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 connective constant mu(N) and amplitude A(N) such that c(n)((N)) is O(mu(n)(N)) for all n and N, and is asymptotically A(N)mu(n)(N) as long as n <= 2(pN) for any fixed p < 1/2. We refer to the regime n << 2N/2 as the dilute phase. We discuss conjectures concerning different behaviors 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 mu(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.