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The distributions $P(X)$ of singular thermodynamic quantities in an ensembleof quenched random samples of linear size $l$ at the critical point $T_c$ arestudied by Monte Carlo in two models. Our results confirm predictions ofAharony and Harris based on Renormalization group considerations. For anAshkin-Teller model with strong but irrelevant bond randomness we find that therelative squared width, $R_X$, of $P(X)$ is weakly self averaging. $R_X\siml^{\alpha/\nu}$, where $\alpha$ is the specific heat exponent and $\nu$ is thecorrelation length exponent of the pure model fixed point governing thetransition. For the site dilute Ising model on a cubic lattice, known to begoverned by a random fixed point, we find that $R_X$ tends to a universalconstant independent of the amount of dilution (no self averaging). Howeverthis constant is different for canonical and grand canonical disorder. We studythe distribution of the pseudo-critical temperatures $T_c(i,l)$ of the ensembledefined as the temperatures of the maximum susceptibility of each sample. Wefind that its variance scales as $(\delta T_c(l))^2 \sim l^{-2/\nu}$ and NOT as$\sim l^{-d}. We find that $R_\chi$ is reduced by a factor of $\sim 70$ withrespect to $R_\chi (T_c)$ by measuring $\chi$ of each sample at $T_c(i,l)$. Weanalyze correlations between the magnetization at criticality $m_i(T_c,l)$ andthe pseudo-critical temperature $T_c(i,l)$ in terms of a sample independentfinite size scaling function of a sample dependent reduced temperature$(T-T_c(i,l))/T_c$. This function is found to be universal and to behavesimilarly to pure systems.Comment: 31 pages, 17 figures, submitted to Phys. Rev.

Self-averaging, distribution of pseudocritical temperatures, and finite size scaling in critical disordered systems