I have some problems understanding one part of a paper (end of page 187 here https://iopscience.iop.org/article/10.1088/0951-7715/26/1/177, for who has access, otherwised the simplified arxiv version here https://arxiv.org/pdf/1503.02676.pdf, but without the problematic bit...)
Ley $p$ a fixed integer and let $n>p$ going to infinity. I have to asymptotically estimate the integral $$(2\pi)^{-2^{p-1}}\int_{[0,2\pi]^{2^{p-1}}} \exp(\log(tr( (Q(\varphi))^n ) )) d\varphi,$$ where $Q(\varphi)$ is matrix containing the $e^{i\varphi_j}$, which have a specific form. In particular the trace is a long sum of the same exponentials, and therefore a trigonometric polynomial. The authors use the saddle point method and so they have to expand the logarithm up to order to 2 around every maximum point. It is easy to see that one global maximum is given when $\varphi_j=0$ for all $j$. However, they claim that the function has other maximum points where at least one of the $\varphi_j$ is non-zero.
They claim that the contribution of these maximum points in of order $O(2^{-n})$ giving the following explanations: the trace is a trigonometric polynomial whose phaces are distributed according to same complicated law, and which can be to same extent replaced by a uniform distribution in $(0,2\pi]$. According to the central limit theorem the contribution of these maxima should be of order $O(2^{-n})$.
I think what they use is the Salem-Zygmund Theorem or central limit theorem for trigonometric polynomial, in order to say that the polynomial converges in distribution to $N(0,1/2)$, but I don't understand how this gives $O(2^{-n})$.
If anybody has a hint, please share it. Thank you!