The three-quarter exponent for honeycomb self-avoiding walk

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.

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)}.

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).

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.

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.

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}.

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)}.