Logarithmic voltage anomalies in random resistor networks

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Abstract
The authors investigate the behaviour of the maximum voltage drop across the bonds in a random resistor network above the percolation threshold. On the basis of numerical simulations on randomly diluted L*L square lattice networks, we find that the average value of the maximum voltage, (Vmax(p;L)), exhibits a peak as a function of the bond concentration, p, which is located above the percolation threshold. This peak value appears to grow logarithmically with L, while the location of the peak appears to approach the percolation threshold very slowly as L increases. To help understand these results, they introduce the 'bubble' model, a quasi-one-dimensional structure in which the system length varies exponentially in the system width. This model is exactly soluble and a rather good description of percolation in systems of greater than one dimension can be obtained. Moreover, within the bubble model the peak value of the maximum voltage increases as In L, while the peak location approaches the percolation threshold as In(In L) In L, in good agreement with our numerical results.