Generalized bootstrap percolation

Details Generalized bootstrap percolation Generalized bootstrap percolation

2012-04-18

arxiv.org/abs/1204.3980v1

Mathematics

Probability

40 pages

Scientific paper

In this paper we introduce generalized bootstrap processes on $\Z^2$ with local, homogeneous, monotone update rules of any kind, and prove the first results about the phase transitions of these processes. Let $\update=\{X_1,...,X_m\}$ be a set of finite, non-empty subsets of $\Z^2\setminus\{0\}$. Under the generalized bootstrap process for $\update$, there is an initial set $A_0\subset\Z^2$ of `infected' sites, and at time $t\geq 1$ the infected set is $A_t=A_{t-1}\cup\{x\in\Z^2:X_i+x\subset A_{t-1} \text{for some} i\in[m]\}$. Let $p_c(\update,t)=\inf\{p:\P_p(0\in A_t)\geq 1/2\}$, where $A_0$ is a random subset of $\Z^2$ in which sites are infected independently with probability $p$. We prove that, under certain weak conditions, there exist constants $\lambda_1,\lambda_2>0$ and integers $\alpha_1,\alpha_2\geq 1$ such that \[(\frac{\lambda_1}{\log t})^{1/\alpha_1} \leq p_c(\update,t) \leq (\frac{\lambda_2}{\log t})^{1/\alpha_2}. \] These are the first results of any kind on bootstrap percolation considered in this level of generality.

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