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 non-zero 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 characterise the event undergoing the phase transition.

区分连​​续相变和一阶相变是随机离散系统中的主要挑战。我们研究了高尔顿-沃森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，令 $\mathcal{T}_2$ 为它包含从其根开始的无限二叉树的事件。这些事件满足相似的递归性质： $\mathcal{T}_1$ 成立当且仅当 $\mathcal{T}_1$ 成立至少一棵由根的孩子发起的树，并且 $\mathcal{T} _2$ 成立当且仅当 $\mathcal{T}_2$ 对于这些树中的至少两个成立。的概率 $\mathcal{T}_1$ 有一个连续的相变，当子分布的均值大于 1 时从 0 开始增加。另一方面， $\mathcal{T}_2$ 的概率具有一阶相过渡，在临界点不连续地跳到非零值。鉴于事件满足的递归属性，我们描述了发生连续相变的关键子分布。在许多情况下，我们还描述了经历相变的事件。