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arXiv 1904.07375math.PR
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基于Galton-Watson树的简单随机游动桥的最大位移

Maximal displacement of simple random walk bridge on Galton-Watson trees

Josh Rosenberg

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摘要：

我们分析了超临界Galton-Watson树上的简单随机游动，其中游动的条件是在时间$2n$返回到根。具体来说，我们建立的渐近阶（常数因子）为$n\to\infty$，从根的最大位移。我们的结果，这是持有几乎肯定的每一个幸存的树$T$（受一些温和的时刻条件），分为两种情况。当后代分布以正概率取小于或等于$1$的值时，桥梁的最大位移约为$n^{1/3}$。相反，当后代分布的最小可能值至少等于$2$（并且是非常数）时，最大位移的阶数小于$n$，但大于$n^γ$（对于任何$γ<1$）。这些情况中的每一个都与规则树的情况相反，在规则树上，桥被认为是扩散的。为了获得我们的结果，我们展示了行走如何倾向于吸引最小程度的顶点的大集群，然后在那里它继续花费大部分时间。此类集群的大小和频率通常取决于后代分布可获得的最小可能值，正是这一事实在很大程度上解释了这两种制度的存在。

英文摘要：

We analyze simple random walk on a supercritical Galton-Watson tree, where the walk is conditioned to return to the root at time $2n$. Specifically, we establish the asymptotic order (up to a constant factor) as $n\to\infty$, of the maximal displacement from the root. Our results, which are shown to hold for almost surely every surviving tree $T$ (subject to some mild moment conditions), are broken up into two cases. When the offspring distribution takes a value less than or equal to $1$ with positive probability, the maximal displacement of the bridge is shown to be on the order of $n^{1/3}$. Conversely, when the offspring distribution has minimum possible value equal to at least $2$ (and is non-constant), the maximal displacement is shown to be of order less than $n$, but greater than $n^γ$ (for any $γ<1$). Each of these cases is in contrast to the case of a regular tree, on which the bridge is known to be diffusive. To obtain our results, we show how the walk tends to gravitate towards large clusters of vertices of minimal degree, where it then proceeds to spend most of its time. The size and frequency of such clusters is generally dependent on the minimum possible value attainable by the offspring distribution, and it is this fact which largely accounts for the existence of the two regimes.

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