  In their seminal work, Azadkia and Chatterjee (2021) initiated graph-based
methods for measuring variable dependence strength. By appealing to nearest
neighbor graphs, they gave an elegant solution to a problem of R\'enyi
(R\'enyi, 1959). Their idea was later developed in Deb et al. (2020) and the
authors there proved that, quite interestingly, Azadkia and Chatterjee's
correlation coefficient can automatically adapt to the manifold structure of
the data. This paper furthers their study in terms of calculating the
statistic's limiting variance under independence and showing that it only
depends on the manifold dimension.
