The lace expansion approach to ballistic behaviour for one-dimensional weakly self-avoiding walks.

Abstract. We prove ballistic behaviour in dimension one for a model of weakly self-avoiding walks where loops of length m are penalized by a factor e[sup -beta/m[sup p]] with p is an element of [0, 1] and beta sufficiently large. Furthermore, we prove that the fluctuations around the linear drift satisfy a central limit theorem. The proof uses a variant of the lace expansion, together with an inductive analysis of the arising recursion relation. In particular, we derive the law of large numbers, first obtained by Greven and den Hollander, and the central limit theorem, first obtained by Konig, for the weakly self-avoiding walk (p = 0 and beta > 0). Their proofs use large deviation theory for the Markov description of the local times of one-dimensional simple random walk. It is the first time that the lace expansion is used to prove behaviour that is not diffusive. It has previously been used by van der Hofstad, den Hollander and Slade to prove diffusive behaviour in dimension d for p is greater than or equal to 0 such that p > 4-d/2 and beta > 0 sufficiently small. The lace expansion presented here compares the above weakly self-avoiding walk to strictly self-avoiding walk in dimension one, obtained when beta = Infinity, and shows that the difference in behaviour is small when beta is large. [ABSTRACT FROM AUTHOR]