The lace expansion for self-avoiding walk in five or more dimensions.

This paper is a continuation of our companion paper [16], in which it was proved that the standard model of self-avoiding walk in five or more dimensions has the same critical behaviour as the simple random walk, assuming convergence of the lace expansion. We prove the convergence of the lace expansion, an upper and lower infrared bound, and a number of other estimates that were used in the companion paper. The proof requires a good upper bound on the critical point (or equivalently a lower bound on the connective constant). In an appendix, new upper bounds on the critical point in dimensions higher than two are obtained, using elementary methods which are independent of the lace expansion. The proof of convergence of the lace expansion is computer assisted. Numerical aspects of the proof, including methods for the numerical evaluation of simple random walk quantities such as the two-point function (or lattice Green function), are treated in an appendix.