The polynomial growth of the infinite long-range percolation cluster

We study independent long-range percolation on ${\mathbb{Z}}^d$ where the nearest-neighbor edges are always open and the probability that two vertices $x,y$ with $\mathrm{\parallel }x-y\mathrm{\parallel }>1$ are connected by an edge is proportional to $\frac{\beta }{\mathrm{\parallel }x-y{\mathrm{\parallel }}^s}$, where $\beta >0$ and $s>0$ are parameters. We show that the ball of radius $k$ centered at the origin in the graph metric grows polynomially if and only if $s\ge 2d$. For the critical case $s=2d$, we show that the volume growth exponent is inversely proportional to the distance growth exponent. Furthermore, we provide sharp upper and lower bounds on the probability that the origin and $n{e}_1$ are connected by a path of length $k$ in the critical case $s=2d$. We use these results to determine the Hausdorff dimension of the critical long-range percolation metric that was recently constructed by Ding, Fan, and Huang [14].