Random sampling of lattice paths with constraints, via transportation

TL;DR

This paper presents a robust MCMC algorithm inspired by optimal transport for efficiently sampling constrained lattice paths, achieving near-uniform samples in polynomial time and bounding mixing times effectively.

Contribution

Introduces a novel transport-inspired approach to bound mixing times of Markov chains for lattice path sampling, improving efficiency and robustness.

Findings

Samples paths of length n in n^{3+ε} steps

Bounds mixing time using contraction properties

Derives bounds for Propp-Wilson CFTP algorithm

Abstract

We discuss a Monte Carlo Markov Chain (MCMC) procedure for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. We show that an approach inspired by optimal transport allows us to bound efficiently the mixing time of the associated Markov chain. The algorithm is robust and easy to implement, and samples an "almost" uniform path of length $n$ in $n^{3+\eps}$ steps. This bound makes use of a certain contraction property of the Markov chain, and is also used to derive a bound for the running time of Propp-Wilson's CFTP algorithm.