Lack of self-averaging in random systems—Liability or asset?

• We find that certain bounded quantities are not self-averaging when the correlation length becomes of the order of the size of the system.
• The lack of self-averaging, expressed in terms of properly chosen signal-to-noise ratios, serves to identify phase boundaries.
• By using such signal-to-noise ratios we identify the boundary of the ferromagnetic phase of the random field Ising system.
• Finally we argue that the signal-to-noise behaves in a generic way, independent of the specific physical quantity for which it is obtained.

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 that the system is large enough. Careful examination suggests that this idea might have a flaw, when the correlation length becomes of the order of the size of the system. We find that certain bounded quantities are not self-averaging when the correlation length becomes of the order of the size of the system. This suggests that the lack 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 of the random field Ising system and compare the findings with more traditional measures.