论文标题
加尔顿 - 瓦特森树上的连续相变
Continuous phase transitions on Galton-Watson trees
论文作者
论文摘要
在随机离散系统中，区分连续和一阶相变是一个主要挑战。我们研究了Galton-Watson树上具有递归结构的事件的主题。例如，令$ \ Mathcal {t} _1 $为Galton-Watson树是无限的事件，然后让$ \ Mathcal {T} _2 $是从其根开始包含一个无限二进制树的事件。这些事件满足类似的递归属性：$ \ Mathcal {t} _1 $在且仅当$ \ nathcal {t} _1 $中保留至少是由根的孩子发起的树，而$ \ nathcal {t} _2 $ holds if且仅在$ \ nathcal concal {twosef and and Culcal { $ \ MATHCAL {T} _1 $的概率具有连续的相变，从0时，儿童分布的平均值高于1。另一方面，$ \ Mathcal {t} _2 $的概率具有一阶相变，不断地跳转到非零的值，以关键的率在非零的情况下。鉴于事件满足的递归属性，我们描述了发生连续相位过渡的关键儿童分布。在许多情况下，我们还表征了进行相变的事件。
Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton-Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton-Watson tree is infinite, and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a nonzero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterize the event undergoing the phase transition.
