Griendt, Jesher, van der (2026) Contact process on the 1-dimensional disk percolation model. Master's Thesis / Essay, Mathematics.
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Abstract
We study the contact process on the random disk graph defined on the integer lattice Z. We assume that the radius of influence follows a geometric distribution on (1, 2, . . .) with parameter p. In this process, the infection either survives indefinitely on the graph or eventually dies out. It is well known that the system undergoes a phase transition at a critical value. In this thesis, we show that the contact process on the random disk graph has the same critical value for both the annealed and quenched processes. We then consider the supercritical regime and condition on survival to obtain our first main result: a shape theorem in one dimension. Although the process is not translation invariant, we prove that a shape theorem still holds under the survival conditioning. We also prove that the speed of the rightmost infected site satisfies an appropriate central limit theorem. Finally, we show that, at the critical value, the process dies out almost surely.
| Item Type: | Thesis (Master's Thesis / Essay) |
|---|---|
| Supervisor name: | Szabo, R. and Trapman, J.P. |
| Degree programme: | Mathematics |
| Thesis type: | Master's Thesis / Essay |
| Language: | English |
| Date Deposited: | 28 Jul 2026 06:36 |
| Last Modified: | 28 Jul 2026 06:36 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/37463 |
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