Kingman's House-of-Cards model: random mutation probabilities and random matrices

TL;DR

This paper investigates Kingman's House-of-Cards model with random mutation probabilities, revealing that the model's core behavior remains dominant even under environmental randomness, using matrix representations for analysis.

Contribution

It introduces a matrix-based approach to compare Kingman's model with variants involving random mutation probabilities, highlighting how randomness influences the model's dynamics.

Findings

Random mutation probabilities do not alter the condensation phenomenon.

The model's fitness distribution remains dominated by Kingman's original model.

Matrix representations are crucial for analyzing the effects of randomness.

Abstract

Kingman's House-of-Cards model is a simple and celebrated model to describe the evolution of population under the competition of selection and mutation. Letting mutation probabilities vary on generations makes the model more realistic and meaningful. This paper considers the condensation phenomenon and limit fitness distribution when Kingman's model is put into a random environment consisting of i.i.d. random mutation probabilities. It turns out that the random model is completely dominated by Kingman's model in terms of condensation and fitness. Another random model where all mutation probabilities are equal to the same random variable is also discussed. The relation between the three models is subtle and shows how extra randomness perturbs Kingman's model. The main tool is the matrix representation of the three models which is discovered in this work and plays crucial roles in the comparison analysis.