Central limit theorems for high dimensional lattice polytopes: symmetric edge polytopes

TL;DR

This paper establishes central limit theorems for symmetric edge polytopes derived from Erdős–Rényi graphs, revealing their probabilistic behavior in high dimensions and identifying unique fluctuation regimes.

Contribution

It provides the first distributional limit theorems for random lattice polytopes, combining combinatorial geometry with probabilistic methods to analyze their structure.

Findings

Derived asymptotics for expectations and variances of polytope edges

Proved central limit theorems with explicit convergence rates

Identified a parameter value with atypical fluctuation behavior

Abstract

We investigate symmetric edge polytopes generated by Erd\H{o}s--R\'enyi random graphs in a high-dimensional regime. These objects provide a natural and largely unexplored model of random lattice polytopes, in which geometric properties are governed by graph-theoretic structure. Focusing on the number of polytope edges and on the number of edges in unimodular triangulations, we derive precise asymptotics for expectations and variances and establish central limit theorems with explicit rates of convergence. Our analysis combines a detailed combinatorial-geometric study of the graph configurations determining the facial structure with the discrete Malliavin--Stein method for normal approximation. In particular, we identify a distinguished parameter value at which the leading variance term cancels, producing an atypical fluctuation regime. To the best of our knowledge, the results obtained here constitute the first distributional limit theorems for random lattice polytopes