Sampling adsorbing staircase walks using a new Markov chain decomposition method

Staircase walks are lattice paths from (0,0) to (2n,0) which take diagonal steps and which never fall below the x-axis. A path hitting the x-axis κ times is assigned a weight of λ^κ, where λ>0. 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 λ=1, and can easily be shown to be rapidly mixing when λ<1. We give the first proof that this Markov chain is also mixing in the more interesting case of λ>1, 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