Sampling Adsorbing Staircase Walks Using a New Markov Chain Decomposition Method

Staircase walks are lattice paths from to which take diagonal steps and which never fall below the -axis. A path hitting the -axis times is assigned a weight of where . A simple local Markov chain which connects the state space and converges to the Gibbs measure (which normalizes these weights) is known to be rapidly mixing when , and can easily be shown to be rapidly mixing when . We give the first proof that this Markov chain is also mixing in the more interesting case of , known in the statistical physics community as adsorbing staircase walks. The main new ingredient is a decomposition technique which allows us to analyze the Markov chain in pieces, applying different arguments to analyze each piece.