εexpansion for transport exponents of continuum percolating systems

Certain classes of continuum percolation problems can be mapped into lattice problems with conducting bonds whose conductivity σ is drawn from a probability density law of the form ${\sigma }^{\mathrm{-}a}$. Such distributions of σ in turn can modify the conductivity exponent t when 0<a<1. It is shown that to first order in ε, the continuum conductivity exponent t¯ is given, for large values of a, by t¯=(d-2)ν+1/(1-a) which agrees with a form proposed by various authors. For small values of a, a new type of crossover to the discrete lattice exponent is predicted. Numerical results are also presented.