Markov chains arising from biased random derangements

TL;DR

This paper studies the cycle structures of biased random derangements through a Markov chain model, analyzing their asymptotic behavior and approximating distributions as the number of children grows large.

Contribution

It introduces a Markov chain framework for biased random derangements and establishes asymptotic and coupling results for their cycle types.

Findings

Derived the asymptotic distribution of cycle counts in biased derangements.

Established bounds on total variation distance between finite and infinite chain models.

Extended coupling techniques to analyze derangements with no 11-patterns.

Abstract

We explore the cycle types of a class of biased random derangements, described as a random game played by some children labeled $1,\cdots,n$. Children join the game one by one, in a random order, and randomly form some circles of size at least $2$, so that no child is left alone. The game gives rise to the cyclic decomposition of a random derangement, inducing an exchangeable random partition. The rate at which the circles are closed varies in time, and at each time $t$, depends on the number of individuals who have not played until t. A $\{0,1\}$-valued Markov chain $ X^n$ records the cycle type of the corresponding random derangement in that any $1$ represents a hand-grasping event that closes a circle. Using this, we study the cycle counts and sizes of the random derangements and their asymptotic behavior. We approximate the total variation distance between the reversed chain of $X^n$ and its weak limit $X^\infty$, as $n\to\infty$. We establish conditional (and push-forward) relations between $X^n$ and a generalization of the Feller coupling, given that no $11$-pattern ($1$-cycle) appears in the latter. We extend these relations to $X^\infty$ and apply them to investigate some asymptotic behaviors of $X^n$.