论文标题
带最近邻居相互作用的吉布斯随机树
A Gibbsian random tree with nearest neighbour interaction
论文作者
论文摘要
如先前工作中所述，我们以最近的邻居相互作用来重新审视随机树模型，从而增强了增长。当基本的自由BienayMé-Galton-Watson（BGW）模型是次临界的时，我们表明具有相互作用的（非马尔科夫）模型表现出子和超临界方案之间的相变。在关键制度中，使用动力学系统中的工具，我们表明该模型的分区功能以$ n^{ - 1} $的限制在生成数字$ n $中。在几乎确定灭绝的关键制度中，我们还证明，在$ n $ n $ n $衰减中，树中的平均外部节点数量如$ n^{ - 2} $。最后，我们给出了随机树的自旋表示，从吉布斯状态理论（包括FKG不平等）的工具开辟了道路。当自由BGW工艺的分支机制定律具有无限的支持时，我们扩展了先前工作的构建。
We revisit the random tree model with nearest-neighbour interaction as described in previous work, enhancing growth. When the underlying free Bienaymé-Galton-Watson (BGW) model is sub-critical, we show that the (non-Markov) model with interaction exhibits a phase transition between sub- and super-critical regimes. In the critical regime, using tools from dynamical systems, we show that the partition function of the model approaches a limit at rate $n^{-1}$ in the generation number $n$. In the critical regime with almost sure extinction, we also prove that the mean number of external nodes in the tree at generation $n$ decays like $n^{-2}$. Finally, we give a spin representation of the random tree, opening the way to tools from the theory of Gibbs states, including FKG inequalities. We extend the construction in previous work when the law of the branching mechanism of the free BGW process has unbounded support.
