Phonon displacement distribution at T=0

We compute the probability distribution function of the displacement squared of linearly coupled quantum oscillators in their ground state. We find it to be one of the extreme-value distributions, the Fisher–Tippett–Gumbel distribution, in d=1, while it is a Gaussian in d=2,3 dimensions. We also discuss the crossover to non-Gaussian distributions (at d=2,3) at finite temperature T. We observe that the quantum effects remain important for sizes ℓ(T)∼aTD/T, where a is the lattice spacing and TD is the Debye temperature.