Percolation of Finite Clusters and Shielded Paths

Bounghun Bock, Michael Damron, C. M. Newman, Vladas Sidoravicius

Research output: Contribution to journalArticlepeer-review


In independent bond percolation on Zd with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite connected component? Grimmett-Holroyd-Kozma used the triangle condition to show that for d≥ 19 , the set of such p contains values strictly larger than the percolation threshold pc. With the work of Fitzner-van der Hofstad, this has been reduced to d≥ 11. We improve this result by showing that for d≥ 10 and some p> pc, there are infinite paths consisting of “shielded” vertices—vertices all whose adjacent edges are closed—which must be in the complement of the infinite cluster. Using values of pc obtained from computer simulations, this bound can be reduced to d≥ 7. Our methods are elementary and do not require the triangle condition.

Original languageEnglish (US)
Pages (from-to)789-807
Number of pages19
JournalJournal of Statistical Physics
Issue number3
StatePublished - May 1 2020


  • High dimensions
  • Percolation
  • Shielded vertices
  • Triangle condition

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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