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

Abstract

Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. 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 $Π_n^K$ and $Π_n^S,$ 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 $Π$ 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.