Patrick M. Lenggenhager, Doruk Efe Gökmen, Zohar Ringel, Sebastian D. Huber and Maciej Koch-Janusz (2018), Physical Review X 10, 011037 (2020) — ARXIV-1809.09632, DOI-10.1103/PhysRevX.10.011037
Key takeaways
- Full capture implies factorization. Take blocks large enough that a finite-range Hamiltonian couples only neighbouring blocks, a block V₀, its neighbours as buffer and the rest as environment. If the coarse variable H₀ keeps all of V₀’s information about the environment, I(H₀ : E₀) = I(V₀ : E₀), then I(E₀ : V₀ | H₀) = 0, and with locality the left and right environments are independent given H₀ (the Lemma, Appendix B). The coarse measure then factorizes across H₀ (Proposition 1), so, barring a fine-tuned cancellation, the renormalized Hamiltonian has no terms coupling the two sides: a perfect RSMI step does not increase the range of interactions. In D dimensions the same argument runs on hyperplanes of blocks, and needs the additional assumption that full capture for a hyperplane amounts to full capture by each of its blocks separately.
- The same for disorder, in 1D. With a product disorder distribution over nearest-neighbour couplings, the optimal coarse-graining of a block and its factorization are unchanged by a disorder change confined to one side’s environment (Proposition 2), so the renormalized disorder distribution has no correlations across the block (Corollary 3). Only the local disorder near a block bears on how that block should be coarse-grained.
- The condition is rarely met, so the rest is numerics. With the number and type of coarse variables fixed in advance, full capture may be impossible, and the authors say so. Their worked case, a block of two spins of the 1D Ising chain coarse-grained into one spin by an RBM rule with parameters (λ₁, λ₂), computes exactly the information kept and perturbatively (cumulant expansion to tenth order) the renormalized couplings. Decimation-like rules keep the most information and give vanishing next-nearest-neighbour and four-spin terms; majority rule keeps the least. Decimation keeps only about half of I(V : E) there (my calculation at their K = 0.1 and one-site buffer, consistent with their Fig. 5), so the theorem’s hypothesis does not hold in the case that illustrates it.
- Not monotonic, but the maximum is right. The ratios of next-nearest to nearest and of four-spin to two-spin couplings fall as information rises along paths of large |Λ|, but Appendix D says this is monotonic only locally; the claim kept is that the global maximum of information is a global minimum of these measures. Some couplings also vanish accidentally away from the maximum.
- Disorder correlations, numerically. For the dilute random Ising chain (16 spins, every disorder realization, ninth-order cumulants), the distance correlation and the KL divergence between neighbouring renormalized couplings, and the weight of couplings outside nearest neighbour two-body terms, all vanish where information is maximized, at decimation.
- Why 1D and 2D prefer different rules. I(H : E) = I(V_Λ : E) − I(V_Λ : E | H), where V_Λ is the linear combination of the block a hidden reads. Majority rule reads a variable with three values that one binary spin cannot encode, so it loses more in encoding than it gains; in 1D, where the environment is two disconnected halves, decimation wins in a four-spin toy model. When the environment is one connected variable, as in a 2D toy model with a 2 × 2 block, averaging gains more than encoding loses, and majority rule wins, matching what is known of the 2D Ising model and what LIT-873’s algorithm found. Maximizing I(V_Λ : E) first and then encoding is not optimal.
Standing in the record
Filed on 2026-10-09 at the owner’s request, as one of the works cited by the batch on hierarchy and hyperbolic geometry (nucleation#113, LIT-865 to LIT-877) that neither record held. The batch work that cites it is Koch-Janusz and Ringel’s Mutual information, neural networks and the renormalization group (LIT-873), which names it as the authors’ sequel and the place to look for a proof beyond 1D. Read on its own merits (NOTE-682); the reading produces THEORY-198.
It supplies part of what LIT-873 and THEORY-194 lacked: a proof, in any
dimension under a stated extra assumption, that keeping all the
information about the far environment forbids the coarse Hamiltonian from
gaining range. It does not meet THEORY-194’s promote_when, which asks
for a proof about the maximizer: the theorem is about perfect capture,
which the maximizer under the usual constraints does not reach, and the
evidence that partial capture still controls the range is two-spin
blocks of 1D chains.
It does not bear on QUESTION-025 beyond the analogy LIT-873 already gives. Its levels are nested spatial blocks, not attributes; there is no co-occurrence, embedding or concept lattice.
In the record it sits beside the information bottleneck (LIT-338), of which the authors call RSMI a realization with a fixed compressed variable; beside the Markov-blanket account THEORY-073 (my connection, not the paper’s), since its central lemma is a conditional-independence statement about a screening region whose placement, like a blanket’s, is chosen before anything is computed; and near Kupiainen’s comment on rigorous RG (LIT-039), read on 2026-10-14 (NOTE-821). The other sequel LIT-873 names, Gökmen, Ringel, Huber and Koch-Janusz, Statistical physics through the lens of real-space mutual information (arXiv:2101.11633), is filed in the same pass as LIT-878.