  We study fluctuations of polynomial linear statistics for discrete
Schr\"odinger operators with a random decaying potential. We describe a
decomposition of the space of polynomials into a direct sum of three subspaces
determining the growth rate of the variance of the corresponding linear
statistic. In particular, each one of these subspaces defines a unique critical
value for the decay-rate exponent, above which the random variable has a limit
that is sensitive to the underlying distribution and below which the random
variable has asymptotically Gaussian fluctuations.
