The critical behaviors and the scaling functions of a coalescence equation

TL;DR

This paper investigates a coalescence equation's critical behaviors, deriving exact solutions and scaling functions that reveal different phase transition types, supported by numerical checks and related to a toy model for depinning transitions.

Contribution

It provides new exact solutions and scaling functions for a coalescence equation, linking critical behaviors to the shapes of critical trees and validating findings numerically.

Findings

Multiple families of exact solutions identified

New conjectures on critical tree shapes proposed

Numerical validation of theoretical results conducted

Abstract

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.