Line percolation


We study a new geometric bootstrap percolation model, line percolation, on the d-dimensional integer grid [n]d. In line percolation with infection parameter r, infection spreads from a subset A ⊂ [n]d of initially infected lattice points as follows: if there exists an axis-parallel line L with r or more infected lattice points on it, then every lattice point of [n]d on L gets infected, and we repeat this until the infection can no longer spread. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine pc(n, r, d), the density at which percolation (infection of the entire grid) becomes likely. In this paper, we determine pc(n, r, 2) up to a multiplicative factor of 1 + o(1) and pc(n, r, 3) up to a multiplicative constant as n→∞ for every fixed r∈N. We also determine the size of the minimal percolating sets in all dimensions and for all values of the infection parameter.

Publication Title

Random Structures and Algorithms