Intermediate-area cylinders in the principal quadratic stratum
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Abstract
Delecroix asked for the area spectrum of cylinders on large-genus square-tiled surfaces, highlighting the normalized-area window [g^(−1/2),2g^(−1/2)]. For the principal holomorphic quadratic stratum, with the number of squares tending to infinity before the genus, we derive a Poisson limit at every intermediate scale s_g→0, g s_g→∞. The rescaled cylinder areas converge to a Poisson point process of intensity dx/(2x); compact-window counts converge with every fixed joint moment. The highlighted count tends to Poisson((log2)/2), and its mean and variance have the same limit. The proof extracts a uniform Gamma/Dirichlet approximation from the published formulas of Delecroix–Liu and Delecroix–Goujard–Zograf–Zorich, then applies classical assembly identities and elementary tail estimates. The weighted permutation model and its unmarked Poisson approximation are prior results. We explicitly credit the earlier microscopic announcements and retain their unresolved normalization discrepancy with the short problem report. AI tools were used extensively; no external human peer review is asserted.
The idea in plain English
Square-tiled surfaces split into maximal horizontal cylinders. This paper studies the fraction of the total surface area occupied by each cylinder in the principal holomorphic quadratic stratum.
First fix the genus and let the maximum number of squares tend to infinity. Then let the genus grow. At every shrinking area scale that is still large compared with the reciprocal genus, the geometric cylinder-area counts have a Poisson limit.
The conclusion includes the whole rescaled point process and every fixed joint moment of counts in finitely many disjoint compact windows. It is stronger than only identifying the mean in one highlighted window.
The precise result
Fix g≥2. Sample uniformly from isomorphism classes of square-tiled half-translation surfaces in Q(1^(4g−4)) made from at most N unit squares, allow all positive cylinder heights, count each maximal horizontal cylinder once, and divide its area by total surface area. First N→∞ gives the law ν_g; the customary automorphism weighting gives the same fixed-genus limit. Only afterward does g→∞.
For every deterministic positive s_g with s_g→0 and g*s_g→∞, the actual geometric point measure Σ_i δ_(X_i/s_g) converges vaguely on (0,∞) to PPP(dx/(2x)). This is the marked normalized-area process, not only the auxiliary unmarked component model.
For any fixed finite family of pairwise disjoint compact intervals [a_l,b_l]⊂(0,∞), the counts converge jointly to independent Poisson variables with means (1/2)log(b_l/a_l), and every fixed joint moment converges.
The count in [g^(−1/2),2g^(−1/2)] converges to Poisson((log2)/2); its mean and variance tend to (log2)/2, and its zero-count probability tends to 2^(−1/2). The vertical marginal has the same law by symmetry.
Why the result holds
Published geometric graph formulas give a uniform comparison with the prior weighted-permutation law a_j=ζ(2j)/(2j). Its area marks are normalized independent Gamma variables, conditionally giving a Dirichlet vector.
Classical assembly identities give the unmarked intermediate-window law. An expected weighted Gamma-mark error, split across small, intermediate and large component sizes, vanishes even when g*s_g tends to infinity arbitrarily slowly.
Concentration of the total Gamma mass gives the normalized-area process. Uniform bounds on higher compact-window count moments, a uniform good-event density comparison and negligible exceptional moments transfer every fixed joint moment to the geometry; total variation alone is not used to transfer unbounded expectations.
Scope and limitations
The theorem concerns the principal holomorphic quadratic stratum with the square-count limit before the genus. It supplies neither a diagonal N(g) limit, arbitrary-stratum extension nor a simultaneous all-direction statement on one surface.
The microscopic regime g*s_g=O(1), the macroscopic regime s_g of order1 and relative accuracy for arbitrary rare events are outside the result. The statement addresses intermediate areas rather than every meaning of a full area spectrum.
The exact Gamma total and Liu’s thesis use 6g−6, while the short problem report prints 4g−4. These give different fixed microscopic laws and remain distinct; the paper makes no new microscopic theorem or correction of that report.
Delecroix–Liu’s earlier microscopic announcements, Liu’s all-multiplicity thesis law and the primitive conjectural slide version are prior intellectual contributions. The complete announced project may contain additional overlapping results; its contents and priority remain uncertain. No first-priority or continued-open-status claim is made.
The established geometric formulas and uniform intersection-number estimates, exact weighted model, selected-component identity and classical unmarked Poisson approximation receive prior credit. The contribution here is the explicit geometric comparison, Gamma-marking transfer and all-fixed-moment consequence.
Status and AI use
Unrefereed preprint, version 1.0, developed with extensive AI assistance. No first-priority claim, external human peer review or formal proof certification is asserted.
AI tools were used extensively in the argument, proof drafting, computation, literature research, internal adversarial checking and preparation. The named author is responsible for the submitted work. Internal AI checking is not human peer review; finite controls do not replace the written limiting proof or the established geometric inputs.
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Cite this paper
Alec Kriebel. (2026). Intermediate-area cylinders in the principal quadratic stratum (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23387774
Background and prior work
- A. Aggarwal, Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials, Invent. Math. 226 (2021), 897–1010: Underlying uniform intersection-number estimates, used through their explicit form in Delecroix–Liu.
- R. Arratia and S. Tavaré, Independent process approximations for random combinatorial structures, Adv. Math. 104 (1994), 90–154: Lemma 5, equation (126): the selected-component factorial identity is prior work.
- R. Arratia, A. D. Barbour and S. Tavaré, Limits of logarithmic combinatorial structures, Ann. Probab. 28 (2000), 1620–1644: Theorem 3.1 already supplies the unmarked Poisson approximation at every intermediate o(n) window rate.
- V. Delecroix, What we do not know(yet) about square-tiled surfaces in large genus?, OWR 41/2024, 2384–2387: Problem 2 gives the highlighted intermediate-area question in the principal ensemble; Problem 3 is the separate diagonal problem. Report published 14 February 2025.
- V. Delecroix, É. Goujard, P. Zograf and A. Zorich, Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves, Invent. Math. 230 (2022), 123–224: Prior weighted-permutation model, geometric correspondence, cylinder-count generating function and uniform low-component graph bounds.
- V. Delecroix and M. Liu, Length partition of random multicurves on large genus hyperbolic surfaces, JEMS 27 (2025), 5093–5131; online 7 June 2024: Exact finite-genus geometric area law, positive graph mixture and uniform coefficient expansion; also credits the earlier announced small-components work.
- M. Liu, Random multi-geodesics on hyperbolic surfaces, doctoral thesis, Université Paris Cité, 2022: Remark 1.5.9, p.22: prior all-multiplicity microscopic law with 6g−6 normalization, distinct from the short report’s 4g−4 display.
- M. Liu, Durham Spectra and Moduli Seminar slides, February 2025: Slides 26 and 195 distinguish the primitive setting and conjectural microscopic presentation with its 1/(2x) tail. No unavailable project contents are assumed.