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Bootstrap percolation models describe systems as diverse as magnetic materials, fluid flow in rocks and computer storage systems. The models have a common feature of requiring not just a simple connectivity of neighbouring sites, but rather an environment of other suitably occupied sites. Different applications as well as the connection with the mathematical literature on these models is presented. Visualizations that show the compact nature of the clusters are provided. I IntroductionIn Bootstrap Percolation (BP) [1] the lattice is occupied randomly with probability p, and then all sites that do not have at least m neighbours are iteratively removed. For m = 1 isolated sites are removed and for m = 2 both isolated sites and dangling ends are removed. For m = 3 and above the situation becomes interesting and lattice dependent and is currently quite hot amongst probabilists. For sufficiently high m, no occupied sites remain for an infinite lattice, and the interest is in the scaling as a function of system size.The conference talk on which this manuscript is based came about as follows: Uri Lev asked for a project topic just as Joan Adler was thinking about BP because Scott Kirpatrick's [2] new application to computer memory storage had reminded her that BP was an interesting topic. (Uri Lev's visualizations and website [3] are described in the final section of this paper.) Other recent activity, also motivated from ref [2] The models have connections to rigidity percolation, [11]. Time developments according to the local rules of Bootstrap Percolation can be viewed as cellular automata [12] , and there are also metastable bootstrap models, not considered further here. Models closely related to canonical BP have been proposed as descriptions of fluid flow in porous media. In these cases crack development [13] and/or advance of the fluid front [14] are assumed to be strongly dependent on the local microstructure. II Applications of Bootstrap PercolationWe show here the importance of the bootstrap principle for several applications. Adler, Palmer and Meyer, [10] showed that environmental effects of a bootstrap type play a crucial role in the orientational ordering process of quadrupoles in solid molecularOrtho hydrogen and para deuterium are quadrupolar molecules, whereas para hydrogen and or- The recent application by Kirkpatrick [2] is of a completely different nature. It relates to the failure of units in a cubic structured collection of computer memories, called a Dense Storage Array. Because it is too expensive to fix individual memory units that fail in such a system, the units are allowed to "fail in place", and it is necessary to ensure that even with a relatively high density of failed units there will still be percolation of data communications. III VisualizationsSince the Computational Physics Group at the Technion developed animation software (AViz) for atomistic simulations it was interesting to apply this also to Bootstrap Percolation. AViz [16] enables a user to draw and animate atoms (or spins, or...

Series expansions for general moments of the bond-percolation cluster-size distribution on hypercubic lattices to 15th order in the concentration have been obtained. This is one more than the previously published series for the mean cluster size in three dimensions and four terms more for higher moments and higher dimensions. Critical exponents, amplitude ratios, and thresholds have been calculated from these and other series by a variety of independent analysis techniques. A comprehensive summary of extant estimates for exponents, some universal amplitude ratios, and thresholds for percolation in all dimensions is given, and our results are shown to be in excellent agreement with the ε expansion and some of the most accurate simulation estimates. We obtain threshold values of 0.2488±0.0002 and 0.180 25±0.000 15 for the three-dimensional bond problem on the simple-cubic and body-centered-cubic lattices, respectively, and 0.160 05±0.000 15 and 0.118 19±0.000 04, for the hypercubic bond problem in four and five dimensions, respectively. Our direct exponent estimates are γ=1.805±0.02, 1.435±0.015, and 1.185±0.005, and β=0.405±0.025, 0.639±0.020, and 0.835±0.005 in three, four, and five dimensions, respectively. Disciplines Physics CommentsAt the time of publication, author A. Brooks Harris was also affiliated with Tel Aviv University. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania.

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