diff --git a/notes/literature-sweep-torus-rank-observables-20260912.md b/notes/literature-sweep-torus-rank-observables-20260912.md new file mode 100644 index 00000000..c4be3c12 --- /dev/null +++ b/notes/literature-sweep-torus-rank-observables-20260912.md @@ -0,0 +1,320 @@ +# Literature sweep: torus rank observables, exact finite-size site polynomials, matching dictionary + +2026-09-12. Executed at the owner's request ("broadly search the related +literature") as a bounded novelty/prior-art sweep around PR #733 and the +#711/#717 dictionary correction. It is a **retrieval note**, not a result. +Nothing here is a claim about our data, and nothing here makes any output +publication-ready. + +## 0. Scope and how this was searched + +Sources: arXiv API (`export.arxiv.org/api/query`, relevance-sorted, +`max_results` 8–10 per query) and general web search. Date: 2026-09-12. + +Query strings actually used (arXiv `all:` / `abs:` fields): + +```text +all:"critical polynomial" AND all:"percolation" +all:"wrapping probability" AND all:"percolation" +all:"percolation" AND all:"homology" AND all:"torus" +all:"site percolation" AND all:"transfer matrix" +all:"matching lattice" AND all:"percolation" +all:"matching polynomial" AND all:"percolation" +all:"Pinson" AND all:"percolation" +all:"percolation" AND all:"conditional sampling" +all:"percolation" AND all:"winding" AND all:"torus" +all:"homological percolation" +all:"percolation" AND all:"Euler characteristic" AND all:"torus" +all:"Arguin" AND all:"percolation" AND all:"torus" +all:"Scullard" AND all:"critical polynomial" +``` + +Plus targeted web queries on: Mertens–Ziff finite matching lattices; torus +wrapping/homology probabilities; Pinson's analytic formula; exact site +percolation probabilities on plane/cylinder/torus; Jacobsen's site-percolation +critical polynomials; homological percolation on a torus; Newman–Ziff wrapping +estimators. + +**Search boundary (must be stated wherever this is cited).** No MathSciNet, +zbMATH, Scopus or Google Scholar citation-graph traversal; no non-English +literature; no books or theses; no patent search; no systematic forward +citation chasing of Pinson (1994) or Arguin (2001). Several entries below are +known only through secondary quoting in arXiv preprints, and are flagged as +such. **`NO_MATCH_FOUND_IN_THIS_SEARCH` below means no match in this sweep, not +a novelty proof.** + +## 1. The one finding that matters most: our rank variable already has a name + +The central observable of this repository is + +```text +r(omega) = rank im[ H_1(K(omega)) -> H_1(T^2) ] in {0,1,2}, X = r - 1. +``` + +The **inclusion-induced map on first homology of the occupied subcomplex into +the ambient torus is exactly the object** studied, under different names, by: + +- **Langlands, Pouliot, Saint-Aubin (1994)** — introduced crossing + probabilities on compact Riemann surfaces; for percolation on a surface `S` + they consider `phi: H_1(X_s) -> H_1(S)` and the probabilities `pi_G` that the + image is a given subgroup `G <= H_1(S)`. + (Quoted in arXiv:1402.0879 and in arXiv:2011.11903v4 §1 as [LPSA94].) +- **H. T. Pinson, "Critical spanning probability on a torus", J. Stat. Phys. 75, + 1167 (1994)** — analytic expressions for those subgroup probabilities as + functions of the torus modular parameter, via an orientation argument on + cluster-boundary curves; rigorous except for the mesh-to-zero limit, which + uses a Nienhuis renormalization-group step. + (Quoted throughout arXiv:0812.2925v2 §1 as [17] and arXiv:0905.3521.) +- **L.-P. Arguin, "Homology of Fortuin–Kasteleyn clusters of Potts models on the + torus", arXiv:hep-th/0111193** — extends Pinson to FK clusters of the + Q-state Potts model, `Q in [1,4]`, with closed forms in Jacobi theta + functions for `Q = 1` (percolation) and `Q = 2`; `Q = 1` is our case. +- **Morin-Duchesne & Saint-Aubin, arXiv:0812.2925v2** — asymptotic behaviour of + `pi({a,b})` for thin tori; exponents tied to weights `h_{r,s}` of the extended + Kac table with half-integer entries as well. +- **Duncan, Kahle & Schweinhart, "Homological percolation on a torus: plaquettes + and permutohedra", arXiv:2011.11903v4; Ann. IHP 61, 2235 (2025)** — for a + random subcomplex `S` of `T^d` they take the natural inclusion + `phi: S -> T^d` and call nontrivial elements of + `im[ phi_*: H_i(S;Q) -> H_i(T^d;Q) ]` **giant cycles**. They prove a sharp + transition (via Friedgut–Kalai), convergence of the threshold function, and + `p_c = 1/2` in the middle dimension `i = d/2`. They explicitly cite + [LPSA94], [Pin94] and [MDSA09] as the 2-dimensional precursors. + +**Dictionary that follows immediately, and that I have not seen written down in +our notes:** + +```text +our P0 <-> pi({0}) (trivial ambient image) +our P1 <-> sum over coprime (a,b) of pi({a,b}) (rank-one primitive family) +our P2 <-> pi(Z x Z) (rank-two / cross topology) +``` + +so that `A_top = = P2 - P0` is a difference of two Pinson/Arguin subgroup +probabilities, and `E_top = P2 + P0` is their sum. If that identification is +correct at the level of events, then **the scaling-limit values of our two +coordinates are already known in closed form (Jacobi theta / Dedekind eta)**, +and the finite-size object we have been fitting is the correction to them. + +Two cautions that must travel with this: + +1. Pinson/Arguin is a **scaling-limit** statement (plus one non-rigorous + renormalization step). Our numbers are **exact finite-size**. Agreement is + expected only after finite-size corrections, and the *shape* of those + corrections is precisely what this repository has been arguing about. +2. The subgroup lattice of `H_1(T^2)` is what our `r in {0,1,2}` is recording. + This is prior art for the *observable*, not for our exact finite-width + computations, and not for the finite-amplitude source work in #733. + +**Verdict: PRIOR_ART_FOUND for the observable definition and (in the continuum +limit) for its distribution. This is the single most important thing to tell +any referee, and it is better coming from us.** + +### 1a. The thin-torus limit is our width-4 regime + +Morin-Duchesne & Saint-Aubin compute the asymptotics of `pi({1,0})` along +`tau_r = 0, tau_i -> infinity`, i.e. exactly the geometry of a **fixed-width +strip whose length grows** — the same aspect-ratio regime as our width-3/width-4 +closures. Their exponents come from Kac-table weights including half-integer +indices. Any claim we make about an orientation-sensitive, aspect-dependent +finite-size exponent should be checked against that asymptotic family before +being called new. + +### 1b. Euler characteristic zero vs homological threshold + +**Bobrowski & Skraba, Phys. Rev. E 101, 032304 (2020), arXiv:1910.10146** — +numerical evidence across four models (site percolation on cubical and +permutahedral lattices, Poisson–Boolean, Gaussian random fields; flat torus, +`d = 2,3,4`) that **the zeros of the expected Euler characteristic curve +approximate the homological-percolation thresholds**, with a discussion of the +approximation error. + +This is directly adjacent to our use of the Sykes–Essam matching polynomial +(`chi(p)`) and of `M_B(p) = P2 - P0` as a threshold functional. It is a +different model class (higher-dimensional / continuum) and it is numerical, but +"EC-zero approximates the topological threshold" is now an explicit, published +heuristic with a named error. **Verdict: PARTIAL_OVERLAP; cite it when we use +an Euler-characteristic zero as a threshold estimator.** + +## 2. Exact finite-size site percolation: who has computed what, to what size + +This is the competitive landscape for PR #733's width-4 exact rank closure and +for #708's automaton. + +| Work | Object | Reach | Relation to us | +|---|---|---|---| +| **Akhunzhanov, Eserkepov, Tarasevich, J. Phys. A 55, 204004 (2022), arXiv:2204.01517** | exact percolation-probability **polynomials**, site percolation on `L x L`: plane (crossing), cylinder (spanning), **torus (wrapping along one direction)** | `L <= 17` plane, `L <= 16` cylinder, **`L <= 12` torus**; dynamic programming + topology-based state reduction; divisibility properties proved; naive FSS gives `p_c = 0.59269` | Closest competitor. They reach full two-dimensional `L x L` tori; we reach **fixed width 4 at arbitrary length** with a rank decomposition they do not make. Their `R^{(e)}, R^{(1)}, R^{(b)}, R^{(h)}, R^{(v)}` are *direction* events; our `r` is an *ambient-rank* event. The map between them is a dictionary exercise we have not done. | +| **Mertens, J. Phys. A 55, 244002 (2022), arXiv:2109.12102** (already in `references.bib`) | exact `R_n(p)` for spanning an `n x n` square (open boundaries) | `n <= 24`, `O(lambda^n)`, `lambda ~ 2.6` | Open boundary, spanning — not torus, not rank. Prior art for the *technique* of exact enumeration + extrapolation, not for our observable. | +| **Newman & Ziff, PRL 85, 4104 (2000), arXiv:cond-mat/0005264** | microcanonical union-find MC; `p_c = 0.59274621(13)` site/square | MC, not exact | The number everyone compares against. | +| **Newman & Ziff, cond-mat/0203496 ("Convergence of threshold estimates")** | exact enumeration of `R_L(p)` for site percolation on `L x L`, crossing | Table up to `L = 7` (L=2..5 from Reynolds–Stanley–Klein; 6,7 from Ziff 1992) | Small-size exact crossing polynomials; superseded in reach but useful as an independent check of any small-L polynomial we produce. | +| **Jacobsen, J. Phys. A 47, 135001 (2014), arXiv:1401.7847** | graph polynomials `P_B(q,v)` and site polynomials `P_B(p)` via a transfer matrix in the **periodic Temperley–Lieb algebra**; "We discuss in detail the role of the symmetries and the embedding of `B`" | bases to 882 edges (bond), site polynomials with up to 243 vertices; `p_c` to ~1e-8..1e-9 | **The symmetry-and-embedding discussion is the direct precedent for our D4/lumping work.** Different algebra (pTL vs our finite automaton), different target (thresholds vs rank decomposition). | +| **Jacobsen, arXiv:1507.03027** (in `references.bib`) | eigenvalue identities in periodic TL; semi-infinite cylinders of circumference `n` | site square `n_max = 21`; `p_c = 0.59274605079210(2)` | Already read for #681/#717. | +| **Scullard, arXiv:1111.1061 ("The percolation critical polynomial as a graph invariant")** | critical polynomial as graph invariant | — | Background for the #717 dictionary question; not yet read at section level here. | +| **arXiv:2010.02887** ("Critical polynomials in the nonplanar and continuum percolation models") | non-planar / continuum extensions | — | Flagged, unread at section level. | +| **arXiv:2205.02734** ("Percolation critical probabilities of matching lattice-pairs") | matching-lattice pair thresholds | — | Flagged, unread at section level; potentially relevant to the `p_c + p_c(NN+NNN) = 1` anchor. | + +**Verdict for this group:** the *exact finite-size* frontier on full `L x L` +tori is `L <= 12` (AET 2022). Our contribution is not "bigger `L`" — it is a +different decomposition (ambient rank) and a fixed-width/arbitrary-length +regime. Say it that way. + +## 3. Matching lattice / critical polynomial dictionary (the #711/#717 channel) + +- **Mertens & Ziff, Phys. Rev. E 94, 062152 (2016), arXiv:1603.07289** — already + PRIMARY_TEXT_READ in `notes/finite-critical-polynomial-dictionary-20260912.md`. + Finite-size generalisation of Sykes–Essam: + `N_L(p) - N̂_L(1-p) - L^2 chi(p) = R_L^x(p) - R̂_L^x(1-p)`. +- **Mertens, Jensen & Ziff, Phys. Rev. E 96, 052119 (2017), arXiv:1602.00644** — + cluster-number universality; uses the Sykes–Essam matching polynomial for + exact lattice/matching-lattice relations. Not yet read at section level. +- **Sykes & Essam (1964)** — in `references.bib`. `chi(p) = p - 2p^2 + p^4` for + the square/NN+NNN pair; `chi_Delta(p) = p - 3p^2 + 2p^3` for self-matching. +- **arXiv:1906.10543** ("Critical p = 1/2 in percolation on semi-infinite + strips") — flagged, unread. + +**Verdict:** the dictionary note in #733 is consistent with the primary texts, +and this sweep found **nothing that contradicts it**. It did find that the +surrounding literature is larger than the three texts we read, in particular +on the matching-lattice-pair side. + +## 4. Symmetry reduction and lumping + +- **May & Wierman, Combin. Probab. Comput. 14, 549 (2005)** — use the graph + **automorphism group** to reduce the computational work of the substitution + method; on `(3,12^2)` bond percolation they cut the bound interval width by + 62%. Methodologically the closest published relative of "reduce by symmetry + before you enumerate" — but it is substitution-method bound improvement, not + a transfer-matrix lumping, and it does not produce state counts. +- **Jacobsen (2014), arXiv:1401.7847** — explicit discussion of the role of + symmetries and of the embedding of the basis in the transfer-matrix + construction. +- No arXiv hit for `abs:"lumping" AND abs:"Markov" AND all:"percolation"`. + +**Verdict:** the *idea* of symmetry reduction is standard; the specific result +in #733 — seven D4 column-probability types, common strong lumpings of size +94/303/179/262/509/303/509, each equal blockwise to the colour-preserving D4 +orbit partition — has **NO_MATCH_FOUND_IN_THIS_SEARCH**. That is a weak +statement and should be re-checked against the transfer-matrix literature +(Jacobsen, Jensen, Enting, Guttmann) before being used in any novelty claim. + +## 5. Site sources and conditioned/rare-sector sampling + +- `all:"percolation" AND all:"conditional sampling"` returned **zero** arXiv + hits. +- Web searches for a point/dipole site-source response of the specific kind in + #733 (zero linear response, nonzero mixed response; four-sign finite-amplitude + extraction by degree-<=2 multiaffinity) returned nothing on topic — the + "dipole" hits are electromagnetic-wave localization, unrelated. +- Exact backward/conditional samplers that condition on a **topological** + final state (our final-rank conditioning) have **NO_MATCH_FOUND_IN_THIS_SEARCH**. + +**Verdict: NO_MATCH_FOUND_IN_THIS_SEARCH for both.** Given the search boundary +in §0, this is the weakest kind of evidence and must not be phrased as novelty. +It does suggest these two are the least contested parts of #733, which is an +argument for writing them up first — not for claiming them. + +## 6. What I would do with this, in priority order + +1. **Check the Pinson/Arguin dictionary numerically before anything else.** + Take the `Q = 1` closed forms (`pi({0})`, `pi(Z x Z)`, `pi({a,b})` in Jacobi + theta functions, as restated with full formulas in arXiv:0905.3521 §"crossing + probabilities", which gives `P_{a,b}(r)` and `P_X(r)` explicitly in terms of + `Z_{m,n}(g,r)` and the Dedekind eta function). Evaluate at our aspect ratios. + Compare with the width-4 exact `P0/P1/P2`. Report the finite-size discrepancy + as the object of interest. **This converts "we think there is a + matching-odd structure" into "here is our measured deviation from the known + continuum answer", which is a much stronger sentence.** +2. **Cross-validate against AET 2022 (arXiv:2204.01517) wherever our geometries + overlap** (their torus polynomials reach `L = 12`; supplemental material + contains the polynomials). Even a single shared small case is worth more than + an internal consistency check. +3. **Read arXiv:2205.02734 and arXiv:1111.1061 at section level** before any + further statement about the matching-`p_c` anchor or about critical-polynomial + novelty. Both are cheap and both sit directly on the #711/#717 channel. +4. **Do not** phrase the rank observable as new. Phrase the finite-width exact + rank decomposition, the source response and the conditioned oracle as the + contributions, and cite Pinson/Arguin/DKS as the definition's origin. + +## 7. BibTeX for the entries not yet in `references.bib` + +Not added to `references.bib` by this note; copied here so the decision to add +is explicit and separate. + +```bibtex +@article{pinson1994, + author = {Pinson, H. T.}, + title = {Critical spanning probability on a torus}, + journal = {Journal of Statistical Physics}, + volume = {75}, pages = {1167--1177}, year = {1994}} + % known to us only via secondary quoting (arXiv:0812.2925, 0905.3521, 1402.0879) + +@article{arguin2001, + author = {Arguin, L.-P.}, + title = {Homology of {Fortuin--Kasteleyn} clusters of {Potts} models on the torus}, + journal = {Journal of Statistical Physics}, year = {2002}, + eprint = {hep-th/0111193}, archivePrefix = {arXiv}} + +@article{mdsa2009, + author = {Morin-Duchesne, Alexi and Saint-Aubin, Yvan}, + title = {Critical exponents for the homology of {Fortuin--Kasteleyn} clusters on a torus}, + eprint = {0812.2925}, archivePrefix = {arXiv}} + +@article{dks2025, + author = {Duncan, Paul and Kahle, Matthew and Schweinhart, Benjamin}, + title = {Homological percolation on a torus: plaquettes and permutohedra}, + journal = {Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques}, + volume = {61}, number = {3}, pages = {2235--2261}, year = {2025}, + eprint = {2011.11903}, archivePrefix = {arXiv}} + +@article{bobrowski2020, + author = {Bobrowski, Omer and Skraba, Primoz}, + title = {Homological percolation and the {Euler} characteristic}, + journal = {Physical Review E}, volume = {101}, pages = {032304}, year = {2020}, + eprint = {1910.10146}, archivePrefix = {arXiv}} + +@article{aet2022, + author = {Akhunzhanov, R. K. and Eserkepov, A. V. and Tarasevich, Yu. Yu.}, + title = {Exact percolation probabilities for a square lattice: site percolation on a plane, cylinder, and torus}, + journal = {Journal of Physics A: Mathematical and Theoretical}, + volume = {55}, pages = {204004}, year = {2022}, + eprint = {2204.01517}, archivePrefix = {arXiv}} + +@article{jacobsen2014, + author = {Jacobsen, Jesper Lykke}, + title = {High-precision percolation thresholds and {Potts}-model critical manifolds from graph polynomials}, + journal = {Journal of Physics A: Mathematical and Theoretical}, + volume = {47}, pages = {135001}, year = {2014}, + eprint = {1401.7847}, archivePrefix = {arXiv}} + +@article{newmanziff2000, + author = {Newman, M. E. J. and Ziff, R. M.}, + title = {Efficient {Monte Carlo} algorithm and high-precision results for percolation}, + journal = {Physical Review Letters}, volume = {85}, pages = {4104}, year = {2000}, + eprint = {cond-mat/0005264}, archivePrefix = {arXiv}} + +@article{newmanziff2002, + author = {Newman, M. E. J. and Ziff, R. M.}, + title = {Convergence of threshold estimates for two-dimensional percolation}, + eprint = {cond-mat/0203496}, archivePrefix = {arXiv}} + +@article{scullard2011, + author = {Scullard, Christian R.}, + title = {The percolation critical polynomial as a graph invariant}, + eprint = {1111.1061}, archivePrefix = {arXiv}} + +@article{wierman2005, + author = {May, William D. and Wierman, John C.}, + title = {Using symmetry to improve percolation threshold bounds}, + journal = {Combinatorics, Probability and Computing}, + volume = {14}, pages = {549--566}, year = {2005}} +``` + +## 8. Standing cautions + +- A missing formula in the texts read is not proof that no prior formula exists. + §5 rests entirely on this negative and should be treated as provisional. +- Everything above is literature positioning. No number in this note has been + recomputed against our own data; §6.1 is the first place where that must + happen. +- This note does not authorize new acquisition, a next-width scan, a new + transfer engine, or any STATUS promotion.