Weakly pinned random walk on the wall: pathwise descriptions of the phase transition

We consider a one-dimensional random walk which is conditioned to stay non- negative and is "weakly pinned" to zero. This model is known to exhibit a p hase transition as the strength of the weak pinning varies. We prove path s pace limit theorems which describe the macroscopic shape of the path for al l values of the pinning strength. If the pinning is less than (resp. equal to) the critical strength, then the limit process is the Brownian meander ( resp. reflecting Brownian motion). If the pinning strength is supercritical , then the limit process is a positively recurrent Markov chain with a stro ng mixing property. (C) 2001 Elsevier Science B.V. All rights reserved.