Fine Grained Analysis of High Dimensional Random Walks

Authors Roy Gotlib, Tali Kaufman



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Author Details

Roy Gotlib
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Tali Kaufman
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

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Roy Gotlib and Tali Kaufman. Fine Grained Analysis of High Dimensional Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.49

Abstract

One of the most important properties of high dimensional expanders is that high dimensional random walks converge rapidly. This property has proven to be extremely useful in a variety of fields in the theory of computer science from agreement testing to sampling, coding theory and more. In this paper we present a state of the art result in a line of works analyzing the convergence of high dimensional random walks [Tali Kaufman and David Mass, 2017; Irit Dinur and Tali Kaufman, 2017; Tali Kaufman and Izhar Oppenheim, 2018; Vedat Levi Alev and Lap Chi Lau, 2020], by presenting a structured version of the result of [Vedat Levi Alev and Lap Chi Lau, 2020]. While previous works examined the expansion in the viewpoint of the worst possible eigenvalue, in this work we relate the expansion of a function to the entire spectrum of the random walk operator using the structure of the function; We call such a theorem a Fine Grained High Order Random Walk Theorem. In sufficiently structured cases the fine grained result that we present here can be much better than the worst case while in the worst case our result is equivalent to [Vedat Levi Alev and Lap Chi Lau, 2020].
In order to prove the Fine Grained High Order Random Walk Theorem we introduce a way to bootstrap the expansion of random walks on the vertices of a complex into a fine grained understanding of higher order random walks, provided that the expansion is good enough.
In addition, our single bootstrapping theorem can simultaneously yield our Fine Grained High Order Random Walk Theorem as well as the well known Trickling down Theorem. Prior to this work, High order Random walks theorems and Tricking down Theorem have been obtained from different proof methods.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Random walks and Markov chains
Keywords
  • High Dimensional Expanders
  • High Dimensional Random Walks
  • Local Spectral Expansion
  • Local to Global
  • Trickling Down

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References

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