Random walks on expanders play a crucial role in Markov Chain Monte Carlo algorithms, derandomization, graph theory, and distributed computing. A desirable property is that they are rapidly mixing, which is equivalent to having a spectral gap $\gamma$ (asymptotically) bounded away from $0$. Our work has two main strands. First, we establish a dichotomy for the robustness of mixing times on edge-weighted $d$-regular graphs (i.e., reversible Markov chains) subject to a Lipschitz condition, which bounds the ratio of adjacent weights by $\beta \geq 1$. If $\beta \ge 1$ is sufficiently small, then $\gamma \asymp 1$ and the mixing time is logarithmic in $n$. On the other hand, if $\beta \geq 2d$, there is an edge-weighting such that $\gamma$ is polynomially small in $1/n$. Second, we apply our robustness result to a time-dependent version of the so-called $\varepsilon$-biased random walk, as introduced in Azar et al. [Combinatorica 1996]. We show that, for any constant $\varepsilon>0$, a bias strategy can be chosen adaptively so that the $\varepsilon$-biased random walk covers any bounded-degree regular expander in $\Theta(n)$ expected time, improving the previous-best bound of $O(n \log \log n)$. We prove the first non-trivial lower bound on the cover time of the $\varepsilon$-biased random walk, showing that, on bounded-degree regular expanders, it is $\omega(n)$ whenever $\varepsilon = o(1)$. We establish this by controlling how much the probability of arbitrary events can be ``boosted'' by using a time-dependent bias strategy.

翻译：扩展图上的随机游走在马尔可夫链蒙特卡洛算法、去随机化、图论和分布式计算中起着关键作用。一个理想的性质是它们能够快速混合，这等价于具有（渐近地）远离0的谱隙$\gamma$。我们的工作包含两个主要方面。首先，我们针对满足利普希茨条件的边赋权$d$-正则图（即可逆马尔可夫链）建立了混合时间鲁棒性的二分性，该条件通过$\beta \geq 1$限制相邻权重的比值。若$\beta \ge 1$充分小，则$\gamma \asymp 1$且混合时间关于$n$是对数阶的。另一方面，若$\beta \geq 2d$，则存在一种边赋权方式使得$\gamma$关于$1/n$是多项式小的。其次，我们将鲁棒性结果应用于由Azar等人[Combinatorica 1996]提出的所谓$\varepsilon$-偏置随机游走的时间依赖版本。我们证明，对于任意常数$\varepsilon>0$，可以自适应地选择偏置策略，使得$\varepsilon$-偏置随机游走在$\Theta(n)$期望时间内覆盖任意有界度正则扩展图，改进了先前的最佳上界$O(n \log \log n)$。我们首次证明了$\varepsilon$-偏置随机游走覆盖时间的非平凡下界，表明在有界度正则扩展图上，当$\varepsilon = o(1)$时，覆盖时间为$\omega(n)$。我们通过控制使用时间依赖偏置策略能够将任意事件的概率“提升”多少来建立这一结果。