Yasuki Isozaki, Nobuo Yoshida
One-sided random walk with weak pinning: pathwise
descriptions of the phase transition
(66K, LaTeX2e)
ABSTRACT. 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.