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Radial transfer estimates and polygon length laws for honeycomb walks

OpenAI · September 26, 2026

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Abstract

We prove critical diameter-tail exponents −2 for unrooted honeycomb polygons and −2/3 for length-weighted polygons, together with a truncated second-length-moment bound of exponent 2/3. Consequently, polygons conditioned to have length at least n have diameter $n^{3/4+o(1)}$ in probability under either weight.

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Cite

@misc{OAI:Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026,
  author = {{OpenAI}},
  title = {{Radial transfer estimates and polygon length laws for honeycomb walks}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf}{OAI:Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/237.md. The Lean statements have not been checked by people against the paper's claims.

Family 237: The three-quarter exponent for honeycomb self-avoiding walk

Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter $n^{3/4+o(1)}$. Its local mass and covering numbers have exponent 4/3. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.

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