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TR22-024 | 17th February 2022 23:58
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#### Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs

**Abstract:**
We study the pseudorandomness of random walks on expander graphs against tests computed by symmetric functions and permutation branching programs. These questions are motivated by applications of expander walks in the coding theory and derandomization literatures. We show that expander walks fool symmetric functions up to a $O(\lambda)$ error in total variation distance. This bound improves upon a line of prior work, which gave bounds that were weaker or applied only in more restricted cases. We extend our analysis to unify it with and strengthen the expander walk Chernoff bound. We then show that expander walks fool permutation branching programs up to a $O(\lambda)$ error in $\ell_2$-distance, and we prove that much tighter bounds hold for programs with a certain structure. We also prove lower bounds to show that our results are tight. To prove our results for symmetric functions, we analyze the Fourier coefficients of the relevant distributions using linear-algebraic techniques. Our analysis for permutation branching programs is likewise linear-algebraic in nature, but also makes use of the recently introduced singular-value approximation notion for matrices (Ahmadinejad et al. 2021).