Bootstrap Percolation

What is this?

Mathematics

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

Theory versus Experiment

&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!

Applications

References

Links:

Analysis of Local Growth Models. Lecture notes by Janko Gravner

Bootstrap Percolation in Eric Weisstein's Mathworld.

Articles:

[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.