Rigorous lattice methods

A finite lattice with compact gauge group gives a well-defined integral. That makes it possible to prove inequalities rather than manipulate a formal path integral. The hard part is preserving the right information when the volume increases and the lattice spacing disappears.

Prerequisites: the rigorous-result lesson and strong-coupling expansions.

After this page: compare a criterion with a verified regime, translate an area bound into lattice units, and identify the missing implication toward a physical continuum gap.

Three kinds of argument

A convergent expansion expresses an observable in a controlled regime and bounds the neglected terms. A correlation inequality relates one quantity to another without necessarily computing either. A dynamical method uses an auxiliary process to obtain estimates for its equilibrium measure. These can be combined, but their conclusions remain different.

For a probability measure $\mu$ on link configurations, a covariance estimate concerns

$$\operatorname{Cov}_{\mu}(f,g) =\int fg\,d\mu-\int f\,d\mu\int g\,d\mu.$$

It must specify allowed functions, support, boundary conditions and constants. “Correlations decay” can mean several inequivalent statements. A bound for one local observable in one box is weaker than a bound uniform over a class of observables, boxes and boundaries.

Two statements that must stay separate

In Chatterjee’s arXiv:2006.16229v5, Definition 2.1 and Theorem 2.2, the specified slab-center condition yields a rectangular-loop bound with a potential that diverges at large separation, for center-charged irreducible representations. Definition 2.3 and Theorem 2.4 use the stronger hypothesis of exponential correlation decay uniform under arbitrary boundary conditions to obtain an area law. A diverging potential need not be linear.

Cao–Nissim–Sheffield’s v2, equations (1.1)–(1.2), Definition 1.4 and Theorem 1.6 establish an area law in an explicit coupling regime for the listed groups. The result includes $SU(N)$, $N\geq2$, in dimension $d\geq2$. It controls rectangular loops on the lattice torus with side lengths at most half the torus size. The dossier retains the full group list and threshold. Its auxiliary dynamics does not become the physical time coordinate.

These statements differ in what is assumed, what is proved and which constants are controlled. A method comparison should begin with that ledger.

Worked example: which potential is implied?

Suppose a hypothetical loop expectation satisfies

$$|\langle W(R,T)\rangle|\leq C e^{-cRT}, \qquad C,c>0,$$

with $C$ independent of $T$. Wherever the logarithm is defined,

$$-\frac1T\log|\langle W(R,T)\rangle| \geq cR-\frac{\log C}{T}.$$

Thus its large-$T$ lower limit is at least $cR$. This is an implication of the displayed bound; identifying that logarithm with a physical static-source energy needs the appropriate transfer-matrix interpretation and overlap assumptions.

Now replace $cR$ by $\log(1+R)$. The potential lower bound still diverges, but no positive linear coefficient follows. This simple comparison explains why “confinement” is not enough information to identify a theorem’s conclusion.

Restore physical units before taking limits

If $R$ and $T$ count lattice steps, then $A_{\mathrm{phys}}=a^2RT$. An area coefficient $c$ in the lattice exponent corresponds dimensionally to $c/a^2$. To discuss a finite physical string tension one must control its scaling, not merely keep the word “positive.”

The same issue appears in a lattice correlation rate $m_{\mathrm{latt}}$: the physical rate is $m_{\mathrm{latt}}/a$. Bounds valid in a strong-coupling region do not automatically cover a continuum scaling trajectory. Nor does a static-source bound alone control the entire vacuum spectrum.

Exercise: test the quantifier

A simulation finds decay under periodic boundary conditions. Can that observation establish the hypothesis of Chatterjee’s Definition 2.3?

Hint: compare both the class of boundary conditions and the kind of evidence.

Solution: no. The definition is a uniform inequality over arbitrary boundaries, cubes and specified bounded local functions. A numerical observation under one family of boundaries supplies neither that universal quantifier nor a rigorous uniform estimate. It can motivate a conjecture or test a regime without being the hypothesis itself.

Read the confinement criterion dossier, then compare lattice bootstrap. Use the notation crosswalk before comparing two parameters both called beta.