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 Ve be the number of vertices. This resolves a fixed-edge limit-distribution question posed by Rüdinger: if βe=Ve/(e+2), then √e(βe−1/2)⇒ N(0,1/32), equivalently Var(Ve)∼ 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(e3/4) window, a quartic correction from the rate function on every o(e5/6) window, precise moderate-tail constants, a quadratic moderate-deviation principle for every ae→∞ with ae=o(√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 6561/32√2(4/27)m. Rooted and unrooted fixed-edge laws are also uniformly exponentially close, with relative discrepancy O(ρe).
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BibTeX
@misc{bridges2026gaussian,
title = {{Gaussian Vertex-Face Balance in Random Convex Polyhedra with Fixed Edge Count}},
author = {Lachlan Bridges},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22402},
eprint = {2609.22402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22402},
}
RIS
TY - UNPB
TI - Gaussian Vertex-Face Balance in Random Convex Polyhedra with Fixed Edge Count
AU - Lachlan Bridges
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22402
UR - https://arxiv.org/abs/2609.22402
AB - 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 √e(β_e−1/2)⇒ N(0,1/32), equivalently Var(V_e)∼ 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→∞ with a_e=o(√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 6561/32√2(4/27)^m. Rooted and unrooted fixed-edge laws are also uniformly exponentially close, with relative discrepancy O(ρ^e).
ER -