Monte Carlo Solution of Bond Percolation Processes in Various Crystal Lattices

We present the outline of an IBM 7090 machine-program for the Monte Carlo estimation of the percolation probability for a variety of space lattices. The underlying theory is briefly summarized. Percolation processes deal with the transmission of a ''fluid" (disturbance, signal, etc.) through a "medium" (material, region, etc.) against impediment by random irregularities situated in the medium.' This paper considers the case where the medium is a regular crystal lattice in two or three dimensions, consisting of "atoms" (the vertices or sites of the lattice) and "bonds" joining specified pairs of atoms. The next section will specify the structure of the lattice more completely. The fluid originates at one or more atoms of the lattice, called the source atoms, and flows from atom to atom along the connecting bonds. However, each bond (independently of all other bonds) has a fixed probability p of being able to transmit fluid and a probability q = 1 -- p of being blocked: these randomly situated blocked bonds constitute the random impediments to the spread of the fluid. We write PN(p) for the probability that the fluid will reach (or "wet", as we shall say) more than N other atoms besides the source atoms; and the problem is to estimate Pip) -- lim*-.* PN(p). We do this by estimating PN(p) for a suitably large value of N: it turns out that N ~ 2000 is sufficient in many cases. The present paper describes the general organization of an IBM 7000 program for obtaining a Monte Carlo estimate of PN(p).