The next meeting of the Applied Mathematics Seminar will be held on November 11, 2025 at 2:00 pm in Room 503 of IMI – BAS.

A talk on

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

will be delivered by Helmut Pitters (Mannheim, Germany).

Abstract: Consider a random permutation of {1, …, ⌊nᵗ²⌋} drawn according to the Ewens measure with parameter t₁, and let K(n, t) denote the number of its cycles, where t ≡ (t₁, t₂) ∈ [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 ⌊nᵗ²⌋ when mutations arrive at rate t₁/2.

Our main result comprises two different couplings of the above models for all parameters n ≥ 2 and t ∈ [0, 1]², such that in both couplings one has weak convergence of processes as n → ∞ to a one-dimensional Brownian sheet. This generalizes and unifies a number of well-known results.