Cutting a unit square and permuting blocks

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 Erd\H{o}s-Tur\'an law to a proper permutation subgroup.