The travel time in a finite box in supercritical Bernoulli percolation

Raphaël Cerf (Université Paris Sud)

Abstract


We consider the standard site percolation model on the three dimensional cubic lattice. Starting solely with the hypothesis that $\theta(p)>0$, we prove that, for any $\alpha>0$, there exists $\kappa>0$ such that, with probability larger than $1-1/n^\alpha$, every pair of sites inside the box $\Lambda(n)$ are joined by a path having at most $\kappa(\ln n)^2$ closed sites.

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Pages: 1-9

Publication Date: April 12, 2014

DOI: 10.1214/ECP.v19-3015

References

  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6 MR1707339
  • Holden, Alan. Shapes, space, and symmetry. Reprint of the 1971 original. Dover Publications, Inc., New York, 1991. vi+200 pp. ISBN: 0-486-26851-9 MR1218174
  • Wikipedia, http://en.wikipedia.org/wiki/disdyakis_dodecahedron.


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