The size of the smallest percolating sets in d-neighbour bootstrap percolation on [n]^d is n^{d-1} for all d ≥ 1, with percolation time at most c_d n^2.
Deterministic bootstrap percolation in high dimensional grids
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abstract
In this paper, we study the k-neighbor bootstrap percolation process on the d-dimensional grid [n]^d, and show that the minimum number of initial vertices that percolate is (1-d/k)n^d + O(n^{d-1})$ when d<=k<=2d. This confirms a conjecture of Pete.
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2019 1verdicts
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Smallest percolating sets in bootstrap percolation on grids
The size of the smallest percolating sets in d-neighbour bootstrap percolation on [n]^d is n^{d-1} for all d ≥ 1, with percolation time at most c_d n^2.