Parallel geodesics and minimal stable length of random groups

We show that for any pair of long enough parallel geodesics in a random group $G_{\mathrm{\ell }}(m,d)$ with $m$ generators at density $d<1\mathrm{/}6$, there is a van Kampen diagram having only one layer of faces. Using this result, we give an upper bound, depending only on $d$, of the number of pairwise parallel geodesics in $G_{\mathrm{\ell }}(m,d)$ when $d<1\mathrm{/}6$. As an application, we show that the minimal stable length of $G_{\mathrm{\ell }}(m,d)$ at $d<1\mathrm{/}6$ is exactly $1$.