The efficient determination of the percolation threshold by a frontier-generating walk in a gradient

The frontier in gradient percolation is generated directly by a type of self-avoiding random walk. The existence of the gradient permits one to generate an infinite walk on a computer of finite memory. From this walk, the percolation threshold pc for a two-dimensional lattice can be determined with apparently maximum efficiency for a naive Monte Carlo calculation (+or-N-12/). For a square lattice, the value pc=0.592745+or-0.000002 is found for a simulation of N=2.6*1011 total steps (occupied and blocked perimeter sites). The power of the method is verified on the Kagome site percolation case.