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(μ_N^n) 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 n2^pN for any fixed p < 1 2 p<12 . We refer to the regime n ≪ 2 N / 2 n2^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(N^-4) . The proofs are based on generating function and Tauberian methods implemented via the lace expansion, for which an introductory account is provided.