2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theorycluster expansion

The cluster expansion — a tool that splits into weakly linked small lumps and rewrites the logarithm as a convergent sum

What stages 4 and 6 need are estimates independent of the size of the box, and quantities far apart becoming nearly independent (formula (9) of 6-01). The problem statement says that because perturbation series usually diverge, expansion methods collectively called "cluster expansions" were developed, and writes 'Refined estimates yield the rate of exponential decay of Green's functions, magnitude of masses, ...' (JW §6.1). Known

The core is a single point. If the blocks are independent of each other, the partition function is a product and the logarithm is a sum.

If it is a sum, quantities per unit volume do not depend on the box, and distant blocks have no effect. Real blocks are weakly linked, and the cluster expansion adds those links in as corrections for each "set of connected blocks" and shows that the sum converges. As a metaphor, it is a thin gas that hardly ever collides. More precisely, two things are needed: (i) the correction linking $n$ of them is of the order of a small number to the power $n$, and (ii) the number of ways of linking, divided by $n!$, is of exponential order (§05).

Example: if each block is independently "empty" or "one particle (weight $z$)", then $Z=(1+z)^N$. $\log Z/N=z-\tfrac{z^2}{2}+\cdots$ converges for $|z|\lt1$ and stops at $z=-1$, where $Z=0$. The reach of the series is decided by where the partition function vanishes for complex $z$.

This example is used throughout the chapter. Place particles at the points $x=1,\dots,N$ ($n_x\in\{0,1\}$), forbid neighbours from being occupied together (hard-core repulsion), and give each particle the weight $z$ (the activity).

If points $N$ and $1$ are also regarded as neighbours to make a ring, one can write $\Xi_N=\operatorname{tr}T^N$ with the transfer matrix $T=\begin{pmatrix}1&\sqrt z\\ \sqrt z&0\end{pmatrix}$ (transfer matrices are in 2-08 §03; the example of the Ising model is in 2-09, and lattice gauge theory in 5-03). The entries are $T_{nn'}=z^{(n+n')/2}$ ($n,n'\in\{0,1\}$; only $n=n'=1$ gives 0). So in $\operatorname{tr}T^N=\sum_{n_1,\dots,n_N}\prod_{x=1}^{N}T_{n_xn_{x+1}}$ ($n_{N+1}=n_1$), the weight of each $n_x$ is picked up twice, half each time, giving $z^{n_x}$, and simultaneous occupation of neighbours vanishes with 0. This is exactly formula (2) on the ring. Of the eigenvalues $\lambda_\pm=\tfrac12(1\pm\sqrt{1+4z})$, the larger survives. We call the following $p(z)$ the pressure (the physical pressure times the inverse temperature, the $\log$ partition function per unit volume).

ComputedTable 1 The coefficients $b_n$ (agree with the closed form up to $n\le24$ by series operations with rational numbers).

The ratio approaches 4, and the radius of convergence is $1/4$. The radius of convergence of a power series is the distance to the nearest point to the origin at which the function stops being analytic in the complex plane (a theorem of complex analysis) Known. Here that point is $z=-1/4$, where the inside of the square root vanishes and the eigenvalues coincide. There is no phase transition for $z\gt0$, and yet the series cannot go beyond $1/4$. The reach of the expansion is decided by a singularity on the side of non-physical negative activity. The zero of $\Xi$ for a chain of length $n$ nearest to the origin is $-1/(4\cos^2\frac{\pi}{n+2})$ (Scott–Sokal 2005, Ex. 3.2), which for $n=5,10,20$ is $-0.3080,-0.2679,-0.2552$. Computed

We extract the structure. Prepare a collection $\mathcal P$ of finite lumps (polymers) $\gamma$, a relation of being incompatible, $\gamma\not\sim\gamma'$ (overlapping, touching and so on; each is incompatible with itself), and complex activities $z_\gamma$, and count only the sets in which every two are compatible.

$\Lambda$ is a finite collection, and $\mathbf 1\{\cdot\}$ is 1 if the condition is true and 0 if false. This formalism is said to have been introduced by Gruber–Kunz (1971) (Fernández–Procacci 2007 §1). Known

Example: in §02, a polymer = one occupied point, incompatible = the same point or a neighbour. In the high-temperature Ising model without external field (2-08), the polymers are connected sets of edges with even degree at every point, they are incompatible if they share a vertex, and the activity is $(\tanh\beta)^{\text{number of edges}}$. Strong coupling in lattice gauge theory (5-04) has connected sets of plaquettes, and continuum fields at weak coupling (§09) have connected sets of cubes. The reason the lumps are small becomes the small parameter of the expansion.

In the device that goes by the name of Mayer (said to go back to Mayer 1937), one writes $\mathbf 1\{\gamma_j\sim\gamma_k\}=1+\zeta_{jk}$ ($-1$ if incompatible, 0 if compatible) and expands the product. Each term corresponds to a graph (points, and edges joining pairs of points) on the points $1,\dots,n$, and in the logarithm only connected graphs remain. The reason is that the weight of a graph splits into the product of the weights of its connected components. Organising by the ways of dividing the points into connected components, between the sums over all $n$ with $1/n!$ attached, $\text{sum over all graphs}=\exp\big(\text{sum over connected graphs}\big)$ (the exponential formula) holds, and taking the logarithm leaves only the connected ones. Known

$G_\gamma$ is the graph joining $j,k$ when $\gamma_j\not\sim\gamma_k$, the sum is over connected subgraphs using all the points, and $|E(G)|$ is the number of edges (1 for $n=1$). $\varphi^{\mathrm T}$ is the Ursell function (unrelated to the field $\phi$), and it is 0 if $G_\gamma$ is not connected. Only sets connected by incompatibility, that is clusters, contribute (Fernández–Procacci (2.3)(2.4). Still a formal series).

Example: in §02, averaging over translations, $b_n=\frac1{n!}\sum_{x_2,\dots,x_n}\varphi^{\mathrm T}(0,x_2,\dots,x_n)$. For $n=2$, each of $x_2=-1,0,1$ gives $-1$ with one edge, so $b_2=-3/2$. The $(0,0)$ forbidden in configurations appears in the logarithm. Summing by brute force up to $n\le6$ agreed with Table 1 in rational numbers. Computed

The sum in formula (5) is huge. When every two are incompatible, $G_\gamma$ is the complete graph $K_n$ (joining all pairs), there are nearly as many connected spanning subgraphs as graphs in all, and if they remained as $\pm1$ the sum would not converge even after dividing by $n!$. But the sum cancels, and only $(-1)^{n-1}(n-1)!$ remains.

ComputedTable 2 $K_n$ (brute force for $n\le6$, a recurrence for $n=10$).

The mechanism is due to Penrose (1967). Dividing the connected spanning subgraphs into intervals $[T,R(T)]$, one for each spanning tree (a subgraph without cycles connecting all the points) $T$, the sum of signs within an interval vanishes by the factor $\prod_{e\in R(T)\setminus T}(1-1)$, and only the trees with $R(T)=T$ (Penrose trees) remain (Fernández–Procacci, Prop. 5). Known

The number of Penrose trees agreed with the absolute value of the alternating sum by brute force for $n\le6$ Computed. Even for $K_n$ there are $n^{n-2}$ spanning trees, and $n^{n-2}/n!$ is of order $e^n$. Since only an exponential remains after dividing by $n!$, (5) can converge if the activities are small. Identities that write the sum in terms of trees or forests from the start were refined from Brydges–Federbush (1978) to the forest formula of Brydges–Kennedy (1987) (Abdesselam–Rivasseau 1994).

ComputedThe numbers in this figure were computed on this machine. The running total for $K_4$ is −16 at 3 edges (16 spanning trees), −1 at 4 (15 of them), −7 at 5 (6 of them) and −6 at 6. $C_4$ gives −3, $P_4$ −1 and $K_3$ +2.

Summing up the tree estimates gives a condition. The widely used condition of Kotecký–Preiss (1986) assigns to each lump a "budget" $a(\gamma)\ge0$ and requires

(the left-hand side includes $\gamma$ itself). If it holds, the series obtained by differentiating $\log\Xi_\Lambda$ with respect to $z_{\gamma_0}$ (the sum of clusters containing the lump $\gamma_0$; the sum per polymer) is at most $e^{a(\gamma_0)}$ independently of $\Lambda$, even with each term replaced by its absolute value (Fernández–Procacci (2.6)(2.7)(2.14)(2.15)). $\log\Xi_\Lambda$ itself grows with the volume, but because this per-polymer upper bound does not depend on $\Lambda$, (5) converges absolutely with estimates uniform in $\Lambda$ (ibid., the sentence right after (2.7)). The change in the logarithm when the lump $\gamma_0$ is removed is also at most $|z_{\gamma_0}|e^{a(\gamma_0)}$ (ibid. (2.11)). Read it as "the total weighted activity of the incompatible partners fits within one's own budget". The conditions of the same type of Dobrushin (1996) and Fernández–Procacci (2007) widen the range of application in that order. Known

ComputedTable 3 The range guaranteed for the gas of §02 (maximum degree $\Delta=2$). Numerical maximisation agrees with the formulas of Fernández–Procacci Table 1.

$1+3\mu+\mu^2$ counts the mutually compatible subsets of three incompatible points (the empty set, the three single points, $\{x-1,x+1\}$), and $(1+\mu)^3$ is coarser by also counting the sets that are not. The $1/4$ in the last row is the exact value $(\Delta-1)^{\Delta-1}/\Delta^{\Delta}$ on the infinite tree with every degree $\Delta$ (Shearer / Scott–Sokal; the starred entry in Fernández–Procacci Table 1) with $\Delta=2$ put in. This tree with $\Delta=2$ is the line of integers itself, so it agrees with the $1/4$ of formula (3). All three conditions above are sufficient conditions.

ComputedThe numbers in this figure were computed on this machine. $|S_N(z)-p(z)|$ for $N=5,20,40$ is 3e−3, 1.5e−5, 6.4e−8 at $z=0.2$ and 0.029, 0.063, 0.88 at $z=0.3$.

If condition (7) holds independently of the volume, three things follow. (i) The limit of the pressure exists and is analytic in the disc. (ii) Only clusters that touch the edge or go round feel the size of the box. (iii) In truncated correlations only clusters touching both $x$ and $y$, that is, connected from $x$ to $y$ by a chain of incompatibility, remain, and each link contributes a factor of order $|z|$ (Ueltschi 2004, Thm 3). Known

Example: counting the gas of §02 on a ring of 14 points, the coefficients of $\log\Xi_{14}/14$ agree with $b_n$ for $n\le13$, and first differ at $n=14$ (going round needs all 14 points). The correction is $\frac1N\log\big(1+(\lambda_-/\lambda_+)^N\big)$, and for small $z$, $\lambda_-/\lambda_+\approx-z$, so it is of order $(-z)^N/N$ (the difference in the coefficient at $n=14$ is exactly $1/14$). The truncated correlation of the density is proportional to $(\lambda_-/\lambda_+)^{|x-y|}$, and the decay rate is $m(z)=\log(\lambda_+/|\lambda_-|)$. Computed

ComputedTable 4 The decay rate $m(z)$ and the estimate from chains $\log(1/|z|)$.

For small $z$, $m\approx\log(1/z)$ and the picture of chains is right; as $z\to-1/4$, $m\to0$ (decay stops at the singularity that fixed the radius). $z=1$ is outside the radius and still has $m\gt0$: convergence is a sufficient condition.

$m(z)$ is the ratio of eigenvalues of the transfer matrix, the "mass gap" of this model (stage 6 of 6-01, 4-11). In field theory too, if truncated correlations in the time direction decay like $e^{-mt}$ uniformly in the volume, and the vectors $\psi$ (with the vacuum component subtracted) made from the observables for which this decay can be shown are dense in the subspace orthogonal to the vacuum, then the Hamiltonian $H$ of the OS reconstruction (4-10) has no spectrum in $(0,m)$. Known

The tool is the spectral measure. For each $\psi$ there is a measure $\mu_\psi$ on $[0,\infty)$ with $\langle\psi,e^{-tH}\psi\rangle=\int e^{-t\lambda}\,\mathrm d\mu_\psi(\lambda)$ (the spectral theorem of 1-10; if there is a basis of eigenvectors $e_k$, $\mu_\psi$ puts the weight $|\langle e_k,\psi\rangle|^2$ on each eigenvalue $\lambda_k$). If $\int e^{-t\lambda}\mathrm d\mu_\psi\le Ce^{-mt}$ holds for all $t$, then looking at $t\to\infty$, $\mu_\psi$ cannot put weight on $\lambda\lt m$. Since this holds for a dense set of $\psi$, no spectrum remains in $(0,m)$. The converse is formula (2) of §5 of the problem statement.

For continuum fields, the covariance $C(x,y)$ of the free field links even distant points. Cut the box into unit cubes $b_i$ (indicator functions $\chi_i$), and multiply the parts straddling cubes by $s_{ij}\in[0,1]$.

If all the $s$ are 1 this is the original covariance, and if all are 0 the cubes are independent (Abdesselam–Rivasseau 1994 (V.A.5)). The two are joined by Taylor's formula (ibid. (III.1)). The index $\ell$ in formula (11) runs over pairs of cubes $\{i,j\}$, and $s_\ell=s_{ij}$. $f(\mathbf s)$ is the partition function with covariance $C(\mathbf s)$, $I$ is a set of pairs, and $\psi_I(h)$ is the point with $s_\ell=h_\ell$ for $\ell\in I$ and $s_\ell=0$ for $\ell\notin I$.

For a single pair, $f(1)=f(0)+\int_0^1f'(h)\,\mathrm dh$, and (11) repeats this pair by pair. $\partial/\partial s_\ell$ ($\ell=\{i,j\}$) brings down one line joining the cubes $b_i,b_j$, of size about $|C|$, and with mass $m_0$ about $e^{-m_0|x-y|}$. The differentiated pairs form lumps, the integral splits lump by lump, and it becomes a gas whose polymers are connected sets of cubes $Y$ (ibid. (V.A.7)(V.A.8)).

However, the interpolated covariance must stay positive definite (the condition for it to make sense as a Gaussian measure), so in practice one uses the forest formula (the Taylor forest formula of Brydges–Kennedy, ibid. Thm III.1). This is the version in which the parameter joining two cubes is set to the minimum of $h_\ell$ along the path joining them in the tree of the forest, and this keeps positive definiteness (ibid. (V.A.9), Thm IV.5). In this form, for any constant $K$, with an ultraviolet cutoff in place and the coupling $g$ taken small enough (depending on $K$), $\sum_{Y\ni0}|A(Y)|K^{|Y|}\le1$ (ibid. (V.A.10); $|Y|$ is the number of cubes in $Y$). It was Glimm–Jaffe–Spencer who brought this type of expansion into constructive field theory (ibid. §I), and the results are in 6-03 §05 and §08. Known

Example: the tool is Gaussian integration by parts (Gaussian distributions are in 1-08 §06). For a standard normal $\phi$, the derivative of the density $e^{-\phi^2/2}$ is $-\phi\,e^{-\phi^2/2}$, so integration by parts gives $\mathbb E[\phi F(\phi)]=\mathbb E[F'(\phi)]$. Extending this to Gaussian variables with mean 0 and covariance matrix $C(s)$ gives $\frac{\mathrm d}{\mathrm ds}\mathbb E_s[H]=\frac12\sum_{a,b}\frac{\mathrm dC_{ab}}{\mathrm ds}\,\mathbb E_s[\partial_a\partial_bH]$ Known. For Gaussian variables $(\phi_1,\phi_2)$ with variance 1 and covariance $sc$, $\mathrm dC_{12}/\mathrm ds=\mathrm dC_{21}/\mathrm ds=c$, so $\frac{\mathrm d}{\mathrm ds}\mathbb E_s[F(\phi_1)G(\phi_2)]=c\,\mathbb E_s[F'(\phi_1)G'(\phi_2)]$. At $s=0$ the two are independent, so integrating in $s$ from 0 to 1 gives

and the correlation of lumps is bounded by the covariance $|c|$ that links them. For $F=G=\cos$ the value is $e^{-1}(\cosh c-1)$ ($0.15932358$ at $c=0.9$), and numerical integration of the left- and right-hand sides agreed to 8 digits. Computed

Strong coupling on the lattice. For small $\beta$ the expansion in lumps of plaquettes works (5-02, 5-04), and the existence and analyticity of the infinite-volume limit and estimates of confinement have been shown (Osterwalder–Seiler 1978). But Chatterjee (2018) cautions that this area law holds for every group including $U(1)$, and is not evidence of non-abelian confinement. Known

Phase-space expansion. When there is no small parameter at the unit scale, the field is split by scale, expanded in the "phase-space cells" of each scale, and the results piled up (Battle–Federbush 1982; the 'phase-space localization' of JW §6.1). In lattice gauge theory, Bałaban's series of papers is this (6-05). Known

The infrared limit. In four dimensions the coupling becomes strong at long distances. Magnen–Rivasseau–Sénéor 1993 §I writes 'there is no easy solvable model of these phenomenons around which to expand', and the problem statement also says that a gap uniform in the volume needs new ideas (JW §6.5). The small $\beta$ where the expansion converges is the shore opposite to the continuum limit $\beta\to\infty$. More precisely, the expansion converges on a finite interval on the side of small $\beta$, and the continuum limit is taken at $\beta\to\infty$ (5-09).

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