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

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 $t_1\mathrm{/}2$. Our main result comprises two different couplings of the above models for all parameters $n\ge 2,$ $t\in [0,1{]}^2$ such that in both couplings one has weak convergence of processes as $n\to \mathrm{\infty }$ \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 $\mathcal{B}$ is a one-dimensional Brownian sheet. This generalises and unifies a number of well-known results.