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Simulating Experiences: Excursions in Programming. Percolation Clustering

Richard J. Gaylord
Organization: University of Illinois at Urbana-Champaign
Department: Department of Material Science and Engineering
Journal / Anthology

Mathematica in Education
Year: 1992
Volume: 2
Issue: 2
Page range: 21-24

Ultimately, the phenomenon of percolation is concerned with connectedness. P.G. deGennes, winner of the 1991 Prize in Physics for his profound work on the theoretical physics of disordered materials (my own field before I was seduced into scientific programming by Mathematica) has described the percolation transition in the following way: “many 'phenomena' are made of random islands and in certain conditions, among these islands, one macroscopic continent emerges.” Areas where percolation-like transitions occur include: chemistry (polymerization and gelation), physics (critical phenomena and phase transitions), materials (electronic and diffusive transport), biology (disease spread) and sociology (inter-organizational interaction). The random site percolation model, which we will work with here, consists of a two-dimensional square lattice where each site is occupied randomly with a probability p, independent of its neighbors. A cluster is defined as a group of occupied nearest-neighbor sites (nearest neighbor sites are those above, below, left or right of a site). There are a number of cluster-related quantities that are interesting to look at, such as the spatial characteristics of clusters (e.g., their fractal dimensions) as a function of p and the 'percolation threshold', which is the value of p at which a spanning cluster (i.e., an uninterrupted path across the lattice) first appears. However, before we can study cluster properties, we need to identify which occupied sites belong to which clusters and that is the subject of this column.

*Mathematics > Discrete Mathematics > Cellular Automata
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