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Percolation of Finite Clusters and Shielded Paths


In independent bond percolation on \({\mathbb {Z}}^d\) 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 \(d \ge 19\), the set of such p contains values strictly larger than the percolation threshold \(p_c\). With the work of Fitzner-van der Hofstad, this has been reduced to \(d \ge 11\). We improve this result by showing that for \(d \ge 10\) and some \(p>p_c\), 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 \(p_c\) obtained from computer simulations, this bound can be reduced to \(d \ge 7\). Our methods are elementary and do not require the triangle condition.

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The research of M. D. is supported by an NSF CAREER Grant. The research of C. M. N. is supported by NSF Grant DMS-1507019. The authors thank an anonymous referee of a previous version of the paper for their helpful comments and suggestions, in particular for pointing out the relation to frozen percolation models, and for suggesting the improved bound in inequality (3.8).

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Correspondence to Michael Damron.

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Communicated by Ivan Corwin.

Numerical Results

Numerical Results

Table 1 Numerical values of \(p_c=p_c^{bond}\) and lower bounds for \(p_{shield}\)

If we use numerical values of \(p_c\), the result can be reduced to \(d=7\). In other words, we can show that \(p_{shield}(d) > p_c(d)\) for \(d \ge 7\). The second column of Table 2 shows numerical values of \(p_c = p_c^{bond}\) for dimensions 5–9. The third column gives lower bounds for \(p_{shield}(d)\) for these dimensions. The fourth gives the maximum of the left sides of (4.14) and (4.15) when setting p equal to the value in the third column. Because this maximum is \(<1\), it shows that the value in the second column is indeed a lower bound for \(p_{shield}\). One can see that the lower bound for \(p_{shield}\) is larger than the value of \(p_c\) for dimensions 7–9.

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Bock, B., Damron, M., Newman, C.M. et al. Percolation of Finite Clusters and Shielded Paths. J Stat Phys 179, 789–807 (2020).

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  • Percolation
  • Shielded vertices
  • High dimensions
  • Triangle condition