Box-Crossings and Continuity Results for Self-Destructive Percolation in the Plane

Abstract

A few years ago (see [1]) two of us introduced, motivated by the study of certain forest-fire processes, the self-destructive percolation model (abbreviated as sdp model). A typical configuration for the sdp model with parameters p and δ is generated in three steps: First we generate a typical configuration for the ordinary site percolation model with parameter p. Next, we make all sites in the infinite occupied cluster vacant. Finally, each site that was already vacant in the beginning or made vacant by the above action, becomes occupied with probability δ (independent of the other sites).

Let ϕ(p,δ) be the probability that some specified vertex belongs, in the final configuration, to an infinite occupied cluster. In our earlier paper we stated the conjecture that, for the square lattice and other planar lattices, the function ϕ(·,·) has a discontinuity at points of the form (p,δ), with δ sufficiently small. We also showed (see [2]) remarkable consequences for the forest-fire models.

The conjecture naturally raises the question whether the function ϕ(·,·) is continuous outside some region of the above-mentioned form. We prove that this is indeed the case. An important ingredient in our proof is a somewhat modified (improved) form of a recent RSW-like (box-crossing) result of Bollobás and Riordan ([4]). We believe that this modification is also useful for many other percolation models.

Part of J. van den Berg’s research has been funded by the Dutch BSIK/BRICKS project.