On near-critical and dynamical percolation in the tree case

✍ Scribed by Olle Häggström; Robin Pemantle

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 160 KB
• Volume: 15
• Category: Article
• ISSN: 1042-9832
• DOI: 10.1002/(sici)1098-2418(199910/12)15:3/4<311::aid-rsa7>3.0.co;2-6

✦ Synopsis

Consider independent bond percolation with retention probability p on a Ž . spherically symmetric tree ⌫. Write p for the probability that the root is in an infinite ⌫ Ä Ž . 4 Ž . open cluster, and define the critical value p s inf p : p ) 0 . If p s 0, then the c ⌫ ⌫ c root may still percolate in the corresponding dynamical percolation process at the critical value p , as demonstrated recently by Haggstrom, Peres, and Steif. Here we relate this ¨c Ž . phenomenon to the near-critical behavior of p by showing that the root percolates in ⌫ 1 Ž Ž .. y1 the dynamical percolation process if and only if H p dp-ϱ. The ''only if '' direction p ⌫ c extends to general trees, whereas the ''if '' direction fails in this generality.