= Alternating arms argument at the self-dual percolation parameter
= Zhang's argument
{c}
{synonym}
= Zhang argument
{c}
{synonym}
At parameter $1/2$, primal and <dual bond percolation> on the <square lattice> have the same <probability distribution>. Suppose an <infinite percolation cluster> exists. Large boxes meet it with <probability> tending to one. The <square-root trick for positively associated events> and quarter-turn symmetry imply that each side has an infinite exterior open arm with <probability> tending to one. The same holds for dual arms. A <union bound> then gives positive <probability> of primal top and bottom arms together with dual left and right arms.
Use a primal box $[-r,r]^2$ and a dual box with sides at $\pm(r+1/2)$. Open only primal <edges> whose endpoints both lie in the primal box, preserving all outward and exterior arm <edges>. The two primal arms are joined inside. Planarity prevents the two alternating dual arms from joining: a proposed dual connection together with the inner boundary arc separates one primal infinite arm from infinity. Thus the modified configuration has two infinite dual clusters. Its positive <probability> by <finite modification of Bernoulli percolation> contradicts <uniqueness of the infinite percolation cluster>. Consequently the <percolation probability> at $1/2$ is zero. This is often called Zhang's argument.
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