Percolation Thresholds of Three Connected Lattices
Authors:Shane Stahlheber, Johnathan Tran, Ted Yoo
Mentor:Alexander Small, Professor, California State Polytechnic University Pomona
Percolation is a phenomenon in statistical physics where a cluster of points in a medium, typically a lattice, spans from one end to the other. These points are occupied with some probability p. The percolation threshold is defined as the probability necessary for a lattice of infinite size to percolate. This value depends on the local geometry of the medium. Percolation phenomena are relevant to a broad range of research topics such as: the study of alloys, the modeling of resistor networks, the spread of disease, and the propagation of forest fires. Though the percolation threshold of many lattices has been found, the value for the (10,3)-a lattice and other related three connected structures have not yet been determined. The (10,3)-a lattice is particularly significant as it has been recently proven to be strongly isotropic and to be the only mathematical relative of the graphene and the diamond lattices due to its highly symmetric nature (Sunada 2008). Along with the (10,3)-a, we have studied (10,3)-b, (12,3)-a and (8,3)-a in order to get a more complete understanding of percolation in three connected structures.
Using the Newman Ziff algorithm (Newman/Ziff 2001) a program was written to simulate percolation. A large lattice was constructed and then filled until percolation was detected. The fraction of occupied sites is recorded and the process repeated for many trials. The fraction occupied for all trials is then convolved with a binomial distribution to find the percolation threshold.
Preliminary results suggest that three connected lattices have the highest percolation thresholds for any three dimensional simple periodic lattice. What is significant is that they are much greater than four connected lattices.