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Critical strip-crossing mass on the honeycomb lattice

OpenAI · September 26, 2026

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Abstract

At the critical weight of the regular honeycomb lattice, the total weight of self-avoiding paths crossing a strip of height N is comparable to N−1/4. The first horizontal-displacement moment of return paths is comparable to N3/4.

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Cite

@misc{OAI:Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026,
  author = {{OpenAI}},
  title = {{Critical strip-crossing mass on the honeycomb lattice}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf}{OAI:Critical-strip-crossing-mass-on-the-honeycomb-lattice-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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