Critical Probabilities of 1-Independent Percolation Models

Source abstract

Given a locally finite connected infinite graph G , let the interval [ p min ( G ), p max ( G )] be the smallest interval such that if p > p max ( G ), then every 1-independent bond percolation model on G with bond probability p percolates, and for p < p min ( G ) none does. We determine this interval for trees in terms of the branching number of the tree. We also give some general bounds for other graphs G , in particular for lattices.