The distributions P(X) of singular thermodynamic quantities in an ensemble
of quenched random samples of linear size l at the critical point Tc are
studied by Monte Carlo in two models. Our results confirm predictions of
Aharony and Harris based on Renormalization group considerations. For an
Ashkin-Teller model with strong but irrelevant bond randomness we find that the
relative squared width, RX, of P(X) is weakly self averaging. RX∼lα/ν, where α is the specific heat exponent and ν is the
correlation length exponent of the pure model fixed point governing the
transition. For the site dilute Ising model on a cubic lattice, known to be
governed by a random fixed point, we find that RX tends to a universal
constant independent of the amount of dilution (no self averaging). However
this constant is different for canonical and grand canonical disorder. We study
the distribution of the pseudo-critical temperatures Tc(i,l) of the ensemble
defined as the temperatures of the maximum susceptibility of each sample. We
find that its variance scales as (δTc(l))2∼l−2/ν and NOT as
∼l−d.WefindthatR_\chiisreducedbyafactorof\sim 70withrespecttoR_\chi (T_c)bymeasuring\chiofeachsampleatT_c(i,l).Weanalyzecorrelationsbetweenthemagnetizationatcriticalitym_i(T_c,l)andthepseudo−criticaltemperatureT_c(i,l)intermsofasampleindependentfinitesizescalingfunctionofasampledependentreducedtemperature(T-T_c(i,l))/T_c$. This function is found to be universal and to behave
similarly to pure systems.Comment: 31 pages, 17 figures, submitted to Phys. Rev.