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The three-quarter exponent for honeycomb self-avoiding walk

Claims random self-avoiding walks of length n on the honeycomb lattice span about n^(3/4), the long-predicted exponent.

13 papers September 26, 2026 Explained by AI

A self-avoiding walk is a path on a grid that never visits the same point twice. Physicists predicted decades ago that in two dimensions a typical walk of n steps spreads over a distance near n to the power 3/4. Thirteen companion papers claim to prove this exponent and related laws on the honeycomb lattice. Lean checks several supporting estimates, such as the strip-crossing mass, but not the 3/4 laws themselves.

First step
1982
Bernard Nienhuis, prediction of the exponent 3/4 from the dilute O(n) model
Significance
Major
The 3/4 size exponent of planar self-avoiding walk is a central open problem in probability and statistical physics.
Trust
Formal proof listed
Field
Probability and statistical mechanics
13 papers

The explanation

Walk on the honeycomb lattice, the pattern of a beehive, where every point has three neighbours. A self-avoiding walk never steps on a point it has already visited. From a fixed start there are 3 walks of one step, 6 of two, 12 of three, then 24 and 48. At six steps the count is 90, not 96, because a walk can close a hexagon and hit its own start. A computer count gives 1,218 walks of ten steps and 2,479,692 of twenty-two. Each extra step multiplies the count by roughly 1.87 at that length. Duminil-Copin and Smirnov proved the true growth factor is √(2 + √2), about 1.848.

Counting is one question; shape is another. How far does a typical walk of n steps spread? A simple random walk, allowed to cross itself, spreads about √n. Self-avoidance pushes the walk outward. Nienhuis predicted in 1982 that in two dimensions the size grows like n to the power 3/4. For n = 10,000 steps that means a diameter near 1,000. The obstacle is that self-avoidance is a global rule. Each step depends on the whole past, so standard random-walk tools do not apply.

The papers claim this exponent for the honeycomb lattice. Pick a walk of large length n uniformly at random. With very high probability, its diameter lies between n^(3/4 − δ) and n^(3/4 + δ), for any small δ. The walk's local mass has exponent 4/3, the reciprocal of 3/4. The statements are about lattice paths; they do not include convergence to a continuum curve. Bridges, walks that cross a strip of height h, have mean length h^(4/3 + o(1)). Here o(1) is a correction that shrinks as h grows. Polygons, closed self-avoiding loops, conditioned to have length at least n have diameter n^(3/4 + o(1)).

Much of the work goes through a critical weight. Each path gets weight x raised to its length, with x = (2 + √2)^(−1/2), about 0.5412. The total weight of paths crossing a strip of height N is comparable to N^(−1/4). Other papers handle walks on cylinders. At loop fugacity two, polygons separating two marks have exponent 1/6. The spatial exponent 3/4 is also claimed for bridges of every large compatible even length.

How sure is this? A computer proof checker called Lean has formal proofs of several supporting results. These include the strip mass of order N^(−1/4), a displacement moment of order N^(3/4), and existence of the free-energy limit. The 4/3 mean-length laws and the fixed-length 3/4 exponents are outside those statements. They rest on thirteen long papers. No specialist review of them has been published.

What it doesn't show. This does not prove convergence to a continuum random curve, a universal result on other lattices, or that the walk's two endpoints are far apart.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. Lean covers supporting strip and bridge estimates, not the 3/4 and 4/3 laws themselves, which rest on thirteen interlocking papers with no published specialist review.

Words used

Self-avoiding walk
A path on a lattice that moves between neighbouring points and never visits a point twice.
Honeycomb lattice
The beehive pattern of hexagons; each point has exactly three neighbours.
Critical weight
The weight per step, here (2 + √2)^(−1/2), at which sums over walks of all lengths sit at the edge of blowing up.
Bridge
A self-avoiding walk that crosses a strip from one side to the other without touching either side elsewhere.
Exponent 3/4
The claim that size grows like length to the power 3/4: ten thousand steps give a span near a thousand.
o(1)
A correction in the exponent that shrinks to zero as the size grows.

The original papers (13)

Lean notes

All quoted lines (9)
  1. The value of x∗x_* is the inverse honeycomb connective constant proved by Duminil-Copin and Smirnov.
    Radial transfer estimates and polygon length laws for honeycomb walks Back to text
  2. Nienhuis's analysis of the dilute O(n)O(n) model predicted the critical point and planar self-avoiding-walk exponents, including the size exponent 3/43/4.
    Radial transfer estimates and polygon length laws for honeycomb walks Back to text
  3. For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/43/4 and simultaneous local-mass and covering exponent 4/34/3, with arbitrary positive exponent slack and arbitrary polynomial failure probability.
    Mass and covering exponents for fixed-length honeycomb walks Back to text
  4. The assertions do not include convergence to a continuum curve, a continuum Hausdorff dimension, a universality theorem on changing lattice, or a fixed-length lower bound on ∣γn−o∣|\gamma_n-o|.
    Mass and covering exponents for fixed-length honeycomb walks Back to text
  5. We prove that, under the all-length critical measure, a honeycomb bridge crossing a strip of hh layers, with its initial port fixed and its terminal port free, has mean length h4/3+o(1)h^{4/3+o(1)}.
    Polynomial vacuum representations and bridge mass for honeycomb walks Back to text
  6. Consequently, polygons conditioned to have length at least nn have diameter n3/4+o(1)n^{3/4+o(1)} in probability under either weight.
    Radial transfer estimates and polygon length laws for honeycomb walks Back to text
  7. At the critical weight of the regular honeycomb lattice, the total weight of self-avoiding paths crossing a strip of height NN is comparable to N−1/4N^{-1/4}.
    Critical strip-crossing mass on the honeycomb lattice Back to text
  8. At fugacity two the cylinder exponent is 1/61/6; the corresponding planar nesting exponent and middle-strip nesting exponent are 1/121/12.
    Cylinder loop weights and planar nesting Back to text
  9. For critical honeycomb self-avoiding walk, we prove spatial exponent 3/43/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length.
    Renewal and changes of law for critical honeycomb walks Back to text

Other sources

  1. openai/math, Lean scope note for family 237 These are supporting summability statements. The paper's h4/3+o(1)h^{4/3+o(1)} mean-length law and its central-visit and hexagonal-chord exponents are outside them.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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