= Three-fold symmetric primal-dual noncoexistence
For independent <bond percolation> on a periodic plane lattice with three-fold rotation symmetry, the primal model at $p$ and the <dual bond percolation> model at $1-p$ cannot both percolate. To adapt <Zhang's argument>, divide a large boundary circle into six arcs, repeating the division in each $120$-degree sector. Vary the division point until the failure probabilities $a,b$ of the two alternating primal arc types are comparable within a factor $(1-p)^{-2}$. Moving past a single crossing edge changes a failure probability by at most a factor $(1-p)^{-1}$, by the <FKG inequality>. If the probability of no primal exterior arm is at most $\varepsilon$, then $a^3b^3\le\varepsilon$ and both failures are at most $(1-p)^{-1}\varepsilon^{1/6}$. For the dual model, at least one arc type has failure at most $\varepsilon^{1/6}$; its three rotations have the same bound. Four alternating primal/dual arms now occur with positive probability, contradicting the <alternating exterior-arm separation lemma> and uniqueness after a finite modification. This extends the four-fold argument to the <martini lattice>.
Back to article page
