Electronic Journal of Probability

Directed polymers in a random environment with a defect line

Kenneth Alexander and Gökhan Yıldırım

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We study the depinning transition of the $1+1$ dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. potential values assigned to each site of $\mathbb{Z}^2$; sites on the positive axis have the potential enhanced by a deterministic value $u$. We show that for small inverse temperature $\beta$ the quenched and annealed free energies differ significantly at most in a small neighborhood (of size of order $\beta$) of the annealed critical point $u_c^a=0$.  For the case $u=0$, we show that the difference between quenched and annealed free energies is of order $\beta^4$ as $\beta\to 0$, assuming only finiteness of exponential moments of the potential values, improving existing results which required stronger assumptions.

Article information

Electron. J. Probab. Volume 20 (2015), paper no. 6, 20 pp.

Accepted: 22 January 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 82D60: Polymers 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

random walk depinning transition pinning Lipschitz percolation

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Alexander, Kenneth; Yıldırım, Gökhan. Directed polymers in a random environment with a defect line. Electron. J. Probab. 20 (2015), paper no. 6, 20 pp. doi:10.1214/EJP.v20-3379. https://projecteuclid.org/euclid.ejp/1465067112

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