Series analysis of randomly diluted nonlinear networks with negative nonlinearity exponent

The behavior of randomly diluted networks of nonlinear resistors, for each of which the voltage-current relationship is |V|=r|I|±, where ± is negative, is studied using low-concentration series expansions on d-dimensional hypercubic lattices. The average nonlinear resistance R between a pair of points on the same cluster, a distance r apart, scales as r(±), where is the correlation-length exponent for percolation, and we have estimated (±) in the range -1±0 for 1d6. (±) is discontinuous at ±=0 but, for ±<0, (±) is shown to vary continuously from max, which describes the scaling of the maximal self-avoiding-walk length (for ±'0-), to BB, which describes the scaling of the backbone (at ±=-1). As ± becomes large and negative, the loops play a more important role, and our series results are less conclusive.