Arithmetic oscillations of the chemical distance in long-range percolation on Zd

We consider a long-range percolation graph on Zd where, in addition to the nearest-neighbor edges of Zd, distinct x,y∈Zd are connected by an edge independently with probability asymptotic to β|x−y|−s, for s∈(d,2d), β>0 and |·| a norm on Rd. We first show that, for all but perhaps a countably many β>0, the graph-theoretical (a.k.a. chemical) distance between typical vertices at |·|-distance r is, with high probability as r→∞, asymptotic to ϕβ(r)(logr)Δ, where Δ−1:=log2(2d/s) and ϕβ is a deterministic, positive, bounded and continuous function subject to log-log-periodicity constraint ϕβ(rγ)=ϕβ(r) for γ:=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 ϕβ is constant which we show to be false by proving that (logβ)Δϕβ tends, as β→∞, 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.