Convexity and exponent inequalities for conduction near percolation

The bulk conductivity σ*(p) of the bond lattice in openZᵈ 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 σ*(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.