Self‐avoiding walk on the hypercube

Abstract

We study the number cn(N)$$ {c}_n^{(N)} $$ of n$$ n $$ ‐step self‐avoiding walks on the N$$ N $$ ‐dimensional hypercube, and identify an N$$ N $$ ‐dependent connective constant μN$$ {\mu}_N $$ and amplitude AN$$ {A}_N $$ such that cn(N)$$ {c}_n^{(N)} $$ is O(μNn)$$ O\left({\mu}_N^n\right) $$ for all n$$ n $$ and N$$ N $$ , and is asymptotically ANμNn$$ {A}_N{\mu}_N^n $$ as long as n≤2pN$$ n\le {2}^{pN} $$ for any fixed p<12$$ p<\frac{1}{2} $$ . We refer to the regime n≪2N/2$$ n\ll {2}^{N/2} $$ as the dilute phase. We discuss conjectures concerning different behaviors of cn(N)$$ {c}_n^{(N)} $$ when n$$ n $$ reaches and exceeds 2N/2$$ {2}^{N/2} $$ , 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$$ {N}^{-1} $$ , with integer coefficients, and we compute the first five coefficients μN=N−1−N−1−4N−2−26N−3+O(N−4)$$ {\mu}_N=N-1-{N}^{-1}-4{N}^{-2}-26{N}^{-3}+O\left({N}^{-4}\right) $$ . The proofs are based on generating function and Tauberian methods implemented via the lace expansion, for which an introductory account is provided.