Shuffling algorithm for boxed plane partitions

TL;DR

This paper presents a Markov chain-based shuffling algorithm for uniformly sampling boxed plane partitions, with analysis of its properties and limiting behaviors of associated point processes.

Contribution

Introduces a novel Markov chain method for uniform sampling of boxed plane partitions and analyzes the limiting correlation functions of the resulting point processes.

Findings

Efficient perfect sampling algorithm for boxed plane partitions.

Derived limiting correlation functions as determinantal point processes.

Analyzed Gibbs properties of the limiting processes.

Abstract

We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from (a x b x c) to ((a-1) x (b+1) x c) or ((a+1) x (b-1) x c). Algorithmic realization of each step involves O((a+b)c) operations. One application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions. Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point process on suitable two-dimensional sections. The limiting correlation functions define a two-dimensional determinantal point processes with certain Gibbs properties.