Box-crossings and continuity results for self-destructive percolation in the plane.

*(English)*Zbl 1173.82329
Sidoravicius, Vladas (ed.) et al., In and out of equilibrium 2. Papers celebrating the 10th edition of the Brazilian school of probability (EBP), Rio de Janiero, Brazil, July 30 to August 4, 2006. Basel: Birkhäuser (ISBN 978-3-7643-8785-3/hbk). Progress in Probability 60, 117-135 (2008).

Summary: A few years ago [see J. van den Berg and R. Brouwer, Random Struct. Algorithms 24, No. 4, 480–501 (2004; Zbl 1054.60105)] 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 \(\delta\) 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 \(\delta\) (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 \(\theta(\cdot,\cdot)\) has a discontinuity at points of the form \((p_c,\delta)\), with \(\delta\) sufficiently small. We also showed [see J. van den Berg and R. Brouwer, Commun. Math. Phys. 267, No. 1, 265–277 (2006; Zbl 1111.60080)] remarkable consequences for the forest-fire models.

The conjecture naturally raises the question whether the function \(\theta(\cdot,\cdot)\) 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 B. Bollobás and O. Riordan [Probab. Theory Relat. Fields 136, No. 3, 417–468 (2006; Zbl 1100.60054)]. We believe that this modification is also useful for many other percolation models.

For the entire collection see [Zbl 1141.82002].

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 \(\theta(\cdot,\cdot)\) has a discontinuity at points of the form \((p_c,\delta)\), with \(\delta\) sufficiently small. We also showed [see J. van den Berg and R. Brouwer, Commun. Math. Phys. 267, No. 1, 265–277 (2006; Zbl 1111.60080)] remarkable consequences for the forest-fire models.

The conjecture naturally raises the question whether the function \(\theta(\cdot,\cdot)\) 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 B. Bollobás and O. Riordan [Probab. Theory Relat. Fields 136, No. 3, 417–468 (2006; Zbl 1100.60054)]. We believe that this modification is also useful for many other percolation models.

For the entire collection see [Zbl 1141.82002].