= Maximum cluster radius under exponential one-arm decay
{title2=$M_n/\log n\to d/\lambda$}
Suppose the <one-arm probability> on the <cubic lattice> satisfies $\beta_r=\exp(-\lambda r+o(r))$ with $\lambda>0$. Define $M_n$ as the largest maximum-norm radius of any cluster rooted in $[-n,n]^d$. Then $M_n/\log n\to d/\lambda$ <convergence in probability>. An upper bound is a <union bound> over $O(n^d)$ roots. For a lower bound, pack $\Theta((n/\log n)^d)$ disjoint radius-$O(\log n)$ boxes and use the independent first-hit connection events inside them. Clusters are finite almost surely under the positive-rate hypothesis.
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