2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerationsthe convexity ceiling, the currency of rarity, candidate rules · Kiichi, Shiori and Rin

Paying with rarity instead of convexity — renormalisation on groups and log-Sobolev

This section is a ten-paragraph summary for readers at [Y3]. The proofs and the conditions of the originals are in §02 onwards.

1. A hierarchical model replaces the lattice by nested boxes so that one step of renormalisation can be written down. For the four-dimensional hierarchical scalar $|\phi|^4$, both the equilibrium properties (the book by Bauerschmidt–Brydges–Slade) and the slowness of relaxation of the stochastic dynamics (Bauerschmidt–Bodineau: the gap vanishes with the same power as for the free field) are theorems Cited. The latter is a statement of the same type as Assumption H.
2. The method of Bauerschmidt–Bodineau takes as its entrance the hypothesis that at each scale "the measure of the fine fluctuations, with the coarse field held fixed, is uniformly convex" Cited. The original itself names convexity at large fields as the main issue in extending the method beyond hierarchical models.
3. The group $SU(2)$ is a closed space (the three-dimensional sphere). On a closed space, imposing convexity bounds the oscillation of the potential by the square of the diameter, and the measure cannot leave the range of a constant multiple of the uniform distribution (Lemma 1 of §03) Proved here. At weak coupling the measure needs to gather near the identity, so the convexity hypothesis cannot hold in principle at weak-coupling scales.
4. For $\phi^4$ on flat space, large fields are convex by nature, and what is not convex is small fields (between the two wells). On the group it is the reverse: near the identity (small fields) is convex, and the far hemisphere (large fields) is necessarily non-convex (§04) Proved here.
5. However, the probability of being on the non-convex side is small, of the order of $e^{-\beta}$ (Proposition 3 of §03) Proved here. The wall does not vanish; it is pushed into a place of small probability.
6. The wall of the first step of Part 10 (that the one-site diagonal at the summit $\tfrac{15}{4}$ falls short of the sum of couplings $4$ by $\tfrac14$) also took the form of paying for the lack of convexity with the rarity of the plateau. The Poincaré inequality is a currency that measures the speed of exit, and what can pay with rarity is the currency of entropy (10-10 Type 4). A step that pays with rarity needs to pass through an inequality of log-Sobolev type.
7. Tools that pay with rarity instead of convexity include Lyapunov conditions (a drift that returns from the bad set) and field-dependent constants. The estimate of the action barrier in Part 10 was an example where the cost of the bistability of the core could be paid by the rarity of the local term (a margin of about 95 times, on the $\hbar=0$ reading).
8. The one-link measure has a single well, and its relaxation constant does not get exponentially worse in $\beta$. A wall of the $\tfrac14$ type of the double well does not come back for one link Estimate.
9. Rewritten in the form of rules, this becomes "the class of measures that pay with rarity" (§07). On the ultraviolet side it seems possible to write a form in which the class closes at each step of coarse-graining, but at the confinement scale the rate of rarity runs out, and a "coupling window" remains between it and the curvature tools of the strong-coupling side. This is a restatement of the second difficulty of the third step (11-07 §06) Estimate.
10. The road of stochastic quantisation (SPDE) uses the same Langevin dynamics, but its aim is the construction of the measure, and the solution theory is subcritical only for $d\lt4$ Cited. It gives tools neither for the four-dimensional H nor for the local H.

Most tools that bound the relaxation constants (the Poincaré constant, the log-Sobolev constant) from above take as their entrance a lower bound on the second derivative of the potential, that is, convexity. We list three.

Bakry–Émery. For a probability measure $\mu=e^{-V}d\mathrm{vol}/Z$ on a Riemannian manifold $M$,

Cited (the single-scale case of Theorem 3.6 of the review by Bauerschmidt–Bodineau–Dagallier). The condition is a lower bound at every point.

Brascamp–Lieb. On $\mathbb R^n$, if $\operatorname{Hess}V\gt0$ then $\operatorname{Var}_\mu(f)\le\mathbb E_\mu\langle\nabla f,(\operatorname{Hess}V)^{-1}\nabla f\rangle$ Cited (the form used by Lemma 2.7 of Bauerschmidt–Bodineau). This too uses convexity at each point.

The hierarchical induction. Bauerschmidt–Bodineau split the Gaussian covariance by scale as $C=C_0+\cdots+C_N$, form the renormalised potential $e^{-V_{j+1}(\phi)}=\mathbb E_{C_j}e^{-V_j(\phi+\zeta)}$ (addition of fields $\phi+\zeta$), and place the following hypothesis Cited (arXiv:1809.02075 §2.3).

Here $Q$ is the projection onto the space of coarse fields, and $\varepsilon_j$ is the notation of the original (different from the lattice spacing $\varepsilon$ in 0+1 dimensions). Under (A1) and two structural hypotheses, the Brascamp–Lieb quadratic form is handed on at each step as $D\le C/(1-\varepsilon)+D_+/(1-\varepsilon)^2$, and the product over all steps is bounded by $\exp\bigl(2\sum_l(\varepsilon_l+O(\varepsilon_l^2))\bigr)$ (Theorem 2.1, Corollary 2.2) Cited. The conclusion is a spectral gap (of Poincaré type), not a log-Sobolev result. The form that flows log-Sobolev over many scales is later work by the same authors (the Polchinski flow, §05). The original writes that "to extend this method beyond hierarchical models, convexity rather than probabilistic estimates is needed, and convexity at large fields is the main issue" Cited.

Three places where it breaks. (a) The summit of the 0+1-dimensional double well. In one row of the Hessian of the action of the core, at the summit the diagonal $4-\tfrac14$ falls short of the sum of the couplings to the neighbours $4$ ((11) of 10-09), and on the face where every site is at the summit, the smallest eigenvalue of a lump of length $k\ge8$ is negative Proved here. Every convexity tool had to pay for this $\tfrac14$ with the rarity of the plateau. (b) The opposite hemisphere of the group. In Proposition 3 of §03 we see that the Bakry–Émery curvature of the one-link Wilson weight is negative on the far hemisphere. (c) (A1) of the hierarchical induction. For group-valued fields, the natural reading of (A1) is "Bakry–Émery for the measure on the fibre with the coarse field held fixed" Estimate, and by Lemma 1 of §03 it cannot hold at weak-coupling scales.

What this section can say: (1)(2) and Theorem 2.1 are copies of the statements of the originals. The reading in (c) (reading (A1) on the group as Bakry–Émery on the fibre) is an estimate, and this book has no exact formulation of the induction on the group.

Proved here Proof. Let $p$ be a maximum point of $V$ ($M$ is compact). For any $q\in M$, take a minimising geodesic $\gamma:[0,L]\to M$ from $p$ to $q$ (unit speed, $L=d(p,q)\le D$) (it exists by the Hopf–Rinow theorem, since a compact manifold is complete). Setting $h(t):=V(\gamma(t))$, since $p$ is a maximum point on a manifold without boundary, $h'(0)=dV_p(\gamma'(0))=0$. Along the geodesic, $h''(t)=\operatorname{Hess}V(\gamma'(t),\gamma'(t))\ge-K$. Integrating gives $h'(t)\ge-Kt$, and integrating once more gives $h(t)\ge h(0)-Kt^2/2$. At $t=L$, $V(q)\ge\max V-KL^2/2\ge\max V-KD^2/2$. Since $q$ is arbitrary, (3) follows. ∎

There is no Lean check (the one-dimensional version "if $h''\ge-K$ and $h'(0)=0$ then $h(0)-h(x)\le Kx^2/2$" is kept as a candidate). The key of the proof is that "the first derivative vanishes at a maximum point", and here the absence of boundary (being closed) is what works. Flat $\mathbb R^n$ has no maximum point, so the same estimate does not follow.

Proved here Proof. Since $\operatorname{Hess}V\ge-\operatorname{Ric}=-2g$, using Lemma 1 with $K=2$ and $D=\pi$ gives $\operatorname{osc}V\le\pi^2$. Since $Z=\int e^{-V}d\mathrm{Haar}\ge e^{-\max V}$, the density is $e^{-V}/Z\le e^{\max V-\min V}\le e^{\pi^2}$. The Haar distribution of class functions is $\frac2\pi\sin^2\theta\,d\theta$ ($\theta\in[0,\pi]$), and from $\sin\theta\le\theta$, $\mathrm{Haar}(\theta\le r)\le\frac2\pi\cdot\frac{r^3}3$. The Wilson weight $V=\beta(1-\cos\theta)$ has $\operatorname{osc}V=2\beta$, so if $2\beta\gt\pi^2$ it contradicts the inequality stated before (4). ∎

How to read (4). At weak coupling, the Wilson measure gathers most of its probability in a ball of radius of order $\beta^{-1/2}$. The right-hand side of (4) gets small like $r^3$, so when $\beta$ is large, a measure with convexity of Bakry–Émery type cannot do that. The content of the lemma is that once convexity is imposed, the measure cannot move further from Haar than a constant factor $e^{\pi^2}$.

Proved here Proof. (i) A function $f$ of $\theta$ alone on the unit sphere has, in geodesic polar coordinates $g=d\theta^2+\sin^2\theta\,g_{S^2}$, $\operatorname{Hess}f=f''\,d\theta^2+f'\cot\theta\,(g-d\theta^2)$ (the radial direction from the second derivative along geodesics, the spherical directions from the second fundamental form $\cot\theta$ of the sphere of constant $\theta$). For $f=\beta(1-\cos\theta)$, $f''=\beta\cos\theta$ and $f'\cot\theta=\beta\sin\theta\cot\theta=\beta\cos\theta$, so $\operatorname{Hess}V_\beta=\beta\cos\theta\,g$. On the unit $S^3$, $\operatorname{Ric}=2g$. The minimum is $2-\beta$ at $\theta=\pi$. (ii) If $\cos\theta\lt-2/\beta\lt0$ then $\theta\gt\pi/2$, and $V_\beta=\beta(1-\cos\theta)\gt\beta+2$. (iii) On $N_\beta$, $e^{-V_\beta}\le e^{-\beta-2}$, and the Haar measure is at most 1, so $\int_{N_\beta}e^{-V_\beta}\le e^{-\beta-2}$. For the denominator, on the ball of $r=\beta^{-1/2}\le1$, $1-\cos\theta\le\theta^2/2$ gives $V_\beta\le\tfrac12$, and for $\theta\le1$, $\sin\theta\ge\theta-\theta^3/6\ge\tfrac56\theta$ gives $\mathrm{Haar}(\theta\le r)\ge\frac2\pi\cdot\frac{25}{36}\cdot\frac{r^3}3=\frac{25}{54\pi}r^3$. Hence $Z_\beta\ge e^{-1/2}\frac{25}{54\pi}\beta^{-3/2}$, and dividing gives (5) ($\frac{54\pi}{25}e^{-3/2}=1.514\cdots$). ∎

(5) is a crude upper bound; near the antipode ($\theta\approx\pi$, $V_\beta\approx2\beta$) the probability is smaller still, of order $e^{-2\beta}$. $\beta\lt2$ in (i) is sharper than the necessary condition $\beta\le\pi^2/2$ that follows from Lemma 1. The role of Lemma 1 is that it is not limited to the Wilson weight: any weight, once convexity is imposed, is confined to the range of (4).

What this section can say: Lemma 1, Corollary 2 and Proposition 3 are statements with proofs placed. What they speak of is the convexity of one measure on one manifold; about the convexity of the conditional measure on the fibre (a measure on the product of many links) they speak only through the estimate of §05.

For $\phi^4$ on flat space (of the type of the one-site $V(\phi)=\lambda(\phi^2-v^2)^2$), $V''(\phi)=4\lambda(3\phi^2-v^2)$ is negative only for $|\phi|\lt v/\sqrt3$. What is not convex is small fields (near the summit between the two wells), and at large fields $V$ grows like $\phi^4$ and is convex. Bauerschmidt–Bodineau wrote "convexity at large fields" as the issue because one needs to show uniformly that the effective potential after renormalisation stays convex at large fields too.

On the group it is the reverse. As in (i) of Proposition 3, near the identity (small fields, $\theta\lt\pi/2$) $\operatorname{Hess}V_\beta\gt0$ and it is close to Gaussian, while on the far hemisphere (large fields) $\operatorname{Hess}V_\beta\lt0$. This is not peculiar to the Wilson weight. A smooth function on a closed manifold has a maximum point, and at a maximum point $\operatorname{Hess}V\le0$. The non-convex place at large fields always exists, for topological reasons Proved here. On the other hand, as in (iii) of Proposition 3, the probability of being there is of order $e^{-\beta}$.

Figure 1  Convex and non-convex places
          (horizontal axis: size of the field; |φ| for φ⁴, angle θ from the identity for the group)

  φ⁴ (flat space):
     |φ| small ――――――――――― |φ| ≈ v ――――――――――――― |φ| large
     non-convex (summit)   bottom of well (convex)   convex (grows like φ⁴)
     → the non-convex place is "small fields".
       Its probability is set by the frequency of crossing between the wells

  SU(2) (closed sphere):
     θ small ――――――――――――― θ = π/2 ――――――――――――― θ = π (antipode)
     convex (near Gaussian)  boundary              non-convex (always, at the maximum)
     → the non-convex place is "large fields".
       Its probability is of order e^(−β) (Proposition 3)

The wall does not vanish; it is pushed into a place of small probability. On flat space, tools that avoid the non-convex summit were conceivable (imposing convexity at large fields, treating small fields separately). On the group the non-convex place cannot be avoided (Lemma 1), but in exchange its probability is exponentially small (Proposition 3). So tools on the group must push with probability, not with convexity Estimate.

An example where the rate of rarity was measured. On the width-2 strip of two-dimensional $U(1)$, the slope of the raw rate of wrapping of a single plaquette (a rare event that crosses a barrier of height $2\beta$) matched the slope $2-1/\beta$ of the Kramers rate $\beta e^{-2\beta}$ to 1–3% ($\beta=3..6$, outer boundary frozen, a post-hoc check, 11-02 §05) Reading. The relation between this number and renormalisation on the group is an estimate.

The smallest example where rarity cannot be paid with local quantities. For the 0+1-dimensional double well, even for the shortest window (a single time) the local ratio is not 1 but exactly $\omega_b/m$ ((7) of 11-03, 11-04 §07) Proved here (Lean LatticeGauge.LocalHDoubleWell.one_slice_ratio, assumptions: $0\lt\varepsilon$, $0\lt m$, $0\lt\omega$. The bundle guarantees only the algebra). The influence of the non-convex summit remains in the ratio however short the window is made. The implication for the group is an estimate Estimate.

What this section can say: that the non-convex place always exists for topological reasons, and the upper bound (5) on its probability, have proofs. "Must be tools that push with probability" is an estimate.

The model. As a four-dimensional hierarchical gauge model, we consider the lattice on which the recursion of the Migdal–Kadanoff approximation holds exactly (Ito 1984 Cited). Setting the couplings of the vertical plaquettes to $0$ or $\infty$ place by place, the recursion of the plaquette weight $g$ can be written down in the form $g\mapsto(g^{*4})^4$ (a convolution gluing four together, and a power stacking four). It is stated that for four-dimensional $SU(N)$ every coupling flows to strong coupling and confines (Ito 1985 Cited (abstract)). This model lacks translation and rotation invariance, and its theorems are not part of the statement of the four-dimensional Yang–Mills problem. That the type of relaxation changes with the choice of coordinates to move was seen in 11-02 §02 (in tree coordinates the gap of the Gaussian approximation is exactly $\beta$, independent of the number of steps; in link coordinates it gets smaller in inverse proportion to the square of the side).

Table 1 carries the parts of the Bauerschmidt–Bodineau induction over, row by row, to this model with values in the group $SU(2)$. "Applies" means the form holds as it is, "changes" means there is a corresponding form but its content is replaced, and "does not apply" means it does not hold; apart from the row of Lemma 1, these are estimates.

In sum. The decomposition of the variance and identities of the type of Lemma 2.8 hold on the group too. What does not hold is the two places that use convexity at each point (Lemmas 2.7 and 2.9), and by Lemma 1 they do not hold at weak-coupling scales however the constants are arranged. The wall of "convexity at large fields" in the original does not vanish on the group but becomes unavoidable, while the probability on the wall drops to $e^{-c\beta}$ Estimate.

The Polchinski flow on the group. Theorem 3.6 of the review by Bauerschmidt–Bodineau–Dagallier, along a flow $C_t$ of Gaussian covariances, derives from the condition

the log-Sobolev constant $2/\gamma$ Cited. $\dot\lambda_t$ may be negative, and non-convex time intervals are paid for as the swelling $e^{-2\lambda_t}$. A version for curved spaces (a form including the flow of the metric and $\operatorname{Ric}$, Theorem 3.20) is also stated, and the original writes that "we currently do not know interesting applications of this generalisation" Cited. It is a candidate home for group-valued fields Estimate. Written on the group, the natural semigroup for this flow is convolution with the heat kernel, $V_t=-\ln(e^{-V}*K_t)$, and what is needed is an estimate of $\inf\operatorname{Hess}V_t$ from below. The estimates of the Migdal–Kadanoff recursion so far (the recursion of the stiffness; the paper record that the number of steps until strong coupling is entered is at most $\beta+4\log\beta+20$ Assumption (no proof placed in this book)) are all estimates from above ($\sup\operatorname{Hess}$), and the quantity that connects the two is the flow of $\inf\operatorname{Hess}V_t$. In weak-coupling time intervals $\inf\operatorname{Hess}V_t\approx-\beta$ (the antipode), and how much $e^{-2\lambda_t}$ swells has not been estimated Estimate. The first number is one step of the one-link weight (question F7, 11-08).

The $\tfrac14$ type does not come back for one link. $e^{(\beta/2)\operatorname{tr}U}d\mathrm{Haar}=e^{\beta\cos\theta}d\mathrm{Haar}$ has a single well ($\theta=\pi$ is the minimum of the density; there is no second local minimum), and the gap is expected to approach the Hessian $\beta$ at the identity for large $\beta$, without getting exponentially worse Estimate (general semiclassical theory. The original on uniform log-Sobolev for Boltzmann measures with a field on spheres is known only bibliographically, and has not been checked). The wall of the $\tfrac14$ type at the summit of the double well (two wells) does not appear for one link. On the fibre, the degenerate family where the coarse value is near $-1$ is expected to produce a polynomial loss (Table 1).

What this section can say: the verdicts of Table 1, apart from the row of Lemma 1, are estimates. The results for hierarchical models have no implication for four-dimensional Yang–Mills under any verdict.

Currency. 10-10 Type 4 of Part 10 said that what the Poincaré inequality measures is the speed of exit, and that what can pay with rarity is the currency of entropy (10-10 Type 4). As a formula, when a set $B$ satisfies $\mu(B)\le e^{-s}$,

Proved here ((3) of 10-09). The ratio of the variance to the Dirichlet form does not change when the function is multiplied by a constant, so if a function supported on a rare set is put in, the rarity cancels between numerator and denominator, and at the Poincaré step rarity has no effect. A step that pays with rarity needs to pass through an inequality of log-Sobolev (entropy) type. This is why the candidate tools on the group are not the Poincaré induction (Brascamp–Lieb at discrete scales) but the type that flows Bakry–Émery at each scale in log-Sobolev form ((6) of §05) Estimate.

The Holley–Stroock perturbation. If $\mu'=e^{-U}\mu/Z$ with $U$ bounded, the Poincaré constant (and the log-Sobolev constant) of $\mu'$ is at most $e^{\operatorname{osc}U}$ times that of $\mu$ Assumption (the standard form as remembered; the original has not been checked in this chapter). It is a tool that pays for the non-convex place with the "oscillation", and the cost is exponential in the oscillation, independent of rarity. Regarding the one-link Wilson weight as a perturbation of Haar, $\operatorname{osc}U=2\beta$, and the loss is $e^{2\beta}$. In Part 10, the plan (R15) to pay only for the summit band with this perturbation put the one-site oscillation $\operatorname{osc}/\hbar=3.7／11／37$ into the exponent, multiplied by the length along the chain, and came out 2000–4900 times outside the allowance (10-10 §03, Type 5) Reading.

Lyapunov conditions. For a generator $L$, if there are $W\ge1$ and a good set $G$ with

(a drift that returns from the bad set to the good set), and a local Poincaré inequality (constant $C_G$) holds on $G$, then the global Poincaré constant is at most $(1+bC_G)/a$ Cited (of the type of Bakry–Barthe–Cattiaux–Guillin 2008. The statement was checked through a restatement in a secondary source, 10-14 §03. The positive control on the one-dimensional Gaussian is row 20 of 10-14 §04). Convexity is used only on the good set $G$, and the bad set is paid for by "the speed of coming back". On the group it is natural to take the good set near the identity and the bad set on the far hemisphere, and the loss is expected to be polynomial Estimate. However, (8) is a Poincaré conclusion, not a form that uses rarity itself. To use rarity, a step that moves the type of (8) into log-Sobolev form (the currency of entropy) is needed Estimate.

Field-dependent constants. Bauerschmidt–Bodineau write in Remark 2.4 that $\varepsilon$ in (A1) may be made to depend on the coarse field, $\varepsilon=\varepsilon(\phi)$ (but uniformly less than 1), and add that it is "hard to use" Cited. It is a direction that allows $\varepsilon$ to be large at bad coarse fields, but as long as it must be uniformly less than 1, it does not get past the ceiling of Lemma 1. To get past it, the condition must be replaced by one in the form of an average weighted by the probability of bad fields Estimate.

An example where rarity paid — the margin of the action barrier. In the first step of Part 10, when the end of the core on the box side of Proposition H4 crossed the summit, a second minimum point appeared on the wall of conditioning, and the Poincaré constant of the core grew like $e^{\Delta S/\hbar}$ (10-11 §06). This is the cost of the non-convex place inside the core. That cost is paid by the rarity $D/(C\xi)^2\le d\,e^{-2S_0/\hbar}$ carried by the local term of Proposition H4 (the rarity of creating a pair of kinks): under the row (ΔScore) $C_P\le c_*\hbar^{-p}e^{\Delta S_*/\hbar}$ and the reading of $\xi$, the local term vanishes like $e^{-(2S_0-\Delta S_*-\delta)/\hbar}$ Proved here ((12) of 10-11. Lean LatticeGauge.CoreActionGap.cloc_term_poly, cloc_term_vanishes, assumptions: (ΔScore), the reading of $\xi$, $\Delta S_*\lt2S_0$). On the $\hbar=0$ reading, $\Delta S^0_*\le1.4\times10^{-2}$, its ratio to $2S_0=4/3$ is at most $1.05\times10^{-2}$, and the margin was about 95 times Reading (Lean LatticeGauge.CoreActionGap.measured_ratio, assumption: the measured upper bound $\Delta S^0\le0.01401$; the conclusion is in the form of the ratio $\le1.06\times10^{-2}$). Rather than removing the non-convex place by convexity, it is paid by a stronger factor of rarity: this is a small real example of the type of this chapter. However, (ΔScore) itself is an assumption row, supported only by the $\hbar=0$ reading and the lower bound on $\hbar\ln C_P$.

What this section can say: (7) and the algebra of the action barrier have proofs; the Lyapunov condition is a type whose statement was checked in a secondary source; Holley–Stroock is the standard form from memory. Which of them works on the group, and with what loss, is an estimate.

Let us try to rewrite what we have seen in this chapter in the form of rules. It is an attempt that follows the phrasing of the question "if new mathematics is to be made, what should be made the rules of the mathematics?", and the rules written here are candidates. Neither consistency nor the ability to restate existing results has been checked. None of them promises a prospect of closing the third step (the continuum limit and the OS axioms) Estimate. Other candidates (axiomatising scale, the algebra of loops, conserved quantities and sectors, an axiom of asymptotic freedom, rules for curvature) and a list of combinations are placed in 11-08 §02.

(Q4) makes the currency type of Part 10 into a rule (10-10 Type 4: Poincaré is the speed of exit, rarity is the currency of entropy).

Candidate first theorems. (a) For one measure: if $\mu\in\mathcal C$, the log-Sobolev constant is at most $F(\kappa_{\rm BE},s,a,b)$, and $F$ is not exponential in $s$. On flat space the type that combines a Lyapunov condition with a local inequality corresponds to this (§06), and the Holley–Stroock perturbation pays $e^{\operatorname{osc}}$. (b) Closure: applying one step of the heat kernel to the one-link Wilson weight of $SU(2)$ stays in the class, with $s'\ge s-O(1)$. This is the first step of question F7 (11-08 §01). For class functions (functions of the single variable, the angle $\theta$ from the identity), convolution with the heat kernel only multiplies the coefficients of the character expansion by exponentials of the Casimir values, so how the lower bound on the curvature on the good set and the weight of the bad set are carried over can be followed by one-variable paper calculation and small numerics Estimate.

Parts that already exist. On flat fields, the weight (regulator) $\tilde G(X,\phi)$ of the appendix of the book by Bauerschmidt–Brydges–Slade has four properties — (i) it factorises over disjoint sets, (ii) $\tilde G(X,\phi+\phi')\le\tilde G^2(X,\phi)\,\tilde G^2(X,\phi')$, (iii) $\mathbb E_+\tilde G^t(X)\le2^{|B(X)|}$, (iv) $\tilde G^t\le\tilde G_+$ — and from them the one-step estimate $\|\mathbb E_+\theta K(X)\|_{\tilde G_+}\le2^{|B(X)|}\|K(X)\|_{\tilde G}$ follows (Lemma A.6.1) Cited. It is a real example of writing the rarity of large fields as a weighted norm and closing it under the expectation of one step of coarse-graining. What is needed on the group is the group version of this "closing in one step", and the place where property (ii) uses addition of fields is replaced by the group product $u_1u_2$. The multiscale Bakry–Émery of Bauerschmidt–Bodineau–Dagallier ((6) of §05, $\dot\lambda_t$ may be negative) and its version for curved spaces (Theorem 3.20) are also candidate parts Cited. The entropy stability lemma (a form that takes an upper bound on the covariance as its condition instead of the pointwise Hessian, Lemma 3.21) uses a linear tilt $e^{(h,\zeta)}$, so it does not carry over directly to the group Estimate.

Combining with one step of coarse-graining — the coupling window. Combining this class with axiomatising scale (a semigroup of coarse-grainings and a flow on it), one can write a form in which the coarse-graining kernel maps the class at each step, $\mathcal C_j=\mathcal C(\kappa_j,s_j,a_j,b_j)$, as $\mathcal C_j\to\mathcal C_{j+1}$. A candidate first theorem is "in a finite box, on the ultraviolet side, as long as the effective coupling $\beta_j$ is at least $\beta_*$, the class closes in the form $s_{j+1}\ge s_j-c$, $\kappa_{j+1}\ge\kappa_j(1-c/\beta_j)$" Estimate. Where it has closed, the effective measure has a uniform log-Sobolev inequality, and from there it can be handed over to the tools of the strong-coupling side (the type that pays with the Ricci curvature of the group; the result of Shen–Zhu–Zhu giving log-Sobolev and the mass gap at small $\beta$ Cited (abstract)). But the rate of rarity $s_j$ is proportional to the effective coupling ($s\approx\beta$ by (5) of Proposition 3), and it runs out at the confinement scale when $\beta_j=O(1)$. The currency of curvature works only in a narrow range. Between the two, a coupling window remains where neither currency is enough. This window is a restatement in the language of rules of the second difficulty of the third step — that there is no expansion connecting the ultraviolet and the infrared — and is not a new obstacle (11-07 §06, 11-07 Table 3) Estimate.

In the hierarchical gauge model (§05), the number of steps before the static flow enters strong coupling is finite, and the three — the ultraviolet side, the window and the strong-coupling side — come together within one model. If the candidate rules are to be tried first, it is on this model. However, the static flow there is a one-variable recursion with no many-body remainder, so crossing the window is easy because of the structure of the model, and the results imply nothing for four-dimensional Yang–Mills Estimate.

Relation to the roots Estimate. Of the three roots of the third step (11-07 §05), the one this class addresses head-on is the third (there is no global gauge fixing, and large fields cannot be controlled by convexity). This is because it places an alternative class where Lemma 1 says "on the group the convexity class is empty". It addresses the first (uniformity in the number of steps) only when combined with axiomatising scale, and the key is whether the recursion of (Q5) gives constants uniform in the number of steps. It does not address the second (connecting the ultraviolet and the infrared) — the coupling window above is the shape of that failure.

What this section can say: the rules are candidates, and the candidate theorems are candidates. The existing parts (the four properties of the weight and Lemma A.6.1, multiscale Bakry–Émery) hold only within the conditions of their originals (flat fields, hierarchical, flows of Gaussian covariances).

Stochastic quantisation is the road that makes the Yang–Mills measure as the long-time limit of a Langevin dynamics (a stochastic partial differential equation) that has it as its invariant measure (6-09). The dynamics is of the same kind as the Langevin dynamics measured by this book's $C_P$, but the aim is the construction of the measure, and the speed of relaxation (H) is not asked about.

The key condition of the solution theory of regularity structures is subcriticality, and it holds only when $d\lt4$ (§1.2 of Chevyrev's review Cited). In two dimensions, a Polish state space on the space of gauge orbits and a Markov process have been made, and the Yang–Mills measure has been shown to be its invariant measure (Chandra–Chevyrev–Hairer–Shen, Chevyrev–Shen Cited (abstract)). In three dimensions, it has reached a new state space, the space of one-forms smoothed by the heat flow, local-in-time solutions and gauge-covariant renormalisation; the three-dimensional Yang–Mills measure is not yet constructed even on the periodic box $\mathbb T^3$ (§1 and §4 of Chevyrev's review Cited).

This road goes not in the direction of extending the lattice renormalisation group, but in the direction of rebuilding where the objects live (the state space) and the means of construction (stochastic dynamics) (11-07 §05) Estimate. However, the condition of subcriticality shows that this new tool too becomes exactly critical in four dimensions. It is one manifestation of the coupling being dimensionless ((c) of 11-07 §01). This road gives tools neither for the four-dimensional H nor for the local H.

What this section can say: the condition of subcriticality and the three-dimensional non-construction are statements in the text of the review; the individual two- and three-dimensional results are copies from abstracts. This book does not use regularity structures as a tool.

(a) Hierarchical models. The results for hierarchical models imply nothing for four-dimensional Yang–Mills (they lack translation and rotation invariance). The verdicts of Table 1 are also estimates within the hierarchical model.

(b) The Polchinski flow on the group. No source formulating the flow by convolution with the heat kernel up to the multiscale Bakry–Émery condition was found within the range searched (this is not "there is none").

(c) The fibre. That the gap in the direction of the degenerate family near coarse value $-1$ drops to $O(1)$, and that long arcs are saddle points, are estimates (by analogy from the Morse index of continuous geodesics).

(d) Dynamics. Statements on the gap or log-Sobolev for Glauber dynamics or Langevin dynamics of hierarchical gauge models were not found within the range searched.

(e) The kind of the original. The induction of Bauerschmidt–Bodineau 2019 is a result of Poincaré type (a spectral gap), not a log-Sobolev result. The form that flows log-Sobolev is later work by the same authors and a theorem of the review.

(f) Candidate rules. The class of §07 is a candidate, and none of consistency, restating existing results, or the candidate theorems has been checked. "The coupling window is a restatement of the second root" is an estimate, and the width of the window has not been measured in any model.

(g) The example of the action barrier. The margin of 95 times is a $\hbar=0$ reading in 0+1 dimensions, and the row (ΔScore) is an assumption row. We say nothing about whether renormalisation on the group has a margin of the same form.

(h) Stochastic quantisation. That four dimensions is critical is not a proof that tools cannot be made in four dimensions. It only copies the statement of the conditions of the original.

What the machine-checking column guarantees is only real algebra and arithmetic; that $\Delta S$ is the action barrier of the core, and that $C_P$ is the Poincaré constant of the path measure, are outside Lean (10-13). Lemma 1, Corollary 2 and Proposition 3 have no machine checking.

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

[01] CIT BE     Bakry–Émery: lower bound on curvature ⇒ log-Sobolev, Poincaré
[02] CIT BB     (A1), Theorem 2.1, Corollary 2.2 of the hierarchical induction   scalar, hierarchical
[03] PRV TOP    one-site diagonal at the summit 15/4 and sum of couplings 4     0+1 (10-09)
[04] PRV L1     Lemma 1: osc V ≤ KD²/2                                         closed manifold
[05] PRV C2     Corollary 2: if convex on SU(2), μ ≤ e^(π²) Haar ⟸ [04]
[06] PRV P3     Proposition 3: one-link curvature (2 + β cos θ), probability (5)
[07] PRV MAX    Hess ≤ 0 at a maximum point of a closed manifold
[08] RDG KR     raw rate on the strip matches the Kramers slope to 1–3%        two-dim. U(1), post hoc
[09] PRV ONE    0+1 one-site ratio ω_b/m                                       Lean one_slice_ratio
[10] EST A1G    (A1) on the group does not hold at weak coupling ⟸ [02][04]
[11] EST TAB    verdicts of Table 1 ⟸ [10]
[12] CIT BBD    multiscale Bakry–Émery, version for curved spaces
[13] ASM MK     number of steps until strong coupling (paper record)
[14] EST NOQ    the 1/4 type does not come back for one link ⟸ [06]
[15] PRV CUR    currency inequality (7)                                        (3) of 10-09
[16] ASM HS     Holley–Stroock cost e^osc                                      from memory
[17] CIT LY     Lyapunov condition (8)                                         secondary source
[18] CIT EPS    field-dependent ε (uniformly less than 1)
[19] ASM DSC    row (ΔScore)                                                   0+1
[20] RDG DSV    ΔS⁰ ≤ 1.4·10⁻², margin about 95 times                          ħ = 0 reading
[21] PRV CLOC   the local term vanishes (12) ⟸ [19]                            Lean cloc_term_poly
[22] EST PROB   tools on the group push with probability ⟸ [05][06][07][15]
[23] CIT BBS    four properties of the weight, Lemma A.6.1                     flat fields
[24] EST CLASS  class of measures that pay with rarity (candidate) ⟸ [04][15][17][23]
[25] CIT SZZ    log-Sobolev and gap at strong coupling                         abstract
[26] EST WIN    the coupling window restates the second root ⟸ [06][24][25]
[27] CIT SPDE   subcriticality d < 4, three dims also not yet constructed
[28] EST NOSQ   stochastic quantisation gives no tools for the 4D H ⟸ [27]

The way to check is to see by eye that on every "⟸" the numbers on the right are smaller than the number on the left. There are no circles. The nodes with proofs placed are [04]–[07], all statements about one measure on one manifold. The nodes on the conclusion side of this chapter, [22], [24], [26], [28], are all Estimate, and there is neither a node for the mass gap nor a node for Assumption H. [21] (an example where rarity paid) is algebra on top of the assumption row [19], and does not enter the premises of [22] onwards. This is because the 0+1-dimensional example is an example of the type, not grounds for a statement on the group.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the convexity ceiling, the lower bound on the Bakry–Émery curvature $\kappa_{\rm BE}$, Brascamp–Lieb, fibre, the reversal of convex and non-convex places, the rate of rarity $s$, currency (entropy), the Holley–Stroock perturbation, Lyapunov conditions, field-dependent constants, the Polchinski flow on the group, the class of measures that pay with rarity, the coupling window, subcriticality. Their meanings are in A-02 Glossary and 11-A1.

Next chapter to read: 11-07 Because it is four dimensions — what works in four dimensions, limited to two points: the dimensionless coupling and topology.

← 11-05 Making time continuous — the Hamiltonian lattice and the vacuum measureContents11-07 Because it is four dimensions — the dimensionless coupling and topology →