1 · The essentials
Imagine tracing a path along the edges of a honeycomb without ever visiting the same point twice. Among paths with a fixed number of steps, choose one uniformly at random and ask how far apart its most widely separated points are. The fixed-length manuscript shows that, for a long path of steps, this distance grows like , allowing a correction to the power that tends to zero. It also describes how much of the path lies in smaller regions and how many smaller regions are needed to cover it. These statements apply to every sufficiently large length, with bounds that work across scales together and with a chance of failure smaller than any chosen inverse power of the length. The family includes a Lean computer-checked formalization, specifically for the companion work on paths joining prescribed boundary points; this should not be read as computer verification of every manuscript.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine laying a cord along the lines of a honeycomb-patterned floor without letting it revisit a junction. The cord's length counts its steps, while the greatest distance between two parts of the cord measures its spread. The result concerns a uniformly chosen allowed layout, not every possible way of laying the cord.
Why it matters
The family addresses the diameter form of Nienhuis's predicted three-quarter exponent for honeycomb self-avoiding walks. Its fixed-length statements describe not only overall spread but also local mass and covering numbers, simultaneously across scales. The distinction between fixed-length sampling and critical weighting over many lengths matters: the companion manuscripts study several such laws, with different boundary conditions and different guarantees.
2 · Going further
A self-avoiding walk on the honeycomb lattice never revisits a vertex. For a uniformly chosen walk of exactly steps, the central question is the size of its diameter, meaning the greatest distance between two of its vertices. Manuscript 4 shows the diameter law , together with local-mass and covering exponents . Its bounds hold at every sufficiently large integer length, with arbitrary positive exponent slack and arbitrary polynomial failure probability, simultaneously across scales. Here denotes a correction to the exponent tending to zero; the assertion is not an exact formula for diameter.
The main manuscript studies closed walks, or polygons, at critical weights. It shows diameter-tail exponents for unrooted polygons and for length-weighted polygons, as well as a truncated second-length-moment bound with exponent . Its stated consequence is that polygons conditioned to have length at least have diameter in probability under either weight. This differs from the uniform, exactly--step walk problem. The abstracts describe an approach through radial transfer estimates, polygon length moments, cylinder weights, and changes of law; they do not supply a complete dependency chain for the fixed-length conclusion.
The Lean-formalized companion, Manuscript 2, concerns critical paths between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon of scale . It gives length in probability, with corresponding statements for half-plane arches, parallel cuts, and nonparallel pure cuts; the half-plane law also has that mean-length exponent. Separate strip statements concern one density-one set of heights and an aligned, critically weighted mixture of even heights in a macroscopic interval. Other companions show strip-crossing mass comparable to and bridge mean length under all-length critical weighting, including a version confined to a fixed multiple of the strip width. Logarithmic bridge-length windows and finite bridge-moment estimates provide further control of lengths and weights.
The cylinder and marked-polygon manuscripts supply related estimates: at loop fugacity two, the cylinder partition exponent is and the planar nesting exponent is . Marked-polygon estimates bound squared-length mass up to diameter by . Hexagon and triangle chord results connect macroscopic diameter restrictions with partition exponent and mean-length exponent . Finally, the renewal manuscript shows spatial exponent for the infinite irreducible-bridge law and for sufficiently large compatible even bridge lengths, along with thermal and near-critical statements. Its unrestricted-walk endpoint lower law is stated only on a common density-one set of lengths, unlike the diameter result at every sufficiently large length.
Useful notions
- Paths on the honeycomb lattice
- Uniform sampling versus weighted sampling
- Power-law growth and small exponent corrections
- Probability tending to one
- Means, moments, and tail bounds
What was known before / what is new
Before
The repository summary identifies the three-quarter exponent as a prediction of Nienhuis. The supplied source gives no further historical account or inventory of earlier established results, so it does not support a more detailed before-and-after history.
New
The fixed-length manuscript shows diameter exponent and simultaneous local-mass and covering exponents for every sufficiently large length, with arbitrary positive exponent slack and arbitrary polynomial failure probability. The main manuscript shows critical polygon diameter-tail and length-moment estimates and derives diameter scaling for polygons conditioned to be sufficiently long. The companion manuscripts develop boundary-path, bridge, cylinder, nesting, and marked-polygon laws, with a Lean formalization identified for Manuscript 2.
Limits and assumptions
These statements concern the honeycomb lattice and the sampling laws, weights, geometries, and boundary restrictions specified in each manuscript. Uniform fixed-length walks, critically weighted paths of all lengths, and polygons conditioned to have length at least n are different settings and must not be interchanged. The scaling laws allow vanishing exponent corrections or arbitrary positive exponent slack; they are not exact size formulas, and in-probability statements do not automatically assert mean laws. The family has a Lean formalization, with the supplied verification scope identified as Manuscript 2 on prescribed-boundary critical chords and its companion statements, not every claim across the family. In particular, the source does not identify Manuscript 4's unrestricted fixed-length result as separately Lean-formalized. The endpoint lower law for unrestricted walks is stated on a density-one set of lengths, and the source does not specify whether an every-large-length endpoint version remains open. No explicit list of open problems is supplied.
Mini glossary
- Self-avoiding walk —
- A lattice path that never visits the same vertex twice.
- Diameter —
- The greatest distance between any two points of a path or polygon.
- Local mass —
- The amount of a walk contained in a specified smaller region.
- Covering number —
- The number of regions at a chosen scale needed to cover the walk.
- Partition sum —
- The total statistical weight of the allowed paths or polygon configurations.
- Density-one set of lengths —
- A set whose proportion among integer lengths up to a growing cutoff tends to one.
Keywords : Honeycomb self-avoiding walk · Three-quarter exponent · Critical path geometry
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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 . 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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Radial transfer estimates and polygon length laws for honeycomb walks
26/09/2026We 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 in probability under either weight.
Critical honeycomb chords with prescribed boundary endpoints
26/09/2026Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length 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 . 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.
Cylinder loop weights and planar nesting
26/09/2026We 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.
Mass and covering exponents for fixed-length honeycomb walks
26/09/2026For 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.
Signed cylinder propagation and marked polygons on the honeycomb lattice
26/09/2026At 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 , the finite critical partition is and the normalized mean length is . 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 .
Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice
26/09/2026We prove a logarithmic window of critical honeycomb bridge lengths: summed through height , bridges of lengths between and have mass at least .
Marked polygon correlations and one-arc bounds
26/09/2026At the critical honeycomb vertex activity, the squared-length mass of simple polygons of diameter at most H, counted modulo translations, is at most . We also prove a quantitative two-mark cylinder estimate and a polynomial one-arc bound uniform even for arbitrarily unequal marked intervals.
Disk transfer representations and confined bridge mass
26/09/2026We prove that critical honeycomb bridges crossing a strip of width R have mean length , 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.
Polynomial vacuum representations and bridge mass for honeycomb walks
26/09/2026We 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 . We also obtain central visit probability and mean length for macroscopic free-boundary chords in a regular hexagon.
Renewal and changes of law for critical honeycomb walks
26/09/2026For 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.
Critical strip-crossing mass on the honeycomb lattice
26/09/2026At 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.
Uniform marked-polygon estimates and sharp finite bridge moments
26/09/2026We 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 , and mass at least on bridges with length at least .
Cap-selected amplitudes and triangle chords for honeycomb walks
26/09/2026At 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 , have partition sum comparable to R3/4 and mean length . We also determine the amplitude selected by two vacuum caps on a cylinder of circumference N and prove that it grows as .