Scaling limits for simple random walks on random ordered graph trees
Scaling limits for simple random walks on random ordered graph trees
复制标题
随机排序图树上简单随机游走的缩放限制
DOI：
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发表时间：
2010
影响因子：
1.2
通讯作者：
D. Croydon
中科院分区：
文献类型：
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作者：
D. Croydon
考虑一类随机有序图树（Tn）n≥1，其中Tn有n个顶点.先前已经建立，如果当适当地重新标度为n → ∞时，相关的搜索深度过程收敛到归一化布朗偏移，则图树上的简单随机游动将布朗连续随机树上的布朗运动作为它们的标度极限。在这里，这一结果是扩展到证明存在的扩散缩放极限时，限制真实的树的体积措施是非原子的，支持的叶子上的限制树，并满足多项式下界的球的体积。进一步，作为推广的一个应用，证明了在后代数为无穷大的条件下，后代服从无穷大方差分布的Galton-Watson树上的简单随机游动，可以重新标度为收敛于相关α-稳定树上的布朗运动.
Consider a family of random ordered graph trees (T n ) n≥1, where T n has n vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as n → ∞, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the limiting tree, and satisfies a polynomial lower bound for the volume of balls. Furthermore, as an application of this generalisation, it is established that the simple random walks on a family of Galton-Watson trees with a critical infinite variance offspring distribution, conditioned on the total number of offspring, can be rescaled to converge to the Brownian motion on a related α-stable tree.