Absence of self-averaging and universal fluctuations in random systems near critical points

The distributions P(X) of singular thermodynamic quantities, on an ensemble of d-dimensional quenched random samples of linear size L near a critical point, are analyzed using the renormalization group. For L much larger than the correlation length xi, we recover strong self-averaging (SA): P(X) approaches a Gaussian with relative squared width R(X) similar to (L/xi)(-d). For L much less than xi we show weak SA (R(X) decays with a small power of L) or no SA [P(X) approaches a non-Gaussian, with universal L-independent relative cumulants], when the randomness is irrelevant or relevant, respectively.