Continuity for self-destructive percolation in the plane

Abstract

A few years ago two of us introduced, motivated by the study of certain forest-fireprocesses, the self-destructive percolation model (abbreviated as sdp model). A typical configuration for the sdp model with parameters p and delta is generated in three steps: First we generate a typical configuration for the ordinary 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 delta (independent of the other sites). Let theta(p, delta) 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 theta has a discontinuity at points of the form (p_c, delta), with delta sufficiently small. We also showed remarkable consequences for the forest-fire models. The conjecture naturally raises the question whether the function theta 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 stronger form of a) recent ingenious RSW-like percolation result of Bollob\'{a}s and Riordan