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Cap-selected amplitudes and triangle chords for honeycomb walks

OpenAI · September 26, 2026

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Abstract

At the critical honeycomb fugacity, self-avoiding port-to-port chords in an equilateral lattice triangle of side R, summed over both boundary endpoints and restricted to diameter at least $R/100$, have partition sum comparable to R3/4 and mean length $R^{4/3+o(1)}$. We also determine the amplitude selected by two vacuum caps on a cylinder of circumference N and prove that it grows as $N^{1/6+o(1)}$.

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Cite

@misc{OAI:Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026,
  author = {{OpenAI}},
  title = {{Cap-selected amplitudes and triangle chords for honeycomb walks}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026/main.pdf}{OAI:Cap-selected-amplitudes-and-triangle-chords-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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