TY - JOUR
T1 - Percolation of Finite Clusters and Shielded Paths
AU - Bock, Bounghun
AU - Damron, Michael
AU - Newman, C. M.
AU - Sidoravicius, Vladas
N1 - Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/5/1
Y1 - 2020/5/1
N2 - 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.
AB - 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.
KW - High dimensions
KW - Percolation
KW - Shielded vertices
KW - Triangle condition
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U2 - 10.1007/s10955-020-02558-4
DO - 10.1007/s10955-020-02558-4
M3 - Article
AN - SCOPUS:85084928223
SN - 0022-4715
VL - 179
SP - 789
EP - 807
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 3
ER -