A new singular behaviour in current distribution of random resistor networks

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Abstract
The authors study the singular behaviour occurring in delta alpha (q)/ delta q of the multifractal spectrum for the current distribution in random resistor networks. The singular behaviour may be understood as the divergent behaviour of specific heat in a thermodynamic phase transition. delta alpha (q)/ delta q shows a peak with the width q0(L)c(L), and the height of which diverges as the system size L increases. However the nature of the singular behaviour in delta alpha (q)/ delta q is very unusual compared with that of the ordinary phase transition in the following ways. First, the width of the peak becomes broader, rather than narrower, with the increasing system size L, because q0(L) to - infinity and qc(L) to 0 as L to infinity . Second, the singularities at q=q0 and q=qc are of different types. The singularity at q=q0 is of power-law type such as approximately Ld(2-m)m/ log L, and the singularity at q=qc is of logarithmic type such as approximately (log L)2m/, where d is spatial dimension and m is measured to be approximately 1.633+or-0.006.