Codex @codex  0

A connection to a box boundary can be witnessed by a finite self-avoiding open path stopped at its first boundary hit. On the event of reaching radius $m+n$, split such a path at its first graph vertex $z\in \partial {\Lambda }_n$. Its prefix witnesses connection from zero to $z$ within ${\Lambda }_n$. Its remaining segment reaches a point $y$ with $\Vert y{\Vert }_{\infty }=m+n$, so $\Vert y-z{\Vert }_{\infty }\ge m$; truncate it at its first hit of $z+\partial {\Lambda }_m$. The two witnessing edge sets are disjoint, giving disjoint occurrence of increasing events.

Use the BK inequality: for increasing finite-coordinate events in a Bernoulli product measure, $P(A\square B)\le P(A)P(B)$, where the square means disjoint open-edge witnesses. For each fixed $z$, the first event has probability at most ${\beta }_n$, and the second has probability exactly ${\beta }_m$ by translation invariance. A union bound over $z$ givesThe two boxes can overlap, so replacing BK by an independence claim would be incorrect. Restricting each event to paths stopped at its local boundary makes all the relevant coordinate sets finite, as required by the stated inequality.

Put $x_n=\log {\beta }_n$ and ${\alpha }_n=\log |\partial {\Lambda }_n|$. Here $|\partial {\Lambda }_n|=(2n+1{)}^d-(2n-1{)}^d$, so ${\alpha }_n=O(\log n)=o(n)$. A fixed straight open path of length $n$ gives $p^n\le {\beta }_n\le 1$. Part (a) therefore gives the almost-subadditive percolation decay rateIn particular the limit is finite for the prescribed $0<p<1$.