By Helga Dankert
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N). 11) We should note that the q th moment of the distributed quantities V is dominated by larger or smaller values of V if q » 1 or q « -1, respectively. 13) The exponent ~o = 2 j v is the fractal dimension of the 0 th moment. The 0 th moment is just the number of bonds. 7). The exponent of the second moment (2 characterizes the distribution of energy dissipations. 13), we have log(5j2) l;2 = = -l;R . 8). 13) indicates that different orders of moments are characterized by different exponents.
L':tt. 76) and z(q) = Df - foo . 77) The following remark should be made. In many cases of statistically distributed measures, the number of boxes possessing the smallest or the largest box measure is unity, which does not depend on L or l. This means that f -00 and f 00 vanish. Next we consider the case of q = O. Lb(l) ::f. Lb(l) is proportional to [-Df, where Df is the fractal dimension of the support. 78) and Do = Df. 79) 52 4. 80) z(O) = 0, and dz(q) I dq q=O = 2a(q)l q=o - 2a(2q)l q =o = O.
This important property gives various scaling relations between exponents, implying that only a finite number of exponents are independent. 13). This implies that the exponents are not simply related. An infinite set of exponents is required to describe the voltage-drop moments. 1 Hierarchical Resistor Network Model 39 a distribution is called multifractal if moments of the distribution are described by an infinite set of independent exponents. In such a case, the usual linear scaling relations between exponents do not hold at all.