## Electronic Journal of Probability

### Directed polymers in a random environment with a defect line

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067112

Digital Object Identifier
doi:10.1214/EJP.v20-3379

Mathematical Reviews number (MathSciNet)
MR3311219

Zentralblatt MATH identifier
1308.82038

Rights

#### Citation

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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