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

• Published in: Stochastic processes and their applications Vol. 96; no. 2; pp. 261 - 284
• Main Authors: Isozaki, Yasuki, Yoshida, Nobuo
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.12.2001; Elsevier Science; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Entropic repulsion; Exact sciences and technology; Fluctuation phenomena, random processes, noise, and brownian motion; Limit theorems; Markov processes; Mathematics; Physics; Probability and statistics; Probability theory and stochastic processes; Random walk; Random walk Weak pinning Wall condition Entropic repulsion Wetting transition Limit theorems; Random walks and levy flights; Sciences and techniques of general use; Statistical physics, thermodynamics, and nonlinear dynamical systems; Wall condition; Weak pinning; Wetting transition; Limit theorems; Entropic repulsion; Wall condition; Wetting transition; Random walk; Weak pinning; Weak solution; Wetting; Entropy; Pinned random walk; Phase transitions; Limit theorem
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/S0304-4149(01)00118-1

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.