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The union bound gives $g_k\le {\sum }_{x\in \partial {\Lambda }_k}{\mathbb{P}}_p(0\leftrightarrow x)$. Hence some boundary graph vertex has percolation two-point connection probability at least $g_k/(8k)$. The rotations and reflections of the square lattice let us choose such a graph vertex as $x=(k,j)$, $-k\le j\le k$. Reflection in the vertical line through $x$ sends $0$ to $e_{2k}=(2k,0)$ and fixes $x$. ThereforeThe Harris-FKG inequality applied to these two increasing events proves the reflection lower bound for two-point percolation:Every graph path from $0$ to $e_{2k}$ reaches $\partial {\Lambda }_{2k}$, so $h_{2k}\le g_{2k}$. Taking roots givesBoth outside expressions have limit of a sequence $\gamma$, proving the even case. For $p>0$, the Harris-FKG inequality with the last horizontal edge gives $h_{2k+1}\ge p{h}_{2k}$, and also $h_{2k+1}\le g_{2k+1}$. The lower bound has rootThe upper bound has the same limit of a sequence. At $p=0$ all positive-distance connection probabilities vanish. Thus ${\lim }_{n\to \infty }{h}_n^{1/n}=\gamma$ for every $p\in [0,1]$.

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.