Independent plaquette variables on a finite planar rectangle

Fix a column $a$. The set $T$ of all horizontal edges together with the vertical edges in column $a$ is a spanning tree: each row is a horizontal path and the selected vertical edges join these row paths in one vertical path, with no other vertical edge in $T$ to close a cycle. Thus $T$ connects every vertex and has no cycle. The only inherited set-theoretic premise is ${\mathrm{A}\mathrm{C}}_{\omega }$ [A1].

In the finite-dimensional gauge-data instance, multiplication is continuous into $G$, so the inverse image of each Borel subset is Borel in the product topology. By [F3], this product Borel sigma-algebra is $\mathcal{B}(G)\otimes \mathcal{B}(G)$, proving the stated joint-measurability condition for that instance. The general compact-group argument below instead uses that condition as supplied data; it does not infer second countability for arbitrary compact groups.

Fix the root $(a,0)$ and set its gauge factor to the identity. Recursively along $T$, each tree-edge condition $h_{s(e)}{U}_e{h}_{t(e)}^{-1}=1$ determines the as-yet-unknown gauge factor at the next vertex uniquely, using multiplication and inversion in $G$. Hence every link configuration has a unique root-fixed gauge transform whose tree links are the identity.

First, inversion preserves $\mu$. For a bounded Borel measurable $f$, the map $(x,y)\mapsto (x^{-1},y)$ is measurable into the product sigma-algebra by [A2]. Composing it with the supplied measurable multiplication and then with $f$ shows that $(x,y)\mapsto f(x^{-1}y)$ is product-measurable. More generally, every finite group word in the coordinates is measurable, by induction using multiplication, inversion and measurable-rectangle generators. Since $\mu \times \mu$ is a probability, its absolute integral is finite by [F5]. Set $I:={\int }_G{\int }_{G}f(x^{-1}y)d\mu (y)d\mu (x)$. Fubini [F6] permits either order of integration. Integrating in $y$ first and using left invariance gives $I={\int }_{G}fd\mu$. Reversing the order and, for fixed $y$, substituting $x=y{x}^{\mathrm{\prime }}$ (again left invariance) gives $I={\int }_{G}f((x^{\mathrm{\prime }}{)}^{-1})d\mu (x^{\mathrm{\prime }})$. Thus $\int f(x^{-1})d\mu (x)=\int f(x)d\mu (x)$, so inversion preserves $\mu$. With root factor $h_{(a,0)}=1$, the inverse coordinate map is $U_e=h_{s(e)}^{-1}{h}_{t(e)}$ on $T$ and $U_e=h_{s(e)}^{-1}{V}_e{h}_{t(e)}$ off $T$. Recovering each $h_v$ recursively along $T$ uses left or right translations and inversion; recovering each off-tree $U_e$ from fixed $h$ and $V_e$ uses left and right translations. The forward and inverse coordinate maps are finite tuples of these measurable group words, hence form a bimeasurable bijection. The translations and inversion preserve each Haar factor, so successive finite Fubini integrations send ${\prod }_{e}d\mu (U_e)$ to ${\prod }_{v\mathrm{\ne }(a,0)}d\mu (h_v){\prod }_{e\mathrm{\notin }T}d\mu (V_e)$; every bounded integrand used here is $L^1$ because all factors are probabilities. The tree coordinates integrate to $1$.

For each face based at $v_0$, substituting the vertex action from [F1] into its ordered boundary product cancels every intermediate factor $h_{v_i}^{-1}{h}_{v_i}$ and gives $U_p(h\cdot U)=h_{v_0}{U}_p(U)h_{v_0}^{-1}$. Since each $w_{i,j}$ is conjugation-invariant by [F2], the full density ${\prod }_{i,j}{w}_{i,j}(U_{p_{i,j}})$ is unchanged by the gauge-fixing transformation.

In tree gauge all horizontal variables are $1$ and $V_{a,j}=1$. The counterclockwise face word gives $P_{i,j}=V_{i+1,j}{V}_{i,j}^{-1}$. For a fixed row $j$, starting at $V_{a,j}=1$, the equations determine $V_{i+1,j}=P_{i,j}{V}_{i,j}$ to the right of $a$ and $V_{i,j}=P_{i,j}^{-1}{V}_{i+1,j}$ to its left. Thus the map from the $M$ non-tree vertical variables in row $j$ to its $M$ plaquette variables is a bijective triangular coordinate change. Both directions are finite group words, so [A2] and the finite-word argument of step 3.1 make it bimeasurable for the product sigma-algebras. Recursively, to the right of $a$ the map from $P_{i,j}$ to $V_{i+1,j}$ is right translation by fixed $V_{i,j}$, while to the left it is inversion followed by right translation by fixed $V_{i+1,j}$; both preserve $\mu$ by bi-invariance and step 3.1. Iterated Fubini therefore shows that this row map preserves the product probability measure. Distinct rows use disjoint vertical variables, so their product gives a measure-preserving bijection from all non-tree vertical variables to all plaquette variables.

By step 3.2 the gauge-fixed unnormalized density is ${\prod }_{i,j}{w}_{i,j}(P_{i,j})$. Each positive continuous $w_{i,j}$ is bounded above and below by positive constants: continuity gives a bounded neighborhood at each group element, and compactness gives a finite subcover; applying the same argument to $1\mathrm{/}{w}_{i,j}$ gives a positive lower bound. Thus there are constants $0<m_{i,j}\le w_{i,j}\le M_{i,j}<\mathrm{\infty }$, so the probability integral gives $0<m_{i,j}\le z_{i,j}={\int }_G{w}_{i,j}d\mu \le M_{i,j}<\mathrm{\infty }$. In particular, each factor and the finite product density are $L^1$; Fubini [F6] applies to factor the partition function. The product Haar measure factors by step 4.1, so the partition function is ${\prod }_{i,j}{z}_{i,j}$; dividing by it yields the product of the displayed one-plaquette probability measures defined by the densities [F4].