# High-Dimensional Expanders from Expanders

## TL;DR

This paper introduces a simple method to convert expander graphs into high-dimensional expanders with rapid mixing properties for high-order random walks, enabling new constructions and sampling models.

## Contribution

It provides an elementary construction technique for high-dimensional expanders from expander graphs and establishes their rapid mixing properties using decomposable Markov chains.

## Key findings

- New construction method for high-dimensional expanders from expander graphs
- Proved rapid mixing of high-order random walks on these complexes
- Developed a probabilistic sampling model for constant degree high-dimensional expanders

## Abstract

We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders. In particular, we show that given an expander graph $G$, adding self loops to $G$ and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by Jerrum et al.