On the structure of random graphs with constant r-balls

We continue the study of the properties of graphs in which the ball of radius r around each vertex induces a graph isomorphic to the ball of radius r in some fixed vertex-transitive graph F, for various choices of F and r. This is a natural extension of the study of regular graphs. More precisely, if F is a vertex-transitive graph and r 2 N, we say a graph G is r-locally F if the ball of radius r around each vertex of G induces a graph isomorphic to the graph induced by the ball of radius r around any vertex of F. We consider the following random graph model: for each n 2 N, we let Gn = Gn(F, r) be a graph chosen uniformly at random from the set of all unlabelled, n-vertex graphs that are r-locally F. We investigate the properties possessed by the random graph Gn with high probability, i.e. with probability tending to 1 as n ! 1, for various natural choices of F and r.We prove that if F is a Cayley graph of a torsion-free group of polynomial growth, and r is sufficiently large depending on F, then the random graph Gn = Gn(F, r) has largest component of order at most n5=6 with high probability, and has at least exp(ni) automorphisms with high probability, where i > 0 depends upon F alone. Both properties are in stark contrast to random d-regular graphs, which correspond to the case where F is the infinite d-regular tree. We also show that, under the same hypotheses, the number of unlabelled, n-vertex graphs that are r-locally F grows like a stretched exponential in n, again in contrast with d-regular graphs. In the case where F is the standard Cayley graph of Zd, we obtain a much more precise enumeration result, and more precise results on the properties of the random graph Gn(F, r). Our proofs use a mixture of results and techniques from geometry, group theory and combinatorics.We make several conjectures regarding what happens for Cayley graphs of other finitely gener-ated groups.