Self-avoiding random walks (SAWs) are studied on several hierarchical lattices
in a randomly disordered environment. An analytical method to determine
whether their fractal dimension D~w is affected by disorder is introduced. Using
this method, it is found that for some lattices, D=w is unaffected by weak
disorder; while for others D=w changes even for infinitesimal disorder. A weak
disorder exponent 2 is defined and calculated analytically [2 measures the
dependence of the variance in the partition function (or in the effective fugacity
per step) v ~ L ~ on the end-to-end distance of the SAW, L]. For lattices which
are stable against weak disorder (2 <0) a phase transition exists at a critical
value v = v* which separates weak- and strong-disorder phases. The geometrical
properties which contribute to the value of 2 are discussed.