A new inductive approach to the lace expansion for self-avoiding walks.

Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on Z[sup d] where loops of length m are penalised by a factor e[sup -beta/m[sup p]] (0 < beta much less than 1) when: (1) d > 4, p is greater than or equal to 0; (2) d less than or equal to 4, p > 4 - d/2. In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d > 4, p = 0. In addition, we prove a local central limit theorem, with the exception of the case d > 4, p = 0. [ABSTRACT FROM AUTHOR]