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Perfect t-embeddings: existence

Primarily AI-generated textHuman understanding: some partsmath.MP — Mathematical Physicsmath-ph — Mathematical Physics

Contributed by Mikhail Basok ↗, Dmitry Chelkak ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

This note gives an affirmative answer to an open question posed in [Chelkak, Laslier, Russkikh'21]. Namely, it demonstrates the existence of perfect t-embeddings for each sufficiently non-degenerate planar bipartite graph GG with positive edge weights. The non-degeneracy condition on GG is formulated in terms of its perfect and almost perfect matchings and does not concern the choice of edge weights. The proof and the main part of the writing were produced by GPT-6 Astra in response to a series of prompts. We manually added Section 1.1 to put this result into a bigger context and slightly edited the introduction in order to improve the readability. We plan to provide a better human-digested version of this result later.

Provenance statement

The proof and the main part of the writing were produced by GPT-6 Astra in response to a series of prompts.

Tools used

OpenAI
Codex

References

  1. Mikhail Basok, Dmitry Chelkak, Benoît Laslier, and Marianna Russkikh, Harmonic functions on Tutte embeddings and linearized Monge-Ampère equation , arXiv:2511.06587 [math-ph], 2025. https://doi.org/10.48550/arXiv.2511.06587.arXivDOI
  2. Mikhail Basok, Benoît Laslier, and Marianna Russkikh, T-embeddings as stationary measures on T-graphs , arXiv:2610.24180 [math-ph], 2026. https://doi.org/10.48550/arXiv.2610.24180.arXivDOI
  3. Tomas Berggren, Matthew Nicoletti, and Marianna Russkikh, Perfect t-embeddings and lozenge tilings , arXiv:2408.05441 [math.PR], 2024. https://doi.org/10.48550/arXiv.2408.05441.arXivDOI
  4. Tomas Berggren, Matthew Nicoletti, and Marianna Russkikh, Perfect t-embeddings of uniformly weighted Aztec diamonds and tower graphs , International Mathematics Research Notices 2024 (2024), no. 7, 5963--6007. https://doi.org/10.1093/imrn/rnad299.DOI
  5. Tomas Berggren, Matthew Nicoletti, and Marianna Russkikh, Perfect t-embeddings of doubly periodic Aztec diamonds , arXiv:2508.04938 [math.PR], 2025. https://doi.org/10.48550/arXiv.2508.04938.arXivDOI
  6. Dmitry Chelkak, Ising model and s-embeddings of planar graphs , Annales Scientifiques de l'École Normale Supérieure 57 (2024), no. 5, 1271--1346. https://doi.org/10.24033/asens.2591.DOI
  7. Dmitry Chelkak, Benoît Laslier, and Marianna Russkikh, Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces , arXiv:2109.06272 [math.PR], 2021. https://doi.org/10.48550/arXiv.2109.06272.arXivDOI
  8. Dmitry Chelkak, Benoît Laslier, and Marianna Russkikh, Dimer model and holomorphic functions on t-embeddings of planar graphs , Proceedings of the London Mathematical Society 126 (2023), no. 5, 1656--1739. https://doi.org/10.1112/plms.12516.DOI
  9. Dmitry Chelkak and Sanjay Ramassamy, Fluctuations in the Aztec diamonds via a space-like maximal surface in Minkowski 3-space , Confluentes Mathematici 16 (2024), 1--17. https://doi.org/10.5802/cml.94.DOI
  10. Pavel Galashin, Amplituhedra and origami, I: tree level , arXiv:2410.09574 [hep-th], 2024. https://doi.org/10.48550/arXiv.2410.09574.arXivDOI
Show all 14 referencesShow fewer references
  1. Pavel Galashin, Amplituhedra and origami, II: loop level , arXiv:2606.03439 [hep-th], 2026. https://doi.org/10.48550/arXiv.2606.03439.arXivDOI
  2. Richard Kenyon, Wai Yeung Lam, Sanjay Ramassamy, and Marianna Russkikh, Dimers and circle patterns , Annales Scientifiques de l'École Normale Supérieure 55 (2022), no. 3, 865--903. https://doi.org/10.24033/asens.2507.DOI
  3. Richard W. Kenyon and Scott Sheffield, Dimers, tilings and trees , Journal of Combinatorial Theory, Series B 92 (2004), no. 2, 295--317. https://doi.org/10.1016/j.jctb.2004.07.001.DOI
  4. David Keating and Hieu Trung Vu, Perfect t-Embeddings of Uniformly Weighted Generalized Tower Graphs , arXiv:2509.18791 [math-ph], 2025. https://doi.org/10.48550/arXiv.2509.18791.arXivDOI

Version history

  1. v1Submitted by Mikhail BasokInitial depositCurrentOct 24, 2026