On the scaling limit of finite vertex transitive graphs with large diameter

Let (X (n) ) be an unbounded sequence of finite, connected, vertex transitive graphs such that |X (n) |=O(diam(X (n) ) (q) ) for some q > 0. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence (X (n) ) converges in the Gromov Hausdorff distance to some finite dimensional torus equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If X (n) is only roughly transitive and |X (n) |=O diam(X (n) (delta) ) for delta > 1 sufficiently small, we prove, this time by elementary means, that (X (n) ) converges to a circle.