On a $\Lambda$-mutation model and the harmonic model

Abstract

We introduce a continuous-time mutation model with two types determined by a finite measure $\Lambda$ on the unit interval. The model satisfies a certain consistency property known from mathematical population genetics and includes so called harmonic models being of interest in mathematical statistical physics. We mainly focus on the situation when the number of particles is equal to some constant $N$. Duality results and scaling limits as $N\to\infty$ for the forward and backward processes are provided leading to a commutative diagram. The stationary distribution of the forward process is studied with an emphasis on the case when $\Lambda$ is a beta distribution. The work bridges particle models from mathematical statistical physics and mutation models from mathematical population genetics.