Locally Self-Avoiding Walks

Abstract

The Edwards model [1] of (possibly weakly) self-avoiding walks plays an important role in polymer theory. In that model, a polymer is represented by a continuous walk subject to a self-avoiding constraint imposed via a δ-potential between any pair of points (the polymer cannot occupy twice the same position in space). Relevant quantities, such as the partition and 2-point correlation functions for a given length T of the polymers, are integrals over all walks of length T starting from a given point or going from points x to y. They are à priori ill-defined in view of short-distance problems, but can be well defined (after renormalization) in space dimensions 2 and 3 [2,3]. Nevertheless, there is so far no result on the main problem of physical interest, namely the behavior of such quantities in the T → ∞ limit. Results on that problem have been obtained only in space dimensions d ≥ 4, in which case some ultraviolet cut-off, regularizing interactions at short distances, has to be introduced. Besides perturbative results [4] and results in the hierarchical approximation [5], some rigorous results have been obtained in that direction in the eighties through probabilistic methods or correlation inequalities [6], More complete results have then been achieved in space dimensions ≥ 5 through lace expansions [7] or, at weak coupling, in the marginal dimension 4 through renormalization group analysis [8]. More precisely, results of [8] apply to a related model, also due to Edwards, in which quantities treated correspond to integrals over all possible lengths, and then give information for large separation of endpoints. Results can probably be established for the model of self-avoiding walks itself in the T → ∞ limit either [9] through an extension of the analysis of [8], or in a direct way [10].