Moments of Moments and Branching Random Walks

Abstract

We calculate, for a branching random walk $$X_n(l)$$ X n ( l ) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable $$\frac{1}{2^{n}}\sum _{l=1}^{2^n}e^{2\beta X_n(l)}$$ 1 2 n ∑ l = 1 2 n e 2 β X n ( l ) , for $$\beta \in {\mathbb {R}}$$ β ∈ R . We obtain explicit formulae for the first few moments for finite n. In the limit $$n\rightarrow \infty $$ 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.

Maxima of log-correlated fields: some recent developments

“…The Riemann hypothesis implies the Lindelöf hypothesis. The similarities between (16) and theorem 1.5 are noteworthy.…”

“…for some constant c 1 (see for example [142]). However, it is also known (unconditionally) that 16 (109)…”

“…Once again, this is a log-correlated process. 16 In (109), f (x) = Ω(g(x)) means f (x) takes the value g(x) infinitely often.…”

“…As N → ∞, that the real and imaginary parts tend independently to Gaussian random variables: Theorem 1.3 (Central Limit Theorem [110]). Take any rectangle B ⊂ C. For fixed θ ∈ [0, 2π), (16) lim…”

“…Large deviations are a focal point of section 2.3. See also (207) and the surrounding discussion for results concerning large deviations of (16).…”