It seems that even the constant in front of the $\sqrt{n}$ is not known, but there are experimental results which seem to describe the distribution pretty well. In particular, it seems that the variance of the shortest path (however they compute it) is essentially $0.$ Anyway, look at the linked paper and the references therein.

It seems that even the constant in front of the $\sqrt{n}$ is not known, but there are experimental results which seem to describe the distribution pretty well. In particular, it seems that the variance of the shortest path (however they compute it) is essentially $0.$ Anyway, look at the linked paper and the references therein.