Read-back: YangMills.two_dimensional_rectangle_law

1. Binders and standing assumptions

The statement is universally quantified over all of the following.

• A type $G$ carrying a group structure, a topology, the assumption that multiplication and inversion are continuous (topological group), the assumption that $G$ is compact, a $\sigma$-algebra on $G$, and the assumption that this $\sigma$-algebra is exactly the Borel $\sigma$-algebra of the topology. Nothing else is assumed about $G$: it is not assumed Hausdorff, connected, non-abelian, second countable, or non-trivial (the one-element group is allowed).
• A natural number $N$ (implicit; $N = 0$ is allowed).
• A group homomorphism $\rho : G \to U(N)$, where $U(N)$ is the group of $N\times N$ complex matrices $A$ with $A^{*}A = AA^{*} = I$ ($A^{*}$ the conjugate transpose). Whenever $\rho(g)$ is written below it is regarded as an $N\times N$ complex matrix.
• A hypothesis $h_\rho$: $\rho$ is continuous (for the subspace topology on $U(N)\subset \mathbb C^{N\times N}$). No definition used in the statement refers to $h_\rho$; it is merely available as an assumption.
• Natural numbers $k, L, T, R$ (each may be $0$).
• A real number $\beta$ (any sign, possibly $0$).
• A hypothesis $h$ described in Section 3 below (the rectangle lies in the box).

Write $M := L\cdot 2^{k}$ throughout. Note $M = 0$ exactly when $L = 0$.

2. The two-dimensional lattice objects

Throughout, the dimension is fixed to $d = 2$. A site is a point $x = (x_0, x_1) \in \mathbb Z^{2}$; $e_0 = (1,0)$ and $e_1 = (0,1)$ are the unit vectors.

Box. The box of radius $M$ is the finite set of lattice points

$$B_M := \{\,x \in \mathbb Z^{2} : -M \le x_0 \le M \text{ and } -M \le x_1 \le M\,\}$$

(a $(2M+1)\times(2M+1)$ square of sites centred at the origin; for $M = 0$ it is $\{(0,0)\}$).

Edges. An edge is a pair $(x,\mu)$ with $x\in\mathbb Z^2$ and $\mu\in\{0,1\}$ (the edge from $x$ to $x+e_\mu$). The box edges are

$$E_M := \{\,(x,\mu) : x \in B_M,\ x + e_\mu \in B_M\,\}.$$

For $M = 0$ this set is empty.

Plaquettes. A plaquette is a triple $(x,\mu,\nu)$; the box plaquettes are those with $\mu < \nu$ (so in dimension two necessarily $\mu = 0,\ \nu = 1$) and with all four corners $x,\ x+e_0,\ x+e_1,\ x+e_0+e_1$ in $B_M$; equivalently

$$P_M := \{\,(x,0,1) : -M \le x_0 \le M-1,\ -M \le x_1 \le M-1\,\},$$

a set of $(2M)^2$ plaquettes, empty when $M = 0$.

Configurations and links. A configuration is an arbitrary function $U : E_M \to G$; write $\mathcal C_M := G^{E_M}$. For an arbitrary edge $e = (x,\mu)$ (not necessarily in the box) the link value is

$$U_{x,\mu} := \begin{cases} U(x,\mu) & \text{if } (x,\mu) \in E_M,\\ 1_G & \text{otherwise}.\end{cases}$$

So edges outside the box silently carry the identity element.

Plaquette variable. For $p = (x,\mu,\nu)$,

$$U_{\partial p} := U_{x,\mu}\; U_{x+e_\mu,\nu}\; U_{x+e_\nu,\mu}^{-1}\; U_{x,\nu}^{-1}.$$

Wilson action and Gibbs weight.

$$S(U) := \sum_{p \in P_M} \operatorname{Re}\operatorname{tr}\rho(U_{\partial p}) \in \mathbb R, \qquad w_\beta(U) := e^{\beta S(U)} > 0 .$$

For $M = 0$ the sum is empty, $S\equiv 0$, $w_\beta \equiv 1$.

Haar probability and configuration measure. Let $\lambda$ be Mathlib's left Haar measure on $G$ normalised by the compact set $G$ itself, i.e. the left-invariant measure with $\lambda(G) = 1$ (this normalisation holds under exactly the assumptions listed in Section 1; no Hausdorff assumption is used). The configuration measure is the product measure

$$\mu_{\mathcal C} := \bigotimes_{e \in E_M} \lambda \quad\text{on } \mathcal C_M = G^{E_M},$$

a probability measure (each factor is a probability measure, hence $\sigma$-finite, so the product is the genuine product measure). For $M = 0$ the index set is empty, $\mathcal C_M$ is a single point, and $\mu_{\mathcal C}$ is the Dirac mass at that point.

Expectation. For $F : \mathcal C_M \to \mathbb C$,

$$\langle F\rangle_\beta := \frac{\displaystyle\int_{\mathcal C_M} F(U)\, w_\beta(U)\; d\mu_{\mathcal C}(U)}{\displaystyle\int_{\mathcal C_M} w_\beta(U)\; d\mu_{\mathcal C}(U)} \in \mathbb C,$$

where $w_\beta(U)$ is regarded as a complex number. Both integrals are Bochner integrals: if an integrand is not integrable with respect to $\mu_{\mathcal C}$ (in particular if it is not almost-everywhere strongly measurable) its integral is defined to be $0$. The quotient is complex division with the convention $z / 0 = 0$; so if the denominator integral is $0$ (which, since $w_\beta$ is bounded and $\mu_{\mathcal C}$ is finite, happens exactly when $w_\beta$ fails to be a.e. strongly measurable), $\langle F\rangle_\beta = 0$ regardless of $F$.

3. The rectangle loop and the hypothesis $h$

Steps. A step is a pair $(\mu, b)$ with $\mu \in\{0,1\}$ and $b$ a Boolean; its displacement is $+e_\mu$ if $b$ is true and $-e_\mu$ if $b$ is false.

The loop. The loop written $\Gamma := \mathrm{rect}_k(T,R;0,1)$ consists of: scale $k$; base point $(0,0)$; and the step list

$$\underbrace{(0,+),\dots,(0,+)}_{T}\ \ \underbrace{(1,+),\dots,(1,+)}_{R}\ \ \underbrace{(0,-),\dots,(0,-)}_{T}\ \ \underbrace{(1,-),\dots,(1,-)}_{R},$$

whose displacements sum to $0$. Concretely this is the boundary of the axis-parallel rectangle $[0,T]\times[0,R]\subset\mathbb R^2$, traversed counter-clockwise from the origin: $T$ unit steps right, $R$ up, $T$ left, $R$ down. If $T = 0$ or $R = 0$ it is a degenerate "rectangle" that retraces a segment; if $T = R = 0$ the step list is empty.

Rescaling to scale $k$. The loop is evaluated "at scale $k$": the loop's own scale is $k$, the natural-number difference $k - k$ is $0$, and refining by $0$ multiplies the base point by $2^{0} = 1$ and repeats each step $2^{0} = 1$ time. So evaluating $\Gamma$ at scale $k$ returns exactly the base point $(0,0)$ and the step list above; no truncated subtraction and no rescaling actually occurs.

Vertices. The vertex list at scale $k$ records the position before each step (the final return to the origin is not listed separately; the origin appears as the first entry). It is the list of $2T + 2R$ points

$$(i,0)\ (0\le i\le T-1);\quad (T,j)\ (0\le j \le R-1);\quad (i,R)\ (i = T, T-1,\dots,1);\quad (0,j)\ (j = R, R-1,\dots,1).$$

Hypothesis $h$. Every vertex in this list lies in $B_M$, i.e. has both coordinates in $[-M, M]$. Since all vertices have coordinates in $[0,T]\times[0,R]$ and the extreme coordinates $T$ and $R$ are attained whenever $T + R > 0$, this hypothesis is equivalent to

$$T \le L\cdot 2^{k}\quad\text{and}\quad R \le L\cdot 2^{k}.$$

When $T = R = 0$ the vertex list is empty and $h$ holds vacuously. When $L = 0$, $h$ forces $T = R = 0$.

4. The Wilson loop observable (left-hand side)

Holonomy. For a configuration $U$ and a starting site $x$, the holonomy along a step list is the ordered product obtained by reading the steps left to right: a step $(\mu,+)$ taken from $x$ contributes the factor $U_{x,\mu}$ and moves to $x+e_\mu$; a step $(\mu,-)$ taken from $x$ contributes $U_{x-e_\mu,\mu}^{-1}$ and moves to $x - e_\mu$; the empty step list has holonomy $1_G$. For $\Gamma$ started at $(0,0)$ this is

$$\operatorname{hol}_\Gamma(U) = \Big(\prod_{i=0}^{T-1} U_{(i,0),0}\Big)\Big(\prod_{j=0}^{R-1} U_{(T,j),1}\Big)\Big(U_{(T-1,R),0}^{-1}\,U_{(T-2,R),0}^{-1}\cdots U_{(0,R),0}^{-1}\Big)\Big(U_{(0,R-1),1}^{-1}\cdots U_{(0,0),1}^{-1}\Big),$$

each product in the indicated order. Every edge occurring here has both endpoints among the vertices of $\Gamma$ (or the origin), so under hypothesis $h$ every such edge lies in $E_M$ and the "identity for edges outside the box" fallback of Section 2 is never triggered. If $T = 0$ or $R = 0$ the forward and backward factors cancel pairwise and $\operatorname{hol}_\Gamma(U) = 1_G$ for every $U$.

Wilson loop. $W_\Gamma(U) := \operatorname{tr}\rho(\operatorname{hol}_\Gamma(U)) \in \mathbb C$. The loop-product observable of the one-element list $[\Gamma]$ is the product of a single factor, namely $W_\Gamma(U)$ itself. (Since $\rho(g)$ is unitary, $|W_\Gamma(U)| \le N$; for $N = 0$ every trace is $0$.)

Left-hand side. The loop correlation of $[\Gamma]$ at parameters $k, L, \beta$ is the expectation, in the box of radius $M = L\cdot 2^k$,

$$\mathrm{LHS} := \langle W_\Gamma\rangle_\beta = \frac{\displaystyle\int_{G^{E_M}} \operatorname{tr}\rho\big(\operatorname{hol}_\Gamma(U)\big)\, e^{\beta S(U)}\; d\mu_{\mathcal C}(U)}{\displaystyle\int_{G^{E_M}} e^{\beta S(U)}\; d\mu_{\mathcal C}(U)},$$

subject to the Bochner "$0$ if not integrable" and complex "$z/0 = 0$" conventions of Section 2.

5. The plaquette measure and the right-hand side

Single-plaquette normaliser. Let

$$Z_\beta := \int_G e^{\beta\, \operatorname{Re}\operatorname{tr}\rho(g)}\; d\lambda(g) \in \mathbb R$$

(a real-valued Bochner integral, so $Z_\beta$ is always a finite real number; it can never be $+\infty$). The integrand is strictly positive and bounded above by $e^{|\beta| N}$, so it is integrable exactly when it is a.e. strongly measurable with respect to $\lambda$; in that case $0 < Z_\beta \le e^{|\beta|N}$. If it is not integrable, $Z_\beta = 0$ by convention.

Plaquette measure. Write $[r]_{+}$ for the extended non-negative real $\max(r,0)$ attached to a real $r$ (so $[r]_+ = r$ when $r \ge 0$). The plaquette measure is the measure on $G$

$$\nu_\beta := \big([Z_\beta]_+\big)^{-1}\cdot \Big(\lambda \text{ with density } g \mapsto \big[e^{\beta \operatorname{Re}\operatorname{tr}\rho(g)}\big]_+ = e^{\beta \operatorname{Re}\operatorname{tr}\rho(g)}\Big),$$

i.e. $\nu_\beta(A) = ([Z_\beta]_+)^{-1}\cdot \int_A e^{\beta \operatorname{Re}\operatorname{tr}\rho(g)}\,d\lambda(g)$ (the inner integral being the lower Lebesgue integral of a $[0,\infty]$-valued function, which is defined for any function, measurable or not), with the inverse taken in $[0,\infty]$ where $0^{-1} = \infty$ and $\infty^{-1} = 0$, and with $\infty\cdot 0 = 0$.

• If the density is a.e. strongly measurable (so $Z_\beta > 0$), $\nu_\beta$ is the probability measure with density $e^{\beta\operatorname{Re}\operatorname{tr}\rho(g)}/Z_\beta$ relative to $\lambda$.
• If it is not, then $Z_\beta = 0$, $([0]_+)^{-1} = \infty$, and $\nu_\beta = \infty\cdot(\lambda$ with that density$)$: a measure assigning $\infty$ to every set of positive weighted $\lambda$-measure and $0$ to null sets; it is neither finite nor $\sigma$-finite.
• The case "$Z_\beta = \infty$" cannot occur, because $Z_\beta$ is a real number.
• For $\beta = 0$ the density is $1$ and $\nu_0 = \lambda$ (if the constant function is measurable, which it is).

Product over $T\cdot R$ plaquettes. Let $\nu_\beta^{\otimes(TR)}$ denote Mathlib's product measure of $T\cdot R$ copies of $\nu_\beta$ on the space $G^{\{0,\dots,TR-1\}}$ of tuples $g = (g_0,\dots,g_{TR-1})$, with the product $\sigma$-algebra. When $\nu_\beta$ is $\sigma$-finite (the probability case above) this is the genuine product measure, satisfying $\nu_\beta^{\otimes(TR)}(A_0\times\cdots\times A_{TR-1}) = \prod_i \nu_\beta(A_i)$. When $\nu_\beta$ is not $\sigma$-finite the construction is still defined (as the measure induced by Mathlib's product outer measure), but the product formula is not asserted by Mathlib. When $T\cdot R = 0$ the index set is empty, the space is a single point (the empty tuple), and the product measure is the Dirac mass at that point (this holds without any $\sigma$-finiteness).

Right-hand side.

$$\mathrm{RHS} := \int_{G^{TR}} \operatorname{tr}\rho\big(g_0\, g_1 \cdots g_{TR-1}\big)\; d\nu_\beta^{\otimes(TR)}(g),$$

where $g_0 g_1\cdots g_{TR-1}$ is the ordered product in $G$ of the tuple entries (the empty product, when $T\cdot R = 0$, is $1_G$). This is again a Bochner integral of a $\mathbb C$-valued function with $|{\cdot}| \le N$: it equals $0$ if the integrand is not integrable with respect to $\nu_\beta^{\otimes(TR)}$ (which includes the case that the measure is not finite while the integrand is bounded away from zero on a set of infinite measure, and the case that the integrand is not a.e. strongly measurable). When $T\cdot R = 0$ the RHS is the integral of the constant $\operatorname{tr}\rho(1_G) = \operatorname{tr} I_N = N$ against a Dirac mass, i.e. $\mathrm{RHS} = N$.

6. The assertion

For every $G$ (with the six structural assumptions of Section 1), every $N$, every continuous homomorphism $\rho : G\to U(N)$, every $k, L, T, R \in \mathbb N$, every $\beta\in\mathbb R$, and assuming $h$ (equivalently $T \le L 2^k$ and $R \le L 2^k$):

$$\boxed{\;\langle W_{\mathrm{rect}_k(T,R;0,1)}\rangle_{\beta,\ M = L2^k} \;=\; \int_{G^{TR}} \operatorname{tr}\rho\big(g_0 g_1\cdots g_{TR-1}\big)\; d\nu_\beta^{\otimes(TR)}(g)\;}$$

as an equality of complex numbers, with the left side unfolded as in Section 4 and the right side as in Section 5. The statement is an equality only; it asserts nothing about the value itself (e.g. no area-law estimate, no positivity, no independence from $L$ or $k$ beyond what the equality implies).

7. Degenerate and edge cases included in the quantification

• $T\cdot R = 0$. The right side is exactly $N$ (Section 5). On the left side the holonomy is identically $1_G$ (Section 4), so the observable is the constant $N$ and the left side is $N\cdot(\int w_\beta\,d\mu_{\mathcal C})/(\int w_\beta\,d\mu_{\mathcal C})$, which is $N$ if the denominator is non-zero and $0$ if the denominator integral is $0$ under the Bochner convention.
• $L = 0$. Then $M = 0$, $h$ forces $T = R = 0$, the box is $\{(0,0)\}$, there are no edges and no plaquettes, the configuration space is a single point, $\mu_{\mathcal C}$ is a Dirac mass, $w_\beta \equiv 1$.
• $N = 0$. All traces vanish: both sides are $0$.
• $\beta = 0$. $w_0 \equiv 1$ and $\nu_0 = \lambda$.
• $\beta$ of either sign is allowed; nothing restricts $\beta > 0$.
• $G$ trivial (one element) is allowed.
• Non-integrable integrands on either side are silently replaced by $0$; a zero denominator on the left makes the left side $0$; a non-measurable single-plaquette density makes $\nu_\beta$ an infinite measure (Section 5).
• Edges outside the box carry the identity, but under $h$ no edge of the rectangle is outside the box.
• Natural-number subtraction appears only in the rescaling $k - k = 0$ and has no truncation effect here.
• $h$ is never impossible: for any $T, R$ it can be satisfied by choosing $L$ or $k$ large enough, and it is vacuous when $T = R = 0$.