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

• Main Author: Pitters, Helmut
• Format: Journal Article
• Language: English
• Published: 09.03.2019
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.1903.04906

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$align* \(K(n, s), S(n, t))-(s_1s_2, t_1t_2) n n, s, tın [0, 1]^2\\( B(s), B(t)), s, tın [0, 1]^2\, align* where$\mathscr B$is a one-dimensional Brownian sheet. This generalises and unifies a number of well-known results.