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We consider a long-range percolation graph on $\mathbb Z^d$ where, inaddition to the nearest-neighbor edges of $\mathbb Z^d$, distinct$x,y\in\mathbb Z^d$ are connected by an edge independently with probabilityasymptotic to $\beta|x-y|^{-s}$, for $s\in(d,2d)$, $\beta>0$ and $|\cdot|$ anorm on $\mathbb R^d$. We first show that, for all but a countably many$\beta>0$, the graph-theoretical (a.k.a. chemical) distance between typicalvertices at $|\cdot|$-distance $r$ is, with high probability as $r\to\infty$,asymptotic to $\phi_\beta(r)(\log r)^\Delta$, where $\Delta^{-1}:=\log_2(2d/s)$and $\phi_\beta$ is a positive, bounded and continuous function subject to$\phi_\beta(r^\gamma)=\phi_\beta(r)$ for $\gamma:=s/(2d)$. The proof parallelsthat in a continuum version of the model where a similar scaling was shownearlier by the first author and J. Lin. This work also conjectured that$\phi_\beta$ is constant which we show to be false by proving that$(\log\beta)^\Delta\phi_\beta$ tends, as $\beta\to\infty$, to a non-constantlimit which is independent of the specifics of the model. The proof revealsarithmetic rigidity of the shortest paths that maintain a hierarchical (dyadic)structure all the way to unit scales.

Arithmetic oscillations of the chemical distance in long-range percolation on $\mathbb Z^d$