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^{2beta X_n(l)}, for beta in {mathbb {R}}. We obtain explicit formulae for the first few moments for finite n. In the limit nrightarrow 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.
Highlights
One of the key ideas that underpins much of the progress outlined above is that the Fourier series representing log PN (A, θ ) exhibits a hierarchical structure typical of problems associated with logarithmically correlated Gaussian fields. This structure is exemplified by the branching random walk
There has again been a good deal of progress in proving the conjecture corresponding to (3) using the analogue for the zeta function of the hierarchical structure exemplified by the branching random walk [1,2,3,4,5,6,21,22,24], and so we see our results for the branching random walk as being of interest in the number theoretical context as well
Where the expectation in (12) is with respect to the Gaussian random variables. These are the moments of moments for the branching random walk
Summary
In recent years there has been significant progress towards understanding the value distribution of the maximum of the logarithm of the characteristic polynomial of a random unitary matrix and of related log-correlated processes [1,2,3,4,5,6,11,17,18,19,20,21,22,24,25,26,27,29].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
