One small-field blocking step, part 1b: the decay rate and the shape of the threshold, and why the weak side’s deficit is ten orders of magnitude

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Part 1 (Gaussian level) left one constant unresolved, the exponential decay rate \(\kappa\) of the fluctuation propagator, and part 2 was to produce the Kotecký–Preiss threshold \(g_{\rm pert}^2\) of the remainder gas. This note settles what can be settled about both without new mathematics, and the result reshapes the weak-side programme. The literature carries no explicit rate. Dimock’s exposition of Balaban’s construction (Rev. Math. Phys. 25 (2013) 1330010, §2.4; read from the arXiv text) proves the decay by a parametrix built from Neumann inverses on cubes of side \(M\) and a random-walk expansion that converges “for \(M\) sufficiently large”, with constants “depending on \(L\) but no other parameters”; the local estimate (80) there has a rate \(\gamma_0\) stated only as \(O(L^{-2})\). Explicit routes give \(\kappa\sim10^{-3}\). The generic Combes–Thomas argument gives \(7\times10^{-4}\) for the hard constraint (part 1); the soft-constraint operator \(-\Delta+aQ^TQ\) is local, has gap \(\tfrac29\) for \(a\ge130\), and gives \(1.7\times10^{-3}\); the analytic-structure route through the folded symbol is set up in Section 2 and reduces to a zero-free strip for one explicit function, which is the defined computation for an order-one rate. The threshold scales as \(\kappa^4\). With the tree-graph organization of the remainder gas, \[g_{\rm pert}\ \simeq\ \frac{1}{e\,C_3\,c_{\rm adj}\,\|C_{\rm fl}\|_{1,\kappa}/g^2} \ \simeq\ \frac{\kappa^4}{2\times10^{4}\,C_3},\] with \(C_3\) the cubic-vertex constant, of order \(10\) to \(30\), so that even at the expected order-one rate \(\kappa\simeq\tfrac12\) the threshold is \(g_{\rm pert}^2\sim10^{-12}\), and at the proven rate it is \(10^{-35}\). Against the physical onset of the perturbative running at \(g^2\simeq1\), the weak side’s explicit deficit is twelve orders of magnitude in \(g^2\) with an ideal propagator and more than thirty with the proven one, where the strong side’s is a factor \(400\) in \(\beta_W\). The sub-target H1a of the research-directions note is therefore closed as a constant-chasing problem: no sharpening of the present method reaches \(g^2\sim1\), and H1 is a methods problem. The estimate of the threshold is a scaling estimate with explicit but crude constants, labelled as such; the decay-rate statements are proved. Constants explicit; nothing promoted.

1. What the literature does, and the soft constraint

Dimock’s Green’s function is \(G_k=(-\Delta+\bar\mu_k+a_kQ_k^TQ_k)^{-1}\): the block averaging enters as a soft Gaussian constraint \(aQ^TQ\) rather than the hard conditioning of part 1, and the operator is then local. The decay is proved by covering the lattice with cubes of side \(M\), inverting with Neumann conditions on each, gluing with a partition of unity, and expanding the error in a random walk whose steps carry a factor \(O(M^{-1})\); the walk converges for \(M\) large and the rate is half the local rate. No numerical value is attached to \(M\), \(C\) or \(\gamma_0\) anywhere in the construction, and Balaban’s original (Commun. Math. Phys. 95 (1984) 17) is of the same kind.

The soft constraint can be made explicit here. With \(Q\) the averaging of part 1 (normalized by \(\tfrac1{16}\)), \(QQ^T=\big(48+8(S+S^{-1})\big)/256\) along \(\mu\) has spectrum in \([\tfrac18,\tfrac14]\), so \(\|Qf\|^2\ge\tfrac18\|f_\perp\|^2\) for the component \(f_\perp\) of \(f\) orthogonal to \(K=\ker Q\). Writing \(f=f_K+f_\perp\) and using \(\|\partial f\|^2\ge\tfrac12\|\partial f_K\|^2-\|\partial f_\perp\|^2\) with \(\|\partial f_K\|^2\ge\tfrac49\|f_K\|^2\) (part 1, Proposition 1) and \(\|\partial f_\perp\|^2\le16\|f_\perp\|^2\), \[\big\langle f,(-\Delta+aQ^TQ)f\big\rangle\ \ge\ \frac29\|f_K\|^2+\Big(\frac a8-16\Big)\|f_\perp\|^2 \ \ge\ \frac29\,\|f\|^2\qquad(a\ge130).\] The operator has range \(2\) along \(\mu\) and \(1\) transversally, with \(\|Q^TQ\|\le\tfrac14\). Conjugating by \(e^{\kappa x\cdot e}\) changes \(-\Delta\) by at most \(2\sinh\kappa\) and \(aQ^TQ\) by at most \(\tfrac a4(e^{2\kappa}-1)\), so the conjugated operator stays invertible with norm \(\le9\) when \(2\sinh\kappa+\tfrac{130}4(e^{2\kappa}-1)\le\tfrac19\), that is \[\kappa\ \le\ 1.7\times10^{-3},\qquad \big|(-\Delta+aQ^TQ)^{-1}(x,y)\big|\le9\,e^{-\kappa|x-y|_\infty}.\] This is a proved rate for the soft-constraint scheme, better than the hard-constraint figure by a factor of two and of the same order: the loss is intrinsic to any argument whose only inputs are the gap and the hopping norm, since their ratio is about \(10^{-2}\).

2. The analytic-structure route, set up

For the hard constraint, the fluctuation covariance is diagonal in the coarse momentum \(\bar k\in[-\tfrac\pi2,\tfrac\pi2]^4\) and couples the sixteen folded fine momenta \(k=\bar k+\pi n\), \(n\in\{0,1\}^4\). The tube weight has the product form \[|\hat w(k)|^2=\prod_{\nu\ne\mu}\cos^2\tfrac{k_\nu}2\ \cdot\ \cos^4\tfrac{k_\mu}2, \qquad\hat k^2=\sum_\nu4\sin^2\tfrac{k_\nu}2,\] and folding exchanges \(\cos^2\leftrightarrow\sin^2\) in the flipped components. With \(s(\bar k)=\sum_n|\hat w(\bar k+\pi n)|^2/\hat k^2(\bar k+\pi n)\), the diagonal entry of the fluctuation covariance at the unfolded momentum is \[C_{\rm fl}(\bar k,\bar k)\ \propto\ \frac{\sum_{n\ne0}|\hat w(\bar k+\pi n)|^2/\hat k^2(\bar k+\pi n)} {|\hat w(\bar k)|^2+\hat k^2(\bar k)\sum_{n\ne0}|\hat w(\bar k+\pi n)|^2/\hat k^2(\bar k+\pi n)},\] in which the massless pole \(1/\hat k^2(\bar k)\) has cancelled identically. Three facts fix where singularities can be. (a) For \(\bar k\to\bar k+i\kappa e\) with \(\cosh\kappa<2\), every flipped denominator satisfies \(\operatorname{Re}\hat k^2(\bar k+\pi n)\ge2-2(\cosh\kappa-1)>0\), so only the unfolded term can have a pole, and it has cancelled. (b) \(|\hat w(\bar k)|^2\ge\tfrac1{32}\) throughout the strip, since each factor has modulus at least \(\tfrac12\) there. (c) The remaining singularities are the zeros of the denominator \(D(\bar k)=|\hat w(\bar k)|^2+\hat k^2(\bar k)R(\bar k)\), with \(R\) the flipped sum, analytic and bounded in the strip. The decay rate of the hard-constraint fluctuation propagator is the width of the largest strip \(|\operatorname{Im}\bar k|<\kappa_*\) in which \(D\) has no zero, and \(\kappa_*\) is expected to be of order one. Establishing it is a bounded computation with four real variables and one explicit function, and it is the defined task for an order-one rate; no part of it is done here.

3. The shape of the threshold

Part 2 organizes the remainder \(S_W-S_2\), the Haar correction and the non-abelian averaging as a gas of polymers on plaquettes, integrated against the fluctuation Gaussian. Its structure is fixed by three ingredients, whatever the details.

Vertices. The cubic term of \(N-\operatorname{Re}\operatorname{tr}U_p\) is \(\operatorname{tr}\big(d\theta\,[\theta,\theta]\big)\)-type; with fluctuations of root-mean-square size \(\tfrac32g\) per link (part 1), \(d\xi\) is of size up to \(6g\) and the cubic vertex per plaquette is of size \(\tfrac2{g^2}\cdot6g\cdot\tfrac94g^2\cdot c_{\rm col}\simeq27c_{\rm col}\,g\), so \(C_3g\) with \(C_3\) of order \(10\) to \(30\); quartic and Haar terms are \(O(g^2)\) and subleading.

Covariance sums. Factorizing the Gaussian integral of a product over a set of plaquettes into connected polymers by the Brydges–Kennedy interpolation puts one fluctuation covariance on each edge of a spanning tree, so every polymer of \(n\) plaquettes carries a factor \(\big(\|C_{\rm fl}\|_{1,\kappa}/g^2\big)^{n-1}\) with \[\frac{\|C_{\rm fl}\|_{1,\kappa}}{g^2}=\sup_x\sum_y\frac{|C_{\rm fl}(x,y)|}{g^2} \le\frac92\Big(\coth\frac\kappa2\Big)^4\ \simeq\ \frac{72}{\kappa^4}\qquad(\kappa\ll1),\] using \(|C_{\rm fl}(x,y)|\le\tfrac92g^2e^{-\kappa|x-y|_\infty}\).

Entropy. Plaquette adjacency \(20\) in four dimensions, so connected sets of \(n\) plaquettes through a given one number at most \((20e)^{n-1}\) (Wilson note §2).

The Kotecký–Preiss condition then holds when the per-plaquette activity times \(e\), the adjacency entropy and the covariance sum is below one, \[e\cdot C_3g\cdot20e\cdot\frac{72}{\kappa^4}\ \le\ 1 \qquad\Longleftrightarrow\qquad g\ \le\ g_{\rm pert}\simeq\frac{\kappa^4}{1.1\times10^4\,C_3},\] which is the form quoted in the abstract.

This is a scaling estimate: the constants \(C_3\), the adjacency, and the tree-graph organization are the standard ones and the form is robust, but no polymer expansion has been carried out here, and a full treatment adds factors of order one to ten in either direction. It does not change the conclusion, because the conclusion is about orders of magnitude.

4. The conclusion for the weak side

The physical onset of the perturbative running is at \(g^2\simeq1\), \(\beta_W\simeq6\), where asymptotic scaling is observed. The explicit small-field step, done with the sharpest propagator rate one can hope for and the standard organization, covers \(g^2\lesssim10^{-12}\); with the proven rate, \(10^{-35}\). Every step between \(g^2_{\rm pert}\) and \(g^2\simeq1\) is, in the rigorous sense, a step without a small parameter, and their number is irrelevant since none is controlled. The strong side’s corresponding deficit, from the proved \(\beta_W=0.0135\) to the physical \(5.7\), is a factor \(400\), and its methods are polymer expansions of the same family. The difference is that the strong-side gas has activities \(\beta_W/6\) per plaquette and a local structure, while the weak-side gas has activities \(C_3g\) dressed by covariance sums that scale like \(\kappa^{-4}\) and vertex constants of order ten.

So H1a, the sharpness sub-target, is closed: sharpening constants within the present method cannot bring \(g_{\rm pert}^2\) within twelve orders of magnitude of the physical onset, and the meeting of the two reaches through constant-chasing is not a viable route. H1 is a methods problem, which is the content of the constructive programme extended through the confinement scale: control of the renormalization steps at couplings where the fluctuation is not small compared with the nonlinearity, with no expansion in any parameter. That is the open problem, stated as precisely as this programme can state it.

5. Consequence for STATE

The decay rate is proved at \(10^{-3}\) per lattice unit by two routes and its order-one value is reduced to a zero-free strip for the explicit function \(D\) of Section 2. The threshold of the remainder gas scales as \(\kappa^4\) and lies at \(g^2\sim10^{-12}\) even for an ideal rate, twelve orders below the physical onset of the running. The weak-side sharpness sub-target is closed as not viable; the open problem is a methods problem at order-one coupling, and the map of this programme ends there.