We show that a coalescence equation exhibits a variety of critical behaviors, depending on the initial condition. This equation was introduced a few years ago to understand a toy model studied by Derrida and Retaux to mimic the depinning transition in presence of disorder. It was shown recently that this toy model exhibits the same critical behaviors as the equation studied in the present work. Here we find several families of exact solutions of this coalescence equation, in particular a family of scaling functions which are closely related to the different possible critical behaviors. These scaling functions lead to new conjectures, in particular on the shapes of the critical trees, that we have checked numerically.

中文翻译：

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合并方程的临界行为和标度函数

我们表明，根据初始条件，合并方程展现出多种临界行为。该方程式是几年前引入的，目的是了解德里达和雷托研究的一种玩具模型，以模拟存在障碍时的固定转变。最近表明，该玩具模型表现出与本工作中研究的方程相同的临界行为。在这里，我们找到了这个合并方程的精确解的几个族，尤其是与不同的可能临界行为密切相关的缩放函数族。这些缩放函数导致新的猜想，尤其是在关键树的形状上，我们已经进行了数值检验。