发表机构
Bar-Ilan University; The Taft School; Wesleyan University; Hebrew University of Jerusalem（巴伊兰大学; 塔夫特中学; 卫斯理安大学; 耶路撒冷希伯来大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究环面上随机几何图的一阶零一律与收敛律，确定固定半径下孪生计数渐近及临界阈值处的泊松收敛，并构造可定义构型。
AI 中文摘要
设$G_D(n;r)$为由$D$维环面上$n$个独立均匀点生成的随机几何图，其邻接关系由环面$L^\infty$距离至多$r$定义。我们研究固定半径和稀疏情形下的一阶零一律与收敛律。在固定半径下，我们确定了每个维度中相邻孪生对期望数的渐近行为。在二维情形中，更一般地，对于每个原点对称凸连接体$K\subset(-1/2,1/2)^2$，孪生计数收敛于均值为$\operatorname{area}(K^\circ)/16$的泊松变量；因此，在$L^\infty$和欧几里得模型中，对所有固定的$0<r<1/2$，零一律均不成立。在每个临界分量阈值$n^k r_n^{D(k-1)}\to a\in(0,\infty)$处，可行连通$k$顶点类型的分量数联合收敛于独立泊松变量，从而得到完整的一阶收敛律。在相邻阈值之间，零一律成立。对于$D\ge3$，我们还构造了一个概率阶为$1/n$的可定义公共邻域构型。
英文摘要
Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics of the expected number of adjacent twin pairs in every dimension. In dimension two, more generally, for each origin-symmetric convex connection body $K\subset(-1/2,1/2)^2$, the twin count converges to a Poisson variable with mean $\operatorname{area}(K^\circ)/16$; hence the zero-one law fails for all fixed $0<r<1/2$ in the $L^\infty$ and Euclidean models. At each critical component threshold $n^k r_n^{D(k-1)}\to a\in(0,\infty)$, the numbers of components of the feasible connected $k$-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For $D\ge3$, we also construct a definable common-neighborhood configuration of probability order $1/n$.
Comments30 pages
