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The induction hypotheses and the bounds on the lace expansion coefficients. In order to advance the induction hypotheses, it is crucial to have bounds on en+1 and gn+1 . 4. In order to apply these results, we need to have bounds on cj 1 z j and cj ∞ z j for all j ≤ n and for all z ∈ In . These bounds follow from the induction hypotheses, as we will now show. 1(c). 1(a). 4, follow from the induction hypothesis. 32) Spread-out oriented percolation above the upper critical dimension 129 when d > 4, where K = 1 + CK3 β ≤ 2 when β is sufficiently small.
11. For nearest-neighbour self-avoiding walk for d ≥ 5, or for sufficiently spread-out self-avoiding walks for d > 4, there exists a constant B and a δ > 0 such that Gzc (x) = B 1 + O(|x|−δ ) . 2) The nearest-neighbour result is proved in , the spread-out result in . These results also apply to the percolation and lattice tree and animal two-point functions. We now discuss the generating function method. To obtain improved asymptotic results, rather than bounds, the second step is to improve the bounds to asymptotics when z is close to the critical value zc , in combination with a Tauberian Theorem to turn asymptotics of the generating functions for z close to zc to asymptotics for the coefficients cn for n large.
Ueltschi studies self-avoiding walk with nearestneighbour attraction, which is a model for a polymer in a repulsive solution. 14) In this model, each nearest-neighbour contact of the path is rewarded with a factor 1 + κ. When κ is small, it can be expected that the model is 136 Remco van der Hofstad close to self-avoiding walk, but for κ large, it may show different scaling behaviour. 15) under the assumption that (1 + κ)2d ≤ 1 + ∆2 . 19) does not satisfy this assumption. 15). 15) must be such that the tails of D decay not faster than exponentially.