Some Results for Range of Random Walk on Graph with Spectral Dimension Two

We consider the range of the simple random walk on graphs with spectral dimension two. We give a form of strong law of large numbers under a certain uniform condition, which is satisfied by not only the square integer lattice but also a class of fractal graphs. Our results imply the strong law of large numbers on the square integer lattice established by Dvoretzky and Erdös (in: Proceedings of Second Berkeley symposium on mathematical statistics and probability, University of California Press, California, 1951). Our proof does not depend on spatial homogeneity of space and gives a new proof of the strong law of large numbers on the lattice. We also show that the behavior of appropriately scaled expectations of the range is stable with respect to every “finite modification” of the two-dimensional integer lattice, and furthermore, we construct a recurrent graph such that the uniform condition holds, but the scaled expectations fluctuate. As an application, we establish a form of law of the iterated logarithms for lamplighter random walks in the case that the spectral dimension of the underlying graph is two.