oai-math-revised beta

Browse › Probability and statistical mechanics › 237 The three-quarter exponent for honeycomb self-avoiding walk

Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice

OpenAI · September 26, 2026

Lean

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

We prove a logarithmic window of critical honeycomb bridge lengths: summed through height $2h(\log h)^{1/64}$, bridges of lengths between $h^{4/3}(\log h)^{-1/8}$ and $2h^{4/3}(\log h)^{1/2}$ have mass at least $h^{3/4}(\log h)^{-C}$.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026,
  author = {{OpenAI}},
  title = {{Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026/main.pdf}{OAI:Cylinder-amplitudes-and-logarithmic-bridge-length-windows-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.

Other papers in this family: