Contribution
LIT-873 showed numerically that the coarse variable of a block chosen to carry the most information about the far environment recovers known renormalization-group variables. This paper supplies a reason. It proves that if a coarse variable keeps all of that information, the coarse probability measure factorizes across the block, which forbids the renormalized Hamiltonian from coupling the two sides, and, for 1D disorder, forbids correlations across the block in the renormalized disorder distribution. It then studies coarse-grainings that keep only part of the information, and decomposes the information lost into a choice-of-mode part and an encoding part, which accounts for why the best rule differs between one and two dimensions.
Key insight
With short-range interactions, a region of finite width mediates every correlation between its two sides. If the region’s summary keeps everything the region knows about the sides, the summary mediates them too, and nothing on one side can depend on the other except through it. That screening is what a short-ranged effective Hamiltonian is, so “keep all the information about the far field” and “do not generate long-range couplings” are, under perfect capture, the same demand.
Assumptions
- Classical lattice systems with finite-range Hamiltonians, at equilibrium, P(X) ∝ e^{K[X]}; blocks taken large enough that only neighbouring blocks interact.
- Block RG. The coarse-graining factorizes over blocks, P(X′|X) = Π_j P_Λ(H_j|V_j). The partition into block, buffer, environment (and an outer region O, for algorithmic reasons) is fixed in advance.
- Full information capture for the theorems: I(H₀ : E₀) = I(V₀ : E₀), the upper bound. The authors concede that with the number and type of coarse variables chosen in advance a rule meeting it need not exist.
- In D dimensions, an extra assumption. The proof runs on (D − 1)-dimensional hyperplanes of blocks, each block still with its own coarse variable; it needs full capture for the hyperplane to be equivalent to full capture by each block separately, “reasonable for a short-ranged Hamiltonian, at least in the isotropic case”.
- No fine-tuning. Factorization of the measure is turned into “no next-nearest-neighbour terms” by excluding the case in which integrating out the buffer exactly cancels pre-existing longer couplings.
- Disorder, for Proposition 2: 1D, a product distribution over nearest-neighbour couplings, and changes confined to one environment.
- For the calculations: the RBM rule P_Λ(h|V) = 1/(1 + e^{−2h Σ λᵢvᵢ}), no bias by Z₂ symmetry; blocks of two spins into one; periodic chains; K = 0.1 and a one-site buffer for the plotted cases; cumulant expansion in powers of K, to tenth order (clean) or ninth (disordered, 16 spins). The physically meaningful rules are the large-|Λ| limit, where the rule becomes deterministic.
Key results
- Bounds (Eqs. 7, 8, 22). 0 ≤ I_Λ(H : E) ≤ H(H); I_Λ(H : E) ≤ I(V_Λ : E) ≤ I(V : E), from the Markov chain E → V → V_Λ → H, where V_Λ is the normalized linear combination Σ λᵢⱼvⱼ a hidden reads.
- Lemma (Appendix B). Full capture gives I(E₀ : X₀ | X₀′) = 0; locality gives I(E_L : E_R | X₀) = 0; together, P(E_L, E_R | X₀′) = P(E_L | X₀′) P(E_R | X₀′). Translation invariance is not used.
- Proposition 1 and Corollaries 1–2. The coarse measure factorizes: P(X′{≤−2}, X′{≥2} | X₀′) = P(X′{≤−2} | X₀′) P(X′{≥2} | X₀′), with the coarse buffer integrated out. Hence no effective terms couple the two sides, and a perfect RSMI step does not increase the range of a finite-range Hamiltonian; in D dimensions under the extra assumption.
- Proposition 2 and Corollary 3. In 1D with product disorder, the optimal rule of a block and the factorization survive any disorder change confined to one environment, so no disorder correlations are generated across the block, and the optimal rule depends only on the disorder near the block.
- 1D Ising, arbitrary two-spin rules (Sec. IV, Figs. 3–5, 10–12). Information kept is largest near (±λ, 0) and (0, ±λ), decimation, and least at majority rule. The rangeness ratio K′₂(2)/K′₂(1) and the m-bodyness ratio K′₄(1,1,1)/K′₂(1) vanish at the maximum for large |λ|; for the RSMI solution, two-point couplings decay exponentially with distance and m-point couplings with m. The perturbative nearest-neighbour coupling converges to the exact decimation value K′ = ½ log cosh 2K as the order rises. The maximum saturates in |λ| (little difference between λ = 3 and λ = 1000). At that maximum decimation keeps only about half of I(V : E): 0.505 by my calculation at K = 0.1 with a one-site buffer, the value Fig. 5’s axis appears to reach. Full capture is not attained.
- Not globally monotonic (Appendix D). Rangeness is a monotonic function of the information only locally; a global maximum of the information is a global minimum of rangeness, possibly with equivalent solutions (for three-spin blocks, decimation to the centre spin is as good for the Hamiltonian as to an edge spin, though the information prefers the edges). Ratios also vanish accidentally elsewhere, so the authors advise watching the information saturate, not a coefficient.
- Dilute random Ising chain (Sec. VI, Fig. 9). Distance correlation and KL divergence between neighbouring renormalized couplings, and the centre of mass of the distribution outside the nearest-neighbour two-body plane, all vanish where the information is maximized, at decimation.
- Encoding loss (Eq. 24, Sec. V). I_Λ(H : E) = I(V_Λ : E) − I(V_Λ : E | H). Majority rule’s V_Λ takes three values, which a binary H cannot encode unless V_Λ = 0 has probability zero. In a four-spin 1D toy model (Eq. 25, Fig. 6) majority rule has the larger I(V_Λ : E) but decimation has the larger I(H : E) throughout; greedy maximization of I(V_Λ : E) followed by encoding is therefore not optimal. Coupling the two environment spins (Appendix E, Fig. 13) narrows the gap. In a 2D toy model (four spins coupled to one environment variable with nine states, Fig. 7), majority rule, and coupling to three spins, keep more than decimation or any two-spin rule at every K_V.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A coarse-graining keeping all of a block’s information about the environment beyond a buffer makes the coarse measure factorize across the block, so a finite-range Hamiltonian gains no range | strong in 1D; moderate in D dimensions | proof, Appendix B; the D-dimensional case adds a per-block capture assumption and excludes fine-tuned cancellation |
| C2 | Under the same condition in 1D with product disorder, no correlations are generated across a block in the renormalized disorder distribution, and the optimal rule depends only on local disorder | strong, for its hypotheses | proof, Proposition 2 |
| C3 | When full capture is not attained, longer-range and many-spin couplings decrease as the information kept increases | weak in general; moderate for two-spin blocks of the 1D Ising chain | perturbative calculation, Figs. 4, 5, 10–12; only locally monotonic |
| C4 | Disorder correlations and couplings outside the nearest-neighbour plane decrease as the information kept increases | moderate for the dilute random Ising chain | perturbative calculation, 16 spins, Fig. 9 |
| C5 | Decimation is the information-optimal two-spin rule in 1D and majority rule in 2D, because a binary coarse spin cannot encode an average, and a connected environment reduces that loss | moderate | exact calculation on four-spin toy models, Figs. 6, 7, 13 |
| C6 | RSMI maximization is a model-independent variational principle defining the optimal RG coarse-graining | weak | framing over C1–C5; the authors call it “strongly” supported |
Method
Analytical throughout. The Lemma and Propositions are proved from the chain rule for mutual information, locality and the block factorization. The effective Hamiltonian of an arbitrary rule is obtained by splitting K into intra- and inter-block parts, writing the renormalized Hamiltonian as an intra-block average of e^{K₁}, and expanding in cumulants, which factorize into three block parameters (a₁, a₂, b) for the Ising chain; the resulting polynomial is brought to a canonical form by eliminating non-canonical terms. The mutual information of an arbitrary rule is computed exactly by transfer matrices in the thermodynamic limit. No learning is done; the RBM is only a differentiable family of rules.
Concepts
- RSMI: I_Λ(H : E), the mutual information between a block’s coarse variables and the environment beyond a buffer, as in LIT-873.
- full information capture: I(H₀ : E₀) = I(V₀ : E₀), the upper bound, called a “perfect” RSMI coarse-graining.
- V_Λ: the linear combination of the block’s spins a hidden couples to under an RBM rule; what the rule chooses to look at, as opposed to how it encodes it.
- mismatch: I(V_Λ : E | H), the information about the environment carried by V_Λ that the coarse variable fails to encode.
- rangeness / m-bodyness: the ratios K′₂(2)/K′₂(1) and K′₄(1,1,1)/K′₂(1), used as measures of the renormalized Hamiltonian’s complexity.
- dCOM: the norm of the mean of the renormalized disorder distribution’s components outside the nearest-neighbour two-body couplings.
Connections
It is the theoretical companion to Koch-Janusz and Ringel (LIT-873, read in NOTE-674), whose saturation argument for 1D and quasi-1D chains it turns into the Lemma and extends; its 1D findings match that paper’s three-filter comparison (decimation best) and its 2D finding matches that paper’s 2 × 2 block spin. It presents RSMI as the information bottleneck (LIT-338) with a compressed variable of fixed type and size and the environment as the relevance variable. It sets itself against fixed schemes, citing van Enter, Fernández and Sokal on the pathologies of real-space RG and Kennedy on majority rule, and, in Appendix F, against perfect actions (Hasenfratz and Niedermayer) and coarse-grained PDE schemes, where optimality is error against a known continuum problem. Rigorous treatment of the RSMI measure is left open; Kupiainen’s comment on rigorous RG (LIT-039) is held here unread.
Bearing on the record
- Produces THEORY-198: full capture forbids range growth, and the theorem’s hypothesis is not met in the cases that illustrate it.
- THEORY-194. Supports it on the point it was thinnest: why the
information-maximizing coarse-graining should give a short-ranged
effective Hamiltonian. It does not meet its
promote_when: the proof is for perfect capture, not for the constrained maximizer, the D-dimensional version adds an assumption, and the partial-capture evidence is 1D. The 2D toy model is consistent with THEORY-194’s majority-rule finding but is a four-spin model, not a lattice. - THEORY-073. The Lemma is a screening statement: the block’s summary plays the role of a blanket between the two environments. The block, buffer and environment are fixed before any information is computed, as THEORY-073 says a blanket’s placement is. My connection, not the paper’s; it neither supports nor weakens THEORY-073.
- THEORY-017. As in NOTE-674: the information is invariant under re-encodings, the partition is not, and here the theorems depend on the partition through the locality assumption.
- QUESTION-025. No direct bearing; no attributes, embeddings or lattices. The encoding-loss decomposition is the nearest thing to a transferable result: what a summary of fixed type can keep depends on whether the quantity it should read fits that type.
- Anthology. No training, no practice instruction; the RBM is a parametrization. Not flagged.
Limitations
- The theorems’ hypothesis is not met by the examples. Decimation of two-spin blocks keeps about half the block’s information, yet gives an exactly nearest-neighbour Hamiltonian; the exactness there comes from the 1D chain’s structure, not from Proposition 1.
- “Monotonic decay” is local. The abstract speaks of decay as a function of information retained; Appendix D says it is monotonic only locally, with accidental zeros elsewhere.
- All partial-capture evidence is 1D, two-spin blocks, weak coupling (K = 0.1), perturbative, with 16 spins for the disordered case. The 2D evidence is a four-spin toy model with a single environment variable.
- The D-dimensional proof rests on an assumption stated as “reasonable”, not proved, and on excluding fine-tuned cancellations.
- Proposition 2 is 1D. Generalization to higher dimensions is said to follow “similarly”, not shown.
- Small inconsistencies in the text. Appendix E calls the 1D toy model “Eq. (6)”, which is the RSMI definition; the toy model is Eq. 25. Appendix D’s “Fig. (4c)” for crossings along paths of fixed |λ| appears to mean Fig. 5. “Course-graining” for coarse-graining twice.
Open questions
- Does the constrained maximizer, not the perfect one, provably bound the range or decay of the effective couplings, in two or more dimensions? The authors name this as what a rigorous result would need.
- Can the number and type of coarse variables themselves be optimized, so that the encoding loss is not fixed by a choice made in advance? The authors pose it.
- Does the per-block capture assumption of the D-dimensional proof hold for any 2D model that can be checked, such as the 2D Ising model with 2 × 2 blocks?