Lack of Self Averaging and Finite Size Scaling in Critical Disordered Systems

We simulated site dilute Ising models in $d=3$ dimensions for several lattice sizes $L$. For each $L$ singular thermodynamic quantities $X$ were measured at criticality and their distributions $P(X)$ were determined, for ensembles of several thousand random samples. For $L \to \infty$ the width of $P(X)$ tends to a universal constant, i.e. there is no self averaging. The width of the distribution of the sample dependent pseudocritical temperatures $T_c(i,L)$ scales as $\delta T_c(L) \sim L^{-1/\nu}$ and NOT as $\sim L^{-d/2}$. Finite size scaling holds; the sample dependence of $X_i(T_c)$ enters predominantly through $T_c(i,L)$.