Self-averaging

At the pure critical point randomness is classified as relevant if, by the standard definition of relevance, it leads to a change in the critical behaviour (i.e., the critical exponents) of the pure system. It has been shown by recent renormalization group and numerical studies that self-averaging property is lost if randomness or disorder is relevant.^[1] Most importantly as N → ∞, R_X at the critical point approaches a constant. Such systems are called non self-averaging. 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. In summary, various types of self-averaging can be indexed with the help of the asymptotic size dependence of a quantity like R_X. If R_X falls off to zero with size, it is self-averaging whereas if R_X approaches a constant as N → ∞, the system is non-self-averaging.

There is a further classification of self-averaging systems as strong and weak. If the exhibited behavior is R_X ~ N⁻¹ as suggested by the central limit theorem, mentioned earlier, the system is said to be strongly self-averaging. Some systems shows a slower power law decay R_X ~ N^−z with 0 < z < 1. Such systems are classified weakly self-averaging. The known critical exponents of the system determine the exponent z.

It must also be added that relevant randomness does not necessarily imply non self-averaging, especially in a mean-field scenario. ^[2] The RG arguments mentioned above need to be extended to situations with sharp limit of T_c distribution and long range interactions.