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It is well known that for a simple random walk on a 2D square lattice extending to infinity the mean square displacement of the walk $\langle \mathbf r^2\rangle \propto N \, :(*)$ with $N$ the number ...

Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...