Bailey E C, Keating J P
School of Mathematics, University of Bristol, Bristol, BS8 1UG UK.
Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK.
J Stat Phys. 2021;182(1):20. doi: 10.1007/s10955-020-02696-9. Epub 2021 Jan 12.
We calculate, for a branching random walk to a leaf at depth on a binary tree, the positive integer moments of the random variable , for . We obtain explicit formulae for the first few moments for finite . In the limit , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.
对于二叉树上深度为(n)处通向叶子节点的分支随机游走，我们计算随机变量(Z_n)（其中(n\in\mathbb{N})）的正整数阶矩。对于有限的(n)，我们得到了前几个矩的显式公式。在(n\to\infty)的极限情况下，我们的表达式与最近关于随机酉矩阵特征多项式矩的矩的猜想和结果一致，这支持了这两个都属于对数相关高斯随机场类别的问题相互关联的观点。
