Burnside process on parking functions and Dyck paths

TL;DR

This paper studies the Burnside process applied to parking functions and Dyck paths, demonstrating rapid mixing and proposing new algorithms for approximately uniform sampling of these combinatorial structures.

Contribution

It introduces the Burnside process on Catalan structures, specifically parking functions and Dyck paths, and proves rapid mixing bounds for these processes.

Findings

Both Burnside processes are rapidly mixing with O(n log n) bounds.

The processes provide novel algorithms for approximately uniformly sampling parking functions and Dyck paths.

Application to sampling triangulations of polygons approximately uniformly.

Abstract

Let $G$ be a finite group acting on a finite set $X$. This group action splits $X$ into disjoint orbits. The Burnside process is a Markov chain on $X$ which has a uniform stationary distribution when the chain is projected to orbits. We initiate the study of the Burnside process on Catalan structures. We consider two special cases: the first where the state space is the set of parking functions of length $n$ and $G = S_n$ is the symmetric group on $[n]$, such that $G$ acts by permuting coordinates, and the second where the state space is the set of labeled Dyck paths of length $2n$ and $G = S_n$ acts by permuting labels. The resulting Burnside processes give novel algorithms for sampling, respectively, an increasing parking function and a Dyck path approximately uniformly at random. Our main result shows that both processes are rapidly mixing, with mixing times upper bounded by $O(n \log n)$. As an application, we show how our Burnside process can be used to sample triangulations of an $(n+2)$-gon approximately uniformly at random.