2026-10-01 · chapter Yang–Mills and the mass gapPart 5 Lattice gauge theoryinfinite-volume limit

The infinite-volume limit and the cluster expansion — at small β the expansion in lumps converges independently of the box, and the limit and a lattice gap follow

Lattice gauge theory is first defined on a box $\Lambda_L$ of side $L$. The weight of the Wilson action (5-01) together with the Haar measure is an ordinary probability measure on a compact space.

The mass gap is a statement about distances stretched to infinity (4-11 §02), so we consider the limit of larger and larger boxes. We call a continuous function determined by finitely many links a local observable. When a probability measure $\mu$ on the whole lattice satisfies

we call $\mu$ an infinite-volume limit. It is weak convergence of measures. Chatterjee (2018) §3 writes “a weak limit of the above probability measures as Λ ↑ Zⁿ” Known.

Example: in two dimensions with free boundary, fixing the gauge along a tree makes the plaquette variables independent, so in every box $\langle W_{R\times T}\rangle=u^{RT}$ (5-02 §04), and the limit is trivial Known.

Existence comes almost free from compactness. $\langle F\rangle_L$ is a bounded sequence, so it has a convergent subsequence, and by a diagonal argument it can be made to converge simultaneously for countably many functions that approximate the local observables Known.

The hard part is uniqueness. Chatterjee (2018) §3 writes “The infinite volume limit may or may not be unique. Indeed, the uniqueness (or non-uniqueness) is in general unknown for lattice gauge theories in dimensions higher than two when β is large.” Known. We look at an example where it fails in the two-dimensional Ising model (2-09). We fix the outer boundary of an $L\times L$ box to $\pm1$ and measure the mean of the central spin.

The right-hand side is the exact formula for the spontaneous magnetisation Known. At $\beta\gt\beta_c$ a different limit appears for each boundary condition (a phase transition; 2-08).

ComputedThe numbers in this figure were computed on this machine (column transfer matrix). If JavaScript is disabled, Table 1 holds the content.

The distance between the centre and the outer boundary is $(L+1)/2$, and at $\beta=0.2$, $\ln(\langle\sigma_0\rangle_{13}/\langle\sigma_0\rangle_{15})=1.219$ is almost the same as the decay rate of correlations $2(K^*-K)=1.223$ (2-09 §06) Computed. The influence of the boundary dies away as fast as correlations do. The tool that shows decay of correlations in gauge theory is the expansion in lumps from §04 on.

At small $\beta$ the weight of each plaquette is almost constant. We split the weight as "constant × (1 + small deviation)" and expand the product.

We split the set $X$ of plaquettes that pick up the deviation into connected parts by the rule they are connected if they share a link. The integrals over parts that share no link are independent, so they factorise.

$|\mathcal P|$ is the number of plaquettes in the box. This is exactly the polymer model of 6-04 §03 Known. Two facts about the activities matter.

The right-hand statement holds because that link appears in only one $f(AU_\ell B)$, and by the invariance of the Haar measure $\int f(AU_\ell B)\,dU_\ell=\int f=0$. What survives are only closed lumps, in which every link is surrounded by two or more plaquettes. The smallest is the surface of a unit cube (6 plaquettes).

Example: for $\mathbb Z_2$, $f=\tanh\beta\cdot U$, and the activity of the cube is $\tanh^6\beta$. For $SU(2)$ we expand each face as $f=\sum_{j\ne0}d_j\,u_j\,\chi_j$ by (3) of 5-02 and integrate by (5) of 5-02; only the terms in which every face carries the same representation $j$ survive. For a closed surface the rule is known that its contribution is $d_j^{\,F-E+V}u_j^{\,F}$ ($F,E,V$ are the numbers of faces, edges and vertices, $F-E+V$ is the Euler characteristic of the surface, and the surface of a cube is a sphere, so $6-12+8=2$) Known. For the cube this can be checked as follows. Gluing adjacent faces 5 times by the formula on the right of (5) ($1/d_j$ each time) turns the 6 faces into a single disc. In the product around its rim, the remaining 7 links cancel in order, starting with those adjacent as in $U_\ell\,U_\ell^{-1}$, and the identity matrix remains. Hence $d_j^{\,6}\cdot d_j^{-5}\cdot\chi_j(1)=d_j^{\,2}$.

At $\beta=1$, $7.682\times10^{-4}$, and direct Monte Carlo integration gives $7.63(8)\times10^{-4}$ Computed.

We fix four words (for details, 6-04 §03 and §04). We call the $X$ of (6) polymers (lumps). Two lumps that share a link are called incompatible. A sequence of lumps joined into one by the incompatibility relation (the same lump may appear any number of times) is called a cluster. The logarithm of the left-hand side of (6) can be rewritten as a sum of terms, one for each cluster, and the question is whether that sum converges absolutely independently of the box Known.

Convergence is a contest between how fast the number of lumps grows and the weight $\varepsilon^n$. Let $c_n$ be the number of sets of $n$ plaquettes, connected through links, containing a root plaquette. A plaquette has 4 links and a link is contained in $2(d-1)$ plaquettes, so the plaquettes that share a link with it number $D=4(2(d-1)-1)=4(2d-3)$ (20 in four dimensions). There is the upper bound of Lemma 2.1(c) of Borgs–Chayes–Kahn–Lovász (2010) Known.

The ratio is about 26 at $n=6$, and the growth rate 53 of the upper bound is about twice that. There are no closed lumps for $n\le5$, and the 4 at $n=6$ are four cubes Computed. To guarantee convergence with this way of counting, $\varepsilon$ must be smaller than the reciprocal of the growth rate of lumps (a value expected to be smaller than $1/26$ from the ratios in Table 2, and larger than $1/53$ from (9)).

As a metaphor, for each plaquette it is a contest between a cost of $\ln(1/\varepsilon)$ and a gain of about $\ln 26$ from the number of choices. Precisely, if the gain wins, the expansion merely becomes unusable; a phase transition need not occur (6-04 §10).

The Kotecký–Preiss condition turns the contest into a theorem (6-04 §06). In the form of a theorem: if there is some $a\gt0$ such that the left-hand inequality of (10) holds for every lump $X$, then the sum over clusters for the logarithm in (6) converges absolutely independently of the box Known. One reads it as giving each lump a budget $a|X|$, and requiring the total weight of its incompatible partners (multiplied by $e^{a|X'|}$) to fit within the budget.

The right-hand side is the count that $X$ has at most $4|X|$ links, and that a lump containing a given link contains one of its $2(d-1)$ plaquettes. $c'_n$ is the number of closed lumps (Table 2 for $n\le6$, (9) for $n\ge7$). Maximising $\varepsilon=x\,e^{-8(d-1)G(x)}$ over $x=\varepsilon e^a$ gives the upper limit $\varepsilon_{\max}$ of the guarantee, and solving $\varepsilon(\beta)=\varepsilon_{\max}$ gives the range of $\beta$.

ComputedThe numbers in this figure were computed on this machine. If JavaScript is disabled, Table 3 holds the content.

What is guaranteed is about $\beta\lt0.016$, and since $\varepsilon\approx\beta$ it hardly depends on the group. This is two orders of magnitude narrower than $\beta\le1$, where the strong-coupling formula agreed with the numbers in 5-02 Table 2. The sources are the supremum norm (for $SU(2)$ at $\beta=0.1$, $\varepsilon=0.104$, while $u=0.025$), the crudeness of (9), and ignoring cancellations Computed. Table 3 is not "where the expansion stops".

When (10) holds independently of the box, the following come out of 6-04 §08 Known.

(a) Analyticity. The limit of $\ln Z_L$ per plaquette is an analytic function of $\beta$ (it converges even when extended to complex $\beta$), and there is no phase transition in that range. (b) Existence of the limit. $\langle F\rangle_L$ converges without taking a subsequence (the influence of the boundary dies away exponentially, as in Table 1). What can be said is the existence of the limit along the sequence with that boundary condition; uniqueness independent of the boundary condition relies on the road of §08. (c) Exponential clustering. In the connected correlation of two local observables only clusters touching both supports survive, and joining a distance $r$ needs about $r$ plaquettes or more.

Example: the smallest lump joining two plaquettes is a tube with $4t$ side plaquettes having those two as its ends (a closed surface together with the ends), and the leading term is $m_0=-4\ln u$ (5-02 §05). (11) justifies this independently of the box.

(d) The lattice gap. We normalise the transfer matrix $T$ (5-03 §05) so that its eigenvalue on the vacuum $\Omega$ is 1. For the state $F\Omega$ made by a gauge-invariant local observable $F$, with its vacuum component subtracted, (11) in the time direction says that $\langle F\Omega,T^tF\Omega\rangle$ decreases at the rate $e^{-mt}$ for every $F$. Such states are dense in the space of all gauge-invariant states orthogonal to the vacuum, so by the spectral theorem the spectrum of $T$ there lies in $[0,e^{-m}]$ (4-11 §02). That is, $\|T\|\le e^{-m}$ on the gauge-invariant states orthogonal to the vacuum, and the gap of $-\ln T$ is at least $m$.

In the literature, Shen–Zhu–Zhu (2023) §1 summarises the content of Osterwalder–Seiler (1978) as “the existence of a mass gap (exponential clustering) follows from convergence of the cluster expansion, so do existence of the infinite volume limit and analyticity of Schwinger functions in the inverse coupling.”, and writes of uniqueness only weakly, that it “should also follow”. Chatterjee (2018) §6 writes that “quark confinement and the existence of a positive self-adjoint transfer matrix” were shown Known. The abstract of the original paper also says “We then prove existence and analyticity of the infinite volume limit of strongly coupled Yang-Mills theories on the lattice and we verify Wilson's confinement bound.” (we have not read the text).

There is also a road that gets there without counting lumps. We prepare three words.

Three words — Langevin dynamics, Ricci curvature, the Hessian

Langevin dynamics is a dynamics that moves the link variables randomly in the form $dQ=-\nabla S_W(Q)\,dt+\sqrt2\,dB$. $\nabla$ is the gradient on the group of the link variables, and $B$ is Brownian motion on the group (an independent fluctuation on each link). It is built so that the balance between the force down the slope and the fluctuations leaves the measure (1) unchanged (has it as invariant measure) (6-09). Ricci curvature is how curved the group is, seen as a curved surface. Where it is positively curved like a sphere, geodesics leaving nearby points gather together. For $SU(N)$ it is the same positive constant everywhere ($N/2$ in the normalisation of the inner product of Shen–Zhu–Zhu). The Hessian is the second derivative of the action $S_W$ taken on the group.

The Poincaré inequality and the Bakry–Émery criterion

The key is the Poincaré inequality (1-10).

That $C_P$ does not depend on the box is what "a uniform estimate" means. In (9) of 1-10, in one dimension $V''\ge\kappa\gt0$ gave $C_P\le1/\kappa$. On a group the curvature adds to this $\kappa$, and the Bakry–Émery criterion takes the following form Known.

Assumption 1.1 of Shen–Zhu–Zhu is $K_S=N/2-8N|\beta_{\mathrm{SZZ}}|(d-1)\gt0$ with $SU(N)$. The term on the right comes from the estimate that the Hessian does not fall below $-8N|\beta_{\mathrm{SZZ}}|(d-1)$. The point is that the Hessian stays of order $|\beta|N$ (an example with one link is in 5-11). As a metaphor, it is the state in which the force pulling back to the bottom of a round bowl is stronger than the bumps of the action. Precisely, what is compared is the curvature $N/2$ with the size $8N|\beta_{\mathrm{SZZ}}|(d-1)$ by which the Hessian can become negative.

The theorem of Shen–Zhu–Zhu

Shen–Zhu–Zhu (2023), under $|\beta_{\mathrm{SZZ}}|\lt1/(16(d-1))$ with $SU(N)$ ($\beta=N^2\beta_{\mathrm{SZZ}}$; conversion in 5-01 Table 4), write in the abstract “The finite volume Yang–Mills measures converge to this unique invariant measure in the infinite volume limit” and “the infinite volume measure has a strictly positive mass gap”. Theorem 1.2 shows that the invariant measure of the Langevin dynamics is unique (1), and that the limit of the measures on periodic boxes agrees with it independently of the subsequence (2). Since the group is compact, the sequence of measures on boxes is always tight, and the “every tight limit” of (2) may be read as a subsequential limit in the sense of §02. Remark 1.3 states that replacing the periodic boundary by Dirichlet or other boundary conditions gives the same invariant measure in the limit. It is one answer, in the strong-coupling range, to the question of §02 Known. For four-dimensional $SU(2)$ this is $\beta\lt1/12$.

The meaning of "infinite-volume limit" changes between the top and the bottom of the table. The limit in this chapter is at fixed lattice spacing. In the continuum limit, $a\to0$ while the box in lattice units also goes to infinity, and the gap in lattice units should close as $m(\beta)\to0$ (5-09) Physics. What is needed is a gap uniform in physical units, and there is no small $\varepsilon$ there. The problem statement §6 writes “no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume. Nor do present methods suggest how to obtain the existence of the infinite volume limit T⁴ → R⁴.” Known.

Also, analyticity says nothing outside the small disc of complex $\beta$ guaranteed by (10). For four-dimensional $U(1)$ the area law fails at large $\beta$ (Guth 1980; according to Chatterjee (2018) §4, the fully rigorous proof is Fröhlich–Spencer 1982; 5-06), while that non-abelian groups have no critical point at finite $\beta$ (the correlation length diverges only as $\beta\to\infty$) is at the level of Chatterjee (2018) §5, “It is believed that … many of the non-Abelian lattice models of interest have βc = ∞.” Physics (6-10).

What this chapter can say

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Next chapter to read: 5-05 Weak coupling and the continuum limit (the opposite shore). For the general theory, 6-04.

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