In this paper we study the random geometric graph $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$ with $L_q$ distance where each vertex is sampled uniformly from the $d$-dimensional torus and where the connection radius is chosen so that the marginal edge probability is $p$. In addition to results addressing other questions, we make progress on determining when it is possible to distinguish $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$ from the Endos-R\'enyi graph $\mathsf{G}(n,p)$. Our strongest result is in the extreme setting $q = \infty$, in which case $\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^\infty_p,p)$ is the $\mathsf{AND}$ of $d$ 1-dimensional random geometric graphs. We derive a formula similar to the cluster-expansion from statistical physics, capturing the compatibility of subgraphs from each of the $d$ 1-dimensional copies, and use it to bound the signed expectations of small subgraphs. We show that counting signed 4-cycles is optimal among all low-degree tests, succeeding with high probability if and only if $d = \tilde{o}(np).$ In contrast, the signed triangle test is suboptimal and only succeeds when $d = \tilde{o}((np)^{3/4}).$ Our result stands in sharp contrast to the existing literature on random geometric graphs (mostly focused on $L_2$ geometry) where the signed triangle statistic is optimal.

翻译：本文研究采用$L_q$距离的随机几何图$\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$，其中顶点从$d$维环面上均匀采样，且连接半径的选取使得边边缘概率为$p$。除回答其他问题外，我们在判定何时能区分$\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^q_p,p)$与Endős-Rényi图$\mathsf{G}(n,p)$方面取得了进展。最强结果出现在极端设定$q = \infty$下，此时$\mathsf{RGG}(n,\mathbb{T}^d,\mathsf{Unif},\sigma^\infty_p,p)$是$d$个一维随机几何图的$\mathsf{AND}$运算。我们从统计物理中推导出类似团簇展开的公式，捕捉每个一维副本中子图的相容性，并据此界定小子图的有符号期望。研究表明，在所有低度检验中，有符号4环计数具有最优性，当且仅当$d = \tilde{o}(np)$时能以高概率成功。相比之下，有符号三角形检验为次优，仅当$d = \tilde{o}((np)^{3/4})$时成功。该结果与现有随机几何图文献（多聚焦于$L_2$度量）形成鲜明对比——后者中有符号三角形统计量是最优的。