Asymptotic Behaviour of an Infinitely-Many-Alleles Diffusion with Symmetric Overdonminance

TL;DR

This paper analyzes the asymptotic behavior of the stationary distribution in an infinitely-many-alleles diffusion model with symmetric overdominance, clarifying the distribution at critical phase transition points.

Contribution

It provides a definitive description of the asymptotic behavior at critical phase transition points, resolving previous uncertainties.

Findings

Identifies the asymptotic behavior at critical points $\lambda=k(k+1)$.

Clarifies the distribution's phase transition structure.

Provides mathematical characterization of the limiting distribution.

Abstract

This paper considers the limiting distribution of $\pi_{\lambda,\theta}$, the stationary distribution of the infinitely-many-alleles diffusion with symmetric overdominance \cite{MR1626158}. In \cite{MR2519357} the large deviation principle for $\pi_{\lambda,\theta}$ indicates that there are countably many phase transitions for the limiting distribution of $\pi_{\lambda,\theta}$, and the critical points are $\lambda=k(k+1), k\geq1$. The asymptotic behaviours at those critical points, however, are unclear. This article provides a definite description of the critical cases.