Until now we have considered systems involving noninteracting walkers. This is an enormous simplification that allows analysis of such problems in terms of a single walker. Reality, however, is more complex and interactions cannot always be neglected. In this chapter we consider an elementary type of hard-core repulsion, known as excluded-volume interactions: walkers, or particles, are not allowed to occupy the same site simultaneously. We describe the dramatic effects that this simple interaction has on diffusion.

A self-avoiding walk (SAW) is a random walk that does not intersect itself. SAWs are a useful model for linear polymer chains: the visited sites represent monomers, and self-avoidance accounts for the excluded-volume interactions between monomers. The study of polymers in random media finds applications in enhancing recovery of oil, gel electrophoresis, gel-permeation chromatography, etc. Flory's theory provides us with a beautiful, intuitive understanding of the anomalous properties of SAWs in regular Euclidean space, and it may be extended to percolation and fractals. The problem of SAWs in finitely ramified fractals can be solved exactly.

Tracer diffusion

Imagine a regular lattice of lattice spacing a with a density c of particles, i.e., each site is occupied with probability c. The particles perform nearest-neighbor random walks, with hopping rate Г, and are subject to excluded-volume interactions: at most one particle may occupy a site at any given moment. Clearly, diffusion of the particles is hindered by these interactions. For example, in the limit c = 1, when all sites are occupied, motion is impossible and the system is frozen.