AI 中文总结
本文研究了随机几何图与Erdős--Rényi图在总变差距离上的可区分性阈值问题，证明了在特定范围内该阈值的精确条件。
AI 中文摘要
球面随机几何图$G(n,d,p)$是通过在单位球$\mathbb{S}^{d-1}\subseteq\mathbb{R}^d$上均匀采样$n$个独立点，并将足够接近的点对连接起来得到的，其中阈值被选择使得边概率为$p$。与此模型相关的核心问题，以及广泛其他模型相关的问题是：当底层几何结构对所得到的图产生影响，使其在总变差距离上可区分于Erdős--Rényi随机图$G(n,p)$时，何时会发生这种情况？这个问题的精确答案最初由Bubeck、Ding、Eldan和Rácz推测，他们预测当$d \gg n^3p^3(\log p^{-1})^3$时，$G(n,d,p)$和$G(n,p)$是不可区分的，并在低维情况下提供了区分这些模型的测试方法。尽管这一猜想吸引了概率论、理论计算机科学和高维统计学研究者的广泛关注，但此前只有在常密度情况下才被完全证明。在本文中，我们解决了该可区分猜想在广泛范围内$1/3 \geq p \geq n^{-1/5} \text{polylog}(n)$的证明。我们证明的关键成分是一个更强的陈述，它给出了$G(n,d,p)$实现指定图$H$的概率的精确渐进行为：在猜想阈值之上，这个概率至多是$G(n,p)$相应概率的$(1+o(1))$倍，其中$H$的带符号三角形计数作为主导修正项出现。
英文摘要
The spherical random geometric graph $G(n,d,p)$ is obtained by sampling $n$ independent points uniformly on the unit sphere $\mathbb{S}^{d-1}\subseteq\mathbb{R}^d$ and joining pairs of points which are sufficiently close, where the threshold is chosen so that the edge probability is $p$. The central question related to this model, and to a broad class of other models, is the following: when does the underlying geometry affect the resulting graph in a way which makes it distinguishable from the Erdős--Rényi random graph $G(n,p)$, as measured in total variation distance? The precise answer to this question was conjectured by Bubeck, Ding, Eldan, and Rácz, who predicted that $G(n,d,p)$ and $G(n,p)$ are indistinguishable precisely when $d \gg n^3p^3(\log p^{-1})^3$, and provided a test for distinguishing these models in the low-dimensional regime. Although this conjecture attracted considerable attention from researchers in probability, theoretical computer science, and high-dimensional statistics, it was previously fully proved only in the constant-density case. In this paper, we resolve the distinguishability conjecture in the broad range $1/3 \geq p \geq n^{-1/5} \text{polylog}(n)$. The key ingredient of our proof is a stronger statement which gives a precise asymptotic formula for the probability that $G(n,d,p)$ realizes a prescribed graph $H$: above the conjectured threshold, this probability is at most $(1+o(1))$ times the corresponding probability for $G(n,p)$, with the signed triangle count of $H$ appearing as the leading correction term.
