The critical behaviors and the scaling functions of a coalescence equation

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.

arXiv commentsubmitted to J. Phys. A

References in corpus (10)

• Fractional moment bounds and disorder relevance for pinning models
• Smoothing effect of quenched disorder on polymer depinning transitions
• Marginal relevance of disorder for pinning models
• Rare-event induced binding transition of heteropolymers
• Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift
• Hierarchical pinning models, quadratic maps and quenched disorder
• The depinning transition in presence of disorder: a toy model
• Genealogies in simple models of evolution
• The free energy in the Derrida--Retaux recursive model
• Parametric statistics of zeros of Husimi representations of quantum chaotic eigenstates and random polynomials