Anisotropic self-avoiding walks

We consider a model of self-avoiding walks on the lattice Z(d) with differe nt weights for steps in each of the 2d lattice directions. We find that the direction-dependent mass for the two-point function of this model has thre e phases: mass positive in all directions; mass identically -infinity; and masses of different signs in different directions. The final possibility ca n only occur if the weights are asymmetric, i.e., in at least one coordinat e the weight in the positive direction differs from the weight in the negat ive direction. The boundaries of these phases are determined exactly. We al so prove that if the weights are asymmetric then a typical N-step self-avoi ding walk has order N distance between its endpoints. (C) 2000 American Ins titute of Physics. [S0022-2488(00)02203-9].