论文标题
Galton-Watson树与第一祖先互动
Galton-Watson trees with first ancestor interaction
论文作者
论文摘要
我们将一组随机的BienayMé-Galton-Watson树一组为有界数的后代和几代人数作为统计力学模型：一棵随机树是最大树的根部子树；如果在随机树中存在节点，则最大树的给定节点上的自旋等于后代的数量，否则等于-1。我们介绍了最近的邻居互动，有利于对邻居的对，这些邻居都有相对较大的后代。然后，我们证明了（1）相关性不平等和（2）用于生成函数，平均外部节点，相互作用能量和相应方差的递归关系。所得的二次动力系统在两个或更大的方面取决于所需的矩数，得出几乎确切的数值结果。后代分布与耦合常数之间的平衡导致了灭绝概率类似物的相图。在过渡线上，在数字上找到$ n+1 $中外部节点的平均数量，以缩放为$ n^{ - 2} $。
We consider the set of random Bienaymé-Galton-Watson trees with a bounded number of offspring and bounded number of generations as a statistical mechanics model: a random tree is a rooted subtree of the maximal tree; the spin at a given node of the maximal tree is equal to the number of offspring if the node is present in the random tree and equal to -1 otherwise. We introduce nearest neighbour interactions favouring pairs of neighbours which both have a relatively large offspring. We then prove (1) correlation inequalities and (2) recursion relations for generating functions, mean number of external nodes, interaction energy and the corresponding variances. The resulting quadratic dynamical system, in two dimensions or more depending on the desired number of moments, yields almost exact numerical results. The balance between offspring distribution and coupling constant leads to a phase diagram for the analogue of the extinction probability. On the transition line the mean number of external nodes in generation $n+1$ is found numerically to scale as $n^{-2}$.
