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We study the equidistribution on spheres of the n-step transition probabilities of random walks on graphs. We give sufficient conditions for this property being satisfied and for the weaker property of asymptotical equidistribution. We analyze the asymptotical behaviour of the Green function of the simple random walk on ℤ2 and we provide a class of random walks on Cayley graphs of groups, whose transition probabilities are not even asymptotically equidistributed.

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Bertacchi, D., Zucca, F. Equidistribution of Random Walks on Spheres. Journal of Statistical Physics 94, 91–111 (1999).

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