3
$\begingroup$

Let $\Lambda$ be a lattice in $\mathbb{R}^d$, $d \ge 2$, thought of as an infinite graph. In percolation theory we consider properties random subgraphs of $\Lambda$. In site percolation $\Lambda^s_p$ every node of $\Lambda$ is included with probability $p$ i.i.d., and we take the induced subgraph on those random nodes. In bond percolation $\Lambda^b_p$ we take every edge with probability $p$.

An event $E$ is a measurable subset of $\{ 0,1 \}^ {V(\Lambda)}$ or $\{ 0,1 \}^ {E(\Lambda)}$. We say $E$ is translation invariant if it is invariant under translations of $\Lambda$.

Is it true that such an event must be a tail event, in which case Kolmogorov's zero-one law applies?

  • 1
    yes, the usual argument applies: Let $\mathcal F_n$ be generated by all nodes within n of the origin, The translation invariant event can be approximated by something in $\mathcal F_n$, for a suitable n, but also by something in $a + \mathcal F_n$ and for large a they are independent2012-11-19

0 Answers 0