Higher dimensional distortion of random complexes

TL;DR

This paper demonstrates that certain random simplicial complexes exhibit significant distortion of filling areas when embedded into Euclidean space, extending the concept of metric distortion from graphs to higher dimensions.

Contribution

It introduces a family of random complexes that show high distortion of filling areas, providing a higher-dimensional analogue to known graph distortion phenomena.

Findings

Large family of complexes with high filling area distortion

Distortion phenomenon extends from graphs to higher-dimensional complexes

Highlights limitations of Euclidean embeddings for certain random complexes

Abstract

Using the random complexes of Linial and Meshulam, we exhibit a large family of simplicial complexes for which, whenever affinely embedded into Euclidean space, the filling areas of simplicial cycles is greatly distorted. This phenomenon can be regarded as a higher order analogue of the metric distortion of embeddings of random graphs.