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.
Highlights
We consider the site percolation model on Z3
Each site is declared open with probability p and closed with probability 1 − p, and the sites are independent
One of the most important problems in percolation is to prove that, in three dimensions, there is no infinite cluster at the critical point
Summary
We consider the site percolation model on Z3. Each site is declared open with probability p and closed with probability 1 − p, and the sites are independent. P every pair of vertices of the box Λ(n) are joined by a path in Λ(n) having at most κ(ln n) closed sites. This result can be recast in the language of first passage percolation. The proof relies essentially on the BK and the FKG inequalities for the probabilistic part (see [1]), and on a tiling of the sphere into 48 spherical triangles for the geometric part The vertices of these triangles are the vertices of a Catalan solid called the disdyakis dodecahedron or the hexakis octahedron (see [2], page 54 top left for a picture, or [3]). The main point is that the spherical triangles have a diameter strictly less than one
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