nucleation

LIT-878: Statistical Physics through the Lens of Real-Space Mutual Information

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StatusActive — read 2026-10-09 (NOTE-681); worth reading as the paper that turns the real-space mutual-information principle of LIT-873 from an iterated coarse-graining into a single-shot extraction of operators: with the mutual information estimated by a neural InfoNCE bound and the coarse-graining a convolutional filter trained against it, the filters that best compress what a block shares with its distant environment are, across the whole phase diagram of the interacting dimer model, the columnar and plaquette order parameters below the BKT transition and the electric fields at high temperature, and the plaquette filter's correlator gives a scaling dimension of 1.00037 against the predicted 1. The maximal information itself, log 4 in the ordered phase and decaying algebraically with the buffer in the critical one, traces the phase diagram. The theorem that the optimal filters are the most relevant operators is cited to another paper, and the model's operators were known beforehand.
Version2
Tagsnatural-sciences, information-theory, representation-learning, anthology-candidate
Cited as “paper” byNOTE-681 — Statistical Physics through the Lens of Real-Space Mutual Information
Cited as “source” byTHEORY-194 — On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers' low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead

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Doruk Efe Gökmen, Zohar Ringel, Sebastian D. Huber and Maciej Koch-Janusz (2021), Physical Review Letters 127(24):240603 — ARXIV-2101.11633, DOI-10.1103/PhysRevLett.127.240603

Key takeaways

  • One step instead of a flow. The real-space mutual-information (RSMI) principle of LIT-873 chooses the coarse variable H of a block V to maximize I_Λ(H : E) with an environment E beyond a buffer B. Here the buffer width L_B plays the role of the RG scale: the filters are optimized once, at large L_B, at each point of the phase diagram, and read as the operators whose correlations survive that distance, rather than iterated as a coarse-graining. That the formal optimum is set by the most relevant operators is cited to Gordon, Banerjee, Koch-Janusz and Ringel (Phys. Rev. Lett. 126, 240601, 2021), not shown here.
  • The estimator is what changed. I_Λ is bounded below by InfoNCE with a separable neural critic f_Θ(h, e) = v(h)ᵀu(e); the coarse-graining is a one-layer CNN followed by a Gumbel-softmax step that keeps H (pseudo-)binary; both are trained together by Adam on Monte Carlo samples. Runs take about 30 s, three to four orders of magnitude faster than LIT-873’s RBM proxy, on systems up to 256 × 256. The bound is usable because RSMI is small: I_Λ ≤ N_V log n by data processing.
  • What it finds on the interacting dimer model (dimers on the square lattice with an aligning interaction; columnar order below a BKT transition at T_BKT = 0.65, a critical phase described by a sine-Gordon theory above it). With a two-bit H: the maximal I_Λ is log 4 below T_BKT (which of the four columnar states the system is in) and decays algebraically with the buffer above it. The optimal filters are columnar and plaquette filters at low T, which coincide with the dimer symmetry-breaking order parameter of Alet et al. and the electric charge operators cos(nφ), sin(nφ) for n = 1, 2, and staggered filters at high T, which read the coarse-grained height gradient, the electric field. The columnar filter disappears above T_BKT, as the n² scaling of the charge operators’ dimensions predicts.
  • Filters as operators. Correlating the trained plaquette filter on 128 × 128 samples at T → ∞ gives a power law with exponent 2.00074, hence a scaling dimension of 1.00037 against the predicted 1; the columnar correlator decays roughly as the predicted r⁻⁸ but is not fitted. Any rotation of the two staggered filters keeps the same information, which the authors read as the free dimer model’s emergent U(1) symmetry.
  • What it rests on, and what is elsewhere. Every operator found was known from the model’s field theory, and the paper says so in using it as a dictionary. The Ising, symmetry and non-equilibrium examples are in the companion paper (arXiv:2103.16887, Phys. Rev. E 104, 064106, 2021), not here.

Standing in the record

Filed on 2026-10-09 at the owner’s request, as one of the works cited by the second part of the batch on hierarchy and hyperbolic geometry (nucleation#113, LIT-865 to LIT-877) that neither record held. It is cited by Koch-Janusz and Ringel’s Mutual information, neural networks and the renormalization group (LIT-873) as the authors’ sequel. Read on its own merits (NOTE-681); it files no THEORY.

It extends the account LIT-873 grounds, THEORY-194, to a third model, across a phase transition, with a different estimator, but on a model whose relevant operators were known in advance, so it does not meet that account’s promote_when. Its mutual-information estimator is the family read in Poole et al. (LIT-246) and MINE (LIT-256), both held here and read; InfoNCE’s source, van den Oord et al., is held in the anthology (ANTH-LIT-589). Its other antecedents are not held: Lenggenhager et al.’s Optimal Renormalization Group Transformation from Information Theory (arXiv:1809.09632) and the optimality theorem of Gordon et al. (Phys. Rev. Lett. 126, 240601).

It does not bear on QUESTION-025 beyond analogy: its levels are spatial scales and its variables those of a lattice model, not attributes of words. NOTE-681 records one parallel worth keeping, that a mutual-information objective here fixes a subspace of filters and leaves the basis within it to symmetry.

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