The $N^{-1/4}$ law for the critical strip-crossing mass of honeycomb self-avoiding walk

Theorem 1.1: for every $N\ge1$ the strip sums are finite, $c\mathcal A_N+\mathcal B_N=1$ with $c=\cos(3\pi/8)$, $m_{N+1}-m_N\asymp\mathcal B_N$, the first rightward displacement moment of arches satisfies $m_N\asymp N^{3/4}$, and the bridge mass satisfies $\mathcal B_N\asymp N^{-1/4}$, nonincreasing in $N$, with constants independent of $N$. The route computes $m_N$ through rhombic transfer matrices and a positive integral, then recovers $\mathcal B_N$ from its increments. Not shown: a limiting constant for $N^{1/4}\mathcal B_N$, statements about walks of fixed length (treated in companions), or other lattices.

Theorem 1.1: for every $N\ge1$ the strip sums are finite, $c\mathcal A_N+\mathcal B_N=1$ with $c=\cos(3\pi/8)$, $m_{N+1}-m_N\asymp\mathcal B_N$, the first rightward displacement moment of arches satisfies $m_N\asymp N^{3/4}$, and the bridge mass satisfies $\mathcal B_N\asymp N^{-1/4}$, nonincreasing in $N$, with constants independent of $N$. The route computes $m_N$ through rhombic transfer matrices and a positive integral, then recovers $\mathcal B_N$ from its increments. Not shown: a limiting constant for $N^{1/4}\mathcal B_N$, statements about walks of fixed length (treated in companions), or other lattices.