Frozen Boundary Percolation on the Triangular Lattice

Abstract

We introduce a new frozen percolation model where we set the freezing condition on open clusters intersecting the boundary of the box Λ(n). Using quite simple arguments, it is easy to show that the probability of the origin freezing does not go exponentially fast to zero nor to one as we let n grow large. We conjecture that in fact the probability of the origin not being contained in a frozen cluster is bounded away from 0, uniformly in n. It turns out that this is rather difficult to prove and we instead focus our attention on a somewhat different model, where not every boundary point initiates the freezing process, but for any epsilon > 0 a boundary vertex is a freezing trigger point with probability n^{-epsilon}. We compare this to a model as in a paper by van den Berg that independently puts holes around points and closes all vertices contained in these holes. The major argument needed in this thesis is a three-arm half-plane stability result, the proof of which follows along the lines of the proof of a similar four-arm full-plane stability statement in van den Berg's paper.