Result 237, Probability and statistical mechanics

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.

Lean formalization Proof

The bigger picture

Why it matters

A path that never revisits a vertex can still fold into many shapes. The manuscripts report a precise law for how far a uniformly chosen long path typically spreads on the two-dimensional honeycomb lattice.

What changes?

For uniform n-step honeycomb walks, which never revisit a vertex, the manuscript reports diameter exponent 3/4: the greatest separation grows like n to that power, allowing any positive exponent tolerance. Local mass, counting vertices in regions, and covering numbers, counting smaller regions needed to cover the path, simultaneously have exponent 4/3 across scales. For any chosen tolerance and polynomial failure bound, the estimates hold at every sufficiently large integer n; the threshold may depend on those choices.

What does that help mathematicians do?

The reported bounds rule out both square-root growth and linear growth as typical diameter scales for these walks. The simultaneous mass and covering estimates also describe how the path occupies space, not just its overall extent. Importantly, diameter measures the greatest separation anywhere along the path. This result alone does not establish the same every-length law for the distance between its two endpoints.

Are there practical applications?

The immediate value is foundational for probability and statistical mechanics: the result quantifies how forbidding repeated visits shapes random lattice paths. Its fixed-length estimates provide geometric information without averaging over different lengths. They also offer precise targets for numerical checks, but the supplied sources do not establish a practical technology or extend this law to other lattices.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

13 manuscripts

Radial transfer estimates and polygon length laws for honeycomb walks

September 26, 2026 100 pages

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 (BibTeX)

@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}
}

Critical honeycomb chords with prescribed boundary endpoints

September 26, 2026 44 pages

Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length $R^{4/3+o(1)}$ in probability, where R is the scale of the hexagon. We prove the corresponding statements for half-plane arches, parallel cuts and nonparallel pure cuts. The half-plane law also has mean length $R^{4/3+o(1)}$ . A separate strip argument gives endpoint mean laws on one density-one set of heights and in an aligned, critically weighted mixture of all even heights in a macroscopic interval.

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Cite (BibTeX)

@misc{OAI:Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026,
  author = {{OpenAI}},
  title = {{Critical honeycomb chords with prescribed boundary endpoints}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf}{OAI:Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026}},
  year = {2026}
}

Cylinder loop weights and planar nesting

September 26, 2026 94 pages

We determine the growth exponent, at every fixed positive loop fugacity, of the critical honeycomb partition function for disjoint polygons separating two prescribed markers on a balanced cylinder. At fugacity two the cylinder exponent is 1/6; the corresponding planar nesting exponent and middle-strip nesting exponent are 1/12.

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@misc{OAI:Cylinder-loop-weights-and-planar-nesting-September-26-2026,
  author = {{OpenAI}},
  title = {{Cylinder loop weights and planar nesting}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026/main.pdf}{OAI:Cylinder-loop-weights-and-planar-nesting-September-26-2026}},
  year = {2026}
}

Mass and covering exponents for fixed-length honeycomb walks

September 26, 2026 66 pages

For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/4 and simultaneous local-mass and covering exponent 4/3, with arbitrary positive exponent slack and arbitrary polynomial failure probability.

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Cite (BibTeX)

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

Signed cylinder propagation and marked polygons on the honeycomb lattice

September 26, 2026 128 pages

At the critical activity and loop fugacity two, we prove a 1/6 partition exponent for disjoint honeycomb polygons separating opposite marks on a balanced infinite cylinder, and a 1/12 planar nesting exponent. For ordered first-exit chords in a regular hexagon of side R, with both boundary ports summed and diameter at least $R/10$ , the finite critical partition is $R^{3/4+o(1)}$ and the normalized mean length is $R^{4/3+o(1)}$ . A separate two-bond estimate on tilted cylinders with controlled site proportions bounds the length-square mass of planar polygons of diameter at most H, modulo translations, by $H^{2/3+o(1)}$ .

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Cite (BibTeX)

@misc{OAI:Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026,
  author = {{OpenAI}},
  title = {{Signed cylinder propagation and marked polygons on the honeycomb lattice}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf}{OAI:Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026}},
  year = {2026}
}

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

September 26, 2026 96 pages

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 $h^{3/4}(\log h)^{-C}$ .

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@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}
}

Marked polygon correlations and one-arc bounds

September 26, 2026 61 pages

At the critical honeycomb vertex activity, the squared-length mass of simple polygons of diameter at most H, counted modulo translations, is at most $H^{2/3+o(1)}$ . We also prove a quantitative two-mark cylinder estimate and a polynomial one-arc bound uniform even for arbitrarily unequal marked intervals.

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Cite (BibTeX)

@misc{OAI:Marked-polygon-correlations-and-one-arc-bounds-September-26-2026,
  author = {{OpenAI}},
  title = {{Marked polygon correlations and one-arc bounds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026/main.pdf}{OAI:Marked-polygon-correlations-and-one-arc-bounds-September-26-2026}},
  year = {2026}
}

Disk transfer representations and confined bridge mass

September 26, 2026 81 pages

We prove that critical honeycomb bridges crossing a strip of width R have mean length $R^{4/3+o(1)}$ , with the same exponent after confinement to a fixed multiple of the strip width. The initial boundary port is fixed, the terminal port is summed, and all finite lengths receive their critical weights. The bridge mass is comparable to R^−1/4; the half-plane arch kernel at endpoint separation m is comparable to m^−5/4.

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Cite (BibTeX)

@misc{OAI:Disk-transfer-representations-and-confined-bridge-mass-September-26-2026,
  author = {{OpenAI}},
  title = {{Disk transfer representations and confined bridge mass}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf}{OAI:Disk-transfer-representations-and-confined-bridge-mass-September-26-2026}},
  year = {2026}
}

Polynomial vacuum representations and bridge mass for honeycomb walks

September 26, 2026 69 pages

We prove that, under the all-length critical measure, a honeycomb bridge crossing a strip of h layers, with its initial port fixed and its terminal port free, has mean length $h^{4/3+o(1)}$ . We also obtain central visit probability $R^{-2/3+o(1)}$ and mean length $R^{4/3+o(1)}$ for macroscopic free-boundary chords in a regular hexagon.

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Cite (BibTeX)

@misc{OAI:Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026,
  author = {{OpenAI}},
  title = {{Polynomial vacuum representations and bridge mass for honeycomb walks}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf}{OAI:Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026}},
  year = {2026}
}

Renewal and changes of law for critical honeycomb walks

September 26, 2026 98 pages

For critical honeycomb self-avoiding walk, we prove spatial exponent 3/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length. We establish the corresponding thermal laws at every large discount scale, including all positive length and spatial moments. For unrestricted uniform walks, the endpoint lower law holds on a common set of lengths of natural density one. We also determine the near-critical exponential correlation scale and small-force free-energy exponent, and prove spatial local lower bounds that permit conditioning on a prescribed terminal vertex.

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Cite (BibTeX)

@misc{OAI:Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026,
  author = {{OpenAI}},
  title = {{Renewal and changes of law for critical honeycomb walks}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf}{OAI:Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026}},
  year = {2026}
}

Critical strip-crossing mass on the honeycomb lattice

September 26, 2026 29 pages

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 N^3/4.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Uniform marked-polygon estimates and sharp finite bridge moments

September 26, 2026 109 pages

We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h^−1/4, first-length mass at most $Ch^{13/12}$ , and mass at least $ch^{-1/4}$ on bridges with length at least $ch^{4/3}$ .

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@misc{OAI:Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026,
  author = {{OpenAI}},
  title = {{Uniform marked-polygon estimates and sharp finite bridge moments}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf}{OAI:Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026}},
  year = {2026}
}

Cap-selected amplitudes and triangle chords for honeycomb walks

September 26, 2026 92 pages

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 R^3/4 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 $N^{1/6+o(1)}$ .

PDF LaTeX source

Cite (BibTeX)

@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's note on what the formalization covers, from lean/docs/237.md.

The three-quarter exponent for honeycomb self-avoiding walk

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The linked formalization establishes finiteness of the critical bridge measures used in the paper. For every strip height $h\ge1$ , the total weight and first length moment of self-avoiding honeycomb bridges from a fixed initial port to a free terminal port are finite. The corresponding sums are also finite when the bridge is confined to the corridor of horizontal width $h(\log h)^2$ , including the normalized first-moment sums.

These are supporting summability statements. The paper's $h^{4/3+o(1)}$ mean-length law and its central-visit and hexagonal-chord exponents are outside them.

The linked formalization proves existence of the infinite-length free energy for the critical honeycomb self-avoiding-walk partition function. For every starting vertex, force direction, and real force parameter, the logarithm of the length- $n$ partition function divided by $n$ converges to its stated free-energy limit.

This selected result supplies the limiting free energy. It does not state the paper's small-force exponent, near-critical correlation scale, or the $3/4$ spatial and moment laws.

For critical self-avoiding walks on the honeycomb lattice, the formalization proves that the total weight of paths crossing a strip of height $N$ is comparable to $N^{-1/4}$ , while the first horizontal-displacement moment of return paths is comparable to $N^{3/4}$ . It also proves finiteness of the strip sums, the exact arch–bridge balance identity, comparison of successive moment increments with bridge mass, and monotonicity of bridge mass. The comparison constants are uniform in the strip height.

Comparator links

Result	Comparator statement
Finiteness of critical bridge mass and first length moments	HoneycombBridgeFiniteness.lean
Existence of the honeycomb free-energy limit	HoneycombFreeEnergy.lean
Critical honeycomb strip mass and displacement moment	CriticalStripMass.lean

At a glance

Subject: Probability and statistical mechanics
Manuscripts: 13
Pages: 1,067
Dated: Sep 26, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Nienhuis's predicted three-quarter exponentProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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