Convexity and exponent inequalities for conduction near percolation

The bulk conductivity o*(p) of the bond lattice in Zd with a fraction p of conducting bonds is analyzed. Assuming a hierarchical node-link-blob (NLB) model of the conducting backbone, it is shown that o*(p) (for this model) is convex in p near the percolation threshold pc, and that its critical exponent t obeys the inequalities 1 <_ t <_ 2 for d = 2 , 3 while 2 <_ t <_ 3 for d >_ 4. The upper bound t = 2 in d = 3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion.