  In this paper, we study the asymptotic behavior of the number of rarely
visited edges (i.e., edges that visited only once) of a simple symmetric random
walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up
to time $n$. First, we evaluate $\mathbb{E}(\alpha(n))$, show that $n\to
\mathbb{E}(\alpha(n))$ is non-decreasing in $n$ and that
$\lim\limits_{n\to+\infty}\mathbb{E}(\alpha(n))=2$. Then we study the
asymptotic behavior of $\mathbb{P} (\alpha(n)>a(\log n)^2)$ for any $a>0$ and
use it to show that there exists a constant $C\in(0,+\infty)$ such that
$\limsup\limits_{n\to+\infty}\frac{\alpha(n)}{(\log n)^2}=C$ almost surely.
