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, for d < d(c) = 4, and, for large disorder for d > 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, L approximately N-zeta with N the number of steps, and the fluctuations in the free energy, DELTA-f approximately L-omega. The bound 1/zeta - omega less-than-or-equal-to d/2 is obtained. Exact results on hierarchical lattices yield zeta > zeta-pure and suggests that this inequality holds for d = 2 and 3, although zeta = zeta-pure cannot be excluded, particularly for d = 2. For d > d(c) there is a transition between strong and weak disorder phases at which zeta = zeta-pure. The strong-disorder fixed point for SAWs on percolation clusters is discussed. The analogy with directed walks is emphasized.