Existence of the free and wired infinite-volume FK limits

Fix $n<m$ and condition the law on $E_m$ on every edge outside $E_n$. By [F1], the conditional law on $E_n$ is an FK law whose boundary partition records connections made by those fixed exterior edges. This partition is at least as wired as the free partition, so averaging the conditional laws gives ${\varphi }_{m,p,q}^0{\mathrm{\mid }}_{E_n}{\ge }_{st}{\varphi }_{n,p,q}^0$. For the wired law on $E_m$, each induced partition on $\mathrm{\partial }{\mathrm{\Lambda }}_n$ is at most as wired as the one-class wired partition, so the same conditional comparison gives ${\varphi }_{m,p,q}^1{\mathrm{\mid }}_{E_n}{\le }_{st}{\varphi }_{n,p,q}^1$.

For each finite edge set $F\subset E_{\mathrm{\infty }}$ and each $S\subseteq F$, the event that every edge in $S$ is open is increasing. Its free probabilities therefore have a bounded monotone limit by step 1.1, and its wired probabilities also have a bounded monotone limit. Every atom on $F$ is recovered from these all-open probabilities by finite inclusion-exclusion: if $O$ is the prescribed open set and $C=F\setminus O$ the prescribed closed set, then $P(\omega {\mathrm{\mid }}_F=O)={\sum }_{T\subseteq C}(-1{)}^{\mathrm{\mid }T\mathrm{\mid }}P({\omega }_e=1\text{ for all }e\in O\cup T)$. Hence every finite-dimensional atom probability converges for both sequences.

The limits in step 2.1 are consistent under restriction, because the finite-volume atom probabilities are consistent whenever the smaller edge set lies inside $E_n$. Under [A1], use a fixed explicit enumeration of the countable edge set to apply the finite-alphabet extension in [F2]. Each limiting consistent family extends uniquely to a probability measure on the cylinder-generated Borel sigma-algebra; the same cylinder limits make it the unique subsequential limit of the corresponding sequence. Therefore the free and wired cylinder limits exist and are unique.