Gaussian Vertex-Face Balance in Random Convex Polyhedra with Fixed Edge Count

Abstract

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 R\"udinger: if $\beta_e=V_e/(e+2)$, then $\sqrt e(\beta_e-1/2)\Rightarrow N(0,1/32)$, 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 $O(\rho^e)$.