Self-avoiding walk on the hypercube

We study the number $c_n^{(N)}$ of $n$-step self-avoiding walks on the $N$-dimensional hypercube, and identify an $N$-dependent \emph{connective constant} $μ_N$ and amplitude $A_N$ such that $c_n^{(N)}$ is $O(μ_N^n)$ for all $n$ and $N$, and is asymptotically $A_N μ_N^n$ as long as $n\le 2^{pN}$ for any fixed $p< \frac 12$. We refer to the regime $n \ll 2^{N/2}$ as the \emph{dilute phase}. We discuss conjectures concerning different behaviours of $c_n^{(N)}$ when $n$ reaches and exceeds $2^{N/2}$, corresponding to a critical window and a dense phase. In addition, we prove that the connective constant has an asymptotic expansion to all orders in $N^{-1}$, with integer coefficients, and we compute the first five coefficients $μ_N = N-1-N^{-1}-4N^{-2}-26N^{-3}+O(N^{-4})$. The proofs are based on generating function and Tauberian methods implemented via the lace expansion, for which an introductory account is provided.

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