Fine Grained Analysis of High Dimensional Random Walks

Roy Gotlib, Tali Kaufman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 [13, 10, 15, 1], by presenting a structured version of the result of [1]. 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 [1]. 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.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023
EditorsNicole Megow, Adam Smith
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772969
DOIs
StatePublished - Sep 2023
Event26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 - Atlanta, United States
Duration: 11 Sep 202313 Sep 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume275
ISSN (Print)1868-8969

Conference

Conference26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023
Country/TerritoryUnited States
CityAtlanta
Period11/09/2313/09/23

Bibliographical note

Publisher Copyright:
© 2023 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

Keywords

  • High Dimensional Expanders
  • High Dimensional Random Walks
  • Local Spectral Expansion
  • Local to Global
  • Trickling Down

Fingerprint

Dive into the research topics of 'Fine Grained Analysis of High Dimensional Random Walks'. Together they form a unique fingerprint.

Cite this