A self-averaging physical property of a disordered system is one that can be described by averaging over a sufficiently large sample. The concept was introduced by Ilya Mikhailovich Lifshitz.
Frequently in physics one comes across situations where quenched randomness plays an important role. Any physical property X of such a system, would require an averaging over all disorder realisations. The system can be completely described by the average [X] where [...] denotes averaging over realisations (âÂÂaveraging over samplesâÂÂ) provided the relative variance R<sub>X</sub> = V<sub>X</sub> / [X]<sup>2</sup> â 0 as NâÂÂâÂÂ, where V<sub>X</sub> = [X<sup>2</sup>] − [X]<sup>2</sup> and N denotes the size of the realisation. In such a scenario a single large system is sufficient to represent the whole ensemble. Such quantities are called self-averaging. 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 R<sub>X</sub> ~ N<sup>−1</sup> thereby ensuring self-averaging. On the other hand, at the critical point, the question whether is self-averaging or not becomes nontrivial, due to long range correlations.
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. Most importantly as N â âÂÂ, R<sub>X</sub> 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<sub>X</sub>. If R<sub>X</sub> falls off to zero with size, it is self-averaging whereas if R<sub>X</sub> approaches a constant as N â âÂÂ, 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<sub>X</sub> ~ N<sup>âÂÂ1</sup> 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<sub>X</sub> ~ N<sup>−z</sup> 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.
The RG arguments mentioned above need to be extended to situations with sharp limit of T<sub>c</sub> distribution and long range interactions.