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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
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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
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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
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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
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of such a system, would require an averaging over all disorder realisations. The system can be completely described by the average where denotes averaging over realisations (“averaging over samples”) provided the
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physical property of a disordered system is one that can be described by averaging over a sufficiently large sample. The concept was introduced by
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as suggested by the central limit theorem, mentioned earlier, the system is said to be strongly self-averaging. Some systems shows a slower
208: < 1. Such systems are classified weakly self-averaging. The known critical exponents of the system determine the exponent
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21:
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There is a further classification of self-averaging systems as strong and weak. If the exhibited behavior is
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that self-averaging property is lost if randomness or disorder is relevant. Most importantly as N → ∞, R
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thereby ensuring self-averaging. On the other hand, at the critical point, the question whether
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243:"Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points"
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288:- S Roy and SM Bhattacharjee (2006). "Is small-world network disordered?".
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approaches a constant as N → ∞, the system is non-self-averaging.
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is self-averaging or not becomes nontrivial, due to long range
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falls off to zero with size, it is self-averaging whereas if R
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150:self-averaging can be indexed with the help of the
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225:distribution and long range interactions.
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82: = − and
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241:-A. Aharony and A.B. Harris (1996).
154:size dependence of a quantity like R
36:one comes across situations where
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69: / → 0 as
320:10.1016/j.physleta.2005.10.105
170:Strong and weak self-averaging
1:
40:plays an important role. Any
267:10.1103/PhysRevLett.77.3700
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136:Non self-averaging systems
22:Ilya Mikhailovich Lifshitz
204:with 0 <
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347:Statistical mechanics
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93:central limit theorem
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312:2006PhLA..352...13R
259:1996PhRvL..77.3700A
38:quenched randomness
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89:extensive quantity
290:Physics Letters A
253:(18): 3700–3703.
143:numerical studies
121:{\displaystyle X}
50:relative variance
42:physical property
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303:cond-mat/0409012
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95:guarantees that
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32:Frequently in
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130:correlations
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73:→∞, where
229:References
152:asymptotic
28:Definition
328:119529257
189:power law
341:Category
275:10062286
308:Bibcode
255:Bibcode
34:physics
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191:decay
158:. If R
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324:S2CID
298:arXiv
271:PMID
316:doi
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164:X
160:X
156:X
147:X
116:X
106:N
101:X
97:R
84:N
79:X
75:V
71:N
66:X
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57:X
53:R
45:X
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