Read-back: YangMills.strong_coupling_area_law

Ambient data and assumptions

The statement is made relative to the following fixed data, all of which are parameters of the theorem (the first two and $N$ are implicit, inferred from the others):

• a natural number $d \in \mathbb{N}$ (the lattice dimension; $d = 0$ and $d = 1$ are allowed);
• 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, and the assumption that this $\sigma$-algebra is the Borel $\sigma$-algebra of the topology. Nothing else is assumed about $G$: it need not be Hausdorff, connected, non-abelian, simple, or non-trivial (the one-point group is admitted);
• a natural number $N \in \mathbb{N}$ ($N = 0$ is allowed);
• a group homomorphism $\rho : G \to \mathrm{U}(N)$, where $\mathrm{U}(N)$ is the group of $N\times N$ complex matrices $A$ with $A^{*}A = AA^{*} = I$;
• a hypothesis $h_\rho$: $\rho$ is continuous (as a map into $\mathrm{U}(N)$ with the topology inherited from $N \times N$ complex matrices). This hypothesis is assumed but does not otherwise appear in the conclusion.

The assertion, with quantifiers in their exact order

There exists a real number $\beta_0$ with $0 < \beta_0$ such that, for every real $\beta$ with $0 < \beta$ and $\beta < \beta_0$, there exist real numbers $\sigma$ and $C$ with $0 < \sigma$ (no constraint at all is placed on $C$) such that, for all natural numbers $k, L, T, R \in \mathbb{N}$ and all indices $\mu, \nu \in \{0, \dots, d-1\}$, if $\mu \neq \nu$ and if the rectangle $\gamma = \mathrm{rect}_k(T, R; \mu, \nu)$ (defined below) satisfies the "in-box" condition $\mathrm{InBox}(\gamma; k, L\cdot 2^{k})$ (defined below), then

$$\bigl\|\; \langle W(\gamma) \rangle_{\beta,\, L\cdot 2^k} \;\bigr\| \;\le\; C \cdot \exp\bigl(-\sigma \cdot (T \cdot R)\bigr),$$

where $T, R$ on the right are cast to real numbers, $\|\cdot\|$ is the modulus of a complex number, and $\langle W(\gamma) \rangle_{\beta, L 2^k}$ is the complex number loopCorrelation ρ k L β [γ] unfolded below.

Dependencies allowed by the quantifier order: $\beta_0$ may depend only on $d, G, N, \rho$. The pair $(\sigma, C)$ may depend on $\beta$ (and on $d, G, N, \rho$) but not on $k, L, T, R, \mu, \nu$: the same $\sigma$ and $C$ must work simultaneously for every scale $k$, every box parameter $L$, every rectangle size $T \times R$, and every pair of distinct directions. The exponent is $-\sigma \cdot T R$ with $\sigma > 0$, so $\exp(-\sigma T R) \in (0, 1]$ and it decreases as the product $T R$ grows; $T$ and $R$ enter only through their product, and there is no rescaling by $2^{-k}$ or by any lattice spacing.

Unfolding the custom definitions

Sites, unit vectors, the box

• A site is a function $x : \{0,\dots,d-1\} \to \mathbb{Z}$, i.e. a point of $\mathbb{Z}^d$. Write $e_\mu$ for the unit vector with a $1$ in coordinate $\mu$ and $0$ elsewhere.
• For $R' \in \mathbb{N}$, the box $B_{R'} = \{x \in \mathbb{Z}^d : -R' \le x_i \le R' \text{ for all } i\}$, a cube of side $2R'+1$ centred at the origin. (For $d = 0$ it is the single empty function.)
• An edge is a pair $(x, i)$ of a site and a direction, read as the segment from $x$ to $x + e_i$. The box edges $E_{R'}$ are the edges $(x,i)$ with $x \in B_{R'}$ and $x + e_i \in B_{R'}$.
• A plaquette is a triple $(x, i, j)$. The box plaquettes $P_{R'}$ are those with $i < j$ and all four corners $x,\ x+e_i,\ x+e_j,\ x+e_i+e_j$ in $B_{R'}$.
• A configuration on the box of half-width $R'$ is a function $U : E_{R'} \to G$ (one group element per box edge).

Links, plaquette products, Wilson action

• The link variable of an arbitrary edge $e$ (not necessarily in the box) is

$$\mathrm{link}_U(e) = \begin{cases} U(e) & e \in E_{R'} \\ 1 \ (\text{identity of } G) & e \notin E_{R'}.\end{cases}$$

So every edge outside the box, or leaving the box, carries the identity.

• The plaquette product for $p = (x,i,j)$ is

$$\mathrm{plaq}_U(p) = \mathrm{link}_U(x,i)\cdot \mathrm{link}_U(x+e_i, j)\cdot \mathrm{link}_U(x+e_j, i)^{-1}\cdot \mathrm{link}_U(x, j)^{-1}.$$

• The Wilson action is the real number

$$S_\rho(U) = \sum_{p \in P_{R'}} \operatorname{Re}\,\operatorname{tr}\bigl(\rho(\mathrm{plaq}_U(p))\bigr),$$

the trace being that of the $N\times N$ complex matrix underlying $\rho(\cdot)$. There is no $1/N$ normalisation and no constant offset.

Measure and expectation

• $\mathrm{haarProb}(G)$ is the Haar measure on $G$ normalised so that the whole space $G$ (taken as the distinguished positive compact set) has measure $1$; i.e. it is a left-invariant probability measure on $G$.
• The configuration measure $\mathbf{P}_{R'}$ on $U : E_{R'} \to G$ is the product measure $\bigotimes_{e \in E_{R'}} \mathrm{haarProb}(G)$ (one independent normalised Haar factor per box edge), hence a probability measure. If $E_{R'}$ is empty (e.g. $d = 0$) the configuration space is a single point.
• The Gibbs weight is $w_\beta(U) = \exp\bigl(\beta \cdot S_\rho(U)\bigr) > 0$, with a plus sign in the exponent.
• The expectation of a complex-valued function $F$ of the configuration is the complex number

$$\langle F \rangle_{\beta, R'} = \frac{\displaystyle\int F(U)\, w_\beta(U)\, d\mathbf{P}_{R'}(U)}{\displaystyle\int w_\beta(U)\, d\mathbf{P}_{R'}(U)},$$

where both integrals are Lebesgue–Bochner integrals of complex-valued functions and the quotient is division in $\mathbb{C}$. Two conventions are in force here: (i) the Bochner integral of a function that is not integrable (not a.e. strongly measurable, or with non-finite integral of its norm) is defined to be $0$; (ii) division by $0$ in $\mathbb{C}$ yields $0$. Consequently, if the denominator integral is $0$ (in particular if the integrand is deemed non-integrable), then $\langle F\rangle_{\beta,R'} = 0$ for every $F$.

Steps, paths, holonomy

• A step is a pair $(i, b)$ with $i$ a direction and $b \in \{\mathsf{true},\mathsf{false}\}$; its displacement vector is $+e_i$ if $b = \mathsf{true}$ and $-e_i$ if $b = \mathsf{false}$.
• The vertices of a path starting at $x$ with step list $s_1, \dots, s_n$ are the $n$ starting points $x,\ x + v(s_1),\ x+v(s_1)+v(s_2),\ \dots$ (the final endpoint is not listed; the empty step list has no vertices).
• The holonomy of a path starting at $x$ with steps $s_1, \dots, s_n$ is the ordered product (first step leftmost)

$$\mathrm{hol}_U(x; s_1,\dots,s_n) = f(x, s_1)\, f(x+v(s_1), s_2)\cdots,\qquad f(y,(i,\mathsf{true})) = \mathrm{link}_U(y,i),\quad f(y,(i,\mathsf{false})) = \mathrm{link}_U(y-e_i,\, i)^{-1},$$

and the holonomy of the empty step list is the identity $1 \in G$.

Loops, scale, refinement

• A loop is a record consisting of a natural number scale, a base site, a list of steps, and a proof that the displacement vectors of the steps sum to $0$.
• $\mathrm{refine}(\gamma, j)$ has scale $\gamma.\text{scale} + j$, base $2^j \cdot \gamma.\text{base}$, and steps obtained by repeating each step of $\gamma$ exactly $2^j$ times consecutively.
• $\mathrm{atScale}(\gamma, k)$ is the pair (base, steps) of $\mathrm{refine}(\gamma,\, k \dot- \gamma.\text{scale})$, where $\dot-$ is truncated natural-number subtraction (if $k < \gamma.\text{scale}$ the exponent is $0$ and no refinement occurs).
• $\mathrm{verticesAt}(\gamma, k)$ is the vertex list of the path $\mathrm{atScale}(\gamma,k)$, and

$$\mathrm{InBox}(\gamma; k, R') \iff \forall x \in \mathrm{verticesAt}(\gamma,k),\ x \in B_{R'}.$$

This is vacuously true when the vertex list is empty.

The rectangle $\mathrm{rect}_k(T,R;\mu,\nu)$

It is the loop with scale $k$, base $0$, and step list

$$\underbrace{(\mu,+),\dots,(\mu,+)}_{T}\;,\;\underbrace{(\nu,+),\dots,(\nu,+)}_{R}\;,\;\underbrace{(\mu,-),\dots,(\mu,-)}_{T}\;,\;\underbrace{(\nu,-),\dots,(\nu,-)}_{R}.$$

Since its scale is exactly the $k$ used in the theorem, $k \dot- k = 0$ and $\mathrm{atScale}(\gamma, k)$ is (base $0$, the same $2T+2R$ steps) with no refinement. Its vertex list (length $2T + 2R$) is, concretely,

$$0,\ e_\mu,\ 2e_\mu,\ \dots,\ (T-1)e_\mu,\quad T e_\mu,\ T e_\mu + e_\nu,\ \dots,\ T e_\mu + (R-1) e_\nu,\quad T e_\mu + R e_\nu,\ (T-1) e_\mu + R e_\nu,\ \dots,\ e_\mu + R e_\nu,\quad R e_\nu,\ (R-1) e_\nu,\ \dots,\ e_\nu .$$

All coordinates are $\ge 0$, so the hypothesis $\mathrm{InBox}(\gamma; k, L 2^k)$ says exactly that every listed vertex has all coordinates $\le L\cdot 2^k$; for $T + R > 0$ and $\mu\ne\nu$ this is equivalent to $T \le L\cdot 2^k$ and $R \le L\cdot 2^k$. For $T = R = 0$ the vertex list is empty and the hypothesis holds for every $k, L$. If $L = 0$ the box is $\{0\}$, so the hypothesis forces $T = R = 0$.

The quantity bounded

loopCorrelation ρ k L β [γ] is the expectation, on the box of half-width $R' = L\cdot 2^k$, of the function

$$U \longmapsto W_\gamma(U) := \operatorname{tr}\Bigl(\rho\bigl(\mathrm{hol}_U(0;\ \text{steps of }\gamma)\bigr)\Bigr) \in \mathbb{C}$$

(the product over the one-element list $[\gamma]$ is just this one factor). Writing $U(y,i)$ for $\mathrm{link}_U(y,i)$ (which is $U$'s value on the edge if the edge is a box edge and $1$ otherwise), the holonomy is

$$\mathrm{hol}_U = \Bigl[\prod_{a=0}^{T-1} U(a e_\mu,\ \mu)\Bigr]\Bigl[\prod_{b=0}^{R-1} U(T e_\mu + b e_\nu,\ \nu)\Bigr]\Bigl[\prod_{a=T-1}^{0} U(a e_\mu + R e_\nu,\ \mu)^{-1}\Bigr]\Bigl[\prod_{b=R-1}^{0} U(b e_\nu,\ \nu)^{-1}\Bigr],$$

with products written left to right in the order indicated. So the asserted inequality is

$$\left\|\ \frac{\int W_\gamma(U)\, e^{\beta S_\rho(U)}\, d\mathbf{P}_{L 2^k}(U)}{\int e^{\beta S_\rho(U)}\, d\mathbf{P}_{L 2^k}(U)}\ \right\| \le C\, e^{-\sigma T R}.$$

Degenerate and edge cases included by the quantifiers

• $d = 0$ or $d = 1$. There are no (resp. no two distinct) directions, so the hypothesis $\mu \ne \nu$ can never be met and the inner universal statement is vacuous; the theorem then asserts only that there exist $\beta_0 > 0$ and, for each $\beta \in (0,\beta_0)$, some $\sigma > 0$ and some $C$, which is trivially true.
• $T = R = 0$. The loop has no steps, the holonomy is $1$, $W_\gamma \equiv \operatorname{tr}(\rho(1)) = \operatorname{tr}(I_N) = N$, the in-box hypothesis is vacuous, and the right-hand side is $C e^{0} = C$. If the denominator integral $Z$ is non-zero and the integrals are genuinely evaluated, the left side equals $\|N \cdot Z / Z\| = N$ and the instance reads $N \le C$, for every $k$ and $L$.
• $T = 0 < R$ or $R = 0 < T$. The steps go out along one direction and straight back; the holonomy telescopes to $1$ (each factor is cancelled by its own inverse in reverse order), so again $W_\gamma \equiv N$ and the exponent is $-\sigma\cdot 0 = 0$; the instance reads $\|\langle N\rangle\| \le C$, i.e. $N \le C$ whenever $Z \ne 0$.
• $N = 0$. $\mathrm{U}(0)$ is the trivial group, every trace is $0$, $S_\rho \equiv 0$, $w_\beta \equiv 1$, $W_\gamma \equiv 0$, and the left-hand side is $0$; the instance reads $0 \le C e^{-\sigma TR}$.
• Zero denominator / non-integrability. If $\int w_\beta\, d\mathbf{P} = 0$ (which by the Bochner convention includes the case where the integrand is not integrable), the correlation is $0$ by the division-by-zero convention and the inequality reduces to $0 \le C e^{-\sigma T R}$, i.e. $C \ge 0$.
• Edges outside the box. No hypothesis prevents the rectangle's edges from leaving $B_{L 2^k}$ other than the vertex condition above; any edge not in $E_{L2^k}$ contributes the identity to the holonomy by the definition of $\mathrm{link}$. (Given the vertex condition, the rectangle's vertices all lie in the box; the edge $(y, i)$ used in a step is a box edge exactly when both $y$ and $y + e_i$ are in the box.)
• Sign of $C$. Since the left-hand side is a modulus, $\ge 0$, the statement can only hold with $C \ge 0$ whenever at least one instance of the hypotheses is satisfiable (i.e. $d \ge 2$); but the statement itself imposes no sign condition on $C$.
• Natural-number subtraction occurs only in $\mathrm{atScale}$ as $k \dot- k = 0$, so no truncation affects this theorem.
• Uniformity. $\sigma$ and $C$ are fixed once $\beta$ is fixed; the bound must then hold for all box sizes $L 2^k$ containing the rectangle and all rectangle sizes, including arbitrarily large $L$, $k$, $T$, $R$.
• No use of $\mathrm{IsCompactSimpleGaugeGroup}$ or of any continuum/admissibility notion from the bundle; only compactness, the topological-group structure, Borel measurability, and continuity of $\rho$ are assumed.