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 $\frac{1}{2^n}{\sum }_{l=1}^{2^n}{e}^{2\beta X_n(l)}$, for $\beta \in \mathbb{R}$. We obtain explicit formulae for the first few moments for finite $n$. In the limit $n\to \infty$, 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.

arXiv comment26 pages, version published in Journal of Statistical Physics

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Cited by in corpus (3)

• On the moments of the moments of $\zeta (1/2+it)$
• Maxima of log-correlated fields: some recent developments
• Hierarchical Structure in the Trace Formula