A self-avoiding walk is a graph-based growth model, whereby a particle or agent walks the graph from its starting point, choosing its route according to either some strategy or randomly, but avoiding any nodes that it has visited before. It therefore leave a trail behind it which it never crosses. Generally the clusters constructed from the trail tend to be "small" as they are limited by the high probability that the walker will eventually become trapped by its own trail. Large trails are possible - just unlikely.
These images are of two-dimensional self-avoiding walks. The walk starts at the centre cell and terminates when it has no route that does not involve visiting a cell that it has already been to. The attrition rate is such that long length trails are very unlikely, but the walk of the left, of length 375 was found by genenrating a largish sample (10000) of self-avoiding walks and picking the largest. In this case the graph is just an x-y mesh, but we can add extra links to make it easier for the walker to find new routes, or damage the lattice by taking some links away.
In high dimensions the trail generated clusters are even less dense. There is a higher possibility of a long trail or large cluster, as it is less likely that the walker will be trapped by its own prior path. The picture below is a 3-dimensional trail of length 450.
These structures can be studied by analysis of the distribution of trail lengths (cluster sizes) as a function of time.