Erdős #529 · geometry / probability

Slade [Sl87] proved that, for $k$ sufficiently large, $d_k(n)\sim Dn^{1/2}$ for some constant $D>0$ (independent of $k$). Hara and Slade ([HaSl91] and [HaSl92]) proved this for all $k\geq 5$.

For $k=2$ Duminil-Copin and Hammond [DuHa13] have proved that $d_2(n)=o(n)$.

It is now conjectured that $d_k(n)\ll n^{1/2}$ is false for $k=3$ and $k=4$, and more precisely (see for example Section 1.4 of [MaSl93]) that $d_2(n)\sim Dn^{3/4}$, $d_3(n)\sim n^{\nu}$ where $\nu\approx 0.59$, and $d_4(n)\sim D(\log n)^{1/8}n^{1/2}$.

Madras and Slade [MaSl93] have a monograph on the topic of self-avoiding walks.

See also [528].

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