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

We consider a long-range percolation graph on Z d where, in addition to the nearest-neighbor edges of Z d , distinct x , y ∈ Z d are connected by an edge independently with probability asymptotic to β | x − y | − s , for s ∈ ( d , 2 d ) , β > 0 and | ⋅ | a norm on R d . 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 ϕ β ( r ) ( log r ) Δ , where Δ − 1 : = log 2 ( 2 d / s ) and ϕ β is a deterministic, positive, bounded and continuous function subject to log-log-periodicity constraint ϕ β ( r γ ) = ϕ β ( r ) for γ : = s / ( 2 d ) . 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 ϕ β is constant which we show to be false by proving that ( log β ) Δ ϕ β 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.