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Let $B_n=\varphi (0\leftrightarrow \partial {\Lambda }_n)$ and $s_n=|\partial {\Lambda }_n|$. This connection event is the union over $x\in \partial {\Lambda }_n$ of $\{0\leftrightarrow x\}$. The union bound therefore supplies a graph vertex with connection probability at least $B_n/s_n$.

Every boundary graph vertex has some coordinate equal to $n$ or $-n$. By applying a permutation and a graph automorphisms, which preserve both the law and the boundary, choose such a graph vertex in the form $x=(n,x_2,\dots ,x_d)$ while retaining its probability bound. Now $e_{2n}-x=(n,-x_2,\dots ,-x_d)$ is a reflection of $x$. Translation and reflection invariance givePositive association of random variables applied to the two connection events, whose intersection implies $0\leftrightarrow e_{2n}$, proves the reflection comparison of random-cluster connections:Also $C_n\le B_n$, since any path to $e_n$ meets the box boundary. Consequentlyand henceHere $s_n=(2n+1{)}^d-(2n-1{)}^d=O(n^{d-1})$, so $\log s_n/n\to 0$. Both outside logarithmic connection rates converge to $\alpha (p,q)$ by part (b). Squeezing gives the rate for every box size, and therefore for the requested even subsequence: