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摘要翻译：
我们证明了在Cayley树上的欧拉行走表现出两种动态相：凝聚相和低密度相。在凝聚相中，自组织区域以致密畴的形式生长。在低密度阶段，自组织（访问）节点的比例从树的一代到下一代迅速下降。我们详细描述了在凝聚相中欧拉游动到根的返回和自组织畴的增长。我们还研究了欧拉游动在两个体系分离点处的临界行为。
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英文标题：
《Euler Walk on a Cayley Tree》
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作者：
A.E. Patrick
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最新提交年份：
2007
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分类信息：
一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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英文摘要：
We show that the Euler walk on a Cayley tree exhibits two regimes (dynamic phases): a condensed phase and a low-density phase. In the condensed phase the self-organized area grows as a compact domain. In the low-density phase the proportion of self-organized (visited) nodes decreases rapidly from one generation of the tree to the next. We describe in detail returns of the Euler walk to the root and growth of the self-organized domain in the condensed phase. We also investigate the critical behaviour of the Euler walk at the point separating the two regimes.
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PDF链接：
https://arxiv.org/pdf/706.3161
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