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

We prove ballistic behaviour in dimension one fora model of weakly self-avoiding walks where loops of length m are penalized by a factor e -ß/mp with p¿ [0, 1] and ß sufficiently large. Furthermore, we prove that the fluctuations around the linear drift satisfy a centrallimit 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 obtainedby Greven and den Hollander, and the central limit theorem, firstobtained by König, for the weakly self-avoiding walk (p = 0 and ß > 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 provebehaviour that is not diffusive. It has previously been used by van der Hofstad, den Hollander and Sladeto prove diffusive behaviour in dimension d for p= 0 such that p > and ß > 0 sufficiently small.The lace expansion presented here compares the above weaklyself-avoiding walk to strictly self-avoiding walk in dimension one, obtained when ß = 8, and shows that the difference in behaviour is small when ß is large.