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