Preprint C148/2012
Coordinate Percolation on Z³
Marcelo Hilário | Richard
Keywords: dependent percolation | Phase transition | Connectivity Decay

We consider the following percolation process defined on the Z³-lattice: For each column that is parallel to one of the coordinate axis we decide whether to remove it or not with a probability (or parameter) depending only on its direction. The columns are removed or not independently, and after establishing the state of each one of them we are left with a random subset of remaining sites called open sites. This model contains infinite-range dependencies that induce interesting properties for the set of open sites. Some of them are not present in the Bernoulli percolation or in percolation models having only local or weaker dependencies. It is proven that, if the columns are removed with high probability then there are no infinite components, almost surely. On the other hand, in case they are removed with low probability, then such components indeed exist. This establishes the phase transition for this model. We also show that the tail distribution for the radius of the open cluster containing the origin decays exponentially fast when at least two of the parameters are fixed to be high. However, if two of the parameters are taken relatively small, then the truncated version for this tail decays, at most, polynomially fast. We also prove that the number of infinite connected components in the supercritical phase is either one or infinite, almost surely.

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