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For independent bond percolation of density $0<p<1$ on the cubic lattice in dimension $d\ge 2$, let ${\beta }_n$ be the one-arm probability for reaching maximum-norm distance $n$. The weighted BK boundary-splitting estimate gives ${\beta }_{m+n}\le |\partial {\Lambda }_n|{\beta }_m{\beta }_n$. Since the logarithm of this boundary size is $o(n)$, the asymmetrically almost-subadditive sequence argument applies to $\log {\beta }_n$. The rate exists and lies in $[0,-\log p]$; positive rate gives exponential decay.

Fix $k\ge 1$ and write $n=qk+r$ with $1\le r\le k$ and $q\ge 0$. Repeatedly use the given inequality with its second index equal to $k$. This yieldsThere are only finitely many possible remainders for fixed $k$, soTaking $k=1$ rules out positive infinity for this upper limit. Let $L=\mathrm{liminf}{x}_n/n$, which may be negative infinity, and choose $k_j\to \infty$ with $x_{k_j}/k_j\to L$. The assumption ${\alpha }_{k_j}/k_j\to 0$ then gives $\mathrm{limsup}{x}_n/n\le L$. This also works when $L=-\infty$, by taking an arbitrarily negative upper bound. Therefore the asymmetrically almost-subadditive sequence hasThe last infimum identity follows from the fixed-$k$ bound and from the convergence of the corrected ratios to $\gamma$. No positivity assumption on ${\alpha }_n$ is required.