Polymer Activities and Normed RG Coordinates

Polymer coordinates separate a finite list of local relevant or marginal monomials from a spatially organized, norm-small remainder. Their usefulness is theorem-level only after three structures work together: factorization over connected components, a field regulator that controls Gaussian shifts, and a large-set weight that beats the number of polymers created by reblocking.

Required background. Finite-Range Decompositions and Multiscale Integration supplies local Gaussian fluctuations. Rigorous RG as a Dynamical System supplies the scale map whose coordinates are defined here.

Helpful background. Cluster Expansions and Correlation Inequalities supplies connected-component combinatorics. Interacting Measures, Stability, and Wick Ordering supplies the large-field stability problem.

Blocks, polymers, and the circle product

Partition $\Lambda_N$ into scale-$j$ blocks of side $L^j$. A polymer $X$ is a union of such blocks; $|X|_j$ is its number of blocks. A polymer activity $K_j(X,\phi)$ is local in $X$, invariant under the model symmetries, and factorizes on connected components:

$$K_j(X,\phi)=\prod_{Y\in\operatorname{Comp}(X)}K_j(Y,\phi).$$

For functions $F,K$ on polymers, the circle product is

$$(F\circ K)(X)=\sum_{Y\subseteq X}F(X\setminus Y)K(Y).$$

Taking $I_j(V,B)=e^{-V_j(B)}$ on one block and multiplying it over polymers gives the exact coordinate $Z_j=(I_j\circ K_j)(\Lambda_N)$. The sum over $Y$ assigns each block either to the local potential or to the remainder, avoiding double counting.

A weighted field norm

For a smooth activity, a $T_\phi$ seminorm records derivatives up to a declared order—in the worked scalar example, order three:

$$\|K(X)\|_{T_\phi^{(3)}} =\sum_{p=0}^{3}\frac1{p!} \sup_{\|f_i\|_{\Phi_j}\leq1} |D^pK(X,\phi)(f_1,\ldots,f_p)|.$$

The test-field norm $\|\cdot\|_{\Phi_j}$ scales derivatives with $L^j$ and the field dimension. A large-field regulator $G_j(X,\phi)\geq1$ grows exponentially in suitable local Sobolev seminorms. With $A>1$, define schematically

$$\|K\|_j=\sup_{X\ \mathrm{connected}} A^{|X|_j}\sup_\phi \frac{\|K(X)\|_{T_\phi^{(3)}}}{G_j(X,\phi)}.$$

Different rigorous implementations use a pair of regulators and scale-dependent amplitudes, but the logical roles are stable. $T_\phi$ controls field Taylor remainders, $G_j$ makes Gaussian translation integrable, and $A^{|X|_j}$ forces exponential decay in polymer size. Brydges and Slade establish the normed algebra and Gaussian estimates in Brydges and Slade 2015, §§ 3–6, pp. 429–455.

Local extraction and one RG step

The localization operator $\operatorname{Loc}_X$ matches a functional against a finite-dimensional space of local polynomials. It is a projection on that space, respects symmetries, and approximates a small-polymer activity with an irrelevant remainder gaining powers of the scale ratio. Its construction and $T_\phi$ estimate are Brydges and Slade 2015, Theorem 1.1 and §§ 1.3–1.5, pp. 464–476.

For a weak lattice $|\phi|^4$ action from Scalar Lattice Actions and Difference Operators, start with $V_0=g_0\sum_x\tau_x^2+\nu_0\sum_x\tau_x$ and $K_0=0$. Integrate $C_1$, expand connected terms, and apply $\operatorname{Loc}$ to the one- and two-block contributions. The extracted quartic, quadratic, derivative, and vacuum monomials define $V_1$; all higher Taylor jets and large connected sets define $K_1$. The single-step theorem bounds

$$\|K_1\|_1\leq Cg_0^3$$

in the declared weighted $C^3$-type field norm for $g_0$ small and the relevant coordinates in their domain Brydges and Slade 2015, Theorems 1.11 and 1.13, pp. 605–613. The power $g_0^3$ depends on how second-order perturbative terms are placed in the improved local coordinate.

The proof combines the product property of finite-range Gaussian expectation, connected reblocking, the localization remainder estimate, and stability of $G_j$ under fluctuation fields. Checking $\operatorname{Loc}^2=\operatorname{Loc}$ independently catches double extraction of a local monomial.

Why localization is needed for contraction

Gaussian integration alone does not make every activity irrelevant. A small-polymer contribution can contain a constant, a mass term, a kinetic term, or a quartic term whose rescaled coefficient stays constant or grows. The linearized map on the unextracted remainder would therefore have noncontracting directions. Localization separates these Taylor jets before the norm estimate:

$$F=\operatorname{Loc}_X F+(1-\operatorname{Loc}_X)F.$$

The first term changes $V_{j+1}$. For the second, vanishing moments against the chosen polynomial test space yield a scale gain in the $T_\phi$ norm; after reblocking and summing connected sets, that gain is what permits a bound $\|\mathcal L_jK\|_{j+1}\leq\kappa\|K\|_j$ with $\kappa<1$. The polynomial space, differentiability order, and scale weights must be matched: extracting too few jets leaves an expanding direction in $K$, while demanding more derivatives than the activity possesses makes $\operatorname{Loc}$ undefined.

There is also an algebra check. For disjoint polymers and compatible multiplicative regulators, the derivative product rule gives

$$\|K(X)K(Y)\|_{T_\phi} \leq \|K(X)\|_{T_\phi}\|K(Y)\|_{T_\phi},$$

while $A^{|X\cup Y|_j}=A^{|X|_j}A^{|Y|_j}$. Thus factorization over connected components is compatible with the large-set weight. This submultiplicativity, together with the convolution sum in the circle product, is the structural reason the coordinate survives repeated products rather than merely one Gaussian integration.

Failure test and status boundary

Remove the large-set factor by setting $A=1$. The number of connected polymers with $n$ blocks through a fixed block grows exponentially in $n$. Even if each activity is pointwise small, the sum over shapes can diverge or lose contraction. Alternatively, weaken $G_j$ so that it does not dominate quartic large-field excursions; Gaussian integration can then magnify rather than suppress the norm. Formal locality survives, but the Banach-map theorem does not.

Polymer coordinates are not unique physical degrees of freedom. Different blocking, regulators, and localization bases can describe the same finite-cutoff integral. Equality of their continuum observables requires a separate comparison theorem.

Exercise

Suppose the number of connected $n$-block polymers containing a fixed block is at most $c^n$ and $|K(X)|\leq\varepsilon^{|X|}$. Which condition makes the absolute sum converge?

Solution

The sum is bounded by $\sum_{n\geq1}c^n\varepsilon^n$, hence converges when $c\varepsilon<1$. In a weighted norm, $A^{|X|}|K(X)|\leq\delta$ gives $|K(X)|\leq\delta A^{-|X|}$, so choosing $A>c$ beats shape entropy. The actual RG estimate also needs compatibility with fields, overlaps, and reblocking.

References

• Brydges, David C., and Gordon Slade. “A Renormalisation Group Method. I. Gaussian Integration and Normed Algebras.” Journal of Statistical Physics 159 (2015): 421–460. DOI; Open PDF.
• Brydges, David C., and Gordon Slade. “A Renormalisation Group Method. II. Approximation by Local Polynomials.” Journal of Statistical Physics 159 (2015): 461–491. DOI; Open PDF.
• Brydges, David C., and Gordon Slade. “A Renormalisation Group Method. V. A Single Renormalisation Group Step.” Journal of Statistical Physics 159 (2015): 589–667. DOI; Open PDF.