Nienhuis's 3/4 exponent for self-avoiding walk on the honeycomb lattice, in diameter form

Theorem 1.1: for every $\delta>0$ and $k>0$ there are $C_{\delta,k}$ and $n_0$ such that for every integer $n\ge n_0$, with $\mathbb P_n$-probability at least $1-C_{\delta,k}n^{-k}$, the diameter satisfies $n^{3/4-\delta}\le D(\gamma)\le n^{3/4+\delta}$, and simultaneously for all visited $z$ and $s\in[1,n]$ the ball mass is within $n^{\pm\delta}$ of $\min\{n,s^{4/3}\}$ and the covering number by $s$-balls within $n^{\pm\delta}$ of $1+ns^{-4/3}$. Corollaries: span at least $n^{3/4-\delta}$ in each lattice normal, covering dimension $4/3$, radius of gyration $n^{3/4+o(1)}$ and $\mathbb E_n[D^p]=n^{3p/4+o(1)}$. Not shown: a fixed-length lower bound for the end-to-end distance (a companion has it only on a density-one set of lengths), an exponent without $n^{o(1)}$ slack, other lattices, or convergence to $SLE_{8/3}$.

Theorem 1.1: for every $\delta>0$ and $k>0$ there are $C_{\delta,k}$ and $n_0$ such that for every integer $n\ge n_0$, with $\mathbb P_n$-probability at least $1-C_{\delta,k}n^{-k}$, the diameter satisfies $n^{3/4-\delta}\le D(\gamma)\le n^{3/4+\delta}$, and simultaneously for all visited $z$ and $s\in[1,n]$ the ball mass is within $n^{\pm\delta}$ of $\min\{n,s^{4/3}\}$ and the covering number by $s$-balls within $n^{\pm\delta}$ of $1+ns^{-4/3}$. Corollaries: span at least $n^{3/4-\delta}$ in each lattice normal, covering dimension $4/3$, radius of gyration $n^{3/4+o(1)}$ and $\mathbb E_n[D^p]=n^{3p/4+o(1)}$. Not shown: a fixed-length lower bound for the end-to-end distance (a companion has it only on a density-one set of lengths), an exponent without $n^{o(1)}$ slack, other lattices, or convergence to $SLE_{8/3}$.