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 $O(\lambda(\frac{p}{\min f})^{O(p)})$ error, where $\lambda$ 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 $O(\lambda p^{O(p)})$ 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 $O(\lambda)$. 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.

翻译：我们利用黏性随机游走扩展了扩张图上的随机游走的伪随机性。基于先前的工作，最近研究表明，扩张图上的随机游走能够在全变差距离下欺骗所有对称函数，其误差上界为$O(\lambda(\frac{p}{\min f})^{O(p)})$，其中$\lambda$是扩张图的第二大特征值，$p$是标记顶点所使用的任意字母表的大小，而$\min f = \min_{b\in[p]} f_b$，其中$f_b$是图中标记为$b$的顶点比例。Golowich和Vadhan推测，对$(\frac{p}{\min f})^{O(p)}$项的依赖并非紧的。本文针对一族扩张图肯定地解决了这一猜想。我们提出了黏性随机游走的一种推广，对于该推广，Golowich和Vadhan使用傅里叶分析方法预测其全变差距离上界为$O(\lambda p^{O(p)})$。对于这族图，我们利用涉及Krawtchouk函数的组合方法，推导出加强的全变差距离上界$O(\lambda)$。此外，我们展示了推广的黏性随机游走之间的等价性，并通过线性代数技术表明，推广的黏性随机游走参数化了一族无限的扩张图。