Away from criticality, when the larger lattice is built from smaller blocks, then due to the additivity property of an extensive quantity, the central limit theorem guarantees that RX ~ N−1 thereby ensuring self-averaging.
It has been shown by recent renormalization group and numerical studies that self-averaging property is lost if randomness or disorder is relevant.
Thus unlike the self-averaging scenario, numerical simulations cannot lead to an improved picture in larger lattices (large N), even if the critical point is exactly known.
If the exhibited behavior is RX ~ N−1 as suggested by the central limit theorem, mentioned earlier, the system is said to be strongly self-averaging.
[2] The RG arguments mentioned above need to be extended to situations with sharp limit of Tc distribution and long range interactions.