We study high-dimensional random geometric graphs (RGGs) of edge-density $p$ with vertices uniformly distributed on the $d$-dimensional torus and edges inserted between sufficiently close vertices with respect to an $L_q$-norm. We focus on distinguishing an RGG from an Erd\H{o}s--R\'enyi (ER) graph if both models have edge probability $p$. So far, most results considered either spherical RGGs with $L_2$-distance or toroidal RGGs under $L_\infty$-distance. However, for general $L_q$-distances, many questions remain open, especially if $p$ is allowed to depend on $n$. The main reason for this is that RGGs under $L_q$-distances can not easily be represented as the logical AND of their 1-dimensional counterparts, as for $L_\infty$ geometries. To overcome this, we devise a novel technique for quantifying the dependence between edges based on modified Edgeworth expansions. Our technique yields the first tight algorithmic upper bounds for distinguishing toroidal RGGs under general $L_q$ norms from ER-graphs for fixed $p$ and $q$. We achieve this by showing that signed triangles can distinguish the two models when $d\ll n^3p^3$ for the whole regime of $c/n<p<1$. Additionally, our technique yields an improved information-theoretic lower bound for this task, showing that the two distributions converge whenever $d=\tilde{\Omega}(n^3p^2)$, which is just as strong as the currently best known lower bound for spherical RGGs in case of general $p$ from Liu et al. [STOC'22]. Finally, our expansions allow us to tightly characterize the spectral properties of toroidal RGGs both under $L_q$-distances for fixed $1\le q<\infty$, and $L_\infty$-distance. Our results partially resolve a conjecture of Bangachev and Bresler [COLT'24] and prove that the distance metric, rather than the underlying space, is responsible for the observed differences in the behavior of spherical and toroidal RGGs.

翻译：本文研究边密度为$p$的高维随机几何图，其顶点均匀分布于$d$维环面上，并根据$L_q$范数在足够接近的顶点间建立连接边。我们聚焦于在两种模型具有相同边概率$p$的情况下，区分随机几何图与Erdős–Rényi随机图。目前大多数研究成果要么针对采用$L_2$距离的球面随机几何图，要么研究$L_\infty$距离下的环面随机几何图。然而对于一般$L_q$距离，特别是当$p$允许依赖于$n$时，许多问题尚未解决。这主要是因为$L_q$距离下的随机几何图无法像$L_\infty$几何结构那样简单地表示为其一维对应模型的逻辑与运算。为突破此限制，我们开发了一种基于修正Edgeworth展开的新型技术来量化边之间的依赖性。该技术首次为固定$p$和$q$情况下区分一般$L_q$范数下环面随机几何图与ER图提供了紧致的算法上界。我们通过证明在$c/n<p<1$的整个参数范围内，当$d\ll n^3p^3$时带符号三角形可以区分这两种模型来实现这一目标。此外，我们的技术还为此任务提供了改进的信息论下界，证明当$d=\tilde{\Omega}(n^3p^2)$时两种分布会收敛，该下界强度与Liu等人[STOC'22]针对一般$p$情况下球面随机几何图目前已知的最佳下界相当。最后，我们的展开式能够精确刻画固定$1\le q<\infty$的$L_q$距离及$L_\infty$距离下环面随机几何图的谱特性。这些结果部分解决了Bangachev与Bresler[COLT'24]的猜想，并证明距离度量（而非底层空间）是导致球面与环面随机几何图行为差异的根本原因。