Cutting a unit square and permuting blocks

TL;DR

The paper develops a two-dimensional generalization of the stick-breaking construction for random partitions of unity by studying block-permutation structures via wreath products. It proves that normalized cycle lengths converge to a square-cutting distribution $\mathcal{P}$, characterized by the self-similarity $\mathcal{P} \stackrel{d}{=} \{U\Sigma,(1-U)\mathcal{P}\}$ with $U\sim U[0,1]$, and derives the distribution and mean of the largest part, $\eta \approx 0.40$, through a convolution equation $\pi(u)=\frac{1}{u}(\pi * \rho)(u)$. A multiplicative-function framework is developed to represent $\pi$ as $\pi(u) = \lim_{x\to\infty} x^{-u} \sum_{n \le x^u} f_x(n)$ with $f_x(p) = \rho\left( \frac{\log p}{\log x} \right)$, and the paper establishes quantitative rates of convergence between the random cycle lengths and $\mathcal{P}$ via coupling techniques. By extending Erdős–Turán laws to wreath-product subgroups, the work connects permutation cycles to fragmentation-type limit processes, situating the square-cutting distribution within the broader Poisson-Dirichlet framework while highlighting its distinct largest-part behavior and number-theoretic representations.

Abstract

Consider a random permutation of $kn$ objects that permutes $n$ disjoint blocks of size $k$ and then permutes elements within each block. Normalizing its cycle lengths by $kn$ gives a random partition of unity, and we derive the limit law of this partition as $k,n \to \infty$. The limit may be constructed via a simple square cutting procedure that generalizes stick breaking in the classical case of random permutations ($k=1$). The expected size of the largest part of this square cutting distribution is approximated to be $0.40$, in contrast with the Golomb-Dickman constant around $0.624$ describing the longest cycle of a uniform random permutation as well as the largest prime factor of a random integer. The distribution function of this largest part is shown to also be the mean of a certain multiplicative function. Along the way we give the first extension of the Erdős-Turán law to a proper permutation subgroup.

Figures (3)

• Figure 1: Stick breaking construction of the Poisson-Dirichlet and square cutting construction of $\mathcal{P}$. Each step of square cutting consists of vertically cutting the leftmost rectangle as well as horizontally cutting the new piece.
• Figure 2: Function approximations for values $0 \le u \le 10$.
• Figure 3: Coupling the square partition defined by the continuous time Poisson process $\Sigma$ (black) with the partition $\Sigma_n$, with break points at multiples of $1/n$, for $n=4$. In reality $\Sigma$ has infinitely many parts a.s. Given a blue rectangle in $\Sigma_n$ with top left corner $y$ we compare it with the black rectangle in $\Sigma$ with top left corner $x$ where $x$ is the rightmost and then bottom most point across from $y$ in the dashed blue square with top left corner $y$.

Theorems & Definitions (23)

• Theorem 1.1: Square cutting
• Theorem 1.2: Mean of a multiplicative function
• Theorem 1.3: Rates of convergence
• Theorem 2.1: Wreath product Erdős-Turán
• proof
• Theorem 3.1: Largest piece from square cutting
• proof
• Lemma 4.1: primedisting*Theorem 23
• Lemma 4.2: Integral approximation
• proof