Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points

The distributions $P\left(X\right)$ 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): $P\left(X\right)$ approaches a Gaussian with relative squared width $R_X\sim (L/\xi {)}^{-d}$. For $L\ll \xi$ we show weak SA ( $R_X$ decays with a small power of $L$) or no SA [ $P\left(X\right)$ approaches a non-Gaussian, with universal $L$-independent relative cumulants], when the randomness is irrelevant or relevant, respectively.