Moments of Moments and Branching Random Walks
Abstract
We calculate, for a branching random walk X_{n}(l ) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable 1/2^{n}∑l_{=1}^{2n}e^{2 β Xn(l )} , for β ∈R . We obtain explicit formulae for the first few moments for finite n. In the limit n →∞ , 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.
 Publication:

Journal of Statistical Physics
 Pub Date:
 January 2021
 DOI:
 10.1007/s10955020026969
 arXiv:
 arXiv:2008.09536
 Bibcode:
 2021JSP...182...20B
 Keywords:

 Branching random walks;
 Moments;
 Logarithmically correlated processes;
 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 26 pages, version published in Journal of Statistical Physics