Berry--Esseen Bounds for Faces in Unimodular Triangulations of Random Polytopes

Consider the number of faces arising from unimodular triangulations of random polytopes associated with Erdős–Rényi graphs. Focusing on two-dimensional faces of regular unimodular triangulations of random symmetric edge polytopes, we derive precise asymptotic expansions for the expectation and variance and then establish a quantitative central limit theorem with an explicit Berry–Esseen bound. The proof combines an algebraically derived characterization of faces in terms of local graph configurations with a detailed combinatorial–geometric study of the relevant configurations. The normal approximation is established using the discrete Malliavin–Stein method. We also determine the asymptotic orders of the expectation and variance and establish a quantitative central limit theorem for the number of faces in each fixed higher dimension. These results identify a common local mechanism underlying the fluctuations of face numbers and extend the distributional theory of random symmetric edge polytopes [Donzelmann et al., arXiv:2603.10650] beyond edge statistics.