On a Markov construction of couplings

TL;DR

This paper constructs a coupling between the distribution of fixed points in a random permutation and a Poisson distribution using Markov chain intertwining, addressing a long-standing open problem with potential applications in random matrix theory.

Contribution

It introduces a novel Markov chain intertwining method to explicitly construct couplings for fixed point distributions, solving a long-standing open problem.

Findings

Successful construction of a coupling for fixed point distribution and Poisson law

Method shows potential for applications in random matrix theory

Provides explicit bounds and construction techniques

Abstract

For $N\in\mathbb{N}$, let $\pi_N$ be the law of the number of fixed points of a random permutation of $\{1, 2, ..., N\}$. Let $\mathcal{P}$ be a Poisson law of parameter 1.A classical result shows that $\pi_N$ converges to $\mathcal{P}$ for large $N$ and indeed in total variation $$\left\Vert \pi_N-\mathcal{P}\right\Vert_{\mathrm{tv}} \leq \frac{2^N}{(N+1)!}$$ This implies that $\pi_N$ and $\mathcal{P}$ can be coupled to at least this accuracy. This paper constructs such a coupling (a long open problem) using the machinery of intertwining of two Markov chains. This method shows promise for related problems of random matrix theory.