Renormalisation-group study of self-avoiding walks on the random lattice

Using a field theoretic method based on the replica trick, it is proved that the three-parameter renormalisation group for an n-vector model with quenched randomness reduces to a two-parameter one in the limit n to 0 which corresponds to self-avoiding walks (SAWS). This is also shown by the explicit calculation of the renormalisation-group recursion relations to second order in epsilon . From this reduction the author finds that SAWS on the random lattice are in the same universality class as SAWS on the regular lattice. By analogy with the case of the n-vector model with cubic anisotropy in the limit n to 1, the fixed-point structure of the n-vector model with randomness is analysed in the SAW limit, so that a physical interpretation of the unphysical fixed point is given. Corrections of the values of critical exponents of the unphysical fixed point published previously are also given.