Self-avoiding walks in quenched random environments

The self-avoiding walk in a quenched random environment is studied using real-space and field-theoretic renormalization and Flory arguments. These methods indicate that the system is described, ford_c=4, and, for large disorder ford>d_c, by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path. This fixed point is characterized by scaling laws for the size of the walk,LN^p withN the number of steps, and the fluctuations in the free energy,fL^p. The bound 1/-d/2 is obtained. Exact results on hierarchical lattices yield>_pure and suggests that this inequality holds ford=2 and 3, although=_pure cannot be excluded, particularly ford=2. Ford>d_c there is a transition between strong and weak disorder phases at which=_pure. The strong-disorder fixed point for SAWs on percolation clusters is discussed. The analogy with directed walks is emphasized.