Bootstrap Percolation

(Scroll down for information about the picture...)

What is this?

We start with a square array of 300 by 300 smaller squares, called "sites". Initially, 5% of the sites are chosen at random to be "infected" - these sites are coloured red. The picture then evolves in discrete time steps according to the following rules. Infected sites remain infected for ever. A currently uninfected site becomes infected at the next time step if at least two of its four nearest neighbours are currently infected. In the particular experiment above, the entire square eventually became infected. The picture shows the time at which sites became infected: initially infected sites are red, subsequent infections are black (earliest), followed by dark blue, light blue, green, yellow, and finally white (latest).


What determines whether the entire square eventually becomes infected? Clearly it depends of the size of the square and the proportion of sites initially infected, but how exactly? In the case of very large squares, the following answer is proved in [6]. Consider a series of experiments in which the size of the square becomes larger and larger indefinitely. Fix a number c, and adjust the initial proportions of infected sites so that when the square is L by L sites, the proportion is c / ln(L). (Here "ln" is the natural logarithm). Then, in the limit as the square become large, the probability that it will become entirely infected approaches 1 (certainty) if c>&lambda, or 0 (certain not to happen) if c<&lambda, where

&lambda = &pi2 / 18 = 0.548311...

Theory versus Experiment

The model seems ideal for computer simulation. Surprisingly however, simulation results can lead to misleading predictions about limiting behaviour (or vice-versa, depending on one's perspective). Simulations in [2] involving squares up to 28,800 by 28,800 (nearly a billion sites) resulted in the prediction

&lambda = 0.245 ± 0.015,

different from the rigorous value above by more than a factor of two! The reason for the discrepancy seems to be that the convergence to the limiting constant 0.548311... is extremely slow, so even very large simulations do not come very close. In fact, combining the simulation results of [2] with the rigorous results of [6], one may (tentatively) estimate that in order to get close to the limiting value, one would need to simulate a square of about

100,000,000,000,000,000,000 by 100,000,000,000,000,000,000 (1020 by 1020) sites,

certainly well beyond the range of any computer!


The above model is called bootstrap percolation, and serves as a mathematically idealised model for phenomena of nucleation and growth. Such models have been applied in the study of crack formation, magnetic alloys, hydrogen mixtures, and computer storage arrays. See the references below for more information. More generally, bootstrap percolation provides an important stepping stone towards understanding other cellular automaton models, which have numerous applications in physics, biology and information technology.



Primordial Soup Kitchen. David Griffeath's cellular automaton page

Analysis of Local Growth Models. Lecture notes by Janko Gravner

Bootstrap Percolation in Eric Weisstein's Mathworld.


[1] J. Adler. Bootstrap percolation. Physica A, 171:453-470, 1991.

[2] J. Adler, D. Stauffer and A. Aharony. Comparison of bootstrap percolation models. Journal of Physics A, 22:L297-L301, 1989.

[3] M. Aizenman and J. L. Lebowitz. Metastability effects in bootstrap percolation models. Journal of Physics A, 22:L297-L301, 1989.

[4] R. Cerf and F. Manzo. The threshold regime of finite volume bootstrap percolation. To appear.

[5] J. Gravner and D. Griffeath. First passage times for threshold growth dynamics on Zd. Annals of probability, 24(4):1752-1778, 1996.

[6] A. E. Holroyd. Sharp metastability threshold for two-dimensional bootstrap percolation. Probability Theory and Related Fields, 125(2):195-224, 2003.

[7] R. H. Schonamnn. On the behaviour of some cellular automata related to bootstrap percolation. Annals of probability, 20(1):174-193, 1992.

Alexander E. Holroyd 2005