Concentration of hitting times in Erdős-Rényi graphs

Pub Date : 2024-05-12 DOI:10.1002/jgt.23119

Andrea Ottolini, Stefan Steinerberger

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Abstract

We consider Erdős-Rényi graphs $G (n, p)$ for $0 < p < 1$ fixed and $n \to \infty$ and study the expected number of steps, $H_{w v}$ , that a random walk started in $w$ needs to first arrive in $v$ . A natural guess is that an Erdős-Rényi random graph is so homogeneous that it does not really distinguish between vertices and $H_{w v} = (1 + o (1)) n$ . Löwe-Terveer established a CLT for the Mean Starting Hitting Time suggesting $H_{w v} = n \pm O (\sqrt{n})$ . We prove the existence of a strong concentration phenomenon: $H_{w v}$ is given, up to a very small error of size $≲ {(\log n)}^{3 ∕ 2} ∕ \sqrt{n}$ , by an explicit simple formula involving only the total number of edges $∣ E ∣$ , the degree $deg (v)$ and the distance $d (v, w)$ .

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厄尔多斯-雷尼图中命中时间的浓度

我们考虑了固定的和的厄尔多斯-雷尼图，并研究了从开始的随机漫步到达。一个自然的猜测是，Erdős-Rényi 随机图是如此同质，以至于它并不真正区分顶点和。Löwe-Terveer 建立了平均起始击球时间的 CLT，表明 .我们证明了一种强集中现象的存在：它是由一个只涉及边的总数、度和距离的显式简单公式给出的，误差很小。

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