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

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 $\lambda =k(k+1),k\ge 1$. The asymptotic behaviours at those critical points, however, are unclear. This article provides a definite description of the critical cases.

arXiv comment25 pages