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.