2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlBackground dependence, Lemma H, local couplings · Kiichi, Shiori and Rin

Background-dependent fluctuations η — absorption step by step

• For fields that take values in a group the measure of the fluctuations depends on the background, and $\eta_k$ accordingly puts a factor $1+\eta_k$ into the one-step estimate. Unless $\sum_k\eta_k\lt\infty$, the constants break down with the number of steps, exactly as in the arithmetic of 15-01 §01 (§01).
• The group version of property 2 of the weight (it splits into a product over a sum of fields) holds on paper in the form background × fluctuation (Lemma R, §02). The group version of properties 3 and 4 can be set down as statements, and assuming them the one-step chain closes in the same form as for flat fields (§03).
• Estimating $\eta_k$ naively from the curvature of the background, the sum diverges as a power of the number of steps or a power of its logarithm, and cannot be carried as an error (§04).
• When the law of the fluctuation does not depend on the background (a product of heat kernels link by link is an example), the chain goes through as it stands with $\eta=0$ (Lemma H, §05). The fluctuation of a genuine step is not of this type.
• For the Wilson action of $SU(2)$, the conditional law of one link is a member of a one-parameter family that does not depend on the background, and the background enters only through the local coupling and the direction of the centre. Using this we set down a proposal to absorb $\eta$ as the shift of local couplings (§06, proposal). The multi-step induction has not been written (§07).

In one step at stage $k$ we split the edge variable into background and fluctuation as $U_l=W_l\,e^{A_l}$ ($W_l\in G$, and $A_l$ an element of the Lie algebra $\mathfrak{su}(N)$), hold the background $W$ fixed and integrate out the fluctuation $A$. We write the law of this fluctuation as $\mu_{k,W}$. The covariance of its Gaussian part is the propagator in a background field $g_k^2C_W$ (the title of the foundational papers in 6-07 Table 1).

For flat fields (a scalar field $\phi=\phi_{\rm 背景}+\zeta$) the law of the fluctuation $\zeta$ is a Gaussian law that does not depend on the background, and property 3 of the weight (closes in one step under the expectation of the fluctuations) is used as the expectation of a function of the fluctuation alone Reading (from reading the chain of 15-02 §04). On the group this is not so. When the background changes, $C_W$ changes, and the law of the fluctuation itself changes. We write the size of this dependence as $\eta_k$ and put it into the one-step estimate as a factor $1+\eta_k$; this is how this chapter sets things up (equation (3)). From the arithmetic of 15-01 §01, for the product of factors $\prod_k(1+\eta_k)$ to be bounded uniformly in the number of steps $K$ one needs $\sum_k\eta_k\lt\infty$, and if $\eta_k$ is of size $g_k^2$ it is not enough Proved here (arithmetic).

Let $l_1,\dots,l_4$ be the edges on the boundary of a plaquette $p$ and $s_i\in\{\pm1\}$ their orientation signs, set $U_{\partial p}=\prod_iU_{l_i}^{s_i}$, and define $W_{\partial p}$ in the same way. The norm is the operator norm.

Lemma R. For $G=SU(N)$,

Proved here Proof. (1) Split each factor into a $W$ part and a fluctuation part. If $s=+1$, $We^{A}=W\cdot Y$ ($Y=e^{A}$); if $s=-1$, $(We^{A})^{-1}=W^{-1}\cdot Y$ ($Y=We^{-A}W^{-1}$). In both cases $Y$ is a conjugate of $e^{\pm A}$. (2) In the product $W_1'Y_1W_2'Y_2W_3'Y_3W_4'Y_4$ ($W_i'=W_{l_i}^{s_i}$), move each $Y_i$ to the right of the $W'$ to its right by $YW'=W'(W'^{-1}YW')$; then $U_{\partial p}=W_{\partial p}\,Y_1''Y_2''Y_3''Y_4''$, and each $Y_i''$ is a conjugate of $e^{\pm A_{l_i}}$. (3) For unitary matrices, repeating $\|XY-1\|\le\|X-1\|+\|Y-1\|$ and $\|hYh^{-1}-1\|=\|Y-1\|$ (equation (8) of 6-05) gives $\|U_{\partial p}-1\|\le\|W_{\partial p}-1\|+\sum_i\|e^{\pm A_{l_i}}-1\|$. (4) $A$ is anti-Hermitian and so unitarily diagonalisable, and for an eigenvalue $e^{i\vartheta}$, $|e^{i\vartheta}-1|=2|\sin(\vartheta/2)|\le|\vartheta|\le\|A\|$. Hence $\|e^{\pm A}-1\|\le\|A\|$. (5) The squares follow from $(x+y)^2\le2x^2+2y^2$ and $(\sum_{i\le4}y_i)^2\le4\sum_iy_i^2$. ∎

Corollary (decomposition of the weight). Setting $G_c(X,U):=\exp\bigl(c\sum_{p\subset X}\|U_{\partial p}-1\|^2\bigr)$ ($c\gt0$), since on a four-dimensional lattice an edge lies on at most 6 plaquettes,

Proved here (sum the second of (1) over $p\subset X$; $8\times6=48$). This is the version of property 2 in which the sum of fields is replaced by "background × fluctuation", and on the fluctuation side the weight is of Gaussian type. It may be a known elementary fact (an estimate of the same form may appear in the large-field papers of the series, but we have not checked the text).

The form that writes the fluctuation with $Y$. Steps (1)–(3) go through with $e^{A}$ replaced by any unitary $Y$. From $\|Y^{-1}-1\|=\|Y-1\|$ and conjugation invariance, $\|(WY)_{\partial p}-1\|\le\|W_{\partial p}-1\|+\sum_{i\le4}\|Y_{l_i}-1\|$, and with $q$ the maximal number of plaquettes on which one link lies (6 in four dimensions), $G_c(X,WY)\le G_c(X,W)^2\exp\bigl(8qc\sum_{l\in\bar X}\|Y_l-1\|^2\bigr)$ Proved here ($\bar X$ are the links on the boundary of the plaquettes of $X$). In this form no small-field condition on the fluctuation is needed (since $\|e^{A}-1\|\le\|A\|$ of step (4) is not used). Lemma H of §05 uses this form.

What Lemma R does not say. (2) speaks only of property 2 and does not touch on closing in one step under the expectation of the fluctuations (property 3).

Let the weight at step $k$ be $G_k:=G_{c/h_k^2}$ ($h_k$ is the field scale at step $k$, for example the small-field threshold $g_kp(g_k)$; $p(g)$ is the factor of the threshold in equation (7) of 6-05), let $\mathcal W_k(X)$ be the set of small-field (regular) backgrounds on $X$, and let $B(X)$ be the set of blocks in $X$. We set down the following three as statements Assumption (statements only; the constant $c,t_0$ and the constants in the definition of $\mathcal W_k$ depend neither on the step nor on the lattice spacing).

• (iii_G) For all $W\in\mathcal W_k(X)$ and $0\lt t\le t_0$, $\mathbb E_{\mu_{k,W}}\exp\bigl(\frac{48tc}{h_k^2}\sum_{l\in X}\|A_l\|^2\bigr)\le2^{|B(X)|}$. The expectation is with respect to the true measure of the fluctuations, not only its Gaussian part.
• (iv_G) For $W$ in the image of the background map, $G_k(X,W)^{2t}\le G_{k+1}(X,W)$ ($0\lt t\le t_0$). Its content is the regularity of the background (the curvature of the fine plaquettes being bounded by the curvature of the coarse plaquettes).
• (A.6.5_G) Let $\theta_WF(W):=F(We^{A})$, and let $T_W(h)$ be the Taylor seminorm built from derivatives in the Lie-algebra directions of the background. For some $\eta_k\ge0$, $\|\mathbb E_{\mu_{k,W}}\theta_WF\|_{T_W(h_{k+1})}\le(1+\eta_k)\,\mathbb E_{\mu_{k,W}}\|F\|_{T_{We^{A}}(h_k)}$, and $\sum_k\eta_k\lt\infty$. This is a version of the original interchange of expectation and seminorm (A.6.5) that allows a factor $1+\eta_k$ for the dependence of the measure on the background.

Assuming these three and Corollary (2) (with $c$ replaced by $c/h_k^2$), the same chain as in Lemma A.6.1 (interchange → definition of the norm → (2) → (iii_G) → (iv_G)) gives, on $W\in\mathcal W_k(X)$,

Proved here (only the chain, assuming the three). Outside $\mathcal W_k$, the large-field factor of (U2) takes over (15-00 §01). The group version of property 3 itself has a form that closes with three things: an estimate of the propagator $C_W$ uniform over regular backgrounds, a uniform estimate of the deviation of $\mu_{k,W}$ from Gaussian, and $cg_k^2/h_k^2$ being small Reading (from the formula that writes Gaussian exponential moments as determinants).

For a product of one-link heat kernels, the group version of property 3 holds for free. When $Y_l=W_l^{-1}U_l$ follows the heat kernel $p_\tau$ independently on each edge, the invariance of Haar measure (invariant on both left and right, since the group is compact) makes the law of $Y$ independent of $W$, and $\mathbb E\exp\bigl(s\sum_l\|Y_l-1\|^2\bigr)=\bigl(\mathbb E\exp(s\|Y-1\|^2)\bigr)^{|X|}$ ($|X|$ is the number of edges of $X$) does not depend on the background Proved here. This is at most $2^{|B(X)|}$ when $s\tau$ is small Reading (from the Gaussian-type upper bound on the heat kernel; the numbers are in Table 1 of §05). The telescoping sum of 15-04 §01 can be read as another face of these "fluctuations independent of the background" Reading.

Estimating from the curvature. A naive estimate of the part by which the measure depends on the background, through the curvature $\|W_{\partial p}-1\|\lesssim g_kp(g_k)$, gives $\eta_k\sim g_kp(g_k)$, and for the sequence of one-loop couplings $g_{K-j}^2\le1/(bj)$ (15-01 §01) $\sum_k\eta_k$ diverges at the order of $K^{1/2}$ (up to logarithmic factors) Proved here (arithmetic, under the naive estimate $\eta_k\sim g_kp(g_k)$).

Measuring with a single plaquette. In the smallest model of a single plaquette (15-06 §01), if we impose a constraint that removes the directions of the generators of the background, the background dependence enters exactly through a single replacement of the coupling $\beta\mapsto\beta\cos(\theta/2)$ ($\mu_W^\beta=\mu_1^{\beta\cos(\theta/2)}$) Proved here. $\eta$ measured by the ratio of covariances is $\theta^2/8$; below the threshold $\theta\le\varepsilon_k=g_kp(g_k)$ it is $\eta_k\lesssim g_k^2p(g_k)^2$, and under the one-loop flow $\sum_{k\le K}\eta_k$ diverges as a power of $\ln K$ Proved here (the arithmetic of the naive estimate). Without the constraint (not taking the background at a critical point), the ratio of covariances is $\beta\theta^2/12$ and grows with $\beta$ Reading. Under either estimate, $\eta$ is not of a size that can be carried as an error, and absorption into the parameters is needed. The single-plaquette example is the smallest example in which that absorption happens exactly.

Which $\eta$? Measuring $\eta$ by a single number changes its type (the ratio of covariances is $\theta^2$, the Kullback–Leibler quantity is $\theta^4$, and the score, i.e. the first-order rate of change of the expectation when the background is moved, is $\theta$) Reading. (iii_G) uses an even quantity and the interchange inequality (A.6.5_G) a first-order one, so the statements must specify "which $\eta$". For a scalar field the measure does not depend on the background, so $\eta_k=0$ and the problem itself does not arise.

There is one case in which the chain of §03 goes through without assumptions: when the law of the fluctuation does not depend on the background.

Setting. $G=SU(N)$, operator norm. A finite set of links $\Lambda$, a partition into blocks, $X$ a union of blocks, $B(X)$ the blocks in it, $\bar X$ the links on the boundary of the plaquettes of $X$. The weight $G_h(X,U):=\exp\bigl((c/h^2)\sum_{p\in X}\|U_{\partial p}-1\|^2\bigr)$. The Taylor seminorm is $\|F\|_{T_U(h)}:=\sum_{n\ge0}\frac{h^n}{n!}\|D^nF(U)\|$, where $\|D^nF(U)\|$ is the supremum over $|X_j|\le1$ of the partial derivative at $t=0$ of right multiplication $F(U e^{t_1X_1}\cdots e^{t_nX_n})$. The weighted norm is $\|K(X)\|_{G_h}:=\sup_U\|K(X)\|_{T_U(h)}/G_h(X,U)$. The fluctuation is a probability measure $\nu$ on $G^\Lambda$, and one step is $\theta_\nu K(W):=\int K(WY)\,d\nu(Y)$. $M_\nu(s,X):=\mathbb E_\nu\exp\bigl(s\sum_{l\in\bar X}\|Y_l-1\|^2\bigr)$.

Lemma H. Assume the following three. (H1) $\nu$ does not depend on the background $W$. (H2) $h_+^2\le h^2/2$. (H3) $M_\nu(8qc/h^2,X)\le2^{|B(X)|}$. Then for every background $W\in G^\Lambda$ and smooth $K(X)$,

Moreover the following hold Proved here (conditions: (H1)–(H3), one step only).

• (a) The interchange is $\eta=0$: for all smooth $F$, $\|\theta_\nu F\|_{T_W(h)}\le\mathbb E_\nu\|F\|_{T_{WY}(h)}$.
• (b) Locality: if $\nu=\otimes_l\nu_l$ (independent link by link), then $\theta_\nu K(X)$ depends only on the restriction of $W$ to $\bar X$, and if $\bar X\cap\bar X'=\emptyset$ it splits into a product. The support does not spread in one step.
• (c) For a product of heat kernels link by link $\nu=p_\tau^{\otimes\Lambda}$, $M_\nu(s,X)=m(s,\tau)^{|\bar X|}$ and $m(s,\tau):=\mathbb E_{p_\tau}e^{s\|Y-1\|^2}$, independent of the background. An elementary upper bound is $m(s,\tau)\le1+(e^{4s}-1)\,\mathbb E_{p_\tau}\|Y-1\|^2/4$. For $SU(2)$ (with the convention for the character coefficients $(n+1)e^{-n(n+2)/(2\beta)}$), $\mathbb E\|Y-1\|^2=2(1-e^{-3/(2\beta)})\le3/\beta$.
• (d) Small fields: $\nu'$, obtained by conditioning $\nu=\otimes\nu_l$ on $\{\|Y_l-1\|\le h'\ \forall l\}$, also satisfies (H1), and $M_{\nu'}\le M_\nu$. If the background is taken in $\mathcal U_\varepsilon:=\{\max_{p\in X}\|W_{\partial p}-1\|\le\varepsilon\}$, the supremum of the norm on the right-hand side need only be taken over $\mathcal U_{\varepsilon+4h'}$.

Proved here Proof. Step 1 (property 2): from "the form that writes the fluctuation with $Y$" of §02, $G_h(X,WY)\le G_h(X,W)^2\exp\bigl((8qc/h^2)\sum_{l\in\bar X}\|Y_l-1\|^2\bigr)$. Step 2 ((a)): inserting $Y^{-1}e^{tX}Y=e^{t\,{\rm Ad}_{Y^{-1}}X}$ between the factors gives $F(We^{t_1X_1}\cdots e^{t_nX_n}Y)=F(WYe^{t_1X_1'}\cdots e^{t_nX_n'})$, $X_j':={\rm Ad}_{Y^{-1}}X_j$, $|X_j'|=|X_j|$. Right multiplication only rotates the direction of differentiation by the adjoint, and the adjoint preserves the operator norm. $F$ is smooth on a product of compact groups, so we can differentiate under the integral (dominated convergence), and since $\nu$ does not depend on $W$, the derivative falls only on the integrand. Hence the derivative of order $n$ is at most $\int\|D^nF(WY)\|d\nu(Y)$; multiply by $h^n/n!$ and sum. Step 3 (the chain): in the order (a) → definition of the norm → Step 1 → (H3) → (H2), $\|\theta_\nu K(X)\|_{T_W(h)}\le\|K(X)\|_{G_h}\,G_h(X,W)^2\,2^{|B(X)|}\le\|K(X)\|_{G_h}\,G_{h_+}(X,W)\,2^{|B(X)|}$. The last follows from (H2): $G_h^2\le G_{h_+}$. (b) follows from Fubini and independence; (c) from independence, the left and right invariance of Haar measure, the convexity of $e^{sz}$ on $z=\|Y-1\|^2\in[0,4]$ (the chord inequality), $\|Y-1\|^2=2-\operatorname{tr}Y$ and $\mathbb E\operatorname{tr}Y=2e^{-3/(2\beta)}$. (d) follows from the fact that conditioning a function increasing in each coordinate on a lower set does not increase its expectation (the coordinates being independent), and from the first inequality of Step 1. ∎

Table 1 is a numerical record of this book's study (trapezoidal quadrature in the angular direction and a sum over characters; as a positive control it agrees with the closed form for $\mathbb E\|Y-1\|^2$ to 9 digits). It agrees with $\ln2/(3n_b)\approx0.0024$ from the Gaussian estimate $(1-2s/\beta)^{-3/2}$. Reading $s=8qc/h^2$, $\beta=4/g^2$, the condition is $c\,g^2/h^2\lesssim1.7\times10^{-4}$ ($q=6$, $n_b=96$) Reading. If $h=g\,p(g)$ then $c/p(g)^2\lesssim1.7\times10^{-4}$, and since $p(g)$ grows as a power of the logarithm, it is satisfied at fine steps.

What is not in Lemma H. This is the precise edge of "the part that can be closed" Reading.

1. All of (H1), that the law of the fluctuation does not depend on the background. The $\mu_{k,W}$ of a genuine step depends on $W$ through three things: the non-quadratic nature of the action (the Wilson action: $\mu_W^\beta=\mu_1^{\beta\cos(\theta/2)}$ for a single plaquette), the constraint of block averaging (the constraint surface depends on $W$), and the propagator in a background field $C_W$ (fluctuations of neighbouring links coupled by plaquette terms that depend on $W$). The place where "the derivative falls only on the integrand" of Step 2 breaks down is $\eta$ itself.
2. Relation to the heat-kernel action. The fluctuation of Lemma H is "a convolution of heat kernels link by link", and the heat-kernel action plaquette by plaquette takes the form of a product over edges only when the plaquette variables become independent coordinates (two dimensions, axial gauge). In three or more dimensions the cube constraints tie plaquettes together (its manifestation is the source of 15-05).
3. One step only. Stacking (H2) over $k$ steps, the weight collapses as $h_k\le2^{-k/2}h_0$. Many steps need the blocking factor, (iv_G), and a contraction of the many-body norm that cancels $2^{|B(X)|}$ (§07).
4. The rarity of large fields does not come out. Since $G_h(X,U)\le e^{4c|X|/h^2}$, this norm does not produce the $e^{-c\,p(g)^2|\Omega^c|}$ of (U2). (d) says only that "a small-field background spreads in one step only to $\varepsilon\to\varepsilon+4h'$".
5. The Wilson action is not treated (the $\eta$ of non-quadratic nature).
6. Correspondence with (U1). Of "the analyticity and decay of localised corrections" in (U1), what Lemma H contains is that the support does not spread in one step ((b)) and that the Taylor norm of radius $h$ is preserved up to a factor $2^{|B(X)|}$ ((4)); it does not contain the uniformity of the decay $e^{-\kappa d(X)}$ or constants uniform in the step.

Lemma H cannot be used once (H1) breaks down. So we consider the direction of splitting the genuine $\mu_{k,W}$ into "a law independent of the background $\nu_k$ (to which Lemma H applies) × a density $\rho_{k,W}$", and extracting the part of the background dependence of the density that can be absorbed into the parameters. The statements of this section are proposals, except for one fact (the basis of the heat-bath method) and one identity.

Fact: the conditional law of one link is a member of a one-parameter family. For the Wilson action of $SU(2)$, let $Q(W)$ be the sum, as quaternions, of the products of each plaquette with the link $U$ removed (the staples), and write $Q=|Q|\hat q$ ($\hat q\in SU(2)$). The conditional law of $U$ with the other links held fixed is $\exp(\beta\langle Q,U\rangle)\,dU$, and setting $U=\hat qY$, the left invariance of Haar measure makes the law of $Y$ equal to $\exp(\beta|Q|\operatorname{Re}Y)\,dY=\mu_1^{\beta|Q|}$. The conditional law is a member of the one-parameter family $\{\mu_1^b\}$, which does not depend on the background, and the background enters only through the local coupling $b(W)=\beta|Q(W)|$ and the direction of the centre $\hat q(W)$ Cited (a basic fact of the heat-bath method for $SU(2)$; as bibliography we give Creutz 1980 and Kennedy–Pendleton 1985, but we have not checked the text, i.e. unconfirmed). The $\beta\mapsto\beta\cos(\theta/2)$ of a single plaquette (§04), and the closed forms of the weights for two and three plaquettes (15-06 §02, §03), can both be read as examples of this one sentence Reading.

Identity: how to split a many-link action. Applying the unit-quaternion identity $\operatorname{Re}(ab)=\operatorname{Re}a\operatorname{Re}b-\operatorname{Im}a\cdot\operatorname{Im}b$ to the rearrangement $U_{\partial p}=W_{\partial p}Y_p$ of Lemma R ($Y_p$ is the product of the fluctuation parts), with $\theta_p$ the rotation angle of $W_{\partial p}$ and $n_p$ its axis,

Proved here (convention for $\frac12\operatorname{tr}=\operatorname{Re}$; quaternion algebra). The first term is an action of background-independent form with local coupling $b_p(W)$, and the second is a term whose leading part is first order in the Lie-algebra fluctuation. In addition there remains a "conjugation mismatch" term, coming from the same $Y_l$ being conjugated by a different partial product of the background in a neighbouring plaquette (of triple-product type).

Proposal η-1: screen the score by the order in $\theta$. We divide the background derivative (the score) of the density $\rho_{k,W}$ according to its order in the background angle $\theta$ Assumption (a reading of the screening; not yet checked around the true critical point for many plaquettes).

• The $\theta^0$ term (a first-order term independent of the background): it vanishes once the centre of the background is moved to the true constrained critical point. For two plaquettes the action becomes even in the fluctuation around the true critical point, and for three plaquettes the weight becomes a function of $|1+W_a+W_b|$ (15-06 §02, §03, both [Proved here]).
• The $\theta^1$ term: the derivative of $b_p(W)$ (the coupling derivative), and the anisotropic part of the quadratic form of the fluctuation left after recentring. The former is absorbed by proposal η-2, and the latter is expected to be absorbable in the same way, as the derivative of a local anisotropic coupling.
• Terms of order $\theta^2$ and higher: the remainder that cannot be absorbed. Multiplying by the Taylor weight $h_k\sim\varepsilon_k$ gives $\eta_k^{\rm res}=O(\varepsilon_k^3)=O(g_k^3p(g_k)^3)$, and the sum converges in the $g^{2+\delta}$ type of 15-01 §01 ($\delta=1$).

Proposal η-2: make the coupling $b$ the zeroth variable of the Taylor norm. We regard the activity $K(X;W,b)$ as a function of both the background and the coupling, and extend the seminorm to $T_{W,b}(h,h_b):=\sum_{n,m}\frac{h^n}{n!}\frac{h_b^m}{m!}\|D_W^n\partial_b^mK\|$ ($h_b\propto b$). If the law of the fluctuation has the form $\mu^{b(W)}$, the background derivative falls on the integrand and on $b(W)$, and the latter is $\partial_Wb\cdot\partial_b\mathbb E_bF=\partial_Wb\cdot{\rm Cov}_b(F,\operatorname{Re}Y)$; that is, what remains of the background derivative is a coupling derivative. If the coupling derivative is put into the norm, (A.6.5_G) is expected to hold without the factor $1+\eta_k$ Assumption (assumption: the central moments of the statistic of the exponential family are bounded uniformly in $b$; the consistency of the norm for a multivariable set of local couplings $\{b_p\}$ with $2^{|B(X)|}$ is not yet settled). In this form, (C) of Lemma F (15-03 §03) would also follow from analyticity in $b$. Our reading is that the precise meaning of "absorbing $\eta$ into the parameters" is "counting the parameters as components of the field" Reading.

Reservations. (5) is specific to $SU(2)$ (only for $SU(2)$ is a sum of staples a constant multiple of a group element). For $SU(N)$, $\operatorname{tr}(WY)$ is a bilinear form in $W$ and $Y$, the local coupling is a matrix rather than a number, and the conditional law does not fall into a one-parameter family (it does for each $SU(2)$ subgroup). Also, a fundamental-plus-adjoint action breaks the form of the first term of (5) (the expansion of $\cos^2$ produces an anisotropic quadratic form that acts only on the components of the fluctuation in the directions of the background) Proved here (quaternion algebra).

Lemma H and proposals η-1 and η-2 are all one-step statements. Stacking steps needs the following three, none of which has been written Assumption.

• (iv_G): for $W$ in the image of the background-field map, the curvature of the fine plaquettes is bounded by $L^{-2}$ times the curvature of the coarse plaquettes plus $O(\varepsilon_k^2)$ (the regularity of the background).
• Contraction of the many-body norm: the one-step loss $2^{|B(X)|}$ is cancelled by the contraction of the irrelevant part of the activity (the group version of the contraction estimate in the appendix for flat fields).
• The remainder after absorbing $\eta$ stays of type $g_k^{2+\delta}$ uniformly in the step. If it is put on the ledger of the transverse contraction of 15-03 §04, the remainder is expected to be bounded by a supremum rather than a sum.

How far in the order the shift of local couplings persists in the smallest system (simultaneous integration of four plaquettes in a row) has not yet been recorded in this book's study; we pose it as a question, as the next check (Table 2).

(a) ① ultraviolet control is not closed. Every statement of this chapter is a one-step statement, a statement set down as an assumption, or a proposal.

(b) Lemma R is only the group version of property 2 of the four properties of the weight, and does not touch property 3, which is the main issue. It may be a known elementary fact.

(c) The group version of properties 3 and 4 and the interchange inequality are only statements and have not been shown. (3) is nothing more than the chain assuming them.

(d) $\eta_k\sim g_kp(g_k)$ and $\eta_k\lesssim g_k^2p(g_k)^2$ are naive estimates, not estimates for the actual measure of the fluctuations.

(e) Lemma H is a statement about a single step in the case where the law of the fluctuation does not depend on the background, and shows nothing about Bałaban's $\mu_{k,W}$, the Wilson action, many steps, or the rarity of large fields. The numbers of Table 1 are a numerical reading with the rough figures $SU(2)$, $n_b=96$.

(f) The fact about the heat-bath method gives its bibliography from memory, and we have not checked the text. Proposals η-1 and η-2 are proposals not yet verified, and $\eta^{\rm res}=O(g^3p^3)$ is a reading of the screening. $SU(N)$ does not have the form of (5).

Dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] PRV PROD   product uniform ⟺ sum finite                            one-loop flow
[02] PRV R      Lemma R, Corollary (2), the Y form                      SU(N)
[03] ASM G3     group version of props 3, 4 and interchange ⟸ [02]      statements only
[04] PRV CHAIN  one-step chain (3) ⟸ [02][03]                           assuming [03]
[05] PRV HAAR   heat-kernel product: fluctuation law background-free
[06] PRV ETA    naive η: the sum diverges ⟸ [01]                        naive estimate
[07] RDG TYPE   the type of η depends on how it is measured ⟸ [06]      single plaquette
[08] PRV H      Lemma H ⟸ [02][05]                                      (H1)–(H3), one step
[09] RDG H3     the numbers for (H3) ⟸ [08]                             SU(2), n_b = 96
[10] CIT HB     one-link conditional law is μ₁^{β|Q|}                   SU(2), bibliography unconfirmed
[11] PRV QUAT   quaternion identity (5) ⟸ [02]                          SU(2)
[12] ASM ABS    proposals η-1, η-2 ⟸ [06][08][10][11]                   proposal
[13] ASM MULTI  three requirements for many steps ⟸ [04][08][12]        statements only

To check it, verify by eye that every number to the right of "⟸" is smaller than the number on the left. There are no cycles. [03], [12] and [13] are assumption rows, and each is a restatement of ① or a question that splits it. Among the nodes of this chapter there is no node saying that ① has been shown.

What this chapter can say

Sources and reproduction

Words used in this chapter — the measure of the fluctuations $\mu_{k,W}$, background dependence $\eta_k$, the four properties of the weight, Lemma R, the group version of properties 3 and 4, the interchange inequality, Lemma H, fluctuations independent of the background, the shift of local couplings, the fact about the heat-bath method, the score, the multi-step induction. Their meanings are in the A-02 glossary.

Next chapter to read: 15-08 The four rows that do not close, and proposals for them — which of the four rows the absorption of $\eta$ in this chapter makes lighter.

← 15-06 Counting the parameters — couplings by loop shape and the attracting classContents15-08 The four rows that do not close, and proposals for them →