Boerma, S. (2014) A sharp threshold for an anisotropic bootstrap percolation model. Bachelor's Thesis, Mathematics.
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Abstract
In bootstrap percolation on the two-dimensional lattice the initial state of any site is infected with probability p and healthy with prob- ability 1 − p. A healthy site can become infected if its neighbors are and an infected site remains infected for all time. The process goes on until all sites are infected or there is a point where no more sites can become infected. If all the sites become infected we say the initial configuration percolates. Balogh and Bollobs [BB03] were able to give a sharp threshold for the standard bootstrap percolation model in a way that can be used for other models. In this bachelor thesis a sharp threshold is given for the anisotropic bootstrap percolation model. The standard and the anisotropic model are compared and it is shown that the fundamental theorem of Friedgut and Kalai [FK96] can be used to give the sharp threshold in both cases. It is shown that the behaviour on the grid and the torus is almost the same. Using a slightly adjusted version of the lemma of Aizenman and Lebowitz [AL88], the length of the epsilon-window for anisotropic bootstrap percolation is determined.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 15 Feb 2018 07:56 |
| Last Modified: | 15 Feb 2018 07:56 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/11617 |
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