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 ℤd where loops of length m are penalised by a factor e−β/m p (0<β≪1) when: (1) d>4, p≥0; (2) d≤4, . 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.
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Received: 29 October 1997 / In revised form: 15 January 1998
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van der Hofstad, R., den Hollander, F. & Slade, G. A new inductive approach to the lace expansion for self-avoiding walks. Probab Theory Relat Fields 111, 253–286 (1998). https://doi.org/10.1007/s004400050168
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DOI: https://doi.org/10.1007/s004400050168