AI 中文总结
研究整数集上受顶点强化的无限临界树索引随机游走，关注强强化情形。通过分析时间依赖广义波利亚瓮过程，对每步抽取次数由序列规定、选球概率与球数函数相关的情况进行研究，有界序列时恢复鲁宾颜色固定特征描述。
AI 中文摘要
我们考虑一类在整数集\(\mathbb{Z}\)上的无限临界树索引随机游走，其中粒子运动受顶点强化影响。主要关注强强化情形，预计过程几乎必然定位于两个位点。分析部分包括研究一个时间依赖的广义波利亚瓮过程，每步抽取次数由任意正整数序列\((\sigma_n)_{n\geq1}\)规定，选取给定颜色球的概率与该颜色球数量的函数成比例。特别对于有界序列\((\sigma_n)_{n\geq1}\)，我们恢复了鲁宾关于一种颜色固定的特征描述。
英文摘要
We consider a class of infinite critical tree-indexed random walks on $\mathbb Z$, where the motion of particles is subject to vertex reinforcement. We mainly focus on the strong reinforcement regime, where we expect the process to localize almost surely on two sites. Part of our analysis includes the study of a time-dependent generalized Pólya urn process, where the number of draws at each step is prescribed by a sequence $(σ_n)_{n\ge 1}$ of arbitrary positive integers, and the probability to pick a ball of a given color is proportional to a function of the {\it number} of balls of that color. In particular for bounded sequences $(σ_n)_{n\ge 1}$, we recover Rubin's characterization for the fixation of one color.
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