A central limit theorem for a generalization of the Ewens measure to random tuples of commuting permutations

We prove a central limit theorem (CLT) for the number of joint orbits of random tuples of commuting permutations. In the uniform sampling case this generalizes the classic CLT of Goncharov for the number of cycles of a single random permutation. We also consider the case where tuples are weighted by a factor other than one, per joint orbit. We view this as an analogue of the Ewens measure, for tuples of commuting permutations, where our CLT generalizes the CLT by Hansen. Our proof uses saddle point analysis, in a context related to the Hardy-Ramanujan asymptotics and the theorem of Meinardus, but concerns a multiple pole situation. The proof is written in a self-contained manner, and hopefully in a manner accessible to a wider audience. We also indicate several open directions of further study related to probability, combinatorics, number theory, an elusive theory of random commuting matrices, and perhaps also geometric group theory.

💡 Research Summary

The paper studies random ℓ‑tuples of pairwise‑commuting permutations in the symmetric group Sₙ and the number Kₗ,ₙ of joint orbits (i.e. the number of H‑orbits where H=⟨σ₁,…,σₗ⟩). For a fixed weight parameter x>0 the authors introduce a probability measure
Pₗ,ₙ,ₓ(σ₁,…,σₗ)= x^{Kₗ,ₙ} / (n! Hₗ,ₙ(x)),
where Hₗ,ₙ(x)=∑_{k=0}^{n} A(ℓ,n,k) x^{k}/n! and A(ℓ,n,k) counts ℓ‑tuples with exactly k joint orbits. When ℓ=1 this reduces to the classical Ewens distribution; when x=1 it is the uniform distribution on commuting ℓ‑tuples.

The main result (Theorem 1.1) states that for any ℓ≥2 and any x>0, as n→∞ the mean and variance of Kₗ,ₙ satisfy
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