Sharp asymptotic for the chemical distance in long-range percolation

Abstract

We consider instances of long-range percolation on ${\mathbb{Z}}^d$ and ${\mathbb{R}}^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $s\in (d,2d)$, and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|\to \infty$. For the model on ${\mathbb{Z}}^d$ we show that, in probability as $|x|\to \infty$, the distance $D(0,x)$ is squeezed between two positive multiples of $(\log r{)}^{\Delta }$, where $\Delta :=1/{\log }_2(1/\gamma )$ for $\gamma :=s/(2d)$. For the model on ${\mathbb{R}}^d$ we show that $D(0,xr)$ is, in probability as $r\to \infty$ for any nonzero $x\in {\mathbb{R}}^d$, asymptotic to $\varphi (r)(\log r{)}^{\Delta }$ for $\varphi$ a positive, continuous (deterministic) function obeying $\varphi (r^{\gamma })=\varphi (r)$ for all $r>1$. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly-exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.

arXiv comment22 pages, 2 figs

References in corpus (4)

• On the scaling of the chemical distance in long-range percolation models
• Transience, Recurrence and Critical Behavior for Long-Range Percolation
• The growth of the infinite long-range percolation cluster
• Graph diameter in long-range percolation

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• Arithmetic oscillations of the chemical distance in long-range percolation on ${\mathbb{Z}}^d$
• The longest edge of the one-dimensional soft random geometric graph with boundaries
• Behavior of the distance exponent for $\frac{1}{|x-y{|}^{2d}}$ long-range percolation
• Graph distances of continuum long-range percolation
• Chemical distance in geometric random graphs with long edges and scale-free degree distribution
• The longest edge in discrete and continuous long-range percolation
• Continuity of the critical value and a shape theorem for long-range percolation