论文标题
关于封面时间波动的普遍性
On the universality of fluctuations for the cover time
论文作者
论文摘要
我们考虑在有限的顶点传递图上随机步行$γ$的有限度。我们找到了一个简单的几何条件，表征了覆盖时间波动：当且仅当$ \ mathrm {diamrm {diam}（γ）^2 = o（n/\ log n）$时，适当归一化的覆盖时间会收敛到标准gumbel变量，其中$ n = |γ| $。我们证明，这种条件还等同于未发现集合的去相关。这些论点依赖于Tessera和Tointon在Gromov定理的多项式增长群体的最佳版本中的最新突破，我们将其利用了强大的热核界限，并对Aldous和Brown和Brown的量化估计进行了完善的定量估计，这些定量估计值是独立利益的命中时间。
We consider random walks on finite vertex-transitive graphs $Γ$ of bounded degree. We find a simple geometric condition which characterises the cover time fluctuations: the suitably normalised cover time converges to a standard Gumbel variable if and only if $\mathrm{Diam}(Γ)^2 = o(n/\log n)$, where $n = |Γ|$. We prove that this condition is furthermore equivalent to the decorrelation of the uncovered set. The arguments rely on recent breakthroughs by Tessera and Tointon on finitary versions of Gromov's theorem on groups of polynomial growth, which we leverage into strong heat kernel bounds, and refined quantitative estimates on Aldous and Brown's exponential approximation of hitting times, which are of independent interest.
