Self‐avoiding walk on the hypercube. Issue 3 (2nd September 2022) - Slade, Gordon
Abstract: We study the number c n ( 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 μ N $$ μ_N $$ and amplitude A N $$ A_N $$ such that c n ( N ) $$ {c}_n^{(N)} $$ is O ( μ N n ) $$ O\left(μ_N^n\right) $$ for all n $$ n $$ and N $$ N $$, and is asymptotically A N μ N n $$ A_Nμ_N^n $$ as long as n ≤ 2 p N $$ n\le {2}^{pN} $$ for any fixed p < 1 2 $$ p<\frac{1}{2} $$ . We refer to the regime n ≪ 2 N / 2 $$ n\ll {2}^{N/2} $$ as the dilute phase . We discuss conjectures concerning different behaviors of c n ( N ) $$ {c}_n^{(N)} $$ when n $$ n $$ reaches and exceeds 2 N / 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 μ N = N − 1 − N − 1 − 4 N − 2 − 26 N − 3 + O ( N − 4 ) $$ μ_N=N-1-N^{-1}-4N^{-2}-26N^{-3}+O\left(N^{-4}\right) $$ . The proofs are based on generating function and Tauberian methods implemented via the lace expansion, for which an introductory account is provided.