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

Source abstract

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.