Read-back: YangMills.strong_coupling_clustering

Fixed data and assumptions

The statement is made for the following data, all of which are universally quantified (they are parameters of the theorem):

• A natural number $d$ together with the assumption $d \neq 0$ (so the index $0$ is a valid coordinate direction).
• A type $G$ carrying: a group structure; a topology; the assumption that multiplication and inversion are continuous; the assumption that $G$ is compact; a $\sigma$-algebra; and the assumption that this $\sigma$-algebra is the Borel $\sigma$-algebra of the topology. No Hausdorff, second-countability, connectedness, non-abelianness, or simplicity assumption on $G$ is made; in particular $G$ may be the trivial group, finite, or abelian.
• A natural number $N$ (possibly $N = 0$).
• A group homomorphism $\rho : G \to U(N)$, where $U(N)$ is the group of $N\times N$ complex unitary matrices; we write $\rho(g)$ also for the underlying $N \times N$ complex matrix.
• The hypothesis that $\rho$ is continuous (for the subspace topology of $U(N)$ inside $N\times N$ complex matrices). This hypothesis is assumed but the conclusion does not refer to it.

Custom definitions, unfolded

Sites, unit vectors, boxes. A site is a point $x \in \mathbb{Z}^d$ (a function from the $d$ coordinate indices to $\mathbb{Z}$). For a coordinate index $\mu \in \{0,\dots,d-1\}$, $e_\mu \in \mathbb{Z}^d$ denotes the standard unit vector. For $R \in \mathbb{N}$ the box is

$$\Lambda_R = \{x \in \mathbb{Z}^d : -R \le x_\mu \le R \text{ for every } \mu\} = [-R,R]^d \cap \mathbb{Z}^d,$$

so $\Lambda_0 = \{0\}$.

Edges and plaquettes. An edge is a pair $(x,\mu)$ with $x \in \mathbb{Z}^d$ and $\mu$ a coordinate index. The finite set of box edges is

$$E_R = \{(x,\mu) : x \in \Lambda_R,\ x + e_\mu \in \Lambda_R\}.$$

A plaquette is a triple $(x,\mu,\nu)$; the finite set of box plaquettes is

$$P_R = \{(x,\mu,\nu) : x \in \Lambda_R,\ \mu < \nu,\ x+e_\mu \in \Lambda_R,\ x+e_\nu \in \Lambda_R,\ x+e_\mu+e_\nu \in \Lambda_R\}.$$

Configurations and links. A configuration on the box of size $R$ is a function $U : E_R \to G$. For an arbitrary edge $e = (x,\mu)$ (not necessarily in $E_R$) the link variable is

$$\ell_U(x,\mu) = \begin{cases} U(x,\mu) & \text{if } (x,\mu) \in E_R,\\ 1 \ (\text{the identity of } G) & \text{otherwise.}\end{cases}$$

So every edge outside the box carries the identity element.

Plaquette variable and Wilson action. For a plaquette $p=(x,\mu,\nu)$,

$$\Pi_U(p) = \ell_U(x,\mu)\cdot \ell_U(x+e_\mu,\nu)\cdot \ell_U(x+e_\nu,\mu)^{-1}\cdot \ell_U(x,\nu)^{-1} \in G,$$

and the Wilson action is

$$S_\rho(U) = \sum_{p \in P_R} \operatorname{Re}\operatorname{tr}\big(\rho(\Pi_U(p))\big) \in \mathbb{R}.$$

There is no factor $1/N$, no constant offset, and no minus sign in this definition. If $N=0$, every trace is $0$ and $S_\rho \equiv 0$.

Measures. The measure $\eta$ on $G$ (called haarProb) is the left-invariant Haar measure constructed from the positive compact set consisting of all of $G$, normalized so that $\eta(G) = 1$ (a probability measure on $G$). The measure $\eta_R$ on configurations (called configMeasure) is the product measure $\prod_{e \in E_R} \eta$ on $G^{E_R}$, with respect to the product $\sigma$-algebra; it has total mass $1$. When $E_R = \emptyset$ (e.g. $R=0$) the configuration space is a single point carrying mass $1$.

Gibbs weight and expectation. For $\beta \in \mathbb{R}$ the Gibbs weight is $w_\beta(U) = \exp\big(\beta\, S_\rho(U)\big) > 0$ (note the positive sign in the exponent). For a complex-valued function $F$ on configurations, the expectation is

$$\langle F\rangle_{R,\beta} = \frac{\displaystyle\int F(U)\, w_\beta(U)\, d\eta_R(U)}{\displaystyle\int w_\beta(U)\, d\eta_R(U)} \in \mathbb{C}.$$

Both integrals are Bochner integrals; by convention a Bochner integral equals $0$ whenever the integrand is not integrable (in particular, whenever it is not almost-everywhere strongly measurable for the product $\sigma$-algebra). Division in $\mathbb{C}$ is total with $z/0 = 0$; hence if the denominator integral is $0$ the expectation is $0$. When $E_R=\emptyset$ the expectation is just $F$ evaluated at the unique configuration.

Steps and holonomy. A step is a pair $s = (\mu, b)$ with $\mu$ a coordinate index and $b$ a Boolean; its displacement is $\vec s = e_\mu$ if $b$ is true and $\vec s = -e_\mu$ if $b$ is false. For a configuration $U$, a starting site $x$ and a finite list of steps, the path holonomy $H_U(x, [\,]) = 1$ and

$$H_U\big(x, (\mu,b)::\text{rest}\big) = \begin{cases}\ell_U(x,\mu)\cdot H_U(x+e_\mu,\text{rest}) & b \text{ true},\\ \ell_U(x-e_\mu,\mu)^{-1}\cdot H_U(x-e_\mu,\text{rest}) & b \text{ false},\end{cases}$$

i.e. the ordered (left-to-right) product of link variables (or their inverses, for backward steps) along the path, with edges outside the box contributing $1$.

Loops. A loop $\gamma$ consists of a natural number $\gamma.\mathrm{scale}$, a base site $\gamma.\mathrm{base} \in \mathbb{Z}^d$, a finite list of steps $\gamma.\mathrm{steps}$, and a proof that the displacements sum to $0$: $\sum_{s \in \gamma.\mathrm{steps}} \vec s = 0$. No other condition is imposed: the step list may be empty, may revisit sites and edges, may backtrack, etc.

Refinement. For $j\in\mathbb{N}$, $\gamma^{(j)}$ (refine γ j) is the loop with scale $\gamma.\mathrm{scale}+j$, base $2^j\cdot\gamma.\mathrm{base}$, and step list obtained by replacing each step of $\gamma$, in order, by $2^j$ consecutive copies of itself.

Loop at scale $k$. For $k \in \mathbb{N}$, $\gamma@k$ (atScale γ k) is the pair (base, steps) of $\gamma^{(k \dot- \gamma.\mathrm{scale})}$, where $\dot-$ is truncated natural-number subtraction ($a \dot- b = 0$ when $a \le b$). Thus:

• if $\gamma.\mathrm{scale} \le k$: base $2^{k-\gamma.\mathrm{scale}}\gamma.\mathrm{base}$, each step repeated $2^{k-\gamma.\mathrm{scale}}$ times;
• if $\gamma.\mathrm{scale} > k$: base and steps of $\gamma$ unchanged (no coarsening takes place).

Translation. For $j\in\mathbb{N}$ and $v\in\mathbb{Z}^d$, $\gamma.\mathrm{translate}(j,v)$ is the loop with scale $\max(\gamma.\mathrm{scale},j)$, base $2^{\max(\gamma.\mathrm{scale},j)-\gamma.\mathrm{scale}}\gamma.\mathrm{base} + 2^{\max(\gamma.\mathrm{scale},j)-j}\, v$, and steps those of $\gamma^{(\max(\gamma.\mathrm{scale},j)-\gamma.\mathrm{scale})}$.

Time translation. $\tau_{j,n}(\gamma)$ (timeTranslate γ j n) is $\gamma.\mathrm{translate}(j, n\,e_0)$ with $n \in \mathbb{N}$ (so $n \ge 0$ only; no backward time shift is expressible). In the case used by this theorem, $\gamma.\mathrm{scale} \le k$, one gets:

$$\tau_{k,n}(\gamma) : \quad \mathrm{scale} = k,\quad \mathrm{base} = 2^{\,k-\gamma.\mathrm{scale}}\gamma.\mathrm{base} + n\,e_0,\quad \mathrm{steps} = \text{each step of } \gamma \text{ repeated } 2^{\,k-\gamma.\mathrm{scale}} \text{ times}.$$

(If instead $\gamma.\mathrm{scale} > k$, the scale and steps would be unchanged and the base shifted by $2^{\gamma.\mathrm{scale}-k}\, n\, e_0$; this case is excluded by hypothesis below.) For $n = 0$, $\tau_{k,0}(\gamma)$ is simply $\gamma$ refined to scale $k$ with no shift.

Wilson loop, loop product, loop correlation. For a configuration $U$ on the box of size $R$, a scale $k$ and a loop $\gamma$,

$$\mathcal W_U^{k}(\gamma) = \operatorname{tr}\rho\big(H_U(\gamma@k)\big) \in \mathbb{C},$$

the trace of $\rho$ applied to the holonomy along the base/steps of $\gamma@k$. For a finite list $A = [\gamma_1,\dots,\gamma_r]$ of loops, $\mathcal P_U^{k}(A) = \prod_{i=1}^r \mathcal W_U^{k}(\gamma_i)$, with the empty product equal to $1$. Finally, for $k, L \in \mathbb{N}$ and $\beta\in\mathbb{R}$, the loop correlation is

$$\langle A\rangle_{k,L,\beta} = \big\langle U \mapsto \mathcal P_U^{k}(A)\big\rangle_{R = L\cdot 2^{k},\,\beta},$$

the expectation above taken on the box $\Lambda_{L 2^k} = [-L2^k, L2^k]^d\cap\mathbb{Z}^d$. Degenerate values: a loop with empty step list has holonomy $1$ and Wilson loop $\operatorname{tr}\rho(1) = N$; if $N = 0$ every Wilson loop is $0$, so $\mathcal P^k_U(A) = 0$ for nonempty $A$ and $=1$ for $A=[\,]$; for $A = [\,]$ the correlation is $\langle 1\rangle_{R,\beta}$, which equals $1$ when the weight integral is a nonzero integrable quantity and $0$ otherwise; for $L=0$ the box is $\{0\}$, there are no edges, and $\langle A\rangle_{k,0,\beta} = N^{\,\mathrm{length}(A)}$. Edges of $\gamma@k$ falling outside $\Lambda_{L2^k}$ contribute the identity to the holonomy. The correlation depends on a loop $\gamma$ only through $\gamma@k$.

The assertion

Under the assumptions listed above, the theorem asserts, with quantifiers in exactly this order:

There exists a real number $\beta_0$ with $0 < \beta_0$ such that for every real $\beta$ with $0 < \beta$ and $\beta < \beta_0$, and for every natural number $k$, there exist a function $W$ from finite lists of loops to $\mathbb{C}$ and a real number $m$ such that

1. $0 < m$;

2. (convergence) for every finite list $A$ of loops such that every loop $\gamma$ in $A$ satisfies $\gamma.\mathrm{scale} \le k$,

$$\lim_{L\to\infty} \langle A\rangle_{k,L,\beta} = W(A) \quad\text{in } \mathbb{C},$$

the limit being along $L \in \mathbb{N}$, $L \to \infty$ (so in particular the limit must exist);

3. (clustering) for every pair of finite lists $A, B$ of loops such that every loop in $A$ has scale $\le k$ and every loop in $B$ has scale $\le k$, there exists a real number $C$ such that for every natural number $n$,

$$\Big\| W\big(A \mathbin{+\!\!+} [\tau_{k,n}(\gamma) : \gamma \in B]\big) \;-\; W(A)\,W(B)\Big\| \;\le\; C\, e^{-m n},$$

where $\mathbin{+\!\!+}$ is list concatenation (the loops of $A$ followed, in order, by the time-translated loops of $B$), $\|\cdot\|$ is the modulus on $\mathbb{C}$, and $n$ is cast to a real number in the exponent.

Remarks on scope and strength (literal consequences of the text)

• Dependencies: $\beta_0$ depends only on $d, G, N, \rho$. $W$ and $m$ may depend on $\beta$ and $k$ (they are chosen after $\beta$ and $k$). $C$ may depend on $A$ and $B$ (and on $\beta, k, W, m$); $m$ does not depend on $A, B$. $C$ is an arbitrary real number, unconstrained in sign or size.
• $W$ is a total function on all finite lists of loops; only its values on lists whose loops all have scale $\le k$ are constrained by clause 2. Every list appearing in clause 3, including $A \mathbin{+\!\!+}[\tau_{k,n}(\gamma):\gamma\in B]$ (whose translated loops have scale exactly $k$), consists of loops of scale $\le k$, so all values of $W$ occurring in clause 3 are also determined by clause 2.
• In clause 3, $W(B)$ is evaluated on $B$ with its loops at their original scales, whereas the concatenated list contains the loops of $B$ refined to scale $k$ and shifted by $n\,e_0$.
• The lists $A$ and $B$ are otherwise arbitrary: they may be empty, may contain repeated loops, loops with empty step lists, non-simple or self-intersecting loops, and loops of $A$ and $B$ may intersect each other; no admissibility, simplicity, disjointness, or positive-time condition appears anywhere in the statement.
• The case $n = 0$ of clause 3 reads $\|W(A \mathbin{+\!\!+} B^{(k)}) - W(A)W(B)\| \le C$, where $B^{(k)}$ is $B$ with each loop refined to scale $k$ and not shifted.
• The case $A = [\,]$ or $B = [\,]$ of clause 3 involves $W([\,])$, which by clause 2 is $\lim_L \langle 1\rangle_{L2^k,\beta}$.
• The theorem says nothing about $\beta \ge \beta_0$, nothing about $\beta \le 0$, nothing about dependence of $m$ or $W$ on $\beta$ or $k$ beyond existence, nothing about uniqueness of $W$ or $m$, and nothing about a continuum limit ($k$ is fixed throughout each instance).
• The only role of the hypotheses $\gamma.\mathrm{scale}\le k$ is to restrict which lists $A, B$ clauses 2 and 3 speak about; for such loops $\gamma@k$ is the genuine refinement to scale $k$ (the truncated subtraction $k \dot- \gamma.\mathrm{scale}$ is an honest difference), and $\tau_{k,n}$ acts as described in the boxed formula above.
• No hypothesis excludes $N = 0$, $\rho$ trivial (constant identity), or $G$ trivial; the statement is asserted for all such cases as well.