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High-Dimensional Expanders from Expanders

Authors Siqi Liu, Sidhanth Mohanty, Elizabeth Yang

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  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • High-Dimensional Expanders
  • Markov Chains

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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., 2004].

Cite As Get BibTex

Siqi Liu, Sidhanth Mohanty, and Elizabeth Yang. High-Dimensional Expanders from Expanders. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 12:1-12:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ITCS.2020.12

Author Details

Siqi Liu
  • EECS Department, UC Berkeley, CA, United States
Sidhanth Mohanty
  • EECS Department, UC Berkeley, CA, United States
Elizabeth Yang
  • EECS Department, UC Berkeley, CA, United States

Funding

  • Liu, Siqi: This research was supported in part by a Google Faculty Award and donations from the Ethereum Foundation and the Interchain Foundation.
  • Mohanty, Sidhanth: Supported by NSF Grant CCF-1718695.
  • Yang, Elizabeth: Supported by NSF Grant DGE 1752814.

Acknowledgements

We thank Tom Gur for introducing us to this intriguing question and for helpful discussions, and we thank Nikhil Srivastava for insightful conversations. We would also like to thank the Simons Institute for the Theory of Computing where a large portion of this work was done. The third author is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1752814.

References

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