AI 中文总结
本文推广随机行列式六阶矩公式至任意分布元素的随机矩阵，通过标记置换表分解方法得到其指数生成函数的闭式形式，并经数值验证结果的正确性。
AI 中文摘要
通过本文提出的标记置换表方法，我们将随机行列式六阶矩的公式推广到元素为任意分布的情况。设$f_6(n) = \text{E}(\text{det } A)^6$，其中$A$是元素独立同分布的$n$阶随机矩阵。我们证明指数生成函数$F_6(t) = \text{sum}_{n=0}^\text{infty} f_6(n)t^n/(n!)^2$是D-有限的，并给出其闭式形式。我们的方法依赖于将标记置换表精心分解为壳、核心和浮动分量，三者分别对$F_6(t)$有独立贡献。分解后只需枚举有限种可能的壳，我们通过高度复杂的计算机程序完成了该枚举。我们采用另一种计算$f_6(n)$的方法，在一般情况验证到$n=7$，在元素仅取两个值的随机矩阵情况验证到$n=9$，验证了所得结果的正确性。
英文摘要
Via the method of marked permutation tables presented in this paper, we generalize the formula for the sixth moment of a random determinant to account for entries with arbitrary distribution. That is, let $f_6(n) = \mathbb{E}(\det A)^6$, where $A$ is an $n$ by $n$ random matrix with independent and identically distributed entries. We show that the exponential generating function $F_6(t) = \sum_{n=0}^\infty f_6(n)t^n/(n!)^2$ is D-finite and we present it in a closed form. Our method relies on carefully decomposing marked permutation tables into a shell, a core, and a floating component, each of which has a separate contribution to $F_6(t)$. After this decomposition, it is sufficient to enumerate over a finite number of possible shells, which we did using a highly intricate computer program. We verified our result up to $n = 7$ in the general case and up to $n = 9$ for random matrices whose entries only take two values by using a different method for computing $f_6(n)$ for these cases.
Comments94 pages, 9 figures, 3 tables
