Anisotropic Self-Avoiding Walks

We consider a model of self-avoiding walks on the lattice Z d with different weights for steps in each of the 2 d lattice directions. We find that the direction-dependent mass for the two-point function of this model has three phases: mass positive in all directions; mass identically – ∞; and masses of different signs in different directions. The final possibility can only occur if the weights are asymmetric, i.e. in at least one coordinate the weight in the positive direction differs from the weight in the negative direction. The boundaries of these phases are determined exactly. We also prove that if the weights are asymmetric then a typical N -step self-avoiding walk has order N distance between its endpoints.