Free, wired, and subsequential infinite-volume FK states on the square lattice

Let $E_{\mathrm{\infty }}$ be the nearest-neighbor edge set of ${\mathbb{Z}}^2$, and let ${\mathrm{\Lambda }}_n=[-n,n{]}^2\cap {\mathbb{Z}}^2$. Let $E_n$ contain the edges with both endpoints in ${\mathrm{\Lambda }}_n$, and let $\mathrm{\partial }{\mathrm{\Lambda }}_n$ be the vertices of ${\mathrm{\Lambda }}_n$ adjacent to a vertex outside it. Fix $q\ge 1$ and $p\in (0,1)$. For each boundary partition ${\xi }_n$ on $\mathrm{\partial }{\mathrm{\Lambda }}_n$, let ${\varphi }_{n,p,q}^{{\xi }_n}$ be the finite FK law on $({\mathrm{\Lambda }}_n,E_n)$ with mass proportional to $p^{\mathrm{\mid }A\mathrm{\mid }}(1-p{)}^{\mathrm{\mid }{E}_n\setminus A\mathrm{\mid }}{q}^{k_{{\xi }_n}(A)}$ for $A\subseteq E_n$, where $k_{{\xi }_n}(A)$ counts components after wiring the partition classes and extend its configurations to $\{0,1{\}}^{E_{\mathrm{\infty }}}$ by setting all edges outside $E_n$ closed.

An infinite-volume FK state at $(p,q)$ is any probability measure on $\{0,1{\}}^{E_{\mathrm{\infty }}}$ that is a cylinder-probability limit of such extended finite-volume laws along a sequence $n_j\to \mathrm{\infty }$ and boundary partitions ${\xi }_{n_j}$. Cylinder-probability convergence means convergence of the probability of every assignment on every finite set of edges. When they exist, the free FK state and wired FK state are the cylinder limits for the identity and one-class partitions, respectively, at every $n$. Their existence and uniqueness as probability-measure cylinder limits are established by a later A-page lemma under its stated measure-extension hypothesis. For either state ${\varphi }_{p,q}^b$, $b\in \{0,1\}$, define ${\theta }^b(p,q):={\varphi }_{p,q}^b(0\leftrightarrow \mathrm{\infty })$, where $0\leftrightarrow \mathrm{\infty }$ means that the open cluster of the origin is not contained in any finite box, and define

$p_c^b(q):=\inf \{p\in (0,1):{\theta }^b(p,q)>0\}\mathrm{.}$

No equality of free and wired states at $p=p_c$ is part of this definition.

Finite FK random-cluster weights for real q and boundary partitions