The strong interaction limit of continuous-time weakly self-avoiding walk

The strong interaction limit of the discrete-time weakly self-avoiding walk (or Domb–Joyce model) is trivially seen to be the usual strictly self-avoiding walk. For the continuous-time weakly self-avoiding walk, the situation is more delicate, and is clarified in this paper. The strong interaction limit in the continuous-time setting depends on how the fugacity is scaled, and in one extreme leads to the strictly self-avoiding walk, in another to simple random walk. These two extremes are interpolated by a new model of a self-repelling walk that we call the “quick step” model. We study the limit both for walks taking a fixed number of steps, and for the two-point function. 1 Domb–Joyce model: discrete time The discrete-time weakly self-avoiding walk, or Domb–Joyce model [6], is a useful adaptation of the strictly self-avoiding walk that continues to be actively studied [1]. It is defined as follows. For simplicity, we restrict attention to the nearest-neighbour model on Zd , although a more general formulation is easy to obtain. David C. Brydges Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2, e-mail: [email protected] Antoine Dahlqvist DMA–Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 5, France, e-mail: [email protected] Gordon Slade Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2, e-mail: [email protected]