Small scale equidistribution of random waves generated by an unfair coin flip

TL;DR

This paper investigates how the small scale equidistribution of random waves is affected when coefficients are generated by an unfair coin, providing explicit conditions for maintaining equidistribution.

Contribution

It introduces explicit criteria on the bias of the coin (deviation from p=0.5) needed to preserve small scale equidistribution of the random waves.

Findings

Equidistribution holds for fair coin coefficients (p=0.5).

Explicit bounds on p deviation ensure equidistribution.

Results extend understanding of randomness influence on wave distribution.

Abstract

In this paper we study the small scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is the coefficients take value $+1$ with probability $p$ and $-1$ with probability $1-p$. Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ($p=0.5$) coin to retain equidistribution.