplices, simplices whose vertices consist of the origin as well as independent random vectors $Y_1^n,\ldots,Y_p^n$ distributed according to $μ^n$. We show that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior, providing a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thäle [Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat. 16, 141--177 (2019)]. Finally, by relating the volumes of random simplices to random determinants, we show that if $A^n$ is an $n \times n$ random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants $c_0,c_1\in(0,\infty)$ and an absolute constant $C\in(0,\infty)$ such that \[\sup_{ s \in \mathbb{R}} \left| \mathbb{P} \left[ \frac{ \log \mathrm{det}(A^n) - \log(n-1)! - c_0 }{ \sqrt{ \frac{1}{2} \log n + c_1 }} < s \right] - \int_{-\infty}^s \frac{e^{ - u^2/2} du}{ \sqrt{ 2 π}} \right| < \frac{C}{\log^{3/2}n}, \] sharpening the $1/\log^{1/3 + o(1)}n$ bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42 (1) (2014), 146--167].
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