Pseudorandomness of the Sticky Random Walk
Emile Anand, Chris Umans
TL;DR
This work extends the pseudorandomness of expander random walks to a generalized sticky random walk with a $p$-ary alphabet, achieving an $O(\lambda)$ total variation distance bound for a family of expanders when $\lambda \le 0.27$, by deploying a Fourier/Krawtchouk analytic framework. It generalizes the sticky random walk to $S(n,p,\lambda)$, derives exact expressions for the expectations of Krawtchouk functions, and uses these to bound deviations from uniform labeling, ultimately showing a tight TVD bound that is independent of the alphabet size for the targeted family. The paper also establishes reductions that connect the generalized sticky walk to an infinite family of expander graphs with spectral expansion $p\lambda$, and provides an alternate complex-domain treatment yielding $O(\lambda p^{O(p)})$ TVD bounds consistent with prior Fourier-based results. The results address conjectures about the tightness of the $(\frac{p}{\min f})^{O(p)}$ factor and open avenues for refining the radius of convergence beyond $0.27$, with implications for derandomization and pseudorandomness on expanders. Overall, the work enhances understanding of how structured Markov walks on labels can fool symmetric tests while preserving strong spectral properties.
Abstract
We extend the pseudorandomness of random walks on expander graphs using the sticky random walk. Building on prior works, it was recently shown that expander random walks can fool all symmetric functions in total variation distance (TVD) upto an $O(λ(\frac{p}{\min f})^{O(p)})$ error, where $λ$ is the second largest eigenvalue of the expander, $p$ is the size of the arbitrary alphabet used to label the vertices, and $\min f = \min_{b\in[p]} f_b$, where $f_b$ is the fraction of vertices labeled $b$ in the graph. Golowich and Vadhan conjecture that the dependency on the $(\frac{p}{\min f})^{O(p)}$ term is not tight. In this paper, we resolve the conjecture in the affirmative for a family of expanders. We present a generalization of the sticky random walk for which Golowich and Vadhan predict a TVD upper bound of $O(λp^{O(p)})$ using a Fourier-analytic approach. For this family of graphs, we use a combinatorial approach involving the Krawtchouk functions to derive a strengthened TVD of $O(λ)$. Furthermore, we present equivalencies between the generalized sticky random walk, and, using linear-algebraic techniques, show that the generalized sticky random walk parameterizes an infinite family of expander graphs.