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The free energy of the random walk pinning model

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  • Nakashima, Makoto

Abstract

We consider the random walk pinning model. This is a random walk on Zd whose law is given as the Gibbs measure μN,Yβ, where the polymer measure μN,Yβ is defined by using the collision local time with another simple symmetric random walk Y on Zd up to time N. Then, at least two definitions of the phase transitions are known, described in terms of the partition function and the free energy. In this paper, we will show that the two critical points coincide and give an explicit formula for the free energy in terms of a variational representation. Also, we will prove that if β is smaller than the critical point, then X under μN,Yβ satisfies the central limit theorem and the invariance principle PY-almost surely.

Suggested Citation

  • Nakashima, Makoto, 2018. "The free energy of the random walk pinning model," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 373-403.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:2:p:373-403
    DOI: 10.1016/j.spa.2017.04.015
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    References listed on IDEAS

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    1. Birkner, Matthias, 2008. "Conditional large deviations for a sequence of words," Stochastic Processes and their Applications, Elsevier, vol. 118(5), pages 703-729, May.
    2. 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.
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