Probabilistic bond percolation in random arrays

Abstract

We consider the problem of percolation in a system having sites distributed at random, but in which only a fractionh of the physical overlaps form viable links. We convert this to a site problem on the covering lattice, and then show that in two dimensionsh ∼- 1/S ⁴ forh ∼- 1, andh ∼- 4)S² forh ≪ 1, whereS is proportional to the critical percolation radius in the original array. This result reproduces the T^−1/3 behavior for log(conductivity) expected of variable-range hopping and found by numerical methods. It also accounts for the region of transition tor-percolation asT → ∞. We make a prediction that in three dimensions,h = 1/8S³ + const/S⁶, but numerical confirmation is lacking for this case. While the argument is not exact, we have demonstrated a novel approach to random systems.