The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order $(\log n)^2$, speeding up the mixing to order $\log n$. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erd\H{o}s-R\'enyi graph is logarithmic.

翻译：快速混合随机游走的理论在现代随机化算法研究中起着基础性作用。通常，混合时间以最差初始位置衡量。众所周知，图中瓶颈的存在会阻碍混合，特别是从狭小瓶颈内部出发会在过程初始阶段显著减缓游走的扩散。平均混合时间定义为从均匀随机顶点出发的混合时间，因此不受这些瓶颈导致的缓慢扩散影响。本文提出一个通用框架，证明具有小瓶颈的图上随机游走的对数平均混合时间。该框架在特定具有异质性质的随机图族中尤为有效。我们在两个已知混合时间阶为$(\log n)^2$的随机模型上展示其适用性，将混合速度提升至$\log n$阶。首先，在连通图的平滑分析背景下，我们证明有界退化度随机扰动图的对数平均混合时间。一个具体实例是纽曼-沃茨小世界模型。其次，我们证明超临界渗流扩张图的对数平均混合时间。当宿主图为完全图时，该应用为超临界埃尔德什-雷尼图巨分支的平均混合时间是对数阶提供了另一种证明。