The number of cycles in a random permutation and the number of segregating sites jointly converge to the Brownian sheet

• Published in: arXiv.org
• Main Author: Pitters, Helmut
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 18.06.2021
• Subjects: Convergence; Genealogy; Mutation; Permutations
• ISSN: 2331-8422

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 \(t_1/2\). Our main result comprises two different couplings of the above models for all parameters \(n\geq 2,\) \(t\in [0, 1]^2\) such that in both couplings one has weak convergence of processes as \(n\to\infty\) \begin{align*} \left\{\frac{(K(n, s), S(n, t))-(s_1s_2, t_1t_2)\log n}{\sqrt{\log n}}, s, t\in [0, 1]^2\right\}\to\{(\mathscr B(s), \mathscr B(t)), s, t\in [0, 1]^2\}, \end{align*} where \(\mathscr B\) is a one-dimensional Brownian sheet. This generalises and unifies a number of well-known results.