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 /spl kappa/ times is assigned a weight of /spl lambda//sup /spl kappa//, where /spl lambda/<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 /spl lambda/=1, and can easily be shown to be rapidly mixing when /spl lambda/>1. We give the first proof that this Markov chain is also mixing in the more interesting case of /spl lambda/<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.