Sharpness of the fluctuation-tail estimates

Determine whether the upper and lower fluctuation-tail estimates for the centered maximal ratio I_L established in Theorem 5 are sharp, namely whether fluctuations of I_L-\mathbb{E}I_L occur at scale \ln^{-1/4}L with the stated exponential right tail and double-exponential left tail.

Background

The paper studies the maximal white-noise-to-perimeter ratio I_L over polygonal curves contained in a ball of radius L and having side lengths at least 1. Theorem 5 gives refined concentration estimates for I_L-\mathbb{E}I_L: an exponential upper tail and a double-exponential lower tail at the scale \ln^{-1/4}L, for deviations above a logarithmic threshold.

The authors explicitly note that the proof does not establish whether these estimates are optimal. Resolving their sharpness would clarify the true fluctuation scale and tail behavior. If the estimates are sharp, the resulting tail asymmetry would support a limiting Gumbel-type description of the fluctuations, analogous to the two-dimensional Gaussian free field.