Revisiting the Theory of Finite Size Scaling in Disordered Systems: $\mathit{\nu }$ Can Be Less than $2/\mathit{d}$

For phase transitions in disordered systems, an exact theorem provides a bound on the finite size correlation length exponent: ${\nu }_{\mathrm{FS}}\ge 2/d$. 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 $\nu \le 2/d$ self-averaging breaks down, disconnecting $\nu$ from ${\nu }_{\mathrm{FS}}$, and the bound applies only for the latter. We illustrate these ideas on two exact examples, with $\nu <2\mathrm{}/d$. We propose a new method of disorder averaging, which is able to capture the intrinsic exponents.