First exit time of termites and random super-normal conductor networks

The resistance of a random super-normal conductor network is calculated through the diffusion of a 'termite', which performs a random walk on the normal bonds and has the same probability to exist from any point on a cluster of superconducting bonds. The first time, T, that the termite exits at a distance r from the origin (averaged over many configurations and runs) is found to behave as T approximately r²k/ for r<< xi ( xi is the percolation correlation length) and as T approximately r²(p_c-p) ^s for r>> xi (( p_c-p)^-s describes the divergence of the conductivity as p to p_c^-, p being the concentration of the superconductor). Scaling arguments are used to show that k=1+s/(2+ theta ), where (2+ theta ) is the fractal dimension of random walks on single clusters at p_c. Numerical simulations at two dimensions (d=2) yields s=1.34+or-0.10 and k=1.34+or-0.03, in agreement with scaling. The authors also show that the probability of return to the origin at time t behaves as t^-dk2/. Preliminary results at d=1 and other calculations methods are also discussed.