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Expander Random Walks: The General Case and Limitations

Authors Gil Cohen, Dor Minzer, Shir Peleg, Aaron Potechin, Amnon Ta-Shma

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Author Details

Gil Cohen
  • Department of Computer Science, Tel Aviv University, Israel
Dor Minzer
  • Department of Mathematics, Massachusetts Institute of Technology, Cmabridge, MA, USA
Shir Peleg
  • Department of Computer Science, Tel Aviv University, Israel
Aaron Potechin
  • Department of Computer Science, University of Chicago, IL, USA
Amnon Ta-Shma
  • Department of Computer Science, Tel Aviv University, Israel

Funding

  • Cohen, Gil: Supported by ERC starting grant 949499 and by the Israel Science Foundation grant 1569/18.
  • Peleg, Shir: The research leading to these results has received funding from the Israel Science Foundation (grant number 514/20) and from the Len Blavatnik and the Blavatnik Family foundation.
  • Ta-Shma, Amnon: The research leading to these results has received funding from the Israel Science Foundation (grant number 952/18)

Acknowledgements

The authors thank Amir Yehudayoff for pointing out that the analysis of the error of the parity function is tight if the second eigenvector of the graph is Boolean. This remark, several years later, matured to the current paper. We thank Venkat Guruswami and Vinayak Kumar for discussing their results with us. We thank Ron Peled for discussions on the CLT in TVD, and for pointing us to results on local convergence for independent processes. We thank Oded Goldreich for the vanishing with t and λ notation.

Cite As Get BibTex

Gil Cohen, Dor Minzer, Shir Peleg, Aaron Potechin, and Amnon Ta-Shma. Expander Random Walks: The General Case and Limitations. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.43

Abstract

Cohen, Peri and Ta-Shma [Gil Cohen et al., 2021] considered the following question: Assume the vertices of an expander graph are labelled by ± 1. What "test" functions f : {±1}^t → {±1} can or cannot distinguish t independent samples from those obtained by a random walk? [Gil Cohen et al., 2021] considered only balanced labellings, and proved that for all symmetric functions the distinguishability goes down to zero with the spectral gap λ of the expander G. In addition, [Gil Cohen et al., 2021] show that functions computable by AC⁰ circuits are fooled by expanders with vanishing spectral expansion. 
We continue the study of this question. We generalize the result to all labelling, not merely balanced ones. We also improve the upper bound on the error of symmetric functions. More importantly, we give a matching lower bound and show a symmetric function with distinguishability going down to zero with λ but not with t. Moreover, we prove a lower bound on the error of functions in AC⁰ in particular, we prove that a random walk on expanders with constant spectral gap does not fool AC⁰.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Expander Graphs
  • Random Walks
  • Lower Bounds

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