  The random geometric graph $\mathsf{RGG}(n,\mathbb{S}^{d-1}, p)$ is formed by
sampling $n$ i.i.d. 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 $p.$ We study the low-degree Fourier coefficients of the
distribution $\mathsf{RGG}(n,\mathbb{S}^{d-1}, p)$ and its Gaussian analogue.
  Our main conceptual contribution is a novel two-step strategy for bounding
Fourier coefficients which we believe is more widely applicable to studying
latent space distributions. First, we localize the dependence among edges to
few fragile edges. Second, we partition the space of latent vector
configurations $(\mathsf{RGG}(n,\mathbb{S}^{d-1}, p))^{\otimes n}$ based on the
set of fragile edges and on each subset of configurations, we define a noise
operator acting independently on edges not incident (in an appropriate sense)
to fragile edges.
  We apply the resulting bounds to: 1) Settle the low-degree polynomial
complexity of distinguishing spherical and Gaussian random geometric graphs
from Erdos-Renyi both in the case of observing a complete set of edges and in
the non-adaptively chosen mask $\mathcal{M}$ model recently introduced by
[MVW24]; 2) Exhibit a statistical-computational gap for distinguishing
$\mathsf{RGG}$ and the planted coloring model [KVWX23] in a regime when
$\mathsf{RGG}$ is distinguishable from Erdos-Renyi; 3) Reprove known bounds on
the second eigenvalue of random geometric graphs.
