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

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. 2020 Mathematics Subject Classification. 58J50.35P20,60B10. Key words and phrases. eigenfunction, plane wave, random wave, random eigenfunction, equidistribution at small scales, unfair coin. The authors would like to acknowledge Jeroen Schillewaert for asking Tacy what would happen to random wave equidistribution if the coefficients were not from a fair coin type distribution. The authors also acknowledge the support of the University of Auckland's Summer Scholar Scheme which funded this project.