ARITHMETIC OSCILLATIONS OF THE CHEMICAL DISTANCE IN LONG-RANGE PERCOLATION ON Zd

We consider a long-range percolation graph on Z(d) where, in addition to the nearest-neighbor edges of Z(d) , distinct x,y is an element of Z(d) are connected by an edge independently with probability asymptotic to beta|x - y|(-s), for s is an element of (d, 2 d) , beta > 0 and |center dot| a norm on R-d. We first show that, for all but perhaps a countably many beta > 0, the graph-theoretical (a.k.a. chemical) distance between typical vertices at |center dot|-distance r is, with high probability as r -> infinity, asymptotic to phi(beta)(r)(log r)(Delta), where Delta(-1) := log(2)(2d/s) and phi(beta) is a deterministic, positive, bounded and continuous function subject to log-log-periodicity constraint phi(beta)(r(gamma)) = phi(beta)(r) for gamma := s/(2d). The proof parallels the arguments developed in a continuum version of the model where a similar scaling was shown earlier by the first author and J. Lin. That work also conjectured that phi(beta) is constant which we show to be false by proving that (log beta)(Delta)phi(beta) tends, as beta -> infinity, to a nonconstant limit that is independent of the specifics of the model. The proof reveals arithmetic rigidity of the shortest paths that maintain a hierarchical (dyadic) structure all the way to unit scales.