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

TL;DR

This paper studies the chemical distance in long-range percolation on  lattices, revealing oscillatory scaling behavior and arithmetic rigidity of shortest paths, with results on the dependence on model parameters.

Contribution

It establishes the asymptotic behavior of chemical distances in long-range percolation, showing oscillations and hierarchical path structures, and disproves the conjecture that a certain scaling function is constant.

Findings

Chemical distance scales as ig(rac{1}{eta}ig)(\, ext{log}\, r)^\, ext{Delta} for large r.

The scaling function _eta(r) is non-constant and depends on eta.

Shortest paths exhibit hierarchical, dyadic structure leading to arithmetic rigidity.

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 $\phi_\beta(r)(\log r)^\Delta$, 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 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 $\phi_\beta$ is constant which we show to be false by proving that $(\log\beta)^\Delta\phi_\beta$ tends, as $\beta\to\infty$, to a non-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.