A NEW INDUCTIVE APPROACH TO THE LACE EXPANSION FOR SELF-AVOIDING WALKS

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 (d) where loops of length m are penalised by a factor e(-beta/mp) (0 < beta much less than 1) when: (1) d > 4, p greater than or equal to 0; (2) d less than or equal to 4, p > 4-d/2. In particular, we derive re sults first obtained by Brydges and Spencer (and revisited by other au thors) for the case d > 4, p = 0, In addition, we prove a local centra l limit theorem, with the exception of the case d > 4, p = 0.