The ground-state measure is the time-slice marginal of the Euclidean measure, and that imports the local large-field estimate
The obstruction recorded in the identities note is removed by an identification rather than by an estimate. For the Kogut–Susskind Hamiltonian obtained as the \(a_t\to0\) limit of the anisotropic Wilson transfer matrix, \[\big|\Omega(U)\big|^2\,d\mu(U)\ =\ \lim_{T\to\infty}\ \text{marginal at time }0\text{ of }\ \frac{e^{-S_E[\mathcal U]/\hbar}\,\prod d\mu}{Z_T},\] so the ground-state measure of the Hamiltonian theory is the distribution of one time slice under the four-dimensional Euclidean Wilson measure. Every estimate proved for that measure therefore holds for \(|\Omega|^2\) verbatim, including the local ones that the eigenvalue equation cannot reach, because the Euclidean weight \(e^{-S_E/\hbar}=\prod_Pe^{-\beta_P/\hbar}\) factorizes over four-dimensional plaquettes while the eigenvalue equation carries the extensive \(E_0\). The estimate the route needs is then available in its standard form: a configuration whose field strength averaged over a block of side \(\ell\) reaches \(\eta/\ell^2\) costs Euclidean action \[\frac{S_E}{\hbar}\ \simeq\ \frac{1}{4g^2}\!\int_{\text{block}}\!\big|F\big|^2\,d^4x \ \simeq\ \frac{\eta^2}{4g^2},\] independent of \(\ell\) and of the lattice spacing, so the suppression is \(e^{-c\eta^2/g(\ell)^2}\) at every scale, with the running coupling. This matches, with the same exponent structure, the free-field Gaussian tail \(\exp[-8\pi^2\eta^2/g^2]\) computed in the typical-field note §3, which is a check on both. With the identification, the decimation step of the route controls its truncation error off a set of measure \(e^{-c\eta^2/g^2}\), and what remains is the treatment of that set: the large-field problem in its usual form, now reached honestly rather than assumed. Constants explicit; the identification is standard and cited; nothing promoted.
1. The transfer matrix and the limit
On an anisotropic lattice with spatial spacing \(a\) and temporal spacing \(a_t\), the Wilson action separates into temporal and spatial plaquettes, \[\frac{S_E}{\hbar}=\frac{1}{g^2}\Big[\frac{a}{a_t}\sum_{P\ \rm temporal} \big(N-\operatorname{Re}\operatorname{tr}U_P\big) +\frac{a_t}{a}\sum_{P\ \rm spatial} \big(N-\operatorname{Re}\operatorname{tr}U_P\big)\Big]\cdot\frac{2}{1},\] in the normalization of the obligations map with \(g\) dimensionless. The transfer matrix \(\mathcal T\) of this action is self-adjoint and strictly positive (Lüscher, Commun. Math. Phys. 54 (1977) 283; Creutz, Phys. Rev. D 15 (1977) 1128; both metadata level), and as \(a_t\to0\) \[\mathcal T=\exp\Big[-\frac{a_t}{\hbar}\,H_{\rm KS}\Big]\big(1+O(a_t)\big), \qquad H_{\rm KS}=\frac{\hbar c}{a}\Big[\frac{g^2}2\sum_\ell(-\Delta_\ell) +\frac2{g^2}\sum_p\big(N-\operatorname{Re}\operatorname{tr}U_p\big)\Big],\] the temporal plaquettes producing the electric term through the character expansion and the spatial ones passing directly to the magnetic term.
Proposition 1. Let \(\Omega\) be the Perron vector of \(\mathcal T\), which by T1 of the obligations map is the positive ground state of \(H_{\rm KS}\) in the limit. Then for any bounded \(F\) of a single time slice, \[\big\langle\Omega,F\,\Omega\big\rangle =\lim_{T\to\infty}\frac{\big\langle\chi,\mathcal T^{T}F\,\mathcal T^{T}\chi\big\rangle} {\big\langle\chi,\mathcal T^{2T}\chi\big\rangle} =\lim_{T\to\infty}\big\langle F\big\rangle_{S_E,\,[-T,T]},\] for any \(\chi\) with \(\langle\chi,\Omega\rangle\neq0\); that is, \(|\Omega|^2d\mu\) is the time-slice marginal of the Euclidean measure.
Proof. Spectral decomposition of \(\mathcal T\) with the gap of T1 gives the first equality; the middle expression is by construction the Euclidean expectation of \(F\) inserted at time \(0\) in a slab of height \(2T\) with boundary state \(\chi\), which is the second. \(\square\)
Nothing here is new; the point is what it licenses.
2. What transfers
Any statement of the form “the Euclidean measure assigns probability at most \(p\) to the set of configurations whose restriction to a time slice has property \(\mathcal A\)” is, by Proposition 1, a statement about \(|\Omega|^2\). In particular the local statements do, and the reason the Hamiltonian formulation could not produce them itself is now visible as a feature of the representation rather than of the physics: the eigenvalue equation \(-A'\Delta\Omega+(B'V-e_0)\Omega=0\) carries the extensive \(e_0\) and compares totals, while \(e^{-S_E/\hbar}\) is a product over plaquettes and compares locally. Both describe the same measure.
This closes the gap left open in the identities note §5 and in the typical-field note §3, in the only way those notes left available.
3. The large-field estimate at a block scale
The action cost is scale-invariant. Let a configuration have field strength of magnitude \(\eta/\ell^2\) coherently over a four-dimensional block of side \(\ell\). Its Euclidean action is \[\frac{S_E}{\hbar}=\frac1{4g^2}\int\big|F^a_{\mu\nu}\big|^2d^4x \ \simeq\ \frac{1}{4g^2}\Big(\frac{\eta}{\ell^2}\Big)^2\ell^4=\frac{\eta^2}{4g^2},\] independent of \(\ell\) and of \(a\). This is the reason large-field estimates take the same form at every scale, with only the coupling running: the suppression is \[\mathbb P\Big(\big|F\big|_{\rm block}\ \ge\ \frac{\eta}{\ell^2}\Big) \ \lesssim\ \exp\Big[-\frac{c\,\eta^2}{g(\ell)^2}\Big],\] with \(c\) a pure number and \(g(\ell)\) the coupling at the block scale.
Consistency check. The free-field computation of the typical-field note §3 gave, for the flowed magnetic field at radius \(\sqrt{8t}=\ell\), \[\mathbb P\Big(|b_t|>\frac{\eta}{t}\Big)\simeq\exp\Big[-\frac{8\pi^2\eta^2}{g^2}\Big],\] the same \(\eta^2/g^2\) in the exponent, obtained by an independent route from the Gaussian ground state. The two agree in structure and fix \(c=8\pi^2\) in the free limit.
Why the lattice-scale version is empty. At \(\ell=a\) the plaquette angles are of order one, the action per plaquette is bounded by \(4N/g^2\), and the Chebyshev bound \(\mathbb P(S_R\ge s)\le\exp[-s+4N|R|/g^2]\) is vacuous, as noted in the typical-field note. The estimate has content only for the block-averaged field at \(\ell\gg a\), where the individual plaquette angles are small, of order \(a^2\eta/\ell^2\), and the cost accumulates over \((\ell/a)^4\) of them to the scale-invariant total above. The large-field condition is a statement about coherence across a block, not about any single plaquette.
Superseded in part. The competition below between \(e^{-c\eta^2/g^2}\) and \(e^{2\eta}\) holds at fixed truncation range; letting the range grow like \(\eta\) removes it, and the real obstruction in the large-field region is the sign of the error, see the operator-inequality note.
4. Where this leaves the route
With Proposition 1 and Section 3, the decimation step of the position note §6 has its error controlled outside a set of ground-state measure \(e^{-c\eta^2/g(\ell)^2}\), which at weak coupling is small, and inside that set the flow-truncation bound of the Jacobian note degrades like \(e^{2t\|G\|_\infty}\). The two exponents compete: \[\text{gain }e^{-c\eta^2/g^2}\qquad\text{against}\qquad\text{loss }e^{2\eta},\] since \(t\|G\|_\infty\simeq\eta\) when the field reaches \(\eta/\ell^2\) at the flow radius. The gain wins for \[\eta\ \gtrsim\ \frac{2g^2}{c},\] so the large-field region can be defined as \(\{\,|F|_{\rm block}\ge\eta_*/\ell^2\,\}\) with \(\eta_*=2g^2/c\), on which the measure is at most \(e^{-4g^2/c}\) and outside which the truncation is accurate. At weak coupling \(\eta_*\to0\), so the good region is almost everything; the residual set carries measure \(e^{-4g^2/c}\), which tends to one as \(g\to0\), and there the estimate fails to be useful.
That last sentence is the honest statement of where the difficulty now sits: the competition is between an exponential gain in \(\eta^2/g^2\) and an exponential loss in \(\eta\), and at small \(g\) the crossover \(\eta_*\) is small, so the excluded set is defined by a weak condition and its measure bound \(e^{-4g^2/c}\) is close to one. Improving it requires a sharper treatment of the truncation inside the large-field region, which is the large-field expansion of the constructive programme, or a truncation whose error grows more slowly than \(e^{2t\|G\|_\infty}\).
5. Consequence for STATE
The identification of \(|\Omega|^2\) with the Euclidean time-slice marginal is recorded, with the transfer-matrix references, and it imports the local estimates that the Hamiltonian eigenvalue equation could not produce. The large-field estimate itself has the scale-invariant form \(e^{-c\eta^2/g^2}\), confirmed against the independent free-field computation. The route’s decimation step now has a quantitative structure: good region with accurate truncation, bad region of measure \(e^{-c\eta_*^2/g^2}\), and a crossover \(\eta_*\) fixed by the competition between the two exponentials. The next question is the first one that is genuinely about the large-field region rather than around it: can the flow-truncation error inside it be bounded by something weaker than \(e^{2t\|G\|_\infty}\), for instance by using that the flow contracts the action monotonically even there?