Percolation one-arm decay rate ($\gamma (p)=\lim g_n^{1/n}$)

For bond percolation on the square lattice, the one-arm probability $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$ has a root limit of a sequence $\gamma (p)\in [p,1]$. The BK boundary-splitting estimate implies that $32n^2{g}_n$ is submultiplicative for positive integers. Applying the Fekete lemma to its logarithm proves existence when $p>0$; at $p=0$ the rate is zero. The same rate is the root limit of the percolation two-point connection probability along a coordinate axis, by the reflection lower bound for two-point percolation.