Polynomial vacuum representations and bridge mass for honeycomb walks

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.