Percolation threshold of a two-dimensional continuum system

Abstract

We discuss the percolation threshold of a two-dimensional continuum system which has conductor only in those regions where a function $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$ is less than a chosen cutoff intensity. An experimental realization of such a system, with $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$ the electric field intensity of a laser speckle pattern, has recently been reported. We carry out a computer study for this case, the results of which are in excellent agreement with the experimental results. Expanding on earlier ideas that it is the saddle points of $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$ which determine the percolation threshold, we introduce an "equivalent network" which has the same threshold as the continuum system. With the use of this result, the computer study constructs the network and then easily finds its threshold. The computer study also finds the densities of maxima, minima, and saddle points of $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$ which are in close agreement with the analytic results of a companion paper. Finally, we use the equivalent network in developing an "effective-lattice" estimate for the percolation threshold.