## Bernoulli

- Bernoulli
- Volume 9, Number 4 (2003), 705-723.

### Directed polymers in a random environment: path localization and strong disorder

Francis Comets, Tokuzo Shiga, and Nobuo Yoshida

#### Abstract

We consider directed polymers in a random environment. Under some mild assumptions on the environment, we prove equivalence between the decay rate of the partition function and some natural localization properties of the path; some quantitative estimates of the decay of the partition function in one or two dimensions, or at sufficiently low temperature; and the existence of quenched free energy. In particular, we generalize to general environments the results recently obtained by Carmona and Hu for a Gaussian environment. Our approach is based on martingale decomposition and martingale analysis. It leads to a natural, asymptotic relation between the partition function, and the probability that two polymers in the same environment, but otherwise independent, end up at the same point.

#### Article information

**Source**

Bernoulli Volume 9, Number 4 (2003), 705-723.

**Dates**

First available in Project Euclid: 15 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1066223275

**Digital Object Identifier**

doi:10.3150/bj/1066223275

**Mathematical Reviews number (MathSciNet)**

MR1996276

**Zentralblatt MATH identifier**

1042.60069

**Keywords**

directed polymers martingales random environment

#### Citation

Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo. Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9 (2003), no. 4, 705--723. doi:10.3150/bj/1066223275. https://projecteuclid.org/euclid.bj/1066223275