Critical length for semi-oriented bootstrap percolation
The author studies two models of two-dimensional semi-oriented bootstrap percolation introduced by \textit{A. M. S. Duarte} [Physica A 157, 1075-1079 (1989)]. Let \(T_ N\) denote the set \([0,N]^ 2\) considered as a torus. Let \(p\) denote the probability of initial occupancy. Put \[ f(p) = \exp \bigl( (\log^ 2 p)/(1000p) \bigr),\;0 < p \leq 1, \quad \text{and} \quad g(p) = \exp \bigl( (1000 \log^ 2 p)/p \bigr),\;0 < p \leq 1. \] The main result of the present paper is the following: \[ \sup_{N \leq f(p)} P^ p (T_ N \text{ ``fills itself up'')} \to 0 \text{ as } p \to 0 \] and \[ \inf_{N \geq g(p)} P^ p(T_ N \text{ ``fills itself up'')} \to 1 \text{ as } p \to 0. \]
scientific article; zbMATH DE number 757527