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

Abstract

We consider a long-range percolation graph on ${\mathbb{Z}}^d$ where, in addition to the nearest-neighbor edges of ${\mathbb{Z}}^d$, distinct $x,y\in {\mathbb{Z}}^d$ are connected by an edge independently with probability asymptotic to $\beta |x-y{|}^{-s}$, for $s\in (d,2d)$, $\beta >0$ and $|\cdot |$ a norm 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 typical vertices at $|\cdot |$-distance $r$ is, with high probability as $r\to \infty$, asymptotic to ${\varphi }_{\beta }(r)(\log r{)}^{\Delta }$, where ${\Delta }^{-1}:={\log }_2(2d/s)$ and ${\varphi }_{\beta }$ is a positive, bounded and continuous function subject to ${\varphi }_{\beta }(r^{\gamma })={\varphi }_{\beta }(r)$ for $\gamma :=s/(2d)$. The proof parallels that in a continuum version of the model where a similar scaling was shown earlier by the first author and J. Lin. This work also conjectured that ${\varphi }_{\beta }$ is constant which we show to be false by proving that $(\log \beta {)}^{\Delta }{\varphi }_{\beta }$ tends, as $\beta \to \infty$, to a non-constant limit which 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.

arXiv comment32 pages, 2 figures, typos corrected, proofs expanded

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Cited by in corpus (2)

• Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on ${\mathbb{Z}}^d$
• Continuity of the critical value and a shape theorem for long-range percolation