Cycle structure of random standardized permutations

In this article, we study a model of random permutations, which we call random standardized permutations, based on a sequence of i.i.d. random variables. This model generalizes others, such as the riffle-shuffle and the major-index-biased permutations. We first establish an exact result on the joint distribution of the number of cycles of given lengths, involving the notion of primitive words. From this result, we obtain various convergence results, most of which are proved using the method of moments. First we prove that the number of small cycles may have either a Poisson limit distribution, or a limit distribution given by a countable sum of independent geometric distributions. Then we establish a limit distribution for large cycles, which is the Poisson-Dirichlet process. Finally we prove a central limit theorem for the total number of cycles.

💡 Research Summary

The paper introduces a broad class of random permutations called “random standardized permutations.” Given a sequence of i.i.d. random variables G₁,…,Gₙ taking values in a countable set I with probabilities p_i, the standardized permutation std(G) is obtained by ranking the values of G (breaking ties by original index) and assigning the ranks 1,…,n accordingly. When the distribution of G is atomless, std(G) is uniformly random; the paper focuses on the discrete case, which encompasses several well‑studied non‑uniform models such as riffle shuffles (uniform on a finite set) and major‑index‑biased permutations (geometric distribution).

The first major contribution is an exact description of the joint distribution of cycle counts. For a cycle of length k whose entries follow a specific pattern i = (i₁,…,i_k)∈Iᵏ, the pattern must be a primitive word (i.e., not a proper power) and considered up to cyclic conjugacy. Let Q_k denote the set of conjugacy classes of primitive words of length k. For each i∈Q_k define D_i as the number of k‑cycles of std(G) whose G‑values follow i. Theorem 1.1 shows that for any collection of pairwise non‑conjugate primitive words i₁,…,i_r and non‑negative integers ℓ₁,…,ℓ_r, \