Poisson limit for the number of cycles in a random permutation and the number of segregating sites

Abstract

Consider a random permutation of $\{1,\dots ,\lfloor n^{t_2}\rfloor \}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n,t)$ denote the number of its cycles, where $t\equiv (t_1,t_2)\in [0,1{]}^2$. Next, consider a sample drawn from a large, neutral population of haploid individuals subject to mutation under the infinitely many sites model of Kimura whose genealogy is governed by Kingman's coalescent. Let $S(n,t)$ count the number of segregating sites in a sample of size $\lfloor n^{t_2}\rfloor$ when mutations arrive at rate $t_1/2$. We show that $K(n,(t_1/\log n,t_2))-1$ and $S(n,(t_1/\log n,t_2))$ induce unique random measures ${\Pi }_n^K$ and ${\Pi }_n^S,$ respectively, on the positive quadrant $[0,\infty {)}^2.$ Our main result is to show that in the coupling of $S(n,t)$ and $K(n,t)$ introduced in~\cite{Pitters2019} we have weak convergence as $n\to \infty$ \begin{align*} (Π_n^K, Π_n^S)\to_d (Π, Π), \end{align*} where $\Pi$ is a Poisson point process on $[0,\infty {)}^2$ of unit intensity. This complements the work in~\cite{Pitters2019} where it was shown that the process $\{(K(n,t),S(n,t)),t\in [0,1{]}^2\},$ appropriately rescaled, converges weakly to the product of the same one-dimensional Brownian sheet.

arXiv admin note: text overlap with arXiv:1903.04906