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

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  • Isozaki, Yasuki
  • Yoshida, Nobuo

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.

Suggested Citation

  • Isozaki, Yasuki & Yoshida, Nobuo, 2001. "Weakly pinned random walk on the wall: pathwise descriptions of the phase transition," Stochastic Processes and their Applications, Elsevier, vol. 96(2), pages 261-284, December.
    • Handle: RePEc:eee:spapps:v:96:y:2001:i:2:p:261-284
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    References listed on IDEAS

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    1. van Leeuwen, J.M.J. & Hilhorst, H.J., 1981. "Pinning of a rough interface by an external potential," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 107(2), pages 319-329.
    2. Caputo, P. & Velenik, Y., 2000. "A note on wetting transition for gradient fields," Stochastic Processes and their Applications, Elsevier, vol. 87(1), pages 107-113, May.
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    Cited by:

    1. Pétrélis, Nicolas, 2006. "Polymer pinning at an interface," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1600-1621, November.
    2. De Coninck, Joël & Dunlop, François & Huillet, Thierry, 2009. "Random walk versus random line," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(19), pages 4034-4040.
    3. Sohier, Julien, 2015. "The scaling limits of the non critical strip wetting model," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3075-3103.
    4. Nakashima, Makoto, 2018. "The free energy of the random walk pinning model," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 373-403.

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