Revisiting the theory of finite size scaling in disordered systems:: ν can be less than 2/d

For phase transitions in disordered systems, an exact theorem provides a bound on the finite size correlation length exponent: nu(FS) greater than or equal to 2/d. it is believed that the intrinsic nu satisfies the same bound. We argue that the standard averaging introduces a noise and a new diverging length scale. For nu less than or equal to 2/d self-averaging breaks down, disconnecting nu from nu(FS), and the bound applies only for the latter. We illustrate these ideas on two exact examples, with nu < 2/d. We propose a new method of disorder averaging, which is able to capture the intrinsic exponents.