Pseudobinomiality of the Sticky Random Walk

Authors Venkatesan Guruswami, Vinayak M. Kumar

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Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Vinayak M. Kumar
  • Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA


V.G thanks Salil Vadhan, Vijay Bhattiprolu, Nicolas Resch, and Mahdi Cheraghchi for valuable discussions about sticky random walks at various points. Thanks to Salil also for suggesting the event underlying Theorem 18. V.K thanks Mayank Pandey for helpful discussions and pointing him to [Gut, 2006]. The authors thank Gil Cohen, Noam Peri, and Amnon TaShma for sharing an early version of their manuscript [Gil Cohen et al., 2020] after this work was posted at [Venkatesan Guruswami and Vinayak Kumar, 2020].

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Venkatesan Guruswami and Vinayak M. Kumar. Pseudobinomiality of the Sticky Random Walk. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 48:1-48:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Random walks on expanders are a central and versatile tool in pseudorandomness. If an arbitrary half of the vertices of an expander graph are marked, known Chernoff bounds for expander walks imply that the number M of marked vertices visited in a long n-step random walk strongly concentrates around the expected n/2 value. Surprisingly, it was recently shown that the parity of M also has exponentially small bias. Is there a common unification of these results? What other statistics about M resemble the binomial distribution (the Hamming weight of a random n-bit string)? To gain insight into such questions, we analyze a simpler model called the sticky random walk. This model is a natural stepping stone towards understanding expander random walks, and we also show that it is a necessary step. The sticky random walk starts with a random bit and then each subsequent bit independently equals the previous bit with probability (1+λ)/2. Here λ is the proxy for the expander’s (second largest) eigenvalue. Using Krawtchouk expansion of functions, we derive several probabilistic results about the sticky random walk. We show an asymptotically tight Θ(λ) bound on the total variation distance between the (Hamming weight of the) sticky walk and the binomial distribution. We prove that the correlation between the majority and parity bit of the sticky walk is bounded by O(n^{-1/4}). This lends hope to unifying Chernoff bounds and parity concentration, as well as establishing other interesting statistical properties, of expander random walks.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Expander graphs and randomness extractors
  • Expander Graphs
  • Fourier analysis
  • Markov Chains
  • Pseudorandomness
  • Random Walks


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