Nontrivial Phase Transitions in a Dependent Parametric Bond Percolation Model

#### M. Harris, R. Meester

1996, v.2, №4, 513-528

ABSTRACT

We consider a discrete bond percolation model on the integer lattice where each vertex connects itself to $k$ of its nearest neighbours, chosen uniformly over the neighbours and independent of all other vertices. We show that no percolation occurs for $k=1$, and that percolation does occur for $k=2$ in two and high dimensions. Further, we investigate what happens in the intermediate model where each vertex connects itself to two neighbours with probability $p$ and to only one neighbour with probability $1-p$. We show, in two and high dimensions, that this model has a phase transition in $p$ at some critical point strictly between zero and one.

Keywords: dependent percolation,nontrivial phase transitions,coupling