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

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