  For a fixed integer $k\geqslant 2$, let $G\in \mathcal{G}(n,p)$ be a simple
connected graph on $n\rightarrow\infty$ vertices with the expected degree
$d=np$ satisfying $d\geqslant c$ and $d^{k-1}= o(n)$ for some large enough
constant $c$. We show that the asymptotical size of any maximal collection of
edges $M$ in $G$ such that no two edges in $M$ are within distance $k$, which
is called a distance $k$-matching, is between $ \frac{(k-1)n\log d}{4d^{k-1}}$
and $ \frac{k n \log d}{2d^{k-1}}$. We also design a randomized greedy
algorithm to generate one large distance $k$-matching in $G$ with asymptotical
size $ \frac{kn\log d}{4d^{k-1}}$. Our results partially generalize the results
on the size of the largest distance $k$-matchings from the case $k=2$ or $d=c$
for some large constant $c$.
