Numerical FK cylinder limits and pressure comparison

If $n<m$, condition the law on $E_m$ on edges outside $E_n$. Its induced partition on $B_n$ refines the wired partition and is coarser than the free partition. By [F2], free laws increase under restriction as volume grows and wired laws decrease. Every all-open event on a fixed finite set therefore has a bounded monotone sequence of probabilities. An atom prescribing open edges $O$ and closed edges $H$ has probability ${\sum }_{T\subseteq H}(-1{)}^{\mathrm{\mid }T\mathrm{\mid }}\varphi (\text{all edges of }O\cup T\text{ open})$. Finite inclusion-exclusion gives the limit of every atom and hence every cylinder event. Limits of finite sums preserve positivity, normalization and all restriction-consistency identities. No measure extension is used.

Write $h=\log (p\mathrm{/}(1-p))$ and $v=e^h$. For $Q_N=\{0,\dots ,N-1{\}}^2$ define the activity sum and its volume normalization $Y_N^b(h)={\sum }_{A\subseteq E(Q_N)}{e}^{h\mathrm{\mid }A\mathrm{\mid }}{q}^{k_b(A)},F_N^b(h)=N^{-2}\log Y_N^b(h)\mathrm{.}$ The finite law is unchanged by replacing the original weights by these activity weights. Since $1\le q^{k(A)}\le q^{N^2}$ and $\mathrm{\mid }E(Q_N)\mathrm{\mid }=2N(N-1)$, one has $0\le F_N^0(h)\le \log q+2\log (1+e^h)$. For fixed $L$, write $N=kL+s$, $0\le s<L$, tile the $kL$-square by $k^2$ disjoint $L$-squares, and leave the remaining vertices as singletons. Restricting the sum to configurations closing every interpiece edge gives $Y_N^0(h)\ge (Y_L^0(h){)}^{k^2}{q}^{N^2-k^2{L}^2}\mathrm{.}$ Closed edges have activity factor $1$ and component factors multiply over the pieces. Hence ${\mathrm{lim\hspace{0.17em}inf}}_N{F}_N^0(h)\ge F_L^0(h)$ for each $L$. The finite bound above and $F_N^0\le {\sup }_L{F}_L^0$ prove that $F(h)={\lim }_N{F}_N^0(h)={\sup }_L{F}_L^0(h)$ exists and is finite.

The same comparison holds for any two nested translated boxes. For a fixed $z\in {\mathbb{Z}}^2$, put $r=\mathrm{\parallel }z{\mathrm{\parallel }}_{\mathrm{\infty }}$; then ${\mathrm{\Lambda }}_{n-r}\subseteq {\mathrm{\Lambda }}_n+z\subseteq {\mathrm{\Lambda }}_{n+r}$. For every fixed increasing cylinder event, the translated free probability is sandwiched between those from the two unshifted boxes, with reverse order for wired probabilities. Their limits coincide by step 1.1. Inclusion-exclusion gives equality for all cylinders. Quarter-turn invariance holds already in each centered finite box. Thus each edge orientation and position has the same limiting open probability $u_b(p,q)$. Parameter monotonicity and ${\pi }^0\le {\pi }^1$ follow from [F2]. For an arbitrary-boundary cylinder limit $\nu$, condition on the exterior of a fixed box ${\mathrm{\Lambda }}_r$, use ${\varphi }_r^0(C)\le {\nu }_j(C)\le {\varphi }_r^1(C)$ for increasing $C$ supported inside it, pass to the given numerical limit, then let $r\to \mathrm{\infty }$. This proves the asserted sandwich.

Wiring boundary vertices lowers component counts by an integer between $0$ and $\mathrm{\mid }\mathrm{\partial }{Q}_N\mathrm{\mid }-1$, so $q^{-(\mathrm{\mid }\mathrm{\partial }{Q}_N\mathrm{\mid }-1)}{Y}_N^0(h)\le Y_N^{\xi }(h)\le Y_N^0(h)\mathrm{.}$ As $\mathrm{\mid }\mathrm{\partial }{Q}_N\mathrm{\mid }\le 4N$, every boundary partition has the same pressure limit $F(h)$. Differentiating the finite sums gives $(F_N^b{)}^{\mathrm{\prime }}(h)=N^{-2}{\mathbb{E}}_{Q_N,h,q}^b\mathrm{\mid }A\mathrm{\mid },(F_N^b{)}^{\mathrm{\prime }\mathrm{\prime }}(h)=N^{-2}{\mathrm{Var}}_{Q_N,h,q}^b(\mathrm{\mid }A\mathrm{\mid })\ge 0.$ Thus these functions and their finite pointwise limit $F$ are convex.

Ordered secant slopes of a finite convex function have finite one-sided limits $F_{-}^{\mathrm{\prime }}(h)\le F_{+}^{\mathrm{\prime }}(h)$, bounded locally by secants to fixed points on either side. If $h<k$ then $F_{+}^{\mathrm{\prime }}(h)\le F_{-}^{\mathrm{\prime }}(k)$. Therefore the nonempty intervals $(F_{-}^{\mathrm{\prime }}(h),F_{+}^{\mathrm{\prime }}(h))$ at nondifferentiability points are pairwise disjoint. Assign each its first rational in a fixed enumeration: this is an injection into a countable set. Let $D_q$ be the image of these at most countably many $h$ under $h\mapsto e^h\mathrm{/}(1+e^h)$. At a differentiability point, for each $t>0$ convexity gives $\frac{F_N^b(h)-F_N^b(h-t)}{t}\le (F_N^b{)}^{\mathrm{\prime }}(h)\le \frac{F_N^b(h+t)-F_N^b(h)}{t}\mathrm{.}$ Pass to lower and upper numerical limits as $N\to \mathrm{\infty }$, then let $t\downarrow 0$; the common pressure limit proves $(F_N^b{)}^{\mathrm{\prime }}(h)\to F^{\mathrm{\prime }}(h)$ for both boundaries.

Finite free edge probabilities in $Q_N$ are at most the infinite-volume numerical value $u_0$, and finite wired ones are at least $u_1$, by nested-box comparison and step 2.1. Thus $(F_N^0{)}^{\mathrm{\prime }}(h)\le (2-2\mathrm{/}N)u_0\le (2-2\mathrm{/}N)u_1\le (F_N^1{)}^{\mathrm{\prime }}(h)\mathrm{.}$ At $p\mathrm{\notin }{D}_q$ both outside quantities tend to $F^{\mathrm{\prime }}(h)$, so $u_0=u_1$. For an arbitrary cylinder event $C$ supported on $H$, use the common-uniform ordered finite heat-bath chains from [F2]. At every time the difference of their $C$ probabilities in absolute value is at most their probability of differing on $H$, which by order and a union bound is at most the sum over $e\in H$ of their mean edge differences. Let time tend to infinity, then the finite box tend to the lattice. This gives $\mathrm{\mid }{\pi }_{p,q}^1(C)-{\pi }_{p,q}^0(C)\mathrm{\mid }\le \mathrm{\mid }H\mathrm{\mid }(u_1-u_0)=0.$ This proves equality of the numerical families, without constructing an infinite coupling or law.

If $p<p^{\mathrm{\prime }}$, select one $t\in (p,p^{\mathrm{\prime }})\setminus D_q$. Such a point exists because a countable set cannot fill a real interval; it requires no sequence of arbitrary choices. For every increasing cylinder event $C$, ${\pi }_{p,q}^1(C)\le {\pi }_{t,q}^1(C)={\pi }_{t,q}^0(C)\le {\pi }_{p^{\mathrm{\prime }},q}^0(C)\mathrm{.}$ Apply this to $D_r$ and take the infimum over $r$ to obtain ${\mathrm{\Theta }}_1(p,q)\le {\mathrm{\Theta }}_0(p^{\mathrm{\prime }},q)$. Finally let $a_n={\varphi }_{n,p,q}^1(D_n)$. For $m\ge r$, truncation of a path at its first smaller-boundary hit and [F2] give $a_m\le {\varphi }_{m,p,q}^1(D_r)\le a_r$. In particular $a_n$ decreases to a real $L$. Letting $m\to \mathrm{\infty }$ gives $L\le {\pi }_{p,q}^1(D_r)\le a_r$. Taking the infimum over $r$ proves $L={\mathrm{\Theta }}_1(p,q)$.