Emergence of condensation in Kingman's model of selection and mutation

TL;DR

This paper analyzes the emergence of condensation in Kingman's model of selection and mutation, showing that the wave of maximal fitness genes follows a gamma distribution, suggesting a universal wave shape in similar models.

Contribution

It introduces a scaling limit theorem describing condensation in Kingman's model and conjectures the universality of the gamma-shaped wave in complex models.

Findings

Wave towards maximal fitness has gamma distribution shape

Condensation occurs as a scaling limit in the model

Evidence suggests universality of wave shape in similar systems

Abstract

We describe the onset of condensation in the simple model for the balance between selection and mutation given by Kingman in terms of a scaling limit theorem. Loosely speaking, this shows that the wave moving towards genes of maximal fitness has the shape of a gamma distribution. We conjecture that this wave shape is a universal phenomenon that can also be found in a variety of more complex models, well beyond the genetics context, and provide some further evidence for this.