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We consider the random geometric graph on $n$ vertices drawn uniformly from a$d$--dimensional sphere. We focus on the sparse regime, when the expecteddegree is constant independent of $d$ and $n$. We show that, when $d$ is largerthan $n$ by logarithmic factors, this graph is comparable to theErd\H{o}s--R\'enyi random graph of the same edge density in the \emph{inclusiondivergence} between the graph laws. This divergence functions in certain wayslike a relaxation of the total variation distance, but is strong enough todistinguish Erd\H{o}s--R\'enyi graphs of different densities with a higherresolution than the total variation distance. To do the analysis, we derivesome exact statistics of the \emph{spherical Wishart matrix}, the Gram matrixof $n$ independent uniformly random $d$--dimensional spherical vectors. Inparticular we give expressions for the characteristic function of the sphericalWishart matrix which are well--approximated using steepest descent.

Random geometric graphs and the spherical Wishart matrix