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 ξ, we recover strong self-averaging (SA): $P(X)$ approaches a Gaussian with relative squared width RX~(L/ξ)^-d. For Lξ we show weak SA ( RX 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.