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Write $\varphi ={\varphi }_{p,q}^1$ and $C_n=\varphi (0\leftrightarrow e_n)$. We use these standard facts about the infinite-volume wired random-cluster measure for $q\ge 1$: it is translation invariant and invariant under coordinate reflections/permutations; it has positive association of the random-cluster model; and every conditional edge-open probability is at leastThe last bound follows because an edge either leaves its component count unchanged on opening, giving probability $p$, or merges two components, giving $p/(p+q(1-p))$. These are standard infinite-volume properties of the limit obtained from the finite wired laws by weak convergence of probability measures. Positive association of random variables applies to connection events by increasing approximation with finite-path events.

Opening the $n$ edges of the straight segment gives $C_n\ge r^n$, so its logarithm is finite. The events $A=\{0\leftrightarrow e_m\}$ and $B=\{e_m\leftrightarrow e_{m+n}\}$ are increasing, and their intersection implies connection to $e_{m+n}$. ThusThe sequence $a_n=-\log C_n$ is a subadditive sequence, with $0\le a_n\le -n\log r$. Part (a) proves the random-cluster inverse correlation length exists and is finite:Since $a_n/n\ge \alpha$ for every $n$, exponentiating yields the requested bound