  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 $\|x-y\|>1$ are connected by an edge is proportional to
$\frac{\beta}{\|x-y\|^s}$, 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 $s\geq 2d$. 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 $ne_1$ 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].
