Convexity and exponent inequalities for conduction near percolation

The bulk conductivity ${\mathrm{\sigma }}^{\mathrm{*}}$(p) of the bond lattice in ${\mathit{openZ}}^{\mathit{d}}$ 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 ${\mathrm{\sigma }}^{\mathrm{*}}$(p) (for this model) is convex in p near the percolation threshold ${\mathit{p}}_{\mathit{c}}$, 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.