Remco van der Hofstad, Frank den Hollander, Gordon Slade.
A new inductive approach to the lace expansion for self-avoiding walks
(86K, Latex 2e)
ABSTRACT. We introduce a new inductive approach to the lace expansion, and apply
it to prove Gaussian behaviour for the weakly self-avoiding walk on
${\Bbb Z}^d$ where loops of length $m$ are penalised by a factor
$e^{-\beta/m^{p}}$ ($0<\beta \ll 1$) when:\\
(1) $d>4$, $p \geq 0$; \\
(2) $d \leq 4$, $p > \frac{4-d}{2}$. \\
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 $d>4$, $p=0$.