Ranks, copulas, and permutons

TL;DR

This paper explores the intersection of discrete mathematics and probability theory by examining the asymptotics of random permutations through Markov chains, pattern frequencies, and boundary theory, with applications to various combinatorial structures.

Contribution

It reviews recent advances connecting permutation asymptotics with Markov chain boundary theory and introduces new applications and perspectives in combinatorics and statistics.

Findings

Use of boundary theory in permutation asymptotics

Connections between pattern frequencies and nonparametric statistics

Applications to other combinatorial families and exchangeability

Abstract

We review a recent development at the interface between discrete mathematics on one hand and probability theory and statistics on the other, specifically the use of Markov chains and their boundary theory in connection with the asymptotics of randomly growing permutations. Permutations connect total orders on a finite set, which leads to the use of a pattern frequencies. This view is closely related to classical concepts of nonparametric statistics. We give several applications and discuss related topics and research areas, in particular the treatment of other combinatorial families, the cycle view of permutations, and an approach via exchangeability.