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 phase transition as the strength of the weak pinning varies. We prove path space limit theorems which describe the macroscopic shape of the path for all 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 strong mixing property.