Random sampling of lattice paths with constraints, via transportation

We investigate Monte Carlo Markov Chain (MCMC) procedures for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. We will see that an approach inspired by optimal transport allows us to efficiently bound 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 n3+ 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 Coupling From The Past algorithm. 1 Lattice Paths with Constraints Lattice paths arise in several areas in probability and combinatorics, either in their own interest (as realizations of random walks, or because of their interesting combinatorial properties: see [1] for the latter) or because of fruitful bijections with various families of trees, tilings, words. The problem we discuss here is to efficiently sample unifo...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Lucas Gerin
• Keywords: Algorithms, CORR 2010, Education, Lattice Paths, Markov Chain