A Self-avoiding Walk with Attractive Interactions

A powerful tool for the study of self-avoiding walks is the lace expansion of Brydges and Spencer [BS]. It is applicable above four dimensions and shows the mean-field behavior of self-avoiding walks, that is, critical exponents are those of the simple random walk. An extensive survey of random walks can be found in [MS]. The lace expansion was originally introduced for weakly self-avoiding walks, and extended to the fully self-avoiding case by Slade and Hara [Sla, HS]. Several improvements, simplifications and alternate approaches have since been proposed, see [GI, KLMS, vHHS, vHS]. A recent work by Bolthausen and Ritzmann [BR] uses a fixed point argument and avoids the difficulties that are present when working in the Fourier space. Its actual range of applicability is limited to the case of small repulsions, but an extension to the self-avoiding case may be possible. The purpose of this article is to show that the lace expansion can also be used when the walk experiences small nearest-neighbor attractions. We consider a model of random walks w = (w0, . . . , wn) with wt ∈ Z d, where the connectivity Cn(x) between 0 and x is defined by