On the supremum of random cusp forms

TL;DR

This paper studies the expected maximum of random cusp forms for the modular group, revealing different growth behaviors near the cusp and within a compact domain, and confirming conjectured bounds.

Contribution

It introduces a random ensemble of cusp forms and determines the order of magnitude of their supremum, confirming conjectured bounds and analyzing behavior near the cusp.

Findings

Expected supremum on compact domains is proportional to √log(k).

Supremum concentrates exponentially around its median.

Global supremum near the cusp grows like k^{1/4} up to a log factor.

Abstract

A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp \sqrt{\log{k}}$, in line with the conjectured bounds. Additionally, the exponential concentration of the supremum around its median is established. Contrary to the compact case, it is shown that the global expected supremum, which is attained around the cusp, grows like $k^{1/4}$, up to a logarithmic factor.