The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions

We prove the existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on ℤ^d. For the critical point, defined as the reciprocal of the connective constant, the coefficients of the expansion are computed through order d⁻⁶, with a rigorous error bound of order d⁻⁷ Our method for computing terms in the expansion also applies to percolation, and for nearest-neighbour independent Bernoulli bond percolation on ℤ^d gives the 1/d-expansion for the critical point through order d⁻³, with a rigorous error bound of order d⁻⁴ The method uses the lace expansion.