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The "true" self-avoiding walk is a natural example of non-Markovian random walks. In these one-dimensional nearest neighbor models, the walker is self-repellent, that is, it is pushed by the negative gradient of its own local time. More informally, the walker prefers places that it has visited fewer times in the past. The site repulsion and edge repulsion models behave similarly; the displacement scales with time to the power 2/3, and the limit is a complicated, but well-defined object, with the local time picture converging to a system of reflected and absorbed Brownian motion. In the directed edge repulsion model, the proper scaling is diffusive, but there is no continuous limit process, and the trajectories are much rougher. The graph of the local times converges to a deterministic triangular shape.
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