Among observables characterizing the random exploration of a graph or lattice, the cover time, namely, the time to visit every site, continues to attract widespread interest. Much insight about cover times is gained by mapping to the (spaceless) coupon collector problem, which amounts to ignoring spatiotemporal correlations, and an early conjecture that the limiting cover time distribution of regular random walks on large lattices converges to the Gumbel distribution in $d\ge 3$ was recently proved rigorously. Furthermore, a number of mathematical and numerical studies point to the robustness of the Gumbel universality to modifications of the spatial features of the random search processes (e.g., introducing persistence and/or intermittence, or changing the graph topology). Here we investigate the robustness of the Gumbel universality to dynamical modification of the temporal features of the search, specifically by allowing the random walker to “accelerate” or “decelerate” upon visiting a previously unexplored site. We generalize the mapping mentioned above by relating the statistics of cover times to the roughness of $1/f^{\alpha }$ Gaussian signals, leading to the conjecture that the Gumbel distribution is but one of a family of cover time distributions, ranging from Gaussian for highly accelerated cover, to exponential for highly decelerated cover. While our conjecture is confirmed by systematic Monte Carlo simulations in dimensions $d>3$, our results for acceleration in $d=3$ challenge the current understanding of the role of correlations in the cover time problem.

中文翻译：

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动态加快覆盖时间

在表征图或晶格随机探索的可观察对象中，覆盖时间（即访问每个站点的时间）继续引起广泛关注。通过映射到（无空间）优惠券收集器问题，可以得到关于覆盖时间的很多见解，这等于忽略了时空相关性，而且早期的猜测是，规则随机游走在大格子上的有限覆盖时间分布会收敛到Gumbel分布中。$d\ge 3$最近被严格证明。此外，许多数学和数值研究指出了Gumbel通用性对修改随机搜索过程的空间特征（例如，引入持久性和/或间断性，或改变图拓扑）的鲁棒性。在这里，我们研究了Gumbel通用性对搜索的时态特征进行动态修改的鲁棒性，特别是通过允许随机步行者在访问之前未开发的站点时“加速”或“减速”。我们通过将覆盖时间的统计量与粗糙度相关联来概括上述映射。$\mathrm{1个}/F^{\alpha }$高斯信号，导致人们猜想Gumbel分布只是覆盖时间分布家族的一种，范围从高加速覆盖的高斯到高减速覆盖的指数。虽然我们的猜想已通过系统的蒙特卡罗模拟进行了验证$d>3$，我们的加速结果 $d=3$ 挑战当前对相关性在掩护时间问题中的作用的理解。