随机几何图中覆盖时间的跳跃
On the jump of the cover time in random geometric graphs
AI总结：
研究超临界d维随机几何图巨分支上简单随机游走的覆盖时间，证明在连通阈值半径处覆盖时间从Θ(n log² n)跳跃到Θ(n log n)。
摘要：
本文研究超临界d维随机几何图（顶点为Poi(n)泊松点过程）巨分支上简单随机游走的覆盖时间。我们证明覆盖时间在连通阈值半径r_c处发生跳跃：设r_g为出现巨分支的阈值，我们证明若半径r满足(1+ε)r_g ≤ r ≤ (1-ε)r_c（其中ε>0任意小），则巨分支的覆盖时间渐近几乎必然为Θ(n log² n)。另一方面，我们证明当r ≥ (1+ε)r_c时，图的覆盖时间渐近几乎必然为Θ(n log n)（此前仅在d=2且半径大一个常数因子时已知）。我们的证明也揭示了r_c附近的一些行为。
英文摘要：
In this paper we study the cover time of the simple random walk on the giant component of supercritical $d$-dimensional random geometric graphs on $\mathrm{Poi}(n)$ vertices. We show that the cover time undergoes a jump at the connectivity threshold radius $r_c$: with $r_g$ denoting the threshold for having a giant component, we show that if the radius $r$ satisfies $(1+\varepsilon)r_g \le r \le (1-\varepsilon)r_c$ for $\varepsilon > 0$ arbitrarily small, the cover time of the giant component is asymptotically almost surely $Θ(n \log^2 n$). On the other hand, we show that for $r \ge (1+\varepsilon)r_c$, the cover time of the graph is asymptotically almost surely $Θ(n \log n)$ (which was known for $d=2$ only for a radius larger by a constant factor). Our proofs also shed some light onto the behavior around $r_c$.
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