Moments of Moments and Branching Random Walks

TL;DR

This paper derives explicit formulas for moments of a sum involving branching random walks and connects these results to conjectures about characteristic polynomials of random matrices, highlighting their shared logarithmic correlation structure.

Contribution

It provides explicit moment formulas for branching random walks and links these to conjectures on random matrix characteristic polynomials, revealing a deep connection.

Findings

Explicit formulas for moments of branching random walk sums.

Asymptotic expressions match conjectures for random matrix moments.

Supports the relation between branching processes and logarithmically correlated fields.

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.