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TR20-163 | 5th November 2020 22:43

Expander Random Walks: A Fourier-Analytic Approach



In this work we ask the following basic question: assume the vertices of an expander graph are labelled by $0,1$. What "test" functions $f : \{ 0,1\}^t \to \{0,1\}$ cannot distinguish $t$ independent samples from those obtained by a random walk? The expander hitting property due to Ajtai, Komlos and Szemeredi (STOC 1987) is captured by the $\mathrm{AND}$ test function, whereas the fundamental expander Chernoff bound due to Gillman (SICOMP 1998), Heally (Computational Complexity 2008) is about test functions indicating whether the weight is close to the mean. In fact, it is known that all threshold functions are fooled by a random walk (Kipnis and Varadhan, Communications in Mathematical Physics 1986). Recently, it was shown that even the highly sensitive $\mathrm{PARITY}$ function is fooled by a random walk (Ta-Shma; STOC 2017).

We focus on balanced labels. Our first main result is proving that all symmetric functions are fooled by a random walk. Put differently, we prove a central limit theorem (CLT) for expander random walks with respect to the total variation distance, significantly strengthening the classic CLT for Markov Chains that is established with respect to the Kolmogorov distance due to Kipnis and Varadhan. Our approach significantly deviates from prior works. We first study how well a Fourier character $\chi_S$ is fooled by a random walk as a function of $S$. Then, given a test function $f$, we expand $f$ in the Fourier basis and combine the above with known results on the Fourier spectrum of $f$.

We also proceed further and consider general test functions - not necessarily symmetric. As our approach is Fourier analytic, it is general enough to analyze such versatile test functions. For our second result, we prove that random walks on sufficiently good expander graphs fool tests functions computed by $\mathbf{AC}^0$ circuits, read-once branching programs, and functions with bounded query complexity.

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