k-neighbor threshold cascade · sharp transition · invasion front
In bootstrap percolation on a grid, initially each site is occupied independently with probability p. Then iteratively: any empty site with ≥ k occupied neighbors becomes occupied. For k=2, the process has a sharp threshold: the final infected fraction jumps from near 0 to 1 near a critical p_c(L) that → 0 logarithmically as L→∞ (Holroyd 2003). Seeds are colored by the iteration round at which they became infected — lighter = earlier.