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The notion of partial geodesic was introduced by Jamshidpey et al. in "Setsof medians in the non-geodesic pseudometric space of unsigned genomes withbreakpoints", 2014. In this paper, we study the density of points onnon-trivial partial geodesics between two permutations $\xi_1^{(n)}$ and$\xi_2^{(n)}$ chosen uniformly and independently at random from the symmetricgroup $S_n$, where $S_n$ is endowed with the breakpoint distance. For apermutation $\pi := \pi_1 \ ... \ \pi_n$, any unordered pair $\{\pi_i ,\pi_{i+1}\}$, for $i=1, ..., n-1$, is called an adjacency of $\pi$. The set ofall adjacencies of $\pi$ is denoted by $\mathcal{A}_\pi$. Denote by $id^{(n)}$the identity permutation, and let $I_n$ be an arbitrary subset of $\mathcalA_{id^{(n)}}$. We classify the set of all adjacencies of a permutation $\pi \inS_n$ into four types, with respect to $I_n$. Then for a permutation $\xi^{(n)}$chosen uniformly at random from $S_n$, we derive a convergence theorem for thenormalized number (after dividing by $n$) of adjacencies of each type in$\xi^{(n)}$ with respect to $I_n$ (for some random or deterministic choices of$I_n$), as $n\rightarrow \infty$. We also see an application of thisconvergence theorem to find the appropriate choices of $I_n$. A geodesic pointof $u$ and $v$ in a pseudometric space $(S,\rho)$ is a point $w$ of the spacethat $\rho(u,w)+\rho(w,v)=\rho(u,v)$. We find an upper bound for the number ofpermutations $x\in S_n$ for which there exists at least one non-trivialgeodesic point between $id^{(n)}$ and $x$, far from both. This partiallyverifies the conjecture of Haghighi and Sankoff stated in "Medians seek thecorners, and other conjectures", 2012, namely we prove that, with highprobability, there is no breakpoint median of two permutations $\xi_1^{(n)}$and $\xi_2^{(n)}$ chosen uniformly and independently at random from $S_n$, farfrom both of them.

Partial geodesics on symmetric groups endowed with breakpoint distance