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.

翻译：随机几何图 $\mathsf{RGG}(n,\mathbb{S}^{d-1}, p)$ 通过如下方式生成：在 $\mathbb{S}^{d-1}$ 上均匀独立采样 $n$ 个向量 $\{V_i\}_{i = 1}^n$，并在满足 $\langle V_i,V_j\rangle \ge \tau^p_d$ 的顶点对 $i$ 和 $j$ 之间放置边，其中 $\tau^p_d$ 使得期望密度为 $p$。我们研究分布 $\mathsf{RGG}(n,\mathbb{S}^{d-1}, p)$ 及其高斯类似物的低次傅里叶系数。我们的主要概念贡献是一种新颖的两步策略，用于界定傅里叶系数，我们认为该策略更广泛地适用于研究潜在空间分布。首先，我们将边之间的依赖关系局部化到少量脆弱边上。其次，我们根据脆弱边集将潜在向量配置空间 $(\mathsf{RGG}(n,\mathbb{S}^{d-1}, p))^{\otimes n}$ 进行划分，并在每个配置子集上定义一个噪声算子，该算子独立作用于与脆弱边（在适当意义上）不相邻的边上。我们将由此得到的界应用于以下问题：1) 解决在观察完整边集以及最近由 [MVW24] 提出的非自适应选择掩模 $\mathcal{M}$ 模型两种情况下，区分球形与高斯随机几何图与 Erdős-Rényi 图的低次多项式复杂度问题；2) 在 $\mathsf{RGG}$ 可与 Erdős-Rényi 区分的参数区域中，展示区分 $\mathsf{RGG}$ 与植入着色模型 [KVWX23] 的统计计算鸿沟；3) 重证随机几何图第二特征值的已知界。