Self-averaging in random systems - liability or asset?
  The study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings, provided the system is large enough. Careful examination suggests that this idea might have a follow, when the correlation length becomes of the order of the size of the system. We find certain bound quantities are not self-averaging when the correlation length becomes of the order of the size of the system. This suggests that the strength of self-averaging, expressed in terms of properly chosen signal to noise ratios, may serve to identify phase boundaries. This is demonstrated by using such signal to noise ratios to identify the boundary of the ferromagnetic phase and compare the findings with more traditional measures. 
Avishay Efrat and Moshe Schwartz