Scale Invariance and Self-averaging in disordered systems

7 pages, 4 ps figuresIn a previous paper we found that in the random field Ising model at zero temperature in three dimensions the correlation lengthis not self-averaging near the critical point and that the violation of self-averaging is maximal. This is due to the formation of boundstates in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensionsnear the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperaturebut the violation of self-averaging is weaker than in the random field case. In the random Ising model we find still weaker violations ofself-averaging and we cannot rule out the possibility of the restoration of self-averaging in the infinite volume limit