Linear embeddings of random complexes

TL;DR

This paper determines precise conditions under which random simplicial complexes can be linearly embedded into Euclidean space, based on their defining parameters.

Contribution

It provides necessary and sufficient inequalities on the parameters for linear embeddability of multiparameter random complexes.

Findings

Derived strict inequalities for embedding parameters

Established conditions for embedding into Euclidean space

Enhanced understanding of geometric properties of random complexes

Abstract

For $X \sim X(n; 1, n^{-\alpha_1}, n^{-\alpha_2}, ...)$ in the multiparameter random simplicial complex model we establish necessary and sufficient strict inequalities on the $\alpha_i$'s to linearly embed the complex into $\mathbb{R}^{2d}$.