Random permutations with cycle weights

TL;DR

This paper investigates how cycle lengths distribute in nonuniform random permutations with weighted cycles, revealing different growth regimes such as linear, fractional, or logarithmic powers of the total elements.

Contribution

It characterizes the cycle length distributions in weighted permutation models, identifying multiple growth regimes based on cycle weights.

Findings

Cycle lengths can grow linearly with the number of elements

Different regimes include fractional and logarithmic growth

The distribution depends on the specific cycle weights used

Abstract

We study the distribution of cycle lengths in models of nonuniform random permutations with cycle weights. We identify several regimes. Depending on the weights, the length of typical cycles grows like the total number $n$ of elements, or a fraction of $n$ or a logarithmic power of $n$.