Random geometric graphs and the spherical Wishart matrix

Abstract: We consider the random geometric graph on $n$ vertices drawn uniformly from a $d$--dimensional sphere. We focus on the sparse regime, when the expected degree is constant independent of $d$ and $n$. We show that, when $d$ is larger than $n$ by logarithmic factors, this graph is comparable to the Erd\H{o}s--R\'enyi random graph of the same edge density in the \emph{inclusion divergence} between the graph laws. This divergence functions in certain ways like a relaxation of the total variation distance, but is strong enough to distinguish Erd\H{o}s--R\'enyi graphs of different densities with a higher resolution than the total variation distance. To do the analysis, we derive some exact statistics of the \emph{spherical Wishart matrix}, the Gram matrix of $n$ independent uniformly random $d$--dimensional spherical vectors. In particular we give expressions for the characteristic function of the spherical Wishart matrix which are well--approximated using steepest descent.