固定边数的随机凸多面体中的高斯顶点-面平衡
Gaussian Vertex-Face Balance in Random Convex Polyhedra with Fixed Edge Count
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中文总结 AI 辅助
本文研究固定边数随机凸多面体的顶点数极限分布，证明其高斯波动并给出大偏差原理，解决Rüdinger提出的问题。
中文摘要 AI 辅助
在具有固定可允许边数 $e$ 的凸三维多面体的组合类型中均匀选择，并令 $V_e$ 为顶点数。这解决了 Rüdinger 提出的一个固定边极限分布问题：若 $\beta_e=V_e/(e+2)$，则 $\sqrt e(\beta_e-1/2)\Rightarrow N(0,1/32)$，等价地 $\operatorname{Var}(V_e)\sim e/32$。利用 Bender 和 Wormald 的经典有根枚举和不对称性结果，我们在每个 $o(e^{3/4})$ 窗口上导出了相对格点局部极限定理，在每个 $o(e^{5/6})$ 窗口上导出了来自速率函数的四次修正，精确的中间尾部常数，对每个 $a_e\to\infty$ 且 $a_e=o(\sqrt e)$ 的二次中间偏差原理，以及具有显式良好速率函数的完整速度-$e$ 大偏差原理。所有固定的标准化矩都收敛。当 $e=3m$ 时，两个极值顶点数具有相等的概率，渐近为 $\frac{6561}{32\sqrt2}(4/27)^m$。有根和无根的固定边分布也均匀指数接近，相对差异为 $O(\rho^e)$。
英文摘要
Choose uniformly among the combinatorial types of convex three-dimensional polyhedra with a fixed admissible number $e$ of edges, and let $V_e$ be the number of vertices. This resolves a fixed-edge limit-distribution question posed by Rüdinger: if $β_e=V_e/(e+2)$, then $\sqrt e(β_e-1/2)\Rightarrow N(0,1/32)$, equivalently $\operatorname{Var}(V_e)\sim e/32$. Using the classical rooted enumeration and asymmetry results of Bender and Wormald, we derive a relative lattice local limit theorem on every $o(e^{3/4})$ window, a quartic correction from the rate function on every $o(e^{5/6})$ window, precise moderate-tail constants, a quadratic moderate-deviation principle for every $a_e\to\infty$ with $a_e=o(\sqrt e)$, and a full speed-$e$ large-deviation principle with an explicit good rate function. All fixed standardised moments converge. When $e=3m$, the two extremal vertex counts have equal probability asymptotic to $\frac{6561}{32\sqrt2}(4/27)^m$. Rooted and unrooted fixed-edge laws are also uniformly exponentially close, with relative discrepancy $O(ρ^e)$.
