Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 13 / 5 / iii

The Harris theorem for square-lattice bond percolation asserts $\theta (1/2)=0$, and hence $p_c\ge 1/2$. Work in the dual square lattice, declaring a dual bond open precisely when its primal bond is closed. At $p=1/2$ this is again independent bond percolation of density one half.

Around a square centered at the origin place four dual rectangles of dimensions $6r$ by $2r$, one on each side, with overlap squares at the corners. Long crossings of all four must meet in the corner squares and their union contains a dual circuit surrounding the central square. The fixed-aspect crossing bound and the Harris lemma give this dual barrier a probability at least $\eta :=c_3^4>0$, independent of $r$.

Take geometrically increasing integer radii, for example with a factor of eight, and shift by half a lattice spacing to lie on the dual lattice. The four-rectangle annuli then have disjoint bond sets. Their barrier events are independent. An infinite primal open path from the origin would have to avoid every closed dual circuit, so the independent annular barriers for percolation giveLetting $j\to \infty$ provesThe extension to smaller $p$ follows by the monotone coupling of Bernoulli percolation. Translation and a countable union also show that no infinite open cluster exists anywhere at $p=1/2$.