  The distribution $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ is formed by sampling
independent vectors $\{V_i\}_{i = 1}^n$ uniformly on $\mathbb{S}^{d-1}$ and
placing an edge between pairs of vertices $i$ and $j$ for which $\langle
V_i,V_j\rangle \ge \tau^p_d,$ where $\tau^p_d$ is such that the expected
density is $p.$ Our main result is a poly-time implementable coupling between
Erd\H{o}s-R\'enyi and $\mathsf{RGG}$ such that $\mathsf{G}(n,p(1 -
\tilde{O}(\sqrt{np/d})))\subseteq \mathsf{RGG}(n,\mathbb{S}^{d-1},p)\subseteq
\mathsf{G}(n,p(1 + \tilde{O}(\sqrt{np/d})))$ edgewise with high probability
when $d\gg np.$ We apply the result to: 1) Sharp Thresholds: We show that for
any monotone property having a sharp threshold with respect to the
Erd\H{o}s-R\'enyi distribution and critical probability $p^c_n,$ random
geometric graphs also exhibit a sharp threshold when $d\gg np^c_n,$ thus
partially answering a question of Perkins. 2) Robust Testing: The coupling
shows that testing between $\mathsf{G}(n,p)$ and
$\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ with $\epsilon n^2p$ adversarially
corrupted edges for any constant $\epsilon>0$ is information-theoretically
impossible when $d\gg np.$ We match this lower bound with an efficient
(constant degree SoS) spectral refutation algorithm when $d\ll np.$ 3)
Enumeration: We show that the number of geometric graphs in dimension $d$ is at
least $\exp(dn\log^{-7}n)$, recovering (up to the log factors) the sharp result
of Sauermann.
