Rigorous Renormalization Group and Continuum Control

A rigorous renormalization group controls a sequence of integrations in explicit norms. This chapter explains how counterterm tuning, remainder estimates, and observable flows turn a formal scale picture into a theorem about a model.

Enter this chapter

Follow the finite-range decomposition through polymer coordinates and trajectory control before interpreting critical exponents or scaling limits. Helpful background. Wilsonian Coarse Graining and Theory Space supplies the physical interpretation of scale integration. Polchinski’s Exact RG Equation gives a complementary continuous-flow formulation. Regulators, Cutoffs, and Continuum Limits distinguishes cutoff stability, regulator removal, and construction of a continuum theory.

Topics

1. Rigorous RG as a Dynamical System — Defines scale maps and the domains on which they are controlled.
2. Finite-Range Decompositions and Multiscale Integration — Localizes fluctuation integration using covariance decompositions.
3. Polymer Activities and Normed RG Coordinates — Measures the nonlocal remainder in weighted polymer norms.
4. Renormalized Trajectories and Counterterm Tuning — Tunes bare parameters to stay on a controlled trajectory.
5. Stable Manifolds and Relevant–Marginal Control — Separates unstable relevant directions from contractive remainders.
6. Universality, Critical Manifolds, and Observable Control — States which observables and limits a universality result compares.
7. Critical φ⁴ Models and Logarithmic Corrections — Derives marginal logarithms within specified scalar models.
8. Weakly Self-Avoiding Walk and Supersymmetric RG — Uses a supersymmetric representation to analyze walk observables.
9. Fermionic Multiscale RG and Fermi-Surface Problems — Adapts multiscale bounds to fermionic momentum-space geometry.
10. Rigorous-RG Scheme Comparison and Continuum-Construction Status — Distinguishes coordinate agreement from a proved continuum construction.

How the ideas fit together

Integrating one scale changes both local couplings and a remainder. A beta function controls only part of that map; a proof must bound the remainder and keep the entire orbit in its domain. Observable insertions require additional estimates Bauerschmidt, Brydges, and Slade 2019, Chapters 3 and 5–10. A critical lattice model also need not yield an interacting continuum limit: the four-dimensional scalar triviality result has a specific model and scaling scope Aizenman and Duminil-Copin 2021, Theorem 1.2, with its cited corrigendum.

Hypotheses and licensed conclusions

Normed RG objects, hypotheses, directional conclusions, and decisive failure tests

Object and domain	Required hypotheses	Licensed conclusion	Excluded converse or upgrade	Adversarial check
Lattice covariance $C=(-\Delta+m^2)^{-1}$ on a torus	A positive finite-range decomposition with uniform derivative bounds and a specified mass-scale regime	Sequential Gaussian integration with exact factorization for sufficiently separated polymers	A momentum-shell partition by itself does not give finite-range independence or positivity of every piece	Put two polymers farther apart than the nominal range and test whether their fluctuation covariance really vanishes
One RG step $R_j:(V_j,K_j)\mapsto(V_{j+1},K_{j+1})$	A scale-dependent domain, localization operator, field and large-set regulators, and differentiable norm estimates	An exact coordinate identity and a contractive bound for the extracted nonperturbative remainder	A truncated beta function neither defines the exact map nor bounds the discarded activity	Delete the large-field regulator and evaluate the norm on a high-amplitude field configuration
All-scale bulk trajectory	Tuned relevant initial coordinates, a controlled marginal recursion, stable-manifold hypotheses, and summable nonlinear errors	An orbit that remains in the small domain and approaches the designated infrared behavior	Forward iteration from generic bare data does not land on the critical manifold, and linear stability is not nonlinear stability	Perturb the bare mass in the unstable direction and check that the deviation grows before the terminal scale
Critical observable with insertions	A controlled bulk orbit, renormalized observable coordinates, coalescence-scale estimates, and convergence in a stated topology	A model-specific two-point asymptotic, susceptibility law, or Gaussian scaling limit	Bulk free-energy control does not automatically prove correlation-function convergence or universality	Insert two sources, follow their mixed coupling, and verify that the claimed amplitude survives the terminal-scale remainder
Weak four-dimensional lattice $|\varphi|^4$ model near criticality	Small positive coupling, dimension four, tuned critical mass, and the lattice and observable hypotheses of the cited theorem	Gaussian critical scaling with proved logarithmic corrections, including the $n$-dependent susceptibility exponent	These results do not yield a non-Gaussian four-dimensional continuum measure or settle every strong-coupling lattice action	Change the dimension or leave the small-coupling domain and test which summability and marginal-flow estimates fail
Supersymmetric walk or sectorized fermionic model	For the walk, an exact boson–fermion representation and supersymmetric localization; for a Fermi surface, regular geometry, sector bounds, and the theorem's temperature range	Walk critical asymptotics or a restricted weak-coupling Fermi-liquid construction with controlled kernels	Scalar polymer estimates do not transfer automatically to Grassmann sectors, nesting, van Hove points, or zero temperature	Move the Fermi level to a singular or nested surface, or break the supersymmetric identity, and locate the first lost estimate
Two regulators or RG coordinate systems	A common microscopic model, explicit maps between domains, uniform error bounds, and separately proved convergence of the same observables	A conditional comparison of selected universal quantities or equivalent limiting correlations	Matching low-order beta coefficients does not prove conjugacy of exact maps, regulator independence, or existence of a continuum QFT	Add an irrelevant term whose accumulated error is not summable and test whether the proposed comparison still closes

Structured table data (JSON).

Review the chapter

A perturbative coupling stays small at every computed scale. What is missing from an all-scale construction?

Solution

Uniform bounds on the omitted coordinates and errors, control of the exact trajectory in its normed domain, and convergence of the intended observables. Smallness of one projected coupling does not bound the accumulated remainder or justify removing the regulator.

Where to go next

Continue to causal perturbation theory for renormalization based on local distribution extension, where the output is generally a formal interacting theory.

References

• Aizenman, Michael, and Hugo Duminil-Copin. “Marginal Triviality of the Scaling Limits of Critical 4D Ising and $\phi_4^4$ Models.” Annals of Mathematics 194 (2021): 163–235. DOI; Open PDF. See also the corrigendum.
• Bauerschmidt, Roland, David C. Brydges, and Gordon Slade. Introduction to a Renormalisation Group Method. Lecture Notes in Mathematics 2242. Singapore: Springer, 2019. DOI; Open PDF.