Above four dimensions

Abstract

The lace expansion has been used to resolve many of the issues concerning the self-avoiding walk in five or more dimensions. Proving convergence of the lace expansion for d = 5 involves a myriad of major technical difficulties, due to the fact that the best bound on the small parameter responsible for convergence of the expansion, namely $\parallel H_{z_c } \parallel _2^2 = \text{B}(z_c ) - 1$, is 0.493. However many of these technical difficulties are not present if the small parameter can be taken to be arbitrarily small, and it is in the context of an arbitrarily small parameter that the proof becomes most transparent. For this reason, in this chapter we give the proof of convergence of the lace expansion and its consequences for the critical behaviour in two contexts: for the nearestneighbour model with large d, and for the “spread-out” self-avoiding walk with steps (x, y) satisfying 0 >║x-y║_∞ ≤ L, for d <4 and large L.