Finite-Range Decompositions and Multiscale Integration

For the massive lattice Green function in dimension $d\geq3$, a finite-range decomposition expresses the covariance as a sum of positive covariances whose kernels vanish beyond their assigned scales and obey uniform derivative bounds. Positivity realizes the field as a sum of independent Gaussian fluctuations; finite range turns geometric separation into exact probabilistic independence at each step.

Required background. Rigorous RG as a Dynamical System supplies the map that consumes the covariance slices. Gaussian Euclidean Fields as Measures supplies Gaussian convolution and covariance positivity.

Helpful background. Domains, Signatures, Supports, and Regularity helps track lattice, torus, and mass domains. Cluster Expansions and Correlation Inequalities explains why exact scale separation improves connected estimates.

Finite-range decomposition theorem

On $\mathbb Z^d$, let

$$C_m=(-\Delta+m^2)^{-1}, \qquad d\geq3,\quad m^2\geq0,$$

where $-\Delta$ is the nearest-neighbor positive lattice Laplacian. For sufficiently large dyadic $L$, Brydges, Guadagni, and Mitter construct positive semidefinite, translation-invariant kernels $C_j$ such that

$$C_m=\sum_{j=1}^{\infty}C_j, \qquad C_j(x,y)=0\quad\text{when }|x-y|\geq cL^j.$$

They also prove scale-covariant difference bounds, schematically

$$|\nabla^\alpha C_j(x,y)| \leq c_{\alpha,k}L^{-(d-2+|\alpha|)(j-1)} \bigl(1+m^2L^{2(j-1)}\bigr)^{-k},$$

for the stated orders and constants. The exact range convention and lattice norm depend on normalization. The theorem, including positivity and the massless scaling limit, is Brydges, Guadagni, and Mitter 2004, Theorem 1.1 and §§ 2–3, pp. 417–432. On a torus $\Lambda_N$, the first $N-1$ pieces are periodized without wraparound at their scales and the remaining long-distance covariance is collected in a final positive term; zero-mode treatment matters when $m=0$ Bauerschmidt, Brydges, and Slade 2019, Chapter 3, pp. 37–50.

The statement is operator-specific. Ellipticity and locality of the lattice Laplacian enter the averaging construction. It does not say that an arbitrary positive covariance possesses compact-range positive slices.

Progressive Gaussian integration

Let $\zeta_j$ be independent centered Gaussian fields with covariances $C_j$. Positivity gives

$$\phi\overset{\mathrm{law}}{=}\sum_j\zeta_j, \qquad \mathbb E_C F =\mathbb E_{C_N}\cdots\mathbb E_{C_1}F.$$

For the Gaussian observable $F_f(\phi)=e^{i\langle f,\phi\rangle}$, one can verify the factorization exactly:

$$\mathbb E_C F_f =\exp\!\left(-\frac12\langle f,Cf\rangle\right) =\prod_j\exp\!\left(-\frac12\langle f,C_jf\rangle\right).$$

This identity is an independent normalization check on the order and signs of progressive integration.

For the four-dimensional torus propagator requested by Lattice Momentum, Propagators, and Cutoff Dispersion, take $d=4$, $m>0$, and $\Lambda_N$. The Fourier covariance is

$$\widehat C_m(p)=\frac{1}{\widehat{-\Delta}(p)+m^2}.$$

The theorem produces positive $C_j$ with spatial range $O(L^j)$. Integrating $F_f$ one slice at a time multiplies its characteristic function by $e^{-\langle f,C_jf\rangle/2}$, and the product reconstructs the lattice propagator exactly. For an interacting observable, the same convolution identity is exact, while its representation by finitely many local couplings plus a small polymer activity is the additional RG theorem.

Separation, scaling, and the zero mode

Finite range gives more than rapid decay. If subsets $X$ and $Y$ satisfy $\operatorname{dist}(X,Y)>cL^j$, then

$$\langle f_X,C_j f_Y\rangle=0$$

for test functions supported in $X$ and $Y$. The Gaussian random vectors $\zeta_j|_X$ and $\zeta_j|_Y$ are therefore independent, since their cross-covariance vanishes. Consequently, for bounded functions depending on the two restrictions,

$$\mathbb E_{C_j}\bigl[F_X(\zeta_j)F_Y(\zeta_j)\bigr] =\mathbb E_{C_j}F_X(\zeta_j)\, \mathbb E_{C_j}F_Y(\zeta_j).$$

This exact factorization is what allows a connected polymer to grow only through a bounded neighborhood in one RG step. Replacing finite range by exponential decay replaces equality by an error estimate, and that error must be included in the polymer norm.

The derivative bound also passes a dimensional check. Below the mass scale, $C_j(x,x)$ has order $L^{-(d-2)(j-1)}$, the square of the canonical fluctuation-field size at scale $j$. When $mL^j$ becomes large, the mass factor gives extra decay, so the long-distance tail is summable. On a finite torus with $m>0$, the constant Fourier mode has eigenvalue $m^{-2}$ and is retained in the final covariance. At $m=0$, however, $-\Delta$ is not invertible on constants. One must either work on the mean-zero subspace, specify a pseudoinverse, or isolate the zero mode. Silently inserting the massless torus covariance into the infinite-lattice theorem changes the operator domain and invalidates the claimed identity.

Proof mechanism and checks

The construction averages local Dirichlet or Poisson extensions over blocks and writes the Green operator as a telescoping sum of positive quadratic forms. Locality gives compact support; rescaling gives derivative bounds; spectral calculus controls $m^2$ uniformly until the mass scale $j_m\approx\log_Lm^{-1}$. Past $j_m$, the factor in the displayed estimate suppresses further slices.

Check four properties separately: $C_j=C_j^*$; $\langle f,C_jf\rangle\geq0$; the exact support radius; and $\sum_jC_j=C_m$ in the claimed operator topology. A decomposition with signed slices may reproduce $C_m$ but does not define independent real Gaussian fields.

Adversarial kernel and continuum boundary

Replace $-\Delta$ by a genuinely nonlocal operator whose kernel has a slow algebraic tail. The averaging step no longer makes the scale covariance vanish outside a block: two polymers at arbitrary separation remain weakly coupled. One may seek a decomposition with rapid or summable decay, but every place that used exact independence must be reproved. Reusing the finite-range polymer contraction unchanged is invalid.

Finite-range decomposition is a multiscale tool, not cutoff removal. It neither tunes relevant parameters nor proves tightness of interacting measures. Those are separate all-scale and observable estimates.

Exercise

Let $C=C_1+C_2$ with $C_i\geq0$. Prove $\mathbb E_C F=\mathbb E_{C_2}\mathbb E_{C_1}F$ first for $F_f=e^{i\langle f,\phi\rangle}$.

Solution

The right side equals $e^{-\langle f,C_1f\rangle/2}e^{-\langle f,C_2f\rangle/2}=e^{-\langle f,Cf\rangle/2}$, the characteristic functional of the centered Gaussian with covariance $C$. Characteristic functionals determine the finite-dimensional distributions. Approximation then extends the convolution identity to suitable measurable integrable $F$.

References

• 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.
• Brydges, David C., Gianfausto Guadagni, and Paul K. Mitter. “Finite Range Decomposition of Gaussian Processes.” Journal of Statistical Physics 115 (2004): 415–449. DOI; Open PDF.